2000 115 libya variograms

Assessment of Local and Global Uncertainty with UncertainVariogram Model: Application to Paleocene Clastics in Libya

Bora Oz ([email protected]) and Clayton V. Deutsch ([email protected])Department of Civil & Environmental Engineering, University of Alberta

Abstract

Heterogeneity characterization and uncertainty analysis are important steps in petroleumengineering. Multiple realizations provide a measure of uncertainity in rock and uid prop-erties. Fluid ow simulation yeild a range of possible reservoir performance predictions.

This paper addresses local and global uncertainty in presence of uncertainty in the vari-ogram itself. An example, with thickness data from 39 wells located in Libya, is presented.Uncertainity in the variogram contributes signicantly to the total thickness uncertainty.

KEY WORDS: Simulation, reservoir characterization, bootstrap, variogram

Introduction

Reservoir heterogeneity has an important aect on uid recovery and displacement pro-cesses. This has led to the use of detailed reservoir models as input to uid ow simulators.Unfortunately, petrophysical and ow properties are only available at few well locations.This lack of detailed knowledge of the reservoir properties leads to uncertainty.

Stochastic modeling techniques are used to introduce heterogeneities in reservoir mod-eling [2]. Multiple equally likely to be drawn gridded images (or realizations) of reservoirattributes are generated on the basis of geological and petrophyscial data. Simulation doesnot aim to produce a best or average image, but to produce a range of images consistentwith the data [6].

The goal of stochastic modeling is to convert uncertainities in the geological descriptionto uncertainty in production performance. Stochastic modeling is useful for heterogeneousreservoirs with widely spaced wells. If the data are sparse, the general geological knowledgeis dominant and uncertainty is large. If more data are available, the data become moredominant and uncertainty decreases [4].

Correct heterogeneity characterization directly aects the reservoir management de-cisions. During the initial evaluation of an oil reservoir, decisions are mainly based onestimates of IOIP (initial-oil-in-place) with associated uncertainty. By generating severalrealizations, via stochastic modeling, it is possible to identify the uncertainty in IOIP. Insome case studies, additional evaluations are performed in order to determine the inuenceof additional wells [4]. A good example for the prediction of IOIP can be found in Linjordetet.al [5]. They studied heterogeneity modeling and uncertainty quantication of the IOIPfor Gulfaks Brent formation. It is important to note that one of the factors aecting the

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calculation of IOIP is the correct estimation of uncertainty in the thickness data (or payzone thickness).

In this study, we model uncertainty due to widely spaced data together with uncertaintyin variogram model. Local and global uncertainty is quantied in presence of uncertaintyin the base-case variogram; then, uncertainty is quantied in presence of low-side and high-side variograms. The example data are thickness measurements from 39 wells from a eldin Libya [3].

Methodology

Obtain the average or expected normal score variogram (base case variogram). For each lag distance there are data pairs that contribute to the calculation of thecorresponding variogram value. Using the pair values and the bootstrap techniqueone can get the distribution of uncertainty in the variogram for each lag distanceaccordingly.

The bootstrap is a statistical re-sampling technique that allows uncertainty in the datato be assessed from the data themselves. The important thing in bootstrap procedureis that sample values are treated as if they were the population and many newbootstrapped samples are resampled with replacement. It assumes that data arerepresentative and all independent. It produces model of uncertainty for simplestatistics and it is good for quick assestment of uncertainty. The assumptions behindthe bootstrap technique are often acceptable when few data are available. Admittedly,it is extreme to assume independence when inferring the variogram; however, wenote that the pairs themselves are scattered over a large area. Of course, the valuesconstituting a pair may show correlation.

Calculate the standard deviation of each uncertainty distribution achieved in theprevious step for each lag distance accordingly.

Establish upper and lower limit variograms by adding and subtracting those standarddeviations from the average variogram value corresponding to the each lag distance.These upper and lower values correspond approximately to specic quantiles of thevariogram uncertainty.

Fit a theoretical model to these variograms (i.e. upper, lower and average). Use each variogram model to generate many equiprobable realizations serving to char-acterize the uncertainty. The number of realizations should be enough to capture thetotal uncertainty (generally at least 50 realizations).

The combined set of realizations may be used to quantify local or global uncertainty.

