Advances in Practical Well-Test Analysis H.J. Ramey Jr., SPE, Stanford U.

Transcription

1 Advances in Practical Well-Test Analysis H.J. Ramey Jr., SPE, Stanford U. Introduction The 1980's produced important advances in well-test analysis. The large number of new and sometimes competitive methods, however, also produced confusion. The main objective of this paper is to consider the current state of practical well-test analysis methods. Often, new studies produce conclusions that time proves incomplete or partly untrue. The storage log-log type curve was initially presented as a method to analyze short-time data. This was later found to be impossible. But the diagnostic value of log-log curves was far more important than short-time analysis. Later, the derivative was added to the type curve, leading to the conclusions that Homer curves were no longer necessary and that short-time analysis was now possible. Neither conclusion is entirely correct, yet the diagnostic value of the derivative remains. Another major development was computer-aided interpretation. The computer was necessary to differentiate data and helped to prepare the large number of graphs required for modem interpretation. An important breakthrough resulted with the development of nonlinear regression for specific models and the ability to consider rate variation. Results of an interpretation could be used to simulate test data, and then simulated and field data could be compared. The regression coefficient or confidence limit provided a quantitative measure of the agreement between field data and the model chosen. Results were also used to determine where a correct straight line would be on a Horner buildup graph. This procedure proved that finding a Horner straight-line slope with precision was difficult. Widespread use of electronic pressure gauges and computer data acquisition created new problems for a well-test analyst. A new problem revealed by type-curve analysis is presented here. Background Well tests were originally performed to determine the quality of a well or to permit estimation of producing rates at different producing pressures. Well-test interpretation Copyright 1992 Society of Petroleum Engineers evolved into a sophisticated field of finding whether poor producing quality was a result of well damage, poor formation permeability, or low formation pressure-thus, a low driving force. Tests were also performed to find static formation pressure for materialbalance studies of gas or oil in place. In the 1950's, Homer 1 and Miller et at. 2 started the first era of modem well-test analysis: straight-line methods. Semilog graphs of shut-in pressure vs. the logarithm of shutin time, or the ratio of producing time plus shut-in time, to shut-in time generated a straight line with a reciprocal slope that was related to effective permeability. The straight line could be extrapolated to long shut-in times to estimate static pressure. Moore et at. 3 and van Everdingen and Hurst 4 had previously presented the concept of wellbore storage and had taught useful analytical methods to solve well-test problems. In 1953, Hurst 5 and van Everdingen 6 presented the concept of the wellbore-skin effect (factor). Finally, Matthews et at. 7 presented an analytical method to correct extrapolated static pressures for a well in various drainage boundaries. By 1967, pressure-buildup methods were so important that the SPE published the first monograph on well-test analysis by Matthews and Russell. 8 This important book is still one of the most popular volumes in the SPE Monograph Series, with 31,542 copies sold between 1967 and The Matthews and Russell monograph, however, signaled the end of the straight-line era of well-test analysis. All constant-rate well-test solutions in a slab reservoir had a sernilog straight line of the same slope, and it was not possible to identify the proper straight line when two or more straight segments appeared in a single well test. Surprisingly, the 1949 van Everdingen and Hurst 4 paper contained a log-log type curve for the wellbore-storage problem that started the second era of well-test analysis. Theis 9 described log-log graphic methods, as well as the Horner semilog pressurebuildup graph. Theis' work was considered historical and was reprinted by the SPE. 10 Like works on straight-line methods, loglog graphing approaches were proposed by two separate research groups at about 650 June 1992 JPT

2 spedistinguished Author SERIES the same time. The log-log procedure allowed comparison of field data and an analytical model throughout the entire time domain and could be used to find the start time of a semilog straight line. New log-log type curves were produced rapidly for a wide variety of flow models (hydraulic fractures, dual-porosity systems, etc.). The second SPE well-test monograph by Earlougher l3 reviews these developments thoroughly. A 1976 study 14 reviewed important findings of the previous decade and stressed practical applications of new methods that were not previously described. At that time, three log-log versions of the storage and skin problem had appeared, and several type curves for fractured wells had been presented. Many well-test analysts threw up their hands at the proliferation of type curves. The 1976 study concluded that a major advantage of the log-log type curve was that it was usually possible to identify the flow model and to find the start of a semilog straight line for the appropriate model. It was recommended that a Homer straight line should still be used as the final basis of analysis where possible. The 1976 study pointed out problems with fracture type curves (short apparent fracture lengths for large jobs) and other existing worries on selection of an appropriate type curve. One major advantage of the understanding that the new methods brought was that it was possible to correct bad test data and, in many cases, to fill in missing data. Problems identified in the 1976 study have