Many natural and man-made materials, such as sand, rock, concrete and bone, are ..... For those idealized materials, mechanical responses do not exhibit size.
NORTHWESTERN UNIVERSITY
A Multi-scale Framework for Modeling Instabilities in Fluid-Infiltrated Porous Solids
A DISSERTATION SUBMITTED TO THE GRADUATE SCHOOL IN PARTIAL FULFILLMENT OF THE REQUIREMENTS for the degree DOCTOR OF PHILOSOPHY
Field of Theoretical and Applied Mechanics
By WaiChing Sun
EVANSTON, ILLINOIS June 2011
UMI Number: 3456707
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© Copyright by WaiChing Sun 2011 All Rights Reserved
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ABSTRACT
A Multi-scale Framework for Modeling Instabilities in Fluid-Infiltrated Porous Solids
WaiChing Sun
Many natural and man-made materials, such as sand, rock, concrete and bone, are multiconstituent, fluid-infiltrated porous solids. The failure of such materials is important for various engineering applications, such as CO2 sequestration, energy storage and retrieval and aquifer management as well as many other geotechnical engineering problems aimed to prevent catastrophic failures due to pore pressure build-up. This dissertation investigates two mechanical aspects of fluid infiltrated porous media, i.e., the predictions of diffuse and localized failures of porous media and the heterogeneous microstructures developed after failures. We define failures as material conditions in which homogeneous deformation becomes unattainable. To detect instabilities, a critical state plasticity model for sand is implemented. By seeking bifurcation points of the incremental, linearized constitutive responses, we establish local criteria that detect onsets of drained soil collapse, static liquefaction and formation of deformation bands under locally drained and undrained conditions. Fully
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undrained and drained triaxial compression simulations are conducted and the stability of the numerical specimens are assessed via a perturbation method. To characterize deformation modes after failures, a multi-scale framework is designed to determine microstructural attributes from pore space extracted from X-ray tomographic images and improve the accuracy and speed of a multi-scale lattice Boltzmann/finite element hierarchical flow simulation algorithm. By comparing the microstructural attributes and macroscopic permeabilities inside and outside a compaction band formed in Aztec Sandstone, our numerical study reveals that elimination of connected pore space and increased tortuosity are the main causes that compaction bands are flow barriers.
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Acknowledgements Studying mechanics at Northwestern is a very humbling and joyful experience. I consider it a special privilege to learn from a number of wonderful people without whom this research would not have been possible. First and foremost I want to express my sincerest gratitude to my advisor Professor Jos´e E. Andrade. Professor Andrade has taught me many valuable lessons which lead me to be a better scientist. His vision and enthusiasm to geomechanics research are motivational for me. His contribution of time and funding is also gratefully acknowledged. At the early stage of my research career, I enjoyed tremendous support from my former advisors Professor Ronaldo I. Borja and Professor Yin Lu Young. Their dedications to science have inspired me to be a mechanician for which I owe much gratitude to. During the last three years at Northwestern, I have been significantly benefited from my co-advisor Professor John W. Rudnicki’s teaching and mentorship. He taught me many wonderful things about mechanics, offered me numerous sound advice and useful guidance, and helped me getting through difficult time. His trustworthy and open minded characters constantly remind me what a dedicated scholar and an effective scientific citizen should be. This dissertation would not be completed without his encouragement and support. Thanks are due to Dr. Nicolas Lenoir for generously offering his expertise to help me analyzing microstructural attributes from tomographic images of the Aztec sandstone specimen. It is my pleasure to work with him. I am also very thankful to Professor
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Richard J. Finno for teaching me many interesting topics on critical state plasticity and offering advice on academic matters whenever needed; to Professor Aaron I. Packman and Dr. Cheng Chen for providing me help and support on implementing the lattice Boltzmann code; to Professor David Salac for fruitful discussion on the level set theory. I am indebted to my friend, Dr. Siu-Chung Yau for his camaraderie and caring, and to my uncle Anthony Siu for his emotional support. I also wish to thank my parents, Hon Chiu Sun and Jenny Siu for providing me a loving environment to grow up. I am very proud to be their son. Lastly, I would like to thank my wife, Heidy Leung, for all her understanding, encouragement and companionship. Her faithful and unconditional support during every stages of my life have always been my main source of strength and happiness.
To her I dedicate this thesis.
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Table of Contents ABSTRACT
3
Acknowledgements
5
List of Tables
10
List of Figures
11
Chapter 1. Introduction
17
1.1. Objectives and statement of the problem
17
1.2. Motivation
18
1.3. Structure of presentation
20
Chapter 2. Background literature
23
2.1. Numerical modeling of poromechanics problems
23
2.2. Static liquefaction and drained diffuse collapse of porous media
27
2.3. Shear and compaction localizations of porous media
29
2.4. Pore-scale flow simulation with lattice Boltzmann method
31
2.5. Geometrical analysis of microstructures of porous media
35
Chapter 3. A simplified method to analyze material instabilities in sands
37
3.1. Abstract
37
3.2. Introduction
37
8
3.3. Drained Responses
40
3.4. Undrained Responses
42
3.5. Critical state plasticity model for sands
45
3.6. Numerical Predictions
49
3.7. Conclusion
58
Chapter 4. Multi-scale Modeling Techniques for Transport Characteristics of Deformation Bands
61
4.1. Abstract
61
4.2. Introduction
62
4.3. Overall architecture
64
4.4. Part 1: Numerical procedures for pore-scale geometrical analysis
66
4.5. Part 2: Two-scale homogenization of permeability using LB and FEM
77
4.6. Representative examples
88
4.7. Conclusion
98
Chapter 5. Connecting microstructural attributes and permeability from 3-D tomographic images of in situ compaction bands using multi-scale computation
100
5.1. Abstract
100
5.2. Introduction
101
5.3. Three dimensional tomographic images and numerical methods
104
5.4. Microstructural attributes
106
5.5. Macroscopic effective permeabilities
108
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5.6. Conclusion Chapter 6. Conclusion and Future Prospective 6.1. Future work
111 113 115
References
118
Vita
126
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List of Tables 3.1
Summary of material parameters used for numerical predictions.
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4.1
Global and local permeabilities obtained from LB simulation scenarios.
88
4.2
Dissipation rates computed via spatial averaging and prescribed values at boundaries.
97
11
List of Figures 2.1
Finite element simulation of the undrained triaxial compression test on Hostun sand. The edge length of the cubic specimen is 0.1m. The color contours represent the pore pressure in kP a. The flow vectors at Gauss points are represented by red arrows. From Sun and Andrade [2010b].
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2.2
Po Shan Landslide, Hong Kong 1972.
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2.3
Relative planar and parallel compaction bands along the northeast flank of Silica Dome. Arrows indicate opposite tips of a single band 62m long and 15mm thick. From Sternlof, Rudnicki and Pollard [2005].
2.4
Examples of discrete velocity vectors for typical 2D and 3D velocity discretizations.
2.5
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Two-dimensional illustration of medial axis determination using BURN algorithm. From Lindquist et al. [1996].
3.1
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Numerical simulation of drained triaxial compression test on Changi sand with initial void ratio = 0.95 (red). (a) simulated deviatoric stress (b) axial strain , (c) void ratio , (d) generalized hardening modulus vs. effective hydrostatic pressure.
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12
3.2
The stress ratio q/p and the sizes of the hardening (M b ) and dilatancy M d limit surfaces vs effective hydrostatic pressure of the contractive Changi sand.
3.3
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Numerical simulation of drained triaxial compression test on Changi sand with initial void ratio = 0.657 (red). (a) simulated deviatoric stress (b) axial strain , (c) void ratio and (d) generalized hardening modulus vs. effective hydrostatic pressure are plotted (red line) and compare with experimental observations (blue dots).
3.4
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The stress ratio q/p and the sizes of the hardening (M b ) and dilatancy M d limit surfaces vs effective hydrostatic pressure of the contractive Changi sand.
3.5
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Numerical simulations of undrained triaxial compression tests on Toyoura sand with initial void ratio = 0.907. (a) simulated deviatoric stress vs. effective hydrostatic pressure and (b) deviatoric stress vs. axial strain. The confining pressure of the three simulations are 100kPa (blue), 1000kPa (red) and 2000kPa (green). Experimental data from Verdugo and Ishihara [1996] used to calibrate the material parameters are plotted in dots.
3.6
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Simulated evolution of hardening modulus (Kp , in blue color) and the corresponding threshold values for pure shear band (Kpsb , in red color) and static liquefaction (Kpliq , in green color) of Toyoura sand with initial void ratio = 0.907 and confining pressure equal to (a) 100kP a (b) 1000kP a and (c) 2000kP a.
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13
3.7
Numerical simulations of undrained triaxial compression tests on Toyoura sand with initial void ratio = 0.735. (a) simulated deviatoric stress vs. effective hydrostatic pressure and (b) deviatoric stress vs. axial strain. The confining pressure of the three simulations are 100kPa (blue), 1000kPa (red), 2000kPa (green) and 3000kPa (cyan). Experimental data from Verdugo and Ishihara [1996] used to calibrate the material parameters are plotted in dots.
3.8
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Simulated evolution of hardening modulus (Kp , in blue color) and the corresponding threshold values for pure shear band (Kpsb , in red color) and static liquefaction (Kpliq , in green color) of Toyoura sand with initial void ratio = 0.735 and confining pressure equal to (a) 100kP a, (b) 1000kP a, (c) 2000kP a and (d) 3000kP a.
4.1
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Flow chart of the numerical procedures used to compute tortuosity, identify connected/occluded pore space and estimate effective permeability at the specimen-scale.
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4.2
A binary image φo and its corresponding edge indicator function g.
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4.3
Two-dimensional pore space example. Left figure is the original binary image, containing only binary data. By applying the level set scheme, the signed distance function is formed inside the pore space as illustrated in the middle figure. The medial axis can be located by interpolating the local maximum point of the signed distance function as illustrated in the right figure.
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4.4
Example evolution of level set function φ.
4.5
Example of corresponding weighted graph for sample medial axis. Medial axis corresponds to example shown in Figure 4.3.
4.6
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Selection of the unit cell size based on the scale of fluctuation. s/L is the ratio between the voxel length and edge length of the specimen.
4.8
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Multiscale numerical scheme used to determine effective permeability in large scale.
4.7
70
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Velocity profiles of lattice Boltzmann simulations on (a) unpartitioned domain (k = 0.015 u2 ), (b) partitioned domain with identified and deactivated occluded porosity (k = 0.013 u2 ), and (c) partitioned domain without any special treatment for occluded porosity (k = 0.0078 u2 ). Where u =lattice unit.
87
4.9
Connected porosity as a function of voxel length in a SC bead packing. 91
4.10
Level set function φ(x, y, z) (represented by the 3D color contour) and the corresponding shortest flow path (represented by the red straight line) as determined by Dijkstra’s algorithm.
4.11
92
(a) Streamline of the simple cubic lattice computed via Stokes finite element model (k = 4.89 × 10−3 R2 ). (b) Velocity profile of the simple cubic lattice obtained via lattice Boltzmann simulation conducted on connected pore space (k = 4.64 × 10−3 R2 ).
4.12
93
(a) Level set functions. (b) Flow paths. (c) Connected pore space for compaction band cell with dimensions 0.54 × 0.54 × 0.54 mm.
94
15
4.13
Energy dissipation rate over the edge length of the cubic samples taken inside compaction band. The energy dissipation rate is normalized with respect to the energy dissipation rate of the largest sample with edge length = 0.75mm.
4.14
96
Results of multiscale effective permeability analysis inside compaction band. (a) Porosity map, (b) vertical velocity field, and (c) pressure map. 98
5.1
Cumulative distribution functions of grain diameters (a) and tortuosities (b) and Scanning-electron microscope cathodoluminescent images outside (c) and inside (d) the shear enhanced compaction band. Thick lines in (b) denote Gamma distribution fits; thin lines represent the tortuosity distributions determined from Dijkstra’s algorithm. In (c) and (d), q:quartz, p:pore. Arrow indicates quartz cement in broken grains.
5.2
103
The shortest flow paths inside (a) and outside (b) and velocity vector field (in blue color) inside (c) and outside (d) the shear enhanced compaction band. Solid grains are colored in gray. The intensity of the blue color in (c) and (d) represents the magnitude of the fluid velocity field.
5.3
109
Occluded and connected porosity of 6 samples taken inside/outside the shear enhanced compaction band in the Aztec Sandstone specimen. Sample labeled as SCB (OUTSIDE) are taken inside (outside) the shear enhanced compaction band.
112
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6.1
The proposed multi-scale hierarchical mixed finite element algorithm.
117
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CHAPTER 1
Introduction 1.1. Objectives and statement of the problem Poromechanics is a broad subject that involves a wide range of materials, from rocks and soils, to bone, tissues, gels and polymer scaffolds. This subject finds its root in the one-dimensional consolidation theory by Karl von Terzaghi [Terzaghi , 1943] and the generalized three dimensional poroelasticty theory developed by Maurice Biot [Biot, 1941] more than half a century ago. The applications of poromechanics are relevant to numerous disciplines, such as geotechnical engineering, geophysics, agricultural engineering, petroleum engineering, biomechanics and material science. This dissertation focuses on the investigation of two mechanical aspects about fluid infiltrated porous media, i.e., the predictions of diffuse and localized failures of porous media and the heterogeneous microstructures developed after failures. To predict the onsets and modes of instabilities, we first incorporate the instability line concepts proposed by Chu, Leroueil and Leong [2003] into a critical state plasticity model to replicate constitutive responses of sand under various drainage conditions. We then perform bifurcation analysis to detect various types of instabilities at the material point level. Since the post-bifurcation mechanical behaviors of porous media are multiscale by nature, the overall constitutive responses at the macroscopic level highly depend on the microstructural deformation mechanism at the grain- and pore-scale. To quantify
18
this interplay across the macro- and micro- scales, we took elements of graph theory, level set interfacial modeling and two-scale lattice Boltzmann/finite element algorithm to form a fully integrated, multi-scale numerical method [Sun and Andrade, 2010a; Sun et al., 2011a]. This method is able to quantify microstructural attributes, such as tortuosity, pore connectivity and the meso- and macro- scale permeabilities of deformation banding specimens in a cost-efficient manner.
1.2. Motivation Strain localization, diffuse collapse and static liquefaction are ubiquitous modes of failure commonly encountered in fluid-infiltrated porous solids. The onset of those failures are governed by two major factors, the load-bearing capability of the porous solid skeleton and the interaction between the pore-fluid and the solid skeleton. During the last three decades, there have been numerous theoretical and numerical models developed to simulate and predict the onset of failures via the mixed-field finite element method [Andrade and Borja, 2007; Pr´evost, 1982b; Schrefler, Sanavia and Majorana, 1996; Zienkiewicz et al., 1999]. These models, however, rely on the assumption that the multi-constituent porous solids can be treated as a continuum at an arbitrary small scale such that material interactions only take place on the level of balance equations, through the exchange of mass, momentum, energy and entropy [Baˇzant and Jir´asek , 2002; Baˇzant and Cedolin , 2003]. Continua described by the description above is coined simple (non-polar) materials [Noll , 1958, 1972]. For those idealized materials, mechanical responses do not exhibit size dependence under fully drained or undrained conditions. Therefore, bifurcation analyses are valid at material point level where material is assumed to deform and respond to
19
load excitations homogeneously [Rudnicki and Rice, 1975; Rudnicki , 2004, 2009]. Coupled with constitutive models to replicate constitutive responses, results from bifurcation analysis can be used to predict onset of instabilities if sufficient information regarding the material properties and loading history is given. Modeling the post-bifurcation responses, however, requires not only the knowledge of the overall material properties at the continuum level, but also the geometrical properties of the microstructure developed after the onset of instabilities. For instance, Finno et al. [1996] have used stereophotogrammetry to study the progressive localization of strain. Their experimental findings provide evidence that steady-state lines (a stress ratio below the critical state under which severe shear strain develops under constant pressure) were different under plane strain and axial symmetric loading conditions. Hall et al, [2010] use X-ray imaging and digital image correlation to track displacement and rotation of individual grains such that the kinematics of shear band formation can be examined at the grain scales. Their results show a rapid increase in acceleration of rotation as the shear band initiates whereas grains elsewhere show relatively constant rotation as a triaxial compression loading progresses. In contrast to the severe rotation observed in shear bands, grain crushing can be a dominating factor that leads to the formation of compaction bands in porous sandstone specimens prepared in laboratories [Baud, Klein and Wong, 2004; Baud, Vajodova and Wong, 2006]. This kinematic feature of compaction bands has attracted wide attention due to the potential of using compaction bands as flow barriers in reservoirs [Holcomb and Olsson, 2003; Holcomb et al., 2007; Sun, Andrade and Rudnicki , 2011b; Taylor and Pollard , 2000]. Since the kinematic feature of compaction bands is developed at the post-bifurcation regime (where length scale plays an important
20
role), the mechanical and hydraulic properties of compaction bands must be studied at the scale where microstructural heterogeneity can be taken into account properly. The purpose of this study is therefore two-fold – (I) to develop a systematic and unified method that predicts onsets of localized and diffuse instabilities at material point level and (II) to examine the hydraulic property of compaction bands across pore- and specimen- scales.
