Improving Infill Development Decision Making With Interval Estimation

Transcription

1 ENERGY EXPLORATION & EXPLOITATION Volume 25 Number pp Improving Infill Development Decision Making With Interval Estimation J. Marcus Jobe 1 and Hutch Jobe 2 1 Professor, 301 Upham Hall, Miami University, Oxford, Ohio 45056, jobejm@muohio.edu 2 Exploration Manager, St. Mary Land and Exploration, 7060 S. Yale, suite 800, Tulsa, Oklahoma , hjobe@stmaryland.com Abstract Following partial completion of infill development in a given field, it becomes necessary to determine whether continued drilling should be pursued in the same field. The usual approaches for deciding whether to continue drilling (Swanson s mean or estimates based on lognormal methods) fail to account for variation and uncertainty. To better inform this decision, we propose implementing statistical confidence intervals (with an accompanying certainty level) for relevant features of interest. Closed form intervals and maximum likelihood based intervals for the average ultimate recovery for all wells in a field, the percent of wells in a field having ultimate recovery values above a certain benchmark value and other important features of interest are considered. Confidence intervals appropriate for both the lognormal and Weibull distribution are given. The proposed methods are illustrated with real data from 64 wells in the Devonian Richfield Dolomite of the Michigan basin (given in the appendix, Table 2). Table 1 summarizes the different confidence intervals. 1. INTRODUCTION Various types of information are available to decision makers which help inform the decision whether to continue drilling or not. One source of information is the recorded performance of wells drilled as part of the infill development. Variables such as total accumulation, initial potential, porosity, pay thickness and recovery factors are used to reflect well performance. Acceptable economical values (already determined) for the average of each of these variables are considered as benchmarks to determine whether continued drilling is economically feasible. Applying typical methods cited in the literature (Swanson s mean) or other lognormal dependent approaches, an estimate of the average for the variable of interest is determined and then compared to the benchmark value leading to a decision whether to continue drilling. Since each of these methods produce a single estimate of the mean, the true mean could be much smaller or much larger than the value given by the estimate, shedding doubt on the decision about continued drilling, i.e., variability is not taken into consideration. Thus, decision making based on whether the single estimate is larger or smaller than some

2 248 Improving Infill Development Decision Making With Interval Estimation benchmark can be very misleading. We address this shortcoming by establishing interval estimates (with accompanying certainty levels) for the quantity of interest. Maximum likelihood methodology has been developed that will give an optimum estimate for an average of interest, i.e., the average ultimate recovery for all wells in the field. Since these maximum likelihood estimates are optimum single value estimates, how close these estimates are to the true average and how sure one can be that they are that close needs to be answered. Maximum likelihood based statistical confidence intervals have been developed to address this important question and are presented in the following sections. Further, interval estimates for the percent of wells having values of the variable of interest above a benchmark value are developed. A level of confidence that the interval contains the percent of wells having values of the variable of interest above the benchmark is also given. We illustrate these methods and intervals with real data from 64 wells in the Devonian Richfield Dolomite of the Michigan basin. In this paper, the variable of interest is ultimate recovery. The reservoir rock in the subject field consisted of tight (less than or equal to 10 millidarcies) dolomites interbedded with sealing anhydrites. There were seven productive dolomite intervals that spanned a stratigraphic section of two hundred sixty feet (80 meters). The dolomite stringers were one to eight meters thick and possessed porosities of six to twenty-four percent. Depth of the reservoir was fourteen hundred (1400) meters. The field in question was fully developed in the early 1950 s on 40 acre spacing. Ninety-nine (99) wells were drilled covering an area of twelve square miles. In the early 1960 s, a nine (9) spot waterflood was consistently implemented throughout the field via converting existing primary producers. The flood was extremely effective and led to additional 40-acre spaced drilling and a gradual change to a five (5) spot injection pattern in the late 1960 s and 1970 s. The 1980 s led to infill drilling which implemented a change to 20-acre spacing. A total of sixty four (64) wells were drilled during these programs with performance of wells gradually decreasing in the later infill stages. This decrease in well performance caused early termination of the infill development in A 1995 review of the total performance of these 64 wells led to the decision to drill six (6) new development wells. Some of the methods presented in this paper significantly influenced the decision (using the Weibull distribution). The 20-acre infill development referred to in this paper was terminated after the Fall 1990 drilling program. This twelve (12) well program produced results far less than expected and thus ended any further development. The performance of this program and the previous seven (1983 through 1989) is given in Table 2 (see the appendix). Poor performance from the 1990 program is attributed to multiple reasons. First of all, in an effort to reduce costs, the entire Richfield interval was fracture stimulated at once rather than partitioning. Secondly, the 1990 wells were concentrated in a single part of the field instead of being evenly distributed. This over-concentration also occurred where there had been an abnormally (greater than 3:1) high secondary to primary recovery ratio showing more complete drainage. Finally, hasty decision making, based on early production performance and incomplete analysis, discounted the entire field potential and thus terminated any further infill drilling until 1996.