since been solved and true breakthroughs presented. The purpose of this study is to present useful, practical methods for welltest analysis and design. Important new information includes selection of an industrystandard storage type curve, development of derivative methods, solution of the finitefracture-conductivity problem, and development of computer-aided interpretation and design. Storage Type Curve The traditional engineering dilemma is that the use of two methods to solve one problem yields different answers. Well-test analysis is replete with this problem. Papers often JPT June 1992 say something like "the data were analyzed by the Smith method yielding 20 md, the Jones method yielding 2 darcies, and the von Schultz type curve yielding 0.15 md. The average is...." Remarkably, the three different methods are usually different graphs of the same solution. The log-log type curves described for the first time in the Earlougher 13 monograph were considered controversial by the SPE Board of Directors. An early SPE board decision was not to publish full-scale log-log type curves because this would indicate SPE approval oflog-iog type curves or a particular type curve. Fortunately, this decision was reversed. Gringarten et al. 15 ended the controversy over the best form of the wellbore-storage and skin-effect type curve in Their type curve of the log of dimensionless pressure vs. the log of tdic D, with C D e 2s as a parameter, became the industry standard. Their original type curve combined radial flow and fracture flow results and indicated the effect of producing time on buildups. The type curve was a remarkable improvement that was accepted immediately. So one problem identified in Ref. 14 was solved. A monumentally important aspect of the Gringarten et al. 15 study is that it came from the service side of the petroleum-industry. Traditionally, the service industry marketed hardware and was not strong in innovation in pressure-transient analysis. The Gringarten et al. study signaled a remarkable change. Part of the change was connected with hardware. The quartz-crystal pressure gauge offered the possibility of perhaps eight-digit pressure measurements. Incredible results were possible. One could detect the moon passing overhead twice a day with a pressure gauge 10,000 ft deep in a shut-in well. The market for sensitive pressure measurements exploded and so did improvements in other types of pressure measurements in wells. Two other factors were also changing well-test analysis. Rapid developments in computer chips made small, large-storage computers possible and perhaps fueled developments in applied mathematics and computer science. A statistical economist in Germany offered a three-page paper on in- verting Laplace transforms. 16 The Stehfest algorithm has had a major impact on petroleum engineering well-test analysis and groundwater-hydrolysis pump-test analysis. Sometimes, it is easier to evaluate a solution in Laplace space than in real space or time. Some current computer programs store models in Laplace form and even do regression in Laplace space. In summary, Gringarten et al. 15 presented a convenient type curve for model and semilog straight-line identification, but a major step remained-combining pressure and pressure-derivative type curves. Derivative Methods Bourdet et al presented a major development in the mid-1980's. They advocated superimposing the log-time derivative of the storage and skin solution on the accepted form 15 of the storage type curve. They observed that the combination of the two type curves could lead to a unique match with field data that eliminated the need for a Homer buildup graph. They also observed that data taken before a semilog straight line could often be interpreted. As is often the case, initial claims for this new method were not entirely correct, and some major advantages of a derivative graph were not yet evident. My initial impression of the derivative procedure was that high-precision data was required to make such a procedure feasible. Actually, the key was development of good numerical procedures to differentiate field data. Bourdon-type data taken in the 1960's, as well as high-precision liquid-level data and gas-purged capillary tube data have been successfully differentiated. Practical aspects of derivative use are discussed later. One final important step made since 1976 concerns computer-aided interpretation. The variety of type curves and semilog graphs involved in well-test analysis and differentiation of field data were perfectly suited to a PC with appropriate software. Computer.Alded Interpretation Probably every research effort involved in well-test analysis began some work on computer-aided interpretation in the 1960's. 651

3 I TABLE 1-SIMULATED BUILDUP DATA k, md q, STBID t p hours rw. ft h, ft /L, cp B, ABlSTB q, PI' psi Ot, psi ~1 s At (hours) o ; PW$ (psi) 1, , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,810 1 x Interpretations ranged from computer plotting of buildup graphs to interactive numerical simulators capable of considering certain formation heterogeneities. The first modern effort to come to my attention was initiated by A.C. Gringarten in He produced a sophisticated program that used log-log type curves for model identification, progressed to Horner analysis, and eventually verified the interpretation by comparing field data with a simulation that used the interpreted parameters. The Gringarten program was developed for field use with a portable computer for wellsite test analysis. This approach started a new oilfield service industry. The next major development was application of nonlinear regression in Laplace space by Rosa and Horne. 