The Application of Sensitivity Analysis to Reservoir Simulation

Abstract

American Institute of Mining, Metallurgical, and Petroleum Engineers, Inc. Abstract The attainment of a pressure history match between field data and the response from a compute model has long been a tedious task. This task has generally been performed by trial and error. The procedure for history matching described in this paper is a semi-automatic technique based on certain useful concepts of sensitivity analysis from formal optimization theory. The procedure is based on a determination of the sensitivity of an objective function to the value of reservoir parameters used in the simulator. Two objective functions were used: the square of the difference between the calculated and measured pressures, and the time integral of this function. The sensitivity determination employs a truncated Taylor's series expansion of the objective function about a point specified by the current estimate of the reservoir parameters. The partial derivative terms of parameters. The partial derivative terms of the expansion are calculated from the finite difference analog of the partial differential equation used in the simulator, differentiated with respect to the current estimate of the reservoir parameters. Implementation of this method requires an initial running of the reservoir simulator, subsequent automatic modification of the permeability map through sensitivity analysis of the permeability map through sensitivity analysis of the objective function, with iteration until a predetermined error in pressure match results. predetermined error in pressure match results. The technique is currently implemented in a semi-automatic form for convenience in continued development. However, the technique is designed to be fully automatic. The technique has been applied to the twenty-seven year history match of the Cotton Valley Gray Sand Reservoir. Pressure data on the seven wells in this field had originally been history matched using manual trial and error procedures. Data were limited in that only procedures. Data were limited in that only twenty-eight measured field pressures were available for matching purposes over the twenty-seven year history. This study produced a significant improvement in the history match. The magnitude of the objective function was reduced by a factor of approximately fifteen from that obtained in the original history match. The computational efficiency of the technique is good in that the sensitivity analysis software package required only 1.5 times as much execution time as the single-phase, gas-flow simulator.