1.3. Structure of presentation The dissertation is structured as follows. After a brief overview of the continuum theory of porous media, Chapter 2 reviews the constitutive models used to replicate responses of porous media, and the related bifurcation analysis used to detect instabilities. Chapter 3 presents a mathematical framework that predicts diffuse and localized instabilities of porous media under fully drained and undrained conditions. By implementing an explicit stress integration algorithm [Jeremi´c and Sture, 1997] for the critical state plasticity model proposed by Manzari and Dafalias [Dafalias and Manzari, 2004; Manzari and Dafalias, 1997], we examine the stability of the constitutive responses of sand under various stress states and loading conditions through numerical simulations. The results of this research confirm that (1) instabilities of fluid infiltrated porous media can be predicted by monitoring the sensitivity of constitutive responses and (2) two families of failure modes corresponding to diffuse and localized eigen-strains are identified through examining the mathematical properties of the linearized tangential elasto-plastic constitutive tensor. These findings provide convenient tools for engineers to detect and ultimately prevent failures of geo-systems, such as foundations and slopes.
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Chapter 4 proposes a number of numerical techniques to predict macroscopic porefluid transport from microstructures with detail as fine as a few microns. These numerical techniques are fully integrated into a scheme to improve accuracy and cost-efficiency. This scheme is specifically designed for analyzing microstructures developed during and after shear- and/or compaction- band formation. The macroscopic transport behavior of deformation bands is of economic importance, because deformation may act as either the flow conduit or barrier that significantly alters the hydraulic properties of reservoirs. To quantify connections between microstructural attributes and macroscopic effective permeability, we first obtain the medial axis of the pore space via a semi-implicit variational level set scheme. Then, we investigate the connectivity of the pore space via graph theory approach similar to that of Lindquist et al. [1996]. Pore-scale lattice Boltzmann flow simulation is conducted in the connected pore space and upscaled to obtain macroscopic permeability via a Darcy’s flow finite element simulations. A number of numerical examples are used to verify the computational framework. Chapter 5 describes an application of the multiscale framework. A multi-scale analysis is performed on tomographic images of a shear-enhanced compaction bands in Aztec Sandstone. The findings of this research reveal that microstructures inside shear-enhanced compaction bands exhibit a significant reduction in connected porosity, but only a moderate reduction in isolated porosity and intrinsic permeability (of less than one order of magnitude). Chapter 6 summarizes the major contributions and findings of this dissertation. Ongoing researches based on the findings of this dissertation are briefly outlined.
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It should be noticed that Chapter 3-5 are intended to be self-contained as they are written for journal publications. As a result, there are repetitions of concepts in these chapters.
23
CHAPTER 2
Background literature 2.1. Numerical modeling of poromechanics problems Previous work on computational geomechanics has placed great emphasis on modeling the coupled diffusion-deformation motion of fluid-infiltrating solid in macroscopic setting with the mixed-field finite element technology [Andrade and Borja, 2007; Brezzi , 1990; Pr´evost, 1982b; Zienkiewicz et al., 1999]. By introducing extra variables such as pore pressure and/or Darcy’s velocity to the governing equation to model the interaction between the pore-fluid and the solid skeleton constituents, the mixed finite element formulation couples the mass and momentum balance equations and discretizes them in a specific way that satisfies the Ladyshenskaja-Babuska-Brezzi (LBB) condition. This condition guarantees that no spurious pressure would be artificially generated due to the improperly discretization of the Galerkin form [Brezzi , 1990]. Examples of mixed finite element formulations that satisfy the LBB condition include the u-v-p (solid displacement - Darcy’s velocity - pore pressure) formulation described in Zienkiewicz et al. [1999], the u-v (solid displacement - Darcy’s velocity) formulation described in Pr´evost [1982b] and the u-p (solid displacement - pore pressure) formulation used in Andrade and Borja [2007]. Due to the coupling between the solid skeleton and the Darcy’s flow, the constitutive response depends on the loading rate and the gradient of the pore pressure. This rate- and pore pressure gradient dependence vanishes if the material is excited at a much
24
higher rate before pore-fluid can diffuse (undrained condition) or if the material is excited very slowly such that pore-fluid remains approximately in steady-state during the loading cycle (drained condition). These two drainage limits provide a convenient way to analyze the uncoupled rate-independent mechanical properties of the solid skeleton. In the 1950s and 1960s, phenomenological plasticity theory suitable for replicating mechanical responses of clay and sand under drained and undrained conditions was developed mainly at Cambridge University [Roscoe, Schofield and Wroth, 1958; Schofield and Wroth, 1968]. This theory is now referred to as critical state plasticity or critical state soil mechanics in the literature. The key features of the critical state theory, which have shown to be consistent with experiential observations [Roscoe, Schofield and Wroth, 1958; Schofield and Wroth, 1968; Taylor , 1948; Wood , 1990; Zienkiewicz et al., 1999] are listed below. (1) There exists a critical state line defined by the ratio between deviatoric and hydrostatic stress and the specific volume (1+void ratio) at which the material may develop unbounded shear at constant deviatoric and hydrostatic stresses without any volumetric changes, i.e.,q/p = M c and 1 + e = Γ − λ log(p ) where M c , Γ and λ are material parameters. (2) The yield surface is sensitive to hydrostatic stress. (3) The plastic dilatancy depends on whether the material is on the dense side or loose side of the critical state line. Materials below the critical state line in the (e, p ) space develop plastic dilatancy upon yielding; materials above the critical state line in the (e, p ) space develop negative plastic dilatancy (contraction) upon yielding.
25
This conceptual model is developed into a implicit stress integration algorithm by Borja and Lee [1990], Andrade and Borja [2006] and Andrade and Borja [2007] using a isotropic hardening rule. Manzari and Dafalias [1997], Li [2002] and Dafalias and Manzari [2004] incorporate the critical state plasticity concept with a purely kinematic hardening rule. In their models, the yield surface is purposely set to be very small compared with the size of the critical state surface in the stress space. The ”difference” between the current stress and critical state surface is then used to manipulate the size of the limit surfaces in order to replicate constitutive responses of soil at the loose and dense states with the same set of material parameters. The multi-yield surface model is another interesting approach designed to handle complex elasto-plastic behavior [Pr´evost, 1982a]. Compared to the limit surface approach used in Manzari and Dafalias [1997], Li [2002] and Dafalias and Manzari [2004], the multi-yield surface model usually involves with a much simpler hardening rule with fewer internal variables. To compensate for the simple hardening law, a collection of yield surfaces is used. Inside the first yield surface, the constitutive response is purely elastic. All yield surfaces are associated with a unique set of generalized plastic modulus, frictional coefficient and plastic dilatancy. While plastic yielding occurs, the yield surfaces consecutively touch and push each other but are not allowed to intersect. The generalized hardening modulus, friction coefficient and plastic dilatancy associated with the outermost yield surface touched by the current stress are used to construct the continuum or consistent elasto-plastic tensor. Figure 2.1 shows the results of a three dimensional globally undrained simulation conducted with a u-p mixed finite element and a critical state plasticity model. In this
26
example, the pore pressure distribution and pore-fluid flow are obtained from an undrained compression simulation on contractive Hostun sand [Sun and Andrade, 2010c]. The onset of shear band is delayed until cavitation of the pore-fluid occurs in the localized zone, which is consistent with the experimental observations described in Mokni and Desrues [1998]. Onset of shear banding in saturated porous media undergoing finite deformation
(a) Flow vectors at end of simulation
(b) Deformed configuration at the end of simulation
Figure 2.1. Finite element simulation of the undrained triaxial compression test on Hostun sand. The edge length of the cubic specimen is 0.1m. The color contours represent the pore pressure in kP a. The flow vectors at Gauss points are represented by red arrows. From Sun and Andrade [2010b].
27
is analyzed by Andrade and Borja [2007] where determinant of the undrained acoustic tensor is computed at each Gauss point of the finite element models. The location with zero determinant is found to be the strain localization zone.
2.2. Static liquefaction and drained diffuse collapse of porous media Static liquefaction and drained diffuse collapses are important failure modes for many geotechnical engineering applications. In engineering practice, a variety of phenomena that involve severe soil deformation caused by monotonic, cycle and transient loads are all coined liquefaction. Kramer [1996] classifies liquefaction into two categories : cyclic mobility and flow liquefaction. Cyclic mobility is a phenomenon that occurs when the shear strength loses due to cyclic excitations caused by earthquakes. This type of liquefaction can be detected by comparing the current shear stress with a Von Mises-type, isotropic failure surface (which is pressure sensitive) defined between the hydrostatic axis and the critical state surface in the stress space. Flow liquefaction is sometimes referred to as static liquefaction [Andrade, 2009; Borja, 2006b; Yamamuro and Lade, 1997]. It occurs with flow failures and strength loss due to pore pressure building up. The 1972 Po Shan landslide (as shown in Figure 2.2) that caused 67 fatalities, 20 injuries and 2 buildings destroyed beyond repair, is a devastating example of flow liquefaction triggered by pore pressure build-up due to heavy rainfall [Schoustra, 1972]. In laboratory settings, this failure is often observed in sand under undrained condition. For instance, undrained triaxial compression tests on Ottawa sand conducted by Yamamuro and Lade [1997] suggest that static liquefaction may occur in
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Figure 2.2. Po Shan Landslide, Hong Kong 1972. medium sand under high confining pressure or in loose sand specimen under both high and low confining pressure. Another type of diffuse failure occurs in drained condition. This failure is commonly observed when the deviatoric stress reaches its peak in sand. Experimental studies on Changi sand by Chu, Leroueil and Leong [2003] have provided physical evidence that drained failures often occur near the critical state line. The onset of instability depends on both the initial confining pressure and void ratio. From a theoretical viewpoint, both static liquefaction and drained diffuse collapses are phenomena in which different amount of strain can develop under the same stress states [Andrade, 2009; Borja, 2006a,b; Darve et al., 2007; Ogden, 1997]. Signified by the singularity of the tangential constitutive tensor, constitutive responses may cease to maintain the one-to-one relation between effective stress and strain and therefore allow uncontrolled deformation to develop without any change in stress.
29
The definitions of static liquefaction described above should not be confused with the static liquefaction defined in Vardoulakis [1985] or the diffuse instability in Rice [1975]. In these studies, the stability of the coupled deformation and diffusion of an initially undrained shear layered infinite domain is analyzed. Unlike the static liquefaction described above, vertical drainage is taken into account in these studies. The results shown that undrained deformation is not stable to perturbations if the drained behavior of the shear layer is not stable.
2.3. Shear and compaction localizations of porous media Using a two-invariant plasticity model, Rudnicki and Rice [1975] provide a bifurcation theory to explain the onset of shear band as a case where homogeneous deformation becomes unattainable due to the singularity of the acoustic tensor. In their analysis, shear banding may occur if a combination of elastic moduli and plastic material parameters (such as hardening modulus H, plastic dilatancy β, frictional coefficient μ) meet a unique criteria. As noted by [Perrin and Leblond , 1993], shear band orientation are continuous over the limits expressed below. (2.1)
(1 − 2ν)N −
√
2 4 − 3N 2 ≤ (1 + ν)(β + μ) ≤ (1 − 2ν) 3
where N is the intermediate principal value of the normalized deviatoric stress Nij and √ √ varies from −1/ 3 for axisymmetric extension to 1/ 3 for axisymmetric compression; ν is the Poisson ratio. If β + μ is within the limits expressed above, shear band forms if the critical value of hardening modulus for shear banding is met [Rudnicki and Rice, 1975],
30
i.e., (2.2)
1+ν 1 1+ν Hc = (β − μ)2 − [N + (β + μ)]2 G 9(1 + ν) 2 3
If (2.1) is not satisfied, then compaction or dilatant band may occur when the hardening modulus met the corresponding critical values [Issen and Rudnicki, 2000], (2.3)
Hkc 1+ν 1 + ν Nk 1 3 = (β − μ)2 − [ − (β + μ)]2 − (1 − Nk2 ) G 9(1 + ν) 1−ν 2 3 4
where k = 1 for the dilatant mode and k=3 for the compaction mode. N1 and N3 are the least and most compressive principal values of Nij . Figure 2.3 shows an array of compaction bands found in Aztec sandstone [Sternlof, Rudnicki and Pollard , 2005]. Scanning electron microscope images of the compaction bands and host rock are taken and a 5% to 8% porosity reduction is observed. A detailed field study was conducted by Eichhubl, Hooker and Laubach [2010] where microstructural attributes, such as grain breakage, length of force chains of compaction and shear bands of the Aztec sandstone found in Valley of Fire are compared. Their findings provide physical evidence that formation of pure compaction bands in the field require a high initial porosity and the most compressive principal stress being on the order of 107 Pa. Interestingly, the naturally formed field specimens found in Valley of Fire and East Kaibab Monocline exhibit comminution much less severe than those reproduced in laboratory [Holcomb et al., 2007]. The reason is not fully understood. One possible explanation is offered by Holcomb et al. [2007] in which the authors speculate that the compaction bands in the field might occur under saturated condition. The interaction of pore-fluid and grains may alter how compaction bands form and propagate.
31
In addition, Tembe, Baud and Wong [2008] suggest that the differences in comminution and damage between compaction bands in field and in lab are due to the different stress state involved. Using the anti-crack model developed in Rudnicki and Sternlof [2005], Tembe, Baud and Wong [2008] found that the stress level is inversely proportional to the compaction band thickness. Since the compaction band of the Aztec sandstone specimens are 2 to 3 orders thicker than the compaction band formed in lab (see Table 5 of Tembe, Baud and Wong [2008]), Tembe, Baud and Wong [2008] predict that the compaction bands in Aztec Sandstone were formed at a much lower stress level than those produced in laboratory. Therefore, they predict higher propagation stresses for the laboratory bands and thus expect the laboratory bands to exhibit more severe damage. This prediction is consistent with the stress level at the onset of compaction banding in Aztec Sandstone estimated by Eichhubl, Hooker and Laubach [2010] (major and minor principal effective stresses are 22 and 7MPa, which are significantly lower than those in Bentheim (far field compressive stress = 340Mpa), Diemelstadt (far field compressive stress= 100-300MPa) and Bleurswiller (far field compressive stress = 100-200MPa) Sandstones) [Tembe, Baud and Wong, 2008].
2.4. Pore-scale flow simulation with lattice Boltzmann method Due to the significant microstructural difference between the deformation bands and the host matrix, many researchs are now exploring the possibility of using deformation bands as flow barriers or conduits. Efforts are given to investigate the transport properties of deformation bands via pore-scale flow simulations [Keehm, Sternlof and Mukerji, 2006]. The overall transport property is then upscaled to specimen-scale [White, Borja and
32
Figure 2.3. Relative planar and parallel compaction bands along the northeast flank of Silica Dome. Arrows indicate opposite tips of a single band 62m long and 15mm thick. From Sternlof, Rudnicki and Pollard [2005].
Fredrich, 2006] and also to field-scale Sternlof et al. [2004] to simulate the overall transport property of reservoirs. In recent years, lattice Boltzmann method (LBM) has shown great promise for simulating fluid flow and modeling mechanical and chemical processes in fluids [Chen and Doolen, 1998; Chen, Packman and Gaillard , 2008; Succi , 2001]. LBM is particularly suited for modeling fluid flow involving complex boundaries. Unlike the traditional macroscopic Navier-Stokes equation where fluid is treated as continuum, the lattice Boltzmann method regards fluids as particles which perform consecutive streaming and collision processes over a discrete lattice mesh. The general form of the lattice Boltzmann equation reads, (2.4)
fi (x + δtci ei , t + δt) = fi (x, t) + Ωi
33
where fi is the distribution function or concentration of particles that travel with velocity ci to the next lattice node at one time step δt. The number of velocity vectors ci is equal to the total number of lattice nodes in the unit cells used to discretize the spatial domain. Figure 2.4 shows the two dimensional 2DQ9 lattice and three dimensional 3DQ19 lattice commonly used in lattice Boltzmann simulation. The term Ωi is called the collision
Figure 2.4. Examples of discrete velocity vectors for typical 2D and 3D velocity discretizations.
operator and determines the scattering rate for a given direction. The choice of Ωi may vary for different applications. For the sake of simplicity, the BGK collision operator, which features a single relaxation time τ , is often used in many engineering applications
34
[Bhatnagar, Gross and Krook , 1954], i.e., Ωi =
(2.5)
fi − fieq τ
where fieq is the local equilibrium distribution function. the macroscopic density ρ and momentum density ρ u can be determined if the distribution function is known, i.e., (2.6)
ρ=
M
fi ; ρ u =
i=1
M
fi ci e
i=1
where M is the total number of lattice nodes. To conserve mass and momentum, the collision operator Ωi must satisfy the following condition, (2.7)
M i=1
Ωi = 0 ;
M
Ωi e = 0
i=1
Equipped with the BGK operator, Equation 2.4 is often computed in two steps as shown below, (2.8)
(2.9)
fit (xi , t + δt) = fi (xi , t) −
fi − fieq (Collision) τ
fi (x + δtci ei , t + δt) = fit (xi , t + δt) (Streaming)
The resultant LBM scheme features an explicit local collision followed by a linear nonlocal streaming process. Unlike the Navier-Stokes equation, the above formulation does not require solving the expensive Poisson problem for pressure and thus significantly reduces the computational cost [Succi , 2001]. Moreover, complex boundary conditions are relatively easy to model, since they can be formulated in terms of simple mechanical rules
35
(e.g., bounce-back, reflections, periodic) of interactions between the particle and the solid wall. These features make LBM very appealing for hydrodynamics simulations, especially for those involving complex boundaries. More advanced topics, such as multiphase flow and lattice Boltzmann in irregular lattices are not covered in this dissertation. Interested readers should refer to, for instance, Chen and Doolen [1998] and Succi [2001] for more details.
