Elliptical Flow In Composite Reservoirs
Authors: Stanislav J.F. , Easwaran C.V. , Kokal S.L. 
Published: March 2010

Abstract
Composite reservoir systems are frequently encountered in production or injection operations. In most of the recent modelling efforts a set of concentric cylinders has been used to describe a composite nature of a reservoir. Some systems, however, exhibit nonisotropical rock heterogeneities.
In this paper, an elliptical flow model accounting for the non isotropic reservoir properties is proposed. The model describes the effects of the wellbore storage the wellbore skin and the front skin. The model sensitivity to the storativity, F, [defined by equation (AJ)] and mobility ratios and the front skin was investigated as well as the effect of these parameter on the flow characteristics

Introduction
Oil and gas reservoirs are frequently featured by heterogeneous rock and fluid properties. Formation heterogeneities may either be of a natural origin, such as vugs, or man-made; for instance, fractures intercepting wells, injected water bubbles, the burned or steam swept regions in thermal oil recovery processes, etc. Formations which are featured by radial or quasi-radial discontinuities in terms of fluid mobilities and hydraulic diffusivities are referred to as composite systems.
The majority of studies on composite reservoirs provide the system description in terms of the radial coordinates; rock isotropy is [hen a necessary requirement.
However, several situations exist when elliptical flow may occur: a case of anisotropic systems; a fracture intercepting a well water influx from an aquifer into an elliptical reservoir and the gravity override and/or underride in case of thermal processes.
In the recent past, various aspects of elliptical flow under the reservoir conditions were investigated by Kucuk and Brigham(l), Obut and Enekin(2) and Stanislav et al.(3). The objective of this study is to determine the system sensitivity to a variety of parameters under the line source inner boundary condition.

Mathematical Model
A two-zone elliptical composite system is considered. Each region is featured by different fluid and rock properties. A well is located at the origin of the coordinate system as shown in Figure I. The model is developed subject to the following assumptions: single phase fluid of constant compressibility is considered; the Formation is characterized by a constant thickness and homogeneous properties in each of the two regions; the two-dimensional elliptical flow takes place in both regions; the front (the interphase between the two regions) is stationary during the test period.
Other features of the model will be introduced in connection with the formulation of the boundary conditions.
The dimensionless form of the flow equation can be written as follows:
region I:
Equation (1) (Available in full paper)
Equation (2) (Available in full paper)
Equation (3) (Available in full paper)
Equation (4) (Available in full paper)
Equation (5) (Available in full paper)
Equation (6) (Available in full paper)
Equation (7) (Available in full paper)
Equation (8) (Available in full paper)
The set of equations, equations (I) to (8), represents a complete mathematical description of the problem considered. All dimensionless quantities are defined in Appendix A.

FIGURE 1. Coordinate system
FIGURE 2, Effects of CO and Sh on P∞D for a homogenous system.