A Three - Dimensional Three - Phase Automatic History Matching Model: Reliability Of Parameter Estimates
Authors: Tan T.B. , Kalogerakis N. 
Published: March 2010

Abstract
An important benefit of using a three-dimensional three-phase automatic history matching simulator is that it is possible to analyze the quality of the match obtained. As the simulator is based on the Gauss-Newton method, it is an easy matter to calculate the variance-covariance matrix of the unknown reservoir parameters using least squares theory- Analyzing this information, as well as the eigenvectors of the Gauss-Newton matrix, parameter zones which are highly correlated can easily be identified in the case of over-parameterization.
Through illustrative examples, this paper provides practical details on how 1o quantify the reliability of the estimated reservoir parameters and how to identify the reservoir parameters and hence the corresponding parameter zones that do not have a significant effect on the match of the measured variables in the field. In addition the effect of having fewer data (limited observability) on the identifiability and reliability of the estimated parameter values is illustrated.

Introduction
The models used in reservoir simulation typically involve several thousand grid cells. The estimation of the reservoir properties of each grid cell is the essence of history matching. The more accurately these parameters can be determined, the more confidence one would place on the predicted results. However, in most cases, the information available from the measured data such as seismic and well data, is not sufficient to identify the reservoir properties of each and every grid cell accurately. This is especially true for newly discovered reservoirs. In a history matching exercise, the engineer would like to know how accurately the parameters have been estimated, whether or not a particular parameter has any influence on the history match and to what degree. The traditional procedure of history matching by varying parameters based on engineering judgment may provide a match, but certainly does not provide the engineer with any estimate of the degree of confidence one may place on the reliability of these parameters. Often the engineer is only too pleased to find a set of parameters that results in a minimally acceptable match.
Least squares (LS) estimation is commonly used to identify the objective function to be minimized and is the basis of many parameter estimation algorithms, Drawing from the extensive body of knowledge that has been developed in the fields of statistical inference and multivariate analysis, it is possible to develop relationships for the variances of estimated parameters, establish confidenceintervals and detect highly correlated parameters from the results of a nonlinear least squares fit. One of the benefits of using a simulator with an automatic history matching capability is the further information provided to [he engineer about the quality of the match and the reliability of the estimates of the parameters.
The techniques used in automatic history matching fall into two major categories. The first order optimal control method proposed by Chen et al.(1) and Chauvent et al.(2) involves the solution of a set of adjoined ordinary differential equations together with the ordinary differential equations of the reservoir model.