3 ENERGY EXPLORATION & EXPLOITATION Volume 25 Number Useful EUR distributions were determined based on volumetrics and decline curve analysis typically associated with conventional reservoir engineering methods. The statistical analysis approach allowed us to make informed predictions as to the most likely recovery size of un-drilled locations. Conclusions from the statistical analysis helped to justify moving forward in a timely fashion with a development drilling program involving multiple wells. Moving forward without unnecessary delay enabled the wells to be drilled, completed and put to sales which increased the net present value. The drilling and completing costs were minimized by project planning rather than planning on a single well basis. Jobe et al. (2000) presented statistical methods to determine whether the Weibull or lognormal distribution would best model ultimate recovery values from all wells in this field. In fact, they showed that for the set of data in Table 2, the Weibull distribution modeled the data better than the lognormal distribution. Hence, for the data in Table 2, credible inference as to whether to drill more wells should be made using methods based on the Weibull distribution, not the lognormal. Jobe et al (2000) presented only preliminary confidence interval methodology useful for determining whether to continue drilling. Building on their work, the following sections establish more extensive confidence interval methodologies. (Some of which were informally used to determine whether to continue drilling which resulted in the significant multimillion dollar gross value-added increase as stated by Jobe et al (2000)). Closed-form confidence intervals for the average ultimate recovery for all wells like the set of sixtyfour (64) and closed-form confidence intervals for the percent of all wells that exceed the economically determined benchmark value for the ultimate recovery variable are given. In general, it is true that not every set of data is best modeled by the Weibull distribution and it is well known the lognormal distribution often accurately models ultimate recovery data. Thus, intervals appropriate for the Weibull and lognormal distributions are given. However, practically speaking, according to Jobe et al (2000) the data in Table 2 is best modeled by the Weibull distribution. Hence, those intervals based on the Weibull distribution are best -most credible. Selected confidence intervals constructed using the maximum likelihood (ML) method for both distributions are illustrated as well. The Weibull and lognormal are the focus of this paper not only because of the reasons mentioned above but because of their ability (flexibility) to model most any distribution. Consider Figures 1 and 2 in sections 2 and 3. It is clear that most distributions can be approximated by either the Weibull or the lognormal. Once the distribution of a set of data is known to be either Weibull or lognormal, powerful confidence interval methodologies, independent of large sample sizes, are available to improve decision making. Of course, the larger the sample size, the smaller the variation in the estimates and the shorter the resulting confidence intervals will be. Several of these methods are illustrated in sections 2 and 3. Using the data from Table 2, a comparison of intervals based on the lognormal versus the Weibull is presented in section 4, Table1. 2. CONFIDENCE INTERVALS ASSOCIATED WITH THE LOGNORMAL DISTRIBUTION Let X be a random variable representing the ultimate recovery from a given well. If the

