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AbstractWe have utili zed sets of simulated well tests in an examplewell/reservoir model with an unfractured vertical well or avertical well with a finite-conductivity (vertical) fracture todemonstrate the dependence of the estimated parameters andtheir confidence intervals on noise in the pressure and flowratedata. Uniformly distributed random noise is introduced intothe pressure data and the resulting well parameter estimatesare obtained using conventional means (analysis/historymatching).

For the cases of a pressure buildup test in an unfractured welland the pressure drawdown cases performed on a fracturedwell , the estimated parameters are almost without bias — andthe confidence intervals grow linearly with the magnitude ofthe noise. For the cases of pressure buildup tests performedon the fractured well , the estimated parameters are insensitiveto the noise — which suggests that this case may not be well -posed for analysis.

Gauge drift and flowrate noises were also considered and theanalysis results from the example well tests demonstrate thatboth of these kinds of noise are handled robustly by conven-tional well test analysis techniques.

Introdu ctionThere are several sources of uncertainty in the analysis andinterpretation of well test data — Horne1 describes these as:

� Physical error in the pressure data (noise, drift,temperature effects and time shift).

� Errors in the flowrate measurement.� The nonunique response of the reservoir and pro-

ducing time effects.� Uncertainty in rock and fluid properties.

This work addresses the impact of the first two issues (randomnoise in the pressure measurements, gauge drift, and flowratemeasurement errors) on well test analysis. The study was con-ducted by generating over 2000 synthetic well test responsessubject to each kind of error being studied. These well testswere then interpreted using nonlinear regression (andoccasionally checked by specialized plots for consistency).The parameter estimates and the confidence intervals2 as-sociated with these estimates were computed and recorded.Horne3 ill ustrates the value of confidence intervals in well testinterpretation — particularly the value of confidence intervalsin assessing the validity of parameter estimations. Forreference, this study considers models for both unfracturedand fractured wells, but only for the case of a homogeneous,infinite-acting reservoir.

In 1994, Horne1 found that if the pressure measurements for awell test include errors which are approximately normallydistributed (for the case of an unfractured vertical well ), theresulting estimates of permeabili ty, skin factor, wellborestorage coefficient and initial pressure are approximately nor-mally distributed. This relatively simple concept forms thethesis for our work. Horne did not address the impact of theuncertainty in the pressure measurements on the confidenceintervals for permeabili ty, skin factor, wellbore storage coeff i-cient and initial pressure, nor did he consider the case of afractured well .

MethodThe rock, reservoir and well properties used for our"unfractured well " cases are given in Table 1. For con-venience, these properties are taken from Bourdet4. Theproperties used for the fractured well case are similar and arepresented in Table 2. The only significant modifications madeto the unfractured well cases are to the flow rate history and tothe skin factor. We acknowledge that these cases are onlyexamples, and that a more general result could be obtained byconsidering additional cases. However, we will note that wedid use an exhaustive number of cases to ensure statisticalvalidity of our results. The conventional "log-log" diagnosticplots for each well test example are given in Figs. 1 and 2.

To assess the effect of random noise imposed on the pressuremeasurements, the synthetic well tests were generated with(systematically) varying amounts of noise, and interpreted(primarily) using nonlinear regression. Unlike Horne's1 work,

SPE 77533

Effects on Well Test Analysis of Pressure and Flowrate NoiseR.A. Archer, University of Auckland, M.B. Merad, Schlumberger, T.A. Blasingame, Texas A&M University

2 R.A. Archer, M.B. Merad, and T.A. Blasingame SPE 77533

in this work the noise introduced into the data was uniformly(not normally) distributed with a zero mean and a varyingampli tude.

For the cases that specifically address the effects of randomnoise in the pressure data, the analysis results are presented intwo formats. The first format is that of a histogram (i.e., aprobabili ty distribution) of the estimated parameters (forspecific (constant) noise level). In the second presentation weuse plots that show the behavior of the average of the para-meter estimates and confidence interval bounds for a particu-lar set of well tests, shown as a function of the noise level.

For the unfractured well cases, 100 synthetic well tests pernoise level were generated and interpreted for noise levels of0.1 to 1.0 psi (by 0.1 increments). In the fractured well case50 well tests were used per noise level.

