The Fenske Conservation Method for Pressure Transient Solutions with Storage

A method for determining pressure transient solutions for wells with storage (and skin) is presented, based upon an approach developed by Fenske in the ground water literature. By specifically conserving the entire mass of the system, including the wellbore or wellbores, it is possible to generate many common type curves. The procedure can be developed in an exact manner, in which case it is equivalent to existing techniques, however, a simple approximation step makes possible the generation of common solutions without recourse to numerical or ornate analytical techniques. In some cases the approximation provides equivalent solutions with substantially less computation, in other cases the Solutions are significantly in error, although perhaps still usable. The achievement of computationally rapid, closed-form solutions is of significant advantage for current developments in computerized interpretation. Examples shown in the paper demonstrate the application of the approximate Fenske method, and its exact generalization. The approximate method gives agreement within 2% of the standard single-well, storage and skin type curves. However, as the storage becomes small, as for example in the cylindrical wellbore case (where it is zero), the accuracy becomes unacceptable at early time. As an example of how the method can be applied to configurations other than those developed by Fenske the derivation of the method for the slug test problem is demonstrated. The solution obtained using the approximate method is within acceptable accuracy. The Fenske method is applicable to a wide variety of problems with storage and skin, probably including problems as yet unsolved. An approximate form is available for fast calculation, and the more correct form is equivalent to standard analytical methods. Introduction From a practical standpoint, the derivation of pressure transient solutions often involves intricate mathematics, even for quite simple configurations. As a result, the development of type curves for a new situation becomes a major research project which may involve the analytic or numerical inversion of complex Laplace transforms, numerical integration of Green's or Source Functions, or finite difference techniques. The use of such methods, although mathematically elegant, also makes it difficult to describe the reservoir behavior in a sufficiently closed form that it may be implemented in a computerized interpretation procedure. Such procedures can be designed to handle tabular data, but can operate procedures can be designed to handle tabular data, but can operate much more effectively if the model response (and its derivatives with respect to the unknown reservoir parameters) can be given in the form of analytical expressions. From a philosophical viewpoint, it must be acknowledged that the diffusion equation, which is almost always used to develop pressure transient solutions, does not, in fact, correctly represent pressure transient solutions, does not, in fact, correctly represent the behavior of the physical system. Pressure responses are not received, even infinitesimally, at great distance when a well is opened. Rather, there is a wave component of the response which transmits the signal through the system. Thus, since the diffusion equation is itself only an approximate solution to the problem, it seems opportune to seek a different approximation, hopefully one which does not involve the complexities of Bessel functions, Hankel transforms and series solutions that may be present in solutions to the diffusion equation. A method presented by Fenske in the groundwater literature in 1977 provides an intriguing possibility in the search for alternative solution methods. This method is particularly useful in problems involving wellbore storage, and, as will be shown in this paper, operates by replacing the Green's function of the problem by a simple approximation. By specifically conserving the mass of the system, the Fenske approach is capable of closely approximating known solutions to wellbore storage problems.