Numerical modelling of negative pore water pressures – influence of water retention parameters
R. Bertilsson
Swedish Geotechnical Institute, Sweden, rebecca.bertilsson@swedgeo.se D. Bendz
Swedish Geotechnical Institute, Sweden
ABSTRACT
Keywords: Suction, silts, slopes.
1 BACKGROUND
For an unsaturated soil, the effect of negative pore water pressures might be a factor highly contributing to the slope stability and therefore a parameter that must be taken into account in stability analysis.
This is in particular true for clayey and silty slopes where high capillary raise is possible and high negative pore water pressures can develop above the ground water table.
The knowledge about the negative pore water pressure profiles in slopes and their variation with time is limited. Results from field measurements of negative pore water pressures are presented for example by Casagli et al. (1999), Fredlund (1995), Lim &
Rahardjo (1994) and Öberg (1995). However, none of these measurements were carried out for a longer period of time than
approximately one year. There is a need of field data and modelling so that the effect of
negative pore water pressure can be fully taken into account in stability analyses.
The Swedish Geotechnical institute is conducting a research project focusing on long time measurements (from 2009 and still on-going) of negative pore water pressures in a steep, silty slope in the outskirts of
Sollefteå, Sweden (www.swedgeo.se). The objective of the project is to increase the level of knowledge of the negative pore water pressure profile and its seasonal variation.
Numerical modelling tools are used to simulate and interpret the behaviour of the pore water pressures measured in field.
In all kinds of analyses that reflect negative pore water pressures, the water retention curve (WRC) of the soil is one of the most important parameters. The WRC describes the relationship between water content and the negative pore water pressure for a particular soil. Studies to increase the understanding of the influence of water retention properties of a soil on the rainfall In all kinds of analyses that reflect negative pore water pressures, the water retention curve (WRC) of the soil is considered to be one of the most important parameters. The objective of this paper was to illustrate what influence the choice of WRC has on the outcome of numerical simulation of negative pore water pressures. WRCs were estimated with three different methods for the same soil samples. It is evident that WRCs that are intended to describe the same material in most cases differ at substantial parts. Simple numerical analyses were also conducted to illustrate the effect these differences in estimated or determined WRCs have on the outcome of a numerical pore pressure analyses. Even minor variations in the WRC might have significant impact on the outcome of the pore pressure calculation. In the cases when negative pore water pressures are considered in pore pressure analyses, it is crucial that correctly determined, site specific water retention data is used.
infiltration process in a slope and the conditions for maintaining negative pore water pressures have been carried out by (among others) L’Heureux et al. (2006), Zahn et al. (2004) and Zhang et al. (2004).
The objective of this paper is to: (i) describe the available alternatives of
estimating the WRC in engineering practice and (ii) demonstrate the influence of the choice of the WRC parameters on the outcome of the numerical simulation of negative pore water pressures for a simple case.
The study is carried out by comparing WRCs estimated by different methods for the same soil samples. It is illustrated how the estimated WRCs may differ depending on the choice of method and available geotechnical information. Simple numerical analyses are conducted to illustrate the effect of these differences on the outcome of a numerical pore pressure analyses.
2 THE WATER RETENTION CURVE 2.1 General
The WRC describes the relationship between water content and the negative pore water pressure for a particular soil, see Figure 1.
Figure 1 Schematically illustrated water retention curve. where
θ [-]= volumetric water content,
θs [-]= saturated volumetric water content,, θr [-]=(at high negative pore water pressures) residual volumetric water content,
u [kPa]= negative pore water pressure ua [kPa]= air entry pressure.
The WRC can be determined in the
laboratory using pressure plate and pressure
membrane extractors. The measured water retention data is commonly fitted to a mathematical expression to simplify
numerical and analytical calculations. Several empirical, mathematical models have been formulated to describe the shape of the WRC (Fredlund and Xing, 1994). One of the most commonly used models is the one formulated by van Genuchten (1980), see Eq. (1).
n m r s
a u
÷÷ ø ö çç
è æ + + -
=
1
r
q q q
q (1)
θs [-] The saturated volumetric water content (water content below the ground water surface and in the saturated capillary zone). This fraction of the total volume is also called effective porosity. With effective porosity the pore volume is considered that is available for water, which is not always equal to the total porosity.