2.5. Geometrical analysis of microstructures of porous media The microstructural features and connectivity of pore space are important factors that influence fluid transport [Carman, 1956; David , 1993]. In additional to calculating homogenized permeability from tomographic images, a number of researchs have focused on analyzing these microstructural attributes through medial axis analysis in geomaterials [Lindquist et al., 1996; Sun, Andrade and Rudnicki , 2011b] as well as biological tissue [Mendoza et al., 2007]. Medial axis analysis is a procedure that analyzes 3D geometry from its spline-like shape skeleton. In the pattern recognition and computer vision research community, this procedure is also known as topological skeletonization [Sirjani and Cross, 1991]. The first recorded skeletonization procedure was developed by Blum [1967]. In his seminal paper, Blum describes an algorithm that ”sets fire” at all grid points on the boundary of the 3D object simultaneously. The simulated fire then propagates inside the interior and the leftover, skeleton-like object is coined the medial axis, as illustrated in 2.5. The algorithm described by Blum is often referred to as BURN algorithm due to its similarity of burning [Lindquist et al., 1996; Sirjani and Cross, 1991]. The medial axis
36
Figure 2.5. Two-dimensional illustration of medial axis determination using BURN algorithm. From Lindquist et al. [1996]. analysis and BURN algorithms have found applications in numerous fields. For instance, Lindquist et al. [1996] and Lindquist and Venkatarangan [1999] modified the BURN algorithm in Blum [1967] to handle geomaterial specimens with complex pore geometry and couple it with Dijkstra’s algorithm [Dijkstra, 1959] to obtain the tortuosity distribution of flow channels. Their studies reveal that lengths of flow channels in sandstone and glass bead specimens both exhibit a Gamma distribution.
37
CHAPTER 3
A simplified method to analyze material instabilities in sands 3.1. Abstract A simplified method to analyze diffuse and localized bifurcations of sand under drained and undrained conditions is presented in this chapter. This method utilizes results from bifurcation analysis and critical state plasticity theory to detect onset of pure and dilatant shear band formation, static liquefaction and drained shear failures under low confining pressure (i.e., less than 10 MPa). Emphasis is given to examine how the presence of pore-fluid may facilitate or delay instability after yielding occurs. The predictions of instabilities are compared with experimental data from triaxial compression tests on Toyoura and Changi sands. 3.2. Introduction Stability of fluid-infiltrated porous solids, such as sand, rock, concrete and bone, is governed by constitutive responses of the solids and the interactions among solid grains and pore-fluid. The predictions of the onset and types of instabilities are both critical for many geotechnical and geomechanical applications, such as slope stability analysis, ground improvement for liquefactible deposit and underground CO2 storage. For instance, Terzaghi [1943] estimates bearing capability of foundation by assuming soil deposits behave like rigid blocks with localized failure surfaces. This assumption, however, may not be relevant if soil deposit underneath the foundation fails in a diffusive mode under static liquefaction.
38
To predict both the onset and types of failures is not a trivial task, primarily because the rate- and pore-pressure gradient- dependence introduced by the coupling between inelastic deformation and pore-fluid flow makes local, homogeneous analysis invalid. However, if the material of interest is at either fully drained or undrained limits, then the presence of pore-fluid becomes only a volumetric constraint to porous solid, which legitimatizes the bifurcation analysis of instantaneous material properties at material point level. By taking full advantage of this simplification at drainage limits, we establish a simple and unified bifurcation analysis approach to assess stability and predict instability modes of sands, without sacrificing the rigor of correct physics. Our goal here is to achieve a balance between simplicity and sophistication of concepts related to instabilities and bifurcations such that the analytical framework can be eventually useful for practicing engineers. The two major building blocks of this unified approach are the critical state plasticity model responsible for replicating constitutive responses of sand [Dafalias and Manzari , 2004; Manzari and Dafalias, 1997] and the bifurcation criteria detecting various types of instability modes from loss of uniqueness [Andrade, 2009; Borja, 2006a,b; Darve et al., 2007; Hill , 1958; Raniecki , 1979; Rudnicki and Rice, 1975; Rudnicki , 2004, 2009; Runesson, Peri´c and Sture, 1996]. To further simplify the analysis, we assume that plastic yielding is primarily a frictional mechanism and that increase in stress under a constant stress ratio is assumed to cause only elastic strain [Dafalias and Manzari , 2004]. As a result, any cap-surface yielding mechanism and the corresponding instability modes, such as compaction banding [Issen and Rudnicki, 2000; Issen, 2002; Rudnicki ,
39
2004] and cataclastic flow [Borja, 2006a, 2007], are not considered here. As a result of this assumption, the pressure-sensitive plasticity sand model developed by Dafalias and Manzari [2004] becomes our obvious choice. While not accounting for any cap-surface yielding behavior, this model incorporates concepts from critical state soil mechanics [Schofield and Wroth, 1968] to derive a kinematic hardening rule capable of replicating stress-strain responses remarkably well and yet maintains its simplicity by using the common two-invariant Drucker-Prager yield surface to predict onset of yielding. The usage of the two-invariant Drucker-Prager yield surface makes it possible to describe incremental elasto-plastic responses and the instability criteria with only five independent scale parameters (i.e., bulk modulus K, shear modulus G, hardening modulus Kp , frictional coefficient μ and dilatancy factor β) and the plastic flow direction. Since the instability criteria are functions of material parameters that can all be measured and extracted in conventional experimental setting (e.g., triaxial, biaxial, simple shear, direct shear tests), we can predict diffuse and localized instabilities without further empirical interpretations. As noted earlier, a more comprehensive stability analysis requires examination of the rate-dependent and non-local constitutive responses as well as the heterogeneous nature of the materials. These elements are beyond the scope of this dissertation. Nevertheless, simplified analyses of stability at drainage limits are useful for giving guidances for a fuller analysis and insights for conservative engineering designs. In the following section, we provide an overview of limit conditions that trigger diffuse and localized instabilities under fully drained and undrained conditions. While the accuracy of the instability predictions strongly depend on the performance of the constitutive
40
model [Issen, 2002], these limit conditions are valid for any two-invariant model without cap surface.
3.3. Drained Responses The fully drained condition is achieved if the time scale of the loading is much larger than that of the pore-fluid flow. In this case, the influence of the pore-fluid flow on the stability of the solid skeleton is negligible. Hence, constitutive relation is assumed to be rate-independent and local, without the need of introducing any poro-elasticity parameter.
3.3.1. Onset of the drained diffuse collapse Assume that the material deforms homogeneously prior to reaching the instability points and that the elastic response is isotropic. Under axisymmetrical loading, the linearized relation between stress and total strain increments can be sufficiently expressed by a elasto-plastic tangential operator with scalar deviatoric and volumetric components, i.e., ⎡
⎤
⎤⎡
⎡
⎤
2
(3.1)
⎢ p˙ ⎥ 1 ⎢ χK − K μβ −3GKβ ⎥ ⎢ ˙v ⎥ ⎦⎣ ⎦ ⎣ ⎦= ⎣ χ ˙s −3GKμ 3χG − 9G2 q˙
σ˙
C ep
˙
where χ = Kp + 3G + Kβμ. The determinant of the elasto-plastic operator C ep can be expressed in terms of Kp , μ, β, G and K, i.e., (3.2)
det(C ep ) =
3KGKp χ
41
where the eigenvalues of C ep are 0 and 3GK(1 + βμ)/χ when Kp = 0. The non-trivial eigen-strain associated with the zero eigenvalue of the elasto-plastic tensor takes the form, 1 [[]] ˙ = βI + 3
(3.3)
3 n 2
where n is the spectral direction of the deviatoric stress increment (same as R in Dafalias and Manzari [2004]). The physical consequence of C ep having a zero eigenvalue is that the material will not experience extra stress if a strain perturbation linearly proportional to the eigen-strain in (3.3) is applied. In other word, the material losses controllability at Kp = 0 under the drained condition.
3.3.2. Onset of shear banding Rudnicki and Rice [1975] and Rudnicki [2009] have shown that any constitutive responses simulated by two-invariant plasticity model has localized bifurcation points at which heterogeneous deformation may replace the initially homogeneous deformation while maintaining a uniform traction field normal to the band. The instability mode is localized and corresponds to a rank-one eigen-strain (strain that is dyadic product of two independent vectors) that lead to zero stress traction across a narrow zone, i.e., (3.4)
⊗ n) = n · [[σ˙ ]] = 0 n · C ep : (m
where n is a unit vector normal to the planar band. A non-trivial solution for m is possible only if the drained acoustic tensor becomes singular, i.e., (3.5)
ep det |ni Cijkl nl | = 0
42
This critical condition that triggers strain localizations can be expressed as a function of hardening modulus H , frictional coefficient μ, plastic dilatancy β and the principal deviatoric stress , i.e., (3.6)
√ Kp K(β − μ)2 K(3 6N2 + (β + μ))2 = − G 4G + 3K 4(G + 3K)
where the scalar N2 is the ratio between the intermediate principal deviatoric stress and the Euclidean norm of the principal deviatoric stresses. For material that does not exhibit cap-type plastic yielding, the plane of localization is orthogonal to the direction of the principal intermediate stress [Issen, 2002; Perrin and Leblond , 1993; Rudnicki and Rice, 1975; Rudnicki , 2004, 2009]. Moreover, the shear band formation precedes the drained diffuse collapse (i.e. shear bands form in the hardening regime) if the following relation is satisfied, (3.7)
(β − μ)2 −
4G + 3K √ (3 6N2 + 3β + μ)2 > 0 4(G + 3K)
Clearly, this is not possible if the flow rule is associative (β = μ).
3.4. Undrained Responses Under the undrained condition, pore-fluid remains trapped inside the pores during the loading cycle and hence constrained the volumetric deformation of the solid skeleton. This volumetric constraint may facilitate or delay instability. Since pore-fluid flow within pores is negligible, the constitutive response is approximately rate- and pore pressure gradientindependent [Rudnicki , 2009]. The constitutive responses of the solids are nevertheless affecting how pore pressure build-up and vice versa.
43
Constitutive relations for undrained responses can be expressed in the same form as (3.1) by replacing drained plasticity parameters with their undrained counterparts if (i) the elastic response remains isotropic, i.e. no coupling between elastic deviatoric and volumetric responses and (ii) the effective stress in Terzaghi form (σij + pδij ) [Terzaghi , 1943] or in Biot form (σij +Bpδij ) [Biot, 1941] and (iii) inelastic increment in the apparent void volume fraction is equal to the inelastic volume strain increment [Rudnicki , 2009].
3.4.1. Onset of static liquefaction The coupled macroscopic constitutive responses of porous solid at the undrained limit can be described by the elasto-plastic material parameters augmented with the poro-elasticity parameters to replicate the coupling between the solid skeleton and pore-fluid. The momentum and mass balance of the uniform undrained solid subjected to axisymmetric setting is [Borja, 2006b; Nur and Byerlee, 1971; Rice and Cleary, 1976], i.e., ⎤
⎡ (3.8)
⎡
2
−B ⎢ p˙ ⎥ ⎢ K − K μβ/χ −3GKβ/χ ⎢ ⎥ ⎢ ⎢ q˙ ⎥ = ⎢ −3GKμ/χ 3G − 9G2 /χ 0 ⎢ ⎥ ⎢ ⎣ ⎦ ⎣ 0 −B 0 −1/M
⎤⎡
⎤
⎥ ⎢ ˙v ⎥ ⎥⎢ ⎥ ⎥ ⎢ ˙ ⎥ ⎥⎢ s ⎥ ⎦⎣ ⎦ f p˙
where B is the Skempton’s coefficient and M is Biot’s modulus. Condensing (3.8) leads to the following relation, ⎡
⎤
⎤⎡
⎡ 2
(3.9)
⎤
2
⎢ p˙ ⎥ 1 ⎢ χ(K + BM ) − K μβ −3GKβ ⎥ ⎢ ˙v ⎥ ⎣ ⎦= ⎣ ⎦⎣ ⎦ χ −3GKμ 3χG − 9G2 ˙s q˙
σ˙
C und
˙
44
The undrained tangential constitutive response expressed in (3.8) ceases to remain stable if C und becomes singular. This happens if the hardening modulus satisfies the following relation, (3.10)
Kpliq = −
B2M Kμβ B2M + K
This result is identical to the static liquefaction criterion in Andrade [2009] (Kp = −Kμβ) if both constituents are incompressible. Under axisymmetric loading, the non-trivial strain increment can be expressed as a tensor whose the volumetric and deviatoric components maintains the following relation, (3.11)
Kβ 1 I+ [[]] ˙ = 2 3B M +K
3 n 2
In cases where the solid grains and pore-fluid are nearly incompressible (compared to the porous solid), M → ∞, B = 1 and the eigen-strain becomes isochoric. This implies that the porous solid can maintain the same deviatoric stress if a non-trivial, pure shear strain perturbation is applied. In other words, the material losses controllability at Kp = −B 2 M Kμβ/(B 2 M + K) ≈ −Kμβ under undrained condition. Since the shear strength is lost at this point, the material would behave like a frictional fluid and hence considered liquefied. 3.4.1.1. Onset of the pure shear band. Following the procedures in Rudnicki and Rice [1975] augmented with the pore-fluid mass balance law, one can derive the undrained
45
limiting hardening modulus which reads, (3.12) √ Kpsb (K und − BM )2 (β − μ)2 (3 6K und N2 + (K und − BM )(β + μ))2 B 2 M Kμβ = − − G K und (4G + 3K und ) 4K und (G + 3K und ) GK und where K und = K + B 2 M is the undrained bulk modulus of the porous media. If both the pore-fluid and the solid grains are incompressible such that M → ∞ and B = 1, then equation (3.12) can be simplified into, (3.13)
Kpsb 9 Kμβ = − N22 − G 2 G
where the band angle is always 45o to the principal stress regardless of the adopted yield criterion [Rudnicki , 2009; Runesson, Peri´c and Sture, 1996]. By comparing (3.13) with (3.10), we predict that undrained shear banding in materials composed of incompressible constituents always occurs in the softening regime and after static liquefaction for both associative and non-associative flow rules.
3.5. Critical state plasticity model for sands We incorporate the instability state line concept [Chu, Lo and Lee, 1993; Chu, Leroueil and Leong, 2003; Lade, 1993] into a critical state plasticity model. The framework of the critical state plasticity theory used here is developed by Manzari and Dafalias [1997] and Dafalias and Manzari [2004] (referred as MD model herein). This model incorporate the bounding surface model concepts to manipulate volumetric and deviatoric hardening such that the simulated elasto-plastic constitutive response is consistent with critical state
46
theory [Schofield and Wroth, 1968]. Interested readers are referred to the original paper cited above for details. The MD model includes an empirical elasticity model which reads, (3.14)
˙ij =
1 1 pδ ˙ ij + s˙ ij 3K(e, p) 2G(e, p)
where sij = σij − p/3δij is the deviatoric part of the Cauchy stress. The bulk modulus K(e, p) and the shear modulus G(e, p) are functions of current mean stress p and void ratio e , i.e., (3.15)
K(e, p) = Ko
(2.97 − e)2 p 1/2 (2.97 − e)2 p 1/2 ( ) ( ) ; G(e, p) = Go 1+e pat 1+e pat
where pat denotes the atmospheric pressure. The elastic domain is enclosed by a purely kinematic hardening Drucker-Prager yield surface which reads,
(3.16)
f (sij , αij p) = [(sij − αij )(sji − αji )]
1/2
−
2 mp = 0 3
where m is a constant and αij is the back stress. The hardening rule of the material is modeled based on the critical state theory [Schofield and Wroth, 1968]. In critical state theory, it is assumed that there exists a unique critical state under which materials shall experience large plastic shear without volume change. In MD model, the critical state is reached if both the stress ratio (deviatoric over hydrostatic) q/p and void ratio e reach their corresponding critical values, i.e., (3.17)
q q = ( )c = M c ; ec = e = eco − λc (p/pat )ξ p p
47
where M c , eo and ξ are scalar parameters determined by interpolating void ratio e, mean stress p and deviatoric stress q = 3/2(sij sij ) at observed critical state. To make the hardening and plastic dilatancy consistent with critical state theory, Dafalias and Manzari [2004] introduce two isotropic limit surfaces and use their distances from the critical state to manipulate the hardening modulus and plastic dilatancy. These two limit surfaces read, (3.18)
Hardening limit surface: q − M b p = 0 ; M b = M c exp [−nb (e − ec )]
(3.19)
Dilatancy limit surface: q − M d p = 0 ; M d = M c exp [−nd (e − ec )]
where nd and nb are scalar constants that control the size of the limit surfaces. They can be determined from the initial yielding points. The hardening modulus Kp and plastic dilatancy factor β are set to be continuous interpolation functions which depend on (1) the distance between limit and critical surfaces and (2) other interpolation functions added to improve the accuracy of the MD model. In axisymmetrical settings, the hardening modulus Kp and the dilatancy factor β are, (3.20)
(3.21)
Kp =
2ph Go ho (1 − ch e) (M b − q/p) ; h = 3 p/pat |q/p − (q/p)in |
β = Ao (1+ < zij nji >)(M d − q/p) ; z˙ij = −cz < − ˙pv > (zmax nij + zij )
(3.22)
nij =
3 1 (sij − αij ) 2 mp
48
where nij is the deviatoric part of the yield surface gradient. h and Ad are functions introduced to improve interpolation of the hardening and dilatancy from experimental observations. ho , ch , cz , zmax are positive scalar material parameters obtained by trialand-error. < x > is the Macaulay bracket, which is equal to x if x > 0 and 0 if x < 0. (q/p)in is the stress ratio at the beginning of the loading process. The tensor zij is called fabric-dilatancy tensor [Dafalias and Manzari , 2004]. It is introduced to take account of the content-normal orientation distribution changes during forward and reverse shearing. Furthermore, the frictional coefficient μ can be obtained from the volumetric component of the yield surface, i.e.,
(3.23)
μ=
3 1 (sij sji − sij αji ) 2 mp2
With the independent scale parameters (i.e., bulk modulus K, shear modulus G, hardening modulus Kp , frictional coefficient μ and dilatancy factor β) determined from the MD model, we have sufficient information to replicate incremental linear tangential responses of the sands and assess the stability of responses.
