A Technique For Assessing And Improving The Quality Of Reservoir Parameter Estimates Used In Numerical Simulators

Abstract

This paper was prepared for the 48th Annual Fall Meeting of the Society of Petroleum Engineers of AIME, to be held in Las Vegas, Nev., Sept. 30-Oct. 3, 1973. Permission to copy is restricted to an abstract of not more than 300 words. Illustrations may not be copied. The abstract should contain conspicuous acknowledgement of where and by whom the paper is presented. Publication elsewhere after publication in the JOURNAL OF PETROLEUM TECHNOLOGY or the SOCIETY OF PETROLEUM ENGINEERS JOURNAL is usually granted upon request to the Editor of the appropriate journal provided agreement to give proper credit is made. Discussion of this paper is invited. Three copies of any discussion should be sent to the Society of Petroleum Engineers office. Such discussion may be presented at the above meeting and, with the paper, may be considered for publication in one of the two SPE magazines. Abstract The sophistication of the current applications of numerical reservoir simulators places great emphasis on the quality of the parameters used therein. The accuracy of parameters, which are often determined by history matching procedures, significantly affects the extent of the applications of a simulator. This paper applies certain useful concepts of sensitivity analysis from formal optimization theory to the assessment of the quality of the estimates of reservoir parameters used in reservoir simulators. The assessment is accomplished by determining the sensitivity of an objective function this case the time integral of the squared error in simulator performance) to the value of reservoir parameters performance) to the value of reservoir parameters used in the simulator. The determination employs a truncated Taylor series expansion of the objective function about a point specified by the current estimate of the reservoir parameters. The partial derivative terms of the expansion are calculated from the finite difference approximation to the differential equations used in the simulator, differentiated with respect to the current estimate of the reservoir parameters. The calculation of the terms in the expansion leads to a complete accounting of all the sources of error in the simulator's performance provided that the true reservoir performance provided that the true reservoir parameters are known. In practice, where the true parameters are known. In practice, where the true values of parameters are not known, the technique can be used in two ways:To provide a complete accounting of the effect of small parameter perturbation.To generate improved estimates of the true parameter values. The method is applied to two reservoir simulators:a radial, slightly compressible single-phase model, anda dimensional slightly compressible, single-phase model. In each case, the model is used to obtain a better estimate of reservoir parameters and a spatial distribution of the sources of the error in performance prediction. performance prediction.