Accounting for Parameter Uncertainty in Reservoir Uncertainty ...

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This method, referred to as the conditional finite-domain (CFD) approach, accounts for the size of the domain and the local conditioning data. It is a stochastic ...
Natural Resources Research, Vol. 18, No. 1, March 2009 (Ó 2008) DOI: 10.1007/s11053-008-9084-7

Accounting for Parameter Uncertainty in Reservoir Uncertainty Assessment: The Conditional Finite-Domain Approach Olena Babak1,2 and Clayton V. Deutsch1 Received 12 June 2007; accepted 27 October 2008 Published online: 3 December 2008

An important aim of modern geostatistical modeling is to quantify uncertainty in geological systems. Geostatistical modeling requires many input parameters. The input univariate distribution or histogram is perhaps the most important. A new method for assessing uncertainty in the histogram, particularly uncertainty in the mean, is presented. This method, referred to as the conditional finite-domain (CFD) approach, accounts for the size of the domain and the local conditioning data. It is a stochastic approach based on a multivariate Gaussian distribution. The CFD approach is shown to be convergent, design independent, and parameterization invariant. The performance of the CFD approach is illustrated in a case study focusing on the impact of the number of data and the range of correlation on the limiting uncertainty in the parameters. The spatial bootstrap method and CFD approach are compared. As the number of data increases, uncertainty in the sample mean decreases in both the spatial bootstrap and the CFD. Contrary to spatial bootstrap, uncertainty in the sample mean in the CFD approach decreases as the range of correlation increases. This is a direct result of the conditioning data being more correlated to unsampled locations in the finite domain. The sensitivity of the limiting uncertainty relative to the variogram and the variable limits are also discussed. KEY WORDS: Geostatistical simulation, multivariate Gaussian distribution, spatial bootstrap.

There are two common approaches to resource uncertainty: (1) statistical analysis of global statistics such as area, thickness, porosity, and saturation and (2) detailed geostatistical modeling of the spatial distribution of structure, facies, porosity, and saturation. The advantage of geostatistical modeling is that detailed realizations are available that can be validated based on geological expertise and used for local decisions as well as global resource uncertainty calculations (Deutsch 2002). Geostatistical modeling of regionalized variables requires statistical parameters such as the mean, a representative distribution, and variogram model. These parameters must be inferred from the available data and then used for all realizations. Practitioners know that resource uncertainty is understated when uncertainty in the parameters is

INTRODUCTION An important task in reservoir management is the quantification of the in situ volume of hydrocarbon (the resource) and the recoverable volume of hydrocarbon (the reserve). Geostatistical tools are often used for modeling resource uncertainty (Journel and Huijbregts, 1981; Goovaerts, 1997). This uncertainty quantification is valuable as supporting information for many management decisions.

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Centre for Computational Geostatistics, Department of Civil and Environmental Engineering, University of Alberta, 3-133 NREF Building, Edmonton AB T6G 2W2, Canada. 2 To whom correspondence should be addressed; e-mail: [email protected]

7 1520-7439/09/0300-0007/0 Ó 2008 International Association for Mathematical Geology

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8 not considered (Wang and Wall, 2003). Uncertainty is reduced because local fluctuations above and below the average cancel out in simulations with constant parameters and the realizations imply a very small uncertainty. The understatement of uncertainty is especially true for large reservoirs with sparse well control. Accounting for uncertainty in the parameters, especially the mean, is important for a realistic assessment of resource uncertainty. Accounting for parameter uncertainty in reservoir modeling is straightforward in principle. Each geostatistical realization can be created using different target statistical parameters (univariate distribution, mean, variogram). The inclusion of parameter uncertainty leads to a set of realizations that better represent resource uncertainty. Geostatistical simulation techniques are well established; it is determining the uncertainty in the input parameters that has not received much attention. Variations of the bootstrap resampling technique (Efron and Tibshirani, 1986, 1993) have been applied for parameter uncertainty. The conventional bootstrap technique assumes the data are independent. This may be a reasonable assumption when layer-average statistics are considered with widely spaced wells; however, reservoir data are often correlated. Spatial correlation can be introduced in the bootstrap method using geostatistical simulation (Journel and Bitanov, 2004; Feyen and Caers, 2006). The spatial bootstrap technique amounts to unconditional simulation at the data locations. The statistical parameters are calculated from the simulated realizations and used for parameter uncertainty. The spatial bootstrap leads to some questionable results. One major problem is that increased spatial correlation leads to parameter uncertainty that is excessively large; the unconditional realizations are very different. Increased correlation between the data causes more variation from one unconditional realization to the next. The spatial bootstrap does not consider the effect of conditioning data or the finite reservoir domain. The aim of this paper is to propose an alternative approach to parameter uncertainty that takes into consideration these factors. The new technique, referred to as the conditional finite-domain (CFD) approach to parameter uncertainty, allows quantification of parameter uncertainty accounting for the conditioning data and the finite domain under investigation. The technique is based on sequential Gaussian simulation (Journel and Kyriakidis, 2004).

