Hydromechanical Modeling Framework for Multiscale

HYDROMECHANICAL MODELING FRAMEWORK FOR MULTISCALE POROUS MATERIALS

a dissertation submitted to the department of civil and environmental engineering and the committee on graduate studies of stanford university in partial fulfillment of the requirements for the degree of doctor of philosophy

Jinhyun Choo August 2016

© 2016 by Jinhyun Choo. All Rights Reserved. Re-distributed by Stanford University under license with the author.

This work is licensed under a Creative Commons AttributionNoncommercial 3.0 United States License. http://creativecommons.org/licenses/by-nc/3.0/us/

This dissertation is online at: http://purl.stanford.edu/cr633zf8014

ii

I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Ronaldo Borja, Primary Adviser

I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Peter Kitanidis

I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Christian Linder

I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Richard Regueiro

Approved for the Stanford University Committee on Graduate Studies. Patricia J. Gumport, Vice Provost for Graduate Education

This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file in University Archives.

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iv

ABSTRACT

Hydromechanical interactions between fluid flow and deformation in porous geomaterials give rise to a wide range of societally important problems such as landslides, ground subsidence, and injection-induced earthquakes. Many geomaterials in these problems possess two-scale porous structures due to fractures, particle aggregation, or other reasons. However, coupled hydromechanical processes in these multiscale porous materials, such as ground deformation caused by preferential flow, are beyond the modeling capabilities of classical frameworks. This thesis develops theoretical and computational frameworks for fully coupled hydromechanical modeling of geomaterials with two-scale porous structures. Adopting the concept of double porosity, we treat these materials as a multiscale continuum in which two pore regions of different scales interact within the same continuum. Three major developments are presented. First, we build a mathematical framework for thermodynamically consistent modeling of unsaturated porous media with double porosity. Conservation laws are formulated incorporating an effective stress tensor that is energy-conjugate to the rate of deformation tensor of the solid matrix. Based on energy-conjugate pairs identified in the first law of thermodynamics, we develop a constitutive framework for hydrological and mechanical processes coupled at two scales. Second, we introduce a novel constitutive framework for elastoplastic materials with evolving internal structures. By partitioning the thermodynamically consistent effective stress into two individual, single-scale effective stresses, this framework uniquely distinguishes proportional volume changes in the two pore regions under finite deformations. This framework accommodates the impact of pore pressure difference between the two scales on the solid deformation, which was predicted by thermodynamic principles. We show that the proposed framework not only improves the prediction of deformation of two-scale geomaterials, but also simulates secondary compression effects due to delayed pressure dissipation in the less permeable pore region. Third, we develop a finite element framework that enables the use of computationally efficient equal-order elements for solving coupled fluid flow and deformation problems in double-porosity media. At the core of the finite element formulation is a new method that stabilizes twofold saddle point problems arising in the undrained condition. The stabilized finite elements allow for equal-order linear interpolations of three primary variables—the displacement field and two pore pressure variables—throughout the entire range of drainage conditions. v

vi

ACKNOWLEDGMENTS

My four years at Stanford have been truly fulfilling, rewarding, and enjoyable. I am so happy that I finally have this opportunity to acknowledge some of the many unforgettable people who have made these years possible and very worthwhile. At this point, I must emphasize that I sincerely thank many more people than I can mention in the following. Firstly, I would like to express my heartfelt gratitude to my Ph.D. adviser, Prof. Ronaldo I. Borja, who is an exemplary scholar, teacher, and individual. Thanks to his unparalleled guidance, support, and insights, I have learned far more than I imagined possible when I arrived here. I also greatly appreciate his constant belief in me—it has made me more hardworking and dedicated. Studying geomechanics under his supervision has been a great honor. I am deeply grateful to Dr. Joshua A. White at Lawrence Livermore National Laboratory. Having such a brilliant and kind person as a collaborator and mentor has been my great fortune. He provided me with his code Geocentric, with which I implemented the computational work in this thesis, and gave me detailed advice whenever needed. I also thank him for recently involving me in the development of Geocentric—I look forward to our continuing collaborations. My sincere appreciation goes to the members of my dissertation committee— Prof. Peter K. Kitanidis and Prof. Christian Linder in our Civil and Environmental Engineering Department and Prof. Richard A. Regueiro at the University of Colorado Boulder—for their time and insightful comments. What I learned from Profs. Linder’s and Regueiro’s courses also directly helped me develop multiple chapters of this thesis. I also thank Prof. Eric M. Dunham in the Department of Geophysics for serving as the chair of my oral examination committee. Financial support for my doctoral studies has been provided by the Fulbright Program, the John A. Blume Earthquake Engineering Center, the Charles H. Leavell Graduate Student Fellowship, the U.S. Department of Energy under Award Number DE-FG02-03ER15454, and the National Science Foundation under Award Number CMMI-1462231. This support is greatly appreciated. My graduate life has also been enriched by many fellow students. I would like to give my special thanks to the former and current members of our geomechanics group—Helia Rahmani, Xiaoyu Song, Martin Tjioe, Kane C. Bennett, Shabnam J. Semnani, Alomir Favero, Júlia Camargo, and Qing Yin. I also shared great times with Nicolò Spiezia from the University of Padua, my former officemate Xiaoxuan Zhang, and other friends in the Blume center and Y2E2. vii

Camaraderie with my Korean friends at Stanford has been an indispensable source of happiness during my Ph.D. I am very grateful to those who studied in the same department—particularly Jung In Kim, Jonghyun Lee, Tae Wan Kim, Yongju Choi, Jinkyoo Park, Hae Young Noh, Jaewook Myung, Soh Young In, Seongwoon Jeong, Sungho Bae, Kangjoo Lee, and Ju-Young Shin. We shared both good and hard times. Deep thanks are extended to the 2013–14 staff members of the Korean Student Association at Stanford, for striving together to make a better environment for Korean people here. I also benefited from time with admirable people in the Korean Catholic Community at Stanford. Many other Korean friends and their families made me more cheerful and relaxed—thank you all. I must express further gratitude to many people in Korea who nurtured me to be a researcher prior to my doctoral studies. Professors at Seoul National University, particularly my Master’s adviser Prof. Choong-Ki Chung as well as Prof. Myoung-Mo Kim and Prof. Junboum Park, deserve my sincere appreciation. I also thank Prof. Young-Hoon Jung at Kyung Hee University and Prof. Wanjei Cho at Dankook University for their guidance in publishing my Master’s research as journal papers. After graduating from my Master’s program, I had the fortune to work with many great researchers at the Korea Institute of Construction Technology, including Dr. Young Seok Kim, Dr. Jangguen Lee, Dr. Ju-Hyong Kim, and Dr. Sam-Deok Cho, to whom I extend my thanks. My deep gratitude also goes to professors at Yonsei University—Prof. Sooil Kim, Prof. Sangseom Jeong, Prof. Junhwan Lee, and Prof. Tae Sup Yun—for their continued assistance and advice. Throughout my life, my family has provided unwavering support, encouragement, and love. I have been blessed to be with my beloved wife Jeong Hun, who has been my dearest lady, closest friend, and ideal companion. In December 2013 we gave birth to our son Seungbin, and he has brought us unbelievable joy and happiness. Thanks to them, I have been resilient in spite of the unavoidable frustrations of research. I am also grateful to my sister and her family, particularly for the memories we made together while they stayed in Los Angeles for two years. I have been fortunate to constantly see my family-in-law and get their encouragement during my time at Stanford. I also would like to express my deep gratitude to extended family members including my grandparents, uncles, aunts, cousins, and nieces. Lastly, I want to especially acknowledge my parents to whom I owe my accomplishments. My father has been a firm supporter of me, not only as a great parent but also as a respectable geotechnical engineer. My mother deserves my eternal gratitude. She has selflessly supported me and our family, taught me great lessons in life, and given me immeasurable happiness and love ever since I was born. The more I live, the more I thank and admire them. To my family I dedicate this thesis.

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CONTENTS

1

introduction 1.1

Motivation and scope . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.2

Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.2.1

Double porosity . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.2.2

Mechanical behavior of aggregated geomaterials . . . . . . .

5

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

1.3 2

mathematical modeling framework

9

2.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

2.2

Conservation laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

2.2.1

Balance of mass . . . . . . . . . . . . . . . . . . . . . . . . . . .

11

2.2.2

Balance of linear momentum . . . . . . . . . . . . . . . . . . .

12

2.2.3

Balance of internal energy . . . . . . . . . . . . . . . . . . . . .

13

2.3

Constitutive framework . . . . . . . . . . . . . . . . . . . . . . . . . .

14

2.4

Mixed finite element formulation . . . . . . . . . . . . . . . . . . . . .

17

2.4.1

Strong form . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

2.4.2

Weak form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18

2.4.3

Time-integrated form . . . . . . . . . . . . . . . . . . . . . . .

19

2.4.4

Matrix form . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20

Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23

2.5.1

Persistent shear band . . . . . . . . . . . . . . . . . . . . . . .

23

2.5.2

Unsaturated slope under rainfall infiltration . . . . . . . . . .

29

Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

40

2.5

2.6 3

1

constitutive modeling framework

41

3.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41

3.2

General formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43

3.2.1

Solid kinematics . . . . . . . . . . . . . . . . . . . . . . . . . .

43

3.2.2

Effective stress and the first law . . . . . . . . . . . . . . . . .

45

3.2.3

Compressibility laws . . . . . . . . . . . . . . . . . . . . . . . .

47

3.2.4

Continuum formulation . . . . . . . . . . . . . . . . . . . . . .

49

Finite element formulation . . . . . . . . . . . . . . . . . . . . . . . . .

51

3.3.1

Conservation laws and strong form . . . . . . . . . . . . . . .

51

3.3.2

Variational equations . . . . . . . . . . . . . . . . . . . . . . . .

53

3.3.3

Linearized variational equations . . . . . . . . . . . . . . . . .

54

3.3

ix

3.4

3.5

3.6 4

57

3.4.1

Stress-point integration . . . . . . . . . . . . . . . . . . . . . .

57

3.4.2

Tangent operators . . . . . . . . . . . . . . . . . . . . . . . . .

60

3.4.3

Yield function . . . . . . . . . . . . . . . . . . . . . . . . . . . .

62


62

3.5.1

Stress-point simulations . . . . . . . . . . . . . . . . . . . . . .

63

3.5.2

1D consolidation with secondary compression . . . . . . . . .

67

3.5.3

Secondary compression beneath a leaning tower . . . . . . .

70

Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77

stabilized finite element framework

79

4.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

79

4.2

Discrete System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81

4.3

Stabilized mixed finite elements . . . . . . . . . . . . . . . . . . . . .

82

4.4


87

4.4.1

Cryer’s sphere . . . . . . . . . . . . . . . . . . . . . . . . . . .

88

4.4.2

Undrained double-porosity sphere . . . . . . . . . . . . . . . .

91

4.4.3

Strip footing . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

93

4.5 5

Discrete formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

closure

105

5.1

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

5.2

Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

bibliography

111

x

LIST OF FIGURES

Figure 1.1

Fluid flow in a single-scale porous medium (left) and a twoscale, fractured porous medium (right). Darker regions are more saturated. In the left-hand figure, the fluid shows a straight wetting front that can be simulated by a classical approach. In the figure on the right, preferential flow takes place while much of the fluid is trapped in the upper region. This type of fluid flow cannot be captured by a classical approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 1.2

Schematic representation of a porous medium with double porosity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 1.3

4

Schematic representation of compression curves of a soil in aggregated and reconstituted states. . . . . . . . . . . . . . .

Figure 2.1

2

Spatial distributions of macropore fractions

ψM.

6

Note that

the overall porosities of the two distributions are the same and uniformly distributed throughout the specimen. . . . .

24

Figure 2.2

Contours of equivalent plastic strain (in percent) at vertical 26

Figure 2.3

compression of 2.7 mm. . . . . . . . . . . . . . . . . . . . . . Contours of mean pore water pressure p¯ (in kPa) for Cases #1 and #2 calculated at vertical compression of 2.7 mm. . . .

27

Figure 2.4

Contours of macropore and micropore water pressures (in kPa) for Case #1 calculated at vertical compression of 2.7 mm. 28

Figure 2.5

Contours of degree of saturation for the macropores at vertical compression of 2.7 mm. Note that the micropores are fully saturated everywhere at this loading stage. . . . . . . .

28

Figure 2.6

Geometry and mesh for the steep hillside slope problem. . .

29

Figure 2.7

Single porosity: Equivalent plastic strain (in percent) in the colluvium at t = 125 minutes. . . . . . . . . . . . . . . . . . .

Figure 2.8

Single porosity: Degree of saturation and pore water pressure (in kPa) in the colluvium at t = 100 minutes. . . . . . .

Figure 2.9

32

Single porosity: Degree of saturation and pore water pressure (in kPa) in the colluvium at t = 125 minutes. . . . . . .

Figure 2.10

31

33

Water retention curves for double-porosity and single-porosity modeling fitted to experimental data from Carminati et al. (2007). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

35

Figure 2.11

Double porosity: Equivalent plastic strain (in percent) in the colluvium at t = 125 minutes. . . . . . . . . . . . . . . . . . .

Figure 2.12

Double porosity: Global degree of saturation and mean pore water pressure (in kPa) in the colluvium at t = 100 minutes.

Figure 2.13

38

Snapshots of pressure differences (in kPa) between the macropores and micropores during the rainfall. . . . . . . . . . . .

Figure 3.1

37

Double porosity: Global degree of saturation and mean pore water pressure (in kPa) in the colluvium at t = 125 minutes.

Figure 2.14

36

39

Reconstructed computed tomography volume of an aggregated silty clay, diameter = 80 mm, height = 35 mm. The aggregates are composed of much smaller silty clay solid particles with intra-aggregate pores. Visible spaces between aggregates are the inter-aggregate pores. After Borja and Koliji (2009). . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 3.2

42

Variation of specific volume with axial stress for aggregated and reconstituted Bioley silt. Experimental data from Koliji et al. (2008b). . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 3.3

63

Variation of macropore void ratio e M with volumetric plastic strain for Bioley silt. Initial value e M0 was calculated at a stress level equal to p¯ c = −100 kPa. Experimental data

from Koliji et al. (2008b). . . . . . . . . . . . . . . . . . . . . . Figure 3.4

64

Variation of specific volume with axial stress for aggregated and reconstituted Corinth marl. Experimental data from Anagnostopoulos et al. (1991). . . . . . . . . . . . . . . . . . . . . .

Figure 3.5

65

Deviator stress versus axial strain for Corinth marl: comparison with Liu and Carter (2002). Experimental data from Anagnostopoulos et al. (1991): open circles are for test at confining pressure of 4 MPa, open squares at 1.5 MPa.. . . . . . . .

Figure 3.6

65

Volumetric strain versus axial strain for Corinth marl: comparison with Liu and Carter (2002). Experimental data after Anagnostopoulos et al. (1991), with open circles for test at confining pressure of 4 MPa, and open squares at 1.5 MPa. 66

Figure 3.7

Deviator stress versus axial strain responses for Corinth marl: comparison with Koliji et al. (2010a). Experimental data from Anagnostopoulos et al. (1991): open circles are for test at confining pressure of 4 MPa, open squares at 1.5 MPa. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xii

66

Figure 3.8

Volumetric strain versus axial strain for Corinth marl: comparison with Koliji et al. (2010a). Experimental data after Anagnostopoulos et al. (1991), with open circles for test at confining pressure of 4 MPa, and open squares at 1.5 MPa. . . .

Figure 3.9

Finite element mesh for 1D consolidation with secondary compression. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 3.10

67

68

Time evolution of vertical displacement at 0.1 m below the top boundary (level A–A) showing separation of scales.Values of α¯ /k m in s/m. . . . . . . . . . . . . . . . . . . . . . . . . . . 69

Figure 3.11

Figure 3.12

Time evolution of preconsolidation stress, relative to the value of p¯ c at time t = 0, at 0.1 m below the top boundary (level A–A). Values of α¯ /k m in s/m. . . . . . . . . . . . . Spatial and temporal evolution of

ψm

69

alongside the finite

element mesh. Level A–A is 0.1 m below the top. Solid curves denote primary consolidation; dashed lines denote secondary compression. . . . . . . . . . . . . . . . . . . . . . Figure 3.13

70

Finite element mesh for foundation of a leaning tower: clay layer is modeled as a double-porosity material with dual permeability. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 3.14

71

Compressibility curves for natural and reconstituted Pancone clay. Experimental data from Calabresi and Callisto (1998). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

72

Figure 3.15

Variation of tower inclination θ with time. . . . . . . . . . . .

73

Figure 3.16

Evolution of macropore pressure p M (in kPa) within the clay layer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 3.17

Evolution of micropore pressure pm (in kPa) within the clay layer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ψm

Figure 3.18

Evolution of micropore fraction

Figure 4.1

Normalized pore pressure p/p0 in Cryer’s sphere with ν =

within the clay layer. . .

0 at time factor T = 0.005. . . . . . . . . . . . . . . . . . . . . Figure 4.2

75 76

90

Comparison of analytical and numerical solutions at the center of Cryer’s sphere. . . . . . . . . . . . . . . . . . . . . .

Figure 4.3

74

90

Excess pore pressures in double-porosity sphere after 0.1 second of undrained loading. Surface pressure varies spatially according to 1 + 0.5 sin( x ). . . . . . . . . . . . . . . . .

Figure 4.4

Schematic illustration for the geometry and boundary conditions of the strip footing problem. . . . . . . . . . . . . . .

Figure 4.5

92

93

Two meshes used in the strip footing example. Mesh #1 has 800 elements while Mesh #2 has 3200 elements. . . . . . . .

xiii

94

Figure 4.6 Figure 4.7 Figure 4.8 Figure 4.9 Figure 4.10 Figure 4.11 Figure 4.12

Pore pressure variations below the center of footing at the initial time step: Case #1 (k M /k m = 2 × 106 ). . . . . . . . . .

96

tial time step: Case #1 (k M /k m = 2 × 106 ) with Mesh #2. . .

97

2 × 106 ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

98

2 × 106 ) with Mesh #1. . . . . . . . . . . . . . . . . . . . . . .

99

Contours of macropore and micropore pressures at the ini-

Time histories of pore pressures at Point A: Case #1 (k M /k m = Time history of displacement at Point A: Case #1 (k M /k m =

Pore pressure variations below the center of footing at the

initial time step: Case #2 (k M /k m = 1 × 102 ). . . . . . . . . . 100 Contours of macropore and micropore pressures at the ini-

tial time step: Case #2 (k M /k m = 2 × 102 ) with Mesh #2. . . 101

Time histories of pore pressures at Point A: Case #2 (k M /k m =

1 × 102 ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

xiv

1

INTRODUCTION

1.1

motivation and scope

In porous materials such as soils and rocks, the flow of the pore fluid can drive the deformation of the solid matrix, and vice versa. These hydromechanical interactions give rise to a wide range of societally important problems that relate to infrastructure, energy, and the environment. Notable examples include landslides, ground subsidence, hydraulic fracturing, and injection-induced earthquakes (Borja et al., 2016; Ellsworth, 2014; Galloway and Burbey, 2011; Zoback, 2007). The classical frameworks for hydromechanical modeling idealize a porous material as a continuum mixture of a solid matrix and pore fluids. Conservation laws are then formulated for the mass and momentum of this mixture, and constitutive relations are introduced to close the formulation and capture the physical behavior. The fluid flow is modeled by a hydrological constitutive relation—typically Darcy’s law—and the deformation is modeled by a mechanical constitutive relation defined with the effective stress in the solid matrix. These classical frameworks implicitly postulate that the material has a singlescale porous structure in which pores exhibit a unimodal size distribution. For example, because Darcy’s law describes fluid flow in a single-scale porous medium, the use of this law implies that the porous structure exists at a single scale. Likewise, because standard forms of effective stress have been validated only for singlescale porous materials (e.g., Borja, 2006; Nur and Byerlee, 1971; Terzaghi, 1943), adopting one of these forms introduces the same postulate. However, many natural and engineered geomaterials possess two-scale porous structures—which exhibit a bimodal pore size distribution that can be viewed as a composite of two coexisting pore regions of different scales. Well-known examples include aggregated soils and fractured rocks. In aggregated soils the two pore regions are the inter-aggregate pores and the intra-aggregate pores, whereas in fractured rocks they are the fractures and the matrix pores. These two-scale porous geomaterials are widespread in a variety of subsurface systems, from hillside slopes and geotechnical structures to hydrocarbon reservoirs and carbon storage sites (Geiger et al., 2013; Gens et al., 2011; Laine-Kaulio et al., 2014; White et al., 2014). Fluid flow in these two-scale porous media is significantly different from its counterpart in single-scale media, primarily because the two pore regions, which

1

have pores of highly contrasting sizes, play very different roles in hydrological processes. For example, in aggregated soils, the inter-aggregate pores form the primary pathways for rapid infiltration, but fluids can also slowly migrate through the aggregates and their contact surfaces (Gerke, 2006; Jarvis, 2007; Šimünek et al., 2006; Yang et al., 2014). As shown in Fig. 1.1, fractures in porous media govern preferential flow patterns, but the matrix pores can trap and transport a major portion of the pore fluids (Barenblatt et al., 1960; Moench, 1984; Warren and Root, 1963). The resulting fluid flow is beyond the modeling capabilities of a classical approach that combines Darcy’s law and the mass balance equation.

Figure 1.1: Fluid flow in a single-scale porous medium (left) and a two-scale, fractured porous medium (right). Darker regions are more saturated. In the left-hand figure, the fluid shows a straight wetting front that can be simulated by a classical approach. In the figure on the right, preferential flow takes place while much of the fluid is trapped in the upper region. This type of fluid flow cannot be captured by a classical approach.

Two-scale internal structures also impact the mechanical behavior of porous materials. As is well known in the geomechanics community, aggregated structures have profound effects on various mechanical properties from strength to compressibility (Burland, 1990; Leroueil and Vaughan, 1990; Mitchell and Soga, 2005). Fractures in the solid matrix also control the overall deformation responses, as evidenced by a number of studies (e.g., Min and Jing, 2003; Tjioe and Borja, 2015, 2016). Mechanical models originally developed for single-scale porous materials are inappropriate for these two-scale materials, because their internal structures evolve by stress conditions. For example, aggregated particles become progressively broken with the increase of compressive stress. Reliable descriptions

2

of two-scale porous materials thus require explicit consideration of the internal structures and of their evolution during the course of loading. Motivated by these distinctive hydrological and mechanical behaviors, a large number of studies have proposed methods for modeling either fluid flow or deformation in two-scale porous materials. Superficially, one can model coupled hydromechanical behavior of two-scale porous materials by combining an existing method for fluid flow with another for deformation. This approach is unreliable, however, because the existing methods for fluid flow are incompatible with current methods for deformation unless the deformation is infinitesimal. For example, while flow models typically require separate descriptions of volume changes in the two pore regions, mechanical models only compute the overall deformation of the entire pore space. We also need to revisit the effective stress because fluid flow in these materials takes place at two scales. These limitations have hindered hydromechanical modeling of two-scale porous materials. This thesis presents three major developments that enable more reliable, efficient modeling of coupled hydromechanical processes in two-scale porous geomaterials. They are: (a) a mathematical framework for thermodynamically consistent hydromechanical modeling of two-scale porous media; (b) a constitutive framework that reliably captures hydromechanical processes under finite deformations; and (c) a computational framework that can solve boundary-value problems efficiently. In this work we pursue continuum mechanics approaches, focusing on the macroscopic level at which the properties of two pore regions can be statistically homogenized in an elementary volume. Specifically, we build this work on the concept of double porosity—a widely used continuum framework for fluid flow in two-scale porous media—which is described in the following section.

