Determination of Brinkman Model Parameters Using Stokes Flow Model

Abstract

© 2019, Springer Nature B.V. The two key parameters of the Brinkman’s model for fluid flow in porous media—permeability and effective viscosity—are determined theoretically. The analytical solution of a 1D problem for the Brinkman equation for the fluid flow in a porous medium between two solid walls is compared with results of the numerical Stokes fully resolved flow model in a two-dimensional periodic cells containing solid inclusions. The boundary value problem for the Stokes flow is solved using the boundary element method. The dependencies of the permeability and effective viscosity on porosity for various types of configurations and shapes of solid inclusions are numerically studied. The product of permeability and effective viscosity is introduced as a new parameter. It is shown that this parameter is almost constant for a wide range of values of porosity excluding the small interval close to the unity value. The analytical estimation confirms the numerically obtained values of the parameter. The obtained dependencies are verified by the solution of the 2D Brinkman flow problem. It is shown that the permeability and effective viscosity determined give good agreement between the pressure fields and flow streamlines of the exact solution of the two-dimensional Brinkman equation and the numerical solution of the Stokes equations. The approximations for the dependencies of permeability and effective viscosity on the porosity are constructed on the basis of the numerical data obtained. The applicability of the Brinkman and Darcy models is discussed.