19 Their approach avoided trialand-error matching of field data with a model simulation and provided a quantitative measure of the results-confidence limits. A major benefit of computer analysis is that it allows us to correct, filter, and select a manageable data set from thousands of data points recorded today and to perform many tedious operations, such as differentiation after numerous time adjustments. The computer frees the analyst to think and to 652! a. <I hlo, , , ,--r , C t (hours) Fig. i-log-iog type curve for simulated buildup.,_,i, ' I r------, , , , 2S00! 2000 a. l! k 37.S PW 2727 I I " 1000L L ~ ~ ~ (t+4t)/4t 1000 Fig. 2-Horner graph for Simulated buildup. k, md s C, ABlpsi TABLE 2-CONFIDENCE LIMITS FOR HORNER ANALYSIS OF SIMULATED BUILDUP DATA or or x or - Mean square deviation = x 10 4 psil! try various possible interpretative models to seek the best possible interpretation. The importance of computer-aided intrepretative methods cannot be overstated. The point is not just that the computer saves labor, but that it permits a quantum jump in the accuracy of the interpretation. I would not interpret a well test today without computer aid simply because I wish to do the best job possible. This cannot be accomplished by finding a semilog straight line on a Horner graph x 10-2 (81.55%) ( Al) (11.74%) Well-test analysis is often referred to as the inverse problem. The answer is 'four," what is the question. Questions might be what is two plus two or what does a golfer yell when he hits a ball into a group? There are many possible answers. Gringarten described a logical approach to the inverse problem for well-test analysis. We seek the simplest model that explains the well-test data. Computer-aided interpretation helps with the nonuniqueness problem. Many possible models can be tested and statistical June 1992 JPT

4 ~OO~ r ' ' , l!loo 1000L L ~ J ~ (HAtyAs ,0 Fig. 3-Test buildup data vs. simulated curve from Horner analysis parameters.! ~ SOO 1000r r----~ r_ _, c TABLE 3-FIELD EXAMPLE q, STBJD 36 t p hours 'w. ft 0.4 h, ft 165 jl. cp B. RBlSTB ti> 0.17 Ph psi \ 2,688 Ct. 13$1-1 : 5)( 10-0 at Pws At Pws (hours) (psi) (hours) (psi) 0 1, , , , , ,157 2, , ,421 2, , ,747 2, , , ', ,637 2, , , , ,982 2, ,988;66 4,892 2, , ,387 2, , , ,, ,602 2, , , , ,938 2, ' 2, ,053 2, , ,169 2, , , , ~, , , , ,035 2, , ,166 2, , ,433 2, ,1~3.29 9, , ,708 2, , ,849 2, , SO SO SOD 1000 As (hours] Fig. 4-Log log type curve for field example. measures used to select the best fit of the models tested. Perhaps the best way to illustrate. the new approach to well-test analysis is through an example. Simulated Buildup The purpose of this example is to show that Homer buildup graphs often yield poor estimates of permeability and skin. Thus, an exact simulation was prepared with known correct formation parameters. Table 1 presents input formation and fluid parameters and simulated pressure-buildup data. Fig. 1 is a log-log type curve showing a unit-slope storage line; a semilog straight line should occur at a shut-in time of about 10 hours. Fig. 2 is a Horner buildup graph where the semilog straight line starts at a shut-in time of 10 hours. The straight line was selected by magnifying the straight line JPT June 1992 i.,o" portion. The last nine buildup points appear to fit a straight line exactly. The results of the Homer analysis are a permeability of 37.6 md, a skin effect of 6.2, and ap* of2,827 psi. These results do not match the entered values of 48 md, skin of 10, and initial pressure of 2,810 psi. However, the results appear reasonable and represent results of a "field" data analysis where correct values are not known. Typecurve analysis indicated that the wellbore storage and skin model was appropriate. The resulting parameters can also be used with the storage and skin model to simulate the test, and the result compared with the data originally interpreted. Fig. 3 shows that such a comparison is not good. Next, the confidence limits of the original interpretation are computed. Table 2 gives the result. The permeability from the Horner interpretation has an error of ± 82 %. Actually, error for the storage model will always be nega- tive. Both permeability and skin are either correct or underestimated. Finally, the buildup data in Table 1 were interpreted with nonlinear regression 19 and the wellbore storage and skin model. The regression produced the input parameters (permeability, skin, and initial pressure) exactly. The results of the preceding exercise are typical of my experience with field data interpretation. For cases that appear to have almost idea110g-10g storage type curves and Horner straight lines, a test of the parameters by simulating the field data often produces poor agreement. Confidence limits of the Horner parameters are often in the 50% to 150% error range with skin-effect errors larger than permeability estimates. Subsequent nonlinear regression produces new parameters with confidence limits reduced 1O-fold. Agreement between field data and model-simulated results is often excellent. The large confidence limits for Homer analysis was a surprise. I previously thought that accuracy of a permeability estimate was determined by the accuracy of graph differentiation or perhaps ± 5 %. The problem is more complex. The apparent Horner straight line is rarely the correct one. Another informative exercise is to plot the correct straight line on the Horner graph 653

