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Unsolicited. This document was submitted to SPE (or its predecessor organization) for consideration for publication in one of its technical journals. This paper has been included in the eLibrary with the permission of and transfer of copyright from the author. The journal publication of this paper is available on OnePetro as SPE-18616-PA. It was published in the 1991 SPE Formation Evaluation, Volume 6, Issue 3.
Computational considerations in obtaining well responses and pressure distributions for several problems discussed in Part I are noted here. In addition, new asymptotic expressions for pressure distributions in closed drainage volumes that are applicable during the boundary-dominated flow period are derived. Interestingly, these expressions are much simpler than the expressions available in the literature and can be used to derive shape factors for a variety of completion conditions (vertical wells, horizontal wells, and vertically fractured wells). Application of constant rate solutions to more complex conditions, such as wellbore storage and constant pressure production are presented.
Hydraulic fracturing has been used extensively over the past fifteen years to stimulate low permeability oil and gas wells. A considerable permeability oil and gas wells. A considerable amount of fractured well performance theory has accumulated during this period. Transient drawdown solutions for vertically fractured liquid wells based on numerical simulation were published in 1964. These solutions established the influence of vertical fractures on transient pressure buildup and drawdown testing. Others have investigated well tests of vertically fractured gas wells using both analytical and numerical models. Recent studies have provided new information for type-curve matching of pressure data obtained from fractured (vertical and horizontal) wells. The objective of this paper is to illustrate the application of numerical simulation in evaluation of fracture stimulation of gas wells. The previously published interpretation methods, such as pressure buildup and drawdown analyses and type-curve pressure buildup and drawdown analyses and type-curve matching, form an extremely important part of the complete analysis. Better and more comprehensive well test interpretation can often be obtained by using the so-called conventional methods and numerical modeling together.
Prats, et.al., originally developed analytical solutions for the performance of vertically fractured reservoirs for the compressible fluid case. They considered both the constant terminal pressure and constant terminal rate cases. In the case of constant rate, however, the early-time pressure transient solutions were not investigated. In 1964, Russell and Truitt published transient pressure solutions for vertically-fractured oil or water wells based on numerical simulation. From their solutions they developed methods of analyzing pressure buildup and drawdown tests with conventional plotting techniques. Clark later applied the Russell-Truitt results in analysis of water-injection well falloff data. Analytical solutions and example applications for vertically fractured wells which produce slightly compressible fluids also were presented by van Everdingen and Meyer.
More recently Gringarten, et.al., have reviewed the work of previous authors and published new solutions especially useful for published new solutions especially useful for type-curve analysis. They illustrated the use of their results (for wells with either vertical or horizontal fractures) in a companion papers.
Although a number of studies have examined the performance of vertically fractured wells, no performance of vertically fractured wells, no analytical study has examined the effect of fracture height on transient flow behavior and production capacity. The purpose of this paper is to close this gap in the knowledge about fractured well behavior.
This paper presents an analytical study of the pressure behavior of a well producing at a constant pressure behavior of a well producing at a constant rate through one vertical fracture from an infinitely large reservoir with impermeable upper and lower boundaries. The fracture height is less than the formation thickness. The characteristics of the dimensionless wellbore pressure drop vs dimensionless time curves are examined for both uniform-flux and infinite-conductivity fractures. Two fracture locations (center and top) in the producing interval are considered. Curves also are presented for calculating the additional pressure drop that results because the fracture height is less than the formation thickness. Other parameters of interest examined here are (1) the ratio of vertical to horizontal permeability and (2) the ratio of fracture height to permeability and (2) the ratio of fracture height to formation thickness. The application of theoretical results to well test analysis also is discussed.
Hydraulic fracturing, introduced in 1949, has provided the petroleum industry with an inexpensive provided the petroleum industry with an inexpensive way to increase the fluid production or injection capacity of wells. The success of many marginal wells can be directly attributed to hydraulic fracturing. Because of the many wells that have been hydraulically fractured, the study of the flow behavior of wells intersecting vertical, horizontal, and inclined fractures has received considerable attention. As a result of these studies, it is possible to predict and analyze pressure behavior of fractured wells and to compute production increases caused by fracture treatments.
All studies on vertical and inclined fractures cited assume that the fracture extends over the entire vertical extent of the formation. Field observations, however, indicate that in Some instances this assumption is not valid. Also, the fracture height through which fluid is actually produced may not be equal to the created fracture produced may not be equal to the created fracture height. There is no discussion in the literature of the transient pressure behavior of a reservoir producing through a well with a fracture that does producing through a well with a fracture that does not extend throughout the vertical extent of the formation.
This paper first examines the effect of fracture height on transient pressure behavior of a vertically fractured well producing at a constant rate. Second, we present information regarding production rate changes as a function of fracture height. Third, we delineate conditions under which it would be possible to recognize (by pressure analysis) that possible to recognize (by pressure analysis) that the thickness and fracture height are different.
REMARKS ABOUT THE NOMENCLATURE
In the petroleum engineering literature, the term, "a partially penetrating well," has been used to describe the situation where a well does not penetrate the entire thickness of the formation. penetrate the entire thickness of the formation. The term, "penetration ratio," has been defined as the ratio of the length of the open interval to the formation thickness. Unfortunately, this term also has been used to define the ratio of the fracture half-length to the drainage length. Among other definitions, the word "penetrate" implies "to enter or pierce; to make way into another body." In this context, using the term, "penetration ratio," to describe both situations is correct. Confusion regarding this terminology has not come up, since all studies on the pressure behavior of vertically fractured wells have assumed that the fracture extends over the entire extent of the formation.
