Pressure Buildup Analysis Research Articles

An Investigation of Pressure-Buildup Tests in Bounded Reservoirs

Two methods of pressure-buildup analysis for bounded reservoirs are analyzed. Pressure-buildup data for a variety of well locations within various rectangular drainage shapes are generated; the results are plotted according to both methods investigated; and the resulting curves are studied for diagnostic features and rules. Introduction One of the most important aspects of formation evaluation is the design, implementation, and interpretation of a pressure-buildup test. Basically, this test requires that a producing well be shut in and that the associated change in bottom-hole pressure be measured as a function of shut-in time. Muskat introduced the first interpretation method for these pressure data in 1937, but its significance was largely overlooked until the late 1940's. At that time, van Everdingen and Hurst revived interest in transient pressure analysis with their paper on the behavior of unsteady-state fluid flow in a porous media. Miller-Dyes-Hutchinson (MDH) and Horner (HOR) in 1950 and 1951, respectively, published fundamental papers on interpretating wellbore pressure buildup papers on interpretating wellbore pressure buildup for the purpose of estimating formation permeability. Both these analysis procedures were similar in that they required a semilogarithmic plot of the test data. Miller et al. suggested that the buildup pressure, pws, be plotted against the logarithm of the time, Delta t, since the well was shut in. On the other hand, Horner suggested that the buildup pressure be plotted as a function of a time ratio, (t + Delta t)/Delta t, where t is the length of producing time before shut in. In both cases, the plotted data should result in a straight line with a slope inversely proportional to the mean formation permeability. Significantly, the Homer technique was developed for the buildup case of a constant production-rate well located in an infinitely large reservoir. production-rate well located in an infinitely large reservoir. The MDH method was developed for the case of a well located in the center of a closed (no-flow outer boundary) circular reservoir and produced to pseudosteady state before shut in. Accordingly, it is commonly believed that the Homer method is more applicable in new wells and that the MDH method is best suited for old wells. In his original work, Horner showed that an extrapolation of the semilog straight line to a time ratio of unity would yield the initial reservoir pressure, pi, provided the production period was short. Moreover, he showed that, production period was short. Moreover, he showed that, for long production periods in a bounded reservoir, the straight line would extrapolate to a pressure p*. It is important to note that p* is not the initial pressure, pi, or the volumetric average pressure, p. However, if production time is short, then pi p p*. In 1954, Matthews-Brons-Hazebroek (MBH) presented correlations that relate p* to p for a variety of bounded reservoir shapes with wells located in a number of positions. To use these correlations to evaluate p accurately, it is necessary that the Homer semilog straight line be identified correctly. The concept of formation damage, or skin factor, was introduced to transient pressure analysis by van Everdingen and Hurst. They discussed its effect on well behavior and presented methods for evaluating its presence. Probably the most reliable estimate of the skin factor is Probably the most reliable estimate of the skin factor is obtained from a pressure-buildup test. Specifically, the evaluation method makes use of the slope of the Homer or MDH semilog straight line. JPT P. 991

Determining Average Reservoir Properties From Gathering-Line Transient Analysis for a Multiwell Reservoir