4 250 Improving Infill Development Decision Making With Interval Estimation ultimate recovery variable can be described as a lognormal random variable, Nelson (1982, pp ) denoted the probability distribution of X to be f(x) = [1/(σx(2π) 1/2 )]exp[-(lnx µ) 2 /2σ 2 ], x > 0 (2.1) where lnx is the natural log of x. Figure 1 portrays several different versions of eqn (2.1) for selected values of σ and µ = 0. More simply, the distribution of Y = lnx is normal with mean E(Y) = µ and variance Var(Y) = σ 2. The mean E(X) and variance Var (X) can be shown to be µ x = E(X) = exp [µ + σ 2 /2] and Var(X) = exp [2µ + 2σ 2 ] exp [2µ + σ 2 ]. (2.2) Figure 1. Graphs of the lognormal distribution for µ = 0 and selected values of σ Since the cumulative distribution function F(x) = P(X < x), the population fraction above x becomes R(x) = 1- F(x) = 1 Φ {[lnx µ]/σ}, x > 0. (2.3) The function Φ is the cumulative distribution function for the standard normal random variable, i.e., Φ (-1) =.1587, Φ (0) =.5, Φ (1) = In the following presentation, application of the six (6) confidence intervals stated in eqns (A.1- A.7) are illustrated. Each interval estimates an important feature of the lognormal distribution representing the ultimate recovery variable X in an infill development context. The data in Table 2 (in the appendix) will be used to illustrate these intervals. Letting Y = lnx and n = 64, important summary statistics from Table 2 are: x = 51.3, s x = 45.29, y = 3.576, s y =.946.

5 ENERGY EXPLORATION & EXPLOITATION Volume 25 Number Note: P(Z > z α/2 ) = α/2, P(t n-1 > t α/2 ) = α/2, x = Σ x /n, s x2 = Σ(x ) 2 /(n-1), y = Σy /n, s y2 = Σ(y y ) 2 /(n-1). The random variable Z is the standard normal random variable and t n-1 is student s t random variable with n-1 degrees of freedom. Example 1: Select a 95% certainty level. The 95% confidence interval for the true ultimate average recovery is given by eqn (A.1). Thus, with 95% confidence, the expression 51.3 ± (1.96)(5.66) or the interval (40.21, 62.39) includes the true average ultimate recovery. For the sake of example, consider another level of confidence, say, 70%. Eqn (A.1) becomes 51.3 ± (1.0364)(5.66). Thus, with 70% confidence, the interval (45.434, ) includes the true average ultimate recovery. The 50 th percentile of ultimate recovery (X) can be estimated with a (1-α)100% confidence interval for any sample size n. Nelson (1982, p. 220) showed it is straightforward to estimate the 50 th percentile with the interval given in eqn (A.3). Example 2: Select a 95% level of confidence. Eqn (A.2) becomes ± (.946)/(64) 1/2. Thus, the interval (3.3397, ) includes the parameter µ with 95% confidence. Eqn (A.3) becomes { exp(3.3397), exp(3.8123) } or (28.21, 45.25). Hence, with 95% confidence, the interval (28.21, 45.25) includes the 50 th percentile of ultimate recovery. The ultimate recovery (X) of a single well to be drilled (like the wells in a sample of size n (large or small)), can be estimated with a (1-α)100% confidence interval presented by Nelson (1982, pp. 224, 225) and given by eqn (A.4). Example 3: Select a 95% confidence level. Eqn (A.4) becomes { exp(1.6708), exp(5.4811) }. Thus, there is a 95% probability the ultimate recovery X from the next well drilled falls in the interval (5.3164, ). Suppose a 70% certainty level is selected. From eqn (A.4), with a 70% probability, ultimate recovery X from the next well drilled will fall in the interval (13.194, ). We turn now to interval estimation of P(X > x o ) the proportion of wells that have an ultimate recovery exceeding a selected benchmark ultimate recovery value x o. Since Y = lnx and the exp() function are monotonic, P(X > x o ) = P(lnX > lnx o ) = P(Y > y o ). Letting P(X > x o ) = R(x o ) = R(y o ) = P(Y > y o ), Nelson (1982, p. 223) presented an interval that includes R(x o ) = R(y o ) with (1-α)100% certainty for n 20. Denoting z o = (y o y )/s y and recalling Y = lnx is a normal random variable, R(x o ) = R(y o ) can be estimated with 1 Φ(z o ). The (1 α)100% confidence interval Nelson (1982, p. 223) presented for R(x o ) = R(y o ) is given in eqns (A.5) and (A.6). Further, the (1 α)100% lower one-sided confidence interval for the percent of all wells having an ultimate recovery X larger than x o is given in eqn (A.7). Example 4: Select a benchmark ultimate recovery value x o = 41, i.e., wells with an ultimate recovery that exceeds 41 are economically profitable. Thus, y o = lnx o = Since s y =.946 and y = 3.576, z o = (y o y )/s y = The estimated proportion of all wells that have an ultimate recovery larger than x o = 41 becomes R(x o = 41) =1 Φ(.1454) =.442. Consider now a 95% confidence interval estimate of R(x o = 41). Substituting n = 64, z o =.1454 and z α/2 =1.96, eqns (A.5) and (A.6) produce R (y o ) =.3476 and R ~ (y o ) = Thus, with 95% certainty the fraction of all wells that have an ultimate recovery amount exceeding x o = 41