3.5.1. Incorporating of shear failure behaviors The constitutive relation developed so far is capable of replicating many important phenomenological characteristics of granular materials [Dafalias and Manzari, 2004; Jefferies and Been, 2006; Manzari and Dafalias, 1997]. However, the constitutive models described above predict perfectly plastic response if e = 1/ch . Since ch is a constant, this implies that perfectly plastic response occurs at a constant void ratio 1/ch regardless of the initial confining pressure. This is inconsistent with the experimental observations on undrained
49
loose sand by Lade and Pradel [1990], Chu, Lo and Lee [1993], Imam et al. [2002], Chu, Leroueil and Leong [2003] and Wanatowski and Chu [2007] of which the onset of instability varies with both the void ratio and the effective stress. Since it is clear that the aforementioned shortcoming is that ch remains fixed for various effective stress, the remedy is very simple. We rewrite ch as a function of effective hydrostatic pressure p. As a result, the expression (1 − ch ) in Equation (3.20) is now replaced by the instability state line obtained from experiments, i.e., (3.24)
Kp =
2ph Go ho (1 − ch (p)e) 1 (M b − q/p) ; h = ; ch (p) = s 3 eo − λ(p/pat )ξ p/pat |q/p − (q/p)in |
L where eso , s = L, D are the zero intercepts of the dilatant (eD o ) and contractive (eo ) insta-
bility state lines in e − p space. The instability state line is underneath the critical state line in the e − p space if the material is dilatant and above the critical state line if the material is contractive.
3.6. Numerical Predictions In this section, we present fully drained and undrained numerical simulations at the material point level. The purpose of performing these simulations is twofold – (1) to compare instability predictions with experimental observations (2) to analyze how presence of pore-fluid affecting stability. As shown in these numerical examples, the modified Manzari-Dafalias model is capable of replicating realistic constitutive responses of sand. This feature is important to ensure the accuracy of the instability predictions. Both drained and undrained triaxial test simulations are conducted with the material parameters presented in Table 1. The material parameters are calibrated with the drained
50
triaxial compression tests conducted by Chu, Leroueil and Leong [2003] (Changi sand) and undrained and drained triaxial compression tests by Verdugo and Ishihara [1996] (Toyoura sand). Symbol Parameter Changi sand Toyoura sand Go Shear coefficient 150 125 ν Poisson ratio 0.05 0.05 Mc Critical stress ratio 1.35 1.25 c eo Critical void ratio parameter 0.89 0.89 λc Critical state parameter 0.04 0.19 ξ Critical state parameter 0.4 0.7 eD instability state parameter 0.72 0.85 o eLo instability state parameter 0.98 1.08 ho Hardening parameter 2.8 7.1 Ao Dilatancy parameter 0.94 0.704 zmax Dilatancy parameter 4.0 4.0 cz Dilatancy parameter 600 600 nb Limit surface size parameter 1.1 1.1 d n Limit surface size parameter 2.8 3.5 Table 3.1. Summary of material parameters used for numerical predictions.
3.6.1. Example 1: Collapses in drained contractive Changi sand This simulation is conducted to replicate the contractive constitutive responses of the DR39 constant shear test reported in Figure 12 of Chu, Leroueil and Leong [2003]. The drained triaxial test was conducted on Changi sand with initial void ratio = 0.95. A 150kP a hydrostatic load is first applied on the Changi sand specimen, then the specimen is loaded in triaxial compression until the confining pressure reaches around 208kP a as shown in Figure 3.1(a). Following this step, the confining pressure decreases under constant deviatoric stress. This decrease in confining pressure causes further axial strain in the specimen and makes the stress state closer to the critical state line as shown in
51
Figure 3.1(a)-(c). Our result shows that the simulated contractive response is in general
400
(a) 300
(b)
0.06 εa
q (kPa)
0.08
CSL
200 100
0.04 Experiment Simulation
0.02
0 0
200 p (kPa)
0 0
400
4
4 Kp (kPa)
void ratio
0.6
(d)
IL CSL
0.8
0
200
x 10
(c) 1
100 p (kPa)
100 p (kPa)
200
2 0 −2 0
Drained Collapse
Kp Kband p 100 p (kPa)
200
Figure 3.1. Numerical simulation of drained triaxial compression test on Changi sand with initial void ratio = 0.95 (red). (a) simulated deviatoric stress (b) axial strain , (c) void ratio , (d) generalized hardening modulus vs. effective hydrostatic pressure. in agreement with the experiments. In particular, our simulation is consistent with the observation that the loose, contractive specimen collapses at stress state below the critical state line in the q-p space. As depicted in Figures 3.1(d) and 3.2, the contractive specimen exhibits perfectly plastic behavior as both the plastic dilatancy and the hardening modulus reduces to zero at q/p = M b = M d (see (3.20) and (3.21)). As a result, a significant amount of shear strain develops without plastic dilatancy and thus leads to the collapse
52
of specimen. This unstable response replicated by the MD model is consistent with the observed rapid increase in axial strain of the specimen occurred before the stress pass through the critical state line as depicted in Figure 7 of Chu, Leroueil and Leong [2003].
1.5
q/p
1
Drained Collapse when β = 0
Mb
0.5
Mc Md q/p
0 150
160
170
180 p (kPa)
190
200
210
Figure 3.2. The stress ratio q/p and the sizes of the hardening (M b ) and dilatancy M d limit surfaces vs effective hydrostatic pressure of the contractive Changi sand.
53
3.6.2. Example 2: Formation of shear band in drained dilatant Changi sand This simulation is conducted to replicate the dilatant constitutive responses of the DR39 constant shear test reported in Figure 12 of Chu, Leroueil and Leong [2003]. A triaxial load is applied under 150kP a confining pressure on Changi sand with a 0.657 initial void ratio. Figure 3.3(a) depicts the stress path. The specimen is first sheared until the deviatoric stress reaches 300kP a. Then the deviatoric stress remains constant while the hydrostatic stress is decreasing. Our result shows that the simulated dilatant response is in general in agreement with the experiments. By monitoring the generalized hardening modulus and its limit value for shear banding, we found that shear band may form in the hardening regime when the effective hydrostatic pressure of the the dilatant specimen is decreasing under constant deviatoric stress. The generalized hardening modulus Kp remains positive but its magnitude decreases as the axial strain develops. This phenomenon is captured in the MD model by shortening the distance between current and instability line (denoted as IL in Figure 3.3(c)) and the distance between the current stress and the isotropic hardening limit surface as illustrated in Figure 3.4. At the onset of shear band formation, the material exhibits plastic dilatancy ( because q/p < M d ). Hence, the strain localization zone may exhibit both shear and dilatancy. Following the onset of shear banding, the modified MD model predicts that the material would become very sensitive to perturbation as the response becomes more and more close to perfectly plastic as depicted in Figure 3.3(b) and (d). Notice that the dense, dilatant specimen collapses at stress state above the critical state line in the q-p space when the current stress reaches the hardening limit surface,i.e., q/p = M b as depicted in Figure
54
3.4. This prediction is consistent with the observed rapid increase in axial strain of the
300
0.06
CSL
(a)
(b)
0.04 εa
q (kPa)
400
200
Experiment Simulation
0.02
100 0 0
200 p (kPa)
0 0
400
100 p (kPa)
200
4
x 10
(c)
0.9
0.7
IL 0
Shear Band
(d)
CSL
0.8
0.6
4 Kp (kPa)
void ratio
1
100 p (kPa)
200
2 0 −2 0
Drained Collapse
Kp Kband p 100 p (kPa)
200
Figure 3.3. Numerical simulation of drained triaxial compression test on Changi sand with initial void ratio = 0.657 (red). (a) simulated deviatoric stress (b) axial strain , (c) void ratio and (d) generalized hardening modulus vs. effective hydrostatic pressure are plotted (red line) and compare with experimental observations (blue dots).
specimen occurred after the stress pass through the critical state line described in Figure 8 of Chu, Leroueil and Leong [2003].
55
2.5 2
q/p
1.5 Drained Collapse when β ≠ 0
1
Mb
Shear Band
Mc 0.5 0
Md q/p 160
180
200 p (kPa)
220
240
260
Figure 3.4. The stress ratio q/p and the sizes of the hardening (M b ) and dilatancy M d limit surfaces vs effective hydrostatic pressure of the contractive Changi sand. 3.6.3. Example 3: Static liquefaction of undrained Toyoura sand in loose state Three monotonic undrained triaxial compression tests on loose Toyoura sand are simulated and compared with the experiments conducted by Verdugo and Ishihara [1996]. Figure 3.5(a) and (b) compares the simulated undrained constitutive response with the experiments. As demonstrated in Figure 3.5 , the simulated constitutive responses are in good agreement with the experiments. Moreover, we found that softening only occurs if the loose Toyoura sand is first consolidated under sufficient confining pressure. In the monotonic undrained triaxial compression test with confining pressure p = 100kP a, both
56
1500
1000 CSL Static Liquefaction
1000 q (kPa)
q (kPa)
800
500 Static Liquefaction
0 0
500
1000 1500 p (kPa)
600 400 200
2000
0 0
0.05
εa
0.1
0.15
Figure 3.5. Numerical simulations of undrained triaxial compression tests on Toyoura sand with initial void ratio = 0.907. (a) simulated deviatoric stress vs. effective hydrostatic pressure and (b) deviatoric stress vs. axial strain. The confining pressure of the three simulations are 100kPa (blue), 1000kPa (red) and 2000kPa (green). Experimental data from Verdugo and Ishihara [1996] used to calibrate the material parameters are plotted in dots. experiential and simulated constitutive response exhibit only hardening. Figures 3.6(a) and (b) compares the hardening modulus predicted by the modified MD model with the the threshold values for static liquefaction (Kpliq ) and shear banding (Kpsb ). Interestingly, we found that the perfectly incompressible constraint imposed by the trapped pore-fluid may stabilize the material and prevent the formation of shear band. Furthermore, this comparison also reveals that static liquefaction may occur either (I) near the critical state line or (II) at the peak of the deviatoric stress. The former case is very similar to a critical state at drained condition in which hardening modulus, frictional coefficient and plastic dilatancy are very small and lead to materials shearing as a frictional fluid at constant volume and thus make material very sensitive to effective stress perturbation as
57
demonstrated by the high condition number of C ep . In the latter case, which is coined as undrained instability in Andrade [2009], the material become unstable and very sensitive to the total stress perturbation due to the singularity of C und . Notice that when max 10
2 1.5
4
5
Kp Ksb p
4
Kliq p
3
1
8
Kp Ksb p
0 -0.5
0
-2
-1.5 -2
-3
-2.5
-4
4
Kp Ksb p Kliq p
4
-1
-1
x 10
6
Kliq p
1 Kp (kPa)
Kp (kPa)
4
2
0.5
-3 0
x 10
Kp (kPa)
2.5
2
0
-2
-4
50 p (kPa)
100
-5 0
500 p (kPa)
1000
-6 0
500
1000 p (kPa)
1500
2000
Figure 3.6. Simulated evolution of hardening modulus (Kp , in blue color) and the corresponding threshold values for pure shear band (Kpsb , in red color) and static liquefaction (Kpliq , in green color) of Toyoura sand with initial void ratio = 0.907 and confining pressure equal to (a) 100kP a (b) 1000kP a and (c) 2000kP a.
terial is in between (I) and (II), strain softening occurs. As pointed out by Verdugo and Ishihara [1996], the deviatoric stress at the onset of (II) strongly depends on the initial confining pressure whereas the onset of (I) does not. One possible explanation is that the stability triggered by peak deviatoric stress is usually associated with non-zero hardening modulus Kp and non-zero frictional coefficient β as indicated in Figure 3.6. This makes the onset of (II) highly dependent of the initial confining pressure.
58
Using the localization criteria expressed in (3.13), we confirm that strain softening does not necessarily guarantee strain localization, which is consistent with the experimental observations of sand made by Chu, Lo and Lee [1993] and Chu, Lo and Lee [1996].
3.6.4. Example 4: Stable constitutive response of undrained Toyoura sand in dense state We simulate undrained triaxial compression test on dense Toyoura sand with confining pressure of 100, 1000, 2000 and 3000kPa. The initial void ratio of this simulation is 0.734. As demonstrated in Figures 3.7(a) and (b), the simulated constitutive response matches closely with the experimental data in Verdugo and Ishihara [1996]. Unlike the loose Toyoura sand, the dense Toyoura sand does not exhibit peak deviatoric stress. As a result, shear failures only occur near the critical state line. Figure 3.8 compares the generalized hardening modulus with the static liquefaction and shear band threshold values. The simulation result demonstrated in Figure 3.8 confirms that there is no shear band formed in the dense Toyoura sand under undrained condition. This observation is consistent with the homogeneous deformation modes observed by Verdugo and Ishihara [1996].
3.7. Conclusion We propose a simple and yet unified method to simulate the onsets of drained collapse, static liquefaction and formation of deformation bands under drained and undrained conditions. This method is based on bifurcation analyses and a two invariant critical state plasticity theory. By comparing numerical simulations with experimental data, not only
59
4000
4000 CSL
3000 q (kPa)
q (kPa)
3000 2000 1000 0 0
Shear failure near CSL
2000 1000
1000
2000 3000 p (kPa)
4000
0 0
0.1
εa
0.2
0.3
Figure 3.7. Numerical simulations of undrained triaxial compression tests on Toyoura sand with initial void ratio = 0.735. (a) simulated deviatoric stress vs. effective hydrostatic pressure and (b) deviatoric stress vs. axial strain. The confining pressure of the three simulations are 100kPa (blue), 1000kPa (red), 2000kPa (green) and 3000kPa (cyan). Experimental data from Verdugo and Ishihara [1996] used to calibrate the material parameters are plotted in dots. do we show that the instability criteria are capable of delivering predictions consistent with experimental observations, but we also provide an physical interpretation on why presence of pore-fluid may facilitate or delay instabilities and how contractive/dilatant state affecting both the onsets and modes of failures. Using experimental data available in the literature, we compare the simulated and observed constitutive responses as well as the predicted and actual onset of instabilities. Our finding confirms that the framework is able to robustly replicate the constitutive response and the onset of various instability modes observed in experiments.
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2
x 10
5
2
Kp Ksb p
1.5
x 10
5
Kp Ksb p
1.5
Kliq p
x 10
1.5
Kliq p
5
2
Kp Ksb p Kliq p
0.5
0.5
0.5
0.5 Kp (kPa)
1
Kp (kPa)
1
0
0
-0.5
-0.5
-0.5
-1
-1
-1
-1
-1.5
-1.5
-1.5
-1.5
1000 2000 p (kPa)
3000
-2 0
1000 2000 p (kPa)
3000
-2 0
1000 2000 p (kPa)
Kp Ksb p
0
-0.5
-2 0
5
Kliq p
1
0
x 10
1.5
1
Kp (kPa)
Kp (kPa)
2
3000
-2 0
1000 2000 p (kPa)
Figure 3.8. Simulated evolution of hardening modulus (Kp , in blue color) and the corresponding threshold values for pure shear band (Kpsb , in red color) and static liquefaction (Kpliq , in green color) of Toyoura sand with initial void ratio = 0.735 and confining pressure equal to (a) 100kP a, (b) 1000kP a, (c) 2000kP a and (d) 3000kP a.
3000
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CHAPTER 4
Multi-scale Modeling Techniques for Transport Characteristics of Deformation Bands This chapter is published as: W.C. Sun, J.E. Andrade, J.W. Rudnicki, A multiscale method for characterization of porous microstructures and their impact on macroscopic effective permeability, International Journal of Numerical Methods in Engineering, doi:10.1002/nme.3220, in press.
