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This paper was downloaded from the Online Library of the International Society for Soil Mechanics and Geotechnical Engineering (ISSMGE). The library is available here: https://www.issmge.org/publications/online-library

This is an open-access database that archives thousands of papers published under the Auspices of the ISSMGE and maintained by the Innovation and Development Committee of ISSMGE. A grain scale model to predict retention properties of unsaturated

Un modèle à échelle de grain pour prédire les propriétés de rétention des sols non saturés

Ehsan Nikooee1,2, Ghassem Habibagahi1, Behrooz Daneshian1 1Department of Civil and Environmental Engineering, Shiraz University, Shiraz, Iran, [email protected]

Thomas Sweijen2, S. Majid Hassanizadeh2 2Department of Earth Sciences, Utrecht University, Utrecht, Netherlands, [email protected]

ABSTRACT: The retention curve (SWRC) is a fundamental relationship in unsaturated soils. Retention properties of unsaturated soils are important for many practical applications. Mechanical and hydraulic properties of soil in unsaturated state depend on the soil saturation. Therefore, knowledge of the retention properties of unsaturated soils and determination of various factors affecting the soil water retention curves are of considerable importance. Experimental methods to measure SWRC are time consuming and costly. To model soil water retention curves, various empirical and physically-based models have been introduced in the literature. There are two types of physically based models which have been recently proposed that capture two phase flow in unsaturated and petroleum engineering, namely: pore scale and grain scale models. Grain scale models can directly consider the presence of grains of different sizes in the sample whereas pore scale models require visualization or estimation of the pore geometry. In this study, a grain scale model is employed for determining soil water retention properties using the discrete element technique and capillary invasion algorithm. KEYWORDS: Discrete element technique, unsaturated soils, soil water retention curve, two phase flow

RÉSUMÉ: La courbe de rétention d'eau (anglais : soil water retention curve SWRC) est une relation fondamentale dans les sols non saturés. Les propriétés de rétention des sols non saturés sont importantes pour de nombreuses applications pratiques. Les propriétés mécaniques et hydrauliques des sols non-saturés dépendent de leur degrée de saturation. Par conséquent, la connaissance des propriétés de rétention d’eauet la détermination de divers facteurs affectant les courbes de rétention d'eau sont d'une importance considérable dans les sols non saturés. Les méthodes expérimentales pour mesurer la SWRC sont généralement longues et coûteuses. Pour modéliser les courbes de rétention d'eau du sol, divers modèles empiriques et physiques ont été introduits dans la littérature. Il existe deux types de modèles physiques proposés récemment qui incluent le flux diphasique dans la poromécanique des sols non saturés et dans l'ingénierie pétrolière, à savoir: les modèles à l'échelle des pores et à celle du grain. Les modèles à échelle du grain peuvent considérer directement la présence de grains de différentes tailles dans l'échantillon tandis que les modèles à l’échelle des pores nécessitent une visualisation ou une estimation de la géométrie des pores. Dans cette étude, un modèle à échelle de grain est utilisé pour déterminer les propriétés de rétention d'eau du sol en utilisant la technique des éléments discrets et l'algorithme d'invasion capillaire. MOTS-CLES: Technique des éléments discrets, sols non saturés, courbe de rétention d'eau, écoulement diphasique

