A finite volume scheme coupled with a hybrid-grid method for the 2-d simulation of two-phase flows in naturally fractured reservoirs

Abstract

Two-phase flows of oil and water in naturally fractured petroleum reservoirs can be described by a system of nonlinear partial differential equations that comprises an elliptic pressure equation and a hyperbolic saturation equation coupled through the total velocity field. Modeling this problem is a great challenge, due to the complexity of the depositional environments, which can include fractures (channels or barriers). In such cases, it is particularly complex to construct structured meshes which are capable of properly modeling the reservoir. In this work, a locally conservative approach to model the oil and water displacements in naturally fractured reservoirs using general unstructured meshes was developed. A cell-centered Finite-Volume Method with a Multi-Point Flux Approximation that uses the so called “diamond stencil” (MPFA-D) was used to solve the pressure equation, coupled with a Hybrid-Grid Method (HyG) to deal with the fractures. The classical First Order Upwind Method (FOUM) was used to solve the saturation equation. The FOUM was applied in two different segregated schemes, in its explicit and implicit versions, respectively the IMPES (IMplicit Pressure and Explicit Saturation) and the SEQ (SEQuential implicit pressure and saturation). The MPFA-D is a very robust and flexible formulation that is capable of handling highly heterogeneous and anisotropic domains using general polygonal meshes. In the HyG, the mesh that discretizes the domain must fit the spatial positions of the fractures, so that they are associated to edges - as 1-D cells in a 2-D mesh -, therefore, the calculation of the fluxes in these edges is dependent on the pressures on fractures and on the adjacent volumes, but, in this strategy, the fractures are expanded, in the computational domain, to the same dimension of the mesh. In this way, it is possible to get, for example, 2-D fracture cells in a 2-D mesh, but avoiding excessive refinement in the fractured regions, in the original mesh. The proposed formulation presented quite remarkable results when compared with similar formulations using classical full pressure support and triangle pressure support methods, or even the with MPFA-D itself when using an equidimensional approach.