Pressure Transient Performance of a Multilayered Reservoir with Crossflow
Authors: Pendergrass J.D. , Berry V.J. 
Published: December 2007

Abstract
Well pressure transient tests provide a means for directly obtaining information about formation pressure and reservoir flow capacity. Such tests have also been proposed for determining presence and location of faults or other reservoir closures and for measuring oil in place. For mathematical convenience, most theoretical studies have considered the reservoirs to be homogeneous. Definitive information is not yet available to show whether the actual presence of nonuniformities will make pressure transient behavior different from that of a uniform reservoir. The conclusions reached from actual transient tests are questionable, therefore, insofar as they rely on the original assumption of homogeneity. One type of nonuniformity commonly assumed to exist is that of stratification. In most reservoirs the strata are thought to be in vertical communication. Equations for the transient flow of a single-phase, compressible fluid in a one-well, bounded, circular reservoir have been solved for several situations involving cross flow between multiple strata of various thicknesses and permeabilities. The results show that except for the very early flow period, which usually is too short to be analyzed, the transient performance observed at the well is substantially identical with that of a homogeneous reservoir having the same dimensions and having the same steady-state flow capacity. Thus, stratification does not adversely affect interpretation of well transient tests. This conclusion holds for all commonly encountered combinations of reservoir thickness and external radius. Deviations are observed for unusually thick reservoirs whose outer radii are relatively small. The results of these studies also show that the presence and the amount of stratification cannot be simply diagnosed from reservoir pressure transient data when there is cross flow between strata.

Introduction
The last decade has brought wide acceptance of the transient well pressure test for determining reservoir parameters. Following the original work of Hurst and van Everdingen, the mathematical theory was thoroughly explored. Numerous authors have suggested how to determine static reservoir pressure, permeability-thickness product, original oil in place and reservoir limits for different reservoir geometry. The same mathematical techniques have been used to predict the transient performance of a reservoir over a long period of time. Most of the theoretical work has been for homogeneous, isotropic systems. Some results have also been presented for a homogeneous, anisotropic reservoir. Petroleum reservoirs are not homogeneous. The deposition process seems to favor creation of a stratified formation. This concept is sufficiently well accepted so that the most natural extension of the transient flow theory beyond the homogeneous case is to a stratified formation. Results for a stratified reservoir with no vertical communication between layers can be obtained from the results for a homogeneous reservoir. Lefkovitz, et al, have given a thorough treatment of the two-layer case. Two recent papers have treated the case of a two-layered reservoir with vertical communication or crossflow, between layers. Russell and Prats find that, after a relatively short time, the two-layered reservoir with cross flow exhibits a simple exponential pressure decline. From this time forward, the behavior is not distinguishable from the behavior of a homogeneous reservoir having the same steady-state flow capacity. The results of Katz and Tek are equivalent. Russell and Prats also speculate that a multilayered reservoir with crossflow will behave as a homogeneous system after long enough production time ".... providing the contrast in kh between layers is not too great". They also suggest that, at intermediate times, ".... the relative positions of the layers with respect to each other will have a great influence on the production behavior and on the time at which the previously mentioned large-time approximation might be valid".Katz and Tek remark upon the mathematical difficulty of treating a reservoir having many layers or strata.
SPEJ
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