Advances in Practical Well-Test Analysis · PDF fileAdvances in Practical Well-Test Analysis H.J. Ramey Jr., ... Bourdet et al. 17.18 presented a major devel ... well-test analysis

650

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 Hurst4 had previously presented the con­cept of wellbore storage and had taught use­ful analytical methods to solve well-test problems. In 1953, Hurst5 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 1991. 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 Hurst4 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 11.12 at about

June 1992 • JPT

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 1978. Their type curve of the log of dimensionless pres­sure vs. the log of tDICD, with CDe2s 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-

sPEDistinguished Author SERIES

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. 17.18 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

TABLE 1-SIMULATED BUILDUP DATA I

k, md q, STBID tp• hours rw. ft h, ft /L, cp B, ABlSTB q, PI' psi Ot, psi ~1 s

At (hours)

o 0.0332947 0.1551056 0.3332977 0.5332947 0.7333069 0.9848022 1.233307 1.533295 1.871201 2.233307 .2.633301· 2;818207 3.433304 3.830093 4.911301 6.068498 7.300705 8.619293

10.02831 11.53360 13.14200 14.86050 16.69659 18.65829 20.75430 22.99380 25.38651 27.94310 30.67450

48 500 150

0 . .25 17

1 1

0.2 .2,810

1 x 10-6

+10 PW$ (psi)

1,094.897 1,122.317 1.217.656 1,344.633 1,471.678 1,584.484 1,708.728 1,814.685 1,924.317 2,027.047 2,117.398 2,198.191 2,229.975 2,315.763 2,358.232 2,439.504 2,491.655 2,5.25.854 2,548.988 2,565.269 2,577.269 2,586.563 2,594.116 2,600.520 2,606.144 2,611.217 2,615.881 2,6.20.230 2,624.321 2,628.196

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 1978. 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

hlO·,..-------,-------,-------,--r--------, C 2.633 •• 01

1000

! a. <I

100

10 0.1

,_ ,I , '

4t (hours)

• I 111.1.1111111

10 100

Fig. i-Log-Iog type curve for simulated buildup.

3000r------,-------,-------,-------------,

2S00

! 2000 a.

l!100 • 127.2 k 37.S9 • 6.22

PW 2727

I I

"

1000L--------L-------~--------~--------~

10 100 (t+4t)/4t

1000

Fig. 2-Horner graph for Simulated buildup.

TABLE 2-CONFIDENCE LIMITS FOR HORNER ANALYSIS OF SIMULATED BUILDUP DATA

k, md s C, ABlpsi

37.5900 6.22000

0.282200 x 10-1

+ or -+ or­+ or -

30.6544 10.547.2

0.331417x 10-2

(81.55%) (169.570Al) (11.74%)

Mean square deviation =0.2109 x 104 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.

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

~OO~----------r-----------'-----------'-----------,

l!lOO

1000L------------L----------~------------J-----------~ 10 100

(HAtyAs 1000 1.,0·

Fig. 3-Test buildup data vs. simulated curve from Horner analysis parameters.

1000r----------------r----~----------r_--------------_,

c 0.4357

! ~ SOO

SO SO SOD 1000

As (hours]

... ..

i.,O"

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

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-

TABLE 3-FIELD EXAMPLE

q, STBJD 36 tp • hours 3.912 'w. ft 0.4 h, ft 165 jL. cp 0.5058 B. RBlSTB 1.333 tI> 0.17 Ph psi Ct. 13$1-1

\ 2,688 : 5)( 10-0

at Pws At Pws (hours) (psi) (hours) (psI)

0 1,812.53 766 2,246.61 20 1,831.43 941 2,285.42 25 1,927.24 1,157 2,332.8 30 1,954.37 1,421 2,384.72 35 1,974.85 1,747 2,434.65 40 1,978.31 2.146 2,484 45 ',980.9 2,637 2,529.93 50 1,983.51 3,240 2.588.86 55 1,986.09 3,982 2,598.66

···60 1,988;66 4,892 2,626 65 1,991.26 7,387 2,656.5 70 1,993.65 7.494 2,656.17 75 ,,996.42 7,602 2,656.14 80 1,999 7,712 2.856.65 90 2,004.16 7,938 2,857.74

100 ' 2,009.29 8,053 2,858.08 110 2,014.43 8,169 2,658.53 120 2,019.56 8.408 2,659.58 130 2,024.69 8.653 ~,660.5 147 2,033.38 8.778 2,660.83 181 2,050.72 9,035 2,660.83 223 2,071.72 9,166 2,660.9 274 2,095.92 9,433 2,661.48 336 2,1~3.29 9,570 2.661.85· 413 2,154.49 9,708 2,662.17 507 2,186.6 9,849 2,662.58 624 2,214.57

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

2800 HOOr-----~-----.----_r-----.--~~r_----,

2tI00

2100

! 2400

.. ! CI. 2000

2200

· .71.1 .900

• 0.1391 2000 • -0.58

PIt 2610 III1III1IIIII11

'BOO

• .0

Fig. 5-Horner graph for field example.