The increased importance of underground gas storage has prompted the development of a new method for finding the average reservoir properties of a multiwell gas reservoir. This method provides the properties of a multiwell gas reservoir. This method provides the same information as a conventional pressure buildup analysis, but eliminates the need for monitoring the pressure buildup of each well. Introduction With the increased importance of underground gas storage, gas companies need accurate statements of gas in place as a function of pressure and as a measurement of reservoir performance. Since gas-storage reservoirs operate all year, the time available to obtain the data for a reservoir analysis is limited to a relatively short time period between the injection and the withdrawal cycles. In the past, an isobaric map of the short shut-in pressure of each well was used to arrive at an average reservoir pressure. This method did not give any indication of pressure. This method did not give any indication of reservoir performance, and, in tight reservoirs, it was seen that the short shut-in pressure would not give the stabilized reservoir pressure. Because of these reasons, a new method was sought to give the needed information. The principal method for estimating a formation's performance and pressure in a gas well is the analysis of performance and pressure in a gas well is the analysis of shut-in bottom-hole pressure-buildup data. Applying this method to a gas well requires that the well be produced to pseudosteady state and that it be shut in for produced to pseudosteady state and that it be shut in for a length of time sufficient to obtain a clearly defined straight line on the plot of bottom-hole pressure vs log (t + Delta t)/t. From the slope of this straight line and from other obtainable data, the effective permeability, flow efficiency, skin factor, and reservoir pressure at infinite shut-in time can be estimated. Using this process as a base, Matthews et al. developed a method for calculating the average reservoir pressure in a bounded reservoir - a reservoir with no water drive. Their procedure requires that each well's rate and pressure buildup be known before applying the method. This would require additional expenditures not normally necessary in a gas-storage field operation. From the foregoing discussion, it is apparent that it is desirable to have an alternative method for obtaining the same data as derived from the conventional buildup analysis. The Matthews et al. method is not practical for a gas-storage reservoir, but it does offer a solution to the problem. Using the same principle used by Matthews et al. for finding the average reservoir pressure, a new and practical method for estimating average reservoir properties is presented in this paper. This new method properties is presented in this paper. This new method does not require that the pressure buildup and rate of each well be known. The required pressure data are obtained by observing the transient behavior of the main header line with all producing wells open to the gathering lines. The new method yields the same information as the conventional pressure-buildup analysis, but avoids the need for shutting in each well and measuring the pressure buildup. Theory Using methods similar to those used by other authors, it is shown in the Appendix that the actual pressure drop at any point in a multiwell reservoir will be the sum of the pressure drops of the individual wells. pressure drops of the individual wells. JPT P. 835

Well-Test Analysis for Wells Producing From Two Commingled Zones of Unequal Thickness

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

Determining Average Reservoir Pressure From Pressure Buildup Tests

Abstract Two simple and equivalent procedures are suggested for improving the calculated average reservoir pressure from pressure buildup tests of liquid or gas wells in developed reservoirs. These procedures are particularly useful in gas well test procedures are particularly useful in gas well test analysis, irrespective of gas composition, in reservoirs with pressure-dependent permeability and porosity, and in oil reservoirs where substantial gas porosity, and in oil reservoirs where substantial gas saturation has been developed. A knowledge of the long-term production history is definitely helpful in providing proper insight in the reservoir engineering providing proper insight in the reservoir engineering aspects of a reservoir, but such long-term production histories need not be known in applying the suggested procedures to pressure buildup analysis. Introduction For analyzing pressure buildup data with constant flow rate before shut-in, there are two plotting procedures that are used the most: the procedures that are used the most: the Miller-Dyes-Hutchinson (MDH) plot and the Horner plot. The MDH plot is a plot of p vs log Deltat, whereas the Horner plot is a plot of p vs log [(t + Deltat)/Deltat]. Deltat is the shut-in time and t is a pseudoproduction time equal to the ratio of total produced fluid to last stabilized flow rate before shut-in. This method was first used by Theis in the water industry. Miller-Dyes-Hutchinson presented a method for calculating the average reservoir pressure, T, in in 1950. This method requires pseudosteady state before shut-in and was at first restricted to a circular reservoir with a centrally located well. Pitzer extended the method to include other Pitzer extended the method to include other geometries. Much later, Dietz developed a simpler interpretation scheme using the same MDH plot: p is read on the extrapolated straight-line section of the pressure buildup curve at shut-in time, Deltat,(1) where C is the shape factor for the particular drainage area geometry and the well location; values for C are tabulated in Refs. 5 and 13. For a circular drainage area with a centrally located well, C = 31.6, and for a square, C = 30.9.Horner presented another approach, which depended on the knowledge of the initial reservoir pressure, pi. This method also was first developed pressure, pi. This method also was first developed for a centrally located well in a circular reservoir.Matthews-Brons-Hazebroek (MBH) introduced another average reservoir pressure determination technique, which has been used more often than other methods: first a Horner plot is made; then the proper straight-line section of the buildup curve is proper straight-line section of the buildup curve is extrapolated to [(t + Deltat)/Deltat] = 1 (this intercept is denoted p*); finally, p is calculated from(2) m is the absolute value of the slope of the straightline section of the Horner plot:(3) pDMBH (tDA) is the MBH dimensionless pressure pDMBH (tDA) is the MBH dimensionless pressure at tDA, and tDA is the dimensionless time:(4) tp k a pseudoproduction time in hours:(5) PDMBH tDA) for different geometries and different PDMBH tDA) for different geometries and different well locations are given in Refs. 6 and 13.The second term on the right-hand side of Eq. 2 is a correction term for finite reservoirs that is based on material balance. Thus, for an infinite reservoir, p = pi = p*, where pi is the initial reservoir pressure. SPEJ P. 55