After a short review of the spatial bootstrap procedure, an explanation and description of the method underlying the CFD approach to parameter uncertainty is presented. Two examples are presented to illustrate the method.

REVIEW OF THE SPATIAL BOOTSTRAP METHOD The spatial bootstrap calculates uncertainty in a statistic of interest, accounting for the spatial correlation between the data (Journel and Bitanov, 2004; Feyen and Caers, 2006). The spatial bootstrap procedure can be summarized by the following steps: 1. Assemble the representative distribution of the Z random variable FZ(z), following a known 3D variogram model c(h), which is the variogram of the normal scores of the Z variable. 2. Using c(h), calculate the n 9 n covariance matrix C between the n samples at hand and themselves. Then calculate its Cholesky decomposition: C ¼ L  LT ; where L is a lower triangular n 9 n matrix. 3. Draw n values (representing n sample data locations) from the representative distribution accounting for spatial correlation. That is, set zi ¼ FZ1 ðpi Þ; where pi = G(ui) with u ¼ ðu1 ; . . . ; un ÞT ¼ L  w;G(Æ) is the standard normal cumulative distribution function, and w is a n 9 1 vector of independent standard Gaussian random numbers, i ¼ 1; . . . ; n: 4. Calculate the statistic of interest from the resampled data set, say, the average above a cutoff a, mZs[a . 5. Return to step 3 and repeat many times, e.g., N = 1000. 6. Assemble the distribution of uncertainty in the calculated statistic. The procedure is straightforward and can be applied for any statistic of interest including the mean and the proportion above some critical cutoff.

Accounting for Parameter Uncertainty in Reservoir Uncertainty Assessment Correlation between the n data values leads to greater uncertainty than would be obtained using the traditional bootstrap approach based on the assumption of independence.

CONDITIONAL FINITE-DOMAIN APPROACH TO PARAMETER UNCERTAINTY METHOD Assume that n data observations of the variable of interest come from a multivariate Gaussian distribution. The mean and covariance matrix (defined by a known variogram model) are the only parameters needed to fully define these probability distribution. Note that if the univariate distribution of the data is not standard normal, the normal score transform can be applied to make it standard normal and then the assumption of multivariate Gaussianity can be made if appropriate (Deutsch, 2002). The results can be back transformed to original data units at any time. If we simulate the full grid of node values conditioned to the available data in the area of interest, we can assess uncertainty in our parameters based on subsets of the full grid; in current practice, we only have a subset of the full multivariate distribution. It follows from our assumption that every set of simulated data (that is, data combination) that have the same configuration as the original data can be considered as an observation from the same underlying distribution as the original data. A data combination will have the same configuration as the original data if it contains the same number of data points and these data points are separated from each other by the same distance vectors found in the original data. Due to stationarity, any data combination with the same configuration as the original data will have the same mean and covariance matrix as the original data, thus, it will also be characterized by the same probability distribution in the multivariate Gaussian setting. The assessment of uncertainty in the statistic of interest based on the simulated data with the same configuration as the original data is reasonable and fair. The procedure for choosing the simulated data from N realizations of the study domain for subsequent uncertainty assessment follows. Based on the full grid of simulated values and by using translation with respect to some center of the original data, one can randomly select any desired number of data combinations with the same