4.1. Abstract Recent technology advancements in X-ray computed tomography (X-ray CT) offer a nondestructive approach to extract complex three-dimensional pore geometries with details as small as a few microns in size. This new technology opens the door to study the interplay between microscopic properties (e.g., porosity) and macroscopic fluid transport properties (e.g., permeability). To take full advantage of X-ray CT, we introduce a multiscale framework that relates macroscopic fluid transport behavior not only to porosity but also to other important microstructural attributes, such as occluded/connected porosity and tortuosity, which are extracted using new computational techniques from digital images of porous materials. In particular, we introduce level set methods, and concepts from graph theory, to determine the tortuosity and connected porosity, while using a Lattice Boltzmann/Finite Element scheme to obtain homogenized effective permeability
62
at specimen-scale. We showcase the applicability and efficiency of this multiscale framework by two examples, one using a synthetic array and another using a sample of natural sandstone with complex pore structure.
4.2. Introduction Understanding the interactions between the microstructural geometry and the macroscopic effective permeability of porous media is a constant challenge of long-standing interest to a variety of technological areas such as hydrology, chemical production processes, geological science and biomechanics. As a result, a vast number of theoretical and numerical studies have attempted to examine how pore-scale geometrical features of flow paths inside porous media influences the fluid diffusion phenomena observed at the macroscale. Many of these studies are oriented toward using capillary tube analogs or statistically reconstructed models to establish correlations or empirical relations between micro-structural attributes (e.g., grain diameters, grain size distribution, pore throat sizes, tortuosities, porosities) and macroscopic permeabilities. While these studies are useful, it is difficult to assess how well the analogs and statistically reconstructed models represent the actual three dimensional pore geometry of real porous media. As a result, it remains unclear whether the findings based on analogs and reconstructed models are truly conclusive and applicable to porous media in general. A more desirable alternative is to directly calculate micro-structural attributes from the pore geometry reconstructed from three dimensional tomographic images and relate them quantitatively to macroscopic variables. This is not a trivial task. Owing to the
63
complexity of the pore geometry in real porous media, the calculations of effective permeabilities often require large scale hydrodynamics simulations, which could be prohibitively expensive. Meanwhile, extracting pore geometry measurements, such as tortuosity and connected/occluded pore space is very difficult without the prior knowledge of a geometrical object called the medial axis Lindquist et al. [1996]. The construction of medial axes, however, is itself another computationally demanding task that requires special numerical treatment to handle complex geometries. To overcome these challenges, we introduce a new computational framework that can rapidly extract the micro-structural attributes (e.g. tortuosity, connected/occluded porosity) and the macroscopic permeabilities in a systematic and computationally affordable manner. The major advantage of this computational framework is its ability to connect different numerical techniques sequentially and re-use outputs of one technique as input for the other ones in order to speed up the calculation and improve the accuracy. In particular, we introduce a new semi-implicit level set scheme to directly extract the three dimensional medial axes that represent the geometry of the pore space. Using these 3D medial axes the starting point, we construct a mathematical object called weighted graph or weighted network to characterize the connectivity of the 3D medial axes. The weighted graph is used to determine shortest flow path via Dijkstra’s algorithm Dijkstra [1959]. These shortest flow paths in return give us the tortuosity. Then, by using the shortest flow path as input, we identify the connected/occluded pore space via a recursive function and use the knowledge of the connected pore space to speed up and improve the accuracy of the multiscale permeability calculations. In the multiscale permeability calculation, we calculate the macroscopic effective permeability via two homogenization
64
procedures, one for micro-to-meso upscaling and one for meso-to-macro upscaling. To assess the accuracy of these homogenization procedures, we introduce a new parameter to quantitatively measure how close the multiscale model satisfies the Hill-Mandel condition [Du and Ostoja-Starzewski, 2006; Hill , 1963]. The organization of this chapter is as follows. First, we explain the computational framework and how various techniques are put together to extract fluid properties and pore geometry measurements. Then, we discuss in more detail how tortuosity and connected/occluded pore space are determined. Following the pore-scale geometrical analysis, we present the multiscale framework to calculate macroscopic effective permeability at a scale relevant to engineering applications. The computational framework is tested by two examples in the application section, followed by the conclusion of this study.
4.3. Overall architecture As depicted in Figure 4.1, the methodology presented in this chapter can be divided into two parts. Part 1 is furnished by a detailed geometrical analysis where two micromechanical attributes are extracted: tortuosity and connected porosity; both key factors influencing macroscopic permeability. Part 2 of the method deals with effectively using the physical attributes extracted during part 1 to compute the macroscopic effective permeability of porous materials using a two-scale lattice Boltzmann/Finite element procedure. The central part of the method is part 1, where three numerical techniques are used to extract the tortuosity and connected porosity. First, the level set method is used to obtained the medial axes of the pore space. Second, a shortest path algorithm is used to
65
FLOWCHART OF THE MULTISCALE FRAMEWORK FOR PORE-FLUID TRANSPORT PROBLEM Geometrical Analysis start with pore geometry
Level Set Method
Direct Pore-Scale Simulations (required significant computational resource)
medial axis of pore space Shortest Path Algorithm
geometrical Tortuosity Region Growing Algorithm
connected / occluded pore space
Homogenization Scheme macroscopic permeability
Finite Element Method
Local grid block permeability
Lattice Boltzmann Method
Figure 4.1. Flow chart of the numerical procedures used to compute tortuosity, identify connected/occluded pore space and estimate effective permeability at the specimen-scale.
extract the tortuosity. Third, a region-growing algorithm is used to identify the connected porosity, and by default the occluded porosity. It must be noted that these processes are performed sequentially, with the output from one serving as the input of the other. Armed with the micromechanical information obtained in part 1, a two-scale homogenization procedure is launched where lattice Boltzmann is implemented in the connected porosity only and the finite element method is then used to take the computations to the macroscopic
66
scale. Domain decomposition and focusing the lattice Boltzmann procedure on the connected porosity differentiates this method from its predecessors and make it very efficient. In the next sections we describe the implementation of the main parts of the method in further detail.
4.4. Part 1: Numerical procedures for pore-scale geometrical analysis Our point of departure is the assumption that a digital representation of the porosity of the material is available. For example, a binary image of the solid phase and porous phase can be furnished using X-ray CT. Once this micro-structure is available, the question is how to use it to make quantitative statements of macroscopic properties such as permeability. To this end, we develop a numerical technique that relies on three interconnected components: development of level sets to determine the three-dimensional medial axes of the pore structure; development of shortest-path algorithm to evaluate the geometric tortuosity; and finally development of a growing-region algorithm to evaluate connected pore space.
4.4.1. Semi-implicit variational level sets for medial axes extraction Medial axes are spine lines that trace the geometrical centers of volume-filling objects, such as flow channels formed by voids in a porous medium. Graphically, medial axes can be thought of as skeletons that inherit the shape of their corresponding volume-filling objects [Kimmel et al., 1995; Lindquist et al., 1996]. Thus, the length of the actual flow channels, and that of the medial axes representing them, are equal. This feature can be
67
used to quantify geometrical properties of the pore space using well-known parameters such as tortuosity and connected porosity. Typically, medial axes are obtained via thinning algorithms, such as the BURN algorithm used in Sirjani and Cross [1991], Kimmel et al. [1995] and Lindquist et al. [1996]. These thinning algorithms share the same key idea: they remove the outer boundary layers of the volume-filling objects until these are thinned into curves. Here, we propose a new thinning algorithm based on the level set method. The main motivation for this new procedure is computational efficiency as the level set method lends itself nicely to capture complex geometries in three-dimensional space. Let us define the binary images describing the microstructure of the porous material as the field φo . This is a bi-variate field, taking values equal to 255 to describe the solid skeleton and 0 to describe pore spaces. The initial field can be easily interpolated using a continuous function and the solid-void interfaces, Γ can be determined by computing the gradient of φo . Here, we define an edge indicator function g such that (4.1)
g=
1+
∇x
1 φo · ∇x φo
where we have made use of the discrete gradient operator ∇x . In a finite difference setting, the discrete gradient operator is finite and proportional to the mesh size. As shown in Figure 4.2, the numerical values of the edge indicator will be close to zero along the solid-void interfaces and equal to one elsewhere. Once the solid-void interfaces have been identified, we can use the level set method to determine the signed distance function.
68
Figure 4.2. A binary image φo and its corresponding edge indicator function g.
A signed distance function φ (x) measures the signed shortest distance between the position x and the interface Γ, i.e.,
(4.2)
φ (x) =
⎧ ⎪ ⎨ − inf y∈Γ x − y if φo (x) = 255 ⎪ ⎩ + inf y∈Γ x − y if φo (x) = 0
where inf denotes the infimum of a function. According to equation (4.2), φ(x) is negative if x is inside the solid phase, positive if x is inside the pore phase, and equal to zero if x ∈ Γ. For example, if we have a two-dimensional pore space resembling a circle, the corresponding signed distance would be a three-dimensional cone with the maximum value of φ at the center of the circle and φ = 0 on the boundary of the circle. Note that since φ(x) is a metric measuring the distance between position x and its closest point on the boundary y ∈ Γ, φ(x) reaches its local maximum if x is located in the middle axes of the
69
volume, i.e., (4.3)
φ (x) = inf x − y = inf x − y y 1 ∈Γ
y 2 ∈Γ
where ∇x φ(x) = 1. Therefore, as shown in Figure 4.3, the medial axes of the pore space can be interpolated from the local maxima of the signed distance.
Figure 4.3. Two-dimensional pore space example. Left figure is the original binary image, containing only binary data. By applying the level set scheme, the signed distance function is formed inside the pore space as illustrated in the middle figure. The medial axis can be located by interpolating the local maximum point of the signed distance function as illustrated in the right figure.
4.4.1.1. Variational formulation. A signed distance function corresponding to the pore space can be determined by a variety of level set evolution equations. A level set evolution equation is a specific type of PDE called Hamilton-Jacobi, which takes the form, (4.4)
∂φ + H (∇x φ) = 0 ∂t
70
where H is called the Hamiltonian. With a properly defined Hamiltonian, the HamiltonJacobi equation can force an arbitrary continuous function to evolve into a signed distance function as a function of time and as illustrated in Figure 4.4
Figure 4.4. Example evolution of level set function φ.
A particular form for the Hamiltonian has been proposed by Li et al. [2005]. This formulation relies on a variational form and circumvents many of the problems associated with traditional level set evolution equations, in particular, re-initialization. An energy functional is introduced to enforce equation (4.4) while other energy functionals are introduced to move the zero level set φ = 0 toward the solid-void interface. Herein, we adopt the formulation of Li et al. and implement it using a new numerical technique. The analytical formulation is briefly summarized below for completeness but further details can be found in Li et al. [2005]. An energy functional E is introduced such that H = ∂E/∂φ, where the last term is defined as the Gateaux derivative of the functional E. The energy is defined as a linear
71
combination of three fictitious energy terms such that E (φ) = μP (φ) + λLg + νAg
(4.5)
where μ, λ, and ν are numerical parameters introduced to control the diffuse rate of the level set function. The energy (internal) P is a penalty functional to drive the level set function to satisfy ∇x φ = 1 such that, P (φ) =
(4.6)
Ω
1 ( ∇x φ − 1)2 dΩ 2
By the same token, the energies (external) Lg and Ag drive φ = 0 along solid-void boundaries. They can be expressed as Lg (φ) =
(4.7)
Ag (φ) =
(4.8)
Ω
gδ(φ) ∇x φ dΩ ˆ g H(−φ) dΩ
Ω
ˆ is the Heaviside function. We note the where δ is the univariate Dirac delta function and H dependence of Lg and Ag on the edge indicator function g. Also, these latter functionals do not affect the value of φ along the solid-void boundaries, but are rather responsible for shrinking the level set function elsewhere such that the local maxima points of the level function coincide with those of the signed distance function. The governing equation (4.4) is then obtained by the principle of least action, where ∂E/∂φ = 0 at steady-state Li et al. [2005], and hence, (4.9)
∂φ ˆ + λδ (φ) ∇x · (g n) ˆ + νgδ (φ) = μ [Δx φ − ∇x · n] ∂t
72
ˆ := ∇x φ/ ∇x φ and is a unit vector in the where Δx is the Laplacian operator and n direction on the gradient of φ. In the next section, we present a new algorithm to integrate the above equation. 4.4.1.2. Semi-implicit integration of variational formulation. In order to obtain a signed distance function for the pore space, it is necessary to integrate equation (4.9) in space and time. In this work, we propose a semi-implicit scheme that can deliver stable, yet economical numerical solutions. Fully explicit solutions become problematic as they require a small time step to conserve stability. In the case of complex pore geometries, such as the ones encountered in natural geologic deposits, a small time step becomes a major handicap to the method. On the other hand, nonlinearities present on the Hamiltonian term (cf., right hand side of equation (4.9)) make fully implicit procedures difficult and expensive since they would require iterations. A good compromise seems to discretize the Hamiltonian semi-implicitly in time, with the Laplacian operator described fully implicitly. This is in contrast to the procedure presented in Li et al. [2005], where the Hamiltonian is integrated fully explicitly. Integrating the governing equation in time using a forward difference scheme, we get (4.10)
¯ φn+1 = φn − ΔtH
where φn+1 corresponds to the φ field evaluated at the discrete time station t = tn+1 and Δt = tn+1 − tn . Also, for a given point in space, we can express the discrete Hamiltonian semi-implicitly by integrating in space using central differencing such that (4.11)
¯ = −μ Δc φn+1 − ∇c · n ˆ n − λδ(φn )∇c · (g n ˆ n ) − νgδ(φn ) H
73
where we have used the discrete central difference operators for the gradient ∇c , the ˆ n = ∇c φn / ∇c φn . We should emphasize divergence ∇c · and the Laplacian Δc , and n that only the first term of the Hamiltonian, corresponding to the Laplacian operator, is expressed implicitly. Consequently, the current semi-implicit scheme does not require iterations, since the nonlinear terms in the Hamiltonian are treated explicitly (evaluated at time tn ). We implement this semi-implicit technique in the numerical examples presented herein and showcase the achieved numerical efficiency of the linear system and the stability afforded by an implicit technique.
4.4.2. Dijkstra’s algorithm for tortuosity extraction Fluid flow in porous media follows complex paths and it’s directed by micro-channels. One parameter that attempts to quantify the complexity of these flow channels is the tortuosity τ , defined as the ratio between the effective length of the shortest flow path Le and the linear distance through the porous medium L [Adler , 1992; Dormieux, Kondo and Ulm, 2006; Lindquist et al., 1996], i.e., (4.12)
τ=
Le L
Everything else being equal, porous media with higher tortuosities require higher pore pressure gradients to achieve the same flow rates, due to the effective elongation of the flow channels. For example, the widely-used Kozeny-Carman equation [Carman, 1956] predicts that the effective permeability of a porous system is inversely proportional to the tortuosity. Because of its important role in the flow properties of porous media, it is often desirable to measure the geometric porosity. One approach used in Lindquist et al.
74
[1996] is to apply a thinning algorithm on the inlet and outlet faces of the porous medium, then apply a shortest path searching algorithm on the entire pore space to determine the effective length. The major drawback of this approach is that it requires significant CPU time and memory usage due to the large number of voxels in typical 3D images of pore structure. In this work, we propose an alternative approach to measure the tortuosity. We apply the shortest path algorithm on the three dimensional medial axis, instead of the entire pore space, obtaining significant savings on computing time. With the medial axis as input, we seek the shortest path connecting opposite edges in a given volume. A key element of the algorithm is the definition of a weighted graph. A weighted graph is a network of nodes connected in between by a unique edge, and each edge has a weight [Marcus, 2008]. As illustrated in Figure 4.5, in our case, each voxel in the medial axes is defined as a node, connected by edges whose weight is given by the Euclidean distance connecting the voxels. The objective of the graph is to provide a three dimensional representation of the network and its connectivity. This will prove crucial for finding the shortest path. Armed with the weighted graph representing the medial axis of the porous medium, Dijkstra’s algorithm [Dijkstra, 1959] is used to determine the effective length and tortuosities of the graph (which correspond to those of the actual porous medium). In our particular problem, Dijkstra’s algorithm works as follows. (1) We locate vertices that are on the inflow and outflow faces of the specimen (volume). (2) Label one of the vertices on the inflow face as the first active vertex and one of the vertices located on the outflow face as the targeted outflow vertex. (3) Use graph connectivity to select unvisited vertices directly connected to the active vertices and that are also connected to the outlet vertex.
75
Weight W Distance e
Vertex Verte V
Voxe el
n
Connected path h
E Edge
Figure 4.5. Example of corresponding weighted graph for sample medial axis. Medial axis corresponds to example shown in Figure 4.3. Selected vertices become part of the active set and the total lengths of paths defined by the active vertices is computed using the weights. Step (3) is repeated until either (i) the targeted outflow vertex becomes an active vertex or (ii) no unvisited vertex is next to an active vertex. In the case of (i), then the length of the shortest path connecting the inflow and outflow vertices is the effective length Le . Dijkstra’s algorithm ranks all lengths of connecting flow channels by ascending order so the shortest path appears first when the algorithm reaches condition (i). If (ii) is reached without (i), the selected inflow and outflow vertices are not connected. We continue the procedure above until all possible pairs of inflow and outflow vertices have been examined. The shortest overall path is selected as the length Le . If there are N1 inflow vertices and N2 outflow vertices, the algorithm is run
76
N 1 × N2 times. The shortest flow paths and the probability distribution of tortuosities are stored before exiting the algorithm. In the example shown in Figure 4.5, the shortest path length is 69 and the algorithm is run twice.