Two-Rate Flow Test, Variable-Rate CaseApplication to Gas-Lift and Pumping Wells

A disadvantage of the familiar two-rate flow test is that the flow rate during the two flow periods must be constant. Here is an extension of the method that can handle variable rate in the second flow period. The test is particularly well suited to wells that have been producing at stable rates for a relatively long time. Gas-lift or pumping wells usually qualify as prime candidates for this approach. Introduction Pressure buildup analyses are widely used to obtain Pressure buildup analyses are widely used to obtain reservoir data such as effective flow capacity of the formation. However, over the years several drawbacks, arising from the procedure of shutting in the well at the wellhead, have become apparent. The mathematics of pressure buildup analysis is based on the assumption that the well is shut in at the sand face, and that no production occurs after the well is shut in. In practice, the well is closed at the surface, and flow into the wellbore continues for a period of time. This is referred to as afterflow. Afterflow in some cases, especially in tight reservoirs, can last for many hours. This requires that the well be shut in for a long period so that data may be obtained that are free of afterflow. When the shut-in period is long, the loss in production may make it period is long, the loss in production may make it uneconomical to conduct a pressure buildup test. Drawdowns are free of afterflow. However, they suffer from two drawbacks. The first is the requirement that pressure be completely stabilized before the test is run. This means a full buildup must precede a drawdown. The second drawback is that the precede a drawdown. The second drawback is that the annulus unloads during the test, and this complicates the analysis. In 1963 Russell published the two-rate flow test, which is designed to eliminate the disadvantages of buildups and drawdowns. The two-rate flow test conducted on a flowing well does not require a shut-in period. Thus it is free of afterflow, and no loss in period. Thus it is free of afterflow, and no loss in production is incurred. The disadvantage of the production is incurred. The disadvantage of the two-rate flow test is that the flow rate during the second flow period must be constant. This is hard to achieve, and a small drift in the rate, if not accounted for, may lead to considerable error in the calculated results. This is so because a small drift in rate affects the slope of the straight line of the pressure vs log time function plot, and the results are very sensitive to the value of the slope, as reported by Selim. To eliminate some of the difficulties encountered in the two-rate test analysis, Selim proposed a modification that consisted of following the rate reduction (second rate) with a rate increase. The modification still requires constant-rate flow and results in lengthening the test time. In this paper we eliminate the disadvantages of the two-rate flow test and maintain its advantages by removing the constant-rate requirement during the second flow period. Thus the test is easier to run, and any fluctuation in rate is accounted for in the calculations. The two-rate flow test, variable-rate case, is suited to wells that have been flowing at stable rates for a period that is relatively long compared with the second period that is relatively long compared with the second flow period. It is also suitable when semisteady-state flow occurs during the first flow period. Pumping and gas-lift wells usually meet this requirement and are prime candidates for the test. In particular, the prime candidates for the test. In particular, the afterflow periods of buildup on these wells are readily analyzed with this approach. JPT P. 93

