Hydro-mechanical Analysis of Rainfall-Induced Landslides 9811507600, 9789811507601

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Lizhou Wu Runqiu Huang Xu Li

Hydro-mechanical Analysis of Rainfall-Induced Landslides

Hydro-mechanical Analysis of Rainfall-Induced Landslides

Lizhou Wu Runqiu Huang Xu Li •

•

Hydro-mechanical Analysis of Rainfall-Induced Landslides

123

Lizhou Wu College of Environment and Civil Engineering Chengdu University of Technology Chengdu, China

Runqiu Huang Chengdu University of Technology Chengdu, China

Xu Li School of Civil Engineering Beijing Jiaotong University Beijing, China

ISBN 978-981-15-0760-1 ISBN 978-981-15-0761-8 https://doi.org/10.1007/978-981-15-0761-8

(eBook)

Jointly published with Science Press The print edition is not for sale in China. Customers from China please order the print book from: Science Press. © Science Press 2020 This work is subject to copyright. All rights are reserved by the Publishers, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publishers, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publishers nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publishers remain neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Foreword

The book entitled “Hydro-mechanical Analysis of Rainfall-Induced Landslides” by Lizhou Wu, Runqiu Huang, and Xu Li is a welcomed addition to the literature that offers rigorous solutions to the challenging issue of rain-induced landslides. Many countries face geological hazards related to slope instability triggered by rainfall. There is a need to bring together a number of physically-based disciplines to provide an engineered approach to address slope failure mechanisms. Because of the complexity of rainfall-induced landslides, the problem has been a challenge for geotechnical engineers for more than a half century. Many researchers in the world are still working on it. Professors Lizhou Wu, Runqiu Huang, and Xu Li are some of them. In this book, the authors present many creative studies on rainfall infiltration principle using coupled analytical analysis, and on slope failure mechanisms using both laboratory model tests and analytical analysis. They have developed a general analytical method for analyzing coupled unsaturated infiltration problems, which was a real challenge in the past and could only be achieved through numerical analysis. This method is applicable not only to landslides, but also to other unsaturated infiltration problems. The influences of rainfall intensity, ponding, layered strata, etc., are clearly described by their analytical solutions. In the model tests, many interesting phenomena have been observed, which deserve further research. I am confident that this book will help researchers, geotechnical engineers, and engineering geologists understand better the issue of rainfall-induced landslides, and promote disaster prevention science and technology.

Zhang Limin Chair Professor The Hong Kong University of Science and Technology, Hong Kong, China

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Preface

Rainfall-induced landslides occur every year in many parts of the world, especially in environments that provide a prolonged and intense rainfall. Rainfall-induced geological disasters including residual or colluvium soil slope failures commonly occur because of rainfall infiltration, particularly in tropical and subtropical areas. Worldwide, rainfall-induced landslides have claimed untold numbers of human lives and have caused economic losses. Rainfall-induced landslides demand greater attention as weather conditions become more extreme because of climate change. Studies on rainfall-caused landslides have been developed tremendously during the past 20 years and have attributed to the advancement of unsaturated soil mechanics. However, a number of complicated mechanisms are involved in the analysis of slope stability subject to rainfall. Studies on rainfall-induced landslides require good knowledge of not only strength properties of the soil, but also hydrologic behavior of the soil governed by the soil seepage properties. A large number of methods including analytical and numerical solutions are developed to forecast landslides based on rainfall infiltration. Pore-water pressure in soil slopes changes due to rainfall infiltration and seepage will cause changes in stresses and, in turn, deformation of a soil. Conversely, stress changes will modify the seepage process because soil hydraulic properties are influenced by the stress changes. Hence the seepage and deformation problems are strongly linked in unsaturated soil slopes under rainfall condition. Many predictive models ignore the coupled effects and cause error compared with practical models. The predicted performance of a slope from a geotechnical model may deviate from reality because of neglecting coupled processes associated with the rainfall infiltration model. The aim of this book is to provide a thorough grounding in rainfall-induced landslides from three aspects: the coupling effect of hydraulic and mechanic; the analytical, numerical and physical simulation methods, and the controlling factors underlying the problem of rainfall-induced landslides. The book clearly presents infiltration analysis and stability analysis methods based on coupled approaches.

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Preface

The analytical and numerical methods which can be used to address the related unsaturated infiltration problems are also presented. This book is an essential reading for researchers and graduate students who are interested in rainfall infiltration, landslides, slope stability, and geohazards in fields of civil engineering, engineering geology, and earth science. The book is written to guide professional engineers and practitioners in slope engineering and geohazard management. This book can enhance their understanding of rainfall-induced landslides, help them analyze a specific problem and prevent landslides and design engineering slopes according to the local soil and climate conditions. Knowledge of unsaturated soil mechanics, analytical and numerical methods, and infiltration is a prerequisite for the readers. This book consists of six chapters. Chapter 1 presents commonly used conceptual models and infiltration into soils and discusses important issues related to analytical solutions and numerical modeling. Chapter 2 presents an analytical solution to 1D coupled infiltration and deformation in unsaturated porous media based rainfall pattern, boundary conditions, and layered structures, respectively. Chapter 3 presents the effects of gravity and hysteresis on 1D unsaturated infiltration. Chapter 4 presents 2D infiltration in unsaturated porous media considering coupled hydro-mechanical models, which is extended to semi-infinite domain. Chapters 5 focuses on physical simulation of rainfall infiltration into unsaturated slopes. Chapters 6 focuses on slope stability analysis based on a coupled approach. In Chap. 6, analytical and numerical solutions are developed to examine the coupled process of water pressure in unsaturated soil slopes during rainfall infiltration. This book also focuses on the major outcomes of the research performed by the authors in the past 10 years. We express our highest gratitude to Professor Limin Zhang (Department of Civil and Environmental Engineering, The Hong Kong University of Science & Technology) for his support and help during all these years. Lizhou Wu Chengdu University of Technology Chengdu, China Runqiu Huang Chengdu University of Technology Chengdu, China Xu Li Beijing Jiaotong University Beijing, China

Contents

1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Physical Properties of Unsaturated Infiltration 1.1.1 Basic Physical Properties . . . . . . . . . . 1.1.2 Soil Suction . . . . . . . . . . . . . . . . . . . 1.1.3 Unsaturated Hydraulic Conductivity . . 1.1.4 Total Head . . . . . . . . . . . . . . . . . . . . 1.1.5 Soil–Water Characteristic Curve . . . . . 1.2 Governing Equation of Water Infiltration . . . . 1.2.1 Richards’ Equation . . . . . . . . . . . . . . 1.3 Shear Strength and Slope Stability . . . . . . . . . 1.3.1 Unsaturated Shear Strength . . . . . . . . 1.3.2 Slope Stability . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2 Analytical Solution to 1D Coupled Infiltration and Deformation in Unsaturated Porous Media . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Effect of Rainfall Pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Governing Equations for Coupled Seepage and Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Analytical Solutions to 1D Coupled Infiltration Problems Before and After Ponding . . . . . . . . . . . . . . 2.2.3 Examples and Analysis Results . . . . . . . . . . . . . . . . . 2.2.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Effect of Boundary Condition . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Coupled Infiltration Equations . . . . . . . . . . . . . . . . . . 2.3.2 Solution for Infiltration into Deformable Soils . . . . . . . 2.3.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Contents

2.4 Effect of Layered Structure . 2.4.1 Governing Equations 2.4.2 Parametric Study . . . 2.4.3 Conclusions . . . . . . . Appendix . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . .

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44 44 47 56 57 58

3 Effects of Gravity and Hysteresis on 1D Unsaturated Infiltration . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Effect of Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Governing Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Boundary and Initial Conditions . . . . . . . . . . . . . . . . . . 3.2.3 The Analytical Solutions for Transient Unsaturated Infiltration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4 The Analytical Solution for Steady Unsaturated Infiltration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.5 Case Study and Analysis . . . . . . . . . . . . . . . . . . . . . . . 3.2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Effect of Hysteresis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Brief Review of Hysteretic Soil–Water Characteristic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Mathematical Formulations for Hysteresis . . . . . . . . . . . 3.3.3 Analytical Solutions Considering Hysteresis . . . . . . . . . 3.3.4 Analysis Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Effect of Semi-infinite Region . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Governing Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Parametric Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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61 61 62 62 64

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4 2D Infiltration in Unsaturated Porous Media . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Analytical Solution to Finite Domain . . . . . . . . . . . . . . . . . . 4.2.1 Governing Equation of Coupled Infiltration Problems 4.2.2 Formulation of the Initial Value Problem . . . . . . . . . 4.2.3 Analytical Solutions to Two-Dimensional Coupled Infiltration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Numerical Solution to Water Table Rise . . . . . . . . . . . . . . . 4.3.1 Governing Equations for the Hydromechanical Processes Involved in Water Table Change . . . . . . . . 4.3.2 Computational Modeling . . . . . . . . . . . . . . . . . . . . .

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Contents

4.3.3 Examples of Water Infiltration . . . . . . . . . . . . 4.3.4 Discussion and Conclusions . . . . . . . . . . . . . . 4.4 Surface Infiltration in Semi-infinite Extent . . . . . . . . . 4.4.1 Governing Equations for Coupled Seepage and Deformation in Unsaturated Soils . . . . . . . 4.4.2 Analytical Solution to the Coupled Infiltration in a Semi-infinite Region . . . . . . . . . . . . . . . . 4.4.3 Computational Modeling of Water Infiltration in a Semi-infinite Unsaturated Region . . . . . . . 4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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5 Physical Simulation of Rainfall Infiltration into Unsaturated Slopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Physical Testing of Rainfall-Induced Loose Deposit Slope Failures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Model Design . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Analysis of the Slope Model Experiments . . . . . . . 5.2.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Laboratory Simulation of Rainfall-Induced Loess Slope Failure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Results and Analysis . . . . . . . . . . . . . . . . . . . . . . 5.3.3 Numerical Analysis . . . . . . . . . . . . . . . . . . . . . . . 5.3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Model Test for Rainfall-Induced Shallow Landslides in Red-Bed Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Geographic and Geologic Setting . . . . . . . . . . . . . 5.4.2 Theoretical Modeling of Rainfall-Induced Shallow Landslides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.3 Laboratory Model Testing of Rainfall-Induced Landslides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.4 Verification of the Infiltration Analysis Model . . . . 5.4.5 Comparative Analysis of the Shallow Landslide Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Slope Stability Analysis Based on Coupled Approach . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Hydromechanical Process of Soil Slope Under Rainfall 6.2.1 Equations Governing Coupled Infiltration and Deformation . . . . . . . . . . . . . . . . . . . . . . . . . .

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6.2.2 The Limit Equilibrium Method for Stability of an Infinite Soil Slope . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3 Analytical Solutions to Rainfall Infiltration in an Infinite Slope . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.4 Examples and Results . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Hydromechanical Coupling of Unsaturated Soil Slope Stability Due to Rainfall Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Numerical Calculation of a Homogeneous Soil Slope . . 6.3.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Three-Phase Coupling of Soil Slope Under Rainfall . . . . . . . . . 6.4.1 The Coupled Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2 Equilibrium Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.3 Analysis of Unsaturated Soil Slope Stability . . . . . . . . . 6.4.4 Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 1

Background

Unsaturated soil is widely distributed on the surface of earth. The soil above groundwater level is in an unsaturated state, and the pores of the soil are filled with both air and water. Many practical engineering such as earth dam, railway embankment, and airport construction involves unsaturated soils. Even beneath or near groundwater level, air cavities or large bubbles can exist between the particles of sedimentary soils. Such soil containing bubbles can also be considered as unsaturated soil. Water infiltration widely occurs in an unsaturated soil. Therefore, the study of water infiltration in unsaturated soil has important theoretical and practical significance. Slope stability issues are commonly encountered in engineering projects. Many soil slope failures are attributed to rainfall infiltration (Huang and Yuin 2010; Cho and Lee 2002; Cai and Ugai 2004; Vallet et al. 2015; Wu et al. 2017a) that increases the pore-water pressure or decreases the matric suction of unsaturated soils (Fredlund and Rahardjo 1993). Matric suction is crucial to the stability of soil slopes because dissipation of matric suction leads to decrease in shear strength of unsaturated soils. Slope failure is closely related to the rainfall-induced transient seepage of slopes in unsaturated soils (Kim 2000). Current researches on rainfall-induced landslides focus mainly on two aspects: (1) studying the processes and mechanisms of infiltration into soil slopes and the internal factors affecting slope stability using conceptual models; and (2) using numerical methods to analyze the relationship between rainfall and landslides. These studies examine slopes and landslides utilizing both experimental and analytical methods (e.g., Jia et al. 2009; Wu et al. 2015, 2017b). Large-scale model experiments are costly and involve limitations including the model size. Many numerical methods used for landslide analysis have examined the landslide failure and the post-failure stages (Bandara and Soga 2015). The complexity involved in landslides can be effectively modeled using numerical approaches that incorporate fully coupled water infiltration and deformation in partially saturated soils (Bandara and Soga 2015; Bandara et al. 2016). Landslide forecasting should take into account rainfall infiltration into © Science Press 2020 L. Wu et al., Hydro-mechanical Analysis of Rainfall-Induced Landslides, https://doi.org/10.1007/978-981-15-0761-8_1

1

2

1 Background

soil slopes. Numerical approaches provide a powerful tool for solving complex, nonlinear infiltration into unsaturated soils. The coupled water infiltration and deformation process in an unsaturated soil are of major interest because of its implications for disaster prevention and environmental issues. Analytical methods have been developed to provide a basic understanding of unsaturated infiltration in terms of the coupling effect (Wu and Zhang 2009; Wu et al. 2012, 2013, 2016a, b; Ho et al. 2014). Numerical models are also effective tools for examining the coupled hydromechanical problem involved in unsaturated porous media (Wu and Selvadurai 2016), where coupled flow and deformation methods are used to analyze slope stability issues (Oka et al. 2010).

1.1 1.1.1

Physical Properties of Unsaturated Infiltration Basic Physical Properties

Soil consists of distinct phases, namely solid, liquid and air phases. Fredlund and Rahardjo (1993) believe that the interfacial properties of air–liquid phase interaction of unsaturated soils are different from water and air, and are the fourth phase called contractile skin. Since the shrink film has only a few molecular layers and can be regarded as a part of the liquid phase, the simplified three-phase structure of the unsaturated soil is shown in Fig. 1.1. We can obtain the physical properties of each phase of unsaturated soil as follows:

Fig. 1.1 Diagram of three-phase unsaturated soil

1.1 Physical Properties of Unsaturated Infiltration

3

Soil density qS : qS ¼

Ms Vs

ð1:1Þ

Ratio of soil density to water density (4 °C, a standard atmospheric pressure) called the relative density Gs, is expressed as Gs ¼

qs qw

ð1:2Þ

qw ¼

Mw Vw

ð1:3Þ

Density of water qw :

where the density of water is usually 1  103 kg/m3. Density of air qa : qa ¼

Ma Va

ð1:4Þ

q¼

M V

ð1:5Þ

qd ¼

Ms V

ð1:6Þ

n¼

Vv V

ð1:7Þ

Total density of soil q:

Dry density of soil qd :

Porosity n:

where Vv = Va + Vw represents the total pore volume. Void ratio e: e¼

Vv Vs

ð1:8Þ

The relationship between the porosity and the void ratio can be obtained as

4

1 Background

n¼

e 1þe

ð1:9Þ

Vw Vv

ð1:10Þ

Degree of saturation S is given by S¼

Moisture content (gravity moisture content) w: w¼

Mw Ms

ð1:11Þ

hw ¼

Vw V

ð1:12Þ

Volumetric moisture content hw :

1.1.2

Soil Suction

Compared with saturated soil, the unsaturated soil pore consists of both pore-water and pore air, and shows a particular mechanical property. The concept of suction comes from soil physics. The original meaning of suction is water absorption, which is the ability of the soil to absorb water. The magnitude of the soil suction reflects the free energy state of the pore-water in a soil (Edlefsen and Anderson 1943). The free energy of water can be measured by the vapor pressure of water, the suction of soil, and the vapor pressure of pore-water. The thermodynamic relationship can be expressed as w¼

RT uv ln vw wv uv0

ð1:13Þ

where w is the suction (total suction) of soil (kPa), R is the molar air constant (8.314 J/(mol K)), T is the absolute temperature (K); vw is the specific volume of water or the density of water Reciprocal (m/kg); wv is the molar mass of water vapor (18.016 g/mol); uv is the vapor pressure of pore-water (kPa); uv0 is the saturated vapor pressure at the same temperature (kPa). The suction defined by relative humidity is usually called total suction and consists of two parts: matric suction and osmotic suction (Aitchison 1965), which is expressed as

1.1 Physical Properties of Unsaturated Infiltration

w ¼ ð ua  uw Þ þ p

5

ð1:14Þ

where ðua  uw Þ is the matric suction; ua is the pore air pressure; uw is the pore-water pressure; p is the osmotic suction. In most cases, we can concern matric suction, regardless of the osmotic suction. However, when the soil salinity changes due to chemical responses, the osmotic suction changes significantly, and its influence should be considered (Fredlund and Rahardjo 1993). Matric suction is related to the capillary phenomenon caused by the surface tension of pore-water. In unsaturated soils, the pore-water pressure is negative, which will cause the pore-water maintaining at an elevation above groundwater level. The rising height of pore-water depends on pore radius and can be calculated by the following formula: h¼

2TS cos a qw gr

ð1:15Þ

where h is the pore-water rise height; Ts is the surface tension of water; a is the contact angle; qw is the density of the liquid; g is the gravitational acceleration, around 9.81 m/s2; and r is the pore radius. The total suction can be obtained by measuring the vapor pressure or relative humidity of the unsaturated soil. The relative humidity of the unsaturated soil can be measured by a hygrometer or a filter paper method (Fredlund and Rahardjo 1993). The hygrometer measures the relative humidity in the soil directly. The filter paper method is used to make the filter paper fully in contact with the unsaturated soil to achieve the suction balance, thereby indirectly determining the relative humidity of the unsaturated soil. The methods for measuring matric suction also include direct and indirect methods. Direct method uses a high intake value ceramic plate to directly measure the negative pore-water pressure of the soil sample. Since the soil sample is connected to the atmosphere, the pore pressure is the atmospheric pressure, whereby the matric suction of the soil sample can be obtained. Indirect measurement uses a sensor to measure matric suction of a soil sample. The electrical and thermal properties of porous media, such as ceramics, are a function of water content. The water content is, in turn, a function of the matric suction. By measuring the electrothermal characteristics of the sensor, the matric suction of the soil around the sensor can be obtained.

1.1.3

Unsaturated Hydraulic Conductivity

The permeability coefficient is also known as the hydraulic conductivity. In an isotropic medium, it is defined as the unit flow rate per unit hydraulic gradient,

6

1 Background

indicating the ease with which the fluid passes through the pore skeleton. It is expressed as kw ¼

qw g K g

ð1:16Þ

where K is the intrinsic permeability of the porous medium, which is only related to the nature of the solid skeleton, kw is the permeability coefficient; η is the dynamic viscosity coefficient. The larger the permeability coefficient, the stronger the permeability of the soil. Since water in unsaturated soils is first discharged from the large pores under the action of suction, the water flow can only flow in the small pores as the suction increases. Therefore, the permeability of the soil from saturated to unsaturated states will fall dramatically. Extending the saturated Darcy’s law to the unsaturated water flow, it is proved that Darcy’s law is also suitable for the flow of water in unsaturated soil. The permeability coefficient of unsaturated soil cannot be assumed to be a constant, and is strongly influenced by the change of pore ratio and degree of saturation of a soil, which is a function of volumetric water content or matric suction. The equation proposed by Brooks and Corey (1964) is ( kw ¼

h ks

ðu u Þ ks ðuaa uww Þb

ig

ðua  uw Þ  ðua  uw Þb ð ua  uw Þ [ ð ua  uw Þ b

ð1:17Þ

where η is the empirical constant and ðua  uw Þb is the intake value, which represents the matric suction that must be achieved when air enters the pores. Another popular equation proposed by Gardner (1958) is kw ¼

1þa

h

ks

ig

ðua uw Þ qw g

ð1:18Þ

in which, a, n is the constants.

1.1.4

Total Head

Total head is often used as a state variable to describe flow phenomena in a soil. The total head concept is generally applicable to both saturated and unsaturated conditions. The use of total head for describing fluid flow stems from consideration of thermodynamic law, which assumes that energy flows from a high position to a low one.

1.1 Physical Properties of Unsaturated Infiltration

7

Fundamentally, total head is the potential of the water retained in the soil pores. For many geotechnical engineering applications occurring on a relatively macroscopic scale (e.g., larger than the particle scale), the total head hw responsible for the flow of pore-water at a given point can be sufficiently represented by the summation of the elevation head he and pressure head hp at that point as follows: hw ¼ he þ hp ¼ z þ

uw qw g

ð1:19Þ

where z is the vertical coordinate distance from a prescribed datum (m); uw is the pore-water pressure (Pa).

1.1.5

Soil–Water Characteristic Curve

The soil–water characteristic curve (SWCC) represents the relationship between matric suction and water content, which is frequently used in the interpretation and prediction of the shear strength–suction relationships (Alonso et al. 1990, 2013; Fredlund and Rahardjo 1993; Gallipoli et al. 2003; Wheeler et al. 2003; Thu et al. 2007; Sawangsuriya et al. 2009; Gens 2010). It is an important part of unsaturated soil mechanics. It plays an important role in understanding the mechanical properties of unsaturated soils and obtaining the parameters. The relationship between the phases can be obtained by using the soil–water characteristic curve, and is used to predict the volume change, shear strength, permeability, and heat conduction of the unsaturated soil. The key characteristics of these relationships can be summarized as follows. The soil–water characteristic curve is shown in Fig. 1.2. First, the SWCC can be divided into three zones; namely, boundary effect, transition, and residual zones (Vanapalli et al. 1999). Within the boundary effect zone, pores in a soil are saturated with capillary water under matric suction. It is likely that there may be some air in the form of occluded bubbles, which is not continuous. As suction exceeds the air-entry suction (AVE), air starts to enter the largest soil pores and subsequently drain the capillary water out of the soil. In this stage of desaturation, both air and water phases are continuous; however, liquid phase flow and capillary effect tends to decrease with increasing matric suction. When matric suction is more than the residual suction, the amount of capillary water and matric suction toward the skeleton constitutive stress become negligible. Water transports in the pores of a soil only in the form of vapor, as the water phase retreats to the micropores and becomes discontinuous in the residual zone of desaturation. The rate of the desaturation of water becomes remarkably slower regardless of the large suction increase in this stage. This characteristic can be recognized from the flattened part of the SWCC at large suction values.

8

1 Background

Fig. 1.2 Soil–water characteristic curve

Volume content of soil θ (%)

θs

θr

Matric suction (kPa)

Second, it is well recognized from experimental investigations that the shear strength properties exhibit nonlinear evolution with suction and water degree of saturation (Escario and Saez 1986; Cunningham et al. 2003; Ng and Yung 2008; Lu and Kaya 2014; Ng and Zhou 2014). Such behavior can be related to the changes of the capillary water and its associated capillary effect and suction, which contributes to the variations in the skeleton constitutive stress and, therefore, the shear strength (Lu and Likos 2006; Alonso et al. 2010). Within the boundary effect zone, the soil system stays in a stage of capillary saturation and can be treated as a continuum medium. The increasing matric suction fully contributes to the constitutive stress of the skeleton, causing a significant increase in the shear strength properties. The contribution of matric suction becomes less efficient with further increasing matric suction and the corresponding drainage of the capillary water in the transition zone, which results in a less significant increase in the shear strength properties. The water retention curves can be characterized by several models, one of them is known as the van Genuchten model (1980): ( hðiÞ ¼

hr þ

hs hr 11=n ½1 þ jahjn 

hs ; h  0

; h\0

ð1:20Þ

where hs is the saturated volumetric moisture content of the soil, hr is the residual volumetric moisture content of the soil; hðiÞ is the soil volumetric moisture content, h is the matric suction, and a and n are the fitting parameters.

1.1 Physical Properties of Unsaturated Infiltration

9

Fredlund and Xing (1994) model is given by hs  n  m hðiÞ ¼   ln e þ ha

ð1:21Þ

where m is an empirical parameter. Gardner et al. (1970) model is written as hðiÞ ¼ hr þ

1.2

hs  hr  n 1 þ ha

ð1:22Þ

Governing Equation of Water Infiltration

Green and Ampt (1912) first established a water infiltration model (Fig. 1.3). During the infiltration process, there is a wetting front. There are two different soil– water content zones on either side of the wetting front. The soil in front of the wetting front is not affected by water infiltration, and the initial water content is kept constant, which is called the drying zone. The soil behind the wetting front is affected by rainfall infiltration, and the soil is saturated, which is called the wetting zone. The negative pore-water pressure on the wetting front remains unchanged. Based on this assumption, the Green–Ampt infiltration model can be expressed as i ¼ ks

zf þ sf þ H dzf ¼ ð hs  hi Þ ks dt

ð1:23Þ

where i is the infiltration rate (m/s); ks is the saturated permeability coefficient (m/s); zf is the wet front depth (m); sf is the wet front matric suction (m); H is the accumulated water depth (m); hs is the saturated volumetric moisture content; hi is the initial volumetric moisture content. For the first time, Horton (1941) established an empirical infiltration model, which can describe the infiltration characteristics of different soils and the corresponding infiltration process. Based on the Horton empirical model, Bouwer (1976)

Fig. 1.3 Green–Ampt infiltration model

θ Wetting zone

Drying zone z

H

10

1 Background

improved the model and established a soil–water infiltration function for unsaturated soils: Z hf ¼ 

hs

hi

kðhÞdh ks

ð1:24Þ

where hf is the suction at the wetting front (kPa); hi is the initial soil suction (kPa); hs is the suction corresponding to the soil surface moisture content (kPa); k(h) is the unsaturated soil hydraulic conductivity (m/s); and h is the soil suction head (m). Philip (1957) used the Boltzmann transformation to obtain the relationship between infiltration rate and time, and proposed the corresponding rainfall infiltration conceptual model. Its water infiltration function is as follows: 1 1 iðtÞ ¼ st2 þ A 2

ð1:25Þ

where i(t) is the infiltration rate (m/s); s is the infiltration rate (m/s); t is the time (s); A is the stable infiltration rate (m/s).

