USE OF NUMERICAL MODELLING TO OPTIMISE LABORATORY TESTING DESIGN FOR THE MEASUREMENT OF THE SOIL WATER RETENTION CURVE

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1 Proceedings of the 13 th International Conference of Environmental Science and Technology Athens, Greece, 5-7 September 2013 USE OF NUMERICAL MODELLING TO OPTIMISE LABORATORY TESTING DESIGN FOR THE MEASUREMENT OF THE SOIL WATER RETENTION CURVE CABARKAPA Z. 1, MAVROULIDOU M 2,* and GUNN, M.J GEOTECHNICAL CONSULTING GROUP LTD, 52a CROMWELL ROAD, LONDON, SW7 5BE, UK 2. FACULTY OF ENGINEERING, SCIENCE & THE BUILT ENVIRONMENT, LONDON SOUTH BANK UNIVERSITY, 103 BOROUGH ROAD, LONDON, SE1 0AA, UK *Corresponding author: M.Mavroulidou; mavroum@lsbu.ac.uk EXTENDED ABSTRACT The quantification of the hydraulic properties of porous media such as the hydraulic conductivity and soil water retention curve (SWRC), i.e. the constitutive relationship between soil suction and the amount of water in the soil, is of paramount importance for the field of geo-environmental engineering (landfills/engineering barriers for waste containment), water quality (water and solute flow in the vadose zone and related groundwater and soil contamination problems), hydrology (runoff, infiltration), vegetation growth linked to water and nutrient uptake from the soil (determined by the storage of water in the soil and related environmental problems linked to soil water balance of vegetated areas) and other environmental hazards (e.g. erosion, slope instability etc.). Reliable measurements are therefore of particular relevance for the accuracy of simulations to solve the above problems, which may well depend on the accuracy of the properties used in the models. A particular problem with determining the SWRC is that conventional SWRC measurement techniques are very long and tedious, as suction equalization times for the measurement of a single curve point at a time may typically last for one to two weeks. Consequently, many months may be needed in order to complete the measurements. This paper discusses how sophisticated numerical modelling can be used prior to laboratory testing to optimize SWRC testing design. The optimization of the testing design is an iterative process and numerical modeling can prove to be a useful tool for the minimization of the testing times, without compromising the quality of the tests. In this study, the validation of the numerical simulator STOMP (Subsurface Transport Over Multiple Phases) in the laboratory environment is first shown. The software is then used prior to testing a) to assess the assumption of a uniform distribution of the degree of saturation throughout the specimen in time (an implicit assumption commonly made when interpreting SWRC testing results) and b) to optimize the test design and testing times e.g. in terms of sample size, imposed flow rates etc. Two different techniques to determine the SWRC are considered and numerically modelled in this paper, namely the filter paper technique and a transient methodology using the triaxial cell, developed by the authors. The suggested methodology is then demonstrated for the measurement of the drying and wetting SWRC of a compacted silt soil (with different initial sample preparation conditions) tested under different confining stresses. The results based on the testing design suggested by the numerical modelling are realistic and show little scatter, giving confidence in the usefulness of the suggested procedure. KEYWORDS: vadose zone and landfill hydraulics, water retention, laboratory measurement, numerical modelling

2 1. INTRODUCTION The soil water retention curve (SWRC) expresses the relationship between water content or degree of saturation and the associated matric suction, i.e. the water storage capacity of a soil at a given soil suction. Physically, the curve indicates how much energy is required at any given moisture content, to remove a small quantity of water from the soil. The curve plays an essential role in characterising the hydro-mechanical behaviour of unsaturated soils, as the unsaturated soil behaviour is highly dependent on the magnitude of soil suction, which in turn is influenced by the moisture content of a given soil. The SWRC of the soil is necessary for the analysis of flow and transport in variably saturated porous media and related engineering problems. It is therefore of paramount importance for various problems in the field of geo-environmental engineering (landfills/engineering barriers for waste containment), water quality (water and solute flow in the vadose zone and related groundwater and soil contamination problems), hydrology (runoff, infiltration), vegetation growth linked to water and nutrient uptake from the soil (determined by the storage of water in the soil and related environmental problems linked to soil water balance of vegetated areas) and environmental hazards such as erosion, slope instability, swelling and shrinking soils etc. In addition to this, methods have been proposed to predict volume change, shear strength, diffusion, adsorption, vapour diffusion, thermal conductivity, and a variety of other properties of unsaturated soils, based