NUMERICAL ANALYSIS OF INFILTRATION
INTO PARTIALLY SATURATED SOIL SLOPES
A thesis submitted to the University of London
for the degree of Doctor of Philosophy and for the Diploma of the
Imperial College of Science, Technology and Medicine
By
Philip Graham Clark Smith
Department of Civil and Environmental Engineering
Imperial College of Science, Technology and Medicine
London, SW7 2BU
June 2003
Abstract
The issue of slope stability is a core concern to geotechnical engineers. Traditional methods of
analysis assume that the soil is fully saturated. However, throughout much of the world, slopes
exist in residual soil deposits. Such soils are often unsaturated, and it has become clear that the
traditional saturated approach to assessing these slopes is inadequate.
Current knowledge of the infiltration processes into unsaturated soil slopes is inadequate to
enable reliable assessment of stability. This thesis constitutes an investigation into these
infiltration processes, using numerical modelling techniques.
A discussion on the general nature of unsaturated soils is followed by a survey of previous
attempts to apply numerical modelling to unsaturated soils. The core equations governing the
fully coupled constitutive behaviour of unsaturated soils are developed and presented, along
with details of how these equations were implemented in numerical form. A conceptual model
that qualitatively assesses the behaviour of unsaturated soils is also presented.
The behaviour of unsaturated soil slopes was investigated through numerical simulations of the
Tung Chung slope in Hong Kong, using the Imperial College Finite Element Program
( ICFEP ), with the numerical predictions compared to field monitoring data of the pore water
pressures / suctions from the site.
It is shown that the variation of pore pressures is very sensitive to the relationship between
suction and degree of saturation, as represented by the Soil Water Characteristic Curve. Also
critical are the permeability parameters of the soil, including the variation of permeability with
suction.
The importance of accurately reproducing the true nature of the rain event is also clearly
demonstrated.
Many areas where further work is required are identified. In particular, there is a need to
establish methods to obtain reliable soil parameters to permit accurate modelling of the
behaviour of unsaturated soils.
2
Acknowledgments
First mention must be made of the Engineering and Physical Sciences Research Council, who
provided the funding that made this project possible. Thanks to the Geotechnical Engineering
Office of the Hong Kong Government ( GEO ), who kindly allowed me access to information
on the Tung Chung slope, and permitted the presentation of this slope as a case study. Thanks
also to Andrew Ridley and Kieran Dineen of Geotechnical Observations, who were responsible
for gathering much of the site data for Tung Chung, and gave practical advise as to the nature of
the site, as well as general support for this project throughout its life.
Next a credit for those in Industry that expressed support for this project at its inception, and
also proved gracious hosts: Dennis Ganendra of Minconsult in Malaysia, to Dr Charles Ng of
Hong Kong University of Science and Technology, John Davis of Arup (HK), and Mr. J. Lam
of GEO.
That this thesis took four years to complete, not three, is perhaps an indication of how pleasant
the working environment is within the soils section. This is in turn a reflection of the staff that
work here. Many fostered my interest in soil mechanics when I was first at Imperial as an
undergraduate. All provided friendly help and advice during my time here as a post-graduate.
While I owe thanks to them all, I must mention three individuals in particular. Dr Lidija
Zdravkovic often gave of her time to guide me through the intricacies of ICFEP without making
me feel ( too ) stupid. Dr Trevor Addenbrooke was my ‘day to day’ supervisor for much of this
project, and helped keep me heading in the right direction more or less to a schedule. Professor
David Potts, my ‘senior’ supervisor, despite myriad claims upon his time ensured that ICFEP
did what I wanted it to do, and always seemed to know the critical question to ask.
If the staff contribute towards the atmosphere, then so to do my fellow researchers, so, as a
record to look back on when memory fades, thanks to Ruth, Shin, Adam, Emilio, Julio, Ralf,
Claudia, Dr. Toyota, Abbas, Angeliki, Felix, Graham, Kirsty, Kostis, Monica, Niki, Peter,
Stuart, Reinaldo, Vincent, Liana, Ni and Rafael. Apologies to the seismologists, and those
researchers more recently starting their studies, whom I never really got to know, and whom I
have excluded from this list.
So to the personnel acknowledgements. Life in College runs parallel to life outside, and
established friendships helped deal with stresses arising in both parts of my life, so thanks to
Joyce and Alex, Tim and Emma, John and Jude, Ian, Stuart, Jon and Mike.
Finally, special thanks to my brother Robert, whose support was invaluable.
3
Contents
Chapter 1: Introduction
16
1.1: Background
16
1.2: Scope of project
17
1.3: Thesis layout
17
Chapter 2: Unsaturated soils
19
2.1: Infiltration into unsaturated soils
19
2.2: Soil suction and air pressure
20
2.2.1: Water content-suction relationship
22
2.2.2: Hysteretic behaviour
25
2.2.3: Air flow and the treatment of the air phase
27
Chapter 3: Slope instability due to infiltration
40
3.1: The infiltration process and wetting front development
40
3.2: Antecedent rainfall
42
3.3: Water flow in an unsaturated soil
45
3.4: The permeability-suction relationship
48
3.5: Suction and soil deformation relationship
51
3.6: Stress-state variables for unsaturated soil
52
3.7: Suction-shear strength relationship
53
Contents
4
3.8: Published work using numerical modelling to analyse the
infiltration process
57
3.9: The Pak Kong failure
78
Chapter 4: The pre-existing capabilities of ICFEP
88
4.1: Introduction
88
4.2: ICFEP version 8.0 general capabilities
88
4.3: Consolidation and permeability models
88
4.3.1: The suction switch
89
4.4: Boundary conditions
89
4.4.1: The precipitation boundary condition
4.5: Modelling of precipitation on to saturated soil
90
93
4.5.1: No suction switch
93
4.5.2: Tension-saturated column with suction switch
96
4.5.3: Element size sensitivity
98
4.5.4: Two-dimensional tension-saturated analysis
99
4.5.5: Conclusions of the tension-saturated analyses
103
Chapter 5: Development of constitutive equations for unsaturated soil
behaviour
133
5.1: Introduction
133
5.2: Qualitative assessment of unsaturated behaviour
133
5.3: Constitutive relation for the soil structure
136
5.4: Constitutive relation for the water phase
141
5.5: Finite element formulation of the constitutive relations for
unsaturated soil
Contents
146
5
5.6: Conceptual model
151
5.6.1: Suction-displacement relationship
152
5.6.2: Water flow-displacement relationship
152
5.6.3: Pore pressure-water flow relationship
155
5.6.4: The conceptual model
157
5.7: Use of linear elasticity
159
5.8: Summary of constitutive equations development
160
Chapter 6: Development of ICFEP models for unsaturated soil analysis
168
6.1: Introduction
168
6.2: The non-linear soil model adopted for unsaturated behaviour
168
6.2.1: Model 16: 3D Mohr-Coulomb model for saturated soil
169
6.2.2: Model 82: Partially saturated 3D Mohr-Coulomb model
169
6.3: The Soil Water Characteristic Curve model adopted for unsaturated
173
behaviour
6.3.1: The Omega variation
176
6.4: Suction dependant permeability
177
6.5: Density variation
178
6.6: Automatic incrementation for the precipitation boundary condition
180
6.7: The precipitation boundary condition as a recharge model
183
6.8: Summary of the new ICFEP unsaturated capabilities
184
Chapter 7: Validation of new coding
200
7.1: Introduction
200
7.2: Water content volume validation
200
Contents
6
7.3: Density variation validation
205
7.4: Consolidation of an unsaturated column
207
Chapter 8: The Tung Chung case study
227
8.1: Introduction
227
8.2: The Tung Chung site
227
8.3: Tung Chung site investigation details
228
8.4: The finite element mesh
230
8.5: Soil properties
232
8.6: Rainfall data
243
8.7: Analyses boundary conditions
244
8.8: Tung Chung field response to rainfall
245
8.9: ICFEP analysis procedure
246
8.10: The ICFEP analysis
247
8.11: Computer resources required
272
8.12: Summary of the Tung Chung study
275
Chapter 9: Conclusions and Recommendations
347
9.1: Introduction
347
9.2: The nature of unsaturated soil
348
9.3: The current state of the art
349
9.4: ICFEP version 8.0
350
9.5: The development of an unsaturated soil model
352
9.6: Unsaturated soil modelling using ICFEP
353
Contents
7
9.7: Valedictory exercises
356
9.8: The Tung Chung case study
357
9.9: Recommendations for further work
359
References
361
Appendices
Appendix A1: Details of ICFEP non-linear material model 16
373
Appendix A2: Details of ICFEP non-linear material model 82
377
Appendix A3: Details of ICFEP SWCC model 2
381
Appendix A4: Details of ICFEP SWCC model 3
383
Appendix A5: The Automatic incrementation procedure in ICFEP
386
Appendix A6: Photographs of the Tung Chung slope, Hong Kong
391
Contents
8
List of illustrations
Figure 2.1: The distribution of subsurface water ( after Bear, 1972 )
32
Figure 2.2: Penetration of air-water interface into soil ( after Childs, 1969 )
33
Figure 2.3: Schematic diagram of hypothetical porous medium ( after White et al, 1970 ) 34
Figure 2.4: The Soil Water Characteristic Curve
35
Figure 2.5: Hysteretic SWCC showing scanning curves ( after Bear, 1979 )
36
Figure 2.6: The raindrop and ink-bottle effects ( after Bear, 1979 )
37
Figure 2.7: Movement of air-water interface within soil pores
38
Figure 2.8: Relative permeability to air and water as a function of saturation
( after Corey, 1957 )
39
Figure 3.1: Lumb wetting front ( after Lumb, 1962a )
80
Figure 3.2: Bodman and Cole wetting front ( after Bear, 1972 )
80
Figure 3.3: Variation in pore water pressure distribution with depth
( after Fredlund and Rahardjo, 1993b )
81
Figure 3.4: Shear strength failure surface for unsaturated soils
( after Fredlund and Rahardjo, 1993a )
82
Figure 3.5: Soil moisture content profiles predicted for rain infiltration
( after Rubin and Steinhardt, 1963 )
Figure 3.6: Comparison of theoretical and experimental data ( after Rubin et al, 1964 )
83
84
Figure 3.7: Effect of AEV on suction profile under infiltration ( after Kasim et al, 1998 ) 85
Figure 3.8: Suction profile showing steady state simulation
( after Fredlund and Barbour, 1991 )
Figure 3.9: Effect of the form of the SWCC ( after Alonso et al, 1995 )
Contents
9
86
87
Figure 4.1: ICFEP permeability-suction switch
105
Figure 4.2: Precipitation boundary condition
106
Figure 4.3: Precipitation boundary condition with large time-step and inflow rate
107
Figure 4.4: Tension-saturated column for ICFEP v8.0 analyses
108
Figure 4.5: Precipitation into tension-saturated soil column, no suction switch
109
Figure 4.6: Precipitation into tension-saturated soil column, no suction switch
110
Figure 4.7: Precipitation into tension-saturated soil column, no suction switch
111
Figure 4.8: Precipitation into tension-saturated soil column, no suction switch
112
Figure 4.9: Precipitation into tension-saturated column, with suction switch
113
Figure 4.10: Precipitation into tension-saturated column, with suction switch
114
Figure 4.11: Precipitation into tension-saturated column, with suction switch
115
Figure 4.12: Precipitation into tension-saturated column, with suction switch
116
Figure 4.13: Element size sensitivity analysis, 2m elements
117
Figure 4.14: Element size sensitivity analysis, 10m elements
118
Figure 4.15: Element size sensitivity analysis, 0.5m elements
119
Figure 4.16: Element size sensitivity analysis, 0.25m elements
120
Figure 4.17: Finite element mesh for tension-saturated slope analyses
121
Figure 4.18a/b: Contours of pore water pressure in tension saturated slope, subject
to precipitation, example 1
122
Figure 4.18c/d: Contours of pore water pressure in tension saturated slope, subject
to precipitation, example 1
123
Figure 4.19: PWP across section of tension-saturated slope, subject to precipitation,
example 1
124
Figure 4.20: PWP across section of tension-saturated slope, subject to precipitation,
example 2
125
Figure 4.21: Contours of pore water pressure in tension saturated slope, subject
to precipitation, example 2
Contents
126
10
Figure 4.22: PWP across section of tension-saturated slope, subject to precipitation,
example 3
127
Figure 4.23: Contours of pore water pressure in tension saturated slope, subject
to precipitation, example 3
128
Figure 4.24: PWP across section of tension-saturated slope, subject to precipitation,
example 4
129
Figure 4.25: Contours of pore water pressure in tension-saturated slope, subject to
precipitation, example 4
130
Figure 4.26: PWP across section of tension-saturated slope, subject to precipitation,
example 5
131
Figure 4.27: Contours of pore water pressure in tension-saturated slope, subject to
precipitation, example 5
132
Figure 5.1: Air/water flow from a saturated soil
162
Figure 5.2: Air/water flow from an unsaturated soil
163
Figure 5.3: Continuity of flow through an element of soil
164
Figure 5.4: The adopted conceptual zonal model
165
Figure 5.5: Conceptual zones
166
Figure 5.6: The modified Newton-Raphson algorithm
( after Potts and Zdravkovic, 1999 )
167
Figure 6.1: Mohr-Coulomb yield surface in principal stress space
( after Potts and Zdravkovic, 1999 )
185
Figure 6.2: Variation of H parameter within ICFEP model 82
186
Figure 6.3: Typical void ratio-suction relationship ( after Toll, 1995 )
187
Figure 6.4: SWCC model 2 – simple, non-hysteretic
188
Figure 6.5: SWCC model 3 – simple, non-hysteretic, non-linear
189
Figure 6.6: Effects of stress state on the SWCC ( after Ng and Pang, 2000 )
190
Contents
11
Figure 6.7: Soil Water Characteristic Surface
191
Figure 6.8: Hysteresis in volumetric water content and permeability relations to suction
( after Liakopoulos, 1965 )
192
Figure 6.9: Water content – permeability relationship
( after Fredlund and Rahardjo, 1993a )
193
Figure 6.10: Variation of density due to changes in saturation
194
Figure 6.11: Schematic operation of existing ICFEP automatic incrementation procedure 195
Figure 6.12: The tolerance zone for the precipitation boundary condition
196
Figure 6.13: Determination of sub-increment size during application of the precipitation
boundary condition
197
Figure 6.14: Schematic operation of existing ICFEP AI procedure for precipitation
198
Figure 6.15: Precipitation boundary condition used to simulate recharge
199
Figure 7.1: Water content volume validation
213
Figure 7.2: SWCCs used for validation testing – filling a 1m cuboid
214
Figure 7.3: Density variation validation
215
Figure 7.4: Wong, Fredlund and Krahn (1998) compression of column
216
Figure 7.5: SWCCs used in consolidation of column
217
Figure 7.6: PWP distribution after 1 second
218
Figure 7.7: PWP distribution after 31 seconds
219
Figure 7.8: PWP distribution after 255 seconds
220
Figure 7.9: PWP distribution after 1023 seconds
221
Figure 7.10: ICFEP mesh for consolidating column analysis
222
Figure 7.11: Vertical displacements after 1 second
223
Figure 7.12: Vertical displacements after 31 seconds
224
Figure 7.13: Vertical displacements after 255 seconds
225
Figure 7.14: Vertical displacements after 1023 seconds
226
Contents
12
Figure 8.1: Location of Tung Chung landslide study area ( after Franks, 1999 )
285
Figure 8.2: Solid geology of Lantau Island, Hong Kong ( after Irfan, 1999 )
286
Figure 8.3: Location of boreholes and trial pits
287
Figure 8.4: Location of pore water pressure monitoring devices
288
Figure 8.5: Depth to rock head
289
Figure 8.6: ICFEP mesh for Tung Chung analysis
290
Figure 8.7: Line of section of ICFEP mesh
291
Figure 8.8: Permeability-suction relationships for volcanic soil
292
Figure 8.9: Suction switch for residual soil
293
Figure 8.10: SWCCs from Anderson, 1984
294
Figure 8.11: SWCCs for decomposed volcanic soil
295
Figure 8.12: SWCCs used for ICFEP analysis
296
Figure 8.13a: Tung Chung monitoring data for SP4, @ 2.73m depth
297
Figure 8.13b: Tung Chung monitoring data for SP5, @ 1.53m depth
298
Figure 8.13c: Tung Chung monitoring data for SP7, @ 2.62m depth
299
Figure 8.13d: Tung Chung monitoring data for SP8, @ 1.00m depth
300
Figure 8.13e: Tung Chung monitoring data for SP9, @ 3.00m depth
301
Figure 8.13f: Tung Chung monitoring data for SP10, @ 1.15m depth
302
Figure 8.14: Tung Chung monitoring data combined plot
303
Figure 8.15a: Tung Chung monitoring data for SP1, @ 2.50m depth
304
Figure 8.15b: Tung Chung monitoring data for SP2, @ 3.00m depth
305
Figure 8.15c: Tung Chung monitoring data for SP3, @ 2.00m depth
306
Figure 8.15d: Tung Chung monitoring data for SP6, @ 3.00m depth
307
Figure 8.15e: Tung Chung monitoring data for TRL129, @ 16.6m depth
308
Figure 8.16a: ICFEP predictions for Run 1
309
Figure 8.16b: ICFEP pore pressure predictions for analysis run 1
310
Contents
13
Figure 8.17: Accumulated flow velocities for Run 1, increment 100
311
Figure 8.18: PWP distribution with depth, Run 1, low section
312
Figure 8.19: PWP distribution with depth, Run 1, mid section
313
Figure 8.20: PWP distribution with depth, Run 1, high section
314
Figure 8.21a: ICFEP predictions for Run 2
315
Figure 8.21b: ICFEP pore pressure predictions for analysis run 2
316
Figure 8.22: PWP distribution with depth, Run 2, mid section
317
Figure 8.23a: ICFEP predictions for Run 3
318
Figure 8.23b: ICFEP pore pressure predictions for analysis run 3
319
Figure 8.24a: ICFEP predictions for Run 4
320
Figure 8.24b: ICFEP pore pressure predictions for analysis run 4
321
Figure 8.25a: ICFEP predictions for Run 5
322
Figure 8.25b: ICFEP pore pressure predictions for analysis run 5
323
Figure 8.26a: ICFEP predictions for Run 6
324
Figure 8.26b: ICFEP pore pressure predictions for analysis run 6
325
Figure 8.27a: ICFEP predictions for Run 7
326
Figure 8.27b: ICFEP pore pressure predictions for analysis run 7
327
Figure 8.28: ICFEP pore pressure predictions for analysis run 7A
328
Figure 8.29a: ICFEP predictions for Run 8
329
Figure 8.29b: ICFEP pore pressure predictions for analysis run 8
330
Figure 8.30a: ICFEP predictions for Run 9
331
Figure 8.30b: ICFEP pore pressure predictions for analysis run 9
332
Figure 8.31a: ICFEP predictions for Run 10
333
Figure 8.31b: ICFEP pore pressure predictions for analysis run 10
334
Figure 8.32a: ICFEP predictions for Run 11
335
Figure 8.32b: ICFEP pore pressure predictions for analysis run 11
336
Figure 8.33a: ICFEP predictions for Run 12
337
Figure 8.33b: ICFEP pore pressure predictions for analysis run 12
338
Contents
14
Figure 8.34a: Tung Chung monitoring data for SP4 with ICFEP Run 12 prediction
339
Figure 8.34b: Tung Chung monitoring data for SP5 with ICFEP Run 12 prediction
340
Figure 8.34c: Tung Chung monitoring data for SP7 with ICFEP Run 12 prediction
341
Figure 8.34d: Tung Chung monitoring data for SP8 with ICFEP Run 12 prediction
342
Figure 8.34e: Tung Chung monitoring data for SP9 with ICFEP Run 12 prediction
343
Figure 8.34f: Tung Chung monitoring data for SP10 with ICFEP Run 12 prediction
344
Figure 8.35a: ICFEP predictions for Run 13(7)
345
Figure 8.35b: ICFEP pore pressure predictions for analysis run 13(7)
346
Figure A5.1: The AI operating procedure
390
Plate A1: Tung Chung slope, view south
392
Plate A2: Tung Chung slope, view south-east
393
Plate A3: Tung Chung slope, view north-west
394
Plate A4: Tung Chung slope, view south-east
395
Plate A5: Tung Chung slope, view south-west
396
Contents
15
CHAPTER 1
Introduction
1.1. Background
Failure of soil slopes, both natural and man-made, during or shortly after rainfall is a common
occurrence in many parts of the world. Such rainfall-related failures are often associated with
tropical areas, where intense rainfall may occur seasonally, and the soils are residual soils derived
from the underlying rock. Under these conditions, infiltration may result in large volumes of water
entering what was initially an unsaturated soil slope. The infiltration may lead to the soil becoming
fully saturated, but equally might lead to an increased degree of saturation, without full saturation
being achieved.
Conventional 'saturated soil mechanics' limit equilibrium approaches to predicting the stability of
slopes seem inapplicable to such unsaturated events, with generated factors of safety bearing little
resemblance to the actual stability of the slope. More detailed numerical analysis approaches
provide a more thorough model of the slope behaviour, but if the soil is modelled fully saturated,
this too results in inaccurate predictions of the slope behaviour. This has been shown by the analysis
carried out into the Pak Kong slope failure, as described later ( see Chapter 3 ).
It is clear that to accurately model the behaviour of unsaturated slopes subject to infiltration, a
dedicated unsaturated approach is required.
Such unsaturated modelling has been undertaken by other researchers, but generally, the approach
has been to model the infiltration process in an unsaturated rigid media, to derive pore
pressures/suctions, then apply this resulting data into a separate analysis to determine the
deformation behaviour of the soil ( for example, Fredlund and Barbour 1992, Ng and Shi 1998 ).
The innovative aspect of this research project is in its attempt to model the infiltration into
unsaturated slopes as a fully coupled process, whereby both infiltration and deformation are
modelled as a single combined process.
Chapter 1
16
1.2. Scope of Project
The aim of this project is to permit the process by which rainfall infiltrates into unsaturated soil to
be better understood, and to enable this process to be modelled through numerical techniques. In
practice, this means developing an infiltration model and incorporating it into the Imperial College
Finite Element Program ( ICFEP ), along with a capability to undertake analysis of unsaturated
soils.
This thesis presents the development of such a model within ICFEP. The newly developed code has
then been used to study the infiltration process in a slope in Hong Kong for which monitoring data
was available. This has established the sensitivity of the process to various factors, such as the value
of the soil permeability relative to the rate of infiltration, the degree of saturation – suction
relationship, and the boundary conditions applied.
Those factors that were most critical in governing the variation due to rainfall of pore pressures
within an unsaturated slope have been identified, and some consideration as to how such factors
might influence the slope stability are presented.
Knowledge of the primary governing factors will be of practical use in rapid risk assessment of
slopes, whether prior to a forecast rainfall event, or after a storm, guiding those tasked with
ensuring slope safety as to the most likely failure locations.
1.3. Thesis Layout
Following this introduction, this thesis is arranged into a number of chapters:
Chapter 2 considers the nature of unsaturated soil.
Chapter 3 reviews the current understanding of the infiltration process, including how infiltration
affects slope stability and previous attempts to use numerical analysis techniques to model the
infiltration process.
Chapter 4 provides an outline of the numerical ( finite element ) capability that was available
within the soil mechanics section at Imperial College at the commencement of this project, with
particular reference to the aspects of ICFEP most relevant to this project; specifically, variable
permeability models and boundary condition options.
Chapter 1
17
Chapter 5 investigates the theory behind fully coupled unsaturated behaviour, and shows how the
equations describing the saturated behaviour of soil are really just a special case of the equations
detailing the more general behaviour.
Chapter 6 provides details of how the theory developed in chapter 5 was integrated into the
existing ICFEP code, to produce a practical tool for the study of infiltration into unsaturated soils.
This includes details of the soil constitutive model developed, the form of the degree of saturationsuction relationship, as given by a Soil Water Characteristic Curve ( SWCC ), and how the
permeability of the soil is described as a function of suction. Also included are those changes
necessary to the boundary conditions, to allow for accurate modelling of precipitation conditions,
and the variation of soil bulk density with varying saturation.
Chapter 7 discusses how the changes to the ICFEP code were validated through a series of simple
analyses.
Chapter 8 presents details of the slope at Tung Chung, on Lantau Island in Hong Kong, which was
used as the case study to investigate the infiltration process. A description of this slope is given,
along with details of the numerical analysis of the slope that was undertaken and the resulting
conclusions.
Chapter 9 concludes this thesis with a summary of the project and its key findings, and provides
recommendations for further study.
Chapter 1
18
CHAPTER 2
Unsaturated soils
2.1. Infiltration into unsaturated soils.
To accurately model the infiltration process and corresponding soil behaviour, it is useful to first
understand the terms and processes involved.
Bear (1979) presents a number of terms useful in considering soil moisture. The field capacity of a
soil is the water content of the soil after all gravitational drainage has ceased. Any water content
above this level, up to full saturation, is referred to as gravitational water. Marinho and Stuermer
(1998) define the field capacity as “the maximum water content a soil can hold or store under a
condition of complete wetting followed by drainage.”
Between the moisture content equal to the field capacity and that achieved when the soil is air-dried
( moisture content equal to the hygroscopic coefficient ), the soil water is referred to as capillary
water. This will not flow under gravity, but can be taken up by plant roots. Once the soil has been
air-dried, only hygroscopic ( or adsorbed ) water remains within it. This water is removed by oven
drying ( at 105° C ), at which point the moisture content is considered to be zero ( zero vapour
pressure ).
Bear (1979) also provides a vertical moisture distribution profile, as shown in figure 2.1. At some
depth lies the groundwater table; beneath this, the soil is fully saturated, and forms the ‘groundwater
zone’, or ‘zone of saturation’. The groundwater table is commonly assumed to be the ‘water
surface’ within the soil, but this is not strictly accurate. More correctly, the groundwater table is the
line of zero pore water pressure ( relative to the atmosphere ); hereafter, the alternate term for the
line of zero pore water pressure, ‘phreatic surface’, will be used, to emphasise this point.
Above the phreatic surface lies the zone of aeration. This is further subdivided into three: the
capillary fringe, the intermediate ( vadose water ) zone, and the soil water zone. Within the capillary
fringe, the soil retains a high degree of saturation ( >75% according to Bear 1979, though most
other references imply 85-90% as the minimum; for example Fredlund & Rahardjo, 1993a;
Chenggang et al, 1998 ), but pore water pressures are negative with respect to the atmosphere.
Significant groundwater flow may occur within the capillary fringe. The capillary fringe is also
Chapter 2
19
known as the tension-saturated zone ( Freeze and Cherry, 1979 ), reflecting the negative ( tensile )
nature of the pore water pressure.
Within the intermediate zone, the water content is normally at or below the field capacity, although
transient gravitational water flow may occur. In the soil water zone the moisture content varies
considerably in response to rainfall, other infiltration, evaporation and plant uptake. The moisture
content can range from fully saturated after heavy rainfall, to almost hygroscopic after a dry period,
with drainage by gravity and plant action.
This moisture profile is obviously idealised, and, while it is not explicitly stated, it seems reasonable
to assume that not all the described regions are necessarily present. A phreatic surface at or close to
the surface will likely inhibit the formation of the less-saturated zones, for example.
2.2. Soil suction and air pressure.
Below the phreatic surface, pore water pressures are greater than atmospheric air pressure, and
considered as positive water pressures. At the phreatic surface the pore water pressure is equal to
the atmospheric pressure, and above the surface, pore water pressures drop below atmospheric and
become negative. Such negative pressures are referred to as soil suction ( or pore water tension ).
Soil suction actually has two components, matric and osmotic ( or solute ) ( Fredlund and Rahardjo,
1993a; Richards, 1967 ). Osmotic suction is a measure of the difference between the partial pressure
of water vapour in equilibrium with pure water to that in equilibrium with the groundwater. It is
generally the result of chemical ( mineral salt ) content. Gardner (1961) states that the osmotic
component of suction is normally neglected. Richards (1967) argues that the effects of osmotic
suction can be very significant, but goes on to state that in the absence of salts, or for uniform
unchanging concentrations, osmotic suction can be neglected. Öberg (1997) provides additional
references to support this hypothesis.
Thus, for ‘straight’ geotechnical engineering problems such as conventional slope stability analysis,
not involving environmental aspects where pollutants and chemical gradients may be present, soil
suction may be considered synonymous with matric suction.
Matric suction is defined as “the difference between the pore-air and the pore-water pressures”
( Fredlund and Rahardjo, 1993b ), or:
ψ = ( ua – uw )
Chapter 2
Eqn 2.1
20
and is associated with capillary action; the process where interface tension between a wetting fluid
( for example water ) and a non-wetting fluid ( for example air ) creates a curve interface boundary
within a narrow opening, leading to a pressure difference between the two fluids. This process is
more fully described in many references ( i.e. Bear 1972, Fredlund and Rahardjo 1993a, Öberg
1997 ).
The magnitude of the pressure difference between the fluids is a function of the width of the gap
between the solid surfaces, be this the diameter of a glass capillary tube used in a laboratory
experiment, or the pore space between soil particles in a soil sample.
The equation for the height of capillary rise, assuming the non-wetting fluid is air, is given by
Lambe and Whitman (1979) as:
hc =
2Ts
cosα
Rγ
Eqn 2.2
Where:
Ts = Surface tension of the liquid
R = Radius of tube
α = contact angle made between the liquid and the tube
γ = unit weight of the liquid
Generally, if a fluid has a contact angle, α, less than 90°, it is called a wetting fluid, while a contact
angle greater than 90° indicates a non-wetting fluid. In a water – air interface, water is the wetting
fluid, with air being the non-wetting fluid. However, as noted by Bear (1972), ‘wettability’ is a
relative term, since it indicates the behaviour of one fluid in relation to that of a second fluid.
Since the height of the capillary rise is dependent on the radius of the void in which said rise is
occurring, it is clear that the capillary rise that will occur within a soil will be affected by particle
size and grading, since this affects the size of the voids ( or pores ) within the soil mass. This has
been illustrated by Lambe and Whitman, 1979 ( presenting data from Lane and Washburn, 1946 ).
Bear (1972) presents similar data.
Fine grained soils, with correspondingly fine grained pore spaces, will be able to sustain large
pressure differences between wetting and non-wetting fluids ( i.e., between pore water and air ), so
will logically permit large capillary rises. Coarse grained soils, with larger voids, will tend to
maintain lower pressure differences between air and water, resulting in a lower capillary rise.
However, the situation is further complicated by the grading of the soil: a uniformly graded soil
Chapter 2
21
would logically have uniform pore size, so will demonstrate a generally uniform capillary rise. A
non-uniformly graded soil will show less predictable and possibly more heterogeneous behaviour.
Possibly, the voids between the larger soil particles will be infilled by finer particles, resulting in a
relatively uniform pore size throughout the soil mass. However, it is more likely that there will be a
distribution of void sizes within the soil mass. In such a case, the capillary rise that occurs will vary
throughout the soil, being dependent on the distribution and connectivity of the pores.
2.2.1. Water content – suction relationship
It is well known that matric suction can be related to the water content of the soil, either as degree
of saturation, or volumetric water content ( see for example Bear 1979, Fredlund and Rahardjo
1993a, Öberg 1997 ). From a fully saturated state, water will initially drain from the larger pores.
As drainage continues, the differential pressure between air and water ( i.e. the matric suction )
increases, and the air-water interface moves into increasingly smaller pores.
The process of desaturation has been explained by numerous authors, for example Bear (1979),
Childs (1969) and White et al (1970), and is illustrated in figure 2.2.
Starting from a fully saturated state, the soil particles are completely submerged under water, with
the air-water interface forming a plane above them ( shown as stage 1 in figure 2.2 ). Since this
interface has no curvature, there can be no pressure difference between the two fluids ( air and
water ). Hence the water pressure on one side of the interface is equal to the air pressure on the
other; that is, the interface is the phreatic surface.
If water is allowed to drain from the sample, the interface will drop towards the soil particles, until
the boundary particles penetrate the interface. This is shown as stage 2 in figure 2.2.
At this point, the air-water interface becomes curved, and a difference in pressure now exists
between the air and water phases. This difference in pressure is a function of the radius of curvature
of the interface, as shown by equation 2.2.
Initially, the pressure difference between air and water is small, with the interface between them
showing a correspondingly large radius of curvature. Thus a meniscus forms between the outer
faces of the boundary soil particles, adjoining the particles at the contact angle.
Since the angle of contact is fixed, and the curvature of the air-water interface is a function of the
pressure difference between air and water ( high pressure difference results in small radius of
curvature ), at low suctions, the interface cannot penetrate deeper into the soil.
Chapter 2
22
However, as drainage continues, the water pressure drops further relative to the air pressure ( i.e.,
suctions increase ), and so the curvature of the interface between the phases increases. This allows
the interface meniscus to penetrate deeper into the soil, and to be no longer limited to the outer
boundary of surface-most soil particles ( see figure 2.2, stage 3 ).
As the water pressure reduces further, the radius of the air-water interface will become increasingly
tight, and so will penetrate into smaller and smaller voids within the soil.
Hence the tendency will be for the larger pores within the soil to drain first, then progressively
smaller ones. However, larger pores at depth within the soil will not drain until the air-water
interface is able to reach that pore.
This is illustrated in figure 2.3, which shows schematically a series of pores or voids within the soil,
all linked by a network of constricted width connecting paths of varying widths. Drainage
commences with void v1, then advances through the soil as suction increases and progressively
narrower constrictions drain. As can be seen from the figure, voids, regardless of their size, only
drain once the suction is sufficient to generate a radius of curvature for the interface meniscus that
is tight enough to penetrate the largest of the connections to that void.
The relationship between matric suction and degree of saturation ( or volumetric water content )
may be presented graphically, in the form of a soil water retention curve ( also known as a soil
moisture retention curve ( Childs 1969 ), or a Soil Water Characteristic Curve, SWCC ). Since
volumetric water content equals ( the degree of saturation multiplied by the porosity ), if the soil is
assumed to be rigid, porosity is constant, and the choice of whether to plot the curve in terms of
saturation or volumetric water content is largely immaterial. More care is needed in dealing with a
consolidating soil, where porosity is not constant, as discussed later in Chapter 6.
The form of these curves tends to be similar, regardless of soil type, and they are generally ‘S’
shaped, although in clayey soils this shape is much less well defined ( see Freeze and Cherry 1979,
and Freeze 1969 ), with no clearly defined residual degree of saturation ( see below ). Figure 2.4a
shows the typical form of the SWCC with principal characteristics, while Figure 2.4b shows some
soil specific data showing the effects of soil grading. It may be noted from Figure 2.4a that the
drainage ( ‘desorption’ ) and rewetting ( ‘adsorption’ ) paths in an SWCC are not the same; this
point is discussed later: see section 2.2.2.
Some initial small soil suction will normally be generated while the degree of saturation remains
high. This corresponds to stage 2 of figure 2.2, with a curved air-water interface that has not yet
truly penetrated into the soil mass. Once the suction is sufficiently high for air to be drawn into the
Chapter 2
23
soil and to become continuous, the degree of saturation will drop sharply as suctions increase. The
suction at which air is drawn in is commonly known as the Air Entry Value ( AEV ), though may
also be known as the air entry pressure or the bubbling pressure ( Brooks and Corey, 1964 ). The
AEV has been defined as “the pressure head necessary to overcome the capillary forces that can be
exerted by the largest pore in the medium.” ( Nicholson et al, 1989 ).
At suctions below the AEV, the air phase within the soil is either absent ( fully saturated soils ) or is
discontinuous, in the form of occluded air bubbles. Such bubbles may form from air trapped by a
re-wetting process, or from air that has come out of solution. At these suctions, airflow within the
soil is not possible other than by diffusion, or as discrete air bubbles carried by the flow of water
( Chenggang et al, 1998 ). There is no significant reduction in the volume of the pore water or in the
area of the soil through which water can flow; hence permeability of the soil is therefore not
significantly changed from the fully saturated case.
At matric suctions greater than the AEV, sufficient air has been drawn into the soil for it to become
a continuous phase. Airflow is now possible. More significantly, the volume of water within the soil
voids is reduced, and since water flow occurs in the water phase of the soil, the soil permeability to
water flow is reduced.
It can be seen that the AEV is comparable to the upper limit of the capillary zone. While Bear
(1979) indicated a degree of saturation of >75% for this point, 85% to 90% seems a more typically
accepted value ( Fredlund and Rahardjo 1993a, Corey 1957 ), though this value is almost certainly
dependent on the soil type ( Corey, 1957 ). Barden (1965) suggested that in compacted clays, the air
voids are continuous up to a degree of saturation of 90%. He also quoted Gilbert (1959), who stated
that in a similar material, air voids were fully continuous at moisture contents up to 4% below
optimum moisture content for compaction, and were fully occluded at a moisture content of 3%
above optimum.
Lambe and Whitman (1979) present an illustrative example, showing a soil column either draining
under gravity or wetting up through capillary rise from a water source at the base. They present
values for the ‘saturated capillary head’ under drainage, and the ‘minimum capillary head’ under
wetting, which are clearly equivalent to AEVs. It is notable that these two values are not equal to
each other. This illustrates an important aspect of unsaturated soil behaviour, namely that of
hysteresis between wetting and drying behaviour. This aspect of behaviour will be addressed later.
Once the magnitude of the suction is above the AEV, the rate at which the degree of saturation
decreases with increasing matric suction depends on the grading of the soil. In a soil where all the
Chapter 2
24
pores are of approximately the same size, the degree of saturation will drop sharply over a small
suction range. Where the soil void size shows a range of values, drainage will commence with the
larger voids, then proceed to the smaller pores as the suction increases ( Childs, 1969 ). Thus the
degree of saturation will decrease more slowly as the suction increases.
Eventually, the degree of saturation will reach some value at which the water phase becomes
discontinuous. This is the irreducible water saturation, or residual degree of saturation, and is
effectively the field capacity of the soil. It should be noted that Loret and Khalili (2000) suggest that
the residual saturation is not actually a true constant soil property, but varies with temperature.
Once the water phase is discontinuous, the permeability of the soil to water is tending to zero, and
little moisture movement can occur; movement of moisture is now through the vapour phase
( Childs, 1969 ). Many references show the soil water retention curve within an ‘engineering range’
of soil suctions, up to several thousand kPa, and plot the curve to be asymptotic to the residual
degree of saturation, implying that saturation can not be reduced below this level. Fredlund and
Xing (1994) showed that further reductions in the degree of saturation are possible, however very
large increases in soil suction are necessary to decrease moisture content further. They showed that
regardless of soil type, a matric suction of approximately 1 000 000 kPa would normally produce
zero moisture content.
2.2.2. Hysteretic behaviour.
In describing the soil suction-water content relationship, the process of desaturation was outlined.
Clearly, a soil that desaturates can also wet up, and given that this thesis is a study of infiltration
into unsaturated soils, this wetting up process needs to be considered further.
The basic features that govern drainage are still applicable to the wetting process. The pressure
difference between air and water phases remains proportional to the radius of curvature of the
interface, and in the same way that air penetrates preferentially the large void spaces and
connections during drainage, wetting and decreasing suction will result in flooding of the smaller
voids first.
However, it is the case that the relationship between water content and suction is not unique, but is
hysteretic, the relationship depending on whether a wetting or drying path is being followed. The
form is typically like that shown in figure 2.5.
Chapter 2
25
Drainage of a soil sample will cause a decrease in water content, and an increase in suction. If
drainage is terminated and wetting commenced the suction will be reduced, but at any given suction
the water content of the soil will now generally be less than that for the same suction on the
drainage path. This can result in a soil with zero suction ( “fully wetted” ) showing less than 100%
saturation. There are a number of reasons for this hysteretic behaviour.
The first is known as the raindrop effect ( see figure 2.6a ), which is the term used to describe how
an advancing interface has a contact angle that is different to that of a receding interface between
the same two fluids ( see Bear 1972 ). While the figure shows the effect as a ‘raindrop’ of water
advancing down-slope in an air atmosphere, this phenomenon applies wherever there is an
advancing or retreating interface between a wetting and a non-wetting phase, including when such
advance is horizontal ( as shown by Bear ).
Generally, the contact angle of an advancing interface ( θ1 in figure 2.6a ) is greater than that for a
receding interface ( θ2 ), and hence an advancing interface ( water advancing / cell infilling ) will
have a greater radius of curvature than a receding ( draining ) interface ( Childs, 1969 ). Hence, in a
pore channel of any given size, the larger radius of an advancing interface implies a smaller value of
suction than would occur at the same spot if the interface was retreating ( which would involve the
interface being correspondingly more tightly curved ).
The second factor is the ink-bottle effect ( see figure 2.6b ).
The air-water interface under drainage conditions penetrates into steadily smaller constrictions, as
the suction increases. As illustrated earlier, if a large pore lies behind a narrow constriction, once
the interface penetrates that constriction, the whole pore will drain ( see also figure 2.7 ).
On re-wetting, the suction will decrease, and the radius of curvature of the interface meniscus
increases. Thus the interface moves ‘back’ towards the wider part of the void ( figure 2.7b ).
However, for the pore to fully flood, the interface must pass back through the original entry
constriction, which requires a tightening of the radius of curvature, which in turn implies that it is
necessary for a local increase in suction ( Bear, 1979 ).
This leads to a third cause of hysteresis. If the pore behind the constriction through which the
interface originally penetrated floods before the void shown in the figure does, then there ceases to
be an air-path out of that void. Any air remaining in that void ( as, for example, is shown in figure
2.7b ) will remain trapped in the soil. Philip (1957b) supports this behaviour.
Thus the process of wetting is not the mirror image of drainage, and can also result in bubbles of
trapped air remaining in the soil ( Childs 1969 ). Hence it is possible for a ‘fully wetted’ soil, with
Chapter 2
26
zero suction, to have a degree of saturation that is less than 100%. With time, the saturation will
tend towards 100%, as the trapped air will escape through diffusion ( Brooks and Corey, 1964 ), or
through being carried by the flow of water ( Bear, 1972 ). Childs and Collis-George, 1950, also
refer to the solution of entrapped air, while Le Bihan and Leroueil’s (2002) work also supports the
hypothesis that entrapped air will be subject to dissolution under water pressure, then be transferred
by water flow.
However, even in a real field situation, it is possible for ‘fully wetted’ soil, beneath the position of
the phreatic surface, to have less than 100% saturation. Bouwer (1964) makes the point that “field
soils are seldom, if ever, saturated below the water table ( except perhaps for long-standing
aquifers )”.
If a series of full wetting and drying cycles are imposed on a soil, the water content-suction will
tend to form a hysteretic loop; while the first re-wetting cycle will tend to produce less than full resaturation, continued drying and wetting will result in a generally closed loop for the water contentsuction relation ( Dineen, 1997 ).
Partial wetting after drainage, followed by a second drainage stage will produce scanning curves
that lie between the wetting and drying plots; see Figure 2.5.
The above has implicitly assumed that the soil structure is rigid. Various authors ( for example,
Bear, 1979 ) have stated that volume change of a soil due to changes in water content can also
contribute towards hysteresis, particularly in the case of cohesive soils. The implications of nonrigid soil behaviour are considered more fully later in this thesis.
2.2.3 Air flow and the treatment of the air phase
From previously, it is clear that in an unsaturated soil, the air phase plays an important part in
governing the soil behaviour.
In a saturated soil, the soil mass is two-phase: the soil solid particles, and a pore fluid, which is
composed entirely of water. In an unsaturated soil, the soil mass becomes three-phase: The soil
solids as before, plus the pore fluid, which now consists of a water phase and an air phase. Within
the unsaturated soil, the air and water phases will be separated by a meniscus, the radius of which
depends on the pressure difference between these two phases, as explained earlier. Some sources
( for example, Fredlund and Hasan, 1979 ) suggest that this meniscus counts as a fourth phase,
though there appears to be no significant advantage, or difference, in counting it as such.
Chapter 2
27
To fully and accurately model unsaturated soils, it is clear that both the air and water phases should
be treated separately, with pressure and flow of both of these phases tracked within the model. In
looking at the behaviour of unsaturated soils, some authors ( e.g. Dakshanamurthy et al, 1984 )
incorporate airflow within the soil, and it is clear that this aspect can be significant to the overall
behaviour of the soil.
However, it is widely accepted that the air phase is free to flow and may be assumed to be at
atmospheric pressure ( Morel-Seytoux, 1973, states that this is “traditional”; see also Freeze and
Cherry, 1979; Lam, Fredlund and Barbour, 1987; Fredlund and Barbour, 1992; Ng and Shi, 1998;
Wong, Fredlund and Krahn, 1998; Zhujiang, 1998 ). There is some indication that the flow of air
out of the soil may affect the flow of water into the soil ( Cabarkapa, 2000 ). While Bear (1973)
reports that the effect of air flow is generally negligible, Morel-Seytoux (1973) states that the flow
of air, escaping from a soil as it wets up, will affect the flow of water within the soil. He shows how
this “counterflow” of air can reduce the rate of infiltration at the surface, with the near surface soil
becoming desaturated because of this air flow.
Also if air can become trapped by the infiltrating water, this can affect both the permeability of the
soil to water flow, and lead to an increase in air pressure ( Freeze and Cherry, 1979 ). Bicalho et al
(2000) discuss how air entrapped on wetting of a soil affects both the SWCC and the hydraulic
conductivity of the soil at high degrees of saturation. Further, Vachaud et al (1974) showed how, in
a vertical column, ponded infiltration produces an advancing wetting front that initially causes
compression of the air phase immediately under the soil surface. This leads to a reduction in the rate
of water infiltration.
The air pressure will increase until it reaches a sufficient value for the air to escape by bubbling to
the surface, this being Morel-Seytoux’s “counterflow”. The flow of air within the soil occurs
through voids that would otherwise be water filled, and so reduces the flow area available for water
flow. Hence, downward infiltration into the column is reduced by the upward flow of air.
Additionally, this upward movement of air may disturb the structure of the soil.
Vachaud et al (1974) demonstrated how this ‘column’ of airflow remains open as the wetting front
penetrates deeper into the column, allowing air flow from the depths of the column to reach the air
boundary surface. However, the flow path available to the air is restricted to a very limited area;
hence the air permeability of the soil is limited. The effect of this is to produce a rise in the air
pressure within the soil, since inflow of water tends to compress it, raising the pressure, quicker
than the pressure can vent through “counterflow”.
Chapter 2
28
Clearly, as shown by equation 2.1, if the air pressure rises, then if the water pressure is unchanged,
matric suction increases. Vachaud et al (1972) considered this issue, and found that “the local soil
air pressure can be very significantly different from the external atmosphere pressure”, and that
such differences would have a large effect on water flow. Further, they found that where air was not
continuous to an atmospheric boundary, or where flow to such a boundary was restricted, the air
pressure would rise, and that this would lead to a direct effect on the suction, which must be taken
into account when considering any suction measurements.
Kisch (1959) considered a multi-layered problem, with an unsaturated sand layer underlying a
( mostly ) saturated clay. He investigated the difference in behaviour between the situation where
the air phase in the sand was at atmospheric pressure, and where it was at some raised value ( equal
to 30cm pressure head of water ). He found that the effect of raising the air pressure was to cause a
corresponding rise in the pore water pressure, such that matric suction was unchanged. Thus the
saturation profile within the soil was also unchanged.
However, the phreatic surface ( the level at which pore water pressure equals atmospheric pressure,
equal to zero; not the level at which pore water pressure and pore air pressure are equal ) rose by
30.6cm. Thus the phreatic surface does not indicate the position of zero suction, if soil air pressure
is greater than atmospheric pressure.
Brooks and Corey (1966) found that the permeability to air flow along an unsaturated stratified test
column was restricted due to some layers remaining “saturated at some capillary pressures”, but air
flow remained possible, since an air-flow permeability was still recorded.
Corey (1957) produced a graph showing relative air ( and water ) permeability, expressed as a
percentage of the maximum permeability ( fully air-saturated and fully water-saturated,
respectively ), in relation to the degree of water saturation ( see figure 2.8 ). This shows, for the
particular sample tested – a very sandy soil – that the relative permeability to air flow does not
become significant until the relative permeability to water tends to zero. That is, the permeability to
air flow is significantly reduced until the water phase becomes discontinuous ( refer back to section
2.2.1 ). Further, Blight (1971) showed air pressure equalisation in a nearly saturated soil took 1000
times longer than when in a dry but otherwise equivalent soil, clearly indicating a impediment to air
flow in the nearly saturated soil.
However, the ability for air to flow freely is clearly dependent on the absolute permeability of the
soil to air flow, not on what percentage of its theoretical maximum it is. Thus, while a soil close to
full water-saturation would be expected to have a low permeability to air flow, it seems likely that
Chapter 2
29
once the suction has increased beyond the AEV and air has been drawn in from an atmospheric
boundary, air flow would be relatively unimpeded. Fredlund and Rahardjo (1993a) present a
compilation of data showing the absolute ( dynamic ) viscosity of water and air at various
temperatures, and this shows that air is, very roughly, just under two orders of magnitude less
viscous than water, and will correspondingly flow that much more readily. Koorevaar et al (1983)
support this point, stating that the air conductivity of a soil is about 50 times its water conductivity
( assuming no change in soil structure ), the ‘times 50’ factor being the difference between air and
water dynamic viscosities, while Forsyth (1988) in his example specifies exactly two orders of
magnitude difference.
Morel-Seytoux (1973) considered both a short soil column, closed at the base, and a long column,
open at the bottom. He found that the assumption of incompressible air was reasonable in the latter
case, and had no significant effect on the water flow behaviour, but that in the former case, water
flow was significantly influenced by both the flow of air, and the increase in air pressure. The
implications of this are that if the air phase within the soil has a flow path laterally, or vertically
down away from the infiltration source, or just if there is sufficient air volume within the soil that a
small volumetric compression causes minimal air pressure change, then restrictions on air flow, and
the effects of increased air pressure may be ignored.
This is in keeping with Vachaud et al (1974), who found that the “counterflow” of air occurred
when lateral air flow was prohibited.
It is questionable whether a real site, probably with an undulating ground surface, and possibly
sloped, when subject to infiltration, would act like a simple test column with no lateral air flow. It
seems more likely that the real world situation would quite readily allow air flow laterally.
Also, in the case of infiltration due to rainfall, the soil surface would not instantaneously be
submerged, as per a ponded case; rather, the infiltration initially will be from discrete raindrops
arriving at the soil surface. Thus at least until the surface layer of soil has become saturated, there
will be a flow path open, through which air within the soil may escape. It should be noted that this
behaviour is not generally modelled; rather, as stated by Rubin and Steinhardt (1963), rain is
normally taken as a surface body of water entering the soil at some prescribed rate, although they do
acknowledge the veracity of the discrete raindrop concept.
The assumption that the air is free to flow allows the air pressure to be set at 0 kPa ( relative to
atmospheric pressure ), and for air flow to be ignored, which simplifies things greatly. As discussed
in section 2.2.1, air is drawn in from an air boundary once the matric suction reaches the AEV,
Chapter 2
30
which is typically around 90% saturation, and above this value of saturation, the soil's behaviour is
likely to be close to the fully saturated case. If it is assumed that the only air in the soil is that which
has been drawn in from an air boundary, free flow of air is a reasonable assumption, as is continuity
with the atmospheric air, hence the soil air pressure may reasonably be set equal to atmospheric
pressure.
In reality, there will be occluded air bubbles within the soil, and these will not necessarily be at
atmospheric pressure. The assumption, however, is that soil bubbles have an insignificant effect on
the overall behaviour of the soil mass.
It is recognised that under certain circumstances, for example in a sludge tip or landfill capping
scheme, where positive gas pressures are being generated within the soil mass, then the assumption
of zero air (gas) pressure will be incorrect. However, this is outside the scope of this research
project.
Chapter 2
31
Figure 2.1: The distribution of subsurface water
( after Bear, 1972 )
32
Stage 1
2
2
3
5
5
4
5
5
5
Figure 2.2: Penetration of air-water interface into soil
( after Childs 1969 )
33
D0 > D1 > D2 > ...
D1
D2
V2
D5
D0
D1
V12
D5
V11
Entry point of
the non-wetting
phase
V10
V13
D3
D5
D4
D1
V14
D5
V9
D1
V5
V4
D5
D2
Capillary pressure high enough
for air-water interface to
penetrate constriction width:
D5
D1
V1
D1
D4
V3
D0
D0
V6
D5
V8
D4
V7
D5
Pores desaturated:
D0
V1
D1
V2 V11
+ V1
D2
V3 V8 V9 V10 V12 V13
+V1 V2 V11
D3
None new
+V1 V2 V3 V8
V9 V10 V11 V12
V13
D4
All
Figure 2.3: Schematic diagram of hypothetical porous medium
( after White et al, 1970 )
34
Figure 2.4a: Typical form of the SWCC ( after Fredlund and Xing, 1994 )
Soil suction
Equivalent
head of water
1: Dune sand
2: Loamy sand
3: Calcareous fine sandy loam
4: Calcareous loam
5: Silt loam derived from loess
6: Young oligotrophous peat soil
7: Marine clay
8: Eutrophous peat soil
9: River basin clay
-100000.0m
-10000.0m
-1000.0m
-100.0m
-10.0m
-1.0m
-0.1m
Volumetric water content
Figure 2.4b: SWCCs for some Dutch soils ( after Koorevaar et al 1983 )
Figure 2.4: The Soil Water Characteristic Curve
35
NOTE: θWO equivalent to the residual water content
Figure 2.5: Hysteretic SWCC showing scanning curves
( after Bear 1979 )
36
θ2
θ1
Figure 2.6A: The raindrop effect
Rewetting
Drainage
Figure 2.6B: The Ink-bottle effect
Figure 2.6: The raindrop and ink-bottle effects ( after Bear 1979 )
37
If this pore re-fills with
water from an alternate
source, air in pore below
becomes trapped
Advance of air into soil pores
Figure 2.7B
Figure 2.7A
Figure 2.7: Movement of air-water interface within soil
pores
38
Notes:
krw indicates relative permeability to water flow.
kra indicates relative permeability to air flow.
Surface indicates test data from soil taken from the ground surface layer
Subsurface indicates test data from soil taken from slightly deeper
Figure 2.8: Relative permeability to air and water as a
function of saturation ( after Corey, 1957 )
39
CHAPTER 3
Slope instability due to infiltration
3.1. The Infiltration Process and wetting front development.
When rainfall infiltrates into a soil slope, it will clearly increase the moisture content of the soil
above the phreatic surface ( and hence reduce the suctions present there ), but as the water flows
downward, it may also result in a rise in the position of the phreatic surface. Such a rise could be
the cause of some slope instability, but as has been proven, for example by Boosinsuk and Yung
(1992), failure may be induced by direct rainfall, rather than by rising groundwater.
The process of infiltration into a soil has been studied by a number of authors, for example Philip
(1957a), Lumb (1962a). They show that the processes at work include diffusion of water due to a
moisture content differential, but also a pressure effect ( arising from gravity potential and
pressure differential ), which is sensitive to the ( saturated ) coefficient of permeability of the soil.
Lumb (1975) suggests that for prolonged infiltration, the diffusion element becomes insignificant
compared to the pressure effect, and so may be ignored. It is stated that for unsaturated soils, "the
rate of infiltration will be greater than the permeability, but will soon decrease". It is assumed that
Lumb is referring to the saturated permeability, and it should also be noted that this was for a flat
surface, not a slope. The ability for infiltration to exceed the saturated permeability is presumed to
be a reflection of the water storage capacity of the soil.
Sun, Wong & Ho (1998) discuss this issue and show more clearly how the influence of the
diffusion element decreases with time. The rate of infiltration through a soil therefore tends to the
fully saturated permeability of the soil. From Sun et al's assessment of Lumb (1962a), the
infiltration process is controlled by time, the fully saturated permeability, the porosity of the soil
and the change in the degree of saturation.
Both Lumb and Sun et al appeared to be interested primarily in the development and advance of a
wetting front, across which there is a marked change in degree of saturation. The wetting front as
proposed by Lumb (1962a) is illustrated in Figure 3.1.
Chapter 3
40
Under infiltration conditions, the soil becomes fully saturated over a very small depth from the
surface, then the degree of saturation drops to a ‘wet value’ of 80 to 90%. This degree of
saturation is maintained until the ‘wetting front’ is reached, at which, the degree of saturation
drops sharply to the original value.
Bodman and Colman (1943), quoted both by Bear (1972), and Youngs (1957), present a similar
form ( see Figure 3.2 ), with a saturated zone extending to approximately 1.5cm below ground
surface. Below this is the transmission zone, in which moisture content is approximately constant,
with a degree of saturation of about 80%. This is followed by the wetting zone, where moisture
content drops, then the wetting front, which “represents the visible limit of moisture penetration
into the soil”.
Youngs (1957) undertook a series of vertical infiltration tests, using either slate dust ( particle size
0.04mm to 0.125mm ), or grade 15 Ballotini ( glass beads, less than 0.1mm in diameter ). He
found that with these materials, there was a uniform degree of saturation within the ‘wet zone’,
with no reduction in saturation over the transmission zone. He suppositioned that this was due to
little air becoming entrapped during the wetting process.
This implies that the exact form of the wetting front profile is partly a function of the grading of
the soil. Referring back to section 2.2.2, it is clear that a uniformly graded soil, tending to have a
uniform pore size, is less likely to entrap air on wetting up than a more varied soil.
Sun et al and Lumb seem less obviously concerned with the implications that the wetting front has
for pore water pressures or suctions beyond whether the soil has become fully saturated or not.
Little attempt is made to relate the wetting front to the overall behaviour of the soil mass, other
than that resulting from the application of saturated pore pressures.
Lumb (1962a) does link the advance of the wetting front to a loss of suction and hence a loss of
strength, through an expression for the shear strength of unsaturated residual soils in Hong Kong
( see Lumb 1962b ). True fully coupled behaviour is not considered.
Sun et al's 1998 work developed the wetting front approach to determine the long-term variation
in suction within a soil profile. They found that 'partially saturated' wetting bands, where the
advance of a wetting front results in the reduction of the magnitude of the suction, without fully
destroying it, were feasible and, depending on the soil properties, even more likely than a fully
saturated wetting front. This point will be considered further in section 3.8. For infiltration flow
Chapter 3
41
into unsaturated soils, they showed that the critical factors governing behaviour are the
permeability-suction relationship, the initial suction condition, and the duration and intensity of
the rainfall.
3.2. Antecedent rainfall
The effects of initial suction and rainfall duration / intensity are interrelated, and so are considered
together here. If the initial suction within the soil is a controlling factor, it would seem likely that
rainfall immediately prior to the storm event being considered would influence behaviour, since
such rainfall would presumably influence the suctions present in the soil.
Various authors ( e.g. Lumb 1975, Fourie 1996 ) have suggested that rainfall induced failures are
related to the duration and intensity of the antecedent rainfall ( up to 15 days prior to the failure
event ), in addition to the intensity of the ‘trigger’ storm. These findings have been contested by
Brand (1985), who pointed out that Lumb’s work was based on daily rainfall data at the Hong
Kong Royal Observatory, not from data collected local to the landslide locations. Brand used both
1 hour and 24 hour rainfall data, from rain gauges near to recorded land slides, and found that
antecedent rainfall had no significant effect on major landslide events in Hong Kong, although he
conceded that 3 to 4 day antecedent rainfall does seem significant to minor landslides. Brand's
data indicates that major events are due to short duration, high intensity rainfall events, with few
major failures if the hourly rainfall is less than 40mm per hour, and most casualty causing
landslides occurring when rainfall exceeded 70mm per hour.
The 24-hour rainfall data also showed reasonable correlation to frequency of slope failure, and
Brand suggested that this data might be a more useful indicator of impending failure. The
occurrence of a single, high, rainfall event ( i.e., the 1-hour rainfall rate ) is difficult to predict but
after several hours of rain, it is possible to estimate an approximate magnitude for the 24-hour
period.
Brand suggested that the 24-hour rainfall data is significant because a high 1-day rainfall will
normally be indicative of a period of high 1-hour rainfall. Hence, it is the high hourly rainfall that
is actually significant, and the 24-hour rainfall data is merely a marker reflecting this, rather than
being significant in its own right. Au (1998) gives data which supports this idea, stating that 1527% of the 24 hour rainfall occurred in 1 hour ( and 50-90% of the 24 hour rainfall occurred in 8
hours ) for a number of ‘cloudburst’ rainstorms.
Chapter 3
42
Where rainfall rates are lesser, Brand found that the antecedent effects built up over 3 or 4 days
may be enough to trigger minor, shallow, failures.
Ng and Shi (1998) carried out a numerical investigation of slope instability due to rainfall. They
found that a “single threshold rainfall intensity” did not provide a reliable warning of land
movement, and that a “critical rainfall duration” proceeding the failure, of between 3 and 7 days,
was also applicable to failure events.
Vaughan (1985) suggested that a decreasing permeability with depth may be partly responsible
for creating threshold event rainstorms which trigger failure in what had previously been
considered safe slopes. This idea does not preclude the possibility of antecedent rainfall also
influencing the slope behaviour, and in his case study from Fiji, Vaughan refers to failure
following “an unusually long period of exceptionally heavy rain” which lasted about 3 days.
Crozier and Eyles (1980) suggest that antecedent rainfall is a controlling factor in slope failure.
Their model generally requires an “antecedent excess rainfall” implying soil saturation must be at
least equal to the field capacity for failure, though an example is given where heavy rain triggered
a slope failure despite high negative pore water pressures. They hypothesised that this was due to
a high infiltration rate combined with low soil permeability, leading to perched groundwater, and
locally high pore water pressures.
Malone and Shelton (1981) looked at landslides in Hong Kong, and concluded that antecedent
rainfall is a relevant factor, in as much as it can affect prevailing soil moisture conditions, and
groundwater recharge. Their work was based on that of Lumb (1975) and Crozier and Eyles
(1980). Kay and Chen (1995) found that both 'hourly' and 'daily' rainfall were significant when
attempting to predict failures of slopes in Hong Kong.
Dai and Lee (2001) looked at rainfall and landslide occurrence in Hong Kong, and found that
‘significant’ landslides ( of 4 cubic metres or greater failure volume ) were best predicted by
reference to the 12 hour cumulative rainfall; however, for ‘large’ landslides ( of 30 cubic metres
or greater failure volume ), the 24 hour cumulative rainfall produced far better predictions. They
make the important point, however, that the occurrence of a landslide is influenced by many
factors other than rainfall, and hence each failure is case specific.
Ayalew (1999) supports this last point. While he found some evidence of a relation between
failure and the ratio of precipitation prior to failure relative to mean annual rainfall, he stated that
Chapter 3
43
“it is well known that rainfall affects the stability of sub-surface materials, but in view of the other
factors that can also influence the mechanisms of landslide generation, the importance of rainfall
is difficult to determine”.
Despite this, Ayalew noted that in Ethiopia, more landslides generally occur in September than in
July, despite July being wetter, while as many failures occur in October as in June, despite “the
amount of rainfall in October is almost negligible”.
This appears to indicate that there may be some time delay between a rainfall event and its effect
on the stability of the ground. Whether this is due to antecedent rainfall leading to a build up of
moisture within the soil, or whether it is a reflection of the movement of water, in some form of
wetting front, is not clear. It is questions such as this that provoked this research project.
Finlay et al (1997) looked at rainfall data for a whole range of time periods, and found that for
Hong Kong, the rainfall over a 1 to 12 hour duration was important for prediction of the number
of failures, with antecedent rainfall also having some influence. They found that the 1 hour
rainfall data gave the best indication of the ‘threshold rainfall’ to cause failure of any ‘single
period of time’ data, but that including the 15 minute, 24 hour, and 30 day rainfall data made their
predictions more accurate.
To determine the number of landslides, the best prediction was achieved using 3-hour rainfall
data, but the estimate was improved if 15-minute through to 30 day data was included. As above,
they found that there is a large degree of uncertainty in predicting landslide failures, due to the
numerous other variables that impact on slope stability, such as slope geometry or the geology.
Dissanayake et al (1999) looked at land sliding in Sri Lanka, and found that there was good
correlation between weekly cumulative movements and one-week cumulative rainfall, particularly
as compared to 2 or 3-week cumulative rainfall. This was, however, for a re-activated slide.
Fredlund and Barbour (1992) presented a model where the suctions developed were dependent on
the initial steady state conditions, and the intensity and duration of the storm event. In this model,
a ‘low intensity’ rainfall was applied over a period of days. This case clearly blurs the distinction
between the trigger event and antecedent rainfall. It raises the question that, if rainfall is
continuous but variable over several days, where is the cut-off between the start of the trigger
event and antecedent rainfall? The Fredlund and Barbour (1992) model also brings in the idea of
Chapter 3
44
‘steady state conditions’. If changes to the soil suctions are to be modelled, it is necessary to first
determine the initial suctions and soil moisture contents.
Fredlund and Rahardjo (1993b) presented an illustrative figure showing how the pore water
pressure / suction profile can vary with depth, reproduced here as Figure 3.3. Given a no-flow
surface boundary, the suction profile will tend in the long term to hydrostatic, in equilibrium with
the position of the phreatic surface. However, the effects of rain, evaporation and transpiration
make no-flow boundaries untypical of real world field situations.
Clearly, the initial pore water pressure / suction conditions selected for an analysis must be
determined on a case by case basis.
3.3
Water flow in an unsaturated soil
The flow of water within a fully saturated soil is normally taken to behave in accordance with
Darcy’s law, which for one-dimensional flow has the form:
vx = − kx
∂h
∂x
Eqn 3.1
Where
vx : Flow ( velocity ) of water in x direction
kx: Coefficient of permeability ( for water flow ) in the x direction
∂h : hydraulic gradient in the x direction
∂x
It is generally accepted ( for example Richards 1967, Childs 1969, Freeze and Cherry 1979,
Koorevaar et al 1983, Ng and Shi 1998 ) that Darcy’s law, and its associated assumptions, remain
applicable to flow through unsaturated soil, except that it now takes the form:
vx = − kx(ψ )
Chapter 3
∂h
∂x
Eqn 3.2
45
Where kx(ψ) is now the coefficient of permeability, as a function of suction. The issue of suctiondependant permeability is discussed further in section 3.4.
Freeze and Cherry (1979) developed an equation for continuity of flow for transient flow through
an unsaturated soil, incorporating Darcy’s law in its unsaturated form, as shown in Eqn 3.3.
∂ 
∂h  ∂ 
∂h  ∂ 
∂h  ∂θ
k (ψ )  +
k (ψ )  +  k (ψ )  =


