Acta Geophysica
vol. 55, no. 1, pp. 23-32
DOI 10.2478/s11600-006-0036-5
Measurements of armour layer
roughness geometry function and porosity
Jochen ABERLE
Leichtweiss-Institute for Hydraulic Engineering, Technical University of Braunschweig
Beethovenstr. 51, 38106 Braunschweig, Germany;
e-mail: [email protected]
Abstract
The roughness geometry function of the interfacial sublayer of a gravel-bed
armour layer was measured directly by filling water stepwise into a laboratory
flume and indirectly from a digital elevation model (DEM) of the surface. The results of both methods are compared and show that the DEM can be used to reliably
estimate the roughness density function for a wide range of the interfacial sublayer.
The direct measurements revealed an absolute minimum of porosity at the level of
the roughness trough which is significantly smaller than porosity in the undisturbed
subsurface and porosity estimates obtained from relationships found in the literature. The significance of the results for hydraulic engineering and ecological applications is highlighted.
Key words: Armour Layer, porosity, digital elevation model, flume study.
1. INTRODUCTION
The roughness geometry of gravel-bed armour layers plays an important role in fluvial
geomorphology and river hydraulics since it determines such flow properties as mean
flow velocity, turbulence, and sediment transport. Its significance for the flow region
between roughness top and trough (interfacial sublayer) becomes apparent from the
double averaged (in temporal and spatial domains) momentum equations (e.g., Nikora
et al. 2001, 2004, 2006a, b, Aberle and Koll 2004, Aberle 2006), which depend on the
roughness geometry function φ. The roughness geometry function is defined as
φ = Vf /V0 , where Vf = volume occupied by fluid within an averaging (total) volume V0.
The definition of φ shows that it can also be interpreted as a statistical measure of ran© 2007 Institute of Geophysics, Polish Academy of Sciences
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J. ABERLE
dom geometry of the bed surface (e.g., Nikora et al. 2001) as well as porosity. The latter is, similarly to φ, defined as the ratio of the space taken up by voids to the total
volume (e.g., Bunte and Abt 2001).
Porosity, on the other hand, is an important parameter for armour layer development, colmation processes, aquatic habitat studies and exchange processes between
free surface and groundwater flows. For example, during armour layer development,
the thickness of the active mixing layer is, for plane beds, controlled by the maximum
grain size and porosity (Marion and Fraccarollo 1997). Hence, modelling bed development with the sediment continuity equation (e.g., Toro-Escobar et al. 1996, Bettes
and Frangipane 2003) or analysing bed material samples extracted by, e.g., waxcoring methods (Marion and Fraccarollo 1997) requires the knowledge of porosity.
Furthermore, porosity is also an important parameter for ecological applications. Fine
sediment that passes the coarse armour layer may accumulate beneath the surface and
during low discharge conditions a compact layer may develop that reduces porosity
and hydraulic conductivity of the streambed. Such a compaction stabilizes the streambed against erosion (Veličković 2005) and is an important link between the sediment
transport regime and the suitability of benthic habitat for stream organisms (Lisle
1989) as well as for exchange processes between free surface and groundwater flows
(e.g., Veličković 2005, Vollmer 2005).
Methods for the measurement of gravel-bed porosity are described in Milhous
(2002) and Carling and Reader (1982). Milhous (2002) developed a simple method for
the determination of porosity below the armoured surface using a wooden frame and
plastic sheets. Carling and Reader (1982) analysed sediment cores sampled with a
freezing technique to relate porosity to the mean grain diameter according to φ =
0.4665 dm–0.21 – 0.0333 (dm in mm) for poorly sorted sediments. A similar relationship
was developed by Komura (1963), φ = 0.245+ 0.0864 dm–0.21 (dm in cm). Both relationships show that, below the surface, porosity is inversely related to mean grain diameter. Taking into account that the uppermost layer of an armour layer is coarser than the
layers below (e.g., Sibanda et al. 2000), porosity is thus expected to increase with
depth into the sediment.
