A macroelement stabilization for multiphase poromechanics

Abstract

Strong coupling between geomechanical deformation and multiphase fluid flow appears in a variety of geoscience applications. A common discretization strategy for these problems is a continuous Galerkin finite element scheme for the momentum balance equations and a finite volume scheme for the mass balance equations. When applied within a fully-implicit solution strategy, however, this discretization is not intrinsically stable. In the limit of small time steps or low permeabilities, spurious oscillations in the pressure field, i.e. checkerboarding, may be observed. Further, eigenvalues associated with the spurious modes will control the conditioning of the matrices and can dramatically degrade the convergence rate of iterative linear solvers. Here, we propose a stabilization technique in which the balance of mass equations are supplemented with stabilizing flux terms on a macroelement basis. The additional stabilization terms are dependent on a stabilization parameter. We identify an optimal value for this parameter using an analysis of the eigenvalue distribution of the macroelement Schur complement matrix. The resulting method is simple to implement and preserves the underlying sparsity pattern of the original discretization. Another appealing feature of the method is that mass is exactly conserved on macroelements, despite the addition of artificial fluxes. The efficacy of the proposed technique is demonstrated with several numerical examples.

Efficient solvers for hybridized three-field mixed finite element coupled poromechanics

“…The stabilization is based on the macro-element theory and the local pressure jump approach originally introduced for Stokes problems [10] and more recently for coupled multiphase flow applications [9]. It has a number of useful features:…”

“…where A M uu , A M up , A M pu are the blocks introduced in ( 8) and ( 12) restricted to M, with homogeneous Dirichlet conditions for the displacements on Γ ∂ M , and A M stab is the matrix form of J in M. The key idea is to set β M such that the non-zero extreme eigenvalues of B M p are not affected by the introduction of the stabilization contribution A M stab . Following the analysis in [25] for 2D problems, we can set β M = (b/2) 2 /(2G + λ), where G and λ are the Lamé parameters on the macro-element M. For 3D problems, a recent analysis has been carried out for a mixed finite element-finite volume formulation of multiphase poromechanis [9]. Extending those results, we can set…”

“…This can result in spurious modes for the pressure, with non physical oscillations of the discrete solution. Different stabilization strategies have been proposed, e.g., [6][7][8][9], but they can negatively impact the algebraic structure of the block discrete problem, with a possible degradation of the computational efficiency. In this work, we address this deficiency by proposing a local pressure jump stabilization technique based on a macro-element approach for the mixed and mixed hybrid formulations, following [9,10].…”

“…Different stabilization strategies have been proposed, e.g., [6][7][8][9], but they can negatively impact the algebraic structure of the block discrete problem, with a possible degradation of the computational efficiency. In this work, we address this deficiency by proposing a local pressure jump stabilization technique based on a macro-element approach for the mixed and mixed hybrid formulations, following [9,10]. Then, we concentrate on the efficient solution of the non-symmetric algebraic systems obtained by the application of the stabilized formulation.…”