Estimation of Mutual Information and Conditional Entropy for Surveillance Optimization
- Duc H. Le (University of Tulsa) | Albert C. Reynolds (University of Tulsa)
- Document ID
- Society of Petroleum Engineers
- SPE Journal
- Publication Date
- August 2014
- Document Type
- Journal Paper
- 648 - 661
- 2014.Society of Petroleum Engineers
- 5.5.8 History Matching, 5.4.1 Waterflooding, 5.1.1 Exploration, Development, Structural Geology
- Uncertainty characterization, Mutual information, Conditional Entropy, Information theory, History matching
- 1 in the last 30 days
- 358 since 2007
- Show more detail
- View rights & permissions
|SPE Member Price:||USD 10.00|
|SPE Non-Member Price:||USD 30.00|
Given a suite of potential surveillance operations, we define surveillance optimization as the problem of choosing the operation that gives the minimum expected value of P90 – P10 (i.e., P90–P10) of a specified reservoir variable J (e.g., cumulative oil production) that will be obtained by conditioning J to the observed data. Two questions can be posed: (1) Which surveillance operation is expected to provide the greatest uncertainty reduction in J? and (2) What is the expected value of the reduction in uncertainty that would be achieved if we were to undertake each surveillance operation to collect the associated data and then history match the data obtained? In this work, we extend and apply a conceptual idea that we recently proposed for surveillance optimization to 2D and 3D water flooding problems. Our method is based on information theory in which the mutual information between J and the random observed data vector Dobs is estimated by use of an ensemble of prior reservoir models. This mutual information reflects the strength of the relationship between J and the potential observed data and provides a qualitative answer to Question 1. Question 2 is answered by calculating the conditional entropy of J to generate an approximation of the expected value of the reduction in (P90–P10) of J. The reliability of our method depends on obtaining a good estimate of the mutual information. We consider several ways to estimate the mutual information and suggest how a good estimate can be chosen. We validate the results of our proposed method with an exhaustive history-matching procedure. The methodology provides an approximate way to decide the data that should be collected to maximize the uncertainty reduction in a specified reservoir variable and to estimate the reduction in uncertainty that could be obtained. We expect this paper will stimulate significant research on the application of information theory and lead to practical methods and workflows for surveillance optimization.
|File Size||1 MB||Number of Pages||14|
Aanonsen, S.I., Nævdal, G., Oliver, D.S. et al. 2009. Review of Ensemble Kalman Filter in Petroleum Engineering. SPE J. 14 (3): 393−412.
Abellan, A. and Noetinger, B. 2010. Optimizing Subsurface Field Data Acquisition Using Information Theory. Mathematical Geosci. 42: 603−630. http://dx.doi.org/10.1007/s11004-010-9285-6.
Barker, J.W., Cuypers, M., and Holden, L. 2001. Quantifying Uncertainty in Production Forecasts: Another Look at the PUNQ-S3 Problem. SPE J. 6 (4): 433−441.
Beirlant, J., Dudewicz, E.J., Györfi, L. et al. 1997. Nonparametric Entropy Estimation: An Overview. International J. of the Mathematical Statistics Sci. 6: 17−39.
Cellucci, C.J., Albano, A.M., and Rapp, P.E. 2005. Statistical Validation of Mutual Information Calculations: Comparison of Alternative Numerical Algorithms. Phys. Rev. E 71: 066,208.
Chapman, D. and Thomas, C.N. 2010. Think Big, Get Small. Hart’s E&P 83 (2).
Coopersmith, E. and Cunningham, P. 2002. A Practical Approach To Evaluating the Value of Information and Real Option Decisions in the Upstream Petroleum Industry. In Proceedings of the SPE Annual Technical Conference and Exhibition, San Antonio, Texas, Richardson, Texas: SPE.
Cover, T.M. and Thomas, J.A. 1991. Elements of Information Theory, New York, New York: Wiley-Interscience.
Darbellay, G. and Vajda, I. 1999. Estimation of the Information by an Adaptive Partitioning of the Observation Space. Information Theory, IEEE Trans. 45 (4): 1315−1321.
