Summary
The control-volume discrete-fracture (CVDF) model is extended to incorporate heterogeneity in rock and in rock-fluid properties. A novel algorithm is proposed to model strong water-wetting with zero capillary pressure in the fractures. The extended method is used to simulate:oil production in a layered faulted reservoir,laboratory displacement tests in a stack of matrix blocks with a large contrast in fracture and matrix capillary pressure functions, andwater injection in 2D and 3D fractured media with mixed-wettability state. Our results show that the algorithm is suitable for the simulation of water injection in heterogeneous porous media both in water-wet and mixed-wettability states. The novel approach with zero fracture capillary and nonzero matrix capillary pressure allows the proper prediction of sharp fronts in the fractures.
Introduction
This work is focused on the numerical treatment of two main physical aspects of multiphase flow in fractured porous media: heterogeneity in rock-fluid properties and reservoir wettability.
In a previous work (Monteagudo and Firoozabadi 2004), a CVDF method was used to discretize the system of equations governing water injection in fractured media with strong-water-wettability state and homogeneous matrix and rock-fluid properties. The method was restricted to a finite contrast in matrix-fracture capillary pressure. In this work, we extend the CVDF model for simulation of water injection in fractured media comprised of heterogeneous rocks and wettability conditions from strong-water-wetting to mixed-wetting conditions. We also present a formulation for infinite contrast in capillary pressures of matrix and fractures (zero capillary pressure in the fracture and finite capillary pressure in the matrix).
The control volume (CV) method, first proposed by Baliga and Patankar (1980), is a finite-volume formulation over dual cells (CVs) of a Delaunay mesh. It is locally conservative and suited for unstructured grids. It has been widely employed for the simulation of multiphase flow in porous media (Monteagudo and Firoozabadi 2004; Verma 1996; Helmig 1997; Helmig and Huber 1998; Bastian et al. 2000; Geiger et al. 2003) and the convergence of the method for two-phase immiscible flow in porous medium has already been proved (Michel 2003).
Numerical treatment of heterogeneity in the framework of the CV method has been extensively studied in the past (Edwards 2002; Edwards and Rogers 1998; Prevost 2000; Aavatsmark et al. 1998a, b). Nevertheless, those works have focused on absolute permeability heterogeneity and anisotropy in single-phase flow. The main concern in those works is the use of full tensor permeability and the accurate generation of streamlines (required by the streamline numerical method). It is well known that the standard CV method produces inaccurate velocity fields around the interfaces of heterogeneous media as the contrast in permeability is increased (Durlofsky 1994). In the standard CV method, Delaunay triangles are locally homogeneous and the polygonal CV cell may be heterogeneous (see Fig. 1a). For accurate streamlines, several authors (Verma 1996; Edwards 2002; Edwards and Rogers 1998; Prevost 2000; Aavatsmark et al. 1998a) have proposed that the polygonal CV cell must be locally homogeneous, implying heterogeneous Delaunay triangles (see Fig. 1b). The latter configuration, however, generates additional problems in the simulation of multiphase flow in porous media. Basically, from mesh generation standpoint, it may not be possible to generate an unstructured mesh where the boundaries of the CV median-dual cell conform to heterogeneous interfaces in the domain. Conforming mesh is important for the discrete-fracture approach. Therefore, it would be necessary to first generate a standard CV cell mesh, and later a homogenization procedure would be required to obtain CV cells with constant permeability. The homogenization or upscaling of permeability is somehow possible, but the same is not true for rock-fluid properties; most challenging is capillary pressure with different endpoints. Therefore, the approach with the homogeneous CV cell may be suitable for single-phase simulation where rock-fluid interactions are not part of the problem. However, rock-fluid interactions have to be taken into account for simulation of multiphase flow in fractured porous medium. Frequently, capillary pressure is disregarded in two-phase flow simulations; however, capillary pressure is of importance for simulation of multiphase flow in fractured porous media (Monteagudo and Firoozabadi 2004; Karimi-Fard and Firoozabadi 2003). Predictions of flow pattern and oil recovery may be severely affected if capillary pressure effect is neglected.
Numerical Simulation of Two-Phase Flows in Heterogeneous Porous Media