However, detailed measurements of roughness geometry function and porosity
within the interfacial sublayer and the subsurface layer of armoured gravel-beds are
time consuming and therefore rare. Nikora et al. (1998, 2001, 2006b), Aberle and Koll
(2004), and Aberle (2006) used topographic data of gravel-bed armour layers obtained
with point gauges and laser scans to estimate φ for the application in the doubleaveraged momentum equations. These studies explicitly concluded that a correction
function for φ is needed for such data, since φ = 0 is obtained at the lowest measured
bed elevation due to the fact that a laser scan of the surface does not include measurements of the pore volume. Thus, in reality φ tends to a non-zero value of the bed porosity below roughness troughs and not to zero.
In this paper, the requirement of a correction function for the roughness geometry
function is investigated by comparing φ-estimates obtained from direct measurements
in a big laboratory flume to φ-estimates derived from laser displacement data.
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ARMOUR LAYER ROUGHNESS GEOMETRY FUNCTION AND POROSITY
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2. EXPERIMENTAL SETUP
The data analysed in this paper were obtained from laboratory experiments in a 20 m
long, 0.90 m wide and 0.60 m high tilting flume in the laboratory of the LeichtweissInstitute for Hydraulic Engineering, Technical University of Braunschweig, Germany.
These experiments were specifically designed to study the interaction between flow
and roughness for flows with low and medium relative submergence based on the
double-averaging approach. The detailed description of the hydraulic measurements is
beyond the scope of this paper as it focuses on the analysis of the topography data.
Preliminary results of the hydraulic measurements can be found in Aberle and Koll
(2004), Aberle (2006) and Nikora et al. (2006b).
In brief, the experimental methodology was based on a subsequent development
of armour layers. At the first step of an experiment, the well mixed sediment (0.63 mm
< d < 64 mm; see Fig. 1 for grain-size distribution) was placed in the flume over
a length Ls = 13.44 m and flattened to a thickness of 0.20 m to ensure that the bed
slope was parallel to the flume slope (S = 0.0027). The distance from the flow straightener at the flume inlet to the sediment bed was 1.00 m and the gap between was
bridged with bricks to avoid extensive scouring at the inlet. The sediment body was
separated from the bricks by a perforated metal plate. At the downstream end the
sediment body was stabilized by a 20 cm high sill. The sill was also constructed of
perforated metal to allow flow in the subsurface layer.
The bed investigated in this study corresponded to the third subsequently developed armour layer which was armoured with a discharge of Q = 220 l/s (with preceding armouring discharges of Q = 120 l/s and 180 l/s, respectively). Each armouring
discharge was run until bed-material transport rate became negligible, indicating a stable armoured bed surface. Photographs of the armour layers were used to obtain the
grain-size distribution of the surface material applying a line by number method (Fehr
1987). The method was calibrated by manually sampling the surface layer for
a selected preceding experiment and adjusting parameters to minimize differences
between a sieve-based grain-size distribution and a distribution from photographs
100
Subsurface Material
90
Armour layer
Percent finer [%]
80
70
60
50
40
30
20
10
0
1
10
d [mm]
100
Fig. 1. View of the flume during laser-scan (left) and grain-size distributions of subsurface material and armour layer (right).
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J. ABERLE
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(Oppermann 2005). Figure 1 shows the grain-size distribution of the subsurface material and of the armour layer under investigation.
The bed topography was measured along the 2.40 m long and 0.716 m wide test
section with an Optimess laser-displacement meter (see Fig. 1). The test section was
located 8 m from the transition bricks-sediment so that neither the scour at the inlet
section nor the downstream sill influenced the development of water worked roughness structure. Longitudinal bed profiles were recorded with a sampling interval ∆x =
1.0 mm, a lateral sampling interval ∆y = 4 mm, and with a vertical resolution of
0.1 mm. The different sampling intervals in x and y directions were due to the footprint of the sampling volume of the laser system (0.2 mm in flow direction and 4 mm
in transverse direction). Figure 2 shows the measured topography of the armour layer
under investigation.
