Estimation of Mutual Information and Conditional Entropy for Surveillance Optimization

Abstract

Summary 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 minus 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 waterflooding 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.

Proxy-Based Work Flow for a Priori Evaluation of Data-Acquisition Programs

“…As is proved in Le et al (2014), the expectation of the prior distribution of J (prior mean) should be the same as the expected value of the means of J over all plausible posterior distributions (the average of posterior means). This relation can be used as a quality-assurance check for the algorithm.…”

“…The larger is the uncertainty range; the further apart are the 90th and 10th percentiles and the larger the U[P(x)]. Although there are other ways to define U[P(x)] in the literature, such as defining U[P(x)] as the Shannon entropy of P(x) (Haber et al 2008;Magnant 2011;Le and Reynolds 2014), the benefit of using Eq. 1 is that the value of U is in the same unit as x so it is intuitive for decision-analysis purposes.…”

“…One can evaluate this metric by performing a history-matching run to establish the posterior distribution of the objective function with the data and compare it with the prior distribution. Some of the few history-matching techniques that are capable of rigorously characterizing the posterior distribution include Markov chain Monte Carlo methods (Metropolis et al 1953;Hastings 1970;Liu and Oliver 2003) and rejection sampling (Caers 2011). These techniques quantify the posterior distribution, and thus the amount of uncertainty reduced, with the observed data from the dataacquisition programs.…”

“…A similar approach also was used by Landa (1997) to quantify the effectiveness of production and seismic data. More recently, Le and Reynolds (2014) introduced the concept of "mutual information (MI)" from information theory to characterize the amount of uncertainty reduction, and derived a formula to approximate the MI on the basis of linear-Gaussian assumption. In their work, the proposed approach was compared with the reference solution from a brute-force method.…”

“…First, as shown in Le and Reynolds (2014), the linear-Gaussian assumption between measurement data and objective functions can have a major impact on the accuracy of the quantification of uncertainty reduction. The method proposed in this work does not make the linear-Gaussian assumption.…”