Emerick, A.A. and Reynolds, A.C. 2011. Combining Sensitivities and Prior Information for Covariance Localization in the Ensemble Kalman Filter for Petroleum Reservoir Applications. Computational Geosci. 15 (2): 251−269.
Emerick, A. and Reynolds, A. 2012a. History Matching Time-Lapse Seismic Data Using the Ensemble Kalman Filter With Multiple Data Assimilations. Computational Geosci. 16 (3): 639−659.
Emerick, A.A. and Reynolds, A.C. 2012b. Combining the Ensemble Kalman Filter Markov-Chain Monte Carlo for Improved History Matching and Uncertainty Characterization. SPE J. 17 (2): 418−440.
Emerick, A.A. and Reynolds, A.C. 2013a. Ensemble Smoother With Multiple Data Assimilations. Computers & Geosci. 55: 3−15.
Emerick, A.A. and Reynolds, A.C. 2013b. Investigation of the Sampling Performance of Ensemble-Based Methods With a Simple Reservoir Model. Computational Geosci. 17 (2): 325−350.
Evensen, G. 1997. Advanced Data Assimilation for Strongly Nonlinear Dynamics. Monthly Weather Rev. 125 (6): 1342−1354.
Evensen, G. 2003. The Ensemble Kalman Filter: Theoretical Formulation and Practical Implementation. Ocean Dynamics 53: 343−367.
Evensen, G. 2007. Data Assimilation: The Ensemble Kalman Filter, Berlin: Springer.
Floris, F.J.T., Bush, M.D., Cuypers, M. et al. 2001. Methods for Quantifying the Uncertainty of Production Forecasts: A Comparative Study. Petroleum Geosci. 7(SUPP): 87−96.
Gao, G., Zafari, M. and Reynolds, A.C. 2006. Quantifying Uncertainty for the PUNQ-S3 Problem in a Bayesian Setting With RML and EnKF. SPE J. 11 (4): 506−515.
Haskett, W.J. 2003. Optimal Appraisal Well Location Through Efficient Uncertainty Reduction and Value of Information Techniques. In Proceedings of the SPE Annual Technical Conference and Exhibition, Denver, Colorado, Richardson, Texas: SPE.
Kraskov, A., Stogbaurer, H., and Grassberger, P. 2004. Estimating Mutual Information. Physical Rev. E 69 (066138): 1−15.
Le, D.H. and Reynolds, A.C. 2012. Optimal Choice of a Surveillance Operation Using Information Theory. In Proceedings of the 13th European Conference of the Mathematics of Oil Recovery, ECMOR XIII, Biarritz, France.
Oliver, D.S., He, N., and Reynolds, A.C. 1996. Conditioning Permeability Fields to Pressure Data. In Proceedings of the European Conference for the Mathematics of Oil Recovery.
Oliver, D.S., Reynolds, A.C., and Liu, N. 2008. Inverse Theory for Petroleum Reservoir Characterization and History Matching, Cambridge, UK: Cambridge University Press.
Quiroga, R.Q., Kraskov, A., and Grassberger, P. 2005. Reply to “Comment on ‘Performance of Different Synchronization Measures in Real Data: A Case Study on Electroencephalographic Signals.’” Phys. Rev. E 72: 063,902.
Reynolds, A.C., He, N., and Oliver, D.S. 1999. Reducing Uncertainty in Geostatistical Description With Well Testing Pressure Data. In Reservoir Characterization-Recent Advances, ed. R.A. Schatzinger and J.F. Jordan, pp. 149−162, American Association of Petroleum Geologists.
Roulston, M.S. 1999. Estimating the Errors on Measured Entropy and Mutual Information. Physica D: Nonlinear Phenomena 125 (34): 285−294.
Shannon, C.E. 1948. A Mathematical Theory of Communication. Bell System Technical J. 27.
Tavakoli, R. and Reynolds, A.C. 2010. History Matching With Parameterization Based On the SVD of a Dimensionless Sensitivity Matrix. SPE J. 15 (12): 495−508.
Ullo, J. 2008. Computational Challenges in the Search for the Production of Hydrocarbons. Scientific Modeling and Simulation SMNS 15 (1−3): 313−337.