Received: 24 March 2018 Revised: 17 June 2018 Accepted: 19 June 2018 DOI: 10.1002/nme.5915

RESEARCH ARTICLE

Large deformation poromechanics with local mass conservation: An enriched Galerkin finite element framework

Jinhyun Choo

Department of Civil Engineering, The University of Hong Kong, Pokfulam, Summary Hong Kong Numerical modeling of large deformations in fluid-infiltrated porous media

Correspondence must accurately describe not only geometrically nonlinear kinematics but also Jinhyun Choo, Department of Civil fluid flow in heterogeneously deforming pore structure. Accurate simulation of Engineering, The University of Hong fluid flow in heterogeneous porous media often requires a numerical method Kong, Pokfulam, Hong Kong. Email: [email protected] that features the local (elementwise) conservation property. Here, we introduce a new finite element framework for locally mass conservative solution of cou- Funding information Seed Fund for Basic Research of The pled poromechanical problems at large strains. At the core of our approach is the University of Hong Kong, Grant/Award enriched Galerkin discretization of the fluid mass balance equation, whereby Number: 201801159010 elementwise constant functions are augmented to the standard continuous Galerkin discretization. The resulting numerical method provides local mass conservation by construction with a usually affordable cost added to the con- tinuous Galerkin counterpart. Two equivalent formulations are developed using total Lagrangian and updated Lagrangian approaches. The local mass conserva- tion property of the proposed method is verified through numerical examples involving saturated and unsaturated flow in porous media at finite strains. The numerical examples also demonstrate that local mass conservation can be a crit- ical element of accurate simulation of both fluid flow and large deformation in porous media.

KEYWORDS finite element method, enriched Galerkin method, poromechanics, large deformation, local mass conservation

1 INTRODUCTION

In deformable porous materials such as soils, rocks, and tissues, the flow of the pore fluid(s) and the deformation of the solid matrix are tightly coupled to each other. Prediction of such a poromechanical process is an important task in many branches of science and engineering. Examples include civil engineering,1-5 energy and environmental technologies,6-10 and biophysics.11-14 This work is motivated by the increasing need for rigorous modeling of strongly coupled poromechanical problems in which the material undergoes moderate to large deformations. Deformations of a porous material entail significant geometric nonlinearity when the applied load or the pore pressure change is high relative to the stiffness of the solid matrix. Such geometrically nonlinear deformations are common in

66 © 2018 John Wiley & Sons, Ltd. wileyonlinelibrary.com/journal/nme Int J Numer Methods Eng. 2018;116:66–90. CHOO 67 many problems of practical interests such as ground settlement, expansion, and collapse to name a few in geotechnical engineering. Accurate modeling of these deformations must rely on large (finite) strain theory for kinematic descriptions. From decades ago to recent years, a number of studies have revealed the importance of large deformation kinematics in a variety of poromechanical problems from consolidation to strain localization to swelling.15-23 Thesestudieshaveshown not only that geometric nonlinearity alone plays a critical role in poromechanical coupling but also that indirect coupling effects (eg, deformation-dependent permeability) can have remarkable effects on the overall flow-deformation responses. Therefore, when the material of interest deforms beyond the infinitesimal deformation range, it is crucial to properly incorporate the direct and indirect consequences of large deformation kinematics. Numerical methods are necessary to solve large deformation poromechanical problems since these problems are inher- ently nonlinear. The continuous Galerkin (CG) finite element method, along with a total Lagrangian or an updated Lagrangian approach, is the most widely used method for the simulation of large deformations in solids. As such, the majority of the existing work on large deformation poromechanics has developed and used CG finite element methods for discretization of the fluid flow (mass balance) equation as well as the solid deformation (linear momentum balance) equation.24-31 The resulting mixed CG finite elements can provide faithful numerical solutions for many problems. How- ever, there also exist a number of poromechanical problems in which the CG method may be a suboptimal choice. Among them, common and practically relevant problems are those that involve a highly heterogeneous permeability field and/or coupling with transport phenomena. In these problems, a CG solution may exhibit nontrivial imbalance of mass at the local (element) level since the CG method enforces the numerical solution to be continuous across elements. For the same reason, a CG solution sometimes manifests nonphysical oscillations as an artifact of continuous approximation of sharp changes in the pressure field. Overcoming these limitations of the CG method requires one to employ an alternative discretization method that pro- vides local mass conservation in the fluid flow problem. This has motivated a number of poromechanical formulations that have combined a locally conservative method for the fluid flow equation with CG discretization of the solid deforma- tion equation. The types of locally conservative methods used for this purpose include the finite volume method,32-36 the Raviart-Thomas mixed finite element method,37-40 the discontinuous Galerkin (DG) method,41 and the enriched Galerkin (EG) method.42 These studies have shown that the use of a locally conservative method enables more robust solution of the fluid flow equation in a poromechanical problem. Nevertheless, almost all of these locally mass conservative formulations for poromechanics have restricted their atten- tion to infinitesimal deformation problems. Few exceptions include the recent work of Kim,43 who has used the finite volume method for total Lagrangian discretization of the fluid flow equation at large strains. The finite volume method has several desirable aspects, but its application to a domain undergoing large deformations demands significant efforts. In particular, as pointed out by Kim,43 one must rigorously treat permeability tensors whose principal directions evolve arbitrarily due to large deformation kinematics. Thus, a sophisticated technique for handling generally anisotropic per- meability such as a multipoint flux approximation method is required even when the material's permeability is isotropic. However, the use of such an advanced finite volume scheme for poromechanical problems can be onerous, especially when the computer implementation is based on a finite element program for large strain problems. In this work, we develop a locally mass conservative finite element framework for poromechanical problems undergo- ing large deformations. The framework builds on our previous work that applies the EG method to discretize the fluid flow equation in small strain poromechanics.42 The EG method, which has emerged as an efficient way to provide local conservation, augments elementwise constant functions to the finite element space of the CG method and uses the vari- ational form of the DG method.44-47 As a consequence, the EG method allows one to enjoy the major advantages of the DG method but with appreciably fewer degrees of freedom as compared with the DG method. It is noted that, for the same mesh size and polynomial order, the optimal convergence rates of EG, DG, and CG methods are identical.44,45 Our recent paper42 has shown that the mixed EG/CG discretization of the flow/deformation equations in poromechanics pro- vide local mass conservation with the same order of accuracy as the standard CG/CG discretization. Furthermore, the paper has also shown that local mass conservation can lead to marked differences in flow-induced deformations such as the timing of strain localization. Here, we generalize the EG/CG discretization to geometrically nonlinear finite elements. To the best of our knowledge, this paper presents the first finite element framework allowing locally mass conservative solution of large deformation poromechanical problems. The remainder of the paper is organized as follows. In Section 2, we describe a mathematical formulation for porome- chanics at large strains and state the problem of interest as a strong form. In Section 3, we propose a finite element framework that uses the EG method for discretization of the fluid flow equation in the large deformation porome- chanical problem. In doing so, we develop two equivalent formulations using total Lagrangian and updated Lagrangian 68 CHOO approaches. In Section 4, three numerical examples are presented to not only verify the correctness and the local mass con- servation property of the proposed finite element formulation but also highlight the importance of local mass conservation for the simulation of flow-driven deformations in porous media. In Section 5, we conclude the work.

2 LARGE DEFORMATION POROMECHANICAL FORMULATION

This section presents a mathematical formulation for fluid-infiltrated porous media undergoing large deformations. We begin by describing a three-phase continuum representation of an unsaturated porous material and its kinematics in the finite deformation range. We then formulate balance equations for the mass and linear momentum of the material in the reference configuration as well as the current configuration. A set of constitutive models is then introduced to complete the formulation. Lastly, we state the initial–bounadry value problem of interest as a strong form.

2.1 Continuum representation and kinematics Consider a porous solid whose pore space is filled with water and air. Using mixture theory, we represent the porous material as a three-phase continuum consisting of the solid matrix (s), water (w), and air (a).Thevolumefractionsofthe three phases are defined as

s w a s w a 𝜙 ∶= dVs∕dV,𝜙∶= dVw∕dV,𝜙∶= dVa∕dV,𝜙+ 𝜙 + 𝜙 = 1. (1)

Accordingly, the porosity is defined as 𝜙 ∶= 1 − 𝜙s = 𝜙w + 𝜙a. (2)

We also define the saturation ratio of the water phase as

S ∶= 𝜙w∕𝜙. (3)

𝜌 𝜌 𝜌 Let s, w,and a denote the intrinsic mass densities of the solid matrix, water, and air phases, respectively. Then, the partial mass densities of the three phases are defined as s s w w a a s w a 𝜌 ∶= 𝜙 𝜌s,𝜌∶= 𝜙 𝜌w,𝜌∶= 𝜙 𝜌a,𝜌= 𝜌 + 𝜌 + 𝜌 , (4) where 𝜌 is the mass density of the total mixture. It is noted that an index is used as a superscript when denoting a partial 𝜌s 𝜌 (volume-averaged) property of a constituent eg, ) and as a subscript when denoting an intrinsic property (eg, s). 𝜑 , The kinematics of the mixture is described by a Lagrangian approach tracing the motion of the solid matrix. Let s(Xs t) denote the motion of the solid matrix at time t, which maps a solid material point Xs in the reference configuration to a point xs in the current configuration. The solid displacement vector is then given by us ∶= xs − Xs. Given the special role played by the solid matrix, we shall drop the subscript (·)s for quantities pertaining to the motion of the solid matrix, 𝜑 𝜑 eg, ∶= s, X ∶= Xs, x ∶= xs, u ∶= us, and so on. The deformation gradient of the solid motion is defined as 𝜕𝜑(X, t) F ∶= . (5) 𝜕X

The Jacobian J is defined as the determinant of F. The Jacobian transforms the reference differential volume dV of the solid matrix into the current differential volume dv as

J ∶= det F = dv∕dV > 0. (6)

The reference differential area d A on the solid matrix is transformed into the current differential area da by Nanson's formula nda = JF−T · NdA = J N · F−1dA, (7) where N and n are unit normal vectors associated with d A and da, respectively. Using the deformation gradient, we also define the left Cauchy-Green deformation tensor b ∶= F · FT. (8) CHOO 69

In this paper, we use “GRAD” and “DIV” to denote the gradient and divergence operators with respect to a point X in the reference configuration, and “∇”and“∇·” to denote these operators with respect to a point x in the current configuration. We also use an overdot to denote the material time derivative following the solid motion. For example, the material time derivative of the Jacobian is given by J̇ = J ∇·u̇ . (9)

Moreover, we use the subscript (·)0 to denote quantities that pertain to the initial condition/reference configuration.

