Abstract
A general mathematic model is derived that may be used to describe fluid movement through naturally fractured reservoirs. The model treats the reservoir as a double-porosity medium consisting of heterogeneous isotropic primary rock matrix blocks and an anisotropic. heterogeneous fracture matrix system. The fractured are assumed to have a general distribution in space and orientation called the fracture matrix function to represent their statistical nature. Simplifying assumptions are made concerning flow in individual fractures and a hemispherical volume integration of microscopic fracture flow equations is performed to arrive at a generalized Darcy-type equation, with a symmetric permeability tensor evolving to describe the flow in the fracture evolving to describe the flow in the fracture matrix. For flow in the primary rock matrix blocks. Darcy's law for an isotropic medium is assumed. Time-dependent porosity equations for the primary rock matrix and the fractures are derived and coupled with the conservation of mass principle for each system to arrive at a governing set of continuity equations. Each resulting continuity equation is coupled further by a fluid interaction term that accounts for fluid movement that can take place between rock matrix blocks and fractures. The resulting equations of continuity and the equations of motion are generalized for multiphase flow through the fractured medium with variable rock and fluid properties. To complete the model formulation, a general set of auxiliary equations are specified, which can be simplified to fit a particular application.
Introduction
Flow of fluid in fractured porous media was recognized first in the petroleum industry in the 1940's. Since that time, many researchers have added to the volume of literature on fractured media. An extensive bibliography on flow in fractured porous media is given in Ref. 1. When attempting to model fluid flow through any type of medium, the researcher must decide which kinds of fluids and the type of flow to model. In the case of fractured porous media where most of the flow takes place through fractures, the flow can become truly turbulent. However, as demonstrated for many encounters with fracture flow, the laminar flow regime probably prevails. The development of fracture flow models has proceeded along two different approaches: the statistical and the fractured rock mass is considered a statistically homogeneous medium consisting of a combination of fractures and porous rock matrix. The fractures are considered ubiquitous, and the system is called statistically homogeneous because the probability of finding a fracture at any given point in the system is considered the same as fining one an any other point. In the enumerative approach, a fractured rock medium is studied by attempting to mode the actual geometry of fractures and porous rock matrix. The locations, orientation, and aperture variations for each individual fracture must be considered in this approach.
Statistical Approach
Many researchers have developed models with the statistical approach. Elkins and Skov used this approach to study anisotropic fracture permeability associated with Spraberry field, TX. Considering the extensive system of orthogonal vertical joints as an anisotropic medium, from a number of drawdown tests they were able to construct permeability ellipsoids whose axes were aligned reasonably well with the observed fractured system. This is called a "one-medium statistical model" because flow in the porous rock matrix was not considered. A two-medium statistical model for transient flow in a fractured rock medium was developed by Barenblatt et al.
SPEJ
P. 669^