1.2

background

1.2.1 Double porosity This work begins by adopting the double-porosity model, a conceptual framework for continuum modeling of fluid flow in two-scale porous media. In essence, this framework conceptualizes a two-scale porous medium as a continuum that possesses two overlapped pore regions of contrasting properties. The two pore regions, or pore scales, interact with each other by means of fluid mass transfer. Depending on the specific materials of interest, the two pore regions have been called by different names in the literature. For terminological consistency, throughout this thesis we shall use macropores to refer to the larger, more permeable pore scale (e.g., inter-aggregate pores or fractures), and micropores to refer to the smaller,

3

less permeable pore scale (e.g., intra-aggregate pores or matrix pores). Figure 1.2 schematically shows a porous medium with double porosity.

= Porous medium with double porosity

+ Solid matrix and micropores

Macropores

Figure 1.2: Schematic representation of a porous medium with double porosity.

Since first proposed by Barenblatt et al. (1960) and Warren and Root (1963), the concept of double porosity has been used extensively in subsurface hydrology and reservoir engineering for a variety of materials ranging from fractured rocks to aggregated soils (e.g., Berkowitz et al., 1988; Carneiro, 2009; Geiger et al., 2013; Lewandowska et al., 2005, 2008; Moench, 1984; Ngien et al., 2012; Trottier et al., 2014; Zimmerman et al., 1993). A key advantage of the double-porosity model is its capability to handle extremely localized and concentrated flows with relatively simple and homogenized parameters that could otherwise be very complicated to describe, let alone, quantify. Broadly speaking, there are two types of double-porosity models in the literature. The first is the dual-porosity model, in which the pore fluids can flow through the macropores but are trapped within the micropores. This model is widely used for describing fluid flow in fractured rocks (Barenblatt et al., 1960; Warren and Root, 1963), and it is essentially identical to the mobile-immobile model that appears in the groundwater literature (Geiger et al., 2013). However, experimental evidence suggests that in some materials, such as aggregated soils, the pore fluids could also migrate within the micropores (Carminati et al., 2007; Gerke, 2006; Jarvis, 2007; Šimünek et al., 2007). For this type of material one can use the dualpermeability model, in which the pore fluid can migrate through the macropores as well as within the micropores. In this thesis, we adopt the double-porosity model in the context of dual permeability. The concept of double porosity is naturally compatible with mixture theory which is a common basis for hydromechanical frameworks. Due to this compatibility, several studies have employed the double-porosity model for coupled hydromechanical modeling purposes (e.g., Callari and Federico, 2000; Elsworth and Bai, 1992; Khalili and Selvadurai, 2003; Koliji et al., 2010a; Lewis and Ghafouri,

4

1997; Zhang and Roegiers, 2005). However, these efforts have been significantly limited with respect to both theoretical and computational aspects. Overcoming these limitations is a recurring theme in this thesis. 1.2.2 Mechanical behavior of aggregated geomaterials One of these limitations of the theoretical and computational aspects of doubleporosity modeling is that no existing constitutive model is able to quantify the evolution of two-scale pore structure under moderate to large deformations. The need for accurate quantification of these two-scale pore evolutions motivates the development of a novel constitutive model in Chapter 3. Because developing a constitutive model requires a sufficient understanding of physical behavior, we focus on aggregated geomaterials, which have been the subject of a vast number of experimental studies (e.g., Burland, 1990; Calabresi and Callisto, 1998; Callisto and Rampello, 2004; Koliji et al., 2008b, 2010a; Leroueil and Vaughan, 1990). The findings of these studies are briefly reviewed in the remainder of this section. Note that these studies sometimes refer to aggregated geomaterials as structured or bonded geomaterials. Since aggregated structures of soils disappear once they are reconstituted, previous studies have investigated how an aggregated soil behaves differently before and after reconstituting it in the laboratory. Reconstituted soils exhibit reproducible responses—which have been extensively characterized by experiments (e.g., Allman and Atkinson, 1992; Choo et al., 2011, 2013a; Cunningham et al., 2003; Jung et al., 2013)—so they can serve as a useful reference in delineating the effect of particle aggregation on mechanical responses. It is worth noting that natural or compacted soils often exhibit a bimodal pore size distribution, whereas reconstituted soils comprised of the same particles exhibit a unimodal pore size distribution (Burton et al., 2015). Figure 1.3 illustrates typical responses of a soil in aggregated and reconstituted states. Compared with its reconstituted counterpart, an aggregated soil exhibits two major differences: (a) larger specific volume and (b) higher compressibility in plastic deformations. Also notable is that, at an elevated stress level, the behavior of an aggregated soil converges with that of its reconstituted counterpart. This converging trend is attributed to the progressive breakage of aggregated structures with increasing stress. These patterns of aggregated and reconstituted soils have been commonly observed in a wide range of natural and engineered geomaterials (Burland, 1990; Callisto and Rampello, 2004; Leroueil and Vaughan, 1990). Measuring separate deformations of macropores and micropores in aggregated soils, Koliji et al. (2008a) shed light on the evolution of two-scale pore struc-

5

Specific volume

Aggregated Reconstituted Confining pressure (log scale) Figure 1.3: Schematic representation of compression curves of a soil in aggregated and reconstituted states.

tures by mechanical loading. Their results show that while the macropores are virtually unaltered before yielding, in the plastic regime dominant deformations take place in the macropores. In other words, while the micropore deformation is responsible for the elastic deformation, it plays a minor role in the plastic compaction of aggregated soils which are much more significant than that of their reconstituted counterparts. They also found that the pore structure of reconstituted soils, which exhibit a single pore scale, resembles the intra-aggregate pore structure of aggregated soils. This implies that an individual aggregate can be analogized to a reconstituted soil. These findings underlie the development of the constitutive framework presented in this thesis.

1.3

overview

The remaining four chapters of this thesis include three chapters that describe theoretical, constitutive, and computational developments, respectively, and a final chapter that concludes the work. The body chapters—Chapters 2, 3, and 4— are adapted from published articles in refereed journals (Borja and Choo, 2016; Choo and Borja, 2015; Choo et al., 2016). Most parts of these three chapters are reproduced directly from the articles, except for the introduction sections. The important background for the entire thesis has been consolidated in Section 1.2, so the introduction sections of the body chapters can concisely highlight specific purposes of those chapters. The subsequent chapters can be outlined as follows:

6

• In Chapter 2, we develop a thermodynamically consistent framework for coupled hydromechanical modeling of variably saturated porous media with double porosity. With an explicit treatment of the two pore scales, conservation laws are formulated incorporating an effective stress tensor that is energy-conjugate to the rate of deformation tensor of the solid matrix. We then establish a constitutive framework for multiscale, multiphysics processes in double-porosity materials on the basis of energy-conjugate pairs identified in the first law of thermodynamics. Numerical simulations of laboratory- and field-scale problems are presented to demonstrate the impact of double-porosity flow on the resulting hydromechanical responses. • Chapter 3 introduces a novel constitutive framework for fluid-infiltrated materials with evolving double-porosity structure. We show that the thermodynamically consistent effective stress is equivalent to the mean of the individual effective stresses in the micropores and macropores weighted according to the pore fractions. This partitioning of the effective stress into two individual effective stresses allows fluid pressure dissipation at the macropores and micropores to be considered separately, with important implications for individual characterization of the hardening responses at the two pore scales. Experimental data suggest that the constitutive framework captures the laboratory responses of aggregated soils more accurately than those previously reported in the literature. Numerical simulations of boundary-value problems reveal the capability of the framework to capture the effect of secondary compression as the micropores discharge fluids into the macropores. • In Chapter 4, we present a stabilized finite element framework for coupled deformation and fluid flow in double-porosity media. Three-field mixed finite elements that interpolate the solid displacement and pore pressures in the macropores and micropores are used for this purpose. In the limit of undrained deformation, the incompressibility constraint causes unstable behavior in the form of spurious pressure oscillation at the two pore scales. To circumvent this instability, we develop a variant of the polynomial pressure projection technique for a twofold saddle point problem. The proposed stabilization allows the use of equal-order (linear) interpolations of the displacement and two pore pressure variables throughout the entire range of drainage condition. • Chapter 5 summarizes the principal contributions of this thesis, and discusses recommendations for future work.

7

8

2

M AT H E M AT I C A L M O D E L I N G F R A M E W O R K

This chapter is adapted from: Choo, J., White, J. A., and Borja, R. I. (2016). “Hydromechanical modeling of unsaturated flow in double porosity media.” International Journal of Geomechanics, D4016002.

2.1

introduction

As explained in Section 1.2, several previous studies have adopted the doubleporosity framework for coupling fluid flow and deformation in two-scale porous media (e.g., Callari and Federico, 2000; Elsworth and Bai, 1992; Khalili and Selvadurai, 2003; Koliji et al., 2010a; Lewis and Ghafouri, 1997; Zhang and Roegiers, 2005). However, there has been significant disagreement in the theoretical formulation of the problem, particularly with the definition of effective stress. The purpose of this chapter is to develop a rigorous and consistent formulation for coupled hydromechanical modeling of unsaturated porous media with double porosity. There are two major challenges in developing a theoretical framework for hydromechanical modeling of double-porosity media. The first is defining the effective stress to accommodate two pore pressure degrees of freedom emanating from the two scales. The second is developing a constitutive framework that captures all the important poromechanical processes at the two pore scales. In this work, we employ the first law of thermodynamics to address these two challenges, motivated by the work by Borja (2004, 2006) and Song and Borja (2014b) for unsaturated porous media with single porosity. Energy-conjugate pairs in the first law of thermodynamics suggest groupings of variables that must be linked via constitutive relations. This is because the same energy-conjugate pairs will appear in the dissipation inequality, and the standard Coleman–Noll procedure will yield that all variables involved in a nondissipative energy-conjugate pair should be related by a constitutive law to ensure non-negative entropy production, i.e., the second law of thermodynamics. As such, identifying energy-conjugate pairs has proven useful to gain insight into constitutive modeling of multiphase porous media. For instance, energy-conjugacy between the effective stress tensor and the rate of deformation tensor of the solid matrix suggests that a mechanical constitutive law must be established in terms of the effective stress, rather than the total stress, or the net stress, or some other measures of stress. A thermodynamically consistent effective stress could lead to

9

many desirable results, including the fact that when reckoned with respect to this stress the position of the critical state line is demonstrably unique (Nuth and Laloui, 2008). For double-porosity media, Borja and Koliji (2009) derived an effective stress tensor that is energy-conjugate to the rate of deformation tensor for the solid matrix with double porosity. By inspecting how the variables are paired in the expression for the rate of change of internal energy, they also identified other constitutive relations for double-porosity media. The constitutive relations of interest in this work concern not only the mechanical laws but also those emanating from fluid flow and coupled hydromechanical processes occurring at the two pore scales. In this chapter, we develop a thermodynamically consistent framework for hydromechanical modeling of double-porosity media where the pores are assumed to be filled with water and air. Thermodynamic consistency is achieved by making use of the aforementioned approaches for the effective stress and multiphysical constitutive relations at the two pore scales. To our knowledge, this is the first time that a poromechanical framework for double-porosity media has been developed from direct application of the principles of thermodynamics. Essential ingredients for coupling unsaturated flow with deformation of the solid matrix having two pore scales are presented in this chapter. Without loss of generality, the theory is developed under the assumption of infinitesimal deformation and the treatment of finite deformation effects is deferred for Chapter 3. This chapter is organized as follows. In Section 2.2, we present conservation laws for double-porosity media incorporating a thermodynamically consistent effective stress tensor. A constitutive framework is developed in Section 2.3 based on energy-conjugate pairs identified in the first law of thermodynamics. In Section 2.4, we develop a three-field mixed finite element formulation for the numerical solution of fully coupled flow and deformation problems. Numerical examples are presented in Section 2.5 to demonstrate the impact of double porosity on the hydromechanical responses of a laboratory-scale specimen and a field-scale slope problem.

2.2

conservation laws

In what follows, we formulate the balance of mass and balance of linear momentum for double-porosity media by distinguishing between the macropore and micropore pressures and by employing a thermodynamically consistent effective stress. Also presented is an expression for the rate of change of internal energy that will prove to be useful for developing other relevant constitutive relations.

10

2.2.1 Balance of mass Consider a mixture of solid, water, and air, in which the solid forms a matrix with two dominant pore scales. The volume fractions are defined in the usual way, φs = dVs /dV ,

φiα = dViα /dV ,

i = M, m ,

α = a, w .

(2.1)

Here, index s pertains to the solid phase, index i denotes either the macropore M or the micropore m, and index α pertains to the pore fluid phase that can either be water w or air a. Indices are used as superscripts when referring to partial properties of a constituent (e.g., volume fractions), and as subscripts when referring to intrinsic properties (e.g., constituent volumes). Volume fractions satisfy the closure relation φs +

∑ ∑

φiα = 1 .

(2.2)

i = M,m α=w,a

Partial mass densities (e.g., ρs ) can be expressed in terms of the intrinsic mass densities (e.g., ρs ) and volume fractions via relations of the form ρs = φs ρs ,

ρiα = φiα ρα ,

i = M, m ,

α = a, w .

(2.3)

The total mass density of the mixture ρ is obtained by summing all the partial mass densities, ρ = ρs +

∑ ∑

ρiα .

(2.4)

i = M,m α=w,a

Pore fractions define the portions of the total pore volume occupied by the two dominant pore regions, ψi =

φi , 1 − φs

i = M, m ,

(2.5)

where φi = φiw + φia . This yields the closure condition ψ M + ψm = 1 .

(2.6)

Accordingly, the local water saturation at pore scale i is given by the relation Si =

φiw φiw = , φi ψ i (1 − φ s )

i = M, m ,

11

(2.7)

which can be thought of as the local degree of saturation at each of the two pore scales. The global (or overall) degree of saturation can then be calculated as S=

φiw = ∑ ψi Si . s 1 − φ i = M,m i = M,m

∑

(2.8)

Without loss of generality and for simplicity, we shall assume in the following developments that water is incompressible and the pore air pressure is assumed to be zero (i.e., atmospheric). These assumptions are realistic for near-surface soil formation applications, and can easily be rectified with the introduction of fluid compressibility and the pore air variables for more general multiphysics applications. The balance of mass for pore water in the macropores and micropores, accommodating for the mass transfer term between the two pore scales, can be expressed as (with a minor correction from Borja and Koliji, 2009) 1 φi S˙ i + Si ψi B ∇ · v + ∇ · qi = ci , ρi

i = M, m ,

(2.9)

where the superimposed dot denotes a material time derivative following the solid motion, B is the Biot coefficient, v is the solid velocity, qi = φiw v˜i are the seepage (Darcy) velocities of pore water in the macropores and micropores, and v˜i = vi − v

is the relative velocity of pore water at pore scale i. The water mass transfer terms are represented by ci appearing on the right-hand side, which satisfy the closure condition c M + cm = 0. 2.2.2 Balance of linear momentum

The balance of linear momentum can be expressed in terms of an effective stress tensor derived by Borja and Koliji (2009), which takes the form ¯ , σ 0 = σ + B p1

(2.10)

where σ and σ 0 are the total and effective Cauchy stress tensor, 1 is the secondorder identity tensor, and p¯ is the mean pore pressure given by p¯ =

∑

i = M,m

h

i ψi Si pi + ψi (1 − Si ) pia ,

(2.11)

in which pi and pia are the pore water pressure and pore air pressure at pore scale i (the index w for the pore water pressure is dropped for simplicity).

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For the case of passive air pressure (i.e., zero pore air pressure), the preceding expression simplifies to the form p¯ =

∑

i = M,m

ψi Si pi .

(2.12)

In words, the effective stress tensor σ 0 can be obtained from taking the mean pore pressure p¯ that is weighted according to the pore fractions and local degrees of saturation of the two pore scales. Under special conditions, this effective stress reduces to some well-known effective stresses. For example, for single-porosity media (i.e., either ψ M = 1 or ψm = 1) this effective stress reduces to the form for single-porosity continua developed by Borja (2006). If the pores are completely saturated with water, it specializes to the effective stress tensor investigated by Nur and Byerlee (1971). Finally, if the Biot coefficient B = 1, then one recovers the Terzaghi effective stress Terzaghi (1943). The balance of linear momentum for the entire mixture now takes the form Borja and Koliji (2009) ¯ ) + ρg = c¯ , ∇ · (σ 0 − B p1

(2.13)

where g is the gravity acceleration vector, and c¯ is the momentum produced by the mass transfer of water between the two pore scales, which takes the form c¯ =

∑

i = M,m

ci v˜i .

(2.14)

Here, the closure condition c M + cm = 0 causes the solid velocity terms to cancel out from its original form given in Borja and Koliji (2009). 2.2.3 Balance of internal energy An expression for the rate of change of internal energy for a solid-fluid mixture with double porosity was presented in Eq. (76) of Borja and Koliji (2009). The following equation is a simplified version of this expression accounting for mechanical powers only (signs of some terms are corrected from the original expression): ρe˙ = σ 0 : d +

∑

i = M,m

−

∑

i = M,m

ci

v˜i · ∇(φiw ) pi −

1 pi − v˜i · v˜i 2

∑

i = M,m

ψi (1 − φs )si S˙ i

− (1 − φs )π ψ˙ m ,

13

(2.15)

where e˙ is the rate of change of internal energy per unit total mass of the mixture, d is the rate of deformation tensor of the solid matrix, si is the local suction at pore scale i defined as si = − pi , and π is the difference between the mean pressures at the macropores and micropores π = S M p M − Sm pm .

To highlight the pairings of the constitutive variables implied by this preceding

expression, we rewrite Eq. (2.15) in a more abstract form as ρe˙ = hσ 0 , di +

−

∑

i = M,m

∑

i = M,m

hv˜i , φiw , pi i −

∑

i = M,m

hψi , (1 − φs ), si , S˙ i i

hc , pi , v˜i i − h(1 − φs ), π, ψ˙ m i , i

(2.16)

where the symbol h◦, . . . , ◦i denotes energy-conjugacy.

This expression for the rate of change of internal energy yields five groups

of constitutive variables suggesting that five constitutive relations must be established relating these variables. Three of these constitutive relations (the terms with the summation) are defined separately at each pore scale. Specific constitutive models for these energy-conjugate pairs are proposed in the following section.

2.3

constitutive framework

Guided by the pairings identified in Eq. (2.16), constitutive laws are now introduced to close the boundary value problem. The first pair on the right-hand side of this equation suggests a constitutive law for the solid matrix of the form σ˙ 0 = C : e˙ ,

(2.17)

where C is a fourth-order tangent tensor that may be obtained, for example, from an elastoplastic constitutive law, and e is the infinitesimal strain tensor of which rate e˙ is equivalent to the rate of deformation tensor d for infinitesimal deformation. The reduced dissipation inequality formulated in Eq. (83) of Borja and Koliji (2009) suggests that a yield function of an unsaturated double-porosity material may depend on the suctions at the macropores and micropores. However, due to the absence of firm experimental evidence, incorporating the effect of two-scale suctions to elastoplasticity, similar to what was done for single-porosity modeling (e.g., Borja, 2004; Borja et al., 2013a; Liu and Muraleetharan, 2012a,b; Song and Borja, 2014b), will not be pursued here. The second pair suggests a constitutive relation among the relative flow vector, volume fraction, and pore water pressure at each pore scale. Noting that qi = φiw v˜i , this is equivalent to a relation between the seepage velocity and the pore pressure at pore scale i. Under the assumptions of laminar flow and isotropic

14

media, we postulate that Darcy’s law is valid for the two pore scales. The seepage velocities of pore water in the macropores and micropores are then given by qiw = −kri

ki · (∇ pi − ρw g ) , µw

i = M, m ,

(2.18)

where kri and k i are the relative and intrinsic permeabilities at pore scale i, and µw is the dynamic viscosity of water. The relative permeability depends on the local degree of saturation and is elaborated further below. Note that porosity may also vary with finite deformation (see Song and Borja, 2014a,b), although it is considered constant in the present infinitesimal formulation. The third pair in the internal energy equation suggests a relation among the local matric suction si and the local degree of saturation Si for a given volume fraction. In this work, we use the well-known van Genuchten equation (van Genuchten, 1980) at each pore scale, which is written as h i − mi Si (si ) = S1i + (S2i − S1i ) 1 + (si /sαi )ni ,

i = M, m,

(2.19)

where S1i and S2i are the residual water saturation and maximum water saturation at pore scale i, respectively; sαi are scaling suctions; and ni and mi are parameters that determine the shape of the water retention curves. Note that the fitting parameter mi is related to the parameter ni according to the relation mi = 1 − 1/ni .

(2.20)

The relative permeabilities of the water phase in the macropores and micropores are given, respectively, by h i2 kri = θ 1/2 1 − (1 − θi1/mi )mi ,

θi =

Si − S1i , S2i − S1i

i = M, m .

(2.21)

Although water retention curves can be developed quite easily for the entire porous material, they are quite challenging to develop for each of the two pore scales. To develop water retention curves appropriate for the macropores and micropores, one can resort to an indirect approach as used in some previous studies (Buscarnera and Einav, 2012; Casini et al., 2012; Durner, 1994). The idea is to superimpose two water retention curves, one for the micropores and the other for the macropores, and calibrate the overall curve with the global water retention data, recalling that the global saturation can be calculated from the weighted average of the local suctions, as shown in Eq. (2.8). Another approach is to ignore suction in the macropores because of their much lower air entry value, as done in Koliji et al. (2010a). However, when the suction in the macropores is not

15

negligible, such as what the experimental results in Romero et al. (1999, 2011) suggest, the latter approach may lead to oversimplification. Last, we note that in some cases the changes in the pore volume could have a significant impact on the water retention curves of double-porosity media (see Della Vecchia et al., 2015; Romero et al., 2011 for some experimental evidence) as well as of single-porosity media (Gallipoli et al., 2003b; Miller et al., 2008). In the finite deformation range, this relationship must be considered whenever experimental data permit. The fourth pair involves mass transfer terms ci that are new to the formulation, i.e., relative to the one presented in Borja and Koliji (2009). For obvious reason, these terms do not enter into the formulation for a single-porosity problem. Note that the terms p and v˜ have already been related by Darcy’s law; thus, we can eliminate the relative velocity term and simply link c and p by an appropriate constitutive law. Also, because of the closure condition c M = −cm , we can interpret

the constitutive law as relating the diffusive mass transfer term to the pressure difference at the two pore scales. In fact, a vast number of studies have advanced constitutive relations of this form (see e.g., Dykhuizen, 1987, 1990; Gerke and van Genuchten, 1993b; Haggerty and Gorelick, 1995; Köhne et al., 2004; Sarma and Aziz, 2006; Warren and Root, 1963). Among them, a first-order mass transfer equation of the following form is common: cM =

α¯ ( pm − p M ) , µw

cm =

α¯ ( p M − pm ) , µw

(2.22)

where α¯ is a dimensionless parameter that depends on the characteristics of the interface between the macropores and micropores, such as permeability, spacing, and shape. Several specific forms of the parameter α¯ have been suggested based on theoretical and experimental results (e.g., Dykhuizen, 1990; Gerke and van Genuchten, 1993b; Warren and Root, 1963). In this work we use an equation proposed by Gerke and van Genuchten (1993b), given by β α¯ = k¯ 2 γ , a

(2.23)

where k¯ is the interface permeability, a is the characteristic length of the macropores spacing, β is a dimensionless coefficient that accounts for macropore geometry, and γ is a dimensionless scaling coefficient that was shown to be 0.4 to fit experimental results. Similar to the constitutive relations developed for the other groups, which are not unique, one can explore other possible constitutive laws for the diffusive mass transfer term that better fit the experimental data. The final pair in the expression for the rate of change of internal energy suggests a constitutive relationship between the mean pore pressure difference and the macropore volume fraction. As noted in Borja and Koliji (2009), this is an open

16

question particularly in the finite deformation range, where the macropore volume fraction may be changing significantly. For the infinitesimal theory where the volume fractions may not be changing significantly, i.e., ψ˙ m ≈ 0, it is reasonable to ignore this yet undetermined constitutive relation.

