A general analytical method is developed to calculate the potential in a rectilinear reservoir for any completion interval, well location, wellbore radius and drainage ratios. By I/sing a transformation rule, the cone shaped no flow boundaries are transformed into a straight line to transfer the reservoir domain into a rectilinear domain. This analytical solution, provides the exact solution for the potential distribution inside the reservoir due to the presence of the cone, the critical production rate the critical Carle height, the shape of the Carle, and the optimum penetration length and location for both single and double coning.


The potential drop due to production from an oil reservoir causes coning of the underlying water or overlying gas towards the well It also causes breakthrough into the producer, if the production rate exceeds a critical value. To prevent breakthrough of the undesired fluids into the well, The production rate must be kept less than the critical rate. The critical production rate is not a unique value; rather it depends on the completion interval as well as the well location. Knowing the potential distribution in the reservoir One can calculate the production rate and cone height. On the other hand, the potential itself depends On the shape of the cone. Once the cone forms, the domain ceases to be rectilinear. Therefore, the potential, which is a solution of Laplace's equation, can not be solved for deformed boundaries. In most analytical solutions, a simplified model has been used. Muskat et al.1 employed the potential for a rectilinear reservoir2 to study the water coning problem. Wheatly3, who first considered the W.D.C. (water and oil contact) as a stream line, developed a method to calculate the critical production rate.

In this investigation, The potential distribution due to any arbitrary well completion interval and location with a constant potential at the wellbore in a rectilinear reservoir is studied. The results and outcomes are presented in Section 1.

To consider the effect of the presence of the cone on the potential, a transformation rule is used to transform The deformed domain into a rectilinear one. Coning studies and the transformation rule are discussed in Section-2.

Coning phenomena, and its basic concepts were first explained by Muskat et al.1. The potential due to a partially penetrating well in a rectilinear domain was used to calculate the critical production rate. This study was only undertaken for a limited number of penetration depths and drainage ratios. The authors did not consider the effect of the presence of a cone on the potential. However, they explained that their results would have been lower if the water cone had been considered.

Previous studies

Arthur4 modified the potential values presented by Muskat1 for wellbore radii and drainage ratios, and presented the production rate calculation through a series of graphs.

Meyer et al.5, assuming that the bead of the cone reaches to the bottom of the producer at breakthrough, employed Hubbert's definition for potential and presented an equation to calculate the critical production rate.

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