5 2800 HOOr-----~ _r ~~r_----, 2tI ! ! CI III1III1IIIII PIt 2610 'BOO.0.00.HOL--- ~ -L ~ ~~ ~ ~ I tOO Fig. 5-Horner graph for field example. Fig. 6-Magnified Cartesian graph of early buildup data. after regression. I've tried this many times in the forlorn hope that I could learn how to pick the correct Homer straight line. My only conclusion is that I do not know how to plot a Homer straight line. It is not wise to depend on straight-line hand methods when regression with a proper model is possible. Another conclusion is that the potential error in permeability and skin from a Homer analysis is often much larger than I previously thought, and both parameters are always underestimated. There are practical occasions when a Homer graph must be used. Perhaps a proper model is not obvious, or computer software is not available, but beware the potential lack of accuracy. Another observation is that regression often changes the skin effect more than the permeability. Sometimes, an apparent negative skin value from a Homer graph will be found to be positive during regression. Finally, limited experience with interpretation of successive tests on the same well indicates that regression reduces variation in apparent permeability from run to run. The confidence limits from regression represent how field data and an analytical model match and how sensitive regression is to a particular parameter. Confidence limits provide a quantitative measure of the match between field data and a model and can be used to rank annual well-test interpretations for a given well. Poor confidence limits have indicated the existence of new flow models not obvious from straight-line or log-log methods. An example of a field test that suggested a new problem is presented next. Field Example This example concerns a low-rate pumping well producing from about 13,000 ft. A pressure-buildup test was run by shutting in the pump and measuring the liquid level in the annulus with a helium-purged capillary tube and precision surface pressure measurement. This is not a typical example of a field test, but it illustrates the power of modem analytical methods. Table 3 presents formation and fluid parameters and buildup data. The purpose of the test was to determine why the oil rate from this well (36 STB/D) was so much lower than that of offset,wells. Fig. 4 is a log-log type curve that is unlike any other I've seen. However, it appears that a semilog straight line probably will occur after about 3,000 hours. Fig. 5 is a Homer graph that apparently has no straight line. The early shape of the buildup is odd and is reminiscent of a humping buildup caused by commingled zones. This was a reasonable interpretation. The thickness of 165 ft is large, and several portions of the interval were perforated. Returning to the log-log type curve (Fig. 4), the shape is similar to that of a dualporosity or communicating layered system. The unit slope storage line, however, is short and breaks far too abruptly for anything except increasing wellbore storage. Furthermore, wellbore storage is usually the first identifiable phenomenon at shut-in, so what caused the rapid pressure rise? Fig. 6 is a magnified Cartesian graph of shut-in pressures. Within the first day, the pressure (liquid level in the annulus) rose rapidly about 200 psi, then rose linearly in time for 300 hours! Thus, there appears to be two wellbore-storage periods in Fig. 6. This strange behavior led to the following analysis. Fig. 6 indicates that the liquid level in the annulus rose rapidly on the first day of shutin. Then there was a period of constant-rate wellbore storage that lasted more than 10 days. What happened? The purpose of the test was to find the cause of a low pumping rate. Pump efficiency had not been checked, and perhaps, was not considered. How could the pressure rise so rapidly at shut-in? Perhaps the pump valves were in bad con-.ooo, ,r--r , 2700 c !lC)0.. '! iOO k "96 p loo 1000 tllsg".900 At (hours).0.00 Fig. 7-Log-log type curve for field example with adjusted flowing pressure P wi + Fig. a-horner graph for adjusted field example buildup. 654 June 1992 JPT

6 I k. md s C. RBlpsi TABLE 4-CONFIDENCE LIMITS FOR HORNER ANALYSIS OF ADJUSTED FIELD EXAMPLE DATA or - + or - + or - Mean square deviation x 10 4 psi n082 ( b) (549.32%) ( b) TABLE 5-RESULTS AND CONFIDENCE LIMITS FOR REGRESSION WITH ADJUSTED FIELD EXAMPLE DATA k, md s C, RB/psi or - + or - + or - Mean square deviation = x 10 2 psi x (17.32%) (36.74%) (4.58%) "Important new information includes selection of an industrystandard storage type curve, development of derivative methods, solution of the finitefracture-conductivity problem, and development of computer-aided interpretation and design." dition and leaking. This could dump the tubing liquid into the annulus. How much of a pressure change would this cause? A useful rule of thumb is that the pipe ID in in. 2 =bbl fluid/l,000 ft. Hence, 13,000 ft of 2-in.-ID tubing would contain 52 bbl of liquid. The annular space between 8-in. and 2.5-in. pipes would contain about 58 bblll,000 ft. Thus, all the tubing liquid could cause a liquid rise in the annulus of about 1,000 ft or a pressure rise of about 300 psi. This rough calculation indicates that the rapid pressure rise was caused by the tubing liquid draining into the annulus, which suggests that both the standing and traveling valves of the pump were bad. This is a pressure-buildup analysis where a major conclusion is that the pump was bad. We have a hypothesis. What do we do about it? If the pump had not been bad, liquid would have risen in the annulus during buildup and all analytical methods would have worked. A leaky pump means that all tubing fluid was dropped into the well at shut-in. This is a slug-test injection at shutin. Known information on slug tests 20 indicates that the slug pressure rise on injection should die rapidly. Perhaps pressure rise should be referenced to the pressure after dumping the tubing fluid. That is, type curves should be the pressure difference of Pws-Pwf+' rather thanpws-pwf' wherepwf+ is the pressure after the sudden rise caused by dumping tubing liquid. Fig. 7 is a log-log type curve where the flowing pressure at 3,912 hours is changed to 1,952 psi (the P wf +), and the first three points after shut-in are deleted. Compared with Fig. 4, Fig. 7 looks like a classic well bore-storage and skin-effect type curve for a homogeneous system. Fig. 8, a Horner graph for the adjusted data, appears to be a classic storage/skin buildup for a homogeneous system. Neither a commingled nor a dual-porosity system is indicated. Because the interpretation seemed reasonable, confidence limits were sought for the Horner analysis. Table 4 gives the result. In Table 4, the large confidence intervals for the