Theory of Waterflooding in Fractured Reservoirs
Authors: de Swaan Abraham 
Published: December 2007

Abstract
This paper presents a new theory of the incompressible flow of two fluids (water displacing oil) in a fractured porous material composed of two distinct media - matrix blocks of low transmissibility embedded in a highly transmissible medium. This general description includes heterogeneous porous media not necessarily of the fractured type.
The theory accounts for an important fact not considered in framer analytical model found in the literature. The blocks downstream in a reservoir subject to waterflood are exposed to a varying water saturation resulting from the water imbibition of the upstream blocks.
Expressions for the water-oil ratio and the cumulative-oil production are derived, allowing a complete economic evaluation of a fractured-reservoir waterflood project. Comparison of experimental curves reported in the literature with curves obtained using this theory show a good fit.

Introduction
Imbibition is a most important mechanism of oil production in the waterflooding of fractured production in the waterflooding of fractured reservoirs. Using the action of capillary forces, it allows the recovery of oil from the interior of blocks that cannot be reached by the externally applied gradients of the waterflood.
Previous papers assume a function to describe the time rate of exchange of oil and water for a single matrix block. In a lineal reservoir, a water table advances as water is injected with the matrix blocks progressively exposed to water, depending on their position. The oil released by the matrix blocks is assumed transferred instantly to the water-oil interphase,. In this way, the oil production is an added function of individual block contributions. An analytical approach to this problem, and a numerical model, use the problem, and a numerical model, use the simplifying assumption of a water front. This may be a sound description in the presence of vertical high-transmissivity fractures where oil may segregate readily, but in fractures with a discrete transmissivity, it is expected that water imbibition and the simultaneous release of oil by these blocks will give rise to a varying saturation in the fractures that will affect the imbibition rates of the downstream blocks.
Braester's analytical approach assumes relative permeabilities of both wetting and nonwetting permeabilities of both wetting and nonwetting phases, intermediate between the fracture's and the phases, intermediate between the fracture's and the matrix's relative permeabilities; these intermediate permeabilities are impossible to measure. The permeabilities are impossible to measure. The model also uses an approximation of the fluid interchange between fractures and blocks. The model may be used for predictions after finding parameters to match observed oil and water parameters to match observed oil and water productions. productions. Kleppe and Morse conducted laboratory experiments on matrix blocks surrounded by fractures and numerical simulations (with rather coarse numerical grids) of Braester's laboratory system. Their numerical simulation computations agree well with the experimental results. This numerical formulation is exact or causalistic; capillary pressures and relative permeabilities are computed pressures and relative permeabilities are computed at every grid block. Their experimental and numerical results are used to test the theory presented here. presented here. Another numerical formulation assumes an approximation for the fluid interchange between fractures and matrix blocks. This approximate formulation did not try to reproduce the exact formulation results of Kleppe and Morse, nor their laboratory experiments.
The theory presented here analitically accounts for varying saturations in the fractures by introducing a convolution. A somewhat similar approach -was used successfully to describe the transient one-phase flow in a fractured reservoir.

THEORY
An outline of the subject theory (developed in the Appendix) includes the following assumed mechanisms and their corresponding mathematical expressions.
SPEJ
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