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configuration as the original data (K). If the phenomenon is isotropic, then rotations with respect to some center of the original data can also be used to select data combinations (Fig. 1). The same K simulated data combinations can be found for all other simulated realizations. Selected combinations can be used to assess and quantify the uncertainty in the statistic of interest. The NK values are then combined with equal probability to create a distribution of uncertainty in the statistic of interest, e.g., the sample mean. This procedure assumes that the univariate distribution is fixed; the parameter uncertainty is assessed with an assumed known input distribution. It is possible to take the results from the proposed approach and repeat the procedure to create a set of N new realizations with input reference distributions taken as a direct output of previous simulation. The input histogram is not assumed to be fixed at the single reference input distribution increasing the uncertainty of these realizations (second pass and so on). The uncertainty will not increase in an unbounded fashion because of the original conditioning data and the finite domain size. When assessing the uncertainty of order one (the first pass) in the statistic of interest, there is no input reference distribution of the variable of interest. The CFD approach to parameter uncertainty proceeds as follows.

Figure 1. Use of centroid and angle in determining new data combination: Conditioning data (circles) are rotated on an angle a anticlockwise around the center (point O) to obtain a new data combination (squares).

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10 Set order of uncertainty k = 1, and (a) Simulate N possible realizations of the variable of interest using a Gaussian simulation technique (for example, sequential Gaussian simulation (SGS)). (b) Calculate and quantify the uncertainty of order 1 in the statistic of interest. Establish the reference distributions to be used in the subsequent assessment of uncertainty in the statistic of interest. The reference distributions are taken as simulated realizations obtained in (a). (c) Set order of uncertainty k = 2. For k ¼ 2; . . . ; 1; perform the following: (a) Use the reference distributions obtained in step k - 1 to create N realizations of SGS using available conditioning data. (b) Calculate and quantify the uncertainty of order k in the statistic of interest. Establish new reference distributions. The reference distributions are taken as simulated realizations obtained in (a). (c) Set order of uncertainty k = k + 1. The CFD approach for the assessment of uncertainty of order k in any statistic of interest is convergent in the sense that the uncertainty reaches a limit and does not change with increased order k. The CFD approach relies on the simulations, it is a stochastic algorithm. Similar to the Markov Chain Monte Carlo approach, there is a burn-in period, when the uncertainty in the parameter of interest will increase or decrease to the point where the parameter uncertainty starts to stabilize. The stabilization phase corresponds to the fluctuation of the limiting parameter uncertainty around some constant value, which defines the expected limiting parameter uncertainty. It should be noted at this point that this paper does not present any theoretical proof of the convergence of the CFD approach. The statement about convergence is based on the fact that in every single case study considered, convergence of the result for parameter uncertainty was always observed. Moreover, the uncertainty is not increasing in an unbounded fashion because of the original conditioning data and the finite domain size. For CFD approach the minimum and maximum possible values of the variable of interest must be specified. This specification is important when there

are few data. The minimum and maximum could be chosen based on expert judgment or analog reservoir information. For example, the porosity within siliciclastic sediments would almost certainly be between 0.0 and 0.4 (Amyx, Bass, and Whiting, 1960). If there is a lack of information on the lower and upper limits of the variable of interest, then it is proposed that they be determined approximately using the following simple procedure: (1) fit a nonlinear regression model to the empirical cumulative distribution function F(z) of the variable of interest and (2) determine the lower and upper tail values as the intersection of the fitted nonlinear regression curve with F(z) = 0 and F(z) = 1, respectively. It worth noting that nonlinear regression is usually applied to provide a method to best fit a model to the data. In the instance of unknown lower and upper limits of the variable of interest, nonlinear regression is used to model the empirical cumulative distribution function (not the data themselves) with the aim of finding the best estimates of the minimum and maximum possible values of the variable of interest. TESTING WITH SYNTHETIC DATA SET Setup As a first illustration of the CFD approach, consider the GSLIB data set cluster.dat. This data set consists of about 100 observations that are sampled on a random stratified grid and 40 observations that are clustered in high valued areas. The clustered data are removed. The distribution of 100 data is approximately lognormal with a sample mean of 2.31 and a sample standard deviation of 3.21. The normal scores variogram is isotropic spherical with range 10 m and contribution of 0.7,  0;  for h ¼ 0 cðhÞ ¼ h 0:3 þ 0:7Sph 10 ; for h[0 The 2D study area is 50 9 50 m2. The mean is the statistic of interest. The uncertainty in the mean is measured by the standard deviation. In this analysis, the number of data and the variogram range vary. Subsets of size 5, 10, and 20 values are considered. The minimum and maximum data values of 0.16 and 5.05 are used in all cases; these values represent the minimum and maximum data values in each data set, respectively. The subsets are inclusive: a subset of 20 data contains the subset of