∂x 
∂x  ∂y 
∂y  ∂z 
∂z  ∂t
Eqn 3.3
Where θ represents the volumetric moisture content ( equal to the volume of water divided by the
total volume of the soil unit ).
Equation 3.3 was derived based on the assumption that the water is incompressible, and that the
porosity of the soil is constant ( that is, the soil is rigid ). Neither of these assumptions is
necessarily true, but Freeze and Cherry argue that changes in either water density or soil porosity
are small compared to changes in the degree of saturation, and may therefore be ignored.
A similar equation is presented by Ng and Shi (1998), as shown in Eqn 3.4.
∂  ∂h  ∂  ∂h 
 kx  +
 ky  + Q =
∂x  ∂x  ∂y  ∂y 
 ∂θw 


 ∂t 
Eqn 3.4
Equation 3.4 is fundamentally the same as equation 3.3 ( the different authors use slightly
different notation ), but it may be noted that Ng and Shi present this equation in its twodimensional form, and also that they introduce the additional term Q, to allow for an applied
boundary flux.
Potts and Zdravkovic (1999) present a form of the continuity equation for saturated soils, as
follows:
∂vx ∂vy ∂vz
− ∂εv
+
+
−Q=
∂x ∂y ∂z
∂t
Substituting Darcy’s law ( equation 3.1 ) gives:
Chapter 3
46
Eqn 3.5
∂  ∂h  ∂  ∂h  ∂  ∂h 
∂εv
kx  +
ky  +  kz  + Q =


∂x  ∂x  ∂y  ∂y  ∂z  ∂z 
∂t
Eqn 3.6
The right hand side of equations 3.5 and 3.6 represents the volumetric strain in the element across
which flow is being measured, and hence, unlike Freeze and Cherry, allows for changes in the soil
porosity ( non-rigid behaviour ). In a rigid soil, this term would go to zero.
Thus it is clear that the fundamental differences in the continuity equation between the saturated
form and the unsaturated form are the change to a suction dependant permeability, and the
addition of a term allowing for changes in the volumetric water content over time.
It is clearly logical that an element of soil within the unsaturated zone can wet up or dry further,
and therefore may be subject to variation in its moisture content ( and hence its degree of
saturation ).
Freeze and Cherry re-wrote equation 3.3 to give equation 3.7, below, in which form it is known as
the Richard’s Equation.
∂ 
∂ψ  ∂ 
∂ψ  ∂ 
∂ψ
 ∂ψ  
k
ψ
+
k
ψ
+  k (ψ ) 
+ 1  = C
(
)
(
)






∂x 
∂x  ∂y 
∂y  ∂z 
∂z
∂t

Eqn 3.7
Where ψ = pore water pressure ( = suction, assuming zero air pressure )
And C = Specific moisture capacity, such that:
C=
∂θ
∂ψ
Eqn 3.8
The right hand side in either equation 3.3 or 3.7 represents the change in water storage within the
soil, and this behaviour is reflected by the slope of the SWCC. The SWCC shows how the water
content ( or degree of saturation, if the curve is in that form ) varies with varying suction. The
slope of the curve indicates the quantity of water that flows in or out of a soil element in response
to a given change in suction ( for example, Lam et al, 1987, Ng and Shi, 1998 ).
Chapter 3
47
In a fully saturated soil, the degree of saturation is constant, and the volumetric water content is
simply equal to the porosity of the soil. Thus, in a rigid element, the volumetric water content is
constant, while in a consolidating analysis, the volumetric water content changes as the soil
porosity does ( assuming solid, incompressible, soil particles ).
It is evident that the saturated continuity of flow equation is really just a specific case of the
unsaturated version of the equation, and that there is no fundamental difference in this aspect of
the flow behaviour.
3.4
The permeability – suction relationship
It has been well established ( for example, Bouwer, 1964, Freeze and Cherry 1979, Ng and Shi,
1998 ) that the coefficient of permeability within the unsaturated region is a function of the pore
water pressure ( i.e. a function of the matric suction ).
As was shown in chapter 2, the desaturation of soil and development of matric suction is
associated with voids within the soil becoming air filled, rather than water filled.
The flow of water in its liquid form through soil occurs through the water filled voids only, so as
the number of air filled voids increase, the available flow path for water flow decreases. Hence,
the effective permeability of the soil to water flow decreases. Bouwer (1964) found that the more
uniform the pore sizes of the soil, the more abrupt is the reduction in permeability with increasing
suction.
Once the water phase within the soil becomes discontinuous, water will no longer be able to flow
in its liquid form; moisture movement will now only be possible in its vapour phase, which occurs
at a much-reduced rate compared to liquid phase flow ( Childs, 1969 ). For this reason vapour
flow is generally ignored while fluid flow is occurring.
Irmay (1954) presented data which supports this, showing that the relative permeability of the soil
to water flow ( that is, the actual permeability divided by the maximum, fully saturated
permeability ) was “practically negligible” below some threshold degree of saturation ( given as
10 to 20% for the data presented ). Irmay further confirmed that flow of this remaining water is
through the vapour phase, requiring an “evaporation process”.
Chapter 3
48
While the general effect of suction on the permeability of a soil is therefore clear, the difficulty
lies in determining the precise relationship between matric suction and water permeability.
Numerous expressions for this relationship have been published, many empirically derived. Such
relationships may give the permeability as a function of suction, a function of degree of
saturation, or a function of a SWCC equation.
For example, Corey (1957) looked at the relation between saturation and permeability ( for both
air and water ) as saturation changed through drainage, in a very sandy sample.
He found that permeability to water flow dropped sharply once air entered the sample, and
became very small at a degree of saturation somewhat greater than zero. This saturation being the
residual degree of saturation, the point at which the water phase becomes discontinuous.
Corey defined the effective saturation, Se, as:
Se = Sw – Sr
1.0 - Sr
Eqn 3.9
Where:
Sw = the actual degree of saturation
Sr = the residual degree of saturation
Corey then proposed an expression for the relative permeability of the soil to water flow using the
effective saturation, this simply being Se raised to the power 4.0. Brooks and Corey (1966)
reviewed this work, and suggested that for a completely uniform pore-size distribution, the
exponent would drop to 3.0. They also reviewed a number of similar expressions developed by
others, including Irmay (1954), who found that the exponent should be 3.0. Brooks and Corey
suggest that the value of the exponent typically lies between 3.0 and 4.0 inclusive.
Brooks and Corey also found that for a medium containing two immiscible fluids ( for example
air and water ), the permeability of the wetting phase is a function of the difference in pressure
between the two phases, the relative permeabilities for air and water flow being given by the
‘Burdine’ equations, which are also functions of effective saturation. While these expressions
were derived mathematically, they are based on assumptions that the soil is isotropic, and can be
modelled as a series of small diameter tubes. Brooks and Corey state that for a more general,
anisotropic situation, the expression put forward by Corey (1957) should be applied.
Chapter 3
49
Brooks and Corey (1964) discuss this expression, and state that “The approximations given by
Corey imply that effective saturation, se, is a linear function of 1/Pc2” ( Pc being the capillary
pressure ). They go on to state that this is not generally true, as has been proven experimentally,
but that the use of Se to give relative permeability is still a reasonable approximation. Hence,
while “permeability is a very sensitive function of capillary pressure”, they found that “the
relative permeability as a function of Se is not very sensitive to changes in the actual slope of the
capillary pressure curve.”
Fredlund, Xing and Huang (1994) reviewed the various published permeability functions for
unsaturated soils, before presenting theory based on a statistical pore size variation which is then
used to develop an unsaturated soil permeability function based on an SWCC. This work assumes
that volume change of the soil structure is negligible, and hence is only applicable for ‘rigid’,
rather than ‘fully coupled’ situations.
Fredlund (1998) further discussed the Fredlund, Xing and Huang permeability function. He
showed that this function extended from zero suction (100% saturation) to 1000000kPa suction
(0% saturation), but qualified this, however, by stating that at saturation below the residual state,
the moisture movement will tend to be vapour flow. From this, he suggests that it might therefore
be more appropriate to set a constant, minimum permeability at the residual saturation. Further
increases in the magnitude of the suction would not result in any further decrease in the
permeability. This is consistent with the ideas presented by Irmay (1954) and Childs (1969), as
given earlier.
The Fredlund approach of providing a very low, but non-zero, permeability for saturations at
residual or below is probably the preferable solution to dealing with water permeability at very
low suctions.
Setting permeability to zero at suctions below the residual degree of saturation would prevent any
flow within this region of soil, and hence would make further changes in degree of saturation
( and thus suction ) impossible unless changes are induced in the soil structure, for example by an
imposed load. Thus, in such a situation, to reproduce the real behaviour of a soil it would be
necessary to model the vapour phase transport of moisture.
Vapour phase flow adds an additional layer of complexity to an already complex issue, with flow
due to air flow, and molecular diffusion as described by Fick’s Law ( Childs 1969 ). This latter
process is highly temperature sensitive ( Gardner 1961, Bear 1972 ). Therefore, to accurately
Chapter 3
50
model this behaviour, it would be necessary to include the air phase as a discrete third phase
( which issue was discussed in section 2.2.3 ), and introduce temperature as an additional variable
throughout the half-space of the analytical problem.
While there are certainly situations in which such detailed assessment would be justified
( typically some form of environmental geotechnical analysis ), for a more general engineering
problem such detail adds considerably to the resources required to solve the problem for little
obvious gain. It also increases the amount / range of data needed to formulate the solvable
‘problem’ from the original engineering issue: for example, if temperature variation affects the
modelled behaviour, then it is necessary to know the initial temperature distribution within the
volume of soil being modelled.
Thus, the Fredlund approach of having a very low cut-off to permeability at the residual degree of
saturation seems to offer a practical engineering approach to dealing with this issue. Typical field
suctions in residual soils appear to be less than 100 kPa ( for example, Sweeney 1982, Abdullah
& Ali 1993, Au 1998, Sun et al 1998, Tsaparas et al 2002 ), although the action of vegetation may
lead to higher suctions developing. For most soils, the suction at residual saturation appears to be
considerably higher than these values ( for example Koorevaar et al 1983, Dineen 1997,
Cunningham, 2000 ), though Koorevaar et al do present data which shows that the suction at
residual saturation for sand can be in the region of 100 kPa.
Hence the permeability of soil at suctions greater than that needed to obtain residual saturation is
likely to rarely be of relevance to an analysis. Even when such permeabilities are applicable, the
use of a very low fluid-flow permeability rather than a diffusion flow seems unlikely to be of
sufficient significance to justify the increased complexity that is required to include diffusion
processes into a general unsaturated flow model.
3.5
Suction and soil deformation relationship
In discussing the effects of suction on permeability, it has generally been assumed that changes in
suction have no effect on the soil structure and hence while the volume of water within the soil
varies, the volume of voids remains constant ( that is, the soil is rigid ). However, this is not
actually the case. Changes to the suction within the soil mass will affect the soil structure.
Chapter 3
51
This aspect of unsaturated behaviour has not been widely investigated. With much of the work
published on unsaturated soils being directed towards agronomy, rather than engineering,
deformation behaviour of the soil is rarely considered.
The suction-deformation relationship is, however, an integral part of unsaturated soil behaviour,
so must be considered in any full-coupled approach to this subject. Such literature as is available
on this aspect of unsaturated behaviour forms the basis for much of the development work
undertaken as part of this project, so is reviewed in detail, later in this thesis, see Chapter 5.
3.6
Stress-state variables for unsaturated soil
In examining the fully coupled behaviour of unsaturated soils, it is necessary to consider the
stress-strain-strength behaviour of the soil, and what are the appropriate stress state variables.
In ‘traditional’ saturated soil mechanics, the principle of effective stress is given to hold:
σ = σ’ + u
Eqn. 3.10
Where σ is the total stress, σ’ is the effective stress, and u is the pore water pressure. The
mechanical behaviour of the ( saturated ) soil is governed by the effective stress.
In an unsaturated soil, the three-phase nature of the soil makes the situation more complex. A
number of modified effective stress laws for unsaturated soil have been proposed at various times,
see for examples Jennings (1961). Fredlund and Rahardjo (1993) also considered in some detail
the various effective stress expressions that have been proposed for unsaturated soils.
Three stress-state variables are presented, of which any two will adequately describe an
unsaturated soil's mechanical behaviour. The three stress-state variables are (σ - ua), (ua – uw) and
(σ - uw), where σ is the total stress, u is the pore fluid pressure, with the subscript ‘a’ indicating
the air phase, and subscript ‘w’ for the water phase.
Throughout the literature on unsaturated soils, the preferred combination of stress state variables
appears to be (σ - ua) and (ua – uw), the matric suction. If these are combined with the assumption
of zero air pressure ( relative to atmosphere ), the two stress-state variables reduce to σ, the total
Chapter 3
52
stress, and – uw, the negative of the pore water pressure ( which is equivalent to the positive
suction ).
While more advanced stress-strain-strength models exist or are in development for unsaturated
soils, this aspect of unsaturated soil behaviour is outside the scope of this thesis. However, to
develop a realistic coupled unsaturated flow model, some attempt must be made to model the
unsaturated mechanical behaviour.
3.7
Suction-shear strength relationship
The reason for undertaking this research project is that saturated soil mechanics approaches are
not adequate for dealing with slope stability problems in unsaturated soil, since they fail to take
into account the effect of soil suction.
One aspect of the soil’s behaviour that suction impacts on is its shear strength: “ A loss of matrix
(sic) suction is followed by a loss of shear strength which can be responsible for shallow
landslides or initiation of debris flow with a failure surface within the unsaturated zone” ( Cojean
et al, 1994 ).
Clearly then, while this thesis is primarily concerned with the infiltration process, some
consideration must be given to the shear strength of the soil.
Generally, it seems to be accepted that suction within the soil contributes towards its shear
strength.
The standard Mohr Coulomb expression for shear strength in a saturated soil is as shown in
equation 3.11.
τ = c + σ ' tan φ
Eqn. 3.11
Where τ = shear strength, c = apparent cohesion, σ’ = normal effective stress, φ = angle of
shearing resistance
To allow for unsaturated soil suction, equation 3.11 has been modified by a number of authors.
Lumb (1962a) suggested:
Chapter 3
53
τ = cs + σ tan φ
Eqn. 3.12
Where τ = shear strength, cs = apparent cohesion, and σ = Total normal stress.
In equation 3.12, the apparent cohesion, cs, is a function of degree of saturation and voids ratio
( Lumb presented graphs for some typical soils ). This approach is undoubtedly simple, and is
consistent with the use of (σ - ua) and (ua – uw) as stress state variables combined with the zero air
pressure assumption.
Lumb presented a more detailed consideration ( Lumb, 1962b ), which showed that the
unsaturated apparent cohesion, cs, could be defined in terms of the saturated apparent cohesion
and the suction:
τ = c1 + (σ + us) tan φ
Eqn. 3.13
Hence:
cs = c1 + us tanφ
Eqn. 3.14
Where
c1 = the saturated apparent cohesion,
us = -u, the negative value of the pore water pressure
( for unsaturated soils, u is negative, hence us is positive ).
Fredlund et al (1978) presented two forms of a shear strength equation based on the MohrCoulomb form.
The first makes use of (σ - uw) and (ua – uw) as the stress state variables, and takes the form shown
in equation 3.15.
τ = c'+ (σ − uw ) tan φ '+ ( ua − uw) tan φ ''
Eqn. 3.15
However, use of equation 3.15 means that any change in the pore water pressure changes both of
the stress state variables. Therefore, Fredlund et al proposed the alternate form, shown as equation
3.16.
Chapter 3
54
τ = c' '+ (σ − ua ) tan φ a + ( ua − uw) tan φ b
Eqn. 3.16
Where
c’’ = cohesion intercept when the two stress state variables are zero,
φa = Friction angle with respect to changes in (σ - ua) when (ua – uw) is held constant,
φb = Friction angle with respect to changes in (u a – uw) when (σ - ua) is held constant.
They show that equation 3.15 and equation 3.16 are related, such that c’’ ≡ c’ and φa ≡ φ’, giving:
τ = c'+ (σ − ua ) tan φ '+ ( ua − uw) tan φ b
Eqn. 3.17
and
tan φ ' = tan φ b − tan φ ' '
Eqn. 3.18
In equation 3.17, only 1 stress state variable responds to changes in pore water pressure, and only
φb is a ‘new’ parameter, beyond ‘standard’ soil properties, which must be determined.
Vanapalli et al (1996) suggest that at high degrees of saturation ( low values of matric suction )
the negative pore water pressure provides a direct increase in effective stress, thus increasing
shear strength in this manner. At lower degrees of saturation, the effect of suction on strength is
more variable, with strength possibly reducing, and must be explicitly allowed for.
They also found that the saturated angle of shearing resistance, φ’, can be assumed constant, at
least for the practical engineering suction range of 0-500 kPa.
They propose an equation for the shear strength of unsaturated soil derived from Fredlund et al’s
(1978) work, as shown below as equation 3.19:
[
τ = [c'+ (σn − ua ) tan φ '] + ( ua − uw ) (Θ K )( tan φ ')
]
Eqn. 3.19
Where:
Θ = the normalised volumetric water content = θ / θs
θ = Volumetric water content
θs = Volumetric water content at 100% saturation.
Chapter 3
55
K = a fitting parameter, used to match predictions to experimental results.
To avoid the need for a fitting parameter, Vanapalli et al proposed alternate variations for their
equation based on the SWCC, in terms of volumetric water content or degree of saturation:

 θ − θr  
τ = c'+ (σn − ua ) tan φ '+ ( ua − uw )  ( tan φ ') 

 θs − θr  

Eqn. 3.20
θr = Residual volumetric water content

 S − Sr  
τ = c'+ (σn − ua ) tan φ '+ ( ua − uw)  ( tan φ ') 

