Upwind discontinuous Galerkin methods with mass conservation of both phases for incompressible two‐phase flow in porous media

Abstract

Discontinuous Galerkin methods with interior penalties and upwind schemes are applied to the original formulation modeling incompressible two-phase flow in porous media with the capillary pressure. The pressure equation is obtained by summing the discretized conservation equations of two phases. This treatment is very different from the conventional approaches, and its great merit is that the mass conservations hold for both phases instead of only one phase in the conventional schemes. By constructing a new continuous map and using the fixed-point theorem, we prove the global existence of discrete solutions under the proper conditions, and furthermore, we obtain a priori hp error estimates of the pressures in L 2 (H 1 ) and the saturations in L ∞ (L 2 ) and L 2 (H 1 ).

Fully implicit hybrid two-level domain decomposition algorithms for two-phase flows in porous media on 3D unstructured grids

“…Since the saturation often changes faster than the pressure, in general, several small saturation time steps are performed immediately after a pressure time step. Enhanced versions of IMPES were proposed in [1,35,36,54] to improve the accuracy and stability by using a semi-implicit scheme for the saturation equation or by introducing a number of iterations in a single pressure-saturation time step. Eslinger [23] presented a decoupled method based on the local discontinuous Galerkin scheme for handling compressible fluids on uniform grids.…”

“…In [29,30,44], decoupled solution using a mixed finite element method for the pressure equation and an explicit or semi-implicit DG method for the saturation equation were studied. In [36], decoupled DG methods with interior penalties and upwinding schemes were applied to the original formulation while the pressure equation is obtained by summing the discretized conservation equations of two phases. It is believed that the most stable scheme for subsurface multi-phase flows is the fully implicit method in which all the coupled nonlinear equations are solved simultaneously [16,42,56,65].…”

A fully mass-conservative iterative IMPEC method for multicomponent compressible flow in porous media

“…Recently, Kou and Sun [13] proposed an improved IMplicit Pressure Explicit Saturation (IMPES) method for the incompressible two-phase flow in porous media. Different from the conventional IMPES methods [12,11,6,7], the pressure equation in [13] was obtained by summing the discretized conservation equations of two phases. This treatment yields a merit that the conservation of mass holds for both phases instead of only one phase in the conventional IMPES methods.…”

“…This treatment yields a merit that the conservation of mass holds for both phases instead of only one phase in the conventional IMPES methods. Inspired by the technique in [13], our objective is to propose an improved IMPEC method for the multicomponent compressible flow with conservation of mass for all components. Fully mass-conservation of multicomponent systems at discrete level should be taken into account when developing some numerical algorithms even at pore-scale simulation (see [9,14,15,16] and references therein).…”

“…Since the unknown parameters are nonlinear and dependent on the molar concentrations, we utilize the iterative approach to update the parameters based on the Peng-Robinson EOS. Similar to the improved IMPES method in [13], the new approach enjoys an appealing feature that the conservation of mass can be obtained for all the components. We call this new improved approach as a fully mass-conservative iterative IMPEC method.…”