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Figure 2. Location maps of the subsamples of size 5, 10, and 20 of the file data set cluster.dat.

10 data and the subset of 10 data contains the subset of 5 data entirely. The data locations and data distributions are shown in Figure 2 for each of the three chosen subsamples. The values of the points in any subset shown in Figure 2 do not closely approximate the lognormal distribution that is observed on the basis of all the data. Four cases are considered for the range of correlation: 5, 10, 20, and 50 m. The amount of local data and the spatial correlation will affect the uncertainty in the sample mean. The uncertainty is the least for the largest data set. The impact of the range of correlation on uncertainty is not very predictable. One might think that the uncertainty should increase with an increase in the range because this is the outcome using the spatial bootstrap approach. Bras and Rodrı´guezIturbe (1985) attribute this behavior to the increased correlation between the data, which causes more variation from one bootstrap realization to the next, a result of reducing the effective number of data with increasing correlation. The increased spatial

correlation in spatial bootstrap leads to greater uncertainty than if the data are more random. Spatial bootstrap does not consider the effect of conditioning data or the finite reservoir domain. With an increase in the range of correlation, the uncertainty should logically decrease because the data values become more correlated to all other data in the area of interest, thus, in estimation and simulation more similar information is used, and the results are more reliable. The data for the CFD approach are simulated on a grid of 100 9 100 locations spaced at 0.5 9 0.5 m2. For each parameter uncertainty assessment step, 100 SGS realizations are used. The combinations for determining uncertainty in the mean are obtained by translation of the original data locations with respect to the randomly selected center o = [ox,oy], where ox,oy  U(0,50). Although the variable is isotropic, no rotations are used to obtain data combinations. All data combinations are constrained to be within the area of 50 9 50 m2. In total, for each

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12 uncertainty calculation, 100 data translations are considered. The limiting uncertainty in the statistic of interest is calculated as the average of the uncertainty values in the stabilization phase. For comparison purposes, the tail options in SGS are chosen first to be the same as the minimum and maximum data values. Then they are chosen to be equal to 0.1 (lower tail) and 10.27 (upper tail); these values are chosen from the same initial data set cluster.dat.

Case Study Results The sensitivity of the limiting uncertainty in the sample mean to the change in the number of data for a range of 10 is shown in Figure 3, which clearly shows the burn-in period and stabilization phase. Results for the sensitivity of the limiting uncertainty in the mean to the change in the number of data for ranges of correlation 5, 20, and 50 demonstrate similar behavior. The effect of the variogram range on the limiting uncertainty in the mean is shown for 5 data in Figure 4. The horizontal lines indicate the limiting uncertainty in Figures 3 and 4 and were obtained as an average of the uncertainty values in the stabilization phase. Note that the indication of the burn-in period and stabilization phase in Figures 3 and 4 is only approximate. In Figure 4, the burn-in period and stabilization phase were indicated to be common for all ranges of correlation for 5 data. The burn-in

period seems to remain invariant to the change in correlation length. The results for the effect of the variogram range on the limiting uncertainty in the mean for 10 data and 20 data appear similar. It is evident from Figure 3 that the limiting uncertainty in the mean decreases with an increase in the number of data. As the range of correlation increases, the limiting uncertainty in the mean decreases; see Figure 4. Looking at Figures 3 and 4, one sees slight stochastic fluctuations in the limiting uncertainty. To decrease the magnitude of these fluctuations, it is necessary to use a larger number of realizations and/or a finer simulation grid. Table 1 summarizes the CFD results obtained for the limiting uncertainty in the mean with respect to the change in the number of data and variogram range. The uncertainty increases for the spatial bootstrap as the range of correlation increases. The uncertainty for the CFD approach decreases as range increases. It can be clearly noted from Table 1 that the uncertainty in the sample mean calculated by the CFD approach is smaller than that predicted by the spatial bootstrap method. Note that for a pure nugget effect variogram (Table 2), the uncertainty predicted by the CFD approach is close to both the uncertainty predicted by spatial bootstrap and the classical (theoretical) result calculated as  ¼ VarðXÞ

