A Nonconforming Finite Element Method for the Biot's Consolidation Model in Poroelasticity
Tufts University（塔夫茨大学）；Universidad de Zaragoza（萨拉戈萨大学）；Penn State（宾夕法尼亚州立大学）；Institute for Mathematics and Informatics, Bulgarian Academy of Sciences（保加利亚科学院数学与信息学研究所）
Xiaozhe Hu, Carmen Rodrigo, Francisco J. Gaspar, Ludmil T. Zikatanov
A stable finite element scheme that avoids pressure oscillations for a three-field Biot's model in poroelasticity is considered. The involved variables are the displacements, fluid flux (Darcy velocity), and the pore pressure, and they are discretized by using the lowest possible approximation order: Crouzeix-Raviart finite elements for the displacements, lowest order Raviart-Thomas-Nedelec elements for the Darcy velocity, and piecewise constant approximation for the pressure. Mass lumping technique is introduced for the Raviart-Thomas-Nedelec elements in order to eliminate the Darcy velocity and, therefore, reduce the computational cost. We show convergence of the discrete scheme which is implicit in time and use these types of elements in space with and without mass lumping. Finally, numerical experiments illustrate the convergence of the method and show its effectiveness to avoid spurious pressure oscillations when mass lumping for the Raviart-Thomas-Nedelec elements is used.
In this manuscript we focus on the question: what is the correct notion of Stokes-Biot stability? Stokes-Biot stable discretizations have been introduced, independently by several authors, as a means of discretizing Biot's equations of poroelasticity; such schemes retain their stability and convergence properties, with respect to appropriately defined norms, in the context of a vanishing storage coefficient and a vanishing hydraulic conductivity. The basic premise of a Stokes-Biot stable discretization is: one part Stokes stability and one part mixed Darcy stability. In this manuscript we rema…
Biot's equations of poroelasticity contain a parabolic system for the evolution of the pressure, which is coupled with a quasi-stationary equation for the stress tensor. Thus, it is natural to extend the existing work on isogeometric space-time methods to this more advanced framework of a partial differential-algebraic equation (PDAE). A space-time approach based on finite elements has already been introduced. But we present a new weak formulation in space and time that is appropriate for an isogeometric discretization and analyze the convergence properties. Our approach is based on a single v…
We develop non-overlapping domain decomposition methods for the Biot system of poroelasticity in a mixed form. The solid deformation is modeled with a mixed three-field formulation with weak stress symmetry. The fluid flow is modeled with a mixed Darcy formulation. We introduce displacement and pressure Lagrange multipliers on the subdomain interfaces to impose weakly continuity of normal stress and normal velocity, respectively. The global problem is reduced to an interface problem for the Lagrange multipliers, which is solved by a Krylov space iterative method. We study both monolithic and s…
In this paper we study the linear systems arising from discretized poroelasticity problems. We formulate one block preconditioner for the two-filed Biot model and several preconditioners for the classical three-filed Biot model under the unified relationship framework between well-posedness and preconditioners. By the unified theory, we show all the considered preconditioners are uniformly optimal with respect to material and discretization parameters. Numerical tests demonstrate the robustness of these preconditioners.