θr [-] The residual volumetric water content (at high negative pore water pressures),
a [kPa] This is a parameter related to the air entry pressure (the pressure at where the soil goes from a saturated to an
unsaturated stage).
n, m [-] These are shape parameters that describe the inclination of the curve. The parameters are related to the pore size distribution and the gradation of the soil.
Larger n gives a steeper curve and reflects a more single graded material. The parameter m is often considered to be a function of the parameter n.
It is known that soils have hysteric water retention properties. The WRC of a drying soil does not coincide with that of a wetting soil. The drying curve (which is the one usually determined in the laboratory) gives an estimation of the maximum likely negative pore water pressure corresponding to a given water content.
Alternative ways of estimating WRC are based on soil classification or particle size distribution. A number of methods, based on theoretical considerations, for estimating the WRC using the grain size distribution are available (Fredlund et al, 2002; Lu and Likos,
θr θs
ua
u [kPa]
θ [-]
2004). The simplest route to estimate the WRC is through visual soil type
classification and comparison with generic WRCs available in databases. However, the WRC is also known to be highly dependent on the soil properties and it is difficult to make generic statements of the WRC for a particular soil.
3 MATERIAL AND METHOD
The soil samples used originated from the silty slope used in the Swedish research
project focusing on long time measurements of negative pore water pressures mentioned in the introduction section. The three samples were collected from different depths and layers in the slope and are classified as a sand, a silt and a silt with clay content. Three alternative methods were used to select WRCs.
Method 1 is the simplest of the methods and based on a visual soil type classification.
The classified soil type was compared to a WRC of a corresponding soil type available in a Swedish database with water retentions curves from almost 400 Swedish soils (Andersson and Wiklert, 1972). The
corresponding WRC chosen in the database was then fitted to the mathematical
expression of van Genuchten (eq. (1)).
Method 2 is similar to method 1, but based on soil type classification by particle size determined in the laboratory. The classified soil type was compared to a WRC of a soil type with corresponding particle size
distribution available in the same database as used for method 1. The corresponding WRC chosen in the database was then fitted to the mathematical expression of van Genuchten (eq 1).
Method 3 is the most accurate method where the WRC was determined in the laboratory. The water retention data achieved was then fitted to the mathematical
expression of van Genuchten (eq 1).
Figure 2 show the WRCs for the three soils estimated according to the three
different methods. It is evident that grain size distribution is not the only governing factor determining the water retention properties of a soil. For the three soils tested here the deviation between the estimated and the measured WRCs was increasing with decreasing grain size.
0,00 0,05 0,10 0,15 0,20 0,25 0,30 0,35 0,40 0,45 0,50
0,1 1 10 100 1000 10000
Volumetric water content, θ
Negative pore water pressure, u [kPa]
Sand 1 Sand 2 Sand 3 Silt 1 Silt 2 Silt 3
Silt with clay content 1 Silt with clay content 2 Silt with clay content 3
Figure 2 WRCs, estimated in three different ways, for three different soils
4 NUMERICAL SIMULATIONS
Simple numerical analyses were carried out to illustrate the effect of the choice of WRC on the outcome of a pore water pressure analysis of a hypothetical slope. The
simulations were not performed to reproduce a real field scenario.
The geometry used in the simulations is show in Figure 3 and consists of a 15 m high slope with a slope angle of 30°. The slope consists of a homogeneous material
corresponding to material Sand 1 or Sand 3 (in Figure 2) in the first set of simulations and to material Silt 1 or Silt 3 in the second set of simulations. In all simulations the materials are assumed to have a saturated hydraulic conductivity of 1e-5 m/s. In the slope there is a fixed inclined water table where the pore water pressure is zero. Below the ground water surface the pore water pressures increase hydrostatically. All simulations started from a hydrostatic condition (also above the ground water surface). Then a precipitation was applied on the ground surface and the slope as a unit flux with the same magnitude as the saturated hydraulic conductivity (which is a high intensity for the precipitation). After certain time intervals the changes in the pore water pressure profile in a section in the middle of the slope was evaluated, see marked section in Figure 3.
Figure 3 Geometry and boundary conditions used in the numerical simulations.
The calculated pore water pressures for the two soil types (silt and sand), evaluated at
certain time intervals, are shown in Figure 4 and Figure 5 respectively. Figure 4 shows that the choice of WRC results in different shape of the calculated pore pressure profile.
