Land Subsidence (Proceedings of the Fifth International Symposium on Land Subsidence,
The Hague, October 1995). IAHS Publ. no. 234, 1995.
313
The step-loading model of subsidence induced by
groundwater level changes with time
LAY SENG YAW & SHIEH MING JYH
Institute of Harbour and Marine Technology Wuchi, Taichung, Taiwan
Abstract This paper tries to develop a step-loading model for simulating
the complex stress of soil deposits, and utilizes the Biot consolidation
theory and superposition concept to analyse subsidence phenomena
induced by changes in groundwater level with time. Finally, the in situ
investigation data of Wang-Kon, which is located in the coastal region of
Chang-Hwa, Taiwan, will be checked in order to prove the usefulness of
this model.
INTRODUCTION
The most serious subsidence problems at Taiwan coastal areas is due to overpumping
groundwater. The groundwater levels fluctuate repeatedly, because of fluctuations in the
amount of groundwater withdrawal, and fluctuations in rainfall. This study looks at the
temporal effects on subsidence of the repeated processes of loading, unloading,
reloading on soil deposits and the cyclic changes in groundwater levels.
Studies of surface consolidation settlement induced by groundwater pumping can be
divided into two categories, one is the decoupled approach method based on Terzaghi's
(Terzaghi, 1943; Taylor, 1948) consolidation theory, the other is the coupled approach
method based on Biot's (Biot, 1941, 1955) consolidation theory. The former first looks
at the pore water pressure distribution, then calculates the strain of the soil deposits and
total settlement of the ground surface by applying the concept of effective stress. The
latter is a couple theory by adopting the displacement of porous media and the pore
water pressure as the basic variables, which is based on the interaction of pore water and
porous media. This couple consolidation theory is generally recognized to be more
reasonable.
Therefore, this paper not only uses the couple consolidation theory of Biot to study
the subsidence problem, but also develops a step-loading model to simulate the complex
stress behaviour of soil deposits due to changing water levels, in order to analyse the
effects on land subsidence with time.
BIOT COUPLE CONSOLIDATION THEORY
The analytical model of land subsidence in this paper is based on the soil couple
consolidation theory. The couple consolidation theory concerning soil was firstly
presented by Biot (1941), then Verruijt (1969), Bear & Corapcioglu (1981) and
Corapcioglu & Bear (1983) etc. studied this theory and derived consolidation models
from the viewpoint of flowing groundwater. Their basic assumptions:
Lay Seng Yaw and Shieh Ming Jyh
314
(a) fully saturated,
(b) pore water is compressible,
(c) solid grain is incompressible,
(d) follows Darcy's law,
(e) isotropic and homogeneous,
(f) small strains,
(g) linearity of stress-strain relations,
(h) the main factor to affect porosity is the effective stress of medium.
According to the above assumptions, the basic equation of this coupled consolidation
theory can be expressed as follows:
GV2u+x
G
ds
•ÈE =o
1 - 2v dx dx
GV 2 « V + _ ^ _ ^ £ - ^
y
GVV
l-2vdy
G ds
l-2vdz
-kV2p + ds
dt
2
dy
_ dp _ 0
(1)
dz
0dp
=0
dt
2
=o
2
ns, V = d ldx + d2/dy2 + d2/dz2, ux, Uy and uz may be interpreted
in these relations,
respectively as the displacement of the soil medium in the x, y, and z direction,
s = Sxx + Syy + szz represents strain amount of medium volume,/) represents the excess
pore water pressure, E, v, and G may be interpreted respectively as Young's modulus,
Poission's ratio and shear modulus, where G = £72(1 + v), and the other constants k,
n and /3 may be interpreted, respectively, as permeability, porosity and compressibility
of pore water.
If we only consider one-dimension consolidation i.e. ux = uv = 0, e = duJdz,
dujdx
duJdy = 0, then the basic equation may be simplified as follows:
2VG
^L
i-SE.
dz dz
=0
2
-k^l +
dz2 dzdt
(2)
dt
= 0
in which n = (1 - v)/(l -2v). In this equation, soil displacement uz and pore water
pressure p appear simultaneously so that it is a one-dimensional couple consolidation
model.