1.2.1

Richards’ Equation

Darcy conducted a penetration test of saturated sand layers and found a quantitative relationship between the seepage velocity of water in the soil and the head loss, namely Darcy’s law: q ¼ K

dH dL

ð1:26Þ

where q is the flux (discharge per unit area, with units of length per time, m/s), K is the saturated permeability coefficient of soil, L is the seepage length, and H is the head height. When the soil is in an unsaturated state, most scholars believe that Darcy’s law can also be used for the analysis of water movement in unsaturated soils. Richards (1931) extended the Darcy’s law of saturated soil to the unsaturated flow, and introduced the continuity equation to derive the equation of motion of the unsaturated soil–water flow, namely the Richards’ equation. The permeability coefficient is expressed as a function of the matric suction (k), and Darcy’s law can be expressed as q ¼ krH

ð1:27Þ

1.2 Governing Equation of Water Infiltration

11

where rH is a hydraulic gradient, including two components of gravity and suction. Continuity equation: @h ¼ r  q @t

ð1:28Þ

@h ¼ r  ½krH  @t

ð1:29Þ

Then:

Substituting Eq. (1.27) into Eq. (1.29), we can obtain: @h @kðhÞ ¼ r  ½kðhÞrðh  zÞ ¼ r  kðhÞrh þ @t @z

1.3 1.3.1

ð1:30Þ

Shear Strength and Slope Stability Unsaturated Shear Strength

Shear strength is a fundamental material property that is required to address a variety of engineering problems including bearing capacity, slope stability, lateral earth pressure, pavement design, and foundation design, among many others. Recently, many researches have focused on the shear strength of unsaturated soils (Fredlund et al. 1996; Vanapalli et al. 1996; Khalili and Khabbaz 1998; Alonso et al. 2010; Lu et al. 2010). Using the concept of Mohr–Coulomb and effective stress, the shear strength of saturated soil can be expressed as sf ¼ c0 þ r0 tan u0

ð1:31Þ

where s (kPa) is the shear strength, c (kPa) is cohesion, rn (kPa) is the normal stress acting on the failure surface, and u (°) is the angle of internal friction. Cohesion and cohesive shear strength are due to chemical bonding between soil particles and surface tension within the water films (Lu and Likos 2006). Frictional shear strength (rn tan u) is owing to internal friction between soil particles that depends on the normal stress acting on the failure surface. Engineering practice shows that the shear strength formula of saturated soil can meet the engineering requirements. The shear strength parameters were also influenced by matric suction. They indicated that with an increase in matric suction, c and u increased which is dependent on soil texture and structure. With the increment of matric suction, total cohesion increased, u did not change, and ub

12

1 Background

decreased nonlinearly. They showed that shear strength significantly increased with an increase in net normal stress or matric suction. However, several phases of unsaturated soils make the shear strength formula of saturated soil difficult to apply. Therefore, some studies on the shear strength criteria of unsaturated soils have been carried out. There are two main representative shear strength criteria here: Bishop (1959) has proposed a shear strength criterion for unsaturated soils: sf ¼ c0 þ ½ðr  ua Þf þ vðua  uw Þf  tan u0

ð1:32Þ

where sf is shear strength and c′ and u′ are the effective cohesion and friction angle, respectively. Some investigations have shown that shear strength can be defined in terms of a simple Bishop-type effective stress, which is very appealing to practicing engineers (Khalili and Khabbaz 1998). The difficulty in this approach, however, lies in quantifying the effective stress parameter, or Bishop’s parameter. In the past, the effective stress parameter has often been set equal to the degree of saturation, which has been shown to cause significant overestimations of the contribution of suction to effective stress. Now, it has been suggested that the effective stress parameter should be defined in terms of an effective degree of saturation that defines the fraction of water (Alonso et al. 2010). Brooks and Corey (1964) defined the parameter as  v¼

S  Sr 1  Sr

0:55=k ð1:33Þ

where S is the degree of saturation, Sr is the residual degree of saturation, and k is the pore-size distribution index. The residual degree of saturation is speculated to represent the largest saturation for which pore-water is held by adsorptive rather than capillary forces. More recently, it has been suggested that the effective stress parameter of the soil should be defined in terms of an effective degree of saturation that defines the fraction of water (Alonso et al. 2010): Khalili and Khabbaz (1998) proposed a mathematical model for estimating the parameter v: (

v¼

ua uw le

0:55

1

for ua  uw [ le for ua  uw  le

ð1:34Þ

where le is the suction marking the transition between saturated and unsaturated states, being the air-expulsion pressure for a wetting process and the air-entry pressure for a drying process. The validity of several forms of v as a function of the degree of saturation was also examined using a series of shear strength test results for statically compacted

1.3 Shear Strength and Slope Stability

13

mixtures of clay, silt, and sand. For matric suction ranging between 0 and 1500 kPa, the following two forms showed a good fit to the experimental results:  k h v¼ hs

ð1:35Þ

where h is volumetric water content, hs is the saturated volumetric water content, and k is a fitting parameter optimized to obtain a best fit between measured and predicted values, and v¼

h  hr hs  hr

ð1:36Þ

where hr is residual volumetric water content and Sr is residual degree of saturation. Based on two stress state variables, Fredlund et al. (1978) proposed the following equation to describe shear strength: sf ¼ c0 þ ½ðr  ua Þ tan u0 þ ðua  uw Þ tan ub

ð1:37Þ

where ub is the internal friction angle due to the distribution of matric suction. The shear strength–suction relationships for unsaturated soils are nonlinear and sensitive to various factors such as the external stress, moisture, soil type, soil structure, and testing technique (Han and Vanapalli 2016). A large number of tests have shown that the shear strength of unsaturated soil is not linearly proportional to the matric suction. The internal friction angle of the same soil sample is not constant, but varies with the suction of the matric. Therefore, the soil–water characteristic curve equation of unsaturated soil was used to determine the nonlinear law of shear strength of unsaturated soil with matric suction, and obtained the following formula: sf ¼ c0 þ ðr  ua Þf tan u0 þ tan u0

Z 0

w

s  sr ½ dðua  uw Þ 1  sr

ð1:38Þ

Lu and Likos (2004) proposed a unified form of shear strength formula: sf ¼ c0 þ vf ðr  ua Þf tan u0 þ vf ðua  uw Þf tan u0 ¼ c0 þ c00 þ ðr  ua Þf tan u0 ð1:39Þ where c00 ¼ vf ðua  uw Þf tan u0

ð1:40Þ

The first two terms in Eq. (1.39), c′ and c″, represent shear strength due to the so-called apparent cohesion in unsaturated soil. In an saturated soil, the third term

14

1 Background

represents frictional shearing resistance provided by the effective normal force at the grain contacts. The apparent cohesion captured by the first two terms includes the classical cohesion c′ representing shearing resistance arising from interparticle physicochemical forces such as van der Waals attraction, and the second term c″ describing shearing resistance arising from capillarity effects. The term c″ is defined as capillary cohesion hereafter. Physically, capillary cohesion describes the mobilization of suction stress v(ua − uw) in terms of shearing resistance. The relationship between capillary cohesion and the maximum suction stress at failure, vf ðua  uw Þf is defined by shear strength also affects the water movement of the soil (Eudoxie et al. 2012).

1.3.2

Slope Stability

Slope failure in unsaturated soil regions triggered by rainfall is a common geological hazard in many parts of the world (Fredlund and Rahardjo 1993; Lu and Likos 2004; Guzzeti et al. 2008). Both rainfall characteristics (rainfall intensity and duration) and soil permeability may influence failure mechanism (Li et al. 2013). Slope failures in unsaturated soils are often induced by rainfall infiltration. The characteristics of water flow, change of pore-water pressure, and shear strength of soils are the major parameters related to slope stability analysis involving unsaturated soils that are directly affected by the flux boundary condition (i.e., infiltration and evaporation) at the soil–atmosphere interface. Generally, unsaturated soil slope failures happen most frequently during or after rain periods. Rainfall infiltration will increase the groundwater level and water pressure and decrease matric suction of unsaturated soils (Hamdhan and Schweiger 2013). Rainfall infiltration caused a decrease in matric suction and an increase in moisture content and hydraulic conductivity in the unsaturated soil. The rainfall intensity and duration, initial water table, and hydraulic conductivity are the parameters that significantly affect slope stability (Ng and Shi 1998; Hamdhan and Schweiger 2013). In general, the failures can be initiated by two mechanisms, i.e., loss of matric suction through the propagation of wetting front and rise of water table. The results showed that the hydraulic responses to rainfall for a homogeneous infinite slope underlain by an impermeable layer can be divided into two stages: (1) the propagation of wetting front and (2) the rise of water table. Based on these hydraulic responses, the type and mechanism of failures were deduced from the analytical analyses. Both the rainfall characteristics and permeability were predominant in controlling the hydraulic responses of soil, and hence the occurrence time, depth of failure plane, and type of surficial slope failures (Li et al. 2013). Although the significance of rainfall infiltration in causing landslides is widely recognized, there have been different conclusions as to the relative roles of rainfall patterns to landslides. The relative importance of soil properties, rainfall intensity,

1.3 Shear Strength and Slope Stability

15

initial water table location, and slope geometry in inducing instability of a homogenous soil slope under different rainfall was investigated through a series of studies. Soil properties and rainfall intensity were found to be the primary factors controlling the instability of slopes due to rainfall, while the initial water table location and slope geometry only played a secondary role. The results from the parametric studies also indicated that for a given rainfall duration, there was a threshold rainfall intensity, which would produce the global minimum factor of safety (Rahardjo et al. 2007). Global climate change, which could lead to more severe fluctuations in rainfall patterns, could trigger deformation of soil slopes resulting in slope instability because of the alteration of intensity, frequency, and quantity of rainfall (Collison et al. 2000; Dixon and Brook 2007). In the geotechnical engineering field, this will lead to shallow landslides, which are a common form of slope instability within many mountainous areas (Jeong et al. 2008; Kim et al. 2012). The influence of climate change on rainfall patterns has the potential to alter the stability of unsaturated soil slopes. Changes in rainfall patterns have a strong influence on the stability of unsaturated soil slopes, which recently have resulted in shallow landslides. Numerical analysis solves for the matric suction distribution in a soil slope while varying permeability, and considering a surface part of a soil with different permeability. Numerical examples demonstrate that change in rainfall patterns may trigger slope failure due to rainfall infiltration. The results can provide an indication of the potential influence of climate change on shallow landslides in many mountainous areas (Kim et al. 2012). Although the effect of rainfall patterns is difficult to include in models of rainfall infiltration, its effect on the development of shallow landslides and slope failure has been investigated using a one-dimensional infiltration model (Tsai 2010). Rainfall patterns have a strong influence on the water infiltration process, the pore-water pressure profiles of soil slopes, and the soil slope stability. Therefore, it is important to take into account the influence of rainfall patterns and incorporate them into landslide prediction systems. Rainwater tends to infiltrate into the shallow layer and flow laterally along the interface of the less permeable rocks. The soil layer would become saturated if rainfall is sufficiently intense, and eventually trigger a landslide. The understanding of the failure mechanism of soil slopes is essential for anticipating the landslide occurrence (Li et al. 2013). Many scholars have proposed equations based on the Green and Ampt model that considers the intensity and the duration of rainfall, the volumetric water content of the soil, and the magnitude of wetting front suction. However, these theoretical equations generally tend to overestimate the factor of safety of soil slopes, resulting in landslides and geological hazards (Kim et al. 2012). For analysis of slope stability in an unsaturated soil, coupled poromechanical analysis has not been widely used to estimate the infiltration deformation of unsaturated soils, although hydraulic properties and shear strength properties affect the slope stability in unsaturated soils during heavy rainfall (Cho and Lee 2002; Kim et al. 2012). The coupled seepage and deformation process in unsaturated soils

16

1 Background

are influenced by soil solid skeleton deformation. Changes in pore-water pressures in unsaturated soils are often a result of precipitation and evaporation events. Pore-water pressure variations due to rainfall seepage will lead to changes in stresses and in turn deformation of a soil mass. Conversely, stress variations will modify the infiltration process as hydraulic properties of a soil are influenced by the changes in stress. Thereafter, the infiltration and deformation problems are strongly linked in unsaturated soil slopes due to water infiltration, and the coupled poromechanical model is preferred to analyze the behavior and stability of unsaturated soils subjected to external loads, particularly rainfall (Kim et al. 2012).

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18

1 Background

Oka F, Kimot S, Takada N (2010) A seepage-deformation coupled analysis of an unsaturated river embankment using a multiphase elasto-viscoplastic theory. Soils Found 50(4):483–494 Philip JR (1957) The theory of infiltration: 4. Sorptivity and algebraic infiltration equations. Soil Sci 84(3):257–264 Rahardjo H, Ong TH, Rezaur RB, Leong EC (2007) Factors controlling instability of homogeneous soil slopes under rainfall. J Geotech Geoenviron Eng 133(12):1532–1543 Richards LA (1931) Capillary conduction of liquids through porous mediums. J Appl Phys 1 (5):318–333 Sawangsuriya A, Edil TB, Bosscher PJ (2009) Modulus-suction-moisture relationship for compacted soils in postcompaction state. J Geotech Geoenviron Eng 135(10):1390–1403 Thu TM, Rahardjo H, Leong EC (2007) Elastoplastic model for unsaturated soil with incorporation of the soil-water characteristic curve. Can Geotech J 44(1):67–77 Tsai TL (2010) Influences of soil water characteristic curve on rainfall-induced shallow landslides. Environ Earth Sci 64(2):449–459 Vallet A, Bertrand C, Mudry J, Bogaard T, Fabbri O, Baudement C, Régent B (2015) Contribution of time-related environmental tracing combined with tracer tests for characterization of a groundwater conceptual model: a case study at the Séchilienne landslide, western Alps (France). Hydrogeol J 23(8):1761–1779 Van Genuchten MY (1980) A closed form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Sci Soc Am J 44:892–898 Vanapalli SK, Fredlund DG, Pufahl DE (1999) Influence of soil structure and stress history on the soil-water characteristics of a compacted till. Géotechnique 49(2):143–159 Vanapalli SK, Fredlund DG, Pufahl DE, Clifton AW (1996) Model for the prediction of shear strength with respect to soil suction. Can Geotech J 33(3):379–392 Wheeler SJ, Sharma RS, Buisson MSR (2003) Coupling of hydraulic hysteresis and stress–strain behaviour in unsaturated soils. Géotechnique 53(1):41–54 Wu LZ, Huang RQ, Xu Q, Zhang LM, Li HL (2015) Analysis of physical testing of rainfall-induced soil slope failures. Environ Earth Sci 73(12):8519–8531 Wu LZ, Selvadurai APS (2016) Rainfall infiltration-induced groundwater table rise in an unsaturated porous medium. Environ Earth Sci 75(2):1–11 Wu LZ, Selvadurai APS, Huang RQ (2013) Two-dimensional coupled hydromechanical modeling of water infiltration into a transversely isotropic unsaturated soil region. Vadose Zone J 12(4) Wu LZ, Selvadurai APS, Zhang LM, Huang RQ, Huang JS (2016a) Poro-mechanical coupling influences on potential for rainfall-induced shallow landslides in unsaturated soils. Adv Water Resour 98:114–121 Wu LZ, Zhang LM, Li X (2016b) One-dimensional coupled infiltration and deformation in unsaturated soils subjected to varying rainfall. Int J Geomech 16(2):06015004 Wu LZ, Zhang LM (2009) Analytical solution to 1D coupled water infiltration and deformation in unsaturated soils. Int J Numer Anal Met 33(6):773–790 Wu LZ, Zhang LM, Huang RQ (2012) Analytical solution to 1D coupled water infiltration and deformation in two-layer unsaturated soils. Int J Numer Anal Met 36:798–816 Wu LZ, Zhang LM, Zhou Y, Li BE (2017a) Analysis of multi-phase coupled seepage and stability in anisotropic slopes under rainfall condition. Environ Earth Sci 76(14):469 Wu LZ, Zhou Y, Sun P, Shi JS, Liu GG, Bai LY (2017b) Laboratory characterization of rainfall-induced loess slope failure. Catena 150:1–8

Chapter 2

Analytical Solution to 1D Coupled Infiltration and Deformation in Unsaturated Porous Media

2.1

Introduction

Unsaturated flow is important in a wide range of engineering applications such as rainfall, irrigation, nuclear waste disposal, and rainfall-induced landslides (Inverson 2000; Wu et al. 2016). The spatial and temporal evolution of flow in an unsaturated medium involves a governing partial differential equation that is expressed by the Richards’ equation (1931). The equation is highly nonlinear because the hydraulic conductivity and the pore-water pressure depend on the moisture content. A number of exact and approximated analytical solutions to the Richards’ equation were derived in past studies (e.g., Parlange et al. 1972, 1997; Broadbridge and White 1988; Warrick et al. 1991; Hogarth et al. 1992; Basha 1999, 2011). While numerical approaches can effectively simulate complex nonlinear infiltrations into an unsaturated porous medium (Tracy 2006; Wu et al. 2016), analytical solutions can verify these numerical procedures. Analytical solutions of the linearized Richards’ equation were derived as an integral (Broadbridge and White 1988; Chen et al. 2001, 2003), as a Laplace transformation (Srivastava and Yeh 1991; Zhan et al. 2013; Wu et al. 2016), and as a Green’s function (Basha 1999; Wu et al. 2016). The analytical solutions are confined to a finite field, whereas the approximate solutions describe the onedimensional (1D) infiltration into a semi-infinite medium for uniform initial conditions under either a time-dependent flux or a constant pore-water pressure boundary condition (Basha 2011). The incorporation of physically based infiltration expressions leads to a better quantification of the infiltration component in hydrologic models and a more reliable prediction of the water infiltration (Basha 2011). Approximate solutions made without incorporating the coupling between infiltration and deformation in unsaturated soils are presented in a simple form, and adapt easily to various boundary conditions (Basha 2011). Variations of negative pore-water pressures in unsaturated soils are often a result of precipitation, irrigation, or evaporation events. Pore-water pressure changes due © Science Press 2020 L. Wu et al., Hydro-mechanical Analysis of Rainfall-Induced Landslides, https://doi.org/10.1007/978-981-15-0761-8_2

19

20

2 Analytical Solution to 1D Coupled Infiltration and Deformation …

to rainfall infiltration or seepage will cause changes in stresses and in turn deformation of a soil. Conversely, stress changes will modify the seepage process because soil hydraulic properties are influenced by the stress changes. Hence, the seepage and deformation problems are strongly linked in unsaturated soils under rainfall condition (Zhan and Ng 2004). The analytical solutions in the literature often do not consider coupling effects between infiltration and deformation in unsaturated soils. However, the coupling effects of infiltration and deformation in unsaturated soils are essential in many cases, e.g., seepage in loose packing colluvium formed by landslides, seepage in municipal solid waste, and consolidation of artificial islands. A typical example is rainfall-induced landslides in expansive soils, or in poorly compacted engineering fills. However, the influence of the deformation of the soil has generally been disregarded in conventional studies of infiltration. To fill this knowledge gap, the coupling effects between deformation and seepage in unsaturated soils is presented in this book. In this section, we will discuss the effects of rainfall pattern, boundary conditions, and layered structure on the pore-water pressure evolution of unsaturated soils, and the coupling effects.

2.2

Effect of Rainfall Pattern

The objective of this section is to propose a conceptual model derived from practical problems and develop a method that can be used to examine the rise of the groundwater table caused by precipitation in an unsaturated, deformable porous medium. Analytical solutions for the hydromechanical coupling problem are obtained using a suitable set of transformations. The authors also examine the relationships between the rainfall-induced rise of the groundwater table and the coupling effect, as well as factors that control the rise of the groundwater table. Exponential water content and hydraulic conductivity relationships are used to linearize the Richards’ equation. By means of a series of transformations and Fourier integral transformation, we present analytical solutions to one-dimensional horizontal and vertical transient saturated and unsaturated infiltration into the soils that allows the consideration of a variety of time-dependent surface fluxes before and after ponding under arbitrary initial conditions. This section also discusses the effect of varying rainfall intensity at the ground surface in analyzing one-dimensional coupled infiltration and deformation in unsaturated soils. Smiles and Raats (2004) studied the effect of deformation (swelling and shrinking) on hydraulic conductivity in unsaturated soils. Three kinds of rainfall cases at the ground surface before and after ponding time are emphasized in analyzing the coupled seepage and deformation in unsaturated soils.

2.2 Effect of Rainfall Pattern

2.2.1

21

Governing Equations for Coupled Seepage and Deformation

In order to analyze the coupled infiltration problem in unsaturated soils effectively, several assumptions have to be made: (1) The soil in each layer is homogeneous, and behaves as an isotropic, elastic material; (2) The soil structure is deformable, but the pore-water is not compressible; (3) The elastic volume change of the soil is due to the wetting or drying of the soil only; no volume change due to total stress changes is considered; (4) The coefficient of permeability at full saturation remains constant regardless of ground deformation; (5) The changes in SWCC and permeability function induced by soil volume change are not considered; (6) Hysteresis in the soil–water characteristics is not considered; i.e., only monotonic wetting or drying is considered; (7) The pore air pressure in the layered soil remains constant. In order to successfully obtain the analytical solution to the coupled problem, some assumptions are needed. Assumptions (4) and (5) are needed to specify a unique relationship between hydraulic conductivity and pore-water pressure of the soil under unsaturated and completely saturated conditions. A unique SWCC not considering hysteresis from assumption (6) simplifies the analysis of the coupled problem. With these assumptions, the coupled governing equations can be linearized and transformed into a solvable form. Using assumption (4), the coupling of the 1D infiltration and the deformation of unsaturated soils can be examined, and the equations that describe the coupling phenomenon can be linearized and solved. In this study, hydraulic conductivity and moisture content both change exponentially with variations in the pore-water pressure head (Raats 1970; Chen et al. 2001). This fact is very helpful for the development of analytical methods. Based on the exponential function, Zhan et al. (2013) employed the permeability curve and the SWCC to effectively and accurately predict the behavior of completely decomposed granite (Fredlund and Rahardjo 1993; Zhan et al. 2013). The parameters involved in the exponential permeability function and soil–water characteristic curve can be determined empirically or fitted to experimental data. At least, in the concerned suction range, the permeability function, and soil–water characteristic curve can be of sufficient accuracy.

2.2.1.1

Coupled Governing Equations

The flow of water in a saturated or unsaturated soil can be described by Darcy’s law under isothermal conditions (Zhang et al. 2005):

2 Analytical Solution to 1D Coupled Infiltration and Deformation …

22




 @ uw qi ¼ ki þx @xi cw

ð2:1Þ

where qi is the flow velocity in the i-direction, u is the pore-water pressure, ki is the coefficient of permeability of pore-water in the i-direction, cw is the unit weight of water, and x is the elevation. Satisfying conservation of fluid mass yields: 

@ @ ðq qi Þ ¼ ðqw nSÞ @xi w @t

ð2:2Þ

in which, S is the degree of saturation, qw is the water density, n is porosity, and h is the volumetric water content. Equation (2.2) can be rewritten as 

@ @q @S @uw @n þ qw S ðq qi Þ ¼ h w þ qw n @xi w @uw @t @t @t

ð2:3Þ

in which the derivative of n with respect to time can be expressed as @n @e ð1  nÞ2 @ev ¼ ð 1  nÞ 2 ¼  @t @t ð1  n0 Þ @t

ð2:4Þ

where n0 is the initial porosity, e is the void ratio, and ev is the volumetric strain. Combining Eqs. (2.1), (2.3), and (2.4), one obtains the 1D governing equation for coupled seepage and deformation of unsaturated soils: 
 @ @ uw h @qw @uw @S @uw @ev k þn  gS þx ¼ @x @x cw qw @uw @t @uw @t @t

ð2:5Þ

where η = (1 − n)2/(1 − n0).

2.2.1.2

Constitutive Relationships for Soil Solids

The force equilibrium equation for a soil mass is rij;j þ bi ¼ 0

ð2:6Þ

where rij is the net total stress tensor and bi is the body force vector. The strain–displacement relations are as follows: eij ¼


 1 @ui @uj þ 2 @xj @xi

ð2:7Þ

2.2 Effect of Rainfall Pattern

23

in which ui and uj are displacements in the i- and j-direction, respectively, and eij is the strain tensor. The stress–strain relationships associated with the normal strains can be written in an incremental form by Fredlund and Rahardjo (1993):  d ð ua  uw Þ d ðrx  ua Þ l   d ry þ rz  2ua þ E E H   d r y  ua l d ð ua  uw Þ dey ¼  d ðrx þ rz  2ua Þ þ E H E dex ¼

dez ¼

 d ðua  uw Þ d ðrz  ua Þ l   d rx þ ry  2ua þ E E H

ð2:8aÞ ð2:8bÞ ð2:8cÞ

where ex, ey, and ez are normal strains in the x-, y-, and z-directions, respectively, rx, ry, and rz are normal stresses in the x-, y-, and z-directions, respectively, (ua − uw) is the soil suction, ua is the pore air pressure, l is Poisson’s ratio, E is the elastic modulus of the soil with respect to changes in net normal stresses, and H is the elastic modulus of the soil with respect to a change in soil suction. Fredlund and Rahardjo’s incremental linear constitutive model for unsaturated soils is selected because the model is simple and can be implemented in the proposed analytical solutions to be presented later. The model has been used to analyze the behavior of a collapsable earth dam and other problems by Pereira (1996) and Zhang et al. (2005). Substitute the stress–strain relations of soil, Eq. (2.8a, 2.8b, 2.8c), and the strain–displacement relations into the force equilibrium equations, the governing equation for stress deformation in 1D condition (ey = 0 and ez = 0) can be obtained as follows:    @ Eð1  lÞ ð1 þ lÞðua  uw Þ ev  þ ½nSqw þ ð1  nÞqs g ¼ 0 ð2:9Þ @x ð1 þ lÞð1  2lÞ ð1  lÞH where qs is the solid density. Based on assumption (3) (i.e., effect of total stress change is not considered) and assumption (6) (i.e., the rate of ua is zero), we can obtain from Eq. (2.9): @ev ð1 þ lÞ @uw ¼ ð1  lÞH @t @t

ð2:10aÞ

@ev ð1 þ lÞcw @h ¼ @t ð1  lÞH @t

ð2:10bÞ

or

in which uw = cwh; cw = qwg.

24

2 Analytical Solution to 1D Coupled Infiltration and Deformation …

Combining Eqs. (2.5), (2.9), and (2.10a, 2.10b) yields: 
 @ @ uw h @qw @uw dS @uw ð1 þ lÞ @uw k ðuw Þ þn þ Sg ð2:11Þ þx ¼ @x @x cw qw @uw @t duw @t ð1  lÞH @t Ignoring the compressibility of the pore-water, the 1D problem on coupled seepage and deformation can be described by the following mathematical model: 
   @ @ uw @S ð1 þ lÞS @uw k ð uw Þ þx ¼ n þg @x @x cw @uw ð1  lÞH @t

ð2:12Þ

When the steady-state of seepage is achieved, Eq. (2.11) reduces to 
 @ @ uw k ð uw Þ þx ¼0 @x @x cw

ð2:13Þ

The hydraulic conductivity of unsaturated soil is expressed as (Gardner 1958)  kðuw Þ ¼

ks ks eawae eauw

wae  uw  0 uw   wae

ð2:14Þ

where ks is the coefficient of permeability at full saturation, a is a desaturation coefficient, and wae is the air-entry value. The physical meaning of the parameters is shown in Fig. 2.1a. The values of these parameters for several soil types are listed in Table 2.1. The relationship between the volumetric water content and soil suction may be described by the Boltzmann model (Ozisik 1989) (Fig. 2.1b):  hð uw Þ ¼

hs hs eawae eauw

wae  uw  0 uw   wae

ð2:15Þ

Based on the relationship between the volumetric water content and the degree of saturation, i.e., S = h(uw)/hs, Eq. (2.15) can be written as  Sðuw Þ ¼

1 eawae eauw

wae  uw  0 uw   wae

ð2:16Þ

Then one can obtain: @Sðuw Þ ¼ @uw



0 aeawae eauw

wae  uw  0 uw   wae

According to Eqs. (2.5), (2.12) and (2.14)–(2.17), we can obtain:

ð2:17Þ

2.2 Effect of Rainfall Pattern

25

(a) 0.45 0.40

θs

Ψ ae

Volumetric water content

0.35 α=0.01kPa -1

0.30 0.25 0.20 0.15 0.10 α=0.1kPa -1

0.05 0.00 0.01

0.1

1

10

100

1000

Suction (kPa)

1.00

ks

Ψ ae

-6

Hydraulic conductivity (10 m/s)

(b)

0.80

α=0.01kPa -1

0.60

0.40

0.20 α=0.1kPa -1

0.00 0.01

0.1

1

10

100

1000

Suction (kPa)

Fig. 2.1 Relationships between the coefficient of permeability, volumetric water content, and soil suction: a SWCC; and b hydraulic conductivity function


 @ @ uw @uw kðuw Þ uw   wae þx ¼ Qhs eawae eauw @x @x cw @t k s @ 2 uw @uw ¼ mv 2 cw @x @t

uw   wae

ð2:18Þ ð2:19Þ

where mv is the corresponding coefficient of volume compressibility, Q = hs[a + η(1 + l)/(1 − l) hsH]. In fact, when −wae < uw, the soil is fully saturated; therefore,

2 Analytical Solution to 1D Coupled Infiltration and Deformation …

26

Table 2.1 Hydraulic conductivity parameters for some soils Soil type

ks (m/s)

wae (kPa)

a (kPa−1)

References

10−10– 20–39 0.016 for Beit Netofa Van Genuchten (1980), Hille 10−8 clay (1998) 7–15 0.043 for a silt loam Silt 10−8– 10−6 1–3 0.106 for a fine sand Sand 10−5– 10−3 Nil 0.081 for Hpgiene Gravel 10−2– 10−1 sandstone Coefficient of permeability at full saturation (ks); air-entry value (wae) and desaturation coefficient (a) Clay

Eq. (2.18) is similar to Terzaghi’s consolidation equation without considering coupling effects. We consider 1D vertical infiltration through a homogeneous soil layer with thickness l. A 1D transient infiltration problem generally involves one initial and two boundary conditions, which is shown in Fig. 2.2. Using dimensionless space and time variables X = ax and T = a2kst/Q, and defining a new variable, W ðX; T Þ ¼ eah  e 2 þ 4 X

T

ð2:20Þ

In which, h = uw/cw. Equation (2.18) can be rewritten as @W @ 2 W ¼ @T @X 2

Fig. 2.2 General initial pore-water pressure conditions for a homogeneous soil profile

ð2:21Þ

X u=u1 Top

l

x u=u0 O

Bottom ui

u

2.2 Effect of Rainfall Pattern Fig. 2.3 General boundary conditions for homogeneous soil profile: flux boundary condition

27

X

q

l

u=u0 O

The initial conditions and boundary conditions, which are controlled by the flux at the ground surface (Fig. 2.3), are as follows: W ðX ; 0Þ ¼ eahi X=2 ¼ FðXÞ   @W q W ¼  eT=4 ¼ f1 ðTÞ @X ks X¼0 W ðL; T Þ ¼ eah0 eL=2 þ T=4 ¼ f2 ðTÞ

ð2:22aÞ ð2:22bÞ ð2:22cÞ

where L is the dimensionless length (al); h0 is the pressure head at the lower bottom, which is 0 when the bottom boundary is at the water table; hi is the initial pressure head at infiltration time t = 0; q is the boundary flux at the ground surface, i.e., X = 0.

2.2.2

Analytical Solutions to 1D Coupled Infiltration Problems Before and After Ponding

Boundary conditions involving gradually increasing rainfall intensity at the top boundary (see Fig. 2.4) are considered here. The analytic solutions to the 1D coupled and uncoupled problems considering these boundary conditions are analyzed.