2.2 Balance laws In the formulations that follow, we make several assumptions plausible for most geologic materials in shallow subsurface systems. They are: (i) the constituent phases do not exchange their masses, (ii) the solid and water phases are incompress- 𝜌 𝜌 𝜌 ible, and (iii) the air pressure is atmospheric (passive air condition). Under these assumptions, s, w,and a are constant, and the balance of solid mass can be written as 𝜙s 𝜙s, 0 = J (10) 𝜙s 𝜙s where 0 is the value of in the initial state. The solid mass balance is conserved by the Lagrangian kinematic description. Moreover, the balance of air mass becomes trivial by the aforementioned assumptions. The balance of water mass in the current configuration may be expressed as 𝜙̇ w + 𝜙w ∇·u̇ +∇·q = 0, (11) where q is the seepage (Darcy) flux vector of the water phase. To reformulate the water mass balance in the reference configuration, we define the pull-back seepage flux vector Q using the Piola transformation of q

Q ∶= J F−1 · q. (12)

According to the Piola identity, DIV Q = J ∇·q. (13)

We then multiply (11) by J and plug (9), (12) and (13) into (11). This gives

𝜙̇ wJ + 𝜙wJ̇ + DIV Q = 0. (14)

The first two terms in (14) can be expressed more succinctly as follows. We define the pull-back of the volume fraction of the water phase as 𝜙w 𝜙w. 0 ∶= J (15)

Its time derivative is given by 𝜙̇ w 𝜙̇ w 𝜙w ̇ . 0 = J + J (16)

Inserting (16) into (14), we get the following Lagrangian form of the mass balance equation:

𝜙̇ w . 0 + DIV Q = 0 (17)

Next, the balance of linear momentum for the mixture in the current configuration may be written as ∇·𝝈 + 𝜌g = 𝟎, (18) where 𝝈 is the total Cauchy stress tensor and g is the gravitational acceleration vector. To translate the momentum balance equation into the reference configuration, we define the first Piola-Kirchhoff stress tensor P and the Kirchhoff stress tensor 𝝉 as P ∶= 𝝉 · F−T and 𝝉 ∶= J𝝈, (19) respectively. Multiplying (18) by J and insert (19) into it, we get

DIV P + 𝜌0 g = 𝟎, (20) 70 CHOO

𝜌 𝜌 where 0 ∶= J is the pull-back mixture density. To sum, the balance equations for the poromechanical problem at hand are given by (17) and (20) in the reference configuration, whereas they are given by (11) and (18) in the current configuration.

2.3 Constitutive models The formulation is completed by introducing constitutive models for three types of physical processes: solid deformation, fluid flow, and water retention. For the constitutive modeling of solid deformation, we adopt the principle of effective stress. The effective stress relationship for an unsaturated porous material may be written as

𝝈 = 𝝈′ − p̄𝟏. (21)

Here, 𝝈′ is the effective Cauchy stress tensor, 1 is the second-order identity tensor, and p̄ is the so-called equivalent pore pressure, which is the generalization of the pore water pressure p in the Terzaghi effective stress. In this work, we choose p̄ ∶= Sp, which makes 𝝈′ energy-conjugate to the rate of deformation tensor of the solid matrix.48 However, it is noted that other forms proposed for p̄ (for example, see other works49-52) can also be used at the modeler's discretion. Once the form of effective stress is defined, a constitutive model developed originally for a dry/drained solid can be introduced. Here, we use formulate solid constitutive models in terms of the effective Kirchhoff stress tensor 𝝉′ ∶= J𝝈′ among several standard and nonstandard choices for finite strain stress measures.53-57 For this purpose, we rewrite the effective stress relationship (21) in the Kirchhoff stress space 𝝉 = 𝝉′ − Jp̄ 𝟏. (22)

Without loss of generality, we consider isotropic hyperelasticity and relate the effective Kirchhoff stress tensor to the strain energy density function W(b) as 𝜕W(b) 𝝉′ = 2b · . (23) 𝜕b In this work, we use two types of strain energy density functions that commonly take two Lamé parameters, namely 𝜆 and G. The first one is ( ) 𝜆 1 2 W (b)= (ln J)2 + G tr ln b , (24) Hencky 2 2 which leads to Hencky elasticity.58 The second one is the strain energy density function for the Neo-Hookean model, which can be expressed as 𝜆 G W (b)= (ln J)2 − G ln J + (tr b − 3). (25) Neo-Hookean 2 2 It is noted that both of the two models reduce to linear elasticity in the infinitesimal deformation range and that 𝜆 and G can be converted into any other set of two independent elasticity parameters, eg, the bulk modulus and Poisson's ratio. As for fluid flow, we adopt the classical multiphase extension of Darcy's law, given by

q = 𝜿 ·(∇p − 𝜌w g), (26) where ( ) k 𝜿 ∶= r k, (27) 𝜇w 𝜇 is the lumped permeability tensor comprised of the relative permeability, kr, the dynamic viscosity of the water, w,and the absolute permeability tensor, k. The value of the absolute permeability may evolve significantly by solid deformation. To incorporate such deformation-induced evolution of permeability, we adopt the Kozeny-Carman equation assuming that the material's permeability is isotropic. Writing the absolute permeability as k = ka1, we relate ka to the porosity as ( ) ( ) 2 3 (1 − 𝜙0) 𝜙 k = k , , (28) a a 0 𝜙3 (1 − 𝜙)2 0 𝜙 𝜙s 𝜙s where the indices 0 are again used to denote initial values. Since = 1 − = 1 − 0∕J, the absolute permeability is a function of J. For the pull-back seepage vector Q, the foregoing Darcy's law can be translated into as follows:

Q = 𝜿0 ·(GRAD p − 𝜌wG). (29) CHOO 71

T In the above expression, G ∶= F · g,and𝜿0 is defined as

−1 −T 𝜿0 ∶= JF · 𝜿 · F . (30)

This shows that the pull-back of the permeability tensor, 𝜿0, can be anisotropic even when 𝜿 is isotropic. Therefore, a total Lagrangian approach to finite strain poromechanics should be able to handle anisotropic permeability tensors in a robust manner. Lastly, we introduce a water retention law that relates the saturation S with the pore water pressure p, which is necessary 𝜙w 𝜙 𝜙w 𝜙 to calculate = S and 0 = J S when the material is unsaturated. It is noted that large deformation in the solid matrix can give rise to dramatic changes in the water retention responses since it changes the pore sizes and thus the air entry pressure.59,60 To accommodate deformation-dependent water retention characteristics, we adopt a model proposed by Gallipoli et al,59 which can be written as ( ) m 1 S = [ ] , (31) s a n 1 + a1(1∕𝜙 − 1) 2 (−p) s where a1, a2, m,andn are material parameters. In case 𝜙 is constant (infinitesimal deformation) and m is chosen to be 1 − n∕1, this model specializes to the well-known van Genuchten model.61 In other words, this model is an extension of the van Genuchten model to capture deformation-induced changes in the air entry pressure. Accordingly, we adopt the relative permeability equation of the van Genuchten model given by [ ] 1∕2 1∕m m 2 kr = S 1 −(1 − S ) . (32)

Once the above constitutive models are inserted into the two balance equations for mass and linear momentum, we arrive at a coupled system of two governing equations in which two primary unknowns are u and p.

2.4 Strong form Based on the foregoing mathematical models, we state the initial–boundary value problem of interest as a strong form. Let d Ω0 ∈ ℝ denote the domain in the reference configuration in the d-dimensional space. The boundary of the initial domain is denoted by 𝜕Ω0, which is suitably decomposed into displacement (Dirichlet) and traction (Neumann) boundaries, 𝜕uΩ0 and 𝜕tΩ0, for the momentum balance equation, and pressure (Dirichlet) and flux (Neumann) boundaries, 𝜕pΩ0 and 𝜕qΩ0, for the mass balance equation. The decomposed boundaries satisfy 𝜕Ω0 = 𝜕uΩ0 ∪ 𝜕tΩ0 = 𝜕uΩ0 ∪ 𝜕tΩ0 and ∅=𝜕uΩ0 ∩ 𝜕tΩ0 = 𝜕pΩ0 ∩ 𝜕qΩ0. The time interval of the problem is denoted by 𝔗 ∶= (0, T] with T > 0. The boundary conditions of the problem are given by ̂ u(X, t)=u(X, t) on 𝜕uΩ0 × 𝔗, (33) ̂ P(X, t)·N(X)=T(X, t) on 𝜕tΩ0 × 𝔗, (34)

p(X, t)=p̂(X, t) on 𝜕pΩ0 × 𝔗, (35) ̂ −Q(X, t)·N(X)=Q(X, t) on 𝜕qΩ0 × 𝔗, (36) where N is the unit outward normal vector in the reference configuration and the hats denote prescribed boundary values.