A Simple Method for Determining Well Pressure in Closed Rectangular Reservoirs

In 1968, Earlougher et al., introduced a "buildingblock" concept, using a constart-rate well producing from the center of a closed square, to generate solutions for pressure behavior in several rectangular systems. Tables of the dimensionless pressure drop function for the square system are presented in Ref. 1, along with illustrations of the technique. Their procedure, while a step toward reducing laborious procedure, while a step toward reducing laborious computations, can sometimes be quite tedious. This note describes a further simplification for obtaining dimensionless pressures for these other rectangular systems at the producing well; other points within these systems are not considered. Conveniently, Earlougher et al. have tabulated the Matthews-Brons-Hazebroek pressure correction function, PDMBH, for 16 closed rectangular systems. These correction functions provide the basis from which the dimensionless wellbore pressure drop functions can easily be obtained. In a recent comprehensive review of pressure buildup analysis, it was shown thatwhereSolving Eq. 1 for pD yieldsIt is clear from Eq. 5 that pD is a function of pDMBH, tDA, and A/rw2. Thus, given pDMBH and A/rw2 for a particular system, pD as a function of tDA can be particular system, pD as a function of tDA can be calculated directly. Columns 2 and 3 of Table 1 present the pD and pDMBH data of Ref. 1 for the case of a well in the pDMBH data of Ref. 1 for the case of a well in the center of a square with it value for A/rw of 2,000. The pD data tabulated in Column 4 are obtained by direct substitution of tDA and pDMBH of Columns 1 and 3, respectively, into Eq. 5. As can be seen, the solutions in Columns 2 and 4 are identical. For dimensionless times less than those given in Table 1, pD can be obtained from pD can be obtained fromFor times greater than those given in Table 1, pD can be obtained bywhere CA is the reservoir shape factor, and is be Euler constant - approximately 1.7811. P. 1305

Concerning the Value of Producing Time Used in Average Pressure Determinations from Pressure Buildup Analysis

Introduction Average drainage volume pressures calculated by the Horner -Matthews, Brons, and Hazebroek (H-MBH) method have frequently been accepted with suspicion because of the apparently arbitrary method for determining the value of producing time that is normally used in the Horner plot. However, it will be shown here that for certain common conditions the value of producing time can be chosen arbitrarily. producing time can be chosen arbitrarily. Horner-Matthews, Brons, and Hazebroek Method As pointed out by Dietz, in the H-MBH method, the production time enters twice: when the buildup production time enters twice: when the buildup pressures are plotted against log [(t + ) to determine pressures are plotted against log [(t + ) to determine p*, and again when the area dimensionless time is p*, and again when the area dimensionless time is used to determine the MBH pressure correction function . It will be shown here that, if semistate flow prevails before shut-in, the use of producing time in these two ways precisely compensates producing time in these two ways precisely compensates for errors that may be made in estimating this time. This means that the correct value of average drainage volume pressure is calculated, even though errors are made in estimating the producing time. One additional requirement is that t >> delta t so that log(t + delta t ) logt, which is almost always the case. After a producing well has been shut in, the bottomhole pressure rise during the transient buildup period is described by the following equation : The derivation of Eq. 1 is based on the assumption that the well has been produced at a constant rate, q, for a period of time, before shut-in. Rate is seldom constant, and Horner' suggested the use of a "corrected time" obtained by dividing the total cumulative production by the last established production rate. This corrected time was suggested for use in place of the theoretically correct but laboriously place of the theoretically correct but laboriously applied principle of superposition. Failure to satisfy the assumption of constant producing rate and the obvious effect of t on p* has resulted in a lack of confidence in drainage volume pressures calculated by the H-MBH method. It will be shown that a lack of confidence for this reason is unfounded if the well was producing at semisteady-state before shut-in. producing at semisteady-state before shut-in. Dietz" has represented the MBH relationship by .............(2) for semisteady-state flow before shut-in. Combining Eqs. 1 and 2, we obtain ..............(3) Eq. 3 represents a combination of the Homer plot and the MBH expression for average drainage volume pressure. This combination eliminates t as a pressure. This combination eliminates t as a parameter, thereby showing that p calculated from the parameter, thereby showing that p calculated from the H-MBH method is independent of the value used for t. An article by Russell contains an example that will demonstrate the lack of dependence of p on t. Example Well C of the Russell paper provides both Horner and extended Muskat plot (Russell's Figs. 9 and 10), reproduced here as Figs. 1 and 2, respectively. JPT P. 1369

Well-Test Analysis for Vertically Fractured Wells

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. Introduction 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. JPT P. 1014

