SPE Uncertainty in rock and fluid properties.

Transcription

1 SPE Effects on Well Test Analysis of Pressure and Flowrate Noise R.A. Archer, University of Auckland, M.B. Merad, Schlumberger, T.A. Blasingame, Texas A&M University Copyright 2002, Society of Petroleum Engineers Inc. This paper was prepared for presentation at the SPE Annual Technical Conference and Exhibition held in San Antonio, Texas, 29 September 2 October This paper was selected for presentation by an SPE Program Committee following review of information contained in an abstract submitted by the author(s). Contents of the paper, as presented, have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material, as presented, does not necessarily reflect any position of the Society of Petroleum Engineers, its officers, or members. Papers presented at SPE meetings are subject to publication review by Editorial Committees of the Society of Petroleum Engineers. Electronic reproduction, distribution, or storage of any part of this paper for commercial purposes without the written consent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgment of where and by whom the paper was presented. Write Librarian, SPE, P.O. Box , Richardson, TX , U.S.A., fax Abstract We have utilized sets of simulated well tests in an example well/reservoir model with an unfractured vertical well or a vertical well with a finite-conductivity (vertical) fracture to demonstrate the dependence of the estimated parameters and their confidence intervals on noise in the pressure and flowrate data. Uniformly distributed random noise is introduced into the pressure data and the resulting well parameter estimates are obtained using conventional means (analysis/history matching). For the cases of a pressure buildup test in an unfractured well and the pressure drawdown cases performed on a fractured well, the estimated parameters are almost without bias and the confidence intervals grow linearly with the magnitude of the noise. For the cases of pressure buildup tests performed on the fractured well, the estimated parameters are insensitive to the noise which suggests that this case may not be wellposed for analysis. Gauge drift and flowrate noises were also considered and the analysis results from the example well tests demonstrate that both of these kinds of noise are handled robustly by conventional well test analysis techniques. Introduction There are several sources of uncertainty in the analysis and interpretation of well test data Horne 1 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 producing time effects. Uncertainty in rock and fluid properties. This work addresses the impact of the first two issues (random noise in the pressure measurements, gauge drift, and flowrate measurement errors) on well test analysis. The study was conducted by generating over 2000 synthetic well test responses subject to each kind of error being studied. These well tests were then interpreted using nonlinear regression (and occasionally checked by specialized plots for consistency). The parameter estimates and the confidence intervals 2 associated with these estimates were computed and recorded. Horne 3 illustrates the value of confidence intervals in well test interpretation particularly the value of confidence intervals in assessing the validity of parameter estimations. For reference, this study considers models for both unfractured and fractured wells, but only for the case of a homogeneous, infinite-acting reservoir. In 1994, Horne 1 found that if the pressure measurements for a well test include errors which are approximately normally distributed (for the case of an unfractured vertical well), the resulting estimates of permeability, skin factor, wellbore storage coefficient and initial pressure are approximately normally distributed. This relatively simple concept forms the thesis for our work. Horne did not address the impact of the uncertainty in the pressure measurements on the confidence intervals for permeability, skin factor, wellbore storage coefficient and initial pressure, nor did he consider the case of a fractured well. Method The rock, reservoir and well properties used for our "unfractured well" cases are given in Table 1. For convenience, these properties are taken from Bourdet 4. The properties used for the fractured well case are similar and are presented in Table 2. The only significant modifications made to the unfractured well cases are to the flow rate history and to the skin factor. We acknowledge that these cases are only examples, and that a more general result could be obtained by considering additional cases. However, we will note that we did use an exhaustive number of cases to ensure statistical validity of our results. The conventional "log-log" diagnostic plots for each well test example are given in Figs. 1 and 2. To assess the effect of random noise imposed on the pressure measurements, the synthetic well tests were generated with (systematically) varying amounts of noise, and interpreted (primarily) using nonlinear regression. Unlike Horne's 1 work,