2 Analytical Solution to 1D Coupled Infiltration and Deformation …

28

Fig. 2.4 Gradually increasing rainfall intensity at the surface boundary

2.2.2.1

Coupled Conditions

Green’s function was employed to solve the partial differential equations (Basha 1999, 2000; Ozisik 1989). Before the ponding time tI, the general solution to Eqs. (2.21) and (2.22a, 2.22b, 2.22c) is 8 2 39  = Z ZT
1 < X=2T=4 4 @G f2 ds5 hðX; T Þ ¼ ln e Gjs¼0  F ðx0 Þ  dx0 þ Gf1  ; a : @X R

s¼0

ð2:23Þ G¼


2  

 1 2X b2m þ L2 bm s X X0 exp 1  1   sin b sin b ð2:24Þ m m L m¼1 b2m þ L2 þ L L2 L L

in which G(X, X′ | T, s) is Green’s function, which describes unsaturated infiltration at X at time T due to surface rainfall; R is the one-dimensional finite region [0, L]. The eigenvalue bm satisfies bmcot(bmL) = −L. When ponding occurs after t > tI, the rainfall will partially infiltrate into soil and the ground surface will remain saturated. Then the boundary will be controlled by the pressure head. Let t′ = t – tI, when t′ = 0, i.e., t = tI, substituting t = tI into Eq. (2.23) gives the initial condition for the solution after tI. The differential equation to the coupled problem after ponding can be written as   @ @h ah @ ks e ðh  xÞ ¼ Peah 0 @x @x @t where P = hsa + cwac/F.

ð2:25Þ

2.2 Effect of Rainfall Pattern

29

The initial condition h0i (t′ = 0) is hðX; t0 Þjt0 ¼0

8 2 39  = Z ZTI
< 1 @G f2 ds5 ¼ ln eX=2T=4 4 Gjs¼0  F ðx0 Þ  dx0 þ Gf1  ; a : @X s¼0

R

ð2:26Þ where TI = a2kstI/P. If ponding occurs at the top boundary, the surface boundary is controlled by the pressure head h1, which is zero as the ponding water does not cumulate on the top surface. T0

Using dimensionless variables X = ax, T′ = a2kst′/P, t′ = t − tI, W 0 ¼ eah  e 2 þ 4 , we obtain the governing equation of 1D coupled infiltration problem with the pressure head boundary at the top surface, i.e., Eq. (2.21). Its initial and boundary conditions are as follows: X

0

W 0 ðX; T 0 ÞjT 0 ¼0 ¼ eahi eX=2 ¼ MðXÞ 0

W 0 ðX; T 0 ÞjX¼0 ¼ eah1 eT =4 ¼ m1 ðT 0 Þ 0

W 0 ðX; T 0 ÞjX¼L ¼ eah0 eL=2 þ T =4 ¼ m2 ðT 0 Þ

ð2:27aÞ ð2:27bÞ ð2:27cÞ

in which h0i is the pressure head profile at t′ = 0, i.e., Eq. (2.26). By using Green’s function, the analytical solution to the coupled governing equation after tI is 8 2 39  = Z ZT 0
< 1 @G @G 0 m1  m2 ds5 hðX; T 0 Þ ¼ ln eX=2T =4 4 Gjs¼0  M ðx0 Þ  dx0 þ ; a : @X @X R

s¼0

ð2:28Þ

2.2.2.2

Uncoupled Conditions

If the coupling effect is ignored and the compressibility of the pore-water is zero, the 1D coupled seepage problem, i.e., Eq. (2.20), can be rearranged as   @ @ @S @h kðhÞ ðh  xÞ ¼ n @x @x @h @t

ð2:29Þ

Using dimensionless space and time variables X = ax and T = akst/hs, and T X defining a new variable, V ¼ eah  e 2 þ 4 , one can derive the differential equation of 1D uncoupled problem, which is similar to the form of Eq. (2.21).

30

2 Analytical Solution to 1D Coupled Infiltration and Deformation …

The analytical solution to the uncoupled governing equation before the ponding  T = akst/hs). time tI is hðX; T Þ, where T is replaced Eq. (2.23) with T( When t > tI, let t′ = t − tI, the initial pressure head profile is h0i (t′ = 0), which is X T 0 derived from Eq. (2.26). Introducing T 0 = akst′/hs, X = ax, and V 0 ¼ eah  e 2 þ 4 , we obtain the governing equation and its initial and boundary conditions, which can be obtained from Eqs. (2.21) and (2.27a, 2.27b, 2.27c). Then the solution to the 1D differential equation of the uncoupled infiltration after ponding time tI is hðX; T 0 Þ. Here T in Eq. (2.21) is replaced with T 0 .

2.2.3

Examples and Analysis Results

In this study, the boundary conditions are prescribed, as usual, at the lower and upper boundaries. Ignoring the variation of the groundwater table, the lower boundary is fixed at x = l, where the pressure head is zero, i.e., h0 = 0. The surface boundary is subject to varying rainfall intensity, i.e., the boundary at the surface boundary varies with time qjX¼0 ¼ q0 ext hjX¼0 ¼ h0

t  ti t [ ti

ð2:30aÞ

where q0 is the initial rainfall intensity; x is a parameter that describes the increasing trend of rainfall intensity with time; tI is the ponding time; h0 is the pressure head that controlled the top boundary after the ponding time. It can vary with ponding depth and is assumed to be zero. Table 2.2 gives the parameters used in the examples. Three rainfall density cases are considered by varying the values of q0 and x, as shown in Fig. 2.4 and Table 2.2. In Fig. 2.4, type A rainfall represents rapidly increasing intensity in a short duration. Type C rainfall is a long-term rain with no obvious increase in intensity. Type B rainfall lies between types A and C. q0 and x are external parameters that influence unsaturated infiltration in unsaturated medium. Soil parameters, ks and a also affect the unsaturated seepage, and are listed in Table 2.2. Therefore, a critical time tI can be defined as tI ¼ f ðq0 ; x; ks ; aÞ

ð2:30bÞ

If t is smaller than tI, no ponding phenomenon will occur. Figure 2.5 shows the pressure head profiles under the uncoupled and coupled conditions before and after tI. The parameters are ks = 1.0  10−4 m/s, q ¼ q0 expðxtÞ, q0 = 0.7  10−4 m/s, x ¼ 4:0  103 s1 , a = 0. 05 cm−1, hs = 0.434, and F = 103 kPa. In Fig. 2.5, the upper boundary is subject to rainfall with its intensity of type B shown in Fig. 2.6. The rate of the change of the pressure head gradually increases before tI and decreases after tI. Figure 2.6 shows that the

2.2 Effect of Rainfall Pattern

31

Table 2.2 Ponding time for several types of soil Soil type

Example no.

Rainfall intensity q0 exp(xt) q0/ x ks (s−1)

Soil parameter aa (cm−1)

Soil parameter kbs (m/s)

Ponding time ti

Coupled (s)

Uncoupled (s)

Clay

a 0.5 1.5e−2 0.01 1.0e−8 806 736 b 1.0e−9 959 889 Silt a 0.5 1.5e−2 0.05 1.0e−6 405 394 b 0.7 4.0e−3 0.05 1.0e−6 1325 1314 c 0.7 4.0e−3 0.02 1.0e−6 1548 1465 Sand a 0.5 1.5e−2 0.1 1.0e−4 268 262 b 0.7 4.0e−3 765 745 c 0.9 1.0e−3 2172 2095 a The parameter a is according to reference Zhan and Ng (2004) and Van Genuchten (1980) b The value of ks is majority referring to Hillel (1998)

ponding times for the coupled and uncoupled conditions are different. The ponding time is 861 s for the coupled condition and tI = 822 s for the uncoupled one. tI for the uncoupled case is more than that for the coupled case because of positive value of F. Water infiltration into the collapse unsaturated soil causes a reduction in the soil volume. The pressure head difference caused by the coupling effect almost disappears when the top surface is controlled by the pressure head, i.e., t > tI. This verifies that change of the top boundary greatly influences the coupling effect. Meanwhile, the analytical solution is in accordance with such numerical method as a computer program FlexPDE (PDE Solutions Inc. 2004), which is implemented to solve the governing equations. Figure 2.6 shows the pressure head profiles under the uncoupled and coupled conditions before and after tI. The parameters are ks = 1.0  10−4 m/s, q ¼ q0 expðxtÞ, q0 = 0.7  10−4 m/s, x = 4.0  10−3 s−1, a = 0. 05 cm−1, others are kept as shown in Fig. 2.5. In Fig. 2.6, the upper boundary is subject to rainfall with its intensity of type B as shown in Fig. 2.7. The rate of the change of the pressure head gradually rises before tI and then decreases after tI. Figure 2.6 shows that ponding time for the coupled and uncoupled conditions is different. The ponding time is 861 s for the coupled condition and tI = 822 s for the uncoupled one. tI for the uncoupled case is more than that for the coupled case because of positive value of F. Water infiltration into the collapse unsaturated soil makes the soil volume shrink. The pressure head difference caused by the coupling effect almost disappears when the top surface is controlled by the pressure head, i.e., t > tI. This verifies that change of the top boundary greatly influences the coupling effect. Figure 2.7 shows the pressure head profiles considering the coupling effect before and after the ponding time. The parameters are as follows: ks = 1.0  10−6 m/s, q0 = 0.7  10−6 cm/min, x = 4.0  10−3 s−1, F = −103 kPa (expansive soil), and a = 0. 02 cm−1. Figure 2.7 indicates that before ponding, the pressure head

32

2 Analytical Solution to 1D Coupled Infiltration and Deformation …

Fig. 2.5 Pressure head profiles under coupled condition before and after tI for 1D vertical infiltration (hs = 0.434, ks = 1.0  10−4 m/s, q ¼ q0 expðxtÞ, q0 = 0.5  10−4 m/s, x = 1.5  10−2 s−1, F = 103 kPa, and a = 0.1 cm−1)

Fig. 2.6 Pressure head profiles under both uncoupled and coupled condition for 1D vertical infiltration (hs = 0.434, ks = 1.0  10−4 m/s, q ¼ q0 expðxtÞ, q0 = 0.7  10−4 m/s, x = 4.0  10−3 s−1, F = 103 kPa, and a = 0.05 cm−1)

difference caused by the coupling effect becomes obvious while the pressure head difference between uncoupled and coupled conditions becomes smaller after ponding. Within the period before the ponding time, the pressure head difference between the coupled and uncoupled conditions gradually increases, and then slowly decreases. It can be seen from Fig. 2.7 that the pressure head profiles considering the coupling effect move faster than those without the coupling effect. This is different from that from Figs. 2.5 and 2.6 because some soils such as expansive soil absorb water due to infiltration and swell, i.e., F < 0. The pressure head difference caused by the coupling effect gradually becomes obvious before tI, and then becomes smaller after tI. Figures 2.8, 2.9, 2.10 and 2.11 demonstrate influence of the parameters (q0/ks, x, ks, a) on the pressure head profiles. In Fig. 2.8 the parameters are ks = 1.0  10−4

2.2 Effect of Rainfall Pattern

33

Fig. 2.7 Pressure head profiles under both uncoupled and coupled condition for 1D vertical infiltration (hs = 0.434, ks = 1.0  10−6 m/min, q ¼ q0 expðxtÞ, q0 = 0.7  10−6 m/s, x = 4.0  10−3 s−1, F = −103 kPa, and a = 0.02 cm−1)

Fig. 2.8 The effect of q0/ks on pressure head profiles under both uncoupled and coupled condition at t = 795 s (hs = 0.434, ks = 1.0  10−4 m/s, q ¼ q0 expðxtÞ, q0/ks = 0.5, 0.7, 0.9, x = 4.0  10−3 s−1, F = 103 kPa, and a = 0.02 cm−1)

m/s, q0/ks = 0.5, 0.7 and 0.9, x = 4.0  10−3 s−1, F = 103 kPa, and a = 0. 02 cm−1. Figure 2.8 represents the pressure head profiles under both coupled and uncoupled conditions at t = 795 s due to different value of q0/ks. At t = 795 s the top ground ponds only for the uncoupled condition when q0/ks is 0.9. No full saturation at the surface boundary occurs for other cases at t = 795 s. The coupling effect seems be more distinct compared with the effect of q0/ks. Figure 2.9 shows the effect of x on the pressure head profiles at t = 301 s. In Fig. 2.9, the key parameters are q0/ks = 0.7, x = 1.5  10−2 s−1, = 4.0  10−3 s−1 and 1.0  10−3 s−1; other parameters are kept unchanged as shown in Fig. 2.8. This parameter x has an important role in the pressure head profiles. The effect of x is greater than the effect of q0/ks. At the same time, this influence is larger than the coupling effect.

34

2 Analytical Solution to 1D Coupled Infiltration and Deformation …

Fig. 2.9 The effect of w on pressure head profiles under both uncoupled and coupled condition at t = 301 s (hs = 0.434, ks = 1.0  10−4 m/s, q ¼ q0 expðxtÞ, q0/ks = 0.5, x = 1.5  10−2, 4.0  10−3, 1.0  10−3 s−1, F = 103 kPa, and a = 0.02 cm−1)

Fig. 2.10 The effect of a on pressure head profiles under both uncoupled and coupled condition at t = 745 s (hs = 0.434, ks = 1.0  10−4 m/s, q ¼ q0 expðxtÞ, q0/ks = 0.7, x = 4.0  10−3 s−1, F = 103 kPa, and a = 0.01, 0.05, 0.1 cm−1)

Figure 2.10 depicts the pressure head profile at t = 745 s under different a values: a = 0.01 cm−1, 0.05 cm−1, and 0.1 cm−1. Other parameters are ks = 1.0  10−4 m/s, q0/ks = 0.7, x = 4.0  10−3 s−1, and F = 103 kPa. At t = 745 s, the top boundary is saturated only for uncoupled condition with a = 0. 1 cm−1. At this time, the pressure head for a = 0. 01 cm−1 is −40 cm at the top surface and the pressure head for a = 0. 1 cm−1 is close to 0 when the coupling effect is considered. The pressure head profiles at different a values show great differences. Under the condition of a = 0. 1 cm−1, the pressure head considering the coupling effect is very close to the pressure head without the coupling effect. The pressure head profiles between the coupled and uncoupled conditions exhibit great difference at a = 0. 01 cm−1.

2.2 Effect of Rainfall Pattern

35

Fig. 2.11 The effect of ks on pressure head profiles under both uncoupled and coupled condition at t = 799 s (hs = 0.434, ks = 1.0  10−4, 1.0  10−5, 1.0  10−6 m/s, q ¼ q0 expðxtÞ, q0/ks = 0.7, x = 4.0  10−3 s−1, F = 103 kPa, and a = 0.01 cm−1)

Figure 2.11 indicates the influence of ks on the pressure head profiles under different cases at t = 799 s. In Fig. 2.11, the key parameters are a = 0.01 cm−1, ks = 1.0  10−4 m/s, 1.0  10−5 m/s, and 1.0  10−6 m/s, and other parameters are the same as shown in Fig. 2.10. The greater the coefficient of permeability at saturation is, the faster the pressure head moves. At t = 799 s, the top boundary ponds for the uncoupled condition when ks = 1.0  10−4 m/s. At this time, the pressure head at the top boundary for the uncoupled condition is about −90 cm when ks = 1.0  10−6 m/s. Rainfall infiltration itself has a significant effect on the pressure head distribution. As for the same q0/ks (= 0.7) in Fig. 2.11, ks = 1.0  10−4 m/s means q0 = 0.7  10−4 m/s, and ks = 1.0  10−6 m/s implies q0 = 0.7  10−6 m/s. Higher rainfall infiltration rate results in much quicker water movement in unsaturated soil. This can explain the great pressure head difference from Fig. 2.11. Based on the above discussion, the parameters play a role in the pressure head profiles. ks and x are the most important factors that greatly influence the pressure head distribution. Parameters a and q0/ks also affect the pressure head profiles (Wu and Zhang 2009). Table 2.2 presents the ponding times in different cases. The parameters in Table 2.2 are based on the references (Hillel 1998; Zhan and Ng 2004). Under the same q0/ks, the smaller the coefficient of permeability at saturation ks is, the larger the ponding time is. The ponding time for the coupled condition is always smaller than that under the uncoupled one because the F-value is positive. The difference range of the pressure head between the uncoupled and coupled conditions is related to a, which is based on Zhan and Ng (2004). When a ¼ 0:05 and 0.1 cm−1, the ponding times for both uncoupled and coupled conditions are very close. With a decreasing to 0.01 cm−1, the difference between the cases becomes greater. The conclusions coincide with the reference (Wu and Zhang 2009). For silt, the ponding time for Case A is 405 s, and increases to 1325 s for Case B when the coupling effect is considered. It can be seen from Table 2.2 that rainfall type has an important

36

2 Analytical Solution to 1D Coupled Infiltration and Deformation …

influence on the ponding time. For identifying dimensionless rainfall intensity, the ponding time for sand is the smallest under the same rainfall pattern. a also influences the ponding time. The larger the a value is, the smaller the ponding time is. For long-term rainfall type, ponding occurs at the top surface after a long infiltration time. This indicates a greater ponding time difference between the coupled and uncoupled conditions. The rainfall pattern plays an important role in the ponding time. According to Table 2.2, the difference between the uncoupled and coupled solutions is significant for clays, and negligible for silts and residual soils.

2.2.4

Conclusions

Analytical solutions to the one-dimensional transient coupled infiltration and deformation in unsaturated soils are obtained using Green’s function. The analytical solutions consider changing rainfall intensity boundary before and after ponding time. The analysis in the case studies indicates that the coupling effect has a significant influence on the pressure head profiles for the transient unsaturated seepage in unsaturated soil if the top boundary is controlled by flux. The coupling effect almost disappears if ponding at the ground surface occurs. The ponding time is different under the coupled and uncoupled conditions. The ponding time is influenced by parameters x and ks.

2.3 2.3.1

Effect of Boundary Condition Coupled Infiltration Equations

Based on Darcy’s law and mass and momentum conservation laws, the equation that governs 1D hydromechanical coupling in unsaturated soils, taking into account rises in the water table, can be obtained from Eqs. (2.9) and (2.12) (Lloret et al. 1987; Kim 2000; Chen et al. 2001; Wu et al. 2012). In which, h is the pressure head (h = uw/cw, cw = qwg); z is the vertical direction; t is the rainfall time; ev (ev = ez for the 1D problem) is the total volumetric strain of the soil mass, which is positive during compression and negative during swelling; k is the unsaturated hydraulic conductivity; n is the percentage of voids; S is the degree of saturation; (ua − uw) is the matric suction; ac = 1 − K/Ks is the hydromechanical coupling coefficient (0  ac  1); K is the bulk modulus of the solid skeleton, and Ks is the bulk modulus of the solid soil (Wu et al. 2012); E is the elastic modulus of the soil that accounts for variations in the net normal stresses (Wu et al. 2012); F is the elastic modulus of the soil due to variations in the matric suction, assumed to be a function of stress; bw is

2.3 Effect of Boundary Condition

37

the compressibility of the fluid; qw is the density of water; qs is the density of the soil phase; and g is the gravitational acceleration. Equations (2.9) and (2.12) govern the coupling of 1D deformation and seepage in unsaturated soils and account for the rise of the groundwater table.

2.3.2

Solution for Infiltration into Deformable Soils

With dimensionless variables, Z = az and T = a2kst/P are introduced into the function W(Z, T) = eah  eZ⁄2+T/4, and Eq. (2.18) can be rewritten as @W @ 2 W ¼ @T @Z 2

ð2:31Þ

The boundaries comprise a lower and an upper boundary (Fig. 2.12). In the literature of analytical solutions, the base boundary is usually assumed to coincide with a stationary groundwater table with the pressure head set to zero (Wu et al. 2012). However, in this study, the authors consider a zero flux at the base boundary. The hydraulic boundary condition is given by k

@h þ k jz¼0 ¼ 0 @z

ð2:32aÞ

The top boundary in Fig. 2.12 is controlled by constant rain intensity (q) at the ground surface, expressed as

Fig. 2.12 Top and base flux boundaries for a soil profile with a finite thickness. The soil layer is between z = 0 and z = D, where z is the vertical coordinate

2 Analytical Solution to 1D Coupled Infiltration and Deformation …

38



@h þ k

¼ qðz; tÞ k @z z¼d

ð2:32bÞ

in which d is the depth in the 1D unsaturated infiltration problem. The initial condition, assuming a hydrostatic profile and boundary equations can be rewritten in the dimensionless form W ðZ ; 0Þ ¼ eahi þ Z=2

ð2:33aÞ

@W W þ jZ¼0 ¼ 0 @Z 2

ð2:33bÞ

@W W q þ jZ¼D ¼  eT=4 @Z 2 ks

ð2:33cÞ

where D(ad) is the dimensionless length, and hi is the initial pressure head when the duration (t) is zero. Based on a Fourier integral transformation (Ozisik 1989), the exact solution to Eq. (2.31) can be derived as follows: 8 2 0 139 ZT = 1 2 2 0 1 < Z=2 X 4K ðbn ; Z Þ  eðbn þ 0:25ÞT @F ðbn Þ þ hðZ; T Þ ¼ ln e ebn t Aðbn ; t0 Þdt0A5 ; a : n¼0 t0 ¼0

ð2:34Þ in which: pffiffiffi K ðbn ; Z Þ ¼ 2

"

b2n þ 0:25 Dðb2n þ 0:25Þ þ 0:5 ZD

F ðbn Þ ¼

#1=2 cos bn Z

K ðbn ; z0 ÞV ðz0 ; 0Þ dz0

ð2:35aÞ

ð2:35bÞ

0

pffiffiffi Aðbn ; t Þ ¼ 2 0

"

b2n þ 0:25 Dðb2n þ 0:25Þ þ 0:5

#1=2 

0

eah0 et =4 þ

qðZ; T Þ 0 cosðbn DÞeL=2 þ t =4 ks



ð2:35cÞ where the eigenvalue bn satisfies bn tan(bnD) = 0.5.

2.3 Effect of Boundary Condition

39

Table 2.3 Parameters used to analyze 1D groundwater seepage Dimensionless rainfall intensity (q/ks)

Saturated coefficient of permeability (ks)

Desaturation coefficient (a)

Volumetric moisture (hs)

Elastic modulus with respect to suction change (F)

Hydromechanical coupling coefficient (ac)

1.0

10−5 m/s

0.01, 0.05, and 0.1 cm−1

0.4

±5  103, and ±105 kPa

1

2.3.3

Example

In this study, a 1D homogeneous soil layer is subjected to rainfall seepage. The flux is set to zero at the base boundary, and to a specific rate at the top boundary (Fig. 2.12). In Fig. 2.12, the displacement at the base boundary is zero, i.e., uz = 0. The analysis parameters are provided in Table 2.3 (Van Genuchten 1980; Tsai and Wang 2011; Zhan et al. 2013). In this study, the authors incorporate the effect of the dimensionless rainfall intensity (q/ks), desaturation coefficient (a), and soil layer height on the pressure head distribution, as well as the coupling response.

2.3.3.1

Influence of Rainfall Intensity

The variations in the dimensionless rainfall intensity over the duration of the rainfall event for both the coupled and uncoupled analyses are shown in Fig. 2.13, for H = 400 cm. The parameters used are ks = 10−5 m/s, hs = 0.4, |F| = 5  103 kPa, and a =

Fig. 2.13 Changes in the pressure head profile over time under coupled and uncoupled states

40

2 Analytical Solution to 1D Coupled Infiltration and Deformation …

0.01 cm−1 (Van Genuchten 1980; Tsai and Wang 2011; Wu et al. 2012). The value of F can be positive or negative; a negative F means an expansive soil where a suction decrease leads to soil volume increase, while a positive F denotes a “collapsible” soil where a suction decrease leads to soil volume decrease (Wu and Zhang 2009). When t = 1 h, the wetting front moves to a depth of 120 cm in both the uncoupled analysis and coupled analysis with F > 0. However, with F < 0, the wetting front reaches 180 cm in the coupled analysis. When t = 1 h, the difference in the pressure head for the coupled (F < 0) and uncoupled analyses is 27.5 cm at the upper boundary, and the maximum difference in the pressure head within the unsaturated zone is 75.1 cm at a soil depth of 340 cm. When t = 15 h, the difference in the pressure head at the top boundary between the coupled (F > 0) and uncoupled analyses is 13.7 cm, and the maximum difference in the pressure head within the soil is 61.8 cm at a depth of 160 cm. The coupling effect becomes more apparent with increasing time, particularly in the lower part of the soil. Compared with the case where the base boundary coincides with the stationary groundwater table (Wu et al. 2012), the case with zero flux at the base boundary leads to more pronounced coupling effects in the lower part of the soil column. It is also noted that groundwater ponding occurs at the base boundary (t = 15 h, Fig. 2.13), particularly in the case of expansive soil (F < 0). For a “collapsible” soil, i.e., the soil volume decreases when wetted (F > 0), groundwater ponding does not occur until very late (t = 28 h). The influence of the dimensionless rainfall intensity on the pressure head profile for the coupled and uncoupled analyses is shown in Fig. 2.14 for t = 20 h. Two values (q/ks = 0.2 and q/ks = 1.0) are used to obtain the results in Fig. 2.14; all other parameters are the same as in Fig. 2.13. As the infiltration time increases and the wetting front moves downwards, the pressure head increases rapidly in the shallow depths of the unsaturated soil. Rainfall intensity plays an important role in the advancement of the wetting front, and the pressure head profile moves more quickly as the rainfall intensity increases. The coupling effect is also closely linked with the rainfall intensity. At t = 20 h, when q/ks = 0.2, the maximum pressure head difference between the coupled (F < 0) and uncoupled analysis results is 28.73 cm in the middle of the soil profile. However, at the base boundary with q/ks = 1.0, the maximum change in the pressure head between the coupled (F < 0) and uncoupled analysis results is 71.94 cm. Other studies reported that the coupling effect is more noticeable in a shallow layer of unsaturated soil (Wu et al. 2012). However, in this study, the coupling effect becomes more noticeable in the base layer, particularly so with increasing time. When q/ks = 1 and t = 30 h, the difference in the maximum pressure head between the coupled (F < 0) and uncoupled analyses is 11.43 cm at the top boundary, and the maximum difference in the pressure head is 71.94 cm at the base boundary. As the rainfall accumulates at the base boundary, the coupled effect becomes more apparent there.

2.3 Effect of Boundary Condition

41

Fig. 2.14 The effect of the dimensionless rainfall intensity on the pressure head profile under coupled and uncoupled states

Fig. 2.15 The effect of a on the pressure head profile under coupled and uncoupled states

2.3.3.2

Effect of the Desaturation Coefficient

Figure 2.15 illustrates the effect of the desaturation coefficient (a = 0.02 and 0.1 cm−1) on the pressure head distribution under coupled and uncoupled states when t = 20 h and q/ks = 1. The desaturation coefficient for clay soils has a significant influence on the pressure head distribution for both the coupled and uncoupled states (Wu et al. 2012). There is a marked difference in the pressure head in the base layer (a = 0.02 cm−1) between the coupled and uncoupled analyses. When a is 0.1 cm−1, the difference in the pressure head between the coupled and uncoupled analyses decreases sharply. For large a (0.1 cm−1), no remarkable difference was observed between the coupled and uncoupled states. The difference in

2 Analytical Solution to 1D Coupled Infiltration and Deformation …

42

the pressure head increases as a decreases. The pressure head distribution is similar for “collapsible” soils (a = 0.1 cm−1) with or without the coupling effect (Fig. 2.15).

2.3.3.3

Effect of F

The elastic modulus of the soil structure for variations in the matric suction is a key parameter when examining seepage into deformable unsaturated porous media (Wu et al. 2012). The effect of F (F = ±5  103 and ±1  105 kPa) on the pressure head profile under coupled and uncoupled conditions is presented in Fig. 2.16, for t = 15 h and q/ks = 1. In the analysis, the desaturation coefficient (a) is set as 0.01 cm−1, and four values of F are used: 105, 5  103, −5  103, and −105 kPa. The other parameters are the same as in Fig. 2.13. At any given time, the difference in pressure heads between the uncoupled and coupled analyses increases as |F| decreases. Indeed, when |F| = 1  105 kPa, the difference between the coupled and uncoupled analyses is minimal. Thus, the effect of F on the coupling effect is significant. The greater the soil stiffness (|F|), the more marked the coupling effect. When F = − 5  103 kPa (i.e., an expansive soil) and t = 15 h, the negative pore-water pressure head dissipates faster in the coupled analysis. When F = 5  103 kPa (i.e., loess-type soil), the pressure head profile for the coupled analysis falls behind that for the uncoupled analysis. With time, the coupling effect may become more apparent in the bottom layer than in the top layer (Fig. 2.16).