Practical Use of Drill-Stem Tests

Abstract Drill-stem testing has advanced from simple sampling of the formation to complete pressure buildup analysis. This paper reviews the mechanical arrangement of the tool, its running and pressure measurements. Various typical drill-stern test pressure charts are shown as an aid in trouble-shooting. Purpose THE ORIGINAL PURPOSE of drill-stem tests included recovery of a sample of the formation fluids and possibly measurement of the formation pressure. Today, the nature and use of the drill-stem test is more sophisticated and most techniques developed for drawdown and buildup tests have been adapted to the shorter drill-stem test. These might include computation of formation permeability, k, skin, s, flow efficiency or damage ratio, D.R., productivity index, P.I., and radius of investigation, ri. At times, reservoir discontinuities such as faults, small reservoirs, two-layer performance, lenses in the pay and fluid contacts may be recognized. A sample of the formation fluids is usually caught at near formation pressure during the test and brought to the surface without release of pressure. If properly removed and transported, such sample might be analysed in a PVT laboratory. At present, the sample usually is only used to establish fluid-type, gas-oil ratio, viscosity, gravity, etc., as measured in the field. In a drill-stem test the pipe above a shut-off valve is filled with air adjusted by water cushion as the pipe is lowered into the mud-filled well. This valve can be intentionally opened and closed after the pipe has been lowered to the desired depth and packer, set to seal off the mud liquid within the annulus of the well. Thus the well may be produced against a controlled amount of back pressure when the tool is initially opened. Test Procedure-Mechanical Equipment Most drill-stem tests taken today are using equipment and procedures developed by major service companies. Some of the tools involved and typical charts for two such companies are included in this paper as exhibits A and H, respectively, to give the reader an appreciation of complexity. Two or more bottom hole pressure gauges are usually employed. The upper gauge measures the pressure inside the drill pipe or tubing string below the shut-in valve. The lower gauge measures the pressure on the outside of the tool. Both gauges are located near the bottom of the equipment. A section of slotted pipe is used to permit entry of the fluids from the well bore below the set packers into the inside of the string of pipe. To reduce buildup of pressure ahead of the unset packer while it is being lowered into the well, fluids are permitted to flow through these slots and various other mechanisms located below the shut-in tool. Such fluids constantly move from below the unset packer through the pipe to the annulus above the packer while the tools are run into the well. In most cases this technique results in a shorter running time without damage to the tools or plugging of the slots or other restrictions in the flow string below the shut-in tool in the drill pipe.

Well-Test Analysis for Wells Producing Commingled Zones

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

A General Pressure Buildup Theory for a Well in a Closed Drainage Area (includes associated paper 6563 )

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.

A Simplified Method of Pressure Buildup Analysis for a Stabilized Well

This method of pressure buildup analysis employs a graphical procedure termed "negative superposition". The technique estimates the effect of one reservoir disturbance when the effects of two or more disturbances are being superimposed upon each other. A special advantage of this method is that it permits the use of much simpler mathematical expressions in the analysis. Introduction This paper presents a new buildup analysis method for a well that has produced long enough to be in pseudosteady-state, or stabilized flow. (These two pseudosteady-state, or stabilized flow. (These two terms are used interchangeably here.) The method, which is equally applicable to infinite-acting wells, is based on the change in pressure caused of oil by the negative rate effect due to shutting the well in. This effect is separated graphically from the continued pressure decline tendency caused by production previous to shutting the well in. production previous to shutting the well in. Although a Homer-type analysis can be used for most infinite-acting well it is inadequate if the pressure buildup has been taken from drillstem tests when pressure buildup has been taken from drillstem tests when the pressure is increasing at the time of shut-in. A later paper is planned for dealing with this problem. Many different techniques have been published to analyze pseudosteady-state pressure buildup. including the Muskat, Miller-Dyes-Hutchinson, and Homer methods. The method of this paper, which for simplicity will be referred to as the "Slider method" has two distinct advantages over those previously published: previously published: 1. When the drainage area approximates radial flow, such as for a well in the center of a square or hexagon, a complete analysis can be accomplished without prior knowledge of the porosity or effective compressibility. 2. The Slider method gives a straight-line pressure plot for much longer periods than do the other methods. plot for much longer periods than do the other methods. This is not an insignificant effect. The plot may be straight for as much as 40 times as long as it is for some of the other methods. There does not seem to be much to say about the first point. To do a complete buildup analysis by other methods it is necessary to know at least the porosity and effective compressibility for the drainage porosity and effective compressibility for the drainage area. The Slider method uses the change in pressure with time before shut-in (which is a function of the porosity and compressibility) to avoid the independent porosity and compressibility) to avoid the independent evaluation of the porosity and compressibility. This is no small advantage since the compressibility of a reservoir below the bubble point is very sensitive to the gas saturation, which in turn is difficult to determine with sufficient accuracy. Ramey and Cobb compared the straight-line portion obtained by various plotting techniques as applied to a square drainage area with the well in the center. Part of their results the time limits on the Part of their results the time limits on the various methods are shown in Table 1, along with the time limit using the Slider plot. Since pseudosteady state begins at about tDA = 0.1 for the pseudosteady state begins at about tDA = 0.1 for the square drainage, smaller producing times were of no significance. Note that the proposed techniques will give an accurate straight-line plot for a period as much as 40 times as long as the other two plots. JPT P. 1155

