Error Analysis in History Matching: The Optimum Level of Parameterization
Authors: Shah Shah , Gavalas G.R. , Seinfeld J.H. 
Published: December 2007

SHAH, SHAH* CALIFORNIA INSTITUTE OF TECHNOLOGY, PASADENA, CALIF. GAVALAS, G.R., CALIFORNIA INSTITUTE OF TECHNOLOGY, PASADENA, CALIF. MEMBER SPE-AIME SEINFELD, J.H., CALIFORNIA INSTITUTE OF TECHNOLOGY, PASADENA, CALIF. MEMBER SPE-AIME

Abstract
The accuracy of the porosity and permeability estimates obtained in reservoir history matching is investigated using covariance analysis. The estimate covariance matrix is obtained for the following cases:estimation of all individual grid properties,parametrization using sensitivity vectors,parametrization by zonation, andBayesian estimation.
The trace of the covariance matrix used as a measure of the over-all accuracy is studied as a /unction of the number of unknown parameters, and a procedure for selecting the parameters, and a procedure for selecting the optimum parametrization is developed. Numerical calculations with a one-dimensional reservoir are used to illustrate the theory.

Introduction
The problem of estimating parameters in mathematical models of petroleum reservoirs (called history matching) is notoriously difficult. Whereas the estimation problem can be posted straight-forwardly, often it is impossible to obtain meaningful parameter estimates. The principal difficulty is that there are usually more unknown parameters than data points and that the data are parameters than data points and that the data are not sensitive enough to changes in the parameters.
Several early and important studies developed the methodology required to treat history matching as a nonlinear regression problem and investigated associated computational problems. More recently, the introduction of methods of optimal control resulted in improved algorithms for automatic history matching. The, actual reliability of the property estimates was approached initially in a property estimates was approached initially in a qualitative fashion and more recently was studied quantitatively. The last two studies treat a small number of unknown parameters from single-well data. In principle, the case of a more detailed reservoir model with data from several wells can be analyzed in the same way, but requires more tedious computations.
A straightforward approach to history matching is to assume that rock properties in each grid block used in the numerical solution of the reservoir simulation equations are unknown parameters. In most practical situations, this approach leads to a large number of unknowns. Evidently, the reservoir model must include only a modest number of unknown parameters for meaningful history matching. parameters for meaningful history matching. Replacing one reservoir model with arbitrarily varying properties with a model in which rock properties are determined by a limited number of properties are determined by a limited number of parameters henceforth will be called parameters henceforth will be called parametrization. The traditional approach to reducing the parametrization. The traditional approach to reducing the number of unknown parameters is zonation of the reservoir, in which properties are assumed the same over regions encompassing several grid blocks. In zonation and other types of parametrization that might be used to reduce the number of unknowns, the error in property estimates has two components. One error is because of the parametrization itself and generally decreases as the number of unknowns increases. The other error is proportional to the measurement error and increases with an increasing number of unknowns. The total error reaches a minimum at some intermediate level of parametrization, which can be regarded as the parametrization, which can be regarded as the optimum level. The problem of selecting an optimum number of parameters has been approached only qualitatively before. One of the main objectives of this study is to develop a practical and quantitative procedure for determining the optimum level of parametrization.
An alternative to parametrization is to use prior geological information in the form of a prior probability density of the reservoir properties probability density of the reservoir properties considered as random variables. A form of Bayesian estimation then can be employed to determine the unknown properties. One question that arises when applying this procedure is the effect of error in the prior statistics employed. In a previous study, prior statistics employed. In a previous study, we studied various practical aspects of Bayesian estimation concerning the reservoir problem and compared the Bayesian method with the zonation method.
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