in part on the information provided by their respective soil-water retention curves (Barbour, 1998). The SWRC is therefore fundamental to predict and interpret the behaviour of unsaturated soils. Computer models incorporating the behaviour of unsaturated soil are now increasingly used by researchers and a few commercial programs incorporating unsaturated soil features are also available. The successful use of such models depends on reliable information on the soil water retention curve of the soil. A number of recent SWRC measurement techniques are available for measuring the SWRC of the soil. Traditionally, the most common techniques have been the filter paper method, the pressure plate or Tempe cell. Such techniques are very lengthy and testing procedure has to be followed carefully for reliable suction measurements. Recent advances suggested alternative methodologies that allow for some flexibility in the boundary conditions in the specimen (see e.g. Mavroulidou et al, 2009 and Mavroulidou et al, 2013). To interpret the results of any of these tests, knowledge of the internal state of the soil sample is required. This is commonly assumed to be uniform. In particular conventional interpretation of filter paper results does not account for the non-uniformity of suction within the sample. However in reality testing involving flow of air and water into or out of the sample may lead to a non-uniform state, as flow implies a pressure gradient throughout the sample. Assumptions as to when equilibrium conditions are achieved throughout the sample may affect the interpretation. This paper shows how numerical modelling can be used to understand the physics of the processes involved during laboratory testing in order to optimise the testing design. In this case this concerns the modelling of common filter paper testing as well as of an advanced SWRC measurement technique using the triaxial cell. Subsequently, we present and evaluate a number of SWRC results produced using the suggested procedure. 2. NUMERICAL ANALYSES 2.1 STOMP software STOMP was developed at the Pacific Northwest National Laboratory to solve a wide variety of non-linear, multiple-phase flow and transport problems in saturated and unsaturated soils under isothermal or non-isothermal conditions. STOMP solves three mass balance equations (water, air and energy) that describe subsurface flow over multiple phases. The aqueous phase primarily comprises liquid water and small quantities of dissolved air. Water transport occurs by advection through the aqueous and gas phases and by diffusion-dispersion through the gas phases. The gas phase comprises variable amounts of air and water vapour. Air transport occurs by advection

3 and diffusion-dispersion through the aqueous and gas phases. Heat transfer occurs by advection of phase mass, diffusion of component mass and thermal diffusion through the fluid and solid phases. The software was extensively verified by its developers and other users of the code through a series of comparisons of the numerical results against known analytical solutions or experimental observations. Prior to the use of STOMP for this study a number of verification exercises were also performed by Cabarkapa (2001). 2.2 Validation of the software in the context of the suggested laboratory testing To further validate the ability of the software to simulate the complex physics involved in the laboratory testing of unsaturated soils, a series of air-drying tests were performed and compared to STOMP predictions. Despite its apparent simplicity the air-drying process is very complex. During this process an initial thermodynamic imbalance occurs between the ambient vapour concentration and that within the specimen. To restore the thermodynamic equilibrium the specimen exchanges water vapour with the environment. Consequently, the vapour pressure within the specimen decreases. For the liquid and vapour to remain at isothermal equilibrium within the specimen, the liquid pressure also decreases and this requires the water to evaporate. The driving force for flow is a gradient in suction, humidity and temperature respectively. Therefore the successful simulation of the evaporation experiment would test the ability of STOMP to accurately model coupled liquid, vapour and heat flow and would give confidence in the use of this model for the simulation of the subsequent laboratory tests. In this study the laboratory tests used commercially available quartz silica flour with angular particles (mean particle size D 50=0.018 mm), mixed with 25% per mass of water. The grain size distribution of the soil is shown in Figure 1a). The software requires some information on the SWRC of the soil in order to perform the analyses. Obviously at the start of a SWRC curve measurement the SWRC of a particular soil is not known as its determination is the actual aim of the testing. An additional difficulty is that the SWRC of a soil varies according to a number of factors (e.g. stress state/history; initial void ratio/initial water content, hydraulic hysteresis etc.) causing differences in the SWRC of the same soil (e.g. Vanapalli et al, 1999; Mavroulidou et al, 