Well-Test Analysis for a Well in a Constant-Pressure Square
Authors: Kumar Anil , Ramey Henry J. 
Published: December 2007

Abstract
Very little information exists for analyzing well tests wherein a part of the drainage boundary is under pressure support from water influx or fluid injection. An idealization is the behavior of a well in the center of a square whose outer boundary remains at constant pressure. A study of this system indicated important differences from the behavior of a well in a square with a closed outer boundary, the conventional system. At infinite shut-in, the well with a constant-pressure boundary will reach the initial pressure of the system, rather than a mean pressure resulting from depletion. It is possible to compute the mean pressure in the constant-pressure case at any time during shut-in. Interpretative graphs for analyzing drawdown and buildup pressures are presented and discussed. This case is also of interest in analyzing well tests obtained from developed five-spot fluid-injection patterns.

Introduction
Moore at. first demonstrated the application of transient flow theory to individual well behavior in 1931. Classic studies by Muskat, Elkins, and Arps in the 1930's and 1940's set the stage for two important papers in 1950 that clearly elucidated the basics of modern well-test analysis. One paper by Horner 5 summarized methods for analyzing transient pressure data from wells in infinite reservoirs (new wells in large reservoirs), and a well in a closed, circular reservoir under depletion (fully developed fields). The second paper by Miller, Dyes, and Hutchinsons considered two cases for wells assumed to have produced a long time before shut-in for pressure buildup. One case assumed a closed circular drainage boundary, and the other case assumed a circular drainage boundary at constant pressure. The former would represent annual well tests for fully developed fields, and the latter would represent wells under full water drive in single-well reservoirs. Since 1950, several hundred papers and a monograph have developed the behavior of a constant-rate well in a closed drainage shape of almost any geometry. Key in this development was a classic study by Matthews, Brons, and Hazebroek. The constant-pressure outer-boundary drainage region problem introduced by Miller-Dyes-Hutchinson was reviewed by Perrine in 1955, discussed by Hazekoek el al. in 1958 in connection with five-spot injection patterns, and mentioned briefly by Dietz in 1965. The only other studies dealing with water-drive conditions (constant-pressure outer boundaries) appear in Ref. 7 (Page 44) and in papers by Earlougher et al., published in 1968. It is clear that this case was eitherconsidered totally unimportant, orstudiously avoided.
Almost all effort was expended on studying closed outer boundary (depletion) systems.Another problem concerned the conventional assumptions involved in developing well-test analysis method. Even for the common closed (depletion) systems, field applications raised the question of the importance of assumptions. Homer method of graphing assumed the well had been produced a short time, whereas the Miller-Dyes-Hutchinson method assumed that production was long enough to reach pseudosteady state -a long time in many cases. Engineers involved in applications were further confused by differences in methods, as well as by the importance of the assumptions required for analytical solutions that established welltest methods. Recently, Ramey and Cobb showed that an empirical approach could be used to avoid assumptions (which were sufficient but unnecessary) inherent in many previous analytical studies. It was decided to apply this method to the limiting case of a well in a full-water-drive, single-well reservoir - a well in a constant-pressure square. This case is a rarity not often seen in practice. It is closely approached by either an injector or a producer in a developed fluid-injection pattern, by a single injector in an aquifer gas-injection storage test, or by some single-well reservoirs in extensive aquifers.The main point is that a well in a constant-pressure square sets a limiting condition similar to a full water drive. The more common case of a well in a partial-water-drive reservoir should lie between this behavior and that of a closed square.
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