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
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- 327 since 2007
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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.
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