Simulation of immiscible two-phase flow in 3-D naturally fractured reservoirs using a locally conservative method, a projection-based embedded discrete fracture model and unstructured tetrahedral meshes

Abstract

Fluid flow in fractured porous media is a very relevant phenomenon, since most of the remaining oil reserves in the world reside in this type of formations, in addition to the fact that fractures are also present in shallower layers of the crust, which makes them influential also in water extraction and waste dispersion. The two-phase flow of oil and water in reservoirs can be mathematically described by a set of non-linear partial differential equations, whose modeling constitutes a great challenge, due to the complexity of the depositional environments, in addition to the presence of fractures. In these cases, it is particularly complex to construct structured computational meshes capable of adequately representing the reservoir. In the present work, a new strategy was developed to simulate immiscible two-phase flow in 3-D fractured porous media, using tetrahedral unstructured meshes. Such a strategy is based on a finite volume method with multipoint flux approximation that uses the so-called "diamond stencil" (MPFA-D), considering a projection-based embedded discrete fracture model (pEDFM) to include the influences of the fractures in the global reservoir model. The MPFA-D is a robust and flexible formulation, capable of handling highly heterogeneous, possibly discontinuous, and anisotropic, even non-orthotropic, diffusion tensors, and which achieves second- order convergence rates for the scalar variable and first-order convergence rates for its gradient. However, like other linear MPFA methods, it does not formally guarantee monotonic solutions or those that respect the Discrete Maximum Principle (DMP) and can produce spurious oscillations in the pressure field for permeability tensors with high anisotropy ratio or for distorted meshes. To deal with this problem and enforce DMP compliance, an alternative non-linear defect correction for MPFA-D was developed. Furthermore, the adopted fracture model avoids the additional complexity of aligning fractures with edges or faces of the computational mesh that discretizes the domain corresponding to the porous medium, making the construction of this mesh more flexible and less susceptible to excessive localized refinements. The saturation terms of the mathematical model are discretized according to the backward Euler method, in the context of a fully implicit numerical scheme. The proposed numerical methods, as well as the simulator composed by them, were tested against problems found in the literature and others elaborated by the author, aiming to demonstrate the robustness and flexibility of the developed computational simulation tool.