The mechanism of fluid flow in the reservoir can be determined through the knowledge of the potential distribution inside the reservoir. Presenting Discrete Flux Element (DFE) Method this paper provides a general analytical solution to potential distribution due to a producer with straight or curvilinear configuration for both uniform flux and uniform potential inner boundary conditions in 3-D. The results consist of:

  1. the equations for productivity indices,

  2. shape of the drainage area for each case.

This study shows that the drainage area for a horizontal well, beyond a certain radius is vertical in the vertical planes (xz and yz). In the horizontal planes it is first elliptical and at points far from the producer it is circular.


Horizontal wells are being used as a better alternative to the vertical wells in view of higher production rates and greater time of water breakthrough into the producer. The steady state potential distribution is a solution of the Laplace's Equation. The direct solution of the Laplace's equation for sources with finite length and straight or curvilinear configuration with respect to all boundary conditions in a rectilinear reservoir, specially for the case of the uniform potential inner boundary condition, is impractical. Presenting DFE Method this paper provides a general analytical solution for partially penetrating wells (horizontal and vertical) and fractures, under steady state condition. Applying DFE Method the case of transient flow also has been studied by Azar-Nejad, Tortike and Farouq Ali.

Previous Studies

Muskat studied the steady state potential distribution due to a vertical partially penetrating well for several cases of penetration depth and wellbore radius. Azar-Nejad and Tortike presented a general analytical solution for any type of geometry of the vertical partially penetrating well for a water and gas coning problem. To the best of author's knowledge, no steady state equation has been presented for potential distribution around a horizontal well or a fracture. However Borisov, presented an equation for productivity index of a horizontal well under steady state condition. This equation has been applied by Giger, et. al. to study some aspect of a horizontal well. Joshi studied the flow into a horizontal well in two vertical planes (xz, yz). Without considering the third dimension (xy), Joshi presented an equation for the productivity index of a horizontal well. A review of the solutions to the horizontal and vertical wells and fractures for transient flow problems can be found in References [1] and [2]. The application of DFE Method in a water and gas coning problem is presented in References [101 and [11].

Statement of the Problem

The aim of this work is:

  1. to determine the potential distribution in the reservoir,

  2. to study the drainage area around a horizontal well, a vertical well, and a fracture under steady state conditions and

  3. to develop the productivity index equation for each case.

The reservoir under study is rectilinear, homogeneous and isotropic. The anisotropic reservoirs can be studied through wellknown transformation rules.

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