4.4.3. Region-growing algorithm for connected pore space extraction Previous research has focused on relating the effective permeability with the total porosity [Lenoir et al., 2010; Sun and Andrade, 2010a]. A major drawback of these approaches is that they do not distinguish the occluded porosity from its connected counterpart. Connected pore channels govern transport properties and, therefore, must be accurately evaluated. Fortunately, both occluded/connected porosities can be measured by using the information we have already acquired, i.e., the graph that represents the porous network and the shortest flow path extracted from the level set and Dijkstra’s algorithms. Since the shortest flow path is inside the connected pore space, we can identify the connected pore space by examining the neighboring voxels and classifying them as connected pore space until reaching the boundaries. To identify the connected/occluded pore space, we first use the flow path obtained from the Dijkstra’s algorithm as seeds planted inside the connected pore space. Then, a recursive function is used to simulate the grow of the spanning tree stemmed from the seeds. This recursive function stops running when all the edges inside the connected pore space are explored and all vertices inside the connected pore space are visited. Since the occluded and connected pore space are mutually exclusive, the region of pore space not visited by the recursive function is therefore the occluded pore space. The pseudo code
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of the computer program used to identify connected/occluded pore space is as given in Box 1. Box 1: Pseudocode of the program used to identify connected pore space (1) Activate all vertices along the flow path as active nodes and mark them as visited vertices (2) While there exists at least one active node (a) call the recursive function MARKNEIGHBOR (3) EXIT FUNCTION MARKNEIGHBOR (1) IF at least one neighbor of the active nodes has not yet been visited (a) Activate the unvisited neighbor vertices (b) Mark them as visited vertices. (c) Deactivate the old active nodes with unvisited neighbor(s). (d) Call the recursive function MARKNEIGHBOR (2) ELSE (a) Deactivate the active nodes with no unvisited neighbor. (3) EXIT
4.5. Part 2: Two-scale homogenization of permeability using LB and FEM The effective permeability of a porous medium can be measured by applying a pore pressure gradient along a basis direction and determining the resultant fluid filtration velocity from pore-scale hydrodynamic simulation. Then the effective permeability tensor can be obtained according to Darcy’s law (4.13)
kij = −
μv vi p,j
where vi is the flow vector in the i-th orthogonal direction, represents volume average, muv us the kinematic viscosity of the fluid, and p,j is the gradient in fluid pressure in the j-th orthogonal direction. The permeability tensor kij is treated here in the standard
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way, where it is assumed to be diagonal and positive definite. Hence, only the diagonal components of the tensor are non-zero and need to be evaluated. One way to calculate the effective permeability in a porous sample is to perform a mesoscale direct numerical simulation (e.g., using lattice Boltzmann) over the entire sample and in this way account for all details in pore geometry. This direct approach, though accurate, requires immense memory and CPU usage times for specimens of sizes relevant to engineering applications. A significantly cheaper way to calculate permeability at the specimen scale is to use a multiscale framework exploiting a lattice Boltzmann/Finite element hybrid scheme [Sun and Andrade, 2010a; White, Borja and Fredrich, 2006]. The key idea of the hybrid scheme is to apply domain decomposition and thereby break down the computationally demanding large simulation into multiple smaller problems that can be handled with less resources, then perform homogenization to obtain the equivalent macroscopic properties, as illustrated in Figure 4.6. We note here that while domain decomposition can reduce computational expenses significantly, it can potentially introduce considerable error if the occluded porosity is interpreted to be connected in the calculations. To avoid this issue, we perform all calculation using the connected porosity only as identified in the previous sections. The two-scale domain decomposition scheme is illustrated in Figure 4.6. The scheme uses lattice Boltzmann as the meso-scale calculation paradigm and finite elements for macroscopic simulations. First, the connected pore space in the sample is decomposed into smaller, manageable domains for direct meso-scale calculation (e.g., using lattice Boltzmann). Permeability tensors are obtained for each subdomain using equation (4.13) utilizing lattice Boltzmann (LB) as the computational paradigm on the connected porosity
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only. Second, each subdomain is represented geometrically by finite elements with permeabilities obtained from the previous step. Finally, finite elements (FEM) are used to estimate the effective permeability of the entire sample, accounting for the heterogeneities implied in each subdomain. This procedure is clearly shown in Figure 4.6 where the sample is decomposed into four subdomains. Permeability calculations are performed in each subdomain by LB and then these values are used in one more macroscale simulation using FEM to estimate the effective permeability of the entire sample.
Figure 4.6. Multiscale numerical scheme used to determine effective permeability in large scale.
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4.5.1. Numerical procedures: lattice Boltzmann and finite elements
The main features of the lattice Boltzmann and finite element procedures used in this work are summarized in this section for the sake of completness. In the lattice Boltzmann procedure, the discrete distribution function fi (x, t) is the main unknown such that the particle distribution satisfies the discrete-velocity Boltzmann equation [He and Luo, 1997; Keehm, Sternlof and Mukerji, 2006; Lenoir et al., 2010; Succi , 2001; Sun and Andrade, 2010a; White, Borja and Fredrich, 2006], i.e., (4.14)
∂fi + ei · ∇x fi = Ci ∂t
where Ci is a collision term that accounts for the net addition of particles moving with velocity ei due to inter-particle collisions. LB is particularly suited to handle complex geometries such as those encountered in natural geomaterials. Also, fluid velocity v and pressure p at lattice node x and time t are both determined from the discrete distribution function, i.e., 1 v= f i ei ; p = c2 ρ ; ρ = fi ρ i=1 i=1 α
(4.15)
α
where α is the number of lattice directions a molecule can move, and c denotes the speed of sound, which is treated as a constant in the Lattice Boltzmann simulations. Using a simple standard technique proposed in Inamuro, Yoshino and Ogino [1995]; Succi [2001], we can reproduce nearly incompressible flows with velocities and pressures that can be used in equation (4.13) to determine the meso-scale value of permeability.
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After extracting the local permeability tensors for all unit cells, we assign the numerical values of these local permeability tensors to the corresponding Gauss points of the finite element model. The finite element model is aimed to simulate the macroscopic diffusion of an incompressible, single-phase pore-fluid. It is based on Darcy’s law augmented with the incompressible constraint, i.e, (4.16)
(4.17)
∇x · v(x) = 0
v(x) = −
1 k(x) · ∇x p(x) v μ
where body forces are neglected and k = kmeso denotes the local permeability tensor obtained from the meso-scale Lattice Boltzmann simulations described above. Combining equations (4.16) and (4.17) yields a single phase pressure equation for steady incompressible flow, i.e., (4.18)
1 x ∇ · (k(x) · ∇x p(x)) = 0 μv
Augmenting equation (4.18) with the pressure prescribed on the corresponding boundaries, we obtain the boundary value problem suitable for finite element discretization. We use the standard Galerkin method to obtain the macroscopic pressure field.
4.5.2. Upscaling effective permeability Accuracy and efficiency of the multiscale hybrid method relies crucially on the size selected for the unit cells. If the unit cells are too large, the speed of the multiscale method
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decreases dramatically; if the unit cells are too small, the multiscale method may fail, since the continuum representation may break down [Sun and Andrade, 2010a; White, Borja and Fredrich, 2006]. One way to strike some balance between accuracy and efficiency is to choose unit cells that are just big enough as to satisfy the continuum requirements. A unit cell that fits this description can be referred to as representative element volume (REV). While there is no universal rule to determine the appropriate size of REV, it is widely accepted that a REV must satisfy the following conditions [Bear , 1972]: (i) the REV must be large enough to contain sufficient statistical information about the microstructure; (ii) the effective constitutive response (e.g., permeability) must be independent of the type of boundary conditions imposed on the REV. The concept of REV, and conditions related to it, have been amply studied in the context of elasticity. Of particular importance is the condition known as the Hill-Mandel condition [Hill , 1963]. In this work, we will extend the Hill-Mandel condition for the application of flow through porous media. Our point of departure is the governing equation (4.18) and Dirichlet or Newmann uniform boundary conditions, analogous to the uniform stress and strain conditions in elasticity [Du and Ostoja-Starzewski, 2006], i.e., (4.19)
v · n = v0 · n
(4.20)
p = p0
on Γv
or
on Γp
where n is the outward normal to the boundary Γv and v 0 and p0 are constant prescribed values of v and p. Using the formulation in Berryman and Milton [1985], one can show that (4.18) can be reformulated via the least action principle. In this case, the solution of the Darcy’s law augmented with continuity equation is the p that minimizes the energy
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dissipation rate D(p), i.e., 1 ∂D(p + αδp) |α=0 = 0 ⇔ v ∇x · (k · ∇x p) = 0 ∂α μ
(4.21)
where the energy dissipation rate reads, (4.22)
D(p) =
1 x ∇ p · k · ∇x p v μ
The role of this energy dissipation rate for the pore-fluid transport problem is analogous to the elastic strain energy for the elasticity problem. Consequently, the necessary condition for both (i) and (ii) for flow in heterogeneous porous media can be regarded as a special form of the Hill-Mandel condition Du and Ostoja-Starzewski [2006]; Hill [1963], which reads, (4.23)
D
macro
1 = Ω
1 ∇ p · v dΩ = Ω Ω x
1 ∇ p dΩ · Ω Ω
x
Ω
v dΩ
where Dmacro is the macroscopic energy dissipation rate. By the divergence theorem, substituting equation (4.17) in equation (4.23), and assuming that the prescribed pressure is applied on the top and bottom of the numerical specimen, we obtain the following expression, (4.24)
D
macro
1 = Ω
Ω
marco 1 x (p2 − p1 )2 kzz meso x ∇ p·k · ∇ p dΩ = μv μv (z2 − z1 )2
where kmeso is the local permeability determined from the pore-scale Lattice Boltzmann simulations and kmacro is the global permeability obtained from the specimen-scale finite element simulations. Also, p2 and p1 and z2 and z1 are the pressures and heights of
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the outlet and inlet faces, respectively. Suffice it to say, the Hill-Mandel’s condition, as stated in equation (4.24), is the criterion that ensures the energy dissipation rate calculated from the boundary values of the macroscopic pressure field being equal to the energy dissipation rate calculated from the volume integration over the numerical specimen. For an arbitrary porous medium which show scale dependency, there may not always exist a definite mesoscale for which the Hill-Mandel’s condition holds exactly. Nevertheless, the difference between the macroscopic energy dissipation rate obtained from the boundary and volume integrations is still a good indicator that quantifies how accurate the homogenization scheme performs. In our calculations, we monitor the difference in these dissipations to ensure the accuracy of the upscaling form the local permeability tensor field kmeso to the global permeability tensor kmacro . In addition to proposing the use of Dmacro to ensure accuracy in going from meso to macro scale, we propose the use of energy dissipation Dmicro to ensure proper transition from micro (from LB) to meso scale. This approach was originally proposed by White, Borja and Fredrich [2006] in which the scale fluctuation of the microscopic energy dissipation rate is used to determine the appropriate size of unit cell for the lattice Boltzmann simulation. The microscopic energy dissipation rate is obtained from the LB simulation, such that, (4.25)
Dmicro = 2μv : ;
1 = (∇x v + (∇x v)T ) 2
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where the local permeability tensor and the microscopic energy dissipation rate over the domain of the unit cell Ωc satisfies the following relation, 1 Ωc
(4.26)
Ωc
Dmicro dΩc = ∇x p · kmeso · ∇x p
To ensure that the lattice Boltzmann simulations are conducted in a unit cell capable of resembling continuum behavior, we conduct a series of simulations on spatial domains with increasing size but fixed centroid. Then, we examine the scale of fluctuation of Dmicro versus the unit cell size and select the unit cell size that ameliorates the local fluctuation in energy dissipation, as illustrated in Figure 4.7.
4.5.3. Remarks on occluded porosity and its impact on homogenized permeability In many situations, particularly in natural porous materials, occluded porosity occupies a significant portion of the pore space. Failure to identify occluded porosity can cause dramatic errors in multiscale modes. Typical situations where significant occluded porosity is expected to play a role include the migration of pore-fill cement into pore space [Almon, Fullerton and Davies, 1976], pore closure in limestones due to CO2 sequestration [Alvarez and Abanades, 2005], and the formation of compaction bands [Sun et al., 2011a]. To illustrate this point, let us consider a two dimensional LB simulation of the sample depicted in Figure 4.8. In this example, our objective is to obtain the vertical global permeability of a sample, discretized using a 30 u × 40 u lattice (u =lattice unit), using three different techniques. In the first technique, LB simulations are conducted on the entire sample, without any domain decomposition. This is equivalent to a direct numerical
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8 7
Pore-scale
Continuum Scale
DISSIPATION
6 5
appropriate size for unit cells
4 3 2 1 0
0
0.1
0.2
0.3 0.4 0.5 0.6 0.7 ELEMENT DIMENSION s/L
0.8
0.9
1
Figure 4.7. Selection of the unit cell size based on the scale of fluctuation. s/L is the ratio between the voxel length and edge length of the specimen.
simulation and is interpreted here as the ‘true’ solution. The second technique uses domain decomposition (sample is split into four parts along the vertical direction) and uses occluded space detection, keeping only connected porosity active. The third technique uses domain decomposition but does not distinguish between connected and occluded porosity. Global permeabilities for the partitioned samples are obtained from the local estimates by Bear [1972],
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n
(4.27)
k=
Li
i=1
n
Li /ki
i=1
where ki are the local values of permeability in each layer of thickness Li , and n = 4 denotes the number of unit cells (layers).
(a)
(b)
(c)
Figure 4.8. Velocity profiles of lattice Boltzmann simulations on (a) unpartitioned domain (k = 0.015 u2 ), (b) partitioned domain with identified and deactivated occluded porosity (k = 0.013 u2 ), and (c) partitioned domain without any special treatment for occluded porosity (k = 0.0078 u2 ). Where u =lattice unit.
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Results for the LB calculations are summarized in Table 4.1. It should be highlighted that the relative error induced by the third procedure with partition but no special treatment of occluded porosity yielded an error four times greater than that of the partitioned method that takes into account occluded porosity. It can be seen from Table 4.1 that the main sources of error come from the mistreatment of occluded porosities in the central partitions. The mistreatment of occluded porosity is not only the source of errors in the estimation of permeability, but it leads to longer calculations as occluded porosity is assigned active lattices. Hence, not accounting for occluded porosity may lead to inaccuracies and inefficiencies. Case 1 2 3 Number of Unit Cell(s) 1 4 4 Occluded Pore Identified? No Yes No Local Permeability, u2 (top) N/A 0.011 0.011 Local Permeability, u2 (2nd top) N/A 0.015 0.0029 Local Permeability, u2 (2nd bottom) N/A 0.015 0.45 Local Permeability, u2 (bottom) N/A 0.014 0.014 Global Permeability, u2 0.015 0.013 0.0078 Relative Error 0 12 % 48 % Table 4.1. Global and local permeabilities obtained from LB simulation scenarios.
4.6. Representative examples In this section, we present two example applications of the proposed framework. The first example deals with a spatially periodic, bi-continuous pore structure furnished by a simple cubic (SC) spherical array. The objective of this example is to verify the accuracy of the proposed method. Owing to the simplicity of the SC pore structure, there is a wealth of theoretical and numerical solutions in the literature that can be used to verify our newly proposed approach. The second example deals with a natural sample of Aztec sandstone
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inside a compaction band that has imaged using synchrotron X-ray CT [Lenoir et al., 2010]. Compaction bands are a type of strain localization known to significantly reduce connected porosity and increase geometric tortuosity, thereby reducing the permeability of sandstones with these formations by orders of magnitude. Because of their implied importance for injection and extraction of fluids, compaction bands have been amply studied [Holcomb et al., 2007; Keehm, Sternlof and Mukerji, 2006; Rudnicki , 2004; Sun and Andrade, 2010a; Sun et al., 2011a; Sun, Andrade and Rudnicki, 2011b; White, Borja and Fredrich, 2006]. However, there are no direct measurement of permeability inside compaction bands from field samples in three dimensions. This example highlights the importance and applicability of the proposed framework in real porous media.