The effect of gauge drift was also considered for gauge driftsof up to and including 0.1 psi/hr. The drift error is treated asdeterministic (rather than a statistically derived estimate) soonly one well test was considered for each level of pressuredrift. The analysis results for well tests with gauge drift errorare plotted against the drift error.

Flowrate noise was considered by creating sets of syntheticwell tests with perfect (i.e., undistorted) pressure data. How-ever, these "perfect" pressure data were then interpreted usinga (synthetic) noisy rate history. For the unfractured verticalwell cases these well tests were interpreted by adding an un-correlated random noise with ampli tude varying from 1 to 9STB/D to the base flowrate of 100 STB/D. For the fracturedwell case the noise ampli tude varied from 10 to 50 STB/D andthe base flowrate was 1000 STB/D. The flowrate noisedistribution had a zero mean, and for each flowrate noise levelwe generated 100 synthetic well tests.

To ensure a uniform analysis/interpretation approach, all welltests were interpreted using nonlinear regression (i.e. manualinterpretation was only used occasionally to check the results).The initial estimates for all parameters were the true (or exact)values for that particular parameter.

Results

Random Noise in the Pressure Measurements

Unfractured Well Case - Buildup Data

Figs. 3 and 4 present the histograms for the permeabili ty andskin factor estimates computed from 100 well tests at the 0.6noise level (unfractured well case). These figures show thatthe estimated values of the permeabili ty and skin factor are(approximately) normally distributed and are centered on thetrue values (k = 10.28 md and s = 2). Figures 5 and 6 showthe behavior of the estimates for a range of noise levels. Eachpoint plotted on these figures represents the average of 100well test interpretations where the pressure data contains agiven level of noise. The noise introduced into the pressuredata is uniformly (not normally) distributed but the resultingpermeabil ity and skin factor estimates are approximatelynormally distributed. This behavior agrees with Horne’s1 ob-

servations and helps to validate our thesis that the results fromcases with pressure error are normally distributed.

Fractured Well Case - Buildup Data

Figures 7-10 present the histograms for the estimates of per-meabili ty, skin factor, fracture half-length, and fracture con-ductivity — estimated from 50 synthetic buildup tests con-ducted at each noise level. These probabilit y distributions aremore irregular than those for the unfractured well case.

The permeabili ty and skin factor are estimated correctly (onaverage) — however, the expected value of the fracture half-length is 143.8 ft, slightly less than the true value of 150 ft.The fracture conductivity is overestimated, at 6920 md-ft (onaverage), compared to the true value of 4500 md-ft. The con-fidence intervals associated with these estimates are shown inFigures 11-14. These results present a very different behaviorcompared to the confidence intervals from the unfracturedwell case — we observe the confidence interval length to beessentially constant, irrespective of the noise level.

For these cases it could be argued that the number ofsimulated well test cases may not be sufficient to establishstatistical validity. However, it is our contention that if this isan issue, it is a minor one — the most likely circumstance isthat the particular case under consideration (i.e., the fracturedwell performance) may not demonstrate sufficient character-istic behavior for the regression algorithm to define eachparameter uniquely. In simple terms, the tests may notprovide the breath of behavior (i.e., flow regimes) for accurateestimates of permeability, skin factor, fracture half-length, andfracture conductivity to be obtained.

It is our contention that this case does il lustrate that, in parti-cular, a complex analysis/interpretation requires that all flowregimes under consideration to be present in order to obtain aunique analysis.

Fractured Well Case - Drawdown Data

For the fractured well case drawdown data were alsoconsidered to see if the behavior observed from the buildupcase would be reproduced. Again 50 synthetic well tests weregenerated and interpreted at each noise level. Figures 15 and16 show histograms of the estimated permeabili ty and fractureconductivity at the 0.6 psi noise level. These distributions canbe approximated by normal distributions.

Interestingly, in these cases the distribution of the fractureconductivity is no longer biased away from the true value of4500 md-ft. The calculated mean of this distribution is 4497.6md-ft. Figures 17 and 18 show the behavior of the confidenceintervals for the permeabili ty and fracture conductivityestimates. These confidence intervals behave in the samefashion as those for the unfractured well case did — i.e. theconfidence interval length grows linearly with the noise in thepressure.