In this study, however, we need to clarify this term, even though we are examining only the pressure behavior of vertically fractured wells in an pressure behavior of vertically fractured wells in an infinite reservoir.
This paper fills in some existing gaps in knowledge of the behavior ofvertically fractured wells under buildup testing conditions. It describes thegeneral characteristics of all common buildup graphs that deal with such wells.The soundness of the best approaches is substantiated: a Horner-type graph issuperior for determining permeability thickness, and a Muskat graph, properlyused, is permeability thickness, and a Muskat graph, properly used, isapplicable for determining fully static pressure.
The pressure behavior of vertically fractured wells is of great interestbecause of the large number of wells that have been hydraulically fractured.Even though much is known about the mechanics of artificial fracturing, theperformance of wells with fractures is imperfectly understood. Studies of thiscase have been based on analytical, analog, and digital methods. Of the severalinvestigations of vertically fractured wells, the most important was conductedby Russell and Truitt, whose primary objective was to provide methods toanalyze performance of vertically fractured wells for transient andpseudosteady-state flow. They considered a homogeneous isotropic reservoir inthe form of a closed square completely filled with a slightly compressibleliquid of constant viscosity. Pressure gradients were assumed to be smalleverywhere, and gravity effects were neglected. The plane of the fracture waslocated symmetrically within the reservoir and parallel to one of the sides ofthe square boundary. The fracture extended throughout the vertical extent ofthe formation, and production was at a constant rate and was assumed to comeonly through the fracture. Russell and Truitt generated dimensionless pressureat the well as a function of dimensinless time and fracture penetration for awell producing in a reservoir like that described above. They analyzed pressurebuildup behavior by means of a Horner graph and found that significantcorrections were required to obtain correct values of permeability-thicknessproduct. The correction became more important as the fracture penetration(length) increased. They recommended that penetration (length) increased. Theyrecommended that the Muskat graph be used to obtain average pressure. Russelland Truitt indicated that they were interested in the effect of producing timeon pressure buildup analysis. However, all of their remarks seem to relateexclusively to wells that have been produced to pseudosteady state. Also, nomention was made of the pseudosteady state. Also, no mention was made of theshut-in times required to obtain the proper straight line. Thus even though theRussell and Truitt paper provides important information in the form ofdimensionless provides important information in the form of dimensionlesspressure-time data, the interpretations provided in their pressure-time data,the interpretations provided in their paper are limited. Fortunately, a recentstudy presents paper are limited. Fortunately, a recent study presents generalempirical methods that can be used to explore all useful characteristics of thecommon buildup analyses graphically. These methods will be applied to determinecorrect analytical procedures for the vertically fractured well.
Transient Flow Information
As already mentioned, Russell and Truitt have presented pressure drawdowndata for a vertically presented pressure drawdown data for a verticallyfractured well in the center of a closed square.
A number of recent studies have resulted in an increased understanding of fractured-well behavior. Two of these studies provide new information on applying log-log type-curve matching procedures to pressure data obtained from fractured wells. This paper compares the applicability of type-curve and conventional semilog methods.
The pressure behavior of fractured wells is of considerable interest because of the large number of wells that intersect fractures. As a result of a number of studies, an increased understanding of fractured-well behavior has been obtained. Although the shape of actual fractures is undoubtedly complicated, most studies assume that real fractures may be ideally visualized as planes intersecting the wellbore. It is generally believed that hydraulic fracturing normally results in one vertical fracture, the plane of which includes the wellbore; however, it is also plane of which includes the wellbore; however, it is also agreed that, if formations are shallow, horizontal fractures can result. The specific orientation of the fracture plane with respect to the wellbore may be subject to debate if the well intersects a natural fracture.
Two recent studies provide new information whereby log-log type-curve matching procedures may be applied to pressure data obtained from fractured (vertical or horizontal) wells. These studies also showed that, under conditions that would appear normal, it is likely that horizontal and vertical fractures would affect well behavior sufficiently such that the orientation, vertical vs horizontal, could be determined. The purpose of this paper is to illustrate the applicability of the results paper is to illustrate the applicability of the results obtained in Refs. 1 and 2.
Vertically Fractured Wells
As mentioned in Ref. 1, new solutions for the transient pressure behavior of a vertically fractured well were pressure behavior of a vertically fractured well were needed because earlier studies were not blended for type-curve analysis. This study examined two boundary conditions on the fracture plane. The first solution, like earlier studies, assumed that the fracture plane is of infinite conductivity. This implies that there is no pressure drop along the fracture plane at any instant in pressure drop along the fracture plane at any instant in time. The second solution, called the uniform-flux solution, gives the appearance of a high, but not infinite, conductivity fracture. (This boundary condition implies that the pressure along the fracture plane varies.) Application of these solutions to field data indicates that the uniform-flux solution usually matches pressure behavior of wells intersecting natural fractures better than does the infinite-conductivity solution. On the other hand, the infinite-conductivity solution often matches the behavior of hydraulically fractured, propped fractured wells better than does the uniform-flux solution.
The Infinite-Conductivity Vertical Fracture in A Square Drainage Region
Gringarten et al. have presented drawdown data for an infinite-conductivity vertical fracture located at the center of a closed-square drainage region and producing a lightly compressible constant-viscosity fluid at a constant rate. The solution for the producing pressure at time t is
kh PwD (tD, Xe/Xf) = (pi - pwf),......(1) 141,2 qB
0.000264 kt tD = .........................(2) c Xf2