A Priori Error Estimates for the Numerical Solution of a Coupled Geomechanics and Reservoir Flow Model with Stress-dependent Permeability

Abstract: In this paper we consider the numerical solution of a coupled geomechanics and a stress-sensitive porous media reservoir flow model. We combine mixed finite elements for Darcy flow and Galerkin finite elements for elasticity. This work focuses on deriving convergence results for the numerical solution of this nonlinear partial differential system. We establish convergence with respect to the L-norm for the pressure and for the average fluid velocity and with respect to the H-norm for the deformation. Estimates respect to the L-norm for mean stress, which is of special importance since it is used in the computation of permeability for poroelasticity, can be derived using the estimates in the H-norm for the deformation. We start by deriving error estimates in a continuous-in-time setting. A cut-off operator is introduced in the numerical scheme in order to derive convergence. The spatial grids for the discrete approximations of the pressure and deformation do not need be the same. Theoretical convergence error estimates in a discrete-in-time setting are also derived in the scope of this investigation. A numerical example supports the convergence results.