The question remains as to why the pressure drawdown casesappear to be more amenable to analysis/interpretation than thepressure buildup cases for this model (fractured well ). The

SPE 77533 Effects on Well Test Analysis of Pressure and Flowrate Noise 3

resolution of this issue lies in the testing sequence — thepressure buildup cases reflect the production history, and assuch, the pressure buildup profile has constraints. In this casethe pressure buildup test duration was twice as long as thedrawdown test, and the data were corrected using the "effec-tive time" function for the radial flow case (the most commonpractice). In summary, the testing sequence and data correc-tion process undoubtedly biased the results of the pressurebuildup case.

Drift Error

The effect of gauge drift was considered by generating andinterpreting an example well test case with varying degrees ofgauge drift, for both the unfractured and fractured well cases.Since the drift is a deterministic error (i.e., such an errorwould be presumed to be fixed, as opposed to being random),only one well test was required at each level of drift error.Figures 19 and 20 ill ustrate the behavior of the estimatedpermeabil ity for the unfractured and fractured well cases — inparticular, these figures show that the permeabili ty estimatesand the length of the associated confidence intervals increaselinearly with the magnitude of the drift error.

The effect of pressure gauge drift error on the estimated per-meabili ty estimated was also addressed analytically in Appen-dix A. Gauge drift means that the observance of a flatpressure derivative function will not always occur for the caseof an infinite-acting reservoir. If the pressure derivative level,∆p'rf, is read near the beginning of radial flow, then Eq. A-5can be used to relate the permeabili ty estimated using non-linear regression to the true reservoir permeabili ty.

Figure 21 shows similar behavior to Figs. 19 and 20 for theestimated fracture conductivity (fractured well case).However, the confidence intervals are wider (in percentageterms). The data considered for this comparison are thebuildup data for the unfractured well case and drawdown datafor the fractured well case.

Flowrate Noise

Figure 22 shows the effect of the flowrate noise on thepermeabil ity estimate for the unfractured vertical well case. InFig. 22 we note that the permeabili ty estimate steadily in-creases with the magnitude of the noise in the flowrate func-tion — we note that this increase lies within tolerable limits,and suggests that flowrate noise (i.e., error) may not adverselyaffect the analysis/interpretation of well test data, at least in apractical sense.

Figures 23 and 24 show the behavior of the estimatedpermeabil ity and fracture half-length parameters for the frac-tured well (drawdown) cases. We note that these estimates donot follow a strictly linear trend — the estimated permeabilit yis essentiall y correct, while the fracture half-length is con-sistently overestimated. Part of this behavior could be due tothe regression analysis (and probably is), but we can note thatin comparing Figs. 22 and 23 it would be difficult to state thatone case has a "better" characteristic profile than the other —

which is somewhat expected for the estimation of permeabili tyusing well test analysis.

DiscussionPut simply, the analysis/interpretation of well test data ob-tained from unfractured vertical wells in infinite-acting reser-voirs is relatively unaffected by random noise in the pressuremeasurements. The pressure noise (i.e., error) does not biasthe parameter estimates, but it does broaden the confidenceintervals associated with the parameters. The broadening ofthe confidence intervals in a linear fashion was expected froman analysis of the confidence interval formulation presentedby Dogru et al.2

Attempts were made to derive a rigorous expression relatingthe change in the confidence interval length to the noise level.A theoretical formulation was developed but is not presentedin this work because the validation of this formulation usingnumerical experiments was not satisfactory. This poorperformance of the theoretical formulation is thought to be dueto the sensitivity of the confidence interval calculation to thespecific set of pressure data points matched by the regressionanalysis. For example, pressure points in the early time regiontended to dominate the calculated confidence intervals.

In the fractured well example we considered the effect ofrandom noise on the analysis/interpretation was substantiallydifferent — depending whether buildup or drawdown datawere being analyzed/interpreted. The pressure buildup casesshowed biased parameter estimates and confidence intervals ofan approximately constant length — in contrast, the drawdowndata showed essentially unbiased parameter estimates and con-fidence intervals that broadened linearly with the noise am-plitude (analogous to the unfractured well cases).

For the cases of pressure drift error we showed that the para-meter estimates vary linearly with the degree of gauge drift.The changes in the estimated parameters were relatively small .For the fractured well case the relative change in the size ofthe confidence interval was much larger than the change in theparameter estimates, indicating that this may have practicalimplications for the analysis of this type of data.