4.6.1. Periodic simple cubic (SC) lattice Simple cubic (SC) cells can be formed by placing the centroid of eight identical spheres at the corners of a cube of equal dimensions. When the spheres are making point contact, it is often called SC bead pack [Saeger, Scriven and Davis, 1995]. In this packing, the total porosity is simply φf = 1 − π/6 and the tortuosity is simply unity, as the shortest flow path is one directly though the center of the cell. Furthermore, as in other simple packings, all porosity is connected. Unlike micro-structural attributes, permeability of SC packings cannot be directly obtained using analytical techniques. Instead, numerical procedures are often employed. The closest analytical solutions are furnished by bounds, such as the lower bound obtained by Dormieux, Kondo and Ulm [2006] where pore spaces are ordered in the sense of inclusions and the permeability of a cylinder with cross-section made up by four circles
90
examined. Since the cylindrical pore space is a subset of that of the SC cell, the permeability of the cylindrical pore space serves as a lower bound for that of the SC cell. The lower bound can be expressed in dimensionless form as k ≥ 4.84 × 10−3 R2 , where R is the radius of the spheres in the SC cell. Naturally, the permeability tensor in the SC cell is isotropic. Additionally, Zick and Homsy [Zick and Homsy, 1982] have analyzed the permeability of the SC bead pack by reducing the Navier-Stokes equation to a set of Fredholm integral equations. They found k = 5.04 × 10−3 R2 . Using the aforementioned studies as backdrop for the accuracy of our proposed method, we perform permeability calculations using the SC bead pack. Our first task is to correctly identify the connected porosity in the sample. The pore geometry is discretized in the usual way using a lattice mesh. The resolution of the lattices, clearly affect the results of the computations. The center of the pore space is selected as the first active lattice and porosity is determined using the region-growing algorithm described in Section 4.4.3. Figure 4.9 shows the estimate of porosity as a function of the lattice resolution. Once the voxel length is smaller that R/50, the numerical solution closely captures the exact solution (1 − π/6). To check if the computational framework can determine the tortuosity, we apply the variational scheme and the Dijkstra’s algorithm as described in Section 4.4.2. Using a resolution of R/50, the resultant level set function and the shortest flow path are illustrated in Figure 4.10. As shown in the figure, in this simple example the tortuosity is unity and Dijkstra’s algorithm is able to obtain this result without any issues. Finally, turning our attention to the effective permeability calculation, we obtain an estimate using lattice Boltzman at the aforementioned lattice resolution. In addition, we carried out a three
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LOG(
)
Figure 4.9. Connected porosity as a function of voxel length in a SC bead packing.
dimensional Navier-Stokes finite element simulation to examine the reproducibility of the permeability calculation. The FE model is composed of 8937 tetrahedral Crouzeix-Raviat elements [Crouzeix and Raviart, 1973] with non-periodic side walls ans prescribed pressures on the top and bottom faces of the cubic domain. The FE model was solved using an open-source differential solver called FEniCS [Logg, 2007]. Figure 4.11 illustrates the results of the LB and FE simulations. The permeability using LB and FE is estimated to be 4.64 × 10−3 R2 and 4.89 × 10−3 R2 , respectively. Since both methods are inherently
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different, and since the calculations are close to the previous values of permeability estimated for SC packings, we consider the 5.1% difference in solutions acceptable. We therefore conclude that the proposed framework to estimate permeability based on level sets and lattice Boltzmann is accurate.
Figure 4.10. Level set function φ(x, y, z) (represented by the 3D color contour) and the corresponding shortest flow path (represented by the red straight line) as determined by Dijkstra’s algorithm.
4.6.2. Permeability calculation on natural complex porosity (e.g., compaction bands) In this section, we demonstrate the applicability of the proposed methodology to a complex porous network furnished by a natural sample of Aztec Sandstone from the Valley of Fire, Nevada. A full tomographic image was obtained from a prismatic sample of 2.25 × 2.25 × 6.00 mm at a resolution of 6 micron. The sample was cored in the field from a naturally formed compaction band and the mean grain size is 0.25 mm. The compaction band
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(a)
(b)
Figure 4.11. (a) Streamline of the simple cubic lattice computed via Stokes finite element model (k = 4.89 × 10−3 R2 ). (b) Velocity profile of the simple cubic lattice obtained via lattice Boltzmann simulation conducted on connected pore space (k = 4.64 × 10−3 R2 ).
sample is studied here because it furnishes a very complex network of connected porosity which has been significantly reduced from the natural rock formation. The total porosity in the compaction band sample is 14% while the total porosity in the host rock is 21%. More information about the sample and the geologic conditions can be found in Lenoir et al. [2010]. Figure 4.12 shows a small cell of dimensions 0.54 × 0.54 × 0.54 mm along with the level set functions, connected flow channels and connected porosity. We can see that the methodology presented above is able to extract a very complex network of porosity and can identify the connected porosity. Also, Dijkstra’s algorithm has been run to obtain
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the shortest flow paths shown in the figure. From these paths, the geometric tortuosity can be extracted, yielding an average tortuosity inside the compaction band of 2.79. This should be contrasted with the tortuosity in the host rock which is around 1.77. So, we can conclude that compaction bands increase the geometric tortuosity considerably.
0.5
z, mm
0.4 0.3 0.2 0.1
0.5 0.4 0.3 0.2 0.1 y, mm
(a)
0.1
(b)
0.2
0.3
0.4
0.5
x, mm
(c)
Figure 4.12. (a) Level set functions. (b) Flow paths. (c) Connected pore space for compaction band cell with dimensions 0.54 × 0.54 × 0.54 mm. Once the shortest flow paths are determined, we use the region-growing algorithm to obtain the connected porosity and differentiate this from the total porosity. As mentioned before, the region-growing algorithm uses voxels located on the shortest flow paths as seeds. Then, the connected pore space region is grown inside the pore space until all connections of the pore network are explored. Figure 4.12 shows the connected pore space for the 0.54 × 0.54 × 0.54 mm unit cell described above. Using this connected pore space, we calculated an average connected porosity in the compaction band of about 7%. This is basically half of the total porosity and should be contrasted with the 19% connected porosity in the host rock. One can conclude that the compaction band reduces the available connected porosity considerably relative to the otherwise intact rock.
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The next step in our analysis is to determine the minimum appropriate size of unit cells to upscale permeability. Our goal is to estimate permeability along the length of the entire 2.5 × 2.5 × 6.00 mm sample. To do this, as described before, we will use domain decomposition and then use LB/FEM to upscale permeability. In order to establish the minimum cell size, we make use of Dmicro as shown in equation (4.25). Figure 4.13 shows the energy dissipation rate as a function of cell size for a typical sample inside the specimen. It can be observed that most fluctuations are eliminated by the time the cell size is greater than 0.4 mm. This result is representative of all other regions in the specimen. This means that any cell size larger than 0.4 mm in dimensions would be appropriate for continuum representation. We select a convenient cell size of 0.75 × 0.75 × 0.75 mm and therefore decompose the entire sample domain using 3 × 3 × 8 cells. Decomposing the domain into 72 cells, we can perform multiscale analysis by estimating permeability in each cell using lattice Boltzmann and then passing the effective permeability of the subregions into a finite element model with 72 brick elements. Figure 4.14 shows the entire sample with the average porosity in each cell. It can be seen that the porosity field fluctuates around a mean of 14%. Furthermore, accuracy of the upscaling is monitored by using Dmacro as described previously. So, the current domain decomposition satisfies the continuum approximation and the accuracy conditions upon scaling. A macroscopic Darcy’s flow simulation is performed to obtain the effective permeability along the longitudinal direction of the sample shown in Figure 4.14. The lateral sides of the specimen are not allowed to pass fluid and the top and bottom faces are prescribed a fluid pressure of 100 and 0 kPa, respectively. From this macroscopic permeability
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1.00E+00 1.00EͲ01 1.00EͲ02
NORMALIZEDDISSIPATION
1.00EͲ03 1.00EͲ04 1.00EͲ05 1.00EͲ06 1.00EͲ07 1.00EͲ08 INSIDECB
1.00EͲ09 1.00EͲ10 0
0.1
0.2
0.3 0.4 0.5 EDGELENGTH,mm
0.6
0.7
0.8
Figure 4.13. Energy dissipation rate over the edge length of the cubic samples taken inside compaction band. The energy dissipation rate is normalized with respect to the energy dissipation rate of the largest sample with edge length = 0.75mm.
calculation, we extract the effective permeability of the sample along the longitudinal direction, 2.5 × 10−13 m2 . This is to be contrasted to the effective permeability outside the compaction band, within the host rock, which is 1.4 × 10−12 m2 , an order of magnitude difference. These results show a smaller drop in permeability than other studies which report drops between two and three orders of magnitude [Baxevanis et al., 2006;
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Keehm, Sternlof and Mukerji, 2006; Sternlof et al., 2004]. This discrepancy may be due to the fact that the permeability in literature such as [Keehm, Sternlof and Mukerji, 2006; Sternlof et al., 2004] are obtained via stochastically reconstructed pore space from 2D SEM images whereas the calculations presented in this example are conducted on 3D tomographic images directly obtained from X-ray CT. To assess the accuracy of the homogenization procedure, we compute the volume-averaged energy dissipation rate and the energy dissipation rate obtained from boundary conditions. The error of the mesoto-macro homogenization procedure for the compaction band specimen is 1.9%, as shown in Table 4.2. These results show that the LBM/FEM multi-scale framework is able to achieve reasonable accuracy, providing that the unit cells are large enough to behave as represenative elementary volumes and a sufficient degree of heterogeneity of the porous medium is preserved.
Volume Averaged Dissipation Rate, J/sec per 1m3 71.1 3 Dissipation Rate obtained from B.C. , J/sec per 1m 69.8 Difference, J/sec 1.3 Homogenization Error 1.9 % Table 4.2. Dissipation rates computed via spatial averaging and prescribed values at boundaries.
It should be noted that all examples shown in this work were conducted in a singleprocessor machine. The permeability calculations for the compaction band samples are not even feasible without resorting to the multiscale LB/FEM technique proposed in this work.
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(a)
(b)
(c)
Figure 4.14. Results of multiscale effective permeability analysis inside compaction band. (a) Porosity map, (b) vertical velocity field, and (c) pressure map. 4.7. Conclusion How does microstructural pore geometry affects the macroscopic fluid transport properties of porous materials? To begin answering this question we have presented a computational framework that quantifies the interplay between microscale geometrical attributes, directly extracted from tomographic images, and the macroscopic pore fluid properties encapsulated in the effective permeability of the material. Specifically, we have incorporated and expanded a variety of numerical techniques, including semi-implicit level sets, graph theory, and lattice Boltzmann coupled with finite element computations exploiting hierarchical multiscale techniques. The numerical techniques proposed are used sequentially with the output form one serving as the input for the next. We have also proposed quantitative criteria to assess the validity of the unit cell size and the accuracy in the
99
multiscale calculations. As a result, and as demonstrated by the numerical examples, the computational framework is able to offer tremendous computational advantages over standard approaches without compromising accuracy.
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CHAPTER 5
Connecting microstructural attributes and permeability from 3-D tomographic images of in situ compaction bands using multi-scale computation This chapter is published as: W.C. Sun, J.E. Andrade, J.W. Rudnicki, P. Eichhubl, Connecting microstructural attributes and permeability from 3-D tomographic images of in situ compaction bands using multi-scale computation, Geophysical Research Letters, doi:10.1029/2011GL047683, in press.
5.1. Abstract Tomographic images taken inside and outside a compaction band in a field specimen of Aztec sandstone are analyzed by using numerical methods such as graph theory, level sets, and hybrid lattice Boltzmann/finite element techniques. The results reveal approximately an order of magnitude permeability reduction within the compaction band. This is less than the several orders of magnitude reduction measured from hydraulic experiments on compaction bands formed in laboratory experiments and about one order of magnitude less than inferences from two-dimensional images of Aztec sandstone. From geometrical analysis, we conclude that the elimination of connected pore space and increased tortuosity due to the porosity decrease are the major factors contributing to the permeability
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reduction. In addition, the multiscale flow simulations also indicate that permeability is fairly isotropic inside and outside the compaction band.
5.2. Introduction Compaction bands are thin tabular zones of localized compactive inelastic deformation and significant porosity reduction. They have been observed in a few field locations, including the upper domain of the Aztec Sandstone at Valley of Fire, Nevada [Eichhubl, Hooker and Laubach, 2010; Hill , 1989; Mollema and Antonellini , 1996; Sternlof et al., 2004], and Navajo Sandstone at the Kaibab monocline, Utah [Solum et al., 2010]. Previous research suggested that compaction bands are much less permeable than the host rock and hence could act as barriers to fluid flow. This feature is important to applications involving injection or withdrawal of pore-fluids, such as CO2 sequestration, energy storage and retrieval, and aquifer management. In this chapter, we use numerical techniques to interpret three-dimensional tomographic images of Aztec sandstone. The cores used here are taken from a band described as a shear-enhanced compaction band (SCB) by Eichhubl, Hooker and Laubach [2010]. Based on field structural and microtextural observations, they inferred that this band accommodated about equal amounts of shear displacement and band-perpendicular shortening, and distinguished this band from pure compaction bands that lack any component of shear displacement. Samples were scanned at the synchrotron APS facility in Argonne National Labs as described by Lenoir et al. [2010]. The presented techniques afford us unprecedented access to determine grain size distributions, occluded and connected porosities and tortuosity of
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samples from the compaction band and the outside matrix. These features are then linked to macroscopic effective permeability tensors using a multiscale lattice Boltzmann/finite element scheme. Keehm, Sternlof and Mukerji [2006] calculated permeabilities from 3D pore geometry statistically reconstructed from 2D images of deformation bands found in the field. A drawback of this approach is that the permeability calculation strongly depends on the quality of pore geometry reconstruction [Adler , 1992]. Hence, the accuracy of the permeability calculation can be compromised if the reconstruction from 2D images does not provide realistic three dimensional structures. Alternatively, Fredrich, DiGiovanni and Noble [2006] used massively-parallel lattice Boltzmann simulations to extract effective permeability directly from 3D tomographic images of Castlegate sandstone. Permeabilities calculated from the lattice Boltzmann simulations were found by Fredrich, DiGiovanni and Noble [2006] to be consistent with laboratory measurements on specimens without deformation bands. The numerical techniques used here are described in more detail in Sun and Andrade [2010a] and Sun, Andrade and Rudnicki [2011b]. They are improved and computationally more efficient versions of those previously used in the literature. An important step to calculate the micro-structural attributes is to obtain the 3D medial axes. Lindquist et al. [1996] described constructions of medial axes and used them to determine tortuosities of several rock types, including a sandstone. Here, we use graph theory and level set-based techniques to calculate 3D medial axes more efficiently.
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1 INSIDE CB OUTSIDE CB
0.8 0.6 0.4 0.2 0
Cumulative probability
Cumulative Probability
1
0.8 0.6 0.4 0.2 0
0.15 0.2 0.25 0.3 Grain Diameter, mm
1.5
2
2.5
3
3.5
Geometrical Tortuosity
(a)
(b)
b
a q
q
f
p
p
f
500 Pm (c)
(d)
Figure 5.1. Cumulative distribution functions of grain diameters (a) and tortuosities (b) and Scanning-electron microscope cathodoluminescent images outside (c) and inside (d) the shear enhanced compaction band. Thick lines in (b) denote Gamma distribution fits; thin lines represent the tortuosity distributions determined from Dijkstra’s algorithm. In (c) and (d), q:quartz, p:pore. Arrow indicates quartz cement in broken grains.
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5.3. Three dimensional tomographic images and numerical methods The Aztec Sandstone is a sedimentary rock composed mainly of weakly cemented, well-sorted, well-rounded quartz grains [Eichhubl, Hooker and Laubach, 2010]. Here, we use tomographic images taken inside and outside a SCB in Aztec sandstone to extract geometrical attributes and effective permeabilities. The converted binary tomographic images used in this study are described by Lenoir et al. [2010]. An efficient method to characterize the pore space is to replace it by a structure of medial axes. This method was first used by Lindquist et al. [1996] to calculate tortuosity. In their application, medial axes are obtained from the pore space eroded by the discrete BURN algorithm, and Dijkstra’s algorithm [Dijkstra, 1959] is applied to compute tortuosity from the medial axes. The discrete BURN algorithm, however, may lead to spurious flow paths that are not medial axes and thus cannot represent the geometry of the pore space efficiently. Moreover, the effective radii of the grains determined by BURN algorithm are not accurate due to the use of integers to store radius data. To overcome the drawbacks of the BURN algorithm, we use a signed distance function to obtain 3D medial axes [Sun and Andrade, 2010a]. The signed distance function φ( x) is a continuous metric function measuring the distance between the position x and its closest point on the boundary between pore and grain y ∈ Γ, so that, (5.1)
φ( x) = S( x) inf || x − y || y ∈Γ
where inf denotes the infimum and S( x) is a step function that is equal to -1 if the position x is occupied by the solid grains and 1 if x is inside pore space. Solving a differential equation by the evolution of a level set function initially taken from the binary image yields
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the signed distance function. In particular, we use the differential equation formulated by Li et al. [2005]. We solve this equation to obtain the signed distance function by introducing an unconditionally stable, semi-implicit 3D level set scheme. More details are given by Li et al. [2005] and Sun and Andrade [2010a]. According to (1), φ( x) reaches its local maximum if x is located on the medial axis of the pore space and φ( x) reaches its local minimum if x is located at the centroid of the solid grains. In the latter case, the value of φ( x) is approximately equal to the effective radii of the solid grains, provided that the shapes of the solid grains are well rounded. Since the signed distance function is continuous, the radii determined from it do not need to be integers as in the BURN algorithm. Furthermore, using the signed distance function does not generate spurious flow paths. Since the shortest-path flow channels must be located inside the connected pore space, we identify all voxels inter-connected to the voxels along the shortest flow path as parts of connected pore space and thereby obtain the occluded and connected porosity. The macroscopic effective permeability is captured by a multiscale hybrid Lattice Boltzmann (LB)/finite element scheme. First introduced to tomographic applications by White, Borja and Fredrich [2006], this hybrid scheme partitions the specimen into small representative elementary volumes (REV) for LB calculation. Finite element simulations of Darcy’s flow are then used to upscale permeabilities of the REV to specimen-scale. This upscaling procedure is the same as used by White, Borja and Fredrich [2006], but here we conducted LB simulations only on the connected pore space of each REV. This eliminates any chance of mistaking occluded pore space as part of the flow network due
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to partition of specimens and thus significantly improves the speed and accuracy of the procedure [Sun and Andrade, 2010a].