^2 r ; n

^2 denotes the estimate of the study domain where r variance. This estimate is taken as the data variance.

Figure 3. Effect of the number of data on the limiting uncertainty in the mean for the range of correlation equal to 10. Crosses, asterisks, and triangles denote the uncertainty in the mean for 5, 10, and 20 data, respectively. Horizontal dotted lines denote the respective limiting uncertainties in the mean. Horizontal dash-dot lines denote approximate separation between burn-in period and stabilization phase.

Figure 4. Sensitivity of the limiting uncertainty in the mean to the change in the range of correlation for 5 data. Circles, dots, squares, and diamonds denote the uncertainty in the mean for the ranges of correlation 5, 10, 20, and 50, respectively. Horizontal dotted lines denote the respective limiting uncertainties in the mean.

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Table 1. Comparison of the Spatial Bootstrap and the CFD Approach to the Limiting Uncertainty in the Mean for 5, 10, and 20 Data and Ranges of Correlation 5, 10, 20, and 50 Uncertainty in the Mean Mean & Std of Data

Method

Range 5

Range 10

Range 20

Range 50

100 data 5 data

2.31 & 3.21 2.48 & 1.67

10 data

2.19 & 1.38

20 data

2.08 & 1.49

– SB CFD SB CFD SB CFD

– 0.62 0.55 0.51 0.39 0.36 0.27

– 0.62 0.52 0.51 0.35 0.39 0.26

– 0.62 0.47 0.55 0.36 0.46 0.25

– 0.78 0.42 0.65 0.34 0.60 0.21

Table 2. Comparison of the Spatial Bootstrap, CFD Approach to the Limiting Uncertainty in the Mean for 5, 10, and 20 Data and Pure Nugget Variogram Uncertainty in the Mean Method

5 data

10 data

20 data

SB CFD Classical

0.62 0.57 0.75

0.49 0.39 0.44

0.36 0.29 0.33

different support of the grid blocks, no volumevariance adjustment was performed; the spacing of the grid nodes was simply changed. Table 3 summarizes the effect of the tails (upper and lower possible values on the variable of interest) on the limiting uncertainty in the mean for the subsets of size 5, 10, and 20 data considered in this case study. The tail values influence the limiting uncertainty. This is especially true when the number of data or range of correlation is small; the effect of the choice of tail values starts to diminish when the number of data and range of correlation increases. Therefore, caution is required when selecting the upper and lower tail values to be used in SGS for assessment of the limiting uncertainty using CFD. In most practical cases, the physical limits of the variable will be known; therefore, this sensitivity is mitigated.

REAL DATA EXAMPLE

Figure 5. Influence of the parameterization for 5 data and range of 10. Black triangles, red crosses, and blue asterisks denote the uncertainty in the mean obtained using simulation on the scale 0.25 9 0.25 blocks, on the scale 0.5 9 0.5 blocks, and on the scale 1 9 1 blocks, respectively.

Figure 5 illustrates that, with the change of grid node resolution, the limiting uncertainty predicted by the CFD approach remains virtually unchanged, that is, the difference between results on the scale 0.25 9 0.25 blocks, on the scale 0.5 9 0.5 blocks, and on the scale 1 9 1 blocks is less than 1.2%. When calculating the limiting uncertainty based on

The following example is based on Chu, Xu, and Journel (1995). A 10,500 9 10,500 ft reservoir layer is considered, with 62 wells in the study area (Fig. 6). The variable of interest is porosity (averaged over the main reservoir layer of interest). The representative (declustered) histogram of porosity is shown in Figure 7. The two horizontal variograms of porosity are shown in Figure 8. Note that Figures 6–8 are shown mainly to familiarize the reader with the chosen porosity data set and to show the anisotropy. The main interest of this study is assessing uncertainty in the mean and variogram of the porosity variable. The CFD approach to parameter uncertainty is aimed at the univariate distribution (summarized by the mean), but the variation of the variogram in the stabilization phase is also of interest since the true variogram model of the porosity is also unknown.