0
200
400
600
0
200
400
600
800
1000 1200 1400 1600 1800 2000 2200 2400
Fig. 2. Armour layer topography of the test section (flow direction from left to right; axes are
scaled in mm).
The laser data were used to estimate φ by assuming that the surface is composed
of pillars with ground area identical to the sampling resolution (1×4 mm) and height of
the corresponding reading. In a next step, the digital elevation model (DEM) was
sliced into 100 vertical segments (with ground area equal to the overall size of the
DEM) and φ was estimated for each segment from the ratio of the volume fraction occupied by the pillars and the total segment-volume. Further results related to the
analysis of roughness structure of armour layers from these experiments are found in
Aberle and Nikora (2006).
In order to investigate the accuracy of the φ-functions obtained from the DEMdata, additional measurements were carried out for this study in which φ was measured
directly. For this purpose, the flume was tilted into a horizontal position and sealed at
the upstream and downstream end. The upstream section was sealed in a distance of
0.155 m to the sediment body by removing a row of bricks and fitting a plate into the
flume cross-section. Another seal-plate was mounted at the downstream end of the
flume in a distance of 0.545 m to the sill. Thus, two small basins with a total length
LB = 0.70 m were created and the total length of the measuring section comprising the
sediment body and the two basins was L = LS + LB = 13.44 m + 0.70 m = 14.14 m.
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ARMOUR LAYER ROUGHNESS GEOMETRY FUNCTION AND POROSITY
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This flume setup was used to measure φ directly by filling stepwise a known
amount of water (measured by a water-meter) into the flume and monitoring corresponding water levels. The latter were measured in the downstream basin and over the
roughness trough (of the total sediment body) using point gauges. Readings were
taken only when the water level was stable. The knowledge of the water volume added
in each step VSt and the corresponding water level increment ∆h enables the calculation of φ = Vf /V0. The volume fraction of water added to the sediment body Vf can be
estimated by subtracting the volume of water added to the basins VB = wLB ∆h (with
w = flume width) from the volume of water added to the flume VSt in the step. The total volume of the averaging region of the sediment bed is known from V0 = wLs ∆h.
Hence, porosity can be calculated by the following equation:
φ=
Vf
V0
=
VSt − VB
V − wLB ∆h
V
L
= St
= St − B .
V0
wLS ∆h
V0 LS
(1)
In total, four additional experiments were carried out to determine φ. Before the
measurements, the sediment body was drained and air dried for a period of 3-4 days.
The experimental runs 1-3 were designed to determine φ solely above the roughness
trough and the additional fourth experiment was carried out to measure porosity also
in the subsurface layer. In runs 1-3, the sealed flume section was slowly filled with
water until water became visible at the roughness trough. From this point onwards,
water was successively filled into the flume in steps of 10-50 litres; 10 l steps were
applied in the lower part of the armour layer (i.e., close to roughness trough) and 50 l
steps in the upper part. For the fourth experiment, water was filled into the flume in
steps of 50 l starting from the flume plastic bottom. Close to the roughness trough, the
step width was reduced to 10 l and increased again to 50 l at some distance to the
roughness trough.
3. RESULTS AND DISCUSSION
Flume measurements
Figure 3 shows the vertical distributions of φ obtained from the flume measurements.
The results in Fig. 3 show that the measurements are characterised by a high degree of
reproducibility, although the beds were only air dried. All curves decrease monotonically from the roughness tops towards the roughness trough. Slightly above the roughness trough, φ is characterised by an absolute minimum value of 6%, indicating at
poor sorting and a densely packed layer. Below the roughness trough, φ increases for
approximately 0.04 m before reaching a constant value (φ ≈ 18-22%) within an approximately 0.06 m thick subsurface layer. The observed successive increase of φ below the roughness trough is associated with a change in the sediment matrix and, thus,
with the finer composition of the subsurface bed material (see Fig. 1). The minor differences in the φ-values within the constant φ-layer are most likely related to the
change in the volume of added water from 50 to 10 l within this particular (fourth) ex-
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J. ABERLE
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periment. Reasonably assuming that the distance from the roughness tops to the constant φ-layer corresponds to the active layer during the armouring process, the active
layer thickness is approximately 2dmax for the bed under investigation.
