Reliable values of static temperature are important for a number of reasons. For example, they are required in designing deep-well cementing programs and in analyzing reservoir fluid properties. Though many engineers hate discouraged the use of recorded log temperatures, these log temperatures can be used to predict static temperature. Introduction Static formation temperature should be determined as accurately as possible for a number of reasons. In-situ sauration distributions computed from resistivity logs require accurate formation water resistivities that depend on temperature. Reliable estimates of bottom-hole temperatures are important in designing deep-well cementing programs and in evaluating reservoir-fluid formation volume factors. Also, the determination of static temperature is necessary for establishing geothermal gradients that can be used to estimate the temperatures of deeper zones. More recently, new exploration techniques have used temperature as a mappable proximity parameter. proximity parameter.Unfortunately, the temperatures recorded during logging operations frequently are not static temperatures. The recorded values are too low. These low temperatures result because the circulating mud temperature frequently is much less than the formation temperature. Schoeppel and Gilarranz reported that early investigators had discouraged the use of bottom-hole temperatures obtained from logs under the assumption that the temperatures would not be correct because the mud and the formation were not in thermal equilibrium. More recently, however, Timko and Fertl suggested that the temperatures recorded while running a series of logs can be interpreted to estimate static formation temperature. They recommend use of a Horner temperature plot, similar to the conventional pressure buildup method, plot, similar to the conventional pressure buildup method, for estimating static formation temperature. Timko and Fertl demonstrate the apparent applicability of the technique with an example. Significantly, we have been unable to find any theoretical justification for an analysis of this type. However, we have seen field cases where the technique gives satisfactory results. The accuracy of this analysis was especially surprising when it was concluded that the two methods - pressure build-up and temperature buildup - are not completely analogous. Thus, the chief objective of the present study was to determine under what conditions the Horner temperature plot can be used to estimate static temperature. Comparison of Temperature And Pressure Buildup Pressure Buildup Pressure Buildup The equation that describes pressure behavior at each point at any time in the well drainage area is point at any time in the well drainage area is(1)2p 1 p c p----- + ---- ---- = ------ ------r2 r r k t Eq. 1 frequently and referred to as the "diffusivity" equation because of its similarity to the diffusivity equation in the heat-transfer literature. Subject to the constraints of an initial condition and a set of boundary conditions, Eq. 1 can be solved. Consider the case of a well producing at a constant rate and located in an infinitely large reservoir. JPT P. 1326
Pressure buildup analysis, when properly done, yields the same results nomatter which of the conventional methods is used. As illustrated here with aclosed square, general interpretation equations may be derived that are correctfor closed drainage regions of any shape. Introduction In 1935, Theis1 showed that buildup pressures in a shut-in waterwell should be a linear function of the logarithm of the time ratio(t+?t)/?t, and that the slope of the line is inverselyproportional to the mean formation effective permeability. Muskat discussedpressure buildup in oil wells in 1937, and proposed determination of staticpressure by a semilog trial-and-error plot that has been found to be applicableto a variety of buildup cases. In the late 1940's, van Everdingen presented aseries of lectures on well test analysis in the U.S. that related to aclassical study of unsteady flow by van Everdingen and Hurst.3 In1951, Horner4 presented a study of pressure buildup that appears tohave summarized fundamental efforts of a number of pioneering researchers inthe Shell companies. Horner also recommended a semilog buildup curve identicalwith the Theis curve, and presented a method for extrapolation to fully builtup static pressure for a closed circular reservoir. This sort of semilogpressure buildup plot is often referred to in the oil industry as a Hornerplot. About the same time, Miller-Dyes-Hutchinson5 presented ananalysis for buildup when the well had been produced long enough to reachpseudosteady, or true steady state prior to shut-in. Their work indicated thatbuildup pressures should plot as a linear function of the logarithm of shut-intime. As in the Horner plot, the slope of the straight line was shown to beinversely proportional to the permeability to the flowing fluid. It isinteresting that about the same time Jacob6 presented a similarapproach to determine aquifer transmissibility for a water well productiontest. Miller-Dyes-Hutchinson presented several important extensions of builduptheory. One was an initial attempt to apply buildup theory to multiphase flow. This problem was later resolved by Perrine.7 More important to thisstudy, an approach to extrapolation to fully built-up static pressure waspresented for outer boundary conditions of either no flow (closed), or constantpressure (water drive). Thus, by the early 1950's, both Horner and Miller-Dyes-Hutchinson hadpresented methods for determining permeability and static pressure from buildupdata. Although there were similaries - both involved semilog plotting - therewere confusing differences between the methods. Perrine7 presented an excellentreview of this theory in 1956 that clearly indicates the state of understandingat that time. He stated that Horner-type plot (which he referred to as the vanEverdingen-Hurst method) was valid for a new well in a large reservoir, butthat the Miller-Dyes-Hutchinson approach was best for older wells in fullydeveloped fields. The latter is a widely held misconception.
