History Matching Three-Dimensional Models Using Optical Control Theory
Authors: Wasserman M.L. , Emanuel A.S. 
Published: March 2010

Abstract
History matching three-dimensional reservoir simulator models by manually changing parameters is a difficult job. In this paper, the authors extend their previously published(Soc. Pet. Eng., Jour., Vol. 14, No.6, p. 593, and Vol. 15, No.4, p. 347) automatic history matching method to handle three-dimensional models. Special attention is given to matching historical pressures which are not single cell pressures, but instead are either wellbore pressuresor volumetric averages of a vertical column of cells. Using these measurements, estimates of vertical conductance and well productivity indices can now be made in addition to horizontal(x and y direction) conductances and cell storage values. A by-product of the matching procedure is on estimate of "flow split" – how much of a well's total flow goes to each faller. Case studies are shown to illustrate productivity index estimation using ellbore pressures and to illustrate three-dimensional conductance and storage estimation using volumetric average-cell column pressures.

Introduction
An important phase of any simulation of reservoir behaviour is history matching. Once the most uncertainreservoir parameters have been estimated by matching calculated and observed performance, the model can be used for predictions.
For many years, researchers have sought means for automating the history matching process. The need for automation became more urgent as models grew in complexity, making matching, by manual adjustment of parameters, difficult, if not impossible. Early approaches(1) involved dividing the reservoir into zones of "uniform" properties and applying some optimization technique to estimate the zonal parameters. The number of zones that could be used was limited, because of computer time limitations. Thus, one had to arbitrarily zonate the reservoir and in the process possibility introduce serious modeling error. Recently, two papers have appeared(2, 3) which represent significant advances in automatic history matching, In these papers, the authors used optimal-control theory to derive the gradient of the measure (a sum of squares) of the deviation between calculated and observed pressures. This gradient is a vector of firstrtial derivatives, of the deviation measure, with respect to the uncertain parameters. Once known, the gradient can be used in a simple optimization scheme (steepest descent) to give improved parameter values which decrease the deviation measure. The authors of this paper subsequently showed (4) how the optimal control theory-matching procedure can be coupled with a two-dimensional multiphase reservoir simulator and illustrated the effectiveness of this procedure on several field examples.
This paper further extends the work in Reference 4. Specifically, we focus on matching three-dimensional multiphase reservoir simulator models. To achieve our goal, we develop methods of matching historical pressures which may be either flowing well-bore values or volumetric average pressures of a vertical column of cells. In addition, we modify our algorithm to treat well "flow-splitting", because we generally know only a well's total flow rate and must somehow determine he amount of fluid going to each layer.
Two illustrations are included. The first is an actual history match of the Kaybob Beaverhill Lake A Pool.