Analysis of gas pressure buildup within a porous catalyst particle which is wet by a liquid reactant

During the startup of a liquid-feed catalytic reactor, liquid reactant can wet the outside surfaces of porous catalyst particles and wick into the pores by capillary flow. Under these conditions, the liquid in the pores of the catalyst particles will prevent the escape of gaseous reaction products until sufficient gas pressure is generated to expel the liquid from the pores. A mathematical analysis of the gas pressure buildup within a porous catalyst particle which is wet by liquid reactant was developed which considers the simultaneous processes of mass transfer, heat, transfer and chemical reaction within the porous particle. Capillary and viscous forces were considered in developing methods to predict the residence time of liquid reactant in the catalyst particle. This liquid penetration model is used to explain the unusually long ignition delays and sudsequent reactor pressure excursions observed during the startup of certain liquid-feed catalytic reactors. In particular, startup characteristics observed with hydrazine catalytic reactors are noted and discussed relative to the computed liquid pore residence times.

Short-Time Well Test Data Interpretation in the Presence of Skin Effect and Wellbore Storage

The log-log type-curve described here shows clearly the presence and duration of wellbore storage as well as the presence and duration of wellbore storage as well as the presence of linear flow due to fracturing. It can be used to presence of linear flow due to fracturing. It can be used to obtain quantitatively the information normally obtained from pressure buildup analyses and to identify the proper straight pressure buildup analyses and to identify the proper straight line in pseudo-radial flow for a fractured well. Introduction Specifications for modem well testing (drawdown or buildup) are usually written in such a way that a well will be tested for a period of time long enough to reach and define a proper "straight line" when test data are plotted in conventional manners. Pressure data obtained before the straight line is reached are not often analyzed, despite the fact that a number of publications have advanced methods for doing so. publications have advanced methods for doing so. One reason for this situation is that many factors are known to affect the short-time data. "Short-time data" signify data obtained before a conventional straight line is reached. Some of the factors are the effects of wellbore storage, perforations, partial penetration, and well stimulation such as fracturing or acidizing. Although the effects of such factors are generally known, the duration and importance have not been clearly defined in all cases particularly when these effects are combined in a well test. However, recent studies have revealed a great deal of potentially useful information concerning the analysis of short-time well test data. Our purpose here is to illustrate the interpretation of short-time well test data through presentation of field examples. Factors to be considered presentation of field examples. Factors to be considered will include wellbore storage, well damage, and fractured wells. Wellbore Storage and Skin Effect The effect of wellbore storage or unloading was originally considered by van Everdingen and Hurst. These studies called attention to the fact that the storage or unloading of fluid contained within the wellbore could cause a significant difference between the surface production rate and the sand-face flow rate in a well immediately following sudden changes in production rate. Gladfelter et al. presented a method production rate. Gladfelter et al. presented a method for correcting pressure buildup data for the changing sand-face flow rate. In another publication a method was presented for estimating the duration of the storage effect, and the Gladfelter et al. correction was generalized to apply to drawdown data. Fundamentally, wellbore storage can occur in several ways. Fluid can be stored by compression of the fluid in a completely filled wellbore, or by movement of a gas-liquid interface. Russell presented a method for analysis of the latter case, pointing out that the virtue of the method was that it was not necessary to know the sand-face flow rate as was the case in the Gladfelter et al. method. One problem with Russell's method was that only a portion of the short-time data was used, yet no criterion for selection of the data was presented. presented. Recently, Agarwal et al. re-examined this problem and presented dimensionless pressure-dimensionless time plots for the case of a well in an infinitely large reservoir, producing at constant surface production rate, and having wellbore storage and a skin effect. Fig. 1 presents a portion of their results. The usual definitions of dimensionless groups were employed. JPT P. 97

