Face-Centered and Volume-Centered Discrete Analogs of the Exterior Differential Equations Governing Porous Medium Flow I: Theory

Abstract

This paper presents a 'physics-oriented' approach to approximate the continuum equations governing porous media flow by discrete analogs. To that end, the continuity equation and Darcy's law are reformulated using exterior differential forms. This way the derivation of a system of algebraic equations (the discrete analog) on a finite-volume mesh can be accomplished by simple and elegant 'translation rules.' In the discrete analog the information about the conductivities of the porous medium and the metric of the mesh are represented in one matrix: the discrete dual. The discrete dual of the blockcentered finite difference method is presented first. Since this method has limited applicability with respect to anisotropy and non-rectangular grid blocks, the finite element dual is introduced as an alternative. Application of a domain decomposition technique yields the face-centered finite element method. Since calculations based on pressures in volume centers are sometimes preferable, a volume-centered approximation of the face-centered approximation is presented too.

Comparison of Two Mathematical Models for 3D Groundwater Flow: Block-Centered Heads and Edge-Based Stream Functions

“…The discrete analog representation has originally been developed in the context of electrical engineering (Bossavit 1998) and is not commonly used in groundwater hydrology; it can be found only in some research articles (e.g., Zijl and Nawalany 2004;Zijl , 2005Zijl , 2007. Hence, the derivation of these fundamental equations is not treated here; only the formulation of the discrete analogs-the block-centered finite difference method and the edge-based stream function method-are briefly presented in the following subsections.…”

“…Hence, the derivation of these fundamental equations is not treated here; only the formulation of the discrete analogs-the block-centered finite difference method and the edge-based stream function method-are briefly presented in the following subsections. Further descriptions can be found in Zijl (2005) and Zijl and Nawalany (2004). …”

“…and (T/L 2 ) may be considered as the discrete analogs of h, ϕ, q, k,ṡ 0 , and γ , respectively (Zijl 2005). Incidence matrix D (−) is a discrete operator that relates the volumes and the faces of grids.…”

“…As demonstrated in the previous section, the and methods are algebraically equivalent and they are finite difference type of methods although they are different from an algorithmic point of view. Apart from a slightly different expression for resistance matrix , they are algebraically equivalent with the mixed hybrid finite element method (Kaasschieter and Huijben 1992;Zijl 2005). However, these three methods are quite different from an algorithmic point of view.…”

“…Discrete analogs are among the computational techniques to get numerical solutions. They are systems of algebraic equations describing flow on a discrete grid in such a way that their solution converges to the continuum solution in the limit of infinitely small cell dimensions (Zijl 2005). Discrete analogs do not only form the basis of numerical simulations, but they also offer conceptual value, especially in the context of direct inverse modeling and the related subjects of regularization and parameter upscaling (Zijl , 2005.…”