Horner analysis are remarkable. Fig. 9 presents field data vs. a simulation with the parameters developed from the Horner analysis. The simulation does not match the field data. Thus, a nonlinear regression 19 on the adjusted field data was made and a good match was found. Table 5 presents the results, and Fig. 10 shows the field data vs. a simulation with the regression parameters. A comparison of Tables 4 and 5 shows a reduction in confidence limits of more than a factor of 10. A comparison of Figs. 9 and 10 shows an important improvement in the agreement of field and simulated results. Fig. 11 is a pressurederivative graph for the same data. It looks reasonable, except that there is a fairly rapid drop to the 0.5 derivative value. More will be said about derivative type curves later. This field example is more complex than discussed here. An analytical justification for the pressure adjustment is presented in the Appendix. But, a conclusion similar to that for the previously discussed simulated example remains: it is mandatory to use interpreted results to simulate test data and to compare the results of the simulation with the field data to be able to place credence in a Horner analysis. Regression and simulation are necessary to perform a proper modern analysis, but regression with determination of confidence limits is of major importance. When I began to train field engineers in well-test analysis 35 years ago, I stressed that the final necessary step was to comment on the quality of the test. Was the rate constant, was the shut-in long enough, etc.? This qualitative comment provides a basis for estimating comparative merit when reviewing results of sequential tests on the same well. The confidence limits provide a quantitative measure of quality. To do a regression analysis and not record the confidence limits is a terrible omission. This confidence-limit review led me to the shocking conclusion ! co! tOO 2tOO.goo.0 ('+4')14' too Fig. 9-Comparison of adjusted field data with results of a simulation made from parameters from a Horner analysis. Fig. 10-Adjusted field data vs. simulation with regression parameters. JPT June

7 100 k c O.IUB 3.03~ ::1 a II, I 1 10 td/cd 100 todd Fig. ii-derivative type curve for adjusted field example data. that I did not know where to draw a semilog straight line on a Homer graph to determine the best interpretation. Adjusting Data Fig. 11 is a type curve for the adjusted Field Example data that can be used to make important observations about the adjustment of buildup data. The pressure (upper) curve and derivative (lower) curve both start along a unit-slope line. This is the wellbore-storage time period when constant flow rate causes a linear relation between pressure and time. The pressure curve is a pressure-difference graph-static pressure less the flowing pressure just before the well was shut in. The flowing pressure is the weak link. 14 Ref. 14 discusses reasons why the flowing pressure may be measured correctly, but not be the proper flowing pressure for the type curve. If the flowing pressure is too low, the pressure difference will be too large and the early field data will approach a unit-slope line from above. If the flowing pressure is too high, the pressure difference will be too low and early data points will approach a unit-slope line from below. Neither event will affect the pressure derivative. Thus, a powerful feature of the derivative type curve is that it aids diagnosis of a flowing-pressure error. The correct flowing pressure can be found by a Cartesian graph of static pressure (Ref. 14). Another problem is that the effective shutin time may be later than the recorded shutin time. Most master valves cannot be closed instantaneously. I have observed gas well shut-ins that required 15 minutes to close the 656 master valve, yet I could still hear noise caused by fluid movement inside the pipe. Sometimes, this problem is obvious. Data from the service company log flowing pressure for some time, and then log the shut-in time. The shut-in pressures continue at the flowing level for some minutes before a rise is apparent. As before, wellbore storage may be used to correct the shut-in time by a linear extrapolation to the flowing pressure. A derivative graph, such as Fig. 11, however, can be of great use to determine whether this problem exists, even if a flowing-pressure history is not available. If the shut-in time is recorded before the effective shut-in, the apparent shut-in pressures do not change for some time. The pressure derivative will be zero. Then, the derivative will increase rapidly toward the correct values on the unit-slope line. Result: a late rapid rise in the early derivative and a sharp maximum usually indicate that the shut-in time requires adjustment. This is another step where computer aid is necessary. Changing the shut-in time requires modification of the producing time, t p ' addition of the correction to all shut-in times, and redifferentiation. This is far too much work to be done by hand. The initial work to produce Fig. 11 to see the problem is too much work to be done by hand. Computer aid is mandatory. Bourdet et ai. 17 originally indicated that proper differentiation was necessary, but they did not describe the method. Obvious differentiation methods such as spline fitting and numerical differentiation did not work well. Simple differencing could produce noise in the differential curve. Finally, ade- quate methods to produce the differential curve were developed, but computer aid was required. Both differentiation and adjustment of data absolutely require computer aid. Computer software is like the slide rule of the 1940's. A good analyst uses it to manipulate data-and not as a black box for interpretation. Combinations of improper flowing pressure and shut-in time exist. Some derivative type curves show the pressure-difference curve approaching a unit slope from above and the derivative type curve approaching a unit