Pressure buildup for two-layer no-crossflow systems has been carefully Pressure buildup for two-layer no-crossflow systems has been carefully studied to determine the proper application of conventional analysis methods. Results of using single-layer buildup plotting forms suggested by Muskat, Miller-Dyes-Hutchinson, and Horner indicate that---under well defined conditions-all three methods can also be applied to two-layer systems. Introduction The three most common graphical techniques used to interpret buildup behavior are the methods of Muskat, Miller-Dyes-Hutchinson, and Homer. Initially, all three methods were developed for a well producing from a reservoir consisting of a single homogeneous layer. (They were lately reviewed by Ramey and Cobb.) In recent years, however, investigators have conducted studies on wells with commingled fluid production from two or more noncommunicating zones. In those cases, fluid is produced into the wellbore from two or more separate layers and is carried to the surface through a common wellbore. The layers are hydraulically connected only at the wellbore. Lefkovits et al., and Duvaut have presented identical rigorous solutions that describe the pressure behavior of a constant-terminal-rate well producing from a bounded, noncommunicating, multilayer reservoir with contrasting properties. Both Lefkovits et al. and Papadopulos have properties. Both Lefkovits et al. and Papadopulos have presented pressure behavior for the infinitely large presented pressure behavior for the infinitely large mulitlayer case. Although much has been published on the behavior of noncommunicating layered systems, the knowledge of well-test applications must be considered elementary. Consequently, pressure buildup for two-layer, no-crossflow systems has been carefully analyzed to determine the proper application of conventional analysis methods to this class of reservoir system. It is remarkable that the duration of transients is often orders of magnitude longer for multilayer systems than for a single layer. The only existing method for determining fully-static pressure for layered systems requires that pseudosteady state be assumed, so one principal objective of this study was to arrive at improved methods of estimating fully static pressure. As a practical step, we decided to limit our attention to systems of only two commingled zones. Finally, it should be emphasized that certain of the following results were first presented by Lefkovits et al. in a pioneering study of pressure buildup in such systems. For example, Ref. 6 clearly describes the extended duration of transients in multilayered systems and thoroughly presents the pressure behavior during drawdown. But only scanty pressure behavior during drawdown. But only scanty information was presented for analysis of pressure buildup by means of the Horner plot, and the method recommended for determining static pressure was the Muskat trial-and-error plot. The Muskat method requires the assumption that the well had been produced long enough to reach pseudosteady state - a very long time as shown by Lefkovits et al. Determination of static pressure for such systems can be exceedingly important. We wish to emphasize that we believe our contribution through this study is a clear definition of the applicability of conventional pressure buildup analysis methods to this important class of problems. JPT P. 27
Established methods of analyzing pressure buildup for two-layer no-crossflow systems are extended here to include the effect of the thickness of each zone for a wide range of permeability ratios. In addition, methods are presented for estimating the permeability of the individual layers. Introduction In 1961, Lefkovitz et al. presented solutions describing the pressure behavior of a well producing at a constant rate from a bounded, producing at a constant rate from a bounded, noncommunicating multilayer reservoir. Their study provided a basis for pressure test analysis of wells producing from commingled zones. They recommended a Homer graph for determining average formation flow capacity but found it unsatisfactory for the evaluation of mean or average reservoir pressure. As a result, they suggested that the pressure. As a result, they suggested that the Muskat graph be used to calculate the static reservoir pressure. More recently Cobb et al., using the results of Lefkovitz et al., examined the pressure behavior of a two-layer reservoir for a wide range of producing and shut-in conditions. They assumed that each zone was of equal thickness and they presented results along lines suggested by the general pressure buildup theory for a wide range of permeability ratios. An important conclusion of Ref. 4 was that the permeability and thickness of each individual layer permeability and thickness of each individual layer cannot be evaluated by the conventional semilog techniques. Accordingly, they recommended that an independent effort be made to determine the individual layer characteristics. The primary objective of this paper is to extend the pressure buildup analysis of wells producing from two commingled zones by including the effect of the thickness of each zone for a wide range of permeability ratios. The results presented in this paper correspond to thickness ratios of 2 and 5. It will be shown that the results may also be used to analyze reservoirs with thickness ratios of 1/2 and 1/5, provided the permeability ratio is less than or equal to 10. The permeability ratio is less than or equal to 10. The second objective of this paper is to present methods for estimating the permeability of the individual layers. These methods may also be applied to earlier work related to two-layer commingled fluid production. We consider here a two-layer reservoir that is horizontal and cylindrical; it is enclosed at the top, bottom, and at the external drainage radius by an impermeable boundary. Each layer is homogeneous and is filled with a fluid of small and constant compressibility. The pressure gradients are small, and gravity effects are negligible in the reservoir. The porosity of the layers is assumed to be equal; the porosity of the layers is assumed to be equal; the permeability and thickness of the two zones are the permeability and thickness of the two zones are the parameters under investigation. The initial reservoir parameters under investigation. The initial reservoir pressure is the same in both layers and the surface pressure is the same in both layers and the surface production rate is constant. Finally, it is also assumed production rate is constant. Finally, it is also assumed that the instantaneous sand-face pressure is identical in both layers. Analysis of Dimensionless Pressure And Time Data For convenience of discussion we shall use the dimensionless variables listed below, defined in English engineering units. JPT P. 1035
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