6 252 Improving Infill Development Decision Making With Interval Estimation is somewhere within the interval (.3476,.5402). From eqn (A.7), a lower one-sided 95% confidence bound for R(x o = 41) (the percent of all wells that have an ultimate recovery larger than x o = 41) becomes Thus, with 95% confidence, at least 36.24% of all wells will have an ultimate recovery of at least x o = 41. If the analyst only needs to be 70% confident, a lower bound for R(x o = 41) becomes Hence, with 70% confidence, at least 41.6% of all wells will have an ultimate recovery of at least x o = 41. Nelson (1982, pp ) presented more sophisticated confidence interval methodologies based on maximum likelihood theory. Maximum likelihood methods are very attractive but specialized software is usually needed for the necessary calculations. Nelson (1982, p. 313) mentioned several advantages of maximum likelihood (ML) methods. He noted ML methods are very versatile, most ML estimators have good properties and these properties usually hold for small samples (small n) as well. Mood et al. (1974, p. 359) stated that under usual conditions and large sample size ML estimators are normally distributed, unbiased and no other unbiased estimators are better. The following three (3) intervals are based on the methods presented by Nelson (1982, pp ). Minitab version 12 software was implemented for the necessary calculations. The 95% two-sided ML confidence interval for R(x o = 41) = R(y o = ) is [.3473,.5398]. With 95% confidence, between 34.7% and 53.98% of all wells have an ultimate recovery that exceeds x o =41. The ML point estimate for the percent of all wells that have ultimate recovery in excess of x o =41 is Thus, about 44.18% of all wells have an ultimate recovery of at least x o =41. See Figure 3 for a graph of the ML point estimates of P(X > x o ) vs. x o. The one-sided 95% ML confidence interval for R(x o =41) is.362. With 95% probability at least 36.2% of all wells have an ultimate recovery that is greater than x o =41. The one-sided 70% ML confidence interval for R(x o =41) is 41.6%. Thus, with 70% probability at least 41.6% of all wells have an ultimate recovery greater than x o =41. Continuing with the ML method, assuming a lognormal distribution best models the ultimate recovery variable X, the 95% ML confidence interval for µ x (the true average ultimate recovery) becomes (42.12, 73.18). The endpoints are calculated based on MLE of µ x = a and the MLE of the variance of a = c (MLE represents the maximum likelihood method produced by specialized software, in our case Minitab 12). The expression a ± 1.96 (c) 1/2 produced the above confidence interval. The 95% ML confidence interval for the 50 th percentile becomes (23.39, 44.97) and is calculated based on specialized software, in our case Minitab 12. Using the ML approach, the 95% confidence interval for the next ultimate recovery value X, is (0, ). The endpoints are calculated based on MLE of µ x = a, the MLE of the variance of X = b and the MLE of the variance of a = c. The expression a ± 1.96 (b + c) 1/2 produced the above confidence interval. In summary, letting X be ultimate recovery best described by a lognormal distribution, eqns (A.1), (A.4) and (A.5) are closed form (1 α)100% confidence interval estimates of important infill development decision-making quantities. Maximum likelihood (1 α)100% confidence interval estimates for important decision-making quantities are also available. Table 1 summarizes all confidence

