Transient Flow in Elliptical Systems
Authors: Kucuk Fikri , Brigham William E. 
Published: December 2007

Abstract
This study presents analytical solutions to elliptical flow problems that are applicable to infinite-conductivity vertically fractured wells, elliptically shaped reservoirs, and anisotropic reservoirs producing at a constant rate or pressure. Type curves and tables are presented for the dimensionless flow rate and the dimensionless wellbore pressure for various inner boundary conditions ranging from K = 1 1, which corresponds to a circle, to K =, which corresponds to a vertical fracture. For elliptical reservoirs, K is the ratio of the major to minor axes of the inner boundary ellipse; for anisotropic reservoirs, it is the square root of the ratio of maximum to minimum permeabilities.

Introduction
Flow in a homogeneous and isotropic porous medium usually will be radial or linear, depending on the shape of the boundary. But in the area surrounding a vertical fracture, an anisotropic formation, or an aquifer with an elliptical inner boundary, flow will be elliptical.The study of elliptical flow in porous media is more recent than the usual radial and linear flow studies, but even elliptical flow studies date back at least several decades. The earliest discussion of steady-state elliptical flow usually is attributed to Muskat. He presented a steady-state analytical solution for the now from a finite-length line source into an infinitely large reservoir.One of the classic papers on elliptic flow by Prats et al. considered flow of compressible fluids from a vertically fractured well in a closed elliptical reservoir producing at a constant pressure. Prats et al. also producing at a constant pressure. Prats et al. also presented a solution for long times for the presented a solution for long times for the constant-rate case.Gringarten et al. found that older studies by Russell and Truitt (where flow is to a vertically fractured well) are unsuitable for short-time analysis. Gringarten et al. presented analytical solutions for fractures with infinite conductivity and with uniform flux. These solutions were for both closed squares and infinite reservoirs produced at a constant rate.In the last few years considerable work has been done on fracture systems, including numerical solutions and a semianalytical solutions for both finite and infinite fracture conductivities. Most of these studies, however, have not used the concept that the fracture is an elliptical flow system. Nevertheless, the results they obtain are important for well testing.Another problem related to elliptical flow is flow through an anisotropic porous medium. For this problem, a line source solution and a long-time problem, a line source solution and a long-time approximation presented by Earlougher are available for the constant-rate case.The purpose of this paper is to study elliptical flow in a broad sense with regard to reservoir engineering problems and to see whether these problems can be problems and to see whether these problems can be solved and whether elliptical problems can be handled in a unified, consistent manner.

Development of Elliptical Flow Models
The flow from an isotropic and homogeneous medium to a map usually will be radial, but lack of homogeneity will distort the radial flow geometry. In particular, flow will be elliptical through a porous particular, flow will be elliptical through a porous medium with directional permeability distribution (simple anisotropy). The inner geometry of a well also can distort radial flow geometry. For example, the flow will be elliptical if the well has an infinite-conductivity vertical fracture. Elliptical flow also will be encountered in flow from an aquifer to a reservoir that has an elliptical boundary at the oil/water contact.
SPEJ
P. 401