Stabilized velocity post-processings for Darcy flow in heterogeneous porous media

TL;DR

The paper tackles accurate velocity approximation for Darcy flow in heterogeneous porous media with conductivity discontinuities at interfaces. It develops stabilized mixed formulations (GLS and adjoint HVM) and unifies global and local post-processing approaches to recover high-quality velocity fields from potential approximations, while exactly or effectively enforcing interface constraints. The methods accommodate equal-order $C^0$ finite elements and both continuous and discontinuous velocity representations, achieving optimal convergence on homogeneous and anisotropic heterogeneous problems. Numerical experiments across layered media demonstrate that incorporating interface conditions yields accurate fluxes and velocity distributions, with convergence orders matching those predicted for smooth problems. The work advances robust, implementable techniques for transport simulations in complex porous media where material discontinuities are present.

Abstract

Stable and accurate finite element methods are presented for Darcy flow in heterogeneous porous media with an interface of discontinuity of the hydraulic conductivity tensor. Accurate velocity fields are computed through global or local post-processing formulations that use previous approximations of the hydraulic potential. Stability is provided by combining Galerkin and Least Squares (GLS) residuals of the governing equations with an additional stabilization on the interface that incorporates the discontinuity on the tangential component of the velocity field in a strong sense. Numerical analysis is outlined and numerical results are presented to illustrate the good performance of the proposed methods. Convergence studies for a heterogeneous and anisotropic porous medium confirm the same orders of convergence predicted for homogeneous problem with smooth solutions, for both global and local post-processings.

Figures (17)

• Figure 1: Domain of the model problem.
• Figure 2: $C^0$ class elements and discontinuous ($C^{-1}$) elements.
• Figure 3: Stable macroelement configurations composed of bilinear quadrilaterals ($\bullet$, nodal points; +, superconvergent points): (a) two homogeneous macroelements composed by an arrange of $2\times1$ bilinear quadrilaterals; (b) a macroelement of $2\times2$ elements with interface of discontinuity. In this last case, the method described in Section \ref{['sec:ppge']} is adopted inside the macroelement
• Figure 4: Domain.
• Figure 5: Approximated potential.