2.3.3.4

Effect of the Soil Layer Height

The effect of the dimensionless soil layer thickness on the pressure head profile under uncoupled and coupled analyses is presented in Fig. 2.17. The following Fig. 2.16 The effect of F on the pressure head profile under coupled and uncoupled states

2.3 Effect of Boundary Condition

43

Fig. 2.17 The influence of the soil layer height on the pressure head profile under coupled and uncoupled states

parameters were used: ks = 10−5 m/s, q/ks = 1, hs = 0.4, |F| = 5  103 kPa, a = 0.01 cm−1, and t = 10 h. H is taken as 200 and 600 cm. To effectively study the influence of the depth of the soil layer, both the soil depth and the pressure head are dimensionless (dimensionless soil depth = depth/H; dimensionless pressure head = pressure head/Pmax; Pmax is the maximum absolute value of the initial pressure head). In Fig. 2.17, the wetting front does not reach the base boundary when H = 6 m at t = 10 h, but does when H = 2 m, showing that the pressure head profile moves faster as the depth of the soil layer decreases. In Figs. 2.13, 2.14, 2.15, 2.16 and 2.17, water accumulates after a prolonged period of rainfall infiltration in swelling soils. Rainwater-induced groundwater ponding occurs earlier in expansive soils than in “collapsible” soils. Swelling soils easily become saturated earlier than collapsible soils under the same infiltration conditions. Therefore, ponding in expansive soils occurs earlier than in collapsible soils.

2.3.4

Conclusions

The authors developed an analytical solution that describes 1D transient groundwater flow in unsaturated soils using a Fourier integral transform, in which a zero-flux boundary allows the groundwater level to move. The results indicate that the lower boundary condition plays an important role in the pressure head profile and that the coupling effect is more pronounced for a zero-flux boundary than for a zero-pressure head boundary. The zero-flux base boundary also leads to groundwater ponding in the lower part of the soil mass. The ponding occurs much earlier in expansive soils than in “collapsible” soils. The coupling effect becomes more noticeable when ponding occurs. The pressure head profile moves faster as the thickness of the soil layer decreases.

44

2.4 2.4.1

2 Analytical Solution to 1D Coupled Infiltration and Deformation …

Effect of Layered Structure Governing Equations

Introducing three new variables, K ¼ eauw , X = acwx, and T = a2cwkst/Q, Eq. (2.18) is transformed to @2K @K @K ¼ þ 2 @X @X @T

uw   wae

ð2:36Þ

We consider the case where the soil profile consists of two distinct soil layers. The reference datum in Figs. 2.18 and 2.19, X = 0, is assumed to be at the interface of the two layers. Several dimensionless parameters are defined here as follows: X ¼ ab cw x for  lb  x  0 so that Lb ¼ ab cw lb

ð2:37aÞ

X ¼ at cw x for 0  x  lb so that Lt ¼ at cw lt

ð2:37bÞ

Kb ¼ K=ksb ; qAb ¼ qA =ksb ; qBb ¼ qB =ksb

ð2:37cÞ

Kt ¼ K=kst ; qAt ¼ qA =kst ; qBt ¼ qB =kst

ð2:37dÞ

T ¼ a2b cw ksb t=Qb

ð2:37eÞ

where ab and at are the desaturation coefficients for the bottom layer and the top layer, respectively; ksb and kst are the coefficients of permeability at full saturation for the bottom layer and the top layer, respectively; qA is the initial steady-state flux at the ground surface that determines the initial pressure distribution along with the pore-water pressure at the bottom boundary (Srivastava and Yeh 1991); qB is the prescribed flux at the ground surface for times greater than 0. The governing equations are then written as @ 2 Kb @Kb @Kb ¼ þ @X 2 @X @T

Fig. 2.18 Sketch of a landfill cover

for  Lb \X\0

ð2:38aÞ

2.4 Effect of Layered Structure

45

Fig. 2.19 General initial conditions for a two-layer soil profile

@ 2 Kt @Kt @Kt ¼b þ @X 2 @X @T where b ¼

for 0\X\Lt

ð2:38bÞ

a2b ksb Qb . a2t kst Qt

The initial condition depends mainly on the rainfall history and the hydraulic conductivities of the two-layer soils. If the historic rainfall intensity is less than ks, the initial pore-water pressure in the zone above the water table is negative. qA is the antecedent infiltration rate. It is the initial steady-state flux at the ground surface. qA along with the pore-water pressure at the bottom boundary determine the initial pore-pressure distribution. The initial pore-water pressure is u0 at the lower boundary, and u1 at the top boundary. At the interface the initial pore-water pressure is continuous. Through a series of transformations, the initial conditions are described using an exponential function. The initial condition in the bottom layer (Fig. 2.20) is Kb ðX; 0Þ ¼ qAb  ðqAb  eab u0 ÞeðLb þ X Þ

ð2:39aÞ

in which, qAb is a dimensionless initial steady-state flux for the bottom layer, qAb = qA/ksb, ksb is the saturated permeability of the bottom layer; u0 is the pore-water pressure of the lower boundary located at the stationary groundwater table.

Fig. 2.20 Flux boundary conditions for a two-layer soil profile

46

2 Analytical Solution to 1D Coupled Infiltration and Deformation …

The initial condition in the top layer (see Fig. 2.20) is n

a =a o Kt ðX; 0Þ ¼ qAt  qAt  qAb  ðqAb  eab u0 ÞeLb t b eX

ð2:39bÞ

in which, qAt is an initial steady-state dimensionless flux for the top layer, qAt = qA/ kst, kst is the saturated permeability of the top layer. The initial condition in the top layer is related with qAb, qAt, and u0. The specified pressure at the lower boundary is (Fig. 2.20) Kb ðLb ; TÞ ¼ eab u0

ð2:40aÞ

The specified flux at the upper boundary is 

@Kt þ Kt @X

 ¼ qBt

ð2:40bÞ

X¼Lt

Continuity of flux at the interface between the two layers requires that: Ksb

    @Kb @Kt þ Kb þ Kt ¼ Kst @X @X X¼0 X¼0 þ

ð2:40cÞ

Equality of pore-water pressure at the interface requires that: 

lnKb ab





ln Kt ¼ at X¼0

 ð2:40dÞ X¼0 þ

After a series of linear transformations, the governing equations of coupled seepage and deformation in unsaturated soils, i.e., Eqs. (2.38a, 2.38b)–(2.40a, 2.40b, 2.40c, 2.40d), are similar to the governing equation under uncoupled conditions. The analytical solution for this type of soil system without considering the coupling effect can be obtained through Laplace’s transformation (Srivastava and Yeh 1991). Then the solution to Eqs. (2.38a, 2.38b)–(2.40a, 2.40b, 2.40c, 2.40d), which considers only coupled seepage and deformation under identical a value (ab = at) for two-layer unsaturated soils, can now be given as follows: Kb ¼ qBb  ðqBb  eab u0 ÞeðLb þ XÞ  4ðqBt  qAb ÞeðLt þ XÞ=2 ½RAb þ RBb or RCb 

for  Lb  X  0



Kt ¼ qBb  qBt  qBb þ ðqBt  eab u0 ÞeLb eX  4ðqBt  qAb ÞeðLt XÞ=2 ½RAt þ RBt or RCt 

for 0  X  Lt

ð2:41aÞ

ð2:41bÞ

where RAb, RBb, RCb, RAt, RBt, and RCt are functions of X and T. Details of these functions are found in Appendix A (Srivastava and Yeh 1991).

2.4 Effect of Layered Structure

47

Where D is a function of k and l, which can be found in Srivastava and Yeh (1991). The pore-water pressure in the two-layer unsaturated soils when ab = at can be now calculated as follows: uðX; TÞ ¼

1 Kb 1n ab ksb eab wae

uðX; TÞ ¼

2.4.2

1 Kt 1n at kst eat wae

for  Lb \X\0

ð2:42aÞ

for 0\X\Lt

ð2:42bÞ

Parametric Study

In the examples, the total thickness of the two soil layers is taken to be 1 m, each layer being 0.5 m thick. The initial pore-water pressure distribution (i.e., initial condition) is characterized by Eq. (2.39a, 2.39b). The lower boundary is assumed to be a stationary groundwater table where the pore-water pressure is u0 = 0. The upper boundary is subjected to a constant rainfall flux (i.e., rainfall intensity).

2.4.2.1

Effect of H

Figure 2.21 indicates changes in the pore-water pressure profile in the two-layer soils with Hb/Ht = 0.1, which considers uncoupled and coupled seepage and deformation in unsaturated soils. In Fig. 2.21, the parameters are qA = 0 m/s, qB = 2.7  10−7 m/s, ab = at = 0.01 kPa−1, ksb = kst = 3  10−7 m/s, Hb = −103 kPa, Ht =

Fig. 2.21 Pore-water pressure profiles in two-layer soils under uncoupled and coupled conditions at Hb/Ht = 0.1 (Hb = −103 kPa, Ht = −104 kPa)

1 0.9

t=1 hr

t=5 hrs

t=20 hrs

0.8

Depth (m)

0.7 0.6 0.5 0.4

Initial condition t=1 hr (uncoupled) t=1 hr (coupled) t=5 hrs (uncoupled) t=5 hrs (coupled) t=20 hrs(uncoupled) t=20 hrs (coupled)

0.3 0.2 0.1

-7

ksb=3×10 m/s -7

kst=3◊10 m/s -1

αb=0.01 kPa 3

Hb=-10 kPa 4

Ht=-10 kPa

0 -10

-8

-6

-4 u (kPa)

-2

0

48

2 Analytical Solution to 1D Coupled Infiltration and Deformation …

−104 kPa, wae = 0 kPa, and hs = 0.4. The sign of H determines whether the soil is collapsible or swelling when the two-layer soils are wetted. H is negative for an expansive soil and positive for a collapsible soil. Figure 2.22 shows that the pore-water pressure profiles considering the coupling effect move faster than those without considering the coupling effect due to the negative values of H. Figure 2.22 shows the pore-water pressure profiles in the two-layer soils with Hb = −104 kPa, Ht = −103 kPa (i.e., Hb/Ht = 10), with other parameters being held the same as in Fig. 2.21. Figure 2.21 further verifies that negative H values lead to faster changes in pore-water pressure when the coupling effect is considered. Soil swelling results in a decrease in the volume of large pores and an increase in the volume of small pores (Zhang and Li 2010), hence a decrease of saturated permeability and an increase of the permeability at high suctions. The rainfall infiltration process is a saturated/unsaturated coupled problem. Hence, the change of pore-water pressure in a soil upon rainfall will depend on the permeability in the infiltrated zone. In the top layer, the change of pore-water pressure will be faster because of the higher unsaturated permeability induced by soil swelling. At t = 20 h, the pressure difference between uncoupled and coupled conditions with Hb/Ht = 10 is about four times as that with Hb/Ht = 0.1 (Fig. 2.22). A smaller absolute value of H in the upper layer soil tends to cause more marked coupling effect. Figure 2.23 illustrates the pore-water pressure changes in the layered soils with different Ht values at t = 10 h. In Fig. 2.23, Hb is constant whereas Ht is varied, i.e., Hb = 103 kPa, Ht = ±103, ±105 kPa. Other parameters are kept identical as those in Figs. 2.21 and 2.22. Figure 2.23 indicates that changes in H have no effect on the pressure profiles if the coupling effect is not considered. When H vanishes the uncoupled conditions occurs. However, variations of H for the upper layer soil clearly influence the pressure profiles if the coupling effect is considered. The smaller the absolute value of H for the upper layer soil is, the greater the pressure difference is. In fact, H is related to the water storage of unsaturated soils subject to

Fig. 2.22 Pore-water pressure profiles in two-layer soils under uncoupled and coupled conditions at Hb/Ht = 10 (Hb = −104 kPa, Ht = −103 kPa)

1 0.9

t=1 hr

t=5 hrs

t=20 hrs

t=50 hrs

0.8

Depth (m)

0.7 0.6 0.5

Initial condition t=1 hr (uncoupled) t=1 hr (coupled) t=5 hrs (uncoupled) t=5 hrs (coupled) t=20 hrs (uncoupled) t=20 hrs (coupled) t=50 hrs (uncoupled) t=50 hrs (coupled)

0.4 0.3 0.2 0.1

-7

ksb=3×10 m/s -7

kst=3◊10 m/s α=0.01 kPa-1 4 Hb=-10 kPa 3

Ht=-10 kPa

0 -10

-8

-6

-4 u (kPa)

-2

0

2.4 Effect of Layered Structure

49

Fig. 2.23 Pore-water pressure profiles in two-layer soils under uncoupled and coupled conditions at different Ht values (Hb = 103 kPa)

1 0.9 0.8

Depth (m)

0.7 0.6 0.5 0.4 Initial condition

0.3

Ht = 1e3 kPa (uncoupled) Ht = 1e3 kPa (coupled)

0.2

Ht = -1e3 kPa (coupled) Ht = 1e5 kPa (uncoupled)

0.1

Ht = 1e5 kPa (coupled) Ht = -1e5 kPa (coupled)

-7

ks=3×10 m/s -1

α=0.01 kPa 3 Hb=10 kPa

0 -10

-8

-6

-4

-2

0

u (kPa)

volume change. The volume of a collapsible soil decreases with matric suction and H is positive. A smaller absolute value of H corresponds to a larger volume reduction and hence a smaller water storage of the soil. The pore-water pressure change has to be larger given the same infiltration amount. In contrast, the volume of an expansive soil increases with matric suction and H is negative. A smaller absolute value of H corresponds to a larger volume increase and hence a larger water storage of the soil. The pore-water pressure change is then smaller given the same infiltration amount. Figure 2.24 presents changes in pore-water pressure with time, where qA = 0 m/ s, qB = 2.7  10−6 m/s, ab = at = 0.01 kPa−1, ksb = 3  10−5 m/s and kst = 3  10−6 m/s in Fig. 2.24a, ksb = 3  10−6 m/s and kst = 3  10−5 m/s in Fig. 2.25b, with other parameters being held at wae = 0 kPa, H = ±103, ±104 kPa, and hs = 0.4. Figure 2.25 also depicts the influence of H on the pore-water pressure profiles under coupled conditions. The value of H has a strong influence on the coupling effect. The pore-water pressure difference caused by coupling is closely related to the H values of the two-layer soils. The pressure profile is convex at the interface of the two soil layers (Fig. 2.24b) when the coefficient of permeability for the bottom layer (ksb) is less than the coefficient of permeability for the top layer (kst), but is concave (Fig. 2.24a) at the same location when ksb is greater than kst.

2.4.2.2

Effect of ks

Typical values of coefficient of permeability at full saturation for several types of soil are shown in Table 2.4. The coefficients of permeability of coarse-grained soils at full saturation are typically larger than those of fine-grained soils. The unsaturated permeability of a coarse-grained soil decreases sharply with increasing matric

2 Analytical Solution to 1D Coupled Infiltration and Deformation …

50

(a)

(b)

1

1 t=0.2 hr

t=1 hr

0.9

0.8

0.8

0.7

0.7

0.6 Initial condition t=0.2 hr (uncoupled) t=0.2 hr (H =10^3 kPa) t=0.2 hr (H =-10^3 kPa) t=0.2 hr (H =10^4 kPa) t=0.2 hr (H =-10^4 kPa) t=1 hr (uncoupled) t=1 hr (H =10^3 kPa) t=1 hr (H =-10^3 kPa) t=1 hr (H =10^4 kPa) t=1 hr (H =-10^4 kPa)

0.5 0.4 0.3 0.2 0.1

Depth (m)

Depth (m)

t=0.2 hr

0.9

0.6 Initial condition t=0.2 hrs (uncoupled) t=0.2 hr (H =10^3 kPa) t=0.2 hr (H =-10^3 kPa) t=0.2 hr (H =10^4 kPa) t=0.2 hr (H =-10^4 kPa) t=2 hrs (uncoupled) t=2 hrs (H =10^3 kPa) t=2 hrs (H =-10^3 kPa) t=2 hrs (H =10^4 kPa) t=2 hrs (H =-10^4 kPa)

0.5 0.4 0.3

ksb=3◊10-5 m/s

0.2

kst=3◊10-6 m/s αb=0.01 kPa-1 αt=0.01 kPa

0.1

-1

0

t=2 hrs

ksb=3◊10-6 m/s kst=3◊10-5 m/s αb=0.01 kPa-1 αt=0.01 kPa-1

0 -10

-8

-6

-4

-2

0

-10

-8

-6

u (kPa)

-4

-2

0

u (kPa)

Fig. 2.24 Effect of H on pore-water pressure profiles for layered soils under uncoupled and coupled conditions with different coefficients of permeability: a ksb = 3  10−5 m/s, kst = 3  10−6 m/s; b ksb = 3  10−6 m/s, kst = 3  10−5 m/s

(a)

(b)

1 t=2 hrs

1 0.9

t=50 hrs

t=10 hrs

0.8

0.8

0.7

0.7

0.6

0.6

Depth (m)

Depth (m)

0.9

0.5 0.4

Initial condition t=2 hrs (uncoupled) t=2 hrs (coupled) t=10 hrs (uncoupled) t=10 hrs (coupled) t=50 hrs (uncoupled) t=50 hrs (coupled)

0.3 0.2 0.1

t=2 hrs

0.4

Initial condition t=2 hrs (uncoupled) t=2 hrs (coupled) t=10 hrs (uncoupled) t=10 hrs (coupled) t=50 hrs (uncoupled) t=50 hrs (coupled)

0.3 ksb=3×10 m/s

0.2

-5

-1

αb=0.01 kPa

0.1

αt=0.01 kPa-1

t=50 hrs

0.5

-7

kst=3◊10 m/s

t=10 hrs

ksb=3×10-5m/s -7

kst=3◊10 m/s -1

αb=0.01 kPa

-1

αt=0.01 kPa

0

0 -10

-8

-6

-4 u (kPa)

-2

0

-10

-8

-6

-4

-2

0

u (kPa)

Fig. 2.25 Pore-water pressure profiles in two-layer soils with ab = at = 0.01 kPa−1: a ksb = 3  10−7 m/s, kst = 3  10−5 m/s; b ksb = 3  10−5 m/s, kst = 3  10−7 m/s

suction and finally becomes lower than that of a fine-grained soil. In Gardner’s model (Gardner 1958), the desaturation coefficient, a, for a coarse-grained soil is larger than that for a fine-grained soil. Figure 2.25 shows the effect of the coefficients of permeability for the two soil layers on the pore-water pressure distributions, where qA = 0 m/s, qB = 2.7  10−7 m/s, ab = at = 0.01 kPa−1, with other parameters being held at wae = 0 kPa, H = 103 kPa, and hs = 0.4. The saturated coefficients of permeability of the two soil layers are ksb = 3  10−7 m/s and kst = 3  10−5 m/s in Fig. 2.25a and ksb = 3  10−5 m/s and kst = 3  10−7 m/s in Fig. 2.25b.

2.4 Effect of Layered Structure

51

In Fig. 2.25, there is a pore-water pressure difference caused by coupling of seepage and deformation in the layered unsaturated soils. The pressure difference is a function of time. In the early stage, the pore-water pressure difference between uncoupled and coupled cases becomes more obvious with time, and the coupling effect is most obvious at t = 10 h in Fig. 2.25. But the pore-water pressure difference gradually becomes smaller and even disappears after t = 10 h. This is because infiltration in the unsaturated soils approaches a steady-state after a long infiltration time, or the right-side item in Eq. (2.12) approaches zero. A certain negative pore-water pressure remains in the soil near the ground surface. This is because the coefficient of permeability at full saturation for the upper soil is greater than the infiltration rate. When the wetting front reaches the top of the bottom layer, most of the water that has infiltrated tends to accumulate due to the relatively low permeability of the bottom layer. So the negative pore-water pressure drops significantly and a turning point occurs at the interface of the two-layer system, as shown in Fig. 2.25. The pressure profile near the interface is convex when kst/ksb = 100 in Fig. 2.25a, and is concave when kst/ksb = 0.01 in Fig. 2.25b. Figure 2.26 shows the influence of saturated coefficients of permeability on pore-water pressure profiles considering the coupling effect at t = 10 h, with ab = at = 0.01 kPa−1. Other parameters are held at wae = 0 kPa, H = 103 kPa, qA = 0 m/s, qB = 2.7  10−7 m/s, and hs = 0.4. In Fig. 2.26a, kst is 3  10−7 m/s; three values of ksb are considered: 3  10−5 m/s, 3  10−6 m/s, and 3  10−7 m/s. In Fig. 2.26b, ksb is 3  10−7 m/s, kst is taken to be 3  10−5 m/s, 3  10−6 m/s, and 3  10−7 m/s, respectively. As shown in Fig. 2.26, ks has a significant influence on the pore-water pressure distributions. The ratio of kst to ksb ranges from 0.01 to 1 in Fig. 2.26a. When kst/ksb = 0.01, 0.1 and 1 the pore-water pressure differences caused by the coupling effect at the interface of the two-layer system increase with kst/ksb. The pore-water pressure profile evolves most quickly under the condition of ksb = kst shown in Fig. 2.26a, whether coupled seepage and deformation is considered or not. As the difference between ksb and kst increases, the pressure profiles evolve slower. In Fig. 2.26b kst/ksb changes from 1 to 100. With the same ksb value, the smaller the value of kst of the top layer is (Fig. 2.26b), the more quickly the negative pore-water pressure near the surface of the top layer decreases in both coupled and uncoupled conditions. When ksb = kst in Fig. 2.26b the coupling effect is most marked. The results show that a large difference between kst and ksb could reduce the coupling effect. Thus a great difference in the coefficients of permeability for the two-layer soil system helps reduce precipitation infiltration into the deeper soil layer.

2.4.2.3

Effect of a

The analytical solution for a two-layer soil system can be obtained only under the condition that the values of a for the two layers are identical. The a values for several types of soil are listed in Table 2.4. Figure 2.27 shows changes in pore-water pressure distributions with time, where ab = at = 0.03 kPa−1 in

2 Analytical Solution to 1D Coupled Infiltration and Deformation …

52

Table 2.4 Saturated coefficients of permeability (ks) and desaturation coefficients (a) for typical soils Soil type

−10

10 – 10−8 10−8–10−6 10−5–10−3

Clay Silt Sand

(a)

ks (m/s)

a (kPa−1)

References

0.016 for Beit Netofa clay

Van Genuchten (1980), Hille (1998)

0.043 for a silt loam 0.106 for a fine sand

(b)

1 t=10 hrs

1

0.8

0.8

0.7

0.7

0.6

0.6

0.5 0.4

Initial condition ks1 = 3◊10-7 m/s (uncoupled) ks1 = 3◊10-7 m/s (coupled) ks1 = 3◊10-6 m/s (uncoupled) ks1 = 3◊10-6 m/s (coupled) ks1 = 3◊10-5 m/s (uncoupled) ks1 = 3◊10-5 m/s (coupled)

0.3 0.2 0.1

t=10 hrs

0.9

Depth (m)

Depth (m)

0.9

0.5 0.4

Initial condition ks2 = 3◊10-7 m/s (uncoupled) ks2 = 3◊10-7 m/s (coupled) ks2 = 3◊10-6 m/s (uncoupled) ks2 = 3◊10-6 m/s (coupled) ks2 = 3◊10-5 m/s (uncoupled) ks2 = 3◊10-5 m/s (coupled)

0.3 0.2

-7

kst=3◊10 m/s αb=0.01 kPa-1

0.1

-1

αt=0.01 kPa

ksb=3◊10-7m/s αb=0.01 kPa-1 αt=0.01 kPa-1

0

0 -10

-8

-6

-4

-2

0

u (kPa)

-10

-8

-6

-4

-2

0

u (kPa)

Fig. 2.26 Effect of ks on pore-water pressure profiles for layered soils under uncoupled and coupled conditions, t = 10 h: a different ksb values; b different kst values

Fig. 2.27a and ab = at = 0.07 kPa−1 in Fig. 2.27b. The remaining parameters in Fig. 2.27 are held at ksb = 3  10−6 m/s, kst = 3  10−5 m/s, wae = 0 kPa, hs = 0.4, H = 103 kPa, qA = 0.45  10−6 m/s, qB = 2.7  10−6 m/s. When ab = at = 0.07 kPa−1, the coupling-induced difference is almost negligible. The coupling effect becomes more marked when the desaturation coefficient decreases to 0.03 kPa−1 from 0.07 kPa−1. The desaturation coefficient has an important effect on the pore-water pressure profiles under both coupled and uncoupled conditions when the desaturation coefficient is low.

2.4.2.4

Effect of Soil Porosity hsb and hst

Figures 2.28 and 2.29 demonstrate the effects of soil porosity on the pore-pressure distributions. Some parameters are kept constant: wae = 0 kPa, H = 103 kPa, qA = 0 m/s, qB = 2.7  10−6 m/s, ab = at = 0.01 kPa−1, ksb = kst = 3  10−6 m/s. Figure 2.28 shows changes in the pore-water pressure profiles in the two soil layers when hsb = 0.4 and hst = 0.6. Figures 2.28 and 2.29 illustrate pore-water pressure profiles in the layered soils with different hsb and different hst values, respectively. In Fig. 2.28, the volumetric water content at saturation (hs) is 0.4 for the upper layer

2.4 Effect of Layered Structure

(a)

(b)

1 0.9

t=5 hrs

t=1 hr

1 0.9

t=20 hrs

0.8

0.8

0.7

0.7

0.6

0.6

Depth (m)

Depth (m)

53

0.5 0.4

Initial condition t=1 hr (uncoupled) t=1 hr (coupled) t=5 hrs (uncoupled) t=5 hrs (coupled) t=20 hrs (uncoupled) t=20 hrs (coupled)

0.3 0.2 0.1 0 -10

-8

ksb=3×10 m/s

0.4

Initial condition t=1 hr (uncoupled) t=1 hr (coupled) t=5 hrs (uncoupled) t=5 hrs (coupled) t=20 hrs (uncoupled) t=20 hrs (coupled)

0.2

kst=3◊10-5 m/s -1

αb=0.03 kPa

0.1

αt=0.03 kPa-1

t=20 hrs

0.5

0.3 -6

t=5 hrs

t=1 hr

ksb =3×10-6 m/s kst =3◊10-5 m/s -1

αb =0.07 kPa

-1

αt =0.07 kPa

0

-6

-4

-2

-10

0

-8

-6

-4

-2

0

u (kPa)

u (kPa)

Fig. 2.27 Pore-water pressure profiles in two-layer soils with ksb = 3  10−6 m/s, kst = 3  10−5 m/s, H = 103 kPa, hs = 0.4, qA = 0.45  10−6 m/s, qB = 2.7  10−6 m/s: a ab = at = 0.03 kPa−1; b ab = at = 0.07 kPa−1

1 t=2 hrs

0.9 0.8 0.7 Depth (m)

Fig. 2.28 Pore-water pressure profiles in two-layer soils with respect to different hsb values (hst = 0.4, ab = at = 0.01 kPa−1, ksb = kst = 3  10−6 m/s, H = 103 kPa, qA = 0, qB = 2.7  10−6 m/s)

0.6 0.5 0.4

Initial condition θsb = 0.2 (uncoupled) θsb = 0.2 (coupled) θsb = 0.4 (uncoupled) θsb = 0.4 (coupled) θsb= 0.6 (uncoupled) θsb = 0.6 (coupled)

0.3 0.2 0.1

-6

ksb=kst=3×10 m/s -1

αb=αt=0.01 kPa θst=0.4 Hb=Ht=103kPa

0 -10

-8

-6

-4

-2

0

u (kPa)

soil and ranges from 0.2 to 0.6 for the bottom layer. Figure 2.28 shows the variations of pore-water pressure with hs for the lower layer soil. Variation of hs of the lower layer soil has some effect on the pressure profiles. However, such variation plays a negligible role in the pressure difference between the uncoupled and coupled cases. A greater hsb means slower movements of the pressure profile. The volumetric water content at saturation (hsb) for the bottom layer soil is 0.4, and the hst value for the upper layer ranges form 0.2 to 0.6 in Fig. 2.29. It can be seen from Fig. 2.29 that changes in hs of the upper layer soil have a marked effect on the pressure profiles. The smaller hst is, the greater the pore-water pressure and the pressure difference are.

