Large Deformation Poromechanics with Local Mass Conservation: an Enriched Galerkin Finite Element Framework

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 continuous Galerkin counterpart. Two equivalent formulations are developed using total Lagrangian and updated Lagrangian approaches. The local mass conservation 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 critical 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 provides 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 deformation 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 permeability 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 undergoing 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 provide 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 poromechanics 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 poromechanical 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 conservation 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

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