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Table 3. Influence of the Tails on the Limiting Uncertainty in the Mean for 5, 10, and 20 Data and Ranges of Correlation 5, 10, 20, and 50 Uncertainty in the Mean Mean & Std of Data

Method

Range 5

Range 10

Range 20

Range 50

100 data 5 data

2.31 & 3.21 2.48 & 1.67

10 data

2.19 & 1.38

20 data

2.08 & 1.49

– SB CFD CFD with tails SB CFD CFD with tails SB CFD CFD with tails

– 0.62 0.55 1.18 0.51 0.39 0.68 0.36 0.27 0.33

– 0.62 0.52 0.98 0.51 0.35 0.46 0.39 0.26 0.30

– 0.62 0.47 0.75 0.55 0.36 0.41 0.46 0.25 0.28

– 0.78 0.42 0.56 0.65 0.34 0.36 0.60 0.21 0.25

Figure 7. Representative (declustered) histograms of porosity. Figure 6. Location map of 62 average porosity data of the data set AmocoData2-D.dat.

The data in the CFD approach are simulated on the grid of 105 9 105 blocks at a spacing of 100 9 100 ft. For each step of parameter uncertainty assessment, 100 SGS realizations are used. The combinations for determination of the subsequent reference distributions are obtained by translation with respect to a randomly selected center o = [ox,oy], where ox,oy  U(0,10500). Due to the strong anisotropy (Fig. 8) only translations and no rotations are used. The data combinations are constrained to be within the area of 10,500 9 10,500 ft. In total, for each uncertainty calculation, 100 successful data translations are considered. The lower and upper tail options in the SGS are set to 4 and 11.4, respectively (Fig. 7). To account for clustering of the data, the declustering weights are used in the calculation of the mean for each of the randomly chosen data combinations. The limiting uncertainty

Figure 8. Two horizontal variograms of porosity. The variogram in North–South direction (direction of major continuity) is shown in light color and the variogram in East–West (direction of minor continuity) is shown in dark color.

in the porosity mean is then calculated as the average of the uncertainty values in the stabilization phase.

Accounting for Parameter Uncertainty in Reservoir Uncertainty Assessment The uncertainty in the mean of porosity as a function of the uncertainty order is shown in Figure 9. Note from Figure 9 that, in the stabilization phase, the mean of the uncertainty in the mean distribution approaches the declustered mean of the data. The uncertainty in the mean increases with the increase in the order of uncertainty during the burnin period. In the stabilization phase, the uncertainty in the mean fluctuates around 0.28; this value corresponds to the limiting uncertainty in the mean (Fig. 9). While results of the SB and CFD for the mean of the uncertainty distribution are virtually the

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same (Fig. 10), the standard deviations of the uncertainty distribution, as measures of uncertainty, are very different. The limiting uncertainty predicted by CFD approach is much smaller than the one predicted by the spatial bootstrap due to the fact that the 62 data show significant spatial correlation with each other. Variograms calculated on the reference distributions are different from the input variogram (Fig. 11). This uncertainty could be considered in subsequent geostatistical modeling. Reservoir models that account for uncertainty in

Figure 9. (a) Mean of the uncertainty in the mean of porosity distribution as a function of the uncertainty order and (b) Uncertainty in the mean of porosity as a function of the uncertainty order.

Figure 10. Limiting uncertainty in the mean obtained based on: (a) spatial bootstrap approach with 10,000 realizations and (b) conditional finite-domain approach.

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Figure 11. Uncertainty in the variogram in the direction of (a) minor and (b) major continuity when calculating the limiting uncertainty in the mean for the Amoco data. Dark lines denote the variogram model used in sequential Gaussian simulation for assessment of the limiting uncertainty; light lines denote the variogram reproduction obtained in simulation.

the variogram along with the uncertainty in mean would result in a more realistic resource uncertainty assessment.

