Robust Approximation of Generalized Biot-Brinkman Problems (original) (raw)
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Numerical Linear Algebra With Applications, 2019
The parameters in the governing system of partial differential equations of multicompartmental poroelastic models typically vary over several orders of magnitude making its stable discretization and efficient solution a challenging task. In this paper, inspired by the approach recently presented by Hong and Kraus [Parameter-robust stability of classical three-field formulation of Biot's consolidation model, ETNA (to appear)] for the Biot model, we prove the uniform stability, and design stable disretizations and parameter-robust preconditioners for flux-based formulations of multiple-network poroelastic systems. Novel parameter-matrix-dependent norms that provide the key for establishing uniform inf-sup stability of the continuous problem are introduced. As a result, the stability estimates presented here are uniform not only with respect to the Lamé parameter λ, but also with respect to all the other model parameters such as permeability coefficients Ki, storage coefficients cp i , network transfer coefficients βij , i, j = 1, • • • , n, the scale of the networks n and the time step size τ. Moreover, strongly mass conservative discretizations that meet the required conditions for parameterrobust stability are suggested and corresponding optimal error estimates proved. The transfer of the canonical (norm-equivalent) operator preconditioners from the continuous to the discrete level lays the foundation for optimal and fully robust iterative solution methods. The theoretical results are confirmed in numerical experiments that are motivated by practical applications.
SIAM Journal on Scientific Computing, 2021
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 properties, is demonstrated through several numerical experiments.
Computational Geosciences, 2007
In this paper, we formulate a finite element procedure for approximating the coupled fluid and mechanics in Biot's consolidation model of poroelasticity. Here, we approximate the pressure by a mixed finite element method and the displacements by a Galerkin method. Theoretical convergence error estimates are derived in a continuous in-time setting for a strictly positive constrained specific storage coefficient. Of particular interest is the case when the lowest-order Raviart-Thomas approximating space or cell-centered finite differences are used in the mixed formulation, and continuous piecewise linear approximations are used for displacements. This approach appears to be the one most frequently applied to existing reservoir engineering simulators.
Computer Methods in Applied Mechanics and Engineering, 2015
In this paper we introduce and analyze an augmented mixed finite element method for the twodimensional nonlinear Brinkman model of porous media flow with mixed boundary conditions. More precisely, we extend a previous approach for the respective linear model to the present nonlinear case, and employ a dual-mixed formulation in which the main unknowns are given by the gradient of the velocity and the pseudostress. In this way, and similarly as before, the original velocity and pressure unknowns are easily recovered through a simple postprocessing. In addition, since the Neumann boundary condition becomes essential, we impose it in a weak sense, which yields the introduction of the trace of the fluid velocity over the Neumann boundary as the associated Lagrange multiplier. We apply known results from nonlinear functional analysis to prove that the corresponding continuous and discrete schemes are well-posed. In particular, a feasible choice of finite element subspaces is given by Raviart-Thomas elements of order k ≥ 0 for the pseudostress, piecewise polynomials of degree ≤ k for the gradient, and continuous piecewise polynomials of degree ≤ k + 1 for the Lagrange multiplier. We also derive a reliable and efficient residual-based a posteriori error estimator for this problem. Finally, several numerical results illustrating the performance and the robustness of the method, confirming the theoretical properties of the estimator, and showing the behaviour of the associated adaptive algorithm, are provided.
Computer Methods in Applied Mechanics and Engineering, 2018
This paper is concerned with the analysis of coupled mixed finite element methods applied to the Biot's consolidation model. We consider two mixed formulations that use the stress tensor and Darcy velocity as primary variables as well as the displacement and pressure. The first formulation is with a symmetric stress tensor while the other enforces the symmetry of the stress weakly through the introduction of a Lagrange multiplier. The well-posedness of the two formulations is shown through Galerkin's method and suitable a priori estimates. The two formulations are then discretized with the backward Euler scheme in time and with two mixed finite elements in space. We present next a general and unified a posteriori error analysis which is applicable for any flux-and stress-conforming discretization. Our estimates are based on H 1 (Ω)-conforming reconstruction of the pressure and a suitable H 1 (Ω) d-conforming reconstruction of the displacement; both are continuous and piecewise affine in time. These reconstructions are used to infer a guaranteed and fully computable upper bound on the energy-type error measuring the differences between the exact and the approximate pressure and displacement. The error components resulting from the spatial and the temporal discretization are distinguished. They are then used to design an adaptive space-time algorithm. Numerical experiments illustrate the efficiency of our estimates and the performance of the adaptive algorithm.