7 ENERGY EXPLORATION & EXPLOITATION Volume 25 Number intervals presented in the previous examples. Economic sensitivity analyses with an accompanying certainty level could be applied using values from the lower endpoint to the upper endpoint of a selected confidence interval. In contrast, an economic sensitivity analysis with an accompanying certainty level is not possible with the single point estimate provided by the usual graphical approaches such as Swanson s Mean (Megill, 1984) and others. 3. CONFIDENCE INTERVALS ASSOCIATED WITH THE WEIBULL DISTRIBUTION Let X be a random variable representing the ultimate recovery from a given well. If the ultimate recovery can be described as a Weibull random variable, Nelson (1982, pp ) described the probability distribution function of X to be f(x) = (β/α β )x β-1 exp[-(x/α) β ], x > 0. (3.1) The parameter β is called the shape parameter and is positive. The parameter α is referred to as the scale parameter. The mean E(X) and variance Var(X ) are µ x = E(X) = α Γ(1 + 1/β) and Var(X) = α 2 [Γ(1 + 2/β) Γ 2 (1 + 1/β)]. (3.2) Figure 2 portrays different versions of eqn (3.1) for selected values of β and α =1. Figure 2. Graphs of the Weibull distribution for α = 1 and different values of β. Continuing, the cumulative distribution function F(x) = P(X < x) is 1- exp[-(x/α) β ], (3.3) and the population proportion above X = x, i.e., 1- F(x) = P(X > x), becomes exp [-(x/α) β ]. (3.4) In the following exposition, application of five (5) confidence intervals are illustrated. Three of these intervals are given by eqns (A.8)-(A.10). Each interval

8 254 Improving Infill Development Decision Making With Interval Estimation estimates an important ultimate recovery feature assuming the distribution of ultimate recovery in an infill development context is best represented by the Weibull distribution. The data in Table 2 (see appendix) will be used to illustrate these intervals. Summary statistics presented in the previous section will be implemented where appropriate. If the sample size is large (n 30), the Central Limit Theorem establishes a reasonable interval estimate of µ x, the average ultimate recovery amount. The form of this interval can be found in eqn (A.8). This interval is the same as that in eqn (A.1) and its interpretation is the same. In fact, for n > 30, eqns (A.1) and (A.8) are appropriate independent of the ultimate recovery distribution. We turn now to interval estimation of P(X > x o ), the proportion of wells that have an ultimate recovery exceeding some selected or benchmark ultimate recovery value x o. Nelson (1982, pp. 227 to 234) presented a single number estimate of P(X > x o ), a two-sided (1 α)100% confidence interval estimate of P(X > x o ) and a one-sided (1- a)100% confidence interval estimate of P(X > x o ). The interval estimates for R(x o ) = P(X > x o ) given by Nelson (1982, pp. 227 to 234) are most appropriate when n 100. However, we will apply these here (n = 64). Letting Y = lnx and y o = lnx o, the point estimate of P(X > x o ) is exp[ -(x o /A) B ] (3.5) where S = [Σ(y i y ) 2 /(n 1)] 1/2, D =.7797(S), L = y (D), U = (y o L)/D, A = exp[l] and B = 1/D. The (1-α)100% two-sided confidence interval for P(X > x o ) is given in equation (A.9). The lower one-sided (1-a)100% confidence bound for P(X > x o ) is given in eqn (A.10). The following example illustrates these intervals. Example 5: Consider a benchmark ultimate recovery value x o = 41(representing an economical well case). Using summary statistics from the previous section, Example 1, Example 4 and the data from Table 2, we see y o = lnx o = , y = 3.576, S =.946, D =.73759, L = , A = , B = and U = From eqn (3.5), the single value estimate of P(X > x o = 41) is Thus, assuming the Weibull distribution models ultimate recovery, it is estimated that 50.83% of all wells will have an ultimate recovery that exceeds 41. Continuing with this example, consider a 95% two-sided confidence interval for P(X > x o ). From eqn (A.9), by substitution, u ~ = , u ~ = and the 95% confidence interval estimate of P(X > 41) becomes [.40448,.60302]. Thus, with 95% confidence, the percent of wells that have an ultimate recovery greater than 41 is somewhere between 40.45% and 60.30%. Consider now construction of a lower one-sided 95% confidence interval for P(X > 41). From eqn (A.10), the u ~ appropriate for the lower one-sided interval becomes and.4215 becomes the lower bound of interest. Thus, with 95% confidence, at least 42.15% of all wells have an ultimate recovery value that exceeds 41. Continuing, if the desired confidence for the lower one-sided interval is 70%, u ~ = Substitution into eqn (A.10) produces a lower bound equal to Hence, with 70% confidence at least 48.12% of all wells have an ultimate recovery amount that exceeds 41. Nelson (1982, pp. 346,347) also presented more sophisticated methodologies for