For the cases of flowrate noise (i.e., error) we noted that theanalysis/interpretation is relatively unaffected by the noise inthe flowrate — for the unfractured and fractured well cases.The associated confidence intervals did however broaden withincreasing noise (as would be expected).

Conclud ing Remarks

1. The difference between the buildup and drawdown results in thefractured well cases raises a question about whether thesuperposition time function used to interpret the data in thebuildup case compounds the effect of pressure and flowratenoise.

2. This research has shown that for the case of random noise inpressure measurements, that the distributions of the parameterestimates may be normally distributed – regardless of whetherthe input noise distribution is uniform (this work) or normal(Horne1).

4 R.A. Archer, M.B. Merad, and T.A. Blasingame SPE 77533

3. In the future we hope to pursue a quantitative approach toexplain the behavior observed in the numerical experimentsshown in this paper. We acknowledge that this study onlyanalyzed two example well tests. We hope to generalize thiswork by considering other examples cases and by analyticallyderiving models to describe the parameter uncertainty. We alsowould like to consider extending the work to other well/ reser-voir models such as horizontal wells, and dual porosity reser-voirs.

NomenclatureB = formation volume factor, RB/STBC = wellbore storage coefficient, STB/psict = total compressibili ty, psi-1

h = net pay thickness, ftk = permeabil ity, mdp = pressure, psiapi = initial pressure, psiaq = production rate, STB/Drw = wellbore radius, fts = skin factor, dimensionlesst = time, hrα = pressure gauge drift rate, psi/hrφ = porosity, fractionµ = viscosity, cp

References

1. Horne, R.N.: "Uncertainty in Well Test Interpretation," paperSPE 27972 presented at the 1994 SPE Annual TechnicalConference and Exhibition, Tulsa, Oklahoma, August 29-31.

2. Dogru, A.H., Dixon, T.N. and Edgar, T.F.: "Confidence Limitson the Parameters and Predictions of Slightly Compressible,Single-Phase Reservoirs," paper SPE 4983 presented at the 1974SPE Annual Technical Conference and Exhibition, Houston,October 6-9.

3. Horne, R.N.: "Advances in Computer-Aided Well Test Inter-pretation," paper SPE 24731 presented at the 1992 SPE AnnualTechnical Conference and Exhibition, Washington, DC, October4-7.

4. Bourdet, D. Ayoub, J.A. and Pirard, Y.M.: "Use of PressureDerivative in Well -Test Interpretation," JPT (June 1989) 293-302.

AcknowledgementsThe first author would like to acknowledge that this work wasperformed at Texas A&M University. The second authorwould like to acknowledge financial support from Schlum-berger.

Appendix A - Effect of Gauge Drift on Permeabili tyEstimation for the Case of an Unfractured Well

The recorded pressure change when the gauge is subject todrift is of the form:

∆pmeas=∆ptrue-α t .......................................................(A-1)

The associated ("well testing") pressure derivative function is:

tt

pt

t

ptp'

rf α−∂

∂=∂

∂= truemeas ÄÄÄ ..........................(A-2)

Even during infinite-acting radial flow this derivative valuewill not be constant with time (due to the pressure drift) —however, if the drift is small the "well testing" pressure deri-vative function (Eq. A-2) would be approximately constant.

The "well testing" pressure derivative function includes the"measured" pressure derivative (in terms of the "estimated"permeabil ity, k') as well as the "true" pressure derivative (interms of the true permeabili ty, k) plus a term that includes thepressure gauge drift (α). Writing Eq. A-2 for the case of anunfractured well producing at a constant rate in an infinite-acting reservoir, we have:

tkh

qB

hk

qBp

''rf αµµ −== 70.6 70.6Ä ...........................(A-3)

Our goal is to define a rigorous relationship between k', k, andα in order to quantify the impact of drif t term (α).

tkh

qB

hk

qB'

αµµ −= 70.6 70.6

Collecting terms, we obtain

µα

qB

th

kk' 70.6

111 −=

Solving for the "pressure drift" term, we have

µα

qB

th

k'k 70.6

111 =− ...................................................(A-4)

Multiplying through Eq. A-4 by k

kqB

th

k

k

µα

70.6

1

'1 =−

Solving for the "estimated permeabil ity" (including the "pres-sure drift" term (α)), we have

k

qB

thk

k

µα

70.6

11

1'

−= ................................................(A-5)

SPE 77533 Effects on Well Test Analysis of Pressure and Flowrate Noise 5

Table 1 – Well test parameters for the unfractured verticalwell case.