The initial conditions of u and p are given by u0(X) and p0(X), respectively, for all X ∈Ω0 at t = 0. ̂ ̂ ̂ ̂ The strong form of this problem is then stated as: given u, T, p, Q, u0,andp0,findu and p such that

DIV P(u, p)+𝜌0(u, p)g = 𝟎 in Ω×𝔗, (37)

𝜙̇ w , , 𝔗. 0 (u p)+DIV Q(u p)=0inΩ× (38)

We remark that the two governing Equations (37) and (38) in this large strain poromechanical formulation are coupled in additional ways than its infinitesimal strain counterpart.42 Specifically, here, the permeability, the porosity, and the saturation do depend on u (more precisely, J), whereas they usually do not depend on u in an infinitesimal deformation formulation. Unless the deformation is uniform throughout the domain, this additional coupling gives rise to a higher 72 CHOO degree of heterogeneity in the fluid flow field. Therefore, the use of a locally conservative method for fluid flow may be even more desirable for poromechanical problems at large strains.

3 FINITE ELEMENT FORMULATION

In this section, we present a finite element framework that uses the EG method for locally mass conservative solution of the large deformation poromechanical problem described in the previous section. We first describe EG discretization of the mass balance equation in the finite deformation range, which is the major contribution of this work. In doing so, we develop two equivalent formulations using total Lagrangian and updated Lagrangian approaches. We then describe time discretization of the mass balance equation and CG discretization of the momentum balance equation. Strategies for the solution and implementation of the proposed formulation are also briefly discussed.

3.1 Enriched Galerkin discretization of the mass balance equation

To begin, we set up the notations and the finite element space for the EG method. Let h denote the set of shape-regular 62 (as defined in Ciarlet ) elements T partitioning Ω0. It is assumed at the outset that the triangulation does not experience severe distortion during the course of loading. Each element has edges (in 2D) or faces (in 3D) at its boundaries, which are commonly referred to as edges hereafter. Let h denote the set of all edges at element boundaries, which is decomposed I  휕 into the set of internal edges h and the set of boundary edges h . The set of boundary edges admits further decomposition 휕  휕p 휕q 휕p 휕q into h = h ∪ h ,where h is the set of edges at the pressure boundary and h is the set of edges at the flux boundary. To construct the finite element space for EG approximation of the pressure variable, we first define the space of elementwise discontinuous polynomials of degree k as

DGk  휓 2 휓| ℚ ,  , h ( h)∶={ ∈ L (Ω0)∶ T ∈ k(T) ∀T ∈ h} (39) 2 where L (Ω0) is the set of functions whose traces on the elements of Ω0 are square integrable and ℚk is the space of DG0  polynomials of degree at most k.Itisnotedthat h ( h) corresponds to the space for elementwise constant interpolation. DGk  ℂ We also define the subspace of h ( h) comprised of elementwise continuous polynomials ( 0)

CGk  DGk  ℂ . h ( h)∶= h ( h)∩ 0(Ω0) (40)

DGk  CGk  In words, h ( h) and h ( h) are the finite element spaces for the DG and CG approximations, respectively, of a scalar variable with kth degree polynomials. We then construct the EG approximation space with kth degree polynomials, EGk 44 denoted by h ,as

EGk  CGk  DG0  . h ( h)∶= h ( h)+ h ( h) (41)

EGk  One can see that h ( h) permits elementwise discontinuous interpolation by adding elementwise constant functions to CGk  the finite element space of the CG method, h ( h). As such, the number of degrees of freedom of the EG approximation is substantially fewer than that of the DG counterpart. For illustration, Figure 1 shows the degrees of freedom in a 2D domain discretized by bilinear (k = 1) CG, DG, and EG approximations. It is noted that, for a sufficiently large number EG1  DG1  of elements, the degrees of the freedom for h ( h) are about 1/2 (in 2D) and 1/4 (in 3D) of those of h ( h).Thereader is referred to the works of Sun and Liu44 and Lee et al45 for extensive details and analyses of the EG method. Weighted interior penalty methods45,63 are used to construct the variational formulation for the EG discretization. Before going on, we introduce notations for the weighted average and the jump of traces of 휁 along an interior edge. For an  I + − interior edge e ∈ h,wedenotebyT and T the two elements sharing it and denote by Ne the unit normal vector oriented from T + to T −. The weighted average and the jump are defined as

{휁}∶=훿e휁|T+ +(1 − 훿e)휁|T− , (42)

⟦휁⟧ ∶= 휁|T+ − 휁|T− , (43) CHOO 73

FIGURE 1 The degrees of freedom for bilinear (ℚ1) continuous Galerkin (CG), discontinuous Galerkin (DG), and enriched Galerkin (EG) approximations of a 2D grid. In (C), the degrees of freedom in the middle of the elements indicate elementwise constant functions enriched 42 to the finite element space of the CG method, after the work of Choo and Lee A, CG-ℚ1;B,DG-ℚ1;C,EG-ℚ1 [Colour figure can be viewed at wileyonlinelibrary.com]

respectively, where 훿e ∈[0, 1] is the weight. For the value of 훿e, a simple but good choice may be 훿e = 0.5, which is used in many DG formulations. In this work, we adopt a different choice presented in Lee and Wheeler46,givenby 휅− 훿 = 0 ,휅± ∶= (N )T · 𝜿 | · N . (44) e 휅+ 휅− 0 e 0 T± e 0 + 0 The EG method is now used to discretize the mass balance Equation (38) in space. Hereafter, the pressure unknown p is understood as spatially discretized without an additional notation. Then, the variational formulation of (38) may be EGk  휓 EGk  written as follows: find p ∈ h ( h) such that for all ∈ h ( h) ∑ ∑ 휓휙̇ w 휓 ∫ 0 dV − ∫ GRAD · QdV T∈ T∈ h T h T ∑ ∑ + ⟦휓⟧{Q · Ne}훿 dA − s ⟦p⟧{(𝜿 · GRAD휓)·Ne}훿 dA + I (휓,p) ∫ e form ∫ 0 e 0 e∈I e∈I h e h e ∑ ∑ 휓 ̂ 𝜿 휓 휕 휓, ̂ + ∫ (Q · Ne)dA − sform ∫ (p − p)( 0 · GRAD )·NedA + I0 ( p − p) 휕 휕 ∈ p ∈ p e h e e h e ∑ 휓 ̂ . − ∫ QdA = 0 (45) 휕 ∈ q e h e 휓, 휕 휓, ̂ In the above, I0( p) and I0 ( p − p) are the interior and boundary penalty terms, which may be defined as ∑ 휓, ⟦휓⟧⟦ ⟧ , I0( p)∶= Λe ∫ p dA (46) e∈ I ∑h e 휕 휓, ̂ 휓 ̂ . I0 ( p − p)∶= Λe ∫ (p − p)dA (47)  휕p e∈ h e

2 Here, Λe is the penalty parameter which may differ by edges, and we choose Λe =(휅̄0k 훼∕he),where휅̄0 is the harmonic 휅+ 휅− 훼 mean of 0 and 0 , k is the the polynomial degree, is a constant penalty value, and he is the edge length. Note that these penalty terms are the same as those of interior penalty methods. As is well known, here, 훼 should be large enough to make the discrete system stable, but too large an 훼 may lead to excessive interelement continuity. Similar to small-strain poromechanical problems studied in our previous work,42 we have experienced that 훼 ≈ 100 is usually a good choice. In addition, the value of sform is chosen for employing one of the following three types of interior penalty methods: sform = 1 for the symmetric interior penalty Galerkin method; sform = 0 for the incomplete interior penalty Galerkin method; and sform =−1 for the nonsymmetric interior penalty Galerkin method. These types of penalty methods also come from the interior penalty methods, and their characteristics have been extensively studied (for example, see other works64-68)and well described in the book by Riviére.69 Notably, for flow and transport problems, symmetric interior penalty Galerkin method can result in an optimal convergence rate in L2 norm, incomplete interior penalty Galerkin method can satisfy the so-called discrete compatibility principle, and nonsymmetric interior penalty Galerkin method can lead to a zero-penalty formulation. In this work, we use the incomplete interior penalty Galerkin method method. 74 CHOO

This variational formulation provides mass conservation locally (ie, in each element) as well as globally (ie, in the whole domain). To show this statement, we define the pull-back seepage flux vectors inside elements and along element interfaces as

Q ∶= −𝜿0 ·(GRAD p − 휌wG), ∀T ∈ h, (48) I Q · N ∶= −{𝜿 ·( p − 휌 G)·N }훿 +Λ⟦p⟧, ∀e ∈  , e 0 GRAD w e e e h (49) 𝜿 휌 ̂ ,  휕p, Q · Ne ∶= − 0 ·(GRAD p − wG)·Ne +Λe(p − p) ∀e ∈ h (50) ̂ , 휕q. Q · Ne ∶= −Q ∀e ∈ h (51)

Now, let us take the test function 휓 as 1 on an element T and 0 on all other elements. The EG solution of Q is then locally conservative since it satisfies ( ) ∑ 휙̇ w ,  . ∫ 0 dV + ∫ Q · NedA = 0 ∀T ∈ h (52) e∈휕T T e The global conservation property can be shown by setting 휓 = 1 throughout the domain, which gives ( ) 휙̇ w . ∫ 0 dV + ∫ Q · N dA = 0 (53) 휕 Ω0 Ω0