9 ENERGY EXPLORATION & EXPLOITATION Volume 25 Number estimating P(X > x o ). These methodologies use the maximum likelihood (ML) approach assuming X has a Weibull distribution. As mentioned previously, in a loose sense this approach produces estimates which are as good as can be determined. A small impediment to this approach with the Weibull distribution is that special software is needed to accomplish the rather extensive calculations. Minitab version 12 software was used for the following examples to produce the ML estimates. Example 6: As in other examples, x o = 41 was selected as the benchmark (in fact, 41 is very close to the actual economically feasible breakeven value for ultimate recovery in the context of the data in Table 2). Using ML estimation, the estimated P(X > x o ) is Thus, ML suggests 50.12% of all wells have an ultimate recovery that exceeds x o = 41. See Figure 3 for a graph of the ML point estimates of P(X > x o ) vs. x o. A (1- α)100% two-sided confidence interval can also be calculated using ML methods. Selecting a 95% confidence level, ML methods produced the interval (.3982,.5957). Thus, with 95% confidence, between 39.8% and 59.57% of all wells have an ultimate recovery exceeding x o = 41. Finally, a (1-α)100% lower one-sided confidence interval for P(X > x o ) can also be constructed using ML methods. A 95% confidence level was selected. The ML method produced.415 as the lower bound. Thus, with 95% confidence, at least 41.5% of all wells have an ultimate recovery greater than x o = 41. If a 70% confidence level is selected, the ML method produced.4743 as the lower bound. Thus, with 70% confidence, at least 47.43% of all wells have an ultimate recovery greater than x o = 41. Continuing with the ML method, assuming a Weibull distribution best models the ultimate recovery variable X, the 95% ML confidence interval for µ x (the true average ultimate recovery) becomes (42.24, 62.83). The endpoints are calculated based on MLE of µ x = α and the MLE of the variance of a = c (MLE represents the maximum likelihood method produced by specialized software, in our case Minitab 12). The expression a ± 1.96 (c) 1/2 produced the above confidence interval. The 95% ML confidence interval for the 50 th percentile becomes (32.62, 51.81) and is calculated based on specialized software, in our case Minitab 12. Using the ML approach, the 95% confidence interval for the next ultimate recovery value X, is (0, ). The endpoints are calculated based on the MLE of µ x = a, the MLE of the variance of X = b and the MLE of the variance of a = c. The expression a ± 1.96 (b + c) 1/2 produced the above confidence interval. In summary, letting X be ultimate recovery best described by a Weibull distribution, eqns (A.8), (A.9) and (A.10) are closed form (1 α)100% confidence interval estimates of important infill development decision-making quantities. Maximum likelihood (1 α)100% confidence interval estimates are given for important decision-making quantities as well. Table 1 summarizes all confidence intervals presented in the previous examples. Economic sensitivity analyses with an accompanying certainty level could be applied using values from the lower endpoint to the upper endpoint of a selected confidence interval. In contrast, an economic sensitivity analysis with an accompanying certainty level is not possible with the single point estimate provided by the usual graphical approaches such as Swanson s Mean (Megill, 1984) and others.

