Block-Centered Finite Difference Methods For General Darcy-Forchheimer Problems

When the fluid in porous media flows under the condition of low velocity and small porosity and permeability, the linear relationship is valid between the Darcy velocity and the gradient of pressure, that is Darcy’s law. In some cases, when the velocity is very high and the flow becomes the turbulent flow, a nonlinear relationship between the Darcy velocity and the gradient of pressure is developed, described by using the general Darcy-Forchheimer equation. In this paper, we consider the general Darcy-Forchheimer problems with Neumann boundary conditions in rectangular domains and present a block-centered finite difference method to solve them.The general Darcy-Forchheimer model with Neumann boundary condition for the one-dimensional problem with Ω=(0,1) can be written as: with the compatibility conditionThe general Darcy-Forchheimer model with Neumann boundary condition for the two-dimensional problem with Ω=(0,1)×(0,1) can be written as: with the compatibility conditionIn order to receive the block-centered finite difference scheme, we esti mate the equations with Taylor expansion. We also show the existence an uniqueness of the solution and the equivalence between block-centered finit difference methods and mixed finite element methods. If the true solution the general Darcy-Forchheimer model is sufficiently smooth, then we demon strate that the convergence of the approximate solution for both velocity an pressure in the discrete L2norm is second-order accurate. Lastly, we presen some numerical experiments using the block-centered finite difference methoc The numerical results show that the convergence rates are second-order for a nonuniform grids, which are in agreement with theoretical analysis.