If we consider that a soil deposit, with thickness H, suffers an instant loading p0,
then by analysing equation (2) its consolidation settlement could be expressed as
equation (3):
4p0H
u
z =
1
X " 1 -exp
E
WriG^b (2n + iy
(2n + l ) n
cvt
2/L
(3)
Step-loading model of subsidence
315
in which vertical consolidation coefficient Cv = &/(«p + l/(2nG)), Hd is the length of
drainage, Hd = H when the drainage condition is one-dimensional, butHd = H/2 when
the drainage condition is two-dimensional.
STEP-LOADING MODEL
Owing to the soil consolidation behaviour (Taiwan Inst, of Harbor and Marine Tech.,
1993) can be divided into two parts: one is permutative deformation of soil particles, the
other is elastic deformation of soil particles and pore water. The former is the main part
of soil deformation, which is not reversible, and the later is reversible and represents a
very small amount of deformation.
In this model, the major consolidation of subsidence analysis is the loading
deformation by the increment of soil effective pressure during the process of lowering
the water level, and the small changes of unloading and reloading deformation are neglected.
The analysis procedure of the step-loading model could be described as follows:
Transfer the groundwater level curve to consolidation water level curve
Although the groundwater level curve, shown in Fig. 1, includes the complex processes
of water level lowering, rising and relowering etc., this model does not consider the tiny
deformation induced by the rising and relowering effects of groundwater level,
therefore, it just only reserves the lowering stages of water level and deletes the other
meaningless factors. It means that this model only selects the lowest water level of each
stage to form a consolidation water level curve, shown as Fig. 2, and this curve looks
like a step-loading process constituted by several consolidation stages.
TAIWAN
TAI-SI
-15 )»9nl.»p»t«^i»fM|^^~i^r.''T"r'T^w"w*w»w-w^
69 70 71 72 73 74 75 76 77 7B 78 80 B1 62 83 84 85 86 87 88 SB 80 81 82
YEAR'S (1969-1892)
Fig. 1 The free water level variety with time.
316
Lay Seng Yaw and Shieh Ming Jyh
TAIWAN
TAi-a
5
4
3
2
1
0
-1 •
-2
- 3
hi i
•
-4
-5
-6
-7
-8
-0
-10
-11
-12
-13
-14
-18
t5 t6
U
I
"Tjl3
-Î"
t
w i ' i 1 1 ' w y w f i||n,i"iii"iiT"i'ww|^w
Ii1«ii|in^^»f.|iiii|.i^i.nu|ifn
•WW
6» 70 71 72 73 74 78 76 77 78 79 80 81 82 83 84 88 88 87 88 88 »0 81 92
YWTS (1SB9-1982)
Fig. 2 The consolidation water level variety with time.
Analysing the settlement curve of each consolidation stage Owing to measured
data of water level from an observation well which represent the changes in water level
in nearby areas, this subsidence type belongs to regional subsidence. Therefore, we
utilized equation (3) which derived from one-dimensional Biot couple consolidation
theory to approach the subsidence variety. If the thickness of the soil deposit is H, the
water unit weight is rw, the lowering amount of consolidation water level is h, which
means the effective stress adds an amount rji, then the settlement variety with time in
each consolidated stage can be calculated by the following formula:
«.-^Ë
ir^n-o
1
x ' 1 -exp
(2n + l) z
(In +1)11
in, di
cj
(4)
in which HpH^pGpC^ may be interpreted, respectively, as soil thickness, drain path
length, Poisson ratio function, shear modulus and consolidation coefficient of No. I soil
deposit. According to the analysis of equation (4), the settlement with time at each
consolidation stage is showed in Fig. 3.
