Coupling of Discontinuous Galerkin Schemes for Viscous Flow in Porous Media with Adsorption

Abstract

Polymer flooding is an important stage of enhanced oil recovery in petroleum reservoir engineering. A model of this process is based on the study of multicomponent viscous flow in porous media with adsorption. This model can be expressed as a Brinkman-based model of flow in porous media coupled to a non-strictly hyperbolic system of conservation laws for multiple components (water and polymers) that form the aqueous phase. The discretisation proposed for this coupled flow-transport problem combines an H(div)-conforming discontinuous Galerkin (DG) method for the Brinkman flow problem with a classical DG method for the transport equations. The DG formulation of the transport problem is based on discontinuous numerical fluxes. An invariant region property is proved under the (mild) assumption that the underlying mesh is a B-triangulation [B. Cockburn, S. Hou, and C.-W. Shu, Math. Comp., 54 (1990), pp. 545-581]. This property states that only physically relevant (bounded and non-negative) saturation and concentration values are generated by the scheme. Numerical tests illustrate the accuracy and stability of the proposed method.

Discontinuous approximation of flow in porous media with adsorption

“…Thus, only physically relevant saturation and concentration values are generated by the scheme. Numerical tests presented in [2] illustrate the accuracy and stability of the proposed method. We here present additional simulations (not included in [2]) of polymer flooding in a reservoir.…”

“…The discretization for (1)-(3) analyzed in [2] combines an existing [5] H(div)-conforming discontinuous Galerkin (DG) method for the Brinkman flow problem (3) with a classical DG method for the transport equations. The DG formulation of the transport problem is based on discontinuous numerical fluxes [4].…”

“…The DG formulation of the transport problem is based on discontinuous numerical fluxes [4]. We refer to [2] for any detail. An important novelty in the numerical approximation of (1), (2) is the use of discontinuous numerical fluxes for the coupled equations that arise for the continuous-in-time, spatially discrete equations that represent the system (1), (2).…”

“…We refer to [2] for any detail. An important novelty in the numerical approximation of (1), (2) is the use of discontinuous numerical fluxes for the coupled equations that arise for the continuous-in-time, spatially discrete equations that represent the system (1), (2). This is achieved by treating each approximate polymer concentration as a discontinuous coefficient entering the numerical flux, see [4, §2.4].…”

“…This is achieved by treating each approximate polymer concentration as a discontinuous coefficient entering the numerical flux, see [4, §2.4]. An invariant region property, stating that the average saturation on each element stays between zero and one and the average polymer concentration only assumes values between those of the average of neighboring or boundary cells, is proved in [2] under the (mild) assumption that the underlying mesh is a B-triangulation [3] plus a suitable time step restriction (CFL condition). Thus, only physically relevant saturation and concentration values are generated by the scheme.…”