document: In this article we consider an extension to the equations of poroelasticity by modelling the flow of a slightly compressible single phase fluid in a viscoelastic porous medium. The constitutive equations therefore allow for the presence of viscoelastic relaxation effects in the porous media (but not the fluid). Fully discrete numerical schemes are derived based on a lagged and non-lagged backward Euler time stepping method applied to a mixed and Galerkin finite element spatial discretization. We show that the lagged scheme is unconditionally stable and give an optimal a priori error bound for it. Furthermore, this scheme is practical and useful in the sense that it can be easily implemented in existing poroelasticity software because the coupling between the viscous stresses and pressures and the elasticity and flow equations is ‘lagged’ by one time step. The required additional coding therefore takes the form of extra ‘right hand side loads’ together with some updating subroutines for the viscoelastic internal variables, but the solver and assembly engines remain intact. This idea of lagging has been used before for nonlinearly viscoelastic diffusion problems in [3,24] but, of course, is not new. Lagging in numerical schemes is discussed more widely by Lowrie in [14].
extractive_keyphrases: assembly engines constitutive equations coupling extension to the equations of poroelasticity flow equations fluid fully discrete numerical schemes mixed and galerkin finite element spatial discretization nonlinearly viscoelastic diffusion problems numerical schemes porous media updating subroutines viscoelastic porous medium viscoelastic relaxation effects
abstractive_keyphrases: lagged and non-lagged backward euler time stepping method modelling the flow of a slightly compressible single phase fluid in a viscoelastic porous medium single phase fluid