2 Analytical Solution to 1D Coupled Infiltration and Deformation …

Fig. 2.29 Pore-water pressure profiles in two-layer soils with respect to different hst values (hsb = 0.4, ab = at = 0.01 kPa−1, ksb = kst = 3  10−6 m/s, H = 103 kPa, qA = 0, qB = 2.7  10−6 m/s)

1 t=2 hrs

0.9 0.8 0.7 Depth (m)

54

0.6 0.5 0.4

Initial condition θst = 0.2 (uncoupled) θst = 0.2 (coupled) θst = 0.4 (uncoupled) θst = 0.4 (coupled) θst = 0.6 (uncoupled) θst = 0.6 (coupled)

0.3 0.2 0.1

ksb=kst=3×10-6m/s -1

αb=αt=0.01 kPa θsb=0.4 3

Hb=Ht=10 kPa

0 -10

-8

-6

-4

-2

0

u (kPa)

2.4.2.5

Effect of qA and qB

Figures 2.30 and 2.31 depict the effect of qA and qB on the pore-pressure distributions. The parameters in Fig. 2.30 are wae = 0 kPa, H = 103 kPa, and qA = 0 m/s, 0.6  10−6 m/s, 1.2  10−6 m/s, qB = 2.7  10−6 m/s, ab = at = 0.1 kPa−1, ksb = 3  10−5 m/s, kst = 3  10−6 m/s, and h = 0.4. The parameters in Fig. 2.31 are wae = 0 kPa, H = 103 kPa, qA = 0.6  10−6 m/s, qB = 0.9  10−6 m/s, 1.8  10−6 m/s, 2.7  10−6 m/s, ab = at = 0.02 kPa−1, ksb = 3  10−6 m/s, kst = 3  10−5 m/s, and hs = 0.4. The rate of rainfall infiltration depends on the rainfall intensity, the water coefficient of permeability and the hydraulic gradient of the surface soil (Pullan 1990). Here, qA and qB are assumed to be constant. qA is the antecedent infiltration

1 t=2 hrs

0.9 0.8 0.7 Depth (m)

Fig. 2.30 Pore-water pressure profiles in two-layer soils with respect to different qA values (ab = at = 0.1 kPa−1, ksb = 3  10−5 m/s, kst = 3  10−6 m/s, H = 103 kPa, hs = 0.4, qB = 2.7  10−6 m/s)

0.6 0.5 Initial condition (qA = 0) Initial condition (qA = 0.6) Initial condition (qA = 1.2) qA =0 (uncoupled) qA = 0 (coupled) qA = 0.6 (uncoupled) qA = 0.6 (coupled) qA = 1.2 (uncoupled) qA = 1.2 (coupled)

0.4 0.3 0.2 0.1

ksb=3◊10-5 m/s kst=3◊10-6 m/s αb=0.1 kPa-1 αt=0.1 kPa-1

0 -10

-8

-6

-4 u (kPa)

-2

0

2.4 Effect of Layered Structure 1 t=3 hrs

0.9 0.8 0.7

Depth (m)

Fig. 2.31 Pore-water pressure profiles in two-layer soils with respect to different qB values (ab = at = 0.02 kPa−1, ksb = 3  10−6 m/ s, kst = 3  10−5 m/s, H = 103 kPa, hs = 0.4, qA = 0.6  10−6 m/s)

55

0.6 0.5 0.4 Initial condition qB = 0.3 (uncoupled) qB = 0.3 (coupled) qB = 0.6 (uncoupled) qB = 0.6 (coupled) qB = 0.9 (uncoupled) qB = 0.9 (coupled)

0.3 0.2 0.1

-6

ksb=3◊10 m/s kst=3◊10-5 m/s αb=0.02 kPa

-1

αt=0.02 kPa-1

0 -10

-8

-6

-4

-2

0

u (kPa)

rate that creates the initial conditions for subsequent infiltration; qB is the subsequent infiltration rate. Previous studies indicate that qA has a significant influence on the initial pore-water pressure distribution, and hence on the pore-water pressure redistribution as a result of subsequent rainfall infiltration (Chen et al. 2001). Figure 2.30 demonstrates that qA influences the initial pore-water pressure profile significantly. qA = 0 represents a zero-flux boundary at the ground surface, with the infiltration rate equal to the evaporation rate. The initial negative pore-water pressure profile for the hydrostatic condition differs significantly from the other cases with nonzero qA values. A smaller qA value means a higher initial negative pore-water pressure in the layered soil. No marked pressure differences between the uncoupling and coupling cases occur because a high a value of 0.1 kPa−1 is adopted. The larger the value of qB is, the more reduction of pore-water pressures in the two-layer soils will be caused by the subsequent rainfall infiltration as shown in Fig. 2.31. Larger qB values also cause more marked pressure differences between the uncoupled and coupled cases. It is found that an increase in initial flux qA has a much more significant effect on the pore-water pressure redistribution than the same increment in prescribed flux qB (Pullan 1990).

2.4.2.6

Effect of Air-Entry Value

Figure 2.32 shows the influence of air-entry value on the pore-water pressure profiles in two-layer soils considering the coupling effect at t = 1 h, with qB = 1.5  10−6 m/s. Other parameters are held at wae = 0 kPa, H = 103 kPa, ab = at = 0.01 kPa−1, ksb = 3  10−6 m/s, kst = 3  10−5 m/s, qA = 0 m/s, and hs = 0.4. In Fig. 2.32, three values of wae, 0 and 5 kPa, are considered so as to illustrate the effects of wae.

2 Analytical Solution to 1D Coupled Infiltration and Deformation …

Fig. 2.32 Pore-water pressure profiles in two-layer soils with respect to different wae values, t = 1 h (ab = at = 0.01 kPa−1, ksb = 3  10−6 m/ s, kst = 3  10−5 m/s, H = 103 kPa, hs = 0.4, qA = 0, qB = 1.5  10−6 m/s)

1 0.9 0.8 0.7

Depth (m)

56

0.6 0.5 0.4 Initial condition (Ψae=0 kPa)

0.3

Initial condition (Ψae=5 kPa) Ψae=0 kPa(uncoupled)

0.2

Ψae=5 kPa(uncoupled)

0.1

Ψae=0 kPa(coupled) Ψae=5 kPa(coupled)

ksb=3◊10-6 m/s kst=3◊10-5 m/s αb=0.01 kPa-1 θs=0.4

0 -15

-10

-5

0

u (kPa)

Figure 2.32 shows the pore-water pressure profiles under uncoupled and coupled conditions at t = 1 h. The initial and transient pore-water pressure profiles are greatly affected by air-entry value wae, as shown in Fig. 2.32. However, changes in wae play an insignificant role in the pressure difference between the uncoupled and coupled cases.

2.4.3

Conclusions

An analytical solution to the linearized coupled governing equation for water infiltration in two-layer soils is obtained using a Laplace transformation. The analytical solutions consider coupling of seepage and deformation, and the situations of time dependence and arbitrary continuous initial pore-water pressure profiles. The analytical solutions describe the transient pore-water pressure distributions in two-layer unsaturated soil systems during one-dimensional, vertical seepage toward a water table. With rainfall infiltration going on, the pore-water pressure difference between uncoupled and coupled cases becomes marked, and then gradually decreases. The smaller the coefficient of desaturation (a) of the two-layer soils is, the greater the pore-water pressure difference caused by the coupling effect is. Variation of H for the two-layer soils plays a marked role in the pressure difference between uncoupled and coupled cases. A smaller absolute value of H of the upper layer soil tends to cause more marked coupling effect. Saturated coefficients of permeability of the top and bottom layers also have a significant influence on the coupled seepage and pore-water pressure profiles. A great difference in the coefficients of permeability for the layered soil system could reduce the degree of coupling between infiltration and deformation. A relatively low permeability cover consisting of layered soils could minimize rainfall infiltration into the soils. Initial

2.4 Effect of Layered Structure

57

flux qA has a much more significant effect on the pore-water pressure redistribution than prescribed flux qB. The smaller hst is, the greater the pressure profile and the pressure difference between the uncoupled and coupled cases are.

Appendix   1 X sin½km ðLb þ XÞ exp  0:25 þ k2m T RAb ¼ Dm m¼1 RAt ¼

1 X m¼1

fsinðkm Lb Þ½sinðlm ðLt  X ÞÞ þ 2lm cosðlm ðLt  X ÞÞ    exp  0:25 þ k2m T  fDm ½sinðlm Lt Þ þ 2lm cosðlm Lt Þg1

ð2:43Þ

ð2:44Þ

in which k and l satisfy ½sinðkLb Þ þ 2k cosðkLt Þ½sinðlLt Þ þ 2l cosðlLt Þ   kst  1 þ 4l2 sinðkLb Þ sinðlLt Þ ¼ 0 ksb l¼


 b  1 0:5 bk2 þ 4

  1 X sinh½km ðLb þ XÞ exp  0:25 þ k2m T RBb ¼ Dm m¼1 RBt ¼

1 X m¼1

fsinhðkm Lb Þ½sinðlm ðLt  X ÞÞ þ 2lm cosðlm ðLt  X ÞÞ    exp  0:25 þ k2m T  fDm ½sinðlm Lt Þ þ 2lm cosðlm Lt Þg1

ð2:45Þ

ð2:46Þ

ð2:47Þ

ð2:48Þ

in which k and l satisfy ½sinhðkLb Þ þ 2k coshðkLt Þ½sinðlLt Þ þ 2l cosðlLt Þ   kst  1 þ 4l2 sinhðkLb Þ sinðlLt Þ ¼ 0 ksb l¼


0:5 b1  bk2 4

ð2:49Þ

ð2:50Þ

2 Analytical Solution to 1D Coupled Infiltration and Deformation …

58

  1 X sin½km ðLb þ XÞ exp  0:25 þ k2m T RCb ¼ Dm m¼1 RCt ¼

1  X

ð2:51Þ

   sinðkm Lb Þ½sinhðlm ðLt  X ÞÞ þ 2lm coshðlm ðLt  X ÞÞ exp  0:25 þ k2m T

m¼1

 fDm ½sinhðlm Lt Þ þ 2lm coshðlm Lt Þg1

ð2:52Þ in which k and l satisfy ½sinðkLb Þ þ 2k cosðkLt Þ½sinhðlLt Þ þ 2l coshðlLt Þ   kst  1  4l2 sinðkLb Þ sinhðlLt Þ ¼ 0 ksb
k¼

1  b l2  4 b

ð2:53Þ

0:5 ð2:54Þ

References Basha HA (1999) Multidimensional linearized nonsteady infiltration with prescribed boundary conditions at the soil surface. Water Resour Res 35(1):75–83 Basha HA (2000) Multidimensional linearized nonsteady infiltration toward a shallow water table. Water Resour Res 36(9):2567–2573 Basha HA (2011) Infiltration models for semi-infinite soil profile. Water Resour Res 47(8):192– 198 Broadbridge P, White I (1988) Constant rate rainfall infiltration: a versatile nonlinear model. I. Analytical solution. Water Resour Res 24(1):145–154 Chen JM, Tan Y, Chen CH (2001) Multidimensional infiltration with arbitrary surface fluxes. J Irrig Drain Eng 127(6):370–377 Chen JM, Tan YC, Chen CH (2003) Analytical solutions of one-dimensional infiltration before and after ponding. Hydrol Process 17(4):815–822 Fredlund DG, Rahardjo H (1993) Soil mechanics for unsaturated soils. Wiley, New York Gardner WR (1958) Some steady-state solutions of the unsaturated moisture flow equation with application to evaporation from a water table. Soil Sci 85(4):228–232 Hillel D (1998) Environmental soil physics. Academic, New York Hogarth WL, Parlange JY, Norbury J (1992) Addendum to “First integrals of the infiltration equation”. Soil Sci 154(5):341–343 Iverson RM (2000) Landslide triggering by rain infiltration. Water Resour Res 36(7):1897–1910 Kim JM (2000) A fully coupled finite element analysis of water-table fluctuation and land deformation in partially saturated soils due to surface loading. Int J Numer Meth Eng 49 (9):1101–1119 Lloret A, Gens A, Batlle F, Alonso EE (1987) Flow and deformation analysis of partially saturated soils. In: Proceedings 9th European conference on soil mechanics, Dublin, vol 2, pp 565–568 Ozisik M (1989) Boundary value problems of heat conduction. Dover, New York, pp 85–87

References

59

Parlange JY (1972) Theory of water movement in soils: VIII. One dimensional infiltration with constant flux at the surface. Soil Sci 114:1–4 Parlange JY, Barry DA, Parlange MB et al (1997) New approximate analytical technique to solve Richards equation for arbitrary surface boundary conditions. Water Resour Res 33:903–906 PDE Solutions Inc. (2004) FlexPDE user guide. PDE Solutions Inc., Antioch, CA Pereira JHF (1996) Numerical analysis of the mechanical behavior of collapsing earth dams during first reservoir filling. PhD thesis, University of Saskatchewan, Canada Pullan AJ (1990) The quasilinear approximation for unsaturated porous media flow. Water Resour Res 26(6):1219–1234 Raats PAC (1970) Steady infiltration from line sources and furrows. Soil Sci Soc Am J 34(5):709– 714 Richards LA (1931) Capillary conduction of liquids through porous mediums. Physics 1(5):318– 333 Smiles DE, Raats PAC (2004) Swelling and shrinking soils. In: Hillel D (ed) Encyclopedia of soils in the environment, vol 4. Elsevier, pp 115–124 Srivastava R, Yeh TCJ (1991) Analytical solutions for one dimension, transient infiltration toward the water table in homogeneous and layered soils. Water Resour Res 27(5):753–762 Tracy FT (2006) Clean two- and three-dimensional analytical solutions of Richards’ equation for testing numerical solvers. Water Resour Res 42(8):W08503 Tsai TL, Wang JK (2011) Examination of influences of rainfall patterns on shallow landslides due to dissipation of matric suction. Environ Earth Sci 63(1):65–75 Van Genuchten MT (1980) A closed form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Sci Soc Am J 44(5):892–898 Warrick AW, Islas A, Lomen DO (1991) An analytical solution to Richards’ equation for time-varying infiltration. Water Resour Res 27(5):763–766 Wu LZ, Zhang LM (2009) Analytical solution to 1D coupled water infiltration and deformation in unsaturated soils. Int J Numer Anal Met 33(6):773–790 Wu LZ, Zhang LM, Huang RQ (2012) Analytical solution to 1D coupled water infiltration and deformation in two-layer unsaturated soils. Int J Numer Anal Met 36:798–816 Wu LZ, Zhang LM, Li X (2016) One-dimensional coupled infiltration and deformation in unsaturated soils subjected to varying rainfall. Int J Geomech 16(2):06015004 Zhan TLT, Ng CWW (2004) Analytical analysis of rainfall infiltration mechanism in unsaturated soils. Int J Geomech 4(4):273–284 Zhan TLT, Jia GW, Chen YM, Fredlund DG, Li H (2013) An analytical solution for rainfall infiltration into an unsaturated infinite slope and its application to slope stability analysis. Int J Numer Anal Met 37(12):1737–1760 Zhang LM, Li X (2010) Micro-porosity structure of coarse granular soils. J Geotech Geoenviron 136(10):1425–1436 Zhang LL, Zhang LM, Tang WH (2005) Rainfall-induced slope failure considering variability of soil properties. Geotechnique 55(2):183–188

Chapter 3

Effects of Gravity and Hysteresis on 1D Unsaturated Infiltration

3.1

Introduction

Unsaturated infiltration is significant in a wide range of engineering applications including rainfall-induced landslides (Zhan et al. 2013). The spatial and temporal evolution of flow in an unsaturated medium involves a governing partial differential equation that is expressed by the Richards’ equation (1931). The equation is highly nonlinear and extremely difficult to obtain because the hydraulic conductivity and the pore-water pressure depend on the moisture content. A number of exact and approximated analytical solutions to the Richards’ equation were derived in past studies (e.g., Broadbridge and White 1988; Warrick et al. 1991; Hogarth et al. 1992; Parlange et al. 1997; Basha 1999, 2011). While numerical approaches can effectively simulate complex nonlinear infiltrations into an unsaturated porous medium (Tracy 2006), analytical solutions can verify these numerical procedures. Analytical solutions of the linearized Richards’ equation were derived as an integral (Broadbridge and White 1988; Chen et al. 2001a, b, 2003), as a Laplace transformation (Srivastava and Yeh 1991; Zhan et al. 2013), and as a Green’s function (Basha 1999). The analytical solutions are confined to a finite field, whereas the approximate solutions describe the one-dimensional (1D) infiltration into a semi-infinite medium for uniform initial conditions under either a time-dependent flux or a constant pore-water pressure boundary condition (Basha 2011). The incorporation of physically based infiltration expressions leads to a better quantification of the infiltration component in hydrologic models and a more reliable prediction of the water infiltration (Basha 2011). Approximate solutions made without incorporating the coupling between infiltration and deformation in unsaturated soils are presented in a simpler form, and adapt easily to various boundary conditions (Basha 2011). Variations of negative pore-water pressures in unsaturated soils are often a result of precipitation, irrigation, or evaporation events. Pore-water pressure changes due to rainfall infiltration and seepage will cause changes in stresses and in turn © Science Press 2020 L. Wu et al., Hydro-mechanical Analysis of Rainfall-Induced Landslides, https://doi.org/10.1007/978-981-15-0761-8_3

61

62

3 Effects of Gravity and Hysteresis on 1D Unsaturated Infiltration

deformation of a soil. Conversely, stress changes will modify the seepage process because soil hydraulic properties are influenced by the stress changes. Hence, the seepage and deformation problems are strongly linked in unsaturated soils under rainfall condition (Zhang et al. 2005). The analytical solutions in the literature often do not consider coupling effects between infiltration and deformation in unsaturated soils. However, the coupling effects of infiltration and deformation in unsaturated soils are essential in many cases, e.g., seepage in loose packing colluvium formed by landslides, seepage in municipal solid waste, and consolidation of artificial islands. A typical example is rainfall-induced landslides in expansive soils, or in poorly compacted engineering fills. However, the influence of the deformation of the soil has generally been disregarded in conventional studies of infiltration. To fill this knowledge gap, the coupling effects between deformation and seepage in unsaturated soils is presented in this book. In this chapter, we will examine the effects of gravity, hysteresis and semi-infinite regions on the unsaturated infiltration behavior.

3.2

Effect of Gravity

3.2.1

Governing Equation

3.2.1.1

1D Vertical Infiltration

The one-dimensional nonlinear differential equation describing water flow in unsaturated soils is expressed as follows:
 @ @ @h @h kðhÞ ðh þ zÞ ¼ @z @z @h @t

ð3:1Þ

where h is the pressure head; h is the volumetric water content; k is the coefficient of permeability for unsaturated soils; @h=@h is the storativity; t is the time; and z is vertical direction (Fig. 3.1a). It is assumed that hydraulic conductivity and water content are functions of the pressure head h: kðhÞ ¼ ks eah and hðhÞ ¼ hr þ ðhs  hr Þeah . Where hs and hr are moisture at saturation and residual moisture, respectively; ks is hydraulic conductivity at saturation; and a is the pore-size distribution parameter. Substituting the exponential function into Eq. (3.1), we can obtain:
  @ @h @h ks eah þ1 ¼ aðhs  hr Þeah @z @z @t

ð3:2Þ

3.2 Effect of Gravity

63

(a) Gravity in the vertical direction

(b) No gravity in the horizontal direction

Fig. 3.1 The boundary conditions of rainfall infiltration: a gravity; b no gravity

We introduce a Kirchhoff variable W ¼ eah so that Eq. (3.2) can be rearranged as ks @ 2 W @W @W ¼ ðhs  hr Þ þ ks @z @t a @z2

ð3:3Þ

Using dimensional variables, T = akst/(hs − hr), Z = az, and defining a new Z T variable, VðZ; TÞ ¼ W  e2 þ 4 . Then, Eq. (3.3) can be further rearranged as @V @ 2 V ¼ @T @Z 2

3.2.1.2

ð3:4Þ

1D Horizontal Infiltration

The one-dimensional Richards’ equation in a horizontal direction is expressed as (Fig. 3.1b)   @ @h @h @h kðhÞ ¼ @x @x @h @t

ð3:5Þ

where x is the horizontal direction. By substituting into Eq. (3.5), we can obtain:
 @ @h ah @h ks e ¼ aðhs  hr Þeah @x @x @t in which gravity is not considered.

ð3:6Þ

64

3 Effects of Gravity and Hysteresis on 1D Unsaturated Infiltration

Introducing Wðx; tÞ ¼ eah , T = akst/(hs − hr), X = ax, Eq. (3.6) is transformed into: @ 2 W @W ¼ @X 2 @T

3.2.2

ð3:7Þ

Boundary and Initial Conditions

We consider 1D vertical and horizontal infiltration through a homogeneous soil layer with thickness l. A 1D transient infiltration problem generally involves one initial and two boundary conditions.

3.2.2.1

Bottom Boundary

The left and bottom boundaries of 1D horizontal and vertical infiltration are expressed as hðx; tÞjx¼0 ¼ h0

ð3:8aÞ

hðz; tÞjz¼0 ¼ h0

ð3:8bÞ

where the pressure head is 0, i.e., h0 = 0. Through a series of transformations, Eqs. (3.8a, 3.8b) becomes:

3.2.2.2

W ðX; T ÞjX¼0 ¼ eah0

ð3:9aÞ

V ðZ; T ÞjZ¼0 ¼ eah0 eT=4

ð3:9bÞ

Prescribed Flux for Surface Condition or Right Boundary

If the infiltration rate in the horizontal and vertical directions is less than ks, the boundaries are subjected to a water flux (q), as follows:



@h k ð hÞ @x

 ¼ qðx; tÞ

ð3:10aÞ

x¼l

 @h þ kðhÞ ¼ qðz; tÞ kðhÞ @z z¼l

ð3:10bÞ

3.2 Effect of Gravity

65

By corresponding transformations, Eqs. (3.10a, 3.10b) can be rewritten as
 @W qðX; TÞ ¼ @X X¼L ks


@V V þ @Z 2

 ¼ Z¼L

qðZ; T Þ L=2 þ T=4 e ks

ð3:11aÞ ð3:11bÞ

in which L = al. 3.2.2.3

Constant Pressure Head for Surface and Right Boundaries

If q is more than ks, the expressions of the corresponding boundaries are as follows: hðx; tÞjx¼l ¼ h1

ð3:12aÞ

hðz; tÞjz¼l ¼ h1

ð3:12bÞ

where h1 is the pressure head at the right boundary in the horizontal direction or the ground surface in the vertical direction, which is zero when all ponding water is assumed to be drained immediately. In another dimensionless form, Eqs. (3.12a, 3.12b) can be transformed as

3.2.2.4

WðX; TÞjX¼L ¼ eah1

ð3:13aÞ

VðZ; TÞjZ¼L ¼ eah1 eL=2 þ T=4

ð3:13bÞ

Initial Conditions

The initial pressure head distributions for horizontal and vertical infiltration (i.e., initial condition) are expressed, respectively, as hðx; 0Þ ¼ hi ðxÞ

ð3:14aÞ

hðz; 0Þ ¼ hi ðzÞ

ð3:14bÞ

where hi is the initial pressure head at t = 0. Through similar transformations, Eqs. (3.14a, 3.14b) becomes: WðX; 0Þ ¼ eahi

ð3:15aÞ

VðZ; 0Þ ¼ eahi eZ=2

ð3:15bÞ

66

3 Effects of Gravity and Hysteresis on 1D Unsaturated Infiltration

3.2.3

The Analytical Solutions for Transient Unsaturated Infiltration

3.2.3.1

The Analytical Solution to 1D Horizontal Infiltration

These methods of the separation of variables and Laplace and Fourier transforms are commonly used to solve the partial differential equation, including heat conduction and infiltration problems (Ozisik 1989; Srivastava and Yeh 1991; Chen and Tan 2005). The Fourier integral transformation was employed to derive solutions of the linearized Richards’ equation with time-dependent surface fluxes (Chen and Tan 2005). These results can be further integrated to obtain closed-form solutions for arbitrary flux before and after ponding. Using Fourier’s integral transform (Ozisik 1989), the general solution to Eq. (3.6) considering 1D horizontal infiltration, i.e., without gravity, is given as 2 39 ( ZT 1 = X 2 2 1 hðX; TÞ ¼ ln K ðbn ; X Þ  ebn T 4Gðbn Þ þ ebn t Bðbn Þdt5 ; a n¼0

ð3:16Þ

t0 ¼0

If a pressure head at the left boundary and a water flux at the right boundary are defined, then the functions in Eq. (3.16) are as follows: rffiffiffi 2 K ðbn ; X Þ ¼ sin bn X L ZL Gðbn Þ ¼

K ðbn ; x0 Þeahi dx0

ð3:17aÞ

ð3:17bÞ

0

rffiffiffi
 2 qðX; T Þ Bðbn Þ ¼ bn eah0 þ sinðbn LÞeah0 L ks

ð3:17cÞ

in which the eigenvalue bn satisfies cos(bnL) = 0. The ponding time tp is assumed. For the water arrival time of tp, the pressure head is zero for the ground surface based on the assumption that all ponding water at the boundary is drained immediately. Therefore, the boundary condition is different before and after ponding. Whenever t > tp, the water infiltration will partially infiltrate into the soil, and the pressure head at the boundary is zero. Let t′ = t − tp based on dimensionless variable transformations T′ = akst′/(hs − hr), X = ax, and introducing Wðx; tÞ ¼ eah ; we can obtain from Eq. (3.6):

3.2 Effect of Gravity

67

@ 2 W @W ¼ @X 2 @T 0

ð3:18Þ

The initial pressure head is expressed as hðX; T 0 ÞjT 0 ¼0

2 39 ( ZTp 1 = X 2 2 1 ¼ ln K ðbn ; X Þ  ebn Tp 4Gðbn Þ þ ebn t Bðbn Þdt5 ð3:19Þ ; a n¼0 t0 ¼0

The general solution to the 1D horizontal saturated and unsaturated infiltration into the soils after ponding, with the two boundaries controlled by the pressure head, is (Ozisik 1989) ( 1 hðX; T 0 Þ ¼ ln a

1 X

2 K ð bn ; X Þ  e

b2n T 0

n¼0

ZT 0

4G0 ðbn Þ þ

e

b2n t0

t0 ¼0

39 = 0 B0 ðbn Þdt5 ;

rffiffiffi 2 K ðbn ; X Þ ¼ sin bn X L 0

ZL

G ðbn Þ ¼

ð3:20aÞ

ð3:20bÞ

0

K ðbn ; x0 Þeahi dx0

ð3:20cÞ

0

rffiffiffi  2  ah0 B ðb n Þ ¼ b e  cosðbn LÞeah1 L n 0

ð3:20dÞ

in which h0i is the initial pressure head at T′ = 0, i.e., Equation (3.19), and the eigenvalue bn satisfies sin(bnL) = 0.

3.2.3.2

The Analytical Solution to 1D Vertical Infiltration

Based on Fourier’s integral transform (Ozisik 1989), the solution to 1D vertical infiltration in unsaturated soils is 8 2 39 ZT 1 = 1 < Z=2 X ðb2m þ 0:25ÞT 4 b2m t0 0 0 hðZ; T Þ ¼ ln e K ðbm ; Z Þ  e F ðbm Þ þ e Aðbm ; t Þdt5 ; a : m¼0 t0 ¼0

ð3:21Þ

68

3 Effects of Gravity and Hysteresis on 1D Unsaturated Infiltration

If a pressure boundary at the bottom and a rainfall flux at the top are defined, then the expressions in Eq. (3.21) are as follows: pffiffiffi K ð bm ; Z Þ ¼ 2

"

b2m þ 0:25 Lðb2m þ 0:25Þ þ 0:5 ZL

F ðbm Þ ¼

0

#1=2 sin bm Z

0

K ðbm ; z0 Þeahi ez =2 dz0

ð3:22aÞ

ð3:22bÞ

0

pffiffiffi Aðbm ; t Þ ¼ 2 0

"

b2m þ 0:25 Lðb2m þ 0:25Þ þ 0:5

#1=2


0

eah0 et =4 þ

qðX; T Þ 0 cosðbm LÞeL=2 þ t =4 ks



ð3:22cÞ with the eigenvalue bm satisfying bmcot(bmL) = −0.5. Equations (3.21)–(3.22a, 3.22b, 3.22c) can be rearranged as ( " #) 1 1 X X 1 Z=2 ðb2m þ 0:25ÞT hðZ; T Þ ¼ ln e B1 sinðbm Z Þ  e þ B2 sinðbm Z Þ a m¼0 m¼0 ð3:23Þ According to the above equations, we can obtain that B1 and B2 are both similar to 1=bm . Then limm! þ 1 B1 ¼ 0, limm! þ 1 B2 ¼ 0 due to L’Hospital’s Rule. If 2 m ! 1, B1 sinðbm Z Þeðbm þ 0:25ÞT ! 0 and B2 sinðbm Z Þ! 0 for any T. Therefore, the analytical solution is convergent in the domain, 0  Z  L, 0  T. After t = tp, the infiltration intensity is more than the saturated permeability coefficient for the soils. Let t′ = t − tp when t′ = 0, i.e., t = tp, substituting t = tp into Eq. (3.14b) gives the initial condition for the solution of this time period. The initial pressure head distribution (i.e., initial condition at t′ = 0) is expressed as 0

hðX; T ÞjT 0 ¼0

8 2 39 ZTp 1 = 1 < Z=2 X ðb2m þ 0:25ÞTp 4 b2m t0 0 05 K ðb m ; Z Þ  e F ðbm Þ þ e Aðbm ; t Þdt ¼ ln e ; a : m¼0 t0 ¼0

ð3:24Þ with Tp = akstp/(hs − hr). After t = tp, the ground surface suffers from the pressure head boundary, and the bottom boundary remains unchanged. Based on the boundary and initial conditions, the solution to 1D vertical infiltration into unsaturated soils after ponding is (Ozisik 1989)

3.2 Effect of Gravity

(

69

2

1 X

2 1 0 hðZ; T 0 Þ ¼ ln eZ=2 K ðbm ; Z Þ  eðbm þ 0:25ÞT 4F 0 ðbm Þ þ a m¼0

ZT 0 t0 ¼0

39 = 2 0 0 ebm t A0 ðbm ; t0 Þdt5 ;

ð3:25Þ where rffiffiffi 2 K ðbm ; Z Þ ¼ sin bm Z L ZL

0

F ðbm Þ ¼

0

0

K ðbm ; z0 Þeahi ez =2 dz0

ð3:26aÞ

ð3:26bÞ

0

rffiffiffi h i 2 0 0 0 0 A ðbm ; t Þ ¼ bm eah0 et =4  cosðbm LÞeL=2 þ t =4 L

ð3:26cÞ

in which the eigenvalue bm satisfies sin(bmL) = 0.