 100 − Sr  

Eqn. 3.21
S = Degree of saturation
Sr = Residual degree of saturation.
The residual values required by equations 3.20 and 3.21 may be determined from the SWCC,
although Vanapalli et al report that these values may be hard to determine for fine grained soils.
Fredlund and Rahardjo (1993) present data on the shear strength of unsaturated sands which
shows strength increasing with suction while the sands are still saturated ( i.e., still in the tension
saturated zone ), but which then show a decrease in strength before becoming constant with
suction. This conforms with Vanapalli et al (1996).
Fredlund and Rahardjo go on to discuss the shear strength equation, presenting the same equation
as shown here as equation 3.17. They note that the parameter φb ( the “angle indicating the rate of
increase in shear strength relative to the matric suction ( ua-uf )f” ) is always less than or equal to
φ’, the ‘saturated’ angle of friction. They present test data which shows φb = φ’ at low suctions,
with φb then steadily decreasing until a steady value, φb << φ’ is obtained.
They state that equation 3.17 reverts to the saturated shear strength equation ( equation 3.11 )
when suction is zero. This can easily be seen from equation 3.17, since the { (u a-uw)tanφ’ } term
will go to zero, and with the zero air pressure assumption, zero suction means u w equals zero
( hence σ = σ’ ).
Chapter 3
56
Both Fredlund and Barbour (1992) and Fredlund and Rahardjo (1993) show how matric suction,
net normal stress and shear strength interrelate to give a three dimensional failure surface for the
Mohr-Coulomb failure envelope, as shown in Figure 3.4.
This figure shows a planar failure surface, although this need not be the case; as stated above, φb
is not ( generally ) constant with suction. Fredlund and Rahardjo present some typical values for
c’, φb and φ’, for various soils worldwide.
In discussing the failure surface, Fredlund and Rahardjo show how the effect of matric suction
can be shown as an increase in cohesion, when shear strength is presented on a two-dimensional
Mohr-Coulomb plot. However, they emphasise that this does not indicate a ‘true’ cohesion effect
from suction, but that this is merely a presentational effect. They go on to discuss the effects of
stress paths and different types of strength testing, which is beyond the scope of this thesis.
Frydman (2000) reviewed work on the shear strength of Israeli soils, which showed that the angle
of friction, for these soils at least, tends to be fairly constant regardless of degree of saturation.
However, the cohesion of the soil was a direct function of saturation, showing a linear increase in
cohesion with a decrease in degree of saturation.
Frydman discussed the mineralogy of Israeli clays, and suggested that this may account for some
of the behaviour, which may render his conclusions inapplicable to other soils. It may also be
noted that he was working with two-dimensional plots, and did not present data on the threedimensional surface suggested by Fredlund and Rahardjo. The apparent linear increase in
cohesion seen by Frydman may well be a reflection of a constant ( for the range of suctions
covered by his test data ) φb angle.
3.8
Published work using numerical modelling to analyse the infiltration
process
Details have been published of a number of attempts that have been made to analyse the
infiltration process using numerical modelling techniques.
Gioda and Desideri (1988) looked at two-dimensional flow through an earth dam on an
impervious base, but with the assumption of a rigid soil ( “assuming…that the deformability of
the soil skeleton can be negated” ). Fundamentally, they approached the problem as a saturated
Chapter 3
57
soil problem, but they did provide for unsaturated flow, in that they incorporated a suctionpermeability relationship that allowed for near-to- but non-zero permeability when pore water
pressures were below atmospheric ( i.e. when PWPs had become suctions ). Their work is of little
direct relevance to this project, since it involves neither infiltration nor consolidating soil, but the
suction-permeability function used is worth noting.
Rubin and Steinhardt (1963) considered rain infiltrating into ( a presumably flat surfaced ) soil.
Their assumptions included that the soil was of “stable structure” ( i.e. rigid ), that the initial
moisture content distribution was uniform, and sufficiently low to give water permeability tending
to zero, and that the air phase was at atmospheric pressure and air was free to flow.
They proposed and then proved using a finite difference technique, that if the infiltration rate was
less than or equal to the fully saturated permeability, all infiltration would enter the soil, and there
would be no surface ponding. Further, if the infiltration rate was less than the fully saturated
permeability, while suctions would be reduced, the soil would remain unsaturated. In this case,
the suctions in the soil after infiltration would correspond to those needed to give a suctiondependent permeability equal to the infiltration rate.
Only if the infiltration rate exceeded the fully saturated permeability of the soil would ponding
occur. Figure 3.5 shows the results of their analysis: clearly, the higher rainfall rate has led to an
increased moisture content in the soil, yet since neither rainfall rate exceeded the fully saturated
permeability, suctions remain, and the soil is still unsaturated.
Despite using a rigid soil, Rubin and Steinhardt’s results are logical and the general form of
behaviour shown is believable.
Rubin et al (1964) followed up Rubin and Steinhardt’s numerical analysis by undertaking
experimental work to confirm their earlier numerical predictions. Using a 5cm diameter test
column of various heights ( typically around 1m ), they experimentally determined the actual
moisture content profile of the column with time when subject to various rates of infiltration.
Their experimental data generally matches the numerical predictions, as shown in figure 3.6,
although for the higher infiltration rates, their numerical analysis tended to over-predict the
moisture content. Also, the numerical model generally gave a less sharp wetting front than was
observed experimentally, while the experimental data also indicated a shallow surface band of soil
tending towards full saturation, regardless of rainfall rate. Thus, the experimental data shows
close correlation with the wetting profile of Bodman and Cole, as shown in Figure 3.2.
Chapter 3
58
Chapuis et al (2001) considered a number of ‘standard’ problems, both saturated and saturatedunsaturated, while assessing verification of numerical codes. They undertook their numerical
analysis using the Seep/W code. Within the code, they state that pore air pressure is atmospheric
( i.e. zero ), which is consistent with the discussion given earlier in section 2.2.3. However, they
also state that the total stress, σ, remains constant, and hence their work cannot be considered to
be fully coupled. The majority of their illustrative examples cover flow to wells or similar, and so
are not directly related to the problem under consideration here, but two significant points can be
noted from their work.
Firstly, they show how in an analysis of flow through a dam numerical convergence problems
may occur, caused by using excessively large elements. This resulted in the program incorrectly
determining permeability and volumetric water content ( as functions of suction ), leading to
excessive capillary flow in the region above the phreatic surface. Adjusting the mesh element size
gave better convergence, and more believable results.
This clearly reinforces the point that non-linear numerical analysis can be highly influenced by
the size / geometry of the elements used, and that care must be taken to investigate the sensitivity
of an analysis to the proportions of the mesh used to solve it.
Secondly, Chapuis et al did consider a ( non-coupled ) analysis featuring infiltration into a slope
with a fissured clay surface layer over less permeable clay. As modelled directly, very high pore
pressures are generated in the fissured clay layer as the infiltration is forced into this layer, but is
unable to drain into the less permeable clay beneath. This is presented as an illustration of a
weakness in a numerical code, causing an inaccurate result to be generated; in this case, the
failure of the code to allow for run-off results in incorrect pore water pressures in the near surface
material.
Chapuis et al suggest that this particular problem be dealt with by adding a surface, highly
permeable gravel layer ( which they justify the presence of by claiming it represents the surface
topsoil and vegetation ), which more readily allows lateral flow, thus preventing the build up of
pore pressures.
While this is a reasonable approach to reproducing a more realistic flow regime for this problem
when the code used is flawed and unable to accurately model the real behaviour, it is not the ideal
solution. It is preferable that the code should be formulated such that the real infiltration / run-off
behaviour is modelled.
Chapter 3
59
Tsaparas et al (2002) also used SEEP/W to analyse seepage in an unsaturated soil slope, then
determined the factor of safety by using the seepage results as input data into the computer
program Slope/W.
They investigated the effects of different intensities and distributions of rainfall, antecedent
rainfall, and the initial groundwater conditions ( depth to phreatic surface and pore water pressure
distribution ).
In undertaking their analyses, they found that a very fine element mesh was needed, with
elements 0.25m square on the slope surface. This was due to numerical difficulties in achieving
convergence in the analyses, due to the non-linearity of the problem.
Tsaparas et al found generally that the longer the period of rainfall lasted, the deeper the wetting
front tended to penetrate. Also, the saturated permeability was important, in that pore pressures
within the soil increased more ( became more compressive ), and at greater depths ( a greater
depth of wetting front ) for the soils with higher saturated permeability.
However, the ratio between the saturated permeability and the rainfall rate was also important. A
high rainfall relative to the saturated permeability produced only a shallow wetting front, since
most of the rain became run-off. From this, it can be seen that Tsaparas et al allowed for the slope
surfaces becoming fully saturated, and hence for run-off to develop. This was done by re-setting
the surface pore water pressure to 0 kPa if at any time this surface pressure became compressive
( more compressive than 0 kPa ). As is discussed in Chapter 4, this approach is flawed, in that it
neglects to ‘correct’ the pore pressures below the surface which are likely to be compressive if the
surface pressure became more compressive than 0 kPa.
Tsaparas et al also considered the effects of antecedent rainfall, and found in general that stopstart rainfall had little long-term effect, particularly where the permeability was relatively high,
since the ground drained during the dry periods. However, prolonged antecedent rainfall would
significantly alter the pore water pressure conditions prior to the main rainfall event, such that the
initial pore water pressure distribution prior to the antecedent rainfall became all but irrelevant.
The effect of rainfall on the stability of the slope seemed to be related to the duration of the
rainfall event, with the longer the rainfall period, the lower the slope factor of safety. Antecedent
rainfall had some impact, especially when continuous, as previously noted, while lower
Chapter 3
60
( saturated ) permeability slopes tended to be more stable, since higher permeabilities permitted
deeper and more rapid penetration of the wetting front, leading to greater loss of suction.
Tsaparas et al make no reference to soil deformation, so it is not clear whether they used a rigidsoil approach, but by determining the seepage pattern than looking at stability, it is clearly not
fully coupled.
Ng et al (2001) used the finite element program FEMWATER to undertake a three-dimensional
analysis of rainfall infiltration into a cut slope in Hong Kong that was prone to failure. While it is
not clear whether the program has a coupled capability, the analyses undertaken treated the
problem as rigid.
The analyses investigated the effects of various patterns and intensities of rainfall on the pore
water pressure distribution in the soil. While the implications of the analytical results on the
stability of the slope are discussed, no attempt is made to analyse the stability numerically, neither
as part of a coupled analysis including the water flow, nor as a separate analysis using the
determined pore pressure regime as input data.
In applying their precipitation data, Ng et al assumed that, on average, 60% of the actual rainfall
intensity infiltrates into the surface of the soil, with the rest taken as run-off ( a statistic applicable
to Hong Kong, and repeated by Tung et al, 1999 ). This is an attempt to reflect the surface
infiltration process as proposed by Mein and Larson (1973). Mein and Larson showed that in an
unsaturated soil, rainfall results in an initially high hydraulic gradient at the surface, which when
combined with the soil’s ability to increase in moisture content and store water, enables the
infiltration rate to initially exceed the fully saturated permeability.
However, as the surface soil becomes fully saturated, the infiltration rate drops to the fully
saturated permeability, and any ‘excess’ precipitation above this value becomes run-off.
Ng et al do not reproduce this effect directly, but rather simulate it indirectly with their ‘60% of
rainfall as infiltration’ approach. However, this approach seems flawed. Their results show that in
some sections of their mesh, under certain precipitation rates, suctions are completely destroyed
and the soil becomes fully saturated at the surface, but elsewhere and / or for other precipitation
patterns, surface suctions remain, and the soil is unsaturated.
Thus, while their approach might give approximately the correct infiltration into the mesh on
average, the actual pattern of inflow does not reproduce the real world situation. The use of a
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61
percentage reduction to the actual precipitation figure to give a surface boundary infiltration rate
seems reasonable if it is necessary to allow for canopy run-off, or any other form of above surface
interception. It seems inappropriate to pre-determine what percentage of the rain reaching the soil
surface actually penetrates into the ground, and in this respect is no better than Chapuis et al’s
approach of inserting a false gravel layer on the surface of the mesh to allow for lateral run-off.
Rather, the actual infiltration should be determined by what the soil ‘wants’ to absorb; this
requires that the code used can calculate this. An approach that enables this to be achieved was
used in this research project, as explained in Chapter 4.
Ng and Shi (1998) undertook a parametric study of rainfall into “a typical unsaturated hillside
with a steep cut slope in Hong Kong”, the slope being colluvium or completely decomposed
granite ( no distinction being made between the two ).
Their study was done using the finite element programme SEEP/W, with the rainfall modelled
through the application of a specified infiltration rate ( equal to the rainfall data ) on the boundary
surface, to determine the pore water pressure distribution. These pore pressures were then used in
a conventional limit equilibrium analysis to calculate factors of safety.
The results of Ng and Shi’s analysis indicate that rainfall causes a rise in the phreatic surface, and
a reduction in the suctions above the phreatic surface. They found that the stability of the slope
was affected by duration and intensity of rainfall. They also noted that the initial hydraulic
boundary condition was significant to stability, thus firmly implying that antecedent rainfall
directly affects slope stability.
It may be noted in the results that Ng and Shi present that pore water pressure becomes greater
than zero ( i.e. compressive pore water pressures ) at some locations on the surface of the mesh in
some analysis, implying that ponding of surface water is occurring. In the ‘cut’ part of the slope,
which has a horizontal surface, this is perhaps reasonable, but some of their results, for example
those given for anisotropic soils, show non-zero, non-suction pore pressures on sloping surfaces.
Such results seem illogical, and are probably the result of the chosen infiltration boundary
condition, which appears to be simply a specified infiltration rate, regardless of the resulting
surface pore water pressure.
Thus while this work provides a useful guide to some aspects of infiltration behaviour, it is
neither fully-coupled, nor does it appear that the infiltration process is being correctly modelled.
Chapter 3
62
Lam et al (1987) presented an approach to modelling “saturated-unsaturated soil systems” using a
numerical technique ( the finite element computer program ‘TRASEE’ ), and included three
example problems, namely flow through a dam after sudden impounding, vertical flow through an
unsaturated soil from a lagoon to the underlying groundwater, and rain-induced infiltration into a
slope in a multi-layered soil ( in the form of a 1m high experimental model ).
No attempt is made to model the strength-strain-stress behaviour of the soil, and while a ‘water
coefficient of volume change’ is incorporated within their analysis, this represents the change of
water storage with suction, and does not reflect consolidation. Moreover, a fixed value is
specified, implying a constant slope to the suction-volumetric water content relationship, which is
clearly not accurate.
Lam et al demonstrate that the phreatic surface is not a flow line, with their first, dam, example
showing flow across the phreatic surface. This example also shows the importance of flow within
the unsaturated region, with a calculated total flow through the dam being 30% greater than that
calculated using a traditional flow net, with flow limited to the saturated zone.
However, despite this, and their ability to reproduce the pore pressure in the slope model
relatively accurately, the Lam et al approach is fundamentally a ‘rigid soil’ approach, and does
not, therefore, accurately portray the behaviour of real soil.
Kasim et al (1998) undertook a study of the effects of steady rainfall on the long term ( steady
state ) suction within the ground, using the SEEP/W computer program.
They looked at both horizontal ground and slopes of three different gradients, and varied the
SWCC and infiltration rate used. However, no allowance for soil displacement was made, with
the analysis being ‘rigid’.
They found that suctions within the soil would not be fully destroyed unless the infiltration rate
matches the saturated soil permeability. Where the infiltration rate is less than the fully saturated
permeability, the steady state suction profile is a function of the infiltration rate, with lower
infiltration resulting in a higher magnitude of suction remaining in the soil.
Further, they showed that increasing the magnitude of the AEV tended to reduce the effect of
infiltration, see Figure 3.7. Two analyses were conducted of a soil column with a horizontal
ground surface. Saturated permeability was the same in both cases, but there was an order of
magnitude difference in the AEVs ( 10 kPa versus 100 kPa ). When an infiltration rate equal to
Chapter 3
63
the fully saturated permeability rate was placed on the soil surface, both examples showed a
complete loss of all suction. However, for lower infiltration rates, the high AEV soil retained
higher ( steady state ) suction at any given depth, for any given inflow rate, than the low AEV
soil.
They suggest that where the AEV is low ( relatively ), the steady state suction is approximately
constant over a finite depth from the surface. However, their analyses were all carried out using
an initial pore water pressure profile that was hydrostatic, with the ‘groundwater level’ ( phreatic
surface ) at 20m below the ground surface. It seems probable that the tendency for the suction
profile to remain approximately constant over some limited depth towards the ground surface
partly reflects this initial pore pressure profile. The same behaviour might well be observed with a
higher magnitude AEV if the initial PWP profile had the phreatic surface deeper.
This idea is supported by the form of the curves in Figure 3.7b, which match that of the ‘nonconstant’ portion of the curves in Figure 3.7a. For example, the ‘0.3 ks’ flux curve becomes
approximately constant at a suction about equal to the AEV in Figure 3.7a. Extrapolating the
same flux value curve in Figure 3.7b, it is apparent that this curve is also tending to become
constant at the AEV. However, because of the depth of the phreatic surface suctions equal to the
AEV are not present within the mesh, and so the curve cannot develop its ‘constant’ portion.
Kasim et al conclude that the AEV is of more influence than the soil desaturation rate parameter
( which is effectively the gradient of the SWCC ) in determining the steady state suction / PWP
profile, but that neither has any great influence on the rise in the phreatic surface that results from
the infiltration process. The response of the phreatic surface is almost totally dependent on the
magnitude of the infiltration relative to the fully saturated permeability.
It should be noted that the results and conclusions obtained by Kasim et al are for final steady
state conditions. It seems certain that during the transient phase which precedes steady-state,
when the water content of the soil is likely to be varying, the gradient of the SWCC, which
reflects the storage capacity of the soil, will be significant in determining the suction or PWP
profile within the soil.
Sun and Nishigaki (2000) used finite element analysis to look at the combined rain and
evaporation process, and how this generates tension cracks within the ground surface.
Chapter 3
64
Their analysis was of a 35° slope in granite and decomposed granite soil. Of particular note is that
their FE mesh uses very thin elements along the slope surface: approximately 0.25m thick. This
compares to the overall slope height of 10m, and a total mesh depth of 30m. The elements within
the mesh get thicker with depth, with the bottom most layer of elements being 3m thick. Sun and
Nishigaki have clearly concentrated elements in the area where they expected the most change in
the pore water pressure.
Sun and Nishagaki also show how incorporating evaporation in the ‘dry’ periods between rainfall
events can significantly alter the near surface PWP or suction, as compared to the PWP profile
that is generated by treating the ‘dry’ periods as ‘zero rain’. They then go on to compare how the
PWP / suction profile within the slope varies if tension cracks of various depths are introduced.
The effects of such cracks on the stability of the slope are considered, with the permeability of the
soil being multiplied by a factor based on the degree of volumetric shrinkage to determine an
increased permeability within the cracked zone, but the stability is not calculated from the FE
analysis. Rather, the numerical analysis assumes a rigid soil, and this analysis produces a PWP
distribution as an output, which is then incorporated into a limit equilibrium slip circle stability
analysis.
Their work suggests that if evaporation occurs between periods of rainfall, the effects can be
marked. However, it is not clear if their rainfall and evaporation data is based on actual field data.
Thus it is not certain how realistic it is to intersperse periods of evaporation between closely
spaced periods of rainfall.
While ICFEP has the capability to model the effects of evaporation and transpiration, the
approach taken in this research project has been to assume that evaporation occurs predominantly
in the dry season, and may therefore be neglected within the analysis undertaken using ICFEP.
Sun and Nishigaki’s work suggests that this may be a mistake, but in the absence of firm data
indicating that significant evaporation can commence the moment rain has stopped, this simpler
approach is justified.
The approach adopted for the ICFEP analyses is supported by Tsaparas et al (2002), who
attempted to model evaporation, but found their results unrealistic compared to field data. They
thus took the decision to exclude the evaporation process, justifying it on the grounds that no
evaporation occurs during a rainfall event, and it was the pore water pressure response to rainfall,
not evaporation, that they were investigating.
Chapter 3
65
Similarly, it is clear from Sun and Nishigaki’s work that the formation of tension cracks may lead
to significant changes in the slope behaviour. This is unsurprising, since the effects of tension
cracks are well known and generally allowed for in ‘traditional’ analysis of saturated soils. The
value of their work is primarily in giving guidance on the extent that such cracks form due to
evaporation. However, if the assumption is made that no evaporation process occurs during the
period covered by an analysis, then this aspect of their work is similarly non-applicable.
Karnawati (2000) used SEEP/W to model the infiltration process in a slope that failed in
Indonesia, with the slope stability analysis undertaken using GEOSTAR. While not explicitly
stated, this approach clearly implies that the pore water pressure response was determined for a
rigid soil, and then ‘plugged into’ the stability analysis.
Karnawati found that in her analysis, the phreatic surface rose in the lower and middle portion of
the slope, while dropping in the higher portion, despite a constant rainfall throughout the analysis.
She suggests that this was due to down-slope flow, primarily within a sand layer.
Presumably, without the rainfall, the drop in phreatic surface in the upper portion of the slope
would have been more pronounced, and the fact that the phreatic surface did drop while under
precipitation conditions suggests that her initial boundary conditions were not representative of a
‘steady state’. Instead, while she states that the specified PWP boundary conditions are typical of
the wet season, they appear to be a ‘snap-shot’ of the situation during transient conditions. This
emphasises the need to accurately determine the true boundary conditions of any problem prior to
analysing it.
It may be noted that Karnawati modelled rainfall as a specified infiltration ( “constant flux” ) on
the surface boundary throughout the analysis. Her results show that the phreatic surface almost
reached the soil surface. Had this occurred, it would no longer have been correct to impose all the
rainfall as infiltration, since some proportion of the rain would have flowed as surface run-off.
Karnawati does not show how she would have allowed for this, and it is probably that she would
have incorrectly retained the fixed-rate infiltration condition.
Fredlund and Barbour (1992) looked at the effects of rainfall on the suctions within a typical
Hong Kong slope, and how this affected the slope stability, using the computer program SEEP/W.
They modelled the rainfall as a specified infiltration flux on the surface boundary, but recognised
the possibility of the soil becoming fully saturated. However, rather than modifying their
Chapter 3
66
boundary condition to allow for run-off, they instead permitted ponding on the upper portion of
the slope.
The lower steeper portion of the slope was covered with chunam, a cement based slope protection
system, and they modelled this by restricting infiltration to 10% of the specified rainfall. This
made it improbable that compressive pore water pressures would be generated at the surface of
this part of the slope, and Fredlund and Barbour’s results confirm this to be the case. However, it
seems unlikely that ponding could occur on such a steep slope, so it is fortunate that this part of
the slope did not become fully saturated. While their results do not suffer because of it, Fredlund
and Barbour’s work does appear flawed by the absence of any capacity to determine run-off.
Fredlund and Barbour started their analysis with a hydrostatic pore water pressure / suction
distribution, then applied a light rainfall, based on the average annual Hong Kong rainfall, to
generate a steady state condition. This resulted in a relatively constant suction from the soil
surface down several metres, at which point the pressure distribution approximately rejoined the
hydrostatic pressure line, see Figure 3.8. This is comparable to the results obtained by Rubin and
Steinhardt (1963), as shown in Figure 3.5. Fredlund and Barbour’s pore water pressure profile is
from approximately mid-slope, just above the chunam covered portion of the slope.
It again shows how a given infiltration rate can lead to a reduction in the suction in a soil, until the
corresponding suction-dependent permeability matches the infiltration rate. At this point the flow
into the soil is able to pass through, without changing the water content of the soil, and a steady
state is achieved.
Fredlund and Barbour used this steady state situation as the starting pressure condition for two
transient analyses. The first was a reproduction of an intense 2-hour storm, while the second
modelled a prolonged 5-day event, with a lower rainfall rate.
In the first case, the specified inflow rate of 118mm/hr slightly exceeded the saturated
permeability of the soil, which resulted in surface ponding. A fully saturated band of soil
developed at the surface, within which the pore water pressure distribution approximately
approached hydrostatic. Since fluid flow is largely driven by pressure head differences, the PWP
distribution determined from this analysis is certainly influenced by Fredlund and Barbour’s
decision to allow surface ponding, which permits corresponding non-zero compressive pore
pressures to occur at the soil surface, and drive the infiltration process. The application of a run-
Chapter 3
67
off boundary condition with a maximum compressive pore pressure of zero at the surface would
logically result in a different pressure regime within the soil.
The effects of the intensive rainfall appear to be limited to a relatively thin band of soil, which
tends to move downward and disperse with time, once the rainfall event had ended.
The second case involved a five-day storm, with rainfall varying on a daily basis. While the
maximum rainfall specified of 7.2mm/hr was considerably less than case 1, the total rainfall for
the event amounted to 275mm, which exceeded that of case 1.
The effect of the longer but less intense storm was less extreme but more widespread. The
suctions within the soil were reduced throughout its depth, such that the general form of the
steady state profile was retained, but with lower magnitude suctions.
The results of the two different storm events modelled by Fredlund and Barbour clearly show that
the suction response to rain is at least in part dependent on the relative magnitude of the rainfall.
Fredlund and Barbour then took the pore water pressure / suction distributions generated by the
two transient analyses, and undertook a limit equilibrium analysis to investigate how the results
affected the factor of safety of the slope. Thus, while their work is of interest, it is still a noncoupled approach. Also notably is that they incorporated into their work variation in the suctiondependent shear strength of the soil, as discussed in section 3.7.
Ng and Pang (2000) looked at the dependence of the SWCC on the applied stress state, and then
considered the implications of this on slope stability.
They undertook a numerical analysis of a “typical cut slope” in Hong Kong, using SEEP/W to
determine the PWP/suction distribution. As they state, the SEEP/W analysis was uncoupled, with
soil deformations ignored. The results of their numerical analysis were then used in a limitequilibrium analysis to determine factors of safety.
To investigate the effects of stress on the SWCC, Ng and Pang undertook two sets of transient
analysis, with a short intense rain and a longer less intense rain in each, using a common multilayered slope mesh. The first assigned a single SWCC to all the soil layers, while the second
allocated different, stress-dependent, SWCCs to each layer. The transient analyses were
commenced from a steady state determined in a similar manner to that used by Fredlund and
Barbour (1992).
Chapter 3
68
Ng and Pang specified an infiltration flux equal to 60% of the actual rainfall data, with the other
40% being assumed to be run-off. It is assumed that this is not representing canopy intercept, but
rather is an attempt to reflect the average amount of rainfall that reaches the ground that will
penetrate into the ground. As such, it is clearly a less than satisfactory approach. During the early
stages of a storm event, when all of the rain should be absorbed into the ground, this approach
will result in insufficient flow entering the soil, while it does not prevent unlimited ponding to
develop later in the event, if the surface soil becomes fully saturated.
The effects of the different SWCCs were immediate, in that the steady state profiles varied. This
difference carried over into the transient analyses. Ng and Pang’s work also showed the same
general difference in pressure response between short intense rain and longer, less intense rain
that was shown by Fredlund and Barbour (1992). This occurred whether stress-dependent SWCCs
were used or not, so clearly both factors are important in determining the response of unsaturated
soil to rainfall.
Clearly, while Ng and Pang were investigating the effects of applied stress on the SWCC and how
this affected the soil’s response, the conclusions of the work could be extrapolated to cover multilayered soils. Ng and Pang effectively showing that a layered soil responds differently than a
homogeneous soil; an unsurprising conclusion. The important aspect of this work is that they
demonstrate stress-dependency in the SWCC ( through laboratory tests ), and then numerically
use this data to show how it may be necessary to treat an apparently homogeneous soil as multilayered.
The fact that the SWCC is affected by applied stress proves that it is not a unique property of the
soil. The SWCC reflects the water storage capacity of the soil, which is obviously a function of
the soil void’s ratio. The fact that the void’s ratio can change when subject to applied stress is
unsurprising, so the idea that the SWCC is affected by stress is quite logical. The nature of the
SWCC is considered further in Chapter 6.
Wong et al (1998) present one of the few attempts to develop a fully-coupled approach to the
behaviour of unsaturated soils. They make use of Biot (1941) and Dakshanamurthy et al (1984) to
generate coupled consolidation equations for unsaturated behaviour. These also form the basis for
the theory developed within this thesis ( see Chapter 5 ).
Chapter 3
69
Wong et al make use of the computer programs SEEP/W and SIGMA/W to undertake their
analyses, the first being used to undertake the seepage analyses while the second deals with stressdeformation behaviour.
They present analyses of an unsaturated triaxial test, to investigate the effects of the gradient of
the SWCC, then go on to model a 2m high column of soil, with the phreatic surface at mid-height,
subjected to an applied surface load. Their results indicate that while applied loads cause a
significant change in the pore water pressure within saturated soil, the effects on the suctions
within the unsaturated soil are all but negligible, the most significant effect being the movement
of the phreatic surface. Conversely, vertical deformations within the unsaturated zone occur
rapidly in response to applied load, whereas in the saturated zone, they occur much more slowly,
as the consolidation process occurs.
Wong et al’s work included a number of simplifications, such as having a constant permeability,
not suction-dependent, and does not attempt to model any part of the infiltration process.
However, it is valuable as an attempt to reproduce coupled behaviour in unsaturated soil, and
provides a comparison against which the new coding within ICFEP can be compared.
De Campos et al (1992) undertook stability analysis of a slope in Brazil, using a computer code
developed at the Catholic University of Rio de Janeiro. They make the point that the information
required to undertake unsaturated analyses, specifically in their case the suction dependent
hydraulic conductivity relationship, is frequently scarce, requiring estimates to be made of such
properties.
Clearly, the accuracy of a numerical analysis is likely to suffer if parameters are estimated, and
this emphasises the need for numerical analysts to work closely with their laboratory-based
colleagues in determining what properties of the soil are needed and ensuring suitable tests are
developed to determine them.
Despite using their own software, de Campos et al still treat the stability problem in two parts,
with an initially rigid soil assumed in which the flow analysis is carried out, the results of which
are then utilised in the second, limit equilibrium analysis, to determine the factor of safety.
Kim (2000) more usefully attempts a fully coupled analysis of an unsaturated soil problem. He
looked at the variation in the position of the phreatic surface within a partly saturated / partly
unsaturated soil column which was subject to a surface load.
Chapter 3
70
His results show that there is an instantaneous rise in the position of the phreatic surface on
application of the load. Depending on the initial depth to the phreatic surface, its position may
continue to rise with time for a short period, before in all cases the position of the surface begins
to drop. The final depth of the phreatic surface lies above its initial position, but below the
minimum depth to which it rose.
Vertical displacements of the column surface also reflect the initial position of the phreatic
surface. The greater the initial depth to the surface, and hence the more of the column that starts
unsaturated, the more rapid is the development of the surface displacements.
Both of these two observations are consistent with the results of Wong et al’s (1998) work.
Kim states that “less final steady-state vertical displacement occurs in the partially saturated soil
columns…These effects of the unsaturated zone are more conspicuous as the initial water table is
located higher and the magnitude of the surface loading increases”.
This and the results he presents seem to suggest that if the column analysed is fully unsaturated,
vertical displacements are apparently equal to those of a fully saturated column, but as the
phreatic surface is raised within the soil, the calculated displacement deviates from the fully
saturated value. The biggest deviation between the calculated value for the saturated and partially
saturated column displacements occurs for the case with the phreatic surface at 2m below the
surface ( i.e., for the case where the phreatic surface is closest to the ground surface, without
actually being at the surface ). Thus the closer the column is to full saturation, the greater the
deviation of the calculated displacement from the fully saturated value.
Kim accounts for the displacement behaviour by saying “the unsaturated zone absorbs a portion
of the mechanical loading stress permanently”, but this does not seem to adequately explain the
behaviour he presents.
He also extends his work to a two-dimensional situation, and showed that the behaviour seen in
the one-dimensional column was also generally applicable to the two-dimensional case.
While Kim demonstrated a fully-coupled analysis, he did not look at the infiltration process, so
while his work is useful in developing the theory necessary to code unsaturated behaviour into
ICFEP, it does not address the central concern of this research project.
Chapter 3
71
Ng and Small (2000) use Biot (1941) to develop the fully coupled equations for unsaturated soil
analysis. However, they are primarily interested in soils that, while unsaturated, have a high
degree of saturation, such that the pore water pressure and the pore air pressure are very close in
value. They state that this is typically at degrees of saturation of 80% or higher.
This appears to lead them to make a number of assumptions, most notably that strain can be
assumed to be small, and hence that porosity of the soil remains approximately constant. Thus any
change in volumetric water content due to a change in suction manifests itself solely as a change
in the degree of saturation, with porosity unchanged. Hence, while the soil is not rigid and may
undergo normal consolidation through loading, it is effectively rigid in response to changes in
matric suction.
Ng and Small’s finite element formulation for unsaturated soil is thus basically the consolidating
full-saturated equations, with the flow matrix modified to allow for suction dependent
permeability, and with an added term to allow for the water storage capacity of the soil. As is
shown in Chapter 5, this does not fully allow for all aspects of unsaturated behaviour, and hence
their formulation is incomplete.
Ng and Small primarily seem to be interested in how ‘almost saturated’ soils respond to imposed
loading, and how this changes the saturation within the soil. They present results of several
analyses using their formulation, coded in FORTRAN, and compare them to laboratory test data,
and there is generally good correlation.
The one example analysis they present that comes close to the situation of infiltration into
unsaturated soils is of rapid draw down affecting a dam. However, in this example they admit that
their formulation for the unsaturated equations is only valid at high degrees of saturation. To
analyse the draw down problem, they are forced to treat the soil as rigid.
Ng and Small’s approach fails to provide the general, fully-coupled unsaturated soil capability
that is necessary to complete this research project. However, it did provide useful guidance in
aspects of the theory developed in Chapter 5.
Forsyth (1988) developed two methods for dealing with transient flow in unsaturated soils,
however neither was fully coupled. His work is notable, however, since while one method made
the common assumption of zero air pressure, the other allowed for the air phase as a separate
phase ( i.e. non-zero air pressures possible ).
Chapter 3
72
Forsyth showed that the zero air pressure assumption often gives much the same results as the full
three phase soil ( two-phase flow approach ), but that under certain circumstances significant
errors may be given if the air phase is not specifically modelled. Such errors occurred when the
air permeability was very low and not changing much, for example when airflow permeability is a
power function of air saturation, and air saturation is very low. They also tended to occur if the
zone of soil suction was small relative to the mesh element size used in the analysis: where the
zone of soil suction lay entirely within the thickness of one element, discrepancies between the
‘standard’ two phase and more detailed three phase approach occurred. If the element size was
adjusted such that the zone of suction was spread over several elements thickness, and hence
changes in the suction between integration points in the mesh were less sharp, both methods gave
comparable results. It is probable, however, that any analysis where the unsaturated zone lay
entirely within the thickness of one element would give results of dubious accuracy. Simple good
practice would suggest that a finer mesh would be required.
Forsyth’s work thus emphasises the need to select element size within a finite element mesh
carefully, when dealing with unsaturated soil behaviour.
The discrepancy between the two methods’ results resulting from low air permeability is less easy
to avoid without actually modelling airflow as a separate phase. However, as discussed in section
2.2.3, it seems reasonable to accept some inaccuracies in the results of the analysis, particularly if
this is limited to a narrow range of saturation when air permeability is low, in return for the
reduction in effort and complexity that can be achieved by maintaining a two-phase approach with
the assumption of zero air pressure.
Lloret and Alonso (1980) were one of the first to attempt to generate a model for unsaturated
behaviour that included consolidation behaviour.
They provided a general formulation for three-dimensional behaviour that allowed for air and
water as separate phases. However, the data for their analyses was determined from onedimensional tests, and their examples are limited to such.
They simulated saturated consolidation, infiltration leading to swelling in an unsaturated soil, and
collapse due to loading then wetting of an unsaturated soil, all one-dimensionally.
Their results show the importance of variable permeabilities in governing deformation in
unsaturated soil. They appear also to support the use of a three-phase approach, with the air phase
Chapter 3
73
explicitly modelled as a separate phase. However, since they did not repeat their analyses with a
two-phase model, it is not possible to ascertain the full significance of the three-phase approach.
Additionally, as discussed in Chapter 2, a one-dimensional analysis is not typical of a field
situation. Hence, a two-phase, zero air pressure assumption remains a credible approach to the
analysis of unsaturated soils.
Li and Zienkiewicz (1992) studied the problem of multi-phase flow in deforming porous media.
They looked at the case of two immiscible fluids, but also accepted the advantages of the zero air
pressure assumption, and presented a simplified single-phase flow approach for unsaturated soils.
However, the examples they present are for two-phase flow, dealing with water injection into an
oil formation.
Their theory is largely a development of Biot (1941), and, as with Ng and Small (2000), provided
useful guidance in the formulation of the unsaturated theory developed in this thesis ( see Chapter
5 ). However, they were clearly addressing a different issue to that covered by this research
project, with no reference being made to precipitation-infiltration or slope stability.
Alonso et al (1988) present a coupled approach for the stress-strain-flow behaviour of partly
saturated soils. While not explicitly stated as such, their work appears to be a progression from
Lloret and Alonso (1980), and again is a full 3-phase approach.
They use their formulation to look at pore water pressure response and deformation in an earth
dam under construction, with either a ‘dry soil’ fill, with an initial degree of saturation of 80%, or
a ‘wet soil’ fill, initially at 90% saturation.
Their results show that the wet soil responds to the construction of subsequent layers above, with
consolidation and an increase in the pore water pressure occurring. This behaviour is not observed
in the dry soil.
Alonso et al (1988) do not give an SWCC for the material used in their analyses, but rather
present state surfaces to show the behaviour of the soil. From this, the AEV appears to be equal to
0 kPa, with the saturation of the soil decreasing noticeably immediately the soil develops suctions.
However, the results of their analyses suggest that the wet soil responds to loading in a manner
reminiscent of saturated soil. In contrast, the dry soil shows markedly different behaviour
compared to a fully saturated soil. This supports the idea that there is some ‘cut-off’ degree of
Chapter 3
74
saturation, below 100%, at which point, the soil’s behaviour changes from a saturated, or at least
pseudo-saturated soil to unsaturated behaviour.
Alonso et al’s (1988) work is also interesting in that to obtain the fully coupled solution they
‘cheat’. Rather than solve the flow and stress-strain behaviour as a single combined equation,
these two aspects are solved by two separate computer programs, which are linked and work
iteratively in tandem to determine the coupled solution.
Additionally, the concept of state surfaces for unsaturated soils provides a useful insight into how
different aspects of an unsaturated soil’s behaviour are inter-related.
As with the other attempts at coupled analysis, Alonso et al (1988) do not address the problem of
rainfall infiltration and slope stability.
However, Alonso et al (1995) use the work of Alonso et al (1988) as the basis to undertake a twodimensional analysis of unsaturated flow due to rainfall in a slope, determining slope stability
through limit equilibrium. Thus they were able to show the evolution of the factor of safety with
time, and how various analysis input factors impacted on the safety factor.
Despite their previous work using a three-phase approach, Alonso et al (1995) make the zero air
pressure assumption for their slope stability analyses. They also distinguish between two different
rainfall boundary conditions. Either a specified pressure boundary condition, with suction equal
zero, is applied, representing a flooded surface with run-off, or a specified infiltration rate is
applied, where the infiltration rate equals the rainfall rate, and is less than the permeability of the
surface soil. However, these two conditions are either/or. Consequently, they do not have a
‘dynamic’ boundary condition that responds to the pore pressures at the boundary surface, and
adjusts the boundary condition to reflect the current pressure.
The results of Alonso et al’s (1995) analyses of a flooded-surface slope revealed the sensitivity to
soil permeability. The higher the permeability, the more rapid was the decline in the factor of
safety, which reflected the quicker penetration of the wetting front into a more permeable soil.
The nature of the SWCC was also shown to be important. If the SWCC shows a sudden drop in
saturation, the factor of safety variation with time would also show a sudden drop. A gentler
SWCC would result in the factor of safety dropping more slowly with time. This is illustrated in
Figure 3.9.
Chapter 3
75
The analyses carried out using a specified infiltration rate demonstrated that the factor of safety of
a slope can continue to decline even after the rainfall event has ended. This is a response to the
continued percolation of the wetting front into the soil, under gravity. This behaviour also reflects
the permeability of the soil, it being noted that the post-rainstorm reduction in Factor of Safety is
more prevalent in lower permeability soils, as the penetration of the wetting front takes longer.
However, the total reduction in factor of safety was greater in the higher permeability soils,
reflecting the greater ability of the water to penetrate into the soil.
Alonso et al (1995) also looked at infiltration into a layered overconsolidated clay slope, with the
‘flooded surface’ boundary condition. It is worth noting that for the ‘non-rain’ period at the end of
their analysis, they assumed negligible evaporation ( in contrast to Sun and Nishigaki (2000), as
described earlier ). Their analysis showed a reduction in safety factor and progressive lateral
displacement with time, particularly in the surface most layer of the soil. Due to differences in
permeability, a perched water table developed in this layer.
This form of lateral spreading, without a clear failure surface, is consistent with some unsaturated
failures, such as Pak Kong ( see section 3.9 ), and suggests that the development of perched water
tables may be highly significant in the behaviour of unsaturated soil slopes.
Neuman (1973) presented an early attempt to codify unsaturated behaviour and used his work to
look at transient seepage through an earth dam
While his work has been generally superseded by more recent publications, Neuman makes an
interesting observation regarding the flow pattern. If the dam is constructed of ‘equally wet’
material, there will be a uniform pore water pressure throughout, and hence a pore water total
head gradient will exist.
Thus, without an external flow being applied, flow will occur as the pore pressures within the
analysis mesh redistribute towards a hydrostatic profile. This flow will be occur regardless of the
imposed boundary conditions.
This again emphasises the need for care in determining initial conditions for a numerical analysis.
Forsyth et al (1995) developed a numerical model for unsaturated flow that included two-phase
flow ( i.e. a three phase – air:soil:water – model ). However, while it allowed for both the water
phase and soil particle compressibility, it dealt only with flow, and was not fully coupled. They
Chapter 3
76
also show how their expression can be developed into the special case of zero air pressure ( two
phase model / one phase flow ).
They were primarily interested in flow in very dry soils, specifically looking at waste cover
designs in arid regions, and they present a number of examples, mostly involving infiltration over
a limited portion of the surface boundary into unsaturated soil. Their examples are run using both
the two-phase flow and one-phase flow versions of their expression.
Their results confirm that generally a one-phase flow analysis gives similar results to the more
detailed two-phase flow approach. The exception was for high degree of saturations ( S = 90% or
higher ), when significant differences occurred, due to the trapping of the air phase. This is
consistent with Forsyth’s earlier 1988 work.
It should also be noted that Forsyth et al (1995) were demonstrating an alternative numerical
technique, and much of their work dealt with numerical aspects, such as required CPU time,
rather than the geotechnical processes at work.
Sun et al (1998) used the SEEP/W program to analyse one-dimensional columns of two different
soils, exposed to various rates of rainfall, and their results were discussed briefly in section 3.1.
A steady state seepage condition for each soil type was generated by applying a low-level
infiltration on the surface boundary. These steady states were then used as the initial pore water
pressure distributions for a series of transient analyses. The transient analyses used three different
rates of infiltration, each lasting for a different duration.
The results of the analyses showed that if the infiltration rate equals or exceeds the saturated
permeability of the soil, the suctions would be fully destroyed, and a 100% saturation wetting
front would penetrate into the soil. A lesser infiltration rate would still generate a wetting front.
However, this would be less than 100% saturation, and hence some amount of suction would
remain in the soil. If the suction left in the soil is compared to the suction-permeability
relationship, the corresponding permeability is found to be equal to the specified infiltration rate.
This is consistent with the work or many others, for example Rubin and Steinhardt (1963) and
Kasim et al (1998), as presented earlier.
Sun et al’s results show that the wetting front tends to penetrate down into the soil quicker for
soils with a higher saturated permeability. However, it is probable that the rate of penetration of
Chapter 3
77
the wetting front reflects the maximum possible infiltration rate: in the less permeable soil, the
maximum infiltration into the soil was set by the saturated permeability, not by the specified
infiltration rate. The difference in speed of penetration by the wetting front for the case where the
infiltration rate was less than the saturated permeability of both soils was less marked.
Sun et al further note that the wetting front will advance into the soil more quickly if the water
storage capacity of the soil is small ( that is, if the gradient of the SWCC is small ). They also
observed that if the supply of water to the soil surface exceeds the saturated permeability, surface
runoff must be generated. No slope stability analyses were undertaken.
3.9
The Pak Kong failure
Part of the ‘inspiration’ for this research project was the failure of a slope at Pak Kong, in Hong
Kong.
Following heavy rainfall in June 1992, a cut slope at the Pak Kong water treatment works
underwent movement, leading to damage at the toe of the slope and to properties in O Long
village at the top of the slope.
A number of limit equilibrium analyses were undertaken, all of which indicated a factor of safety
for the failed slope of greater than 1.0, when ‘realistic’ water pressures, based on the field data,
where used. Only by applying a considerable increase in the height of the phreatic surface,
leading to almost full saturation, could failure be simulated ( Watson Hawksley, 1993 ).
It was noted that the actual failure was more of a lateral spreading, and that it moved
intermittently, apparently in response to rainfall events, with the movement stopping of its own
accord. Hence it did not appear to be a clear cut, ‘rigid body’ slip moving along a well-defined
slip surface. This may account for the inability of the limit equilibrium analyses to correctly
predict failure, and because of this, a series of numerical analyses were undertaken.
The numerical analyses were undertaken using the Imperial College Finite Element Program,
ICFEP ( GCG, 1993 ). At the time, while tensile pore water pressures could be incorporated into
an analysis, the soil could only be treated as saturated.
Chapter 3
78
It was recognised that to accurately model the deformations of the slope it would be necessary to
model the initial condition as accurately as possible. To this end, the stresses in the slope were
determined by allowing them to form from the self-weight of the soil in the original natural slope
profile. The excavation to form the cut slope was then modelled.
The initial phreatic surface was determined from the field observations of water level in the slope.
The effects of precipitation, however, were not modelled directly. Rather, the phreatic surface was
raised by various amounts, to simulate the effects of rainfall.
The analyses were conducted in the form of a parametric study, and failure of the slope was only
achieved when the angle of friction of one of the soil layers was reduced to below that which was
likely, and the phreatic surface was raised to the slope surface.
In all the analyses, there was a tendency for lateral spreading to occur as the phreatic surface was
raised to the soil surface. However, this spreading was of lesser magnitude than was observed in
the real slope, was generally more deep seated, and tended to involve rigid body movement, rather
than the more dispersed strains implied by the field data from the real slope.
The ICFEP analyses of the Pak Kong slope emphasised the need for a better understanding of the
infiltration process into unsaturated soil, and through the unsuccessful attempt to reproduce the
field observed deformations, demonstrated why a fully coupled approach is required. In this
respect, it was the ‘ancestor’ of this research project.
Chapter 3
79
Sf 1.0
So
0
So = Initial saturation
Depth
Sf = Final degree of
saturation
h
Figure 3.1: Lumb wetting front ( after Lumb, 1962a )
Ground surface
Saturated zone
n
Moisture content, c
Transmission
zone
n = porosity
Wetting zone
Saturation
Depth
Air dry soil
Transition zone
Wetting front
’
Z
Figure 3.2: Bodman and Cole wetting front ( after Bear, 1972 )
80
Steady state
evaporation
+ qwy
Steady state
flow upwards
Cover
Steady state
infiltration
- qwy
+ qwy
Steady state flow
downward
Vadose
zone
- qwy
Static equilibrium with water
table ( i.e. ‘hydrostatic’ )
( qwy = 0 )
Datum
( Negative porewater pressures )
y
Water table ( Phreatic surface )
x
Figure 3.3: Variation in pore water pressure distribution with
depth ( after Fredlund and Rahardjo, 1993b )
81
on
,
M
(u atr
a - ic
u su
w)
ct
i
φb
Extended Mohr-Coulomb
failure envelope
φ’
Shear strength, τ
φb
(ua - uw)f tanφb
c’
φ’
c’
0
Net normal stress, (σ - ua)
Figure 3.4: Shear strength failure surface for unsaturated soils
( after Fredlund and Rahardjo, 1993a )
82
Water content, cm3 / cm3
.00
.02
.04
.06
.08
.10
.12
.14
0
.16
.18
5625
10
20
.22
.24
2400
9000
3600
12375
4800
15750
X, cm
.20
6000
30
7200
19125
40
8400
22500
50
60
For Rehovot sand.
Numbers on curves indicate infiltration duration, in seconds.
The profile series on the right ( 2400 to 8400 seconds ) is for rain intensity of 0.001306
cm/sec.
The profile series on the left ( 5625 to 22500 seconds ) is for rain intensity of 0.0003528
cm/sec.
Figure 3.5: Soil moisture content profiles predicted for rain infiltration
( after Rubin and Steinhardt 1963 )
83
Water content cm3/cm3
.000 .060
.120
.180 .000 .060
.120 .180 .240
.300
6
12
Soil depth, cm
18
24
30
36
42
48
54
A
B
60
Data for Rehovot sand.
A represents infiltration of 13.4 mm/hr
B represents infiltration of 47.0 mm/hr
Crosses represent experimental data from short infiltration durations
Circles represent experimental data from long infiltration durations
Continuous lines represent the theoretical curves.
Rehout sand porosity = 0.40; therefore at 100% saturation,
volumetric water content = 0.40
Figure 3.6: Comparison of theoretical and experimental
data ( after Rubin et al, 1964 )
84
20
3.7a:
AEV: 10 kPa
Surface flux
Elevation (m)
15
10
No flux
1 ks
0.3 ks
0.1 ks
5
0.01 ks
0.001 ks
0
-200
-150
-100
-50
0
50
100
150
Pore-water pressure (kPa)
20
3.7b:
AEV: 100 kPa
Elevation (m)
15
Surface flux
No flux
10
1 ks
0.3 ks
0.1 ks
0.01 ks
5
0.001 ks
0
-200
-150
-100
-50
0
50
100
150
Pore-water pressure (kPa)
ks for both 3.7a and 3.7b = 1E-05 m/s
Figure 3.7: Effect of AEV on suction profile under infiltration
( after Kasim et al, 1998 )
85
80
75
70
65
60
ELEVATION ( m )
55
50
45
40
35
Hydrostatic pressure
profile
30
Simulated steady
state case
25
20
-30
-20
-10
0
SUCTION ( meters of water )
10
20
Figure 3.8: Suction profile showing steady state simulation
( after Fredlund and Barbour 1992 )
86
106
Fig 3.7a: SWCC
Curve A
Suction (Pa)
105
104
Curve B
103
102
0
0.2
0.4
0.6
Degree of saturation
0.8
1.0
2.2
Fig 3.7b: FofS
Global safety factor
2.0
1.8
Curve B
Curve A
1.6
1.4
1.2
0
0.2
0.4
0.6
6
Time ( 10 seg )
0.8
1.0
Figure 3.9 Effect of the form of the SWCC
( after Alonso et al, 1995 )
87
CHAPTER 4
The pre-existing capabilities of ICFEP
4.1. Introduction
ICFEP is a finite element computer program, developed by Professor David Potts at Imperial
College, specifically for geotechnical engineering analysis. The capabilities of the program have
been progressively updated and expanded. At the commencement of this project, version 8.0 of
ICFEP was in use. Development work for this project and a number of other simultaneous
research projects introduced several new capabilities to ICFEP, the most relevant being an ability
to undertake unsaturated soil analysis. Implementation and validation of these new capabilities
led to the formal designation of ICFEP version 9.0.
Before presenting the development work that was undertaken as part of this research project, it is
necessary to outline the pre-existing capabilities of ICFEP version 8.0.
4.2. ICFEP version 8.0 general capabilities
ICFEP uses four or eight nodal quadrilateral elements to resolve finite element analyses, whether
two-dimensional plane strain, two-dimensional plane stress, or three-dimensional axi-symmetric
problems. Three-dimensional problems may also be analysed through the application of Fourier
series aided analysis.
Analysis can be drained, undrained or partly drained ( i.e. coupled ), and a large range of
constitutive soil models is available. However, within version 8.0, all such models are for
saturated soils. A number of consolidation / permeability options and a variety of boundary
conditions, several of which are relevant to this project, are available in version 8.0 of the
program.
4.3. Consolidation and permeability models
A notable strength of ICFEP is its ability to undertake fully coupled consolidation.
Chapter 4
88
To calculate the consolidation behaviour of a soil, it is necessary to simulate its permeability
behaviour. Prior to this project, ICFEP had the capability to simulate the permeability of a soil to
water flow through a number of models. Options existed for isotropic or anisotropic permeability,
with or without spatial variation. Two methods of generating stress-dependent permeability were
available, as was a voids ratio dependent model, and perhaps most relevantly, suction-dependent
permeability behaviour could also be modelled. This ‘suction switch’ may be combined with
either of the stress-dependent models.
4.3.1. The suction switch
The suction dependent permeability model –‘the suction switch’ – within ICFEP is a relatively
simple one, and is illustrated in Figure 4.1.
When using the suction switch with otherwise homogeneous isotropic permeability, a constant
( fully saturated ) permeability is maintained for compressive values of pore water pressure, and
up to a specified tensile pore water pressure ( p1 in the figure ). Permeability then drops
logarithmically with a linear increase in the tensile PWP, until some predetermined residual
permeability is obtained ( kmin at p2 in the figure ), at which point permeability again become
constant regardless of further changes in the PWP.
If the suction switch is combined with a stress-dependent model, then the initial constant
permeability is actually determined from the stress-dependent behaviour, with the residual
permeability affected proportionally, but otherwise the model works as described above.
A number of other researchers, for example Li and Griffiths (1988), and Giorda and Desideri
(1988) have used similar suction-permeability relationships to the suction switch within ICFEP.
The use of a constant residual permeability is also consistent with the ideas of Fredlund (1998).
4.4. Boundary conditions
ICFEP permits a wide variety of boundary conditions to be applied. All the boundary conditions
may be specified for particular increments of the analysis, and hence are not fixed for the entire
analysis, but may be varied throughout the problem.
Chapter 4
89
Such boundary conditions include the option to specify forces, moments, rotations or
displacements of boundary nodes. Normal and / or tangential stresses may also be applied, and
body forces, such as the self-weight of elements, may be specified.
The boundary conditions particularly applicable to infiltration problems are those concerned with
flow and consolidation. Options include a fixed pore water pressure ( variable flow ) boundary, or
a specified flow ( variable pore water pressure ) boundary, which includes the possibility of a noflow boundary. Non-zero specified flows may be in or out of the analysis mesh. Additionally, any
node within the mesh may be specified as a source or sink, adding or removing water
( respectively ). Most relevantly to this project, ICFEP contains a precipitation boundary
condition.
4.4.1. The precipitation boundary condition
The precipitation boundary condition within ICFEP v8.0 enables the simulation of rainfall on a
ground surface. It acts either as an infiltration ( specified flow ) condition, or as a constant
pressure ( variable flow ) condition.
The operation of the precipitation boundary condition is illustrated in Figure 4.2. The boundary
condition requires that an infiltration rate ( i.e. the rainfall rate at the ground surface ) be
specified, along with some maximum threshold value ( THV ) of the pore water pressure at the
surface boundary.
If at the start of an increment of the analysis, the pore water pressure at the surface boundary is
below ( that is, is more tensile than ) the THV, then ICFEP applies an infiltration boundary, using
the specified infiltration rate.
If the pore water pressure at the boundary exceeds ( that is, is more compressive than ) the THV
at the start of the increment, ICFEP re-sets the boundary pore water pressure to be equal to the
THV, and maintains a constant pressure equal to this value.
The constant boundary pressure is maintained by applying an inflow that is some portion of the
specified infiltration rate. The ‘excess’ portion of the specified infiltration is disregarded. If the
specified infiltration rate is reduced after the boundary has switched to a constant pressure
boundary, then it may switch back to being an infiltration boundary if the new maximum inflow
rate is insufficient to maintain the THV pressure.
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In applying the precipitation condition, the specified infiltration rate is normally taken as the
actual rainfall for the site under analysis. If allowance is required for canopy intercept, this must
be done by inputting a reduced rainfall rate. However, no allowance needs to be made for run-off:
the boundary condition automatically determines the portion of the specified inflow that enters
the mesh and treats the remainder as run-off, based on the THV chosen.
The proportion of the infiltration that becomes run-off is not, however, explicitly modelled.
Rather, it is simply discounted from the analysis, since this flow occurs outside of the analysis
mesh.
Typically for a slope analysis, the THV would be set to 0 kPa. Thus the soil could develop an allcompressive ( ‘fully saturated’ ) pore water pressure profile, but a compressive pore water
pressure greater than zero could not build up at the ground surface. Non-zero THVs may also be
specified: compressive pore pressures greater than zero may be specified for the THV, to allow
surface ponding to occur. The maximum depth of ponding that can be achieved will thus be
determined by the value of the THV specified. Alternately, a tensile THV may be specified,
which prevents total loss of suctions at the ground surface.
It should be noted, however, that care is required in dealing with ponding. The precipitation
boundary condition applies the specified infiltration at the soil surface, and the resulting changes
in the pore water pressure reflect this. In the ‘real’ field condition, once ponding occurs, rainfall
enters into a free water surface, which will be some height above the soil surface.
Within ICFEP, if ponding is permitted, the increase in pore pressure at the ground surface will
imply a certain depth of ponded water ( for example, a THV of –9.81 kPa would imply a
maximum depth of ponding of 1m ). From the infiltration ( rainfall ) rate specified it is possible
to determine the time that should be required for such a depth of ponding to build up, assuming
that all the rainfall ponds and none enters the soil. Since some of the rainfall does penetrate into
the soil, the actual time required to achieve a given depth of ponding will be somewhat greater
than this. However, ICFEP generates ponding water pressures much quicker than this would
indicate, and the surface pore water pressure, once above ( more compressive than ) 0 kPa, is
inconsistent with the total volume of water applied as rainfall. That is, while the combination of
precipitation rate and elapsed time indicate that, say, 50mm of water has fallen on the ground
surface, the calculated rise in PWP at the ground surface ( when starting from a fully saturated
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condition with the phreatic surface at the ground surface ) is invariably greater than 50mm waterequivalent pressure.
This is a reflection of the fact that the rainfall is applied straight to the surface of the soil, and not
to the ponded water surface. Inflow into the soil, which is what ICFEP models, involves adding
water into a volume of space already occupied fully by a mixture of soil particles and water. Thus
the PWP responds rapidly to even small volumes of infiltration. To account for ponding, the
infiltration should be into a free water surface in some imaginary element above the soil surface,
which is empty of soil particles or ( prior to ponding ) any water, and in which only hydrostatic
water pressures can develop.
However, as long as there is no requirement for surface ponding to be accurately modelled, the
precipitation boundary condition works accurately.
Since this project is concerned with rainfall into slopes, the assumption will always be made that
no ponding is possible, and the maximum pore water pressure at the surface will be limited to
0 kPa ( that is, the phreatic surface can be co-located with the soil surface, but no free standing
water can exist on top of the soil ). As such, this inconsistency between rainfall rates and depths
of ponding is not significant. However, further development of ICFEP is required to address this
issue before any analyses involving ponding is attempted.
In investigating the capabilities of ICFEP v8.0, a further problem was determined with the
precipitation boundary condition, which did need to be dealt with as part of this project. As stated
above, the boundary pore pressure is adjusted back to the THV if at the start of the increment the
pressure exceeds the THV as a result of the previous increment’s infiltration. Where inflows are
relatively small and the increment time step is short, the amount by which the pore pressure
exceeds the THV is likely to be small, and this method of operation is reasonable.
However, it was found than that if the infiltration rate was high ( relatively ) and the time step was
large, very high compressive pore pressures could be generated at the surface on the last
increment in which an inflow boundary condition exists. This is illustrated by Figure 4.3.
‘Increment 0’ represents some pre-existing pore water pressure distribution. Precipitation is
applied from increment 1, and the precipitation rate is high relative to the permeability of the soil,
while the time step of the increment is relatively long. The pore water pressure distribution at the
end of increment 1 is as shown in the figure. While on the next increment the surface pore water
pressure is corrected, the shallow sub-surface pore pressure distribution is in error: this is shown
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as the ‘increment 2’ plot line in the figure. The increment 2 PWP distribution shown is obviously
unrealistic given that the surface PWP should not be able to exceed 0 kPa. Clearly, it was
necessary to modify the boundary condition to limit the amount by which the THV could be
exceeded before the condition switched from inflow to constant pressure. This was done as part of
the development work, through the use of an automatic incrementation procedure, and is
described in Chapter 6.
4.5. Modelling of precipitation on to saturated soil
While ICFEP v8.0 is unable to model unsaturated behaviour, the combination of the precipitation
boundary condition and the suction switch does enable some study to be made of the pore water
pressure response to rainfall.
While the unsaturated soil behaviour model was under development ( as detailed in Chapters 5
and 6 ), a limited study was undertaken of infiltration into a tension-saturated ( tensile PWP, but
with a fully saturated soil model ) soil, principally through a series of column analyses.
The initial set of analyses looked at how differences in the specified infiltration rate ( within the
precipitation boundary condition ), the saturated permeability and the parameters on the suction
switch affected the resulting pore water pressure distribution within the column.
The column modelled consisted of a 100m high column in plane strain, with no-flow boundaries
along the sides and base, and a hydrostatic PWP profile with an initial phreatic surface 20m
below the ground surface ( see Figure 4.4 ). Analysis was carried out normally using 100
increments, each of 12 seconds. Soil behaviour was modelled as linear elastic, with a Young’s
modulus of 25000 kPa and a Poisson’s ratio of 0.25. Thus the volume of the column was able to
expand elastically as water entered the surface.
4.5.1. No suction switch
Precipitation occurring at various rates was applied to the surface of the soil column. The soil
permeability was set to be independent of the pore water pressure ( no suction switch applied ) at
either 3.4E-6 m/sec or 3.4E-5m/sec, and was isotropic and homogeneous.
Where the inflow rate was set less than the permeability of the soil, there was no significant
change in the pore water pressures. For both permeabilities, with the inflow rate set at two orders
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of magnitude less than the permeability, the change in the pore water pressure at the column
surface after 50 increments of analysis ( at which point the analysis was terminated ) amounted to
less than 1 kPa ( or less than 0.5% of the initial tensile pressure ).
The more permeable column was also analysed with an inflow of only 1 order of magnitude less
than the permeability, and this produced a change in the surface PWP of about 9 kPa ( or less than
5% of the original tensile PWP ).
The depth of the slight variation in the pore pressure that did occur seemed to be dependent on the
permeability of the soil, but independent of the inflow rate. For the less permeable soil, a small
reduction in pore water tension occurred over the top 6m of the column, whereas in the more
permeable soil, this zone of effect extended to 30m below the column surface. Where the
specified inflow rate was varied, the higher the inflow rate the greater the variation in pore
pressure, but the depth of the zone effect appeared to remain the same. It is believed that this
behaviour is due to an elastic change in the soil structure caused by inflow, since no change in the
PWP would be expected for the case where the inflow rate is less than the soil permeability in a
rigid soil.
When the inflow rate was increased to be exactly equal to the permeability of the soil, there was a
clear, gentle reduction in the pore water tensions within the soil, as shown in Figure 4.5.
The figure shows the response of the more permeable column, but both columns exhibited similar
behaviour. It may be noted that while the tensile value of the pore water reduces, the highest
tension remains at the soil surface throughout. Thus, the pore water pressure always becomes
more compressive with depth.
This behaviour was also sensitive to the permeability of the soil, with the changes in pore water
pressure being more rapid in the higher permeability soil. Obviously, since inflow was set equal
to permeability, this may simply be a reflection of the fact that the inflow rate is higher.
For the lower permeability soil, the effects were much the same, just of a smaller magnitude.
The modelled behaviour was felt to be slightly surprising. With inflow equal to permeability, it
was assumed inflow would pass straight through the soil. With no-flow side and bottom
boundaries, it was expected that under these conditions, there would be a uniform increase in
water pressure, with the PWP profile remaining hydrostatic, but with the phreatic surface steadily
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rising. Again, it is believed that the divergence between expected and actual behaviour is a
reflection of the ability of the soil to deform elastically.
While not shown in the figure, when the inflow ( precipitation ) was stopped but the analysis
continued on, internal drainage occurred within the mesh, leading to the development of the
expected profile. The PWP profile became approximately linear but slightly steeper than
hydrostatic quite rapidly after cessation of infiltration, then slowly reverted towards the
hydrostatic state, but with pore pressures more compressive than the initial pre-infiltration
condition.
Once the inflow rate specified exceeded the soil permeability, it was expected that noticeable
effects would occur. This was found to be the case, and there was a significant difference between
this behaviour and that described immediately above.
With inflow exceeding permeability, there was a rapid change in the surface pore water pressure,
with it becoming much more compressive. The precipitation threshold value ( specified as 0 kPa )
was rapidly reached, with the boundary condition then switching to the constant pressure
condition. Figure 4.6 illustrates the modelled behaviour. As can be seen, unlike the case where
inflow equals the permeability, in this case the pore pressure does not always become more
compressive with depth. Rather, the initial tensile pressures tend to remain at shallow depth in the
column, despite the compressive changes at the surface.
It was found that increasing the rainfall inflow ( while keeping the soil permeability constant )
had the effect of increasing the rate at which the tensile pore pressure became compressive.
Similarly, for a given inflow rate, increasing the permeability of the soil leads to a more rapid loss
of pore water tension. The influence of the soil permeability seemed to be greater than that of the
specified inflow rate.
These points are illustrated by the results presented in Figures 4.6, 4.7 and 4.8.
It can be seen from these results that ( when using a tension-saturated model ) without some form
of suction-dependent permeability it is not possible to produce a true wetting front effect.
Certainly, it seems impossible to reproduce the situation where precipitation reduces the tensile
water pressure, while still preserving some degree of tension in the pore water. It was believed
after this work that the application of the suction switch would enable this form of behaviour to
be reproduced.
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4.5.2. Tension-saturated column with suction switch
The analyses undertaken without a suction-dependent permeability ( as detailed above ) were then
repeated with the suction switch applied.
It was found that the application of a suction-dependent permeability had a significant effect on
the pore water pressure response to infiltration. This is illustrated by Figures 4.9, 4.10, 4.11 and
4.12. These present the results of analyses using the more permeable soil column, with a suction
switch set to give a kmax / kmin ratio of 100, between tensile PWPs of 10 kPa ( ‘p1’ ) and 100 kPa
( ‘p2’ ) ( as defined in Figure 4.1 ). It may be noted that Figure 4.11 is the equivalent to Figure 4.7,
and Figure 4.12 relates to Figure 4.8, in respect to flow rates.
As can be seen, a distinct wetting front profile is developed when a suction-dependent
permeability is applied. Notable also is that where the inflow exceeds the maximum permeability,
the infiltration tends to reduce pore water tension within the ‘wetted zone’ ( the area above the
wetting front ) to zero ( as shown in Figures 4.11 and 4.12 ). This can be compared to the case
where the infiltration rate is specified as being equal to the maximum permeability, as shown in
Figure 4.10, or where infiltration is less than the maximum permeability, as in Figure 4.9.
Where inflow equals the maximum permeability, the pore water tension is rapidly reduced by
infiltration to a tensile value of 10 kPa. This corresponds to the ‘p 1’ value set on the suction
switch. Hence the soil permeability at the surface of the column is now equal to its specified
maximum value, and remains constant unless the infiltration rate drops allowing the pore pressure
to become more tensile again.
However, deeper within the column, the water tensions are higher, and permeability is
correspondingly lower. Thus, while the wetting front continues to develop deeper in the soil, the
near surface behaviour is flow through a constant permeability soil, and becomes as described for
the no-suction switch case, with a constant pore water pressure at the soil surface. The
permeability of the soil at the surface is now equal to the imposed inflow, so flow in equals flow
out, and an approximately steady condition is achieved within this zone.
The case where the infiltration rate was less than the maximum permeability is illustrated by
Figure 4.9. Here, the effect of the suction switch is to give a suction-dependent permeability at the
surface that is initially less than the inflow rate.
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As infiltration proceeds, the tensions in the pore water are reduced, and the permeability of the
soil increases correspondingly, until it matches the inflow rate. At this point, a pseudo-steady
state is achieved. Due to the side and base boundary conditions, the soil column tends to ‘fill up’
with water, leading to a gradual rise in the phreatic surface later in the analysis.
It may be noted from Figure 4.9 that the steady pore tension is 55 kPa; reference to Figure 4.1,
and the application of the specified suction switch parameters, shows that the permeability rate
which corresponds to 55kPa suction is 3.4E-6 m/min. Thus it can be seen that the steady state is
achieved once the suction dependent permeability has risen to a value equal to the applied
infiltration rate.
The behaviour of the soil column with the suction switch applied was generally similar regardless
of the specified maximum permeability, although the effect of lowering the maximum
permeability was to increase the reduction in tensile pore pressure at the column surface. That is,
the compressive change in pore water pressure that occurred at the column surface was less for
the more permeable column than for the less permeable column.
It was also found that changing the ratio of permeabilities in the suction switch ( i.e. kmax / kmin )
had some small effect. The smaller the ratio used the smaller the change in pore water pressure
that occurred at the column surface. This is consistent with the influence of permeability. A small
suction switch ratio will result in the soil permeability at the surface being slightly reduced from
the specified maximum, whereas a high suction switch ratio will produce a much-reduced near
surface permeability. The soil will respond to infiltration accordingly.
Clearly, then, the application of the suction switch produces significant effects when modelling
infiltration into a tension-saturated soil. For the range of values used, the exact parameters applied
within the switch seem to be less significant than the act of applying the switch, although this may
not be the case generally. More over, while it was possible to reproduce the type of wetting front
that is believed to occur in unsaturated soil using the suction switch combined with a tensionsaturated soil, this is not sufficient to claim that unsaturated soils may be modelled. For example,
the tension-saturated approach does not allow for the storage of water and variation in volumetric
water content that occurs in an unsaturated soil. The development work set out in chapters 5 and 6
is still required.
This tension-saturated work does show that some of the response of unsaturated soil to infiltration
is likely due to the variations in permeability, rather than the direct effects of variable water
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content and any related volume change. For example, the resulting pore water pressure
distributions shown in Figures 4.9 to 4.12 are comparable to the work of Rubin et al (1964),
Kasim et al (1998) and Sun et al (1998).
4.5.3. Element size sensitivity
It is notable that in Figures 4.9 to 4.12, the plotted pore water pressure distribution is fairly
angular for many of the cases shown.
As described by Chapuis et al (2001) and Forsyth (1988), there is likely to be some element size
dependency in the results of a finite element analysis of infiltration into unsaturated ( or tension
saturated ) soil, and it was believed that the angular distribution plots were due to this aspect.
Thus, while the development work for unsaturated behaviour was underway, a small study was
undertaken into the effects of element size on the resulting pore water pressure distribution
generated by ICFEP, for the case of infiltration into a tension saturated soil column.
The element size sensitivity study was carried out using the same basic 100m high tensionsaturated column as previously described, except that the number and size of elements in the top
20m of the column were varied.
For this study, the maximum permeability of the soil was specified as 3.4E-9 m/s, with inflow
equal to 3.4E-7 m/s. One increment of the analysis represented 1200 seconds. The suction switch
was applied, with permeability varying between water tension values of 1 kPa and 1000 kPa. The
permeability ratio was varied, being either 10, 1000, or 1000,000, to ensure any effects observed
were due to element size, and were not simply due to an extreme permeability ratio.
Selected results of some of the sensitivity study analyses ( all with a permeability ratio of
1000,000 ) are presented in Figures 4.13 to 4.16. It should be noted that these figures only show
the PWP distribution for the top 40m of the soil column, since this is the region were the water
pressure was responding to infiltration.
Figure 4.13 shows the results obtained when the top 20m of the column was composed of 2m
thick elements (in all cases, the elements below 20m depth were 10m thick ). As can be seen, the
effects of infiltration are limited to near the surface of the column, and the resulting PWP
distribution is not smooth, but demonstrates a ‘spike’ just under the wetting front.
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Figure 4.14 shows the results when the near surface element thickness was increased to 10m. the
apparent response to infiltration shows a less immediate reaction at the surface, but with a deeper
area of effect. The ‘blocky’ nature of the PWP distribution clearly corresponds to the element size
chosen, and demonstrates that there is element size sensitivity.
Figure 4.15 shows the results when the top 20m of the column was modelled using 0.5m thick
elements, while Figure 4.16 shows 0.25m thick elements. In both cases, the resulting PWP
distribution profile is relatively smooth, and the two plots are basically the same. The implication
is that for this particular infiltration problem, a surface element size of 0.5m is necessary to avoid
element size dependency in the results. Further reduction in the element size does not appear to
provide any further benefit, and by increasing the number of elements in the mesh, increases the
computational effort required to obtain a solution.
This conclusion does not seem to be affected by the choice of the permeability ratio ( although the
results for the other ratios are not presented here ).
‘Good practice’ in finite element analysis is to use smaller elements where the greatest change is
occurring, and it is clear from the results so far presented from tension-saturated ICFEP analyses,
and from the literature ( see Chapter 3 ), that the near surface elements tend to experience the
greatest change in PWP. Thus the conclusion of the element size study – that relatively small
elements should be placed in the near surface position – is consistent with what might be
expected.
Experience gained during this project through extensive use of ICFEP and the precipitation
boundary condition suggests that for analysis of ‘field-scale’ problems, the surface element layer
should generally be less than 1m thick.
4.5.4. Two-dimensional tension-saturated analysis
This project is principally concerned with the effects of precipitation on unsaturated soil slopes.
The one-dimensional ( column ) analysis of a tension-saturated soil gave results that conform to
some degree to the expected behaviour for unsaturated soils ( based on the literature review ).
From this, it was felt that a full two-dimensional analysis of rainfall into a tension-saturated slope
might prove useful to understand the likely behaviour of an unsaturated soil slope. The analysis
undertaken was carried out using ICFP v8.0, but with the automatic incrementation procedure
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( described in Chapter 6 ) adopted to deal with the precipitation boundary condition problem
detailed in section 4.4.
The FE mesh used for the analyses is shown in Figure 4.17. A range of permeabilities were
applied, but the results presented here are all for a maximum permeability of 3.4E-7 m/s unless
stated otherwise. The suction switch was applied, with the parameters used varying according to
the analysis, as did the specified precipitation rate. The incremental time step was set at 300
seconds. The sides of the mesh were given no-flow boundary conditions, while the base had a
fixed pore water pressure boundary condition. The precipitation boundary condition was applied
across the whole of the upper surface of the mesh, including the horizontal sections as well as the
actual slope.
Figures 4.18 and 4.19 show the results of an analysis, with precipitation at 6.8E-8 m/s. The
suction switch parameters had permeability varying between water tension values of 1 kPa and
10 kPa, with a permeability ratio of 100.
Figure 4.18 ( a to d ) shows the evolution of the contours of PWP within the slope. As can be
seen, the effect of the rainfall is to quite rapidly reduce the pore water tension in the near surface
soil, but within the core of the slope there appears initially to be minimal effects. Additionally, the
effects of rainfall seem to be greater lower down the slope, with a broad zone of soil wetting up.
In contrast, the soil at the crest shows a much narrower wetting band.
With time, the drop in tensile PWP percolates deeper into the mass of the soil, but it is notable
that significant compressive water pressures do not build up. Also, the effect of gravity drainage
appears to be visible, in that slight tensions are retained in the upper portion of the slope. This is
more clearly illustrated by Figure 4.19, which presents the pore water pressure along a horizontal
cross section through the slope, 4m below the crest ( as shown in Figure 4.17 ).
The initial pressure along this section is a constant tensile value, consistent with the section’s
height above the initial phreatic surface.
The effect of rainfall is to produce a ‘wetted band’ of soil on the surface of the slope, which
gradually migrates in towards the core. The ‘wetted band’ retains some slight pore water tension.
Given the suction switch parameters, a tension value of 1 kPa would be logical, based on the onedimensional analyses described earlier, but as can be seen, slightly higher tension values are
retained, suggesting that some two-dimensional flow effect may be present. It may also be noted
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that while the inflow rate is greater than the initial ( tension-dependent ) permeability of the soil
across most of the mesh surface, it is less than the maximum permeability of the soil. Hence some
amount of pore water tension beyond the ‘p1’ value of 1 kPa is not entirely illogical.
Increasing the specified infiltration rate had the logical effect of creating a more rapid reduction
in the pore water tensions, with a sharp wetting front, as shown in Figure 4.20.
Two ‘spot-value’ contour plots for this case are presented in Figure 4.21. As can be seen in Figure
4.21a, the effect of heavier rainfall is to destroy the tensile pore pressures along the entire surface
of the mesh, with no zone of slightly tensile pore pressures being retained towards the top of the
slope.
Figure 4.21b shows what is probably approaching a steady state condition, with a phreatic surface
that is co-located with the slope surface, and no tensile pore pressures anywhere in the slope. The
resulting pore water pressure contours are certainly influenced by the prescribed boundary
conditions, with a fixed PWP along the base, and no lateral flow permitted through the sides of
the mesh. However, it is also clear that the slope has ‘wetted up’ to a much greater degree under
the heavier rainfall than it did in the first case presented.
In this case, the pore water tension at the surface is rapidly reduced to zero due to the rainfall rate,
which is generally very much greater than the tension-dependent permeability. However, since
the inflow rate is less than the maximum permeability, and particularly given the apparent twodimensional effects detailed for example 1 ( see Figure 4.18 ), it would seem likely that the true
long term steady state condition would see the redevelopment of a slight surface pore water
tension, such that the permeability associated with that tension ( through the suction switch )
equalled the inflow rate. Reference to the suction switch equation on Figure 4.1 indicates that this
would mean a tensile value of just over 1 kPa at the surface.
It may also be noted that the ‘-25 kPa’ contour shown in Figures 4.21a and 4.21b is apparently
unchanging where it lies under the horizontal ground surface at the base of the slope. However,
comparison of these contours with Figure 4.18a, which shows the initial condition for all the
slope analyses, shows that the pore pressure has become more compressive in this zone, as would
be expected.
Figure 4.22 shows the situation when the precipitation rate was not changed, but instead the
permeability ratio in the suction switch was increased by two orders of magnitude ( with the ‘p2’
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value in the suction switch also increased, by one order of magnitude ). The wetting front
becomes sharper, but penetrates into the soil more slowly. Also of note is that the ‘residual’
tension in the near-surface soil is higher. Figure 4.23 shows the related contour plots.
Clearly, increasing the permeability ratio results in a reduction in the actual permeability when
pore water tensions equal or exceed the ‘p1’ value in the suction switch. Since the initial condition
at the height of the section shown in Figure 4.22 is of a tension value greater than the specified
‘p2’ value ( which is more tensile than ‘p1’), the initial permeability in example 3 is considerably
less than in example 1. Thus the observed behaviour is a reflection of the actual permeability of
the soil, and is as might be expected, with a slower rate of penetration of the wetting front into the
soil.
It can be seen that if the specified precipitation rate of example 3 is applied to the suction switch
formula, there is a corresponding water pressure of approximately 18 kPa, which is consistent
with the ‘residual’ water tension shown in Figure 4.22.
If the permeability ratio is held constant, but the range of pore water pressure over which the
permeability varies is increased, the effect appears markedly different. This was the case in
example 4, presented in Figure 4.24, where there appears to be no proper wetting front. Instead, it
seems that the whole soil mass is wetting up evenly, such that the pore water pressure gradually
becomes more compressive, but the contours of PWP remain approximately horizontal ( as shown
in Figure 4.25 ).
As described for example 2, the contours of pressure shown ( in Figure 4.25 ) indicate little
change in the pore water pressure distribution under the horizontal ground surface at the base of
the slope. However, again, the figures show increments 40 and 120. Comparison between these
results and the initial condition, given in Figure 4.18a, shows that there has been a pore pressure
response to infiltration in this region of the mesh.
In this case, the range over which the suction switch applied was considerably larger than the
range of tensile pore water pressures within the slope. Thus, there is little variation in
permeability of the soil, so the soil’s behaviour tended towards that of a soil with a homogeneous
permeability ( i.e. a soil in which there was no suction-switch applicable ). Hence the behaviour is
reminiscent of the tension-saturated soil column with no suction switch shown in Figure 4.5.
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However, it appears that the soil permeability, infiltration rate and suction switch parameters all
interact. Figures 4.26 and 4.27 illustrate the development of the pore pressures when both the
range over which the suction switch operated and the permeability ratio were increased.
Additionally, however, both the imposed rainfall and the maximum permeability of the soil were
decreased ( as shown on the figure ). While there is still no clear wetting front developing, some
sensitivity to the source of the infiltration is revealed.
Although the range of tensile pore water pressures over which the soil permeability varies extends
considerably beyond the maximum tension in the soil, the permeability ratio is sufficiently large
that there is some significant pore pressure dependent variation in permeability within the slope
mesh.
This example illustrates that, at least in tension-saturated soils, it is incorrect to simply divide
examples into ‘wetting front’ and ‘non-wetting front’ cases. Rather, the development of a wetting
front is clearly dependent on a number of parameters, which results in a gradual variation in the
observed response to infiltration.
4.5.5. Conclusions of the tension-saturated analyses
From the results of the tension-saturated analyses, it appears that a significant variation in
permeability with direction of flow is required to produce a distinct wetting front. The more
homogeneous the permeability, the less distinct the wetting front becomes, until, when
permeability is more or less constant, there is no discernible wetting front, but just a general
wetting up.
Since this behaviour appears to be controlled by the permeability variation, it is logical to assume
that it could be mimicked through the use of a permeability model that treated permeability as a
function of depth or effective stress, thus avoiding the need for a dedicated suction switch.
However, it should now be clear that with pore water pressure dependent permeability, the
permeability and pore pressures interact. Changing one of these parameters produces a change in
the other, which affects the whole flow regime within the soil. Attempts to mimic this behaviour
with a simple depth dependent model would be unable to show this interactive behaviour. If the
total stress is known, then substituting an effective stress permeability model for the suction
switch would seem acceptable. However, as discussed later ( see Chapter 6 ), for an unsaturated
soil, changes in the volumetric water content of a unit element of soil will produce corresponding
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changes in the weight of the soil element, as well as the change in the pore pressure reflected by
the SWCC.
Thus the total stress in an unsaturated soil subject to infiltration or other variation in water content
is not constant. It seems clear therefore, that pore water pressure dependent permeability should
be modelled explicitly if the correct flow regime is to be reproduced with any accuracy, and the
ICFEP permeability-suction switch provides a way in which this can be done.
The results of the tension-saturated slope analysis, while not using an accurate model of
unsaturated soil behaviour, and based also on a fictitious slope, provide a useful insight into how
variable the behaviour of an unsaturated slope is likely to be. The results show how several
factors may be expected to impact on the resulting behaviour.
Notably, it appears that the variation in permeability is critical, and it is through affecting
permeability that the controlling factors affect behaviour. However, further conclusions about the
behaviour of an unsaturated slope require a proper unsaturated soil model, since clearly, the
tension-saturated analyses cannot reproduce the variation in water content that will occur within
an unsaturated soil. To determine the significance of this factor, a true unsaturated approach must
be developed.
Chapter 4
104
Log k
K max
K min
P1
P2
Pore water pressure
(Tension +ve)
Log k = log kmax - {(p-p1)/(p2-p1)}log(kmax/kmin)
Figure 4.1: ICFEP permeability - suction switch
105
PWP distribution at start of
increment
Low rain fall intensity:
infiltration ( surface PWP more
tensile than THV )
THV
Tensile,
+ve
Compressive,
-ve
High rain fall intensity: surface
ponding (THV compressive) or runoff ( surface PWP = THV )
PWP distribution at start of
increment
Tensile,
+ve
THV
Compressive,
-ve
THV: Threshold Value as specified in precipitation boundary condition
Figure 4.2: Precipitation boundary condition
106
Increment 0
Increment 1
Increment 2
Pore Water Pressure
Compression
Tension
Depth, z
THV
THV: Threshold Value as specified in
precipitation boundary condition
Figure 4.3: Precipitation boundary condition with large
timestep and inflow rate
107
5m
+196.2 kPa
Initial ( hydrostatic )
pore water pressure
distribution
100m
-784.8 kPa
Figure 4.4: Tension-saturated soil column for ICFEP v8.0
analysis
108
Increment 0
0
Increment 5
10
Increment 25
Increment 50
20
Increment 80
Increment 100
30
40
50
Depth (m)
60
70
80
90
100
200
100
0
-100
-200
-300
-400
Accumulated Fluid Stress ( kPa )
-500
-600 -700
-800
( tension = positive )
ks = 3.4E-5 m/min
Infiltration = 3.4E-5 m/min
Figure 4.5: Precipitation into tension-saturated soil
column, no suction switch
109
0
Increment 25
10
Increment 50
20
Increment 80
30
Increment 100
40
50
Depth (m)
60
70
80
90
100
200
100
0
-100
-200
-300
-400
Accumulated Fluid stress ( kPa )
-600 -700
-500
-800
( tension = positive )
ks = 3.4E-6 m/min
Infiltration = 3.4E-4 m/min
Figure 4.6: Precipitation into tension-saturated soil
column, no suction switch.
110
0
Increment 25
10
Increment 50
20
Increment 80
30
Increment 100
40
50
Depth (m)
60
70
80
90
100
200
100
0
-100
-200
-300
-400
Accumulated Fluid stress ( kPa )
-600 -700
-500
-800
( tension = positive )
ks = 3.4E-5 m/min
Infiltration = 3.4E-4 m/min
Figure 4.7: Precipitation into tension-saturated soil
column, no suction switch.
111
Increment5
0
Increment10
10
Increment 25
20
Increment 50
30
Increment 80
40
Increment 100
50
Depth (m)
60
70
80
90
100
200
100
0
-100
-200
-300
-400
Accumulated Fluid stress ( kPa )
-600 -700
-500
-800
( tension = positive )
ks = 3.4E-5 m/min
Infiltration = 3.4E-3 m/min
Figure 4.8: Precipitation into tension-saturated soil
column, no suction switch.
112
0
Increment 0
Increment 100
Increment 300
Increment 600
Increment 800
Increment 1000
10
20
30
40
50
Depth (m)
60
70
80
90
100
200
100
0
-100
-200
-300
-400
Accumulated Fluid stress ( kPa )
-600 -700
-500
-800
( tension = positive )
ks = 3.4E-5 m/min
Infiltration = 3.4E-6 m/min
Suction switch ratio, kmax/kmin = 100
Figure 4.9: Precipitation into tension saturated column,
with suction switch.
113
0
Increment 0
Increment 25
Increment 50
Increment 80
Increment 100
10
20
30
40
50
Depth (m)
60
70
80
90
100
200
100
0
-100
-200
-300
-400
Accumulated Fluid stress ( kPa )
-600 -700
-500
-800
( tension = positive )
ks = 3.4E-5 m/min
Infiltration = 3.4E-5 m/min
Suction switch ratio, kmax/kmin = 100
Figure 4.10: Precipitation into tension saturated column,
with suction switch.
114
Increment 0
Increment 10
Increment 25
Increment 50
0
10
Increment 80
Increment 100
20
30
40
50
Depth (m)
60
70
80
90
100
200
100
0
-100
-200
-300
-400
Accumulated Fluid stress ( kPa )
-600 -700
-500
-800
( tension = positive )
ks = 3.4E-5 m/min
Infiltration = 3.4E-4 m/min
Suction switch ratio, kmax/kmin = 100
Figure 4.11: Precipitation into tension saturated column,
with suction switch.
115
0
Increment 0
Increment 10
Increment 25
Increment 50
Increment 80
Increment 100
10
20
30
40
50
Depth (m)
60
70
80
90
100
200
0
100
-100
-200
-300
-400
Accumulated Fluid stress ( kPa )
-600 -700
-500
-800
( tension = positive )
ks = 3.4E-5 m/min
Infiltration = 3.4E-3 m/min
Suction switch ratio, kmax/kmin = 100
Figure 4.12: Precipitation into tension saturated column, with
suction switch.
116
100
Increment 0
Increment 1
Increment 6
92
Increment 18
84
Depth, m
76
68
60
200
100
0
-100
Accumulated fluid stress
-200
-300
( tension positive )
Figure 4.13: Element size sensitivity analysis, 2m elements.
117
100
Increment 0
Increment 1
Increment 6
92
Increment 18
84
Depth, m
76
68
60
200
100
0
-100
Accumulated fluid stress
-200
-300
( tension positive )
Figure 4.14: Element size sensitivity analysis, 10m elements.
118
100
Increment 0
Increment 1
Increment 6
92
Increment 18
84
Depth, m
76
68
60
200
100
0
-100
Accumulated fluid stress
-200
-300
( tension positive )
Figure 4.15: Element size sensitivity analysis, 0.5m elements.
119
100
Increment 0
Increment 1
Increment 6
92
Increment 18
84
Depth, m
76
68
60
200
100
0
-100
Accumulated fluid stress
-200
-300
( tension positive )
Figure 4.16: Element size sensitivity analysis, 0.25m elements.
120
121
24m
4m
4m
24m
Figure 4.17: Finite element mesh for tension-saturated slope
analyses.
60m
Position of sections shown in later figures
16m
0k
Pa
co
unt ntour
il th
a
is p t surf
oin ace
t
+100 kPa
+150 kPa
0 kPa
+25 kPa
Note: Tension positive
+50 kPa
-25 kPa
Contour values
122
Figure 4.18: Contours of pore water pressure in tensionsaturated slope, subject to precipitation, example 1.
NOTE: Tension positive
Precipitation rate 6.8E-8 m/s ( THV = 0.0 kPa )
Suction switch varies p1 = 1 kPa, p2 = 10 kPa, with permeability ratio of 100
kmax : 3.4E-7 m/s
Figure 4.18b: Increment 40
Figure 4.18a: Increment 0
+100 kPa
+150 kPa
0 kPa
+25 kPa
Note: Tension positive
+50 kPa
-25 kPa
Contour values
123
Figure 4.18: Contours of pore water pressure in tensionsaturated slope, subject to precipitation, example 1.
NOTE: Tension positive
Precipitation rate 6.8E-8 m/s ( THV = 0.0 kPa )
Suction switch varies p1 = 1 kPa, p2 = 10 kPa, with permeability ratio of 100
kmax : 3.4E-7 m/s
Figure 4.18d: Increment 120
0k
Pa
co
unt ntour
il th
a
is p t surf
oin ace
t
Figure 4.18c: Increment 80
0k
Pa
co
unt ntour
il th
a
is p t surf
oin ace
t
1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0
Distance (m)
Increment 120
Increment 80
Increment 0
Increment 40
124
Figure 4.19: PWP across section of tension-saturated slope,
subject to precipitation, example 1.
NOTE: Tension positive
Precipitation rate 6.8E-8 m/s ( THV = 0.0 kPa )
Suction switch varies p1 = 1 kPa, p2 = 10 kPa, with permeability ratio of 100
kmax : 3.4E-7 m/s
200
170
140
110
80
50
20
-10
-40
-70
Accumulated
fluid stress
( kPa ) -100
1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0
9.0 10.0 11.0 12.0
Distance (m)
Increment 120
Increment 80
Increment 0
Increment 40
125
Figure 4.20: PWP across section of tension-saturated slope,
subject to precipitation, example 2.
NOTE: Tension positive
Precipitation rate 3.3E-7 m/s ( THV = 0.0 kPa )
Suction switch varies p1 = 1 kPa, p2 = 10 kPa, with permeability ratio of 100
kmax : 3.4E-7 m/s
200
170
140
110
80
50
20
-10
-40
-70
Accumulated
fluid stress
( kPa ) -100
+100 kPa
+150 kPa
0 kPa
+25 kPa
Note: Tension positive
+50 kPa
-25 kPa
Contour values
126
Figure 4.21: Contours of pore water pressure in tensionsaturated slope, subject to precipitation, example 2.
NOTE: Tension positive
Precipitation rate 3.3E-7 m/s ( THV = 0.0 kPa )
Suction switch varies p1 = 1 kPa, p2 = 10 kPa, with permeability ratio of 100
kmax : 3.4E-7 m/s
Figure 4.21b: Increment 120
sur
fac
e
sur
fac
e
oun
d
oun
d
Phr
eat
ic s
urf
ace
at g
r
Figure 4.21a: Increment 40
Phr
eat
ic s
urf
ace
at g
r
1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0
9.0 10.0 11.0 12.0
Distance (m)
Increment 120
Increment 80
Increment 0
Increment 40
127
Figure 4.22: PWP across section of tension-saturated slope,
subject to precipitation, example 3.
NOTE: Tension positive
Precipitation rate 6.8E-8 m/s ( THV = 0.0 kPa )
Suction switch varies p1 = 1 kPa, p2 = 100 kPa, with permeability ratio of 10000
kmax : 3.4E-7 m/s
200
170
140
110
80
50
20
-10
-40
-70
Accumulated
fluid stress
( kPa ) -100
Note: Tension positive
+25 kPa
0 kPa
-25 kPa
Contour values
+150 kPa
+100 kPa
+50 kPa
128
Figure 4.23: Contours of pore water pressure in tensionsaturated slope, subject to precipitation, example 3.
NOTE: Tension positive
Precipitation rate 6.8E-8 m/s ( THV = 0.0 kPa )
Suction switch varies p1 = 1 kPa, p2 = 100 kPa, with permeability ratio of 10000
kmax : 3.4E-7 m/s
Figure 4.23b: Increment 120
Figure 4.23a: Increment 40
1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0
9.0 10.0 11.0 12.0
Distance (m)
Increment 120
Increment 80
Increment 0
Increment 40
129
Figure 4.24: PWP across section of tension-saturated slope,
subject to precipitation, example 4.
NOTE: Tension positive
Precipitation rate 6.8E-8 m/s ( THV = 0.0 kPa )
Suction switch varies p1 = 1 kPa, p2 = 1000 kPa, with permeability ratio of 100
kmax : 3.4E-7 m/s
200
170
140
110
80
50
20
-10
-40
-70
Accumulated
fluid stress
( kPa ) -100
+100 kPa
+150 kPa
0 kPa
+25 kPa
Note: Tension positive
+50 kPa
Contour values
-25 kPa
130
Figure 4.25: Contours of pore water pressure in tensionsaturated slope, subject to precipitation, example 4.
NOTE: Tension positive
Precipitation rate 6.8E-8 m/s ( THV = 0.0 kPa )
Suction switch varies p1 = 1 kPa, p2 = 1000 kPa, with permeability ratio of 100
kmax : 3.4E-7 m/s
Figure 4.25b: Increment 120
Figure 4.25a: Increment 40
1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0
9.0 10.0 11.0 12.0
Distance (m)
Increment 120
Increment 80
Increment 0
Increment 40
131
Figure 4.26: PWP across section of tension-saturated slope,
subject to precipitation, example 5.
NOTE: Tension positive
Precipitation rate 3.3E-8 m/s ( THV = 0.0 kPa )
Suction switch varies p1 = 1 kPa, p2 = 1000 kPa, with permeability ratio of 1000
kmax : 3.4E-8 m/s
200
170
140
110
80
50
20
-10
-40
-70
Accumulated
fluid stress
( kPa ) -100
+100 kPa
+150 kPa
0 kPa
+25 kPa
Note: Tension positive
+50 kPa
-25 kPa
Contour values
132
Figure 4.27: Contours of pore water pressure in tensionsaturated slope, subject to precipitation, example 5.
NOTE: Tension positive
Precipitation rate 3.3E-8 m/s ( THV = 0.0 kPa )
Suction switch varies p1 = 1 kPa, p2 = 1000 kPa, with permeability ratio of 1000
kmax : 3.4E-8 m/s
Figure 4.27b: Increment 120
Figure 4.27a: Increment 40
CHAPTER 5
Development of constitutive equations for unsaturated soil
behaviour
5.1
Introduction
As detailed in Chapter 4, ICFEP v8.0 was able to undertake analysis of a soil section containing
tensile pore water pressures subject to rainfall induced infiltration. The results of such analyses,
as presented earlier, demonstrate some of the features expected from infiltration into unsaturated
soils, based on the literature review. However, such results are clearly not an accurate indication
of the unsaturated soil case, since the soil was treated as fully saturated, regardless of pore water
pressure.
The aim of this research project is to model the effects of infiltration into unsaturated soils, so it
was necessary to modify the constitutive equations currently within ICFEP, which assume a
fully saturated soil, to allow for a varying degree of saturation.
5.2
Qualitative assessment of unsaturated behaviour
In a fully coupled analysis, the application of a stress will produce volumetric water content
changes. In the fully saturated case ( with soil particles considered incompressible ), the pore
voids are completely filled with pore water, which is also considered incompressible. Thus, the
application of stress will produce a change in the volume of voids within the soil, leading to a
corresponding change in the volume of the soil water of equal magnitude.
In the unsaturated case, the pore fluid is now two-phase, combining water and air. The air,
whether as occluded bubbles within the water, or as a continuous phase, is compressible and can
flow. Hence the volumetric change of the pore voids will no longer be equal to the volumetric
water content change. Rather, a given change in the volume of the soil mass would result in a
change to the volume of the water within the soil of lesser magnitude, with the difference being
the volume change of the air phase.
Additionally, there are other causes of water change in unsaturated soils that do not occur in the
fully saturated case. Volumetric water content is related to the matric suction present ( this
relationship is the basis of the Soil Water Characteristic Curve ). Therefore, a change in matric
Chapter 5
133
suction produces a direct change in the volumetric water content of the soil ( and vice versa ).
Further, a change in matric suction will also act like a change in the applied stress, in that it will
produce a change in the soil structure, hence leading to variation in the volume of voids ( as
discussed by Toll, 1995 ). This will produce a further change in the volumetric water content of
the soil, as discussed above.
To incorporate these factors, the finite element constitutive model for the saturated soil needs a
number of additions/adjustments, which include the input of new soil parameters.
An examination of the literature covering consolidation behaviour of unsaturated soils ( for
example, Biot, 1941; Fredlund and Morgenstern, 1976; Dakshanamurthy et al, 1984, Fredlund
and Rahardjo, 1993a ) provides the theoretical basis for these parameters.
Biot (1941) provides the basic theory on which the later literature is based. He makes a number
of assumptions in support of his work, including isotropy of the material and that Darcy’s law is
applicable.
Notably, Biot assumes that the water within the soil pores is incompressible, but that the water
may contain air bubbles, which, although not explicitly stated, are compressible, giving a
compressible pore fluid within the soil. Thus it may be inferred from this that Biot’s work is
applicable to saturated soils, or soils containing small, discreet quantities of air, that generally
behave saturated, being either truly saturated or tension saturated. While the water may contain
air bubbles, it appears that Biot did not allow for the case of a continuous air phase, and as such,
his work does not specifically deal with ‘true’ unsaturated soils.
In developing his theory, Biot found it necessary to define three new parameters to describe the
soil behaviour in the general case ( i.e. when the soil is not necessarily fully saturated ), and in
doing so, he actually identified the parameters governing unsaturated soil behaviour.
The first of Biot’s parameters was used in the definition of stress induced strain in a cubic
element of soil.
Biot gave:
ex =
σx ν
σ
− (σy + σz ) +
E E
3H
Eqn 5.1
( and similar for the y and z axis )
Chapter 5
134
where:
ex : strain in x direction
σx : total stress in x direction
σy : total stress in y direction
σz : total stress in z direction
ν : Poisson’s ratio
E : Young’s modulus
σ : the increment of water pressure
H : “An additional physical constant”
While not specifically stated as such by Biot, the stress and strain terms ( ex, σx, etc ) are clearly
actually incremental values, for consistency with the water pressure term.
From equation 5.1, the H parameter clearly relates a change in matric suction to the resulting
change in soil structure.
Biot further considered how the soil’s water content varied with the stresses acting, and
proposed the relationship:
θ=
1
σ
(σx + σy + σz) +
3H 1
R
Eqn 5.2
Where
“H1 and R are two physical constants”
θ: the increment of water content.
It is clear from equation 5.2 that the R parameter links volumetric water content change to water
pressure change, and must therefore be some function of the gradient of the SWCC. The
parameter H1 relates stress to a volumetric water content change. Since the stresses in the soil
largely govern the soil’s deformation, it can be seen that the H1 parameter may also affect the
relationship between soil structure deformation and volumetric water content.
Thus, Biot introduces three new parameters to describe the behaviour of a consolidating soil that
may contain air.
However, he shows that for the specific isotropic stress state case, H1 and H are equal and can
be defined as a single variable, H, which Biot describes thus: "the coefficient 1/H is a measure
of the compressibility of the soil for a change in water pressure". Biot's work is the basis for all
Chapter 5
135
the later papers, and this reduction to a single variable ( i.e. H1 = H ) is common to all, except
Fredlund and Rahardjo (1993a). In their work, they retain H1 and H as separate properties
( though they are termed Ew and H respectively ): this aspect is explored in further detail in
section 5.4.
The fully coupled constitutive relationship for finite element analysis is a combination of the
constitutive relation for the soil structure, and that for the water phase. To modify the finite
element code to allow for unsaturated soils, it is necessary to re-formulate these two relations.
5.3
Constitutive relation for the soil structure.
The development of the constitutive relationship for the soil structure is based on the linearelastic stress-strain work presented by Biot (1941), and developed by Dakshanamurthy,
Fredlund and Rahardjo (1984) and Fredlund and Rahardjo (1993a).
The basic stress strain relation for saturated soils ( assuming linear elastic behaviour ) has been
described by Fredlund and Rahardjo (1993a):
εx =
(σx − uw)
E
−
ν
(σy + σz − 2uw)
E
Eqn 5.3
And similar for εy and εz
Where:
εx : Strain in x-direction
σx : Total direct stress in the x direction
σy : Total direct stress in the y direction
σz : Total direct stress in the z direction
E : Modulus of elasticity or Young’s modulus for the soil structure
ν : Poisson’s ratio.
Shear deformations are given by Fredlund and Rahardjo as:
γxy =
τxy
G
Eqn 5.4
And similar for γyz and γzx
Where G = Shear modulus [ = E/2(1+ν) ]
Chapter 5
136
Combining equation 5.3 ( with similar expressions for εy and εz ) and equation 5.4 ( including
expressions for γyz and γzx ) gives:
 ∆ εx 
 1
 ∆ εy 
−ν



