The Mechanical Behavior of a Poroelastic Medium Saturated with a Newtonian Fluid

In this paper we systematically derive, via the theory of homogenization, the macroscopic equations for the mechanical behavior of a deformable porous medium saturated with a Newtonian fluid. The derivation is first based on the equations of linear elasticity in the solid, the Stokes equations for the fluid, and suitable conditions at the fluid-solid interface. A detailed comparison between the equations derived here and those by Biot is given. The homogenization approach determines the form of the macroscopic constitutive relationships between variables and shows how to compute the coefficients in these relationships. The derivation is then extended to the nonlinear Navier-Stokes equations for the fluid in the deformable porous medium for the first time. A generalized Forchheimer law is obtained to take into account the nonlinear inertial effects on the flow of the Newtonian fluid through such a medium. Both quasi-static and transient flows are considered in this paper. The properties of the macroscopic coefficients are studied. The computational results show that the macroscopic equations predict well the behavior of the microscopic equations in certain reasonable test cases.