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 P. 117

A formulation for pressure transients in terms of the intrinsic, or core, properties of the two media that compose the fractured reservoir establishes the influence of these properties on-and their corroboration from-the pressure/time relationship observed in well and interference tests. The influence of the following reservoir characteristics are analyzed: the area of fractures transverse to flow; the dimensions, shape, and properties of rectangular parallelepiped matrix-rock blocks; and permeability reduction in the block surfaces.A restatement of the so-called quasisteady-state intermedia flow provides Warren and Root's a and A parameters with the physical meaning they lacked and allows direct determination of the blocks' minimum dimensions.

This note presents a new formulation of the growth of the --average~as saturation, fram zero to its critical value, in a producing, originally undersaturated,oil reservoir. The increase of the gas saturt ion is described by an integral equation involving the fluid properties.The new theory has 2een ap~lied to field data, giving the same good re-sults as a former procedure based on the material balance equation.

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A model has been developed to describe the behavior of naturally fractured reservoirs with black oil in which high transmissibility in the fractures and low oil production rates allow the gravitational segregation of the gas, oil and water phases. The influxes of these fluids from the matrix rock to the fractures are described by functions dependent on the properties of the reservoir fluids and the matrix rock blocks. These properties are measurable by laboratory tests on cores and formation studies in the reservoir. This approach contrasts with a common trend in fractured reservoir simulation favoring the use of parameters to account for the matrix-fracture interaction; these parameters cannot be explicitly linked to the reservoir characteristics.The presented formulation results in a system of three simultaneous integra-differential equations where the dependent variables are: the dephts of the gas-oil and water-oil contacts, and the pressure at the datum. Observed and calculated variables, obtained applying the model, are in good agreement throughout the forty-five years long production history of an actual reservoir that includes a period with null production, while all the wells were closed, that resulted in an unusual fluctuation of the variables.