Performance Matching With Constraints
Authors: Carter R.D. , Kemp L.F. , Pierce A.C. , Williams D.L. 
Published: December 2007

Abstract
This paper describes two new iterative reservoir performance matching techniques for the performance matching techniques for the single-phase compressible flow case. The influence coefficients employed in these procedures are computed by Jacquard's method - very economical one when the number of reservoir parameters to be estimated exceeds the number of observation locations. A derivation of the method based on the properties of the diffusion equation is presented properties of the diffusion equation is presented Sample results are given for the two new procedures and the magnified diagonal method of Jacquard and Jain. It is concluded that all the procedures discussed in this paper are about equally effective in reducing the difference between computed and observed pressure. However, the new procedures that employ pressure. However, the new procedures that employ linear programming methods can reduce the vector of pressure differences to an acceptably low level while guaranteeing that computed values of the reservoir parameters are within predetermined constraint intervals, whereas the method of Jacquard and Jain, being unconstrained, can lead to unacceptable values of the reservoir parameters. On the other hand, the (magnified diagonal) procedure of Jacquard and Jain requires less computing time per iteration. per iteration

Introduction
Performance matching is the process of varying reservoir characteristics in a reservoir simulator until the performance predicted by the simulator agrees, within some acceptable tolerance, with a set of observed performance data, while at the same time the parameters meet some criterion of reasonableness. This paper presents performance matching techniques that are applicable to single-phase isotropic compressible flow governed by Darcy's law. Thus, the basic equation is(1)
The performance matching problem is to estimate M(x, y) and V(x, y) given a complete description of Q and usually an incomplete and sometimes inaccurate description of p. (Variables are described in the Nomenclature.) Many schemes have been presented for automatically solving this problem. The papers of Jacquard and of Jacquard and Jain stand out because they present an ingenious convolutional method for computing the influence coefficients of a linear system that relates the differences between calculated and observed pressures to changes in reservoir properties. The calculation requires a number of simulations per iteration equal to one plus the number of observation points. The value of plus the number of observation points. The value of their technique may have been overlooked by later authors. The papers by Jacquard and Jain also present a magnified diagonal variant of the classical least-squares procedure for solving an underdetermined, exactly determined, or overdetermined linear system. In the case of underdetermined systems, the magnified diagonal method interpolates between corrections based on steepest descent and corrections based on linearity. The Jacquard and Jain procedure also scales corrections to prevent the calculation of negative values of M and V. Dupuy applied the methods of Refs. 6 and 7 to additional problems. Jahns then presented a method for the single-phase compressible flow problem based on the same perturbation principle as given by Jacquard and perturbation principle as given by Jacquard and Jain, but differing from their method as follows. First, the influence coefficients are calculated by difference quotients based on numerical simulation results. Second, the reservoir description is increased in detail as the calculation progresses. Third, statistical measures of the reliability of the calculated reservoir properties are provided. The direct method of obtaining influence coefficients employed in Jahns' work requires a number of simulations per iteration equal to one plus the number of reservoir parameters to be determined.
SPEJ
P. 187