In this chapter a general model for the two-phase fluid flow in porous media is presented, together with its simplified form, known as the Richards equation, which is applicable (under specific assumptions) to describe water flow in the vadose zone. In each case the governing equations are formulated at the Darcy scale, using the capillary pressure–saturation relationship and an empirical exten...

We develop a finite element discretization and multigrid solver for a Darcy-Stokes system of three dimensional vuggy porous media, i.e., porous media with cavities. The finite element method uses low order mixed finite elements in the Darcy and Stokes domains, and special transition elements near the Darcy-Stokes interface to allow for tangential discontinuities implied by the Beavers-Joseph bo...

We study the validity of the quasistatic approximation in the fully evolutionary Stokes-Darcy problem for the coupling of groundwater and surface water flows, as well as the dependence of the problem on the specific storage parameter. In the coupled equations that describe the groundwater and surface water flows for an incompressible fluid, the specific storage, S0, represents the volume of wat...

This paper presents a two-grid method for solving systems of partial differential equations modelling free flow coupled with porous media flow. This work considers both the coupled Stokes and Darcy as well as the coupled Navier-Stokes and Darcy problems. The numerical schemes proposed are based on combinations of the continuous finite element method and the discontinuous Galerkin method. Numeri...

A general form of the double porosity model of single phase flow in a naturally fractured reservoir is derived from homogenization theory. The microscopic model consists of the usual equations describing Darcy flow in a reservoir, except that the porosity and permeability coefficients are highly discontinuous. Over the matrix domain, the coefficients are scaled by a parameter representing the s...

The effects of double diffusive natural convection heat and mass transfer along a vertical plate embedded in a power-law fluid saturated Darcy porous medium in the presence of Soret and Dufour effects are studied. The governing partial differential equations are transformed into ordinary differential equations using similarity transformations and then solved numerically. A parametric study of t...

A problem of flow and heat transfer in a non-Newtonian Maxwell liquid film over an unsteady stretching sheet embedded in a porous medium in the presence of a thermal radiation is investigated. The unsteady boundary layer equations describing the problem are transformed to a system of non-linear ordinary differential equations which is solved numerically using the shooting method. The effects of...

steady state natural convection of al2o3-water nanofluid inside a square cavity filled with a porous medium is investigated numerically. the temperatures of the two side walls of the cavity are maintained at th and tc, where tc has been considered as the reference condition. the top and the bottom horizontal walls have been considered to be insulated i.e., non-conducting and impermeable to mass...

The double bracket dissipation approach is applied to the Vlasov kinetic equation. The Vlasov equation is then transformed by a Poisson map to moment dynamics, leading to a nonlocal form of Darcy’s law. Next, kinetic equations for particles with anisotropic interaction are considered and also cast into the double bracket dissipation form. The moment dynamics for these double bracket kinetic equ...