2.4

mixed finite element formulation

This section formulates a three-field mixed finite element solution to an initial boundary value problem of coupled solid deformation and fluid diffusion with double porosity. We begin by presenting strong and weak forms based upon the theoretical developments described in the previous sections. Then, we develop a three-field matrix form for solving initial boundary value problems. 2.4.1 Strong form Consider a closed domain denoted by Ω = Ω ∪ Γ, where Ω is an open domain and

Γ is the boundary of Ω. The boundary Γ is assumed to be suitably decomposed

as follows: displacement and traction boundaries for solid, Γu and Γt , respectively; pressure and flux boundaries for water in the macropores, Γ p M and Γq M , respectively; and pressure and flux boundaries for water in the micropores, Γ pm and Γqm , respectively. The boundaries are subject to the following set relations: Γ = Γu ∪ Γt

Γ = Γ p M ∪ Γq M

Γ = Γ pm ∪ Γ qm

and ∅ = Γu ∩ Γt ,

(2.24)

and ∅ = Γ pm ∩ Γqm ,

(2.26)

and ∅ = Γ p M ∩ Γq M ,

(2.25)

where ∅ is the null set and the overline denotes a closure. The strong form of the initial boundary value problem can be stated as follows. Given u, ˆ tˆ, pˆ M , qˆ M , pˆ m , and qˆm , find u, p M , and pm such that ¯ ) + ρg = c¯ , ∇ · (σ 0 − B p1

(2.27)

and 1 φi S˙ i + Si ψi B ∇ · v + ∇ · qi = ci , ρi

i = M, m ,

(2.28)

subject to boundary conditions u = uˆ

on

Γu

(2.29)

17

n · σ = tˆ

on

Γt

(2.30)

p M = pˆ M

on

Γ pM

(2.31)

−n · q M = qˆ M

on

Γq M

(2.32)

pm = pˆ m

on

Γ pm

(2.33)

−n · qm = qˆm

on

Γ qm ,

(2.34)

and initial conditions u = u0 ,

p M = p M0 ,

(2.35)

pm = pm0

for all ( x, t) ∈ (Ω × t = 0). Here, t is the traction vector and symbols with the hat denote the prescribed boundary conditions. 2.4.2 Weak form To develop the weak form of the problem, we define the spaces of trial functions as

Su = {u | u ∈ H 1 , u = uˆ on Γu } ,

(2.36)

S p M = { p M | p M ∈ H 1 , p M = pˆ M on Γ p M } ,

(2.37)

1

S pm = { pm | pm ∈ H , pm = pˆ m on Γ pm } ,

(2.38)

where H 1 denotes a Sobolev space of order one. Imposing homogeneous conditions on the Dirichlet boundaries, we define corresponding spaces of weighting functions as

Vu = {η | η ∈ H 1 , η = 0 on Γu } ,

(2.39)

1

V p M = {ω M | ω M ∈ H , ω M = 0 on Γ p M } ,

(2.40)

V pm = {ωm | ωm ∈ H 1 , ωm = 0 on Γ pm } .

(2.41)

The weak form of the problem may be stated as follows. Find {u, p M , pm } ∈

Su × S p M × S pm such that for all {η, ω M , ωm } ∈ Vu × V p M × V pm the following equations are satisfied: (a) Balance of linear momentum Z

s

Ω

0

¯ ) dΩ = ∇ η : (σ − B p1

Z Ω

η · (ρg + c¯) dΩ +

18

Z Γt

η · tˆ dΓ .

(2.42)

(b) Balance of mass in the macropores Z Ω

ω M φ M S˙ M dΩ +

Z Ω

ω M S M ψ M B ∇ · v dΩ −

=−

1 ρi

Z Ω

Z Ω

ω M c M dΩ +

∇ ω M · q M dΩ

Z Γq M

ω M qˆ M dΓ .

(2.43)

(c) Balance of mass in the micropores Z Ω

ωm φm S˙ m dΩ +

Z Ω

ωm Sm ψm B ∇ · v dΩ −

=−

1 ρi

Z

Z Ω

ωm cm dΩ +

Ω

∇ ωm · qm dΩ

Z Γqm

ωm qˆm dΓ .

(2.44)

2.4.3 Time-integrated form For the time integration of the variational equations for the mass balance, the first-order accurate, unconditionally stable backward Euler scheme is used to take advantage of its high-frequency numerical damping that suppresses spurious oscillations in time. It is noted that this feature is particularly advantageous for double-porosity problems in which the permeability contrast between the two pore scales could possibly be very significant. In other words, a time step that is acceptable for diffusion in the macropore region may not be suitable for diffusion in the micropore region, resulting in pressure oscillation with respect time unless the backward Euler scheme is used. At time tn+1 , the time-integrated variational equation for the balance of mass for the macropores reads

R pM =

Z Ω

ω M φ M (S M − SnM ) dΩ +

Z

∆t − ∆t ∇ ω M · q M dΩ − ρw Ω Z

Ω

ω M Sm ψ M B ∇ · (u − un ) dΩ

Z

M

Ω

ω M c dΩ − ∆t

Z Γq M

ω M qˆ M dΓ (2.45)

= 0. For the micropores, the equivalent expression reads

R pm =

Z Ω

ωm φm (Sm − Snm ) dΩ +

− ∆t

Z Ω

∇ ωm · qm dΩ −

Z

∆t ρw

Ω

ωm Sm ψm B ∇ · (u − un ) dΩ

Z Ω

ωm cm dΩ − ∆t

Z Γqm

ωm qˆm dΓ

(2.46)

= 0, where index n denotes quantities at time tn , and ∆t is the time increment. For brevity, the index “n + 1” has been omitted for terms pertaining to time tn+1 .

19

h have been defined after multiplying the original timeNote that the residuals R(·)

integrated equations by the time increment ∆t. 2.4.4 Matrix form

For spatial discretization we perform the standard Galerkin approximation and interpolate the displacement and two pore pressure fields using a three-field mixed finite element. It is worth noting that the interpolation functions for displacement and pore pressures are not arbitrary, and only a limited number of possible combinations work in practice. In the limit of fully saturated and undrained deformation, the interpolation functions must satisfy the inf–sup condition for twofold saddle point problems Howell and Walkington (2011); otherwise, the pore pressure solution will exhibit spurious oscillations in space similar to those studied in Murad and Loula (1994) and White and Borja (2008). With the standard shape function matrices, we represent the trial functions as uh = N u d + Nˆ u dˆ

     

phM = N p p M + Nˆ p pˆ M  ,     h p p ˆ pm = N pm + N pˆ m

(2.47)

where superscript h denotes a spatially discretized function; N is the shape function matrix with superscripts indicating the fields being interpolated; d, p M , pm are the nodal displacement vector, nodal macropore pressure vector, and nodal micropore pressure vector, respectively; and the hats pertain to contributions from the essential boundary conditions. The gradient and divergence of the primary variables are interpolated as follows:

∇ uh = Bd + Bˆ dˆ ∇ · uh = bd + bˆ dˆ

         

  ∇ phM = Ep M + Eˆ pˆ M       h ∇ pm = Epm + Eˆ pˆ m 

.

(2.48)

Following the standard Galerkin approximation, we use the same shape functions are used for the variations. The matrix form of the problem is developed by inserting the finite elements approximations into the Galerkin form of the mixed variational equations (2.42), (2.45), and (2.46). The residual equations take the vector forms as follows:

20

(a) Balance of linear momentum:

Ruh

=−

Z

0

Ω

Z

h

( B σ − b B p¯ ) dΩ + T

T

u T

Ω

( N ) (ρg + c¯) dΩ +

Z Γt

( N u )T tˆ dΓ

→ 0.

(2.49)

(b) Balance of mass in the macropores:

Rhp M =

Z Ω

( N p )T φ M (S M − SnM ) dΩ +

− ∆t

Z Ω

E · qhM dΩ −

∆t ρw

Z Ω

Z Ω

( N p )T S M ψ M B ∇ · (uh − unh ) dΩ

( N p )T c M dΩ − ∆t

Z Γq M

( N p )T qˆ M dΓ

→ 0.

(2.50)

(c) Balance of mass in the micropores:

Rhpm

=

Z

p T m

Ω

( N ) φ (S

− ∆t

Z Ω

E

h · qm

m

− Snm ) dΩ

∆t dΩ − ρw

Z

Z Ω

( N p )T Sm ψm B ∇ · (uh − unh ) dΩ p T m

Ω

( N ) c dΩ − ∆t

→ 0.

Z Γqm

( N p )T qˆm dΓ

(2.51)

Here, the equalities to zero in the previous variational equations are replaced with rightward arrows tending to zero to imply that a Newton–Raphson iteration h to zero (for nonlinear scheme may be involved in driving the residual vectors R(·)

problems).

The residuals defined above are generally nonlinear with respect to the primary variables d, p M , and pm due to material and/or geometric nonlinearities. We thus employ a Newton–Raphson iteration to solve the problem. The linearized problem is defined by a Jacobian matrix with a 3 × 3 block structure of the form   ∆d A B C  1 1       B2 D E1  ∆pM      ∆p C2 E2 F m 

        

=−

  h    Ru

Rhp M     Rh pm

    

,

(2.52)

   

where ∆d, ∆pM , and ∆pm are the relevant search directions. The submatrices of the Jacobian matrix are given by A=−

Z Ω

BT Ck B dΩ ,

(2.53)

21

B1 =

"

Z

M

b B

S ψ

T

Ω

M

∂S M + h phM ∂p M

! u T

+ (N )

∂ρ ∂c¯h g + ∂phM ∂phM

!# N p dΩ , (2.54)

C1 = B2 =

Z Ω

Z Ω

bT B S m ψ m +

∂Sm h ∂pm

h pm

+ ( N u )T

∂ρ g+ h h ∂pm ∂pm

( N p )T BS M ψ M b dΩ ,

N p dΩ ,

(2.55)

( N p )T

Z

( N p )T BSm ψm b dΩ , Ω ! Z ∆t ∂cm p T E2 = − (N ) N p dΩ , ρw Ω ∂phM Z i ∂Sm h F = ( N p )T h φm + ψm B ∇ · (uh − unh ) N p dΩ ∂pm Ω Z ∂κm T h p E κm E + h (∇ pm − ρw g ) N dΩ + ∆t ∂p Ω m m Z ∆t ∂c − ( N p )T N p dΩ , h ρw Ω ∂pm

C2 =

(2.56)

i ∂S M h M M h h φ + ψ B ∇ · ( u − u ) N p dΩ n ∂phM Ω " # Z ∂κ M T h p + ∆t E κ M E + h (∇ p M − ρw g ) N dΩ ∂p M Ω ! Z M ∂c ∆t ( N p )T − N p dΩ , ρw Ω ∂phM M Z ∆t ∂c N p dΩ , ( N p )T E1 = − h ρw Ω ∂pm D=

Z

∂c¯h

(2.57) (2.58) (2.59) (2.60)

(2.61)

where Ck is the consistent stress-strain matrix (k denotes an iteration counter) and κ(·) is the matrix of permeability coefficients kr(·) k (·) /µw . Note that the Jacobian matrix is generally nonsymmetric. Numerous strategies for the solution of the linearized system are available. Direct methods for nonsymmetric matrices could be employed, but for large systems they could be prohibitively expensive because of the time-consuming factorization step and large memory requirements involved in the calculations. Here we employ an iterative solver based on Krylov subspace methods. To make use of this iterative solver, the block-preconditioning technique proposed by White and Borja (2011) has been extended to accommodate the 3 × 3 block-partitioned structure of the

Jacobian matrix.

22

2.5

numerical examples

This section presents two numerical examples that demonstrate the impact of double porosity on the ensuing hydromechanical responses of geotechnical structures. The examples compare the structural responses predicted by double-porosity simulations with those obtained by equivalent homogenized single-porosity simulations. The first example involves the development of a persistent shear band in a laboratory-scale plane strain compression of an aggregated soil with imposed heterogeneity in the pore fractions. The second example deals with the impact of double porosity on fluid flow and deformation of a field-scale unsaturated slope subjected to rainfall infiltration. 2.5.1 Persistent shear band The first example involves a rectangular specimen of an aggregated soil with a uniform overall porosity and subjected to plane strain compression. The flow condition is locally drained but globally undrained, i.e., water cannot flow through the exterior boundaries, but is otherwise free to migrate within the specimen. Air pressure is assumed to remain zero within the specimen. The top and bottom boundaries of the specimen are free to slide horizontally, whereas the two vertical boundaries are subjected to a uniform external pressure. Under this condition, a single-porosity formulation would predict that stresses, pore water pressures, and deformations would be uniformly distributed throughout the specimen. No persistent shear band would form, and at the bifurcation point two conjugate shear bands could possibly form through any point within the specimen. In other words, from a modeling standpoint, the specimen may be considered an element and not a structure. Previous studies reveal that imperfections in the form of quantified heterogeneity can trigger a persistent shear band in unsaturated porous materials. Two types of heterogeneity have been considered in the past: one arising from nonuniform density, and another emanating from inhomogeneous degree of saturation (Andrade and Borja, 2006, 2007; Borja and Andrade, 2006; Borja et al., 2013a,b; Song and Borja, 2014a,b). In this example, we investigate a third form of heterogeneity that is unique to double-porosity formulation: a spatially varying pore (or void) fraction. The following simulations consider an unsaturated porous material with uniform overall density and uniform global degree of saturation, but with a spatially varying pore fraction. A rectangular specimen 0.05 m wide and 0.10 m tall with a uniformly distributed total porosity of 0.4 was modeled with 1,040 quadrilateral mixed ele-

23

ments. The following material parameters were assumed in the simulations: intrinsic permeabilities k M = 2 × 10−10 m2 and k m = 1 × 10−13 m2 ; dynamic viscosity of water µw = 10−6 kPa·s; diffusive mass exchange parameters k¯ = 10−4 × min(krM k M , krm k m ), a = 10−3 m, β = 4, and γ = 0.4 for Eq. (2.23), which are somewhat similar to the parameters used in Gerke and van Genuchten (1993a). For the water retention curves, the van Genuchten model was used for the two pore scales, with S1M = 0.1, S2M = 1.0, n M = 4 and sαM = 2.5 kPa for the macropores and S1m = 0.2, S2m = 1.0, nm = 2 and sαm = 25 kPa for the micropores. This difference

in the water retention parameters was motivated by the fact that the macropores typically have a smaller air entry value than the micropores, see Romero et al. (1999, 2011).

Case #1

Case #2

Figure 2.1: Spatial distributions of macropore fractions ψ M . Note that the overall porosities of the two distributions are the same and uniformly distributed throughout the specimen.

For the solid matrix we used a non-associative Drucker–Prager plasticity model, with friction angle of φ = 30◦ and dilation angle of ψ = 5◦ . Cohesion of aggregated soils can be significantly high (Fuentes et al., 2013), but it may quickly decrease when aggregated structures degrade by plastic strain. To consider this aspect, we assumed initial cohesion of 15 kPa and adopting a cohesion softening law presented in Borja (2013), which is given by c = c0 exp[−(λ/k1 )2 ] ,

(2.62)

24

where c and c0 are the current and initial cohesion, k1 is a positive constant, and λ is the cumulative plastic strain defined as λ=

Z t

ke˙ p k dt ,

(2.63)

where ep is the plastic part of the infinitesimal strain tensor. This softening law was used with k1 = 0.01. For the numerical integration of this plasticity model the readers are referred to Chapter 4 of Borja (2013). Note that, although the yield surface does not explicitly take the suction stress as a parameter, the plastic response does depend on the suction because it is incorporated into the definition of the effective stress. Isotropic linear elasticity was assumed inside the yield surface, with elastic bulk modulus K = 5 MPa and Poisson’s ratio ν = 0.3. The solid grains were assumed to be incompressible, resulting in a Biot coefficient of B = 1. The gravity load was neglected in the simulations. The soil was initially unsaturated with a uniform suction of 5 kPa in the macropores and micropores, resulting in a uniformly distributed global degree of saturation of 0.58. Two macroporosity distributions were generated statistically using a normal distribution function with a specified mean of 0.2 and a standard deviation of 0.01. Two realizations, labeled Case #1 and Case #2 and having macropore fractions ranging from a low value of 0.41 to a high value of 0.59, are shown in Fig. 2.1. Due to significant difference in the water retention characteristics, the initial local saturations in the two pore regions were nearly at the extreme ends of the spectrum: 0.11 for the macropores and 0.98 for the micropores. This means that most of the pore water was initially inside the micropores. After the initial conditions were established, the specimen was consolidated by an isotropic pressure of 100 kPa, and then compressed by the top boundary moving downwards by 2 × 10−5 m for each time increment. Note that whereas the mechanical model

is rate-independent, the overall hydromechanical response is rate-dependent due to the presence of fluid flow. Figure 2.2 portrays the equivalent plastic strain

√

2/3λ for the two cases at

vertical compression of 2.7 mm. Note that for each of the two initial macropore fraction distributions, a unique shear band develops, one is the conjugate of the other, and that the locations of the two persistent shear bands are uniquely identified. The orientations of the two shear bands are approximately 47◦ to 48◦ from the horizontal, which is fairly close to the Roscoe angle of 45◦ + ψ/2. This result indicates that heterogeneity in the pore fractions can also be an effective trigger of a shear band, and more specifically, a persistent shear band that has unique orientation and position in an otherwise homogeneously deforming body. It is clear that this result cannot be generated from a modeling framework that relies solely

25

on single-porosity formulation, which is not capable of distinguishing between the properties of the macropore and micropore regions.

Case #1

Case #2

Figure 2.2: Contours of equivalent plastic strain (in percent) at vertical compression of 2.7 mm.

Figure 2.3 shows contours of the mean pore pressure p¯ at the same vertical ¯ which accommodates both the macropore compression of 2.7 mm. Note that p, and micropore water pressures, is positive everywhere, although lower values are detected in the zone of the persistent shear band due to greater plastic dilatancy predicted by the Drucker–Prager model in this region. However, this does not imply that both the macropore and micropore pressures are positive inside the specimen. In fact, Fig. 2.4 shows that the macropore water pressures barely changed from their initial value of −5 kPa, but the micropore water pressures dramatically increased as a result of the imposed specimen deformation (induced primarily by the very low permeability of the micropores). Because the mean pore pressure p¯ is

calculated from the two pressures weighted according to their pore fractions and local degrees of saturation, and because the micropore water pressures are well above zero, the resulting overall pore pressures inside the specimen are positive for both cases. An interesting consequence of the fact that the macropore water pressures barely changed from their initial negative values is that the macropores remain essentially unsaturated even as the micropores reach full saturation. Figure 2.5 affirms this statement: the degrees of saturation in the macropores barely increased from their initial value of 0.11 even as the micropores reached full saturation for both cases. Because the global degree of saturation is the weighted average of the 26

Case #1

Case #2

Figure 2.3: Contours of mean pore water pressure p¯ (in kPa) for Cases #1 and #2 calculated at vertical compression of 2.7 mm.

individual degrees of saturation, then the specimen as a whole remains unsaturated even as the mean pore pressure is positive. Of course, a positive overall pressure on an unsaturated porous medium cannot be explained in the context of single-porosity formulation—only by treating the two pore regions explicitly can one explain how this unique combination of hydrologic states can occur. On a related note, an approach often used to reconcile the aforementioned unique combination in the context of single-porosity formulation is to ignore suctions in the macropores and simply take the microsaturation as the global saturation (Koliji et al., 2010a). However, this may be an oversimplification for materials that can have non-trivial suctions in the macropores, such as those studied in Romero et al. (1999, 2011).

27

Macropores

Micropores

Figure 2.4: Contours of macropore and micropore water pressures (in kPa) for Case #1 calculated at vertical compression of 2.7 mm.

Case #1

Case #2

Figure 2.5: Contours of degree of saturation for the macropores at vertical compression of 2.7 mm. Note that the micropores are fully saturated everywhere at this loading stage.

28

2.5.2 Unsaturated slope under rainfall infiltration The second example involves a steep hillside slope subjected to rainfall infiltration. The problem is similar to the one simulated in Borja and White (2010) and Borja et al. (2012a,b): a steep slope made up of a thin layer of colluvium approximately 1–2 m thick and underlain by a rigid but highly fractured bedrock was subjected to rainfall infiltration. Rainfall was applied on the surface of the slope in the form of a flux that infiltrated into the colluvium, saturating it and causing the slope to form localized plastic shear zones. Different drainage conditions into the bedrock have been investigated in the previous work (Borja and White, 2010; Borja et al., 2012a,b), and here, for simplicity, the bedrock will be assumed to be impermeable. To investigate the impact of multiscale properties of the colluvium, this example assumed the colluvium to be a double-porosity medium and compared the ensuing hydromechanical response with that obtained from a single-porosity medium with the same overall material properties. 26.27

Elevation (m)

20 colluvium

10 bedrock

0

0

10

20 Distance (m)

30

36.19

Figure 2.6: Geometry and mesh for the steep hillside slope problem.

2.5.2.1

Single-porosity modeling

The colluvium was first considered as a single-porosity material. Figure 2.6 portrays the geometry and finite element mesh for the slope, modeled in this example as a plane strain problem. The colluvium domain was discretized with 649 quadrilateral mixed elements, and its bottom layer was fixed to the bedrock, which was assumed to be rigid and impermeable. This boundary condition means that sliding of the colluvium could only take place in the form of plastic bulk shearing

29

of elements attached to the bedrock. The top and bottom vertical boundaries of the colluvium are no-flow boundaries (i.e., impervious walls). These boundary constraints could impact the flow patterns during advanced periods of sustained rainfall, as elaborated later. Intrinsic mass densities of ρs = 2.6 t/m3 were assumed for the solid grains and ρw = 1.0 t/m3 for water. The porosity of the solid matrix was 0.39, which is similar to that of the aggregated soil tested in Carminati et al. (2007). For the mechanical modeling, we employed a thermodynamically consistent effective stress tensor derived by Borja (2006). A Biot coefficient of B = 1 was assumed, which is reasonable for soils. An elastic-perfectly plastic Drucker–Prager model was used with the following parameters: bulk modulus K = 10 MPa, Poisson’s ratio ν = 0.35, cohesion c = 2.5 kPa (derived primarily from plant roots), friction angle φ = 38◦ , and dilation angle ψ = 25◦ . These parameters are similar to those used in previous simulations conducted by the authors (Borja and White, 2010; Borja et al., 2012a,b). For the water retention model a van Genuchten curve was fitted to the experimental data for an aggregated soil presented in Carminati et al. (2007). Reference data obtained for an assembly of aggregates where pore water was retained in both the inter- and intra-aggregate pores were used to determine the homogenized properties for an equivalent single-porosity material. The following parameters were obtained for the equivalent single-porosity material: residual saturation S1 = 0.2, maximum saturation S2 = 1.0, shape parameter n = 1.92, and scaling suction sα = 25 kPa. For fluid diffusion, we assigned an intrinsic permeability of k = 10−11 m2 and a dynamic viscosity of water µw = 10−6 kPa · s. An initial uniform suction of 25 kPa was assumed throughout the slope.