slope from below. A major feature of the Bourdet et al. 17 derivative type curve is that both the pressure-difference and pressure-derivative curves start with a unit slope, which they must do for the wellborestorage and skin-flow model. Ref. 14 indicates that similar corrections may be made for fracture flow models. Static pressure should be a linear function of the square root of time. This is correct for the simple linear flow fracture models. Fractured Wells The 1976 study 14 reported that tests on hydraulically fractured wells often matched the early fracture type curves 21 but yielded apparent fracture lengths of 10 ft when design lengths were more than 1,000 ft. An early study by Holditch and Morse 22 attributed this to high-velocity flow in a.fracture. But work by Ramey et al. 23 indicated that finite-fracture permeability-width was a more likely cause of the problem. They used a finite-element solution. Prats 24 provided the key to this problem in a classic June 1992 JPT

8 f "The confidence limits provide a quantitative measure of quality. To do a regression analysis and not record the confidence limits is a terrible omission." (b) t p+ Fig. A 1-Leaky pump problem. study on steady-state flow. In Aug. 1978, Cinco-Ley et al. 25 presented a truly classic semianalytic solution of this problem that has become a standard for verification of finite-difference solutions. Another important study was offered the next year by Agarwal et al. 26 The Cinco-Ley et al. 25 study solved the fracture well-test problem cited in Ref. 14. It is a magnificent study. Conclusions A number of important findings since 1976 have brought well-test analysis closer to fulfilling the expectations of analysts since the 1950's. Type curves, including pressurederivative and finite-conductivity fracture type curves, and computer-aided interpretation have given modern analysts the tools to do remarkable interpretative work. Artificial intelligence programs for model selection are already running. 27 Regression with confidence limits on model parameters and final simulation of a test provide proof that an interpretation was reasonable. As a result, the following conclusions appear justified. 1. It is often impossible to find a correct and observable Horner straight line that provides parameters that will generate the field data reasonably when simulated. Regression with all data and a proper model is necessary, and confidence limits should be recorded as a measure of the quality of the analysis. 2. Pressure-buildup data frequently require adjustment of the flowing pressure, the shut-in time, or both. JPT June Pressure-derivative type curves are sensitive throughout the time domain and permit identification of events not evident on either log-log or semilog pressure graphs. 4. Modern analytical procedures use computer-aided methods for the following steps: (1) to inspect pressure and derivative type curves and to filter and adjust data as needed; (2) to find a Horner straight line for nonlinear-regression input parameters; (3) to find confidence limits and to simulate the test with the final parameters after regression; (4) to compare field data and model simulation in dimensionless coordinates; and (5) if the result is not good, to select the next model in order of complexity and repeat previous steps. Artificial intelligence software can also aid in the selection of proper models. Nomenclature B = FVF, RB/STB C t = system compressibility, psi - 1 C = wellbore-storage constant, ft3 CD = dimensionless wellbore-storage constant de = casing JD, ft d t = tubing OD, ft h = formation thickness, ft k = permeability, md L1 = Height 1 in Fig. A-I, ft L2 = Height 2 in Fig. A-I, ft m = slope of semilog line, psi/cycle P = pressure, psi p* = Horner extrapolated pressure, psi IIp = Pws -Pwj' psi P D = dimensionless pressure change defined by Eq. A-7 P DSL = dimensionless pressure function defined by Eq. A-8 Pi = initial pressure, psi Pwj = flowing wellbore pressure, psi Pwj+ = flowing wellbore pressure at shut-in, psi Pws = shut-in wellbore pressure, psi q = surface flow rate, STBID r w = wellbore radius, ft s = skin, dimensionless t = time, hours Ilt = shut-in time, hours td = dimensionless time IltD = dimensionless shut-in time tp = producing time before shut-in, hours /L = viscosity, cp Po = density, Ibm/ft3 rf> = porosity, fraction Acknowledgments Portions of this work were funded by U.S. Department of Energy Grant No. DE AS07-841D Computer time was provided by Stanford U. Software was provided by Scientific Software-Intercomp Inc. and Garret Computing Systems Inc. References I. Horner, D.R.: "Pressure Build-Up in Wells," Proc., Third World Pet. Cong., E.J. Brill (ed.), Leiden (1951) II, Miller, C.C., Dyes, A.B., and Hutchinson, C.A. Jr.: "Estimation of Penneability and Reservoir Pressure from Bottom-Hole Pres- 657

9 sure Build-Up Characteristics," Trans., AIME (1950) 189, Moore, T.V., Schilthuis, R.J., and Hurst, W.: "The Determination of Permeability from Field Data," API Bulletin 211, API, Dallas (1933). 4. van Everdingen, A.F. and Hurst, W.: "Application of the Laplace Transformation to Flow Problems in Reservoirs," Trans., AIME (1949) 186, Hurst, W.: "Establishment of the Skin Effect and Its Impediment to Fluid Flow Into a Wellbore," Pet. Eng. (Oct. 1953) 25, B van Everdingen, A.F.: "The Skin Effect and Its Influence on the Productive Capacity of a Well," Trans., AIME (1953) 198, Matthews, C.S., Brons, F., and Hazebroek, P.: "A Method for Determination of Average Pressure in a Bounded Reservoir," Trans., AIME (1954) 201, Matthews, C.S. and Russell, D.G.: Pressure Buildup and Flow Tests in Wells, Monograph Series, SPE, Richardson, TX (1967) Theis, C. V.: "The Relationship Between the Lowering of the Piezometric Surface and the Rate and Duration of Discharge of a Well Using Ground-Water Storage," Trans., AGU (1935) II, Theis, C.V.: "The Relationship Between the Lowering of the Piezometric Surface and the Rate and Duration of Discharge