In this paper we propose and analyze a staggered discontinuous Galerkin method for a five-field formulation of the Biot system of poroelasticity on general polygonal meshes. Elasticity is equipped with stress-displacement-rotation formulation with weak stress symmetry for arbitrary polynomial orders, which extends the piecewise constant approximation developed in (L. Zhao and E.-J. Park, SIAM J. Sci. Comput. 42 (2020), A2158-A2181). The proposed method is locking free and can handle highly distorted grids possibly including hanging nodes, which is desirable for practical applications. We prove…
We present a mixed finite element method for a five-field formulation of the Biot system of poroelasticity that reduces to a cell-centered pressure-displacement system on simplicial and quadrilateral grids. A mixed stress-displacement-rotation formulation for elasticity with weak stress symmetry is coupled with a mixed velocity-pressure Darcy formulation. The spatial discretization is based on combining the multipoint stress mixed finite element (MSMFE) method for elasticity and the multipoint flux mixed finite element (MFMFE) method for Darcy flow. It uses the lowest order Brezzi-Douglas-Mari…
Water injection in the aquifer induces deformations in the soil. These mechanical deformations give rise to a change in porosity and permeability, which results in non-linearity of the mathematical problem. Assuming that the deformations are very small, the model provided by Biot's theory of linear poroelasticity is used to determine the local displacement of the skeleton of a porous medium, as well as the fluid flow through the pores. In this continuum scale model, the Kozeny-Carman equation is commonly used to determine the permeability of the porous medium from the porosity. The Kozeny-Carm…
This work proposes a mixed finite element method for the Biot poroelasticity equations that employs the lowest-order Raviart-Thomas finite element space for the solid displacement and piecewise constants for the fluid pressure. The method is based on the formulation of linearized elasticity as a weighted vector Laplace problem. By introducing the solid rotation and fluid flux as auxiliary variables, we form a four-field formulation of the Biot system, which is discretized using conforming mixed finite element spaces. The auxiliary variables are subsequently removed from the system in a local h…
Linear poroelasticity models have a number of important applications in biology and geophysics. In particular, Biot's consolidation model is a well-known model that describes the coupled interaction between the linear response of a porous elastic medium and a diffusive fluid flow within it, assuming small deformations. Although deterministic linear poroelasticity models and finite element methods for solving them numerically have been well studied, there is little work to date on robust algorithms for solving poroelasticity models with uncertain inputs and for performing uncertainty quantifica…
This work is concerned with the iterative solution of systems of quasi-static multiple-network poroelasticity (MPET) equations describing flow in elastic porous media that is permeated by single or multiple fluid networks. Here, the focus is on a three-field formulation of the problem in which the displacement field of the elastic matrix and, additionally, one velocity field and one pressure field for each of the $n \ge 1$ fluid networks are the unknown physical quantities. Generalizing Biot's model of consolidation, which is obtained for $n=1$, the MPET equations for $n\ge1$ exhibit a double …
The classical Biot's theory provides the foundation of a fully dynamic poroelasticity model describing the propagation of elastic waves in fluid-saturated media. Multiple network poroelastic theory (MPET) takes into account that the elastic matrix (solid) can be permeated by one or several ($n\ge1$) superimposed interacting single fluid networks of possibly different characteristics; hence the single network (classical Biot) model can be considered as a special case of the MPET model. We analyze the stability properties of the time-discrete systems arising from second-order implicit time stepp…
In this work, we introduce a novel abstract framework for the stability and convergence analysis of fully coupled discretisations of the poroelasticity problem and apply it to the analysis of Hybrid High-Order (HHO) schemes. A relevant feature of the proposed framework is that it rests on mild time regularity assumptions that can be derived from an appropriate weak formulation of the continuous problem. To the best of our knowledge, these regularity results for the Biot problem are new. A novel family of HHO discretisation schemes is proposed and analysed, and their performance numerically eva…
In this work, we consider a hybrid mixed finite element method for Biot's model. The hybrid P1-RT0-P0 discretization of the displacement-pressure-Darcy's velocity system of Biot's model presented in \cite{C. Niu} is not uniformly stable with respect to the physical parameters, resulting in some issues in numerical simulations. To alleviate such problems, following \cite{V. Girault}, we stabilize the hybrid scheme with face bubble functions and show the well-posedness with respect to physical and discretization parameters, which provide optimal error estimates of the stabilized method. We intro…
We develop robust solvers for a class of perturbed saddle-point problems arising in the study of a second-order elliptic equation in mixed form (in terms of flux and potential), and of the four-field formulation of Biot's consolidation problem for linear poroelasticity (using displacement, filtration flux, total pressure and fluid pressure). The stability of the continuous variational mixed problems, which hinges upon using adequately weighted spaces, is addressed in detail; and the efficacy of the proposed preconditioners, as well as their robustness with respect to relevant material properti…
