Presenting Discrete Flux Element (DFE) Method this paper provides the solution to the diffusivity equation for horizontal wells with regular and irregular geometries. DFE method is used to derive the equation of potential and its derivative with uniform potential as well as uniform flux Inner Boundary Conditions (IBC). The results showed that the equivalent pressure point moves in time and is not the same as the equivalent derivative point. Pressure derivative with respect to ln(tD) reflects the wellbore length and wellbore distance to the no-flow boundary.

Introduction

Potential distribution around a partially penetrating well, either horizontal or vertical, is to be obtained through solving the diffusivity equation in 3-D. Solution to the diffusivity equation for the sources that are fully penetrated, can be found directly. However in the cases where the sources are partially penetrated and/or they have irregular geometry, direct solution is impractical. This paper presents a new method as Discrete Flux Element (DFE) that permits calculating potential distribution inside the reservoir for partially penetrating wells with irregular geometry. Gravity is not neglected therefore this solution can be used to study special flow problems such as: coning, where gravity plays an important role in modeling the physics of the problem (Azar-Nejad, Tortike and Farouq Ali and Azar-Nejad and Tortike). Therefore the term potential is used throughout the paper. However, if one neglects the gravity effect one can use pressure instead. The reservoir under study is a rectilinear reservoir i.e. an infinite horizontal slab. The other type of boundary conditions can be constructed by the Method of Images. The reservoir is assumed isotropic, however, anisotropy can be introduced through well-known transformation rules. All dimensions are made dimensionless with respect to 2ht, where ht is the reservoir height. Therefore radial flow is represented by a unit slope line in a plot of potential (pressure) against ln(tD).

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