A Method for Estimating the Interporosity Flow Parameter in Naturally Fractured Reservoirs
Authors: Uldrich D.O. , Ershaghi Iraj 
Published: December 2007

Abstract
For naturally fractured reservoirs of the double-porosity type, Warren and Root defined two parameters characterizing such systems, F ft, and epsilon. While F ft may be obtained easily from the straight lines of the buildup or drawdown plot, no explicit method for estimating epsilon was plot, no explicit method for estimating epsilon was suggested in the original paper.This paper presents a method whereby the coordinates of the inflection point on a buildup or drawdown plot may be used to estimate epsilon. We show that for pressure drawdown tests, epsilon can be estimated under certain conditions, while for pressure buildup tests, only the ratio of the interporosity flow parameter to the total system porosity/compressibility parameter to the total system porosity/compressibility product, epsilon/[(phi ct)f + (phi Ct)ma], is obtained. product, epsilon/[(phi ct)f + (phi Ct)ma], is obtained. By using the concept of inflection points, an equation is derived where F ft may be obtained from a pressure buildup or drawdown test when no early- or late-time data are available.

Introduction
Warren and Root presented a solution to the problem of radial flow of a slightly compressible problem of radial flow of a slightly compressible fluid in a naturally fractured reservoir. They assumed that flow occurs only in the fractures and that the matrix blocks, assembled as a uniformly distributed source, deliver the fluid to the fracture system. They characterized such a system by two parameters related to the properties of the reservoir. One of these parameters, the fluid capacitance coefficient, F ft, parameters, the fluid capacitance coefficient, F ft, is used to represent the ratio of the porosity/ compressibility product for the fractures to that for the entire system: (phi ct)f/[(phi ct)f + (phi ct)ma]. The second parameter, epsilon, is defined as the interporosity flow parameter, which indicates the degree of interporosity flow between the matrix blocks and the fracture system. As shown by Kazemi, the fluid capacitance coefficient may be obtained from the following equation:
F ft = antilog (-delta p/m)...................(1)
where, delta p = vertical separation of the two straight lines on a buildup or drawdown test plot, psi (kPa); and m = slope of the straight lines on a buildup or drawdown test plot, psi/cycle (kPa/cycle). While the computation of F ft from this equation is straight-forward, no clear method of finding epsilon has yet been proposed. In the original paper, Warren and Root did not elaborate on a suitable method for the determination of epsilon. Kazemi discussed the use of interference test data to find the total system porosity/compressibility product, (phi ct) f + (phi ct)ma. Using this information, a trial-and-error procedure may be applied to the pressure buildup equation procedure may be applied to the pressure buildup equation to obtain a satisfactory answer for epsilon. The optimum epsilon was defined as the value resulting in the best curve fit of the theoretical equations to the field data.
SPEJ
p. 324