Quantification of Prediction Uncertainty: What Does It Mean and How Does It Evolve with History Matching

Abstract

The history matching problem in reservoir simulation generally does not have a unique solution, and a probabilistic framework - often based on Monte Carlo simulation - is usually employed. This probabilistic data integration process involves the construction of an ensemble of realizations of the Reservoir Characterization Model (RCM), whereby each realization is consistent with (1) the statistical description (e.g., permeability correlation structure) of the RCM, (2) static and dynamic measurements (e.g., permeability, pressure, saturation, production rates), and (3) the governing equations and associated constitutive relations.

Even under the strong assumption that the statistical properties of the RCM are known with a high degree of certainty, we will never know the one unique ‘realization’ that represents the real system, which includes the RCM and the flow performance predictions. Nevertheless, one can quantify the relationship between the Unconditional (U) and Conditional (C) uncertainty spaces - unconditional and conditional refer to pre- and post-history-matching. Here, we discuss the relationship between U and C in the context of Kriging-based inversion employed with the Statistical Moment Equations (SMEs) for flow in heterogeneous reservoirs. Since SME-based Kriging and the Ensemble Kalman Filter (EnKF) method are identical with respect to model updating (e.g., conditioning the permeability and the porosity fields), the analysis presented here is applicable to both approaches.

The inversion process uses measurements to modify (update) U into C. The modification consists of (1) reduction in the size of the uncertainty space and (2) translation, or shift, of the centroid of the distribution. The ratio of the determinants of two covariance matrices is used to compare the relative change in size between two probability distributions. The distance measure is based on the separation between the locations of the distribution centers normalized by their covariance matrices, and it is used as a measure of confidence in U. We prove that the size of C will not increase compared with that of U, and we show that the degree of translation depends on whether the unconditional uncertainty space (U) is a correct representation of the true space. If the assumptions are correct, C will be bounded by U. If the assumptions on U are incorrect, the inversion process will yield a C that is not fully confined within U. For such cases, the computed C (i.e., conditional uncertainty space) should not be accepted, and the input statistical model should be reassessed.

We use a simple example with two saturation measurements. The Kriging-based inversion method is then applied in a sequential manner, i.e., one saturation measurement is applied at a time. The result demonstrates that the sequential updating scheme ends up honoring both measurements. This behavior is consistent with the fact that the conditional space (C) is guaranteed to be bounded when the unconditional space (U) is correct.

In another example, two pressure measurements are used for conditioning. The two measurements are obtained from two locations that are close to each other. One measurement is high; the other is significantly lower. When the two measurements are taken separately, each measurement is considered to be plausible/realizable for the assumed U. However, when the two measurements are considered together, the distance measure indicates that the probability of occurrence is quite low; thus casting doubt on the correctness of either the measurements or the U.

The relative changes in size and distance between distributions provide a self-consistent framework to define the uncertainty space and quantify the relationship between distributions pre and post data integration.