We present an embedded-hybridizable discontinuous Galerkin finite element method for the total pressure formulation of the quasi-static poroelasticity model. Although the displacement and the Darcy velocity are approximated by discontinuous piece-wise polynomials, $H(\text{div})$-conformity of these unknowns is enforced by Lagrange multipliers. The semi-discrete problem is shown to be stable and the fully discrete problem is shown to be well-posed. Additionally, space-time a priori error estimates are derived, and confirmed by numerical examples, that show that the proposed discretization is f…
A posteriori error estimates for approximate solutions of Barenblatt-Biot poroelastic model
University of Bergen（卑尔根大学）；Bergen University College（卑尔根理工学院）；V.A. Steklov Institute of Mathematics in St.-Petersburg（圣彼得堡V.A.斯捷克洛夫数学研究所）；University of Iceland（冰岛大学）
In this work, we consider the Biot problem with uncertain poroelastic coefficients. The uncertainty is modelled using a finite set of parameters with prescribed probability distribution. We present the variational formulation of the stochastic partial differential system and establish its well-posedness. We then discuss the approximation of the parameter-dependent problem by non-intrusive techniques based on Polynomial Chaos decompositions. We specifically focus on sparse spectral projection methods, which essentially amount to performing an ensemble of deterministic model simulations to estim…
Weakly imposed symmetry and robust preconditioners for Biot's consolidation model
University of Oslo（奥斯陆大学）；The Institute for Computational and Engineering Sciences, University of Texas Austin（德克萨斯大学奥斯汀分校计算与工程科学研究所）；Simula Research Laboratory（西幕拉研究实验室）
We consider the systematic numerical approximation of Biot's quasistatic model for the consolidation of a poroelastic medium. Various discretization schemes have been analysed for this problem and inf-sup stable finite elements have been found suitable to avoid spurios pressure oscillations in the initial phase of the evolution. In this paper, we first clarify the role of the inf-sup condition for the well-posedness of the continuous problem and discuss the choice of appropriate initial conditions. We then develop an abstract error analysis that allows us to analyse some approximation schemes …
The Biot problem of poroelasticity is extended by Signorini contact conditions. The resulting Biot contact problem is formulated and analyzed as a two field variational inequality problem of a perturbed saddle point structure. We present an a priori error analysis for a general as well as for a $hp$-FE discretization including convergence and guaranteed convergence rates for the latter. Moreover, we derive a family of reliable and efficient residual based a posteriori error estimators, and elaborate how a simple and efficient primal-dual active set solver can be applied to solve the discrete G…
Poroelasticity theory models the dynamics of porous, fluid-saturated media. It was pioneered by Maurice Biot in the 1930s through 1960s, and has applications in several fields, including geophysics and modeling of in vivo bone. A wide variety of methods have been used to model poroelasticity, including finite difference, finite element, pseudospectral, and discontinuous Galerkin methods. In this work we use a Cartesian-grid high-resolution finite volume method to numerically solve Biot's equations in the time domain for orthotropic materials, with the stiff relaxation source term in the equati…
In this paper, we consider the numerical solution of some nonlinear poroelasticity problems that are of Biot type and develop a general algorithm for solving nonlinear coupled systems. We discuss the difficulties associated with flow and mechanics in heterogenous media with nonlinear coupling. The central issue being how to handle the nonlinearities and the multiscale scale nature of the media. To compute an efficient numerical solution we develop and implement a Generalized Multiscale Finite Element Method (GMsFEM) that solves nonlinear problems on a coarse grid by constructing local multisca…
In this paper, we consider the numerical solution of poroelasticity problems that are of Biot type and develop a general algorithm for solving coupled systems. We discuss the challenges associated with mechanics and flow problems in heterogeneous media. The two primary issues being the multiscale nature of the media and the solutions of the fluid and mechanics variables traditionally developed with separate grids and methods. For the numerical solution we develop and implement a Generalized Multiscale Finite Element Method (GMsFEM) that solves problem on a coarse grid by constructing local mul…
We consider the quasi-static Biot's consolidation model in a three-field formulation with the three unknown physical quantities of interest being the displacement $\boldsymbol{u}$ of the solid matrix, the seepage velocity $\boldsymbol{v}$ of the fluid and the pore pressure $p$. As conservation of fluid mass is a leading physical principle in poromechanics, we preserve this property using an $\boldsymbol{H}(\operatorname{div})$-conforming ansatz for $\boldsymbol{u}$ and $\boldsymbol{v}$ together with an appropriate pressure space. This results in Stokes and Darcy stability and exact, that is, p…
We develop a mixed finite element method for the coupled problem arising in the interaction between a free fluid governed by the Stokes equations and flow in deformable porous medium modeled by the Biot system of poroelasticity. Mass conservation, balance of stress, and the Beavers--Joseph--Saffman condition are imposed on the interface. We consider a fully mixed Biot formulation based on a weakly symmetric stress-displacement-rotation elasticity system and Darcy velocity-pressure flow formulation. A velocity-pressure formulation is used for the Stokes equations. The interface conditions are i…
