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This paper was presented at the University of Oklahoma-SPE Production Research Symposium in Norman, Okla., April 29–30, 1963, and is considered the property of the Society of Petroleum Engineers. Permission to published is hereby restricted to an abstract of not more than 300 words, with no Illustrations, unless the paper is specifically released to the press by the Editor of the Journal of Petroleum Technology or the Executive Secretary. Such abstract should contain conspicuous acknowledgement of where and by whom the paper is presented. Publication elsewhere after publication in the JOURNAL OF PETROLEUM TECHNOLOGY or the SOCIETY OF PETROLEUM ENGINEERS JOURNAL is usually granted upon request providing proper credit is given that publication and the original presentation of the paper.

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A finite difference approximation to the two- dimensional partial differential equation for unsteady-state gas flow that is capable of describing flow behavior through widely varying permeability configurations has been developed. Additionally, it is capable of describing flow through various types of boundary conditions such as open-hole completions, partially penetration wells and single-plane fractures. Inclusion of the variable flow rate boundary equations makes it possible to study periods of pressure build-up as well as draw-down tests. While the treatment of the permeability functions is not entirely correct as verified by the increasing material balance error for permeability discontinuities at large distances away from the wellbore, it does provide a much greater detailed description of the possible permeability variations than heretofore presented.

Through use of the model we have established the existence of a relatively short transient period following each change in flow rates for all geometries other than uniform permeability with an open-hole completion. This period constitutes the time required for the pressure gradients to traverse the distance from the perforated interval or fracture plane to both tile upper and lower boundaries and to establish flow paths such that the entire vertical distribution of reserves contributes its proportionate share of the production. Theoretically, it is not possible to accurately analyze results obtained during this period using one-dimensional techniques. The length of this interval can be a matter of minutes or hours for thin permeable formations, or days in-the case of some thick low-permeability reservoirs. Specifically, the time duration depends upon the completion interval, formation thickness, well stimulation horizontal and vertical permeability distributions, and porosity.

The succession of steady-state approximation method has been shown to give reliable predictions when using properly selected data. This means that routine computational methods can be used for reliable predictive calculations if the data from a meaningful test are properly used. General criteria for this selection of test and data have been included. Basically, these depend on the duration of the vertical component of flow to be expected.


There has, been considerable effort in recent years on the development of mathematical reservoir models.

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