A multipoint flux approximation finite volume method for the numerical simulation of the stokes-brinkman problem in 2-D using unstructured meshes

Abstract

Fractures, vugs, cavities and caves can significantly alter the fluid flow on carbonate petroleum reservoirs, making the computational simulation even more challenging due to the existence of free flow (Stokes) and porous media flow (Darcy). Different strategies can be devised to handle this problem, including the triple porosity and single permeability, the triple porosity and triple permeability etc. A much more accurate modeling can be done by using the so-called Darcy-Stokes model. In this approach, the Darcy ́s Law is used in the porous media domain and the Stokes equation is used in the free flow region. However, for carbonate petroleum reservoirs the use of this model is extremely complex due to the difficulties of accurately rep- resenting the geometrical intricacies of the naturally fractured media. An alternative approach is to use the Stokes-Brinkman (S-B) model. In this formulation, a single equation, is used to represent the fluid flow in both, the free flow region and in the porous media. This approach avoids the explicit modeling of the interface between the two domains, by transitioning be- tween them automatically. In this context, in the present thesis, we have used the S-B model to represent the fluid flow in the entire reservoir rock. To solve the system of equations, we have used a cell-centered finite volume scheme based on harmonic points (MPFA-H) and the Rhie and Chow’s interpolation to ensure the coupling between the variables. The MPFA-H is a robust method that can handle heterogeneous and anisotropic media on unstructured polyg- onal meshes. Finally, we have compared the SIMPLEC (Semi Implicit Method for Pressure Linked Equations Consistent) and the monolithic approaches to solve the system of equations. The proposed formulation presented good results, especially regarding the physical represen- tation of the Stokes and Darcy models in the so-called free-flow and porous medium domains, respectively, demonstrating proximity to data available in the literature, considering the same flow conditions.