A Hybrid High-Order Method for Multiple-Network Poroelasticity
2020
We develop Hybrid High-Order methods for multiple-network poroelasticity, modelling seepage through deformable fissured porous media. The proposed methods are designed to support general polygonal and polyhedral elements. This is a crucial feature in geological modelling, where the need for general elements arises, e.g., due to the presence of fracture and faults, to the onset of degenerate elements to account for compaction or erosion, or when nonconforming mesh adaptation is performed. We use as a starting point a mixed weak formulation where an additional total pressure variable is added, that ensures the fulfilment of a discrete inf-sup condition. A complete theoretical analysis is performed, and the theoretical results are demonstrated on a complete panel of numerical tests.
A stabilized finite element method for finite-strain three-field poroelasticity
Computational Mechanics, 2017
We construct a stabilized finite-element method to compute flow and finite-strain deformations in an incompressible poroelastic medium. We employ a three-field mixed formulation to calculate displacement, fluid flux and pressure directly and introduce a Lagrange multiplier to enforce flux boundary conditions. We use a low order approximation, namely, continuous piecewise-linear approximation for the displacements and fluid flux, and piecewise-constant approximation for the pressure. This results in a simple matrix structure with low bandwidth. The method is stable in both the limiting cases of small and large permeability. Moreover, the discontinuous pressure space enables efficient approximation of steep gradients such as those occurring due to rapidly changing material coefficients or boundary conditions, both of which are commonly seen in physical and biological applications.
A fully coupled scheme using virtual element method and finite volume for poroelasticity
Computational Geosciences, 2019
In this paper, we design and study a fully coupled numerical scheme for the poroelasticity problem modelled through Biot's equations. The classical way to numerically solve this system is to use a finite element method for the mechanical equilibrium equation and a finite volume method for the fluid mass conservation equation. However, to capture specific properties of underground media such as heterogeneities, discontinuities and faults, meshing procedures commonly lead to badly shaped cells for finite element based modelling. Consequently, we investigate the use of the recent virtual element method which appears as a potential discretization method for the mechanical part and could therefore allow the use of a unique mesh for the both mechanical and fluid flow modelling. Starting from a first insight into virtual element method applied to the elastic problem in the context of geomechanical simulations, we apply in addition a finite volume method to take care of the fluid conservation equation. We focus on the first order virtual element method and the two point flux approximation for the finite volume part. A mathematical analysis of this original coupled scheme is provided, including existence and uniqueness results and a priori estimates. The method is then illustrated by some computations on two or three dimensional grids inspired by realistic application cases.
In this project, the evolution and fundamentals of Theory of Porous Media are studied. The field equations governing the dynamic response of a fluid-saturated elastic porous medium with intrinsically incompressible solid and fluid constituents are derived and analyzed. The results of the governing field equations are treated numerically using Standard Galerkin procedure and the Finite element method. This class of problems comes under volumetrically coupled problems due to the solid-fluid momentum interaction that involves a coupling between the momentum equations and the incompressibility constraint. In this concern, two numerical examples are considered (one-and two-dimensional wave propagation example). These examples are tested numerically using multi-field formulations feasible in porous media dynamics. The tests are carried out under different combination of approximation order of the individual unknowns with several refinement levels of mesh and different time-step sizes. Here the degree of coupling is controlled by the permeability parameter, so each test cases are carried out with both large and small permeability values. In the first numerical example, firstly the analytical solution of the fluid-saturated poroelastic medium is derived using Laplace transformation and the solutions of the unknown variables are found using MAPLE programming. Then, the solutions of different formulations,which are analyzed numerically, are compared with the analytical solution for solid displacement and pore-fluid pressure through error calculation. These error values of each test cases are plotted graphically against different time-step sizes and different mesh sizes. In the second numerical example, the propagation of solid displacement and pore-fluid pressure waves inside a porous medium is studied and the results are plotted graphically. The in-plane motion of the displacements and the pressure oscillations are compared between different formulations and polynomial degrees combinations. In conclusion, the best formulation with appropriate test cases that suits for solving such porous media problems is recommended and the wave propagation inside the porous media is discussed shortly.