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1 Prediction of Oil Production With Confidence Intervals* James Glimm 1,2, Shuling Hou 3, Yoon-ha Lee 1, David H. Sharp 3, Kenny Ye 1 1. SUNY at Stony Brook 2. Brookhaven National Laboratory 3. Los Alamos National Laboratory *Supported in part by the Department of Energy and National Science Foundation
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2 Application of Prediction Theory Reservoir Development Choices, for example Sizing of Production Equipment Location of Infill Drilling During Early Development Stages Risk high Payoff high
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3 Basic Idea: I History match with probability of error Start with geostatistical probability model for permeability, etc. Observe production rates, etc.
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4 Basic Idea: II Multiple simulations from ensemble (Re)Assign probabilities based on data, degree of mismatch of simulation to history
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5 Basic Idea: III Redefine probabilities and ensemble to be consistent with: (a) data (b) probable errors in simulation and data
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6 Basic Idea: IV New ensemble of geologies = Posterior Prediction = sample from posterior Confidence intervals come from - posterior probabilities - errors in forward simulation
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7 New Idea: Arrival Time Error Models Formulate solution error model in terms of arrival times, rather than solution values errors are equi-distributed relative to solution gradients, ie relative to changes in solution values:
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8 Arrival Time Model is Simple, Robust Regard as the new independent variable and t as the new dependent variable. Thus and the error is equidistributed in units ofThis makes the error robust. We compare arrival time and solution based error models
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9 Illustration: Posterior reduces choices and uncertainty
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10 New Result: Predict outcomes and risk Risk is predicted quantitatively Risk prediction is based on - formal probabilities of errors in data and simulation - methods for simulation error analysis - Rapid simulation (upscale) allowing exploration of many scenarios
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11 Problem Formulation Simulation study: Line drive, 2D reservoir Random permeability field log normal, random correlation length
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12 Simple Reservoir Description in unit square constant
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13 Ensemble 100 random permeability fields for each correlation length lnK gaussian, correlation length
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14 Upscaling Solution from fine grid 100 x 100 grid Solution by upscaling 20 x 20, 10 x 10, 5 x 5 Upscaled grids
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15 Upscaling by Wallstrom, Hou, Christie, Durlofsky, Sharp 1. Computational Geoscience 3:69-87 (1999) 2. SPE 51939 3. Transport in Porous Media (submitted) Upscaling References
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16 Examples of Upscaled, Exact Oil Cut Curves Scale-up: Black (fine grid) Red (20x20) Blue (10x10) Green (5x5)
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17 Select one geology as exact. Observe production for Assign revised probabilities to all 500 geologies in ensemble based on: (a) coarse grid upscaled solutions (b) probabilities for coarse grid errors. Compared to data (from “exact” geology) Design of Study
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18 Bayes Theorem Permeability = geology Observation = past oil cut posterior; prior
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19 Errors and Discrepancies usually but implies geology FineCoarse
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20 Example Fig. 1 Typical errors (lower, solid curves) and discrepancies (upper, dashed curves), plotted vs. PVI. The two families of curves are clearly distinguishable.
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21 Mean error Sample covariance Precision Matrix Gaussian error model: has covariance C, mean
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22 In Bayes Theorem, assume is exact. Then, is an error, probability
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23 For arrival time error models, the formulation is identical, except that the independent variables s and t now denote the solution values, and not the time values, while the error e(s) denotes an error in the time of arrival of the solution value s.
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24 Model Reduction: Limited data on solution errors Don’t over fit data Replace by finite matrix
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25 Prediction based on (a)Geostatistics only, no history match (prior). Average over full ensemble (b)History match with upscaled solutions (posterior). Bayesian weighted average over ensemble. (c)Window: select all fine grid solutions “close” to exact over past history. Average over restricted ensemble. Three Prediction Methods
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26 Window prediction is best, but not practical -uses fine grid solutions for complete ensemble -tests for inherent uncertainty Prior prediction is worst - makes no use of production data. Comparing Prediction Methods
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27 Prediction error reduction, as per cent of prior prediction choose present time to be oil cut of 0.6 Error Reduction
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28 Window based error reduction:50% (fine grid:100 x 100) Upscaled error reduction: 5 x 523% 10 x 1032% 20 x 2036% Error Reduction
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29 Confidence Intervals 5% - 95% interval in future oil production Excludes extreme high-low values with 5% probability of occurrence Expressed as a per cent of predicted production
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30 s 0 = oil cut at present time. t 0 = present time. Compute 5%--95% confidence intervals for future oil production, based on posterior and forward prediction using upscaled simulation. Result is a random variable. We express confidence intervals as a percent of predicted production, and take mean of this statistic. Confidence Intervals
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31 Confidence intervals in percent for three values of present oil cut s 0 and three levels of scaleup with fine grid values included. s 0 100x10020x20 10x10 5x5 0.8[-13,22][-21,36][-24,35][-27,34] 0.6[-14,20][-18,20][-22,22][-29,25] 0.4[-14,17][-18,18][-24,21][-33,23] Confidence Intervals
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32 Arrival Time Error Analysis Error Model defined by 5 solution values: s = 1- (Breakthrough), 0.8, 0.6, 0.4, 0.2. Covariance is a 5 x 5 matrix, diagonally dominant, and neglecting diagonal terms, thus has 5 degrees of freedom. Thus it is simple. Covariance is basically independent of the geology correlation length. Thus it is robust.
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33 Covariance Matrix: 10x10 Scaleup
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34 Correlation matrix =
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35 Histogram of off diagonal elements of the correlation matrices, 5x5, 10x10, 20x20 scaleup
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36 Diagonal covariance matrix elements, three levels of scaleup, averaged over all correlation lengths
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37 Error covariance for arrival time error model is proportional to the degree of scale up
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38 Diagonal covariance matix elements for 10x10 scaleup, showing general lack of dependence on correlation length (except for s = 0.2 entry)
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39 Covariance matrix diagonal entries for arrival time error model are independent of correlation length, except for final (s = 0.2) entry.
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40 Confidence intervals for arrival time error model (%)
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41 Arrival time error model vs. solution value model: confidence intervals (%) for s = 0.6 and 10x10 scaleup
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42 Summary and Conclusions New method to assess risk in prediction of future oil production New methods to assess errors in simulations as probabilities New upscaling allows consideration of ensemble of geology scenarios Bayesian framework provides formal probabilities for risk and uncertainty
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43 References J. Glimm, S. Hou, H. Kim, D. H. Sharp, “A Probability Model for Errors in the Numerical Solutions of a Partial Differential Equation”. Computational Fluid Dynamics Journal, Vol. 9, 485-493 (2001). J. Glimm, S. Hou, Y. Lee, D. H. Sharp, “Prediction of Oil Production with Confidence Intervals”, SPE reprint SPE66350 (2001). J. Glimm, S. Hou, H. Kim, D. H. Sharp, K. Ye, W. Zhu, “Risk Management for Petroleum Reservoir Production”, J. Comp. Geosciences, to appear. J. Glimm, Y. Lee, K. Ye, “A Simple Model for Scale Up Error” Cont. Math. 2002 (to appear).
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