The initialization phase consisted of applying static gravity load to the colluvium to establish the initial stress conditions at the quadrature points, and then re-setting the nodal displacements to zero. Then, rainfall was simulated by applying a normal water flux of qˆ = 50 mm/hr on the slope surface, a natural boundary condition in the terminology of the finite element method. An important consideration in rainfall infiltration simulation is that the pore water pressure cannot be greater than zero on the slope face. To ensure that this flow physics is satisfied, we monitored any exfiltration of water from the slope face and switched the boundary condition to zero pressure, an essential boundary condition, once a positive pore pressure was detected on the slope face. The fluid flux was applied until the slope developed localized deformation patterns. Figure 2.7 shows a snapshot of plastic strains developing within the slope after 125 minutes of rainfall. The slope exhibited a multiple block failure mechanism analogous to the results reported in Borja and White (2010) and Borja et al. (2012a)

30

Elevation (m)

and explained further in Varnes (1978). Development of zones of localized deformation took place progressively. After 100 minutes of rainfall a localized zone emerged on the slope face (primary), followed by another zone at 120 minutes (secondary), and then a third zone at 125 minutes (tertiary). Note that deformation in the tertiary zone was more intense than the one that developed in the secondary zone. As the tertiary zone of localized deformation developed, the global Newton–Raphson iteration showed a much slower (sub-linear) convergence rate. Such behavior of the iterative solution is quite common and is often interpreted as a sign of an impending slope failure, although such inference is not made in this work. It is noted that a similar sub-linear convergence rate was observed as the persistent shear band developed progressively in the rectangular specimen of the previous example.

26.27

26.27

100 min (primary)

Elevation (m)

20

20

100 min (primary) 120 min (secondary)

125 min (tertiary)

10

120 min (secondary) 0

0

10

20 Distance (m)

30

36.19

Figure 2.7: Single porosity: Equivalent plastic strain (in percent) in the colluvium at t = 125 minutes.

125 min (tertiary

Figure 2.8 shows contours of degree of saturation and pore water pressure in

10

the colluvium after 100 minutes of rainfall, which is the time instant at which the

primary plastic zone emerged on the slope face. It is clear that the location of the intense plastic zone correlates well with the zone of full saturation in which the pore water pressures increased to a value greater than zero. This implies that loss of suction was the primary reason for the development of the localized plastic zone. On the other hand, Fig. 2.9 depicts the degree of saturation and pore water

pressure after 125 minutes of rainfall. At that time instant, the entire colluvium domain was nearly fully saturated, resulting in the development of the secondary and tertiary plastic zones.

0

31

0

10

20

30

26.27 Saturation Elevation (m)

20

10

0

0

10

20 Distance (m)

30

36.19

26.27 Pore pressure

Elevation (m)

20

10

0

0

10

20 Distance (m)

30

36.19

Figure 2.8: Single porosity: Degree of saturation and pore water pressure (in kPa) in the colluvium at t = 100 minutes.

32


20

10

0

0

10

20 Distance (m)

30

36.19

26.27 Pore pressure

Elevation (m)

20

10

0

0

10

20 Distance (m)

30

36.19

Figure 2.9: Single porosity: Degree of saturation and pore water pressure (in kPa) in the colluvium at t = 125 minutes.

33

It is noted that the zone of full saturation in the colluvium domain developed from both the top and bottom end boundaries of the mesh and converged somewhere in the middle of the slope. It is obvious that the vertical wall at the bottom end of the mesh blocked fluid flow, causing the saturated zone to propagate uphill under the sustained rainfall. However, the vertical wall at the top end of the mesh did not affect the development of a saturated zone that propagated downhill; rather, it is the combined effects of rainfall intensity, colluvium geometry, and permeability that were responsible for saturating the top end of the mesh. This flow dynamics will be revisited in the context of double-porosity simulation discussed next. 2.5.2.2

Double-porosity modeling

The next step of the analysis entails converting the colluvium domain into a double-porosity medium. To have a meaningful comparison with the single-porosity simulation, the overall porosity of the double-porosity medium must be the same as that of the single-porosity medium. This means that the total porosity of the double-porosity medium must somehow be distributed to the two pore scales, and here we arbitrarily assigned 0.09 of the porosity to the macropores and 0.3 to the micropores, for a total porosity of 0.39. The next step was to assign water retention characteristics to the two pore scales such that the overall water retention characteristics are comparable to those of the single-porosity medium. To this end, the experimental data reported in Carminati et al. (2007) were used as a reference and their data were calibrated by superimposing water retention curves for the two pore scales. This entails a trial and error procedure and the results are shown in Fig. 2.10. In this figure, the two dashed curves are the water retention curves for the two pore scales; the red curve is the superposition of the two dashed curves. The blue curve is the water retention curve used in the single-porosity simulation. The superposition of the two water retention curves was based on the definition of global degree of saturation for a double-porosity medium, see Eq. (2.8). We have used the fact that the air entry value for the micropores is typically much higher than that for the macropores. A similar superposition approach was employed in the literature (Buscarnera and Einav, 2012; Casini et al., 2012; Durner, 1994). The calibration described in the preceding paragraph yielded the following van Genuchten parameters. For the macropores, S1M = 0.05, S2M = 1.0, n M = 3

and sαM = 12 kPa; for the micropores, S1m = 0.2, S2m = 1.0, nm = 2.2 and sαm = 50 kPa. The corresponding permeabilities for the macropores and micropores are 4.33 × 10−11 m2 and 4.33 × 10−15 m2 , respectively.

The mechanical properties used for the solid component were the same as

in the single-porosity simulation. Note that whereas the elastoplastic parameters

34

100

Suction (kPa)

Micropores

75

Single porosity

Double porosity

50 Macropores

25

Experimental data

0

0

0.2

0.6 0.4 Saturation

0.8

1

Figure 2.10: Water retention curves for double-porosity and single-porosity modeling fitted to experimental data from Carminati et al. (2007).

were the same as in the previous case, the total suction was obtained from the combination of the local macropore and micropore suctions. Finally, the following values were assumed for the diffusive mass transfer: a = 0.01 m, β = 11, γ = 0.4 and k¯ = 0.01 × min(krM k M , krm k m ).

Simulations were conducted in exactly the same manner as in the single-porosity

case. The same initial suction of 25 kPa was specified at the two pore scales so that the initial stress condition of the slope remained the same and pore water pressures were in local equilibrium. As in the previous example, the macropores were much drier than the micropores under the same suction because of their water retention characteristics. Care must be taken to ensure a realistic modeling of the flux boundary condition. Here, the rainfall was allowed to infiltrate the macropores only but not the micropores, owing to the very low permeability of the micropores. In other words, the flux boundary condition was set to zero for the micropores, but was set equal to the full rainfall intensity value of qˆ = 50 mm/h for the macropores. This assumed that the macropores were permeable enough to accommodate all of the rainfall volume without surface runoff. These flow conditions are similar to the numerical experiments conducted in Gerke and van Genuchten (1993a). We first investigate the failure pattern in the slope predicted by the doubleporosity simulation. Figure 2.11 shows the contour of plastic strain after 125 minutes of rainfall. We observe that the simulation predicted only one zone of intense deformation, unlike the single-porosity simulation which predicted three localized yield zones. This plastic zone is located between the secondary and tertiary zones of the single-porosity simulation. The fact that this zone of intense deformation

35

emerged at almost the same time instant as the moment when the single-porosity simulation developed a tertiary localized zone may be attributed to the fact that the colluvium was made to accommodate approximately the same volume of rain water that saturated the lower end of the mesh. However, there was no primary or secondary localized shear zone predicted by the double-porosity simulation.

26.27

Elevation (m)

20

125 min 125 min

10

0

0

10

20 Distance (m)

30

36.19

Figure 2.11: Double porosity: Equivalent plastic strain (in percent) in the colluvium at t = 125 minutes.

To explain the difference between the mechanisms of deformation predicted by the single- and double-porosity simulations, we plot the contours of overall degree of saturation and mean pore water pressure at time instants t = 100 minutes and

10

t = 125 minutes in Fig. 2.12 and Fig. 2.13, respectively, and compare with those

20 =Distance (m)

30

36.19

calculated from the single-porosity simulations portrayed in Fig. 2.8 and Fig 2.9, respectively. Recall that the mean pore water pressure p¯ determines the effective stress, which in turn determines the plastic yield zone. Figure 2.12 shows that, at t

100 minutes, the colluvium domain remained unsaturated for the most

part, except near the soil-bedrock interface, which reached full saturation. This result could be attributed to the much higher permeability of the macropores that allowed the rainwater to flow more freely underneath the slope. In contrast, the permeability of the single-porosity medium was not high enough to allow the rain water to flow as freely, and so a bottleneck formed that caused the upper segment of the slope to become fully saturated. This result leads to an important conclusion that, in a double-porosity medium, it is the permeability of the macropores, and not the average permeability, that determines the resulting flow pattern.

36


20

10

0

0

10

20 Distance (m)

30

36.19

26.27 Pore pressure

Elevation (m)

20

10

0

0

10

20 Distance (m)

30

36.19

Figure 2.12: Double porosity: Global degree of saturation and mean pore water pressure (in kPa) in the colluvium at t = 100 minutes.

37


20

10

0

0

10

20 Distance (m)

30

36.19

26.27 Pore pressure

Elevation (m)

20

10

0

0

10

20 Distance (m)

30

36.19

Figure 2.13: Double porosity: Global degree of saturation and mean pore water pressure (in kPa) in the colluvium at t = 125 minutes.

38

The difference in deformation and flow patterns in double-porosity formulation may be attributed to the preferential/non-equilibrium flow. Indeed, this is a major feature that distinguishes double-porosity flow from single-porosity flow (Gerke, 2006; Jarvis, 2007; Šimünek et al., 2006). The occurrence of nonequilibrium flow can be examined by checking whether the macropore and micropore pressures are different at the same local point. Figure 2.14 shows the pressure difference p M − pm in the slope at four time instants. We confirm that local

non-equilibrium existed throughout the entire rainfall period, suggesting that the

flow physics was different from the single-porosity case. This difference impacted the evolution of pressure/saturation and resulted in a different failure mechanism. It also highlights the fact that an equivalent single-porosity modeling of a double-porosity material may not be justified in the context of a boundary value problem. In other words, conversion of a double-porosity medium into an equivalent single-porosity medium is a gross simplification, and the approach could generate misleading results when applied to boundary value problems. 26.27

26.27 t = 50 min

t = 75 min 20 Elevation (m)

Elevation (m)

20

10

0

0

10

20 Distance (m)

30

10

0

36.19

26.27

0

10

20 Distance (m)

t = 125 min 20 Elevation (m)

20 Elevation (m)

36.19

26.27 t = 100 min

10

0

30

0

10

20 Distance (m)

30

10

0

36.19

0

10

20 Distance (m)

30

36.19

Figure 2.14: Snapshots of pressure differences (in kPa) between the macropores and micropores during the rainfall.

39

2.6

closure

This chapter has presented a hydromechanical framework for unsaturated porous media with two dominant pore scales. The framework relies on a thermodynamically consistent definition of effective stress, a measure of stress that is energyconjugate to the rate of deformation of the solid matrix. The same thermodynamic framework has enabled identification of other state variables that must be linked via constitutive laws, including the transfer of fluid mass between the two pore scales that must be related via a constitutive law to the difference in pore pressures at the two scales. A mixed finite element formulation was used to solve boundary value problems for unsaturated porous media with double porosity. In previous papers, it has been shown that heterogeneity in density and degree of saturation can have a first-order effect on the development of a persistent shear band in porous materials. The present work has shown that a third form of heterogeneity, emanating from the spatial distribution of the pore fraction, can also have a significant impact on the development of a persistent shear band. Two numerical examples have been used to demonstrate the impact of double porosity on the deformation and flow patterns. The first example demonstrates a unique condition that is not possible with single-porosity theory: an unsaturated mixture of solid, water, and air with an overall positive pore water pressure. The second example suggests that failure mechanisms in slopes can be significantly altered by an explicit treatment of the two pore scales. While the results from the first example may be intuitive, the results from the second example are more quantitative in the sense that the new failure patterns can only be captured by a robust computational model that accounts for the two pore scales.

40

3

CONSTITUTIVE MODELING FRAMEWORK

This chapter is adapted from: Borja, R. I. and Choo, J. (2016). “Cam-Clay plasticity, Part VIII: A constitutive framework for porous materials with evolving internal structure.” Computer Methods in Applied Mechanics and Engineering, 309, 653–679.

3.1

introduction

The modeling framework developed in Chapter 2 assumes that the internal structure of double-porosity materials remains unchanged by deformations. While this assumption is plausible when the deformation is infinitesimal, it becomes increasingly inadequate for moderate to large deformations. Thus, much work needs to be done with respect to mathematically delineating the evolution of internal structure and how it impacts fluid flow through the micropores and macropores of materials with two dominant pore scales. The aim of this chapter is to develop a constitutive framework that can capture the proportional changes in the micro- and macro-porosities during the course of loading. Here we focus more specifically on the behavior of aggregated soils depicted by the computed tomography volume of Fig. 3.1. The volume contains aggregates of reconstituted soil with much smaller intra-aggregate pores (micropores), as compared to the much larger inter-aggregate pores (macropores) that are very visible in the figure. For the sake of clarity, we use the concept of dual permeability in which fluid may flow not only through the larger macropores but also through the smaller micropores, as well as between contacting aggregates through their contact areas (Gerke, 2006; Šimünek et al., 2006). Fluid may also be exchanged between the micropores and macropores through the surface of the aggregates exposed to the inter-aggregate pores (Carminati et al., 2008). The macropores define the internal structure of the volume. As the volume is compressed both the macropores and micropores can compact, but the compaction of the macropores is more dominant during the early stages of compression. As the macropores are squeezed the internal structure disappears (Burland, 1990), leaving behind a volume with only the micropores and having a similar material behavior to that of the reconstituted soil. A challenge with modeling double-porosity media is that there are two compressibility responses that must be captured within the same constitutive framework— that of the reconstituted material with an evolving internal structure, and that of

41

Figure 3.1: Reconstructed computed tomography volume of an aggregated silty clay, diameter = 80 mm, height = 35 mm. The aggregates are composed of much smaller silty clay solid particles with intra-aggregate pores. Visible spaces between aggregates are the inter-aggregate pores. After Borja and Koliji (2009).

the reconstituted material itself (Anagnostopoulos et al., 1991; Koliji et al., 2010a,b). Because the internal structure disappears with collapse of the macropores, the response of the system with internal structure must necessarily approach the response of the reconstituted material itself at high compressive loads. This requires a nonlinear continuum mechanics approach accommodating an evolving configuration in general and an evolving macro-porosity in particular. If the macropores and micropores are filled with fluids that are free to migrate within and outside their pore scales, then the constitutive framework must also accommodate two volume constraints imposed by the two pore scales. Furthermore, in the context of critical state theory the preconsolidation pressures at the two pore scales must also be tracked concurrently with fluid pressure dissipation. In this chapter, we take a nonlinear continuum mechanics approach to accommodate an evolving internal structure by partitioning the overall effective stress of double-porosity media into two individual effective stresses appropriate for the macropores and micropores. In doing so, fluid pressure dissipation at the two pore scales as well as the accompanying hardening of the preconsolidation stresses can be tracked individually. The core aspect of the formulation is the expression for the rate of change of internal energy for double-porosity media (Borja and Koliji, 2009), which determines the energy-conjugate pairing of the relevant constitutive variables. The constitutive model is cast within a mixed finite element framework formulated in the reference configuration. Without loss of generality, throughout this chapter we assume that the two pore scales are fully saturated with the same type of fluid (e.g., water). This chapter is organized as follows. In Section 3.2, we develop a general constitutive framework for porous materials with evolving double-porosity structures.

42

In Section 3.3, we formulate conservation laws and mixed finite elements in Lagrangian form, into which we cast the developed constitutive framework. For a nonlinear finite element implementation of this framework, in Section 3.4 we develop algorithmic formulations necessary for the local stress-integration and global tangent operators. We validate the proposed framework and present its features in Section 3.5 through numerical examples ranging from an element testing to a field-scale boundary-value problem.

3.2

general formulation

This section focuses on the kinematics of deformation of a porous solid with double porosity. The general notion of compressibility in the context of double porosity is also introduced, along with a formulation of elastoplasticity in terms of the effective Kirchhoff stress that incorporates an evolving internal structure, herein represented by the micropore fraction ψm . 3.2.1 Solid kinematics Consider an aggregated soil represented by a double-porosity continuum. The motion of the solid is defined by the deformation x = φ( X, t), with deformation gradient F = ∂φ/∂X. The solid deformation carries with it both the macropore and micropore volumes quantified by the void ratios em ( X, t) =

dVvm , dVs

e M ( X, t) =

dVvM , dVs

(3.1)

where dVvm and dVvM are the volumes of the micropores and macropores, respectively, and dVs is the volume of solid contained in the elementary volume. We also define the following volume fractions φs ( X, t) =

dVs , dV

ψm ( X, t) =

dVvm , dVv

(3.2)

where dVv = dVvm + dVvM is the total volume of the void within the total volume dV of the mixture. In this last equation, φs is the solid volume fraction while ψm is the fraction of the void occupied by the micropores. The specific volume of the micropores is given by vm ( X, t) = 1 + em ( X, t) ,

(3.3)

43

which is the total volume of the aggregate per unit solid volume. The overall specific volume is v( X, t) = vm ( X, t) + e M ( X, t) ,

(3.4)

which is the total volume of the mixture per unit solid volume. In aggregated soils, the aggregate is typically associated with a reconstituted material with no internal structure such as clay. On the other hand, the internal structure of the soil is defined by the macropores. We note that v( X, t) → vm ( X, t) as e M ( X, t) → 0,

which means that the internal structure of the soil disappears when the macropores collapse. If the solid does not exchange mass with the fluid, then its motion should be mass conserving. Dropping the spatial and temporal arguments X and t for brevity, this means that Jφ˙s ρs = 0 ,

(3.5)

where J = det( F ) is the Jacobian of the solid motion, ρs is the intrinsic mass density of the solid, and the superimposed dot denotes a material time derivative with respect to the solid motion. Without loss of generality we shall assume that the solid constituent is incompressible and take ρs = constant, which yields the following evolution of the solid volume fraction φs =

φ0s , J

(3.6)

where φ0s is the volume fraction of the solid when J = J0 = 1. Thus, the evolution of φs can be determined from the variation of the Jacobian J alone. The evolutions of the specific volumes are given by v=

J 1 = s φs φ0

(3.7)

for the macropores, and vm = 1 + ψm

1 − φs ψm J M = ψ + φs φ0s

(3.8)

for the micropores. Note that the expression for v is determined by J alone, whereas the expression for vm also contains the micropore fraction ψm that must

44

be determined separately. The difference between the two specific volumes is the void ratio of the macropores, e M = v − vm = ψ M

J 1 − φs M = ψ − 1 . φs φ0s

(3.9)

3.2.2 Effective stress and the first law For double-porosity media whose macropores (denoted by index M) and micropores (denoted by index m) are saturated with incompressible fluid (e.g., water), the rate of change of internal energy per unit volume takes the form (Borja and Koliji, 2009; Choo et al., 2016) ¯ di + Jρe˙ = hτ,

∑

i = M,m

hv˜i , φi , pi i −

∑

i = M,m

hci , pi , v˜i i − h(1 − φs ), π, ψ˙m i ,

(3.10)

where the symbol h◦, . . . , ◦i denotes energy-conjugate pairing in product form. The notations are as follows: e˙ is the rate of change of internal energy per unit total mass of the mixture, ρ is the total mass density of the mixture, τ¯ is the effective Kirchhoff stress tensor, d is the rate of deformation tensor, v˜i is the relative velocity of fluid in pore scale i with respect to solid, φi is the volume fraction of pore scale i, pi is the intrinsic Kirchhoff fluid pressure in pore scale i, ci is a mass exchange term satisfying the closure condition c M + cm = 0, φs is the volume fraction of solid (defined previously), ψm is the void fraction for the micropores (also defined previously), and (3.11)

π = p M − pm

is the Kirchhoff fluid pressure jump between the two pore scales. The Kirchhoff stresses and pressures differ from the Cauchy definitions by the factor J, i.e., τ¯ = J σ¯ and pi = Jθi , where σ¯ is the Cauchy effective stress tensor and θi is the Cauchy fluid pressure in pore scale i. The foregoing expression for the rate of change of internal energy yields four groups of constitutive variables that imply four types of constitutive relations. We shall elaborate these pairs of constitutive relations in a later section, but for now we focus on the effective stress σ¯ and the fourth term in the energy equation involving the pressure jump π. The effective Kirchhoff stress tensor is given by the expression (Borja and Koliji, 2009) ¯ , τ¯ = τ + p1

(3.12)

45

where τ is the total Kirchhoff stress tensor, 1 is the second-order identity tensor, and p¯ is the Kirchhoff mean fluid pressure given by p¯ = ψm pm + ψ M p M .

(3.13)

Here, we take the Biot coefficient equal to unity following the earlier assumption of an incompressible solid constituent. Rewriting the total Kirchhoff stress tensor as τ = ψm τ + ψ M τ, we find that τ¯ = ψm τ¯m + ψ M τ¯ M ,

(3.14)

where τ¯m = τ + pm 1 ,

τ¯ M = τ + p M 1

(3.15)

are the single-porosity effective Kirchhoff stress tensors at each pore scale (Borja, 2006). Note that these two stress tensors are not independent, but instead are related by the equation τ¯m = τ¯ M − π1 .

(3.16)

In other words, the two single-porosity effective stresses are simply translated on the hydrostatic axis by the fluid pressure jump π. We now consider an isotropic state of stress and denote the mean normal effective stresses by the symbols τ¯m = tr(τ¯m )/3 and τ¯M = tr(τ¯ M )/3. Equation (3.16) thus reduces to π = τ¯M − τ¯m .

(3.17)

When the macropores and micropores are both yielding, the effective mean normal stresses may be replaced by their respective preconsolidation stresses, i.e., τ¯m = pcm and τ¯M = pcM . The overall preconsolidation stress for the aggregated soil is then equal to τ¯ =

1 tr(τ¯ ) = ψm τ¯m + ψ M τM := p¯ c , 3

(3.18)

where p¯ c = ψm pcm + ψ M pcM .

(3.19)

46

When the material is yielding the fluid pressure jump is also equal to the jump in the preconsolidation pressures, i.e., π = pcM − pcm .

(3.20)

The fourth energy-conjugate term on the right-hand side of Eq. (3.10) suggests compressibility laws of the form pcm = pcm (φs , ψm ) and pcM = pcM (φs , ψm ), which give π = pcM (φs , ψm ) − pcm (φs , ψm ) ≡ g(φs , ψm ) .

(3.21)

For incompressible solid constituents, variations in void fraction φs are associated with changes in the volume of the pores. If such changes are irreversible, or if the reversible component is insignificant, then compressibility laws of the following form may be used π = g˜ ( J p , ψm ) ,

(3.22)

where J p is the plastic component of J. Indeed, compressibility laws of this form have been developed in the literature for some aggregated soils such as Bioley silt (Koliji et al., 2008b) and Corinth marl (Anagnostopoulos et al., 1991). The next section elaborates how these compressibility laws impact the evolution of the internal structure of a material with two pore scales. 3.2.3 Compressibility laws Consider a “normally consolidated” porous material where both the macropores and micropores are yielding such that the internal structure evolves according to the following compressibility laws, vm = f ( pcm ) ,

e M = g( pcM ) .

(3.23)

The inverse relations are given by pcm = f −1 (vm ) ,

pcM = g−1 (e M ) .

