Comparison of Sensitivity Coefficient Calculation Methods in Automatic History Matching

The efficiency of automatic history matching algorithms depends on two factors: the computation time needed per iteration and the number of iterations needed for convergence. In most history matching algorithms, the most time-consuming aspect is the calculation of the sensitivity coefficients – the derivatives of the reservoir variables (pressure and saturation) with respect to the reservoir properties (permeabilities and porosity). This paper presents an analysis of two methods – the direct and the variational – for calculating sensitivity coefficients, with particular emphasis on the computational requirements of the methods. If the simulator consists of a set of N ordinary differential equations for the grid-block variables (e.g., pressures) and there are M parameters for which the sensitivity coefficients are desired, the ratio of the computational efforts of the direct to the variational method is R = N ( M + 1 ) N ( N + 1 ) + M . Thus, for M<N the direct method is more economical, whereas as M increases, a point is reached at which the variational method is preferred.