Nonlinear and non-local analytical solution for Darcy–Forchheimer flow through a deformable porous inclusion within a semi-infinite elastic medium

Abstract

Permeability is a nonlinear and non-local function of the intimate coupling between pore fluid flow and solid deformation in porous media. A class of related problems involves the effect of fluid injection or withdrawal on the transport properties of geomaterials. This paper presents an analytical solution for the nonlinear and non-local problem of fluid flow through a disk-shaped porous elastic inclusion below the free surface of an elastic half-space. The solution accounts for the following nonlinear mechanisms: (i) variations in the permeability coefficient due to the flow-induced deformation of the inclusion; (ii) the inertial losses of the pore fluid flow. The former and latter mechanisms are formulated using the Green's function for a dilatation centre in an elastic half-space and the Darcy–Forchheimer equation for fluid flow through porous media, respectively. An analytical perturbation solution to the considered problem is developed and validated against the numerical finite element solution to the same problem. The described nonlinear mechanisms are represented by two dimensionless parameter groups. The extreme values of these dimensionless groups govern the solution asymptotic behaviours mimicking the special-case solutions in which either mechanism is forced to vanish. The applied aspects of the solution are demonstrated through the wellbore performance index parameter that quantifies the subsurface rock ability to deliver the pore fluid toward or away from a wellbore in a porous reservoir. Unlike the linear models, the presented nonlinear solution captures the observed dependence of the performance index on the wellbore flow rate.