Application to Thickness of the Paleocene Clastic in Libya

The study area is a sandstone formation in Libya. There are 39 thickness data available inthe 30 km by 30 km study area. The location map of the eld is given in Figure 1. The

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thickness values are lower in the central and north parts of the eld. There are, however,less data in the Northern part of the eld.

The histogram of the data is presented in Figure 2. No weights are used in this histogram.Most of the data fall into the range of 1700-3500 feet and the mean value is 2489.9 feet.

In Figure 3, the declustered mean versus cell size was plotted. This is the typical resultwhen low values are over sampled, (i.e. clustered in low valued areas). A Cell-size of 10 kmwas chosen for declustering (where the declustered mean becomes stable). These weightswill be used in Sequential Gaussian Simulation.

A new histogram with the declustering weights is presented in Figure 4; the frequencyof high values is increased due to the declustering weights.

Uncertainty in the Experimental Variogram

The variogram model itself carries uncertainty; therefore, for each lag distance, upper andlower variogram values will be used as one approach to characterize this uncertainty. Theupper, base and lower experimental variograms can be obtained by performing bootstrapon all variogram pairs for each lag distance.

The procedure, that how we got the upper and lower case variograms for thickness datain this study, was briey given below.

A modied GAMV program from GSLIB [1] is used to calculate the variogram valuesfor each lag distance. All pair values that go into each lag calculation are stored.

The bootstrap technique permits determination of the distribution of uncertainty inthe variogram value for each particular lag distance. The distributions, obtained afterbootstrap for each lag distance, are given in Figure 5.

By adding and subtracting one standard deviation, which are obtained from the boot-strap distributions for each lag distance (see Figure 5), from the average variogramvalues, upper and lower limits of the experimental variogram can be obtained. Forexample, for lag distance 1.883 km, average variogram value is 0.179, the upper limitis 0.179+0.135 = 0.314 and the lower limit is 0.179 0.135 = 0.044. In this example,the value of 0.135 is the standard deviation obtained for lag distance 1.883 km frombootstrap(see Figure 5). The resultant variograms (i.e. upper and lower cases) alongwith the base case average variogram are given in Figure 6.

The low values taken all together is not exactly a low variogram; it is quite unlikely;however, it does serve as an extreme bound for uncertainty quantication. The samecould be said for the upper.

A variogram model is then tted to the upper, lower and base case variograms. Thesetted models are shown in Figure 7.

Cross Validation and Simulation by SGSIM

To compare these three experimental variograms, cross validation was applied. Althoughthese three variogram models are all possible, it is clear from Figure 8 that the correlation

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coecient for the lower case limiting variogram is higher than the other two. We mayconclude that the real semivariogram values might be nearer to the lower case variogrammodel.

Two realizations, obtained by using the program SGSIM from GSLIB [1], for each caseare given in Figure 9. the heterogeneity changes consistently with the variogram model.

Local and Global Uncertainty Analysis

Two points were chosen to perform local uncertainty analysis; Point-1, with coordinates ofx=21 km, y=12 km, was chosen where there are more data and Point-2, with coordinatesof x=13 km, y=27 km, was chosen from north part of the study area, where there are lessdata.

50 realizations with the base variogram model leads to local distributions of thickness.Simulation with the upper and lower case variogramsleads to a total of 150 realizations,see Figure 10. It is clear that uncertainty for the thickness at Point-1 is smaller thanthe uncertainty for the thickness at Point-2. The range of uncertainty can be identiedby checking the standard deviation (or variance) of the distibution. Uncertainty increasessignicantly when we take into account upper and lower case variograms.

The square region shown in Figure 1 was divided into dierent volumes. First, 1 km by1 km block averaging (100 blocks) was considered, see the top of Figure 11. Then, applyingthe same procedure for dierent volumes, upscaling was performed by 5 km versus 5 kmand 10 km versus 10 km blocks. The resultant distributions for these two volumes are alsopresented in Figure 11. It is clear that as the averaging volume increases, heterogeneity andthe variance decrease, however, for all cases the mean does not change.

For the global uncertainty analysis, logically similar study was performed as in case oflocal uncertainty. Uncertainty for the whole eld was studied. Again using the data fromprevious 150 realizations, 1km by 1km, 5km by 5km and 10km by 10km block averagingswere applied and upscaled. The results are presented in Figure 12. Again decrease invariance is clear as the averaging volume increases.

