Modeling of convection-dominated thermoporomechanics problems using incomplete interior penalty Galerkin method

Abstract

In order to obtain converged and accurate numerical solutions, in general, small time steps are required for time-dependent problems. For popular continuous finite elements, however, the use of small time steps often results in oscillatory pressures and temperatures for thermoporomechanics problems with low conductivities. Moreover, convection-dominated thermal poromechanics problems often have very sharp temperature fronts that propagate over spatial domains. When applied to these problems, continuous elements present serious deficiencies in that large error induced in coarse mesh zones will propagate and pollute the solutions in fine mesh zones. Such oscillatory and polluted solutions are nonphysical and must be precluded. We propose discontinuous Galerkin methods to be alternatives to handle these challenges. This paper focuses on the finite element formulation of the incomplete interior penalty Galerkin method on thermoporoelasticity problems with heat convection. In this formulation, we propose discontinuous spaces for all three field variables. To deal with the convection effect, an upwinding scheme is simply and elegantly implemented through element interfaces, which is favored by the discontinuous Galerkin framework. In addition, the inner pressure boundary conditions important to many engineering problems are also taken into account in our discontinuous Galerkin formulation. Excellent performance of this discontinuous Galerkin method on precluding temperature oscillations and on blocking error propagation is demonstrated through solving a convection-dominated oil well-injection problem.