Calculating the settlement curve for a single soil stratum If there are k
consolidation stages in the No. I soil stratum, we can use superposition theory to
calculate the settlement variety with time Sft) by the following formula:
Sft)
AH-
k
n\.<v=i
,
i
AE
„-o (2« + l)
z
1 -exp
g » + i)n
2HM
cjf-tj)
'u(t-tp
(5)
Step-loading model of subsidence
317
TAIWAN
TAI-S
papa,; ¥ 1 ' l |ii'V'T l ^"l'f"l'"q'n-l''W"l"TT"l
l i W W W f l ' W W P W ^ ' n ' W
69 7D 71 72 73 74 75 76 77 78 79 80 «1 B2 83 84 85 88 87 8B 89 90 91 92
YEAR'S (1989-1992)
Fig. 3 The settlement of each consolidation stage.
TAIWAN
T«-a
69
• • l " ! " 1 ! " " ! ' " ! li"I1111!1"!"!"!""!"1!"1!1"")""!""!1 I 11 ! 1 \"l"\ ' l ' l " l l"l 'I 1 t'"l»
7 0 7 1 Tl. 7 3 7 4 7 5 7 8 7 7 7 8 7 9 8 0 81 8 2 8 3 8 4 8 5 8 8 8 7 8 8 8 9 9 0 9 1 9 2
YEAR'S ( 1 9 8 9 - 1 9 9 2 )
Uod«! predict
+
R«al data
Fig. 4 The settlement changes with time in Tai-si.
in which tp /z- represent respectively the initial time and the lowering amount of
consolidation water level in the No. J consolidation stage, the u(t - tp is the Heaviside
step function, u(t - tp - 0 when t < tp u(t - tj} = 1 when t > tp Fig. 3 is the
settlement curve in each consolidation stage. By equation (5), we can obtain the
settlement changes with time of the No. I soil stratum. This result is shown in Fig. 4.
318
Lay Seng Yaw and Shieh Ming Jyh
Fig. 5 Soil profiles and monitoring depth in Wang-Kon.
Step-loading model of subsidence
319
Calculating the subsidence curve for the whole soil deposits If there are m soil
strata, we can use superposition theory again to calculate the total subsidence Sj(t) by
the following formula:
m
s
i(t) = E w
^
1=1
EXAMPLE ANALYSIS
The application of the model is illustrated by the in situ observed data of Wang-Kon,
which is situated on the Chang-Hwa coast of Taiwan. Here an investigation station of
groundwater level and settlement was established for each separated soil stratum by the
Water Conservancy Bureau in May 1989 (Taiwan Water Conservancy Bureau, 1993).
According to the observed data, the soil profiles, the groundwater level and the
settlement of each stratum are shown in Figs 5, 6 and 7, respectively. From the
groundwater level changes of each stratum in Fig. 6, we can find that the change in free
water level in the shallow stratum is so small that the settlement of shallow stratum is
tiny, that can also be explained from the settlements of No. ll-No.14 in Fig. 7.
Additionally, we can find that the major settlement range is located at No. 5-No. 11,
which means that the major settlement is distributed at depths of 44.5-125.5 m.
Therefore, the subsidence analysis in this paper aims mainly at the range of depth 44.5125.5 m.
From Fig. 6, we can find that the changes of confined water level at depth 76 m, 82
m, 106 m are almost the same, therefore, our subsidence analysis is based on the variety
of water level at depth 106 m. Through the trial and error method, we find the
Preconsolidation Head is -8.2 m, and the initial water level in May 1989 is - 9 . 3 m.
Because the stratum at depth 44.5-50.4 m is a part of the upper confining bed and the
stratum at depth 123.9-125.5 m is a part of the lower confining bed, their settlement
when calculated by equation (5), had to be reduced appropriately. As the other parts at
2
5
WAN0-KON
7
-*-
Î Zl V
î v
""'1
AJLK
WLA
%
A
T
;
ïv\\\ /hi b^^i i
- 1 3 ~ n r i i i i i i I Ï Ï i i r 11 tT r r t 11 r i i 11 i T I i i r i i r i i r i i i i VT i r r r i t~A
89/5
90/1
00/7
81/1
91/7
S2/1
92/7
93/1
93/7
YEAR'S (19B9/5-1993/6)
76U
A
S2M
Fig. 6 The water level changes of each stratum in Wang-Kon.