2 2 R.A. Archer, M.B. Merad, and T.A. Blasingame SPE in this work the noise introduced into the data was uniformly (not normally) distributed with a zero mean and a varying amplitude. For the cases that specifically address the effects of random noise in the pressure data, the analysis results are presented in two formats. The first format is that of a histogram (i.e., a probability distribution) of the estimated parameters (for specific (constant) noise level). In the second presentation we use plots that show the behavior of the average of the parameter estimates and confidence interval bounds for a particular set of well tests, shown as a function of the noise level. For the unfractured well cases, 100 synthetic well tests per noise level were generated and interpreted for noise levels of 0.1 to 1.0 psi (by 0.1 increments). In the fractured well case 50 well tests were used per noise level. The effect of gauge drift was also considered for gauge drifts of up to and including 0.1 psi/hr. The drift error is treated as deterministic (rather than a statistically derived estimate) so only one well test was considered for each level of pressure drift. The analysis results for well tests with gauge drift error are plotted against the drift error. Flowrate noise was considered by creating sets of synthetic well tests with perfect (i.e., undistorted) pressure data. However, these "perfect" pressure data were then interpreted using a (synthetic) noisy rate history. For the unfractured vertical well cases these well tests were interpreted by adding an uncorrelated random noise with amplitude varying from 1 to 9 STB/D to the base flowrate of 100 STB/D. For the fractured well case the noise amplitude varied from 10 to 50 STB/D and the base flowrate was 1000 STB/D. The flowrate noise distribution had a zero mean, and for each flowrate noise level we generated 100 synthetic well tests. To ensure a uniform analysis/interpretation approach, all well tests were interpreted using nonlinear regression (i.e. manual interpretation 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 permeability and skin factor estimates computed from 100 well tests at the 0.6 noise level (unfractured well case). These figures show that the estimated values of the permeability and skin factor are (approximately) normally distributed and are centered on the true values (k = md and s = 2). Figures 5 and 6 show the behavior of the estimates for a range of noise levels. Each point plotted on these figures represents the average of 100 well test interpretations where the pressure data contains a given level of noise. The noise introduced into the pressure data is uniformly (not normally) distributed but the resulting permeability and skin factor estimates are approximately normally distributed. This behavior agrees with Horne s 1 observations and helps to validate our thesis that the results from cases with pressure error are normally distributed. Fractured Well Case - Buildup Data Figures 7-10 present the histograms for the estimates of permeability, skin factor, fracture half-length, and fracture conductivity estimated from 50 synthetic buildup tests conducted at each noise level. These probability distributions are more irregular than those for the unfractured well case. The permeability and skin factor are estimated correctly (on average) however, the expected value of the fracture halflength is ft, slightly less than the true value of 150 ft. The fracture conductivity is overestimated, at 6920 md-ft (on average), compared to the true value of 4500 md-ft. The confidence intervals associated with these estimates are shown in Figures These results present a very different behavior compared to the confidence intervals from the unfractured well case we observe the confidence interval length to be essentially constant, irrespective of the noise level. For these cases it could be argued that the number of simulated well test cases may not be sufficient to establish statistical validity. However, it is our contention that if this is an issue, it is a minor one the most likely circumstance is that the particular case under consideration (i.e., the fractured well performance) may not demonstrate sufficient characteristic behavior for the regression algorithm to define each parameter uniquely. In simple terms, the tests may not provide the breath of behavior (i.e., flow regimes) for accurate estimates of permeability, skin factor, fracture half-length, and fracture conductivity to be obtained. It is our contention that this case does illustrate that, in particular, a complex analysis/interpretation requires that all flow regimes under consideration to be present in order to obtain a unique analysis. Fractured Well Case - Drawdown Data For the fractured well case drawdown data were also considered to see if the behavior observed from the buildup case would be reproduced. Again 50 synthetic well tests were generated and interpreted at each noise level. Figures 15 and 16 show histograms of the estimated permeability and fracture conductivity at the 0.6 psi noise level. These distributions can be approximated by normal distributions. Interestingly, in these cases the distribution of the fracture conductivity is no longer biased away from the true value of 4500 md-ft. The calculated mean of this distribution is md-ft. Figures 17 and 18 show the behavior of the confidence intervals for the permeability and fracture conductivity estimates. These confidence intervals behave in the same fashion as those for the unfractured well case did i.e. the confidence interval length grows linearly with the noise in the pressure. The question remains as to why the pressure drawdown cases appear to be more amenable to analysis/interpretation than the pressure buildup cases for this model (fractured well). The