Pressure Buildup Behavior in a Two-Well Gas-Oil System

Abstract In analyzing pressure buildup tests for field wells producing both oil and gas, the common practice is to use a modification of single-phase flow theory. Validity of such an approximation has been demonstrated for single-well solution-gas-drive systems. This paper indicates that such approximations are also valid for two-well solution-gas-drive systems, which infers that the technique can be used for multiple-well systems. A computer was used to simulate the behavior of a two-well solution-gas-drive reservoir to test the validity of the above type of analysis. Simulation results indicate that pressure buildup tests in such a system can be analyzed within engineering accuracy for formation permeability and pressure. A rule of thumb is given for estimating the length of time a well must be shut in for the pressure in a nearby producing well to increase significantly. To observe such an increase, the shut-in well would have to be left shut in much longer than normal. INTRODUCTION An important technique for obtaining data concerning a producing petroleum reservoir is the pressure buildup test. Such a test, when properly conducted and analyzed, provides information on the average reservoir pressure and the permeability in the major drainage area of a well. The theory upon which the analysis of a pressure buildup test is based assumes that the behavior of the fluid in the reservoir is adequately described by the diffusivity equation.1 Use of the diffusivity equation implies that a single fluid of small and constant compressibility is flowing. To analyze a pressure buildup test in situations which involve more than one fluid phase, the single-fluid analysis is extended. This paper verifies the extension of pressure buildup analyses to two-phase, two-well systems. Miller, Dyes and Hutchinson1 have presented a technique for analyzing pressure buildup tests in circular bounded reservoirs. Their method requires that the reservoir be producing at pseudo-steady state (constant pressure-gradients) prior to shutin. An alternate reservoir model, presented by Horner,2 assumes that the reservoir is infinite. The Homer model does not assume a pseudosteady state prior to shut-in, but the assumption that the reservoir is infinite implies that the average pressure can be determined only if the total production prior to shut-in is small compared to the total fluid originally present. Limitations imposed by the pseudo-steady state and infinite reservoir assumptions are avoided by the technique proposed by Matthews, Brons and Hazebroek.3 Their method can be used to calculate both the permeability and the average pressure in bounded single-well systems which are not necessarily at steady state. They also suggested determining the average pressure of a multi-well reservoir producing from pseudo-steady state by volumetrically averaging the static pressures of the individual wells. Matthews and Lefkovits4 used numerical simulation techniques to verify that this method does produce adequate results for single-phase, multiple-well systems. Perrine5 proposed modifications to the single-phase theory so that pressure buildup tests in multiple-phase systems could be analyzed. He suggested replacing the single-phase mobility (k/µ) by the sum of the mobilities of the individual phases, and replacing the fluid compressibility by an average compressibility weighted by the saturations of the separate phases. By using numerical techniques to solve the two-phase flow problem for a single-well system, he concluded that this approximation gives results within engineering accuracy. More recently, Weller6 has shown that, for a gas-oil system with a single well at the center of a circular reservoir, pressure buildup tests can be analyzed adequately by using the modification proposed by Perrine. Weller's results also indicate that, as the gas saturation increases, this analysis becomes less accurate.

Some Considerations on Bottom Hole Pressure Build-Up Phenomena-(III)

The general aspects of the present study are summarized in "Abstract" of the first instalment.This instalment deals with the explanation of the author's new method of bottom hole pressure build-up analysis for exponentially declining flow rate case. (to be continued)

Some Considerations on Bottom Hole Pressure Build-up Phenomena-(II)

The general aspects of the present study are summarized in “Abstract” of the preceding instalment. This instalment deals with the explanation of the author's new method of bottom hole pressure build-up analysis and also the discussions on the existing conventional methods, referring to numerical and field examples. (to be continued)

Some Considerations on Bottom Hole Pressure Build-up Phenomena-(I)