10 256 Improving Infill Development Decision Making With Interval Estimation 4. CONCLUSIONS Upon deciding which probability distribution best models the realized ultimate recovery data (see Jobe et al., 2000), appropriate confidence interval estimates of quantities of interest should be implemented. Interval estimates include information about variability revealed in the data. Further, these intervals locate the feature of interest and communicate how sure one can be that the feature of interest has been located. The usual approaches (such as Swanson s mean (Megill, 1984) and other single point estimates) fail to account for the important concept of variability. The extensive statistical developments associated with the lognormal and Weibull distribution can account for variability and need to be part of the decision making process associated with infill development. Economic sensitivity analyses with an accompanying certainty level could be applied using values from the lower endpoint to the upper endpoint of a selected confidence interval. In contrast, an economic sensitivity analysis with an accompanying certainty level is not possible with the single point estimate provided by the usual graphical approaches. The statistical developments presented in our paper provide interval estimates for important features of interest directly related to whether future drilling should be pursued. Confidence intervals for the following important features of interest associated with the lognormal distribution have been highlighted: the mean ultimate recovery, the ultimate recovery from the next well drilled, the ultimate recovery value such that 50% of all wells have an ultimate recovery that is less, and the proportion of all wells that have an ultimate recovery greater than a selected benchmark (x o ) representing an economical breakeven point. We have illustrated a total of ten (10) confidence intervals that directly apply to these features of the lognormal distribution. In addition to closed-form type intervals, ML methods have also been presented. Since the intervals associated with ML are generally accepted as best, these should be pursued. The availability of appropriate computer software makes the ML methods quite appealing. Eight (8) confidence intervals have also been recommended when the Weibull distribution best models the realized ultimate recovery data. The following were identified as features of interest: the mean ultimate recovery value and the proportion of all wells that have an ultimate recovery greater than a selected benchmark (x o ) representing an economical breakeven point. Closed-form versions of corresponding confidence intervals have been presented and illustrated. ML intervals were also calculated for the mean ultimate recovery, 50 th percentile of the ultimate recovery, the next value of ultimate recovery variable X and the proportion of all wells that have an ultimate recovery greater than the selected benchmark (x o ). As in the lognormal setting, since the intervals associated with ML are generally accepted as best, these should be pursued. (See Figure 3. for ML estimated P(X > x0) vs. x0 assuming a lognormal or Weibull model. Since the Weibull model was judged to best model the data, the corresponding curve Circle estimates P(X > x0) vs. x0 most reliably). Finally, the multitude of statistical developments associated with the Weibull and lognormal distributions can significantly improve the decision making facet of infill development because both variation and selected certainty levels are incorporated into the corresponding estimation methods (confidence intervals).

11 ENERGY EXPLORATION & EXPLOITATION Volume 25 Number Figure 3. Plot of ML Estimated P(X > x0) vs. x0. Table 1 given below summarizes the confidence intervals developed in this paper. It is especially noteworthy to see how the intervals differ dependent on what type of distribution is assumed. Since Jobe et al (2000) determined the Weibull distribution better modeled the data in Table 2, confidence intervals in Table 1 corresponding to the Weibull distribution should be chosen to improve decision-making because they are the most accurate, i.e., the most informative. If the data had been determined to have come from a lognormal distribution, confidence intervals in Table 1 corresponding to the lognormal distribution would be preferred. REFERENCES 1. Jobe, J.M. and Jobe, T.H. (2000). A Statistical Approach For Additional Infill Development, Energy Exploration and Exploitation, vol. 18, no. 1, pp Nelson, Wayne (1982). Applied Life Data Analysis, John Wiley & Sons, New York, New York, pp , 220, , , and Megill, R.E. (1984). An Introduction to Risk Analysis, 2 nd edition, Tulsa, Oklahoma, Penwell Publishing Co., pp Mood, A.M., Grabill, F.A. and Boes, D.C. (1974). Introduction to the Theory of Statistics, 3 rd edition, McGraw-Hill, New York, New York, p Minitab Inc., Version 12 (1997). State College, Pennsylvania.