(3.24)

Note that pcm and pcM vary with φs and ψm through vm and e M , respectively, see Eqs. (3.8) and (3.9). Now, assume that there is no pore pressure jump between the micropores and macropores so that π = 0. This condition holds if both pore scales are dry, or if the load is applied at a sufficiently slow rate that no pore pressure difference

47

develops between the two pore scales. Setting pcm = pcM ≡ p¯ c determines the

evolution of the micropore fraction ψm for a given evolution of φs (or J). Note

that the compressibility laws are generally of nonlinear form, so a closed-form expression for ψm may not be readily available. However, in principle one can always calculate ψm numerically (by iteration, for example). Having determined the variation of ψm with J, one can then substitute it back into the expression for the preconsolidation stress to obtain the corresponding variation of p¯ c as a function of J. Thus, both ψm and p¯ c can be determined from the given time history of J. Now, suppose that instead of J we are given the time history of the effective isotropic stress τ¯ ≡ p¯ c , and assume once again that π = 0. The goal is to find ψm and J such that

pcm = f −1 (vm (ψm , J )) ≡ p¯ c ,

(3.25)

pcM = g−1 (e M (ψm , J )) ≡ p¯ c .

(3.26)

and

We have two equations in two unknowns, so we can determine the variations of J and ψm from the given time history of p¯ c . If π 6= 0, then its time history must be prescribed to determine the evolution of the micropore fraction ψm .

The specific compressibility laws of interest are of the following form. For the micropores, we have v˙ m p˙ cm = −cc , vm pcm

ε˙ ev = −cr

p˙ cm , pcm

(3.27)

where cc and cr are total and elastic compressibility indices, respectively, and εev is the elastic logarithmic volumetric strain. Direct time integration gives h ln(v /v ) − (εe − εe ) i m mn v vn pcm = ( pcm )n exp − , c c − cr

(3.28)

where subscript n pertains to reference value at time tn , and vm = vm (ψm , v) from Eq. (3.8). This equation is equivalent to the bilogarithmic compressibility law proposed by Butterfield (1979) for reconstituted soils. For the macropores, we consider a similar bilogarithmic compressibility law of the form e˙ M p˙ = −c M cM , eM pcM

(3.29)

48

where c M is the total compressibility index determined from the macropore compressibility curve, which is the difference between the total and micropore compressibility curves. Integrating yields h ln(e /e ) i M Mn . pcM = ( pcM )n exp − cM

(3.30)

This law predicts an asymptotic decay in the sense v → vm as − pcM → ∞. Later we shall validate these compressibility laws against experimental data on aggregated soils. 3.2.4 Continuum formulation This section formulates the rate constitutive equations for double-porosity continua with an evolving internal structure represented by the scalar variable ψm . The formulation is done in abstract form to elucidate the essential ingredients of the theory. More specific elements of the theory are presented in the next section. Our point of departure is the rate constitutive equation in effective Kirchhoff stress space of the form ∂G τ¯˙ = αe : d − λ˙ , ∂τ¯

(3.31)

where d is the symmetric part of the velocity gradient tensor l, called the rateof-deformation tensor, G is the plastic potential function, λ˙ ≥ 0 is the plastic consistency parameter, and αe is a rank-four elastic tangent tensor analogous to the elastic constitutive tensor ce in infinitesimal theory. We should note that αe is amenable to spectral decomposition, and contains the initial stress terms necessary to make the transformation objective with respect to rigid-body rotation. Because τ¯ is a symmetric tensor, the plastic flow direction ∂G/∂τ¯ is also a symmetric tensor, which means that the plastic spin is ignored. ¯ p¯ c ) ≤ 0. Assuming the Next, we consider a yield function of the form F (τ,

yield condition F = 0 is satisfied, the consistency condition for continued yielding takes the form F˙ = 0

⇒

∂F ∂F : τ¯˙ + p¯˙ c = 0 , ∂τ¯ ∂ p¯ c

(3.32)

where p¯˙ c =

∂ p¯ c ˙ m ∂ p¯ c ˙p ψ + p J ≡ α1 ψ˙ m + α2 λ˙ . ∂ψm ∂J

49

(3.33)

In this last equation, we take J˙p ∝ λ˙ and α2 is some scalar coefficient of the consis˙ tency parameter λ. We now eliminate ψ˙ m by appealing to Eq. (3.22). Taking the rates gives π˙ =

∂ g˜ ∂ g˜ m ψ˙ + p J˙p ≡ β 1 ψ˙ m + β 2 λ˙ . ∂ψm ∂J

(3.34)

Here, we note that π˙ is a given function representing the rate of fluid jump between the two pore scales. Provided that β 1 6= 0, we can solve for ψ˙ m and substitute into Eq. (3.33) to obtain p¯˙ c =

α β α1 1 2 π˙ − − α2 λ˙ . β1 β1

(3.35)

To determine the consistency parameter, we substitute the constitutive equation (3.31) and hardening law (3.35) into the consistency condition (3.32) and solve ˙ The result reads for λ. ∂F α1 1 ∂F : αe : d + π˙ , λ˙ = χ ∂τ¯ ∂ p¯ c β 1

(3.36)

where χ=

∂G ∂F : αe : +H, ∂τ¯ ∂τ¯

H=

∂F α1 β 2 α2 − . ∂ p¯ c β1

(3.37)

Substituting the consistency parameter back into the constitutive equation (3.31) and simplifying yields τ¯˙ = αep : d + θπ˙ ,

(3.38)

where αep = αe −

1 e ∂G ∂F α : ⊗ : αe χ ∂τ¯ ∂τ¯

(3.39)

and θ=−

1 ∂F α1 e ∂G α : . χ ∂ p¯ c β 1 ∂τ¯

(3.40)

This result indicates that the effective Kirchhoff stress is driven not only by the solid velocity gradient but also by the fluid pressure jump between the two pore scales, which in turn arises from the difference in the preconsolidation stresses between the macropores and micropores.

50

3.3

finite element formulation

This section presents a u/p M /pm finite element formulation for porous materials with double porosity, where u = solid displacement vector, p M = macropore fluid pressure, and pm = micropore fluid pressure. The formulation follows the standard lines of hyromechanical coupling (Borja and Alarcón, 1995; Borja et al., 1998; Choo and Borja, 2015; Choo et al., 2016; Sun, 2015), except that it now treats the micropore fraction ψm as an internal variable related to the independent variables by a certain constitutive law. The section concludes with a discussion of the stress-point integration algorithm accommodating the evolution of the micropore fraction in the finite deformation range. 3.3.1 Conservation laws and strong form Before presenting the relevant conservation laws, we first recall some preliminary results pertinent to finite deformation analysis (see Song and Borja, 2014a,b). The first Piola-Kirchhoff stress tensor is P = τ · F −T and the corresponding effective stress tensor is P¯ = τ¯ · F −T . The effective stress equation in terms of these stress tensors takes the form ¯ −T , P = P¯ − pF

(3.41)

where p¯ is the Kirchhoff mean fluid pressure defined in Eq. (3.13). The total mass density is the sum of the partial mass densities, ρ = ρs + ρ M + ρm .

(3.42)

The partial mass densities are given by ρs = φs ρs ,

ρ M = (1 − φ s ) ψ M ρ w ,

ρ m = (1 − φ s ) ψ m ρ w ,

(3.43)

where the intrinsic mass densities ρs and ρw are assumed constant. The pullback mass densities are ρ0 = Jρ = ρ0s + ρ0M + ρ0m .

(3.44)

We recall that ρ˙ 0s = Jρ˙ s = 0 if the solid does not exchange mass with the fluid, see Eq. (3.5). Thus, ˙ ˙ w − J (1 − φs )ψ˙ m ρw , ρ˙ 0M = Jρ M = ψ M Jρ ˙ + J (1 − φs )ψ˙ m ρ . ρ˙ m = Jρ˙ m = ψm Jρ 0

w

w

51

(3.45) (3.46)

The relative fluid velocities in the macropores and micropores are defined with respect to the solid velocity v and take the form v˜ M = vM − v ,

v˜m = vm − v ,

(3.47)

where vM and vm are the fluid velocities in the macropores and micropores, respectively. These relative velocities give rise to fluid mass fluxes q M = ρ M v˜ M ,

qm = ρm v˜m ,

(3.48)

with Piola transforms Q M = JF −1 · q M ,

Qm = JF −1 · qm .

(3.49)

We now consider a fluid-saturated body B with boundary ∂B in the reference

configuration, and assume that the solid matrix possesses two pore scales. The boundary ∂B admits the decomposition ∂B = ∂Bu ∪ ∂Bt ,

∅ = ∂Bu ∩ ∂Bt ,

(3.50)

where ∂Bu and ∂Bt are Dirichlet and Neumann boundaries with displacement and tractions prescribed. For quasistatic loading the balance of linear momentum takes the form (Borja and Koliji, 2009) DIV(P ) + ρ0 G = c0 (v˜m − v˜ M )

in B ,

(3.51)

where DIV is the Lagrangian divergence operator, G is the gravity acceleration vector, and c0 = Jc is the pullback fluid mass transfer coefficient, with c being the value in the current configuration, reflecting transfer of fluid mass into or from the micropores. The relevant boundary and initial conditions are u = u∗ on ∂Bu , P · N = t on ∂Bt , and u( X ) = u0 ( X ) at time t = 0 and for all X ∈ B . We also consider the following decomposition of ∂B ∂B = ∂B p M ∪ ∂Bq M ,

∅ = ∂B p M ∩ ∂Bq M

(3.52)

∅ = ∂ B pm ∩ ∂ B qm

(3.53)

for the macropores, and ∂ B = ∂ B pm ∪ ∂ B qm ,

52

for the micropores, where p and q are pressure and flux boundary conditions, respectively. The fluid mass conservation equations in the macropores and micropores take the form (Borja and Koliji, 2009) ρ˙ 0M + DIV( Q M ) = −c0 ρ˙ 0m + DIV( Qm ) = c0

in B ,

(3.54)

in B .

(3.55)

The boundary conditions are as follows: p M = p∗M on ∂B p M , pm = p∗m on ∂B pm ,

Q M · N = Q M on ∂Bq M , and Qm · N = Qm on ∂Bqm . The initial conditions are: p M ( X ) = p M0 ( X ) and pm ( X ) = pm0 ( X ) at time t = 0 and for all X ∈ B .

3.3.2 Variational equations To develop the weak form, we need the space of configurations denoted by Cu = {u : B → Rnsd | ui ∈ H 1 , u = u∗ on ∂Bu } ,

(3.56)

and the space of variations Vu = {η : B → Rnsd | ηi ∈ H 1 , η = 0 on ∂Bu } ,

(3.57)

where H 1 is the Sobolev space of functions of order one. The variational form of linear momentum balance is given by Z B

Z GRAD η : P − ρ0 η · G + c0 (v˜m − v˜ M ) dV −

∂Bt

η · t dA = 0

(3.58)

for all η ∈ Vu .

Next, we consider the spaces of pore pressures as C pi = { pi : B → R | pi ∈ H 1 , pi = pi∗ on ∂B pi } ,

i = M, m .

(3.59)

and the spaces of variations Vφi = {φi : B → R | φi ∈ H 1 , φi = 0 on ∂B pi } ,

i = M, m .

(3.60)

The variational form of balance of fluid mass takes the form (no sum on i) Z B

φi (ρ˙ 0i − a0 ) dV −

Z B

GRAD φi · Qi dV +

53

Z ∂ B qi

φi Qi dA = 0

(3.61)

for all φi ∈ Vφi , where a0 =

 

c0 ,

if i = M ,

 −c , 0

if i = m .

(3.62)

Alternatively, the variational equation for the balance of linear momentum can be written in terms of the symmetric Kirchhoff effective stress tensor as Z B

∇s η : τ¯ dV −

Z B

∇ · η p¯ dV −

Z B

ρ0 η · G dV +

Z B

c0 (v˜m − v˜ M ) dV =

Z ∂Bt

η · t dA

(3.63)

for all η ∈ Vu . As for the variational equation for fluid mass conservation, we can

use the backward implicit scheme to write the variational equation in the timeintegrated form Z B

i φi (ρ0i − ρ0n ) dV − ∆tρw

Z B

∇ φi · J v˜i dV − ∆t

Z B

φi a0 dV = −∆t

Z ∂ B qi

φi Qi dA ,

for all φi ∈ Vφi , where ∆t = t − tn is the step size. 3.3.3 Linearized variational equations Before developing the linearized variation equations, we first introduce Darcy’s law so as to reduce the finite element equations to u/p M /pm form. For doubleporosity media, Darcy’s law takes the form J v˜ M = −K M · ∇ U M ,

J v˜m = −Km · ∇ Um ,

(3.64)

where U M and Um are the fluid potentials in the macropores and micropores, respectively, given by

UM =

pM +z, ρw g

Um =

pm +z, ρw g

(3.65)

where z is the elevation axis and g is the gravity acceleration constant. Assuming isotropy in permeability, the pullback permeability tensor K can be expressed in terms of effective grain diameter D and Jacobian J through an equation of the form K M = K ( D M , J )1 ,

K m = K ( Dm , J ) 1

54

(3.66)

where D M is the effective diameter of the aggregates and Dm is the effective diameter of the grains in the micropores. An example of an empirical expression for the scalar function K ( D, J ) is given by the well-known Kozeny–Carman equation (Bear, 1972) K ( D, J ) = Jk( D, J ) ,

k ( D, J ) =

ρw g D2 ( J − φ0s )3 , µw 180 J (φ0s )2

(3.67)

where µw is the dynamic viscosity of water and φ0s is the volume fraction of the solid when J = 1. Alternatively, the Kozeny–Carman expression can be written in normalized form relative to the initial value of hydraulic conductivity and evolving according to J in a fashion similar to the equation above. In the examples discussed in this chapter, the evolutions of hydraulic conductivities are expressed in terms of initial values k M0 and k m0 . A second constitutive law leading to the u/p M /pm formulation determines the mass transfer term c = c0 /J in terms of the fluid pressure jump. A vast number of studies have advanced constitutive relations for this term (see e.g., Dykhuizen, 1990; Gerke and van Genuchten, 1993b; Haggerty and Gorelick, 1995; Köhne et al., 2004; Sarma and Aziz, 2006; Warren and Root, 1963). Here, we adopt the constitutive law used in Chapter 2 of the form c=

α¯ ( p M − pm ) , µw

(3.68)

where α¯ is a dimensionless parameter that depends on the characteristics of the interface between the macropores and micropores, such as permeability, spacing, and shape. This parameter can be determined either by an equation that takes permeability and pore geometry into account (e.g., Gerke and van Genuchten, 1993b), or by inversion of test data (e.g., Trottier et al., 2014). We now consider a constitutive equation for the effective Kirchhoff stress of the form τ¯ = τ¯ (u, p M , pm ) .

(3.69)

Strictly speaking, τ¯ is also a function of the micropore fraction ψm , but this variable also depends on the independent variables u, p M , and pm . Linearizing this stress tensor yields δτ¯ = α : ∇ δu + a M δp M + am δpm ,

(3.70)

55

where α is a tangential moduli tensor discussed in the next section, and ai = ¯ ∂τ/∂p i for i = M, m. Furthermore, from Eq. (3.44) the pullback total mass density has the linearization δρ0 = δρ0M + δρ0m = ρw δJ ,

(3.71)

where δJ = J ∇ · δu. This results in the following linearized variational equation for balance of momentum Z

∇ η : a : ∇ δu dV +

BZ

−

B

δρ0 η · G dV + δ

Z

Z B B

Z

Z

B

η : a M δp M + am δpm dV −

c η · ( J v˜m − J v˜ M ) dV =

∂B

δ(∇ · η) p¯ + ∇ · η δ p¯ dV

η · δt dA ,

(3.72)

where a = α − τ¯ 1, δ(∇ · η) = − ∇ η : δu ∇, and c = c0 /J, see the discussion

surrounding Eq. (3.51). The stress term in the tangent moduli has components (τ¯ 1)ijkl = τ¯il δjk . Eliminating the relative fluid velocities using Darcy’s law, and linearizing the

associated volume integral, yields δ

Z B

c η · ( J v˜m − J v˜ M ) d =

Z

c η · δ K M · ∇ U M − Km · ∇ Um dV

BZ

−

B

δc η · K M · ∇ U M − Km · ∇ Um dV .

(3.73)

Disregarding the subscripts, we note that δ K · ∇ U ) = δK · ∇ U + K · δ(∇ U ) ,

(3.74)

where δK = K 0 ( J )δJ 1 ,

δ(∇ U ) = ∇ δp − ∇ p · ∇ δu .

(3.75)

Furthermore, δc =

α¯ δp M − δpm . µw

(3.76)

See Song and Borja (2014b) for further details on these derivatives. Next we consider the linearization of the time-integrated variational equation for the balance of fluid mass. For pore scale i = M, m, we have Z B

φi δρ0i dV

− ∆tρw δ

Z

i

B

∇ φ · Ki · ∇ U dV − ∆t

56

Z

i

B

φ δa0 dV = −∆t

Z ∂ B qi

φi δQi dA ,

(3.77)

where δa0 follows from Eq. (3.76) and δρ0i follows from Eqs. (3.45) and (3.46). The flux term has the linearization δ

Z

i

B

∇ φ · Ki · ∇ U dV =

Z

i

δ(∇ φ ) · Ki · ∇ U dV +

BZ

+

B

Z

∇ φi · δKi · ∇ U dV ,

B

∇ φi · Ki · δ(∇ U ) dV (3.78)

where δ(∇ φi ) = − ∇ φi · ∇ δu .

(3.79)

It remains to determine the tangent operators α, a M and am in Eq. (3.72); these are developed in the next section.

3.4

discrete formulation

This section describes the discrete evolution of the micropore fraction ψm , which measures the internal structure in a double-porosity material, as well as its impact ¯ For constitutive modeling of the solid material, on the effective stress tensor τ. we employ the classic three-invariant hyperelasto-plasticity and solve the local unknowns in principal elastic strain space, see Borja (2013). 3.4.1 Stress-point integration We recall the classic stress-point integration for finite deformation hyper-elastoplasticity consisting of an elastic predictor for the elastic left Cauchy-Green deformation tensor be , followed by a plastic corrector in elastic logarithmic principal strains (Borja, 2013). The elastic predictor is calculated from the relative deformation gradient f = ∂x/∂xn and the current value of the elastic left Cauchy-Green deformation tensor ben , after which a spectral decomposition is performed, i.e., be tr = f · ben · f T =

3

∑ (λeAtr )2 m( A) ,

(3.80)

A =1

where λeAtr is the trial elastic principal stretch with an associated principal direction ¯ we have n( A) and spectral direction m( A) = n( A) ⊗ n( A) . By coaxiality of be tr and τ, τ¯ =

3

∑

A =1

τ¯A m( A) ,

(3.81)

57

where τ¯A is a principal value of the effective Kirchhoff stress tensor obtained from the hyperelastic constitutive equation τ¯A =

∂Ψ e , ∂εeA

εeA = log(λeA ) ,

(3.82)

in which Ψ e is the stored energy function and εeA is the elastic logarithmic principal

strain determined from the logarithm of the elastic principal stretch λeA . The stress-

point integration algorithm entails the determination of the evolution of εeA along

with those of the internal state variables including the micropore fraction ψm .

The local stress-point integration is as follows. Given the following starting values at tn : (τ¯A )n for A = 1, 2, 3; ( p M )n , ( pm )n , micropore fraction ψnm , and preconsolidation pressures ( pcM )n and ( pcn )n ; and given increments ∆u, ∆p M , and ∆pm ; find the principal stresses τ¯A ≡ (τ¯A )n+1 for A = 1, 2, 3, and micropore frac-

tion ψm ≡ ψnm+1 . To accommodate nonlinear hyperelasticity we use the notion of residuals and perform a return mapping in principal logarithmic elastic strain space as follows: r A = εeA − εeAtr + ∆λ

∂F → 0, ∂τ¯A

A = 1, 2, 3,

(3.83)

where εeAtr = log(λeAtr ) and ∆λ is the incremental plastic consistency parameter. Next, we impose the yield criterion r4 = F (τ¯1 , τ¯2 , τ¯3 , p¯ c ) → 0 ,

(3.84)

where p¯ c is the mean preconsolidation pressure defined from the hardening law r5 = p¯ c − ψm pcm ψm ,

3

∑

A =1

εeA − ψ M pcM ψm ,

3

∑

A =1

εeAtr → 0 .

(3.85)

The final residual term is the thermodynamic restriction r6 = pcm ψm ,

3

∑

A =1

εeA ,

3

∑

A =1

εeAtr − pcM ψm ,

3

∑

A =1

εeAtr − ( pm − p M ) → 0 , (3.86)

where p M = ( p M )n + ∆p M and pm = ( pm )n + ∆pm . It must be noted that in the above equations, the evolutions of εeAtr (for A = 1, 2, 3), p M , and pm are all given,

and thus, we have a total of six residual equations in six unknowns, namely, εeA (A = 1, 2, 3), ∆λ, p¯ c , and ψm .

58

The six unknowns are solved by a local Newton iteration. Setting     r  1            r   2       r   3 r= ,  r    4            r5          r6

  εe1       εe2      εe 3 x=   ∆λ       p¯ c     m ψ

            

A = r 0 ( x) ,

,

(3.87)

           

the problem is then to find the solution x∗ such that r = 0. To develop the elements of the local tangent operator r 0 ( x), we first note the tangential elastic constitutive equation δτ¯A =

3

∑ aeAB δεeB ,

(3.88)

B =1

where aeAB = ∂2 Ψ e /∂εeA ∂εeB is the tangential elasticity matrix in principal axes. Consider the local residual expression r = r ( x), where the components of r are given in Eqs. (3.83)–(3.86) and those of x are given in Eq. (3.87). We note that for the local problem the values of εeAtr , p M , and pm are given. The local tangent operator A = r 0 ( x) has components given below. First, we consider the following derivatives with respect to x A = εeA for A = 1, 2, 3: A AB = δAB + ∆λ

3

∂2 F e a , ∂τ¯A ∂τ¯C CB C =1

∑

B = 1, 2, 3 .

(3.89)

The remaining derivatives for A = 1, 2, 3 are A4A =

3

∂F e a , ∂ τ¯B BA B =1

∑

A5A = −ψm

∂pcm , ∂εeA

A6A =

∂pcm , ∂εeA

(3.90)

where ∂pcm /∂εeA = ∂pcm /∂εev . Next we consider derivatives with respect to x4 = ∆λ. For A = 1, 2, 3, we have A A4 =

∂F , ∂τ¯A

A44 = A54 = A64 = 0 .

(3.91)

The derivatives with respect to x5 = p¯ c are of the form (for A = 1, 2, 3) A A5 = ∆λ

∂2 F , ∂τ¯A ∂ p¯ c

A45 =

∂F , ∂ p¯ c

59

A55 = 1 ,

A65 = 0 .

(3.92)

Finally, the derivatives with respect to x6 = ψm are of the form (for A = 1, 2, 3, 4) ∂p ∂pcm − pcm − ψ M cM + pcM , ∂ψm ∂ψm ∂pcm ∂p = − cM . m ∂ψ ∂ψm

A56 = −ψm

A A6 = 0 ,

A66

(3.93)

We note from the chain rule that ∂pcM ∂p ∂e = cM Mm , ∂ψm ∂e M ∂ψ

∂pcm ∂pcm ∂vm = . ∂ψm ∂vm ∂ψm

(3.94)

3.4.2 Tangent operators In this section we develop expressions for the global tangent operators accommodating the variation of the micropore fraction ψm . The key aspect to obtaining these tangent operators is the consistent linearization of the local residual vector r in Eq. (3.87) through the local tangent operator A, which is available in closed form as shown above. Consider the functional representation of τ¯ in terms of u, p M , and pm as indicated in Eq. (3.69), and its spectral representation shown in Eq. (3.81). We note that the spectral directions of τ¯ depend on u alone through be tr , whereas its principal values depend not only on εeAtr but also on p M and pm , all of them through the

principal elastic logarithmic strains εeA for A = 1, 2, 3. Therefore, we can write the variation of τ¯ as 3

δτ¯ =

∑

A =1

δτ¯A m( A) + τ¯A δm( A) ,

(3.95)

where δτ¯A =

∂εeC ∂εeC ∂τ¯A 3 ∂εeC e tr δε + δp + δp ∑ ∂εe ∑ ∂εe tr B ∂p M M ∂pm m C B =1 B C =1 3

3

=

∂τ¯A

∂τ¯A

∑ a AB δεeBtr + ∂p M δp M + ∂pm δpm ,

(3.96)

B =1

and a AB =

3

∑

C =1 3

aeAC

∂εeC , ∂εeBtr

∂εe ∂τ¯A = ∑ aeAC C , ∂pm ∂pm C =1

3 ∂εe ∂τ¯A = ∑ aeAC C , ∂p M ∂p M C =1

aeAC =

∂τ¯A ∂2 Ψ e = . ∂εeC ∂εeA ∂εeC

60

(3.97)

Furthermore, the spin of τ¯ is the same as the spin of be tr , and so, 3

∑

A =1

τ¯A δm( A) =

3

∑ ∑

A =1 B 6 = A

ω AB (τ¯B − τ¯A )m( AB) ,

(3.98)

where ω AB is the spin of be tr , and m( AB) = n( A) ⊗ n( B) .