Discussion and Conclusions

It is important to characterize the uncertainty in the thickness distribution of a reser-voir for reserve estimation and decision making.

A sensitivity analysis on the experimental variogram has been performed by deter-mining upper and lower limiting variograms. We assume that these lower and uppervariograms are the limiting variograms.

If there are few data available, it is reasonable to consider uncertainty in the experi-mental variogram. The average variogram may not inadequate to capture the fulluncertainty in the variable.

Uncertainty (i.e., the variance of the distribution) increased approximately 65 percentfor Point-1 and increased 40 percent for Point-2, when we consider uncertainty in

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the variogram. This shows the necessity of considering the uncertainty of variogrammodel itself.

Global uncertainty by using dierent averaging volumes are obtained and presented inFigure 12. These distributions can be used to estimate IOIP and be useful in makingdecisions on future development of the reservoir.

References

[1] C. V. Deutsch and A. G. Journel. GSLIB: Geostatistical Software Library and UsersGuide. Oxford University Press, New York, 1992.

[2] H. H. Haldorsen and E. Damsleth. Stochastic modeling. Journal of Petroleum Technol-ogy, pages 404412, April 1990.

[3] Michael Edward Hohn. Geostatistics and Petroleum Geology, Second Edition. WestVirginia Geological Survey.

[4] Holden Lars, Henning Omre, and Hakon Tjelmelond. Integrated reservoir description.SPE European Petroleum Computer Conference, pages 15-23, Stavanger, Norway, 25-27May 1992. Society of Petroleum Engineers. SPE paper # 24261.

[5] A. Linjordet, P. E. Nielsen, and E. Siring. Heterogeneities modeling and uncertaintyquantication of the gullfaks brent formation in-place hydrocarbon volumes. SPE/NPF3-D Reservoir Modeling Conference, pages 167-173, Stavanger, Norway, 16-17 April1996. Society of Petroleum Engineers. SPE paper # 35497.

[6] J. F. McCarthy. Reservoir characterication: Ecient random-walk methods for up-scaling and image selection. SPE Asia Pacic Oil and Gas Conference and Exhibi-tion, pages 159-171, Singapore, 8-10 February 1993. Society of Petroleum Engineers.SPE paper # 25334.

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Figure 1: Location map of the thickness of Paleocene clastics. thickness, feet.

Freq

uenc

y

Thickness, ft

1400. 2400. 3400. 4400..000

.040

.080

.120Number of Data 39

mean 2489.9std. dev. 586.6

coef. of var .24maximum 3941.0

upper quartile 2859.7median 2436.0

lower quartile 2116.0minimum 1439.0

Figure 2: Histogram of the thickness of Paleocene clastic.

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Figure 3: Declustered Mean versus Cell Size, km

Freq

uenc

y

Thickness, ft

1400. 2400. 3400. 4400..000

.040

.080

.120

Number of Data 39mean 2669.0

std. dev. 661.0coef. of var .25

maximum 3941.0upper quartile 3078.3

median 2572.3lower quartile 2157.6

minimum 1439.0weights used

Figure 4: Histogram of the thickness(weights used).

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Freq

uenc

y

Avg

.00 .40 .80 1.20.000

.050

.100

.150

.200

.250

Bootstrap Dist. for lag=1.883Number of Data 1000

mean .1799std. dev. .1349

coef. of var .7499maximum .6590

upper quartile .3340median .1790

lower quartile .0240minimum .0100

Freq

uenc

y

Avg

.00 .40 .80 1.20.000

.040

.080

.120Bootstrap Dist. for lag=4.709

Number of Data 1000mean .6585

std. dev. .1013coef. of var .1538

maximum 1.0920upper quartile .7225

median .6520lower quartile .5865

minimum .3780

Freq

uenc

y

Avg

.00 .50 1.00 1.50 2.00.000

.040

.080

.120




upper quartile 1.0065median .9180


Freq

uenc

y

Avg

.00 1.00 2.00 3.00.000

.100

.200

.300






Freq

uenc

y

Avg

.00 1.00 2.00 3.00.000

.050

.100

.150

.200

.250






Figure 5: Bootstrap results for the variogram value for the 5 lags used in variogram calculation.

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Figure 6: Uncertainty in Experimental Variogram: Upper Case Variogram, Base Case Variogram(average), and Lower Case Variogram

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Figure 7: Variogram modeling results: upper, base and lower case variograms.