of a Well Using Ground-Water Storage," Pressure Transient Testing Methods, Reprint Series, SPE, Richardson, TX (1980) 14, McKinley, R.M.: "Wellbore Transmissibility From Afterflow-Dominated Pressure Buildup Data," JPT (July 1971) ; Trans., AIME, Agarwal, R.G., Al-Hussainy, R., and Ramey, H.J. Jr.: "An Investigation of Wellbore Storage and Skin Effect in Unsteady Liquid Flow: 1. Analytical Treatment," SPEJ (Sept. 1970) ; Trans., AIME, Earlougher, R.C. Jr.: Advances in Well Test Analysis, Monograph Series, SPE, Richardson, TX (1977) Ramey, H.J. Jr.: "Practical Use of Modern Well Test Analysis," Pressure Transient Testing Methods, Reprint Series, SPE, Richardson, TX (1980) 14, Gringarten, A.C. et al.: "A Comparison Between Different Skin and Wellbore Storage Type Curves for Early-Time Transient Analysis," paper SPE 8205 presented at the 1979 SPE Annual Technical Conference and Exhibition, Las Vegas, Sept Stehfest, H.: "Algorithm 368: Numerical inversion of Laplace Transforms," Communications of the ACM (Jan. 1970) 13, No.1, Bourdet, D. et al.: "A New Set of Type Curves Simplifies Well Test Analysis," World Oil (May 1983) Bourdet, D. et al.: "Interpreting Well Tests in Fractured Reservoirs," World Oil (Oct. 1983) Rosa, A.J. and Horne, R.N.: "Automated Type-Curve Matching in Well Test Analysis Using Laplace Space Determination of Parameter Gradients," paper SPE presented at the 1983 SPE Annual Technical Conference and Exhibition, San Francisco, Oct Ramey, H.J. Jr., Agarwal, R.G., and Martin, I.: "Analysis of 'Slug Test' or DST Flow Period Data," J. Cdn. Pet. Tech. (July-Sept. 1975) Gringarten, A.C., Ramey, H.J. Jr., and Raghavan, R.: "Applied Pressure Analysis for Fractured Wells," JPT (July 1975) ; Trans., AIME, Holditch, S.A. and Morse, R.A.: "The Effects of Non-Darcy Flow on the Behavior of Hydraulically Fractured Wells," JPT (Oct. 1976) Ramey, H.J. Jr. et al.: "Pressure Transient Testing of Hydraulically-Fractured Wells," Proc., American Nuclear Soc. Topical Meeting on Energy and Mineral Resource Recovery, Colorado School of Mines, Golden (1977). 24. Prats, M.: "Effect of Vertical Fractures on Reservoir Behavior-Incompressible Fluid Case," SPEJ (June 1961) ; Trans., AIME, Cinco-Ley, H., Samaniego-V., F., and Dominguez-A., N.: "Transient Pressure Behavior for a Well With a Finite-Conductivity Vertical Fracture," SPEJ (Aug. 1978) Agarwal, R.G., Carter, R.D., and Pollock, C.B.: "Evaluation and Performance Prediction of Low-Permeability Gas Wells Stimulated by Massive Hydraulic Fracturing," JPT (March 1979) ; Trans., AIME, Allain, O.F. and Horne, R.N.: "Use of Artificial Intelligence in Well-Test Analysis," JPT (March 1990) Appendix- The Leaky Pump Problem Fig. A-I is a schematic of the leaky pump problem. Fig. A-la shows conditions at the instant of shutting in the well by stopping the pump at time tpo The tubing is full, the bottornhole flowing pressure is Pwf' and there is a liquid level in the annulus. Fig. A-I b shows conditions a moment after time tpo Both the standing and traveling valves in the pump leak, and oil drains rapidly from the tubing into the annulus. The liquid level in the tubing drops L j feet, and the liquid level in the annulus rises Lz feet. A volumetric balance is 11" Z _7r Z Z -d t L j --(de -d t )Lz.... (A-I) 4 4 Let Po be the density of wellbore oil. The pressure gradient in the liquid is dpldl=p o l144 psi/ft... (A-2) The wellbore storage during drawdown is based on the annular volume, while wellbore storage during pressure buildup after pump leakage is based on the total annulus and tubing storage, Cz = (~d,;' )r( PoI144)..... (A-4) The dimensionless forms of these storage coefficients are 18(d,;'-d/) pohcpctr;... (A-5) and C O2 = 18d,;' pohcpctr~... (A-6) The sudden drainage of liquid from the top L j feet of tubing is a slug injection into the well at the instant of shut-in. During drawdown, producing pressures are given by the wellbore-storage and skin solution with a storage coefficient of CD!. Because of the sudden emptying of the tubing, wellbore storage changes abruptly from CD! to C DZ at shut-in. However, it is necessary to superpose a slug injection of the quantity of liquid drained from L j feet of tubing at shut-in. The conventional wellbore-storage and skin dimensionless wellbore pressure is ---(Pi-P), 141.2qBj.t... (A-7) and the slug dimensionless pressure is (Pwf+ -p) PDSL(s'CD'tD)=----=~~...::...:.- (Pwf+ -Pwf)... (A-8) The shut-in wellbore pressure is (Pi -Pws) = I41.2qBj.t x [PD(S, CD!,tp +At) -PD(S,CDzAt)] -(Pwf+ -Pwf)PDSL(S,Cm,AtD),... (A-9) and the flowing pressure at shut-in is ----(Pi -Pwf) =PD(s,CDI,tp)' I41.2qBj.t... (A-1O) The conventional log-log type curve for buildup is based on the pressure difference (Pws-Pwf): ----(Pws -Pwf )=PD(s,CDI,tp) 141.2qBj.t -PD(S,CDj,tp +At) +PD(s,Cm,At) + (Pwf+ -Pwf)PDSL(S,Cm,At). I41.2qBj.t... (A-Il) Ordinarily, the first two terms on the right in Eq. A-II are assumed to be equal because shut-in times are restricted to less than 10% of the producing time. If this were done, the result would still be a superposition of the wellbore-storage and skin problem am! an instantaneous slug injection. There would be an instantaneous jump in pressure at shut-in. Let us derive the pressure from the difference between the static pressure and P wf + ' an instant after shut-in. June 1992 JPT