The relevant tangent operators of interest, defined in (3.70), then take the fol-

lowing forms 3

α=

3

∑ ∑

A =1 B =1

a AB m( A) ⊗ m( B) +

3

τ¯A − τ¯B e tr e tr 2 β AB 2 A =1 B 6 = A ( λ B ) − ( λ A )

∑ ∑

,

β AB = (λeBtr )2 m( AB) ⊗ m( AB) + (λeAtr )2 m( AB) ⊗ m( BA) , 3

3

aM =

∂τ¯A ∑ ∂p M m( A) , A =1

am =

∂τ¯A ( A) m . ∂pm A =1

∑

(3.99)

Complete development of the tangent operators relies on closed-form expressions for the derivatives of εeA . Consider the converged residual equation of the local problem, r ( x∗ ) = 0. We recall that this residual equation relies on the assumption that εeAtr , p M , and pm are all given. Thus, the residual vector may be viewed as a function of z as well, i.e., r = r (z, x), where    εe1 tr      e tr    ε2 z := εe3 tr      pM      p m

          

and

         

∂x ∂r + A ∗ = 0. ∂z x∗ ∂z x

(3.100)

In the foregoing equation, it is understood that r (z, x∗ ) = 0, and thus, all the derivatives must be evaluated at x = x∗ . This equation then yields the tangent matrix B :=

∂x ∂r = − A −1 . ∂z ∂z x

(3.101)

The derivatives of interest can be obtained from the elements of the matrix B, ∂εeI = BI J , ∂εeJ tr

∂εeI = B I4 , ∂p M

∂εeI = B I5 , ∂pm

(3.102)

for I, J = 1, 2, 3. Note that only elements in the upper 3 × 5 block of the matrix B are used.

61

Lastly, we develop the elements of the matrix D := ∂r/∂z necessary for constructing the global tangent operator. The components are D AB = ∂r A /∂z B , where the ranges are A ∈ [1, 6] and B ∈ [1, 5]. For A, B = 1, 2, 3, we have D AB = −δAB ,

D4B = 0 ,

∂p ∂pcm − ψ M ecM , ∂εeBtr ∂ε Btr ∂pcm ∂p = e tr − ecM . ∂ε B ∂ε Btr

D5B = −ψm D6B

(3.103)

Continuing, we evaluate D A4 = D A5 = 0 ,

for A = 1, . . . , 5 ,

D64 = 1 ,

D65 = −1 .

(3.104)

3.4.3 Yield function We use a three-invariant plasticity model derived from enhancing the two-invariant modified Cam-Clay theory. The three invariants of the effective Kirchhoff stress tensor τ¯ are r 3 tr(ξ 3 ) 1 1 √ cos 3θ = q= kξ k , , (3.105) τm = tr(τ¯ ) ≤ 0 , 3 2 tr(ξ 2 )3/2 6 where ξ = τ¯ − τm 1. The yield function is of the form (see Borja, 2013) F = ζ2

q2 + τm (τm − p¯ c ) ≤ 0 , M2

(3.106)

where M > 0 is the slope of the critical stress line, and ζ is the scaling function of the form (Argyris et al., 1974; Gudehus, 1973) ζ (ρ, θ ) =

(1 + ρ) + (1 − ρ) cos 3θ . 2ρ

(3.107)

The parameter ρ should be in the range 7/9 ≤ ρ ≤ 1 to guarantee smoothness and convexity. An associative flow rule is assumed throughout.

3.5

numerical examples

The objectives of this section are twofold: (a) to show that the proposed doubleporosity formulation can predict the experimental observations well, and (b) to

62

demonstrate the robust aspects of the algorithm in handling the concurrent transient fluid pressure dissipation and an evolving internal structure. 3.5.1 Stress-point simulations As a first example, we conduct stress-point simulations of the compression of two aggregated soils, namely, Bioley silt and Corinth marl. The goal of the experiments was to investigate the compressibility characteristics of these two aggregated soils under “drained” condition where p¯ c = pcM = pcm . Because there is no transient pore pressure dissipation to consider, it suffices to treat the soil sample as a homogeneously deforming element and conduct stress-point simulations on this element. Koliji et al. (2008b) investigated the compressibility characteristics of an aggregated Bioley silt under a 1D compression test in which the lateral strain was held fixed while the sample was compressed vertically. The experiment entailed measuring the compressibility indices cc and cr for the reconstituted soil, as well as the compressibility index c M for the aggregated (structured) soil. Figure 3.2 shows the relevant experimental data for the structured soil with a preconsolidation pressure of p¯ c = −100 kPa, as well as for the reconstituted soil (Koliji et al., 2008b). From these data, the following compressibility indices were determined: cc = 0.085, cr = 0.009, and c M = 0.4. Note that the macropores do not exhibit elastic deformations, and so all recoverable deformations can be attributed solely to the micropore deformations. 2.8

Specific volume

2.5

Aggregated

2.2 1.9 Reconstituted 1.6

1.3 -0.001

Experiment Calibration -0.01 -0.1 Axial stress (MPa)

-1

-5

Figure 3.2: Variation of specific volume with axial stress for aggregated and reconstituted Bioley silt. Experimental data from Koliji et al. (2008b).

63

Koliji et al. (2008b) then used neutron tomography to investigate the evolution of the macropore void ratio e M as a function of the overall plastic volumetric strain for the same Bioley silt sample. Figure 3.3 shows the experimental data along with the evolution of e M predicted by the proposed constitutive formulation. Very good agreement can be observed. 1

e M /e M0

0.9 0.8 0.7 0.6 0.5

Experiment Simulation 0

0.05 0.1 Volumetric plastic strain

0.15

Figure 3.3: Variation of macropore void ratio e M with volumetric plastic strain for Bioley silt. Initial value e M0 was calculated at a stress level equal to p¯ c = −100 kPa. Experimental data from Koliji et al. (2008b).

Next we consider a similar 1D compression test conducted by Anagnostopoulos et al. (1991) on an aggregated sample of Corinth marl. Their testing program also included triaxial testing on the same soil in addition to 1D compression testing. Predicting the triaxial responses of this soil using parameters calibrated from the 1D compression test is a challenge since the two stress conditions are vastly different. The following discussions compare the triaxial responses predicted by the proposed constitutive framework with those reported previously in the literature. Figure 3.4 shows the experimental data from 1D compression tests on aggregated and reconstituted samples of Corinth marl (Anagnostopoulos et al., 1991). From these data, the following compressibility indices have been determined for this soil: cc = 0.046, cr = 0.004, and c M = 0.8. Using these parameters, the triaxial test was then simulated as a stress-point problem and the results are shown in Figs. 3.5–3.8. We observe that the proposed constitutive formulation predicted both the deviator stress–axial strain and volumetric strain–axial strain responses reasonably well. In contrast, the model proposed by Liu and Carter (2002) significantly overpredicted the deviator stress at a confining pressure of 4 MPa (Fig. 3.5), and also overpredicted the volumetric strain at a confining pressure of 1.5 MPa (Fig. 3.6). On the other hand, the constitutive framework proposed by Koliji et al. (2010a) captured the deviator stress-axial strain responses just as accurately as the 64

1.7 Aggregated

Specific volume

1.6

1.5

1.4

1.3 -0.01

Experiment Calibration

Reconstituted

-0.1 -1 Axial stress (MPa)

-10

-50

Figure 3.4: Variation of specific volume with axial stress for aggregated and reconstituted Corinth marl. Experimental data from Anagnostopoulos et al. (1991).

proposed constitutive model (Fig. 3.7); however, their model underpredicted the volumetric compression at both confining pressures considerably (Fig. 3.8). For the two soils considered, the proposed constitutive formulation appears to adapt to other stress conditions more accurately.

Deviator stress (MPa)

10

Liu and Carter (2002) Simulation

8 6 4 2 0

0

0.05

0.1 Axial strain

0.15

0.2

Figure 3.5: Deviator stress versus axial strain for Corinth marl: comparison with Liu and Carter (2002). Experimental data from Anagnostopoulos et al. (1991): open circles are for test at confining pressure of 4 MPa, open squares at 1.5 MPa..

65

Volumetric strain

0

−0.02 −0.04 −0.06 −0.08 −0.1

Liu and Carter (2002) Simulation 0

0.05

0.1 Axial strain

0.15

0.2

Figure 3.6: Volumetric strain versus axial strain for Corinth marl: comparison with Liu and Carter (2002). Experimental data after Anagnostopoulos et al. (1991), with open circles for test at confining pressure of 4 MPa, and open squares at 1.5 MPa.

Deviator stress (MPa)

10

Koliji et al. (2010) Simulation

8 6 4 2 0

0

0.05

0.1 Axial strain

0.15

0.2

Figure 3.7: Deviator stress versus axial strain responses for Corinth marl: comparison with Koliji et al. (2010a). Experimental data from Anagnostopoulos et al. (1991): open circles are for test at confining pressure of 4 MPa, open squares at 1.5 MPa.

66

Volumetric strain

0

−0.02 −0.04 −0.06 −0.08 −0.1

Koliji et al. (2010) Simulation 0

0.05

0.1 Axial strain

0.15

0.2

Figure 3.8: Volumetric strain versus axial strain for Corinth marl: comparison with Koliji et al. (2010a). Experimental data after Anagnostopoulos et al. (1991), with open circles for test at confining pressure of 4 MPa, and open squares at 1.5 MPa.

3.5.2 1D consolidation with secondary compression Several investigators have argued that secondary compression in clays can be attributed in part to delayed dissipation of excess pore pressures in the micropores of multiple porosity media (Cosenza and Korosak, 2014; Murad et al., 2001; Navarro and Alonso, 2001). Considering that many aggregated soils such as peat show pronounced secondary consolidation, it is interesting to see if the proposed double-porosity formulation could indeed replicate such phenomena. Figure 3.9 shows the finite element mesh considered in the simulation of onedimensional consolidation process in a soil with double porosity. The domain is 0.1 m wide and 1.0 m tall, and discretized into 20 quadrilateral mixed elements with biquadratic displacement interpolation of displacement and bilinear interpolation of the macropore and micropore pressures. The material parameters for the solid were assumed to be the same as those of Bioley silt presented in the previous example. The initial hydraulic conductivities for the macropores and micropores were taken as follows: k M0 = 10−5 m/s and k m0 = 10−9 m/s. The mass transfer term α¯ was assumed to be a linear function of the micropore hydraulic conductivity, with two values considered to investigate its impact on the hydromechanical response: (a) α¯ /k m = 102 s/m, and (b) α¯ /k m = 103 s/m. The initial condition was established, first, by subjecting the material to an isotropic compressive stress of 5 kPa, after which it was loaded to a vertical compressive stress of 200 kPa under fully drained condition on both the macropores and micropores. Then, an additional downward load of w = 10 kPa was applied on top of the mesh while allowing fluid in the macropores to drain to the top

67

surface and fluid in the micropores to drain into the macropores. In other words, the top surface was assumed to be a zero-pressure boundary for the macropores and a no-flow boundary for the micropores, which means that the micropores could only drain into the macropores but not through the top drainage boundary. The consolidation phase began with a time increment of 1 s and with subsequent increments magnified by a factor of 1.01, i.e., ∆tn+1 = 1.01∆tn .

w A

A

solid displacement macro- and micropore pressures

Figure 3.9: Finite element mesh for 1D consolidation with secondary compression.

Figure 3.10 shows the time evolution of the vertical displacement at a point located 0.1 m from the top boundary (level A–A in Fig. 3.9). The shape of the curves suggests two stages of consolidation: a primary consolidation in which the macropores drain through the top boundary, followed by a secondary compression in which the micropores drain into the macropores. Note that for a sufficiently large mass transfer coefficient (¯α/k m = 103 s/m) the secondary compression is not so pronounced; however, for a sufficiently small mass transfer coefficient (¯α/k m = 102 s/m) the two stages of consolidation become very distinct. In the latter case, the first stage of consolidation had enough time to finish even before the second stage of compression could begin, producing a clear separation of scales in time domain. For the same point at level A–A, Fig. 3.11 depicts the time evolution of the preconsolidation stresses p¯ c , pcM , and pcm , relative to the initial value of p¯ c . The macropore preconsolidation stress pcM shows early hardening as the transient macropore water pressure p M dissipates during the first stage of consolidation. However, the hardening response on pcM is followed by mild softening as the transient micropore water pressure pm dissipates and the micropore preconsolidation stress pcm undergoes hardening during the second stage of consolidation. Between t = 105 s and t = 106 s, all three preconsolidation stresses approach a common value as the transient pore water pressures in both the macropores and micropores dissipate. Results of this example indeed show that the proposed 68

Displacement (mm)

0

α¯ /k m = 102 α¯ /k m = 103

−1 −2 −3 −4 −5

0

1

2 3 4 Logarithm of time (s)

5

6

Figure 3.10: Time evolution of vertical displacement at 0.1 m below the top boundary (level A–A) showing separation of scales.Values of α¯ /k m in s/m.

double-porosity formulation can capture the secondary compression attributed to the fluid dissipation in the micropores.

Relative preconsolidation (kPa)

-10 -8 pcM -6 -4 p¯ c

-2 0 2

α¯ /k m = 102 α¯ /k m = 103

pcm 0

1

2 3 4 Logarithm of time (s)

5

6

Figure 3.11: Time evolution of preconsolidation stress, relative to the value of p¯ c at time t = 0, at 0.1 m below the top boundary (level A–A). Values of α¯ /k m in s/m.

Figure 3.12 portrays the spatial and temporal evolution of the internal structure variable ψm for α¯ /k m = 102 s/m. Between t = 102 s and t = 103 s, the spatial distribution resembles what one would expect from a typical one-dimensional consolidation curve, i.e., as the macropores compact, the micropore fraction ψm increases. At t = 103 s, the primary consolidation is essentially complete, and ψm is nearly uniform with depth. Beyond t = 103 s, secondary compression begins, producing nearly uniform compaction of the micropores with depth and a decrease in 69

the value of ψm . Secondary compression appears to end at approximately t = 106 s. Once again, separation of scales is noted, in which the primary consolidation and secondary compression take place sequentially, due to the relatively small value of the mass transfer coefficient α¯ used in the simulation. 1 0.8

t = 102 s t = 103 s t = 104 s

Height (m)

t = 105 s

0.6

t = 106 s

0.4 0.2 0 0.686

0.691 Micropore volume fraction

ψm

Figure 3.12: Spatial and temporal evolution of ψm alongside the finite element mesh. Level A–A is 0.1 m below the top. Solid curves denote primary consolidation; dashed lines denote secondary compression.

3.5.3 Secondary compression beneath a leaning tower The third and final example illustrates the effect of secondary compression on the time-dependent tilting of a seven-storey tower. The finite element mesh for this example accommodates a condition of plane strain and is shown in Fig. 3.13. The foundation of the tower consists of a 10 m-thick sand layer underlain by a 30 mthick clay layer. Under the clay layer is a rigid, impermeable bedrock. The ground water table is located at the top of the sand layer. The tower is assumed to be rigid with a distributed weight of W = 6.6 MN/m at 23.2 m from the base. We should note that this example is only a hypothetical problem, and that the finite element mesh shown in Fig. 3.13 should not be interpreted as a model for any actual tower. The sand layer is assumed to be a Neo-Hookean material with a bulk modulus of 12 MPa, Poisson’s ratio of 0.3, an effective (buoyant) unit weight of 1.0 t/m3 , and a very high permeability. Hence, this layer may be assumed to be incapable of generating excess pore water pressure. The clay layer, on the other hand, is assumed to be a double-porosity material with similar properties to Pancone clay in Pisa, Italy, see Calabresi and Callisto (1998). Figure 3.14 shows the natural compressibility of this clay, as well as the compressibility of the reconstituted material.

70

The fact that the two compressibility curves are not one on top of the other implies that this particular clay exhibits double porosity to a certain extent. Based on these compressibility curves, we have calibrated the material parameters for the clay as cc = 0.176, cr = 0.022, c M = 0.6, φ M = 0.12, and φm = 0.52. The intrinsic mass densities for the solid and water are assumed to be 2.65 t/m3 and 1.0 t/m3 , respectively. The initial hydraulic conductivities of the macropores and micropores are assigned as k M0 = 10−8 m/s and k m0 = 10−11 m/s, respectively, whereas the mass transfer coefficient has the value α¯ /k m = 100 s/m.

23.2 m

W

water table sand

clay

bedrock

Figure 3.13: Finite element mesh for foundation of a leaning tower: clay layer is modeled as a double-porosity material with dual permeability.

To begin the simulation, gravity load was first imposed on the foundation soil under “drained” condition, i.e., with no excess pore fluid pressure developing. Since the Cam-Clay model does not allow a stress-free state, a very small initial isotropic compressive stress of 10 kPa (effective) was first specified at all Gauss integrations points. Once the soil was in static equilibrium under gravity, the overconsolidation ratio of the clay was set to 1.6 (i.e., the size of the yield surface at yield was multiplied by a factor of 1.6) to define a preloaded state. Then, the tower load W was applied under “undrained” condition (i.e., no fluid flow). The tower was given a small geometrical tilt in the beginning so that by the time all of its weight was applied its inclination was around θ ≈ 4.9◦ from the vertical,

see Fig. 3.13. Under this sustained tower weight, the fluid pressure in the soil was allowed to dissipate, causing the tower to tilt further with time. Figure 3.15 shows the time-evolution of the angle θ after all of the tower weight

has been applied. For comparison, we also present the result obtained with singleporosity formulation in which the clay was assumed to have a hydraulic conductiv71

Specific volume

3

2.5

2

1.5 -0.02

Natural Reconstitutied Calibration -0.2 Axial stress (MPa)

-2

Figure 3.14: Compressibility curves for natural and reconstituted Pancone clay. Experimental data from Calabresi and Callisto (1998).

ity equal to the volume-average of the hydraulic conductivities of the macropores and micropores. Observe that the double-porosity simulation predicts faster rate of fluid pressure dissipation, with the tower reaching a steady-state tilt at t ≈ 103

days. This is to be expected since the higher permeability in the macropores would make drainage easier for the same drainage path, whereas the micropores would have a shorter effective drainage path since they could drain directly into the macropores. Also note that separation of scales in time is evident even on the inclination-time response, with the onset of secondary compression in the form of rapid change in slope noted at time t ≈ 20 days. The double-porosity simulation also predicted slightly different tower tilt at steady state, resulting primarily from

the fact that porosity changes in the macropores and micropores were tracked separately whereas the single-porosity simulation did not account for these changes explicitly. Figures 3.16 and 3.17 show the time-evolution of the macropore and micropore fluid pressures, p M and pm , respectively. Observe that p M appears to decrease monotonically with time everywhere (Fig. 3.16), whereas pm exhibits the so-called Mandel–Cryer effect in the vicinity of the tower base. This effect is characterized by a momentary increase in the fluid pressure (note the more intense red region at t ≈ 1 day) before it subsequently dissipates (Fig. 3.17). On the other hand,

Fig. 3.18 depicts the time-evolution of the internal structure variable ψm . The figure suggests a significant increase of the micropore fraction in the vicinity of the tower base at time t ≈ 40 days due to compression of the macropores, followed

by a subsequent decrease at time t ≈ 400 days due to compression of the micro-

pores. In general, this sequential compression of the macropores and micropores

72

θ (degrees)

5.2

Double porosity Single porosity

5.1

5

4.9 0.1

1

10 100 1000 Time (days)

10000

Figure 3.15: Variation of tower inclination θ with time.

is similar to the phenomenon of primary/secondary compression observed in the 1D consolidation example of the previous section.

73

0.1 day

1.0 day

20 days

40 days

400 days

0

60

120

Figure 3.16: Evolution of macropore pressure p M (in kPa) within the clay layer.

74

0.1 day

1.0 day

20 days

40 days

400 days

0

60

120

Figure 3.17: Evolution of micropore pressure pm (in kPa) within the clay layer.

75

0.1 day

1.0 day

20 days

40 days

400 days

0.795

0.805

0.815

Figure 3.18: Evolution of micropore fraction ψm within the clay layer.

76

3.6

closure

We have presented a novel constitutive framework for solid deformation and fluid flow in a porous material exhibiting two pore scales. A unique feature of the framework is the separate treatment of hardening in the macropores and micropores arising from fluid flow through these two pore scales. This allows the evolution of the internal structure, herein represented by the micropore fraction ψm , to be defined concurrently with fluid flow and solid deformation. The formulation has been validated against experimental data on aggregated soils, and has been implemented into a finite element code utilizing a u/p M /pm formulation. Many aggregated soils such as peat and highly compressible clays show pronounced secondary consolidation. This chapter demonstrates that such phenomenon may be interpreted in the context of double-porosity problem as secondary compression attributed in part to delayed dissipation of excess pore pressures in the micropores, which is captured well by the proposed framework.

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78

4

S TA B I L I Z E D F I N I T E E L E M E N T F R A M E W O R K

This chapter is adapted from: Choo, J. and Borja, R. I. (2015). “Stabilized mixed finite elements for deformable porous media with double porosity.” Computer Methods in Applied Mechanics and Engineering, 293, 131–154.

4.1

introduction

Finite element formulation for a deformable solid with double porosity requires an explicit treatment of the macropore and micropore pressures. This is a natural consequence of the fact that the weighted sum of the two pore pressures represents the overall pore pressure that determines the effective stress (Borja and Koliji, 2009). In Chapters 2 and 3, we formulated three-field mixed finite elements with three primary independent field variables, namely, the solid displacement field, the macropore pressure, and the micropore pressure. Following the standard Galerkin approximation to develop the finite element matrix equation, we arrived at a global coefficient matrix possessing a 3 × 3 block structure, in which

the pore pressure contributions from the macropores and micropores occupy 2 × 2

separate matrix blocks.

A form of instability arises in the spatial interpolations of the field variables during undrained deformation when there is no relative flow between the solid and fluid. In the single-porosity formulation, this problem gives rise to a coefficient matrix with a 2 × 2 block structure form-identical to that encountered in the

solution of Stokes flow and incompressible elasticity problems. Instability manifests itself in the form of spurious pressure oscillation, and is widely regarded as the result of the mixed finite element not satisfying the inf–sup condition (Babuška, 1973; Brezzi, 1974; Brezzi and Bathe, 1990). For a thorough discussion of the inf– sup instability in computational poromechanics, we refer the readers to Murad and Loula (1994). Equal-order interpolations of displacement and pressure variables are known to exhibit this type of instability, which historically has been the motivation behind the use of higher-order elements such as the Taylor–Hood elements that employ one order higher interpolation for displacement compared to pressure (Taylor and Hood, 1973). However, such higher-order interpolation inevitably leads to dramatic increase in the problem size, inhibiting large-scale computations.