3 SPE Effects on Well Test Analysis of Pressure and Flowrate Noise 3 resolution of this issue lies in the testing sequence the pressure buildup cases reflect the production history, and as such, the pressure buildup profile has constraints. In this case the pressure buildup test duration was twice as long as the drawdown test, and the data were corrected using the "effective time" function for the radial flow case (the most common practice). In summary, the testing sequence and data correction process undoubtedly biased the results of the pressure buildup case. Drift Error The effect of gauge drift was considered by generating and interpreting an example well test case with varying degrees of gauge drift, for both the unfractured and fractured well cases. Since the drift is a deterministic error (i.e., such an error would 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 illustrate the behavior of the estimated permeability for the unfractured and fractured well cases in particular, these figures show that the permeability estimates and the length of the associated confidence intervals increase linearly with the magnitude of the drift error. The effect of pressure gauge drift error on the estimated permeability estimated was also addressed analytically in Appendix A. Gauge drift means that the observance of a flat pressure derivative function will not always occur for the case of an infinite-acting reservoir. If the pressure derivative level, p' rf, is read near the beginning of radial flow, then Eq. A-5 can be used to relate the permeability estimated using nonlinear regression to the true reservoir permeability. Figure 21 shows similar behavior to Figs. 19 and 20 for the estimated fracture conductivity (fractured well case). However, the confidence intervals are wider (in percentage terms). The data considered for this comparison are the buildup data for the unfractured well case and drawdown data for the fractured well case. Flowrate Noise Figure 22 shows the effect of the flowrate noise on the permeability estimate for the unfractured vertical well case. In Fig. 22 we note that the permeability estimate steadily increases with the magnitude of the noise in the flowrate function we note that this increase lies within tolerable limits, and suggests that flowrate noise (i.e., error) may not adversely affect the analysis/interpretation of well test data, at least in a practical sense. Figures 23 and 24 show the behavior of the estimated permeability and fracture half-length parameters for the fractured well (drawdown) cases. We note that these estimates do not follow a strictly linear trend the estimated permeability is essentially correct, while the fracture half-length is consistently overestimated. Part of this behavior could be due to the regression analysis (and probably is), but we can note that in comparing Figs. 22 and 23 it would be difficult to state that one case has a "better" characteristic profile than the other which is somewhat expected for the estimation of permeability using well test analysis. Discussion Put simply, the analysis/interpretation of well test data obtained from unfractured vertical wells in infinite-acting reservoirs is relatively unaffected by random noise in the pressure measurements. The pressure noise (i.e., error) does not bias the parameter estimates, but it does broaden the confidence intervals associated with the parameters. The broadening of the confidence intervals in a linear fashion was expected from an analysis of the confidence interval formulation presented by Dogru et al. 2 Attempts were made to derive a rigorous expression relating the change in the confidence interval length to the noise level. A theoretical formulation was developed but is not presented in this work because the validation of this formulation using numerical experiments was not satisfactory. This poor performance of the theoretical formulation is thought to be due to the sensitivity of the confidence interval calculation to the specific set of pressure data points matched by the regression analysis. For example, pressure points in the early time region tended to dominate the calculated confidence intervals. In the fractured well example we considered the effect of random noise on the analysis/interpretation was substantially different depending whether buildup or drawdown data were being analyzed/interpreted. The pressure buildup cases showed biased parameter estimates and confidence intervals of an approximately constant length in contrast, the drawdown data showed essentially unbiased parameter estimates and confidence intervals that broadened linearly with the noise amplitude (analogous to the unfractured well cases). For the cases of pressure drift error we showed that the parameter 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 of the confidence interval was much larger than the change in the parameter estimates, indicating that this may have practical implications for the analysis of this type of data. For the cases of flowrate noise (i.e., error) we noted that the analysis/interpretation is relatively unaffected by the noise in the flowrate for the unfractured and fractured well cases. The associated confidence intervals did however broaden with increasing noise (as would be expected). Concluding Remarks 1. The difference between the buildup and drawdown results in the fractured well cases raises a question about whether the superposition time function used to interpret the data in the buildup case compounds the effect of pressure and flowrate noise. 2. This research has shown that for the case of random noise in pressure measurements, that the distributions of the parameter estimates may be normally distributed regardless of whether the input noise distribution is uniform (this work) or normal (Horne 1 ).

