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: (1) oil production in a
layered faulted reservoir, (2) laboratory displacement tests in a stack of
matrix blocks with a large contrast in fracture and matrix capillary pressure
functions, and (3) water 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.
© 2007. Society of Petroleum Engineers
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History
- Original manuscript received:
16 July 2005
- Revised manuscript received:
10 November 2006
- Manuscript approved:
17 November 2006
- Version of record:
20 September 2007
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