These local and global conservation properties can be rephrased in the current configuration using (6), (7), and (9). The resulting expressions are [ ] ∑ 휙̇ w 휙w ̇ ,  , ∫ ( )+( )∇ · u dv + ∫ q · ne da = 0 ∀T ∈ h (54) e∈휕T T e [ ] 휙̇ w 휙w ̇ , ∫ ( )+( )∇ · u dv + ∫ q · nda = 0 (55) Ω 휕Ω where the Ω and 휕Ω denote the current (deformed) domain and its boundary, q ∶= (1∕J)F · Q is the push-forward of Q, and ne and n are the respective counterparts of Ne and N in the current configuration. In an infinitesimal deformation setting, (52) and (53) become identical to (54) and (55), respectively, as well as to the local and global mass conservation statements presented in Choo and Lee.42 It is noted that these local and global conservation properties are attained mainly by the following two features: (i) the test function 휓 can be set as 1 on an element and exactly as 0 elsewhere and (ii) the Dirichlet boundary conditions are enforced weakly. These two features are common in other closely related conservative methods using finite volume or DG discretization schemes. Indeed, the weighted interior penalty formulation (45) can also be used for DG discretization of the same equation. The foregoing variational Equation (45) may be called a total Lagrangian formulation. As is well known, one can dis- cretize the same problem using an updated Lagrangian approach whereby the derivatives and integrals are reckoned with respect to the current configuration.70 The EG method, of course, can also be applied to develop an updated Lagrangian formulation. To wit, it is not overly difficult to reformulate (45) as ∑ ∑ 휓 휙̇ w 휙w ̇ 휓 ∫ [ + (J∕J)]dv − ∫ ∇ · qdv T∈ T∈ h T h T ∑ ∑ + ⟦휓⟧{q · n }훿 da − s ⟦p⟧{(𝜿 ·∇휓)·n }훿 da + I(휓,p) ∫ e e form ∫ e e e∈I e∈I h e h e ∑ ∑ 휓 𝜿 휓 휕 휓, ̂ + ∫ (q · ne)da − sform ∫ p( ·∇ )·ne da + I ( p − p) 휕 ∈ p e∈I e h e h e ∑ 휓 ̂ , − ∫ qda = 0 (56) 휕 ∈ q e h e 휕 ̂ 휕 ̂ where I, I ,andq are the counterparts of I0, I0 ,andQ in the current configuration, respectively. One can arrive at an equivalent variational formulation by EG discretization of the mass balance equation in the current configuration (11). It is noted that the total Lagrangian and updated Lagrangian formulations are theoretically equivalent provided that appropriate constitutive approaches are used. CHOO 75

3.2 Time discretization of the mass balance equation The semidiscrete variational mass balance equation, either (45) or (56), needs to be further discretized in time. In this work, we use the implicit Euler method, which is unconditionally stable and commonly used in the computational poromechanics literature. Let us denote by Δt ∶= tn +1 − tn the time increment from time tn to tn +1 and add the sub- 휙w 휙 scripts (·)n and (·)n +1 to denote respective values at these time instances. The time derivative of 0 ∶= J S in (45) can be approximated as ( ) ( ) 휙w − 휙w 휙̇ w ≈ 0 n+1 0 n . (57) 0 Δt In case (56) is used, the time discretization may be performed as

w w ̇ (휙 )n+ −(휙 )n J ̇ (ln J) + −(ln J) 휙̇ w ≈ 1 and = ln J ≈ n 1 n . (58) Δt J Δt

It is noted thatJ̇∕J is discretized after being converted into the time derivative of ln J,soastopreventtheJacobianJ becoming zero or negative. See Sun et al27 for a discussion of this aspect. It is also noted that the direct integration of 휙w is also required for mass conservation, see Celia et al.71 The discretization procedure is completed by inserting (57) into (45) for a total Lagrangian formulation or inserting (58) into (56) for an updated Lagrangian formulation and evaluating all other variables at time tn +1. The fully discrete form of the mass balance equation will be presented later in this section.

3.3 Continuous Galerkin discretization of the momentum balance equation The remaining part is to discretize the momentum balance Equation (37) in space. Similar to the mass balance equation, CG discretization of the momentum balance equation would not guarantee elementwise equilibrium (though nodal forces are balanced). This aspect, however, is usually not as crucial as local mass conservation, particularly when the solid material is compressible and does not exhibit significant jump in the displacement field. For this reason, here, we simply use the classic CG method for discretization of the deformation problem. Since the CG discretization is more or less standard, we only describe its essence in the sequel. For notational brevity, we again understand that the variables in the following are discretized in space. The finite element spaces of the trial and test functions may be written as

 CGk  1 | ̂ 휕 , | ℚ ,  , h ( h)∶={u ∈ H (Ω0) u = u on uΩ0 u T ∈ k(T) ∀T ∈ h} (59)

CGk  𝜼 1 |𝜼 𝟎 휕 , 𝜼| ℚ ,  , h ( h)∶={ ∈ H (Ω0) = on uΩ0 T ∈ k(T) ∀T ∈ h} (60) where H1 is the Sobolev space of order one. The discrete momentum balance equation in a total Lagrangian form can be developed as 𝜼 𝜼 휌 𝜼 ̂ . − ∫ GRAD ∶ P dV + ∫ ·( 0 g)dV + ∫ · T dA = 0 (61) 휕 Ω0 Ω0 tΩ0 Using (19) and (22), we can obtain an equivalent form of (61) as

𝜼 𝝉′ ̄𝟏 𝜼 휌 𝜼 ̂ . − ∫ ∇ ∶( − Jp )dV + ∫ ·( 0 g)dV + ∫ · T dA = 0 (62) 휕 Ω0 Ω0 tΩ0

The above form allows one to use a constitutive model formulated with respect to the effective Kirchhoff stress tensor 𝝉′. When updated Lagrangian discretization is used as in (56), it would be desirable to reformulate (62) further in the current configuration as ( ) 𝜼 𝝈′ ̄𝟏 𝜼 휌 𝜼 ̂ , − ∫ ∇ ∶ − p dv + ∫ ·( g)dv + ∫ · tda = 0 (63) 휕 Ω Ω tΩ where t̂ is the counterpart of T̂ in the current configuration. In this case, the effective Cauchy stress tensor may be cal- culated as 𝝈′ =(1∕J)𝝉′ from a constitutive model formulated with 𝝉′. This type of updated Lagrangian approach is also common in large deformation modeling, see the works of de Souza Neto et al54 and Coombs72 for example. 76 CHOO

We remark that when undrained deformation/incompressible flow is expected in the problem, the polynomial degree EGk  CGk+1 CGk+1 for the CG discretization needs to be one order higher than its EG counterpart (ie, h must be used with h ∕ h for k ≥ 1) for inf-sup stability.42,73 We use this type of mixed finite elements in the following numerical examples. A stabilized formulation for the mixed EG/CG finite elements such as those for the mixed CG/CG finite elements27,74-76 will be presented in a future publication.

3.4 Solution and implementation strategies Before closing this section, we describe some strategies used in this work for the solution and implementation of the present finite element formulations. First, we use Newton's method to solve the formulation as it is always nonlinear. Linearization of finite strain poromechanical formulations is extensively discussed in the literature,25,77,78 so we do not repeat the same linearization procedure herein. For reference, we simply write some expressions that can be useful for the linearization. Denoting by 훿(·) the linearization operator, they are

훿F =∇훿u · F, (64)

훿F −1 =−F −1 ·∇훿u, (65)

훿J = J∇·훿u, (66)

훿(∇ · 𝜼)=−∇𝜼 ∶∇T훿u, (67)

훿(∇휓)=−∇휓 ·∇훿u. (68)

At each Newton iteration, one must solve a linear Jacobian system in which the matrix can be strongly ill conditioned. To facilitate solution of this linear system, we use an iterative solver in conjunction with a block-partitioned precondi- tioner explained in the work of White et al.79,80 This solver has been shown to be efficient and scalable for two-field (u/p) poromechanical formulations at small strains. We have observed that this approach continues to perform well for large deformation poromechanical problems. It is noted that, for the EG formulation at hand, the nested solver for the pressure block may be further advanced by exploiting its potentially blocked structure as done in Lee et al.45 A detailed discussion on this aspect is provided in our previous work on small strain poromechanics.42 Also noted is that a sequential implicit method such as the fixed stress method studied in the works of Kim et al33 and Mikelic´ and Wheeler81 may be favored when separate code is used for the fluid flow and solid deformation problems. The EG formulation can be implemented through a standard algorithm for implementing an interior penalty method, which is explained in detail in Riviére,69 among others. As compared with CG implementation, the major difference is that one needs to assemble local contributions not only from elements but also from edges. Specifically, once the mass balance equation of the updated Lagrangian form (56) is fully discretized in space and time, the contribution of an interior element T to the mass residual vector is given by (after multiplying Δt) [ ] i 휓i 휙w 휙w 휓i 휙w 휓i , [Rmass]T ∶= ∫ ( )n+1 −( )n dv + ∫ ( )n+1 [(ln J)n+1 −(ln J)n] dv −Δt ∫ ∇ · qn+1dv (69) T T T where i denotes the shape function index. The contribution of an interior edge e is given by

i i i i [R ] ∶= Δt ⟦휓 ⟧{q · n }훿 da − s Δt ⟦p⟧{(𝜿 ·∇휓 )·n }훿 da +Δt (Λ ) ⟦휓 ⟧⟦p ⟧da. (70) mass e ∫ n+1 e e form ∫ e e ∫ e n+1 n+1 e e e

Translating the above equations to total Lagrangian versions as well as extending them to boundary elements and edges are straightforward. For implementing EG, an important point to note is that a function in the EG space is additively 휓 EGk  decomposed into a CG part and an elementwise constant part. To wit, for any ∈ h ( h),

CG DG 휓 휓CGk 휓DG0 ,휓CGk  k  ,휓DG0  0  . = + ∈ h ( h) ∈ h ( h) (71) CHOO 77

Recognizing this property is essential for an efficient implementation of the EG method because it allows one to eliminate terms that indeed vanish by definition. For example, see

GRAD 휓 = GRAD 휓CGk + GRAD 휓DG0 = GRAD 휓CGk . (72) ⏟⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏟ =0

The above equation means that the flux terms inside elements, ie, the second summation in (69), remain unchanged from CG implementation. Furthermore, it is noted that

⟦휓⟧ = ⟦휓CGk ⟧ + ⟦휓DG0 ⟧ = ⟦휓DG0 ⟧ (73) ⏟⏟⏟ =0 since 휓CGk is continuous across edges. This fact greatly simplifies the implementation of edgewise contributions (70). For this reason, the implementation of the EG method may be no more difficult than other conservative methods of similar performance. In this paper, we have presented both total Lagrangian and updated Lagrangian formulations, although they are the- oretically equivalent. The motivation was that the reader might prefer one of them from an implementation point of view, particularly when a specific approach is already used in the existing code. For fresh implementation of the updated Lagrangian formulations (56) and (63), it would be advisable to construct an isoparametric map between the parent element and the current (deformed) element rather than moving the mesh. Details of total Lagrangian and updated Lagrangian approaches and their implementations are well explained in Belytschko et al,70 among others. We have implemented and tested both of (45) and (56) and observed that their solutions are virtually identical.