Height over plastic bottom [m]
0.25
0.2
Roughness trough
0.15
1st Run
2nd Run
0.1
3rd Run
4th Run
0.05
0
0
0.2
0.4
0.6
0.8
1
1.2
φ = Vf /V0
Fig. 3. Estimates of roughness density and porosity for the experimental runs 1-4.
Below the constant φ-layer and in a distance of 0.06 m to the flume plastic bottom, φ-values increase again with depth into the sediment. Bearing in mind that the
maximum grain size corresponds to dmax = 0.064 m, it can be concluded that the matrix
in this region is influenced by the plastic bottom and dmax. The plastic bottom prevents
thorough mixing of the particles and, thus, larger pore volumes may develop in this
region. Such an increase of φ can also be observed for densely packed, single layered
gravel particles of the same diameter (Stephan 2001). For this case φ reduces from the
gravel tops to approximately half the particle diameter where the single-layered bed is
characterised by the maximum volume of solids. In the region below, the volume of
solids reduces again and hence φ increases towards the impermeable flume bottom
(φ → 1), where finally φ = 0 due to the impermeable surface. Thus, the observed increase of φ towards the plastic flume bottom in Fig. 3 may be associated with a similar
mechanism. The sediment matrix may act as a filter for larger particles close to the
flume bottom, leading to increased pore volumes. However, smaller particles such as
sand and fine gravel particles may still pass this filter region and settle at the bottom.
Such a behaviour would explain the observed decrease of φ for the measurement point
closest to the plastic bottom in Fig. 3, although further data is needed to explore this
behaviour in more detail.
A closer examination of the results shown in Fig. 3 reveals that the φ-curves are
characterised by smaller scatter (with values φ > 1) in the region of the absolute roughness top. This scatter is associated with measurement accuracy. In this region the volume of solids is small compared to the volume of added water since it is composed
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ARMOUR LAYER ROUGHNESS GEOMETRY FUNCTION AND POROSITY
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only by the tops of a few protruding particles. Hence the corresponding φ-estimates
are prone to measurement errors. In fact, an inaccuracy in the point gauge reading of
0.0001 m results in an error for the φ-estimate of approx. 3% in this region. However,
it is worth mentioning that the error is reduced in regions further below the roughness
tops; the same measurement error corresponds to an error in φ-estimates of approximately 1% in the mid-region and 0.2% close to the roughness trough.
The measurements can also be used to analyse bulk porosity values. For runs 1-4,
the bulk porosity, estimated from the plastic bottom to roughness trough, corresponds
to 22.2, 24.1, 23.2, and 23.3%, respectively. An additional measurement, where the
bed material was placed in a 10 l bucket, resulted in an estimate of bulk porosity
φ = 28%. This difference can be attributed to sample preparation in the bucket, in
which the sediment was not compacted in the same way as in the flume nor characterised by an armour layer.
Using the relationships of Carling and Reader (1982) and Komura (1963), bulk
values for φ of 26.4% and 33.4%, respectively, are obtained for the subsurface material. These values are reasonably close to the bulk and bucket estimates of φ as well as
to estimates found in the literature (φ = 30-40% for static armour layers, e.g., Marion
and Fraccarollo 1997, Bettes and Frangipane 2003). However, using the grain-size distribution of the surface layer results in estimates of φ = 20.9 and 31.8%, respectively,
being significantly larger than the observed minimum value of φ = 6% at the roughness trough. Thus, this result shows that the minimum value of porosity of the surface
layer may be significantly overestimated using bulk relationships.
Laser displacement data
Figure 4 shows the φ-function estimated from the DEM together with the φ-function
obtained from the 2nd experimental run of the flume measurements. The latter has
been included to directly compare the φ-estimates of both methods. The laser-data
φ-curve decays similarly to the flume-curves monotonically from φ = 1 at the roughness tops towards the roughness trough. However, in contrast to the flume measurements, φ = 0 is obtained at the roughness trough although φ = 6% would be expected.