3.2.4

The Analytical Solution for Steady Unsaturated Infiltration

3.2.4.1

Vertical Infiltration

When the steady state of infiltration is achieved, Eq. (3.2) reduces to:
  @ ah @h ks e þ1 ¼0 @z @z

ð3:27Þ

For steady-state infiltration in vertical directions, variables k and u in Eq. (3.27) are independent of time, hence: kðhÞ

@ ðh þ zÞ ¼ constant @z

ð3:28Þ

The solution for h is as follows: 1 h ¼ lnðc1 eaz þ c2 Þ a

ð3:29Þ

where c1 and c2 are nonzero coefficients that depend on the infiltration boundaries. With the substitution of the boundary conditions (surface flux boundary conditions

70

3 Effects of Gravity and Hysteresis on 1D Unsaturated Infiltration

and h0 = 0 at the bottom boundary) defined in Eqs. (3.8b) and (3.10b) into Eq. (3.29), we obtain: 1 h ¼ ln a


   q az q 1 e þ ks ks

ð3:30aÞ

Equations (3.30a, 3.30b) indicates that the pressure head profile at the steady state depends on q/ks and a. For a given soil, the steady-state pressure head is a unique function of q/ks. For the constant pressure head at the ground surface, by substituting the pressure boundary conditions described in Eqs. (3.8a) and (3.10a) into Eq. (3.29) and assuming h0 = 0, we obtain the solution to the steady pressure head:  ah1  1 e  1 az eL  eah1 e þ L h ¼ ln L a e 1 e 1

ð3:30bÞ

in which Eq. (3.30b) indicates that the steady-state pressure head profile under the pressure boundary condition depends on h1, L and a. The pressure head of the solution is zero due to h1 = 0.

3.2.4.2

Horizontal Infiltration

When infiltration reaches the steady state, Eq. (3.6) can be rewritten as   @ ah @h ks e ¼0 @x @x

ð3:31Þ

For steady-state infiltration in the horizontal directions, we can obtain from Eq. (3.31): kðhÞ

@h ¼ constant @x

ð3:32Þ

Substituting the boundary conditions (right boundary conditions and h0 = 0 at the left boundary) from Eq. (3.32), we obtain:   1 q h ¼ ln ax þ 1 a ks

ð3:33aÞ

 ah1  1 e 1 ax þ 1 h ¼ ln a L

ð3:33bÞ

Equation (3.33a) indicates that the steady-state pressure head profile before ponding depends on q/ks and a. Equation (3.33b) shows that the steady-state

3.2 Effect of Gravity

71

Table 3.1 The boundary conditions for rainfall infiltration analysis Series

Figure Figure Figure Figure Figure Figure Figure

The boundary conditions One boundary (left or bottom) 3.2 3.3 3.4 3.5 3.6 3.7 3.8

u0 u0 u0 u0 u0 u0 u0

at at at at at at at

the the the the the the the

bottom left bottom left bottom left or bottom left or bottom

The other boundary (right or top) t > tp t < tp Flux q at the top u0 at the top Flux q at the right u0 at the right Fixed q at the top Fixed q at the right Fixed q at the top Fixed q at the right or top Constant pressure (u0 = 0) at the right or top

pressure head profile under the pressure boundary condition depends on h1, L, and a (Table 3.1).

3.2.5

Case Study and Analysis

In the examples, the thickness of the homogeneous soil is taken to be 100 cm in the horizontal and vertical directions. The initial pressure head distribution (i.e., initial condition) is characterized by the linear form. The lower and left boundaries are assumed to be a zero pressure head, i.e., h0 = 0. The upper or right boundary is subjected to a water flux q. Through the case study, the influence of gravity on the pressure head profiles under different boundary conditions is analyzed. In Figs. 3.2 and 3.3, hs = 0.4, hr = 0.04, ks = 10−5 m/s, q ¼ q0 expðxtÞ, q0 = 0.9  10−5 m/s, x = 4  10−5 s−1, a = 0.01 cm−1, and tp = 2634.0 s. These parameters are mainly from Van Genuchten (1980) and Hillel (1998). Figures 3.2 and 3.3 represent the pressure head profiles with time under a gradually increasing infiltration rate. Figure 3.2 depicts vertical infiltration into unsaturated soils, i.e., the gravity potential is considered. In Fig. 3.2, the surface boundary is flux before tp and is zero pressure head (h0 = 0) after tp. 1D horizontal infiltration, which means no gravity potential is considered, is represented in Fig. 3.3. Before tp, the right boundary is controlled by flux, and the zero pressure head (h1 = 0) occurs at the right boundary after tp in Fig. 3.3. The bold line represents the pressure head profiles at t = tp, when the water infiltration rate is equal to ks in Figs. 3.2 and 3.3. The pressure head profiles between the initial linear line and the tp line (black bold line), which are calculated by Eqs. (3.21) and (3.25), respectively, correspond to the flux boundary at the ground surface. The pressure profiles on the right of the black bold line (tp line) are calculated according to the zero pressure head at the surface boundary and the new initial condition of the black bold line (tp line). At t = 2634.0 s, the pressure head for vertical infiltration is −46.3 cm at z = 0 m

3 Effects of Gravity and Hysteresis on 1D Unsaturated Infiltration

Fig. 3.2 Pressure head profiles before and after tp for 1D vertical infiltration (hs − hr = 0.36, ks = 10−5 m/s, q ¼ q0 expðxtÞ, q0 = 0.9  10−5 m/s, x = 4  10−5 s−1, a = 0.01 cm−1)

100 90 80 70

L (cm)

72

60 50 40 30 20 10 0 -100

Initial condition t = 300 s t = 1200 s t = tp t = 6000 s t = 24000 s t = 48000 s t = 90000 s

-80

-60

-40

-20

0

-40

-20

0

h (cm)

100 90 80 70

L (cm)

Fig. 3.3 Pressure head profiles before and after tp for 1D horizontal infiltration (hs − hr = 0.36, ks = 10−5 m/s, q ¼ q0 expðxtÞ, q0 = 0.9  10−5 m/s, x = 4  10−5 s−1, a = 0.01 cm−1)

60 50 40 30 20 10 0 -100

Initial condition t = 300 s t = 1200 s t = tp t = 6000 s t = 24000 s t = 48000 s t = 90000 s

-80

-60

h (cm)

(Fig. 3.2), and the pressure head for horizontal infiltration is −21.3 cm at x = 0 m in Fig. 3.3. The calculated results show that the pressure head profiles for horizontal infiltration move faster than those for vertical infiltration in unsaturated soils under the same initial and flux boundary conditions. The gravity potential greatly influences transient seepage in unsaturated soils when the boundary is controlled by flux. The effect of gravity has to be taken into account when solving the Richards’ equation for arbitrary surface boundary conditions (Parlange et al. 1997). Figures 3.4 and 3.5 show the pressure head profile changes with time under a constant infiltration rate. In Fig. 3.4, the water infiltration rate (q) is 0.8  10−5 m/s at the surface boundary for vertical infiltration, and the zero pressure head (h1 = 0) is at the right boundary for horizontal infiltration in Fig. 3.5. The corresponding parameters are as follows: hs = 0.46, hr = 0.03, ks = 10−5 m/s, a = 0.1 cm−1. The steady pressure head for the vertical infiltration (considering the gravity potential) is −1.1 cm at z = 0 m in Fig. 3.4, which is calculated by Eq. (3.30a). The steady-state

3.2 Effect of Gravity

73

Fig. 3.4 Pressure head profiles of 1D vertical infiltration under a fixed flux boundary (hs − hr = 0.43, ks = 10−5 m/s, q = 0.8  10−5 m/s, a = 0.1 cm−1)

100 90 80

L (cm)

70 60 50 40 30 20 10 0 -100

Initial condition t = 300 s t = 1800 s t = 3600 s t = 6000 s t = 12000 s t = 24000 s Steady state

-80

-60

-40

-20

0

20

-20

0

20

h (cm)

100 90 80 70

L (cm)

Fig. 3.5 Pressure head profiles of 1D horizontal infiltration under a fixed flux boundary (hs − hr = 0.4, ks = 10−4 m/s, h1 = 0 cm, a = 0.1 cm−1)

60 50 40 30 20 10 0 -100

Initial condition t = 300 s t = 1800 s t = 3600 s t = 6000 s t = 12000 s t = 24000 s Steady state

-80

-60

-40

h (cm)

pressure head distribution of the horizontal infiltration from Eq. (3.33a) is plotted in Fig. 3.5, and the pressure head is 23 cm at x = 0 m. Figure 3.4 indicates that the pressure head changes quickly at the early stage of water infiltration, then the variation of the pressure becomes smaller with time and becomes close to the steady state. In Fig. 3.4, the steady-state pressure head for the vertical infiltration shows negative pressures, which are close to the zero value. However, the steady-state pressure head for the horizontal infiltration is always more than zero, as shown in Fig. 3.5. Figure 3.6 shows a comparison between analytical and numerical solutions considering the gravity effect. In Fig. 3.6, hs = 0.4, hr = 0.04, ks = 10−5 m/s, q = 0.8  10−5 m/s, and a = 0.04 cm−1. The numerical solution is obtained by a MATLAB-implemented program. Figure 3.6 indicates that the analytical solution is highly precise compared with the numerical solution.

74

3 Effects of Gravity and Hysteresis on 1D Unsaturated Infiltration

Fig. 3.6 Comparison between analytical and numerical solutions (hs − hr = 0.36, ks = 10−5 m/s, q = 0.8  10−5 m/s, a = 0.04 cm−1)

100 90 80 70

L (cm)

60 50 40 30

Initial condition t = 1200 s (numerical) t = 1200 s (analytical) t = 3600 s (analytical)

20

t = 3600 s (numerical)

10

t = 7200 s (analytical)

0 -100

t = 7200 s (numerical)

-80

-60

-40

-20

0

h (cm)

Figure 3.7 shows the effect of gravity potential on the pressure head distribution with one boundary of a flux and the other of zero pressure. a = 0.01 cm−1, hs = 0.4, hr = 0.04, ks = 10−5 m/s, and q = 0.5  10−5 m/s in Fig. 3.7. In Fig. 3.7, the real line denotes the 1D vertical infiltration, i.e., considering the gravity, and the dashed line depicts the horizontal infiltration, i.e., no gravity potential. The pressure head difference induced between gravity and no gravity increases more and more with time. The gravity potential greatly slows water infiltration into unsaturated soils when the flux boundary at the ground surface is applied, as shown in Fig. 3.7. The pressure head difference between considering gravity and no gravity is 9.8 cm at t = 300 s of infiltration, 21.4 cm at t = 1500 s, and 49.1 cm at t = 9000 s. The analysis demonstrates that the pressure head profiles considering gravity move slower than those considering no gravity if the surface boundary is controlled by a flux (less that ks).

Fig. 3.7 The effect of gravity on the pressure head profiles under constant flux (hs − hr = 0.36, ks = 10−5 m/s, q = 0.5  10−5 m/s, a = 0.01 cm−1)

100 90 80

L (cm)

70 60 50 40 30 20 10 0 -100

Initial condition t = 300 s (vertical) t = 300 s (horizontal) t = 1500 s (vertical) t = 1500 s (horizontal) t = 9000 s (vertical) t = 9000 s (horizontal)

-80

-60

-40

h (cm)

-20

0

3.2 Effect of Gravity

75

Fig. 3.8 The effect of gravity on the pressure head profiles under constant pressure head (hs − hr = 0.36, ks = 10−5 m/s, h1 = 0 cm, a = 0.01 cm−1)

100 90 80

L (cm)

70 60 50 40 30

Initial condition t = 300 s (vertical) t = 300 s (horizontal) t = 1500 s (vertical)

20

t = 1500 s (horizontal)

10

t = 9000 s (vertical)

0 -100

t = 9000 s (horizontal)

-80

-60

-40

-20

0

h (cm)

Figure 3.8 shows that the pressure head changes with time under the condition of constant pressure head (h1) at one boundary and with zero pressure head (h0 = 0) at the other boundary. The other parameters are kept the same as in Fig. 3.7. For a given time, the pressure profiles for the vertical infiltration get close to those for the horizontal infiltration, as depicted in Fig. 3.8. The calculated results indicate that gravity seems to play no role in the pressure head profiles induced by transient infiltration into unsaturated soils if the two boundaries are determined by the pressure head. Therefore, the pressure head difference between the vertical infiltration and the horizontal infiltration in unsaturated soils mainly depends on the boundary conditions.

3.2.6

Summary

The analytical solutions to the one-dimensional, horizontal and vertical transient infiltration in unsaturated soils using Fourier integral transformation can consider varying flux and pressure head boundaries. The analytical solution is based on the assumption of constant diffusivity. The steady-state solutions for horizontal and vertical infiltration into unsaturated soils are derived. The solutions should provide a useful means to validate various numerical solutions. The calculated results in a case study indicate that the effect of gravity potential on the pressure head profiles induced by water infiltration into unsaturated soils depends on the boundary conditions. The analysis shows that the pressure head profiles in the horizontal directions move faster than those in the vertical directions if one flux boundary and the other pressure head boundary are determined. The results demonstrate that the gravity potential greatly influences the pressure head distribution caused by unsaturated seepage if a water flux at the top boundary and a water table at the lower boundary are assumed. However, gravity plays no marked role in the pressure

76

3 Effects of Gravity and Hysteresis on 1D Unsaturated Infiltration

head profiles induced by transient infiltration if the two boundaries are controlled by the pressure head. The infiltration-induced pressure head difference between considering gravity and no gravity can be estimated by the steady-state solutions to water infiltration into unsaturated soils in the horizontal and vertical directions.

3.3

Effect of Hysteresis

An exponential model is used to describe the hysteresis in SWCC and permeability functions and to derive a 1D analytical solution to the governing equation for unsaturated porous media considering hysteresis. Parametric studies are also carried out to evaluate the influences of hysteresis on pore-water pressure responses in the soil. Considerable effort has been made into the analysis and description of hysteretic soil hydraulic properties. This has led to numerous models for describing hysteresis in h (Gillham et al. 1976; Mualem 1974, 1984; Kool and Parker 1987; Hogarth et al. 1988; Li 2005; etc.). Most models presented in the literature require, directly or indirectly, at least two boundary hysteresis curves to predict the scanning curves (Hogarth et al. 1988; Philip 1964; Hank et al. 1969; Dane and Wierenga 1975; Jaynes 1985; Nimmo 1992; Kawai et al. 2000). The significance of hysteresis has also been demonstrated in several numerical studies (Russo et al. 1989; Gillham et al. 1976). Although recognized as being important, hysteresis is not usually considered in seepage studies because of its complexity. Therefore, a simple model for including hysteresis in seepage analysis is preferred (Pham et al. 2003).

3.3.1

Brief Review of Hysteretic Soil–Water Characteristic Models

A SWCC is hysteretic, i.e., the water content at a given suction along a wetting path is less than that along a drying path. There are an infinite number of scanning curves inside the hysteresis loop (Fig. 3.9). The difference in water content between the wetting and drying processes is believed to be caused by (Klausner 1991): (i) Irregularities in the cross sections of void passages or the “ink-bottle” effect (Haines 1930); (ii) The contact angle being greater in an advancing meniscus than in a receding meniscus; (iii) Different volumes of the entrapped air when the soil suction is increasing or decreasing. The hysteretic nature of the SWCC has been known for a long time. In many routine engineering and agriculture applications, however, the SWCCs are often assumed to be nonhysteretic since the measurement of a complete set of hysteretic SWCCs is time consuming and costly, and it is difficult to represent these curves in

3.3 Effect of Hysteresis

77

Fig. 3.9 Hysteresis in soil– water characteristic curves

0.4

Volumetric Water Content

0.35 0.3 0.25 0.2 0.15 0.1 0.05

Primary Curve Wetting-drying Curve Drying-wetting Curve

0 0.01

0.1

1

10

100

1000

Suction (kPa)

a simple mathematical form for use in numeric analysis. Several models of hysteretic SWCCs have been developed. Four of these are briefly reviewed and compared in terms of their suitability for use in analytical solutions. (1) Hogarth et al. hysteretic model (1988) Most existing hysteretic models predict scanning SWCCs when the wetting and drying boundaries are known. Hogarth et al. (1988) reformulated a simple hysteresis model on the basis of the Brooks and Corey SWCC equation (Brooks and Corey 1964). In principle, the method requires the knowledge of a drying curve (boundary or primary) to predict the wetting boundary and all scanning curves. (2) Kacimov and Yakimov hysteretic model (1998) Kacimov and Yakimov (1998) conducted a qualitative analysis of similar moisture profiles advancing in dry soils during a constant rate infiltration. The Richards’ partial differential equation is reduced to an ordinary one that can be integrated. Using simple kinematic relations, nonmonotonic saturation-depth profiles traveling both in 1D case and in 2D finger-type flows are interpreted as a consequence of hysteresis of conductivity as a function of water content. The estimations obtained allow the description of fingering as a result of hysteresis. The mean velocity of hysteretic finger-type flow can be higher than the velocity of 1D hysteretic flow, whereas the latter advances faster than the usual monotonic front without hysteresis. (3) Li hysteretic model (2005) Li presented a phenomenological approach for modeling SWCC, which includes smooth hysteretic responses to arbitrary wetting/drying paths (Li 2005). To describe the hysteretic behavior of SWCC, Li (2005) introduces an image suction, s of the matric suction, s, and a projection center, a. It can be seen that a wetting/drying state is uniquely associated with a set of degree of saturation, a, s, and s. This conceptual model was implemented in FORTRAN and used to calculate the response of the drying–wetting scanning curves (Li 2005).

78

3 Effects of Gravity and Hysteresis on 1D Unsaturated Infiltration

(4) Pedroso and Williams model (2010) A novel approach for modeling the nonlinear saturation-suction response with hysteresis is presented by Pedroso and Williams (2010), where a simple differential equation is introduced to describe the shapes of the curves. The great advantage of this new technique is easy to determine the parameters. The implementation of the resulting equations into fully hydromechanical models for numerical analyses is straightforward. The technique is simple, yet versatile due to the rational basis used in the deduction of the equations, which allows for future extensions to soils displaying more complex unsaturated behavior (Pedroso and Williams 2010). Experimental observations showed that the Hogarth et al. hysteretic model (1988) is simple, accurate and general. However, many parameters are needed and these parameters may hinder the development of analytical solutions considering hysteresis. Although Kacimov and Yakimov (1998) put forward a hysteretic loop, the model does not involve hysteretic relationship between water content and pore-water pressure. The smooth hysteretic SWCCs of arbitrary wetting/drying paths are presented by Li (2005). However, a computer program is needed to implement the model. It is difficult to obtain an analytical solution with this model.

3.3.2

Mathematical Formulations for Hysteresis

In 1D analytical solution of unsaturated soils seepage, exponential functions were usually used. Srivastava and Yeh (1991) assumed that the relationships between water coefficient of permeability (K) and pore-water pressure (uw) and between volumetric water content (h) and uw are both exponential functions, k ¼ ks eauw and h ¼ hr þ ðhs  hr Þeauw . In which ks is the saturated coefficient of permeability; hr is the residual water content; hs is the saturated water content; a is the desaturation coefficient; u is the pore-water pressure; K is the coefficient of permeability at pore-water pressure uw. The hysteresis region (the so-called loop) is defined by the boundary, or the main drying and wetting curves. Hysteresis of SWCC can also be expressed using the exponential functions, as shown in Fig. 3.9. The main wetting and drying curves depend on the desaturation coefficient, the residual water content, and saturation water content. The scanning curves that begin at a main curve can be written as (Fig. 3.9)

hw ¼ hr þ ðhs  hr Þ eaw ðu þ mw Þ þ nw

ð3:34aÞ



hd ¼ hr þ ðhs  hr Þ ead ðu þ md Þ þ nd

ð3:34bÞ

where subscript d denotes a drying process and w denotes a wetting process; aw and ad are factors controlling the gradients of the scanning curves; m and n are separate

3.3 Effect of Hysteresis 1.1 1

Coefficient of Permeability(10-6 m/s)

Fig. 3.10 Relationship between permeability coefficient and matric suction

79

ks

0.9

αd

0.8 0.7 0.6

αw

0.5 0.4 0.3 0.2 0.1 0 0.001

Main drying curve Main wetting curve 0.01

0.1

1

10

100

1000

10000

Suction (kPa)

parameters for adjusting the slope and intercept of the curve. These curves start from initial water content along different depths. Equations (3.34a, 3.34b) can be rearranged as h ¼ h0r þ Dheauw

ð3:35Þ

where the desaturation coefficient (a) is determined by the curves, as shown in Fig. 3.9; Dh = (hs − hr) eaw mw for wetting and Dh = (hs − hr)ead md for drying; hr′ = hr +(hs − hr)n. Hysteresis has been found in both h(u) and K (Kool and Parker 1987; Haines 1930; Staple 1969). Studies suggest, however, that there be little hysteresis in K or it is so slight as to be masked by the error of measurements and can be ignored (Likos and Lu 2004; Topp 1971). According to exponential functions of volumetric water content and pore-water pressure, k ¼ ks eauw and h ¼ hr þ ðhs  hr Þeauw , the corresponding hydraulic conductivity considering hysteresis is given as k ¼ ks eauw . where a is described as shown in Fig. 3.10. The subscript w denotes a wetting process, and the subscript d denotes a drying process in Fig. 3.10. For a wetting–drying or drying–wetting loop, values of a should follow aw \a0w \0d \ad .where ad and aw are the parameters for drying and wetting boundaries, respectively, ad′ and aw′ are those for drying and wetting scanning curves, respectively.

3.3.3

Analytical Solutions Considering Hysteresis

The hysteresis in the soil–water characteristics and hydraulic conductivity is considered. Other assumptions are mentioned above. The coefficient of permeability at full saturation remains constant regardless of hysteresis. With no water infiltration for the upper boundary, the gravity potential makes the soil–water content pass downward so that the upper part of the soil profile becomes dry and the lower one

80

3 Effects of Gravity and Hysteresis on 1D Unsaturated Infiltration

Fig. 3.11 The flowchart of the critical depth calculation

The parameter input for hysteresis model Start from t0 (t0 < t < t1) Using Eq. (22) with wetting and drying scanning curves respectively

Hysteresis model

Solve uw and ud at t

No

|uw–ud| < e Yes

The intersection point at t Calculating the critical depth (l0)

of the soil profile becomes wetter. The drying region in the upper part becomes scale with time. Then at a given time there is a critical point between the wetting region and the drying one in the soil profile. The critical depth is not constant, and moves downward with simulation time (Tan et al. 2009). The critical point for wetting and drying processes at time t is determined by the intersection point between the two analytical solutions using ad′ and aw′, which represent monotonic wetting and drying scanning curves, respectively. The flowchart of the critical point calculation combined with a hysteresis model is shown in Fig. 3.11. Indeed, the main drying or wetting curves are employed to account for hysteresis. However, the scanning curves truly reflect hysteresis in SWCC. Based on drying and wetting scanning curves the effect of hysteresis on 1D seepage in unsaturated soils is concerned.

3.3.3.1

1D Analytical Solution

According to Eq. (3.35) and k ¼ ks eauw , Richards’ equation can be rewritten into:
  @ @ uw @uw ks eauw þx ¼ aDheauw @x @x cw @t

ð3:36Þ

We consider 1D vertical infiltration through a homogeneous soil. The thickness of the unsaturated soil layer, l, is defined in Fig. 3.12. A 1D transient infiltration

3.3 Effect of Hysteresis

81

Fig. 3.12 General initial and boundary conditions: a initial condition; b boundary condition

problem generally involves one initial and two boundary conditions. The initial pore-water pressure distribution (i.e., initial condition) is conventionally expressed as an exponential function: uw ðx; 0Þ ¼

ln½wi  ðwi  eau0 Þeacw x  a

ð3:37aÞ

 þ where wi ¼ eL au1  eau0 =ðeL  1Þ; L = acwl; u0 is the pore-water pressure at the bottom boundary; u1 is the pore-water pressure at the ground surface in Fig. 3.12a. In the study, the boundary conditions are composed of the lower and upper boundaries in Fig. 3.12. The lower boundary is given as uw ð0; tÞ ¼ u0

ð3:37bÞ

The lower boundary is usually set at the initial groundwater table, where the pore-water pressure is u0 = 0. In general, the upper boundary is subjected to a varying rainfall flux (q) (Fig. 3.12) as follows:
 @uw k ð uw Þ þ k ð uw Þ ¼ qðx; tÞ @z x¼l

ð3:38Þ

We introduce a Kirchhoff variable K ¼ eauw , so that Eq. (3.36) can be rearranged as

82

3 Effects of Gravity and Hysteresis on 1D Unsaturated Infiltration

ks @ 2 K @K @K ¼ Dh þ ks 2 @x @t acw @x

ð3:39Þ

Using dimensionless variables, T = acwkst/(Dh), X = acwx, and at the same time, X T defining a new variable, V(X, T), which is given as V ¼ K  e 2 þ 4 , Eq. (3.35) can then be further rearranged as @V @ 2 V ¼ @t @X 2

ð3:40Þ

The initial and boundary conditions are expressed in terms of V and X:   VðX; 0Þ ¼ w1  ðwi  eau0 ÞeX eX=2 Vð0; TÞ ¼ eau0 eT=4
 @V V qðX; TÞ L=2 þ T=4 þ ¼ e @X 2 X¼L ks

ð3:41aÞ ð3:41bÞ ð3:41cÞ

where wi ¼ ðeL þ au1  1Þ=ðeL  eau0 Þ. Using Fourier’s integral transform (Ozisik 1989), the general solution to Eqs. (3.40)–(3.41a, 3.41b, 3.41c) (t < t0) is 8 2 39 ZT 1 = 2 2 0 1 < X=2 X uðX; TÞ ¼ ln e Kðbm ; XÞ  eðbm þ 0:25ÞT 4Fðbm Þ þ ebm t Aðbm ; t0 Þdt05 ; a : m¼0 t0 ¼0

ð3:42Þ where pffiffiffi Kðbm ; XÞ ¼ 2

"

b2 þ 0:25  2 m L bm þ 0:25 þ 0:5 ZL

Fðbm Þ ¼

#1=2

Kðbm ; x0 ÞVðx0 ; 0Þdx0

sin bm X

ð3:43aÞ

ð3:43bÞ

0

pffiffiffi Aðbm ; t Þ ¼ 2 0

"

b2 þ 0:25  2 m L bm þ 0:25 þ 0:5

#1=2


0

eau0 et =4 þ

q1 ðX; TÞ 0 cosðbm LÞeL=2 þ t =4 ks



ð3:43cÞ

3.3 Effect of Hysteresis

83

Fig. 3.13 Rainfall infiltration history (q1 > 0, q2 = 0)

q (m/h)

q1

q2

0 t0

0

t1

t (hrs)

Fig. 3.14 Wetting–drying scanning curves in the soil profile

0.4

Volumetric Water Content

0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0.01

Primary curve Wetting scanning curve Drying scanning curve 0.1

1

10

100

1000

Suction (kPa)

in which the eigenvalues, bm, satisfy bmcot(bmL) = −0.5. The upper boundary suffers from rainfall intensity jump. q1 is the rainfall intensity before t0, q2 is the rainfall intensity after t0. For an upper boundary with a rainfall intensity jump in Fig. 3.13, the desaturation coefficient should vary with depth after t > t0, which corresponds to different scanning curves in Fig. 3.14. Due to hysteresis effect, the parameters of SWCCs in Fig. 3.6 are slightly modified according to the silt loam by Van Genuchten (1980). In Fig. 3.14, the parameters for a single wetting scanning curve are aw′ = 0.08 kPa−1, hr′ = 0.24, and Dh = 0.18. The parameters for three drying scanning curves are ad′ = 0.02 kPa−1, hr′ = 0.23, Dh = 0.16, 0.15, and 0.13 from bottom to top, respectively. If the initial conditions are defined as a pressure profile at t = t0, a general solution to Eq. (3.40) can be obtained using Fourier integral transformation (Ozisik 1989):

84

3 Effects of Gravity and Hysteresis on 1D Unsaturated Infiltration

( " 1 X 2 1 X=2 uðX; TÞ ¼ ln e A sinðbm XÞeðbm þ 0:25ÞT a m¼0 þ

1 X



ðb2m þ 0:25ÞT

B sinðbm XÞ 1  e



m¼0

þ

1 X



ðb2m þ 0:25ÞT

C sinðbm XÞ 1  e



#)

m¼0

ð3:44Þ where a depends on drying and wetting paths.  ZL 2 b2m þ 0:25 A¼  2 sinðbm XÞVðX; 0Þdx L bm þ 0:25 þ 0:5

ð3:45Þ

0

in which VðX; 0Þ ¼ eaui þ X=2 . ui(x) can be the pore-water pressure at t = t0, which is considered as the initial condition. If the flux (q2) at the upper boundary is defined after t > t0, coefficients B and C are given by 2bm eau0 B¼  2 L bm þ 0:25 þ 0:5 C¼

2eL=2 sinðbm LÞ   2  q2 ks L bm þ 0:25 þ 0:5

ð3:46aÞ

ð3:46bÞ

in which the eigenvalues, bm, satisfy bmcot(bmL) = −0.5; q2 is the rainfall intensity after t0.