Approximately linear or exponential flow rate decline is often reported at the conventional DST or Flow Test. At the present study, a new practical method based on theoretical considerations is brought forward for pressure build-up analysis in such cases. Its characteristic feature exists in plotting the observed pressure data vs the appropriate functions different from that of logarithmic type as in the conventional methods, which are derived, taking the flow period, shut-in period and decline rate into considerations, and the numerical values of which can be easily known on the prepared charts. This new method provides with the simultaneous evaluation of prevailing reservoir pressure, transmissibility and flow decline rate, even though the last is previously unkown, and it seems to the author that better results can be obtained by the new method than by other conventional ones. The evaluation of skin factor is also treated at the present study. As a matter of course, the skin factor formula for variable rate case differs from the conventional one for constant rate case and the accuracy of results by the latter is theoretically discussed. It is considered that the modification for the skin factor formula has its practical meaning especially at DST, because the final flow rate can't generally be known. The full contents will appear in instalments.

Extensions of Pressure Build-Up Analysis Methods

Extensions of Pressure Build-Up Analysis Methods

Unsteady-State Behavior of Naturally Fractured Reservoirs

Abstract A simplified model was employed to develop mathematically equations that describe the unsteady-state behavior of naturally fractured reservoirs. The analysis resulted in an equation of flow of radial symmetry whose solution, for the infinite case, is identical in form and function to that describing the unsteady-state behavior of homogeneous reservoirs. Accepting the assumed model, for all practical purposes one cannot distinguish between fractured and homogeneous reservoirs from pressure build-up and/or drawdown plots. Introduction The bulk of reservoir engineering research and techniques has been directed toward homogeneous reservoirs, whose physical characteristics, such as porosity and permeability, are considered, on the average, to be constant. However, many prolific reservoirs, especially in the Middle East, are naturally fractured. These reservoirs consist of two distinct elements, namely fractures and matrix, each of which contains its characteristic porosity and permeability. Because of this, the extension of conventional methods of reservoir engineering analysis to fractured reservoirs without mathematical justification could lead to results of uncertain value. The early reported work on artificially and naturally fractured reservoirs consists mainly of papers by Pollard, Freeman and Natanson, and Samara. The most familiar method is that of Pollard. A more recent paper by Warren and Root showed how the Pollard method could lead to erroneous results. Warren and Root analyzed a plausible two-dimensional model of fractured reservoirs. They concluded that a Horner-type pressure build-up plot of a well producing from a factured reservoir may be characterized by two parallel linear segments. These segments form the early and the late portions of the build-up plot and are connected by a transitional curve. In our analysis of pressure build-up and drawdown data obtained on several wells from various fractured reservoirs, two parallel straight lines were not observed. In fact, the build-up and drawdown plots were similar in shape to those obtained on homogeneous reservoirs. Fractured reservoirs, due to their complexity, could be represented by various mathematical models, none of which may be completely descriptive and satisfactory for all systems. This is so because the fractures and matrix blocks can be diverse in pattern, size, and geometry not only between one reservoir and another but also within a single reservoir. Therefore, one mathematical model may lead to a satisfactory solution in one case and fail in another. To understand the behavior of the pressure build-up and drawdown data that were studied, and to explain the shape of the resulting plots, a fractured reservoir model was employed and analyzed mathematically. The model is based on the following assumptions:1. The matrix blocks act like sources which feed the fractures with fluid;2. The net fluid movement toward the wellbore obtains only in the fractures; and3. The fractures' flow capacity and the degree of fracturing of the reservoir are uniform. By the degree of fracturing is meant the fractures' bulk volume per unit reservoir bulk volume. Assumption 3 does not stipulate that either the fractures or the matrix blocks should possess certain size, uniformity, geometric pattern, spacing, or direction. Moreover, this assumption of uniform flow capacity and degree of fracturing should be taken in the same general sense as one accepts uniform permeability and porosity assumptions in a homogeneous reservoir when deriving the unsteady-state fluid flow equation. Thus, the assumption may not be unreasonable, especially if one considers the evidence obtained from examining samples of fractured outcrops and reservoirs. Such samples show that the matrix usually consists of numerous blocks, all of which are small compared to the reservoir dimensions and well spacings. Therefore, the model could be described to represent a "homogeneously" fractured reservoir. SPEJ P. 60ˆ