 ∆ εz  1  − ν

= 
 ∆ γ xy  E  0
 0
 ∆ γ yz 



 0
 ∆ γ zx 
−ν
−ν
1
−ν
−ν
1
0
0
0
0
0
0
  ∆ (σ x − uw) 
0
0
0   ∆ (σ y − uw) 


0
0
0   ∆ (σ z − uw) 


2(1 + ν )
0
0   ∆ τ xy 
0
2(1 + ν )
0   ∆ τ yz 


0
0
2 (1 + ν )  ∆ τ zx 
0
0
0
Eqn 5.5
Which leads to:

 1
 ν

 ∆ (σ x − uw )
 1− ν
 ∆ (σ y − uw )
 ν


 ∆ (σ z − uw ) 
E (1 − ν )  1 − ν


=
 ∆ τ xy  (1 + ν )(1 − 2ν )  0

 ∆ τ yz 



 0
 ∆ τ zx 

 0

ν
1− ν
ν
1− ν
ν
1− ν
0
0
0
0
1
0
0
0
0
1 − 2ν
2 (1 − ν )
0
0
0
0
1 − 2ν
2(1 − ν )
0
0
0
0
1
ν
1− ν



0   ∆ εx 
  ∆ εy 

0  
 ∆ εz 


0   ∆ γ xy 
  ∆ γ yz 

0   ∆ γ zx 
1 − 2ν 

2(1 − ν ) 
0
Eqn 5.6
This can be further simplified to:
{ ∆ σ } − {m} ∆ uw = [ D] ∆ ε
Eqn 5.7
Chapter 5
137
Where {m}T = ( 1 1 1 0 0 0 }, and the [D] matrix is as defined by Potts and Zdravkovic (1999):

 1
 ν

1− ν
 ν
E (1 − ν )  1 − ν
[ D] =

(1 + ν )(1 − 2ν )  0


 0

 0

ν
1− ν
ν
1− ν
ν
1− ν
0
0
0
0
1
0
0
0
0
1 − 2ν
2(1 − ν )
0
0
0
0
1 − 2ν
2 (1 − ν )
0
0
0
0
1
ν
1− ν



0 

0 

0 


0 
1 − 2ν 

2(1 − ν ) 
0
Eqn 5.8
Equation 5.7 then leads to:
{ ∆ σ } = [ D]∆ ε + {m}( ∆ uw)
Eqn 5.9
Dakshanamurthy et al (1984) used Biot’s work as a basis to produce a full three dimensional,
three phase consolidation theory, such that:
εx =
(σx − ua ) ν
E
−
E
(σy + σz − 2ua ) + 
ua − uw 

H1 
Eqn 5.10
( and similar for the y and z axis ).
Where:
εx : strain in the x direction
σx : direct stress in the x direction
σy : direct stress in the y direction
σz : direct stress in the z direction
ua : the pore air pressure
Chapter 5
138
uw : the pore water pressure
(ua-uw) : the matric suction
E : Young’s modulus for the soil structure with respect to (σ - ua)
H1 : the elastic modulus for the soil structure with respect to (ua-uw)
ν : Poisson’s ratio
Fredlund and Rahardjo (1993a) provide the same form of expression, and also give it in
incremental form, for changes of stress and strain, though in both cases use ‘H’ rather than ‘H1’.
It can be seen that equation 5.10 is fundamentally the same as equation 5.1 ( with Biot’s “3H”
replaced by “H1” ), but with the addition of the air pressure terms. More over, the unsaturated
form of the soil constitutive equation ( equation 5.10 ) is clearly a simple extension of the
saturated form, presented in equation 5.3.
In matrix notation, equation 5.10 ( once expanded out to include the equivalent variants for εy
and εz and including the shear deformation terms ) may be written:
0
0
0   ∆ (σx − ua )
 ∆ εx 
 1 −ν −ν
1 0 0
 ∆ εy 
−ν 1 −ν



0 1 0
0
0
0
∆ (σy − ua )






 ∆ εz  1  − ν − ν 1
0
0
0   ∆ (σz − ua )  1  0 0 1

= 
+ 

0
0   ∆ τxy  H  0 0 0
 ∆ γxy  E  0 0 0 2(1 + ν )
0 0 0
0 0 0
 ∆ γyz 
0
2(1 + ν )
0   ∆ τyz 






0
0
2(1 + ν )   ∆ τzx 
 ∆ γzx 
0 0 0
0 0 0
0 0 0  ∆ (ua − uw) 
0 0 0  ∆ (ua − uw) 


0 0 0  ∆ (ua − uw) 


0 0 0  ∆ (ua − uw) 
0 0 0  ∆ (ua − uw) 


0 0 0  ∆ (ua − uw) 
Eqn 5.11
Where H =
"an additional physical constant" ( Biot ); Note that Biot actually uses 3H, not H.
"the elastic modulus for the soil structure with respect to (ua-uw)" ( Dakshanamurthy et al, ‘H1’).
"modulus of elasticity for the soil structure with respect to a change in matric suction (ua-uw)"
( Fredlund and Rahardjo ).
Chapter 5
139
Rearranging the above equation gives:

 1
 ν

 ∆ (σ x − ua )
1− ν
 ∆ (σ y − ua )
 ν


 ∆ (σ z − ua ) 
E (1 − ν )  1 − ν


=
 ∆ τ xy  (1 + ν )(1 − 2ν )  0

 ∆ τ yz 



 0
 ∆ τ zx 

 0

ν
1− ν
ν
1− ν
ν
1− ν
0
0
0
0
1
0
0
0
0
1 − 2ν
2(1 − ν )
0
0
0
0
1 − 2ν
2(1 − ν )
0
0
0
0
1
ν
1− ν



ua − uw
0   ∆ (ε x − H ) 


ua − uw 
 ∆ (ε y − H ) 

0 
ua − uw
 ∆ (ε z − H ) 


0   ∆ γ xy 
  ∆ γ yz 

0   ∆ γ zx 


1 − 2ν 

2(1 − ν ) 
0
Eqn 5.12
With the [D] matrix defined as for saturated soils, this may be reduced to:
{ ∆ σ } − { ∆ ua} = [ D]( ∆ ε −
∆ ( ua − uw )
)
H
Eqn 5.13
Hence:
{ ∆ σ } = [ D ] ∆ ε − [ D] (
∆ (ua − uw)
) + { ∆ ua}
H
Eqn 5.14
Assuming ua =0 at all times ( that is, assuming that the air is free to flow and cannot increase in
pressure above atmospheric pressure, as discussed in section 2.2.3 ), and introducing:
{mH}T = ( 1/H 1/H 1/H 0 0 0 )
Eqn 5.15
then
Chapter 5
140
{ ∆ σ } = [ D]∆ ε + [ D]{m }( ∆ uw)
H
Eqn 5.16
Hence comparison of equation 5.9 and equation 5.16 shows that the general, unsaturated case
degrades to the specific, saturated case when [D]{mH} = {m}, and therefore, for the saturated
soils case, H = E/(1-2ν).
5.4
Constitutive relation for the water phase.
The development of the constitutive relation for the water phase is based on the same sources as
that for the soil structure.
Dakshanamurthy et al (1985) re-arranged equation 5.10 to give:
(σx − ua ) = 2G( εx + αε ) − β ( ua − uw)
Eqn 5.17
( and similar for σy and σz )
with:
β=
E
1
2G (1 + ν )
=
H 1 (1 − 2ν ) H 1 (1 − 2ν )
Eqn 5.18a
α=
ν
1 − 2ν
Eqn 5.18b
G = Shear modulus:
G=
E
2(1 + ν )
Eqn 5.18c
Chapter 5
141
Volumetric strain equals the sum of the direct strains:
ε = εx + εy + εz
Eqn 5.18d
They then give an expression for the volumetric water content:
θw =
(σx + σy + σz − 3ua ) ( ua − uw)
3H 11
+
R
Eqn 5.19
Equation 5.19 clearly reverts to the original Biot expression ( given as equation 5.2 ) if the air
pressure, ua, is assumed to be always zero.
Dakshanamurthy et al’s term H11 ( equal to Biot’s H1 ) is defined by them as “the elastic
modulus for the water phase with respect to (σ-ua)”, while R is “the elastic modulus for the
water phase with respect to (ua-uw)”. They state, but do not show, that substitution of equation
5.17 ( and its equivalent σy and σz variants ) into equation 5.19 gives:
θw =
β
ε + γ ( ua − uw )
3
Eqn 5.20a
Where
γ =
1
β
− 1
R H1
Eqn 5.20b
Clearly, the summation of equation 5.17 for σx, σy and σz gives:
Chapter 5
142
(σx + σy + σz − 3ua ) = 2G(εx + εy + εz + 3αε ) − 3β ( ua − uw)
Eqn 5.21
Substituting into equation 5.19, and allowing for equation 5.18d:
θw =
2G( ε + 3αε ) − 3β ( ua − uw) ua − uw
+
3H 11
R
Eqn 5.22a
Hence:
θw =
2Gε (1 + 3α ) 3β ( ua − uw) ua − uw
−
+
3H 11
3 H 11
R
Eqn 5.22b
Substituting for α ( equation 5.18b ):
θw =
3ν   1
3β 
2 Gε 
 + −
 ( u a − u w)
1 1+
3 H 1  1 − 2ν   R 3H 11 
Eqn 5.22c
This may be expanded out to:
θw =
2 Gε  1 + ν 

 + γ ( ua − uw)
3H 11  1 − 2ν 
Eqn 5.22d
Which from equation 5.18a, if H11 and H1 are assumed the same, becomes:
θw =
ε
β + γ ( ua − uw )
3
(Eqn 5.20a)
Chapter 5
143
There is thus a dependence on Biot’s proof that “H1 and H are equal” ( see section 5.2 ).
However, even with this assumption, there is an inconsistency, since while:
H11 ( Dakshanamurthy et al, Eqn 5.19) ≡ H1 ( Biot, Eqn 5.2 )
H1 ( Dakshanamurthy et al, Eqn 5.10) ≡ 3H ( Biot, Eqn 5.1 )
So if H=H1 in Biot’s work, correctly, H1=3H11 in Dakshanmurthy et al’s work.
It appears, therefore, that Daksnahamurthy et al’s work is in error, and that equation 5.20a is
hence incorrect. This is borne out by the work of Wong, Fredlund and Krahn (1998).
Wong et al show that the change in the volumetric water content of the soil, θw, may be
expressed thus:
θw = βε + ω(ua-uw)
Eqn 5.23a
where
β = E/H(1-2ν)
Eqn 5.23b
ω = (1/R) - (3β/H)
Eqn 5.23c
with H, as before, being "the elastic modulus of the soil structure with respect to (ua-uw)", and R
being "a modulus relating a change in volumetric water content to change in matric suction (uauw)".
The above expression is stated by Wong et al as being from Dakshanamurthy et al (1985).
Working through the development of the expression, the Wong et al version appears to have
correctly substituted the various ‘H’ parameters. It may be noted also that this work was
published later than the Dakshanamurthy et al material, and one of its authors was also coauthor of the Dakshanamurthy paper.
All the expressions for θw are basically developments of work by Biot(1941), whose original
expression for the water phase was given in equation 5.2, and is repeated below:
θ=
1
σ
(σx + σy + σz) +
3H 1
R
(Eqn 5.2)
where σ is "the increment of water pressure", and σi is the increment of stress.
Chapter 5
144
As previously stated, Biot proves that for an isotropic stress state, H1 in the above expression
equals H, the "additional physical constant" in the soil structure relationship. Based on this, Biot
goes on to develop the volumetric water content relation:
θw = α +σ/Q
Eqn 5.24a
where
1/Q = (1/R) - (α/H)
Eqn 5.24b
and
α = E/(3(1-2ν)H)
Eqn 5.24c
Hence the later developments of Biot's expression really only amount to replacing the increment
of water pressure, σ, with the change in the matric suction, (ua-uw), and redefining the notation,
such that where Biot uses '3H', this is instead written as just 'H'.
However, Fredlund and Rahardjo (1993a) also develop an expression for the water phase, which
is fundamentally the same initially:
d (ua − uw)
dVw
3
=
d (σ mean − ua ) +
Hw
Vo
Ew
Eqn 5.25
Where:
Hw = "water volumetric modulus associated with a change in (ua-uw)", and hence equivalent to
R, as given in Equations 5.2 and 5.19.
Ew is given as the "water volumetric modulus associated with a change in (σ - ua).
The Ew term is clearly the same as Biot's 3H1. However, while Biot links the soil and water
consolidation behaviour by showing “H1=H” ( for a specific stress case ), Fredlund and
Rahardjo do not appear to rely on this proof, and clearly specify Ew as a separate parameter.
From the definitions of Ew and the H parameter from the soil structure constitutive relation, it
appears that these two parameters are representing fundamentally different aspects of the
soil/water behaviour, and there is no real reason why the two should be identical, under all
circumstances. Thus the Biot proof of “H1=H” ( and the similar corresponding assumptions
made by Dakshanamurthy et al and Wong et al ) seems unreliable.
Chapter 5
145
Using Ew as given by Fredlund & Rahardjo, and assuming that it is not generally equal to H,
then the water content expression can be developed in a similar manner as by Dakshanamurthy
et al, from:
θw =
(σ x + σ y + σ z − 3ua ) (ua − uw)
+
Ew
R
Eqn 5.26
To
θw =
Hβε
 1 3β 
+ (ua − uw ) −

 R Ew 
Ew
Eqn 5.27a
Where
β =
E
1
H (1 − 2υ )
Eqn 5.27b
5.5
Finite element formulation of the constitutive relations for
unsaturated soil
The constitutive relation for the soil structure for an unsaturated soil was given in section 5.3 as:
{ ∆ σ } = [ D] ∆ ε + [ D]{m }( ∆ uw)
H
(Eqn 5.16)
While that for a saturated soil was:
{ ∆ σ } = [ D]∆ ε + {m}( ∆ uw)
(Eqn 5.9)
Potts and Zdravkovi (1999) showed that for the non-consolidating case ( i.e. if the pore
pressures are ignored ) equation 5.9 reduces to:
{∆σ} = [D]{∆ε}
Eqn 5.28
They go on to develop this into a FE formulation:
Chapter 5
146
{∆RG}= [KG]{∆d }
Eqn 5.29
They later develop the full Finite Element expression, containing the water pressure term, as:
[KG]{∆d}nG + [LG]{∆p f}nG = {∆RG}
Eqn 5.30
Where:
[KG] = the global stiffness matrix
{∆d}nG = the vector of global nodal displacements ( for the entire FE mesh )
{∆RG} = the vector of global nodal forces.
{∆pf}nG = the vector of global pore fluid pressures
[LG] = the off-diagonal ( cross-coupling ) sub-matrix
With:
[ KG ] =
∑
N
i =1


 ∫ [ B] T [ D'][ B]dVol 
 vol
i
Eqn 5.31
[ LG ] =
∑


T

{
}
[
]
m
B
[
N
p ]dVol 
∫
i =1
 vol
i
N
Eqn 5.32
And
[D’] = Effective stress constitutive matrix
[B] = Strain matrix
[Np] = matrix of pore fluid pressure interpolation functions
{m} is defined as for equation 5.7
If the incremental pore fluid pressure is allowed a degree of freedom at every node of a
consolidating element, [Np] is the same as [N], the matrix of displacement shape functions.
It appears, therefore, that to convert the ICEP constitutive relationship for the soil structure into
the general case for saturated or unsaturated soils, the {m} term within the [LG] expression
should be replaced by [D]{mH}, as detailed above in section 5.3. This is confirmed by Wong,
Chapter 5
147
Fredlund & Krahn (1998). Hence the modified form for the ICFEP constitutive equation for
unsaturated soils is:
[KG]{ ∆d } + [Ld]{∆pf}nG = {∆RG}
Eqn 5.33
where
[ Ld ] =
∑
N
i =1


 ∫ [ D]{ mH}[ B] T [ Np ]dVol
 vol
i
Eqn 5.34
The constitutive relation for the water phase of an unsaturated soil was given as equation 5.27a:
θw =
Hβε
 1 3β 
+ (u a − u w)  −

 R Ew 
Ew
(Eqn 5.27a)
Wong et al used a slightly different, and apparently flawed, version of this equation, as shown in
equation 5.23a:
θw = βε + ω(ua-uw)
(Eqn 5.23a)
and from this, they show how the constitutive relation for the water phase may be formulated in
finite element terms. The constitutive relation for the water phase becomes:
 ∆t



1
β [ Lf ]{ ∆ δ } −  [ Kf ] + ω [ MN ] { ∆ uw} = ∆ t  {Q} t + ∆ t + [ Kf ]{ uw} t 
γw
 γw



Eqn 5.35
With {∆δ} being the nodal displacements, equivalent to {∆d}nG, and [MN] being the mass
matrix, where:
[MN] = NTN, with N being the row vector of shape functions.
The form of the constitutive relation for the water phase used within ICFEP for a saturated soil
is presented by Potts and Zdravkovic (1999), and is given below:
Chapter 5
148
(
)
[ LG ] T { ∆ d } nG − β∆ t [ Φ G ]{ ∆ pf } nG = [ n ] + Q + [ Φ G ]({ pf } n )1 ∆ t
G
G
Eqn 5.36
With notation as for equations 5.30, 5.31 and 5.32, plus:
β = a time stepping factor reflecting the variation of pf with time.
∆t = the time increment
[ΦG] = the permeability sub-matrix
Q = the flow due to sources / sinks
and
[n ] =
G
∑ (∫
N
i =1
vol
[ E ] T [ k ]{i } dVol
G
)
i
Eqn 5.37
where:
[k] = the permeability matrix
{iG} = the vector defining the direction of gravity
[E] = the matrix containing the pore fluid pressure interpolation functions
Clearly equations 5.35 and 5.36 are very similar, with the differences mainly in the choice of
notation. The definition given by Wong et al for their [Lf] term matches that given here ( in
equation 5.32 ) for [LG]T in the ICFEP formulation, while other apparent differences are because
Wong et al work in terms of pressure head, rather than pressures. With the assumption of zero
air pressure, matric suction (ua-uw) and the negative of the pore water pressure (pf) are
effectively the same.
Allowing for these factors, it is apparent that the non-cosmetic differences between these
equations are directly related to the use of the water phase constitutive relationship for
unsaturated soil. In the saturated equation (5.36), the left-hand side is composed of two
segments:
[ LG] T { ∆ d } nG
Accounts for flow due to soil structure displacement ( changes in the volume of
the voids )
− β∆ t [ Φ G ]{ ∆ pf } nG
Chapter 5
Accounts for flow due to consolidation ( changes in the stress regime )
149
In equation 5.35, the left-hand side may be similarly broken down, with:
β[ Lf ]{ ∆ δ }
Being the equivalent to the soil structure displacement term, and
 ∆t

−  [ Kf ] + ω[ MN ] { ∆ uw}
 γw

Combining the consolidation term with that necessary to reflect
the changes in the stored water content within the soil, and the
effect of changing matric suction on the soil structure.
Since ICFEP already incorporates a β term, as the time stepping factor ( which Wong et al
assume to be 1.0 ), it is necessary to adjust the notation used.
However, it is also necessary to allow for the ‘Ew’ parameter which exists within equation
5.27a, but which was omitted by Wong et al.
Therefore, a new parameter, Ω, is introduced, such that:
Ω =
E
1
Ew (1 − 2ν )
Eqn 5.38
The Ω parameter replaces the use of β, such that the water phase constitutive equation (5.27a)
thus becomes:
 1 3Ω 
θw = Ω ε + (ua − uw) −

R H
Eqn 5.39
( It may be noted from equations 5.38 and 5.27b that Ω = β.H/Ew ).
Following the work of Wong et al, this leads to a FE formulation for the water phase of an
unsaturated soil:
(
(
T
Ω [ LG ] { ∆ d } nG − ( β ∆ t[ Φ G ] + ω [ MN ]){ ∆ pf } nG = [ nG ] + { Q} + [ Φ G ] { ∆ pf } nG
) )∆ t
1
Eqn 5.40
Chapter 5
150
Where:
 1 3Ω 
ω = −

R H
Eqn 5.41
Equations 5.30 and 5.36 together give the form of the ICFEP governing equation for a saturated
soil:
 [ KG ]
T

[ LG ]
{ ∆ RG}
[ LG]   ∆ d  
 =  [ n ] + Q + [ Φ G ] { pf }

(
− β∆ t[ Φ G ]  ∆ pf  




nG








 nG


(
G
nG
))
1

∆ t 

Eqn 5.42
Combining equation 5.33 and equation 5.40 gives the proposed ICFEP governing equation for
unsaturated soils the following form:
 [ KG ]
T

 Ω [ LG ]
{ ∆ RG}
[ Ld ]
  ∆ d  
=
 

− β∆ t [ Φ G ] − ω[ MN ]   ∆ pf   [ n ] + Q + [ Φ G ]({ pf }




 nG








nG


(
G
nG
))
1

∆ t 

Eqn 5.43
5.6
Conceptual model
Equation 5.43 presents the form of the constitutive equation that has been implemented into
ICFEP to enable the analysis of unsaturated soil problems. However, the aim of this project is
not simply to solve a mathematical problem, but to produce a reasonably accurate method of
predicting a real soil’s behaviour in a field situation.
It therefore seems sensible to qualitatively assess the behaviour of an unsaturated soil in some
detail ( beyond that done in section 5.2 ), and to consider how the new ICFEP equation reflects
this behaviour. This assessment leads on to the development of a conceptual model that was
found to be useful in understanding the behaviour of unsaturated soil. It also provided guidance
during the coding necessary to introduce the required new unsaturated soil parameters into
ICFEP.
Chapter 5
151
5.6.1 Suction-displacement relationship
As shown by equation 5.33, nodal reactions within a FE mesh are related to the displacements
within the mesh ( via the stiffness matrix ) and the pore water pressures ( through the crosscoupling matrix ).
In an unsaturated soil, as discussed in section 5.3, changes to the suction may result in changes
to the overall soil structure. Thus, it is necessary to relate pore pressures to displacements. Since
these are the two ‘unknowns’ that are calculated within the FE analysis, there is no direct link
between them, and so it is necessary to link them indirectly, which has been done through the
top right-hand side ( the cross-coupling matrix ) within the main constitutive matrix.
−
−

X   D  R
=
−   P   − 
The cross-coupling term, X, in the unsaturated equations is modified from its saturated form,
and generates the reactions, R, necessary to produce the actual deformations, D, induced by
changes in suction ( pore pressures, P ).
5.6.2 Water flow-displacement relationship
In a saturated soil, the voids within an element of soil are full of water. Therefore, any
compression of the soil element, for example through the application of an external load, causes
consolidation. Water flows out of the element such that the volume of water expelled equals the
total volume change of the element. This is illustrated in Figure 5.1.
In an unsaturated soil, the situation changes. The volume of voids is now the sum of the volume
of water plus the volume of air within the element. If the soil is compressed, the volume of
voids will again reduce, as shown in Figure 5.2. This volume change will be accompanied by a
flow of both air and water. However, the question that arises is “What are the relative
proportions of the air and water flow volumes?”
From either equation 5.42 or equation 5.43 it can be seen that the relationship between water
flow and soil deformation is modelled by ICFEP, through the lower left-hand term of the main
constitutive matrix:
−
X

Chapter 5
−  D  − 
=
−   −   F 
152
In the saturated case, the water flow, F, resulting from the deformations, D, is determined
through the cross-coupling term, X, and the flow produced will be equal to the volumetric
deformations.
In the unsaturated case, the total air and water flow equals the volumetric deformations.
Therefore, the water flow lies somewhere between the total deformations ( all water flow / no
air flow ) and zero ( all air flow / no water flow ). As shown by equation 5.43, this is modelled
in ICFEP through the inclusion of the Ω parameter in the lower left-hand side of the equation:
 −
Ω X

−   D  − 
=
−   −   F 
From the physical behaviour that is being modelled, Ω must lie between 0.0 and 1.0 inclusive.
Further, Ω = 1.0 recovers the saturated situation. From the definition of Ω giving in
equation 5.38, the value of E w can be assessed.
For saturated soil ( Ω = 1.0 ):
Ew =
E
(1 − 2ν )
Eqn 5.44
As the soil becomes increasingly unsaturated, it might reasonably be expected that Ω becomes
smaller, and hence Ew becomes progressively larger. Ω equal to zero represents the no water
flow ( all air flow ) case, and implies that Ew has become infinitely large.
The variation of the Ω parameter is however uncertain. As discussed earlier, Wong et al (1998)
and Dakshanamurthy et al (1984) both present a flawed definition of Ω ( termed “β” in their
notation ), which incorporates the H parameter rather than Ew.
Wong et al (1998) show that H increases with an increase in matric suction, thus resulting in
their Ω-equivalent term decreasing with an increase in suction, which is consistent with the
previously stated expectation.
Kim (2000) in his work had an equivalent to the Ω parameter, this being the degree of saturation
multiplied by Biot’s hydromechanical coupling co-efficient, αc, which is defined as:
Chapter 5
153
αc = 1 −
K
Ks
Eqn 5.45
Where:
K = the bulk modulus of the solid skeleton
Ks = the bulk modulus of the solid
Potts and Zdravkovic (1999) suggest that the bulk modulus of the solid soil particles ( Ks ) is
generally very much greater than the bulk modulus of the solid skeleton ( K ), which implies
that αc tends towards 1.0. Thus, in Kim’s work, the equivalent to the Ω parameter tends towards
the soil saturation. As discussed below, this is not unreasonable for soils with a fairly high
degree of saturation, when any air in the soil is in the form of occluded bubbles. It seems a more
dubious result when the air phase is continuous.
Ng and Small (2000) also presented a saturated soil constitutive FE relationship then modified it
to allow for unsaturated behaviour. However, their modification appears to only include the
addition of a “storage matrix”, with no equivalent to the Ω parameter, which thus is effectively
set to 1.0 in all cases. Again, this seems unrealistic.
It appears, therefore, that the available literature offers little reliable guidance to the nature and
variation of the Ω parameter. Hence, an assessment of the actual soil behaviour is clearly
sensible.
For simplicity of writing, it will be assumed that an element of soil is undergoing compression.
The standard assumption that soil particles are incompressible is also made. It is also assumed
that air is free to flow, and does not increase in pressure. The case for zero air pressure / free air
flow has been explained earlier.
As stated previously, for a fully saturated soil, the water flow out of an element of soil will
equal the change in the volume of the element of soil, so Ω equals 1.0.
If the soil contains bubbles of occluded air, it is reasonable to assume that such bubbles will
flow with any water flow, so water and air will flow in a ratio equal to their relative degrees of
saturation ( that is, Ω equals degree of water saturation ). The situation once air is present as
more than discrete air bubbles ( that is, once suctions exceed the AEV ) is less clear.
Figure 5.2 illustrates the case of an unsaturated soil subject to compression. Is it sensible to
expect that a mostly saturated soil ( say, S = 90% ), in which the air phase may be discontinuous
will give up 90% water and 10% air? ( acting like the ‘occluded bubbles’ case? ) If the
saturations are 50/50, would flow be in the same ratio, even though air generally flows much
more readily? And if the degree of saturation is sufficiently low ( say 10% ) that the water phase
has become discontinuous, is it still reasonable to expect any water flow?
Chapter 5
154
It seems probable ( Dineen 2000, Colmenares 2000 ) that once the degree of saturation drops
sufficiently for the air phase to become continuous, little or no water flows if the soil is
compressed by an external load. Further, once the water phase is discontinuous, no flow will
occur under such circumstances. Comments by Bear (1972) also support this idea.
If this is the case, then any external compression of an unsaturated soil ( with a discontinuous
water phase ) will cause a reduction in the volume of the voids of the soil element, with a
corresponding flow of air. While the overall volume of the soil would have reduced, the volume
of water would have remained constant, so the volumetric water content ( or in other words the
degree of saturation ) will have increased. This would require that the matric suction decrease
( that is, that the pore water pressure becomes more compressive ), which seems mechanically
consistent with the application of an external compressive load.
As the soil decreases in volume, the water phase will occupy an increasingly large proportion of
the voids, so will become more continuous and will increasingly tend to flow. Thus
compression of an element of unsaturated soil will initially tend to drive out only air, but as the
compression continues, more and more water will flow. Eventually, continued compression will
drive all the air from the soil, inducing a fully saturated condition, and only water will then
flow. While an unproven hypothesis, the simple mechanics of this behaviour suggest that it is
basically correct.
The implication of this is that Ω varies from 1.0 when fully saturated to 0.0 at the point at which
the water phase becomes discontinuous. The variation between these two points is not clear,
though a relatively rapid initial drop in the value of Ω once suction exceeds the AEV ( and any
air within the water is no longer only occluded ) seems likely, with Ω becoming less than the
degree of saturation for any given suction beyond the AEV.
5.6.3 Pore pressure-water flow relationship
Section 5.4 showed the constitutive relationship for the water phase in an unsaturated soil, and
in equation 5.27a demonstrated how the volumetric water content is directly ( in part )
dependent on the matric suction. As shown by that equation, changes in matric suction induce
changes in the volumetric water content through two aspects of behaviour. Simplifying equation
5.27a:
θw = f(ε) + f(ψ)a + f(ψ)b
Eqn 5.46
Where
Chapter 5
155
f(ε) indicates a function of strain,
f(ψ) indicates a function of matric suction, with subscripts a and b indicating different functions.
f(ε)gives the water volume change per soil volume change, and was just discussed in section
5.6.2.
f(ψ)a is the change in suction multiplied by the water volume content change per unit change in
suction, and reflects the relation between soil suction and volumetric water content. Hence it
deals with the changes in water storage within an unsaturated soil. This is illustrated by Figure
5.3. As such ( as can be seen from the definition of the parameter R, given earlier ) it is
dependent on the gradient of the SWCC.
If the soil were completely rigid, then a change in suction would change the water content of the
soil, but have no effect on the soil structure. However, in the fully coupled case, the soil
structure can deform, as discussed in section 5.6.2.
Thus in addition to the flow generated due to the relationship between suction and water
content, there may also be flow generated due to the changes in the volume of the element of
soil.
This flow ( water volume change ) is given by f(ψ)b.
In the saturated form of the ICFEP constitutive matrix, changes in the pore pressure, P, lead to a
time-dependent flow, F, through the application of the permeability sub-matrix, X. This is the
standard process of consolidation.
−
−

−  −   − 
=
X   P   F 
Clearly, in the unsaturated case, the suction induced flows must be generated in the same
portion of the equation, since this is where flow and pore pressures interact. Thus:
−
−