5.4. Microstructural attributes Using the computational framework described above, we calculate grain size distributions, tortuosities, and occluded/connected porosities using the 3D signed distance function φ(x, y, z) of the tomographic images. Figure 5.1(a) depicts the estimated grain size distributions of the samples taken inside and outside the SCB. The mean grain diameter is 0.26 mm inside and outside the band, with a variance of 0.013 and 0.021, respectively. Although there are slight differences in the content of fines, both the SCB and outside matrix display well sorted grains of similar sizes. This similarity in grain sizes may be explained by the partially cemented microfractures produced during band formation, which may offset the higher degree of grain breakages observed inside the bands, as shown in Figure 5.1(c) and 5.1(d). Furthermore, the average grain volume (1.6 × 107 μm3 ) is five orders of magnitude larger than that of the tomographic image voxel (216μm3 ) and the mean grain diameter is 42 times larger than the voxel length. These measurements suggest that the tomographic images have enough resolution to render the grain geometry accurately. Figures 5.2(a) and 5.2(b) show the shortest flow paths for 0.6 × 0.6 × 0.6 mm3 Aztec sandstone specimens from inside and outside the SCB. As illustrated, the pore space is much less interconnected and has fewer flow channels inside the band than in the host rock. The flow channels inside and outside the band are both highly three dimensional,
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suggesting that reconstructions from two dimensional slices may not give an accurate picture. Next, we calculate tortuosities of larger specimens, 2.25 × 2.25 × 6 mm3 , that contain sufficient flow channels to be statistically representative. Geometrical tortuosity τ is defined as the ratio between the length of the actual flow path l and the linear distance d, i.e., τ = l/d [Adler , 1992; Lindquist et al., 1996]. Figure 5.1(b) illustrates the distribution of tortuosities inside and outside the SCB. The mean value is 2.5 inside the band and 2.0 outside. More specifically, more than 30% of the flow channels in the band have tortuosities greater than 2.8, while less than 1% of flow channels in the outside matrix have tortuosities greater than 2.8. These results quantatitively demonstrate that the grain re-arrangement and pore collapse inside the band lead to significantly more tortuous flow channels. The distribution of tortuosity, both inside and outside the band, is fit well by the Gamma distribution [Figure 5.1(b)], as noted by Lindquist et al. [1996] for Berea sandstone. The variance of the tortuosity is 0.18 inside and 0.06 outside the SCB indicating a much wider distribution of the lengths of flow channels inside the band. From the tomographic images, we calculate an average total porosity of 20% inside the band and 14% outside. These results are consistent with other measurements in Aztec sandstone by Eichhubl, Hooker and Laubach [2010] and Sternlof et al. [2004]. However, we are able to calculate not only the total porosity but also the occluded and connected porosities. The occluded (connected) porosity is the total volume of the occluded (connected) pore space divided by the total volume of both pore space and solid grains. The sum of the occluded and connected porosities is equal to the total porosity.
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Figure 5.3 depicts the occluded and connected porosities of 6 cubic samples, 3 from the shear enhanced compaction band (labeled as SCB) and 3 from the outside matrix (labeled as OUTSIDE). The side lengths are all 0.75 mm. As illustrated in the figure, three equally spaced samples traverse the width of the SCB, about 1 cm. They are taken from images between 3.66 mm and 6.66 mm beneath the top band boundary. The other three samples are from outside the band at successively larger distances from the band boundary. The occluded porosity is about the same, an average of 5.8% for specimens inside the band and 6.4% for those outside. The 7.7% connected porosity inside the band is only roughly half of the 15% connected porosity outside. Thus, the reduction occurs primarily in the connected porosity which contributes to the formation of a flow barrier.
5.5. Macroscopic effective permeabilities The directional permeabilities of two hundred 0.75 × 0.75 × 0.75 mm3 samples (half inside, half outside the band) are calculated via lattice Boltzmann simulations. The volume of the samples is selected based on an energy dissipation criterion proposed in Sun and Andrade [2010a] to minimize size effects and obtain representative permeability calculations. We estimate an average permeability of 2.1 × 10−13 m2 inside the band and 1.3 × 10−12 m2 outside, and a variance of 0.26 and 0.18, respectively. The increase on the ratio between standard deviation and mean value signifies that the pore geometry of the band is less homogeneous than the outside matrix. The macroscopic effective permeabilities of two 2.25 mm×2.25 mm×6 mm samples are determined via a lattice Boltzmann/finite element scheme [Sun and Andrade, 2010a; White, Borja and Fredrich, 2006]. Figures 5.2(c) and 5.2(d) shows the magnitude of the velocity field generated
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0.4 z, mm
z, mm
0.4 0.2 0 0.5
0.5
y, mm 0 0
x, mm
0.2 0 0.5
0.5
y, mm 0 0
(a)
(b)
(c)
(d)
x, mm
Figure 5.2. The shortest flow paths inside (a) and outside (b) and velocity vector field (in blue color) inside (c) and outside (d) the shear enhanced compaction band. Solid grains are colored in gray. The intensity of the blue color in (c) and (d) represents the magnitude of the fluid velocity field.
from the lattice Boltzmann method by imposing a pore pressure gradient on two opposite faces of the samples. The solid grains are plotted in gray color whereas the blue shaded region represents occurrence of fluid flow. Higher intensity of the blue color indicates higher fluid flux. Figures 5.2 (c) and 5.2(d) confirms that fluid flow inside a SCB is confined to only a small portion of the entire pore space, whereas the fluid flow in the
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outside matrix distributes more evenly over the specimen. This finding is consistent with the higher connected/occluded porosity ratio inside SCB. The effective permeability is obtained according to, (5.2)
μv 1 kij = − p,j VΩ
Ω
vi ( x)dΩ
where μv is the dynamic viscosity of the fluid occupying the spatial domain of the porous medium Ω. The effective permeabilities reported above based on lattice Boltzmann/finite element simulations, pertain to the direction normal to the SCB. The 0.77 order of magnitude difference is close to the 1.1 order permeability reduction predicted from the modified Kozeny-Carman equation [Carman, 1956] using connected porosities φf , which reads
(5.3)
τout kin = kout τin
φfin φfout
3
1 − φfout
2
1 − φfin
Nevertheless, the effective permeability reduction obtained from multiscale simulations and the modified Kozeny-Carman equation are both less than the several orders of magnitude inferred for compaction bands in laboratory sandstone specimens [Holcomb and Olsson, 2003; Vajdova, Baud and Wong, 2004] and the 2.5 order of magnitude permeability reductions inferred by Keehm, Sternlof and Mukerji [2006]. Presumably, the larger permeability reduction for the laboratory specimens is due to more intense comminution in lab specimens compared to compaction bands collected in the field (see Figures 5.1(c) and 5.1(d)). The estimates of Keehm, Sternlof and Mukerji [2006], and similar estimates by Solum et al. [2010] are based on pore volume reconstructions from two-dimensional
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images. As mentioned earlier, this seems likely to overestimate the reduction in permeability. Whether the difference in methodology is sufficient to account for the order of magnitude difference is unclear but illustrates the need for further studies of this type. The procedure use here also makes it possible to estimate the permeability parallel to the band. We find that the permeability in parallel direction is roughly the same as perpendicular to the band, with a 0.74 reduction of effective permeability along the axis parallel to the band. This nearly isotropic transport property is consistent with the orientation of maximum compressive principal stress (45-50 degree to the shear-ehanced bands) inferred by Eichhubl, Hooker and Laubach [2010] based on field structural and microtextural observations. Furthermore, the results suggest that reduction of connected pore space and increases of tortuosity also decrease permeability parallel to the band. These mechanisms seem to be less likely, than, for example, grain crushing and comminution to have a strong directional dependence. This is an important issue since deformation bands that have experienced relatively large shear (compared with compaction) may develop through-going slip surfaces. This could increase the permeability parallel to the slip surface and create a preferential flow paths.
5.6. Conclusion Using a computationally efficient method, we compare macroscopic permeabilities and microstructural attributes of a shear-enhanced compaction band and host rock. Our results reveal that increased tortuosity and elimination of connected pore space, not simply
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a reduction in total porosity, are the major factors causing reduced permeability. Furthermore, the results suggest that permeability is reduced not only for flow perpendicular to the band but also for flow along the band.
2.25mm
CB1
9.0mm mm
.75mm .75mm
CB2
CB3
OUTSIDE1
OUTSIDE2
6.3mm 6 3m
.75mm .75mm
OUTSIDE3 0
0.05
OCCLUDED POROSITY
0.1
0.15
0.2
CONNECTED POROSITY
Figure 5.3. Occluded and connected porosity of 6 samples taken inside/outside the shear enhanced compaction band in the Aztec Sandstone specimen. Sample labeled as SCB (OUTSIDE) are taken inside (outside) the shear enhanced compaction band.
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CHAPTER 6
Conclusion and Future Prospective We have proposed a simple method to analyze shear failures of porous media at drained and undrained limits, and a multi-scale scheme to analyze connections between microstructural attributes and the macroscopic transport of deformation bands formed in field. In order to detect onset of instabilities under drained and undrained conditions, we incorporate the instability state line concepts [Chu, Leroueil and Leong, 2003] to derive hardening rules for the Manzari-Dafalias critical state plasticity model [Dafalias and Manzari, 2004]. This improvement prevents the simulated constitutive response failed at a fixed void ratio under the drained condition. Using this critical state plasticity model as our starting point, we predict onsets of diffuse instabilities by analyzing the perturbation sensitivity of the drained and undrained continuum elasto-plastic tensor, and onsets of localized instabilities by monitoring the drained and undrained acoustic tensors. Experimental data are used to calibrate the critical state plasticity model [Chu, Leroueil and Leong, 2003; Verdugo and Ishihara, 1996]. The simulated and recorded constitutive responses are compared and the onset of instability predicted numerically are compared with observations made in literature. Our results show that the simple framework we proposed is capable of both replicate the drained and undrained constitutive response and deliver consistent predictions on both the onsets and modes of instabilities.
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To examine how microstructural deformation inside deformation bands affecting macroscopic transport at the post-bifurcation regime, we introduce a new multi-scale flow simulation scheme. This multi-scale scheme is composed of two component – (1) a geometrical analytical scheme incorporating concepts from level set and graph theory to determine tortuosity and connectivity of pore space and (2) a multi-scale lattice Boltzmann/finite element flow simulation procedure which take advantage of the connectivity information from (1) to improve accuracy and speed of the macroscopic permeability calculation. We apply the multi-scale flow simulation scheme to analyze X-ray tomographic images from a Aztec sandstone specimen taken from Fire Valley State Park, Nevada [Eichhubl, Hooker and Laubach, 2010; Sternlof et al., 2004]. Inside this Aztec sandstone specimen, a shear-enhanced band with a 5 − 8% total porosity reduction has been found. To analyze the microstructural and transport property of this shear-enhanced compaction band, we first converted the tomographic images provided by Dr. Lenoir [Lenoir et al., 2010] into binary data via a open source software called ImageJ [Abramoff, Magelh˜ aes and Ram, 2004]. This binary data discretize the 3D pore space into a domain composed of bi-phase 3D voxels suitable for lattice Boltzmann calculations. Following this step, we computed the tortuosity, connected and occluded porosity, and permeability at specimen scale. Our numerical analysis reveals a new discovery that increased tortuosity and elimination of connected pore space, not simply a reduction in total porosity, are the major factors causing the reduced permeability. Furthermore, the results also suggest that permerability is reduced for flow perpendicular to and along the shear-enhanced compaction band.
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6.1. Future work The multi-scale transport framework will be incorporated with a discrete element model that simulates grain fragmentation. The objective of the extension is two-fold – (1) to simulate micromechanics of compaction localization observed in laboratory with a costefficient grain fragmentation model; (2) to explore the relation between the evolution of solid skeleton deformation and fluid transport in porous rock. To simulate the severe grain crush commonly observed in laboratory, we propose a new method that (I) incorporate tetrahedral geometric algorithm developed by Jerier et al. [2009] to control the grain size distribution of the fractured grains and (II) graph representation theory to update the grain connectivity changes induced by fracture [Mota, Knap and Ortiz , 2008]. To further our understanding on the interplay between solid skeleton deformation and fluid transport behaviors, we will simulate tri- and bi- axial compression tests via DEM and record all the geometrical data of the solid skeleton (i.e., grain position, radius..etc) as the loading progresses. These geometrical data are used to reconstruct 3D pore geometry at various time step. By conducting geometrical analysis and multi-scale flow simulations on the pore space before, during and after the formation of deformation bands, we examine how grains rotation, re-arrangement and crushing inside the localization zone affect the tortuosity, connected and occluded porosities and ultimately lead to the macroscopic permeability reduction. Another proposed extension of the current research includes a hierarchical framework that incorporates the pore-scale lattice Boltzmann method, the grain-scale discrete element method and a macroscopic coupled diffusion-deformation mixed field finite element. The goal is to simulate the multi-physics, multi-scale nature of the coupled pore-fluid
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diffusion and solid skeleton deformation at the post-bifurcation regime. Figure 6.1 showcases the design the multi-scale scheme. In the multi-scale scheme, a three dimensional mixed finite element model is used as a vehicle for the material subroutine that can switch from phenomenological-based to micro-mechanical-based, whenever deemed necessary. In the stable region, the material subroutine will use phenomenological models to determine constitutive responses of porous media. On the other hand, micro-mechanics models will be employed in locations where material instability is detected. In the unstable region, the stress-strain relation of the solid skeleton is acquired from the discrete element simulations and the permeability is estimated via the lattice Boltzmann scheme. Both DEM and LBM are conducted in unit cells assoicate4d with a fixed number of Gauss points inside each finite element. The communication mechanism is as follows. First, the macroscopic strain history of each Gauss point in the finite element is projected as boundary conductions for the corresponding unit cell. Then, the DEM simulates the granular motion of the unit cell, return the essential material parameters to assembly the elasto-plastic tensor to the finite4 element and bypasses the fabric change to the LBM subroutine. The LBM subroutine computes the permeability based on the geometric information obtained from the DEM. Finally, the tangential elasto-plastic and permeability tensors obtained from the micro-mechanics models are used to assemble the finite element. In the finite element level, the multi-scale framework propagates the solution in the same way a convention finite element scheme does.
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Proposed Multi-scale Scheme for Coupled diffusionDeformation Simulations
TIME STEP LOOP GLOBAL ITERNATION LOOP
MICRO-MECHANICS UPDATE
MIXED FINITE ELEMENT LOOP
PHENOMENOLOGICAL UPDATE
UPDATE Cep WITH DEM
GAUSS INTEGRATION
UPDATE Cep WITH CRITICAL STATE PLASTICITY
Yes es UPDATE K WITH LBM
No N BIFURCATION?
UPDATE K WITH COZENY-CARMAN EQUATION
UPDATE Cep and K CONVERAGE?
END INTERATION
Figure 6.1. The proposed multi-scale hierarchical mixed finite element algorithm.
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Vita WaiChing Sun Department of Civil and Environmental Engineering Northwestern University Evanston, IL 60208
Education PhD. Theoretical and Applied Mechanics, Northwestern University, 2008-2011 M.A. Civil Engineering, Princeton University, 2007-2008 M.S. Civil Engineering (Geomechanics), Stanford University, 2005-2007 B.S. Civil Engineering,University of California, Davis, 2002-2005
Selected Journal and Conference Publications 1. W.C. Sun, J.E. Andrade, J.W. Rudnicki, P. Eichhubl, Connecting microstructural attributes and permeability from 3-D tomographic images of in situ compaction bands using multi-scale computation, Geophysical Research Letter, doi:10.1029/2011GL047683, in press. 2. W.C Sun, J.E. Andrade, J.W. Rudnicki, A multiscale method for characterization of porous microstructures and their impact on macroscopic effective permeability, International Journal of Numerical Methods in Engineering, doi:10.1002/nme.3220, in press. 3. W.C. Sun and J.E. Andrade, Capturing the effective permeability of field compaction band using hybrid lattice Boltzmann/Finite element simulations, Proceedings of 9th World Congress of Computational Mechanics/APCOM 2010, Sydney, Australia, 2010. 4. W.C. Sun and J.E. Andrade, Diffuse bifurcations of porous media under partially drained conditions, Proceedings of International Workshop on Multiscale and Multiphysics Processes in Geomechanics, Stanford, California, 2010. 5. R.I. Borja and W.C. Sun, Co-seismic sediment deformation during the 1989 Loma Prieta Earthquake, Journal of Geophysical Research, doi:10.1029/2007JB005265, 2008. 6. R.I. Borja and W.C. Sun, Estimating inelastic sediment deformation from local site response simulations, Acta Geotechnica, 2(3) :183-195, 2007.
