The potential distribution around a partially penetrating source is a solution of the diffusivity equation. Presenting a new method "Discrete Flux Element" (DFE) this paper provides the solution to the diffusivity equation with uniform potential as well as uniform flux inner boundary conditions (IBC). This solution provides the ability for modeling any sources with arbitrary geometry. The pressure derivative equations have been derived for uniform potential as well as uniform flux inner boundary conditions. The pressure derivative reflects the wellbore length and the distance of a horizontal well to the no-flow boundary.
The potential distribution due to any sources is a solution of the diffusivity equation. A solution can easily be found for the sources that are fully penetrated and perpendicular to the no-flow boundaries. However, there are many cases where these conditions do not exist, such as horizontal wells, vertical partially penetrating wells and fractures. In this paper sources of this type are called Sources with Finite Length (SFL). A simple mathematical equation does not exist for SFL. Sources that are called horizontal or vertical do not always consist of straight line, rather they can have irregular geometry. Approximating the geometry of the sources with a straight line sometimes could cause error especially when dealing with pressure analysis.
This paper presents a new analytical method, called here Discrete Flux Element (DFE). Because of the nature of the method it is possible to take the geometry of the sources into account without losing accuracy.
The potential due to a source is to be obtained by integrating a point source solution over the length of the source. This integration is possible by the assumption of uniform flux IBC and straight configuration. Both uniform flux and uniform potential IBC cannot be satisfied at the same time. Muskat studied the potential distribution under steady state conditions due to a vertical partially penetrating well. He showed that a uniform flux IBC solution calculated at a certain point is identical to the solution of a uniform potential IBC. Ref. 3 for a fracture and Refs. 4 through 6 for a horizontal well, using a uniform potential solution at an equivalent pressure point. simulated a uniform potential IBC (infinite conductivity) solution under unsteady state conditions.
In this paper we show that:
the equivalent pressure point moves in time and only at late times this point will be stabilized,
even at late times the location of the equivalent pressure point does not have a unique value for wellbores with different lengths,
the pressure derivative with respect to time at this point does not represent the pressure derivative of the uniform potential IBC.
In this study we use potential instead of pressure to include the gravity. DFE Method provides the ability to study the problems, such as coning, where gravity plays an important role. Azar-Nejad and Tortike and Farouq Ali applying DFE Method studied the reservoirs with bottom water or a gas cap. Refs. 7 and 8 showed the ability of the DEF Method for solving a moving boundary problem.