A Theoretical Study of Pressure Distribution and Fluid Flux in Bounded Stratified Porous Systems with Crossflow
Authors: Katz Marvin L. , Tek M. Rasin 
Published: December 2007

Abstract
The flow of fluids in stratified porous systems, in which adjacent layers of the system are in communication over part or all of their common interface, in many instances involves cross flow between layers. A theoretical study has been made of the unsteady-state flow of slightly compressible fluids during depletion of bounded stratified porous systems. Consideration has been restricted to single-phase flow, with negligible capillary and gravity effects. Analytical expressions have been derived for pressure distribution and for fluxes at the producing face in two-layer systems with complete fluid communication between layers. These expressions occur in the form of double Fourier and Fourzer-Bessel series. The method of extending the solution to any number of layers is demonstrated by deriving the equation for pressure distribution in a three-layer system. On the basis of the exact results for two-layer systems, certain generalizations regarding the behavior of stratified systems in general are presented. The performance of stratified systems, in terms of cumulative flux as a function of time, is shown to lie at all times between bounds established from single-layer theory. The upper bound is given by treating the system as a single layer with arithmetic average physical properties. The corresponding lower bound is the summation of the fluxes from each layer as treated individually. Variation in performance of stratified systems between the upper and lower bounds is determined by the extent to which cross flow occurs in the system. The principal parameters affecting the amount of cross flow are the vertical permeability of the system, the ratio of system thickness to "radius of drainage" and, for radial systems, the ratio of system thickness to interior radius. At very early times, the system performance is near the lower bound. As time and the radius of drainage increase, the performance tends toward the upper bound

Introduction
One of the principal problems confronting the reservoir engineer today is the incorporation of the effect of reservoir heterogeneity into predictions of reservoir performance. The first part of this problem is the characterization of the actual reservoir heterogeneity in a mathematically useful manner. It is then necessary to calculate the effect of this heterogeneity on the system performance. It is frequently possible to estimate this effect by comparison with the known behavior of an idealized system. One such idealized system is the stratified or "layercake" model. Many studies have been made of the behavior of stratified systems, from both a theoretical and experimental standpoint. The great majority of the theoretical studies have been concerned with systems of separated layers, with no fluid crossflow. Among the more noteworthy of such theoretical papers is that recently published by Lefkovits, et al, in which a comprehensive analysis was presented for the behavior of bounded stratified systems which are unconnected except at the wellbore. Three recent references have dealt with stratified systems including the effect of crossflow. Jacquard has presented mathematical derivation s pertaining to depletion of two layer systems producing at constant rate. A doctoral dissertation by Katz contains analytical solutions and tables of results for pressure distribution and fluid flux during constant-terminal- pressure depletion of two-layer and three-layer systems with crossflow, together with tables of eigenvalues necessary in the computations. The dissertation also presents results from experimental studies of stratified systems using a heat-transfer analog model. This paper is largely based on the material in this dissertation.
SPEJ
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