A Tall Block Model for Immiscible Displacement in Naturally Fractured Reservoirs
Authors: Paes-Leme Paulo Jorge , Nunes Neci P. , Douglas Jim , Arbogast Todd 
Published: December 2007

Abstract
A model for two-phase immiscible, incompressible flow in a naturally fractured reservoir is derived via the technique of homogenization for a new geometry for the fractures. Two families of parallel planes normal to the bedding of the reservoir describes the fracture geometry. A finite difference scheme is introduced to approximate the solution of the model. The method involves a local-in-time linearization of the water and capillary potentials on the blocks in a fashion that allows an implicit treatment of the fracture potentials simultaneously with the matrix block boundary potentials. The algorithm leads to natural parallelization of the block calculations.

Introduction
The choice of the appropriate model to simulate the flow in a naturally fractured reservoir is strongly dependent on the geometry of the fractures. This choice is determined by the general size of the matrix blocks. Of course the complexity of the model and consequently the computational requirements increase with the block size.
In a new way of treating the interaction between the fluids in the fractures and those in the matrix based on applying the proper boundary conditions to the partial differential equations describing the flow in the matrix is studied. The resulting double porosity models incorporate three important scales of length the pore scale (microscopic) the fracture spacing and the reservoir length (macroscopic). The were derived first on the basis of physical intuition and with the exception of the limit model of, they were also derived by the mathematical theory of homogenization.
For naturally fractured reservoirs, the homogenization process, starting from the microscopic description of the flow by means of a standard model with physical parameters that vary rapidly in space, leads to a macroscopic model on a two-sheeted covering over the reservoir, with flow in the fracture being modelled on one sheet. The second sheet contains information of the intermediate length scale, that is, the behavior of the flow in each matrix block.
In this paper, a model to simulate the flow of an immiscible displacement (waterflooding) in a naturally fractured petroleum reservoir is studied. The three-dimensional physical situation in mind is of a relatively thin field without fractures parallel to the bedding plane in which the size of the matrix blocks is approximately equal to the size of the reservoir orthogonal to the bedding place. Over each line, orthogonal to the bedding place, there is an attached matrix block. The flow in the fractures and in the blocks is modelled by correct boundary conditions on the faces of each block. The model is an example of a large block situation in fractured reservoirs. The model is a version of a double porosity/double permeability concept. but not one that has been considered previously. The paper is organized in the following way. First, the model is derived via homogenization, the a numerical method for approximating the solution of the associated initial-boundary value problem is described briefly. Finally, results of some simulations on a problem is described briefly. Finally, results of some simulations on a test problem are presented.

Derivation of the Model
Consider the simple case of reservoir consisting of a horizontal slab fractured by two families of parallel, equally spaced, vertical fractures. Assume the matrix blocks, are translates of, where the interval I corresponds to the height of the reservoir (see Figure 1). This is, they form a periodic structure such that, are disjoint and, = m Denote by f= - m the fracture spacing Let Q be a period, or cell, of this structure containing a matrix block Qm and half of surrounding fractures (see Figure 1).
As in let and be the porosity and absolute permeability of an individual fracture, that is on the scale of the pores. For simplicity assume they do not depend explicitly on space and time. Let denote = K Kraf (s) = and = Similarly define analogous quantities for the matrix blocks. To exploit the natural symmetry of the equations = 1 and =-1. Let= Paf - P and = Pom - be the potentials of phase in the fractures and in the matrix respectively. Following phase in the fractures and in the matrix respectively. Following Darcy's law is assumed to hold on the macroscopic (pore) scale. At this scale, the equations describing the flow are
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