1 INTRODUCTION In pore network models, the connectivity among different pores is accounted for. In a bundle of capillary tubes model, it is Methods that are commonly used in unsaturated soil mechanics assumed that all pores of a certain size will be drained at a to determine soil water retention curve based on the soil grain given matric suction level. In the real soil porous network, size distribution fall into three categories. however, this assumption does not hold as it does not happen in real materials due to the presence of throats of various size In the first category, a statistical method is employed to among different pores. The presence of smaller diameter throats estimate soil water retention curve based on the volume-mass among pores would prevent the complete drainage of all pores properties and particle size distribution (Gupta and Larson 1979, of the same diameter and only some of them will undergo the Ghosh 1980, Aberg 1996). In the second class of models, the drainage depending on the throat size spatial distribution. artificial intelligence based methods such as genetic Therefore, a pore network model which consists of both throats programming, multi-agent genetic programming and neural and pores provides a more realistic picture of soil porous networks have been used to predict the soil water retention data structure than the simple bundle of capillary tube model (Schaap and Bouten 1996, Pachepsky et al. 1996, Koekkoek (Nikooee et al. 2014). and Booltink 1999, Johari et al. 2006, Garg et al. 2014 a,b). The Achilles heel of the pore network models is their need for a In the third class of approaches, a physically based and correct characterization of the soil porous structure. This is not conceptual model of soil is presented. Such models aim to easily accessible and most often needs sophisticated methods determine the pore size distribution from grain size distribution such as micro-tomography imaging. One alternative is to use based on which the soil water retention curve is then predicted. grain scale models. The main idea is to build a packing made of The major difference among the models presented in the last grains of different size and then extract the pore network category is the mathematical conceptualization considered for directly from the packing. The basic soil properties such as the particles and pores. Some of them portray soil porous particle size distribution and are often employed to structure as a bundle of capillary tubes (Arya and Paris 1981) build the packing. where as there are models which depict soil porous structure as a network of pores and throats (Rostami et al. 2013, Khaksar et al. 2013, Joekar-Niasar 2016).

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There are various mathematical procedures to connect the soil porous structure to its texture and/or to directly estimate pore size distribution from grain size distribution. For instance, fractal techniques have been used for this purpose (Tyler and Wheatcraft 1990, Rieu and Sposito 1991). But the problem with fractal techniques is that usually such mathematical idealizations do not consider the presence of different elements in the soil pore network, namely, pores and throats (i.e., the presence of throats is commonly overlooked in the fractal techniques). Therefore, to extract a rather realistic depiction of soil porous structure preserving real connectivity among different pores, a mathematical tessellation technique which can distinguish pores and throats is a better alternative. Having noted the merits of a physically based method for the determination of soil water retention curve (SWRC), in this study a grain scale model is presented. A mathematical tessellation technique is employed to obtain different pore units and throats. Finally, a capillary invasion algorithm is pursuit to obtain the soil water retention curve. In the following section, details of the computational procedure are presented. Finally, the model is assessed against the experimental data available in the literature and compared to present state-of-the-art techniques.

2 COMPUTATIONAL PROCEDURE

2.1 Scope and assumptions The soil grains were considered to be spherical and rigid. Figure 1. (a) One pore unit surrounded by four particles; (b) Scalene Furthermore, the simulation was quasi-static (we modelled triangle forming pore throat and its inscribed cricle (c, d) Inscribed equilibrium experimental data-points), and interaction of water sphere in a pore body with radius Ri. and soil microstructure was overlooked. We focused solely on modelling the drying branch of the soil water retention curve, and air-water contact angle was set to zero.

2.2 Soil Packing generation Table 1. The basic properties of soil # 11605 of soil vision database

USCS Texture Poorly graded In order to construct the packing, an open source discrete element code YADE-DEM was used (Šmilauer 2010). The Soil Counter 11605 model domain was a cubic region with dimensions 0.02m x 0.02m x 0.02m, where 2000 randomly generated particles were Cu 1.4733 distributed. Particle size distribution of two soil samples (soil # Cc 1.0021 11605 and soil #10700 of soil vision database) were used in order to construct the soil packing (see Table 1-4 for the basic , n 0.382 properties of these two soil samples) Void Ratio, e 0.618

Values of mechanical parameters of the particles used in our Specific Gravity, Gs 2.67 simulations are reported in Table 3. Linear contact mechanics was used, using the parameterization as presented in Belheine et al. (2009). Table 2. The particle size distribution of soil # 11605 of soil vision Particle Diameter (mm) Percent Passing The initial porosity of the randomly placed particles was larger than 0.99. To compact the cloud of particles, a confining 2 0.989 of one kPa was applied on all domain boundaries. An 0.417 0.954 arbitrary large angle of 30° was used in this stage. The compaction was pursuit which resulted in a loose packing. The 0.297 0.654 packing was further compacted by a second phase during which the friction angle was decreased incrementally until the porosity 0.246 0.386 of soil samples was reached. 0.149 0.013

2.3 Extraction of the pore unit assembly

From the packing of spheres, an assembly of pore units (or tetrahedron) was extracted using a regular tessellation technique

(see Chareyre et al. 2012). Vertices of each tetrahedron were located at the centre of four neighboring particles (see Figure 1). The scalene triangle between two adjacent tetrahedra was considered as the pore throat between them.