79

To avoid this computational challenge, various stabilization schemes that allow equal-order interpolations of displacement and pressure fields have been developed, including the Brezzi–Pitkäranta scheme (Brezzi and Pitkäranta, 1984), the Galerkin least-squares approach (Hughes et al., 1986), the variational multiscale method (Hughes et al., 1998), and the polynomial pressure projection (PPP) technique (Bochev et al., 2006; Dohrmann and Bochev, 2004). Successful application of these stabilization schemes has been reported for poromechanics of single-porosity media (Sun et al., 2013; Truty and Zimmermann, 2006; Wan, 2002; White and Borja, 2008). However, whereas a few studies have advanced the finite element formulation for double-porosity poromechanics (e.g., Choo et al., 2016; Elsworth and Bai, 1992; Khaled et al., 1984; Lewis and Ghafouri, 1997; Zhang and Roegiers, 2005), none of them has addressed the issue of stabilization. Due to the presence of two distinct pore scales, stabilization of mixed finite elements for double-porosity media is a challenging endeavor. The micropores typically exhibit a very low permeability, so they generally deform in nearly undrained fashion; however, the macropores can deform in either undrained or drained mode. A prototype example is fissured rock: the low-permeability rock matrix typically inhibits drainage and only recharges fluids from the fissures, where most of the flow takes place. However, when the rate of loading is fast, both the macropores and micropores can deform in an undrained fashion. With the passage of time, drainage can occur much faster in the macropores than in the micropores, and as a consequence, pressure jumps, or very steep gradients between the macro- and micropore pressures, could develop. These pressure jumps enhance the rate of mass transfer between the two pore regions, thereby also affecting their mechanical behavior. Because the two pore pressure fields are treated separately in the proposed double-porosity framework, higher-order elements, such as the Taylor–Hood elements, are now even less desirable to use. Low-order mixed finite elements are almost a necessity, thus further motivating the development of robust stabilization techniques for such elements. In this chapter, we develop stabilized mixed finite elements for deformable porous media with double porosity. At the core of our stabilization is a variant of the PPP technique, which was initially developed for Stokes flow (Bochev et al., 2006; Dohrmann and Bochev, 2004) and later successfully applied to other singleconstraint problems including poromechanics (Bochev and Dohrmann, 2006; Liu and Borja, 2010; Sun et al., 2013; White and Borja, 2008). To focus on the topic at hand, namely, stabilization, we shall limit the solid behavior to linear elasticity in the infinitesimal setting. This chapter is organized as follows. We begin in Section 4.2 by presenting a discrete system emanating from the mixed finite element formulation developed

80

in Chapter 2. In Section 4.3, we describe a variant of the PPP technique developed for dual treatment of the two pressure constraints in the undrained limit that leads to a twofold saddle point problem. The performance and efficacy of the proposed stabilization scheme are demonstrated in Section 4.4 through several numerical examples involving various combinations of drainage regimes in the two scales.

4.2

discrete system

In this chapter we consider the double-porosity formulation for fully saturated cases, in which instability can arise under undrained conditions. The presence of air in the pore space induces some compressibility in the solid matrix, thus the unsaturated case is not so critical with respect to this type of instability. Recall the three-field mixed finite element formulation developed in Chapter 2. At each Newton–Raphson update, we need to evaluate the following Jacobian system (for notational simplicity, hereafter we shall drop the superscript h from the residual R)      Ru R= R pM     R pm

    





 A B1 C1    R ( X ) =  B2 D E1  ,   C2 E2 F 0

,

   

(4.1)

where X is the solution vector with components ∆d, ∆pM , and ∆pm . Note that the Jacobian matrix has a 3 × 3 block structure.

In fully saturated cases, the individual matrices comprising the Jacobian matrix

written in Eqs. (2.53)–(2.61) become as follows: A=−

Z Ω

BT Ck B dΩ ,

(4.2) !

∂c¯h N p dΩ , ∂phM Ω Ω h Z Z ∂c¯ T m p u T C1 = b Bψ N dΩ + ( N ) N p dΩ , h ∂pm Ω Ω B1 =

Z

B2 =

Z

bT Bψ M N p dΩ +

Ω

D = ∆t

Z Ω

( N u )T

( N p )T Bψ M b dΩ , Z Ω

E κ M E dΩ − ∆t

Z

p T

Ω

(N )

∂c M h ∂pm

(4.3) (4.4) (4.5)

T

E1 = −∆t C2 =

Z

Z Ω

( N p )T

N p dΩ ,

( N p )T Bψm b dΩ ,

∂c M ∂phM

! N p dΩ ,

(4.6) (4.7) (4.8)

81

E2 = −∆t F = ∆t

Z

Z

p T

Ω

(N )

!

E κm E dΩ − ∆t T

Ω

∂cm ∂phM

N p dΩ , Z

p T

Ω

(N )

(4.9)

∂cm h ∂pm

N p dΩ.

(4.10)

We remark that when the diffusive mass transfer is zero, all the derivatives of c M and cm with respect to the two pore pressure variables vanish. In this case, E1 = E2 = 0, B1 = B2T , and C1 = C2T , leading to a symmetric Jacobian matrix having a form similar to the 2 × 2 block matrix arising in single-porosity formu-

lation (White and Borja, 2008). We also remark that in the limit of undrained

deformation, which occurs either when ∆t → 0 (fast rate of loading) and/or when κ M , κm → 0 (impermeable solid), combined with the condition of no diffusive

mass transfer, the entire 2 × 2 lower right-hand side block consisting of submatrices D, E1 , E2 , and F vanishes, resulting in a coefficient matrix that potentially could be either singular (i.e., problem is not solvable) or unstable in the sense of the inf–sup condition for twofold saddle point problems, see Howell and Walkington (2011). Double-porosity formulation could also lead to another interesting scenario, and that is when κm → 0, which leads to F → 0. The result is an incompressible

micropore but a compressible macropore (if drainage in the macropore is allowed).

While this condition may appear to be somewhat similar to that encountered in incompressible elasticity or Stokes flow, in which a diagonal submatrix block also vanishes, the physics of the present problem is not quite the same because of the dual nature of porosity. In any case, whether it is the micropore alone that is incompressible, or it is the entire mixture that cannot change in volume, instabilities in the undrained limit will naturally impact the performance of many mixed finite elements. Therefore, this issue must be addressed with a suitable stabilization.

4.3

stabilized mixed finite elements

The subject of the present stabilization study is the class of low-order mixed finite elements employing equal-order (linear) interpolations of displacement and pressure fields. These elements do not satisfy the inf–sup condition and are known to exhibit pronounced pressure oscillation in the undrained limit. We should note, however, that this class of mixed finite elements generally perform well under drained conditions. Therefore, we seek an effective stabilization that circumvents the pressure oscillation in the undrained limit but preserves the acceptable performance in drained calculations. The added challenge in double-porosity stabilization is the high contrast in the permeabilities of the macropores and micropores.

82

The stabilization needs to address potential oscillation in the micropore pressures even if the macropore pressure responses seem stable. Before considering the double-porosity stabilization, we first review the general stability condition for mixed finite elements in single-constraint problems such as Stokes flow, incompressible elasticity, and single-porosity poromechanics, for which the stabilization technique extended in this work was originally proposed. The discrete inf–sup condition for velocity and pressure interpolations reads (Brezzi and Bathe, 1990)

R sup

Ω

vh ∈Suh

ph ∇ · vh dΩ ≥ C k p h k0 h k v k1

(4.11)

for all ph ∈ S ph , where Suh and S ph are the discrete spaces for velocity and pressure

interpolations, respectively, k · kk is the standard Sobolev norm with order k, and

C is a positive constant independent of element size h. This condition determines

the bound for an intrinsically stable pair of finite element spaces Suh and S ph .

Unfortunately, equal-order linear pair elements do not satisfy the condition

described in the preceding paragraph. However, Bochev et al. (2006) showed that they do satisfy a weaker inf–sup condition given by

R sup vh ∈Suh

Ω

ph ∇ · vh dΩ ≥ C1 k ph k0 − C2 k ph − Πph k0 k v h k1

(4.12)

for all ph ∈ S ph , where Π : L2 (Ω) → R0 is a projection operator from spaces of L2 to piecewise constants R0 , and C1 and C2 are positive constants independent

of h. Comparing Eqs. (4.11) and (4.12), we find that the last term, C2 k ph − Πph k0

in Eq. (4.12), quantifies the deficiency of the equal-order linear pair elements. The idea behind the PPP technique is then to penalize this deficiency by augmenting a stabilization term to the discrete variational equation. The projection operator Π for the discrete pressure field ph can be defined as Πph |Ωe =

1 Ve

Z Ωe

ph dΩ ,

(4.13)

where Ωe and V e denote the domain and volume of an element, respectively. In words, the operator Π projects a given field to its average value within the element. For single-constraint problems, specific stabilization terms for penalizing the deficiency have essentially taken the same form except for the constants. We emphasize that in all cases the stabilization term should be sufficiently large to penalize the deficiency, but not too large as to be overly diffusive and induce undesirable smoothing.

83

For single-porosity poromechanics, a stabilization term has been proposed by White and Borja (2008) for the mass residual term of the form

M=

Z Ω

1 ˙ (ω h − Πω h )( ph − Πph ) dΩ , 2G

(4.14)

where G is the shear modulus of the solid matrix and the superimposed dot denotes a time differentiation. After temporal and spatial discretization, the expression becomes 1 M = ∆t 0

Z Ω

1 [ N p − Π( N p )][( ph − pnh ) − Π( ph − pnh )] dΩ , 2G

(4.15)

where ω and N p are the weighting and shape functions for the pore pressure, respectively. We note that the stabilization term originally proposed by Bochev et al. (2006) is inherently parameter-free, much like Eqs. (4.14) and (4.15). However, White and Borja (2008) introduced a parameter τ to avoid over-diffusion that may occur when steep pressure gradients are involved, such as in localized zones (e.g., faults) where the permeabilities between adjacent material zones are orders of magnitude different. Recalling that the double-porosity framework treats such heterogeneity as different fields, it is very unlikely that similar over-diffusion in the pressure field would be encountered. Therefore, we shall revert back to the original parameter-free PPP technique. To accommodate two pore pressure constraints, we consider a twofold saddle point problem of the generic form (see Howell and Walkington, 2011 for details): Find (u, p1 , p2 ) ∈ Su × S p1 × S p2 such that a(u, v) + b1 ( p1 , v) + b2 ( p2 , v) = f (v)

(4.16)

b1 (q1 , u) = g1 (q1 )

(4.17)

b2 (q2 , u) = g2 (q2 ) ,

(4.18)

for all v ∈ Su , q1 ∈ S p1 , and q2 ∈ S p2 , where a, b1 , and b2 are bilinear operators.

As noted in Howell and Walkington (2011), this can be viewed as a single saddle point problem on Su × (S p1 × S p2 ), with bilinear form b(( p1 , p2 ), u) = b1 ( p1 , u) + b2 ( p2 , u) .

(4.19)

The relevant inf–sup condition is given by (see Howell and Walkington, 2011) sup

v∈Su

b1 ( p1 , v) + b2 ( p2 , v) ≥ C (k p1 kS p1 + k p2 kS p2 ) k v k Su

for all ( p1 , p2 ) ∈ S p1 × S p2 , with C > 0.

84

(4.20)

We are interested in the condition where b1 = b2 ≡ b, and with k · kS p1 =

k · kS p2 ≡ k · kS p , which is motivated by the same order of Sobolev spaces for the two pressure fields. In this case, the relevant inf–sup condition, using the triangle inequality, is given by sup

v∈Su

b ( p1 + p2 , v ) ≥ C (k p1 kS p + k p2 kS p ) k v k Su

≥ C (k p1 + p2 kS p )

(4.21)

for all ( p1 , p2 ) ∈ S p × S p , with C > 0. One can, of course, make the substitutions p1 ← (1 − φ) p1 and p2 ← φp2 , where 0 ≤ φ ≤ 1, so that the inf–sup condition

can simply be phrased as a single saddle point problem in the mean of the two pressures. This motivates the following stabilization for the problem at hand. Consider the problem of double porosity as a twofold saddle point problem. The inf–sup condition in the discrete space takes the form

R sup

Ω

vh ∈Suh

h ) ∇ · v h dΩ B(ψ M phM + ψm pm h k0 ) ≥ C (k Bψ M phM k0 + k Bψm pm h k v k1 h ≥ C (k B(ψ M phM + ψm pm )k0 )

(4.22)

h ∈ S h , with C > 0 independent of h. for all phM ∈ S ph and pm p

Comparing Eq. (4.22) with Eq. (4.11), we find that the pressure constraint p in

single-constraint problems is now replaced by the mean pore pressure weighted according to the pore fractions (and accommodating the Biot coefficient B as well, to make the representation more physically meaningful): h p¯ h = B(ψ M phM + ψm pm ).

(4.23)

Note that both p for single porosity and p¯ for double porosity emanate from the effective stress equation. Following Bochev et al. (2006), we now treat the twofold saddle point problem as a single saddle point problem in the mean pressures, and write the corresponding weaker condition as

R sup vh ∈Suh

Ω

p¯ h ∇ · vh dΩ ≥ C1 k p¯ h k0 − C2 k p¯ h − Π p¯ h k0 k v h k1

(4.24)

h ∈ S h . Once again, the last term C k p h ¯ h k0 quantifies for all phM ∈ S ph and pm 2 ¯ − Πp p

the deficiency of the equal-order interpolations.

The stabilization term to penalize the resulting deficiency can be readily developed. The trial function for the stabilization term is p¯ defined in Eq. (4.23).

85

Accordingly, the Galerkin formulation dictates a similar form for the weighting function, h ω¯ h = B(ψ M ω hM + ψm ωm ).

(4.25)

This gives rise to the following stabilization term patterned after the expression proposed by White and Borja (2008): 1 ˙ (ω¯ h − Πω¯ h )( p¯ h − Π p¯ h ) dΩ Ω 2G Z 1 ˙ Bψ M (ω hM − Πω hM )( p¯ h − Π p¯ h ) dΩ = Ω 2G Z 1 ˙ h h Bψm (ωm + − Πωm )( p¯ h − Π p¯ h ) dΩ . Ω 2G

¯ = M

Z

(4.26)

¯ naturally splits into components associObserve that the expression for M

ated with the two weighting functions for the macropore and micropore pressures. These components provide the necessary stabilization terms for the mass balance equations for the macropores and micropores. In terms of the residual vectors, we obtain (after temporal and spatial discretization of Eq. (4.26) and multiplying it by ∆t)

Ω

1 Bψ M [ N p − Π( N p )][( p¯ h − p¯ nh ) − Π( p¯ h − p¯ nh )] dΩ , 2G

Ω

1 Bψm [ N p − Π( N p )][( p¯ h − p¯ nh ) − Π( p¯ h − p¯ nh )] dΩ . (4.28) 2G

¯ p = Rp + R M M

Z

¯ p = Rp + R m m

Z

(4.27)

¯ p and R ¯ p have been stabilized, the Jacobian Because the residual vectors R m M

matrix also must be stabilized. The 2 × 2 lower right-hand submatrix block now

contains the following additional terms

2 1 Bψ M [ N p − Π( N p )][ N p − Π( N p )] dΩ , Ω 2G Z 1 B2 ψ M ψm [ N p − Π( N p )][ N p − Π( N p )] dΩ , E¯ 1 = E1 + Ω 2G Z 1 E¯ 2 = E2 + B2 ψm ψ M [ N p − Π( N p )][ N p − Π( N p )] dΩ , Ω 2G Z 2 1 F¯ = F + Bψm [ N p − Π( N p )][ N p − Π( N p )] dΩ . Ω 2G ¯ = D+ D

Z

86

(4.29) (4.30) (4.31) (4.32)

Consequently the new Jacobian matrix now takes the form 

   A B C A B C 1 1  1 1        ¯ E¯ 1  .  B2 D E1  →  B2 D     ¯ ¯ C2 E2 F C2 E2 F

(4.33)

Note that the additional stabilization terms form a symmetric matrix as a result of the Galerkin formulation. Before closing this section, we briefly address the issue of solvability of the undrained problem with double porosity. The extreme case of interest occurs when the matrices D, E1 , E2 , and F are all zero, which corresponds to fully undrained condition at the two pore scales and without stabilization. Ignoring the mass transfer terms, the coefficient matrix becomes symmetric when A = AT , i.e., 

  T T A B C 1 1    A B2 C2     B2 D E1  →  B2 0 0    C2 E2 F C2 0 0

    T A G   , →  G 0

(4.34)

where GT = [ B2T C2T ]. We see that undrained loading in a double-porosity medium creates a condition in which the width of the coupling matrix GT increases about two times relative to that emanating from a single-porosity problem. This means that the null submatrix block on the diagonal also doubles in size relative to the single-porosity problem. The solvability aspects of a problem having a coefficient matrix of the form given by Eq. (4.34) have been addressed by Brezzi and Bathe (1990), who in turn credited Arnold (1981) for an “elegant presentation.” In a nutshell, the undrained problem is solvable provided that: (a) the size of A is greater than the size of the null submatrix block on the diagonal, thus ensuring that the kernel set of G is not an empty set; (b) the square matrix formed from G by eliminating the elements of the kernel space of G is not singular, and (c) the square submatrix of A associated with the kernel space of G is not singular. We refer to Gatica (2002) and Howell and Walkington (2011) for further discussions on this aspect.

4.4

numerical examples

This section presents numerical examples demonstrating the performance and efficacy of the proposed stabilized mixed finite elements. We begin by verifying our finite elements through Cryer’s sphere, a classic single-porosity poromechanics

87

problem for which an analytical solution exists. Subsequently, we perform a comparative investigation into the stabilized low-order mixed finite elements with intrinsically stable high-order and unstable low-order elements. For this purpose, we first consider an undrained double-porosity sphere under spatially varying loading. We then introduce a plane strain strip footing problem and examine the efficacy of the proposed stabilization scheme when the permeabilities of the two pore scales are orders of magnitude different. This difference not only affects the pressure difference between the macropores and micropores, but also controls the combinations of drainage regimes in the two scales. The following notations are used for referring to the mixed finite elements of interest. The interpolation for displacement is denoted by Q and for pressure by P, followed by a number representing the order of interpolation. For example, Q2P1P1 is a mixed element employing second-order (quadratic Lagrangian) interpolation for displacement and first-order (linear) interpolation for the two pressures, while Q1P1P1 adopts the same linear interpolation for displacement and the two pressure fields. We also use a lower case “s” to denote stabilization, i.e., Q1P1P1s refers to the stabilized mixed finite elements investigated in this chapter. Throughout this section we have employed the same level of interpolation for the two pressure fields, and used the same pressure nodes for the micropore and macropore pressure interpolations. 4.4.1 Cryer’s sphere The objective of our first example is to verify the formulation and implementation of our mixed finite elements for double-porosity poromechanics. We are not aware of a closed-form analytical solution to any problem related to fully coupled double-porosity poromechanics, so we have selected Cryer’s sphere problem (Cryer, 1963) as a test example. Cryer’s sphere is a classic single-porosity problem that can be solved by Laplace transform. In this problem a saturated poroelastic sphere is compressed by applying a uniform pressure on its outer surface. The analytical solution of this problem indicates that the excess pore pressure initially rises before it begins to dissipate, a phenomenon often referred to in the literature as the Mandel–Cryer effect. This initial pressure increase is a signature feature of coupled responses between solid deformation and fluid diffusion. Cryer’s sphere problem has served as a useful reference in the past to verify various formulations and implementation of coupled poromechanics solutions. For verification we performed single-porosity simulations of Cryer’s sphere problem with the proposed double-porosity mixed finite elements. This was done by assigning a null porosity to all integration points of the micropores, and zero

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micropore pressures for the initial and boundary conditions at the nodes. Mass transfer was ignored as well. By comparing results from single-porosity finite elements, we confirmed that this setting successfully emulates a single-porosity condition by maintaining zero pressures in the micropores throughout all time steps. We also checked to verify that the double-porosity mixed elements could capture the analytical solution by suppressing the macropore pressures and activating only the micropore pressure degrees of freedom. We conducted the simulations using three types of mixed finite elements: Q2P1P1, Q1P1P1, and Q1P1P1s. We modeled an octant of a unit sphere taking advantage of symmetry. For Q2P1P1 the domain was discretized with 3456 quadrilateral elements, resulting in a total of 98245 degrees of freedom (90099 for displacement and 4073 for each pressure type); for Q1P1P1s the same number of elements resulted in a total of 20365 degrees of freedom (12219 for displacement and 4073 for each pressure type). Note that the total number of degrees of freedom generated by Q2P1P1 elements is much higher than that generated by the lower-order elements. In addition, higher-order elements also require a greater number of integration points. The potentially significant reduction in computing effort is the primary motivating factor for stabilizing these elements. The solution of Cryer’s sphere problem is a function of Poisson’s ratio for the solid matrix. Therefore, we considered three values of Poisson’s ratio: ν = 0, 0.25 and 0.4, and fixed all other parameters at constant values except for the permeability coefficient that was adjusted to normalize the time axis. The rest of the parameters were assigned the following values: bulk modulus K=1000 kPa, ρs = ρ f = 1.0 t/m3 , φ M = 0.5 or φm = 0.5 with φ M + φm = 0.5, µ f = 10−6 kPa · s,

and k f = 3.33 × 10−12 to 8.18 × 10−12 m2 . We assumed B = 1, which is a realistic

value for soils. We simulated the sphere problem until the non-dimensionalized time factor T = cv t/r2 has reached a value of 0.2 after using 40 time steps, where cv is the coefficient of consolidation, t is time, and r is the drainage length. In what follows, we express the solutions in terms of normalized pore pressure p/p0 , where p is the pore pressure normalized with respect to the boundary pressure p0 . Figure 4.1 shows the distribution of normalized pore pressure p/p0 at the initial time step T = 0.005 when ν = 0. The three types of element gave virtually the same results, so we only present the solutions from the Q1P1P1 simulations in this figure. Also, the double-porosity simulations obtained by activating either only the macropores or only the micropores gave identical solutions; therefore, the figure does not distinguish between the two solutions. We find that even without stabilization equal-order linear interpolations do not exhibit pressure oscillations, since this problem deals with drained conditions right from the initial time step

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(note that the focus of this example is only on the verification of the finite element implementation, and not on stabilization during undrained condition). In Figure 4.2 we show the evolutions of p/p0 at the center of the sphere predicted by the analytical and numerical solutions. The three mixed elements provided essentially the same results that agree with the analytical solutions. These results thus verify the formulation and implementation of the mixed finite elements under drained loading conditions.

Figure 4.1: Normalized pore pressure p/p0 in Cryer’s sphere with ν = 0 at time factor T = 0.005. 2

p/p0

1.5

ν=0 ν = 0.25

1

ν = 0.4 Analytical Q2P1P1 Q1P1P1 Q1P1P1s

0.5

0

0

0.05

0.1 T

0.15

0.2

Figure 4.2: Comparison of analytical and numerical solutions at the center of Cryer’s sphere.