Chapter 5
−
 −   − 
=
( X − ω )   P   F 
156
Where ω is as defined in equation 5.41 and incorporates the two functions of matric suction
from equation 5.46 that give flow components.
5.6.4 The conceptual model
The above sections have shown how the changes made to the saturated form of the ICFEP
constitutive equation ( equation 5.42 ) to generate the unsaturated form, equation 5.43, all
represent some real physical process within the soil. They also emphasised how the behaviour
of an unsaturated soil reflects aspects of its air and water content.
It is therefore possible to extend the above to give a full conceptual model for the behaviour of
unsaturated soil, as presented in Figure 5.4.
The conceptual model presented draws heavily on the work of White et al (1970), and divides
the soil into four principle zones, one of which is further sub-divided into two.
Within zone 1, the soil is fully saturated. No air is present, as shown by Figure 5.5, which
illustrates a typical two-dimensional slice through an element of soil. Zone 1 extends from
0 kPa to all compressive pore water pressures, and for all soil in this zone, ‘conventional’
saturated soil mechanics applies.
Zone 2 represents the tension-saturated zone. Air may be present within the soil as occluded
bubbles, having come out of solution. Air will also have started to penetrate into the soil from
air-boundaries, but will have not yet penetrated past the outer-most soil particles, as shown in
Figure 5.5.
It is assumed that the behaviour of soil in this condition is still largely that of a saturated soil, so
saturated soil mechanics continues to apply within zone 2 of the conceptual model.
The boundary between zone 2 and zone 3 is set by the Air Entry Value. In zone 3, the air phase
will have penetrated significantly into the soil.
Zone 3a distinguishes the situation where, although the air phase is continuous from any point at
which it is present within the soil back to an air boundary, it is not continuous all the way across
the element.
Thus the permeability of the element to airflow across the entire width of the element remains
restricted. The water phase, however, is continuous across the element, and water is still able to
flow freely in all directions. Within zone 3a, it is likely that both air and water will flow if the
Chapter 5
157
element is subject to external compression. The Ω parameter will therefore be less than 1.0, but
greater than 0.0.
The switch to zone 3b occurs when the air phase becomes continuous across the element ( see
Figure 5.5 ). However, the water phase is also still continuous ( Figure 5.5 shows a twodimensional slice through a three-dimensional element, so in the figure it appears that the water
phase has become discontinuous: this is not actually the case ).
Within the conceptual model, it is assumed that the switch from zone 3a to zone 3b occurs at the
point of contraflexure within the SWCC.
Since both air and water are now continuous, both will flow within zone 3b. However, as
previously discussed, it is to be expected that air will flow much more readily than water, so the
Ω parameter, while still greater than zero, will increasingly tend towards zero.
Once the soil element is largely desaturated, and the water remaining within it is discontinuous,
the model enters zone 4. Moisture movement in the real soil is now limited to vapour phase
flow, which is assumed to be negligible in the FE model ( as discussed earlier ). Hence
compression of the soil will cause only air flow ( Ω is now equal to zero ).
The conceptual model continues to make use of the core assumptions, namely that Darcy’s law
applies, solid particles are incompressible and that the air phase is free to flow, with a constant
zero air pressure as a result. Additionally, as can clearly be seen, the model is non-hysteretic,
with a unique degree of saturation for any value of suction.
This later assumption is obviously inaccurate, as was shown by the earlier discussion of
hysteresis in unsaturated soils. Since hysteresis results in significant variations between the
drying-SWCC and the wetting-SWCC, there will obviously be variations in the position of the
zonal boundaries for wetting and drying if the conceptual model were to be formulated as a
hysteretic model. For example, as has been shown by Bear (1972), it is possible that on wetting
up, suctions can be reduced to zero, and pore pressures can go compressive, with less than
100% water saturation. Hence the soil would fall into the ‘tension saturated’ zone 2, while
actually having some amount of compressive pore water pressure.
However, fundamentally, the division of the soil’s behaviour into zones based on the relative
connectivity of the air and water phases across a soil element is possible whether wetting or
drying. Since it is this that appears largely to govern at least some aspects of behaviour, the
zonal model, as a concept, remains valid. It may further be noted that another member of the
Imperial College soils section research team is undertaking work into the hysteretic behaviour
Chapter 5
158
of unsaturated soils concurrently with this project. The assumption here of non-hysteretic
behaviour prevents duplication of effort.
5.7
Use of linear elasticity
The basis of the theory presented in this chapter is the work of Biot (1941), who made the
assumption of linear elastic behaviour. This assumption has been accepted by all later authors,
and also extends to the work presented here. However, the development of the theory from a
linear elastic assumption does not preclude its use for non-linear problems.
It is clear that the unsaturated soil behaviour that is to be modelled is highly non-linear. As
mentioned in Chapter 3, as air enters the soil and the degree of saturation reduces, so the
permeability of the soil to water flow reduces. The existing permeability-suction model ( the
‘suction switch’ ) incorporates a linear variation in permeability, and as such is a non-linear
relationship. As is discussed in Chapter 6, the actual relationship generally lacks even the linear
variation, and it is likely that future development of this work will see a more accurate
relationship being developed.
Further, even though the SWCC currently adopted is non-hysteretic, it clearly shows that the
relationship between degree of saturation and suction is far from linear. Additionally, while both
the H parameter’s relationship to suction and the Ω parameter’s variation are currently modelled
with linear variations, the fact that these parameters do depend on the current value of suction
shows that they are modelling non-linear behaviour. Also, the use of a Mohr-Coulomb soil
material model means that even the basic stress-strain behaviour is non-linear.
It is therefore clear that the behaviour being modelled is highly non-linear, which brings into
question the core assumption of linear elasticity, upon which the theory presented here is based.
However, the linear approach can be justified. In much the same way that a circle can be
approximated by a regular polygon with very many small sides, it is possible to approximate
non-linear behaviour using a linear approach, if the incremental changes in each step of the
analysis are kept small.
As was shown earlier in this chapter, the modified ICFEP code, with an unsaturated capability,
is largely just an extension of the pre-existing saturated code. Thus the use of linear elasticity as
the basis of the core theory extends to the original code. However, the ICFEP code has
demonstrated over many years of use that it can accurately model behaviour involving soils
with non-linear stiffness and permeability.
Chapter 5
159
ICFEP achieves this by using a modified Newton-Raphson solution strategy incorporating an
error controlled sub-stepping stress point algorithm.
Figure 5.6 shows the general manner in which this algorithm functions. Incremental
displacements calculated using the incremental stiffness matrix are used to determine
incremental strains. The soil constitutive model is then used to convert the strains into stresses,
via the stress point algorithm. These are then integrated and compared to the actual applied
forces, and the difference becomes the residual load, which is, in effect, the ‘error’ generated by
using a linear model to reflect non-linear behaviour. However, the algorithm functions
iteratively, generating additional displacements, until the residual load is reduced to some
acceptably small value. This approach to modelling non-linear behaviour is more fully
discussed by Potts and Zdravkovic (1999).
In the case of the modified code, the unsaturated soil parameters H, R and Ω have all been
introduced, all of which are non-linear, varying with respect to suction. Each can also be
prescribed separately from the others; for example, it is possible to specify the H parameter as a
constant value within an analysis, while still having a variable R and Ω.
ICFEP therefore uses a modified version of the sub-stepping stress point algorithm to deal with
pore pressures and associated fluid flow in a manner similar to that used to deal with stresses
and strains, until the residual flow is reduced to an acceptable size. Since the three non-linear
unsaturated parameters are each independent of the others, but are all functions of suction
( pore pressure ), ICFEP needs to individually recalculate each for each sub-step of the
algorithm operation.
Thus it can be seen that it is possible to reproduce non-linear behaviour through the application
of a linear model, whether saturated or unsaturated, using theory developed from linear
elasticity, although doing so requires some effort to be expended to incorporate the required
algorithm.
5.8
Summary of constitutive equations development
This chapter has shown how the pre-existing constitutive equations within ICFEP have been
modified to allow for unsaturated soil analysis. As shown by the conceptual model, each of the
changes made corresponds clearly to an aspect of the actual mechanical behaviour of a real soil.
This work has shown that three additional soil parameters must be determined to undertake
unsaturated analysis.
Chapter 5
160
The first, H, is the elastic modulus for the soil structure with respect to matric suction, and so
affects the deformation of the soil structure that occurs when suction changes.
The second, R, is the gradient of the SWCC, and governs the changes in water storage within
the soil due to a change in suction.
The third parameter in its ‘true’ form is Ew, the water volumetric modulus with respect to
changes in total stress ( assuming the air pressure, ua, is always zero ). In practice, this last
parameter is subsumed into Ω, which governs directly the water flow resulting from a change in
the soil voids volume.
The issue of determining values for these parameters is discussed more fully in Chapter 6, but in
brief, the use of Ω offers a practical advantage over the use of Ew in that it has a fixed range
( 0.0 to 1.0 ), and the suctions at which these occur can be readily estimated from the SWCC.
The conversion of the theory presented within this chapter into a practical tool for the analysis
of unsaturated soils requires the development of a number of analytical models and these are
now presented in Chapter 6.
Chapter 5
161
In a saturated soil,
volume change of an element = water flow into / out of the element
( Assuming soil particles are incompressible )
Vol = 1.0m3; e = 0.6
Vol = 0.8m3; e = 0.28
S = 100%; n = 0.375
S = 100%; n = 0.21875
Vsoil = 0.625m3; Vvoid = 0.375m3
Vsoil = 0.625m3 ; Vvoid = 0.175m3
Vwater = 0.375m3; Vair = 0.0m3
Vwater = 0.175m3 ; Vair = 0.0m3
Overall volume change = 0.2m3 ; Flow of water (out) = 0.2m3
Since there is no air in the element, there is no air flow.
Figure 5.1: Air / water flow from a saturated soil
162
In an unsaturated soil,
volume change of an element = water flow into / out of the element PLUS
air flow into / out of the element.
( Assuming soil particles are incompressible )
Vol = 1.0m3; e = 0.6
Vol = 0.8m3; e = 0.28
S = 60%; n = 0.375
S = ???; n = 0.21875
Vsoil = 0.625m3; Vvoid = 0.375m3
Vsoil = 0.625m3 ; Vvoid = 0.175m3
Vwater = 0.225m3; Vair = 0.15m3
Vwater = ??? m3 ; Vair = ??? m3
Overall volume change = 0.2m3 ; But what volume of water flows?
Figure 5.2: Air / water flow from an unsaturated soil
163
B
C
A
Flow in = A + B + C
Z
equals
Flow out = X + Y + Z
X
Y
( Assumes soil is rigid, with no change in porosity )
Figure 5.3a: Saturated soil
Flow in = A + B + C
B
equals
C
Flow out = X + Y + Z
plus
Change in volumetric
water content = n.dS
A
± dVol
Z
Where n = porosity, and dS =
change in degree of saturation
( Assumes soil is rigid, with no
change in porosity )
X
Y
Figure 5.3b: Unsaturated soil
Figure 5.3: Continuity of flow through an element of soil
164
Zone 1
Zone 2
Zone 3
100%
3b
Degree of saturation
3a
Zone 4
“AEV”
Suction
Figure 5.4: The adopted conceptual zonal model
165
1000000 kPa
Zone 1
Zone 2
Zone 3a
Zone 3b
Zone 4
Figure 5.5: Conceptual zones
166
True solution
Load
ψ1
∆Ri
∆d1
∆di
Displacement
∆d1 = estimate of displacements from first iteration
∆di = Final estimate of displacements for entire increment of analysis
ψ1 = the residual load from iteration 1
∆Ri = Actual reactions ( boundary condition forces ) applied during
the increment of analysis
Figure 5.6: The modified Newton-Raphson algorithm,
( after Potts and Zdravkovic, 1999 ).
167
CHAPTER 6
Development of ICFEP models for unsaturated soil analysis
6.1. Introduction
The basic theory behind unsaturated soil behaviour has been presented in Chapter 5, along with
an explanation of how this theory has been incorporated into the ICFEP finite element
constitutive equations.
However, as stated, unsaturated analysis requires a number of new soil parameters, and this
requires that a number of new soil behavioural models be formulated. These models are
presented here.
Further, the effects on the density of the soil of changing the degree of saturation are considered.
Additionally, as detailed in section 4.4.1, the existing ICFEP precipitation boundary condition
was not accurate enough for the present work, and required the application of an automatic
incrementation procedure, which is also described in this chapter.
6.2. The non-linear soil model adopted for unsaturated behaviour
The model adopted to describe the material behaviour of an unsaturated soil was derived from
an existing saturated soil model. ICFEP non-linear soil model 16 enables soil to be modelled in
accordance with three-dimensional Mohr-Coulomb theoretical behaviour.
In developing the unsaturated soil model as a simple extension of a saturated soil model, it was
realised that the resultant would be somewhat simplified. However, some form of unsaturated
soil model was required to permit the analysis necessary to investigate flow in unsaturated soils.
A considerably more complex model for unsaturated soil behaviour, based on the Barcelona
model, which in turn is derived from critical state soil mechanics, has been developed by other
researchers at Imperial College concurrent with this project. However, it was clear early on that
this would not be available within the time available for completion of this project.
The basic soil behavioural model developed is therefore another of the assumptions and
simplifications necessary to enable progress to be made with this research.
Chapter 6
168
6.2.1. Model 16: 3D Mohr Coulomb model for saturated soil
The ICFEP model 16 enables soil behaviour to be modelled using effective stress in accordance
with the standard Coulomb failure criteria:
τf = c′ + σ ′nf tan φ ′
Eqn 6.1
Where:
τf = shear stress at failure ( on the failure plane )
c’ = effective cohesion
σnf’ = effective normal stress ( on the failure plane ) at failure
φ’ = effective angle of shearing resistance ( “angle of friction” ).
As shown by Potts and Zdravkovic (1999), this gives the soil yield surface the shape of a
hexagonal cone in principal effective stress space ( σa’,σb’ and σc’ ), see Figure 6.1.
The basic data required for the operation of this model are the cohesion, c’, the angle of
shearing resistance, φ’, and the angle of dilation, ν. The angle of dilation forms part of the flow
rule that governs plastic volumetric straining. Model 16 also incorporates the capacity to deal
with soil tensions. Appendix A1 presents details of model 16, as taken from the ICFEP user’s
manual.
The choice of model 16 as the basis of the developed unsaturated soil model was made because
of its relative simplicity. To undertake a finite element analysis, it is obviously necessary to
determine suitable soil behaviour parameters. With relatively few, commonly measured, soil
parameters required for model 16, only the determination of the unsaturated parameters was
likely to be of difficulty.
Additionally, since the base model is relatively simple, it was felt that this would make it easier
to understand the influence of the unsaturated aspects of behaviour.
6.2.2. Model 82: Partially saturated 3D Mohr-Coulomb model
The ICFEP model 82 is simply model 16, with the H parameter introduced as required in the
cross-coupling matrix, [Ld] ( see section 5.5 ).
Chapter 6
169
Additionally, once the magnitude of suction exceeds the AEV, σ’nf ( the effective stress term in
equation 6.1 ) is changed to σnf. That is, ICFEP switches to operating in total stresses. This is
consistent with the work of Fredlund et al (1978), as presented in equation 3.17. However, it
should be noted that ICFEP model 82 makes no allowance for the φb term in Fredlund et al’s
equation, which reflects the effect of matric suction ( as shown in Figure 3.4 ). Thus, effectively,
model 82 assumes φb ( and hence the (ua-uw)tanφb term in equation 3.17 ) equals zero in all
circumstances. Great care should therefore be exercised before using this model in a more
general case.
As shown by Wong et al (1998), H is a function of matric suction. It was recognised that there
would likely be little experimental data available to permit the determination of H. Therefore,
rather than implement an overly detailed model, it was decided to reflect H through a ‘tri-linear’
curve, as shown in Figure 6.2.
As can be seen in Figure 6.2, three values of H are specified, each with an accompanying value
of suction also required. Between these points, H varies linearly. At suctions beyond the
maximum specified, H becomes constant, while at suctions with a magnitude less than the AEV,
no value for H is given, since in this range, the soil is assumed to behave as saturated soil ( as
described in the discussion on the conceptual model in Chapter 5 ).
This general form of variation for H appears to be consistent with what little available data there
is for this parameter ( namely, Wong et al 1998 ), though the idea of H becoming constant at
higher suctions is pure speculation. The use of three points to define H does, however, have
some theoretical justification.
From an approximation for the volumetric strain and the slope of the void ratio versus suction
curve, Wong et al state that “the slope of a void ratio versus matric suction curve is {3/(1-n)H}.
This may be re-written as:
H=
3 dψ
(1 − n) de
Eqn 6.2
Where:
n : porosity
dψ/de : inverse of the gradient of voids ratio versus matric suction curve.
H is thus a function of the voids ratio – matric suction relationship ( as could be predicted, based
on its definition – see Chapter 5 ). Clearly, this relationship is not one that is commonly
Chapter 6
170
determined for soils, and H is not a routinely determined parameter, so a simplified distribution
for H is appropriate.
The effects of changes to matric suction on the void ratio have been investigated by Toll (1995).
He considered both the standard voids ratio, e, and what he termed the “equivalent void ratio”,
ew, such that:
ew =
ew =
volume of water
volume of soil particles
θw
= e.Sr
1+ e
Eqn 6.3
Where Sr is the degree of saturation.
The behaviour of both e and ew with changing suction are shown in Figure 6.3.
In the figure, point A represents some initial, fully saturated condition. Point B indicates the
point of desaturation, which is effectively the AEV. Between these two points, the soil is
effectively saturated ( actually, tension-saturated ), and behaves as such, which conforms to the
conceptual model presented in Chapter 5.
At suctions beyond point B, some pores in the soil have become air-filled, and no longer
respond significantly to changes in matric suction. Therefore, the overall change in e tends to
reduce, whereas ew, being a function of degree of saturation as well as e, tends to vary even
more then previously.
This behaviour continues until the shrinkage limit, at point C, at which point the soil is
sufficiently desaturated that only relatively minor changes in e can now occur. The equivalent
void ratio, ew, however, continues to decrease until point D is reached. This is the point at which
the water phase becomes discontinuous. As was discussed in Chapter 3, once the water phase
ceases to be continuous, little further moisture movement can occur. Hence the degree of
saturation will not vary significantly with any further increase in matric suction.
This behaviour matches the conceptual model presented, with point D clearly corresponding to
the change from zone 3B to zone 4, while point C seems to conform to the boundary between
zones 3A and 3B, though this is more of an assumption.
Toll’s theoretical model was investigated by Dineen (1997), whose experimental data generally
confirms Toll’s work.
Chapter 6
171
The model presented by Toll provides some theoretical basis for the distribution of H adopted
for use in ICFEP model 82. With three distinct points dividing the soil’s behaviour into distinct
zones that match the conceptual model, the three values of suction required by the model, AEV,
u1 and u2, are readily identified for what they represent, even if assessing a numerical value for
them is difficult.
Also, it can be seen from Toll’s model and Dineen’s experimental data that de, the change in
void’s ratio, tends to reduce as suction increases. Hence, through equation 6.2, H becomes
larger as suctions increase, which is as represented by ICFEP model 82. The use of a constant H
value beyond the suction value u2 is not specifically supported by the theoretical or
experimental evidence, but represents a convenient assumption for dealing with a suction range
which is unlikely to be entered for most practical field-based problems.
Appendix A2 includes the extract from the ICFEP user’s manual for Model 82. It may be noted
that the basic soil strength parameters are still input as effective stress parameters, even though
the basic theory was developed in terms of total stress ( actually total stress minus air pressure,
but air pressure is assumed to be zero ), and matric suction. Since ICFEP calculates effective
stress from the total stresses and pore water pressures at each gauss point, there is no problem in
switching from an effective stress to a total stress approach, with the switch from using effective
stresses to using total stresses occurring at the AEV.
While, as discussed in Chapter 3, there have been a number of modified effective stress
relationships proposed for unsaturated soils, ICFEP model 82 continues to use the saturated
effective stress relationship, as given in equation 3.10. Again, it must be emphasised that model
82 is not presented as a perfect model for reproducing unsaturated behaviour. Rather, it is
designed to enable the basic response of unsaturated soil to be studied. Hence, the assumed
effective stress behaviour is felt to be reasonable.
When pore pressures are more compressive than the AEV ( that is, the soil is saturated or
tension saturated ), ICFEP treats the soil as saturated. The H parameter is not required in this
situation, and Model 82 effectively devolves to Model 16. Thus, it can be seen from this that
Model 82 is effective for both unsaturated soils analysis ( where all the soil within the analysis
is unsaturated ) and for partially saturated soil analysis ( where the analysis includes some
unsaturated soil and some saturated soil ). Within the partially saturated case, an element of soil
may change from being saturated to being unsaturated, or vice versa.
Chapter 6
172
While Model 82 inputs the value of the H parameter, along with the more basic strength
parameters of the soil, it is also necessary to detail values for the gradient of the SWCC and for
the Ω parameter. These are input using a soil water characteristic curve model.
6.3. The soil water characteristic curve model adopted for unsaturated
behaviour.
During the course of this project, two SWCCs were developed for use in unsaturated analysis.
The first of these was a simple ‘quad-linear’ curve, as shown in Figure 6.4. This enabled the
gradient of the curve to be defined, but is clearly very simplified. Additionally, it was found that
the kinks in the curve could induce numerical convergence problems during analysis. This
SWCC model 2 curve was only ever intended as a temporary model, and use of it was
discontinued once SWCC model 3 was available ( Note that in ICFEP, SWCC model 1
indicates no SWCC required ).
As has been previously stated, a number of other researchers were investigating unsaturated
soils concurrent with this project, and amongst this other work was an investigation of the
hysteretic behaviour of unsaturated soils. As part of this work by others, SWCC model 3 was
produced, details of which are shown in Figure 6.5. As can be seen, this is somewhat more
realistic than the model 2 curve, being non-linear, though still non-hysteretic, and it is based on
the work of Van Genuchten (1980). Attempts to produce a hysteretic SWCC model were not
completed in time to be of use to this research project. Extracts from the ICFEP users’ manual
for each of the two SWCCs are included in Appendices A3 and A4.
It can be seen from Figure 6.5 that with SWCC model 3, the minimum possible saturation may
be greater than 0%; thus it is possible to model a residual degree of saturation.
Both models 2 and 3 are defined in part by a suction value specified as being the “suction at the
shrinkage limit”. It should be noted that this is incorrect terminology. The shrinkage limit and
its position on the conceptual model were discussed in section 6.2. The suctions referred to in
the SWCC models actually correspond to the suctions at which ‘long term’ high-suction
behaviour commences. As such, they are more accurately the suctions at which the water phase
becomes discontinuous, and so represent the boundary between zone 3b and zone 4 in the
conceptual model.
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173
If the SWCC had been plotted in terms of the volumetric water content instead of the degree of
saturation, then the suction value specified as being the ‘suction at the shrinkage limit’ would
actually be the shrinkage limit. The change in gradient in the curve identified by this point
denotes the point at which the voids ratio becomes constant, beyond this, changes in volumetric
water content are due to changes in degree of saturation only.
A conscious decision was made that the model curve should be specified in terms of degree of
saturation, not volumetric water content. Within the conceptual model it is assumed that if the
pore water pressures are compressive, then the soil is fully saturated ( and for tensile water
pressures less than the AEV, the soil is tension saturated, so behaves fully saturated, and may be
assumed to have a degree of saturation of 100% ). This being the case, then the zero-suction
‘start point’ of the SWCC is always precisely known, and is not affected by changes in the void
ratio ( whether such changes occur saturated or unsaturated ).
This last point introduces an aspect of the SWCC that must be considered further.
The SWCC may be presented as either a volumetric water content versus suction curve
( SWCC-vwc ), or as a degree of saturation versus suction curve ( SWCC-sat ). It might be
expected that one unique curve of each type exists for any soil. Given the relationship between
volumetric water content and degree of saturation, the two curves would be related, and
logically be similar in appearance. However, Ng and Pang’s (2000) work suggests that the idea
of a unique SWCC for any given soil is incorrect.
Ng and Pang (2000) considered the effect of stress state on the SWCC, and showed that with the
SWCC plotted in terms of volumetric water content, the SWCC depends on the applied total
stress ( see Figure 6.6 ). Since the volumetric water content ( which is equal to porosity times
degree of saturation ) shows applied stress dependency even at zero suction ( actually 0.1 kPa in
the figure ), at which value the degree of saturation will be constant at 100% in all stress cases,
this is actually showing that the porosity ( and hence the voids ratio ) of the soil is stress
dependent. This is neither a new nor surprising idea. However, it does have implications for the
SWCC.
This idea can be developed by considering an element of soil that is in an unsaturated state, such
that it lies at some point on an SWCC, with some value of suction.
If the element of soil is compressed by an external stress, the void ratio will change. Chapter 5
discussed the possible variation in saturation that may arise from such compression, but if it is
assumed that the compression is accompanied by some form of water pumping, or injection of
air, then it becomes possible for the soil element to maintain a constant degree of saturation.
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174
If the saturation remains constant, then on an SWCC-sat, the soil element would not move,
indicating that suction was unchanged. However, since the void ratio had changed, there would
be a corresponding variation in volumetric water content. Thus the soil’s position on an SWCCvwc would alter, and a change in suction would be required. This is clearly contradictory.
Either the described behaviour is physically impossible, which seems unlikely to be the case, or
the idea of a single SWCC in either saturation of water content terms, is wrong.
Given that volumetric water content is a function of saturation and void ratio ( more
specifically, porosity ), and from Ng and Pang’s work showing void ratio ( stress ) dependency
in the SWCC-vwc, it is clear that the SWCC-vwc cannot be defined for a particular soil as a
single curve. It in fact requires a whole series of nested curves, each for a particular value of
void ratio.
It is more credible that the SWCC-sat may be unique, but it is likely that this too is incorrect.
Certainly, a fully saturated soil, at zero suction, has S=100% regardless of void ratio.
However, it seems unlikely that the saturation-suction relationship remains unique for a soil if
the void ratio of the soil is changed. The ability to sustain suction is largely based on the size of
voids within the soil – this was explained earlier, in Chapter 2. If the soil is compressed, then
the voids within the soil will get smaller, and thus correspondingly higher suctions will be
required to obtain a given degree of desaturation. This is effectively why clay and silt soils
( with very small pore spaces ) tend to sustain higher suctions at any given degree of saturation
compared to sands.
The logical conclusion of this is that there is no unique SWCC for a given soil ( in whatever
form the SWCC is presented ); rather, there is actually a Soil Water Characteristic Surface
( SWCS ), which can be plotted in either degree of saturation or volumetric water content terms
( see Figure 6.7 ).
However, while the idea of an SWCS seems more accurate than that of the SWCC, it is of little
immediate benefit for this project. There is only limited data available on SWCCs, most of
which seems to have been determined at zero applied stress. While formulating an SWCS model
for use in ICFEP would seem more theoretically sound than using a simple SWCC independent
of applied stress, it would be of little practical benefit, since the parameters to input into it
would be unlikely to be available.
The use of SWCC model 3 as presented is therefore the best practical approach that could be
adopted for this project. If a stress-dependent SWCC is required for an analysis, this can be
Chapter 6
175
simulated by representing the soil within the analysis mesh as a series of layers, each with its
own SWCC.
With the saturation – suction relationship being influenced by changes in the void ratio, the
assumption of a unique SWCC-sat will be in error. However, since the volumetric water content
is a multiple of porosity times saturation, adopting a unique SWCC-vwc would be even more
incorrect, since both porosity and saturation would be varying. The use of saturation rather than
volumetric water content in model 3 is therefore further validated.
The general failure of other authors to pick up on the effects of void ratio change on the SWCC
is unsurprising, since, as discussed in detail in Chapter 3, much of the existing literature on
unsaturated soil assumes rigid behaviour. In rigid soils, void ratio changes obviously do not
occur.
Clearly, however, there is scope for further research into the nature of the SWCS.
6.3.1. The Omega variation
As discussed in section 5.6.2, the Ω parameter ( defined by equation 5.38 ) governs the volume
of water that flows for a given change in the volume of voids of the soil. It is therefore
necessary to define this parameter, including how it varies, and the suction over which it varies.
As was discussed, Ω equals 1.0 when the soil is fully saturated. Therefore, in keeping with the
conceptual model, if the pore pressure is equal to or more compressive than the AEV, Ω equals
1.0. Further, once the water phase becomes discontinuous, the conceptual model shows that Ω
reduces to 0.0.
Both SWCC models ( models 2 and 3 ) in ICFEP allow the suction at which Ω becomes zero to
be input as an independent variable, while in both cases, Ω equals 1.0 at the AEV.
There appears currently to be no experimental data to indicate how omega varies between these
two suction conditions, so the simple assumption of a linear variation has been made within the
ICFEP models.
This assumption is justified only because some form of distribution is required, and there is no
evidence for anything better. Hence, this is yet another area of unsaturated soil behaviour that
requires further research beyond this project. Since Ω is really just the volume of water flow per
unit volume change of a soil element, it seems feasible that a laboratory test could be
Chapter 6
176
established to permit such research, enabling Ω to be determined for a given soil at a variety of
suction values.
6.4. Suction dependent permeability
Chapter 4 detailed the existing suction dependent permeability model in ICFEP. Since such a
model already existed, further development was not required. However, some consideration was
given to developing an alternate model.
Fredlund and Rahardjo (1993a) state that the same hysteresis that is present in the SWCC is also
present in the permeability-suction relationship, and illustrate this by reference to Liakopoulos
(1965), see Figure 6.8.
However, the permeability-volumetric water content shows little hysteresis, as shown in Figure
6.9. Fredlund and Rahardjo list further references in support of this idea, which also extends to
include the permeability-degree of saturation relationship being of similar form. The use of a
soil permeability model based on either volumetric water content or degree of saturation would
therefore seem preferable to the current ICFEP option.
However, this only really applies if hysteresis is to be taken into account. Since no attempt is
made within this project to allow for the hysteretic behaviour of unsaturated soils – a
simplification that is generally made, as has been previously discussed – there is no need to
cater for the effects of hysteresis on the permeability of the soil.
Clearly, once a working hysteretic SWCC ( or SWCS ) is developed, it will be necessary to
allow for this behaviour in the permeability model, since the assumption of a single
permeability value for any given value of suction will be incorrect.
If hysteretic behaviour is being modelled, then the existing suction-permeability model ( the
‘suction switch’ ) will be inadequate, and would require modification to reflect the hysteresis in
this behaviour. Effectively, a new permeability model would need to be developed and input
into ICFEP.
A “working hysteretic SWCC” implies that the volumetric water content ( or degree of
saturation ) can be determined accurately, and the difference between wetting and drying
behaviour at any given suction clearly distinguished. In such a case, it would clearly make more
sense to take the volumetric water content ( or degree of saturation ) and use it to determine
permeability through a new water content ( or saturation ) – permeability model, rather than
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177
attempting to use a modified, hysteretic, version of the existing soil permeability model. The
ICFEP permeability model is thus an area that is likely to require future development.
6.5. Density variation
In a saturated soil, all the voids within the soil are full of water, and this remains true regardless
of changes in the pore water pressure.
In an unsaturated soil, as the pore pressure ( suction ) changes, the degree of saturation changes,
and thus ( almost by definition ) the volume of water within the voids must have changed. Since
water has mass, if the volume of water within an element of soil changes, it follows that the
mass of that element must also have changed. This can be illustrated through equation 6.4:
ρ=
Gs + Sr . e
. ρw
1+ e
Eqn 6.4
Where:
ρ: Bulk density
Gs: specific gravity of the soil particles
Sr: degree of saturation
e: void ratio
ρw: Density of water ( 1000 kg/m3 )
In a saturated, deforming soil, as deformation of an element occurs, the void ratio, e, changes,
leading to a water flow into or out of the element. This causes a change of mass, and so should
be reflected in a change in the stress conditions beneath the element ( unless, for example, the
water flows out of the element but then ponds on its surface ). Typically, the water flowing from
a consolidating element in a saturated analysis will be considered to have ‘drained away’, and
disappears from the analysis once it passes through a mesh boundary, while no allowance will
normally be made for the resulting loss of mass. However, the deformations and resulting flowinduced mass changes are, in relative terms, small, and so can reasonably be disregarded.
In an unsaturated soil, things will be different; a change in saturation changes the density of the
soil. Assuming a rigid soil for clarity, the element’s overall volume remains unchanged.
However, if the saturation increases, water has entered the soil, so added mass, making the
element denser; if the saturation decreases, water has left the element, reducing the mass, and
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178
making the element less dense ( see Figure 6.10 ). This will change the stresses in the soil
below, and depending on the void ratio of the soil and the range over which the saturation
varies, this stress change may be significant.
Further, during the wetting up process that will result from rainfall, the surface-most soil layers
will increase in water content and hence in mass, which will impact on slope stability, as this is
likely to increase the mass driving any failure. In this situation, changes of density due to
wetting up are likely to be most significant. Clearly, the changes in stress induced by water
content variation will be relatively small in absolute terms. Hence at depth within the soil, the
effect of water volume changes will be insignificant compared to the total stresses imposed by,
say, 20m depth of soil. However, within a few metres of the ground surface, the total vertical
stress due to soil self weight will amount to a few tens of kilopascals. Since the ground surface
is where evaporation and precipitation occur, it is this surface band of soil that will generally
undergo the greatest variation in water content (and hence mass ). Given the relatively low
‘background’ vertical stress, the density variation due to water movement is likely to have a
significant effect.
Clearly then, to accurately model unsaturated soil behaviour, it is necessary to reflect this
variation in density of the soil as degree of saturation alters.
Within ICFEP, this capability was added as part of this project. From the SWCC model, ICFEP
can determine the degree of saturation corresponding to the matric suction ( or compressive pore
water pressure ) present at each gauss point in the mesh.
Since the initial voids ratio of the soil must be specified, and changes in the void ratio are
determined throughout the analysis to enable deformations to be calculated, ICFEP can also
determine the porosity of each element at any point during the analysis. Thus ICFEP was easily
modified to enable the determination of the volumetric water content at each gauss point.
However, in addition to determining the volumetric water contents for the current increment,
ICFEP was adjusted so that it records the values for the previous increment, and is thus able to
determine the incremental change in water content for each element. Converting a volumetric
content change to an actual volume change, and thence to a mass change is relatively
straightforward.
Having calculated the mass change due to flow for each gauss point, ICFEP reproduces the
change in density of each element by applying a body force to the element, equivalent to the
calculated mass change. The procedure used is fundamentally the same as that used to introduce
self-weight in newly ‘built’ elements when construction processes are being simulated.
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179
The density change body force is applied using the sub-stepping algorithm detailed in
section 5.7. An initial estimate of the body force is made, and then in each sub-step a correction
is applied. In this way, ICFEP calculates a density change force over the increment of the
analysis that is both consistent with the suction-degree of saturation relationship and which
avoids generating out of balance forces within the mesh.
Thus, unless it is specifically input into an analysis that density change is not required, ICFEP
automatically adjusts the mass of each element within the analysis to reflect the changing water
content. This leads to the inducement of whatever stress changes in the analysis mesh that such
a change in mass provokes.
It is believed that this capability to reflect density variation due to changes in saturation is
unique to ICFEP.
6.6. Automatic incrementation for the precipitation boundary condition
The existing ICFEP precipitation boundary condition was discussed in Chapter 4, and this
discussion included reference to a limitation with the boundary condition that enabled
unrealistic pore water pressure profiles to be generated. This limitation was corrected through
the application of an automatic incrementation ( AI ) procedure.
The ICFEP AI capability already existed, but was a relatively new capability, and existed only
for stress-strain behaviour. It did not initially address pore water pressures. The AI was based on
work by Abbo and Sloan (1996) and Sheng and Sloan (1999), and enabled the size of a finite
load step to be automatically adjusted to reduce the error generated during non-linear finite
element analysis over an increment of the analysis to within an acceptable tolerance. It does this
by breaking the increment down into a number of sub-increments.
Appendix A5 presents full details of the AI procedure implemented into ICFEP, but the basics
are summarised here. In the stress-strain case, for each sub-increment, two estimates of the
displacement are made through integrating the relationship between load and displacement in
different ways. If the difference between these two is too great, the sub-increment is reduced in
size before the actual load-step is undertaken, thus avoiding the need for repeated iterations. If
the two estimates are similar, the analysis will generally be reproducing the non-linear
behaviour of the soil accurately. The results of the actual load step will generally still produce a
residual load, which will be checked against a tolerance value, as discussed in Chapter 5.
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180
The use of the AI procedure ensures that in simulations of load-displacement problems, the soil
closely follows the true load-displacement curve, and so offers an improved method of
modelling non-linear behaviour.
The AI procedure also aids the accurate determination of the failure load. Each sub-increment
lasts for some portion of the overall time of the full increment, and any and all loads applied
during the full increment are broken up and applied proportionally to the sub-increments. As an
example, the application of a surface load may lead to failure in a foundation problem. With the
AI procedure invoked, ICFEP would automatically repeat the increment of failure, but break it
up into a series of sub-increments. Each sub-increment would represent some fragment of the
time of the original, full, increment, and the load applied during each sub-increment would be
the same fraction of the full load.
Each time a sub-increment produced failure, ICFEP would continuously refine the size of the
sub-increment until the user-defined minimum size of sub-increment was reached. From the
proportion of the load that could be applied without causing failure, and the slightly greater
proportion of the load that resulted in failure, an accurate indication of the failure load can be
determined. This procedure is illustrated schematically by Figure 6.11. While the figure shows
each sub-increment as being half the size of the preceding one, this does not have to be the case.
The sub-increment size is calculated by ICFEP individually for each sub-increment. This point
is further illustrated below, in the discussion on the AI’s application to the precipitation
boundary condition.
The AI can also be useful pre-failure, in that the sub-increment size used by ICFEP gives a good
indication of the rate of change in the load-displacement curve. A small sub-increment size
( and correspondingly large number of sub-increments ) is indicative that there is significant
change occurring in the load-displacement curve.
The AI procedure was modified as part of this project, to operate in conjunction with the
precipitation boundary condition.
With the AI operating, a tolerance is specified around the precipitation threshold value ( THV )
( For a THV of 0.0 kPa, the tolerance normally used is ±0.1 kPa ), see Figure 6.12. Should the
boundary pore pressure remain more tensile than the THV and lie outside the tolerance, the
boundary condition remains an infiltration condition. If the boundary pore water pressure
becomes more compressive, such that it lies within the tolerance zone surrounding the THV,
then ICFEP accepts this as being equal to the THV. The boundary condition is changed to a
Chapter 6
181
pore pressure condition, with the pressure being set at exactly equal to the THV on the next
increment.
However, if the calculated pore pressure on the boundary at the end of an increment is more
compressive than the THV and lies outside of the tolerance specified, then the program rejects
that increment. Instead, it will automatically calculate some smaller sub-increment, and recalculate the pore pressures over that shorter period of time. The new sub-increment size is
calculated as a proportion of the failed one.
ICFEP compares the difference between the boundary PWP at the start of the sub-increment and
the THV to the change of the PWP at the boundary calculated over the failed sub-increment.
From this, and assuming a linear variation, the new sub-increment size is determined as a
proportion of the old one, see Figure 6.13. Since non-linear behaviour is being modelled, this
linear proportion method rarely gives a sufficiently accurate result immediately, but will
progressively reduce the error.
The AI will continue to adjust the size of the sub-increments used until the boundary pore water
pressure is approximately equal to ( within the specified tolerance ) or less than the THV, at
which point the sub-increment is accepted. Further sub-increments will be applied until the
THV is reached, at which point the boundary condition is changed. What remains of the full
increment will then be applied, with a pore water pressure boundary condition. This is often
completed in a single sub-increment, but may be broken into several sub-increments. This
procedure is illustrated schematically by Figure 6.14. Again, for simplicity, this shows each subincrement as being half the size of the preceding one.
When applied to the saturated soil column model discussed in Chapter 4, the ICFEP AI was
found to operate effectively in controlling the precipitation behaviour and restricting the
boundary pore water pressures generated to values not exceeding the THV.
As implied by this last statement, the use of the AI procedure with the precipitation boundary
condition is independent of the unsaturated behaviour models, and the capability should be used
in any analysis involving the precipitation boundary condition, whether the soils are saturated or
unsaturated.
Chapter 6
182
6.7. The precipitation boundary condition as a recharge model
The precipitation boundary condition enables a flow rate to be specified to a boundary unless
and until the pore pressure on that boundary becomes more compressive such that it reaches a
user-prescribed value.
This capability can be used to model other processes beyond precipitation.
One issue that needs to be addressed in all slope stability problems is the presence of
groundwater, and specifically, how to model the phreatic surface.
While it may be appropriate to place an impermeable boundary along the base of a slope mesh
in some situations, this is not generally the case. Additional, the head and foot of an analysis
mesh will rarely if ever be impermeable boundaries. It is therefore generally the case that some
degree of flow needs to be permitted through these boundaries.
While a fixed pressure boundary would enable flow through the boundaries to develop freely,
such boundaries place artificial restraints on the pore water pressure response to other stimuli,
such as precipitation. The alternative, of a specified flow boundary condition, leaves the pore
water pressure free to vary, but can instead result in an unrealistic build up of water pressure,
since accurately determining the flow rate is difficult, especially since it may well vary
throughout the duration of the analysis.
The precipitation boundary condition provides an alternative to these options, by providing a
form of ‘recharge’ into the analysis when the boundary condition is specified along the base of
the analysis mesh. The inflow rate, instead of being based on rainfall data, is set equal to the
fully saturated permeability of the soil, specifically, the permeability of the soil underlying the
mesh, and therefore outside the analysis. The THV may be set to give the maximum permissible
compressive pore pressure at the base of the analysis. This can be set to be consistent with the
maximum height of the phreatic surface above the base ( assuming a hydrostatic profile ),
during ‘normal’ conditions. The effect of this is illustrated in Figure 6.15.
Flow boundary conditions for the sides may be set as specified flow or specified pore pressure
boundaries. However, since the aim is to allow the phreatic surface freedom to move, it is
clearly preferable to set no-flow side boundaries, which will provide no restraint on the response
of the phreatic surface, while also avoiding the imposition of potentially unrealistic inflows
across these boundaries.
Chapter 6
183
During dry periods, the slope will tend to dry out as water drains down and out of it under
gravity, but continuous recharge from the greater part of the soil mass that is not explicitly
modelled will maintain a deep phreatic surface.
Wetter periods will tend to raise the phreatic surface, as the precipitation rate begins to match
the drainage rate, and may do so sufficiently to switch the base boundary condition to a fixed
pressure condition. Under extreme rainfall, transient perched water tables or non-hydrostatic
pore water pressure profiles are free to develop.
Hence use of the precipitation boundary condition as a recharge condition on the base of the
analysis enables a variable pressure boundary condition to be maintained, which is relatively
sensible and realistic, and does not restrict the pore pressure response to precipitation events.
6.8. Summary of the new ICFEP unsaturated capabilities
As developed for this project, ICFEP now has the capability to undertake unsaturated soil
analysis.
This capability is currently limited, since the models used to reflect unsaturated behaviour are
clearly simplistic, but the models do reflect real soil behaviour. Moreover, this project does not
stand alone, but is one of several underway at Imperial College investigating different aspects of
unsaturated behaviour. The simple models used and presented here provide a basic framework
which, it is intended will be developed further by others, and onto which later models may be
fitted in.
In addition to developing new unsaturated models, several existing aspects of ICFEP, applicable
to saturated behaviour, have been developed further as a result of the demands of this project,
most notably the use of automatic incrementation.
It may also be of interest to note at this point that, while this project assumes that periods of
time between rainfall events are dry, and may be modelled numerically as ‘no flow’ increments,
research by another member of the Imperial College soil’s section has been ongoing to model
evaporation and transpiration processes. When the two projects are complete, there will be the
potential to integrate the two, to fully reflect the hydrological cycle’s impact on soil behaviour.
Chapter 6
184
σb’
Space diagonal
’
’= σ c
σ
b
’=
σa
σa’
σc’
Figure 6.1: Mohr-Coulomb yield surface in principal
stress space (after Potts and Zdravkovic, 1999 )
185
H3 > H2 > H1
u2 > u1 > AEV ( using tension positive notation )
H
H3
H2
H1
AEV
u1
u2
Figure 6.2: Variation of H parameter within
ICFEP model 82
186
suction
VCL : Virgin Consolidation Line
e, ew
A
B
C
e
VCL
ew
D
0
Log (ua-uw)
Figure 6.3: Typical void ratio - suction relationship
( after Toll, 1995 )
187
Degree of
saturation, S (%)
100%
Saturation
at air entry
suction
Saturation at
shrinkage
limit
Suction
BDSUCT
AESUCT
Suction at
shrinkage
limit
Suction at
S=0%
BDSUCT: Suction at the beginning of desaturation
AESUCT: Air entry suction
Figure 6.4: SWCC model 2 - Simple, non-hysteretic
188
Degree of
saturation, S (%)
100%
S0
Udes
Uae
Usl
U0
Udes : Suction at the beginning of de-saturation
Uae : Suction at air-entry value
Usl : Suction at shrinkage limit
U0 : Suction in long term
S0 : Degree of saturation in long term
Figure 6.5: SWCC model 3 - Simple, non-hysteretic,
non-linear
189
Suction
0.44
Volumetric water content
0.42
0.40
0.38
0.36
0.34
0.32
0.30
0.1
1
10
100
Matric suction (kPa)
CDV-N1 (0kPa)
CDV-N2 (40kPa)
CDV-N3 (80kPa)
Figure 6.6: Effects of stress state on the SWCC
( after Ng and Pang, 2000 )
190
1000
Void ratio decreasing
S=100%
Degree of
saturation, S
S=100%
Void ratio
increasing
Matric suction
Figure 6.7a: SWCC-sat varies with void ratio
Void ratio decreasing
Volumetric water
content, θ (= n.S )
Void ratio
increasing
Matric suction
Figure 6.7b: SWCC-vwc varies with void ratio
Figure 6.7: Soil Water Characteristic Surface
191
Volumetric water content, θw
0.35
0.30
Saturated volumetric
water content = porosity
of soil, n = 30%
0.25
0.20
0.15
Drying
0.10
Wetting
0.05
0
5
5
30
35
Saturated coefficient
of permeability,
ks = 4.3 x 10-6 m/s
4
(x 10-6 m/s)
Water coefficient of permeability, kw
10
15
20
25
Matric suction, (ua-uw) (kPa)
Figure 6.8a: SWCC-vwc
0
3
2
Drying
1
Wetting
0
0
5
10
15
20
25
Matric suction, (ua-uw) (kPa)
30
35
Figure 6.8b: water coefficient of permeability
versus matric suction
Figure 6.8: Hysteresis in volumetric water content
and permeability relationships to suction
(after Liakopoulos, 1965 )
192
Water coefficient of permeability,
kw (x10-6 m/s)
5
4
3
Wetting
2
1
Drying
0
0
0.10
0.20
0.15
0.05
Volumetric water content, θw
0.25
0.30
Figure 6.9: Water content - Permeability relationship
( after Fredlund and Rahardjo 1993a )
193
 Gs + Sr . e 
ρ= 
 . ρw
 1+ e 
Assuming:
Specific gravity of soil particles, Gs = 2.7
Void ratio is constant (rigid soil), e = 0.8
Wetting
up
Sr = 0.10
Sr = 1.00
. x 0.8 
 2.7 + 01
ρ= 
 x1000
 1 + 0.8 
. x 0.8 
 2.7 + 10
ρ= 
 x1000
 1 + 0.8 
= 1544.44 kg/m3
= 1944.44 kg/m3
Figure 6.10: Variation of density due to changes in
saturation
194
195
Apply
Increment
Time 100%
Apply SubIncrement 1
Time 50%
Re-apply SubIncrement 2
Time 25%
Re-apply SubIncrement 3
Time 12.5%
Failure
Load 100%
Load 50%
Load 25%
Analysis
converges
Apply SubIncrement 2
Time 50%
Analysis
converges
Apply SubIncrement 3
Time 25%
Failure
Load 12.5%
Failure
Load 50%
Failure
Load 25%
Minimum subincrement size reached
Without AI, a load of 100% causes failure
With AI, a load of 50%+25% = 75% is okay, but 75%+12.5% = 87.5% causes failure. Therefore, the actual failure load is between 75% and 87.5%.
A smaller minimum sub-increment size would allow a more precise answer.
Figure 6.11: Schematic operation of existing ICFEP Automatic Incrementation procedure
Threshold value:
THV
Tension (+ve)
THV+tol
B
Compression (-ve)
THV-tol
D
C
A
PWP distribution
with depth
Depth
PWP distribution A gives a boundary surface PWP that is more compressive than the THV and is outside
the tolerance zone, so will be rejected.
PWP distribution B gives a boundary surface PWP that is more tensile than the THV, so will be accepted.
Since it lies outside the tolerance zone, no change in boundary condition is invoked.
PWP distribution C gives a boundary surface PWP that is more compressive than the THV and is outside
the tolerance zone, so will be rejected.
PWP distribution D gives a boundary surface PWP that lies within the tolerance zone, so will be accepted.
Since it lies within the tolerance zone, the boundary condition is changed to a constant pressure condition.
Shows alternate possibility for distribution D
Figure 6.12: The tolerance zone for the precipitation boundary
condition
196
Y
THV
PWP tension
X
PWP compression
PWP distribution at the end of
the rejected sub-increment
PWP distribution at the
start of the sub-increment
Depth
Old ( rejected ) sub-increment size = 100%
New sub-increment size = 100% * (X/Y)
X = difference between boundary PWP at the start of the subincrement and the THV.
Y = change in the PWP at the boundary, calculated over the rejected
sub-increment
Figure 6.13: Determination of sub-increment size during
application of the precipitation boundary condition
197
Apply
Increment
Time 100%
Apply SubIncrement 1
Time 50%
198
Re-apply
SubIncrement 2
Apply SubIncrement 3
inflow 100%
inflow 50%
Time 25%
inflow 25%
Time 25%
pressure BC
Boundary PWP < THV-tol,
increment rejected:
A
B PWP > THV+tol,
sub-increment ok:
B
Apply SubIncrement 2
Time 50%
B PWP < THV+tol,
B PWP > THV-tol:
D
Set B PWP equal
to THV
Boundary condition is now a specified
pressure boundary, not a specified
inflow rate boundary
inflow 50%
B PWP < THV-tol,
increment rejected:
C
Increment
completed
Note that negative values of pressure indicate compressive PWP, positive values of pressure indicates tensile PWP; hence compression < tension.
Letters A, B, C and D relate to the PWP distributions shown on Figure 6.12
Figure 6.14: Schematic operation of existing ICFEP AI procedure for precipitation
Low Rainfall
Little or no
Precipitation
Actual Phreatic
Surface
THV based on hydrostatic
PWP distribution
assuming a ‘normal’
Phreatic surface.
Drainage
THV
Inflow
High Rainfall
Precipitation
Actual Phreatic Surface, can rise or fall
in response to surface precipitation
Phreatic surface
free to move
Drainage
THV
Fixed pressure boundary
Figure 6.15 Precipitation boundary condition used to
simulate recharge
199
CHAPTER 7
Validation of new coding
7.1. Introduction
Chapter 6 presented the development work necessary to permit ICFEP to be modified such that
it can model unsaturated soil. However, before the modified code could be used to investigate a
practical real-world situation, it was necessary to undertake a number of validatory exercises to
confirm that the coding changes had been implemented correctly.
As should now be clear, there is little existing literature on the fully coupled behaviour of
unsaturated soil, and no convenient closed form solution could be located against which the
modified code could be checked. The validation exercises undertaken were therefore selected
such that an aspect of the real physical behaviour of the soil was being reproduced.
ICFEP was, at the start of this project, operating in version 8.0. As such, the majority of the
code has been extensively tested and used, and no further validation work was undertaken on
the pre-existing sub-routines, even where, like the precipitation boundary condition or the
suction switch permeability model, they were central to this research.
It was necessary to undertake some checking of the automatic incrementation procedure, but
this only involved confirmation that the operation of the AI did not interfere with the operation
of any other portion of the code, and that the ‘tolerance zone’ around the Threshold value
( when the AI is used with the precipitation boundary condition ) functioned correctly. This
checking was done by analysing an extensive series of basic problems, all using saturated
behaviour, since the AI work predated the implementation of the unsaturated soil model. Since
no problems were encountered, and the AI procedure functioned correctly, no details of this
work are presented here.
7.2. Water content volume validation
The first of the validation exercises undertaken was used to confirm that ICFEP correctly
calculated changes in the volume of water within a simulation involving unsaturated soil.
Chapter 7
200
The exercise involved filling a 1m cube of soil ( actually an infinitely long cuboid, since the
analysis was undertaken using plane strain ) with water. The cuboid was given a known
moisture content at the start of the analysis, and from this, the additional volume of water
needed to fully saturate and fill it could be determined. Using the precipitation boundary
condition with the AI procedure, ICFEP was used to calculate the actual water that entered the
cuboid mesh prior to it becoming fully saturated, and the two results compared. The mesh used
and basic procedure followed is illustrated in Figure 7.1.
The initial pore water pressure distribution for the cuboid was in all cases from 0 kPa at the base
to +9.81 kPa at the cuboid surface ( noting that tension is positive ), as shown in Figure 7.1. The
validation tests were undertaken using a number of SWCCs, identified by letters ‘A’ to ‘C’, as
shown in Figure 7.2. Simpson’s rule was used to integrate the SWCCs between 0 kPa and
9.81 kPa, to determine the mean average degree of saturation for the test cuboid for each
SWCC. The mean average degree of saturation is thus the area under the SWCC between 0 and
9.81 kPa suction, divided by 9.81.
An initial voids ratio of 0.6 was prescribed, giving a porosity of 0.375. Hence the initial water
content of the cuboid was 0.375*the mean value of S ( per metre length of the cuboid ). From
this, the volume of water needed to theoretically fill the cube is equal to 0.375 minus the initial
water content. This assumes that no deformation takes place; this is not the case, and this point
is covered shortly.
The fully saturated permeability of the cuboid was set at 3.4E-7 m/s, with no suction
dependency ( that is, permeability was constant, regardless of suction ). The precipitation rate
applied to the top surface of the cuboid was 1.5E-7 m/s. This was chosen as being sufficiently
low that the surface of the cuboid did not fully saturate before elements deeper in the mesh.
Rather, the pore water pressure distribution remained approximately hydrostatic throughout the
simulation. The threshold value in the precipitation boundary condition was set at 0.0 kPa.
Hence when the boundary condition changed from an inflow condition to a fixed pressure
boundary, the phreatic surface within the soil would be co-located with the surface of the mesh,
and the whole mesh would have just become saturated.
The sides and base of the cuboid were set as no-flow boundaries, so that the cuboid acted as a
bucket, slowly filling up with water. The base of the mesh was given a fixed zero displacement
boundary condition in both horizontal and vertical planes. The sides of the mesh were given a
zero horizontal displacement boundary condition, but were permitted to move vertically. Hence,
all deformation of the mesh was limited to one dimensional ( vertical ) movement. As such, the
Chapter 7
201
overall volume change of the cuboid could be determined readily from the average surface
vertical deformation.
Over the series of simulations, the soil was given a variety of stiffness parameters, and
precipitation was applied in various time increments, with the AI procedure in use. The total
volume of water entering the mesh was determined by multiplying the inflow rate ( of 1.5E7 m/s ) by the time increment by the number of increments required to cause the boundary
condition at the surface to switch from an inflow to a pressure condition.
Generally, the switch in the boundary condition occurred not at the end of a full increment, but
at some point within an increment. However, when the AI is in use, the size of each subincrement as a proportion of the total increment is given. Thus, by noting in which subincrement the boundary condition change occurred, an accurate estimate of the size of the prechange portion of the full increment could be obtained.
Results of six tests are given below. For each test, the details of the stiffness parameters are
given in Table 7.1. The time step and SWCC details are specified for each test.
Test 1
This used SWCC A. A constant time step of 10,000 seconds was applied.
The surface precipitation boundary condition changed during increment 188, after 26.141% of
the increment.
Thus actual flow into the mesh, A is:
A = 187.26141*10000sec*1.5E-7 m/s = 0.280892m3 ( per metre length of the cuboid ).
The theoretical volume of water needed to fill the mesh depends on the volume needed to raise
the mean degree of saturation to 100%, which is dependent on the SWCC. From the initial voids
ratio, this gives the volume of water assuming no deformation ( a rigid soil ), so an additional
volume of water, equal to the overall volume deformation of the mesh is also required. Since the
cuboid is 1m wide, the mean surface deformation, in metres, is also equal to the deformation
volume, in m3, per metre length of the cuboid.
In test 1, the mean surface displacement at the end of increment 188 was 0.17362E-04m, hence
the theoretical volume of water to fill the cuboid, T, is given as:
T = 0.282094+0.17362E-04 = 0.282111m3 ( per metre length of the cuboid ).
Chapter 7
202
Hence actual divided by theoretical, A/T = 99.57%.
Test 2
This used SWCC B. A constant time step of 5,000 seconds was applied.
The surface precipitation boundary condition changed during increment 426, after 69.568% of
the increment.
Thus actual flow into the mesh, A is:
A = 425.69568*5000*1.5E-7 m/s = 0.319272m3 ( per metre length of the cuboid ).
The mean surface displacement at the end of increment 426 was 0.34033E-04m, hence:
T = 0.319541+0.34033E-04 = 0.319575m3 ( per metre length of the cuboid ).
Hence actual divided by theoretical, A/T = 99.91%.
Test 3
This used SWCC C. A time step of 10,000 seconds was applied during increments 1 to 185
inclusive, then steps of 900 seconds were used thereafter.
The surface precipitation boundary condition changed during increment 211, after 12.642% of
the increment.
Thus actual flow into the mesh, A is:
A = [(185*10000)+({210.12642-185}*900)]*1.5E-7 m/s = 0.280892m3 ( per metre length of the
cuboid ).
The mean surface displacement at the end of increment 426 was 0.18324E-04m, hence:
T = 0.282094+0.18324E-04 = 0.282112m3 ( per metre length of the cuboid ).
Hence actual divided by theoretical, A/T = 99.57%.
Test 4
This used SWCC C. A time step of 10,000 seconds was applied during increments 1 to 185
inclusive, then steps of 900 seconds were used thereafter.
Chapter 7
203
The surface precipitation boundary condition changed during increment 213, after 29.306% of
the increment.
Thus actual flow into the mesh, A is:
A = [(185*10000)+({212.29306-185}*900)]*1.5E-7 m/s = 0.281185m3 ( per metre length of the
cuboid ).
The mean surface displacement at the end of increment 213 was 0.39254E-03m, hence:
T = 0.282094+0.39254E-03 = 0.282487m3 ( per metre length of the cuboid ).
Hence actual divided by theoretical, A/T = 99.54%.
Test 5
This used SWCC C. A time step of 10,000 seconds was applied during increments 1 to 185
inclusive, then steps of 900 seconds were used thereafter.
The surface precipitation boundary condition changed during increment 211, after 10.028% of
the increment.
Thus actual flow into the mesh, A is:
A = [(185*10000)+({210.10028-185}*900)]*1.5E-7 m/s = 0.280889m3 ( per metre length of the
cuboid ).
The mean surface displacement at the end of increment 211 was 0.10754E-03m, hence:
T = 0.282094+0.10754E-03 = 0.282202m3 ( per metre length of the cuboid ).
Hence actual divided by theoretical, A/T = 99.53%.
Test 6
This used SWCC C. A time step of 10,000 seconds was applied during increments 1 to 185
inclusive, then steps of 900 seconds were used thereafter.
The surface precipitation boundary condition changed during increment 211, after 5.168% of
the increment.
Thus actual flow into the mesh, A is:
Chapter 7
204
A = [(185*10000)+({210.05168-185}*900)]*1.5E-7 m/s = 0.280882m3 ( per metre length of the
cuboid ).
The mean surface displacement at the end of increment 211 was 0.35644E-04m, hence:
T = 0.282094+0.35644E-04m = 0.282130m3 ( per metre length of the cuboid ).
Hence actual divided by theoretical, A/T = 99.56%.
Thus it may be seen from the results of these typical tests that ICFEP correctly calculates the
volume of water required to saturate a cuboid of initially unsaturated soil, and hence cause the
boundary condition to change when using the precipitation boundary condition.
The actual volume of water that was required to cause the precipitation boundary condition
change from inflow to constant pressure was invariably very slightly lower than the theoretical
volume. It should be noted however, that the switch in boundary conditions occurs once the
surface pore water pressure value falls within a tolerance zone around the specified threshold
( the default tolerance of 0.1 kPa was used throughout ). It is therefore possible that the
boundary condition change was occurring when the surface pore pressure was still very slightly
tensile.
These results show that the modified form of ICFEP does reliably calculate changes to the water
volumes within an unsaturated soil.
7.3. Density variation validation
Having determined that ICFEP correctly calculates water flow volumes within an unsaturated
soil, the next validation exercise undertaken was to confirm that the density variation function
operated correctly.
This was carried out in a similar manner to the previously described procedure, using a 1m
plane strain cuboid. Again, the cuboid had an initial pore water pressure distribution such that
the pressure at the base was 0 kPa, and at the surface 9.81 kPa ( positive equals tension ), see
Figure 7.3. The deformation and flow boundary conditions were also as specified for the water
volume validation tests, and the stiffness parameters used were as for test 6, as described in
Chapter 7
205
section 7.2, with SWCC C, as shown in Figure 7.2. However, a constant time step, of 10,000
seconds was applied during this exercise.
The cuboid was filled with water in the same manner as before, and the total volume of water
required to fully saturate the mesh was recorded. Multiplying this volume by the density of
water, 9.81 kN/m3, gave the increase in mass of the cuboid, and hence the theoretical increase in
the total stress, D, acting at the base of the mesh was determined.
The boundary condition change occurred after 24.007% of increment 188, given an inflow of:
187.24007*10000*1.5E-7 = 0.280860 m3 .
This produces a value of D of –2.755238 kPa.
This was compared to the actual mean total stress change, F, calculated by ICFEP, determined
by summing over the nodes at the base of the mesh the change in total vertical nodal force
occurring between the initial condition and the point at which the mesh became fully saturated.
This gave a compressive increase in total stress of –2.756154 kPa.
It can be seen that on wetting up, the theoretical stress change, D, based on the volume of water
flowing into the mesh, is a very close approximation to the actual stress change calculated by
ICFEP, F, such that:
D/F = -2.755238 / -2.756154 = 99.97%.
Again, it should be noted that while the use of the AI gives a very accurate estimate of the flow
into the mesh prior to the precipitation boundary condition change, the need for a tolerance zone
does allow for a small deviation from the true value ( with default tolerance values being used ).
The process was then reversed, with the same cuboid given a fully saturated initial pore water
pressure distribution. The water pressure along the base was then reduced ( made more tensile )
incrementally, in steps no greater than +1 kPa, until a hydrostatic profile with 0 kPa at the
surface was achieved. The stress change calculated from the changes in nodal forces along the
base given by ICFEP was found to be +2.756196 kPa.
This is effectively the negative of the stress change that occurred during the wetting-up process
( Fwetting = 99.9984% of Fdrying ), indicating that the density variation function works equally well
whether wetting or drying.
Chapter 7
206
Thus, it appears that the variable density function correctly adjusts the density of an unsaturated
soil subject to either wetting or drying, and that the changes in vertical stress induced by these
density variations are accurately predicted by ICFEP.
7.4. Consolidation of an unsaturated column
While no rigorous theoretical solution was located against which the modified ICFEP code
could be validated, it was possible to recreate analysis previously published by others.
In particular, Wong et al (1998) undertook the analysis of a column undergoing consolidation.
As detailed in section 3.8, this work did not involve any form of precipitation or infiltration
boundary condition, but was based on similar theory to that used for this project, and did follow
a coupled approach.
It therefore seemed reasonable to recreate the work of Wong et al using the modified ICFEP
code, to permit comparison of results.
The analysis involved loading a 2m high column with a 10 kPa surface load, as shown in Figure
7.4. The initial pore water pressure distribution was hydrostatic, with the phreatic surface at 1m
depth.
Young’s modulus was set at 6000 kPa, with a Poisson’s ratio of 0.33. The H parameter was
constant with suction, at 1.765E+04, and permeability was similarly non-suction dependent, at
3.6E-06 m/s. The SWCC used by Wong et al is shown in Figure 7.5a.
Wong et al’s work showed that the immediate pore water response to the imposition of the load
was for the pressure to increase by an amount equal to the load applied ( i.e. 10 kPa ) within
most of the saturated soil beneath the phreatic surface. However, little or no pore water response
was initially noted in the bulk of the unsaturated soil above the phreatic surface. The soil in the
mid-column region showed a varying pore pressure response, acting as an interface region
between the two forms of response.
With time, the consolidation process causes the pore water pressure within the saturated soil to
drop. Conversely, within the unsaturated soil, the pore pressure increases ( becomes less
tensile ), indicating that flow is tending to occur upwards, from the saturated towards the
unsaturated region.
Chapter 7
207
This initially produces a steady rise in the position of the phreatic surface. However, given
sufficient time, the highly non-linear pore pressure distribution that resulted immediately from
the imposition of the load tends to revert back towards a linear hydrostatic profile. Wong et al’s
results show that this results in a slight drop back in the position of the phreatic surface, though
it remains above the initial position. The results of Wong et al’s analysis, for consolidation
periods of 1 second, 31seconds, 255 seconds and 1023 seconds are shown in Figures 7.6 to 7.9
respectively.
To recreate the analysis of Wong et al, the mesh shown in Figure 7.10 was used. It is not clear
how many elements, and of what size, Wong et al used, but from their data points on the plots
of pore water pressure against depth, it is believed that the mesh used for the ICFEP analysis is
comparable. While not expressly stated by them, Wong et al appear to use 8-noded quadrilateral
elements, which can also be used in ICFEP, and which were adopted in this case for the ICFEP
analysis.
Similarly, the boundary conditions applied in the ICFEP analysis were those assumed to have
been applied by Wong et al, namely no horizontal displacement on the sides, no horizontal nor
vertical displacement along the base, and a no-flow condition along all three of these
boundaries.
The SWCC used in the ICFEP analysis was as shown in Figure 7.5b. Comparing this to Figure
7.5a, it can be seen that the curves are not identical, even allowing for the fact that the Wong et
al curve is in terms of volumetric water content, while the ICFEP curve is formulated relative to
the degree of saturation.
This inability to exactly reproduce the Wong et al curve is because the ICFEP SWCC model
generates the curve from an equation, rather than by connecting distinct points individually
input. The equation parameters used in this case was determined by using the Microsoft ‘excel’
spreadsheet program to generate a best-fit line to the Wong et al data points between 0 kPa and
15 kPa, using the ICFEP SWCC model 3 equation.
From this, it is evident that a new SWCC model for ICFEP would be useful, such that the curve
could be input as a series of distinct suction values, with the corresponding degrees of saturation
( or volumetric water contents ) also specified. While such a model provides a possible option
for future development of ICFEP, for this current work, the currently available SWCC model 3
was used.
However, such a model would to a degree resemble the model 2 SWCC originally developed
for ICFEP ( see section 6.3 and Figure 6.4 ), in that the curve would actually consist of a series
Chapter 7
208
of straight lines. Each distinct data point would tend also to mark a ‘kink’ in the curve. As stated
in section 6.3, it was found that such sharp changes in the gradient can induce numerical
convergence problems.
Further, the parameter R is derived from the gradient of the SWCC. Each ‘kink’ point on the
curve would mark the instantaneous change of the gradient, which would thus be indeterminate
at that point. An assumption as to the value of the gradient at such points would therefore need
to be made. For SWCC model 2, it was assumed that the gradient of the curve at the ‘kink’
points was equal to the gradient of the curve at a suction just less than the suction of the ‘kink’
point.
As previously stated, the ICFEP curve shown in Figure 7.5b is not a good match for the full
Wong et al curve. However, assuming that the volumetric water content of 39% ( at zero
suction ) on Wong et al’s curve equates to 100% saturation, with zero volumetric water content
being 0% degree of saturation, then both curves can be re-plotted for the 0 to 20 kPa range, as
shown in Figure 7.5c.
As can be seen, within this limited range, the curves are quite close matched. Given that the
maximum suction within the column is initially 9.81 kPa, and, as can be seen from Figures 7.6
to 7.9, the magnitude of the suction never increases above this value, only this initial portion of
the curve is applicable to the analysis. Hence the degree to which the ICFEP and Wong et al
curves match for any value of suction beyond 10 kPa is irrelevant.
The unique capability of the density variation function was not used, nor, since there was no
precipitation boundary condition, was the AI capability deployed. The responses of the pore
water pressure to the imposed load over time, as predicted by ICFEP, are included on Figures
7.6 to 7.9.
As can be seen, the basic response pattern is the same, and the actual numerical values of pore
pressure are close. In particular, it can be seen that the ICFEP predictions also show a peak
height to the phreatic surface after 31 seconds, with a drop back after 255 seconds. However,
the phreatic surface after 31 seconds is lower in the ICFEP prediction than in the Wong et al
work, so the difference in phreatic surface height is less marked.
Given that the SWCCs are not precisely the same, and that, as described in section 5.4, Wong et
al use a ‘β’ term ( see Equation 5.23b ), rather than the ‘Ω’ term used in ICFEP, it is evident that
ICFEP is reproducing much the same behaviour as Wong et al. From the values used by Wong
et al for the H parameter, it is evident that their β term was 1.0 throughout the analysis,
regardless of the value of suction. To reproduce their work as accurately as possible, the suction
Chapter 7
209
at which Ω became zero in the ICFEP analysis was specified as 1000,000 kPa. Hence the value
of Ω remained close to 1.0 throughout the analysis. Differences between Wong et al’s results
and those predicted by ICFEP are therefore likely due primarily to the differences in SWCCs.
Wong et al also predicted the vertical displacement throughout the column with time, as shown
in Figures 7.11 to 7.14 ( noting that initial displacement at 0 seconds was zero throughout the
depth of the column ). The corresponding ICFEP predictions are also shown on these figures.
Again, it can be seen that the basic behaviour predicted by ICFEP matches that predicted by
Wong et al, although the discrepancies between the precise values of displacements, while still
small, are slightly greater than occurred for the pore pressure results. The differences in the
predicted results can again be explained primarily by differences in the SWCCs used, and to a
lesser degree by the effect of the Ω parameter.
As also was shown by Wong et al, it is apparent from the ICFEP analysis that the settlement
within the unsaturated zone is effectively instantaneous. After increment 1 the gradient of the
displacement curve remains constant within the unsaturated zone, and merely moves to the left
as settlements within the underlying saturated soil develop. As the saturated soil consolidates,
water must be flowing up into the unsaturated zone, as shown by the drop in suction that occurs.
However, the high value of H used means that no significant swelling results from this water
movement.
The most serious discrepancy is in the predicted displacements within the saturated zone of the
soil. The Wong et al work indicates a vertical displacement of 1.1mm at 1m above the base of
the column after 1023 seconds, which they suggest is about the end of the consolidation period.
This displacement value is consistent with linear elastic deformation, except that it appears to
take no account of the effect of changes in the pore water pressure ( see equation 5.3 ).
Incorporating the change in pore pressure results in linear elastic deformations that match the
ICFEP prediction, these being slightly lower ( about 92% ) than that predicted by Wong et al at
the 1m mark.
Thus it is evident that ICFEP ( with no imposed flow or precipitation boundary condition, and
no density variation ) is able to reproduce the general behaviour predicted by Wong et al. Since
the fundamental theory being used is basically the same, this implies that ICFEP is correctly
implementing this theory.
It should be noted, though, that Wong et al utilise the β term which, as detailed in Chapter 5, is
not a reliable approach to modelling unsaturated behaviour. Further, their use of a value of 1.0
Chapter 7
210
for this term implies an assumption, made because of a lack of reliable data. In reproducing their
work using ICFEP, the Ω parameter was set to be approximately equal to 1.0 throughout the
analysis, even though this required a variation of Ω with suction that was unrealistic and
generally inconsistent with the SWCC used.
For more general cases, while it is currently necessary to use an assumed variation in the Ω
parameter with suction, the value of this parameter can be closely related to the physical state of
the soil, as shown by the conceptual model. The current ICFEP model enables this to be
modelled at least in part, and as such, it provides a superior approach to that given by Wong et
al.
While Wong et al undertook analysis of consolidation of an unsaturated soil column, they did
not extend their work beyond this, to truly investigate flow into unsaturated soil. While their
work provides a comparison against which results from ICFEP may be compared, it stops short
of investigating the effects of precipitation into unsaturated soil. Having established the
reliability of the newly modified ICFEP code, this project can now undertake such an
investigation.
Chapter 7
211
Table 7.1
Test
Young’s
Poisson’s
modulus, E
ratio, ν
(kPa)
H1 ( at 0 kPa )
H2 ( at 20 kPa )
H3 ( at 1000,000
kPa )
(kPa)
(kPa)
(kPa)
1
500,000
0.0
500,000
1000,000
10,000,000
2
200,000
0.3
500,000
1000,000
10,000,000
3
500,000
0.0
500,000
1000,000
10,000,000
4
10,000
0.3
25,000
2500,000
25,000,000
5
50,000
0.3
125,000
1250,000
12,500,000
6
200,000
0.1
250,000
2500,000
25,000,000
Note that parameters H2 and H3 have no effect on the results of the analysis, since the pore
water pressure throughout the mesh remains between 0.0 kPa and H1 at all times during the
simulation.
Chapter 7
212
Precipitation, at
1.5E-7m/s
Plane
strain
No flow
boundary
1m
1m
No flow
boundary
+9.81 kPa
0 kPa
Initial PWP
distribution
1m
final PWP
0 kPa
-9.81 kPa
Volume of water in ‘cube’, V = n * Sav
where n = porosity, and Sav = average degree of saturation
Fully saturated volume of water in ‘cube’, W = n * 1.0
( S = 100% )
Theoretical water inflow needed to fully saturate ‘cube’, T = W - V
Actual flow in, A = Time per increment x flow per increment x No. of increments.
A / T = 99.53% to 99.91%
( SWCC and stiffness parameters varied ).
Figure 7.1: Water content volume validation
213
α = 1.28; n = 2.1; m = 1.0; sr = 0.15.
100
PWP at which Ω = 0.0: 9.81kPa.
Saturation (%)
80
Mean saturation between 0.0 and 9.81 kPa =
24.775%
60
Hence volume of water required to fill cuboid (
assuming rigid behaviour ) = 0.28209m3 ( per
metre length )
40
20
0
0
1
2
3
4
100
6
7
8
9
10
α = 1.28; n = 1.5; m = 1.0; sr = 0.0.
PWP at which Ω = 0.0: 9.81kPa.
80
Saturation (%)
5
suction ( kPa)
SWCC-A
Mean saturation between 0.0 and 9.81 kPa =
14.789%
60
Hence volume of water required to fill cuboid (
assuming rigid behaviour ) = 0.31954m3 ( per
metre length )
40
20
0
0
1
2
3
4
6
7
8
9
10
α = 1.28; n = 2.1; m = 1.0; sr = 0.15.
100
PWP at which Ω = 0.0: 0.01kPa.
80
Saturation (%)
5
suction ( kPa)
SWCC-B
Mean saturation between 0.0 and 9.81 kPa =
24.775%
60
Hence volume of water required to fill cuboid (
assuming rigid behaviour ) = 0.28209m3 ( per
metre length )
40
20
0
0
1
2
3
4
5
6
7
8
9
10
suction ( kPa)
SWCC-C
α, n, m and sr parameters are used to define the curve, in accordance with Van Genuchten (1980)
Figure 7.2: SWCCs used for validation testing filling a 1m cuboid
214
Initial PWP
Precipitation, at
1.5E-7m/s
+9.81 kPa
No flow
boundary
0 kPa
No flow
boundary
Final PWP
0 kPa
-9.81 kPa
∆σ
calculated from the sum of
the nodal force changes
A . 9.81 = Theoretical change in total stress, D, where A = Actual inflow
Compare to actual stress change, F ( Calculated from base nodal forces )
D / F = 99.97%
Figure 7.3: Density variation validation.
215
tension
compression
10 kPa
-9.81 kPa
2m
9.81 kPa
Initial hydrostatic PWP distribution:
Note that Wong et al use a tension-negative notation for pore water
pressure
Figure 7.4: Wong, Fredlund and Krahn (1998)
compression of column
216
Volumetric water content (%)
40
35
30
25
20
15
10
5
0
-100
-80
-40
-60
-20
0
20
Pore water pressure (kPa)
Figure 7.5a: Wong , Fredlund and Krahn (1998) SWCC
100
α = 0.009
Saturation (%)
80
n = 1.45
60
m = 96.5
40
sr = 0.15
20
0
0
20
40
60
80
100
Suction (kPa)
Figure 7.5b: SWCC used in ICFEP analysis
100
Figure 7.5c: Comparison of SWCCs over
0-20 kPa suction range
80
Note that the Wong et al data is now given
relative to suction rather than pore water
pressure.
60
0
10
20
Figure 7.5: SWCCs used in consolidation of column
217
Height above base of column ( m )
2.0
1.5
1.0
0.5
0.0
-10
-5
0
5
10
15
Pore-water pressure ( kPa )
Wong, Fredlund and Krahn (1998) data:
ICFEP simulation:
Figure 7.6: PWP distribution after 1 second
218
20
Height above base of column ( m )
2.0
1.5
Max. water table
1.0
0.5
0.0
-10
-5
0
5
10
15
Pore-water pressure ( kPa )
Wong, Fredlund and Krahn (1998) data:
ICFEP simulation:
Figure 7.7: PWP distribution after 31 seconds
219
20
Height above base of column ( m )
2.0
1.5
1.0
0.5
0.0
-10
-5
0
5
10
15
Pore-water pressure ( kPa )
Wong, Fredlund and Krahn (1998) data:
ICFEP simulation:
Figure 7.8: PWP distribution after 255 seconds
220
20
Height above base of column ( m )
2.0
1.5
Final water
table
1.0
0.5
0.0
-10
-5
0
5
10
15
20
Pore-water pressure ( kPa )
Wong, Fredlund and Krahn (1998) data:
ICFEP simulation:
Figure 7.9: PWP distribution after 1023 seconds
221
10 kPa imposed load
2m
No flow boundary
1m
Figure 7.10: ICFEP mesh for consolidating column
analysis
222
Height above base of column, m
2.0
1.5
1.0
0.5
0.0
-2.5
-2.0
-1.5
-1.0
-0.5
0.0
Vertical displacement, mm.
Wong, Fredlund and Krahn (1998) data:
ICFEP simulation:
Figure 7.11: Vertical displacements after 1 second
223
Height above base of column, m
2.0
1.5
1.0
0.5
0.0
-2.5
-2.0
-1.5
-1.0
-0.5
0.0
Vertical displacement, mm.
Wong, Fredlund and Krahn (1998) data:
ICFEP simulation:
Figure 7.12: Vertical displacements after 31 seconds
224
Height above base of column, m
2.0
1.5
1.0
0.5
0.0
-2.5
-2.0
-1.5
-1.0
-0.5
0.0
Vertical displacement, mm.
Wong, Fredlund and Krahn (1998) data:
ICFEP simulation:
Figure 7.13: Vertical displacements after 255 seconds
225
Height above base of column, m
2.0
1.5
1.0
0.5
0.0
-2.5
-2.0
-1.5
-1.0
-0.5
0.0
Vertical displacement, mm.
Wong, Fredlund and Krahn (1998) data:
ICFEP simulation:
Figure 7.14: Vertical displacements after 1023 seconds
226
CHAPTER 8
The Tung Chung case study
8.1. Introduction
To investigate the effects of rainfall induced infiltration into a slope composed of unsaturated
soil, it was decided that a real slope, actually existing in the field, should be used rather than
some fictional ‘representative slope’.
As was discussed in section 3.9, the Pak Kong slope in Hong Kong suffered movement
apparently as a result of rainfall. Attempts were made to model this failure using the saturated
soil models within ICFEP, but with limited success. Initially, it was planned to use this slope as
the case study. However, there was no data available for this site regarding the unsaturated soil
properties applicable and little reliable information on the pore water pressures within the slope.
During the duration of this project, colleagues at Imperial College became involved in
monitoring of another Hong Kong slope, at Tung Chung. While no movement occurred in this
slope during the period over which data was collected, extensive records of rainfall and pore
water pressures/suctions at different plan locations and depths within the slope were obtained.
There was still a problem in obtaining reliable unsaturated soil parameters. The availability of
the pore pressure response to rainfall, on a day by day basis, meant that it was feasible to
simulate the slope and the effects of the rainfall using the unsaturated soil models within ICFEP.
It was felt that an investigation of the factors that influenced the changes in pore water pressure
due to rainfall would be more useful than simply trying to reproduce the movement in a given
slope, and so the Tung Chung case study was accordingly adopted.
8.2. The Tung Chung site
The Tung Chung slope is a natural slope close to Tung Chung New Town, on Lantau Island, in
the Hong Kong Special Administrative Region of the People’s Republic of China, as shown in
Figure 8.1.
Chapter 8
227
The slope lies within a natural terrain landslide study area designated by the Geotechnical
Engineering Office ( GEO ) of the Hong Kong government, overlooking the North Lantau
Highway, which forms the principal road link between Tung Chung New Town and Chek Lap
Kok ( Hong Kong international ) airport, and the heart of Hong Kong.
The slope is fairly steep, with several gullies and old landslide scars, and is covered with a mix
of grass and low scrub vegetation. Photographs of the slope are included in Appendix A6.
The majority of the west and central portion of Lantau Island, including the study area, is
underlain by rocks of the Lantau Volcanic group, with intrusive granitoids elsewhere: rhyolites
of the Kwai Chung suite form the north end of the island and granites of the Lamma suite occur
mostly to the south east corner ( Sewell et al, 2000 ). These granitoids do extend along the north
coast of the island, but do not extend significantly in from the coastal strip. Hence the bedrock
underlying the Tung Chung site is volcanic, see Figure 8.2.
The soil on the site is generally derived from decomposed bedrock, and so as would be
expected, the soil present on the slope is classified as decomposed volcanic material. Such
material has been described in detail from the geological point of view by Irfan (1999), but his
work is less useful for engineering purposes.
Colluvium deposits are also present, being quite extensive along the bases of the slopes within
the area. Slopes in the study area are generally less than 30° in gradient, but tend to continue
outside of the area where they become steeper ( Franks, 1998 ). Ultimately, the ground rises to
869m above sea level, at Sunset Peak, towards the centre of the island.
8.3. Tung Chung site investigation details
The Tung Chung study area was subjected to a limited site investigation carried out under the
auspices of the GEO. Five boreholes were sunk into the slope within the study area, with a
further three boreholes close by. Full details of these boreholes, including the drilling techniques
used, were not available for this project. Six trial pits were also dug, for which logs were
available.
To provide information on the groundwater regime, seven standpipe piezometers and ten
tensiometers were installed in the slope at relatively shallow depths, supplemented by one deep
standpipe. The locations of the boreholes, trial pits and pore water pressure monitoring devices
Chapter 8
228
are shown in Figures 8.3 and 8.4. Figure 8.4 also shows the locations of the two rain gauges
( R1 and R2 ) that were installed on site for the duration of the instrumented monitoring period.
Potential movement of the slope was monitored using wires anchored on to the surface of the
slope, and attached to a pressure sensor sealed within water filled tubes set further up the slope.
Any differential movement between the anchor point and the tube would cause the sensor to
change height within the tube, resulting in a change of pressure being recorded by the sensor.
During the monitoring period, no movement of the slope was detected.
From the investigation work undertaken, it is evident that beneath the topsoil layer, which was
of variable thickness, but typically less than 0.5 metres, the soil was either colluvium or a
residual soil. The colluvium was present over much of the slope, not just the base, though the
site investigation logs that are available show that differentiating between the two soil types was
not easy. Given that the colluvium was created by deposits from debris flows, this is
unsurprising, since the colluvium is in effect ‘displaced residual soil’.
The Colluvium
The colluvial deposits are generally described as soft or soft to firm, orangish- or reddishbrown, sandy, clayey silt. This is sometimes described as gravelly, else with occasional gravel.
The gravel appears to be sub-angular to sub-rounded, of fine to medium particle size, and to be
a coarse ash tuff, highly to completely decomposed. Irfan (1999) discussed the description of
rock decomposition and the use of grades for describing weathering, and this point is considered
more fully below.