CONCLUSIONS A new method for assessing uncertainty in reservoir statistics of interest is proposed. The CFD approach to parameter uncertainty directly accounts for the conditioning data and size of the domain. This method is the first known approach that allows direct incorporation of these two important factors. The CFD approach is a stochastic approach based on a multivariate Gaussian model to determine the limiting (expected) uncertainty in the statistic of interest. The proposed method is convergent, design independent, and invariant under parameterization. The effect of varying the number of data and the range of correlation on the limiting uncertainty in the sample mean was investigated using CFD approach based on a synthetic data set. Uncertainty decreases quickly as the number of data increases. The uncertainty in the statistic of interest also decreases as the range of correlation increases. This is due to the fact that the conditioning data are more correlated with each other and more correlated to the locations being simulated. Note that the spatial bootstrap approach predicts that the uncertainty would increase as the range of correlation increases.

Variogram uncertainty can be investigated from the limiting uncertainty realizations. This uncertainty could be transferred into subsequent modeling by applying SGS each time not only with a different reference distribution, but also with a different input variogram corresponding to that reference distribution. The impact of the lower and upper tail values on the limiting uncertainty value obtained using the CFD approach was investigated. It was shown that the limiting uncertainty is sensitive to the lower and upper tail values. Often, the lower and upper tail values based on the variable of interest are known. If this is not the case, a method for calculating the lower and upper possible values of the variable of interest was proposed. Future work should consider different statistics of interest, the impact of the parameter uncertainty on the uncertainty in reserves, more detailed investigation of variogram uncertainty, and detailed 3D analysis.

ACKNOWLEDGMENTS This research was partially supported by Alberta Ingenuity Foundation, the Natural Sciences and Engineering Research Council of Canada, the University of Alberta, and industry sponsors of the Centre for Computational Geostatistics.

Accounting for Parameter Uncertainty in Reservoir Uncertainty Assessment REFERENCES Amyx, J. W., Bass, D. M., and Whiting, R. L., 1960, Petroleum reservoir engineering: McGraw-Hill, New York. Bras, R. L., and Rodrı´guez-Iturbe, I., 1985, Random functions and hydrology: Addison-Wesley Publishing Co., Reading, MA, 557 p. Chu, J., Xu, W., and Journel, A. G., 1995, 3-D implementation of geostatistical analyses—the Amoco case study, in Yarus, J. M., and Chambers, R. L., eds., Stochastic modeling and geostatistics: principles, methods, and case studies. AAPG computer applications in geology: 3AAPG, Tulsa, OK, p. 201–216. Deutsch, C. V., 2002, Geostatistical reservoir modeling: Oxford University Press, New York, 376 p. Efron, B., and Tibshirani, R. J., 1986, Bootstrap methods for standard errors, confidence intervals and other measures of statistical accuracy: Stat. Sci., v. 1, p. 54–77.

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Efron, B., and Tibshirani, R. J., 1993, An introduction to the bootstrap: Chapman and Hall, New York, 436 p. Feyen, L., and Caers, J., 2006, Quantifying geological uncertainty for flow and transport modeling in multi-modal heterogeneous formations: Adv. Water Resour., v. 29, no. 6, p. 912–929. Goovaerts, P., 1997, Geostatistics for natural resources evaluation: Oxford University Press, New York, 483 p. Journel, A. G., and Bitanov, A., 2004, Uncertainty in N/G ratio in early reservoir development: J. Petrol. Sci. Eng., v. 44, no. 1– 2, p. 115–130. Journel, A. G., and Huijbregts, Ch. J., 1981, Mining geostatistics: Academic Press, London, 600 p. Journel, A. G., and Kyriakidis, P. C., 2004, Evaluation of mineral reserves: a simulation approach: Oxford University Press, New York, 216 p. Wang, F. J., and Wall, M. A., 2003, Incorporating parameter uncertainty into prediction intervals for spatial data modeled via a parametric variogram: J. Agric. Biol. Environ. Stat., v. 8, no. 3, p. 296–309.