4 4 R.A. Archer, M.B. Merad, and T.A. Blasingame SPE In the future we hope to pursue a quantitative approach to explain the behavior observed in the numerical experiments shown in this paper. We acknowledge that this study only analyzed two example well tests. We hope to generalize this work by considering other examples cases and by analytically deriving models to describe the parameter uncertainty. We also would like to consider extending the work to other well/reservoir models such as horizontal wells, and dual porosity reservoirs. Nomenclature B = formation volume factor, RB/STB C = wellbore storage coefficient, STB/psi c t = total compressibility, psi -1 h = net pay thickness, ft k = permeability, md p = pressure, psia p i = initial pressure, psia q = production rate, STB/D r w = wellbore radius, ft s = skin factor, dimensionless t = time, hr α = pressure gauge drift rate, psi/hr φ = porosity, fraction µ = viscosity, cp References 1. Horne, R.N.: "Uncertainty in Well Test Interpretation," paper SPE presented at the 1994 SPE Annual Technical Conference and Exhibition, Tulsa, Oklahoma, August Dogru, A.H., Dixon, T.N. and Edgar, T.F.: "Confidence Limits on the Parameters and Predictions of Slightly Compressible, Single-Phase Reservoirs," paper SPE 4983 presented at the 1974 SPE Annual Technical Conference and Exhibition, Houston, October Horne, R.N.: "Advances in Computer-Aided Well Test Interpretation," paper SPE presented at the 1992 SPE Annual Technical Conference and Exhibition, Washington, DC, October Bourdet, D. Ayoub, J.A. and Pirard, Y.M.: "Use of Pressure Derivative in Well-Test Interpretation," JPT (June 1989) Acknowledgements The first author would like to acknowledge that this work was performed at Texas A&M University. The second author would like to acknowledge financial support from Schlumberger. Appendix A - Effect of Gauge Drift on Permeability Estimation for the Case of an Unfractured Well The recorded pressure change when the gauge is subject to drift is of the form: p meas = p true -α t... (A-1) The associated ("well testing") pressure derivative function is: p p p ' Ä rf = t meas Ä Ä = t true α t... (A-2) t t Even during infinite-acting radial flow this derivative value will not be constant with time (due to the pressure drift) however, if the drift is small the "well testing" pressure derivative function (Eq. A-2) would be approximately constant. The "well testing" pressure derivative function includes the "measured" pressure derivative (in terms of the "estimated" permeability, k') as well as the "true" pressure derivative (in terms of the true permeability, k) plus a term that includes the pressure gauge drift (α). Writing Eq. A-2 for the case of an unfractured well producing at a constant rate in an infiniteacting reservoir, we have: ' Ä p rf = 70.6 = 70.6 αt... (A-3) k ' h kh Our goal is to define a rigorous relationship between k', k, and α in order to quantify the impact of drift term (α) ' k h = 70.6 αt kh Collecting terms, we obtain hαt = k' k 70.6 Solving for the "pressure drift" term, we have hαt =... (A-4) k k' 70.6 Multiplying through Eq. A-4 by k k 1 = k' hαt k Solving for the "estimated permeability" (including the "pressure drift" term (α)), we have 1 k' = k... (A-5) 1 hαt 1 k 70.6