The colluvium layer appears typically to be 1m to 3m thick, although it becomes a little thicker
towards the base of the slope ( for example around TRL130; see Figure 8.3 ).
Below the colluvium, and in some places directly below the topsoil, lies the residual soil.
The residual soil
The residual soil is often an uncertain description, with the difference between residual soil and
colluvium marginal.
The residual soil is described as a soft or soft to firm, sandy, clayey silt. Generally reddishbrown in colour, with yellowish-brown mottling, it also contains inclusions or bands of pale
grey, completely decomposed coarse ash tuff.
Chapter 8
229
This material grades with depth into an extremely to very weak, completely decomposed tuff.
Colour remains reddish-brown, mottled yellowish-brown, with pale grey, sometimes greenish.
Some structure is also visible. In accordance with the British Standards Institution classification
scheme for rock, which is referred to by Irfan (1999), the residual soil appears to vary from
grade VI, ‘residual soil’ to grade V ‘completely weathered rock’.
The investigation logs made available for this project do not extend to sufficient depth to show
further details of the weathering pattern present. However, a plan showing the depth to rockhead was made available, and is reproduced as Figure 8.5. In this instance, rock-head is defined
as the top of grade III material. Grade III material is described as ‘moderately weathered’,
meaning that less than 50% of the rock has weathered to soil.
GEO also produced a number of interpretative sections, which showed the slope composed of a
band of surface ‘soil’, with underlying rock. The rock represented material of grade III or better,
while the soil was listed as either grade IV to V ( ‘Highly weathered’: more than 50% material
decomposed to soil, to ‘completely weathered’: all material decomposed to soil, but structure
still present ), or as residual soil ( grade VI ) or colluvium. This breakdown of materials in the
slope, coupled with Figure 8.5, provided the basis upon which the ICFEP mesh was generated.
The numerical parameters selected for the soil properties for use in the ICFEP analysis are
discussed in section 8.5.
8.4. The finite element mesh
To undertake an analysis of this slope using ICFEP, it was first necessary to generate a finite
element mesh with which to represent the slope.
The mesh developed for the analysis used 8-noded quadrilateral elements, and is shown in
Figure 8.6. As can be seen, the mesh is relatively simplified, with a constant gradient for the
surface boundary, and again a ( different ) constant gradient base boundary.
The mesh breaks the slope into 2 material types. The upper layer represents the colluvium ( and
topsoil ), while the lower material represents the ‘residual soil’, actually the grade IV to grade
VI material. The base of the mesh marks the demarcation between Grade IV and grade III
material, and as such, represents ‘rock head’.
The gradient used is felt to be typical of the slope. Clearly, as can be seen from both the surface
contours and the isopachytes of regolith shown in Figure 8.5, in reality both surface gradient
Chapter 8
230
and ‘rock-head’ gradient are highly variable. There is a similar variability in the thickness of the
colluvium deposits. It was thus impossible to produce a detailed section, with each change in
gradient accurately reproduced, which would be applicable to more than one tightly defined line
of section.
Pore water pressure / suction data was available for tensiometers SP4, SP5, SP7, SP8, SP9 and
SP10. As can be seen from Figure 8.4, these are relatively closely grouped in one part of the
study area, but do not lie in a straight line. However, the surface contours in this region do
provide a relatively uniform gradient. The mesh was therefore limited to a section of the study
area, including that part containing the piezometers, as shown in Figure 8.7. Data was also
available for tensiometers SP1 SP2, SP3 and SP6, and piezometer TRL129, but none of these
instruments were included within the ICFEP mesh, as discussed in section 8.8.
The length of the mesh used ensures any boundary effects that may result from the mesh sides
are sufficiently far from the locations of the piezometers as to have no significant effect.
Prior to forming the mesh presented in Figure 8.6, a number of one-dimensional column
analysis were undertaken to further investigate the sensitivity of the simulation to element size,
effectively extending the work described in section 4.5.3, but using Tung Chung specific
parameters.
Summarising the results of section 4.5.3, it was found that for the general case, surface elements
should be less than 1.0m thick, and preferably around 0.5m thick. However, reducing the
element thickness to 0.25m appeared to be overly precise, giving relatively little benefit for the
increase in computational time required.
The Tung Chung specific column analysis confirmed that a minimum thickness of 0.5m for the
surface most elements was advisable ( and since the results effectively duplicate those shown in
Chapter 4, they are not reproduced here ).
The need to use fine elements in the Tung Chung finite element mesh extended not just to the
slope surface, but also to the interface between the colluvium and the residual soil. The
necessity to maintain accuracy at this internal boundary made the use of 0.5m minimum
thickness elements for the full depth of the colluvium necessary.
However, the variation in thickness of the colluvial layer meant that such an approach produced
much finer elements at the top of the slope within the mesh, with colluvium elements only
0.25m thick. The mesh size sensitivity study suggests that this is overly precise. It would have
been feasible to simply reduce the number of elements in this portion of the mesh, but it was felt
Chapter 8
231
at the time the mesh was generated that the use of a slightly ‘over detailed’ mesh would not
significantly affect the computational effort required for the analyses. ( Given the time required
to complete some of the analyses, as detailed later, this may well have been an incorrect
assumption ).
8.5. Soil properties
As discussed in the previous section, to undertake the analysis of the Tung Chung slope, the soil
was divided into three materials. The first of these was the colluvium layer, which formed the
surface band of material within the ICFEP mesh. Below this was the ‘residual soil’, this being
the insitu weathered material that was comprised of at least 50% soil (weathering grade IV or
higher ). This material formed the lower section of the mesh.
The third grouping was the intact ( grade III or less ) material, which, while underlying the
residual soil and hence outside of the analysis mesh, still needed to be considered, since its
properties could influence the applicable boundary conditions.
Relatively little information was available regarding the properties of the soil, with only grading
curves, liquid and plastic limits, and in some cases moisture contents available. It was therefore
necessary to draw on the published literature to obtain values for most of the soil parameters.
Plastic properties
An indication of the grading of the soil samples recovered from the site is given by the
descriptions given in section 8.3.
The colluvium was found to have a plastic limit of 21% to 28%. The liquid limit for
‘undisturbed’ colluvium was from 47% to 54%, but material recovered from areas of previous
landslides at the site showed a greater range, from 38% to 60%. Moisture content was in the
range 22% to 26%.
‘True’ ( grade VI ) residual soil had a plastic limit of 21% to 27%, a liquid limit of 43% to 51%,
with a moisture content of 22% to 23%. For the completely weathered ( grade V ) material,
which within the ICFEP analyses was also considered a residual soil, the plastic limit ranged
from 18% to 28%, with a liquid limit of 28% to 54%. Moisture content was typically lower than
the other soil types, being 17% to 23% in most cases, although two samples gave lower values
of 16% and 11%.
Chapter 8
232
Irfan (1999) cautions against relying on plastic indices and soil grading to predict the behaviour
of residual volcanic soils, citing the effects of structure and mineralogy. It should be noted that
ICFEP makes no direct use of these properties.
Elastic properties
Values of Young’s modulus, E, and Poisson’s ratio, ν, for the colluvium and residual soil were
not available from the site data, nor could they be located from the general literature.
However, as stated earlier ( see Section 3.9 ), it was originally intended to analysis the Pak
Kong slope failure, and some data regarding the soils there was available.
The soil at Pak Kong was also a completely decomposed volcanic soil, so parameters for this
soil should be comparable to those at Tung Chung, though it is recognised that this might be an
inaccurate assumption.
A saturated finite element analysis was undertaken of the Pak Kong slope ( GCG, 1993 ),
involving a parametric study, with the soil there divided into three layers, at different depths.
Values of Poisson’s ratio of 0.25 were generally taken for all layers, throughout the study
( although a slightly higher value of 0.3 was applied to the two upper layers in one instance ).
The Young’s modulus for the deepest of the three layers was in all cases set at 50,000 kPa,
while the other two layers were given values of either 50,000 kPa or 25,000 kPa ( in any
particular case, both layers had the same value ).
With no better source of data, the Pak Kong analysis was adopted as the source of the elastic
parameters for the Tung Chung analysis.
Since the real Pak Kong slope underwent movement, the parameters were selected from the
parametric study that generated failure of the slope. Hence a Young’s modulus of 25,000 kPa
was selected, along with a Poisson’s ratio value of 0.25. These values were applied to both the
colluvium and residual soil layers, since there was nothing to justify or quantify any differential.
Since the ‘intact’ rock below the mesh lay outside the area of the analysis, it was not necessary
to determine these parameters for this material ( but see also the discussion on boundary
conditions, below ).
Chapter 8
233
Voids Ratio
Since unsaturated flow involves changes in the quantity of water stored in the ( voids of the )
soil, the initial voids ratio must be determined. However, no data on this property was available
from the site investigation.
Lau et al (1998), for a completely decomposed Tuff, gave a voids ratio value of 0.4.
Lumb (1962a) gave some data for the porosity of decomposed volcanic material, which equated
to a voids ratio of 0.57 to 0.82, but this was for remoulded samples, so is of limited applicability
here. However, in a later work ( Lumb, 1975 ), he gave a range for insitu voids ratios of 0.25 to
1.5 for decomposed volcanics near the ground surface. With depth, the values tended toward the
middle of that range.
Irfan (1999) confirmed a range of 0.3 to 1.2 for the voids ratio of volcanic soils, while Dai et al
(1999) suggested a porosity for colluvium of 0.494, which gives a void ratio of 0.976.
Ng and Pang (2000) presented some stress-dependent SWCCs for undisturbed samples of
completely decomposed volcanic soil, presented in volumetric water content/matric suction
format.
From the relationship: ‘Volumetric water content = porosity * degree of saturation’, the porosity
( and hence the voids ratio ) at full saturation ( suction = 0.1 kPa ) could be determined from this
data, for each of the curves presented by Ng and Pang.
Their data for a 0 kPa stress curve shows an initial porosity of 0.435, giving a voids ratio of
0.77. For the curve under 40 kPa stress, initial porosity was 0.425, giving e = 0.74, and for the
80 kPa stress curve, porosity was 0.410, with a voids ratio of 0.695.
Sun et al (1998) also provide an SWCC for a volcanic soil ( this being identified as from Chai
Wan, suggesting that it was formed from Repulse Bay volcanic group rocks, not the Lantau
group present at Tung Chung ). Using the same principle as above, the initial porosity is 0.45,
giving a voids ratio of 0.82.
It was decided that the 40 kPa and 80 kPa stress conditions would be reasonable approximations
of the stresses in the two ICFEP soil layers. Further, Ng and Pang’s work was also used in
determining suitable SWCCs for the ICFEP analyses ( see later ). Hence the initial voids ratios
adopted were 0.74 for the colluvium, and 0.695 for the residual soil.
However, for one analysis ( Run 8 ), the initial voids ratio for the colluvium was increased to
1.00 ( based on the data from Dai et al, 1999 ), to determine what, if any, effects this had.
Chapter 8
234
Bulk density
Values for bulk density were available from both the general literature, and as site specific data.
Franks (1999) quoted data suggesting that the insitu bulk density of the colluvium within the
Tung Chung study area was 2.2 to 2.3 Mg/m3 ( 21.58 to 22.56 kN/m3 ).
Dai et al (1999) suggests a dry density for the colluvium in different study areas on Lantau
Island ( to the south and south-west ) of 1.35 Mg/m3 ( 13.24 kN/m3 ), while Lau et al (1999)
give the dry density for a completely decomposed Tuff as 1.5 Mg/m3 ( 14.72 kN/m3 ).
Assuming the initial voids ratios given above, these dry densities would lead to fully saturated
bulk densities of 17.4 kN/m3 and 18.7 kN/m3 respectively ( Based on bulk density = dry density
+ [porosity*density of water] ). However, if Lau et al’s void ratio of 0.4 is taken, this implies a
fully saturated bulk density of 17.5 kN/m3.
Fredlund and Barbour (1992) presented an example of a cut slope in Hong Kong, but in
decomposed Granite. They gave 19.6 kN/m3 as the unit weight for all the granite-derived soils,
regardless of degree of decomposition, and also used the same value for the colluvium deposits
( which presumably were also granite-derived ). Thus the density of the colluvium is not
necessarily different to that of the residual soil.
Yim and Yuen (1998) gave details of a soil in the north-east of the New Territories, for which
both the colluvium and the completely decomposed volcanics were given a bulk density of
19 kN/m3, again confirming that the two soil types can have identical densities. More generally,
Lumb (1962b) gave a range for the fully saturated density of Hong Kong volcanically derived
soils, of 110 to 140 lb/cu.ft. ( 17.3 to 22.0 kN/m3 ).
Data from the Pak Kong analysis ( GCG, 1993 ) showed the bulk density of the volcanic soil
there was 17.0 kN/m3 for the surface layer, then 17.5 kN/m3 for the intermediate zone, and
18.2 kN/m3 for the deepest layer of soil. It may be noted that Pak Kong is in the east of the New
Territories, and does not overlie Lantau group volcanic rock, but rather Repulse Bay volcanic
group rock.
Field data from the Tung Chung site was determined using the sand replacement method in a
series of trial pits, at depths of 1.0m, 2.0m or 3.0m. Some of the data is for pits not shown in
Figure 8.3, and whose precise location was not made available for this thesis. All the bulk
density data that was available appeared to apply to the colluvium. It was found to have a quite
variable density, mostly in the range 1.8 to 1.9 Mg/m3 ( 17.7 to 18.6 kN/m3 ), but extended from
1.63 to 1.93 Mg/m3 ( 16.0 to 18.9 kN/m3 ).
Chapter 8
235
From this, it was clear that there is considerable variation in the possible values of bulk density,
both in the limited field data, and from the more general literature. This is entirely consistent
with the highly variable nature of these soils, as has been noted by many authors ( for example,
Lumb 1975 ).
Since higher soil densities were more likely to induce slope movements during the analyses, and
no such movements were observed in the actual slope, it was decided to apply bulk density
values towards the lower end of the likely range. Accordingly, a value of 17.0 kN/m3 was
applied for the colluvium, and 18.2 kN/m3 was taken as the bulk density of the residual soil.
However, both these values are for fully saturated soil. In the analyses, the initial conditions for
both soil layers involve the existence of small suctions ( see section 8.8 ). Hence the soils
initially are slightly less than fully saturated, and are accordingly slightly lighter.
Since the stresses in the soil at the start of the analyses are to be determined from the self weight
of the soil ( again, see later ), it was necessary to determine the actual bulk density of the soils,
at the degrees of saturation that applied in each layer.
The initial bulk density was hence calculated as:
Initial bulk density = Saturated bulk density – [ ( 1-Sr )*n*9.81 ],
Where Sr was the degree of saturation of the soil layer at the start of the analysis, and n was the
porosity of the layer ( In the case of Run 8, the initial bulk density was not corrected to allow for
the change in voids ratio of the colluvium ).
This gave initial bulk density values of 16.08 kN/m3 for the colluvium, and 17.71 kN/m3 for the
residual soil.
Further changes in the soil density, reflecting the changes in degree of saturation, would be
automatically determined through the application of the new density variation capability in
ICFEP.
Non-linear soil parameters ( strength and deformation parameters )
The non-linear material properties of the soils were modelled using ICFEP model 82 ( see
section 6.2 ). Parameters specified within this model are the cohesion, c’, and the angle of
shearing resistance, φ’. Additionally, the H parameter, governing deformations due to changes
in matric suction, is also specified in this model.
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Since there was no reliable data as to the value or variation of the H parameter, this was set to
be constant, regardless of suction, for both the colluvium and the residual soil.
As discussed in section 5.3, for consistency between the saturated state and general
( unsaturated ) case of the constitutive equations, H = E/(1-2ν) at full saturation. This
established the correct value of H at the AEV, and it was therefore this value that was taken to
apply for all values of suction. With the elastic properties as specified earlier, it can be seen that
the value of H adopted was 50,000 kPa.
There was a similar lack of site-specific data for the values of the strength parameters, but in
this case, the general literature provided some guidance.
Lumb (1962b) provided data for the shear strength of decomposed volcanic soil, stating that
cohesion is generally small when saturated, but is much greater, and the soil becomes mainly
cohesive, when unsaturated.
In the saturated case, cohesion is simply the drained cohesion c’, but unsaturated cohesion is c’
plus the effects of matric suction. Lumb suggests a shearing angle of 30° to 35° for decomposed
volcanics, with a cohesion of 0-10,000 lb/sq.ft. ( 0-481 kPa ) possible. This gives an indication
of an appropriate range for φ’, but the cohesion range given is not just for c’, but includes the
effect of suction. Lumb’s only suggested guide for the value of c’ is ‘quite low’.
In a later work ( Lumb, 1975 ), he provides more complete details of sensible ranges for the
drained strength parameters of decomposed volcanic soil. For φ’, the range given is 25.6° to
37.8°, with 31.7° on average. For drained cohesion, Lumb gives values for either ‘wet’ ( fully
saturated ) or ‘dry’ ( 75% saturation ).
For the wet material, the range given is 0 to 78 ( averaging 23.2 ) kPa, while for the dry soil the
range is 2 to 110 ( averaging 56.5 ) kPa. Lumb puts a number of qualifiers on the use of these
values, but they are reliable as general indicators.
Lau et al (1998), for their completely decomposed Tuff, give parameters for the intact strength
of c’ in the range of 0 to 10 kPa, and φ’ being 30 to 35°, which are consistent with the details
given by Lumb (1962b).
Lau et al also gave values for discontinuity strength. This raised the question of whether any
attempt needed to be made within the ICFEP analyses to reproduce the effects of structure.
Given the limited data available, the highly variable nature of the soil, and the complexity of the
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unsaturated analyses, it seemed unwise to attempt to add a further layer of complexity into the
problem. Therefore no attempt to model specific soil structural effects was made.
Yim and Yuen (1998) gave details of the drained strength of the soils they encountered in the
New Territories, with the completely decomposed volcanics having 0 kPa cohesion and 45°
angle of friction. For the colluvium, they gave 2 kPa cohesion, and 36° angle of friction. These
values were used as design parameters for a soil nailing project.
Yim and Yuen’s values of friction angle appear a little higher than have otherwise been
suggested. Conversely, Au (1998) suggests a value of 30° might be more accurate, though this
is given as being a ‘lower-bound’ value, and he states that this value is dependent on the degree
of decomposition ( with a higher friction angle resulting if there is less decomposition ).
From the Pak Kong analyses ( GCG, 1993 ), the friction angle of the decomposed volcanic soil
was generally given as 40°. However, for many of the runs, this parameter was reduced to 35°
for the middle soil layer. This appeared to be an attempt to reproduce the movement recorded at
that site, rather than being based on specific soil data.
Cohesion was initially specified as 10 kPa for all three soil layers, but changed from the second
run to increase with depth, 5 kPa-10 kPa-20 kPa. Again, this change appeared to have been
implemented solely in an attempt to reproduce the recorded movements at Pak Kong.
As with the choice of bulk density, it was decided that since no movement of the actual Tung
Chung slope had occurred, the values adopted for the strength parameters should be selected
such that, while within the range of probable values, they reduced the probability of a slope
failure occurring during the analyses. Accordingly, the strength parameters were towards the
higher end of the range of probable values.
For the residual soil, a friction angle of 40° was specified, with a drained cohesion, c’, of
30 kPa. The colluvium was assumed to be slightly weaker, and hence was specified as having a
friction angle of 35°, and a c’ value of 20 kPa. These values were constant for all runs.
It is likely that the adopted cohesion values in particular are actually a little high, but they are
not excessively so. It was also noted that the soil descriptions from the Tung Chung site
frequently refer to clayey silt, and in some cases the soil is actually described as a clay,
suggesting that a slightly higher cohesion might be appropriate for this site.
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Soil permeability
If the pore water pressure response in the Tung Chung slope is to be reproduced, it is clearly
necessary to have a reasonably accurate assessment of the permeability of the soils there. This
requires knowledge of both the fully saturated permeability and the permeability – suction
relationship. As for most of the other soil parameters, there was no site-specific data available.
Dai et al (1999) gave a saturated permeability of 2*10-5 m/s for colluvium, noting this to be “a
comparatively lower value”. No suction dependency details were given.
Lumb (1962a) suggested that for decomposed volcanic soil, a saturated permeability of 3.0
inches/day ( approximately 3*10-7 m/s ) was typical. Again, no details of the effects of suction
were given. Elsewhere ( Lumb, 1962b ), he confirms that the range of permeabilities that can
occur for these soils is large, and illustrates this by giving a range for the likely permeabilities of
decomposed volcanic soil. This range was 5*10-6 inches/min to 1*10-3 inches/min ( 2*10-9 m/s
to 4*10-7 m/s ).
However, Lumb later ( Lumb, 1975 ) points out how the field permeability of decomposed
volcanic soil depends on the soil structure, which is unlikely to be included in small samples
taken for laboratory testing. He illustrated this by giving data showing that the mean value of
saturated permeability as measured in laboratory testing was 1.5*10-8 m/s, while the mean value
obtained from insitu field tests was 1.5*10-6 m/s ( which implies that his earlier values were
obtained from testing laboratory samples ). However, he still provides no guidance as to the
relationship between permeability and matric suction.
Anderson (1984) presented data from a range of sites in Hong Kong, with saturated
permeabilities for colluvium of 1*10-5 m/s to 2*10-5 m/s. These values were for colluvium
associated with decomposed granite, decomposed volcanics, and for simply ‘colluvium’, where
no associated soil type was given. The fully saturated permeability given by Anderson for
completely decomposed volcanic soil lay in the range 1*10-6 m/s to 2*10-6 m/s.
Au (1998) gave data showing that mass permeability of decomposed volcanic rock was of the
order of 10-5 to 10-7 m/s, while for colluvial deposits, the highly variable nature of the soil
allows “natural soil pipes” to develop, and mass permeability ranges from 10-4 to 10-7 m/s.
However, he also gave examples of flow through specific zones of natural pipes where
permeabilities of 2*10-1 m/s to 2*10-2 m/s were recorded.
Sun et al (1998) gave data for the permeability – suction relation for a volcanic soil ( from Chai
Wan, Hong Kong island), which is reproduced in Figure 8.8a. This shows a fully saturated
permeability of 1*10-5 m/s.
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Ng and Pang (2000) presented similar data for the completely decomposed volcanic soil they
used to investigate stress dependent behaviour of the SWCC, as shown in Figure 8.8b. This data
shows that the fully saturated permeabilities they recorded were dependent on stress, but in all
cases were less than that given by Sun et al. Ng and Pang’s data is for a decomposed Tuff from
Shatin, in the New Territories. Hence both their soil and that from Sun et al (1998) are not
derived from exactly the same material as at Tung Chung.
Fredlund and Barbour (1992) gave data for the permeabilities of decomposed granite soils.
While not directly applicable to Tung Chung’s volcanic derived soil, their data showed the
completely decomposed granite to have a saturated permeability of 7*10-6 m/s, while the
overlying colluvium had a permeability of 3*10-5 m/s. This is consistent with Anderson (1984)
showing the colluvium to be about an order of magnitude more permeable than the decomposed
soil it is derived from.
The analysis of the Pak Kong slope ( GCG, 1993 ) made use of a saturated permeability of
5*10-6 m/s for all the soil within the slope, except for in one run, where the permeability of the
surface-most layer was increased to 5*10-5 m/s as part of the parametric study.
Based on this data, the base value fully saturated permeability adopted for the residual soil was
5*10-6 m/s. For the colluvium, the fully saturated permeability was initially ( run 1 ) specified as
3*10-5 m/s, and this value was also used in run 3. However, a higher value of 3*10-4 m/s was
investigated in run 2, and this greater value was then standardised upon from run 4 onwards.
The permeability specified above for the residual soil was described as the ‘base’ value because
this permeability was taken as being stress dependent.
Vaughan (1985) discussed how “most residual soils exhibit a permeability which decreases with
depth”, and the use of a stress dependent permeability for the residual soil in the ICFEP
analyses was intended to reflect this behaviour.
ICFEP includes the option to make the ( fully saturated ) permeability a function of mean
effective stress, using the relationship shown below ( Equation 8.1 ).
k = ko. e( a. p ')
Eqn 8.1
Where:
k = permeability ( at some mean effective stress, p’ )
ko = Base permeability, at zero mean effective stress
a = a constant
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p’ = mean effective stress ( Tension positive ).
The constant ‘a’ was determined using Ng and Pang’s (2000) data, see Figure 8.8b. Taking ko as
1*10-6 m/s, the permeability of 1*10-7 m/s at 40 kPa stress gives a value of 0.0575 for the
constant ‘a’. Again using a ko of 1*10-6 m/s, at 80 kPa stress, the permeability of 5*10-8 m/s
implies the constant ‘a’ is 0.03744.
Since the two non-zero stress values given by Ng and Pang are in the region of the stresses
likely to occur within the residual soil in the ICFEP analyses, the mean average of the two
derived ‘a’ values was taken; hence a = 0.0475 was used.
The intention was that the base permeability value ( of 5*10-6 m/s ) should apply at
approximately the top of the residual soil layer, which lies 1m to 2m below the ground surface.
Accordingly, a different value of ko was required, such that k at the top of the residual soil ( at
non-zero mean effective stress ) would equal this base value. Assuming an average of 1.5m of
colluvium, imposing a mean effective stress of about 16 kPa per metre depth, ko should have
been set to 1.6*10-5 m/s. However, an error was made in calculating this value, and this was not
noted until after several runs had been completed. The actual value of ko input was 1*10-6 m/s.
Such a value for the saturated permeability of a decomposed volcanic soil ( at zero mean
effective stress ) is still comparable with the data given by the general literature, and so this
value continued to be used for all further runs as well.
The use of stress dependent permeability affects the saturated permeability of the soil, but does
not reproduce the effects on permeability of suction. However, the two permeability functions
are not mutually exclusive, and the suction dependent permeability was reproduced using the
standard ICFEP suction switch ( see section 4.3.1 and Figure 4.1 ).
For the residual soil, P1 was set at 1 kPa, P2 at 40 kPa, and the ratio kmax/kmin as 20,000, as
shown in Figure 8.9. This gave a little over four orders of magnitude decrease in permeability
between 1 and 40 kPa suction, which is consistent with Ng and Pang (2000), and only a little
more extreme than Sun et al’s (1998) data would suggest.
For the colluvium, the suction switch details were varied over different runs. The P1 parameter
was set at either 0.1 or 1.0 kPa, whichever was consistent with the AEV on the SWCC ( see
below ). For runs 1 to 9 inclusive, P2 was set at 60 kPa, while for the later runs ( 10 to 12 ), the
value was reduced to 5 kPa, giving a much more extreme change in permeability with suction.
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The permeability ratio adopted was also varied, from 500 to 500,000. Details of the suction
switch parameters applied to the colluvium for each run are shown in Table 8-1. The
permeability ratios used for the two soil types were initially guided by the Sun et al (1998) and
Ng and Pang (2000) data, but were later changed freely as attempts were made to reproduce the
actual field data.
The P2 values selected for both materials’ suction switches were initially chosen to be just
slightly greater than the maximum suction recorded on site ( disregarding the last few days of
monitoring, when suctions became very large ). However, for the colluvium, this parameter was
adjusted for runs 10 to 12, to investigate the response. This variation was not based on any
specific soil data.
The permeability parameters of the colluvium represented one of the principal variables in the
Tung Chung analyses.
The Soil Water Characteristic Curve
The SWCCs used for the Tung Chung analyses were developed from the literature, much like
the permeability details. However, generally there was less data available.
Anderson (1984) presents some basic SWCC data for colluvium and for a volcanic soil ( Tai Po
Volcanic ), for suctions up to 100 kPa, see Figure 8.10.
Sun et al (1998) gave a full curve, to suctions of 1000,000 kPa, for their Chai Wan soil, as
shown in Figure 8.11a, while Ng and Pang (2000) gave three curves, reflecting the stress
dependency they determined ( see Figure 8.11b ).
Based on this limited data, a separate SWCC was developed for each of the colluvium and the
residual soil, as shown in Figure 8.12. However, the colluvium SWCC was only used for runs 1
and 2. From run 3 onwards, the colluvium was modelled using the same SWCC as was
developed and used for the residual soil.
Included on Figure 8.12 are the parameters required to define the curve ( see section 6.3 ), and
the parameters required for the ICFEP SWCC model 3 which are derived from the curve ( see
Figure 6.5 ).
ICFEP SWCC model 3 also requires the input of the suction at which Ω becomes zero. For the
colluvium, this was set at 0.2 kPa. Hence effectively no flow would occur as a result of
deformation changes in this material while unsaturated.
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For the residual soil, Ω became zero at a suction of 400 kPa ( approximately the point of
contraflexure of the SWCC ). This marks the boundary between zones 3a and 3b in the
conceptual model ( see section 5.6.4 ), so is actually inconsistent with the model. The
conceptual model suggests that Ω should become zero at the boundary between zones 3b and 4
( see Figure 5.4 ). However, given the relatively low suctions recorded by the field monitoring
( see Section 8.8, below ), and the high suctions that are necessary to reach ‘zone 4 conditions’
with the residual soil SWCC, it was felt that the specified value of suction for zero Ω was
appropriate in this case. This serves to confirm further the need for more complete data on the
parameters applicable to unsaturated soils to be available.
It may also be noted that the soil is reasonably stiff, and not subject to any applied loading, so
deformations were expected to be relatively small. Hence little deformation-induced flow was
expected.
8.6. Rainfall data
The rainfall that occurred at the site was simulated in the analysis through the use of the
precipitation boundary condition ( see section 4.4.1 ).
The analyses were run with one increment representing one day. Accordingly, the specified
infiltration rate for each increment was the average rainfall for that day as determined from the
two on-site rain gauges ( as shown in Figure 8.4 ). Details of the recorded rainfall are shown in
Table 8-2.
Since there are no significant trees or heavy bushes at the site, no allowance was necessary for
canopy run-off. Hence 100% of the rainfall was taken as infiltration ( “100% rainfall” being the
mean average daily rainfall of the two rain gauges ).
The threshold value used with the precipitation boundary condition was taken as 0.0 kPa, with a
standard tolerance zone of ±0.1 kPa. Thus the ground could become fully saturated, but no
ponding would be able to occur.
The analyses were run with the automatic incrementation capability in use, thus ensuring that
any switch from infiltration to constant pressure boundary condition occurred accurately.
It should be noted that the rainfall data for 15 July and 1 August was incorrectly input into the
ICFEP data file for analyses runs 1 through 6 inclusive. The 15 July value was mistakenly input
as 18.675mm, while the data for 1 August was incorrectly taken as 97.5mm. Correct values for
rainfall on these dates were applied from analysis run 7 and onwards. The results of each ICFEP
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run are discussed later, but these errors did not appear to have any significant effect on the
ICFEP results or their interpretation.
8.7. Analyses boundary conditions
To undertake a finite element analysis, it is necessary to specify the boundary conditions that
apply to the analysis, since these largely define the problem, and can have a significant effect on
the analysis results. Boundary conditions can be grouped into two types, displacement
conditions and hydraulic conditions.
Displacement boundary conditions applied
For the Tung Chung analyses, the base of the mesh was considered to be ‘rock head’, and to
provide a solid base to the mesh. Accordingly, a zero vertical displacement boundary was
specified along the bottom of the mesh.
The site monitoring showed no sign of movement during the monitoring period. To reflect this,
a zero horizontal displacement condition was applied to both sides of the mesh, and to the base
boundary.
The top ( ground ) surface of the mesh was left free to move both horizontally and vertically,
while the sides were able to deform vertically, thus permitting consolidation or swelling of the
soil.
Hydraulic boundary conditions applied
The hydraulic boundary condition applied to the upper surface of the analysis mesh was the
precipitation boundary condition, as detailed above in section 8.6.
The sides of the mesh were given no-flow boundary conditions. This allowed the phreatic
surface to move freely at each end of the mesh, ensuring no artificial restriction was placed on
the response of the pore water pressure to rainfall. However, it also prohibited any down-slope
recharge into the mesh through the up-slope side from further up the hillside.
Such flow would have been difficult to model accurately anyway, and to address this potential
source of inflow, the precipitation boundary condition was applied to the base of the mesh, to
represent recharge into the mesh from higher ground.
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Reflecting the initial pore water pressure conditions in the soil ( see section 8.8 ), the threshold
value for the base precipitation was set to vary from 0 kPa at the toe of the slope to –39.0 kPa
( compressive water pressure ) under the head of the slope.
For most of the analyses run, an infiltration rate of 1.5*10-6 m/s was applied to the base
precipitation boundary condition, this being based on an estimate of the permeability of the
underlying rock. Noting the error made in calculating the stress dependent permeability, it is
evident that this inflow rate would generally have exceeded the fully saturated permeability of
the residual soil at the base of the mesh, and so would have led to the pressure reaching the
threshold value rapidly.
Partly as a response to this, for runs 9, 11 and 12, the infiltration rate applied to the base
precipitation boundary condition was changed to 0.0 m/s. Thus no flow in or out of the mesh
would occur across the boundary while pressures were more tensile than the threshold value.
However, if infiltration led to a rise in the pore pressure such that the threshold value was
reached, whatever cross-boundary flow necessary to prevent the threshold being exceeded
would occur.
With the precipitation boundary condition applied to the base of the mesh, the steady state
phreatic surface is limited to a set height above the base of the mesh. However, a further
transient rise in the phreatic surface ( with a non-hydrostatic pressure profile ) is still possible.
8.8. Tung Chung field response to rainfall
The Tung Chung field data for pore water pressures / suctions showed the response of the pore
water pressure to rain over the duration of the monitoring period.
Six of the tensiometers for which data was available were clustered as pairs: SP4 and 5, SP7 and
8, and SP9 and 10 ( see Figure 8.4 ), with one of each pair being shallow ( about 1.0m to 1.5m )
in the colluvium, and the other being deep ( 2.5m to 3.0m ), in the residual soil. The data from
these tensiometers was recorded on an hourly basis, from around midday on 3 March, 2001,
until early afternoon, 12 November, 2001. An exception to this was SP4, for which the data was
available only up to 15 September. The field data from these six tensiometers is presented in
Figure 8.13 ( a to f ), and as a combined plot in Figure 8.14.
It should be noted the SP7 initially showed little apparent response, leading to fears by the
monitoring team that it was not functioning correctly. Accordingly, this tensiometer was flushed
on 6 April, causing the break in the field data and sharp compressive rise in indicated pore
pressure.
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As can be seen in Figure 8.4, there were an additional four tensiometers, SP1, SP2, SP3, and
SP6, plus a deep stand-pipe piezometer, TRL129, which was approximately co-located with
SP1. The monitoring data from these instruments was collected over the same period as the
pairs of instruments mentioned above, and is shown in Figures 8.15 ( a to e ). As can be seen,
both SP3 and SP6 were also flushed, like SP7. However, the recovery of these two instruments
was much more rapid. The reason for this is unknown.
The plan position of these additional instruments meant that it would be impossible to include
more than one of these tensiometers within the ICFEP analyses section. Additionally, with only
one tensiometer at each plan location, these instruments gave no indication of the distribution of
pore pressure with depth at their locations. Because of this, none of these later instruments were
included in the ICFEP analyses. On reflection, it would have been relatively simple to allow for
SP3 to have been included in the ICFEP model, and it was an error not to have done so.
As is evident from the tensiometer plots, all six of the instruments incorporated into the ICFEP
analyses indicated pore water tensions at the start of the monitoring/simulation period.
However, it is not easy to distinguish the point at which the instruments commence recording
‘true’ insitu suctions, rather than showing installation effects.
It appears that suctions within the colluvium are initially between about 5 kPa and 20 kPa, while
for the residual soil, the ‘true’ initial suctions seem to range from 10 kPa to around 30 kPa.
Accordingly, in the ICFEP simulations, the colluvium was given an initial pore water pressure
of +10 kPa ( suction ), while the residual soil had an initial pore water pressure of +25 kPa
( suction ).
8.9. ICFEP analysis procedure
The ICFEP analyses were undertaken in a series of steps.
First, the mesh ( see Figure 8.6 ) was generated.
Then, for each separate run, an initial stress file, giving the initial pore water pressures and
voids ratios, was applied. At this point, the soil within the mesh was weightless.
Stresses due to soil self weight were then generated by ‘turning on’ gravity, with the specified
unit weights of the soil layers being used to generate the mass of each element, and hence to
induce the initial stresses in the mesh. During this stage, no consolidation or density change was
permitted, so the initial suctions specified in the initial stress file ( as detailed in section 8.8 )
were still applicable to this stage.
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The analysis proper was then undertaken, with each increment of the analysis representing one
day. Twelve ‘unsaturated’ runs were undertaken in total. Specific differences between these
runs are detailed below, in section 8.10. Additionally, a single analysis was undertaken using
only the saturated soil modelling capability, to demonstrate the improved accuracy of modelling
obtained through the use of an unsaturated soil capability.
8.10.
The ICFEP analyses
The ICFEP analyses involved twelve separate analyses using the unsaturated ICFEP models
presented earlier, plus a thirteenth analysis, reproducing run 7, but restricted to using saturated
soil behaviour models.
The general details of the analyses were given earlier in this chapter, along with run-specific
variations in the parameters used. A summary of the differences between each run is given in
Table8-3.
Each analysis was actually broken down into four segments, each run as separate, consecutive
data files. The first, ‘spring’, covered the period of March and April, ‘summer1’ dealt with May
and June, then ‘summer2’ ( July and August ), and finally ‘autumn’, which covered the period
of September and that part of October for which there was rainfall data, with the analysis ending
on 18 October.
For runs 1 to 9, no numerical problems were encountered during the analyses, and each data file
could be left to run through to completion. Runs 10 to 12, however, suffered numerous
problems in achieving numerical convergence, and as discussed later, each data file for these
runs was subject to a number of restarts.
For each run, the pore water pressure / suction was listed then plotted for six individual points,
approximately located to represent SP4, 5, 7, 8, 9 and 10. Thus SP4, which is one of the ‘low’
cluster of tensiometers on the real slope, and which lies within the residual soil ( at 2.73m
depth ), is represented by the data from a finite element node 90m in from the toe of the mesh, at
a depth of 2.66m.
In theory, each tensiometer is represented by the node in the slope mesh nearest to its real
position. However, while the analyses were undertaken using 8-noded elements, the ICFEP
analyses were run with pore pressures calculated at the four corner nodes only. This enforced a
linear pore pressure distribution across each element, making the pore pressure distribution
consistent with the stress distribution. ICFEP does have the option to calculate pore pressures at
all 8 nodes, but this leads to a quadratic variation in pore pressure across the elements, which
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matches the form of the displacements, but is correspondingly at variance with the stress
distribution. While this is theoretically permissible, it would make comprehension of the results
more complex. Accordingly, ICFEP was set to calculate pore water pressures only at the four
corner nodes of each element. Therefore, it was necessary to restrict the selection of nodes to the
corner nodes only.
Further, it was decided to use the same lateral position in the mesh for both the ‘deep’ ( residual
soil ) and ‘shallow’ ( colluvium ) tensiometers. For the mid-height pair in particular, it was
difficult to find a position in the mesh where there was a vertical line of corner-nodes, two of
which were at the approximate depth of tensiometers SP9 and 10.
It would probably have been better to select node positions individually, such that each was as
close as possible to the ‘correct’ depth, but not necessarily inline vertically with its ‘pair’.
However, given the number of assumptions that were necessary to determine the soil parameters
used, it was clear from the start of the analyses that it was unlikely that the field data would be
perfectly predicted by the ICFEP analyses. For this reason, it was decided to select nodes to
represent the tensiometers such that they were vertically in line.
Table 8-4 shows the relative position and depth of each of the six tensiometers, and the
corresponding position and depth of the ICFEP nodes simulating them. Note that it is assumed
that each shallow tensiometer of a pair is co-located ( in plan ) with the deep instrument of that
pair.
ICFEP Run 1
The predictions of pore pressure with time for Run 1 are shown in Figure 8.16. Figure 8.16A
shows separate predictions for each of the three sections through the slope, with the two sets of
data for each section presented together. Figure 8.16B consolidates all six predictions in one
figure, for ease of comparison to each other, and to the field data summary figure
( Figure 8.14 ). This presentation format is generally used for the majority of the ICFEP
predictions presented within this chapter.
Compared to the consolidated field data ( Figure 8.14 ), it is clear that the ICFEP predictions for
Run 1 bear little resemblance to the actual field response. There is, however, some trend to the
predictions reflecting the daily precipitation rates, which the field data more clearly shows.
The first month of the ICFEP simulation shows that the suctions in the colluvium are tending to
increase during zero rainfall days, which is clearly because of simple gravity drainage, since the
initial pore pressures specified are not consistent with a hydrostatic condition. Moreover, the
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pressures in the colluvium show a sharp response to significant rainfall events, such as occur on
25 March and 4 April. Such storms cause an immediate compressive increase in pore pressures,
but are followed by an immediate resumption of drainage and increasing suction. In this respect,
the ICFEP simulation is reproducing the behaviour seen in the field, but the actual numerical
values of the pressures are incorrectly predicted.
In contrast, the deeper residual soil appears to be unaffected by rainfall events within the ICFEP
analysis. Suctions here remain approximately constant for a period. The first of the deep nodes
to show variation in suction is that lowest down the slope, RL, and the decrease in suction
shown cannot be clearly assigned to a particular rainstorm.
It seems likely that what is actually being shown is the effect of downslope drainage within the
FE mesh, under gravity, as the soil tries to reach a steady-state condition. This is consistent with
the loss of suction in the residual soil layer occurring first and to the greatest degree in the down
slope node ( RL ), and then later occurring to a lesser degree at the mid-slope node ( RM ).
Further, the responses of the nodes within the colluvium support this. CL, the node lowest down
the slope, appears to gain the least suction, while node CH, towards the head of the slope
undergoes drainage and increases in suction at the greatest rate of any of the nodes in the
colluvium.
An examination of the accumulated flow velocities generated showed that flow was
predominately occurring within the colluvium layer, and was approximately parallel with the
slope gradient, but with a slight tendency to flow more vertically downward towards the base of
the colluvium layer. Figure 8.17 shows a typical flow pattern, in this case for Run 1,
increment 100 ( 8 June ), at the approximate location of the mid-slope section. Given that the
colluvium saturated permeability was 3*10E-5 m/s, whereas the residual soil permeability was
no greater than 1*10E-6 m/s ( and most likely somewhat less: see section 8.5 ), this general flow
pattern was entirely predictable without recourse to numerical modelling, being the process of
interflow ( or subsurface storm flow ) as described for example by Freeze and Cheery (1979).
The first truly heavy rainfalls occur around mid-June, and at this point, the simulation shows a
significant response at all six nodes. A rapid loss of suction is predicted, with pore pressures
becoming compressive, and close to a hydrostatic profile, with a phreatic surface at the ground
surface. Thus the soil, at least to the depth of the nodes being monitored, becomes fully
saturated.
In fact, as can be seen in Figure 8.18 to 8.20, the mesh is fully saturated throughout its depth at
the three sections through the monitoring node locations. These figures again show the effect of
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down slope drainage, with the ‘high’ section ( Figure 8.20 ) maintaining lower ( more tensile )
pore water pressures in general than the others, during the earlier part of the rainstorm.
The effects of the base precipitation boundary condition and the specified threshold value can
also be seen in these figures.
In each section, the 10 June data shows a deep pore water pressure distribution that is
approximately hydrostatic with the threshold value at the base. The effects of rainfall prior to
this date have changed the profile in the shallower soil, even to the extent of producing a small
perched water table in the colluvium ( Figure 8.18, 8.19 and 8.20 ), with slightly compressive
pore pressures at the low and mid-section positions.
The 15 June sections show that the soil has now become fully saturated under several days’
heavy rainfall. In the lower sections, the base boundary condition is now acting as a drain,
restricting the development of compressive pore pressures at shallower depths, and resulting in
a steeper-than-hydrostatic profile developing. However, the ‘high’ section ( Figure 8.20 ) still
shows a hydrostatic profile at depth.
By 20 June ( after several days of lesser rainfall ), the low and mid-sections show the same
profile as on 15 June. Any reduction of inflow due to the lesser rain appears to have been
counteracted by the effects of down slope flow within the mesh sustaining the compressive pore
pressures. The high section, however, has changed considerably. The surface pore water
pressure has again become tensile, but the pressures deeper within the section are now much
more compressive, as a wetting front has passed vertically into the ground.
The implication of this is that under low to moderate rainfall, all flow within the soil will occur
preferentially in a lateral down slope manner, through the most permeable strata ( ‘interflow’ ).
However, under high rainfall, the capability of the soil to drain solely through flow in this
manner is exceeded, and the infiltrating rain water begins to pond. This leads to both surface
run-off ( with a phreatic surface co-located with the ground surface), and to the development of
truly vertical flow. As this component of flow develops, suctions deeper within the soil are
destroyed and the soil becomes more permeable, making it easier for this flow to be sustained.
As can be seen both from the changes in pore water pressure distribution in Figure 8.20 and the
fluctuation pore pressures with time ( Figure 8.16 ), this period of vertical flow appears to be
short, with suctions rapidly re-establishing themselves in the upper layers of the soil. Once this
occurs, permeability again drops, and the tendency towards lateral down slope flow returns.
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The ICFEP predictions in Figure 8.16 show constant pore pressures for all nodes for a period of
time during the two ‘wet periods’ of mid-June to end of July and early September. In both these
periods, the ‘high’ section nodes are first to show a response to drier interludes, whereas the
‘low’ section nodes are last to start responding to the rainfall pattern. This is consistent with a
predominantly lateral down slope flow regime, with vertical flow only occurring under severe
rainfall events.
In such a case, the idea of a critical threshold rainstorm that could cause slope instability
( through increasing the compressive pore water pressure at depth in the slope ), much as
suggested by Brand (1985), is credible.
However, comparison of Figure 8.16 to Figure 8.14 shows that the ICFEP predictions from
Run 1 were far from an accurate reflection of the actual ground response, implying that either
the soil parameters chosen or the boundary conditions applied were incorrect. Accordingly, the
parameters used were modified, and the analysis was repeated.
Run 2
Run 2 was a repeat of Run 1, but with the saturated permeability of the colluvium increased to
3*10E-4 m/s. It was expected that this would allow the colluvium layer to drain more rapidly,
particularly during the ‘wet periods’, and so would give a pore water pressure with time that
more closely reflected the rainfall data. The resulting ICFEP prediction is shown in Figure 8.21.
As can be seen, the effects of the change in colluvium permeability did increase the rate at
which pore pressures drained in this layer. However, this increased drainage rate applied
regardless of the suction within the soil, leading to a rapid build up of suctions post-rainfall. The
effect of this was to prevent a significant reduction in suction from being maintained, and as a
result, only at point CL was a compressive pore water pressure predicted at any stage. Even
then, only very low compressive pressures were generated.
The general evidence of lateral down-slope flow is still present, and if anything is more
pronounced than in Run 1. The higher permeability of the colluvium permits greater flow, so
even during the ‘wet periods’, no significant downward infiltration into the deeper residual soil
appears to occur.
This is illustrated by Figure 8.22, showing the pore pressure distribution with depth across the
slope mid-height section, for a series of increments.
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The 31 May plot shows a relatively hydrostatic distribution, with a slight ‘bulge’ around the top
of the residual soil layer, caused by May rainfall. The significant rainfall on 5 June acts to
reduce the suctions in the colluvium, but the pressure distribution within this layer still remains
approximately hydrostatic. No noticeable effects from this day’s rain can be seen in the pore
pressures within the residual soil.
The rainfall from 6 to 10 June continues this pattern of change in the pore water pressure
distribution, as can be seen from the 10 June plot. However, the 15 June data shows that, despite
the major rain event on 11 June, the next four days of low rainfall allow suctions to re-establish
themselves in the colluvium. This process continues while rainfall remains low, and the profile
attempts to revert to a hydrostatic distribution throughout its height.
Throughout this period of varying rain and fluctuating suctions within the colluvium, the
pressures in the residual soil remain largely unchanged.
Of some interest is that the low slope residual soil point, RL, maintains a slightly higher suction
than the mid-section point, RM, from around the beginning of June. However, this is primarily
due to the greater depth below ground surface of RM compared to RL. The suction at the
surface of the low section remains less ( that is, the pore pressures are more compressive ) than
at the position of the mid-height section, which in turn has a lower surface suction than occurs
at the high-section.
The results of Run 2 show that given a highly permeable ( relative to the underlying soil )
surface soil layer, no wetting front develops. Instead, internal flow within the slope is wholly
lateral down-slope flow.
This is consistent with the results of Run 1, and does not disprove the possibility of a critical
threshold rainstorm. Rather, it suggests that the level of the threshold event is dependent on the
saturated permeability of the surface-most soil layers. Clearly, however, the ICFEP predictions
from Run 2 are still not closely matched to the actual field data.
Run 3
Run 3 involved repeating the Run 1 analysis, but with the SWCC developed for the residual soil
applied also to the colluvium. This would ensure that for any given value of suction, the degree
of saturation of the colluvium would be greater in this run than occurred in Run 1.
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Further, for any given flow ( change in degree of saturation ), a larger change in suction will
tend to occur in this run than was seen in Run 1. Hence it was expected that the pore pressures
in the colluvium would be more responsive to individual rainfall events.
Figure 8.23 shows the ICFEP predictions of pore pressures with time for Run 3. As can be seen,
the expected behaviour did occur, with more rapid changes in suction within the colluvium for
both wetting and drying than was observed in Run 1. However, the basic pattern of the
prediction is largely unchanged. Moreover, the ‘wet periods’ during which the soil remains fully
saturated are of greater duration in Run 3 than the equivalent periods were in Run 1.
Since a larger variation in suction now results from a given change in degree of saturation, the
results of Run 3 show larger suctions developing in the colluvium at the end of the dry periods,
and lower ( more compressive ) suctions, or even compressive pore water pressures, after the
low intensity periods of rain, when compared to Run 1.
With higher suctions being developed, the effect of the suction switch begins to be seen more
directly, particularly in the response of point CH. Whereas in Run 1 the suction at this point
develops fairly linearly, in Run 3 the rate at which suctions build tends to decrease as the
suctions themselves increase. The higher suctions developed in Run 3 will result in a
correspondingly lower permeability in the colluvium, through the effect of the suction switch.
This will tend to retard further suction increases, since the rate at which drainage can occur will
be reduced, and so gives the non-linear response seen in Figure 8.23 for point CH.
The field data shows a similar behaviour for all points, but only within the low suction range
( of about 0 kPa to –10 kPa ). Below –10 kPa ( that is, at higher suctions ) this ceases to be the
case, and suctions tend to develop in a more linear pattern, or even at an increasing rate, as
suctions increase. This aspect of the field behaviour was followed up in Runs 10 to 12.
The pore water pressure for the deeper residual soil from Run 3 shows much the same response
as shown in Run 1. The only significant variation appears to be in the low section data ( point
RL ), where two distinct reductions in suction occur, each shortly after moderate rainstorms.
The first of these is towards the start of April, while the second is around mid-May. These
responses were not seen in the two earlier runs, but match closely the form, if not the exact
magnitude, of the field data.
However, Run 1 did show a more prolonged drop in suction, such that this run and Run 3 gave
approximately the same suction at RL from mid-April until late May ( with Run 1 giving
slightly more compressive values ).
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In the case of Run 1, this behaviour appears to have been the simple tendency for the pore
pressure distribution within the soil to come to a steady state equilibrium condition, with the
pressure and height differential between the colluvium and residual soil driving a low-rate
vertical flow, independent of rainfall.
In the case of Run 3 and the field data, the pore pressure responses are much sharper, and
clearly relate to rainfall events, implying a causal link.
Within the ICFEP simulation, the drop in suction in the residual soil commences immediately
after the peak compressive pore water pressure is reached in the overlying colluvium. The field
data shows identical behaviour for the first of the two rain events, but not for the second. There,
the response of the suction in the colluvium to this second storm is less marked, since the
suctions are already very low, and the corresponding response in the residual soil is somewhat
more drawn out.
The sharp nature of the responses in the residual soil layer implies that they are caused by an
equally sharp increase in compressive pore pressures in the colluvium, producing a hydraulic
gradient sufficient to overcome the effects of the relatively low permeability of the residual soil.
Hence, while the response in the residual soil at RL is still a result of vertical gravity-driven
drainage, it is the availability of a near-surface water supply that enables this flow to develop.
The change in SWCC for the colluvium between Run 1 and Run 3 results in more pronounced
changes in suction within the colluvium layer under the imposed rain, with suctions being less
( more compressive ) and compressive pore water pressures being generated more often. With
the application of the suction switch, this will give the colluvium a higher permeability in Run 3
than is achieved in Run 1, with the fully saturated permeability being achieved at point CL on
two occasions during the spring and early summer period.
From this, it might be expected that lateral down-slope flow would occur in the colluvium more
readily in Run 3 than in Run 1, which should reduce the tendency for water to drain vertically
into the residual soil ( since it would flow within the colluvium instead ).
However, more compressive pore pressures in the colluvium give an increased hydraulic
gradient to drive the vertical flow. Moreover, the increase in colluvium permeability applies
across the length of the slope ( as shown indirectly by the pressures predicted for points CM and
CH ). Hence while water may be able to flow laterally away from point CL more readily, there
is also increased flow towards this point from further up-slope.
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Comparison of Figure 8.23 with Figure 8.16 shows that points RM and RH maintain higher
suctions in Run 3 than they did in Run 1 during this early period of the analysis. This implies
that water that permeated vertically at these locations in Run 1 instead flowed laterally through
the colluvium in Run 3.
Thus the ICFEP Run 3 results show little that has not already been demonstrated by the earlier
two runs, other than to illustrate the influence of the SWCC on the soil’s response to a given
infiltration.
Run 4
In an attempt to improve the accuracy of the ICFEP predictions relative to the field data, Run 4
combined the changes made in Runs 2 and 3. That is, the saturated permeability of the
colluvium was increased, and the SWCC used for this material was that developed for the
residual soil. The resulting plot of pore pressure over time is shown in Figure 8.24.
From Figure 8.24, it can be seen that the behaviour modelled is close to that predicted in Run 2
( see Figure 8.21 ), but the influence of the different SWCC has resulted in a yet sharper
response to rainfall, followed by more rapid drainage. Further, the variation in predicted
suctions within the colluvium is now more extreme than in any previous run.
Other than this, however, the general behaviour follows the previously seen patterns. Of note is
that, like in Run 2, the sensitivity of the pore pressures to quite small changes in water content
gives a highly variable suction in the colluvium throughout the ‘wet periods’, unlike the
approximately constant, saturated, condition seen in Runs 1 and 3.
Within the residual soil, point RH shows no significant response to the rainfall, much as in
Run 2. The relatively small perturbations in pressure at this point also match the small
fluctuations present at the equivalent point in the field data ( SP7 ), although the field data gives
suctions about 12 kPa smaller. Given that SP7 is 35cm deeper than point RH, a difference
between the two of around 3.5 kPa can be accounted for simply by the depth difference. Since
both field data ( for SP7 ) and ICFEP prediction ( for RH ) remain approximately constant from
their initial value, the remaining difference may be indicative of a discrepancy in the initial
pressure condition specified for the ICFEP analysis.
Lower down the slope, the responses to rainstorms in April and May predicted in Run 3 for
point RL are no longer present. With much larger magnitude suctions being predicted within the
colluvium, these rainstorms produce pore pressures in the colluvium that are only slightly more
compressive than the pressures in the residual soil, so there is evidently insufficient hydraulic
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gradient developed to cause flow into the residual soil. This remains the case until the start of
the main ‘wet period’ around mid-June, at which point the colluvium saturates.
The ‘high’ residual soil point, RH, still fails to respond to the rainfall, suggesting that lateral
down-slope flow is freely occurring at that section in the slope. However, both RM and RL
show a sudden compressive change in pore pressure immediately following the extreme rain
event on 11 June, with point RL showing the greater change. Also, the response of RL is
instantaneous to the rainstorm, whereas the response of RM appears to be delayed by two days,
and then to be spread out over two to three days.
This part of the ICFEP prediction resembles the field data in general form, but shows suctions
of greater magnitude than where actually recorded. The high suctions in the colluvium have
resulted in no significant change in the suctions at RL and RM from their initial condition until
the June rain event. As a result, the ICFEP simulation shows suctions at these points that are of
too high a magnitude as compared to the field data at the start of the June storms. Even though
ICFEP quite accurately picks up the storm response, since the magnitude of these suctions is
wrong prior to the rainstorms, it remains incorrect afterwards.
Further, with lower magnitude suctions present prior to the June storms, the field measurements
show an earlier response to the moderately heavy rain commencing from 5 June. However, the
relative time of the responses matches that seen in the ICFEP simulation, with SP4 ( simulated
by point RL ) reacting immediately to the rainstorms, whereas SP9 ( equivalent to RM ) takes
about two days to respond, commencing on 8 June, and has a response dragged out over two to
three days. Thereafter, the pattern of pressure variation predicted by ICFEP for RM and RL is
quite similar to the corresponding field data for points SP9 and SP4, although the magnitudes of
pressure remain different.
Nothing in the ICFEP prediction for Run 4 indicates any ‘new’ behaviour is occurring. Rather,
the behaviour continues to follow the pattern that was explained for the earlier runs. The most
notable result for Run 4 seems to be the confirmation that no significant change in the suction
within the residual soil will occur in response to rainfall unless that rainfall produces
compressive pore pressures or very low magnitude suctions in the overlying colluvium. That is,
unless a significant vertical hydraulic gradient develops, flow occurs preferentially in a lateral
down-slope manner ( ‘interflow’ ).
Run 5
The results of Run 4 showed that suctions were being predicted in the colluvium that were much
greater in magnitude than the field data. This suggested that the permeability of this layer within
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the ICFEP model was too great, allowing flow to occur too freely from the soil. However, the
fully saturated permeability used in the ICFEP simulation for the colluvium appeared to be
correct, since the rate of change in pore pressures seemed quite close to the actual field
behaviour once the colluvium layer became saturated.
To address this, the suction switch for the colluvium was modified, such that the ratio between
fully saturated permeability and ‘minimum’ permeability ( occurring at 60 kPa tensile pore
water pressure ), was increased from 500.0 to 5000.0. The plot of the resulting pore pressure
predictions with time is given in Figure 8.25.
As can be seen, this had the desired effect of reducing the magnitude of the suctions developed
in the colluvium, while still allowing rapid drainage of compressive pore pressures. Other than
that, the fluctuations of compressive pore pressure / suction predicted by ICFEP in Run 5 match
closely those determined in Run 4.
The response at point RM in Run 5 to the early June rainfall is a little greater than in Run 4.
Given that the colluvium above this point is unsaturated at the start of this rain event, the
permeability of the colluvium will be less in Run 5 than in Run 4. This inhibits flow within the
colluvium, and as a result, the build up of compressive pore pressures at point CM is very
slightly greater in Run 5 as compared to Run 4. It is likely that this provides sufficient driving
force to the vertical flow to cause the greater response seen at point RM.
Beyond the early June rain events, the residual soil plots indicate that the change to the
colluvium suction switch has generated a slight reduction in the magnitude of suction developed
in the residual soil, of a few kilopascals. This is logical, since the colluvium pore pressure is
now less tensile, and the hydraulic gradient driving flow is correspondingly increased. However,
the basic pore pressure response is largely unchanged from Run 4, indicating that the lateral
down-slope flow regime is still predominant.
Of additional note is that not only did the change to the suction switch in Run 5 reduce the
maximum magnitude of suctions developed, it also tended to ‘compress’ the range of suctions.
For example, in Run 5, the suction at point CL varied from a magnitude of 35 kPa to 17 kPa on
25 March, whereas in Run 4, the value changed from 48 kPa to 27 kPa.
Since the fully saturated permeability of the colluvium was not adjusted, the changes made to
the suction switch will obviously manifest themselves more as the soil becomes increasingly
unsaturated ( develops higher magnitude suctions ).
It is therefore unsurprising that the
reduction in suctions in Run 5 is greater at high magnitude suctions, and that the range of
suctions predicted becomes compressed.
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While Run 5 improved on the accuracy of Run 4, in attempting to reproduce the field data it is
still some way off getting the magnitude of pressures correct.
Run 6
Run 6 was a progression from Run 5, with the same input parameters except the suction switch
ratio for the colluvium was increased further to 50,000.0. This was expected to exaggerate the
changes already observed between Run 4 and Run 5. The ICFEP prediction for Run 6 is shown
in Figure 8.26, and as can be seen, the expected behaviour was observed.
The colluvium response is generally to show lower magnitude suctions, with a smaller range of
pressures, while the residual soil layer again shows a small reduction in the magnitude of
suctions developed. The behaviour of the soil at low magnitude suctions or compressive pore
water pressures shows no significant change.
Comparison to the field data ( Figure 8.14 ) shows that Run 6 results in the ICFEP predictions
becoming more accurate than previous runs, but the Finite Element analysis is still failing to
correctly generate the magnitude of suctions. However, since the change made to the suction
switch was improving the accuracy this process was continued.
Run 7
With Run 6 improving the accuracy of the ICFEP prediction, but still failing to match closely
the field data, the change applied to the colluvium suction switch ratio was made more extreme,
with a permeability ratio of 500,000.0 applied for Run 7. This resulted in the ICFEP predictions
shown in Figure 8.27.
As was expected, this run merely confirmed the trend of Runs 4 to 6, with the colluvium suction
being of lower magnitude and more concentrated, and a slight reduction in the residual soil
suction, while the peak compressive pore pressures generated were generally unchanged.
It should be noted that prior to Run 7, the rainfall specified in the analysis data file was
inaccurate for two increments ( days ) of the analysis ( see Tables 8.2 and 8.3 ). This error was
corrected in Run 7 ( and thereafter ), hence the mid-July compressive peak in pore pressure is
generated, while the early August spike seen in Runs 1 to 6 is much reduced.
Also of note in the ICFEP Run 7 predictions is the response observed at point RH, the residual
soil point at the highest section of the slope. Not only are the magnitudes of suction at this point
reduced compared to those of the earlier runs, but there is also a clear period over which the
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suctions change, this being approximately the month of July. Re-examining the Run 6 data
shows that a similar, but considerably smaller, change occurred in that run.
In Run 7, there are three definite ‘peaks’ in the RH data between late June and the end of July.
The first, on 25 June, appears to be a response to the 11 June rain event, indicating a 14 day
delay between rainfall and pressure response.
Another relatively sharp response at this point occurs on 13 July. This could be a reaction to the
6 July rainstorm, implying a much more rapid effect, but might equally be due to the storm of
27 June, which gives a response period of 16 days.
The greatest of the ‘peaks’ in the RH data occurs on 24 July, when suctions are reduced to less
than 16 kPa. This follows on shortly after the storm of 15 July, but it seems more likely that this
is the result of the most intense rain event of the year, on 6 July, giving a 18 day time delay
between rainfall and maximum pore pressure response. In this latter case, there is no clear
response at RH to the 15 July rainstorm, but since this was of a smaller nature than the 6 July
event, this is not unreasonable.
A similar, but more drawn out, change in suction occurs at RH around 12 September.
Examination of the rainfall data shows that a period of moderate to very heavy rainfall started
on 27 August, lasting until 3 September, with a single intense storm following on 5 September.
The rainfall pattern and ICFEP predictions for September when viewed together are consistent
with an approximate two week time lag between rainfall and pressure response.
Comparing this with the field data ( Figure 8.14 ), it can be seen that the equivalent point in the
actual slope, SP7, shows a similar pattern of apparently delayed response to the main rain
events. However, while the effect of the June rainstorm seems to take around two weeks to
manifest itself at SP7, the response to the July events is closer to a month before it occurs.
Interestingly, within the ICFEP prediction there also appears to be minor perturbations in the
pressure with time plot for RH ( and also RL and RM ) which relate directly to rain events and
the near instantaneous responses predicted within the colluvium. These are very small
( generally less than 1 kPa ), and show a drop followed by an equal rise in the magnitude of
suction in the residual soil, exactly coinciding with similar but much larger changes in the
colluvium.
This is probably showing the effect of the density variation capability of ICFEP. The rainfall
that causes the reduction in suction in the colluvium does so by changing the degree of
saturation, and so also increases its mass. This leads to a compressive increase in total stress
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deeper within the soil, at least part of which appears to be reflected as a small compressive
change in the deep soil pore water tensions.
Such minor variations are not clearly visible in the field data, but there is a general degree of
variation within the field data that would mask such effects. However, Run 6 and earlier ICFEP
predictions also show the minor perturbations in the deep pore tensions.
Also, as mentioned earlier, Run 6 does show a steady change in suction for point RH over the
July period, from about 25 kPa pore tension to around 21 kPa. However, it is less easy to
discern clear peaks of response in the Run 6 predictions. Further, the 15 July rainfall data was
incorrectly specified for Run 6 and before, being under represented by an order of magnitude.
This likely had some impact on the predicted pore pressure responses. However, based on the
behaviour that it is believed that Run 7 shows, this rain event is the least significant of the major
storms, its influence being largely overshadowed by the larger storm nine days earlier.
While point RH appears to take around two weeks or a little over to respond to rainfall, both in
the ICFEP simulations and the recorded field data, this does not seem to be the case for the
residual soil points lower down the slope.
As discussed in the section on Run 4, the RL point responds near instantaneously to rainfall,
while RM appears to take around two to three days to respond, and to show a more drawn out
response. In both cases, this only applies after the first of the heavy June rains, and this applies
to both Run 6 and Run 7. However, this behaviour is less obvious in the field data.
While SP4 ( equivalent to RL ) shows the rapid response predicted, the ICFEP predictions
consistently under estimate the compressive pore pressures generated. Further, SP9 ( the
equivalent to RM ) shows little delay in its response to rain events, and is also much sharper in
its response than the ICFEP predictions for RM.
A logical explanation for the difference between predicted and field behaviour at point RM
( SP9 ) would be that the pressures predicted in the overlying colluvium were different, and that
this was affecting the vertical hydraulic gradient that developed.
The Run 7 ICFEP predictions for point CM ( equivalent to SP10 ) show a fair degree of
similarity to the SP10 field data. Once the ‘wet period’ of June commences, the prediction
closely matches the field data in form. The predictions correctly predict the maximum
compressive pressures, while give maximum tensions that are about 15 kPa too high. The
exception to this is the period of August, where the ICFEP predictions show a response to three
moderate rainstorms, which appear to have had an insignificant effect on the field responses.
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Comparing the field data to the results of ICFEP Run 7 it can be seen that the ‘base’ suctions at
the start of each rain event for the field data are of lower magnitude than in the ICFEP
predictions for the low and mid-height section points ( CL, RL, CM, RM ). With permeability a
function of suction, the permeability of the soil in the ICFEP simulation will therefore be
reduced compared to what would apply if the field data suctions were present in the simulation.
It is thus logical that the time lag in the response time between rainfall and pressure response in
the ICFEP analyses is greater than in the actual data.
The difference between the actual observed pressures and the ICFEP simulation predictions
during August are less obviously accounted for.
Of all the field monitoring points, only SP6 ( see Figure 8.15d ) and possibly SP5 ( figure
8.13b ) show any sign of a response to these rain events. However, all the ICFEP predictions
indicate that these events affect the pressure in the colluvium.
It is suspected that this may show the influence of rainstorm duration. The ICFEP predictions
are made assuming all the rainfall that occurs in a day is distributed evenly throughout the day.
Clearly, this assumption will in many cases be wrong, but it is the only practical assumption.
While rainfall data can be acquired at more precise time intervals than 24-hours ( data for every
5 minute period is not unusual ), only the daily rainfall figures were available for the Tung
Chung site. While it is unlikely that the rainfall that occurred did so exactly evenly throughout
each day, there is no evidence that would justify using any other form of distribution.
Further, assuming each day’s rainfall occurs evenly throughout the day enabled the ICFEP
simulations to be run using 1 increment of analysis for each day of the period simulated. Thus
232 increments were required to simulate the period from March until late October. If some
other form of rainfall distribution had been adopted, with the rainfall concentrated into a few
hours within each day, additional increments of analysis would have been required to simulate
the ‘non-raining’ parts of each day. At very least, this would have doubled the number of
increments required, considerably adding to the time and computer resources needed to run the
analyses.
It is believed that the failure of the field data to reflect the moderate August storms indicates
that these storms occurred in a manner significantly different to the assumed ‘constant rainfall
throughout the day’ pattern used in the ICFEP work. Instead, it is suspected that these storms
may have been intense, short duration events, perhaps occurring in only five or ten minutes.
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In such a case, it would be likely that the surface soil layer would saturate almost immediately,
generating a sharp wetting front. The low permeability of the still unsaturated soil at quite
shallow depth would impede the penetration of the wetting front, and the result would be that
the process of ponding would commence. Hence much of the rainfall would be shed from the
slope as surface run-off.
The marked difference in permeability between the saturated upper band of colluvium, and
deeper but still unsaturated colluvium, would encourage what water that had entered the soil to
flow preferentially in a lateral direction, through the saturated surface layer. The saturated band
would then drain in the same manner, and suctions would be restored. Meanwhile, the deeper
colluvium, and the entire residual soil layer, would be unaffected by the rain event.
To investigate this hypothesis, Run 7 was re-run for the period of August ( as Run 7A ).
The ‘summer2’ period was re-run unchanged until the end of July. Thereafter, the previous
approach of using one increment of analysis per day of the simulation, with the daily rainfall
distributed evenly throughout the day, was discontinued. Instead, the number and duration of
the increments was set such that on each day that rain fell, the rainfall occurred at the very start
of the day, over a period of 0.01 days ( just under 15 minutes ). Increments of varying durations
were used between rain events to model the dry periods. Table 5 shows the increment lengths
used.
Details of pore pressure / suction were taken for each of the six monitoring points at the end of
each day ( not the end of each rainfall ), and plotted as shown in Figure 8.28. Note that some dry
( non-rain ) periods were actually greater than 1 day in duration, in which case the data was for
the end of the dry period ( up to 5.99 days long ). Further, this analysis was terminated at the
end of August.
As can be seen from Figure 8.28, the results of this analysis appear to prove the earlier
hypothesis. The pore pressure data from the simulation shows a steady increase in the
magnitude of the suction throughout August, with no identifiable response to the rainfall prior to
28 August. Only the quite extensive rain events on 29 and 30 August appear to have induced
any noticeable change in the pore pressures ( and the field data responded to these events as
well ).
This has quite major implications for the analysis of infiltration into unsaturated soil. Changing
the period of rainfall clearly has a quite dominant effect on the predicted pore water response in
the slope.
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Data that was obtained for the Pak Kong study ( Data provided by the Hong Kong Observatory,
for N15 automatic rain gauge at Sun Tsun secondary school, in Sai Kung ) shows that rainfall
may be quite variable throughout the day.
For example, the data from that site for 13 June 1992 shows a total rainfall over the day of
207.5mm. Hourly rainfall through the day varied from 0mm to 57.5mm, while over one 15minute period, 20.5mm of rain fell, and the peak 5-minute rainfall that day was 8.5mm.
The assumption that such highly varied rainfall can be modelled by simply using the sum total
for the day in a 1-day increment is clearly not accurate, based on the results of ICFEP Run 7 and
Run 7A. However, attempting to produce an accurate rainfall distribution model leads to
numerous problems.
While it might be feasible to attempt to predict in advance the daily rainfall pattern at some
location, using some form of statistical technique, the idea of making meaningful predictions of
the 5-minute rainfall distribution ( not just for a day, but possibly for several months ahead )
seems beyond any current weather forecasting techniques. Simulating past events is obviously
less of a problem in this respect, as long as the rainfall data was collected with sufficient
precision.
However, even if the rainfall data is available at 5-minute intervals, the numerical analysis
remains less than easy. As stated earlier, the ICFEP analyses undertaken for this project used
232 increments of analysis ( at 1 increment represents 1 day ) to simulate the period of March to
late October. If 1-hour rainfall data had been available, the same period would have required
5568 increments of 1-hour duration; reducing the increment size to 5 minutes would have
pushed the number of increments required up to 66,816. Clearly, the more precisely the rainfall
pattern is modelled, the greater the burden placed on the computer resources used to run the
numerical analyses.
From the results of Run 7 and Run 7A, some patterns of behaviour in the response of the pore
pressures can be identified, most of which were present, but less clear, in the earlier runs.
It seems evident that the deep soil layer higher up the slope only responds to quite intense
storms, capable of fully saturating the surface colluvium, and does so with a time lag of two
weeks or more between the rain event and the observed response. This suggests a slow moving
wetting front is percolating down in this area, and implies that the critical time for slope stability
might not be during or directly after the rainfall, but may instead be some days or weeks later,
once suctions at depth have been destroyed.
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Lower down the slope, the deeper soil seems to wet up more readily and with less of a delay.
The changes in the mass of the upper layer of soil due to changes in saturation appear to affect
the pore pressures deeper in the soil, but the effect appears to be relatively insignificant.
However, most significantly, it can be seen that the rainfall distribution pattern used to model
the rainfall is critical, and unless this is correct, there is no realistic chance of reproducing field
behaviour in a numerical analysis.
Clearly there is scope for further investigation of this aspect of behaviour, but Run 7A was not
undertaken until after Run 12, and the limitation of time prevented additional study of the
effects of rainfall distribution pattern.
The analyses conducted after Run 7 continue to use the original ‘1 day represented by 1
increment’ approach throughout each run. While this likely ensures that the results will not
match the field data, it is consistent with the earlier runs, and enables the influence of specific
changes in parameters to be readily observed.
Run 8
Run 8 was a repeat of Run 6, except that the initial voids ratio of the colluvium was increased to
1.00 ( rather than 0.74 ), it being felt that the permeability ratio used for the colluvium was more
realistic in Run 6 than in Run 7. The resulting pore pressure predictions are shown in Figure
8.29.
Since the volumetric water content of a soil is a function of the porosity ( and hence the voids
ratio ), it seemed likely that changing the initial voids ratio value would significantly affect the
predicted behaviour.
It was expected that the higher voids ratio would result in a greater volume of water being
required to enter or leave the colluvium to obtain any given change in suction. As such, it was
believed that the range of pressure values predicted by ICFEP for the colluvium would tend to
narrow.
As can be seen from figure 8.29, this did occur, but it is clear that the effect is very small. A
more extreme increase in the voids ratio would no doubt have generated a greater effect, but
compared to other factors, the influence of initial voids ratio appears to be insignificant.
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Run 9
Run 9 was again a repeat in general of Run 6. However, in this case, the base ‘recharge’
precipitation boundary condition was changed. The infiltration rate was set to 0.0 m/s. Thus the
base boundary became a no-flow boundary while the pressure there remained more tensile than
the threshold value ( which was unchanged from that used in Run 6 ).
However, if precipitation on the slope surface followed by downward flow led to a build up of
compressive pore pressure at the base boundary, the condition there would switch to a fixed
pressure boundary. Drainage would occur out of the mesh, and the boundary pore pressure
would remain at the specified threshold value.
The constant pressure boundary condition would then remain unless lateral flow within the
mesh caused pressures at the base to become more tensile than the threshold value, at which
point a specified flow boundary condition ( of zero inflow ) would be restored.
The results of ICFEP Run 9 are shown in Figure 8.30. As can be seen through comparison with
Figure 8.26, the change to the base boundary condition had no effect on the predicted pore
pressures within the colluvium.
There was some fairly small effect on the predictions of pressure within the residual soil layer.
This seems to have affected the response at point RM the most.
However, it can be seen that this effect is small, amounting to a difference in the predicted pore
pressures of a few kilopascals at most, and there is no significant change in the basic pattern of
the pore pressure response to rainfall.
This suggests that the base pore boundary condition is relatively unimportant, at least if this
boundary is far enough away from the points of interest. The mesh used in the ICFEP analysis
varies in thickness between 6m and 10m, being around 8m to 9m thick at the sections
containing the six points under consideration.
Three of these points lie within the colluvium, and are within 1.5m of the mesh upper surface,
while even the three points within the residual soil layer are within 3.5m of the mesh upper
surface. Thus, the base boundary avoids being the closest boundary to any of these points, all of
which are much nearer the surface than the base.
Further, although the lower portion of the mesh tends to be more saturated than the upper half,
the surface colluvium layer has a higher saturated permeability than the residual soil, while the
residual soil permeability is stress-dependent, and so decreases with depth. These factors mean
Chapter 8
265
that the permeability of the deeper soil may be less than that in the shallower deposits. Thus
elements within the mesh are more sensitive to flow through the upper boundary than the lower
boundary, unless the elements are very close to the base.
This effect can be seen in the results from Run 1, in Figures 8.18 to 8.20, where the drainage
action of the bottom boundary condition is only significant over the lower 1.5m to 2m of the
mesh, which is well clear of the monitoring points.
This single run comparison does not constitute a full investigation of the effects of variation in
the base boundary condition on the pore response throughout the mesh. It does, however,
indicate that the choice of boundary condition did not significantly alter the numerically
predicted behaviour pattern, such that this boundary condition became the dominant factor
governing the predicted response.
Run 10
A constant feature of the results of Runs 1 to 9 is that the predicted pore pressure responses
have tended to be too variable, as compared to the field data. While the field data does show
sharp responses to the extreme rain events, the more gentle rainfall in between appears to have
relatively little effect. The ICFEP predictions, however, show a more marked response to these
smaller rainstorms.
In an attempt to improve the accuracy of the ICFEP predictions, the suction switch parameters
for the colluvium were changed. The ratio between saturated permeability and minimum
permeability was kept the same, but the suction range over which the change occurred was
altered from a range of 1 kPa to 60 kPa, to one of 1 kPa to 5 kPa. With this change made,
Run 10 was commenced, being otherwise the same as Run 6.
Figure 8.31 shows the resulting pore pressure predictions. As can be seen, the analysis was
incomplete. It was found that ICFEP had great difficulty in achieving numerical convergence
for this highly non-linear problem. The analysis was terminated by ICFEP during the increment
representing 10 May ( a ‘zero’ rain day, as shown by Table 8-2 ), when it was no longer
possible to reduce the size of the sub-increment used by the AI procedure. ( ICFEP compares
the size of the sub-increment to the remaining portion of the increment, and to the maximum
number of sub-increments as specified by the input file, to determine whether the maximum
number of sub-increments is likely to be exceeded. If this is the case, the analysis is
terminated ).
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It is notable that failure of the analysis occurred not during a severe storm, with very high
inflow rates, but actually in the reverse situation of zero rainfall, with correspondingly low flow
rates within the mesh. ICFEP’s principal difficulty was in obtaining convergence for the pore
pressures. Section 5.7 discussed how ICFEP makes use of a linear approach to model non-linear
behaviour, with a residual load ‘error’ being generated that is iteratively reduced to some small,
acceptable value, this being defined by the tolerance value. The same basic process also applies
to determination of the pore pressures, but in Run 10, with standard tolerances applied, ICFEP
was unable to achieve numerical convergence for the pore pressures.
While it would have been possible to adjust the tolerance values applied and attempt to force the
analysis through the increment of failure, this was not felt to be necessary.
The results achieved prior to failure ( see Figure 8.31 ) clearly showed that the change to the
suction switch parameter was having the desired effect, with the range of pore pressures
predicted within the colluvium layer very much less than in previous runs, and much closer to
that shown by the field data ( it being assumed that the first month or so of the field data reflects