4 NUMERICAL EXAMPLES

This section presents three numerical examples that verify the proposed finite element formulation and highlight the criti- cal role of local mass conservation in some poromechanical problems. The first example uses a classic benchmark problem in the literature to verify the formulation and demonstrate the effects of large deformation kinematics on poromechanical processes. The second and third examples simulate deformations in porous media induced by fluid injection and extrac- tion, respectively, under saturated and unsaturated conditions. In these two examples, the proposed EG/CG discretization of the flow/deformation equations is compared with the conventional CG/CG discretization of the same equations. For brevity, in this comparison, we will simply call the EG/CG discretization of the flow/deformation equations “EG” and call the CG/CG discretization “CG.” At this point, it is noted that the present numerical examples are designed to demonstrate aspects that have not been covered in our previous work at small strains.42 In the previous work, we have performed a convergence study for the pore pressure solution using an analytical problem available at small strain and shown that the EG discretization can significantly alleviate pressure oscillation due to a permeability jump. Demonstration of these aspects are not repeated herein. It is also noted that all simulations are conducted using the same finite strain formulation, so every difference between the EG and CG results can be attributed to the property of local mass conservation or lack thereof. The following numerical examples are prepared using Geocentric, a massively parallel finite element code for geomechanics used in a number of previous studies of the author and co-workers.79,80,82-84 This code is built upon the deal.II finite element library,85,86 p4est mesh handling library,87 and the Trilinos project.88

4.1 Cryer's problem Our first example is Cryer's 3D consolidation problem,89 which is a classic benchmark problem in poromechanics at small strains. In this problem, a poroelastic sphere is instantaneously subjected to a uniform boundary pressure. As soon as the boundary pressure is applied, excess pore pressure develops inside the sphere, and so, the pore fluid starts to drain through the outer boundary. The boundary pressure is maintained constant throughout. Nevertheless, the pore pressure around the center of the sphere initially rises for a while, becoming higher than the boundary pressure, before it starts to decreases. The initial rise of the pore pressure is referred to as the Mandel-Cryer effect in the literature. It is well known that the Mandel-Cryer effect can only be captured by a mathematical model that correctly takes into account the two-way coupling between fluid flow and solid deformation, ie, it cannot be reproduced by a one-way or loosely 78 CHOO coupled model. For this reason, the analytical solution for Cryer's problem has been widely used for verifying tightly coupled poromechanical formulations.75,90 Although the original problem considers infinitesimal deformation, Gibson and co-workers have studied its extension to the finite deformation range, first assuming constant permeability91 and later incorporating evolving permeability.92 As such, this problem is selected for two purposes: (i) to verify the implementation and (ii) to demonstrate the effects of finite strain kinematics and deformation-dependent permeability. The new EG finite element formulation is used to simulate this problem in a sphere of radius 1 m. Taking advantage of symmetry, we discretize an octant of the sphere by 3456 hexahedral elements having 90 999 unknowns for the CG-ℚ2 approximation of the displacement field and 7529 unknowns for the EG-ℚ1 approximation of the pore pressure field. The 휙 material is homogeneous, and it is fully saturated and free of gravitational forces throughout. The initial porosity 0 is set as 0.5. The solid behavior is modeled by Hencky elasticity, defining Poisson's ratio 휈 ∶= 휆∕(2휆 + 2G) and the bulk modulus K ∶= 휆 + 2G∕3. The solution of the original Cryer's problem depends only on Poisson's ratio 휈, and we select 휈 = 0, which gives the highest rise in the pore pressure. The bulk modulus K is set as 1 MPa. The hydraulic parameters 2 are then assigned such that the coefficient of consolidation cv = 0.003 m /s, which ultimately gives the nondimensional consolidation time ̃t = 0.003t. We consider the time domain from ̃t = 10−3 to ̃t = 1 and discretize it by 120 time steps with a uniform interval in the logarithmic space of ̃t.

Three cases of boundary pressures p0 are assigned as ratios to the bulk modulus K, namely p0∕K = 0.001, 0.1, and 0.5. . Here the highest pressure case p0∕K = 0 5 will result in fairly large deformation since the maximum possible p0∕K for −1 91 this material is calculated to be 0.693 (from the equation ln (1 − 휙0) in Gibson et al ). It is noted that, while the evolu- tion of the normalized pore pressure p∕p0 in the original Cryer's problem is independent of the boundary pressure, this is no longer the case when the problem is generalized to finite strains. The higher the boundary pressure, the more the behavior would deviate from the solution assuming infinitesimal strains. In addition, we compare two cases of perme- ability responses: (i) constant permeability and (ii) evolving permeability according to the Kozeny-Carman Equation (28). The latter case would be more realistic.

Figure 2 shows the normalized pressure p∕p0 at the center of the sphere during consolidation, simulated with the three values of p0∕K in both the constant and Kozeny-Carman permeability cases. The small strain analytical solution is also plotted for comparison. It can be seen that, when the boundary pressure is fairly small relative to the bulk modulus (the . case of p0∕K = 0 001), the large strain finite element formulation gives a result nearly identical to the analytical solution irrespective of the permeability relationships. This is because the geometric nonlinearity and the porosity evolution are marginal when the external load is very small. To confirm this, in Figure 3, we plot the Jacobian J at the sphere center versus the normalized time. Note that the small strain formulation assumes J = 1 as indicated in the figure. It can be found . . . that J ≈ 1 throughout when p0∕K = 0 001. However, under higher boundary pressures ( p0∕K = 0 1andp0∕K = 0 5), J becomes significantly lower than 1 during the process of consolidation. In such cases, the consolidation behavior deviates from the analytical solution at small strain. In case the permeability is assumed to be constant, the consolidation process becomes slightly faster at a higher boundary pressure, manifesting the virtually same degree of the Mandel-Cryer effect. The faster consolidation rate can be attributed to the shortened drainage length (the sphere radius) explicitly taken into account in the large strain formulation. Conversely, when the permeability evolves by porosity change, a higher boundary

FIGURE 2 Comparison of normalized pore pressure values at the center of the sphere for the constant and Kozeny-Carman permeability cases [Colour figure can be viewed at wileyonlinelibrary.com] CHOO 79

FIGURE 3 Comparison of Jacobian values at the center of the sphere for the constant and Kozeny-Carman permeability cases [Colour figure can be viewed at wileyonlinelibrary.com]

. ̃ . FIGURE 4 Pore pressure distributions in the constant and Kozeny-Carman permeability cases when p0∕K = 0 5andt = 0 1 pressure leads to a slower consolidation process and a less pronounced Mandel-Cryer effect. This result shows that the deformation-induced reduction of permeability plays a critical role in coupled poromechanical responses. To illustrate the difference between the constant and Kozeny-Carman permeability cases, the pore pressure distributions in these . ̃ . two cases are drawn in Figure 4 when p0∕K = 0 5andt = 0 1. The figure indicates that the consolidation process is significantly delayed throughout the domain when the permeability is modeled by the Kozeny-Carman equation. On a related note, these results are consistent with the findings of Gibson et al91,92 who solved Cryer's problems at finite strains with and without permeability evolution. The results of this example have demonstrated that the proposed finite element formulation appropriately cap- tures fully-coupled poromechanical responses and their transition by finite deformation effects. The importance of deformation-induced permeability has also been highlighted. Lastly, it is noted that the CG solution of this problem is vir- tually the same as the EG solution presented above, presumably because the permeability field is initially homogeneous and it does not become significantly heterogeneous during the problem. In such cases, the standard CG method may be favored since it involves fewer degrees of freedom than the EG for the same mesh and polynomial order. However, EG and CG solutions are not always very close, as will be shown in the following examples.