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

.HOL---__ ~ ____ -L ____ ~ ____ ~~ ____ ~ ____ ~

3900 4000 4tOO 4200 4300 40400 4500 .00

__ I

Fig. 6-Magnified Cartesian graph of early buildup data.

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 hump-

ing 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 3.277

2!lC)0

! 2300 .. .. '

2iOO .. 171.1 k 0.1398 • -1."96

p. 269 •

• 0 tllSG" .900 1000 .0 lOO

At (hours) .0 .00

Fig. 7-Log-log type curve for field example with adjusted flow- Fig. a-Horner graph for adjusted field example buildup. ing pressure P wI + •

654 June 1992· JPT

TABLE 4-CONFIDENCE LIMITS FOR HORNER ANALYSIS OF • ADJUSTED FIELD EXAMPLE DATA

k. md s

• C. RBlpsi

0.141300 -1.53000 3.30100

+ or -+ or -+ or -

0.296678 8.40485 1.n082

(209.96°1b) (549.32%) (53.6401b)

Mean square deviation .. 0.3455 x 104 psi 2

I

TABLE 5-RESULTS AND CONFIDENCE LIMITS FOR REGRESSION WITH ADJUSTED FIELD EXAMPLE DATA

k, md s C, RB/psi

0.111805 -1.73821 3.03355

+ or -+ or -+ or -

0.193642 x 10-1

0.638584 0.139022

(17.32%) (36.74%) (4.58%)

Mean square deviation =0.2934 x 102 psi 2

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 tests20 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

2700 ...

2500

! 2300 ..

2tOO

.goo .0

('+4')14'

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

2700

2500

co ! 2300 ..

2tOO

1900

• 00

"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."

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

•• 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 1992 655

1::1 a..

100

10

1

0.1

k c •

II, 1 1

1• 1

I •

0.1

O.IUB 3.03~

-1.738

1 10 td/Cd

100

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, tp ' 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-

327.8

tODD

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 curves21 but yielded apparent fracture lengths of 10 ft when de­sign lengths were more than 1,000 ft. An early study by Holditch and Morse22 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. Prats24 pro­vided the key to this problem in a classic

June 1992 • JPT

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 1992

f

(b) t p+

3. 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 dt = 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

"The confidence limits provide a quantitative measure of quality. To do a regression analysis and not record the confidence limits is a terrible omission."

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. DE­AS07-841D12529. 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, 503-22.

2. Miller, C.C., Dyes, A.B., and Hutchinson, C.A. Jr.: "Estimation of Penneability and Reservoir Pressure from Bottom-Hole Pres-

657

sure Build-Up Characteristics," Trans., AIME (1950) 189,91-104.

3. 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, 305-24.

5. Hurst, W.: "Establishment of the Skin Ef­fect and Its Impediment to Fluid Flow Into a Wellbore," Pet. Eng. (Oct. 1953) 25, B-6.

6. van Everdingen, A.F.: "The Skin Effect and Its Influence on the Productive Capacity of a Well," Trans., AIME (1953) 198, 171-76.

7. Matthews, C.S., Brons, F., and Hazebroek, P.: "A Method for Determination of Average Pressure in a Bounded Reservoir," Trans., AIME (1954) 201, 182-91.

8. Matthews, C.S. and Russell, D.G.: Pressure Buildup and Flow Tests in Wells, Monograph Series, SPE, Richardson, TX (1967) 1.

9. 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, 519-24.

10. 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, 27-32.

11. McKinley, R.M.: "Wellbore Transmissibility From Afterflow-Dominated Pressure Buildup Data," JPT (July 1971) 863-72; Trans., AIME,251.

12. 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) 279-90; Trans., AIME, 249.

13. Earlougher, R.C. Jr.: Advances in Well Test Analysis, Monograph Series, SPE, Richard­son, TX (1977) 5.

14. Ramey, H.J. Jr.: "Practical Use of Modern Well Test Analysis," Pressure Transient Testing Methods, Reprint Series, SPE, Richardson, TX (1980) 14, 46-67.

15. 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. 23-26.

16. Stehfest, H.: "Algorithm 368: Numerical in­version of Laplace Transforms," Communi­cations of the ACM (Jan. 1970) 13, No.1, 47-49.

17. Bourdet, D. et al.: "A New Set of Type Curves Simplifies Well Test Analysis," World Oil (May 1983) 95-106.

18. Bourdet, D. et al.: "Interpreting Well Tests in Fractured Reservoirs," World Oil (Oct. 1983) 77-87.

19. Rosa, A.J. and Horne, R.N.: "Automated Type-Curve Matching in Well Test Analysis Using Laplace Space Determination of Pa­rameter Gradients," paper SPE 12131 presented at the 1983 SPE Annual Technical Conference and Exhibition, San Francisco, Oct. 5-8.