stabilisation of the results, post-installation ). Accordingly, the run was allowed to terminate,
and Run 11 was commenced instead.
Run 11
Run 11 was basically the same as Run 10, except that the base boundary condition was changed,
as per Run 9. It was suspected that this boundary condition may have been the cause of the
convergence problems encountered in Run 10, and that changing it might therefore enable the
analysis to be completed.
As can be seen from Figure 8.32, this was not entirely the case. While the analysis progressed
further, it again failed in the same manner, this time during the increment representing 12 June.
Comparison of the results from Run 10 and 11 tends to confirm the conclusion from Run 9 that
the change to the base boundary condition has no noticeable effect on the pressures within the
colluvium, and no significant effect on the response in the residual soil. However, there
obviously was some influence from the changed boundary condition, since the analysis
progressed considerably further purely as a result of this change, as compared to Run 10.
Comparison of the Run 11 results to those of Run 9 ( Figure 8.30 ) and the field data ( Figure
8.14 ) shows that the sharper suction switch within the colluvium has increased the accuracy of
the predictions for the colluvium. From April onwards, the predictions of pressure within this
layer given by Run 11 are generally quite close to the field data. The residual soil predictions
Chapter 8
267
are less good, showing a more rapid loss of suction early on than was seen in the field data, but
the analysis failed before any of the significant rain events occurred. The residual soil pressure
predictions were, however, quite close to the field data at the point at which the analysis failed.
Run 12
Since the choice of base boundary condition was clearly not the cause of the convergence
problems, it was decided to try a further run, with this boundary condition returned to its
standard setting. Instead, the specified saturated permeability of the residual soil layer was
increased to 5*10E-5 m/s, as noted in Table 8-3.
However, due to an operator error in establishing the data file, this change was applied only to
the seepage properties parameters, but not also to the variable permeability properties. As a
result, the actual permeability used by ICFEP was unchanged, so Run 12 was effectively a
repeat of Run 10.
Again, numerical convergence problems were encountered during this run. However, in this
case, the convergence tolerances were adjusted as necessary to force the analysis through each
increment, where problems occurred. However, care was taken to ensure that the tolerances
used remained as small as possible. The iterative residuals are compared against both the
incremental change and the accumulated ( since the start of the analysis ) change in the variable,
with default tolerances being 2% in each case. To complete Run 12, it was necessary to increase
the incremental tolerance values considerably, for a few increments up to 99%, though generally
the increased incremental tolerance values were kept below 80%. The accumulated tolerances
were not loosened in any way. As detailed later, many of the numerical convergence problems
occurred because the incremental change in a variable was very small. Hence ICFEP was
attempting to reduce the residual to default-2% of a very small number.
Even with looser tolerances being permitted, it was still found that the increment representing
26 June could not be converged. To enable this day to be simulated, this increment was replaced
by five increments, each of 0.2-day duration. The pore pressures for this day were taken as those
obtained from the last of these increments. Each of the five increments received the same daily
rate of rainfall, so received one-fifth the total rainfall for that day.
It should have been possible to achieve the same result through the application of the AI
procedure, with the maximum number of sub-increments increased, to allow for smaller subincrements. However, experience with the AI procedure showed that it could be slow to increase
the size of sub-increments used once the cause of a problem had been passed, thus considerably
increasing the run time required.
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Additionally, with highly non-linear problems like that being undertaken, the AI procedure is
better able to estimate the ‘ideal’ sub-increment size when the overall increment length is
shorter. A larger main increment tends to result in many more iterations to determine a subincrement size that can be converged, because the effect of non-linearity becomes more
pronounced over the longer time scale.
The results of the ICFEP simulation for Run 12 are shown in Figure 8.33.
As can be seen, this run provides the predictions that come closest to the field data of any of the
runs, particularly as regards the colluvium. However, equally clear is that while the general
form of the predicted responses matches that of the field data, the magnitudes of the predicted
pressures do not precisely equate to the actual field values ( Figure 8.34 shows the Run 12
results superimposed onto the field data plots ).
In particular, the middle and low sections are predicted as becoming saturated during the early
to mid July period, which results in the residual soil pressures at the middle section being
predicted as about 10 kPa too compressive for much of the monitoring period. Conversely, the
predictions for the colluvium at the low section are approximately 10 kPa more tensile than the
field data.
Interestingly, the ‘spikes’ in the pore pressures responses resulting from the August rains that
were seen in Runs 1 to 9 are much less pronounced in Run 12. Also of note in relation to this, is
that the ‘base’ pressures in the colluvium during this period are suctions of 5 kPa to 7 kPa,
indicating that this soil is at minimum permeability, which from the fully saturated permeability
value and the parameters on the suction switch ( see Table 8-1 ), can be seen to be 6*10E-9 m/s.
A rainfall of 25 mm/day approximately equals a flow rate of 2.9*10E-7 m/s. Thus the typical
minor storms of August ( see Table 8-2 ) would give an inflow rate greater than the permeability
of the colluvium in Run 12. Hence the soil surface will have saturated rapidly, the surface pore
pressure will have reached the threshold value, and much of the rainfall will have become runoff, rather than penetrate into the soil. The days following the minor storms are either dry or
have very low rainfall rates, so no wetting front develops.
By contrast the earlier runs had a ‘base’ suction prior to the August rainfall events of around
30 kPa to 40 kPa suction. However, the looser suction switch parameters would have given prerainfall permeabilities of around 1.5*10E-6 to 2.5*10E-7 m/s. Thus the full rainfall could
penetrate into the soil, with little or no run-off.
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The inference from all the above is that the permeability function applicable to the upper-most
soil layer is quite critical to the predicted behaviour: an unsurprising conclusion. Whether it is
solely the actual value of permeability that matters, or whether the rate of change of
permeability ( the gradient of the suction switch ) that applies is also important is less clear.
Repeating Run 12 with a third set of suction switch parameters would have been a useful
exercise, but the time available for this project meant that this was unfeasible. As discussed
below ( see Section 8.11 ), this Run was particularly time-consuming to complete, both due to
the actual run-time, and the requirement to constantly adjust tolerances to enable this highly
non-linear problem to be analysed. Since Run 12 is effectively Run 6, other than the change to
the suction switch, both of these factors result from the rapid change in permeability that occurs
for relatively little change in suction resulting from the modified suction switch.
Run 3 noted that the field data tended to show suctions that increased linearly or at an increasing
rate, once the suction exceeded 10 kPa magnitude. It was expected that the modified suction
switch, with the minimum permeability occurring for suctions of 5 kPa or more tensile pore
pressures, would tend to show similar behaviour ( at least giving a linear change in suction
beyond the 5 kPa point ). However, the suctions in Run 12 did not develop sufficiently for it to
be possible to determine if this is the case.
The results from ICFEP Runs 1 to 12 have shown how changes to the different unsaturated soil
parameters result in variations of the predicted pore pressure response to rainfall.
Prior to this project, any attempt to simulate the same problem using ICFEP could only have
been undertaken using a saturated soil model ( with suction dependent permeability, since the
suction switch was pre-existing ). Having developed an unsaturated approach that appears able
to reproduce something approaching real insitu behaviour, the question arises as to whether
similar results could have been achieved using the saturated soil model. Accordingly, Run 13
was undertaken.
Run 13
Since Run 12 gave fairly good agreement with the field data, it was originally intended to repeat
this run using a saturated soil model ( as Run 13(12) ).
However, when attempted the analysis suffered from the same numerical convergence problems
as afflicted the unsaturated analysis, and also took excessively long to run. Having suffered a
numerical convergence failure on the increment representing March 25, it was decided to abort
this run.
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270
Instead, a saturated analysis using the Run 7 parameters was attempted, this being the closest
prediction to the field data of Runs 1 to 9. In this Run 13(7), the soil strength parameters, linear
material properties and permeability details (including the suction switch ) were identical to
those used in Run 7. However no Soil Water Characteristic Curve was applied, and no value of
the H parameter was input, with the analysis data file set up to indicate a saturated, rather than
unsaturated analysis. Additionally the density variation capability is not used during saturated
analysis.
Again, the analysis run suffered a numerical convergence failure, on the increment representing
12 June. However, the pattern of the predictions prior to this increment seemed clear, and from
the time taken to process the analysis that far, it was expected that the analysis would require at
least four weeks further run time to complete. Accordingly, the saturated analysis was
terminated at this point.
The results of Run 13(7) are shown in Figure 8.35.
As can be seen from comparison to the field data ( figure 8.14 ) and the Run 7 results ( Figure
8.27 ), the saturated approach predicts a pore pressure response markedly different from both
the unsaturated predictions and the field measurements. The pattern of the saturated analysis
seems to indicate an approximately constant ‘base’ suction, with a sharp deviation from this
value occurring due to every rain event. Only under the extreme June rainfall does the base
suction value change significantly.
Both the compressive increase in pressure following rain and the drainage that restores the
suctions afterwards occur much more rapidly than the equivalent behaviour shown by the
unsaturated model, or as is seen in the real prototype. It seems likely that this reflects the effect
of water storage variability in unsaturated soil. In the saturated soil, only a very small inflow of
water is needed to produce a rapid and large compressive increase in the pore water pressures.
However, in the unsaturated case, any inflow mainly results in a change in the degree of
saturation, with a relatively small change in suction ( pore pressure ), depending on the SWCC
relationship. Quite considerable volumes of water may be required to produce large changes in
the degree of saturation, before inflow generates significant pore pressure changes in an
unsaturated soil.
From the limited data obtained from the original, aborted, Run 13(12) ( the saturated re-run of
Run 12 ), it appears that this run was showing similar behaviour, though with the ’base suction’
values in the colluvium being 15 kPa to 20 kPa lower in magnitude.
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271
8.11.
Computer resources required
The ICFEP analyses were run using a SUN ULTRA 10 333MHz workstation. This was a state
of the art machine at the commencement of this project in 1999, but is now dated, with much
faster machines being available.
Clearly, the analyses undertaken were of a highly non-linear problem, and this was reflected in
the time required to complete each run. Full records of the duration required to complete each
analysis were not kept, but a reasonably comprehensive listing was made.
Generally, it was found that the sum time required to run all four parts (‘spring’, ‘summer1’,
‘summer2’ and ‘autumn’ ) of each analysis was between 100 and 200 hours for each of the
earlier unsaturated runs (up to Run 9 ).
Once the more extreme suction switch parameters were introduced ( run 10 onwards ), the
analysis became much more time consuming. Run 10 took 141 hours to complete the ‘spring’
phase, and the numerical failure during ‘summer1’ occurred after 24 hours. Run 11 required 135
hours for ‘spring’ and took 405 hours to reach numerical failure during ‘summer1’.
For Run 12, the frequent numerical convergence problems resulted in a ‘stop-start’ approach to
the analysis, and the tolerances used were not always the same as applied in other runs
( generally being ‘looser’ making the run quicker ). Further, much of this run was done in small
runs of a few, or even a single, increment and the times required for many of these short
sections of the run were not recorded. As a result, the exact time taken is not known, but it was
considerably in excess of 1680 hours ( that is, greater than 70 days continuous run time ).
This was despite an attempt to speed up the analysis. When ICFEP is running through the
iterative process to obtain convergence for a non-linear solution, both the minimum and
maximum number of iterations used by the program may be user-defined.
Increasing the minimum number forces ICFEP to iterate further, even if the solution has
converged, and generally results in a further reduction in size of the residual values ( thus
making the increment more accurate, and improving the chance that later increments will
converge ).
The maximum number of iterations marks the point at which ICFEP ‘gives up’ and either
attempts a smaller sub-increment, if using the automatic incrementation procedure, or
terminates the analysis due to numerical convergence failure, if no AI, or if the maximum
number of sub-increments has been exceeded.
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272
To reduce the time taken by the analysis, both these values were reduced in Run 12. Further, in
Run 13(7), the minimum and maximum iterations values were identical to those used in Run 7
for ‘spring’, but were reduced for ‘summer1’ and onwards, again in an attempt to reduce the
time taken by the analysis. The resulting time taken for ‘spring’ in Run 13(7) was 231 hours,
while the numerical failure during ‘summer1’ occurred after 348 hours.
Generally, it appears that increasing the permeability ratio for the colluvium led to an increase
in the time taken by the analysis. The effect of this change is to increase the rate of change of
permeability due to a change in suction, so makes the analysis more non-linear. Hence the
increased duration of the analysis run is to be expected.
Conversely, a single change to the boundary conditions ( for example to the initial void ratio or
base pore pressure condition ) appears to have no real effect on the time required to run these
analysis.
Runs 10 to 12, however, demonstrated an order of magnitude increase in the time taken to
complete the analysis. In these runs, the suction switch parameters for the colluvium were
altered to such an extent that very large changes in permeability occurred for very small changes
in suction. Thus this aspect of behaviour was extremely non-linear, and the time taken for the
analysis is a reflection of this.
However, this does not appear to be a result of using an unsaturated soil model. Rather, it
appears to be purely due to the permeability variation, since the unsuccessful attempt to repeat
Run 12 using a saturated soil model, as Run 13(12), took a great deal of time, before failing
numerically. Run 13(7), the saturated repeat of Run 7, also took a long time to run, even after
the data file was modified to speed it up. In fact, the saturated analyses runs were the most time
consuming of all the runs.
It is evident from this that the use of an unsaturated soil approach does not in itself significantly
increase the length of time required to run an analysis. However, the more non-linear the
problem is, the greater the time that will be required.
This is borne out by the fact that the saturated analyses were generally longer to run than the
unsaturated simulations. It may be noted that the suction switch parameters of Run 13(7) are the
same as used in Run 7, yet the time required for the analysis is very much greater. The
predictions for the saturated analyses show that the pore pressures are changing much more
rapidly than in the unsaturated cases. Since permeability is a function of pore pressure, the
permeability will also vary much more rapidly. Thus running the analyses using a saturated soil
model tends to make the problem more non-linear ( as well as being ‘wrong’ ).
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273
The time taken to undertake the analyses could have been reduced a little if a less precise
analysis were undertaken.
As mentioned in Chapter 6, the operation of the Automatic Incrementation procedure with the
precipitation boundary condition makes use of a tolerance zone around the specified threshold
value. This has a default value of ±0.1 kPa, which was used throughout the Tung Chung
simulations.
Clearly a looser tolerance would have made it easier for ICFEP to switch from an inflow to a
constant pressure condition at the boundary, and should therefore have reduced the number of
sub-increments ( and hence the time ) needed for each increment of the analysis in which this
boundary switch occurred. An examination of the pore pressure variation indicated that the
precipitation boundary condition switched from an inflow condition to a fixed pressure
boundary at some point or points along the surface boundary on approximately 40 increments of
the analysis ( this for Run 12 ). Thus, close to 20% of the analyses’ increments would have been
affected by this aspect of the analysis. The number of sub-increments required to solve a given
increment varied quite widely, but in some cases several hundred sub-increment steps were
required.
However, any analysis involving the AI risks becoming time consuming. Operation of the AI
requires two estimates of displacement be made. Each estimate requires a ( separate ) inversion
of the stiffness matrix. This is a fairly complex task and requires significant computer effort in
any case, but becomes much more complex and time consuming when the stiffness matrix is
non-symmetric. A more detailed explanation of the operation of the AI procedure is given in
Appendix A5.
For coupled unsaturated analysis, the full consolidation stiffness matrix ( as presented in
Chapter 5 ) will always be non-symmetric, hence it is inevitable that such analysis will be
relatively time consuming.
ICFEP does have the capability to permit an analysis to be undertaken using either a symmetric
or a non-symmetric solver, at the operator’s discretion. Clearly, the non-symmetric approach
was required for the unsaturated analyses. Since the saturated simulations, Runs 13(12) and
13(7) were intended to be as close to their unsaturated equivalents as possible without using the
unsaturated soil models, these runs too were undertaken using the non-symmetric solver, this
being necessary due to the non-linear soil properties used. However, it would have been
possible to change the soil parameters used slightly, and then apply the symmetric solver to
these simulations, which would no doubt have reduced the time required notably. However, in
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274
such a case the saturated analysis results would not then have been fully comparable to the
unsaturated runs.
From the ICFEP predictions, it appears that realistic soil parameters ( that is, parameters that
give predictions close to the real field data ) may well be highly non-linear, making analysis of
any real unsaturated slope problem highly time consuming. Attempting to substitute a saturated
soil model would appear not to help to reduce the computer time required while, based on the
ICFEP simulations, rendering the results much less accurate.
8.12.
Summary of the Tung Chung study
The attempt to simulate the Tung Chung slope using ICFEP revealed a number of features of
unsaturated analysis.
First, it was found that without site specific data, it was quite difficult to select reliable values
for the various soil parameters. While the available literature generally provided a reasonable
guide, there is a fairly broad range of values that may be applicable for most of the soil
properties.
Having selected what were felt to be appropriate parameters for the soil, it was found that the
results of the ICFEP predictions were quite sensitive to the values chosen, with the saturated
permeability, Soil Water Characteristic Curve and suction-switch parameters all having
significant influence on the results.
Conversely, variation of the initial voids ratio of the soil seemed to be fairly insignificant, while
the base boundary condition appeared to affect the predicted pressures only within a narrow
band close to that boundary.
Significant vertical flow into the deeper, residual, soil layer seemed often to be dependent on the
surface colluvium saturating. Otherwise, rain infiltration tended to enter the colluvial layer, then
flow laterally through this layer without penetrating into the deeper soil.
It was found that the maximum increment size that is used to model rainfall is critical to the
prediction of the pore pressures that result. Significant variation in the predictions was shown
when the daily rainfall was compressed into a 15-minute period, rather than being distributed
evenly throughout the day. This implies that accurate modelling of an unsaturated slope subject
to infiltration requires rainfall data to at least 1-hour detail, and preferably even more precise.
Chapter 8
275
An attempt to reproduce the Tung Chung analysis using a saturated soil model gave predictions
that were a lot less accurate and moreover the saturated analysis was found to be numerically
slower and ‘more difficult’ than the equivalent unsaturated analysis.
Thus the adoption of an unsaturated model has significantly improved the modelling capability
for infiltration into unsaturated soils.
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276
Table 8-1
Suction switch parameters for the colluvium
Run
P1 ( kPa )
P2 ( kPa )
Kmax/Kmin
1
0.1
60
500
2
0.1
60
500
3
1.0
60
500
4
1.0
60
500
5
1.0
60
5000
6
1.0
60
50,000
7
1.0
60
500,000
8
1.0
60
50,000
9
1.0
60
50,000
10
1.0
5
50,000
11
1.0
5
50,000
12
1.0
5
50,000
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277
Table 8-2.
Raingauge Rainfall Data at Tung Chung East (Tai Tung Shan)
Date
Rain Gauge R2
Rain Gauge R1
Rainfall Total (mm)
Rainfall Total (mm)
11-Jan-01
0.5
0
0.25
13-Jan-01
4.0
1.0
2.5
25-Jan-01
4.5
4.0
4.25
26-Jan-01
36.0
34.5
35.25
27-Jan-01
6.5
6.5
6.5
28-Jan-01
10.5
10.5
10.5
1-Feb-01
0
0.5
0.25
2-Feb-01
1
0.5
0.75
3-Feb-01
0.5
1
0.75
4-Feb-01
1
1.5
1.25
11-Feb-01
0.5
0.5
0.5
25-Feb-01
17
15.5
16.25
26-Feb-01
0
0.5
0.25
28-Feb-01
1.5
1
1.25
3-Mar-01
0
0.5
0.25
8-Mar-01
3
3
3
11-Mar-01
5
2
3.5
12-Mar-01
1.5
0.5
1
25-Mar-01
61
8
34.5
27-Mar-01
1.5
1.5
1.5
29-Mar-01
3
2.5
2.75
4-Apr-01
50.5
41
45.75
5-Apr-01
21.5
17
19.25
6-Apr-01
1.5
0.5
1
9-Apr-01
7
0.5
3.75
10-Apr-01
0.5
0
0.25
11-Apr-01
8
2.5
5.25
18-Apr-01
12.5
0
6.25
19-Apr-01
4.5
0
2.25
21-Apr-01
1.5
0
0.75
22-Apr-01
10
3
6.5
Chapter 8
278
Mean average
rainfall, mm
Date
Rain Gauge R2
Rain Gauge R1
Mean average
rainfall, mm
24-Apr-01
1
0.5
0.75
25-Apr-01
3.5
0.5
2
1-May-01
8.5
0.5
4.5
3-May-01
4
0
2
9-May-01
37
3
20
11-May-01
0.5
0
0.25
17-May-01
8.5
0
4.25
18-May-01
46.5
1
23.75
19-May-01
0.5
0
0.25
21-May-01
51.5
16.5
34
22-May-01
3
0
1.5
30-May-01
4.5
0
2.25
2-Jun-01
2
0
1
3-Jun-01
0.5
0
0.25
4-Jun-01
5.5
0
2.75
5-Jun-01
63.5
0
31.75
6-Jun-01
38
0
19
7-Jun-01
80
54.5
67.25
8-Jun-01
73
35.5
54.25
9-Jun-01
37
17
27
10-Jun-01
6.5
0
3.25
11-Jun-01
209
198
203.5
12-Jun-01
60.5
57.5
59
13-Jun-01
28
22
25
14-Jun-01
1.5
1.5
1.5
15-Jun-01
8.5
3
5.75
16-Jun-01
22.5
3.5
13
17-Jun-01
4.5
0
2.25
20-Jun-01
1
0
0.5
21-Jun-01
3.5
4.5
4
23-Jun-01
2
0.5
1.25
24-Jun-01
12
13.5
12.75
25-Jun-01
56.5
50.5
53.5
26-Jun-01
56.5
31
43.75
Chapter 8
279
Date
Rain Gauge R2
Rain Gauge R1
Mean average
rainfall, mm
27-Jun-01
167.5
169
168.25
28-Jun-01
1
1.5
1.25
29-Jun-01
0.5
0.5
0.5
30-Jun-01
11
9
10
1-Jul-01
57.5
50
53.75
2-Jul-01
8.5
7.5
8
3-Jul-01
2.5
1
1.75
5-Jul-01
13
8
10.5
6-Jul-01
243
227.5
235.25
7-Jul-01
15.5
18
16.75
8-Jul-01
14
13.5
13.75
13-Jul-01
62.5
73.5
68
15-Jul-01
187.5
186
186.75
16-Jul-01
3.5
2.5
3
17-Jul-01
54
54
54
18-Jul-01
57.5
56.5
57
19-Jul-01
2.5
2.5
2.5
20-Jul-01
0.5
0.5
0.5
21-Jul-01
53.5
53
53.25
22-Jul-01
1.5
1
1.25
24-Jul-01
15
19.5
17.25
25-Jul-01
12
10.5
11.25
26-Jul-01
6
6.5
6.25
27-Jul-01
2
2
2
1-Aug-01
8.5
11
9.75
2-Aug-01
25
27
26
3-Aug-01
2
2
2
6-Aug-01
2
2.5
2.25
10-Aug-01
4
4
4
11-Aug-01
14.5
16
15.25
12-Aug-01
3.5
4
3.75
13-Aug-01
0
2
1
14-Aug-01
1.5
0
0.75
16-Aug-01
26.5
27.5
27
Chapter 8
280
Date
Rain Gauge R2
Rain Gauge R1
Mean average
rainfall, mm
19-Aug-01
0
0.5
0.25
27-Aug-01
18.5
20.0
19.25
28-Aug-01
14.5
16.0
15.25
29-Aug-01
82.5
84.0
83.25
30-Aug-01
125.0
131.5
128.25
31-Aug-01
3.5
3.0
3.25
1-Sep-01
182.0
165.0
173.5
2-Sep-01
52.5
58.5
55.5
3-Sep-01
64.5
64.0
64.25
5-Sep-01
106.0
105.5
105.75
6-Sep-01
0.5
12.0
6.25
7-Sep-01
8.5
6.5
7.5
8-Sep-01
6.5
6.0
6.25
15-Sep-01
8.0
0
4
20-Sep-01
22.5
21.0
21.75
21-Sep-01
34.0
33.0
33.5
4-Oct-01
42.0
39.5
40.75
9-Oct-01
4.5
3.5
4
18-Oct-01
0
0.5
0.25
Where a date is absent from this table, then zero rainfall was recorded at both rain gauges that day.
Chapter 8
281
Table8-3
Summary of ICFEP analyses
Run
Details
1
First run, using general parameter, etc
2
As Run 1, except permeability of colluvium increased to 3*10E-4 m/s
3
As Run 1, except residual soil SWCC now also used for colluvium
4
As Run 3, except permeability of colluvium increased to 3*10E-4 m/s
5
As Run 4, except suction switch k-ratio for colluvium increased to 500.0
6
As Run 4, except suction switch k-ratio for colluvium increased to 50,000.0
7
As Run 4, except suction switch k-ratio for colluvium increased to 500,000.0
8
As Run 6, except initial voids ratio of colluvium increased to 1.0
9
As Run 6, except base recharge precipitation BC specified with inflow of
0.0 m/s
10
As Run 6, except colluvium suction switch operates from 1 kPa to 5 kPa
suction
11
As Run 10, except base recharge precipitation BC specified with inflow of
0.0 m/s
12
As Run 11, except saturated permeability of residual soil 5*10E-5 m/s
13
As Run 7, except run using saturated soil models only.
Note 1: For Runs 1 to 6 inclusive, the rainfall on 15 July was incorrectly input as 18.675mm,
instead of 186.75mm, and on 1 August rainfall was input as 97.5mm instead of 9.75mm.
Note 2: The change to saturated permeability made in Run 12 was made to the seepage
properties only, but not also in the variable permeability ( stress – dependent permeability )
condition. Since the actual permeability used by ICFEP was determined from the parameters
included in the stress-dependent permeability condition, Run 12 was effectively just a repeat of
Run 10. However, as described in the text, Run 10 was incomplete, while Run 12 was taken to
completion.
Chapter 8
282
Table 8-4
Relationship between tensiometers and ICFEP nodes used to simulate them
SP
Tensiometer
Tensiometer
position along
Depth
ICFEP node
line of section
ICFEP node
ICFEP node
position along
depth
line of section
4
190m
2.73m
RL
190m
2.66m
5
190m
1.59m
CL
190m
1.48m
7
242.5m
2.62m
RH
244m
2.27m
8
242.5m
1.00m
CH
244m
0.88m
9
214m
3.00m
RM
208m
3.37m
10
214m
1.15m
CM
208m
1.04m
Note that the toe of the ICFEP mesh is 100m along the line of section. Hence point RL is 90m
in from the toe of the ICFEP mesh.
Chapter 8
283
Table 8-5
Run 7A increment size for August
Increment Number
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
Chapter 8
Increment duration ( days )
0.01
0.99
0.01
0.99
0.01
2.99
0.01
1.99
2.00
0.01
0.99
0.01
0.99
0.01
0.99
0.01
0.99
0.01
1.99
0.01
2.99
0.01
5.99
1.00
1.00
0.01
0.99
0.01
0.99
0.01
0.99
0.01
0.99
0.01
0.99
284
Date represented
01/08 rain event
01/08
02/08 rain event
02/08
03/08 rain event
03-05/08
06/08 rain event
06-07/08
08-09/08
10/08 rain event
10/08
11/08 rain event
11/08
12/08 rain event
12/08
13/08 rain event
13/08
14/08 rain event
14-15/08
16/08 rain event
16-18/08
19/08 rain event
19-24/08
25/08
26/08
27/08 rain event
27/08
28/08 rain event
28/08
29/08 rain event
29/08
30/08 rain event
30/08
31/08 rain event
31/08
0
285
Figure 8.1: Location of Tung Chung landslide study area
( after Franks, 1999 )
5 km
scale
286
Figure 8.2: Solid Geology of Lantau Island, Hong Kong
( after Irfan, 1999)
N
287
TRL128
DH-TCE-1
DH-TCE-2
TRL129
D122
TP1
TP4
TP3
TP5
TP2
TP6
Figure 8.3: Location of boreholes and trial pits
D123
TRL127
EL81
TRL130
Study Area
Trial Pit
Borehole
Scarp
Legend:
288
TRL 129
SP1
A7
SP2
A6
SP3
SP4
SP5
A5
A4
SP6
SP7
SP10
SP9 A2
A3
SP8
A1 R1
Study Area
Figure 8.4: Location of pore water pressure monitoring devices
R2
289
16 14
10
12
8
6
4
2 2
10
8
24
2
22
14
16
18
20
26
6
6
4
8 10
Figure 8.5: Depth to rock Head
16
14
12
10
10
12
10
12
8
14
10
12
10
4
8 6
4
8
10
10
8
4
2
2
2
4
8
6
2
Study Area
4
8
8
4
8
6
6
6
4
290
6m
1m
2m
22m AOD
Residual
soil
viu
al
idu
Res
l
s oi
lu
col
m
( analysis points
CH and RH )
High section
6m
2m
1m
1m
106m AOD
Enlargement of top end of mesh, showing
dimensions
( analysis points
CM and RM )
Figure 8.6: ICFEP mesh for Tung Chung analysis
colluvium
( analysis points
CL and RL )
Low section
Mid section
174m ( 87 columns of elements, each 2m wide )
10m
2m
291
Figure 8.7: Line of section of ICFEP mesh
1E-2
1E-4
1E-6
Permeability (m/sec)
1E-8
1E-10
1E-12
1E-14
1E-16
1E-18
1E-20
1E-22
0.1
1
10
100
1000
10000 100000 1000000
Suction (kPa)
Coefficient of permeability (m/sec)
Figure 8.8a: Permeability - suction relationship for Chai Wan volcanic soil
(after Sun et al, 1998 ).
1E-5
1E-6
1E-7
1E-8
1E-9
1E-10
1E-11
CDV-N1 (0kPa-wetting)
CDV-N2 (40kPa-wetting)
CDV-N3 (80kPa-wetting)
CDV-N1 (0kPa-drying)
1E-12
1E-13
1E-14
0.1
10
100
Matric suction (kPa)
1
1000
Figure 8.8b: Permeability - suction relationship for volcanic soil
(after Ng and Pang, 2000 ).
Figure 8.8: Permeability-suction relationships for volcanic
soils.
292
Log k
kmax= Fully saturated permeability, = 5E10-6 m/s
K max
K min
P1 = 1 kPa
P2 = 40 kPa
Pore water pressure
(Tension +ve)
Log k = log kmax - {(p-p1)/(p2-p1)}log(kmax/kmin)
(kmax/kmin) = 20,000
Figure 8.9: Suction switch for residual soil
293
294
Volumetric water content, θ
0
20
60
100
0
Figure 8.10: SWCCs from Anderson, 1984
Suction. kPa
0
80
0
40
.1
.2
.3
.4
.5
.6
.7
.1
.2
.3
.4
.5
.6
.7
(a) Tai Po Volcanic
Volumetric water content, θ
20
60
Suction. kPa
40
(b) Colluvium
80
100
0.55
Volumetric water content
0.50
0.45
0.40
0.35
0.30
0.25
0.20
0.15
0.10
0.05
0.00
0.1
1
10
100
1000
10000
100000
1000000
Suction ( kPa )
Figure 8.11a: Chai Wan Volcanic soil ( after Sun et al, 1998 )
0.44
Volumetric water content
0.42
0.40
0.38
0.36
0.34
0.32
0.30
0.1
CDV-N1 (0kPa)
CDV-N2 (40kPa)
CDV-N3 (80kPa)
10
1
100
Matric suction ( kPa )
Figure 8.11b: Volcanic soil SWCCs ( after Ng and Pang, 2000)
Figure 8.11: SWCCs for decomposed volcanic soil
295
1000
100
Saturation (%)
80
60
40
20
0
0
50
100
150
200
250
suction ( kPa)
Curve defining parameters:
α = 0.044, n = 0.996, m = 1.0, Sr = 36.5%.
Parameters derived from curve, for ICFEP SWCC model:
Udes = 0.0, Uae = 0.1, Uo = 1000000 ( all in kPa )
Figure 8.12a: SWCC for Colluvium
100
Saturation (%)
80
60
40
20
0
0.01
0.1
1
10
100
1000
10000
100000
suction ( kPa)
Curve defining parameters:
α = 0.00225, n = 0. 685, m = 1.0, Sr = 0.0%.
Parameters derived from curve, for ICFEP SWCC model:
Udes = 0.0, Uae = 1.0, Uo = 1000000 ( all in kPa )
Figure 8.12b: SWCC for Residual soil
Note that different scales are used for suction in the two curves
Figure 8.12: SWCCs used for ICFEP analysis
296
01-Mar-01 31-Mar-01 30-Apr-01 30-May-01 29-Jun-01 29-Jul-01 28-Aug-01 27-Sep-01 27-Oct-01 26-Nov-01
30
20
10
297
kPa
0
-10
-20
-30
-40
Figure 8.13a: Tung Chung monitoring data for SP4, @ 2.73m depth
01-Mar-01 31-Mar-01 30-Apr-01 30-May-01 29-Jun-01 29-Jul-01 28-Aug-01 27-Sep-01 27-Oct-01 26-Nov-01
20
10
0
298
kPa
-10
-20
-30
-40
-50
Figure 8.13b: Tung Chung monitoring data for SP5, @ 1.53m depth
01-Mar-01 31-Mar-01 30-Apr-01 30-May-01 29-Jun-01 29-Jul-01 28-Aug-01 27-Sep-01 27-Oct-01 26-Nov-01
10
0
-10
kPa
299
-20
-30
-40
-50
Figure 8.13c: Tung Chung monitoring data for SP7, @ 2.62m depth
01-Mar-01 31-Mar-01 30-Apr-01 30-May-01 29-Jun-01 29-Jul-01 28-Aug-01 27-Sep-01 27-Oct-01 26-Nov-01
0
-20
300
kPa
-40
-60
-80
-100
Figure 8.13d: Tung Chung monitoring data for SP8, @ 1.00m depth
01-Mar-01 31-Mar-01 30-Apr-01 30-May-01 29-Jun-01 29-Jul-01 28-Aug-01 27-Sep-01 27-Oct-01 26-Nov-01
10
0
301
kPa
-10
-20
-30
-40
-50
Figure 8.13e: Tung Chung monitoring data for SP9, @ 3.00m depth
01-Mar-01 31-Mar-01 30-Apr-01 30-May-01 29-Jun-01 29-Jul-01 28-Aug-01 27-Sep-01 27-Oct-01 26-Nov-01
0
-20
kPa
302
-40
-60
-80
-100
Figure 8.13f: Tung Chung monitoring data for SP10, @ 1.15m depth
01-Mar-01
40
31-Mar-01
30-Apr-01
30-May-01
29-Jun-01
29-Jul-01
28-Aug-01
27-Sep-01
30
20
10
303
Suction, kPa
0
-10
-20
-30
-40
-50
SP5 - Low (-1.53m)
SP4 - Low (-2.73m)
SP10 - Mid (-1.15m)
SP9 - Mid (-3.00m)
SP8 - High (-1.00m)
SP7 - High (-2.62m)
-60
Figure 8.14: Tung Chung monitoring data combined plot
27-Oct-01
26-Nov-01
1-Mar-01
31-Mar-01
30-Apr-01
30-May-01
29-Jun-01
29-Jul-01
28-Aug-01
27-Sep-01
27-Oct-01
26-Nov-01
10
0
304
kPa
-10
-20
-30
-40
-50
Figure 8.15a: Tung Chung monitoring data for SP1, @ 2.50m depth
01-Mar-01
31-Mar-01
30-Apr-01
30-May-01
29-Jun-01
29-Jul-01
28-Aug-01
27-Sep-01
27-Oct-01
10
0
305
kPa
-10
-20
-30
-40
-50
Figure 8.15b: Tung Chung monitoring data for SP2, @ 3.00m depth
26-Nov-01
01-Mar-01 31-Mar-01 30-Apr-01 30-May-01 29-Jun-01 29-Jul-01 28-Aug-01 27-Sep-01 27-Oct-01 26-Nov-01
10
piezometer flushed, no air
0
-10
kPa
306
-20
-30
-40
-50
-60
Figure 8.15c: Tung Chung monitoring data for SP3, @ 2.00m depth
01-Mar-01 31-Mar-01 30-Apr-01 30-May-01 29-Jun-01 29-Jul-01 28-Aug-01 27-Sep-01 27-Oct-01 26-Nov-01
10
pie zomete r flushed, no air
0
-10
307
kPa
missing
data
-20
-30
-40
-50
Figure 8.15d: Tung Chung monitoring data for SP6, @ 3.00m depth
3.5
3
2.5
308
head (m)
2
1.5
1
0.5
0
01-Mar-01
31-Mar-01
30-Apr-01
30-May-01
29-Jun-01
29-Jul-01
28-Aug-01
27-Sep-01
27-Oct-01
26-Nov-01
Figure 8.15e: Tung Chung monitoring data for TRL129, Standpipe at 16.6m depth
Date
01-Mar
31-Mar
30-Apr
30-May
29-Jun
29-Jul
28-Aug
27-Sep
27-Oct
26-Nov
27-Sep
27-Oct
26-Nov
27-Oct
26-Nov
30
Suction, kPa ( compressive PWP shown positive )
20
1.48m bgl ( CL )
10
2.66m bgl ( RL )
0
-10
-20
-30
-40
-50
-60
Figure 8.16a(i): ICFEP results for low section
Date
01-Mar
31-Mar
30-Apr
30-May
29-Jun
29-Jul
28-Aug
30
Suction, kPa ( compressive PWP shown positive )
20
1.04m bgl ( CM )
10
3.37m bgl ( RM )
0
-10
-20
-30
-40
-50
-60
Figure 8.16a(ii): ICFEP results for mid section
Date
01-Mar
31-Mar
30-Apr
30-May
29-Jun
29-Jul
28-Aug
27-Sep
30
Suction, kPa ( compressive PWP shown positive )
20
0.88m bgl ( CH )
10
2.27m bgl ( RH )
0
-10
-20
-30
-40
-50
-60
Figure 8.16a(iii): ICFEP results for high section
Figure 8.16a: ICFEP predictions for Run 1
309
Date
01-Mar
31-Mar
30-Apr
30-May
29-Jun
29-Jul
28-Aug
27-Sep
30
310
Suction, kPa ( compressive PWP shown positive )
20
1.48m bgl ( CL )
1.04m bgl ( CM )
0.88m bgl ( CH )
10
2.66m bgl ( RL )
3.37m bgl ( RM )
0
2.27m bgl ( RH )
-10
-20
-30
-40
-50
-60
Figure 8.16b: ICFEP pore pressure predictions for analysis run 1
27-Oct
26-Nov
Colluvium
Residual
soil
No significant flow within the
residual soil layer
eo
Ba s
fm
esh
Flow arrow scale: 0.5 metres/day
accumulated average flow
Figure 8.17: Accumulated flow velocities for Run 1,
increment 100 ( 8 June ).
311
Height above base of mesh, metres
8.09
7.28
6.47
5.66
4.85
4.05
3.24
2.43
10 June
1.62
15 June
20 June unchanged from 15 June
50.0
40.0
0.81
-10.0
-20.0
30.0
20.0
10.0
0.0
Pore water pressure ( compression negative ), kPa
-30.0
-40.0
Depth of section: 8.09m
Figure 8.18: PWP distribution with depth, Run 1, low section.
312
Height above base of mesh, metres
8.48
7.63
6.78
5.94
5.09
4.24
3.39
2.54
10 June
1.70
15 June
20 June unchanged from 15 June
50.0
40.0
0.85
20.0
30.0
10.0
0.0
-20.0
-10.0
Pore water pressure ( compression negative ), kPa
-30.0
-40.0
Depth of section: 8.48m
Figure 8.19: PWP distribution with depth, Run 1, mid section.
313
Height above base of mesh, metres
9.33
8.40
7.46
6.53
5.60
4.66
3.73
2.80
10 June
1.87
15 June
0.93
20 June
50.0
40.0
30.0
0.0
20.0
10.0
-10.0
-20.0
Pore water pressure ( compression negative ), kPa
-30.0
-40.0
Depth of section: 9.33m
Figure 8.20: PWP distribution with depth, Run 1, high section.
314
Date
01-Mar
31-Mar
30-Apr
30-May
29-Jun
29-Jul
28-Aug
27-Sep
27-Oct
26-Nov
27-Sep
27-Oct
26-Nov
27-Oct
26-Nov
10
1.48m bgl ( CL )
0
2.66m bgl ( RL )
Suction, kPa
-10
-20
-30
-40
-50
-60
Figure 8.21a(i): ICFEP results for low section
Date
01-Mar
31-Mar
30-Apr
30-May
29-Jun
29-Jul
28-Aug
10
1.04m bgl ( CM )
0
3.37m bgl ( RM )
Suction, kPa
-10
-20
-30
-40
-50
-60
Figure 8.21a(ii): ICFEP results for mid section
Date
01-Mar
31-Mar
30-Apr
30-May
29-Jun
29-Jul
28-Aug
27-Sep
10
0
0.88m bgl ( CH )
2.27m bgl ( RH )
Suction, kPa
-10
-20
-30
-40
-50
-60
Figure 8.21a(iii): ICFEP results for high section
Figure 8.21a: ICFEP predictions for Run 2
315
Date
01-Mar
31-Mar
30-Apr
30-May
29-Jun
29-Jul
28-Aug
27-Sep
10
1.48m bgl ( CL )
316
Suction, kPa ( Compressive PWP shown as positive )
1.04m bgl ( CM )
0
0.88m bgl ( CH )
2.66m bgl ( RL )
-10
3.37m bgl ( RM )
2.27m bgl ( RH )
-20
-30
-40
-50
-60
Figure 8.21b: ICFEP pore pressure predictions for analysis run 2
27-Oct
26-Nov
Height above base of mesh, metres
8.48
7.63
6.78
5.94
5.09
4.24
3.39
31 May
5 June
2.54
10 June
1.70
15 June
0.85
20 June
50.0
40.0
20.0
30.0
10.0
0.0
-20.0
-10.0
Pore water pressure ( compression negative ), kPa
-30.0
-40.0
Figure 8.22: PWP distribution with depth, Run 2, mid section.
317
Date
01-Mar
31-Mar
30-Apr
30-May
29-Jun
29-Jul
28-Aug
27-Sep
27-Oct
26-Nov
27-Sep
27-Oct
26-Nov
27-Oct
26-Nov
30
20
10
Suction, kPa
0
-10
-20
-30
1.48m bgl ( CL )
-40
-50
2.66m bgl ( RL )
-60
Figure 8.23a(i): ICFEP results for low section
Date
01-Mar
31-Mar
30-Apr
30-May
29-Jun
29-Jul
28-Aug
30
20
10
Suction, kPa
0
-10
-20
-30
1.04m bgl ( CM )
-40
-50
3.37m bgl ( RM )
-60
Figure 8.23a(ii): ICFEP results for mid section
Date
01-Mar
31-Mar
30-Apr
30-May
29-Jun
29-Jul
28-Aug
27-Sep
30
20
10
Suction, kPa
0
-10
-20
-30
-40
-50
0.88m bgl ( CH )
2.27m bgl ( RH )
-60
Figure 8.23a(iii): ICFEP results for high section
Figure 8.23a: ICFEP predictions for Run 3
318
Date
01-Mar
31-Mar
30-Apr
30-May
29-Jun
29-Jul
28-Aug
27-Sep
30
319
Suction, kPa ( Compressive PWP shown as positive )
20
10
0
-10
-20
-30
1.48m bgl ( CL )
-40
1.04m bgl ( CM )
0.88m bgl ( CH )
2.66m bgl ( RL )
-50
3.37m bgl ( RM )
2.27m bgl ( RH )
-60
Figure 8.23b: ICFEP pore pressure predictions for analysis run 3
27-Oct
26-Nov
Date
01-Mar
31-Mar
30-Apr
30-May
29-Jun
29-Jul
28-Aug
27-Sep
27-Oct
26-Nov
27-Sep
27-Oct
26-Nov
27-Oct
26-Nov
Suction, kPa ( Compressive PWP shown as positive )
20
10
1.48m bgl ( CL )
0
2.66m bgl ( RL )
-10
-20
-30
-40
-50
-60
Figure 8.24a(i): ICFEP results for low section
Date
01-Mar
31-Mar
30-Apr
30-May
29-Jun
29-Jul
28-Aug
Suction, kPa ( Compressive PWP shown as positive )
20
10
1.04m bgl ( CM )
0
3.37m bgl ( RM )
-10
-20
-30
-40
-50
-60
Figure 8.24a(ii): ICFEP results for mid section
Date
01-Mar
31-Mar
30-Apr
30-May
29-Jun
29-Jul
28-Aug
27-Sep
Suction, kPa ( Compressive PWP shown as positive )
20
10
0.88m bgl ( CH )
0
2.27m bgl ( RH )
-10
-20
-30
-40
-50
-60
Figure 8.24a(iii): ICFEP results for high section
Figure 8.24a: ICFEP predictions for Run 4
320
Date
01-Mar
31-Mar
30-Apr
30-May
29-Jun
29-Jul
28-Aug
27-Sep
20
321
Suction, kPa ( Compressive PWP shown as positive )
1.48m bgl ( CL )
10
1.04m bgl ( CM )
0.88m bgl ( CH )
0
2.66m bgl ( RL )
3.37m bgl ( RM )
2.27m bgl ( RH )
-10
-20
-30
-40
-50
-60
Figure 8.24b: ICFEP pore pressure predictions for analysis run 4
27-Oct
26-Nov
Da te
01-Mar
31-Mar
30-Apr
30-May
29-Jun
29-Jul
28-Aug
27-Sep
27-Oct
26-Nov
27-Sep
27-Oct
26-Nov
27-Oct
26-Nov
20
Su ction, kPa ( Co mpressive PWP is sho wn as po sitive )
CL : 1.48m bgl ( SP5 )
10
RL : 2.66m bgl ( SP4 )
0
-10
-20
-30
-40
-50
-60
Figure 8.25a(i): ICFEP results for low section
Date
01-Mar
31-Mar
30-Apr
30-May
29-Jun
29-Jul
28-Aug
Suction, kPa ( C ompressive PWP is shown as positive )
20
CM : 1.04m bgl ( SP10 )
10
RM : 3.37m bgl ( SP9 )
0
-10
-20
-30
-40
-50
-60
Figure 8.25a(ii): ICFEP results for mid section
Date
01-Mar
31-Mar
30-Apr
30-May
29-Jun
29-Jul
28-Aug
27-Sep
Suction, kPa ( C ompressive PWP is shown as positive )
20
10
0
CH : 0.88m bgl ( SP8 )
RH : 2.27m bgl ( SP7 )
-10
-20
-30
-40
-50
-60
Figure 8.25a(iii): ICFEP results for high section
Figure 8.25a: ICFEP predictions for Run 5
322
Date
01-Mar
31-Mar
30-Apr
30-May
29-Jun
29-Jul
28-Aug
27-Sep
20
CL : 1.48m bgl ( SP5 )
323
Suction, kPa ( Compressive PWP is shown as positive )
RL : 2.66m bgl ( SP4 )
10
CM : 1.04m bgl ( SP10 )
RM : 3.37m bgl ( SP9 )
0
CH : 0.88m bgl ( SP8 )
RH : 2.27m bgl ( SP7 )
-10
-20
-30
-40
-50
-60
Figure 8.25b: ICFEP pore pressure predictions for analysis run 5
27-Oct
26-Nov
Date
01-Mar
31-Mar
30-Apr
30-May
29-Jun
29-Jul
28-Aug
27-Sep
27-Oct
26-Nov
27-Sep
27-Oct
26-Nov
27-Oct
26-Nov
Suction, kPa ( C ompressive PWP is shown as positive )
20
CL : 1.48m bgl ( SP5 )
10
RL : 2.66m bgl ( SP4 )
0
-10
-20
-30
-40
-50
-60
Figure 8.26a(i): ICFEP results for low section
Da te
01-Mar
31-Mar
30-Apr
30-May
29-Jun
29-Jul
28-Aug
Su ction, kPa ( Co mpressive PWP is sho wn as po sitive )
20
CM : 1.04m bgl ( SP10 )
10
RM : 3.37m bgl ( SP9 )
0
-10
-20
-30
-40
-50
-60
Figure 8.26a(ii): ICFEP results for mid section
Da te
01-Mar
31-Mar
30-Apr
30-May
29-Jun
29-Jul
28-Aug
27-Sep
Su ction, kPa ( Co mpressive PWP is sho wn as po sitive )
20
10
0
CH : 0.88m bgl ( SP8 )
RH : 2.27m bgl ( SP7 )
-10
-20
-30
-40
-50
-60
Figure 8.26a(iii): ICFEP results for high section
Figure 8.26a: ICFEP predictions for Run 6
324
Date
01-Mar
31-Mar
30-Apr
30-May
29-Jun
29-Jul
28-Aug
27-Sep
325
Suction, kPa ( Compressive PWP is shown as positive )
20
10
0
-10
-20
-30
-40
CL : 1.48m bgl ( SP5 )
RL : 2.66m bgl ( SP4 )
CM : 1.04m bgl ( SP10 )
-50
RM : 3.37m bgl ( SP9 )
CH : 0.88m bgl ( SP8 )
RH : 2.27m bgl ( SP7 )
-60
Figure 8.26b: ICFEP pore pressure predictions for analysis run 6
27-Oct
26-Nov
Da te
01-Mar
31-Mar
30-Apr
30-May
29-Jun
29-Jul
28-Aug
27-Sep
27-Oct
26-Nov
27-Sep
27-Oct
26-Nov
27-Oct
26-Nov
20
PW P, kPa ( Comp ressive PW P sh own as p ositive )
10
0
-10
-20
-30
-40
CL : 1.48m bgl ( SP5 )
-50
RL : 2.66m bgl ( SP4 )
-60
Figure 8.27a(i): ICFEP results for low section
Da te
01-Mar
31-Mar
30-Apr
30-May
29-Jun
29-Jul
28-Aug
20
PW P, kPa ( Comp ressive PW P sh own as p ositive )
10
0
-10
-20
-30
-40
CM : 1.04m bgl ( SP10 )
-50
RM : 3.37m bgl ( SP9 )
-60
Figure 8.27a(ii): ICFEP results for mid section
Da te
01-Mar
31-Mar
30-Apr
30-May
29-Jun
29-Jul
28-Aug
27-Sep
20
PW P, kPa ( Comp ressive PW P sh own as p ositive )
10
0
-10
-20
-30
-40
CH : 0.88m bgl ( SP8 )
-50
RH : 2.27m bgl ( SP7 )
-60
Figure 8.27a(iii): ICFEP results for high section
Figure 8.27a: ICFEP predictions for Run 7
326
Date
327
PWP, kPa ( Compressive PWP shown as positive )
01-Mar
20
31-Mar
30-Apr
30-May
29-Jun
29-Jul
28-Aug
27-Sep
10
0
-10
-20
-30
CL : 1.48m bgl ( SP5 )
-40
RL : 2.66m bgl ( SP4 )
CM : 1.04m bgl ( SP10 )
RM : 3.37m bgl ( SP9 )
-50
CH : 0.88m bgl ( SP8 )
RH : 2.27m bgl ( SP7 )
-60
Figure 8.27b: ICFEP pore pressure predictions for analysis run 7
27-Oct
26-Nov
Date
01-Mar
31-Mar
30-Apr
30-May
29-Jun
29-Jul
28-Aug
27-Sep
20
328
PWP, kPa ( Compressive PWP shown as positive )
10
0
-10
-20
-30
CL : 1.48m bgl ( SP5 )
-40
RL : 2.66m bgl ( SP4 )
CM : 1.04m bgl ( SP10 )
RM : 3.37m bgl ( SP9 )
-50
CH : 0.88m bgl ( SP8 )
RH : 2.27m bgl ( SP7 )
-60
Figure 8.28: ICFEP pore pressure predictions for analysis run 7A
27-Oct
26-Nov
Da te
01-Mar
31-Mar
30-Apr
30-May
29-Jun
29-Jul
28-Aug
27-Sep
27-Oct
26-Nov
27-Sep
27-Oct
26-Nov
27-Oct
26-Nov
20
PW P, kPa ( Comp ressive PW P sh own as p ositive )
10
0
-10
-20
-30
-40
CL : 1.48m bgl ( SP5 )
-50
RL : 2.66m bgl ( SP4 )
-60
Figure 8.29a(i): ICFEP results for low section
Da te
01-Mar
31-Mar
30-Apr
30-May
29-Jun
29-Jul
28-Aug
20
PW P, kPa ( Comp ressive PW P sh own as p ositive )
10
0
-10
-20
-30
-40
CM : 1.04m bgl ( SP10 )
-50
RM : 3.37m bgl ( SP9 )
-60
Figure 8.29a(ii): ICFEP results for mid section
Da te
01-Mar
31-Mar
30-Apr
30-May
29-Jun
29-Jul
28-Aug
27-Sep
20
PW P, kPa ( Comp ressive PW P sh own as p ositive )
10
0
-10
-20
-30
-40
CH : 0.88m bgl ( SP8 )
-50
RH : 2.27m bgl ( SP7 )
-60
Figure 8.29a(iii): ICFEP results for high section
Figure 8.29a: ICFEP predictions for Run 8
329
Date
01-Mar
31-Mar
30-Apr
30-May
29-Jun
29-Jul
28-Aug
27-Sep
20
330
PWP, kPa ( Compressive PWP shown as positive )
10
0
-10
-20
-30
CL : 1.48m bgl ( SP5 )
-40
RL : 2.66m bgl ( SP4 )
CM : 1.04m bgl ( SP10 )
RM : 3.37m bgl ( SP9 )
-50
CH : 0.88m bgl ( SP8 )
RH : 2.27m bgl ( SP7 )
-60
Figure 8.29b: ICFEP pore pressure predictions for analysis run 8
27-Oct
26-Nov
Date
01-Mar
31-Mar
30-Apr
30-May
29-Jun
29-Jul
28-Aug
27-Sep
27-Oct
26-Nov
27-Sep
27-Oct
26-Nov
27-Oct
26-Nov
20
PWP, kPa ( Compression shown as positive )
10
0
-10
-20
-30
-40
CL : 1.48m bgl ( SP5 )
-50
RL : 2.66m bgl ( SP4 )
-60
Figure 8.30a(i): ICFEP results for low section
Date
01-Mar
31-Mar
30-Apr
30-May
29-Jun
29-Jul
28-Aug
20
PWP, kPa ( Compression shown as positive )
10
0
-10
-20
-30
-40
CM : 1.04m bgl ( SP10 )
-50
RM : 3.37m bgl ( SP9 )
-60
Figure 8.30a(ii): ICFEP results for mid section
Date
01-Mar
31-Mar
30-Apr
30-May
29-Jun
29-Jul
28-Aug
27-Sep
20
PWP, kPa ( Compression shown as positive )
10
0
-10
-20
-30
-40
CH : 0.88m bgl ( SP8 )
-50
RH : 2.27m bgl ( SP7 )
-60
Figure 8.30a(iii): ICFEP results for high section
Figure 8.30a: ICFEP predictions for Run 9
331
Date
01-Mar
31-Mar
30-Apr
30-May
29-Jun
29-Jul
28-Aug
27-Sep
20
332
PWP, kPa ( Compression shown as positive )
10
0
-10
-20
-30
CL : 1.48m bgl ( SP5 )
-40
RL : 2.66m bgl ( SP4 )
CM : 1.04m bgl ( SP10 )
-50
RM : 3.37m bgl ( SP9 )
CH : 0.88m bgl ( SP8 )
RH : 2.27m bgl ( SP7 )
-60
Figure 8.30b: ICFEP pore pressure predictions for analysis run 9
27-Oct
26-Nov
Date
01-Mar
31-Mar
30-Apr
30-May
29-Jun
29-Jul
28-Aug
27-Sep
27-Oct
26-Nov
27-Sep
27-Oct
26-Nov
27-Oct
26-Nov
20
PWP, kPa ( Compressive PWP shown as positive )
10
0
-10
-20
-30
-40
CL : 1.48m bgl ( SP5 )
-50
RL : 2.66m bgl ( SP4 )
-60
Figure 8.31a(i): ICFEP results for low section
Date
01-Mar
20
31-Mar
30-Apr
30-May
29-Jun
29-Jul
28-Aug
PW P , k P a ( C o m p re ss iv e P W P sh o w n a s p o s it iv e )
10
0
-10
-20
-30
-40
CM : 1.04m bgl ( SP10 )
-50
RM : 3.37m bgl ( SP9 )
-60
Figure 8.31a(ii): ICFEP results for mid section
Date
01-Mar
20
31-Mar
30-Apr
30-May
29-Jun
29-Jul
28-A ug
27-Sep
P W P , k P a ( C o m p re s s iv e PW P s h o w n as p o s itiv e )
10
0
-10
-20
-30
-40
CH : 0.88m bgl ( SP8 )
-50
RH : 2.27m bgl ( SP7 )
-60
Figure 8.31a(iii): ICFEP results for high section
Figure 8.31a: ICFEP predictions for Run 10
333
Date
01-Mar
20
31-Mar
30-Apr
30-May
29-Jun
29-Jul
28-Aug
27-Sep
334
PWP, kPa ( Compressive PWP shown as positive )
10
0
-10
-20
-30
CL : 1.48m bgl ( SP5 )
-40
-50
RL : 2.66m bgl ( SP4 )
CM : 1.04m bgl ( SP10 )
RM : 3.37m bgl ( SP9 )
CH : 0.88m bgl ( SP8 )
RH : 2.27m bgl ( SP7 )
-60
Figure 8.31b: ICFEP pore pressure predictions for analysis run 10
27-Oct
26-Nov
Date
01-Mar
31-Mar
30-Apr
30-May
29-Jun
29-Jul
28-Aug
27-Sep
27-Oct
26-Nov
27-Sep
27-Oct
26-Nov
27-Oct
26-Nov
20
PWP, kPa ( Compressive PWP shown as positive )
10
0
-10
-20
-30
-40
CL : 1.48m bgl ( SP5 )
-50
RL : 2.66m bgl ( SP4 )
-60
Figure 8.32a(i): ICFEP results for low section
Date
01-Mar
20
31-Mar
30-Apr
30-May
29-Jun
29-Jul
28-Aug
PW P , k P a ( C o m p re ss iv e P W P sh o w n a s p o s it iv e )
10
0
-10
-20
-30
-40
CM : 1.04m bgl ( SP10 )
-50
RM : 3.37m bgl ( SP9 )
-60
Figure 8.32a(ii): ICFEP results for mid section
Date
01-Mar
20
31-Mar
30-Apr
30-May
29-Jun
29-Jul
28-A ug
27-Sep
P W P , k P a ( C o m p re s s iv e PW P s h o w n as p o s itiv e )
10
0
-10
-20
-30
-40
CH : 0.88m bgl ( SP8 )
-50
RH : 2.27m bgl ( SP7 )
-60
Figure 8.32a(iii): ICFEP results for high section
Figure 8.32a: ICFEP predictions for Run 11
335
Date
01-Mar
31-Mar
30-Apr
30-May
29-Jun
29-Jul
28-Aug
27-Sep
20
336
PWP, kPa ( Compressive PWP shown as positive )
10
0
-10
-20
-30
CL : 1.48m bgl ( SP5 )
-40
RL : 2.66m bgl ( SP4 )
CM : 1.04m bgl ( SP10 )
-50
RM : 3.37m bgl ( SP9 )
CH : 0.88m bgl ( SP8 )
RH : 2.27m bgl ( SP7 )
-60
Figure 8.32b: ICFEP pore pressure predictions for analysis run 11
27-Oct
26-Nov
Date
01-Mar
31-Mar
30-Apr
30-May
29-Jun
29-Jul
28-Aug
27-Sep
27-Oct
26-Nov
27-Sep
27-Oct
26-Nov
27-Oct
26-Nov
PWP, kPa ( Compressive pore pressures shown as positive )
30
20
10
0
-10
-20
-30
CL : 1.48m bgl ( SP5 )
-40
-50
RL : 2.66m bgl ( SP4 )
-60
Figure 8.33a(i): ICFEP results for low section
Date
01-Mar
31-Mar
30-Apr
30-May
29-Jun
29-Jul
28-Aug
PWP, kPa ( Compressive pore pressures shown as positive )
30
20
10
0
-10
-20
-30
CM : 1.04m bgl ( SP10 )
-40
-50
RM : 3.37m bgl ( SP9 )
-60
Figure 8.33a(ii): ICFEP results for mid section
Date
01-Mar
31-Mar
30-Apr
30-May
29-Jun
29-Jul
28-Aug
27-Sep
PWP, kPa ( Compressive pore pressures shown as positive )
30
20
10
0
-10
-20
-30
-40
-50
CH : 0.88m bgl ( SP8 )
RH : 2.27m bgl ( SP7 )
-60
Figure 8.33a(iii): ICFEP results for high section
Figure 8.33a: ICFEP predictions for Run 12
337
Date
01-Mar
31-Mar
30-Apr
30-May
29-Jun
29-Jul
28-Aug
27-Sep
338
PWP, kPa ( Compressive pore pressures shown as positive )
30
20
10
0
-10
-20
-30
CL : 1.48m bgl ( SP5 )
-40
RL : 2.66m bgl ( SP4 )
CM : 1.04m bgl ( SP10 )
RM : 3.37m bgl ( SP9 )
-50
CH : 0.88m bgl ( SP8 )
RH : 2.27m bgl ( SP7 )
-60
Figure 8.33b: ICFEP pore pressure predictions for analysis run 12
27-Oct
26-Nov
Tung Chung East - Hong Kong Natural Terrain Study 2001
SP4 @ 2.73m
01/03/2001
31/03/2001
30/04/2001
30/05/2001
29/06/2001
29/07/2001
28/08/2001
27/09/2001
27/10/2001
26/11/2001
30
20
10
kPa
339
0
-10
-20
ICFEP prediction, Run 12 ( RL: 2.66m bgl )
-30
-40
Figure 8.34a: Tung Chung monitoring data for SP4, @ 2.73m depth,
with ICFEP Run 12 prediction
Tung Chung East - Hong Kong Natural Terrain Study 2001
SP5 @ 1.53m
01/03/2001
31/03/2001
30/04/2001
30/05/2001
29/06/2001
29/07/2001
28/08/2001
27/09/2001
27/10/2001
20
10
0
kPa
340
-10
-20
ICFEP prediction, Run 12 ( CL 1.48m bgl )
-30
-40
-50
Figure 8.34b: Tung Chung monitoring data for SP5, @ 1.53m depth,
with ICFEP Run 12 prediction
26/11/2001
Tung Chung East - Hong Kong Natural Terrain Study 2001
SP7 @ 2.62m
01/03/2001
31/03/2001
30/04/2001
30/05/2001
29/06/2001
29/07/2001
28/08/2001
27/09/2001
27/10/2001
10
0
341
kPa
-10
-20
-30
ICFEP prediction, Run 12 ( RH : 2.27m bgl )
-40
-50
Figure 8.34c: Tung Chung monitoring data for SP7, @ 2.62m depth,
with ICFEP Run 12 predictions.
26/11/2001
Tung Chung East - Hong Kong Natural Terrain Study 2001
SP8 @ 1.00m
01/03/2001
31/03/2001
30/04/2001
30/05/2001
29/06/2001
29/07/2001
28/08/2001
27/09/2001
27/10/2001
0
-20
kPa
342
-40
ICFEP prediction, Run 12 ( CH : 0.88m bgl )
-60
-80
-100
Figure 8.34d: Tung Chung monitoring data for SP8, @ 1.00m depth,
with ICFEP Run 12 predictions.
26/11/2001
Tung Chung East - Hong Kong Natural Terrain Study 2001
SP9 @ 3.00m
01/03/2001
31/03/2001
30/04/2001
30/05/2001
29/06/2001
29/07/2001
28/08/2001
27/09/2001
27/10/2001
30
20
10
kPa
343
0
-10
-20
ICFEP prediction, Run 12 ( RM : 3.37m bgl )
-30
Figure 8.34e: Tung Chung monitoring data for SP9, @ 3.00m depth,
with ICFEP Run12 predictions.
26/11/2001
Tung Chung East - Hong Kong Natural Terrain Study 2001
SP10 @ 1.15m
01/03/2001
31/03/2001
30/04/2001
30/05/2001
29/06/2001
29/07/2001
28/08/2001
27/09/2001
27/10/2001
0
-20
kPa
344
-40
ICFEP prediction, Run 12 ( CM : 1.04m bgl )
-60
-80
-100
Figure 8.34f: Tung Chung monitoring data for SP10, @ 1.15m depth,
with ICFEP Run 12 predictions.
26/11/2001
Date
01-Mar
31-Mar
30-Apr
30-May
29-Jun
29-Jul
28-Aug
27-Sep
27-Oct
26-Nov
27-Sep
27-Oct
26-Nov
27-Oct
26-Nov
PWP, kPa ( Com pressive po re p ressu res sh own as po sitive )
30
20
10
0
-10
-20
-30
-40
CL : 1.48m bgl ( SP5 )
-50
RL : 2.66m bgl ( SP4 )
-60
Figure 8.35a(i): ICFEP results for low section
Date
01-Mar
31-Mar
30-Apr
30-May
29-Jun
29-Jul
28-Aug
PWP, kPa ( Com pressive po re p ressu res sh own as po sitive )
30
20
10
0
-10
-20
-30
-40
CM : 1.04m bgl ( SP10 )
-50
RM : 3.37m bgl ( SP9 )
-60
Figure 8.35a(ii): ICFEP results for mid section
Date
01-Mar
31-Mar
30-Apr
30-May
29-Jun
29-Jul
28-Aug
27-Sep
PWP, kPa ( Com pressive pore pressures sh own as positive )
30
20
10
0
-10
-20
-30
CH : 0.88m bgl ( SP8 )
-40
-50
RH : 2.27m bgl ( SP7 )
-60
Figure 8.35a(iii): ICFEP results for high section
Figure 8.35a: ICFEP predictions for Run 13(7)
345
Date
01-Mar
31-Mar
30-Apr
30-May
29-Jun
29-Jul
28-Aug
27-Sep
27-Oct
346
PWP, kPa ( Compressive pore pressures shown as positive )
30
20
10
0
-10
-20
-30
CL : 1.48m bgl ( SP5 )
RL : 2.66m bgl ( SP4 )
-40
CM : 1.04m bgl ( SP10 )
RM : 3.37m bgl ( SP9 )
CH : 0.88m bgl ( SP8 )
-50
RH : 2.27m bgl ( SP7 )
-60
Figure 8.35b: ICFEP pore pressure predictions for analysis run 13(7)
26-Nov
CHAPTER 9
Conclusions and Recommendations
9.1. Introduction
Human activity inevitably results in people living, travelling and working adjacent to sloping
ground. Efficient use of land, particularly in locations with limited level ground such as Hong
Kong, require that in many cases such slopes are man-made, or at least artificially steepened.
The stability of such slopes, whether natural or man-made, is vital, since failure can lead to loss
of life and severe disruption to the economy of the region or even the country affected.
The issue of slope stability is a core subject within the field of soil mechanics, and is addressed
in even the most basic of textbooks. However, the theories governing it, like most ‘established’
soil mechanics, are based on the assumption that the soil is fully saturated.
However, over much of the world’s surface, particularly in regions of tropical climate, the soil is
often unsaturated, at least for some part of the year. Tropical regions are also subject to
rainstorms that can be brief but extreme in rainfall rate, as illustrated by the rainfall recorded at
Tung Chung, Hong Kong, in 2001 ( presented in Chapter 8, Table 8-2 ). Further, many of the
nations that lie within the tropical regions of the globe are developing nations, which can illafford the cost and disruption that may follow a large-scale slope failure.
Attempts to use saturated soil mechanics based techniques to analyse slope stability problems in
unsaturated soils have typically failed to accurately reproduce the failure, even when advanced
numerical modelling methods have been used ( for example, Pak Kong ).
It is clear then, that the approach required to address the issue of stability of slopes composed of
unsaturated soils cannot simply be the same as for saturated soils, but must be specific to the
unsaturated case.
Since failure of unsaturated slopes is most closely associated with heavy rainfall, the first step in
understanding and accurately modelling the slope-stability process in unsaturated material is to
obtain an understanding of the infiltration and flow patterns that occur in such soils under
typical rain events.
Chapter 9
347
It is this first step that this thesis has sort to address.
9.2. The nature of unsaturated soil
Chapter 2 considered the nature of unsaturated soil, and defined some of the basic terms used
when discussing its behaviour.
The two different components of soil suction, osmotic and capillary ( or matric ), were
described, and it was established that in most situations involving rainfall infiltration the
osmotic element is negligible. Further details were then given of the processes that govern the
generation of capillary suction.
This aspect was extended to detail the relationship between the ( matric ) suction in the soil and
the water content of the soil ( whether given as volumetric water content or degree of
saturation ), a relationship which may be shown graphically through the Soil Water
Characteristic Curve for the soil. Consideration was also given to the variation in permeability
of the soil as its water content ( or degree of saturation ) varies.
It was shown that the relationship between water content and suction is not unique, but is
actually hysteretic, depending on whether the soil is undergoing a wetting or drying action, and
the physical causes of this phenomenon were presented.
Finally, consideration was given to the air phase within an unsaturated soil. The behaviour of
the air phase was discussed, and it was established that in most cases it is not necessary to
explicitly model the flow of air within the soil, nor attempt to predict the air pressure. Rather,
the air may be assumed to be at zero pressure ( relative to atmosphere ), and be free to flow,
without significantly reducing the accuracy of the predicted pore water flow or pore water
pressure distribution.
Such an assumption considerably simplifies any attempt to model unsaturated soil behaviour,
since only the flow of pore water need now be reproduced. Any model developed to predict
unsaturated soil behaviour needs only be a two-phase model, rather than three-phase. For a
numerical model, this makes the development of the model more straightforward, and by
reducing the number of unknown variables for which values must be determined, also reduces
the computational effort required to reach a solution.
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348
9.3. The current state of the art
Chapter 3 provided a survey of the literature published that deal with rainfall infiltration into
unsaturated soil, and the concept of the wetting front was discussed. Consideration was then
given to the effects of antecedent rainfall, as argued by various other authors.
The basic theory governing flow in an unsaturated soil was discussed, and it was shown that it is
generally accepted that Darcy’s Law ( as established for saturated soil ) applies equally to the
unsaturated case. Equations governing the continuity of flow were presented, including that
known as the Richard’s Equation. From these equations it could be seen that the continuity of
flow within an unsaturated soil is largely the same process as in a saturated soil. However, it
must also allow for a variable permeability that is a function of suction ( pore pressure ), and
also for the variable volume of water that may be stored within an element of unsaturated soil.
The suction dependent permeability was then discussed in more detail, with reference being
made to some of the published relationships. However, more consideration was given to the
behaviour of the soil at very high suctions, and the effects this has on permeability. It was
established that for practical purposes, it is reasonable to assume that at or beyond the residual
degree of saturation, permeability is constant ( and very low ). Any vapour phase moisture
movement is assumed to be incorporated within the specified permeability rate, so there is no
need to specifically model this mode of moisture transport.
Following on from the discussion of permeability within unsaturated soil, the stress-state
variables applicable, which are required to enable fully coupled behaviour to be modelled, were
considered. Thereafter, the relationship between suction and shear strength of the soil, as
presented by others in the literature, was reviewed. A number of different published suctionstrength relationships were presented and discussed, but without a preference being established
for any of them.
A detailed discussion was then presented of the previous attempts by others to utilise numerical
modelling techniques to analyse the infiltration process into unsaturated soil. It was shown that,
while many attempts have been made, the majority are restricted to a rigid-soil case. Those few
that have attempted some form of coupled analysis have generally had to make assumptions that
leave their work flawed in some way.
It is clear from the literature review that this project is attempting to advance the understanding
of unsaturated soils, and significantly improve the existing capability to model their behaviour.
Chapter 9
349
Chapter 3 concluded with a brief consideration of the slope failure at Pak Kong, in Hong Kong.
This slope was previously analysed using ICFEP, but this analysis was undertaken using a
saturated soil model, and it failed to accurately recreate the failure that was observed at the site.
It was this illustration of the limitations of the existing modelling capability that largely inspired
this project.
9.4. ICFEP version 8.0
Prior to the commencement of this project, the Imperial College Finite Element Program,
ICFEP, existed in its version 8.0 form. Chapter 4 presented the capabilities and limitations of
this version of the program.
Broadly, ICFEP v8.0 provided a capability to undertake fully coupled finite element analysis
using a range of saturated soil behavioural models.
Despite the limitation of only having a fully saturated capability, an option for applying a
suction-dependent permeability model ( the ‘suction switch’ ) existed, and this matched the
form of this relationship suggested by some authors as being applicable to unsaturated soil.
A variety of boundary conditions were also available, most notably a precipitation boundary
condition that allows for rainfall induced infiltration, and can automatically adjust the inflow
rate into the soil based on the maximum permitted depth of surface ponding. A potential
problem with the functioning of this boundary condition was identified, the solution to which
was described later in Chapter 6.
A study was made of the effects of rainfall into a column of saturated soil. It was found that
without the suction switch, if the rainfall rate was relatively low, pore tensions were slowly
reduced, but the most tensile pore pressure remained at the column surface.
Higher rainfall rates caused complete loss of pore water tension at the surface, reducing the
pressure here to the threshold value specified on the precipitation boundary condition, while
leaving more tensile pressures present deeper in the soil. However, thereafter there was a
general increase in the compressive nature of the pore pressures throughout the column.
There was no distinct wetting front below which the pore pressures remained unchanged by the
infiltration process.
Introducing the suction switch changed this behaviour, and produced a form of wetting front.
Even though the entire column was saturated, the effect of suction-dependent permeability was
to restrict the rate of downward percolation. Rainfall entered into the column surface then
tended to ‘perch’ in the upper layer of soil, only slowly penetrating deeper into the column. As
Chapter 9
350
it did, the pore pressures increased ( became more compressive ) at a deeper level, so increasing
permeability, allowing the water to flow a little deeper.
Moreover, without the suction switch, if the rainfall rate were sufficiently high to cause
response in the pore pressure ( meaning an inflow rate greater than the permeability ) then given
sufficient time, all pore water tension within the column would be destroyed ( assuming the
threshold value for the precipitation card were set at 0 kPa ).
With the suction switch present, it was found that some tensile pore pressures tended to remain
in the column above the wetting front. Because the permeability was a function of the ( tensile )
pore pressure, it was possible for the rainfall rate to exceed the surface permeability while being
less than the maximum permeability. In such a case, the infiltration would cause a compressive
change in the surface pore water pressure, until the permeability associated with that pressure
( on the suction switch ) equalled the infiltration rate. At this point a steady state condition
would be achieved, and no further reduction in the tensile pressures would occur.
With no suction switch present, there is only a single permeability. The inflow rate is either
insufficient to cause a change in the pore water pressure distribution, or exceeds the maximum
permeability so increases the pressure until a steady state hydrostatic profile, consistent with the
specified threshold value at the surface, is achieved.
Thus it was shown that some elements of expected unsaturated behaviour, such as the
development of a wetting front and the possibility that suctions can be retained at a reduced
magnitude in the near surface soil during infiltration, can be predicted even with a saturated soil
model.
Part of the column study involved a consideration of element size sensitivity. It was found that
the choice of element size did have some effect on the predicted pore pressure profiles resulting
from infiltration, and that where significant changes in the profile were expected, a maximum
element thickness of 1m was recommended.
The column analysis was then extended to a two-dimensional case of a slope. This was
undertaken using the modified precipitation boundary condition ( that resolved the previously
encountered problem ), and the suction switch.
This two-dimensional saturated analysis confirmed the tendency for a wetting front to occur,
with the possibility of suctions being reduced in magnitude but not completely destroyed at the
soil surface.
The two dimensional nature of the situation was reflected in the predicted pore pressure
distributions, which implied flow occurring partly in a ‘down slope’ manner, not just vertically.
This would be consistent with the pore pressures in the surface soil becoming less tensile,
Chapter 9
351
resulting in a surface layer of soil becoming more permeable. Lateral flow would then be easier
than purely vertical flow.
Increasing the ratio between maximum and minimum permeabilities on the suction switch
tended to produce a sharper but shallower wetting front. Conversely, increasing the range of
suction over which the suction switch operated tended to reduce the rate of change of
permeability with suction. The larger the suction range applied, the closer the soil came to
behaving as if it has a uniform permeability, and the wetting front became increasingly
indistinct.
It was shown by the work presented in Chapter 4 that the existence of a form of wetting front is
possible even when the soil is fully saturated, since its formation depends on the presence of a
variable permeability within the soil that responds to changes in pore pressure. The actual
behaviour observed is dependent on the manner in which permeability varies with suction, and
also on the applied infiltration rate.
9.5. The development of an unsaturated soil model
Chapter 4 concluded with the statement that a true unsaturated approach must be developed if
infiltration into an unsaturated soil slope were to be modelled accurately. In Chapter 5, such an
approach was developed.
The constitutive relations governing the behaviour for both the soil structure and the water
phase were presented. The theoretical development work indicated that three new parameters
are required to reflect unsaturated soil behaviour. One of these is the gradient of the Soil Water
Characteristic Curve ( linking water volume change to the change in suction ), the second relates
water volume content within the soil to total stress, while the third ( the ‘H’ parameter ) links
the deformation of the soil structure to matric suction.
It was shown that previous attempts by other authors to apply these relationships tended to make
the assumption that these last two parameters were identical to each other. However, this
assumption was based on a specific case, and it is not sound generally. The approach taken
within this thesis therefore presented a more accurate method of modelling unsaturated
behaviour.
The manner in which the theoretical relationships could be converted into the necessary finite
element formulation was demonstrated, at which point a new parameter, Ω, was introduced.
This parameter incorporated the water content-total stress parameter. The advantage of this
approach was that limit values of 0.0 and 1.0 exist for Ω at suction values that may be readily
determined from the soil’s SWCC, whereas the parameter Ew that relates water volume content
to total stress is not a property of the soil that is generally known.
Chapter 9
352
To assist in visualising the behaviour of the real soil that this theory seeks to model, a
conceptual model was developed and presented. This utilised a series of behavioural zones into
which the soil would fall, depending on the current suction within the soil, relative to that soil’s
SWCC.
Each of the new unsaturated aspects of the finite element formulation was examined in turn, and
the physical processes that they reflected were detailed. In this way it was clearly shown that the
theoretical model presented was a true reflection of the real behaviour of an unsaturated soil.
It was through developing the conceptual model that the parameter Ω was developed, and
through which the advantage of using this substitute parameter was demonstrated.
In developing the conceptual model, the possible variation of Ω was discussed, and in doing so,
its limit values of 0.0 to 1.0 were established.
It was shown how this parameter is in fact the fundamental controlling factor in governing flow
into / out of an unsaturated soil element. Consideration was given to previous work by others,
but it was found that no other author had accurately reproduced this aspect of unsaturated soil
behaviour.
Chapter 5 terminated with a discussion on the use of linear elasticity.
The theory developed to model unsaturated soil was initially based on the soil behaving linear
elastic. Clearly, in reality unsaturated soil behaviour is non-linear. It was shown, however, that
it is possible to extend the linear elastic based theory to account for non-linear behaviour with
reasonable accuracy. By using small increments of loading, this process can be extended such
that the non-linear behaviour of a real-world prototype may be predicted with good accuracy.
9.6. Unsaturated soil modelling using ICFEP
Having established the constitutive equations governing the behaviour of unsaturated soil,
Chapter 6 demonstrated the models that were developed to enable ICFEP to apply these
constitutive equations.
The existing, relatively basic, three-dimensional Mohr Coulomb model for saturated soil was
adopted as the basis of the non-linear material properties model. This was modified for use as an
unsaturated model by including within its modified form a simple distribution with suction of
the H parameter ( this being the parameter that relates soil deformation to matric suction ).
The Soil Water Characteristic Curve provided an all new input model which from its gradient
allows for the change in stored water within the soil as a function of matric suction.
Chapter 9
353
Two forms of SWCC were developed, the first being a ‘quad-linear’, very simplified, form used
for early development work only. Problems with numerical convergence were encountered in
some instances when using this, so when the second form of SWCC ( developed by a colleague
within the soil mechanics section ) became available, this later version was preferred.
Both curves are non-hysteretic, while the improved second form was based on the equations of
Van Genuchten. Further work is underway by others in the Imperial College Soils section to
develop this into a fully hysteretic curve.
The SWCC input module also incorporated the data controlling the variation of Ω with suction.
Currently, this is input only in terms of the suctions that define the two limit values, with a
linear variation assumed between these points.
The behaviour that the Ω parameter is reflecting does not appear to have been investigated to
any degree by any other authors. This is largely a reflection of the fact that much if not most
research into unsaturated soils to date has assumed a rigid soil.
However, in general, accurate modelling of engineering problems involving unsaturated soils
requires that the soil be modelled fully coupled, and it is therefore necessary that further
research be undertaken to determine the true nature of the Ω parameter and how it varies.
The SWCC as used was formulated in terms of matric suction against degree of saturation, as
opposed to volumetric water content. The use of degree of saturation rather than volumetric
water content was chosen since at zero suction ( full saturation ) the starting point of the curve is
known ( S = 100% ). ( With the current SWCC models in ICFEP, S=100% also extends to the
AEV suction ). If the curve were formulated in terms of volumetric water content, then this
value would be equal to the porosity of the soil at full saturation, so would vary according to
soil type and applied stress. Further, S = 100% is an exact, known value, whereas the porosity of
the soil, although possible to measure, is not always known, and any value determined for it will
be subject to experimental error.
In discussing the relative merits of using degree of saturation or volumetric water content for the
SWCC, it was shown that there is not a unique SWCC for any given soil that can be plotted
either in terms of saturation or water content.
Rather, the behaviour shown by the SWCC is also sensitive to stress changes, since a change in
stress will typically cause a change in the voids ratio ( porosity ) of the soil. It is possible that a
stress change could leave the degree of saturation unchanged. Thus a constant degree of
saturation would imply no change in suction, but the same constant degree of saturation
combined with a change in porosity gives a change in volumetric water content, which indicates
that a change in suction should occur.
Chapter 9
354
From this, it was determined that for any soil there are actually a series of nested SWCCs, each
for a different stress, which together form a Soil Water Characteristic Surface ( SWCS ).
It is evident, however, that there has been little work previously published on this aspect of
behaviour. Given the current lack of data, the approach taken within this project was to accept
the inaccurate premise of the SWCC, and this was accordingly the basis of the ICFEP SWCC
module. Within an ICFEP analysis, stress dependency in the SWCC can be broadly modelled by
reproducing the situation to be modelled with a mesh containing multiple soil layers, each
representing a narrow range of depth ( stress ), and assigning a unique SWCC to each layer.
Nevertheless, it is clear that continued use of ‘the SWCC’ is a fundamentally incorrect
approach. There is a corresponding need for further research to enable the use of ‘the SWCS’ to
become first practical, and then standard.
The addition of a new unsaturated data module and a non-linear soil material property model
did not require modification to the existing pressure-dependent permeability model ( the suction
switch ). However, it was recognised that the permeability-suction relationship is also
hysteretic, which is not currently reproduced. Development of a hysteretic SWCC ( or SWCS )
would then generate a requirement to develop a hysteretic suction switch.
The development of the unsaturated capability within ICFEP also included the ability for the
program to automatically adjust the mass of a soil element as the degree of saturation of that
element varies. It is believed that ICFEP is unique in having this ability.
The case study examined, of Tung Chung, suggests that the importance of this element density
variation is not great, since it did not appear to have any significant effect for that site. However,
this capability reflects a real physical process that is without doubt occurring, and the ability to
reproduce it may prove important more generally. The full implications of density change due to
changes in degree of saturation were probably not revealed by the Tung Chung study, since no
movement of the slope occurred, either in the analyses or in the real prototype.
As mentioned earlier, the existing precipitation boundary condition within ICFEP was found to
have a limitation in its operation. It was found that the switch from an inflow condition to a
constant pressure boundary condition was not modelled with sufficient accuracy for the needs of
this project. Accordingly, this was addressed during the formulation of the unsaturated modules
for ICFEP.
An automatic incrementation process was already available within ICFEP for application to
load-displacement problems, and as part of this project, this AI process was developed further to
apply to pore pressures as well. In summary the AI process instructs ICFEP to repeat any
increment of the analysis during which numerical convergence could not be obtained, or where
Chapter 9
355
the predicted displacements are too far from the true load-displacement curve, but to do so using
a series of smaller sub-increments.
This was coupled with a modification to the precipitation boundary condition, which subtly
adjusted the manner in which this boundary condition was applied. The result ensured that the
precipitation boundary condition produced realistic pore pressure distributions.
Further, the application of the AI procedure to pore pressures proved vital during the case study,
where the extremely non-linear permeabilities encountered would have rendered the analysis
unsolvable unless very small time increments were applied. The AI procedure effectively
applies such very small time increments automatically, but only at those parts of the analysis
that require this.
However, while the AI was effective in its operation, the case study did reveal a problem. It was
found that the duration of time required for the analysis tended to be very large. This occurred
because with the type of highly non-linear problem being dealt with, the AI tends to break most
increments of analysis down into a series of very small sub-increments. The incremental
changes in flow, pore pressures etc in each of these sub-increments are therefore also very
small. ICFEP then tries to obtain numerical convergence of the residuals based on a small
percentage ( default: 2% ) of these incremental changes, which are correspondingly small
values, and which tend to require much computer effort to achieve.
This brought into question the nature of the convergence tolerances used within ICFEP. Since
the current form of the tolerances had been well used and was therefore functional and reliable
( if time consuming, when used with the AI and highly non-linear problems ), no attempt to
revise the operation of the tolerances was made within this project. However, this aspect of the
program’s operation is under review.
During the work to modify the precipitation boundary condition it also became apparent that
this boundary condition could be used to represent a base ‘recharge’ condition, thus providing a
less artificially restricted pore pressure / flow boundary to the sides and base of the analysis
mesh.
9.7. Valedictory exercises
Having introduced new modules and material property models into ICFEP, it was necessary to
prove that they reliably and accurately modelled the behaviour of the soil.
The limited theoretical work available forced the validation exercises to consist firstly of flow
( water volume ) balance and mass balance exercises using a 1m cuboid of soil. The exercises
presented demonstrated that ICFEP was correctly maintaining continuity of flow and stress, and
Chapter 9
356
that associated pore pressure distributions ( whether tensile or compressive ) were sensible and
matched expectations.
The second method by which the reliability of the modifications to ICFEP were proven was by
matching ICFEP predictions against those from the only reasonable published attempt at
coupled analysis that could be located. It was shown that the published work made some
assumptions that were not necessarily correct, and that the attempt to repeat the work using
ICFEP was not definitely an exact repeat of the process used. However, the ICFEP analysis was
still able to produce results that closely resembled the published work.
These two sets of exercises together were taken as a good indication that the modifications to
ICFEP did result in the program producing reasonably accurate predictions of the pore pressure
variation in a ( coupled/deformable ) unsaturated soil.
9.8. The Tung Chung case study
With ICFEP now modified to undertake unsaturated analyses, the program was used to carry out
the core aim of this project: to investigate the effects of infiltration into an unsaturated soil
slope.
The Tung Chung slope in Hong Kong was selected, since it had been instrumented throughout
the wet season of 2001, and so good data existed for the variation in pore pressures / suctions
within the slope.
The results of the analyses demonstrated that the predicted pore pressure response is highly
dependent on the unsaturated soil parameters selected. In particular, the choice of SWCC, fully
saturated permeability and suction switch parameters has a great influence on the analyses
results. Conversely, as long as the mesh used is large enough to ensure that the points of
principal interest are not close to them, the side and base boundary conditions appear to have a
limited effect.
Similarly, the initial voids ratio seemed to be fairly insignificant. Further, it was noted also that
the effects of the initial pore pressure were mostly limited to the first few increments of the
simulation, after which the applied precipitation coupled with normal drainage processes tended
to govern the pore pressure responses. However, the initial pore water pressures were selected to
be consistent with the pore pressure boundary conditions, and the choice of the base boundary
condition may have limited the range to which suctions were able to develop at a few points in
the analyses.
A further point that was clearly demonstrated is that the predicted pore responses are highly
dependent on the nature of the rainfall distribution assumed. The response depends not just on
how much rain falls, but also on how quickly it falls. It was shown that a long duration ( 24
Chapter 9
357
hour ) storm may produce a level of response that will not occur if the whole rainfall occurs in a
much shorter time ( 15 minutes ).
The results predicted for the short duration rainfall showed similarities to those obtained when
the normal daylong rainfall was applied with suction switch parameters giving very low
permeabilities at small magnitude suctions. In both these cases, the soil surface was seen to fully
saturate very rapidly, leading to most of the rainfall being shed as surface runoff.
This appears to be one of the most significant variables in the behaviour of the infiltration
process. Significant changes in pore pressure / suction will only occur if the rain that falls can
enter the ground and penetrate deep into the soil. Where most of the rain becomes runoff,
suctions below the surface strip of soil remain largely unaffected by the precipitation event.
It was also shown that the flow and pore pressure distribution within the slope will reflect any
layering of different materials. While in a horizontally layered soil water may be expected to
perch within an upper layer and slowly percolate into a lower, less permeable material, where
the layers are inclined a lateral down-slope flow develops which is largely within the upper
layer.
The analysis undertaken demonstrated that the pore pressures’ response time to precipitation
was generally short ( near instantaneous ) in the surface colluvium layer. However, response in
the deeper residual soil tended to show a time lag, with it taking anything from a few days to
about a month for the wetting front resulting from a rain event to reach a given depth.
Thus it was shown that the critical time for slope stability for shallow failures will be at or
immediately after the rain event that is triggering them. However, deeper-seated failure may
occur some time after the trigger event.
Further, it was shown that the pore pressure response to a given rainstorm depends in part on
what the pore pressure or suction is at the start of the rainstorm. This pressure governs the point
in the soil’s position on the suction switch, and hence determines the permeability at the start of
the storm.
From this, it is clear that antecedent rainfall is a significant factor in the pore pressure response,
and hence must also be a governing factor in the stability of unsaturated slopes subjected to
rain-induced infiltration.
The last analysis undertaken of Tung Chung reverted to an original saturated soil model,
existent prior to the commencement of this project. This generated predictions for the pore
pressures within the slope considerably different, and much less accurate, to any predicted using
the unsaturated capabilities. In doing so, this demonstrated that the new capabilities have
Chapter 9
358
significantly improved the ability to predict pore pressure responses to rainfall in unsaturated
slopes.
9.9. Recommendations for further work
The work undertaken for this project has helped increase the understanding of the behaviour of
infiltration processes in unsaturated soil.
It was hoped at the start of this project that a definitive guide to the critical factors influencing
these processes could be produced, and that a clear understanding of how such factors affect
slope stability would be obtained. In this respect, this project has failed to achieve its full aims,
since numerous questions remain. However, in striving for these ambitious targets, the areas in
which further work is necessary have been highlighted.
It is well known that the SWCC and the suction switch relationship are hysteretic, and some
work has been done to develop models that reflect this. Whether this behaviour is significant to
the response to rainfall in a slope is uncertain, and until such models are available, this cannot
be determined.
The issue of the SWCC itself needs to be addressed. It was clearly shown that the SWCC is not
unique for a soil, and that the correct approach should be to use a Soil Water Characteristic
Surface ( SWCS ). While various equations have been published for the SWCC, the same does
not hold true for the SWCS. Apart from some recently published work by Gallipoli et al (2003)
there appears to have been no attempt to produce a working SWCS model. Continued reliance
on the ‘wrong’ approach of using the SWCC would seem to be merely laying the foundation for
future problems when this inaccuracy is finally addressed. It is surely better that the ‘correct’
approach be adopted as soon as is practical.
The most likely reason for the lack of research into the SWCS is the general lack of data that
could be used to plot such a surface. This reflects the situation applicable to most aspects of
modelling unsaturated behaviour: that there is little reliable data to permit determination of the
soil parameters.
Determining the suction-dependent permeability or the SWCC/SWCS does not yet appear to be
a standard practice when undertaking site investigations of unsaturated soils, yet without the
( site specific ) data, any analysis will never produce anything better than a general guide to the
expected behaviour. Thus there needs to be greater care taken to ensure that site investigation
data ( obviously including laboratory tests on recovered samples ) is sufficiently comprehensive.
While there is at least some data on permeability and the suction – degree of saturation
relationship, there appears to have been no attempt to rigorously investigate the relationship
Chapter 9
359
between soil deformation and flow in an unsaturated soil ( as modelled by ICFEP using the Ω
parameter ). There is hence a clear need for laboratory investigation of this aspect of behaviour,
to better determine the true nature and variation of Ω. Similarly, the deformation caused to the
soil by a change in suction ( reflected in the H parameter ) is also an area where further
laboratory investigation is required.
The case study further demonstrated how the duration of a rain event can impact upon the pore
pressure response. Lack of time in this project precluded a detailed investigation of this aspect,
but it should be investigated further. This will require another fully instrumented slope to be
monitored through a wet season, with the rainfall data being collected at as precise a time
interval as possible ( probably in 5 minute long periods ). Numerical predictions of the pore
pressure response would then be made, using increments of 5 minutes, but also with the rainfall
data consolidated to permit increments of say 1 hour and 1 day. This would show the sensitivity
of the predicted pore pressure responses to the increment of rainfall size used.
This thesis therefore represents not the conclusion of the work investigating unsaturated soils
begun at Imperial College, but the basic framework from which further progress can be made.
Chapter 9
360
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APPENDIX
A1:
Details of ICFEP non-linear material model 16.
373
*NONLINEAR MATERIAL PROPERTIES – Model 16
MODEL 16 : 3D Mohr Coulomb
YIELD SURFACE
Note: Tension +ve convention
J
F (σ ) =
(
c'
− p ' ). g (ϑ )
tan φ '
−1 = 0
g (ϑ ) =
sin φ '
sin ϑ . sin φ '
cosϑ +
3
PLASTIC POTENTIAL (FLOW RULE)
An angle of dilation is specified and the shape of the plastic potential in the deviatoric plane is
again given by a hexagon.
DATA CARD ENTRIES
MNOS
P2
P3
P4
P5
P6
P7
P8
P9
P10
P11
P12
P13
P14
16.0
c’
φ’>0
υ
TIPTOL
NTS
TENSVAL
xG
yG
zG
Gxt
Gyt
Gzt
AXVAL
cohesion
angle of shearing resistance
angle of dilation
apex tolerance (see notes)
no tension switch (see notes)
allowable tensile stress (see notes)
x coordinate of t0
y coordinate of t0
z coordinate of t0
Δt/Δxm change in t with material coordinate xm
Δt/Δym change in t with material coordinate ym
Δt/Δzm change in t with material coordinate z m
this refers to the card number in the*AXIS
374
*NONLINEAR MATERIAL PROPERTIES – Model 16
P15
ELEM
VALUES input module, which defines the
transformation angles between the material xm, ym
(,zm) and the global axes xG, yG (,zG).
Number of element for which calculated t is to be
printed out
Note: A minimum of four parameters must be specified
ADDITIONAL PARAMETERS CALCULATED BY ICFEP
P27
P28
P29
P30
P31
cos(φ’)
sin(ν)
sin(φ’)
tan(φ’)
Set to 1.0 if spatially varying allowable tension t.
Otherwise set to zero.
NOTES
1. A tension positive convention is adopted in the program, i.e. +ve values refer to tensile
quantities. However parameters should be input assuming a compression +ve sign
convention. The program will make the appropriate conversions necessary for the
tension +ve sign convention that it adopts.
2. If the stress state approaches the apex of the yield surface it will be pushed back to the
point A in the diagram above (i.e it will be pushed back a distance of TIPTOL from
the apex). This prevents:
a)
the singularity of the apex causing problems, especially with the
flow steps.
b)
the stress state flipping onto the mirror image of the yield surface.
If TIPTOL is not specified or is set to 0.0 a default TIPTOL=TOLY*J mc is assumed.
3. If the no tension switch NTS=0.0 (or is not set) tension will be permitted if it occurs
below the yield surface. If NTS=1.0 any tensile stresses which exceed ‘TENSVAL’
375
*NONLINEAR MATERIAL PROPERTIES – Model 16
will be redistributed assuming the material to have negligible stiffness. If NTS=2.0 the
No Tension plasticity model will be invoked to redistribute any tensile stresses in
excess of ‘TENSVAL’. See notes for MODEL 1.
If the allowable tensile stress varies spatially then values must be input for xG, yG, (zG),
Gxt, Gyt (Gzt) and AXVAL. The allowable tension t at each integration point is then
calculated as:
t = TENSVAL + Gxt.Δxm + Gyt.Δym + Gzt.Δzm
t = TENSVAL + Gxt.Δxm + Gyt.Δym
for 3D analyses
for all other analyses
where Δxm, Δym (and Δzm) are changes in material coordinates xm, ym (and Δzm) from
the position given by xG, yG (and zG) (which are the global coordinates of the point at
which t=TENSVAL). For plane stress, plane strain, axi-symmetric and Fourier Series
aided analyses the values of zG and Gzt are not used and any values input will be
ignored. Consequently for such analyses set zG=Gzt=0.0.
NOTE: If the No Tension plasticity model is invoked (i.e. NTS=2.0) it is used in
combination with the 3D Mohr Coulomb model using a double yield surface approach.
4. If the material and global axes coincide then set AXVAL=0.0. Otherwise AXVAL
refers to the card number in the *AXIS VALUES input module which defines the
transformation angle (2D analyses) or the transformation matrix(3D analyses) between
the material and global axes. Note a real number (i.e. 3.0) must be set for AXVAL.
5. A non-zero value of the angle of shearing resistance, φ’ must be specified to avoid
potential ambiguities in the yield function. Small values are allowed but if the user
wishes to model a Tresca material then Model 19 should be used.
376
APPENDIX A2
Details of ICFEP non-linear material model 82
377
*NONLINEAR MATERIAL PROPERTIES - Model 82
MODEL 82 : Partially Saturated 3D Mohr Coulomb
YIELD SURFACE
Note: Tension +ve convention
J
F (σ ) =
(
c'
− p ' ). g (ϑ )
tan φ '
−1 = 0
g (ϑ ) =
sin φ '
sin ϑ . sin φ '
cosϑ +
3
PLASTIC POTENTIAL (FLOW RULE)
An angle of dilation is specified and the shape of the plastic potential in the deviatoric plane
is again given by a hexagon.
BULK MODULUS ASSOCIATED WITH SUCTION CHANGES
H
H3
H2
H1
Suction
AENT
u3
u2
378
*NONLINEAR MATERIAL PROPERTIES - Model 82
DATA CARD ENTRIES
MNOS
P2
P3
P4
P5
P6
P7
P8
P9
P10
P11
82.0
c’
φ’>0
υ
AENT
H1
U2
H2
U3
H3
TIPTOL
cohesion
angle of shearing resistance
angle of dilation
Air entry suction
Bulk modulus at AENT
Suction value 2
Bulk modulus at U2
Suction value 3
Bulk modulus at U3
apex tolerance (see notes)
0
>AENT
>0
>U2
>0
Note: A minimum of ten parameters must be specified
ADDITIONAL PARAMETERS CALCULATED BY ICFEP
P27
P28
P29
P30
cos(φ’)
sin(ν)
sin(φ’)
tan(φ’)
NOTES
1. A tension positive convention is adopted in the program, i.e. +ve values refer to
tensile quantities. However parameters should be input assuming a compression
+ve sign convention. The program will make the appropriate conversions
necessary for the tension +ve sign convention that it adopts.
2. If the stress state approaches the apex of the yield surface it will be pushed back to
the point A in the diagram below (i.e it will be pushed back a distance of TIPTOL
from the apex). This prevents:
a)
the singularity of the apex causing problems, especially with
the flow steps.
b)
the stress state flipping onto the mirror image of the yield
surface.
If TIPTOL is not specified or is set to 0.0 a default TIPTOL=TOLY*J mc is
assumed.
379
*NONLINEAR MATERIAL PROPERTIES - Model 82
380
APPENDIX A3
Details of ICFEP SWCC model 2.
381
*SOIL CHARACTERISTIC CURVES - Model 2
MODEL 2 : SIMPLE NON-HYSTERETIC
DESCRIPTION
In this model the above non-hysteretic soil characteristic curve is assumed for both drying and
wetting.
DATA CARD ENTRIES
MNOS
P2
P3
P4
P5
P6
P7
P8
2.0
BDSUCT
Sae
AESUCT
Ssl
Usl
U0
OMSUCT
model number
Suction at the beginning of desaturation
Degree of saturation at air entry
Air entry suction
Degree of saturation at shrinkage limit
Suction at shrinkage limit
Suction in long term ( 0% saturation)
Suction at which Ω becomes zero
ADDITIONAL NOTES
1. The parameter Ω appears in the cross coupling (bottom left) and the main consolidation
(bottom right) sub matrices of the partly saturated coupled consolidation stiffness matrix. It is
assumed to be 1.0 for suctions smaller than the air entry value (AESUCT), to vary linearly
from 1.0 at the air entry value (AESUCT) to 0.0 at the transition suction (OMSUCT) for
suctions between AESUCT and OMSUCT, and to be 0.0 for suctions greater than OMSUCT.
The program checks that OMSUCT>AESUCT.
382
APPENDIX A4
Details of ICFEP SWCC model 3
383
*SOIL CHARACTERISTIC CURVES - Model 3
MODEL 3: SIMPLE NON-HYSTERETIC NON-LINEAR MODEL
DESCRIPTION
In this model the above non-hysteretic and non-linear soil water characteristic curve is assumed
for both drying and wetting. The equation that defines this curve is as follows:
m