12 258 Improving Infill Development Decision Making With Interval Estimation Table 1. Confidence Intervals for Characteristics of Interest Assuming a Weibull or Lognormal Distribution for the Ultimate Recovery Variable X. Quantity to be estimated Lognormal Weibull Restriction/Method µ x (40.21, 62.39) (40.21, 62.39) n > 30 (42.12, 73.18) (42.24, 62.83) ML, n > 30 Median (28.21, 45.25) no closed form any n > 1 (23.39, 44.97) (32.62, 51.81) ML, n > 30 X (single observation) (5.31, 240.1) no closed form any n > 1 (0, ) (0, ) ML, n > 30 P(X > x o =41) (.3476,.5402) (.404,.603) n lognormal >20 n Weibull > 100 (.3473,.5399) (.3982,.5957) ML, n > 30 Lower One-sided for (.3624 to 1) (.4125 to 1) n lognormal >20 P(X > x o =41) n Weibull > 100 (.362 to 1) (.415 to 1) ML, n > 30 *Note: All confidence intervals are at the 95% confidence level; ML corresponds to the maximum likelihood method and n is sample size. Table 2. Estimate Ultimate Recovery (1000s barrels) for the original n = 64 wells

13 ENERGY EXPLORATION & EXPLOITATION Volume 25 Number APPENDIX Consider X distributed as lognormal, Y = lnx distributed as normal. For large sample size (n 30), a (1 α)100% two-sided interval estimate for m x is x ± z α/2 s x / n (A.1) A (1 α)100% two-sided interval estimate of µ for any sample size n is y ± tn 1;α/2 s y / n (A.2) Extending (A.2), a (1 α)100% two-sided interval estimate of the 50 th percentile of the ultimate recovery variable X is {exp[y t n 1;α/2 s y / n], exp[y + t n 1;α/2 s y / n] }. (A.3) The (1 α)100% two-sided interval estimate for the ultimate recovery X of the next well is {exp[y t n 1;α/2 s y / (1/n)+1] ], exp[y + t n 1;α/2 s y / (1/n)+1] ]}. (A.4) The (1 α)100% two-sided interval estimate for the proportion of wells with an ultimate recovery that exceeds a selected value x o, i.e., P(X > x o ), is where and [ ~ R(y o ), R ~ (y o )], (A.5) R(y ~ o ) = 1 Φ[z o + (z -/2 / n){1 + [z o2 (n/2)/(n-1)]} 1/2 ] R ~ (y o ) =1 Φ[z o - (z -/2 / n){1 + [z o2 (n/2)/(n-1)]} 1/2 ]. (A.6) Extending (A.5) and (A.6), the following is a (1 α)100% lower one-sided confidence bound for P(X > x o ): R(y ~ o ) = 1 Φ[z o + (z - / n){1 + [z o2 (n/2)/(n-1)]} 1/2 ]. (A.7) Consider now X distributed as Weibull. For large sample size (n 30), a (1 α)100% two-sided interval estimate for µ x is x ± z α/2 s x / n. (A.8) The (1 α)100% two-sided interval estimate for the proportion of wells with an

14 260 Improving Infill Development Decision Making With Interval Estimation ultimate recovery X that exceeds a selected value x o, i.e., P(X > x o ), is Lower bound: exp[-exp(u ~ )] Upper bound: exp[-exp( ~ u)] (A.9) where u ~ = U + z -/2 [( U 2 (1.1) U(.1913))/n] 1/2 ~ u = U z -/2 [( U 2 (1.1) U(.1913))/n] 1/2. The (1 α)100% lower one-sided confidence bound for P(X > x o ) is exp[ exp(u ~ )] (A.10) where u ~ = U + z α [( U 2 (1.1) U(.1913))/n] 1/2.