2009). However for the purposes of preliminary numerical simulations of the experimental procedure used to optimise the testing design an approximate curve can be used for this stage. This can be based on typical measured curves for similar types of soil (e.g. using databases such as UNSODA, see Nemes et al, 2001), correlations with the PSD curve of the material (e.g. Arya & Paris, 1981; Nasta et al, 2009) or preliminary measurements of SWRC based on other conditions. The SWRC of the silica flour in terms of degree of saturation versus matric suction, used in the numerical analyses, is shown in Figure1b). It was described mathematically by Van Genuchten s (1980) model as: 1 = r ( s r) m [1+ (a ) n ] (1) where: ψ =the pressure head; θ, θ s and θ r = the volumetric water content, saturated volumetric water content and residual volumetric water content respectively; a, n and m = curve-fitting parameters. The least-squares method computer program FIT created by Cabarkapa (Cabarkapa, 2001) with a non-linear curve-fitting algorithm was used to fit the required model parameters to the experimental results of SWCC. The constraint m=1-1/n was used during curve-fitting. According to FIT the Van-Genuchten model parameters of the assumed drying curve of the silica flour soil were: a=0.l, n=1.8, θ r = The variation of the aqueous phase permeability k l as a function of saturation and the gaseous phase permeability k g were respectively expressed as (Mualem 1976) :

4 k l (S e) = k sat (S e) 0.5 1/m [1- (1 S e ) m ] 2 (2) k g (S e) = k sat (S g) 0.5 1/m [(1- S e ) m ] 2 (3) where: k sat =the saturated soil aqueous phase permeability (k sat=0.5 cm/h for the present soil), S e = the effective saturation S e= (S r S r, res) /(1 S r, res) in which S r,res is the residual degree of saturation, m = coefficient defined above (van Genuchten s model) and is taken here to be m=1-1/n. The effective thermal conductivity of the silica flour used was computed as (White & Oostrom, 1996): k e = k s (1 - n T ) + k l [n T - n D (1 - S l)] (4) where: k e = the equivalent thermal conductivity tensor, W/m K; k s =porous media thermal conductivity tensor, W/m K; n D, n T =the porosity (diffusive and total respectively); S l =the actual aqueous saturation; k l =the permeability to the aqueous phase. The thermal conductivity and specific heat of the soil particles were respectively 2.3 W/mK and J/kgK. Statically compacted specimens of 50 mm diameter and 100 mm height were created. After compaction, the specimen had a degree of saturation of 75.06%, porosity n= and volumetric water content θ= The specimen was placed on an electronic balance and was allowed to dry freely for a period of three days. Evaporation was allowed from the exposed surfaces of the specimen under controlled laboratory conditions (air temperature 22.5±0.5 C and relative humidity of 35). The change in mass of the specimen and hence the water content was continuously monitored during the test. A relative humidity sensor with an accuracy better than ±2% at 20ºC was used for relative humidity (RH) measurements. Conventional thermocouples of an accuracy of ±0.25ºC were installed along a vertical profile and on the top surface. An infrared camera was also used to measure temperature distribution at the surface of the specimen based on thermal energy radiation from the specimen during the air-drying. Data acquisition software was used to automatically record temperature, relative humidity and the mass of the specimen. For the numerical analysis the 100mm height x 50 mm diameter cylindrical specimen was represented by an axisymmetric 10x25 rectangular finite difference mesh (250 grid points; radial spacing 2.5 mm; vertical spacing 4.0 mm). Figure 2 shows the comparative non-isothermal flow numerical analysis results from STOMP in terms of temperature and mass evolution in time, against the respective measurements. Overall the numerical and experimental results were in good agreement which gave confidence in the use of STOMP for the SWRC testing simulation. (a) (b) Figure 1. Silica flour soil characteristics: (a) PSD; (b) Trial SWRC used in STOMP

5 Figure 2. Validation of STOMP software in a laboratory experiment situation (specimen air-drying) 2.3 Use of STOMP for the optimization of the testing processes Simulation of air-drying /filter paper testing for the drying SWRC The filter paper method is based on the principle that if a wet soil specimen is placed in contact with a dry filter paper water will flow from the sample to the filter paper until equilibrium is established. It is assumed that when equilibrium is reached the suction in the sample can be related with water content in the filter paper. Understanding the air drying process is of great importance for filter paper measurements of suction, where the sample is usually left on the bench for a couple of hours and then the filter paper is placed on the sample until equilibrium is achieved. STOMP was used to study of the movement of the drying front and the time-dependent distribution of suctions /degree of saturation within a drying soil sample. The effects of different types of boundary condition on the evaporative fluxes from