20. 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) 1-11.

658

21. Gringarten, A.C., Ramey, H.J. Jr., and Raghavan, R.: "Applied Pressure Analysis for Fractured Wells," JPT (July 1975) 887-92; Trans., AIME, 259.

22. Holditch, S.A. and Morse, R.A.: "The Ef­fects of Non-Darcy Flow on the Behavior of Hydraulically Fractured Wells," JPT (Oct. 1976) 1169-78.

23. 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) 105-17; Trans., AIME,222.

25. 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) 253-64.

26. 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) 362-72; Trans., AIME, 267.

27. Allain, O.F. and Horne, R.N.: "Use of Ar­tificial Intelligence in Well-Test Analysis," JPT (March 1990) 342-49.

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 -dt L j --(de -dt )Lz . ...... (A-I) 4 4

Let Po be the density of wellbore oil. The pressure gradient in the liquid is

dpldL=P ol144 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)

18d,;' and CO2 = .......... (A-6)

pohcpctr~

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 CDZ 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

kh ---(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

I41.2qBj.t (Pi -Pws) =

kh

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

kh ----(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):

kh ----(Pws -Pwf )=PD(s,CDI ,tp) 141.2qBj.t

-PD(S,CDj,tp +At) +PD(s,Cm ,At)

kh + (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

kh kh -------(p~-Pwj+)=-------141.2qBJ.! 141.2qBJ.!

X [(Pws-Pwj)-(Pwj+ -Pwj)]'

............... (A-12)

With Eqs. A-8, A-lI, and A-12,

kh ---(Pws-Pwj+)=PD(S,CDl,tp) 14 1. 2 qBJ.!

-PD(s,CDl ,tp +~t)+PD(s,CD2,~t)

kh + (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:

-kh ---(Pwj+ -Pwj) = 141.2qBJ.!

(Pwj+ -Pwj)

141.2qBJ.!

kh 2.303

. . . . . . . . . . . . . . . (A-14)

qBJ.! where m=162.6- . ......... (A-IS)

kh

Consider the last term on the right in Eq. A-14. The slug test dimensionless pressure, PDSL (s,CD2,~t), is less than 0.001 for tDICD ;?103 (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

2.303

For large values of CDe2s , say ;?1Q10, PDSL -1 changes from 0 to -1 in the three

JPT • June 1992

log cycles before tDICD = 103 . 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 tDICD 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 1.589 873 cp x 1.0* ft x 3.048*

md x 9.869233 psi x 6.894 757

psi -1 x 1.450 377

'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 20592. 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. 1981-Dec. 1983. 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

1952-63, with an a.salgnment to the Chinese Petroleum Corp. In Taiwan In 1982, and for Texsa A&M U. during 1963-66. He holds as and PhD degrees In chemical engineering from Purdue U. Ramey served on the 1972-15 SPE Board .of DIrectors and the 1990-91 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).

659

We have read "Advances in Practical Well­Test Analysis" by H.J. Ramey Jr. (June 1992 J PT, Pages 650-59) 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 = + 10.

.~ • Q.

E-82

E+83

E+82

E-81

Discussion of Advances in Practical Well·Test Analysis Roberto Aguilera, SPE, and Michael C. Ng, Servipetrol Ltd.

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

k=48

Delt. Time, hr

E+ge

SPE26134 compared with the simulated data2 with ex­cellent reslllts (Fig. D-3).

Also excellent is the comparison of the calculated permeability vs. permeability in­put in the simulator (48.2 vs. 48.0 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.

Servipetrc!

E+G1

Fig. D-1-Diagnostic log-log crossplot of ilp vs. ilt.

278 March 1993 • JPT

References 1. Bourdet, D. et af.: "A New Set of Type

Curves Simplifies Well Test Analysis," World Oil (May 1983) 95-106.

2. Aguilera, R. and Song, S.J.: "Well Test-NFR: A Computerized Process for Transient Pres-

E+81

~ .. ~ c . " a , II. ;> ti 0 , ~

e+ee

E-81 E+.e

sure Analysis of Multiphase Reservoirs With Single, Dual or Triple Porosity Behavior," paper ClM No. 88-39-52 presented at the 1988 Annual Technical Meeting ofthe Pet. Soc. of ClM, Calgary, June 12-16.

k=48

E+81 E+82

Fig. D·2-Type curve for homogeneous reservoirs.

k=48

Tl" = 8.2E+82

\.,I = 1. IE.'8

L", 1. 8E+ee

1:+'1

Cd =

E+ee

E-81 E+ee E+81 E+82

Fig. D.3-Comparison of a single type curve and derivative with simulated data.

JPT • March 1993

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) JPT

Copyright 1993 Society of Petroleum Engineers

Sarvi .trcl

E+83

E+83

279