4.2 Swelling by radial water injection The second example simulates the problem of swelling of a porous material by radial water injection. We consider a cylindrical domain of radius 1.5 m containing a hole of radius 0.5 m in its center. Taking advantage of symmetry, we model a quarter of the domain and discretize it by 594 quadrilateral elements as depicted in Figure 5. Plane strain condition is assumed. As for material properties, we assign the water retention parameters as a1 = 0.027, a2 = 8.433, m = 0.5, 80 CHOO

FIGURE 5 Setup of the swelling problem [Colour figure can be viewed at wileyonlinelibrary.com]

. 휙s . −13 2 n = 2 0, and 0 = 0 5. The initial absolute permeability ka,0 is set as 5 × 10 m , and the dynamic viscosity of water 휇 −6 휆 is set as w = 10 kPa·s. The solid matrix is modeled as a Neo-Hookean material with = 500 kPa and G = 750 kPa. The material is initially unsaturated with a suction of 200 kPa. Gravitational force is neglected. The problem begins by applying an inward flux of 0.01 mm/s on the hole boundary as illustrated in Figure 5. We run the simulation with a time increment of Δt = 1 minute until the injection time reaches 35 hours. For comparison, we solve the problem using both the proposed EG discretization and the conventional CG discretization. The CG discretization results in 644 degrees of freedom for approximating the pressure field, and 594 (the number of elements) degrees of freedom are added to the EG discretization. The number of the displacement degrees of freedom is 1288 for both discretization methods. Figure 6 shows the EG and CG results of the time evolution of the pore pressure fields during the process of flow-induced swelling. It is noted that the ranges of the color bars are adjusted as time proceeds to clearly show the pressure field at each time instance. As expected, the pore pressure builds up from the inner boundary. Until 21 hours of injection, the EG and CG solutions of pore pressure appear not very different. Nevertheless, they still have some important differences. To better illustrate the differences, the EG and CG pore pressure solutions at 9 hours are magnified in Figure 7. The figure shows that the CG solution manifests nonphysical oscillation around the water infiltration front, whereas such oscillation is virtually absent in the EG solution. The propagation of the infiltration front is also slightly retarded in the CG solution. As the water is further injected from 21 hours, the EG and CG solutions become increasingly different. It can be seen that from 25 hours, the pore pressure is significantly higher in the EG solution than in the CG solution. This growing difference may be attributed to that the permeability field becomes increasingly heterogeneous by finite deformations. The EG and CG solutions are further compared with respect to their local mass conservation properties. To do this, we define the residual of the mass balance at an element T as

[ ] ∑ | 휙w 휙w 휙w , Rmass T ∶= ∫ ( )n+1 −( )n dv + ∫ ( )n+1 [(ln J)n+1 −(ln J)n] dv + Δt ∫ qn+1 · ne da (74) e∈휕T T T e where the subscripts (·)n +1 and (·)n denote quantities at the current and previous time steps, respectively. Note that this is a time-discrete version of the local mass balance statement (54). When this residual is calculated for the EG solution, the variables 휙w and q are computed from the EG pore pressure solution such that (74) becomes the residual of (54). Likewise, for the local mass residual of the CG solution, these variables are computed from the CG pore pressure field. Figure 8 presents these local mass residuals of the EG and CG solutions in terms of their norm, ||Rmass||, at the same time instances as in the previous figure. It is confirmed that the EG solution is locally mass conservative throughout in that all −10 −8 the values of ||Rmass|| are smaller than 10 (on a related note, the tolerance of Newton iterations was 10 ). However, the CG solution exhibits orders of magnitude larger local mass imbalance than those of the EG solution. Interestingly, when the material is unsaturated (until 21 hours), local mass balance residuals are higher around the water infiltration front. After the material becomes fully saturated, the distribution of the mass balance residuals is more uniform. CHOO 81

FIGURE 6 Pore pressures in the swelling problem simulated by the enriched Galerkin (EG) and continuous Galerkin (CG) methods 82 CHOO

FIGURE 7 Magnified pore pressure fields in the enriched Galerkin (EG) and continuous Galerkin (CG) solutions at 9 hours in the swelling problem [Colour figure can be viewed at wileyonlinelibrary.com]

We then examine how the difference in the EG and CG results of the fluid flow can affect the solid deformation response. For this purpose, we compute increments in the natural log of the Jacobian, which equals the true (logarithmic) volumetric strain, from the beginning of the fluid injection. Figure 9 shows these values of the EG and CG results at the same time instances as in the previous figures. The volumetric deformations of the EG and CG results manifest patterns similar to the pore pressures shown in Figure 6: they become increasingly different as the material continues to swell. This is a natural consequence in that the swelling process in this problem is driven by the buildup of the pore pressure. Recall that the higher the pressure, the more the effective stress becomes tensile. As a result, the CG solution underestimates the amount of swelling as compared with the EG solution. Therefore, the results of this example have shown that local mass conservation can be critical to the simulation of a problem dealing with an initially homogeneous domain because the heterogeneity in the permeability field can be amplified by poromechanical processes during the problem. Also shown is that the local imbalance of fluid mass alone can have a significant impact on the volumetric deformation of an elastic porous material due to fluid injection.

4.3 Land subsidence due to groundwater withdrawal As our final example, we perform poromechanical simulation of land subsidence due to groundwater withdrawal. Figure 10 illustrates the setup of the problem. We consider a 150-m wide and 30-m high rectangular domain under plane strain condition. At the center of the domain is a 20-m long pumping well, from which groundwater is extracted at depths of 15-20 m below the top surface. For simplicity, the well is modeled as a surface, and the process of groundwater pumping is modeled by prescribing an outward flux on this surface. No-flow conditions are imposed on all other boundaries. With respect to solid deformation, the lateral boundaries are supported by rollers, the bottom boundary is fixed by pins, and the other boundaries are traction-free. It is noted that the geometry and boundary conditions of this example are simplified from those of a real subsidence problem in 3D. However, as will be shown, the 2D setup still allows us to reproduce the major characteristics of water table drawdown due to pumping. Thus, the simplified configuration may be good enough for our purpose, namely an investigation of the performance of numerical methods. The material in this problem is heterogeneous from its initial state. We first generate a heterogeneous porosity field using a Gaussian random field model with a mean porosity of 0.35 as shown in Figure 10. We then correlate the porosity field −13 2 휙 . with the absolute permeability field using the Kozeny-Carman Equation (28), with ka,0 = 5 × 10 m and 0 = 0 35. The heterogeneous porosity values also affect the water retention Equation (31). All other parameters are homogeneous and constant throughout. The remaining parameters of the water retention model are assigned as a1 = 0.038, a2 = 3.491, m = 0.632, and n = 0.718, which are adopted from Song and Borja78 who fitted this model with the experimental data 93 휌 . 3 휌 . of a clayey silty sand in Salager et al. The intrinsic densities of the solid and water are s = 2 6t/m and w = 1 0 3 휇 −6 t/m , respectively, and the dynamic viscosity of water is w = 10 kPa·s. As for the solid constitutive model, we use the Neo-Hookean model with 휆 = 25 MPa and G = 37.5MPa. CHOO 83

FIGURE 8 Residual norms of local mass balance in the swelling problem simulated by the enriched Galerkin (EG) and continuous Galerkin (CG) methods 84 CHOO

FIGURE 9 Natural log of the Jacobian in the swelling problem simulated by the enriched Galerkin (EG) and continuous Galerkin (CG) methods CHOO 85

FIGURE 10 Setup of the land subsidence problem. Drawn inside the domain is the initial porosity field generated from a random field model. Blue arrows denote where groundwater is withdrawn

FIGURE 11 Pore pressures in the land subsidence problem simulated by the enriched Galerkin (EG) and continuous Galerkin (CG) methods

As in the previous example, we simulate this problem using both the EG and CG formulations. The domain is dis- cretized by 2880 quadrilateral elements, and this discretization results in 6082 displacement degrees of freedom for both formulations. The numbers of the pressure degrees of freedom are 5921 for the EG discretization and 3041 for the CG discretization. The domain is assumed to be an unconfined aquifer, and it is initialized such that the water table is located several meters below the ground surface. For the initialization, we assign a uniform initial suction of 10 kPa throughout the domain and then run a number of gravity loading steps (without groundwater pumping) until the fluid flow reaches a steady state. At this steady state, the water table is located about at a depth of 9 m, and the top surface is unsaturated manifesting a suction of 111 kPa and a water saturation ratio of 0.745. As the depth increases from the top, the suction decreases, ie, the pore pressure increases), and the water saturation increases. Once the initial steady state is generated in this way, we begin the groundwater pumping process by applying an outward flux of 7 mm/h on the well boundary. With a constant time increment of Δt = 6 hours, we simulate the problem until the groundwater pumping time reaches 240 days. Figure 11 presents changes in the pore pressure fields during the process of groundwater withdrawal, simulated by the EG and CG methods. Similar to the previous example, the two pore pressure fields show increasing differences with time. In general, the drawdown of the water tables is somewhat more localized in the EG solution than the CG solution. This difference may be attributed to the fact that the EG method allows pressure and velocity fields that are discontinuous across element boundaries. In Figure 12, we show the residual norms of the local mass balance of the EG and CG solutions of this problem. It can be seen that the EG solution has virtually no errors in elementwise mass balances as it did in the previous example. Conversely, the local mass residuals in the CG solution is much more significant than those in the previous example. In 86 CHOO

FIGURE 12 Residual norms of local mass balance in the land subsidence problem simulated by the enriched Galerkin (EG) and continuous Galerkin (CG) methods

FIGURE 13 Vertical displacements in the land subsidence problem simulated by the enriched Galerkin (EG) and continuous Galerkin (CG) methods

particular, the mass residual norms around the pumping zone are up to the order of 10−1, which is about three orders higher than the maximum residual norm in the previous example. Lastly, Figure 13 compares the EG and CG simulation results of the changes in the vertical displacement during the groundwater withdrawal process. Throughout the simulation time, larger and wider subsidences are predicted by the CG formulation than the EG formulation. This trend is consistent with the wider drawdown of the water table in the CG solutions shown in Figure 11. It is also noted that the differences between the EG and CG solutions become more evident as time proceeds. Taking the observations from this and the previous examples together, it can be concluded that the proposed EG formu- lation provides local mass conservation in poromechanical problems in the finite deformation range. The examples have also demonstrated that violation of local mass conservation can lead to either an overestimation or an underestimation of flow-induced deformations in porous materials. CHOO 87

