Staggered grids discretization in three-dimensional Darcy convection

TL;DR

This paper develops a mimetic finite-difference scheme on staggered grids for simulating three-dimensional Darcy convection in porous media, analyzing stability and steady state bifurcations.

Contribution

It introduces a novel discretization method using staggered nonuniform grids and special schemes for nonlinear terms in Darcy convection problems.

Findings

Detected branching of steady states under certain boundary conditions

Analyzed instability scenarios of the fluid's resting state

Developed a scheme applicable to complex porous media convection simulations

Abstract

We consider three-dimensional convection of an incompressible fluid saturated in a parallelepiped with a porous medium. A mimetic finite-difference scheme for the Darcy convection problem in the primitive variables is developed. It consists of staggered nonuniform grids with five types of nodes, differencing and averaging operators on a two-nodes stencil. The nonlinear terms are approximated using special schemes. Two problems with different boundary conditions are considered to study scenarios of instability of the state of rest. Branching off of a continuous family of steady states was detected for the problem with zero heat fluxes on two opposite lateral planes.