Effective equations for fluid-structure interaction with applications to poroelasticity

Modeling of fluid-solid interactions in porous media is a challenging and computationally demanding task. Due to the multiscale nature of the problem, simulating the flow and mechanics by direct numerical simulation is often not feasible and an effective model is preferred. In this work, we formally derive an effective model for Fluid-Structure Interaction (FSI). In earlier work, assuming infinitesimal pore-scale deformations, an effective poroelastic model of Biot was derived. We extend this model to a nonlinear Biot model that includes pore-scale deformation into the effective description. The main challenge is the difference in coordinate systems of the fluid and solid equations. This is circumvented by utilizing the Arbitrary Lagrange-Eulerian (ALE) formulation of the FSI equations, giving a unified frame in which to apply two-scale asymptotic techniques. In the derived nonlinear Biot model, the local cell problem are coupled to the macroscopic equations via the effective coefficients. These coefficients may be viewed as tabular functions of the macroscopic parameters. After simplifying this dependence, we assume the coefficients depend on macroscopic pressure only. Using a three dimensional pore geometry we calculate, as a proof-of-concept example, the effective permeability and Biot coefficients for various values or pressure. We observe that, for this geometry, a stronger pressure dependence on flow quantities than on mechanically based effective quantities.