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True

Estimate

Cross Validation-upper case variogram

0. 1000. 2000. 3000. 4000.0.

1000.

2000.

3000.

4000.Number of data 39Number plotted 39

X Variable: mean 2454.1std. dev. 281.7

Y Variable: mean 2489.9std. dev. 586.6

correlation .48rank correlation .51

True

Estimate

Cross Validation-base case variogram

0. 1000. 2000. 3000. 4000.0.

1000.

2000.

3000.





True

Estimate

Cross Validation-lower case variogram

0. 1000. 2000. 3000. 4000.0.

1000.

2000.

3000.





Figure 8: Cross validation results: upper,base and lower variograms.

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Simulation Result-upper case

East,km

Nor

th, k

m

0.0 30.00.0

30.0

1250

1500

1750

2000

2250

2500

2750

3000

3250

3500

3750

4000

Simulation Result-upper case

East, km

Nor

th, k

m

0.0 30.00.0

30.0

1250

1500

1750

2000

2250

2500

2750

3000

3250

3500

3750

4000

Simulation Result-base case

East, km

Nor

th, k

m

0.0 30.00.0

30.0

1250

1500

1750

2000

2250

2500

2750

3000

3250

3500

3750

4000

Simulation Result-base case

East, km

Nor

th, k

m

0.0 30.00.0

30.0

1250

1500

1750

2000

2250

2500

2750

3000

3250

3500

3750

4000

Simulation Result-lower case

East, km

Nor

th, k

m

0.0 30.00.0

30.0

1250

1500

1750

2000

2250

2500

2750

3000

3250

3500

3750

4000

Simulation Result-lower case

East, km

Nor

th, k

m

0.0 30.00.0

30.0

1250

1500

1750

2000

2250

2500

2750

3000

3250

3500

3750

4000

Figure 9: Realizations by SGSIM: upper, base and lower case variograms.

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Freq

uenc

y

Thickness,ft

1250. 1750. 2250. 2750. 3250..000

.100

.200

.300

Local Uncertainity Analysis (Point#1)Number of Data 50

mean 2617.0std. dev. 187.1




Freq

uenc

y

Thickness, ft

1250. 1750. 2250. 2750. 3250..000

.040

.080

.120

.160

Local Uncertainity Analysis (Point#2)Number of Data 50

mean 2037.2std. dev. 374.1




Freq

uenc

y

Thickness, ft

1250. 1750. 2250. 2750. 3250..000

.050

.100

.150

.200

Local Uncert.Anal.all cases(point#1)Number of Data 150

mean 2698.5std. dev. 265.3




Freq

uenc

y

Thickness, ft

1250. 1750. 2250. 2750. 3250..000

.040

.080

.120

.160Local Uncert.Anal.all cases(point#2)





minimum 1262.0

Figure 10: Uncertainty distributions for Point-1 and Point-2: top gures: base case variogram;bottom gures: values are obtained by combining upper, base and lower case variograms.

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Freq

uenc

y

thickness ,ft

1200. 2200. 3200. 4200..000

.040

.080

.120

.160

Uncertainity in Thickness (1kmx1km)Number of Data 15000

mean 2073.2std. dev. 419.6




Freq

uenc

y

thickness, ft

1200. 2200. 3200. 4200..000

.040

.080

.120

.160Uncertainity in thickness- 5kmx5km





minimum 1396.3

Freq

uenc

y

thickness, ft

1200. 2200. 3200. 4200..000

.100

.200

.300

Uncertainity in thickness- 10kmx10kmNumber of Data 150

mean 2073.2std. dev. 139.0




Figure 11: Uncertainty analysis for the region shown in Figure 1, for dierent volumes averaging.

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Freq

uenc

y

thickness, ft

1200. 2200. 3200. 4200. 5200..000

.040

.080

.120

Uncertainity in Thickness (1kmx1km)Number of Data 135000

mean 2516.2std. dev. 637.8




Freq

uenc

y

thickness, ft

1200. 2200. 3200. 4200. 5200..000

.020

.040

.060

.080

.100


mean 2516.3std. dev. 542.8




Freq

uenc

y

thickness, ft

1200. 2200. 3200. 4200. 5200..000

.040

.080

.120


mean 2516.3std. dev. 392.2




Figure 12: Uncertainty Analysis for the whole eld, for dierent volume averaging volumes.

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