1
S =
⋅ (1 − S o ) + S o
n 
1
+
[(
U
−
U
)
α
]
des


The slope of the curve at any point is defined as:
R=
∂S
= −m n α(1 − S o )
∂U
[(U −U des ) α] n −1
[1 +[(U −U
384
des ) α]
n
]
m +1
*SOIL CHARACTERISTIC CURVES - Model 3
DATA CARD ENTRIES
MNOS
P2
P3
P4
P5
P6
P7
P8
P9
3.0
Udes
Uae
Uo
So
ALFA
N
M
UΩ
Model number
Suction at the beginning of de-saturation
Suction at air-entry value
Suction in long term
Degree of saturation in long term
α - fitting parameter (see note 2)
n - fitting parameter
m - fitting parameter
Suction at which Ω becomes zero
P10
Usl
Suction at shrinkage limit. If not set, this will be
automatically set equal to P4. This may cause a
warning message if this value is not compatible with
the value set in model 82.
Optional:
ADDITIONAL NOTES
1.
The parameter Ω appears in the cross-coupling (bottom left) and main consolidation
(bottom right) sub-matrices of the partially saturated coupled consolidation stiffness matrix.
It is assumed to be 1.0 for suctions smaller than the air entry value (Uae), to vary linearly
from 1.0 at the air entry value to 0.0 at the transition suction (UΩ) for suctions between Uea
and UΩ, and to be 0.0 for suction greater than UΩ. The program checks that UΩ>Uae.
2.
The dimensions of α have to be compatible with the dimensions of pore pressure, so that
the product [(U-Udes)α] in the above equation becomes non-dimensional. This implies
α=[m2/kN]. The other two fitting parameters n and m are non-dimensional.
If the fitting of data is done in terms of a S-U diagram, then α will automatically have
appropriate dimensions. However, if data fitting is done in terms of a S-h diagram (i.e.
suction vs. pressure head), then α will have dimensions of [1/length]. This value first has to
be converted into units of [1/m] (if the head was not in metres), and then divided by γw (i.e.
9.81kN/m3).
385
APPENDIX A5: The Automatic Incrementation procedure in ICFEP
386
The Automatic Incrementation procedure in ICFEP
The AI procedure implemented into ICFEP is based primarily on the work of Abbo and Sloan
(1996).
Like all non-linear finite element programs, ICFEP solves problems using discrete ( finite )
loading increments. While the smaller the increment size used the smaller the ‘step’ within the
predicted load-displacement relationship, and the closer to the true behaviour are the ICFEP
predictions, the true load-displacement behaviour actually follows a continuously varying
smooth curve. Hence there is inevitably some error between the numerical predictions and the
true behaviour.
When numerical analysis of a problem involving non-linear materials is undertaken, any of a
number of approaches may be taken to solve an increment of analysis, as discussed by Potts and
Zdravkovic (1999),
The simplest of these approaches is the tangent stiffness method. In this technique, the stiffness
is determined as the tangent of the load-displacement curve at the start of the increment of
analysis. This stiffness is used with the incremental load to determine the calculated incremental
displacement. As is shown by Potts and Zdravkovic, this can generate steadily accumulating
errors in the predicted displacements.
ICFEP avoids this inaccuracy when operating without the AI procedure by using the modified
Newton-Raphson algorithm.
The current position on the soil load-displacement curve is located (based on the current load ),
and the corresponding tangent stiffness is determined. This stiffness is used to determine the
displacement resulting from loads applied during the increment, as per the tangent stiffness
method. However, as discussed in section 5.7 and illustrated by Figure 5.6, the modified
Newton Raphson algorithm is used to ensure that the true load-displacement path is closely
followed. The increment is hence solved through a series of iterations, but in each iteration the
stiffness matrix used is the tangent stiffness matrix from the start of the increment.
It is often not feasible to use very small increments of analysis, and in such cases significant
errors may be generated even with the application of the modified Newton Raphson algorithm,
particularly where the problem is highly non-linear. The implementation of the AI procedure
helps to minimise these errors.
With the AI procedure in operation, the solution procedure followed by ICFEP is modified.
387
The use of increments of analysis is retained, and the operator specifies on which increment or
increments the AI procedure is to apply. When an increment is specified as having the AI in
operation, ICFEP initially sets the first sub-increment size equal to the full increment ( subincrement size = 100% ). Using the tangent stiffness of the load-displacement curve for the load
applicable at the start of the sub-increment, an estimate of the change in displacements due to
the load applied during the sub-increment is determined ( ∆δ1 ). For consolidation analyses, the
sub-incremental change in pore pressures ( ∆PWP1 ) is also determined. From the changes in
displacements, the changes in stress and strain are calculated.
The changes in stress and strain are then used to calculate the new position on the loaddisplacement curve, and the tangent stiffness at this location is determined. This stiffness is then
used to calculate a second estimate of displacements for the sub-increment ( ∆δ2 and ∆PWP2 ).
That is, the calculation of sub-incremental displacements is repeated, but using an alternate
‘estimate’ of the stiffness matrix.
If the tangent stiffnesses at the start and end of the sub-increment are substantially the same,
then ∆δ1 and ∆δ2 will be approximately equal ( similarly for ∆PWP1 and ∆PWP2 ). However,
more likely, particularly for severely non-linear problems, the two estimates of displacement
will differ.
From the two estimates of changes in displacement, an assessment of the dimensionless error is
determined, see equation A5.1. A similar error term is determined for the pore water pressure
during consolidation analysis.
n = node
0.5
δer =
∑ (∆ δ
2
− ∆ δ 1)
2
I =1
n = node
∑ (δ + 0.5( ∆ δ
+ ∆ δ 2) )
2
1
I =1
Eqn A5.1
Where:
δ = displacement at start of sub-increment
δer = dimensionless error
This error is compared to a ( user-specified ) tolerance. If the error is less than the tolerance,
then the step size is accepted, and the analysis continues to the next sub-increment. However, if
the error exceeds the tolerance the sub-increment is rejected. A smaller sub-increment size is
determined, and ICFEP then repeats the process.
388
Once the AI procedure confirms the acceptability of a sub-increment size, the sub-increment is
solved using the modified Newton-Raphson approach, which gives the actual changes in strain,
stress, pore pressures, etc. However, at this point it is possible that the increment may fail to
converge numerically, or, for analyses involving precipitation, pore pressures may be given on
the precipitation boundary that lie outside and more compressive than the specified threshold
value tolerance zone. In either of these last two cases, the sub-increment will be rejected, and a
new, smaller, sub-increment will be attempted.
Regardless of whether a sub-increment is accepted or rejected, the next sub-increment size is
automatically determined by the AI process, rather than being pre-determined ( though the subincrement size obviously cannot be set to be larger than the remaining portion of the
increment ). The sub-increment size is determined as a function of the specified tolerance, and
of the dimensionless relative error, this being the estimate of the absolute error divided by the
absolute value of displacement from the sub-increment that has just been completed, as shown
in equation A5.1.
Where the sub-increment was rejected due to an illegal pore pressure on a precipitation
boundary, the new sub-increment size is determined as described in section 6.6, and shown in
Figure 6.12.
If the initial sub-increment = 100% is solved in one sub-increment, ICFEP moves on to the next
increment ( or terminates the analysis, if this was the last increment ), and again checks to see if
the AI procedure is to be applied to the following increment.
If the initial sub-increment = 100% is rejected, ICFEP determines a smaller sub-increment size,
then repeats the process as described above. ICFEP then uses as many sub-increments as
necessary to solve the full increment. However, there are a maximum number of sub-increments
that ICFEP will use ( user specified, but defaulting to 1000 ). If the sub-increment size is
reduced to such a level that this maximum is likely to be exceeded, ICFEP terminates the
analysis and generates an error message.
The operation of the AI procedure is illustrated in Figure A5.1 ( in terms of displacements only,
for clarity ).
The ICFEP AI process provides a robust method of ensuring that the load-deformation
behaviour of a soil is accurately reproduced, and, as stated in the main text, has been
successfully extended to improve the accuracy of the precipitation boundary condition.
389
Specify increments of analysis, with AI operational
First sub-increment = 100% of full increment
Start sub-increment
Use initial Tangent stiffness plus incremental load to calculate incremental
displacement, ∆δ1
Use end of increment load to determine alternate Tangent stiffness. Use alternate
Tangent stiffness plus incremental load to calculate incremental displacement, ∆δ2
From ∆δ1 and ∆δ2, calculate dimensionless error, δer
Error exceeds tolerance?
Yes
No
Adjust subincrement size
Accept subincrement size
Undertake ‘normal’ modified Newton-Raphson analysis
Fails to
converge
Converges
Pore pressure on boundary meets
precipitation boundary condition
Reject subincrement
No
Yes
Accept sub-increment
Further sub-increments remaining for this increment?
Set size of next
sub-increment
Yes
No
Commence next increment
(or terminate analysis if all
increments complete )
Figure A5.1: The AI operating procedure
390
Appendix A6
APPENDIX A6: Photographs of the Tung Chung site
391
Appendix A6
Plate A1: Tung Chung slope. View approximately south, showing relic slip and location of
Piezometer A6 and tensiometer SP2.
392
Appendix A6
Plate A2: Tung Chung slope. View south-east, from approximately 20m SE of the location of
SP1.
393
Appendix A6
Plate A3: Tung Chung slope. View north-west, from approximately 40m SE of SP2. North
Lantau highway visible at base of the slope.
394
Appendix A6
Plate A4: Tung Chung slope. View south-east, from south-east boundary of study area. Hence
plate shows slope above ( outside of ) the study area.
395
396
Plate A5: Tung Chung slope. View south-west, from south-east boundary of study area. Plate hence shows slope outside of study area, but the
general profile of the slope ( including within the study area ) is revealed.
Appendix A6
Notes
397