Performance of Layered Reservoirs with Crossflow--Single-Compressible-Fluid Case
Authors: Russell D.G. , Prats M. 
Published: December 2007

Abstract
The performance of a well in a bounded, layered reservoir with interlayer cross flow has been investigated mathematically. The system studied comprises a centrally located well in a bounded cylindrical reservoir composed of two layers of contrasting physical properties. The reservoir contains a single fluid of small and constant compressibility. In the case of a well producing at constant pressure, the production rate of the cross flow system declines exponentially after the early transient behavior dies out. For wells which produce with constant rate, the pressure-time relationship is linear following the transient. For very early production time, the production rate of the two-layer cross flow system behaves as that of the two-layer stratified system (the case in which the layers are in communication at the well only).Except for early time, in most practical cases the performance of a two-layer reservoir with crossflow can be duplicated by that of a single-layer reservoir. The equivalent single-layer reservoir must have the same pore volume and the same drainage and wellbore radii as in the two-layer case. In addition, the equivalent reservoir must have a "kh" (permeability x thickness) product equal to the sum of the "kh" products of the respective layers in the cross flow system. If the porosity and permeability distributions are known, the production and pressure behavior of a well with and without cross flow can be predicted. From a comparison of the observed pressure or production performance of a well and the theoretical behavior, the occurrence of interlayer cross flow can be detected.

Introduction
When flow is possible across the bedding plane from a low-permeability layer into an adjacent layer of moderate or high permeability, the earlier depletion of the more permeable layer will cause such flow to occur. If this situation exists, the oil recovery from tight layers can be appreciable; thus, the economic ramifications of interlayer crossflow are apparent. The possible importance and extent of flow between adjacent and connected layers - whether resulting from capillary, gravitational, or viscous forces - have been the subject of considerable speculation for many years. Jacquard presented a rigorous analysis of the crossflow due to single-fluid expansion occurring between two adjacent and connected layers when the well produces at constant rate. His presentation was entirely mathematical, however, and no numerical results from evaluation of the mathematical solutions were presented. Our objectives in the preparation and presentation of the present paper are essentially twofold -to analyze the behavior of a well which produces a single compressible fluid from a bounded, layered reservoir with interlayer crossflow, andto present practical numerical results and simplified formulas to facilitate engineering analysis of reservoirs with crossflow.
The constant-terminal-pressure case was analyzed rigorously, whereas the constant-terminal-rate case was treated only for semi steady-state conditions.

DESCRIPTION OF THE IDEALIZED TWO-LAYER RESERVOIR
The reservoir to be studied is assumed to be bounded and cylindrical in shape, with a completely penetrating well at its center. The reservoir is composed of two adjacent horizontal layers between which fluid is free to flow. Each layer is assumed to be homogeneous, isotropic and completely filled with a fluid of small and constant compressibility.
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