This paper presents new analytical solutions to unsteady state and pseudo-steady state flow in naturally fractured reservoirs whose boundaries can better be approximated by elliptical rather than linear or radial coordinates. While elliptical flow may occur in several geometric systems, fractured reservoirs are believed to exhibit elliptical flow behavior, particularly during the transitional period. Directional permeability inherent in fractured reservoirs creates reservoir anistropy. Flow from anistropic reservoir towards a circular wellbore will be elliptical rather than radial.

In this study the pressure pattern of naturally fractured reservoir is considered to develop under the assumption that allows matrix to fracture cross flow to result from a diffusion mechanism of fluid transfer through the matrix. The constant rate pressure distribution equations developed in this study were coupled to skin and wellbore storage parameters.

Type curves which can be used for well test analysis were generated with results of various analytical solutions. In the second part of this study, presented in the part two paper several field pressure data were analyzed using these type curves and the results compared with those obtained from type curves generated using radial or linear model solutions.


A significant percentage of wells tested each year have fractures within their drainage area. While most of these fractures are the result of using hydraulic fracturing as a method of stimulation, other fractures are less expected, having occurred naturally as a result of tectonic stresses within the earth. An almost infinite variety of naturally fractured system is possible but fractures at depth greater than 3000 ft are generally believed to be vertical and most reservoirs of well test analysis interest are deeper than 3000 ft. Many theoretical studies of naturally fractured systems have been made but, because of the diversity of these reservoirs no single method always applies. Part of the difficulty in developing a global model lies in lack of adequate published pressure transient data from naturally fractured systems.

Many methods in the literature for analyzing pressure data are based upon ideal radial or linear flow models which assume among others, homogeneous and isotropic porous medium. A fracture is a heterogeneity and any reservoir heterogeneity which represents a significant departure from these assumed conditions will cause measured formation face pressures to deviate from the behavior predicted by these models.

When a reservoir is fractured, the resulting pressure behavior can no longer be described by conventional radial flow theory. instead, it has been shown that pressures exhibit linear flow behavior at early test times. Later, during the transitional flow period, pressure transients move away from the fracture with a shape that is elliptical or semi-elliptical in geometry. However, as the pressure waves continue to propagate, they begin to approach a radial geometry, thereby, marking the pseudoradial flow period.

The studies of elliptical flow in porous media are much fewer and far between than those of linear and radial geometries. Muskat presented the earliest study of elliptical flow during steady-state period for the flow from a finite conductivity line source into an infinitely large reservoir. Prats et al. used elliptical geometry to simulate the effect of vertical fractures on compressible fluid flow in reservoir.

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