a drying sample were also investigated. Initially a sample of dimensions 100 mm height and 50 mm diameter was used for the simulations. The conditions during air drying were then simulated. Due to space limitations only indicative results are shown for each analysis type. The first simulation is of "top drying", i.e. assuming that the sample would be kept laterally wrapped during drying (this would be necessary for very wet samples) and left on an impermeable surface to dry out from the top only for a certain period before starting the subsequent filter paper measurement. In this case there is liquid water migration towards the bottom of the sample because of gravitational body forces and water vapour migration to the top surface because of the lower relative humidity at the top boundary. From the results (Fig 3a) it can be seen that after 30 days there is a transient state with continuing drying. Both the aqueous and gas pressures increase with depth and the effects of vapour pressure lowering, keep the top of the sample from drying to the residual saturation. The second simulation is of "top and bottom" drying relevant to a situation where the sample was left to dry on a coarse grid, so that drying from both surfaces could happen. From Figure 3b) it can be seen that levels of degree of saturation at the middle of the sample (i.e. corresponding to the drainage path) similar to those represented in Figure 3a) at the bottom of the sample (the furthest points from the drying surface in this case) are now reached in 15 days. This simulation eventually reached steady state conditions after 30 days (not shown here) as the soil dried to equilibrium conditions slightly above the residual saturation because of vapour pressure lowering. During the transient state stages, the bottom of the sample remained wetter than the top, because of gravitational body forces causing water to migrate downwards. For drying from all surfaces (i.e. if the sample was now taken out of the laterally wrapping) as in the previous simulations, the water initially migrated toward the bottom of the sample due to gravitational body forces causing a skewed saturation. As drying proceeded capillary forces became stronger making for a more uniform distribution of soil moisture. From Figure 3c) it can be seen that the mid-point of the

6 sample (i.e. the most point remote from the drying surfaces) reached similar degrees of saturation as in Figures a) and b) within 2.5 days only and equilibrium conditions (not shown here) after only five days. The above findings are very interesting as in the laboratory the sample is usually left to dry for a few hours only before being wrapped again with filter paper, which means that the suction conditions between the bottom and top of the sample would not be uniform. Even if suction measurements were taken at the mid-point of the sample, these would not necessarily be representative of the suction state throughout the sample. It should also be noted that for samples without true or apparent cohesion it would be difficult to ensure all-round drying with samples of the dimensions simulated, as the samples would collapse and would be difficult to handle. It can therefore be concluded that short samples should be used for filter paper testing. Further analyses, not shown here for the sake of brevity, indicated that the use of samples of 25mm height or less would be necessary to ensure uniformity in terms of suction, within a reasonable drying time. Samples of standard oedometer dimensions (20 mm height and 75mm diameter) were therefore used and an all-round drying procedure was adopted to air-dry the sample prior to filter paper testing Simulation of SWRC testing using a triaxial cell testing procedure The simulated procedure concerned a triaxial system used for the SWRC as described in Mavroulidou et al (2013). A commercially available stress path triaxial system modified to accommodate the axis translation technique for unsaturated soil testing was used. For this, the lower coarse porous disc was replaced by a high air entry value ceramic disc (semi-permeable membranes can also be used), acting as a barrier for air flow, so air must flow out from the top of the specimen as it is pushed aside by infiltrating water. To conduct a test, the specimen was first saturated and then air pressure was applied on the top of the specimen through the upper (coarse) porous stone, using an air pressure source. The water pressure was controlled /measured at the bottom of the specimen. The difference between the air and water pressures is the matric suction in the specimen. The following instrumentation was used for SWRC measurements: (a)pressure transducers to measure the cell pressure, pore air pressure at the top of specimen and pore water pressure at both the base and at mid-height of the specimen; (b)an external volume gauge to measure water volume changes; (c) Miniature submersible linear variable differential transformers (LVDT) to measure axial and radial strain internally; (d) A temperature sensor to measure temperature near the triaxial cell. To conduct a SWRC test, a computer-controlled pump was used to control the