Parameter Value UnitsB 1.06 RB/STBCs 0.01 STB/psict 4.2x10-6 psi–1

h 107 ftk 10.28 mdpi 3900 psiaq 174.0 (0 - 15.33 hr)

0.0(15.33 - 45.33 hr)STB/DSTB/D

rw 0.29 fts 8 (dimensionless)φ 0.25 (fraction)µ 2.5 cp

Table 2 – Well test parameters for the fractured vertical wellcase.

Parameter Value UnitsB 1.06 RB/STBCs 0.01 STB/psict 4.2x10-6 psi–1

h 107 ftk 10.28 mdkfw 4500 md-ftpi 3900 psiaq 1000.0 (0 - 50 hr)

0.0(50 - 150 hr)STB/DSTB/D

rw 0.29 fts 2 (dimensionless)xf 150 ftφ 0.25 (fraction)µ 2.5 cp

Table 3 – Statistics of parameter estimates — unfracturedwell case (0.6 psi no ise level).

Parameter MeanStandardDeviation

InputValue Units

Cs 0.009995 1.31x10-5 0.01 STB/psik 10.2801 0.08327 10.28 mds 8.0008 8.0093 8 (dim-less)pi 3900.029 0.2667 3900 psia

Figure 1 – Diagno stic plot for the unfractured vertical wellcase.

Figure 2 – Diagno stic plot for the fractured vertical well case(finite conductiv ity vertica l fracture).

Figure 3 – Distribution o f permeabili ty estimates, unfracturedvertical well case (0.6 psi no ise).

Figure 4 – Distribution o f sk in factor estimates, unfracturedvertical well case (0.6 psi no ise).

6 R.A. Archer, M.B. Merad, and T.A. Blasingame SPE 77533

Figure 5 – Confidence intervals for the estimated permeabili ty,unfractured vertical well case.

Figure 6 – Confidence intervals for the estimated sk in factor,unfractured vertical well case.

Figure 7 – Distribution o f permeabili ty estimates, fracturedvertical well case (0.6 psi no ise, buildup case).

Figure 8 – Distribution of sk in factor estimates, fractured verti -cal well case (0.6 psi noise, buildup case).

Figure 9 – Distribution o f f racture half-length estimates, frac-tured vertical well case (0.6 psi noise, buildupcase).

Figure 10– Distribution of f racture condu ctiv ity estimates, frac-tured vertical well case (0.6 psi noise, buildupcase).

SPE 77533 Effects on Well Test Analysis of Pressure and Flowrate Noise 7

Figure 11– Confidence intervals for the estimated permeabili ty,fractured vertical well case.

Figure 12– Confidence intervals for the estimated pseudoradialflow sk in factor, fractured vertical well case.

Figure 13– Confidence intervals for the estimated fracture half-length, fractured vertical well case.

Figure 14– Confidence intervals for the estimated fracture con-ductivity, fractured vertical well case.

Figure 15– Distribution o f permeabili ty estimates, fracturedvertical well case (0.6 psi no ise, drawdown case).

Figure 16– Distribution of f racture condu ctiv ity estimates, frac-tured vertical well case (0.6 psi noise, drawdowncase).

8 R.A. Archer, M.B. Merad, and T.A. Blasingame SPE 77533

Figure 17– Confidence intervals for the estimated permeabili ty,fractured vertical well case (drawdown).

Figure 18– Confidence intervals for the estimated fracture con-ductivity, fractured vertical well case (drawdowncase).

Figure 19– Estimated permeabili ty and confidence interval, un-fractured vertical well case (drift case).

Figure 20– Estimated permeabili ty and confidence interval,fractured vertical well case (drift case).

Figure 21– Estimated fracture condu ctivity and confidenceinterval, fractured vertical well case (drift case).

Figure 22– Estimated permeabili ty and confidence interval, un-fractured vertical well case (rate noise case).

SPE 77533 Effects on Well Test Analysis of Pressure and Flowrate Noise 9

Figure 23– Estimated permeabili ty and confidence interval,fractured vertical well case (rate noise case).

Figure 24– Estimated fracture half-length and confidence inter-val, fractured vertical well case (rate no ise case).