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Table 3. The basic properties of soil # 10700 of soil vision Silty Sand Soil Counter 10700 Cu 7.3761 Cc 2.0641 Porosity, n 0.394 Void Ratio, e 0.65 Specific Gravity, Gs 2.64

Table 4. The particle size distribution of soil # 10700 of soil vision Particle Diameter (mm) Percent Passing 2 1.000

1 0.994

0.5 0.921 Figure 2. The result of grain scale method as compared to the 0.25 0.735 experimental data as as Fredlund et al. (1997) estimation for sample # 11605 of soil vision database 0.106 0.279

Table 5. Input micromechanics parameters Property Value Unit

Stiffness ratio* 0.04 -

Particle normal stiffness 9.6 x108 N/m

Friction angle 30 °

Number of particles 2000 - * The ratio of tangential stiffness to normal stiffness

2.4 Capillary invasion algorithm

Following Yuan et al. (2016) and Nikooee et al. (2016), the Mayer-Stowe-Princen (MSP) algorithm was used to determine throat entry pressure. A throat was considered to be invaded provided that the suction imposed on the boundaries of the domain was greater than the throat entry pressure, and as long Figure 3. The result of grain scale method as compared to the as a continuous drainage path could form. experimental data as well as the Fredlund et al. (1997) estimation for soil sample # 10700 of soil vision database 2.5 Comparison with available techniques As it can be seen both Fredlund technique and grain scale The predicted soil water retention curves were compared with method are of a good agreement with the experimental data. For predictions obtained from other available procedures. For this soil # 11605 however the grain scale approach better matches purpose, the method of Fredlund et al. (1997) which also the residual degree of saturation (while Fredlund et a. method predicts soil water retention curves based on the grain size tends to further decrease). For soil #10700, both approaches distribution was employed. Fredlund et al. (1997) assume soil cannot capture the soil air entry value quite well. For this soil, particle size distribution can be divided into a number of equal Fredlund et al. (1997) method results, however, in a closer size group of particles. A packing porosity and SWRC are assumed for each group of particles. The sum of sub-SWRCs match. produces the targeted soil water retention curve. In order to use Fredlund approach, the associated module in soil vision 4 CONCLUDING REMARKS software was used. In the current study, a grain scale method for determining 3 RESULTS AND DISCUSSION SWRCs based on the grain size distribution and void ratio was presented. The method showed reasonable accuracy. The Figure 2 presents the predicted SWRCs versus Fredlund et al. (1997) extension of the approach to imbibition curves and inclusion of estimation as well as experimental data-points for soil # 11605. Figure more sophisticated phenomena such as contact angle hysteresis 3 presents the predicted SWRCs versus Fredlund et al. (1997) estimation as well as experimental data-points for soil # 10700. and dynamic two phase flow are the subject of ongoing research of the authors.

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5 ACKNOWLEDGEMENT network investigation on hysteresis phenomena and influence of state on the SWRC. International Journal of , 15(5), p.04014072. Ehsan Nikooee, Thomas Sweijen and Majid Schaap M.G. and Bouten W. 1996. Modeling water retention Hassanizadeh would like to thank the European Research curves of sandy soils using neural networks. Water Resources Research, 32(10), 3033-3040. Council for ERC advanced research funding under the Šmilauer V., 2010. Cohesive particle model using the discrete European Union's Seventh Framework Programme (FP/2007- element method on the yade platform (Doctoral dissertation, Université 2013) / ERC Grant Agreement no. 341225 which has supported de Grenoble; and Czech Technical University in Prague. them during this study. Thomas Sweijen gratefully Tyler S.W. and Wheatcraft S.W. 1990. Fractal processes in soil acknowledges financial support from the Technology water retention. Water Resources Research, 26(5), 1047-1054. STW, the technological branch of the Netherlands Yuan, C., Chareyre, B. and Darve, F., 2015. Pore-scale simulations Organisation of Scientific Research, NWO, and the Dutch of drainage in granular materials: finite size effects and the ministry of Economic Affairs under contract no. 12538, entitled representative elementary volume. Advances in Water Resources, in press. Interfacial effects in ionized media.

6 REFERENCES

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