The differences in the shape and velocity of the modelled water fronts that propagate down the soil profile reflect the water
retention properties of the soils. Figur 2 show that the deviation of Sand 1 and Sand 3 WRCs are limited. However, the difference between them have an significant impact on the outcome of the pore water pressure calculations and how the pore water profile develops with time (Figure 5).
The results from the simple simulations show that the calculated pore water pressures are very sensitive to how the water retention properties of the simulated soil are estimated.
The calculated pore water pressure profiles
a) b)
Y (m)
Pore-Water Pressure (kPa) 8
10 12 14 16 18
-20 -40 -60 -80
-100 0 20
Pore-Water Pressure (kPa) 8
10 12 14 16 18
-20 -40 -60 -80
-100 0 20
Figure 5 Calculated pore water pressures for the soil types a) Sand 1 and b) Sand 3.
a) b)
Y (m)
Pore-Water Pressure (kPa) 8
10 12 14 16 18
-20 -40 -60 -80
-100 0 20
Pore-Water Pressure (kPa) 8
10 12 14 16 18
-20 -40 -60 -80
-100 0 20
Figure 4 Calculated pore water pressures for the soil types a) Silt 1 and b) Silt 3.
Hydrostatic water pressure Impermeable
Precipitation
Fixed ground water table Section for evalutation
hydrostatic 0,5 d 1 d 1,5 d 2 d 2,5 d
hydrostatic 0,5 d 1 d 1,5 d 2 d 2,5 d
may differ both in shape and in the rate at which they propagate through the soil profile even at seemingly small deviations in WRCs used.
The simulations and the calculated pore water pressures are to be considered as only some illustrative examples of differences that might arise. It is important to stress that the outcome of the simulations are dependent on initial water content in the soil and the boundary conditions.
5 CONCLUSIONS
In the cases when negative pore water pressures are considered in pore pressure analyses, it is crucial that correctly
determined, soil type specific water retention data is used. Even minor variations in the WRC might have significant impact on the outcome of the pore pressure calculation and the stability analyses. It is crucial that
accurate and experimentally determined, site specific water retention data is used.
6 REFERENCES
Andersson, G. & Wiklert, P. (1972).
Markfysikaliska undersökningar i odlad jord. XXIII.
Om de vattenhållande egenskaperna hos svenska jordar. Grundförbättring, vol. 25, s 53-143.
Casagli, N., et al. (1999). Pore water pressure and streambank stability: Results from a monitoring site on the Sieve River, Italy. Earth Surface Processes and Landforms 24(12): 1095-1114.
Fredlund, D.G. (1995). The stability of slopes with negative pore-water pressures. Ian Boyd Donald symposium. Monash University, Melbourne, Australia.
M.D. Fredlund, G.W Wilson, D.G. Fredlund (2002) Use of the grain-size distribution for estimation of the soil water characteristic curve, Can Geotech. J, 39, 1103-1117.
D.G. Fredlund and A. Xing (1994) Equations for the soil-water characteristic curve, Can. Geotech. J., 31, 521-532
Geo-Slope International Ltd. 2009.SEEP/W - User's manual, fourth edition. Calgary, Alberta.
L'Heureux, J. S., Hoeg, K., and Hoydal, O. A.
2006. Numerical analyses and field case study of slope subjected to rainfall. Geotechnical special publication, 147, 2279-2290.
Lim, T.T., and Rahardjo, H. (1994). Field
measurement of matric suction in a residual soil slope.
International conference on landslides, slope stability and the safety of infra-structures. Kuala Lumpor, Malaysia: 227-234.
Lu, N. and , Likos, W. J. (2004). Unsaturated soil mechanics. John Wiley & Sons.
Van Genuchten, M. T. (1980). Closed-form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Science Society of America Journal, 44:5, 892-898.
Zhan, T. L. T., and Ng, C. W. W. 2004. Analytical analysis of rainfall infiltration mechanism in
unsaturated soils. International journal of geomechanics, 4:4, 273-284.
Zhang, L. L., Fredlund, D. G., Zhang, L. M. and Tang, W. H. 2004. Numerical study of soil conditions under which matric suction can be maintained.
Canadian Geotechnical Journal, 41:4, 569-582.
Öberg, A.-L. (1995). Negative pore pressures - Seasonal variation and importance in slope stability analysis. 1st international conference on unsaturated soils, UNSAT ’95, Paris.
http://www.swedgeo.se/upload/SGI- tjänster/pdf/Siltslänter.pdf