This difference is due to the fact that the laser scan of the surface does not include
measurements of the pore volume. Nonetheless, both curves are almost identical for
φ > 12% (z > 0.018 m above roughness trough).
For distances to the roughness trough larger than half the roughness range
(φ > 60%), the laser data result in slightly underestimated values of φ compared to the
values obtained from the flume measurements. These differences are associated with
(i) the measurement accuracy of the flume experiments and (ii) the different ground
areas of both methods. The deviation of the curves becomes apparent at the level
above the roughness trough where the volume of added water was increased from 10
to 50 l. Thus, the aforementioned inaccuracies in point gauge readings may become
more significant. The second explanation is related to the different measurement scale.
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J. ABERLE
30
0.25
Height over plastic bottom [m]
2nd Run
Laser Displacement Meter
0.2
0.15
0.1
0
0.2
0.4
0.6
0.8
1
1.2
φ = Vf /V0
Fig. 4. Comparison of roughness density function obtained from laser data and obtained from
direct measurements of run 2.
For the flume measurements, φ was estimated for the total sediment body while, for
the indirect estimates using the laser data, φ-estimates are restricted to the ground area
of the DEM. The absolute maximum and minimum elevation of the sediment body
was not located within the DEM-section. This becomes apparent from the different
maximum and minimum bed elevations for both methods in Fig. 4. Furthermore, differences in bed geometry such as the scour at the flume inlet or the accumulation of
bed material at the downstream sill may slightly bias the results from the flume measurements.
Nonetheless, the above results show that the laser scan provides an accurate estimate of the roughness geometry function for almost the total range of the interfacial
sublayer. Thus, it can be concluded that a correction function for φ for gravel-bed armour layers is required only for the region close to the roughness trough. However,
the derivation of an appropriate correction function requires more data sampled over
different armour layers.
4. CONCLUSIONS
In this paper, a direct and an indirect method for estimation of the roughness geometry
function φ within the interfacial sublayer of a gravel-bed armour layer were compared.
For the direct method, water was filled stepwise into a big laboratory flume and φ was
calculated from the volume of added water and the associated increment of water levels. The indirect method was based on the analysis of an armour layer DEM. The
comparison of both methods showed that DEM data can be used to accurately estimate
φ for a wide range of the interfacial sublayer (φ >12%), although the pore volume is
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ARMOUR LAYER ROUGHNESS GEOMETRY FUNCTION AND POROSITY
31
not reflected in DEM data. It was concluded that a correction function for φ estimated
from DEM’s is required only for the region close to the roughness trough.
The direct measurements of the roughness geometry function and subsurface porosity revealed an absolute minimum value of φ at the level of the roughness trough of
φ = 6% for the bed under investigation. This low value of porosity was associated with
the dense packing of the surface particles due to static armour. Bearing in mind that
this study was carried out with particles larger than 0.63 mm, it can be concluded that
armour layers in the field may be characterised by even lower values of φ. Furthermore, the observed minimum value of φ is significantly smaller than φ-values estimated from relationships found in the literature. Thus, the results of this study are important for applications concerning armour layer development, exchange processes between subsurface and surface flows as well as benthic habitat studies, where porosity
is an important parameter. More data on this topic would be desirable to generalise the
findings of this study. However, such data is not available because simultaneous direct
measurements of porosity and bed geometry are time consuming and therefore rare.
A c k n o w l e d g m e n t s . The research was conducted under contract DI-651/4-3
from DFG (Deutsche Forschungsgemeinschaft). U. Ecklebe and M. Oppermann provided technical support. The author is grateful for useful discussions and suggestions
to V. Nikora, D. Pokrajac, K. Koll, A. Dittrich and to all participants of the “doubleaveraging” workshops.
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Received 19 September 2006
Accepted 17 October 2006
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