10 (p~-pwj+)= qBJ.! 141.2qBJ.! X [(Pws-Pwj)-(Pwj+ -Pwj)]'... (A-12) With Eqs. A-8, A-lI, and A-12, ---(Pws-Pwj+)=PD(S,CDl,tp) qbj.! -PD(s,CDl,tp +~t)+pd(s,cd2,~t) + (Pwj+ -Pwj) 141.2qBJ.! X[PDSL(S,CD2,~t)-1].... (A-B) The last term initially would be zero and would approach the constant value: - ---(Pwj+ -Pwj) = 141.2qBJ.! (Pwj+ -Pwj) 141.2qBJ.! (A-14) qbj.! where m= (A-IS) Consider the last term on the right in Eq. A-14. The slug test dimensionless pressure, PDSL (s,cd2,~t), is less than for tdic D ;?10 3 (see Ref. 20). This is approximately the time that the sernilog straight line starts for the wellbore-storage and skin case. Thus, [PDSL(S,CD2,~t)-t]""'-1... (A-16) and is constant by the time of the start of the semilog straight line. Thus, the skin effect found by use of Pwj + as a new flowing pressure should be increased by (Pwj+ -Pwj) ~P s = (A-17) 2m For large values of C D e2s, say ;?1Q10, PDSL -1 changes from 0 to -1 in the three log cycles before tdic D = This is the maximum and decline period of the pressure derivative with respect to log time. Thus, the slug injection should affect the derivative type curve on the approach to 0.5. The derivative should drop rapidly. This effect would probably not be noticeable because the slug-test dimensionless pressure drops rapidly to less than 0.1 for tdic D between 10 and 100. In most cases, the skin contribution caused by use of Pwj + would be constant long before the semilog straight line starts. Finally, some comments on the effect of a leaky pump are required. During production, bad check valves would reduce pump efficiency and make the surface oil rate less than the pump displacement rate. This would not affect anything else. The wellbore storage would still be based on the annular volume, and the skin effect would be the correct skin effect. A bad pump would cause a low oil rate even though the skin effect was negative. Although this problem was identified by seeking an explanation for the strange loglog type curve in Fig. 4, it could have been recognized by reflection on pumping well problems. But in my experience, well-testanalysis research often requires experiments at full scale. Do not reject anomalous data. Seek causes. Loren Krase and R.G. Agarwal have advised me that the derivation presented here is approximate. I agree. A Discussion that provides the rigorous derivation will be presented in the near future. SI Metric Conversion Factors bbl x cp x 1.0* ft x 3.048* md x psi x psi -1 x 'Conversion factor is exact. E-Ol = m' E-03 = Pa s E-Ol = m E-04 = I"m2 E+OO = kpa E-Ol = kpa- 1 This paper is SPE Distinguished Author Series articles are general, descriptive presentations that summarize the state of the art in an area of technology by describing recent developments for readers who are not specialists in the topics discussed. Written by individuals recognized as experts in the area. these articles provide key references to more definitive work and present specific details only to illustrate the technology. Purpose: To inform the general readership of recent advances in various areas of petroleum engineering. A softbound anthology, SPE Distinguished Author Series: Dec Dec is available from SPE's Book Order Dept. JPT Author Henry J~ Ramey Jr. la the Keleen and Carlton Beal professor of petroleum engineering at Stanford U. and an International consultant. Before Joining Stanford In 1966, he worked for Mobil 011 during , with an a.salgnment to the Chinese Petroleum Corp. In Taiwan In 1982, and for Texsa A&M U. during He holds as and PhD degrees In chemical engineering from Purdue U. Ramey served on the SPE Board.of DIrectors and the Westem RegIonal Meeting planning committee. A Distinguished Member since 1983, Ramey has received the Lester C. Uren Award (1973), the John Franklin Carll Award (1975), and the Anthony F. Lucas Gold Medal (1983). JPT June

11 Discussion of Advances in Practical Well Test Analysis Roberto Aguilera, SPE, and Michael C. Ng, Servipetrol Ltd. SPE26134 We have read "Advances in Practical Well Test Analysis" by H.J. Ramey Jr. (June 1992 J PT, Pages ) with a great deal of interest. As is always the case with Ramey's papers, his ideas are exciting and thought-provoking. In the case of the present paper, we have extended the analysis to demonstrate that type curve and derivative analysis 1 can lead to reasonable results. We have concentrated on analyzing Ramey's simulated buildup data because, in this case, we know the correct answers-permeability =48 md and skin = Our first step included a diagnostic loglog crossplot of f:..p vs. f:..t. Like Ramey, we obtained an early slope of 1.0 as Fig. D-I shows. Next, we placed the Fig. D-l data on top a type curve for homogeneous reservoirs and included the pressure derivative (Fig. D-2). Our best match indicates that the test was not long enough to reach the correct straight line in a Horner plot. From our best match, we calculated a transmissibility of about 820 md-ft/cp (permeability =48.2 md) and a skin of +9. The match was validated by generating a single type curve and derivative, which we compared with the simulated data 2 with excellent reslllts (Fig. D-3). Also excellent is the comparison of the calculated permeability vs. permeability input in the simulator (48.2 vs md) and calculated skin vs. skin input in the simulator (10 vs. 9). Thus, this is one instance in which the type curve and derivative analysis provide a reasonable quantitative result in lieu of a regression analysis. Furthermore, the derivative suggests that a Horner analysis should not be used in this case because the correct semilog straight line has not been reached. k=48 E+83.~ Q. E+82 Delt. Time, hr Servipetrc! E-82 E-81 E+ge E+G1 Fig. D-1-Diagnostic log-log crossplot of ilp vs. ilt. 278 March 1993 JPT

12 References 1. Bourdet, D. et af.: "A New Set of Type Curves Simplifies Well Test Analysis," World Oil (May 1983) Aguilera, R. and Song, S.J.: "Well Test-NFR: A Computerized Process for Transient Pres- sure Analysis of Multiphase Reservoirs With Single, Dual or Triple Porosity Behavior," paper ClM No presented at the 1988 Annual Technical Meeting ofthe Pet. Soc. of ClM, Calgary, June SI Metric Conversion Factors cp x 1.0* E-03 Pa's ft x 3.048* E-OI = m Conversion factor is exact. (SPE 26134) Copyright 1993 Society of Petroleum Engineers JPT k=48 ~.. ~ c. " a, II. ;> ti 0, ~ E+81 e+ee E-81 E+.e E+81 E+82 E+83 Sarvi.trcl Fig. D 2-Type curve for homogeneous reservoirs. k=48 Tl" \.,I = 8.2E+82 = 1. IE.'8 1:+'1 L", 1. 8E+ee Cd = E+ee E-81 E+ee E+81 E+82 E+83 Fig. D.3-Comparison of a single type curve and derivative with simulated data. JPT March