continuous flow of water into or out of the soil specimen (for the case of wetting and drying SWRC tests respectively), while changes in pressure and suction applied to the specimen were measured. The pump has the ability to supply or withdraw water at a variable rate, and this feature was used to ensure uniform saturation conditions in the triaxial specimen during testing, according to the findings of the numerical modelling. Water content measurements were continually obtained during the drying or wetting of the specimen. Using overall volume changes of the specimen S r is determined as: S r Gsw0 (1 e0 ) w e - (1 e ) 0 0 v (5) where: G s = the specific gravity of the soil particles; e 0 =the initial void ratio; w 0 = the initial water content; ε v = the total volumetric strain; ε w = the water volumetric strain The key parameter in the suggested technique for the SWRC measurement is the rate of withdrawal: if too slow, the test will be lengthy; if too fast, the degree of saturation will become non-uniform within the specimen. The uniformity of the specimen is an implicit assumption in the interpretation of results and it needs to be ensured for accurate predictions. Numerical analyses prior to testing in the form of parametric studies were therefore performed using STOMP, to assess the optimal pumping or water injection rate

7 for this particular test. These assumed different rates of water flow in or out of the specimen, selected as a decreasing fraction of the saturated water permeability k sat of the soil. Figure 4(a) and (b) show indicative numerical analysis results based on a drying SWRC test on a compacted silica flour specimen of 50 mm height. The results in Figure 4(a) represent the evolution of degree of saturation with time at three points in the specimen (located at different heights) for a constant, fast flow rate equal to k sat/5. It can be clearly seen that after a certain time the degree of saturation at the top and bottom of the specimen differs. This is because at lower degrees of saturation, the water permeability of the specimen also decreases and hence if the pumping rate is maintained constant and fast, this soil drainage rate could not match the pumping rate. The problem could be resolved by using low rates of pumping equal to k sat/250, leading to constant degrees of saturation throughout the specimen (see Fig. 4(b)). However conducting the test at low pumping rates would be time-consuming. For this reason further STOMP analyses were then performed using variable pumping rates (allowing for faster water extraction initially and slower water extraction at later saturation stages so that the degree of saturation throughout the specimen was uniform during the measurements). These enabled the selection of appropriate variable flow rates for the silica silt soil, which were subsequently used in the tests. The exact flow rate values to be used for each soil would of course depend on the soil properties and specimen geometry. Again the shorter size of the specimen for the SWRC testing was also selected based on prior numerical analyses (not shown here) as these showed that it was difficult to obtain uniform conditions with longer specimens. (a) (b) (c) Figure 3. Air drying simulations evolution of degree of saturation S r with time: (a) top drying, t=30 days; (b) top and bottom drying, t=15 days; (c) All-boundary drying: 2.5 days

8 Figure 4 Variation of S r of the specimen with time in triaxial system (numerical results). 3. APPLICATION TO THE MEASUREMENT OF THE SWRC OF A SILICA FLOUR SOIL Figure 5 shows some indicative SWRC results for the compacted silica flour specimens, obtained from the triaxial cell method with prior numerical modeling to optimize the testing. Desaturation /resaturation of the specimen was achieved by withdrawing (or injecting) water from the specimen using a digital pressure/volume controller and measuring the pressure difference between top and bottom of the specimen. At the same time the pressure transducer measured the corresponding water pressure at the bottom of the specimen. During testing, the pore pressures were checked at the mid-height of the specimen using a miniature pore pressure transducer: Discrepancies between the porewater pressure at the base and middle of the specimen (which were not captured in the numerical model results) were observed only rarely. This confirmed the assumption that for the purposes of the laboratory simulations to obtain ranges of flow rates, an approximate SWRC was sufficient. In the rare cases of discrepancies between the porewater pressure at the base and middle of the specimen the abstraction (or injection) rate was reduced. The trends from the experimental SWRC results are as expected and give confidence in the suggested technique: namely it can be clearly seen that specimens subjected to higher net stresses p-u a showed higher air-entry values than specimens subjected to lower net stresses and that the rate of drying (determined by the slope of the curves) was reduced with increased net stress. This can be explained by the decrease in the voids ratio of the specimen under higher applied load and can be supported by the