5 SPE Effects on Well Test Analysis of Pressure and Flowrate Noise 5 Table 1 Well test parameters for the unfractured vertical well case. Parameter Value Units B 1.06 RB/STB C s 0.01 STB/psi c t 4.2x10-6 psi 1 h 107 ft k md p i 3900 psia q ( hr) 0.0( hr) STB/D STB/D r w 0.29 ft s 8 (dimensionless) φ 0.25 (fraction) µ 2.5 cp Table 2 Well test parameters for the fractured vertical well case. Parameter Value Units B 1.06 RB/STB C s 0.01 STB/psi c t 4.2x10-6 psi 1 h 107 ft k md k fw 4500 md-ft p i 3900 psia q (0-50 hr) 0.0( hr) STB/D STB/D r w 0.29 ft s 2 (dimensionless) x f 150 ft φ 0.25 (fraction) µ 2.5 cp Figure 2 Diagnostic plot for the fractured vertical well case (finite conductivity vertical fracture). Table 3 Statistics of parameter estimates unfractured well case (0.6 psi noise level). Parameter Mean Standard Deviation Input Value Units C s x STB/psi k md s (dim-less) p i psia Figure 3 Distribution of permeability estimates, unfractured vertical well case (0.6 psi noise). Figure 1 Diagnostic plot for the unfractured vertical well case. Figure 4 Distribution of skin factor estimates, unfractured vertical well case (0.6 psi noise).

6 6 R.A. Archer, M.B. Merad, and T.A. Blasingame SPE Figure 5 Confidence intervals for the estimated permeability, unfractured vertical well case. Figure 8 Distribution of skin factor estimates, fractured vertical well case (0.6 psi noise, buildup case). Figure 6 Confidence intervals for the estimated skin factor, unfractured vertical well case. Figure 9 Distribution of fracture half-length estimates, fractured vertical well case (0.6 psi noise, buildup case). Figure 7 Distribution of permeability estimates, fractured vertical well case (0.6 psi noise, buildup case). Figure 10 Distribution of fracture conductivity estimates, fractured vertical well case (0.6 psi noise, buildup case).

7 SPE Effects on Well Test Analysis of Pressure and Flowrate Noise 7 Figure 11 Confidence intervals for the estimated permeability, fractured vertical well case. Figure 14 Confidence intervals for the estimated fracture conductivity, fractured vertical well case. Figure 12 Confidence intervals for the estimated pseudoradial flow skin factor, fractured vertical well case. Figure 15 Distribution of permeability estimates, fractured vertical well case (0.6 psi noise, drawdown case). Figure 13 Confidence intervals for the estimated fracture halflength, fractured vertical well case. Figure 16 Distribution of fracture conductivity estimates, fractured vertical well case (0.6 psi noise, drawdown case).

8 8 R.A. Archer, M.B. Merad, and T.A. Blasingame SPE Figure 17 Confidence intervals for the estimated permeability, fractured vertical well case (drawdown). Figure 20 Estimated permeability and confidence interval, fractured vertical well case (drift case). Figure 18 Confidence intervals for the estimated fracture conductivity, fractured vertical well case (drawdown case). Figure 21 Estimated fracture conductivity and confidence interval, fractured vertical well case (drift case). Figure 19 Estimated permeability and confidence interval, unfractured vertical well case (drift case). Figure 22 Estimated permeability and confidence interval, unfractured vertical well case (rate noise case).

9 SPE Effects on Well Test Analysis of Pressure and Flowrate Noise 9 Figure 23 Estimated permeability and confidence interval, fractured vertical well case (rate noise case). Figure 24 Estimated fracture half-length and confidence interval, fractured vertical well case (rate noise case).