3.3.3.2

Hysteresis Analysis

Figure 3.13 describes a rainfall intensity jump. The rainfall time domain can be divided into two stages as follows: (a) First stage (0 < t < t0): The ground surface is subjected to rainfall infiltration during the period. The lower boundary condition is a constant pore-water pressure condition. The soil follows the wetting curve described in Fig. 3.14. The whole soil profile is being wetted before t = t0, and hence the wetting scanning curve is unique. By substituting parameters Dhð¼ ðhs  hr Þeaw mw Þ and aw′ into Eqs. (3.42)–(3.43a, 3.43b, 3.43c), we can obtain the analytical solution. If no rainfall infiltration at the surface before t = t0 occurs, we derive the analytical solution with Dhð¼ ðhs  hr Þead md Þ and ad′. (b) Second stage (t0 < t < t1): The surface boundary condition is changed into q = 0 during the period. The upper soils are being dried, and the soil–water moves downward during the period due to gravity potential. However, the lower soils

3.3 Effect of Hysteresis

85

are being wetted from water infiltration of former q1 because the soil is always unsaturated under both transient and steady seepage conditions. Drying and wetting processes occur simultaneously in the soil profile. Therefore, there is a critical point, i.e., a wetting–drying interface. The soil below the critical point is under wetting, whereas a monotonic drying process occurs above the critical point. The critical depth is not constant, but moves downward with time (Tan et al. 2009). The lower soil is still depicted using a single wetting scanning curve, but different drying scanning curves (dashed lines) along depths denoting the upper soil drying are shown in Fig. 3.14. The value of a for a wetting scanning curve is usually more than that for a drying scanning curve. The determination of the critical depth, i.e., the distance from the ground surface to the critical point, is a key to the analytical solution considering hysteresis. A reversal criterion for drying and wetting is met when the pore-water pressure increment changes its sign at any node and time step, which is easy to identify in a numerical program (Tan et al. 2009). At t′ (t0 < t′ < t1) the solution using monotonic drying curves is plotted in Fig. 3.15. Different drying scanning curves described at different depths above the critical point can only obtain an analytical solution because the expressions of different drying scanning curves have the same parameter, ad′. The solution for pore-water pressure profiles at t΄ is given in Fig. 3.15. In Fig. 3.15, the blue line depicts the pore-water pressure distribution at t′ (t0 < t′ < t1) calculated using the parameter, aw′, and the red line represents the pressure profile of the analytical solution with the parameter, ad′. The blue and red lines describe the monotonic wetting and drying scanning curves shown in Fig. 3.14, respectively. The intersection of the blue and red lines in Fig. 3.15 determines the critical point between upper drying part and lower wetting part at t = t′. The soil profile below the critical point completely follows a monotonic wetting process, and its scanning curve is unique. At t = t′ the pore-water pressure profiles above the critical point lies between the blue and red lines because of the gradual increase of the critical depth with time. During the second stage

Fig. 3.15 The pore-water pressure profile fitted according to linear relation, at t′ (t0 < t′ < t1)

10 9 8

Depth (m)

7 6 5 4 3 2 1 0 -80

Monotonic wetting curve Monotonic drying curve Hysteretic curve -70

-60

-50

-40

u (kPa)

-30

-20

-10

0

86

3 Effects of Gravity and Hysteresis on 1D Unsaturated Infiltration

(q = 0), the ground surface keeps drying all the time, and hence, the pore-water pressure at the ground should be kept the same as that of red line at the surface if hysteresis is considered. During the second stage, the pore-water pressure considering hysteresis at the critical point is the intersection point of the blue and red lines. Below the critical point, the pore-water pressure of the hysteretic soils at t = t′ (during the second stage) is described by the blue line in Fig. 3.15, which comes from the analytical solution with aw′ for a complete wetting scanning curve. In order to calculate the pore-water pressure considering the hysteresis, a linear relationship is assumed uh ¼

x  l0 lx ud þ uw l  l0 l  l0

ðl0 \ x \ lÞ

ð3:47Þ

where x is the distance from the water table. uh is the pore-water pressure considering hysteresis; ud is the pore-water pressure, that is calculated using completely different drying scanning expressions in Fig. 3.14; uw is the pore-water pressure corresponding to the result with the unique wetting scanning curve; l0 is the distance from the bottom groundwater table to the critical point. Figure 3.15 shows that the hysteretic profile above the critical point lies between the two boundary curves by assuming a linear relation. The profile considering hysteresis below the critical point is obtained by the solution using the monotone wetting scanning curve. The linear relationship between pore pressure and depth is assumed because the nonlinear relationship between pore pressure and depth may lead to deviation outside the domain defined by the two blue and red lines from the analytical solutions in terms of monotonic drying or wetting scanning curves.

3.3.4

Analysis Examples

The thickness of the homogeneous soil is taken to be l = 10 m in the examples. The initial pore-water pressure distribution (i.e., initial condition) is conveniently characterized by an exponential form (Fig. 3.12a). The lower boundary is located at the stationary groundwater table, where the pore-water pressure is u0 = 0 as shown in Fig. 3.12. The upper boundary is subjected to a step-wise rainfall infiltration process (q1 > 0 at t < t0; q2 = 0 at t > t0 in Fig. 3.13). The parameters for several soil types are listed in Table 3.2. The parameters of the hysteresis model in Table 3.3, which are slightly modified according to the silt loam (Van Genuchten 1980) due to hysteresis, are used to study the influence of the SWCC hysteresis on the pore-water pressure profiles. Figure 3.16 represents pore-water pressure distribution of 1D soil profile suffering from unsaturated infiltration jump (q = 2.7  10−6 m/s for t < 50 h; q = 0 for t > 50 h). In Fig. 3.16, the parameters are ks = 3.0  10−6 m/s, hs = 0.4, and hr = 0.2. q/ks is 0.9 before t0 = 50 h, and q/ks is zero after t0. The desaturation coefficient ad′ = 0.02 kPa−1 for the drying scanning curve, and aw′ = 0.08 kPa−1 for the

3.3 Effect of Hysteresis

87

Table 3.2 Coefficient of permeability at full saturation (ks), and desaturation coefficient (a) for several types of soil Soil type

ks (m/s)

a (kPa−1)

References

Clay Silt

10−10–10−8 10−8–10−6

0.016 for Beit Netofa clay 0.043 for a silt loam

Sand Gravel

10−5–10−3 10−2–10−1

0.106 for a fine sand 0.081 for Hpgiene sandstone

Van Genuchten Van Genuchten (1989) Van Genuchten Van Genuchten

(1980), Hillel (1998) (1980), Hills et al. (1980), Hillel (1998) (1980), Hillel (1998)

Table 3.3 Parameters for 1D solution of unsaturated soil seepage aw (kPa−1)

ad (kPa−1)

hs

hr

Dh((hs − hr)eam)

hr′(hr + (hs − hr)n)

ks (10−6 m/s)

0. 1

0.01

0.4

0.2

0.13–0.2

0.205–0.25

3.0

Fig. 3.16 Pore-water pressure profiles during wetting–drying process considering hysteresis

10 9 8

Depth (m)

7 6 5 4 3 2 1 0 -100

Initial condition t=20 h(hysteresis) t=50 h(hysteresis) t=60 h(hysteresis) t=80 h(hysteresis) t=100 h(hysteresis) -80

-60

-40

-20

0

u (kPa)

wetting scanning curve. The pore-water pressure at the bottom boundary is zero. There is a monotonic wetting process in the whole soil profile in 50 h. Drying begins to occur from the top at t = 50 h. After t = 50 h, drying and wetting for the whole soil profile occur at the same time. At t = 60 h (in 10 h), the critical point between wetting and drying processes moves downwards by about 3 m. However, the point moves downwards about 4.5 m deep from the ground surface at t = 80 h (in 30 h), and moves downwards about 5.5 m far from the ground surface at t = 100 h (in 50 h). The dry front or the critical point moves downwards more slowly with time.

88

3 Effects of Gravity and Hysteresis on 1D Unsaturated Infiltration

Fig. 3.17 Comparison between hysteresis and no hysteresis conditions (a = 0.1 kPa−1 for nonhysteresis; hs − hr = 0.2; m = 0)

10 9 8 7

Depth (m)

6 5 4 3 2 1 0 -100

Initial condition t=20 h(hysteresis) t=20 h(nonhysteresis) t=50 h(hysteresis) t=50 h(nonhysteresis) t=80 h(hysteresis) t=80 h(nonhysteresis) t=100 h(hysteresis) t=100 h(nonhysteresis)

-80

-60

-40

-20

0

u (kPa)

Figure 3.16 also shows the effect of hysteresis on the pore-water pressure distributions during rainfall infiltration jump (q > 0 for t < t1; q = 0 for t > t1). The pressures under the flux boundary of q > 0 change faster with time than those under the condition of q = 0. According to the hysteresis model, higher desaturation coefficient means wetting process, and lower desaturation coefficient shows drying conditions. The greater the value of a, the faster the pressure profile moves. The pressure profile with a drying scanning curve changes more slowly than that with a wetting curve if hysteresis is considered. It is noted that during rainfall infiltration process the pore-water pressure response are affected by the hysteresis of SWCC in Fig. 3.16. The pressure profiles considering hysteresis are different from those using the main drying or wetting curve. Figure 3.17 shows the large differences in the pressure profiles when hysteresis is and isn’t considered. The infiltration rate (q) is 2.7  10−6 m/s before t < 50 h, q is zero after t > 50 h. In Fig. 3.17, the parameters are hs = 0.4, and hr = 0.2, others are kept as Fig. 3.16. A wetting boundary curve (aw = 0.1 kPa−1) is used to represent the nonhysteresis condition, and the hysteresis curves are shown in Fig. 3.14. At a particular time t > t0, there is a critical intersection point between the upper drying region and lower wetting one. The critical point moves downwards with time. The pore-water pressure distributions for wetting front curves considering hysteresis apparently differ from nonhysteretic flow and the variation increases with time. Figure 3.18 also indicates large differences between hysterestic and nonhysterestic flows. A drying boundary curve with ad = 0.01 kPa−1 is used, and other parameters are kept invariable in Fig. 3.18. In the early completely wetting stage, there is some difference between the two conditions. However, the difference becomes more marked when the upper boundary condition is changed, i.e., q = 0. Figures 3.16, 3.17 and 3.18 show the great influence of the desaturation coefficient (a). Figure 3.19 shows the influence of parameter m on the pressure profiles,

3.3 Effect of Hysteresis

89

Fig. 3.18 Comparison between hysteresis and no hysteresis conditions (a = 0.01 kPa−1 for nonhysteresis; hs − hr = 0.2; m = 0)

10 9 8 7

Depth (m)

6 5 4 3 2 1 0 -100

Initial condition t=20 h(hysteresis) t=20 h(nonhysteresis) t=50 h(hysteresis) t=50 h(nonhysteresis) t=80 h(hysteresis) t=80 h(nonhysteresis) t=100 h(hysteresis) t=100 h(nonhysteresis) -80

-60

-40

-20

0

-40

-20

0

u (kPa)

Fig. 3.19 The effect of m on pore-water pressure distribution (a = 0.02 kPa−1; hs − hr = 0.2)

10 9 8

Depth (m)

7 6 5 4 3 2 1 0 -100

Initial condition t=5 h(m=0) t=5 h(m=-1) t=5 h(m=-2) t=20 h(m=0) t=20 h(m=-1) t=20 h(m=-2) t=50 h(m=0) t=50 h(m=-1) t=50 h(m=-2)

-80

-60

u (kPa)

other parameters are q = 2.7  10−6 m/s, ks = 3.0  10−6 m/s, hs = 0.4, and hr = 0.2. The difference of the pressure profiles between hysteresis and nonhysteresis becomes slightly marked with time, and is minor and negligible. Parameter n is removed deriving the derivative operations. Therefore, n does not play a role in the pressure profiles. From Figs. 3.17, 3.18 and 3.19, the effect of desaturation coefficient (a) on the pore pressure profile is found to be the most significant.

90

3.3.5

3 Effects of Gravity and Hysteresis on 1D Unsaturated Infiltration

Conclusions

A new hysteresis model is presented using exponential SWCC and permeability function to describe permeability function and a series of hysteresis scanning curves. The parameter desaturation coefficient (a), Dh, and hr′ in the hysteresis model are determined from the main drying and wetting curves. The hysteresis model is effective in developing the analytical solution of unsaturated seepage considering hysteresis. An analytical solution to the governing equation considering hysteresis characteristics is derived using the Fourier integral transform. A constant or a step-wise rainfall infiltration process can be also considered. In 1D soil profile, nonmonotonic wetting or drying process can be calculated using Eqs. (3.44)–(3.47). The calculated results demonstrate that the critical point between wetting and drying processes moves downwards slowly with time, and that there is a great difference between the hysteresis and nonhysteresis conditions. The influence of hysteresis should be considered in analyzing infiltration and evaporation processes in unsaturated soils. The influence of the hysteresis model parameters, n, m, and a, are also analyzed. The desaturation coefficient, a, is the most important factor influencing pore pressure distributions. n plays no role in the pressure profile. The influence of m on pore pressure profiles may be negligible.

3.4 3.4.1

Effect of Semi-infinite Region Governing Equation

In this book, the governing equations for coupled seepage and deformation in deformable two-layer unsaturated soils are derived on the basis of conservation of fluid mass, Darcy’ law and Fredlund’s constitutive model of unsaturated soils (Fredlund and Rahardjo 1993). Instead of directly solving the complex nonlinear coupled formulations for two-layer unsaturated soils, these formulations are first linearized and transformed into an uncoupled form. This uncoupled form of governing equations can then be solved using Laplace transformation developed by Srivastava and Yeh (1991) for single or layered soil systems. Parametric studies are carried out to investigate the influence of several key model parameters on the transient pore-water pressure responses during water infiltration considering the coupling effects.

3.4 Effect of Semi-infinite Region

3.4.1.1

91

Coupled Governing Equations

Based on Darcy’s Law under isothermal conditions and conservation of 1D fluid mass, the 1D governing equation for coupled seepage and deformation of unsaturated soils in a semi-infinite domain can be derived from Eq. (3.20a) (Kim 2000; Wu and Zhang 2009), where pw is the pore-water pressure; k is the coefficient of permeability of the pore-water; cw is the unit weight of water; z is the depth (Fig. 3.1); t is the infiltration time; h is the volumetric water content; Sr is the degree of saturation; qw is the water density; n is porosity; ev is the volumetric strain; ac = 1 − K/Ks is the hydromechanical coupling coefficient (0  ac  1), K is the bulk modulus of the solid skeleton, Ks is the bulk modulus of the soil solid (Kim 2000); and bw is the fluid compressibility. 3.4.1.2

Analytical Solutions to 1D Semi-infinite Coupled Transient Infiltration

In many analytical solutions, the relatively simple exponential hydraulic conductivity function of Gardner (1958) is employed (Zhu and Mohanty 2002). Zhan et al. (2013) used an exponential function to effectively predict the permeability coefficient and the soil–water characteristics curve with high accuracy. In the suction range relevant to the authors’ problem, the volumetric water content and soil–water characteristic curve can be sufficiently accurate. The exponential functions, which represent the water content, are used to linearize the equation governing the coupled infiltration. Both the coefficient of permeability (Raats 1970) and water content change exponentially with pore-water pressure. This formulation provides an efficient way of deriving the analytical solution. The hydraulic conductivity of unsaturated soil and the relationship between the volumetric water content and soil suction may be expressed as kðuw Þ ¼ ks eauw and hðuw Þ ¼ hs eauw , where ks is the coefficient of permeability at full saturation and a is the desaturation coefficient. If the coupled condition is ignored, Eq. (2.18) can be transformed into:
  @ uw @uw auw @ ks e ; z ¼ ahs  eauw @z @z cw @t

ð3:48Þ

The initial pore-water pressure distribution, and a boundary condition, as shown in Fig. 3.1. That is uw ðz; 0Þ ¼ ur

ð3:49aÞ

92

3 Effects of Gravity and Hysteresis on 1D Unsaturated Infiltration

and uw ðz; 0Þjz!1 ¼ ur ;

ð3:49bÞ

where ur is a constant that is negative in a semi-infinite domain. Using dimensional space and time variables, Z = acwz, T = a2cwkst/Q, Q = hsa + Z T ac (1 + l)/(1 − l) H. and defining a new variable, W(z, t), W ¼ eauw  e 2 þ 4 . Substituting the variables into Eq. (3.48), the authors obtain: @ 2 W @W ; ¼ @Z 2 @T

ð3:50Þ

W ðZ; 0Þ ¼ eaur eZ=2 ;
 @W W q þ  ¼ eZ=2 þ T=4 ; @Z 2 z¼0 ks

ð3:51aÞ ð3:51bÞ

and W ðZ; T ÞjZ!1 ¼ eaur eT=4 :

ð3:51cÞ

The general solution to the problem of infiltration coupled with deformation in unsaturated soils can be obtained following Ozisik (1989): Z W ðZ; T Þ ¼

1

b¼0

2  ðbÞ þ K ðb; Z Þ  eb t 4F 2

ZT

3 0 db; eb t Aðb; t0 Þdt5 2 0

ð3:52Þ

t0 ¼0

where Aðb; t0 Þ ¼ K ðb; zÞjz¼0 f1 ðt0 Þ; Z 1  ðbÞ ¼ F K ðb; z0 ÞF ðz0 Þ dz0 ;

ð3:53aÞ ð3:53bÞ

z¼0

and rffiffiffi 2 b cos bZ þ 0:5 sin bZ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K ðb; Z Þ ¼ : p b2 þ 0:25

ð3:53cÞ

The pore-water pressure (uw) can be obtained by substituting Eqs. (3.52) and Z T (3.53a, 3.53b, 3.53c) into W ¼ eauw  e 2 þ 4 .

3.4 Effect of Semi-infinite Region

93

q

Fig. 3.20 One-dimensional model of rainfall infiltration incorporating the deformation of semi-infinite unsaturated porous media

0

pw(t=0)=ur

Z w=0

3.4.2

+∞

Parametric Study

In the parametric study described in this book, the authors used a model of a semi-infinite domain (Fig. 3.20). The upper flux is specified when z approaches infinity, and the pore-water pressure is constant (ur), as defined in Eq. (3.49a). In this section, the authors discuss the effect of some key parameters on the pore-water pressure and the coupling effect.

3.4.2.1

Effect of the Dimensionless Rainfall Intensity: q/ks

Figure 3.21 shows the effect of the dimensionless rainfall intensity on the pore-water pressure profile at t = 10 h. The parameters used are ks = 10−5 m/s, a = 0.01 cm−1, hs = 0.4, ur = −100 kPa, l = 0.3, and H = ±103 kPa (Van Genuchten 1980; Wu and Zhang 2009; Zhan et al. 2013). The authors used two values of q/ks (q/ks = 0.3 and 0.9) to investigate the influence of the dimensionless rainfall intensity on the pressure distribution in a semi-infinite domain. The coupling effect is more pronounced when the higher dimensionless rainfall intensity values were used. This indicates that the dimensionless rainfall intensity is closely related to the coupling between water infiltration and deformation in a semi-infinite unsaturated porous medium.

94

3 Effects of Gravity and Hysteresis on 1D Unsaturated Infiltration

Fig. 3.21 Pore-water pressure profile in a 1D semi-infinite region under two values of rainfall intensities at t = 10 h

3.4.2.2

Effect of the Desaturation Coefficient: a

The effects of the desaturation coefficient on the pore-water pressure profile in a semi-infinite domain for both coupled and uncoupled conditions are shown in Fig. 3.22. The parameters used are ks = 10−5 m/s, q/ks = 0.3, hs = 0.4, ur = −100 kPa, l = 0.3, and H = ±103 kPa. The authors used two a values (a = 0.01 and 0.1 kPa−1) to examine the influence of the desaturation coefficient on the pore-water pressure distribution in a semi-infinite domain. For a = 0.1 kPa−1, the pore-water pressure is easy to obtain close to the saturated condition on the ground surface of the semi-infinite domain. However, the pore-water pressure for a = 0.01 kPa−1 changes more slowly in the shallow layers, and increases more rapidly in the relatively deep regions. The pore-water pressure for a = 0.01 kPa−1 changes more quickly than that for a = 0.1 kPa−1. Thus, the Fig. 3.22 Influence of the desaturation coefficient on the pore-water pressure profile in a semi-infinite domain at t = 10 h

3.4 Effect of Semi-infinite Region

95

pore-water pressure distribution differs greatly under the two desaturation coefficient values. The value of a, i.e., the soil type (a = 0.01 kPa−1 for clay, a = 0.1 kPa−1 for sand) influences the coupling effect. A smaller a (clay) corresponds to a stronger coupling effect.

3.4.2.3

Effect of the Initial Pore-Water Pressure: ur

Figure 3.23 depicts the effect of the initial pore-water pressure (ur) on the pore-water pressure profile in a semi-infinite domain for both coupled and uncoupled states at t = 20 h. The authors used two values of ur (ur = −100 and −200 kPa) to examine the influence of ur on the pore-water pressure distribution in a semi-infinite domain. Other parameters remained the same for the analysis (Fig. 3.22). The pore-water pressure for the case of ur = −200 kPa changes more quickly than that for ur = −100 kPa. A lower ur value denotes faster movement of the pore-water pressure. Although the initial pore-water pressure influences the pore-water pressure profile, it has no effect on the coupling between water infiltration and soil deformation in a semi-infinite unsaturated porous medium. The initial pore-water pressure plays a considerable role in the coupling effect (Wu et al. 2009, 2012), which is based on a limited domain involving flux at the top boundary and zero pore-water pressure at the bottom boundary. The elastic model for obtaining analytical solutions to coupled infiltration problems employed in previous studies (Wu et al. 2009, 2012, 2016a, b) was used in this study for a semi-infinite domain. The results indicate that the initial pore-water pressure is not related to the elastoplastic model of unsaturated soils. It can be closely linked with the semi-infinite dimension.

Fig. 3.23 Dimensionless pore-water pressure in a 1D semi-infinite domain at t = 20 h

96

3.4.3

3 Effects of Gravity and Hysteresis on 1D Unsaturated Infiltration

Conclusions

Using Fourier integral transformation, analytical solutions to a coupled water infiltration and deformation problem in a unsaturated porous medium are extended from a finite-width domain to a semi-infinite region. The analytical solution indicates that dimensionless rainfall intensity has a marked influence on the coupling effect. Clay (with a lower a value) shows an obvious coupling effect. Although the initial pore-water pressure influences the pore-water pressure profile, it plays no role in the coupling between water infiltration and deformation in a semi-infinite domain consisting of a unsaturated porous medium.

References Basha HA (1999) Multidimensional linearized nonsteady infiltration with prescribed boundary conditions at the soil surface. Water Resour Res 35(1):75–83 Basha HA (2011) Infiltration models for semi-infinite soil profile. Water Resour Res 47:W08516 Broadbridge P, White I (1988) Constant rate rainfall infiltration: a versatile nonlinear model. 1. Analytic solution. Water Resour Res 24(1):145–154. https://doi.org/10.1029/ WR024i001p00145 Brooks RH, Corey AT (1964) Hydraulic properties of porous media. Hydrology, Paper No. 3. Colorado State University, Fort Collins, Colorado Chen JM, Tan YC, Chen CH, Parlange JY (2001a) Analytical solutions for linearized Richards equation with arbitrary time-dependent surface fluxes. Water Resour Res 37(4):1091–1093 Chen JM, Tan YC, Chen CH (2001b) Multidimensional infiltration with arbitrary surface fluxes. J Irrig Drain Eng (ASCE) 127(6):370–377 Chen JM, Tan YC, Chen CH (2003) Analytical solutions of one-dimensional infiltration before and after ponding. Hydrol Process 17(4):815–822 Chen JM, Tan YC (2005) Analytical solutions of infiltration process under ponding irrigation. Hydrol Process 19(18):3593–3602 Dane JH, Wierenga PJ (1975) Effect of hysteresis on the prediction of infiltration, redistribution and drainage of water in layered soil. J Hydrol 25(3–4):229–242 Fredlund DG, Rahardjo H (1993) Soil mechanics for unsaturated soils. Wiley, New York Gardner WR (1958) Some steady-state solutions of the unsaturated moisture flow equation with application to evaporation from a water table. Soil Sci 85:228–232 Gillham RW, Klute A, Heerman DF (1976) Hydraulic properties of a porous medium: measurement and empirical representation. Soil Sci Soc Am J 40(2):203–207 Haines WB (1930) Studies in the physical properties of soil. V. The hysteresis effect in capillary properties and the modes of water distribution associated therewith. J Agric Sci 20:97–116 Hank RJ, Klute A, Bresler E (1969) A numerical method for estimating infiltration redistribution, drainage, and evaporation of water from soil. Water Resour Res 13:992–998 Hillel D (1998) Environmental soil physics. Academic, New York Hills RGB, Hudson IP, Wierenga PJ (1989) Modeling one-dimensional infiltration into very dry soils. Water Resour Res 25:1271–1282 Hogarth W, Hopmans J, Parlange JY, Haverkamp R (1988) Application of a simple soil-water hysteresis model. J Hydrol 98(1–2):21–29 Hogarth WL, Parlange JY, Norbury J (1992) Addendum to “First integrals of the infiltration equation”. Soil Sci 154:341–343 Jaynes DB (1985) Comparison of soil–water hysteresis models. J Hydrol 75:287–299

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Kacimov AR, Yakimov ND (1998) Nonmonotonic moisture profile as a solution of Richards’ equation for soils with conductivity hysteresis. Adv Water Resour 21(8):691–696 Kawai K, Karube D, Kato S (2000) The model of water retention curve considering effects of void ratio. In: Proceedings Asian conference on unsaturated soils, pp 329–334 Kim JM (2000) A fully coupled finite element analysis of water-table fluctuation and land deformation in partially saturated soils due to surface loading. Int J Numer Meth Eng 49:1101– 1119 Klausner Y (1991) Fundamentals of continuum mechanics of soils. Springer, New York Kool JB, Parker JC (1987) Development and evaluation of closed-form expressions for hysteretic hydraulic properties. Water Resour Res 23(1):105–114 Li XS (2005) Modelling of hysteresis response for arbitrary wetting/drying paths. Comput Geotech 32(2):133–137 Likos WJ, Lu N (2004) Hysteresis of capillary stress in unsaturated granular soil. J Eng Mech ASCE 130(6):646–655 Mualem Y (1974) A conceptual model of hysteresis. Water Resour Res 10:514–520 Mualem Y (1984) Prediction of the soil boundary wetting curve. Soil Sci 137:379–389 Nimmo JR (1992) Semi-empirical model of soil water hysteresis. Soil Sci Soc Am J 56:1723–1730 Ozisik M (1989) Boundary value problems of heat conduction. Dover, New York, pp 85–87 Parlange JY, Barry DA, Parlange MB, Hogarth WL, Haverkamp R, Ross PJ, Ling L, Steenhuis TS (1997) New approximate analytical technique to solve Richards equation for arbitrary surface boundary conditions. Water Resour Res 33:903–906 Pedroso DM, Williams DJ (2010) A novel approach for modelling soil-water characteristic curves with hysteresis. Comput Geotech 37(3):374–380 Pham HQ, Fredlund DG, Barbour SL (2003) A practical hysteresis model for the soil-water characteristic curve for soils with negligible volume change. Geotechnique 53(2):293–298 Philip JR (1964) Similarity hypothesis for capillary hysteresis in porous materials. J Geophys Res 69(8):1553–1562 Raats PAC (1970) Steady infiltration from line sources and furrows. Soil Sci Soc Am J 34(5):709– 714 Richards LA (1931) Capillary conduction of liquids through porous mediums. Physics (NY) 1:318–333 Russo D, Jury WA, Butters GL (1989) Numerical analysis of solute transport during transient irrigation. 1. The effect of hysteresis and profile heterogeneity. Water Resour Res 25(10):2109– 2118 Srivastava R, Yeh TCJ (1991) Analytical solutions for one-dimensional, transient infiltration toward the water table in homogeneous and layered soils. Water Resour Res 27(5):753–762 Staple WJ (1969) Comparison of computed and measured moisture redistribution following infiltration. Soil Sci Soc Am Proc 33:840–847 Tan YC, Ma KC, Chen CH et al (2009) A numerical model of infiltration processes for hysteretic flow coupled with mass conservation. Irrig Drain 58(3):366–380 Topp GC (1971) Soil-water hysteresis: the domain theory extended to pore interaction conditions. Soil Sci Soc Am Proc 33:219–225 Tracy FT (2006) Clean two- and three-dimensional analytical solutions of Richards’ equation for testing numerical solvers. Water Resour Res 42:W08503 Van Genuchten MT (1980) A closed form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Sci Soc Am J 44:892–898 Warrick AW, Islas A, Lomen DO (1991) An analytical solution to Richards’ equation for time-varying infiltration. Water Resour Res 27(5):763–766 Wu LZ, Zhang LM (2009) Analytical solution to 1D coupled water infiltration and deformation in unsaturated soils. Int J Numer Anal Meth Geomech 33:773–790 Wu LZ, Zhang LM, Huang RQ (2012) Analytical solution to 1D coupled water infiltration and deformation in two-layer unsaturated soils. Int J Numer Anal Meth Geomech 36:798–816