We note two important observations from this example. First, under drained condition the equal-order unstabilized element, Q1P1P1, performs well and does not exhibit any instability. Second, the PPP technique does not introduce any numerical artifacts that significantly alter the drained solution. This latter feature is important because, as noted earlier, poromechanics problems typically cover the entire range of drainage conditions, from fully undrained to fully drained. There90

fore, a robust stabilization scheme should not alter the solution in the regime where no stabilization is needed. The next example shows how the proposed stabilization scheme can improve the performance of an inherently unstable mixed element in the undrained regime. 4.4.2 Undrained double-porosity sphere Having verified our mixed finite elements, we now turn our attention to undrained deformation in double-porosity materials and consider the same unit sphere as a next example. In this example, all boundaries of the sphere were treated as noflux boundaries. The modified problem does not allow an analytical solution, but it does provide a simple setting for investigating the efficacy of the proposed stabilization technique in a 3D double-porosity problem. We assumed a double-porosity material with φ M = 0.3 and k M = 6 × 10−14

m2 for the macropores, and φm = 0.1 and k m = 6 × 10−17 m2 for the micropores,

resulting in a permeability ratio of k M /k m = 103 . The two pore scales were cou-

pled with mass transfer terms in which a = 0.1 m, β = 3.0, and γ = 0.4. For the solid matrix we assigned K = 1000 kPa and ν = 0.25. All other parameters remain the same as in the previous example. To impose pressure variations in the domain, we applied a spatially varying pressure 1 + 0.5 sin( x ) kPa on its outer surface. This pressure variation perturbed the stress field from the isotropic condition considered by Cryer to a full 3D stress field within the sphere. Figure 4.3 compares the excess pore pressures obtained using the three mixed elements after 0.1 second of loading. From the results of simulations with Q2P1P1 elements, we first observe that pressure distributions in the micropores are more localized than those in the macropores due to the much lower permeability of the micropores. The micropore pressures predicted by the unstabilized Q1P1P1 elements exhibit a checkerboard pattern, which is a typical spurious mode associated with numerical instability. We also observe some mild macropore pressure oscillations produced by the unstabilized Q1P1P1 elements in places with the pressure gradients are high. In contrast, the reference solution provided by the higher-order Q2P1P1 elements is smooth. Furthermore, the results obtained from the stabilized Q1P1P1s elements also show a smooth pattern that is nearly identical to those produced by the higher-order Q2P1P1 elements. We note that the stabilization successfully eliminated oscillation at the two scales, resulting in nearly identical solutions to those provided by the higher-order elements.

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Macropores

Micropores

Q2P1P1

Q1P1P1

Q1P1P1s

Figure 4.3: Excess pore pressures in double-porosity sphere after 0.1 second of undrained loading. Surface pressure varies spatially according to 1 + 0.5 sin( x ).

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4.4.3 Strip footing Our next example is a plane strain strip footing problem on a foundation material with double porosity. We adopted a typical configuration of strip footing for this example. Figure 4.4 illustrates the specific geometry and boundary conditions. Considering symmetry, we modeled only one-half of a 4 m wide footing resting on the ground. The fluid boundary conditions were identical for both pore scales except for the ground surface, which was assumed to be a homogeneous Dirichlet boundary for the macropores (i.e., p M = 0) and a Neumann boundary (no flux) for the micropores. The reason for suppressing drainage through the micropore ground surface is to eliminate unnecessary oscillations occurring near the drainage boundary of a low-permeability medium (see Murad and Loula, 1992 for example). This oscillation is due to sharp pressure gradients, or shocks, which is not related to the spurious pressure oscillation stemming from the inf–sup instability. Shocks in the form of sharp pressure gradients can be treated by other techniques employing discontinuous pressure interpolation, such as the finite volume method. Stabilization of such oscillation near drainage boundaries is a topic of other studies (e.g., Preisig and Prévost, 2011) and will not be covered in this work. To fully focus on the instability at hand, we assumed the micropore scale as a globally undrained medium. Regarding the displacement field, we fixed the vertical displacements at the bottom boundary and the horizontal displacements at the two vertical sides. We selected Point A, which is located 0.5 m below the center of the footing, as a reference point for reporting the evolutions of the two pore pressures and displacement with time.

A

Drainage for macropores / No flux for micropores

Symmetry

2m

y Vertical roller / No flux

Horizontal roller / No flux

Traction

5m

x 10 m

Figure 4.4: Schematic illustration for the geometry and boundary conditions of the strip footing problem.

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We investigated the mesh sensitivity issue by employing two finite element meshes depicted in Fig. 4.5. The total number of degrees of freedom with Q2P1P1 and Q1P1P1(s) mixed elements are 8364 and 3444 for Mesh #1, and 32724 and 13284 for Mesh #2, respectively. We note that even in 2D analysis, the reduction in the total number of degrees of freedom engendered by the low-order stabilized elements is still fairly significant, although it is not as significant in this case as in the 3D problem of the previous example.

Mesh #1

Mesh #2

Figure 4.5: Two meshes used in the strip footing example. Mesh #1 has 800 elements while Mesh #2 has 3200 elements.

We examined the performance of the proposed stabilization scheme by considering the permeabilities of the macropores and micropores (k M and k m , respectively) to be several orders of magnitude different. For the micropores, we assumed the following values: φm = 0.1 and k m = 5 × 10−17 m2 . For the macropores, we assumed φ M = 0.05 and k M = 1 × 10−10 m2 for Case #1, and decreased

the permeability to 5 × 10−15 m2 for Case #2. This leads to a drop in the perme-

ability ratio k M /k m from a value of 2 × 106 for Case #1 to a value of 1 × 102 for

Case #2. As will be shown later, this difference in the permeability ratio also alters the combination of drainage conditions from the initial step. The remaining

parameters were identical for both cases. We activated the diffusive mass transfer between two pore scales, with a = 5 × 10−2 m, γ = 0.4 and β = 11.0. We assumed the following properties for the

pore fluid: ρ f = 1.0 t/m3 and µ f = 1 × 10−6 kPa · s; and for the solid matrix: ρs = 2.6 t/m3 and B = 0.9. We considered the foundation medium to be linearly

elastic with bulk modulus K = 2000 kPa and Poisson’s ratio ν = 0.2. We applied the footing load as a ramp function that increases from an initial value of zero to a final value 20 kPa over a period of 180 s, after which it was held constant at 20 kPa for the next 180 s. Time steps taken were ∆t = 5 s during the ramp-loading phase, and ∆t = 10 s thereafter. In what follows, we consider the variation and distribution of the excess pore pressure fields induced by the footing load. We first focus on the two pore pressure fields at the initial time step when the system is deforming in undrained fashion within the entire time scale. In particular, we focus on the pressure variations 94

below the center of the footing (i.e., along the symmetry line). We then present time histories of macropore and micropore pressures at Point A illustrated in Fig. 4.5. 4.4.3.1

Case #1: Higher permeability contrast

We first consider the case in which the macropores are 2 × 106 times more permeable than the micropores. This condition is analogous to that of a highly fissured

rock, where the permeability of the fissures is several orders of magnitude higher than that of the rock matrix. Figure 4.6 compares the macropore and micropore pressures below the center of the footing at the initial step. Significant difference can be observed between the two pore pressure fields both quantitatively and qualitatively. Due to the much lower drainage within the time interval, the micropore pressures are around two orders of magnitude higher than the macropore pressures. Also, the micropore pressures decrease from the top drainage surface to the bottom, while the macropore pressures slightly increase with depth. Setting the foregoing results as the reference solution, we next focus on the stability of the Q1P1P1 mixed elements with Mesh #1. The macropore pressures calculated with these unstable elements are nearly the same as those calculated with Q2P1P1 elements. However, the micropore pressures exhibit pronounced spurious oscillations. With Mesh #2, on the other hand, these oscillations are somewhat suppressed but not completely eliminated. In contrast, the stabilized Q1P1P1s elements provide smooth solutions that are nearly identical to those obtained with the Q2P1P1 elements. As the mesh is refined (Mesh #2), the stabilized solutions became even closer to those calculated by the higher-order elements. In Fig. 4.7 we compare the two pore pressure fields generated at the same time instant using Mesh #2. As expected, the two pore pressure fields show completely different distributions throughout the entire domain. With respect to stability, the unstabilized equal-order pair Q1P1P1 elements exhibit spurious micropore pressure oscillations, although the macropore pressure distribution shows a smooth, stable pattern. These features are unique to double-porosity formulation and is not encountered in single-porosity poromechanics, i.e., stability on one scale does not necessarily imply stability on another scale. More importantly, the proposed stabilization technique completely circumvents instabilities irrespective of scale — it suppresses instabilities where they would otherwise arise, and preserves the correct response at the scale where no stabilization is needed. We next examine the time-evolution of pore pressures at Point A shown in Fig. 4.8. Not surprisingly, all three mixed elements yield essentially the same macropore pressure-time responses. The micropore pressure response calculated by the unstabilized Q1P1P1 elements still deviates slightly from that calculated by

95

5

5

4

4

3

3

y (m)

y (m)

(a) Mesh #1

2 Q2P1P1 Q1P1P1 Q1P1P1s

1 0

0

0.002 0.006 0.004 Macropore pressure (kPa)


1 0

0.008

0

0.3 0.6 0.9 Micropore pressure (kPa)

1.2

5

5

4

4

3

3

y (m)

y (m)

(b) Mesh #2


1 0

0



1

0.008

0

0


1.2

Figure 4.6: Pore pressure variations below the center of footing at the initial time step: Case #1 (k M /k m = 2 × 106 ).

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Macropores

Micropores

Q2P1P1

Q1P1P1

Q1P1P1s

Figure 4.7: Contours of macropore and micropore pressures at the initial time step: Case #1 (k M /k m = 2 × 106 ) with Mesh #2.

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the higher-order elements, but the deviation is not significant. No oscillation in the time domain is observed, a testament of the high-frequency numerical damping of the backward Euler method. Note that both the macropore and micropore pressure histories show a decreasing trend during the constant footing load period: the macropore pressure decreases as fluid drains into the ground surface, while the micropore pressure decreases as the micropore fluids drain into the macropores (recall that the micropore fluids cannot drain into the exterior boundaries because of the no-flux boundary condition on all exterior surfaces prescribed for the micropore scale). On the other hand, Figure 4.9 shows that the displacement at Point A remains nearly unaffected by the pressure instabilities. (a) Mesh #1 25 Micropore pressure (kPa)

Macropore pressure (kPa)

0.16

0.12

0.08

0.04

0

Q2P1P1 Q1P1P1 Q1P1P1s 0

90

180 Time (s)

270

20 15 10

0

360

Q2P1P1 Q1P1P1 Q1P1P1s

5

0

90

180 Time (s)

270

360

(b) Mesh #2 25 Micropore pressure (kPa)


0.16

0.12

0.08

0.04

0


90

180 Time (s)

270

360

20 15 10 Q2P1P1 Q1P1P1 Q1P1P1s

5 0

0

90

180 Time (s)

270

360

Figure 4.8: Time histories of pore pressures at Point A: Case #1 (k M /k m = 2 × 106 ).

4.4.3.2

Case #2: Lower permeability contrast

Next we investigate the instability patterns and examine the performance of the proposed stabilization scheme when the ratio of the two permeabilities is lower, at

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Displacement (mm)

0

Q2P1P1 Q1P1P1 Q1P1P1s

-5

-10

-15

0

90

180 Time (s)

270

360

Figure 4.9: Time history of displacement at Point A: Case #1 (k M /k m = 2 × 106 ) with Mesh #1.

k M /k m = 1 × 102 . Figure 4.10 shows the pore pressure variations below the center

of the footing at the initial time step. We see that the macropore and micropore

pressures calculated by the Q2P1P1 elements are now fairly close to each other, unlike in the previous simulations where they were orders of magnitude different. This is attributed to extensive diffusive mass transfer that allowed the two pore pressures to be nearly the same. Figure 4.10 shows that simulations with Q1P1P1 elements along with the coarser Mesh #1 now exhibit spurious oscillations in both the macropore and micropore pressures. The oscillations have been somewhat alleviated with the finer Mesh #2, but not completely circumvented particularly on the micropore scale. However, the proposed stabilization scheme completely eliminates these oscillations at both scales, suggesting the efficacy of the technique when simultaneous multiscale stabilization is required. Furthermore, the results show that instabilities do not depend on the magnitude of the pressure per se, but rather, on the degree of permeability of the material. This can be gleaned from the fact that the calculated macropore and micropore pressures are nearly the same for this example, yet the magnitudes of oscillation are significantly different. Figure 4.11 shows the overall distributions of the macropore and micropore pressures at the initial time step obtained from the finer Mesh #2. Qualitatively, the pore pressure distributions within the two scales are somewhat similar, although the micropore pressure distribution is a little bit more concentrated below the footing. The unstabilized Q1P1P1 elements once again exhibit pore pressure oscillations at both scales, although oscillations at the macropore scale concentrate more in places with sharp pressure gradients, and are generally milder. In

99

5

5

4

4

3

3

y (m)

y (m)

(a) Mesh #1


1 0

0



1 0

1

0


1

5

5

4

4

3

3

y (m)

y (m)

(b) Mesh #2


1 0

0



1

1

0

0


1

Figure 4.10: Pore pressure variations below the center of footing at the initial time step: Case #2 (k M /k m = 1 × 102 ).

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contrast, the stabilized Q1P1P1s elements generate smooth pressure distributions throughout, and essentially reproduce the results generated by the higher-order Q2P1P1 elements.

Macropores

Micropores

Q2P1P1

Q1P1P1

Q1P1P1s

Figure 4.11: Contours of macropore and micropore pressures at the initial time step: Case #2 (k M /k m = 2 × 102 ) with Mesh #2.

Figure 4.12 shows the time histories of the macropore and micropore pressures at the reference Point A. We see that the Q1P1P1 solutions with Mesh #1 generated larger numerical errors in the micropore pressures, although errors in the macropore pressures are much smaller. Even though the permeability of the micropore region is the same as in the first case considered, the errors in time histories become much larger in the present case. This can be explained by extending the reasoning provided in the previous case: since the permeability contrast is much lower in the present case, fluids in the micropores cannot drain as easily into the macropores, because the permeability of the macropores is now closer to that of the micropores. The virtually constant post-peak pressure, which was decreasing in the previous case, is evidence of this drainage difference. However, these errors seem to vanish when the finer Mesh #2 is used. Also, in all cases, the time histories of Q2P1P1 and Q1P1P1s are nearly the same, suggesting that stabilization is indeed effective. While not presented in this chapter, we also point out that the

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spurious oscillations observed in the pore pressure fields have not impacted the stability of the displacement field. (a) Mesh #1 20 Micropore pressure (kPa)


20

15

10

5

0


90

180 Time (s)

270

15

10

5

0

360


90

180 Time (s)

270

360

(b) Mesh #2 20 Micropore pressure (kPa)


20

15

10

5

0


90

180 Time (s)

270

360

15

10

5

0


90

180 Time (s)

270

360

Figure 4.12: Time histories of pore pressures at Point A: Case #2 (k M /k m = 1 × 102 ).

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4.5

closure

This chapter has presented stabilized low-order mixed finite elements for coupled solid deformation and fluid diffusion in a material exhibiting two pore scales. The stabilization is a variant of the PPP technique developed for dual treatment of two distinct pore pressure constraints. One appealing aspect of the proposed technique is the physical motivation behind the stabilization: the mean trial pore pressure is a weighted mean determined according to the pore fractions, and so the mean weighting function should also be a weighted mean of the pore pressure variations determined according to the same pore fractions. Remarkably, this physical motivation is backed up by mathematical developments, in which a twofold saddle point problem is used to develop an equivalent single saddle point problem in terms of the weighted mean pore pressures. This leads to a parameter-free PPP stabilization technique for the equivalent single saddle point problem in the mean pore pressures, which allows equal-order interpolations of the solid displacement and the two pore pressure fields. Numerical examples have demonstrated the efficacy of the proposed stabilization technique under various drainage conditions, from fully undrained to fully drained conditions, and from 2D to 3D loading and drainage configurations. Results of these studies can have a significant impact on the stabilization of numerical algorithms related to similar multiphysical and multiscale problems in science and engineering, such as those arising in contact problems.

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5

CLOSURE

5.1

summary

This thesis has developed theoretical and computational modeling frameworks for coupled hydromechanical processes in two-scale porous materials, such as aggregated soils and fractured rocks. To capture fluid flow through two-scale pore spaces, we conceptualized these materials as a double-porosity continuum composed of two coexisting pore regions of different scales. We then focused on modeling deformations in double-porosity media and their interactions with two-scale fluid flow, with the following objectives: establishing a rigorous theoretical underpinning for coupled hydromechanical modeling, developing a reliable constitutive model for two-scale deformations, and improving the computational efficiency of finite element solutions. With these objectives in mind, in Chapter 2 we first developed a theoretically consistent framework for coupled hydromechanical modeling of unsaturated porous media with double porosity. The development of the framework relied on continuum principles of thermodynamics that require variables in an energy-conjugate pair to be related by a constitutive law. Honoring these principles, we adopted an effective stress that is energy-conjugate to the rate of the deformation tensor of the solid matrix. Then, by interpreting the implication of each energy-conjugate pair for constitutive modeling, we established a constitutive framework for coupled hydromechanical processes in two-scale porous media. We then cast the resulting framework into a three-field mixed finite element formulation. Through numerical simulations, we demonstrated how two-scale fluid flow controls the failure patterns of unsaturated soils from laboratory- to fieldscale problems. Specifically, we showed that pore-scale heterogeneity can trigger a persistent shear band, and that preferential flow can alter the timing and location of slope failure, both phenomena that cannot be captured by a classical single-porosity framework. Subsequently, in Chapter 3 we focused on modeling the deformation of porous solids with evolving two-scale internal structures. This modeling tackled the unresolved challenge of capturing separate volume changes in two pore scales at finite strains. Our solution to this challenge was to decompose the overall effective stress into two individual stresses, each of which was responsible for the compression of the corresponding pore scale. The upshot of this approach was

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that it enabled geometrically exact quantification of multiscale evolution of pore structures, a necessary task for coupling double-porosity flow and finite deformations. This approach gave rise to an equation that related the pore pressure jump to local pore fractions, which in turn ensured thermodynamic consistency in the entire range of deformations and fluid flow. Through stress-point simulations, we showed that the novel constitutive model is not only uniquely compatible with the finite deformation kinematics of double-porosity materials, but it also has better predictive capabilities than the models in the literature have. Two-scale hydromechanical simulations based on this framework revealed that secondary compression can be in part due to the delayed pore pressure dissipation in the micropores, confirming a widely accepted hypothesis. We also demonstrated that such secondary compression can lead to separation of time in the settlement of a superstructure. In the last part, Chapter 4, we turned our attention to reducing the computational cost by developing a stabilized mixed finite element formulation that allows the use of equal-order linear interpolations. The discrete system emanating from the double-porosity formulation posed two types of saddle point problems. One was a single saddle point problem in which the micropore pressure was the constraint, and another was a twofold saddle point problem in which both the macropore and micropore pressures were constraints. The major challenge was to circumvent the twofold saddle point problem for which no stabilization method had been proposed. Rephrasing the stability condition at hand, we developed a stabilization method by extending the polynomial pressure projection technique to the twofold saddle point problem. The new stabilization method inherited all of the salient advantages of the original technique, including its parameter-free nature and minimal efforts required for implementation. We demonstrated the performance and efficacy of the proposed stabilization method through various numerical examples that cover two-scale drained, single-scale undrained, and two-scale undrained conditions. We showed that the stabilization terms effectively circumvent the instability conditions in the undrained pore scales, whereas they do not introduce any anomalies in the drained pore scales. Therefore, the stabilized finite element formulation can offer significant computational advantages—including a great reduction in the number of unknowns and Gauss points—throughout the entire range of drainage conditions. The novel contributions of this thesis can be summarized as follows: • A thermodynamically consistent framework for modeling coupled fluid flow and deformation in double-porosity media. • A constitutive framework for fluid-infiltrated porous materials with evolving double-porosity structures. 106

• A numerical simulation of secondary compression due to delayed pore pressure dissipation in the micropores. • A stabilized mixed finite element method that allows the use of equal-order interpolations for a twofold saddle point problem.

5.2

future work

Lastly, we would like to make some recommendations for improving this work, with respect to mechanical modeling, hydrological modeling, computational methods, and applications. On the mechanical modeling In developing the constitutive framework in Chapter 3, we have restricted our focus on fully saturated materials in which the pore space is occupied by a single type of fluid. However, many porous materials such as unsaturated soils are infiltrated with two or more types of fluids. The presence of multiple fluids, which results in suction stress (or capillary pressure), has significant impacts on the mechanical responses from the effective stress to the hardening behavior. While incorporating the impact on the effective stress into the framework is straightforward, the impact on the hardening behavior requires much work. The major challenge is how to develop two individual relationships between the suction and preconsolidation stresses in the macropores and micropores. One possible approach is to extend the idea behind the “bonding variable” proposed by Gallipoli et al. (2003a). The bonding variable is originally introduced to incorporate the effect of suction on the preconsolidation stress of single-porosity materials. Central aspects of the bonding variable include that (a) the number of water menisci per unit volume can be represented by the degree of saturation; and that (b) the ratio between the void ratio of the normal compression line in the unsaturated state to the corresponding void ratio in the fully saturated state is a unique function of the bonding variable. These aspects should be extended to double-porosity materials where two degrees of saturation and two void ratios are defined. This extension, however, requires new experimental data from which we can derive separate equations for the bonding effects at two scales. Resolving this issue, or devising another approach, is an interesting open problem.

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On the hydrological modeling In this work we have adopted the double-porosity model originally proposed by Barenblatt et al. (1960). However, researchers have observed that this original model is unsatisfactory in many cases, and improved it for more realistic descriptions of fluid flow in multiscale porous media. An important improvement is the multirate mass transfer model (Geiger et al., 2013; Haggerty and Gorelick, 1995), which introduces non-unique mass transfer rates for immobile, microporous zones. Accommodating the multirate mass transfer model into poromechanics appears to pose several challenges, but it can certainly improve the accuracy for fluid flow. The evolution of the permeability by finite deformations also deserves further investigation. In Chapter 3 we modeled such permeability evolution by employing the Kozeny–Carman equation; however, this equation was actually developed from experimental results of packed spheres where the pore structure can differ significant from real geomaterials’ pore structure. Also, for simplicity isotropic permeability has been assumed throughout this work. However, anisotropy is not only inherent in many geomaterials but also can evolve by the deformation of the solid matrix (Choo et al., 2013b; Kang et al., 2013). Therefore, developing and incorporating an anisotropic permeability–deformation relationship is a necessary enhancement. Also, an important task is to take into account the impact of evolving macroand micro-porosity on water retention characteristics, as mentioned in Chapter 2. Accommodating this aspect into a double-porosity formulation is challenging because the relationship between porosity and water retention at each pore scale has not yet been characterized by experiments. One feasible approach may be to overlap two water retention surfaces—such as those proposed in Gallipoli et al. (2003b) and Salager et al. (2010)—for experimentally obtainable overall behavior. This approach is similar to how we calibrated two water retention curves in Chapter 2. Then, we can determine the parameters for the micro-retention surface from reconstituted materials, similar to how we calibrated the micro-compressibility in Chapter 3. On the computational methods To reduce computational costs, one can explore options to enhance the linear solver for double-porosity poromechanics. One promising option is to tailor the sub-preconditioners for the mass balance blocks to the macropores and micropores. For finding other options, we refer to the recent work of White et al. (2016)

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which introduces a unified framework for developing efficient solution methods for single-porosity poromechanics problems. Extending this framework may allow us to arrive at a nearly optimal solution strategy for the linear system emanating from the double-porosity formulation. On the applications The outcome of this work can be applied to advance our understanding and prediction of a wide range of engineering problems in which hydromechanical interactions take place at multiple length and time scales. This thesis has shown that explicit consideration of two-scale hydromechanical coupling enhances our modeling capabilities for diverse problems, including persistent shear bands, hydrologically driven slope failures, and long-term settlements. We believe that this is also the case for many other engineering problems. For example, two-scale hydromechanical processes can be crucial to the performance of embankments because they are often made of compacted soils exhibiting two scales of porosity. Also, since hydrocarbon reservoirs often contain double-porosity materials such as fractured rocks or carbonate rocks, the developed framework can significantly improve the assessment of wellbore behavior in energy production. We look forward to seeing the contribution of this thesis to many engineering technologies that make the built environment more resilient and sustainable.

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