An Approximate Solution Of Linear Flow In Naturally Fractured Reservoirs
Authors: Aguilera R. 
Published: March 2010

Abstract
Equations are presented for evaluation of linear flow in naturally fractured reservoirs represented by dual-porosity systems. The problem has not been treated previously in the petroleum engineering literature. This situation might happen, for example, in the case of drawdowns or buildups in dual-porosity systems which are hydraulically fractured or semi-infinite dual-porosity systems which are bounded by two parallel linear discontinuities.
The equations apply only for oil systems. The extension to gas wells, however, is straightforward. The model assumes flow from the matrix into the fractures. There is no flow from a matrix block into another matrix block.
It is shown that a conventional cross plot of log of delta p vs log of time should result in two parallel straight lines with a slope equal to 0.5 and a transition period which depends on the type of inter-porosity flow (pseudo steady-slate, transient or gradient) and the shape of the matrix blocks. This plot allows determination of the storativity ratio, omega, fracture porosity and distance between natural fractures.
A conventional cross plot of delta p vs square root of time on Cartesian coordinates should result in two distinct straight lines. The first straighr line has a slope bigger than the second straight line. The ratio of these slopes squared gives the storativity ratio omega. The analysis also permits calculation of the hydraulic fracture length.

Introduction
Naturally fractured reservoirs have been the object of intensive research during the last few years in the geologic as well as the engineering fields. Transient pressure analysis has received particular attention.
Barenblatt and Zheltov(1) and Warren and Root(2) handled naturally fractured reservoirs by assuming pseudo steady-state inter-porosity flow in a model made out of cubes with spaces in between. Flow toward the wellbore was assumed to be radial via the natural fractures. Their work led to the conclusion that a conventional cross plot of pressure vs log of time should result in two parallel straight lines with a transition period in between. The separation of the two straight lines allowed calculation of the storativity ratio omega, i.e. the fraction of the total storage within the natural fractures.
Kazemi(3) used a numerical model of a finite reservoir with a horizontal fracture under the assumption of unsteady state inter-porosity flow and substantiated Warren and Root's conclusion with respect to the two parallel straight lines. The transition period, however, was different due to the unsteady rather than pseudo steady-state inter-porosity flow assumption.
de Swaan(4) developed a diffusivity equation and analytical solutions to handle the first and last straight lines. His method, however, could not analyze the transition period. Najurieta(5,6) developed analytical solutions of de Swaan's radial diffusivity equation which could handle the transition period as well as the first and last straight lines.
Streltsova(7) used a gradient flow model and indicated that the transition period should yield a straight line with a slope equal to ½ the slope of the early and late straight lines.