5 CLOSURE

We have presented a new finite element framework for locally mass conservative simulation of coupled poromechanical problems at large strains. The key element of this framework is the use of the EG method for discretization of the fluid mass balance equation. The EG discretization has allowed us to attain local mass conservation by augmenting additional degrees of freedom whose number is equal to the number of elements. This additional cost is expected to be affordable in most cases, and it is substantially lower than the cost that might be added by another locally conservative finite element method. In this regard, the proposed finite element formulation is believed to be an appealing option for simulation of fluid flow in largely deforming porous media in a variety of practical applications. Through numerical examples, we have shown that local mass conservation can be important for accurate prediction of deformation as well as flow in porous media at large strains, extending our findings from poromechanical modeling at small strains.42 Results of the numerical examples also suggest that local mass conservation can be increasingly important as the solid deformation departs from the infinitesimal range. This is a natural consequence of that finite deformation gives rise to significant changes in the permeability and the water retention responses, making the material more hetero- geneous. Despite these amplified heterogeneities, the EG method has performed well for flow in porous media at finite strains. The development of this work can be readily extended to address more complex processes in fluid-infiltrated porous media undergoing large deformations. One example is consolidation-induced transport of contaminants in soils.94-96 The proposed finite element formulation can serve as a useful basis for advanced modeling of this type of problem since local mass conservation is highly desired, if not essential, for numerical simulation of coupled flow and transport in porous media. Other examples include numerical modeling of hydrogels, whose swelling and shrinking responses in simple geometry have been remarkably well described by poromechanical theory.97 We leave these extensions as topics of future research.

ACKNOWLEDGEMENTS The author thanks Professor Sanghyun Lee of Florida State University for his helpful comments on the enriched Galerkin method as well as on the paper. Financial support for this work was provided by the Seed Fund for Basic Research of The University of Hong Kong (No. 201801159010).

ORCID

Jinhyun Choo http://orcid.org/0000-0002-5861-3796

REFERENCES 1. Gambolati G, Teatini P, Baú D, Ferronato M. Importance of poroelastic coupling in dynamically active aquifers of the Po river basin, Italy. Water Resour Res. 2000;36(9):2443-2459. 2. Teatini P, Ferronato M, Gambolati G, Gonella M. Groundwater pumping and land subsidence in the Emilia-Romagna coastland, Italy: modeling the past occurrence and the future trend. Water Resour Res. 2006;42(1). 3. Sun W, Chen Q, Ostien JT. Modeling the hydro-mechanical responses of strip and circular punch loadings on water-saturated collapsible geomaterials. Acta Geotech. 2014;9(5):903-934. 4. Soga K, Alonso E, Yerro A, Kumar K, Bandara S. Trends in large-deformation analysis of landslide mass movements with particular emphasis on the material point method. Géotechnique. 2016;66(3):248-273. 5. Borja RI, Choo J, White JA. Rock moisture dynamics, preferential flow, and the stability of hillside slopes. In: Multi-Hazard Approaches to Civil Infrastructure Engineering. Cham, Switzerland: Springer; 2016:443-464. 6. Zoback MD. Reservoir Geomechanics. Cambridge, UK: Cambridge University Press; 2007.

7. Rutqvist J. The geomechanics of CO2 storage in deep sedimentary formations. Geotech Geol Eng. 2012;30(3):525-551.

8. White JA, Chiaramonte L, Ezzedine S, et al. Geomechanical behavior of the reservoir and caprock system at the In Salah CO2 storage project. Proc Natl Acad Sci. 2014;111(24):8747-8752. 9. Juanes R, Jha B, Hager BH, et al. Were the May 2012 Emilia-Romagna earthquakes induced? A coupled flow-geomechanics modeling assessment. Geophys Res Lett. 2016;43(13):6891-6897. 10. Celia MA. Geological storage of captured carbon dioxide as a large-scale carbon mitigation option. Water Resour Res. 2017;53(5):3527-3533. 11. Cowin SC. Bone poroelasticity. JBiomech. 1999;32(3):217-238. 12. Charras GT, Yarrow JC, Horton MA, Mahadevan L, Mitchison TJ. Non-equilibration of hydrostatic pressure in blebbing cells. Nature. 2005;435(7040):365-369. 88 CHOO

13. Skotheim JM, Mahadevan L. Physical limits and design principles for plant and fungal movements. Science. 2005;308(5726):1308-1310. 14. Franceschini G, Bigoni D, Regitnig P, Holzapfel G. Brain tissue deforms similarly to filled elastomers and follows consolidation theory. J Mech Phys Solids. 2006;54(12):2592-2620. 15. Gibson RE, England GL, Hussey MJL. The theory of one-dimensional consolidation of saturated clays. Géotechnique. 1967;17(3):261-273. 16. Gibson RE, Schiffman RL, Cargill KW. The theory of one-dimensional consolidation of saturated clays. II. Finite nonlinear consolidation of thick homogeneous layers. Can Geotech J. 1981;18(2):280-293. 17. Fox P, Berles J. CS2: a piecewise-linear model for large strain consolidation. Int J Numer Anal Methods Geomech. 1997;21(7):453-475. 18. Borja RI, Tamagnini C, Alarcon E. Elastoplastic consolidation at finite strain. Part 2: finite element implementation and numerical examples. Comput Methods Appl Mech Eng. 1998;159(1-2):103-122. 19. Armero F. Formulation and finite element implementation of a multiplicative model of coupled poro-plasticity at finite strains under fully saturated conditions. Comput Methods Appl Mech Eng. 1999;171(3-4):205-241. 20. Li C, Borja RI, Regueiro RA. Dynamics of porous media at finite strain. Comput Methods Appl Mech Eng. 2004;193(36-38):3837-3870. 21. Uzuoka R, Borja RI. Dynamics of unsaturated poroelastic solids at finite strain. Int J Numer Anal Methods Geomech. 2012;36(13):1535-1573. 22. Macminn CW, Dufresne ER, Wettlaufer JS. Large deformations of a soft porous material. Phys Rev Appl. 2016;5(4):1-30. 23. Auton LC, MacMinn CW. From arteries to boreholes: steady-state response of a poroelastic cylinder to fluid injection. Proc Royal Soc A: Math Phys Eng Sci. 2017;473(2201): 20160753. 24. Sanavia L, Schrefler BA, Steinmann P. A formulation for an unsaturated porous medium undergoing large inelastic strains. Comput Mech. 2002;28(2):137-151. 25. Andrade JE, Borja RI. Modeling deformation banding in dense and loose fluid-saturated sands. Finite Elem Anal Des. 2007;43(5):361-383. 26. Regueiro RA, Ebrahimi D. Implicit dynamic three-dimensional finite element analysis of an inelastic biphasic mixture at finite strain. Part 1: application to a simple geomaterial. Comput Methods Appl Mech Eng. 2010;199(29-32):2024-2049. 27. Sun W, Ostien JT, Salinger A. A stabilized assumed deformation gradient finite element formulation for strongly coupled poromechanical simulations at finite strain. Int J Numer Anal Methods Geomech. 2013;37(16):2755-2788. 28. Song X, Borja RI. Finite deformation and fluid flow in unsaturated soils with random heterogeneity. Vadose Zone J. 2014;13(5). 29. Borja RI, Choo J. Cam-Clay plasticity, Part VIII: a constitutive framework for porous materials with evolving internal structure. Comput Methods Appl Mech Eng. 2016;309:653-679. 30. Krischok A, Linder C. On the enhancement of low-order mixed finite element methods for the large deformation analysis of diffusion in solids. Int J Numer Methods Eng. 2016;106(4):278-297. 31. Spiezia N, Salomoni VA, Majorana CE. Plasticity and strain localization around a horizontal wellbore drilled through a porous rock formation. Int J Plast. 2016;78:114-144. 32. Prévost JH. Two-way coupling in reservoir–geomechanical models: vertex-centered Galerkin geomechanical model cell-centered and vertex-centered finite volume reservoir models. Int J Numer Methods Eng. 2014;98(8):612-624. 33. Kim J, Tchelepi HA, Juanes R. Stability and convergence of sequential methods for coupled flow and geomechanics: fixed stress and fixed-strain splits. Comput Methods Appl Mech Eng. 2011;200(13-16):1951-1606. 34. Jha B, Juanes R. Coupled multiphase flow and poromechanics: a computational model of pore pressure effects on fault slip and earthquake triggering. Water Resour Res. 2014;50(5):3776-3808. 35. Yoon HC, Kim J. Spatial stability for the monolithic and sequential methods with various space discretizations in poroelasticity. Int J Numer Methods Eng. 2018;114(7):694-718. https://doi.org/10.1002/nme.5762 36. Choo J, Sun W. Cracking and damage from crystallization in pores: coupled chemo-hydro-mechanics and phase-field modeling. Comput Methods Appl Mech Eng. 2018;335:347-349. 37. Jha B, Juanes R. A locally conservative finite element framework for the simulation of coupled flow and reservoir geomechanics. Acta Geotech. 2007;2(3):139-153. 38. Ferronato G. A fully coupled 3-D mixed finite element model of Biot consolidation. J Comput Phys. 2010;229(12):4813-4830. 39. Castelletto N, Gambolati G, Teatini P. A coupled MFE poromechanical model of a large-scale load experiment at the coastland of Venice. Comput Geosci. 2015;19(1):17-29. 40. Castelletto N, White JA, Ferronato M. Scalable algorithms for three-field mixed finite element coupled poromechanics. J Comput Phys. 2016;327:894-918. 41. Liu R, Wheeler M, Dawson C, Dean R. Modeling of convection-dominated thermoporomechanics problems using incomplete interior penalty Galerkin method. Comput Methods Appl Mech Eng. 2009;198(9-12):912-919. 42. Choo J, Lee S. Enriched Galerkin finite elements for coupled poromechanics with local mass conservation. Comput Methods Appl Mech Eng. 2018. Accepted for publication. 43. Kim J. A new numerically stable sequential algorithm for coupled finite-strain elastoplastic geomechanics and flow. Comput Methods Appl Mech Eng. 2018;335:538-562. 44. Sun S, Liu J. A locally conservative finite element method based on piecewise constant enrichment of the continuous Galerkin method. SIAM J Sci Comput 2009;31(4):2528-2548. 45. Lee S, Lee YJ, Wheeler MF. A locally conservative enriched Galerkin approximation and efficient solver for elliptic and parabolic problems. SIAM J Sci Comput 2016;38(3):A1404-A1429. 46. Lee S, Wheeler MF. Adaptive enriched Galerkin methods for miscible displacement problems with entropy residual stabilization. J Comput Phys. 2017;331:19-37. CHOO 89