320
Lay Seng Yaw and Shieh Ming Jyh
TAIWAN
0 -J»l|»llgBgj|gB*A«BB*
—*° ~t I 7 I U
88/5
B
WANG-KON
» * » « A j 4 | | f g j a j f f ^ | f g = j 3 3 $ j ^ g g ' HHo. 2 ,
J.sl
i ii i r i T l
I I I 1 I I I 1 I I I 1 I I II I I ! I I I I 1 1 I J I I 1 1 1 I I I I ! J 1 M
83/7
90/1
»0/7
«1/1
»1/7
92/1
82/7
93/1
YEAR'S ( 1 9 B 8 / 5 - 1 9 9 3 / 6 )
Fig. 7 The settlement changes of each stratum in Wang-Kon.
TAIWAN
WANG-KON
^
OOQtlon
™*° ~f I i i i i i i i i i i i i i i i i i i i i i i I I i I i i i i i i i i i i i I i r i i i i i i
69/5
BO/1
90/7
91/1
91/7
92/1
92/7
03/1
Ù
Real data
1 1 1 ITTl
93/7
YEAR'S ( 1 9 8 9 / 5 - 1 9 9 3 / 6 )
'
Model predict
Fig. 8 Total settlement at 44.5-125.5 m depth in Wang-Kon.
Table 1 Soil characteristics in Wang-Kon.
Soil class.
CL
ML
CL-SM
SM,SP
SM-GM
Cv (m2year_1)
E (t m"2)
450
700
600
1000
1300
0.35
0.35
0.35
0.30
0.30
15
10
30
40
Step-loading model of subsidence
321
depth 50.4-123.9 m belong to the confined aquifer or interbed, their settlement when
calculated by equation (5) need not be reduced.
Owing to the above considerations and the subsidence analysis of step-loading
model, after inputting the data of Table 1 we can obtain the total settlement variety with
time at the depth of 44.5-125.5 m (Fig. 8), and these results match the in situ observed
data very well.
CONCLUSION
(a) This paper uses Biot couple consolidation theory, a step-loading model and the
superposition concept to analyse the subsidence induced by changes in groundwater
level. The analytical results correspond to the data observed in situ.
(b) It is very important to distinguish between confined aquifer, confining bed and
interbed for making a reasonable analysis, and the parameters of settlement
characteristics in each stratum are also the key of reliability of analytical results.
Additionally, the supposition of preconsolidation head and initial groundwater level
also affect the analysis result of subsidence.
REFERENCES
Bear, J. &Corapcioglu,M. J. (1981) Land subsidence due to pumping, 1. Integrated aquifer subsidence equation based on
vertical displacementonly. Wat. Resour. Res. 17(4), 937-946.
Biot, M. A. (1941) General theory of three-dimensional consolidation./. Appl. Phys. 12(2), 155-164.
Biot, M. A. (1955) Theory of elasticity and consolidation for a porous anisotropic solid. J. Appl. Phys. 26(2), 182-185.
Corapcioglu, M. J. & Bear, J. (1983) A mathematical model for regional land subsidence due to pumping, 3. Integrated
equation for a phreatic aquifer. Wat. Resour. Res. 19(4), 895-908.
Terzaghi, K. (1943) Theoretical Soil Mechanics. John Wiley & Sons, New York.
Taylor, D. W. (1948) Fundamental of Soil Mechanics. John Wiley & Sons, New York.
Taiwan Inst, of Harbor and Marine Tech. (1993) The behavior and prediction of land subsidence at Taiwan coast.
Taiwan Water Conservancy Bureau (1993) The study of monitor groundwater level and settlement for each separated soil
stratum in the Wang-Kon of Taiwan.
Verruijt, A. (1969) Elastic storage of aquifers. In: Flow Through Porous Media (ed. by R. J. M. de Weist), 331-376.
Academic Press, New York.