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Wu LZ, Zhang LM, Li X (2016a) One-dimensional coupled infiltration and deformation in unsaturated soils subjected to varying rainfall. Int J Geomech 16(2). https://doi.org/10.1061/ (ASCE)GM.1943-5622.0000535 Wu LZ, Selvadurai APS, Zhang LM, Huang RQ, Huang J (2016b) Poro-mechanical coupling influences on potential for rainfall-induced shallow landslides in unsaturated soils. Adv Water Resour 98:114–121 Zhan TLT, Jia GW, Chen YM, Fredlund DG, Li H (2013) An analytical solution for rainfall infiltration into an unsaturated infinite slope and its application to slope stability analysis. Int J Numer Anal Met 37(12):1737–1760 Zhang LL, Zhang LM, Tang WH (2005) Rainfall-induced slope failure considering variability of soil properties. Geotechnique 55(2):183–188 Zhu JT, Mohanty BP (2002) Analytical solutions for steady state vertical infiltration. Water Resour Res 38(8):1145

Chapter 4

2D Infiltration in Unsaturated Porous Media

4.1

Introduction

Coupling between water infiltration and mechanical deformation in partially saturated porous media is central to many natural and man-made systems in civil and environmental engineering. During water infiltration, the pore-water pressure is redistributed, on one hand, by the hydraulic properties of the partially saturated soil including retention characteristics and permeability, and, on the other hand, by the external loading due to climate conditions (rainfall intensity, duration, and evapo-transpiration rate). Changes in the pore-water pressure are generated by infiltration, which in turn modifies the hydraulic domain and induces deformations of the partially saturated porous medium. Alternatively, any variation in the mechanical loading can exert an effect on the infiltration process. It is indeed the hydromechanical coupled response of a partially saturated porous medium that is responsible for the most common instabilities associated with water infiltration: landslides and excessive settlements, due to collapse or shear strength reduction (Thorel et al. 2011; Rotisciani et al. 2015). The prediction of the hydromechanical behavior of geomaterials due to rainfall infiltration plays an important role in addressing geotechnical and geo-environmental issues related to ground movement or landslides and other geologic disasters. Examples of the effects of saturation variation induced by wetting on the soil deformation has been reported at both laboratory and in situ scales (Castelblanco et al. 2011; Casini 2012). The analytical solutions to infiltration in unsaturated porous media are a basis for understanding the influence of the infiltration mechanisms. An analytical solution of the governing equations involving unsaturated infiltration can also assist in the evaluation and calibration of the accuracy of the computational solutions to practical problems. Many researchers have examined infiltration problems using analytical solutions (e.g., Srivastava and Yeh 1991; Basha 1999, 2000). The analytical solutions to one-, two-, and three-dimensional infiltration in unsaturated soils in © Science Press 2020 L. Wu et al., Hydro-mechanical Analysis of Rainfall-Induced Landslides, https://doi.org/10.1007/978-981-15-0761-8_4

99

100

4 2D Infiltration in Unsaturated Porous Media

single- and two-layer systems usually use methods based on either an integral transform technique (Basha and Selvadurai 1998) or Green’s function techniques (Srivastava and Yeh 1991; Basha 1999, 2000; Chen et al. 2001; Zhan et al. 2013). Laplace transform techniques are used to obtain analytical solutions to one-dimensional infiltration in single- and two-layer unsaturated soils (Srivastava and Yeh 1991). Using the same method, Zhan et al. (2013) proposed an analytical solution for rainfall infiltration into an infinite unsaturated slope irrespective of whether the slope is homogenous or two layered. This solution can simulate the infiltration process where there is a stable groundwater level in the region. Chen et al. (2001) used a Fourier integral transformation and a three-dimensional linearized Richards equation to derive analytical solutions for the volumetric water content distribution in single-layer unsaturated soils. Infiltration and flow in saturated and unsaturated heterogeneous porous media have been examined using analytical solutions (Srivastava and Yeh 1991; Basha and Selvadurai 1998; Basha 1999, 2000; Chen et al. 2001; Zhan and Ng 2004; Selvadurai et al. 2014; Wu et al. 2012). Analytical solutions to the coupled infiltration problem for homogenous and two-layer porous media have been obtained using Fourier integral and Laplace transformations (Wu and Zhang 2009; Wu et al. 2012). Analytical solutions can also be developed for the one-dimensional coupled problem. During water infiltration into unsaturated porous medium, the porosity in the porous medium changes with the level of saturation. At the same time, stress modification in the unsaturated porous medium leads to porosity changes. Deformation in the unsaturated porous medium leads to variations in the porosity, which influences the water flow in the unsaturated porous medium. The deformation-seepage approach is of significance because it is helpful to better understand the natural process of water infiltration into an unsaturated porous medium. Analytical solutions to 1D and 2D coupled unsaturated infiltration and deformation problems during rainfall have received increased attention in recent years (Wu and Zhang 2009; Wu et al. 2012, 2016, 2017b). Analytical solutions to transient coupled infiltration problems have been obtained using Fourier integral and Laplace transform techniques (Wu and Zhang 2009; Wu et al. 2012). Numerical techniques are increasingly being used to examine coupled seepage and deformation in unsaturated porous media. Numerical procedures are necessary because real-life problems of rainfall infiltration involve complicated geometries, soil heterogeneity, and complex initial conditions and boundary conditions (Kim 2000; Oka et al. 2010; Garcia et al. 2011). Solutions to such practical situations can rarely be obtained using analytical approaches (Garcia et al. 2011). Many numerical techniques that consider the hydromechanical processes involved in water infiltration into unsaturated porous media have been presented in the literature in soil science and geotechnique. Recent advances and references to historical findings are given by Mansuco et al. (2012). As an example, Griffiths and Lu (2005) examined the slope stability in an unsaturated elastoplastic medium due to rainfall infiltration,

4.1 Introduction

101

where the Bishop’s effective stress concept for an unsaturated porous medium and a one-dimensional suction theory were employed. Ehlers et al. (2004) employed a coupled seepage-deformation technique to examine the deformation and the localization of strains in an unsaturated porous medium due to infiltration. Kim (2000) presented a fully coupled numerical model for the water table fluctuation and land deformation in a partially saturated soil due to surface loading. Cho and Lee (2001) employed the net stress concept in a seepage-deformation coupled approach to examine the development of instability in unsaturated soil slopes. Using an unsaturated coupled hydromechanical model, Alonso et al. (2003) computed the deformations and the variation of the safety factor with time for an unstable slope in a profile of weathered over-consolidated clay. Multiphase coupled elasto-viscoplastic finite element analysis formulations have also been used to numerically investigate the generation of pore-water pressure and deformations during rainfall seepage into a one-dimensional soil column (Oka et al. 2010; Garcia et al. 2011). The interactions between infiltration and deformation in porous media have been modeled computationally using various multiphysics schemes. Numerical techniques for analyzing coupled water infiltration and deformation in partially saturated porous media are being increasingly utilized because rainfall infiltration involves complicated geometries, soil heterogeneity, and complex spatial and temporal boundary conditions (Wu et al. 2017a, b). Analytical approaches to such practical situations are difficult (Garcia et al. 2011). Many numerical approaches that incorporate the coupled seepage and deformation behaviors involved in water infiltration into a partially saturated porous medium have been presented in the literature in both the soil mechanics and geotechnical disciplines (Khalili and Zargarbashi 2010). Numerical models have also been developed to investigate the coupled hydromechanical behavior of partially saturated soils where ground deformation occurs as a result of the water table fluctuation and surface loading (Ehlers et al. 2004; Rotisciani et al. 2017) or the development of soil slope instability (Cho and Lee 2001; Alonso et al. 2003). Multiphase, coupled elasto-viscoplastic finite element analysis has been used to examine the evolution of pore-water pressure and deformations due to rainwater infiltration into a one-dimensional soil column (Garcia et al. 2011). In this section, we will develop analytical and numerical solutions to coupled infiltration and deformation in unsaturated soils. Meanwhile, surface infiltration in semi-infinite extent in unsaturated soils will be studied.

4.2

Analytical Solution to Finite Domain

The objective of this study was to develop an analytical solution for two-dimensional coupled unsaturated seepage and deformation in a transversely isotropic, unsaturated porous medium. Using a Fourier transform technique and a suitable set of transformations, we obtained analytical solutions to the

102

4 2D Infiltration in Unsaturated Porous Media

two-dimensional coupled infiltration in an unsaturated medium that was assumed to exhibit transverse isotropy in the hydraulic and mechanical properties. The influence of the transverse isotropy on the coupled infiltration was investigated.

4.2.1

Governing Equation of Coupled Infiltration Problems

To analyze the coupled infiltration problem analytically, several assumptions have to be made: 1. The mechanical behavior of the porous medium corresponds to that of a transversely isotropic medium under plane strain conditions. 2. The liquid in the porous medium obeys Darcy’s law. 3. The pore-air pressure in the porous medium remains constant. 4. The volume change of the soil is due to the wetting or drying of the soil only, and the volume change due to total stress change is not considered. 5. The soil structure is deformable, but the pore-water is not compressible. 6. Both hydraulic conductivity and saturation vary exponentially with the pressure head. To successfully obtain the analytical solution to the two-dimensional infiltration, some additional assumptions are required. Assumption 6 simplifies the analysis of the two-dimensional coupled problem, and the coupled governing equations can be linearized and transformed into a solvable form. Based on the mass conservation law and Darcy’s law under isothermal conditions (assumption 2), the two-dimensional governing equation for coupled infiltration and deformation of unsaturated soils can be obtained in the form (Kim 2000; Wu and Zhang 2009; Wu et al. 2012): r2 h ¼

h @qw @h @S @h @ev þn  gS qw @h @t @h @t @t

ð4:1Þ

where h is the pressure head (h = uw/cw); ev is the volumetric strain; η = (1 − n)2/ (1 − n0); n0 is the initial porosity; h is the volumetric water content; S is the degree of saturation; qw is the density of water; and n is the porosity. In this study, attention was given to two-dimensional problems in hydraulically and elastically orthotropic media under plane strain conditions (eyy = 0). In this case, the elastic stress–strain relation that takes into account the influence of pore-air and pore-water effects can be rewritten as (Fredlund and Rahardjo 1993) exx ¼

1  m2x mx þ 1 ð1 þ mx Þ ðrxx  ua Þ  mz ðrzz  ua Þ þ ðua  uw Þ Ez Hx Ex

ð4:2Þ

4.2 Analytical Solution to Finite Domain


 1  mx mz mx þ 1 1 mx ezz ¼ ðrzz  ua Þ  mx ðrxx  ua Þ þ þ ðua  uw Þ Ex Hz Hx Ez

103

ð4:3Þ

where Ex and Ez are the elastic moduli of the soil with respect to changes in the net normal stresses in the x- and z-directions, respectively; Hx and Hz are the elastic moduli of the soil with respect to a change in soil suction in the x- and z-directions, respectively; vx and vz are the Poisson’s ratios of the soil; (ua − uw) is the soil suction; ua is the pore-air pressure; and uw is the pore-water pressure. Equations (4.2) and (4.3) are based on assumptions 1 and 2, which makes it easier to analyze two-dimensional coupled infiltration in the porous medium; these equations are obtained from the elastic constitutive model given by Fredlund and Rahardjo (1993) and assumption 1. The assumption of elasticity simplifies the complex properties of the porous medium. Because the total stresses are assumed to remain unchanged and the air pressure ua is assumed to be constant, the changes in total stresses are not considered and we obtain from Eqs. (4.2) and (4.3): @exx 1 þ mx @uw ¼ @t Hx @t
 @ezz 1 mx @uw ¼ þ Hz Hx @t @t

ð4:4Þ ð4:5Þ

Porous media have complex mechanical properties (Desai and Siriwardane 1984; Davis and Selvadurai 2002; Pietruszczak 2010). Elastic properties for the same porous medium vary according to the degree of saturation. In assumption 3, the porous medium is elastic and the air pressure is kept constant. This simplifies the analysis of the two-dimensional coupled problem. Considering two-dimensional plane strain states in the soil, we have ev ¼ exx þ ezz . Then we can obtain the total volumetric strain rate as
 @ev 1 þ 2mx 1 @h ð4:6Þ ¼ cw þ Hz @t @t Hx where cw = qwg (g is the gravity). In this study, the fluid transport characteristics of the porous medium were considered to be transversely isotropic and the coordinate directions were assumed to coincide with the principal directions of elasticity. Hydraulic transverse isotropy in porous media has been extensively documented in the literature in geomechanics and groundwater engineering, and references can be found in Selvadurai (2003). If we also restrict attention to incompressible pore fluids and substitute Eq. (4.6) into Eq. (4.4.1), the two-dimensional problem for coupled infiltration and deformation in unsaturated soils can be obtained in the following form:

104

4 2D Infiltration in Unsaturated Porous Media

    
 @ @h @ @h @ @S 1 þ 2mx 1 @h kx kz þ gcw S þ þ þ ½ kz  ¼ n @x @x @z @z @z @h Hz @t Hx

ð4:7Þ

where kx and kz are the permeability coefficients of the unsaturated porous medium in the x- and z-directions, respectively. For the hydraulic conductivity of a transversely isotropic unsaturated medium in the x- and z-directions, we adopt the variations that are in the form of exponential functions (Gardner 1958): kx ¼ ksx ekah

ð4:8:1Þ

kz ¼ ksz eah

ð4:8:2Þ

where ksx and ksz are the principal coefficients of hydraulic conductivities at full saturation in the x- and z-directions; a is the desaturation coefficient; and k is a dimensionless parameter describing transversely isotropic properties of infiltration. The orthotropic medium becomes uniform if k = 1. The volumetric water content of the unsaturated medium is expressed as hðhÞ ¼ hs eah . Based on the relationship between the volumetric water content and the degree of saturation, i.e.: SðhÞ ¼

hð hÞ hs

ð4:9Þ

Equation (4.9) can be written as SðhÞ ¼ eah . McKee and Bumb (1984) suggested an exponential function for the relationship between the normalized water content and suction. This is also referred to as the Boltzmann model. Based on a statistical study of the experimental data, Sillers and Fredlund (2001) provided fitting parameters for the McKee and Bumb (Boltzmann) model for eight different types of soils. The exponential form, which was used by McKee and Bumb (1984) to describe the soil–water characteristic curve, gives the best results if the pore-size distribution of the soil is close to a c distribution. From Eqs. (4.7)–(4.9), we obtain the two-dimensional governing equation for the coupled infiltration:     @ kah @h @ ah @h @ @h ksx e e þ ksz þ ksz ½eah  ¼ P  eah @x @x @z @z @z @t

ð4:10Þ

where P = ahs + ηacw[1/Hz + (1 + 2vx)/Hx] and hs is assumed to be constant. In Eq. (4.10), P is composed of two parts, one of which is from the uncoupled part while the other is a contribution from the deformations. The effects of volume change in the soil on the saturation and permeability are not considered.

4.2 Analytical Solution to Finite Domain

105

Introducing a change in the dependent variable designated by (i.e., the Kirchhoff variable): w ¼ eah

ð4:11Þ

Equation (4.10) can be rearranged as
 ksx @ @w ksz @ 2 w @w @w wk1 ¼P þ ksz þ @x @z @t a @x a @z2

ð4:12Þ

It is difficult to obtain an analytical solution from Eq. (4.12) when k 6¼ 1. Therefore, by assuming k = 1, which implies that a has the same value in the x- and z-directions, Eq. (4.12) can be reduced to: ksx @ 2 w ksz @ 2 w @w @w ¼P þ þ ksz 2 2 @z @t a @x a @z

4.2.2

ð4:13Þ

Formulation of the Initial Value Problem

Two-dimensional infiltration through a transversely isotropic unsaturated soil applicable to a rectangular region of width a and height b, as shown in Fig. 4.1, was examined. The two-dimensional transient infiltration problem applicable to the region generally involves one initial condition and four boundary conditions (Ozisik 1989; Selvadurai 2000), which are as follows: hðx; z; 0Þ ¼ hi ðx; zÞ  @hðx; z; tÞ ¼0 @x x¼0  @hðx; z; tÞ ¼0 @x x¼a hðx; z; tÞjz¼0 ¼ h0   @h þ kz kz ¼ qðx; tÞ @z z¼b

ð4:14:1Þ ð4:14:2Þ ð4:14:3Þ ð4:14:4Þ ð4:14:5Þ

where hi is the initial condition for the equilibrium case; h0 is the pressure head at the lower boundary, which is zero if the lower boundary (z = 0) is located at the stationary groundwater table; and q is a function that varies with position (x) and time (t).

106

4 2D Infiltration in Unsaturated Porous Media

Fig. 4.1 Two-dimensional infiltration model in a transversely isotropic unsaturated porous medium

q0

q0

h =0 x

h =0 x

b

z h=0

x

a

Using dimensionless variables, Z = az, X = ax(ksz/ksx)1/2, T = akszt/P, and defining a new variable, V(X, Z, T): V ¼ w  eð 2 þ 4 Þ Z

T

ð4:15Þ

Equation (4.13) can be rearranged as @2V @ 2 V @V þ ¼ @X 2 @Z 2 @T

ð4:16Þ

This is a classical heat conduction-type equation; the boundary and initial conditions are as follows: Z

VðX; Z; TÞjT¼0 ¼ FðX; YÞ ¼ wi  e2

ð4:17:1Þ

 @V  ¼0 @X X¼0

ð4:17:2Þ

 @V  ¼0 @ X X¼A

ð4:17:3Þ

T

V jZ¼0 ¼ e 4   @V V q B T þ ¼ e2 þ 4 @Z 2 Z¼B ksz in which A = aa and B = ab.

ð4:17:4Þ ð4:17:5Þ

4.2 Analytical Solution to Finite Domain

4.2.3

107

Analytical Solutions to Two-Dimensional Coupled Infiltration

Using a Fourier integral transform (Ozisik 1989), the general solution to the initial boundary value problem posed by Eqs. (4.16) and (4.17) can be written as ( 1 X 1 h X 2 2 1 hðX; Z; TÞ ¼ ln eZ=2 K1 ðbm ; XÞK2 ðkn ; ZÞ  eðbm þ kn þ 0:25ÞT a m¼0 n¼0 0 139 ð4:18Þ ZT = 2 2 0
m ; kn Þ þ  @Fðb eðbm þ kn Þt Hðbm ; kn ; t0 Þdt0A5 ; t0 ¼0

where ( K1 ðbm ; XÞ ¼ " pffiffiffi K2 ðkn ; ZÞ ¼ 2

p1ffiffiffi A 1 pffiffiffi cosðb XÞ m A

k2n þ 0:25 2 Bðkn þ 0:25Þ þ 0:5

for m ¼ 0 for m ¼ 1; 2; 3; . . .

ð4:19:1Þ

#1=2 n ¼ 1; 2; 3; . . .

sinðkn ZÞ

ð4:19:2Þ

The eigenvalue bm = mp/A, and kn satisfies the transcendental Eq. (4.2) kncot (knB) + 1 = 0, and  dK2 ðkn ; ZÞ Hðbm ; kn ; t Þ ¼  dZ

ZA

0

ZA  x0 ¼0


m ; kn Þ ¼ Fðb

ZA

z¼0

x0 ¼0

 K2 ðkn ; ZÞ K1 ðbm ; x Þe dx þ 0:5  0

T 4

0

Z¼B

ð4:19:3Þ

q B T K1 ðbm ; x0 Þ e2 þ 4 dx0 ksz ZB

z0

K1 ðbm ; x0 ÞK2 ðkn ; z0 Þ  wi  e 2 dx0 dz0

ð4:19:4Þ

x0 ¼0 z0 ¼0

Equations (4.19.3) and (4.19.4) can be rewritten as "

k2n þ 0:25 Hðbm ; kn ; t Þ ¼ 2 2 Bðkn þ 0:25Þ þ 0:5 0

#1=2

  T e 4 sinðbm AÞ q B pffiffiffi kn þ e2 sinðkn BÞ ksz bm A ð4:20Þ

108

4 2D Infiltration in Unsaturated Porous Media

and "
m ; kn Þ ¼ 2 Fðb

k2n þ 0:25 Bðk2n þ 0:25Þ þ 0:5

#1=2

sinðbm AÞ ð0:5  aÞ sinðkn BÞeð0:5aÞB  kn ½eð0:5aÞB cosðkn BÞ  1 pffiffiffi bm A k2n þ ð0:5  aÞ2 ð4:21Þ Equation (4.18) can be rewritten as ( 1 X 1 h X 2 2 1 hðX; Z; TÞ ¼ ln eðX þ Z Þ=2 B1 cosðbm XÞ sinðkn ; ZÞ  eðbm þ kn þ 0:25ÞT a m¼0 n¼0 ) i þ B2 cosðbm XÞ sinðkn ; ZÞ ð4:22Þ From L’Hospital’s rule, when m and n approach infinity, both B1 and B2 will approach zero, i.e., limm! þ 1 B1 ¼ 0 and limm! þ 1 B2 ¼ 0. Accordingly, both items in Eq. (4.22) will approach zero when m and n approach infinity for all times. Here, the analytical solution is convergent in the domain. The analytical solution given by Eq. (4.18) has a series form. It can be shown that the inclusion of m, n = 150 is sufficient to ensure convergence of the results.

4.2.4

Numerical Results

The analytical results presented above can be used to develop numerical results. Table 4.1 lists the parameters used in the development of the numerical results for certain idealized situations. In the analysis, the boundaries were kept unchanged. Figure 4.2 shows the pressure head profiles of two-dimensional coupled infiltration in an isotropic porous medium. The values of a, which ranges from 0.01 to 2.6 cm−1, are those given by McKee and Bumb (1984). The parameters used to develop the results shown in Fig. 4.2 (a = b = 200 cm) are as follows: hs = 0.4, ksx = ksz = 10−5 m s−1, η = 0.8, q0 = 0.8  10−5 m s−1, a = 0.01 cm−1, mx = 0.3, and Hx = Hz = 103 kPa (Hx and Hz are the moduli associated with suction in the xand z-directions, respectively). At the same depth, the pressure head on the right-hand side varies more rapidly than the pressure head at the left because evaporation occurs at the right while rainfall infiltration is present on the left. The change in the pressure head gradually decreases in the deeper zones. Figure 4.3 shows the pressure head profiles for two-dimensional coupled infiltration in an orthotropic porous medium. The parameters used in the development

4.2 Analytical Solution to Finite Domain

109

Table 4.1 The parameters used in the analysis of the two-dimensional coupled infiltration problem for the transversely isotropic unsaturated medium Figures

Case

q0 m s−1

Hx kPa

Hz

ksx m s−1

ksz

– 0.8  10−5 103 103 10−5 10−5 −4 3 4 −5 – 0.8  10 10 10 10 10−4 −5 3 −5 −5 A 0.8  10 1  10 10 10 10−5 3 −6 B 0.5  10 10 C 5  103 10−4 q0 is maximum infiltration rate at surface boundary; Hx and Hz are the elastic moduli of the soil with respect to a change in soil suction in the x- and z-directions, respectively; ksx and ksz are the principal coefficients of hydraulic conductivities at full saturation in the x- and z-directions 4.2 4.3 4.4

(a) t = 0 hour

(b) t = 5 hours

(c) t = 15 hours

Fig. 4.2 The pressure head profiles for the two-dimensional coupled infiltration in an isotropic porous medium (hs = 0.4, ksx = ksz = 10−5 m/s, η = 0.8, q0 = 0.8  10−5 m/s, a = 0.01 cm−1, mx = 0.3, and Hx = Hz = 103 kPa)

(a) t = 0 hour

(b) t = 5 hours

(c) t = 15 hours

Fig. 4.3 The pressure head profiles for the two-dimensional coupled infiltration in a transversely isotropic porous medium (hs = 0.4, ksx = 10−5 m/s, ksz = 10−4 m/s, q0 = 0.8  10−4 m/s, a = 0.01 cm−1, mx = 0.3, Hx = 103 kPa, and Hz = 104 kPa)

110

4 2D Infiltration in Unsaturated Porous Media

of the numerical results are as follows: hs = 0.4, ksx = 10−5 m s−1, ksz = 10−4 m s−1, q0 = 0.8  10−4 m s−1, a = 0.01 cm−1, mx = 0.3, Hx = 103 kPa, and Hz = 104 kPa. As is evident from Fig. 4.3, the pressure head changes more rapidly than in Fig. 4.2. Figure 4.4 illustrates the influence of the transversely isotropic elasticity properties of the soil on the pressure head profiles (case A: Hx = 103 kPa, ksx = 10−5 m/s; case B: Hx = 0.5  103 kPa, ksx = 10−6 m/s; case C: Hx = 5  103 kPa, ksx = 10−4 m/s). The parameters (Hz = 103 kPa, ksz = 10−5 m/s) are kept unchanged. In Fig. 4.4, at time t = 15 h, the dotted line represents the pressure head for Case A (Hx = 103 kPa, ksx = 10−5 m s−1), the solid line depicts Case B (Hx = 0.5  103 kPa, ksx = 10−6 m s−1), and the dashed line is Case C (Hx = 5  103 kPa, ksx = 10−4 m s−1). The parameters in the z direction were kept unchanged (Hz = 103 kPa, ksz = 10−5 m s−1). In these three cases, the boundary conditions were unchanged (q0 = 0.8  10−5 m s−1). Obvious changes in the pressure head in the surface zones can be seen; in the deeper zones, where there is no water infiltration, the influences of the pressure head are negligible. When there is a smaller difference between the horizontal and vertical coefficient of permeability (i.e., ksx  ksz), infiltration into the unsaturated medium is more rapid. When the porous medium is hydraulically isotropic, the pressure head changes occur much faster. Figure 4.5 gives a comparison of the water pressure head calculated using a numerical solution and the analytical solution. The analytical solution was obtained using Eq. (4.17.1), and the FlexPDE software was used to obtain the numerical results, where the convergence criterion was 10−4. The error between the numerical and analytical solutions is 1), the factor of safety of the unsaturated slope is greater under the same rainfall condition.

6.4.5

Conclusions

Based on the analysis in this book, the following conclusions can be drawn: (1) Advancement of the wetting front in a slope consisting of unsaturated porous medium is closely related to pore-air discharge from the slope. The pore-water pressure increases over infiltration time. The pore-air pressure of the soil slope

6.4 Three-Phase Coupling of Soil Slope Under Rainfall 1.7

1.6

Factor of safety

Fig. 6.32 Effect of soil anisotropy on the factor of safety: A, three-phase coupling; B, air–liquid coupling; C, solid–liquid coupling

229

1.5

1.4

1.3

1.2

1.1 A

B

C

varies with depth. The maximum pore-air pressure occurs in the deep layer of the slope and is difficult to dissipate. (2) The influence of soil anisotropy on the slope seepage varies not only with the depth of the soil slope, but also with changes in the total stresses in the slope. When parameter b (Ex/Ey) is smaller and parameter η = (kx0/ky0) is larger, the rainwater infiltrates faster into the shallow layers of the soil slope. The situation is opposite in the area far away from the top of the unsaturated soil slope. (3) The stability of the soil slope is greater under the conditions of a smaller b (Ex/ Ey) and a larger η = (kx0/ky0). The anisotropy of the soil properties such as elastic modulus and coefficient of permeability has a marked effect on the soil slope stability.

Appendix
Eq. (6.20) becomes Applying the Laplace transform and replacing W by W,

@2W @W
þ W0 ¼ 0  sW þ 2 @z @z

ð6:60Þ

The bottom boundary condition (Eq. (6.21b)) and surface top boundary condition (Eq. (6.21c)) for the soil slope are given by

230

6 Slope Stability Analysis Based on Coupled Approach


ð 0Þ ¼ e W

aw0

s




@W
¼ qB  sW

@z s z¼L

ð6:61Þ ð6:62Þ

The general solution to Eq. (6.60) with the boundary conditions that are described by Eqs. (6.61) and (6.62) is expressed as
¼ W0 ðzÞ þ ðqB  qA ÞeðLzÞ=2 GðsÞ W s

ð6:63Þ

in which h i 0:5 sinh z ð s þ 0:25 Þ 1 h i h i GðsÞ ¼ s ðs þ 0:25Þ0:5 cosh Lðs þ 0:25Þ0:5 þ 0:5 sinh Lðs þ 0:25Þ0:5

ð6:64Þ

The inversion of G(s) is achieved by using the residue theorem as the sum of residues of eSlG(s) at the poles of G(s). The residue at s = 0 is e–(L−z)/2 − e−(L+z)/2. The other poles are acquired by assigning (s + 0.25)1/2 as a complex number. All the poles are at pure imaginary values of (s + 0.25)1/2 that is a function of i k. We can obtain k that satisfies the positive roots of the characteristic equation as follows: tanðkLÞ þ 2k ¼ 0;

ð6:65Þ

and the residue at kn , which is the nth root of Eq. (6.65), can be written as 2 4 sinðLkn Þ sinðkn zÞeð0:25 þ kn Þt ¼ : 1 þ 0:5L þ 2Lk2n

ð6:66Þ

The equation for W, i.e., Equation (6.22), is therefore obtained.

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