New Skin and Wellbore Storage Type Curves for Partially Penetrated Wells Fikri J. Kuchuk, * SPE, Sohio Petroleum Co. Paul A. Kirwan, * * Sohio Petroleum Co. Summary. An analytic solution is derived for the transient pressure behavior of a partially penetrated well when wellbore storage and skin effects are significant. New type curves, generated by this solution, enable the engineer to analyze well test data from partially penetrated wells and to discriminate between near-wellbore damage and pseudoskin caused by partial penetration. Introduction Partial completion of wells to inhibit gas and water coning has been common practice in the petroleum industry for many years. Numer- ous studies, predicated on a variety of assumptions, have inves- tigated the theoretical pressure response and productivity of such wells. However, very few of these studies have addressed the prob- lem of interpreting early-time transient pressure data from a par- tially penetrated well when wellbore storage is significant. The value of early-time analysis for fully penetrated wells has been discussed by several authors. 1-5 There are a number of rea- sons why a reliable model for interpreting early-time data is espe- cially needed in partially penetrated wells. The transition period preceding radial flow in a partially completed well can display markedly different characteristics from its fully completed coun- terpart, even if the apparent total skin in the two systems is identi- cal. In other words, pseudoskin, caused by partial penetration, does not have the same effect on early-time data as damage skin, which is conceptually associated with an infinitesimally thin region around the wellbore. This is an important observation that provides the practicing engineer with means of identifying the various compo- nents oftotal skin, and of verifying conclusions drawn from analy- sis of data obtained after radial flow is established if such data are available. In many instances, however, radial flow is not established with- in the duration of a well test. When wellbore storage is significant, the time it takes to reach the end of the afterflow effects is depend- ent on the total skin of the system. Because of the large total skin apparent in many partially completed wells, the time to the end of afterflow can be prohibitively long. Cost or operational restrictions can make it impractical to run tests long enough to determine radi- al flow. In these circumstances, early-time analysis is the only ap- proach available. Analysis of early-time data may prove to be of particular value in interpreting results from tests in saturated or near-saturated reser- voirs. Pressure drawdown in a flowing well causes the evolution of solution gas within the vicinity of the wellbore. Oil mobility then becomes a function of distance from the well. The Horner analy- sis, if feasible, will yield the total skin resulting from well bore damage, completion geometry, and the modified transmissibility around the wellbore caused by the presence of free gas. Only by including an interpretation of early-time data can one hope to iden- tify the different components of total skin. Moreover, the problem oftest interpretation in saturated reser- voirs is often confounded by the presence of a gas cap, which can cause two complicating factors. First, a large standoff that inhibits gas coning can lead to very low penetration ratios. Second, if a well is in direct communication with a gas cap, a Horner plot of buildup data may exhibit a pronounced curve flattening that may adversely affect the semilog straight-line period. 6 At best, this can 'Now with Schlumberger·Doll Research. "Now with BP Petroleum Development. Copyright 1987 Society of Petroleum Engineers 546 make the selection of a straight line very difficult; at worst, it can represent a breakdown of the radial flow assumption, which is an integral part of the Horner method. Some of these problems were recognized and discussed in two case studies 7 ,8 of buildup test in- terpretations in Prudhoe Bay field in Alaska. The purposes of this study are (1) to present an analytic solution that describes the transient pressure behavior of a partially penetrated well flowing at a constant rate in an infinite reservoir, and that takes into account wellbore storage, skin, and permeability anisotropy; (2) to present a set of new type curves for partially penetrated wells in the Gringarten format; (3) to compare the above results with previous studies and to emphasize certain salient differences; and (4) to show by example how the new type curves may be used to estimate the various components of skin in a saturated reservoir. Previous Work Streltsova 6 used the McKinley format to present type curves for partially penetrated wells with wellbore storage. As in the original McKinley approach, 3 all calculations were made for zero wellbore skin and a single value of the diffusivity parameter, krltjJp.c t • on the basis that this parameter exerts less influence on pressure response than transmissivity, krhlp.. The Streltsova solution was obtained by assuming a uniform-flux condition at the wellbore and using the McKinley wellbore storage (afterflow) algorithm.3 Bilhartz and Ramey 9 and Gringarten and Ramey 10 have argued that a uniform-potential (infinite-conduc- tivity) inner-boundary condition is a better representation for the partially penetrated wells than a uniform-flux condition. The differ- ence in early-time pressure behavior resulting from these different inner-boundary conditions will be seen later when the Streltsova solution is compared with results from this study. In 1975, Gringarten and Ramey 10 investigated the infinite- conductivity boundary condition in some detail. By considering the flux distribution along the perforated interval in a partially penetrated well after radial flow is established, they were able to define an "equivalent average pressure point." This represents the location in the vertical direction at which, after radial flow is established, the pressure obtained by assuming uniform flux at the wellbore is equal to the expected wellbore pressure for a uniform-potential con- dition. Bilhartz and Ramey 9 used a finite-difference model to inves- tigate the combined effects of wellbore storage, skin, partial penetra- tion, and vertical anisotropy on transient pressure behavior, assuming uniform potential at the wellbore. These numerical simu- lation results were in good agreement with the analytic solution presented here. Chu et al. 11 recently presented a method for analyzing transient pressure data dominated by wellbore storage and skin. They sug- gested that available type curves for fully penetrated wells can be used for partially penetrated wells by use of a simple coordinate transformation. The implications of this transformation are discussed later. SPE Formation Evaluation, December \987