47. Lee S, Wheeler MF. Enriched Galerkin method for two phase flow in porous media with capillary pressure. J Comput Phys. 2018;367:65-86. 48. Borja RI. On the mechanical energy and effective stress in saturated and unsaturated porous continua. Int J Solids Struct. 2006;43(6):1764-1786. 49. Coussy O. Poromechanics. Hoboken, NJ: John Wiley & Sons; 2004. 50. Gray WG, Schrefler BA, Pesavento F. Work input for unsaturated elastic porous media. J Mech Phys Solids. 2010;58(5):752-765. 51. Fuentes W, Triantafyllidis T. On the effective stress for unsaturated soils with residual water. Géotechnique. 2013;63(16):1451-1455. 52. Kim J, Tchelepi HA, Juanes R. Rigorous coupling of geomechanics and multiphase flow with strong capillarity. SPE J. 2013;18(6):1123-1139. 53. Simo JC, Hughes TJR. Computational Inelasticity. New York, NY: Springer-Verlag New York; 1998. 54. de Souza Neto EA, Peric D, Owen DRJ. Computational Methods for Plasticity: Theory and Applications. Hoboken, NJ: John Wiley & Sons; 2008. 55. Borja RI. Plasticity Modeling & Computation. Berlin, Germany: Springer-Verlag Berlin Heidelberg; 2013. 56. Bennett KC, Regueiro RA, Borja RI. Finite strain elastoplasticity considering the Eshelby stress for materials undergoing plastic volume change. Int J Plast. 2016;77:214-245. 57. Bennett KC, Borja RI. Hyper-elastoplastic/damage modeling of rock with application to porous limestone. Int J Solids Struct. 2018;143:218-231. 58. Anand L. On H. Hencky's approximate strain-energy function for moderate deformations. J Appl Mech. 1979;46(1):78-82. 59. Gallipoli D, Wheeler S, Karstunen M. Modelling the variation of degree of saturation in a deformable unsaturated soil. Geotechnique. 2003;53(1):105-112. 60. Miller GA, Khoury CN, Muraleetharan KK, Liu C, Kibbey TCG. Effects of soil skeleton deformations on hysteretic soil water characteristic curves: experiments and simulations. Water Resour Res. 2008;44(5). 61. van Genuchten MT. A closed-form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Sci Soc Am. 1980;44(5):892-898. 62. Ciarlet PG. The Finite Element Method for Elliptic Problems. Philadelphia, PA: SIAM; 1978. 63. Ern A, Stephansen AF, Zunino P. A discontinuous Galerkin method with weighted averages for advection–diffusion equations with locally small and anisotropic diffusivity. IMA J Numer Anal. 2009;29(2):235-256. 64. Baumann CE, Oden JT. A discontinuous hp finite element method for convection–diffusion problems. Comput Methods Appl Mech Eng. 1999;175(3):311-341. 65. Rivière B, Wheeler MF, Girault V. A priori error estimates for finite element methods based on discontinuous approximation spaces for elliptic problems. SIAM J Numer Anal. 2001;39(3):902-931. 66. Arnold DN, Brezzi F, Cockburn B, Marini LD. Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J Numer Anal. 2002;39(5):1749-1779. 67. Sun S, Wheeler MF. Discontinuous Galerkin methods for coupled flow and reactive transport problems. Appl Numer Math. 2005;52(2):273-298. 68. Sun S, Wheeler MF. Symmetric and nonsymmetric discontinuous Galerkin methods for reactive transport in porous media. SIAM J Numer Anal. 2005;43(1):195-219. 69. Riviére B. Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations. Philadelphia, PA: SIAM; 2008. 70. Belytschko T, Liu WK, Moran B, Elkhodary KI. Nonlinear Finite Elements for Countinua and Structures. Hoboken, NJ: John Wiley & Sons; 2014. 71. Celia M, Bouloutas ET, Zarba RL. A general mass-conservative numerical solution for the unsaturated flow equation. Water Resour Res. 1990;26(1):1483-1496. 72. Coombs WM. Finite Deformation of Particulate Geomaterials: Frictional and Anisotropic Critical State Elasto-Plasticity [PhD thesis]. Durham, UK: Durham University; 2011. 73. Boffi D, Cavallini N, Gardini F, Gastaldi L. Local mass conservation of Stokes finite elements. J Sci Comput. 2012;52(2):383-400. 74. White JA, Borja RI. Stabilized low-order finite elements for coupled solid-deformation/fluid-diffusion and their application to fault zone transients. Comput Methods Appl Mech Eng. 2008;197(49-50):4353-4366. 75. Choo J, Borja RI. Stabilized mixed finite elements for deformable porous media with double porosity. Comput Methods Appl Mech Eng. 2015;293:131-154. 76. Sun W. A stabilized finite element formulation for monolithic thermo-hydro-mechanical simulations at finite strain. Int J Numer Methods Eng. 2015;103(11):798-839. 77. Borja RI, Alarcon E. A mathematical framework for finite strain elastoplastic consolidation. Part 1: balance laws, variational formulation, and linearization. Comput Methods Appl Mech Eng. 1995;122(1-2):145-171. 78. Song X, Borja RI. Mathematical framework for unsaturated flow in the finite deformation range. Int J Numer Methods Eng. 2014;97(9):658-682. 79. White JA, Borja RI. Block-preconditioned Newton-Krylov solvers for fully coupled flow and geomechanics. Comput Geosci. 2011;15(4):647-659. 80. White JA, Castelletto N, Tchelepi HA. Block-partitioned solvers for coupled poromechanics: a unified framework. Comput Methods Appl Mech Eng. 2016;303:55-74. 90 CHOO

81. Mikelic´ A, Wheeler MF. Convergence of iterative coupling for coupled flow and geomechanics. Comput Geosci. 2013;17(3):455-461. 82. Choo J, White JA, Borja RI. Hydromechanical modeling of unsaturated flow in double porosity media. Int J Geomech. 2016;16(6):D4016002. 83. Choo J. Hydromechanical Modeling Framework for Multiscale Porous Materials [PhD thesis]. Stanford, CA: Stanford University; 2016. 84. Choo J, Sun W. Coupled phase-field and plasticity modeling of geological materials: from brittle fracture to ductile flow. Comput Methods Appl Mech Eng. 2018;330:1-32. 85. Bangerth W, Hartmann R, Kanschat G. deal. II—a general-purpose object-oriented finite element library. ACM Trans Math Softw. 2007;33(4):1-27. 86. Arndt D, Bangerth W, Davydov D, et al. The deal. II library, version 8.5. JNumerMath. 2017;25(3):137-146. 87. Burstedde C, Wilcox LC, Ghattas O. p4est: scalable algorithms for parallel adaptive mesh refinement on forests of octrees. SIAM J Sci Comput 2011;33(3):1103-1133. 88. Heroux MA, Willenbring JM. A new overview of the Trilinos project. Sci Program. 2012;20(2):83-88. 89. Cryer CW. A comparison of the three-dimensional consolidation theories of Biot and Terzaghi. Q J Mech Appl Math. 1963;16(4):401-412. 90. Bandara S, Soga K. Coupling of soil deformation and pore fluid flow using material point method. Comput Geotech. 2015;63:199-214. 91. Gibson RE, Gobert A, Schiffman RL. On Cryer's problem with large displacements. Int J Numer Anal Methods Geomech. 1989;13(3):251-262. 92. Gibson RE, Gobert A, Schiffman RL. On Cryer's problem with large displacements and variable permeability. Géotechnique. 1990;40(4):627-631. 93. Salager S, El Youssoufi M, Saix C. Definition and experimental determination of a soil-water retention surface. Can Geotech J. 2010;47(6):609-622. 94. Lee J, Fox PJ, Lenhart JJ. Investigation of consolidation-induced solute transport. I: effect of consolidation on transport parameters. J Geotech Geoenviron Eng. 2009;135(9):1228-1238. 95. Lee J, Fox PJ. Investigation of consolidation-induced solute transport. II: experimental and numerical results. J Geotech Geoenviron Eng. 2009;135(9):1239-1253. 96. Pu H, Fox PJ. Consolidation-induced solute transport for constant rate of strain. I: model development and simulation results. JGeotech Geoenviron Eng. 2015;141(4):04014127. 97. Bertrand T, Peixinho J, Mukhopadhyay S, MacMinn CW. Dynamics of swelling and drying in a spherical gel. Phys Rev Appl. 2016;6(4):064010.

Howtocitethisarticle: Choo J. Large deformation poromechanics with local mass conservation: An enriched Galerkin finite element framework. Int J Numer Methods Eng. 2018;116:66–90. https://doi.org/10.1002/nme.5915