Analytical Solutions For Vertical Fractures In A Composite System
Authors: Stanislav J.F. , Easwaran G.V. , Kokal S.L. 
Published: March 2010

Abstract
A method is proposed to interpret the pressure transient data of injection wells in secondary arid tertiary recovery processes. A composite system is considered in which the inner region represents a swept volume and the outer region extends to infinity. The two regions are thus featured by different hydraulic diffusivities and mobility's. The well is vertically fractured and the fracture is assumed to exhibit an infinitely large conductivity. The flow behavior in the system is described by use of elliptical geometry.
The model was tested against several special cases for which analytical solutions are available and very good matches were obtained.

Introduction
In general, oil and gas deposits are characterized by rock and fluid heterogeneities. Although the pressure transient theory based on the assumption of system homogeneity is frequently successful in interpreting the well test data, there are also well and reservoir conditions which do not permit such a simplification: formation damage or improvement near the wellbore, bubble of the natural gas injected into the formation, water bank in a waterflood project, the burned zone in an in-situ combustion project and the swept zone in steam flooding or steam stimulating scheme. In all these situations the system properties undergo an abrupt change at some distance from the wellbore. A composite system consisting of an inner region surrounding the well and the outer region characterized by unaltered reservoir properties must therefore be considered and the pressure transient theory be modified accordingly. Several studies of the composite reservoirs have already been presented in literature(1–4).
In elliptical shaped reservoirs, anisotropic reservoirs and formations surrounding the vertically fractured wells, an elliptical flow geometry is assumed to occur.
It is the subject of this study to investigate the transient pressure behavior in a composite system with vertically fractured wells using elliptical coordinates. Only one study has addressed this problem so far(4).

The Model Development
The main features of the model include:—Artificial vertical fracture of infinitely high conductivity;—Two regions, inner and outer, (within themselves considered isotropic and homogeneous) characterized by different rock properties;—The flood front of an infinitesimal thickness is stationary during the testing period; and—Single-phase fluid of constant compressibility.
The two-dimensional single-phase flow equations for constant compressibility fluids in elliptical coordinates are thus considered. The dimensionless form of the flow equation for the two regions can be written as follows:
Region 1:
Equation (1) (Available In Full Paper)
Region 2:
Equation (2) (Available In Full Paper)
Initial pressure distribution in both regions in uniform and equal to P1:
Equation (3) (Available In Full Paper)
Equation (4) (Available In Full Paper)
At the inner boundary (fracture) a constant pressure is specified as:
Equation (5) (Available In Full Paper)
Region 2 is assumed to be infinitely large with respect to any pressure disturbance caused by the well:
Equation (6) (Available In Full Paper)
The two following continuity requirements must be satisfied at the two regions interface:
Equation (7) (Available In Full Paper)
Equation (8) (Available In Full Paper)