Optimal $$L^2$$ A Priori Error Estimates for the Biot System

Abstract

Coupled subsurface fluid flow and geomechanics is receiving growing research interests for applications in geothermal energy, unconventional oil and gas recovery and geological CO\(_2\) sequestration. A key model characterizing these processes is the Biot system. In this paper, we present optimal \(L^2\) error estimates for the Biot system. The flow equation for the pressure is discretized in time by a backward Euler scheme and in space by a continuous Galerkin scheme, while the elastic displacement equation is discretized at all time steps by a continuous Galerkin scheme. We prove optimal \(L^2\) a priori error estimates in space for the resulting Galerkin scheme, provided the domain is a convex polygon or polyhedron according to the dimension and the data and solution spaces have sufficient regularity. The key idea is to introduce suitable auxiliary elliptic projections in the error equations and to use one such projection to approximate the given initial pressure. These theoretical results are confirmed by numerical experiments performed with a fixed-stress split algorithm.