A new mathematical model is developed to predict the transient pressure behavior of a partially-penetrating well PPW). The model is derived by solving the two-dimensional (2-D) diffusivity equation. The Laplace transformation and the separation of variables technique are used. A convolution technique is used to incorporate the wellbore storage and skin factor. The solution has the form of the infinite Fourier-Bessel series and can be easily implemented using a personal computer.

Using the new model, the pressure behavior of a PPW is analyzed, and the approach of current interpretation techniques is investigated. Currently, it is claimed that a plot of pressure versus log (time) displays an early- and a late-time straight time. It has also been assumed that there is a spherical flow period between the early-time and late-time radial flow regimes. This investigation shows that there is neither early-time straight line nor spherical flow period. What may appear to be a straight line on these plots is merely the result of an inflection point. Therefore, the interpretation techniques based on the existence of an early-time radial flow period and spherical flow in the transition period are questionable. Further study has also shown that the type-curve matching methods provide a non-unique solution.

Pressure-derivative and pressure-integral behavior of a PPW are also examined. Derivative and integral type curves were constructed, and their ability to interpret the test data is investigated.

A new interpretation technique is proposed to alleviate the short-comings of existing methods. The new method provides unique solutions to horizontal and vertical permeability's and skin effect. The method is based on the de-convolution of measured pressure and flow rate data. The limitations of the proposed interpretation technique are also outlined.


Often, only a portion of hydrocarbon-bearing formations is perforated. Occasionally the borehole does not completely penetrate the entire formation. Such well completions are referred to as restricted-entry, limited-entry, or partially penetrating wells. The transient flow behavior of a PPW is different from that of a fully penetrating well (FPW), as illustrated by Fig. 1. In the case of a FPW, adequately described by one-dimensional radial flow, a plot of pressure versus logarithm of time yields a straight line whose slope is related to formation capacity. Transient pressure response is more complex in the case of a PPW.

Transient pressure response of a PPW has been extensively studied.1–32 Several models were derived by solving the two-dimensional (2-D) diffusivity equation with various mathematical techniques. Among the mathematical methods used are Greens' function, integral transformations, and finite differences. In most studies, the final solution is not in closed form and requires long computation time. Usually, wellbore storage and skin factor were not included in the models. Most of the studies present equations either to predict the pressure response of a PPW to fluid withdrawal or to determine the pseudo skin factor for a given set of reservoir data. Interpretation of measured pressure data to determine reservoir parameters such as permeability and skin factor due to damage has been attempted only in a few studies.12,19,20,31

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