findings of other researchers (e.g. Vanapalli et al, 1999). Hysteresis (a well-known feature of the SWRC) can also be noted for specimens subjected to reversals in the drying/wetting conditions. The results of each SWRC branch in Fig. 5a were obtained in a matter of days which is one important advantage of the suggested methodology. (The exact length of the testing will obviously depend on soil type /state, specimen dimensions etc.). Figure 5b shows indicative filter paper results of statically compacted samples of silica silt with appropriate dimensions and equilibration periods as indicated by the numerical analyses. These were subject to atmospheric drying and tested using Whatman No 42 filter paper. The matric suctions for each measurement point was calculated using the calibration by Chandler and Gutierrez (1986) for this filter paper type. Volume measurement of the soil sample was performed using calipers. The results show realistically and with little scatter the expected effect of the initial state of the sample in terms of void ratio, as induced by the different compactive forces. Namely that the higher the initial (after compaction) void ratio the lower the water retention of the soil, translated as lower degrees of saturation throughout the tested suction ranges, while the shape of

9 the curves shows similarity despite the shift in the y- direction due to the respective initial void ratios. 4. CONCLUSIONS This paper showed how numerical modelling can be used to assist laboratory testing design for efficient and reliable SWRC measurements. This was demonstrated for two different SWRC tests, namely filter paper testing and the determination of SWRC in the triaxial cell, using a transient state methodology. Prior numerical simulation of these tests was used to select appropriate sample sizes and (in the case of SWRC testing in the triaxial cell) variable (adjustable) flow rates, ensuring uniform saturation throughout the specimen during SWRC testing, while at the same time shortening the testing period. After validation, the technique was successfully applied for the measurement of drying and wetting curves of a silica flour soil subjected to different levels of confinement. For the triaxial cell tests, the numerical simulations showed that the flow rate should be carefully selected for reliable measurements. The numerical analyses also showed that the measurement accuracy can be further improved using a smaller specimen height to improve the uniformity in the degree of saturation throughout the specimen during testing. For the triaxial testing methodology testing times were considerably reduced compared to conventional SWRC testing due to the optimisation of the water extraction/supply rate. Note that, once set up and validated, the software can be a useful tool in the experimental design and interpretation of tests on other soils, overall increasing the testing efficiency and accuracy. (a) (b) Figure 5. Examples of experimental SWRCs: (a) triaxial testing; (b) filter paper testing Acknowledgements The authors would like to thank Dr. Mark White from Pacific Northwest National Laboratory, U. S. Department of Energy for providing STOMP software REFERENCES Arya, L.M. and Paris, J.F., 1981, A physico-empirical model to predict the soil moisture characteristic from particle size distribution and bulk density data, Soil Science Society of America Journal, 45: Barbour, S.L. (1998) Nineteenth Canadian Geotechnical Colloquium: The soil-water characteristic curve: a historical perspective. Canadian Geotechnical Journal, 35, Cabarkapa, Z., 2001, Mechanical behaviour and modelling of unsaturated soil. PhD thesis. London: LSBU Chandler, R.J. & Gutierrez, C.I The filter paper method for suction measurement, Géotechnique 36 (2): Mavroulidou, M, Zhang, X. Cabarkapa, Z. and Gunn MJ (2009) A study of the laboratory measurement of the Soil Water Retention Curve, 11th International Conference on the

10 Environmental Science and Technology (CEST2009), 3 5 September 2009, Chania, Greece, pp. A907-A915; ISBN: Mavroulidou, M., Cabarkapa, Z. and Gunn, MJ (2013) Efficient laboratory measurements of the Soil Water Retention Curve, Geotechnical Testing Journal, 36(1), DOI: /GTJ Mualem, Y. 1976, A new model for predicting the hydraulic conductivity of unsaturated porous media, Water Resources Research, 12: Nasta, P., Kamai, T., Chirico, G.B., Hopmans, J.W. and Romano, N., 2009, Scaling soil water retention function using particle-size distribution, Journal of Hydrology, 37(3-4): Nemes, A., Schaap, M. G., Leij, F. J., and Wösten, J. H. M, 2001, Description of the unsaturated soil hydraulic database UNSODA version 2.0, Journal of Hydrology, 251(3-4): Vanapalli, S.K., Fredlund, D.G. and Pufahl, D.E, 1999, The influence of soil structure and stress history on the soil-water characteristics of compacted till, Géotechnique, 49 (2): Van Genuchten, M.T., 1980, A closed form equation for predicting the hydraulic conductivity of unsaturated soils, Soil Science Society of America Journal, 44: White, M.D. & Oostrom, M., 1996, STOMP-Subsurface Transport Over Multiple Phases, Theory Guide, Pacific Northwest National Laboratory, Washington.