Abstract
The efficiency of automatic history matchingalgorithms depends on two factors: the computationtime needed per iteration and the number of iterations needed for convergence. In most historymatching algorithms, the most time-consumingaspect is the calculation of the sensitivitycoefficientsthe derivatives of the reservoir variables(pressure and saturation) with respect to the reservoirproperties (permeabilities and porosity). This paper presents an analysis of two methodsthe direct andthe variationalfor calculating sensitivitycoefficients, with particular emphasis on thecomputational 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 forwhich the sensitivity coefficients are desired, the ratioof the computational efforts of the direct to thevariational method is
N(M + 1)R = .N(N + 1) + M
Thus, for M less than N the direct method is moreeconomical, whereas as M increases, a point isreached at which the variational method is preferred.
Introduction
There has been considerable interest in thedevelopment of automatic history matching algorithms.Although automatic history matching can offer significant advantages over trial-and-errorapproaches, its adoption has been somewhatlower than might have been anticipated when thefirst significant papers on the subject appeared. Oneobvious reason for the persistence of thetrial-and-error approach is that it does not requireadditional code development beyond that already involvedin the basic simulator, whereas automatic routinesrequire the appendixing of an iterative optimization routine to the basic simulator. Nevertheless, theinvestment of additional time in code developmentfor the history matching algorithm may be returned many fold during the actual history matchingexercise. In spite of the inherent advantages ofautomatic history matching, however, the automatic adjustment of the number of reservoir parameterstypically unknown even in a moderately sizedsimulation can require excessive amounts ofcomputation time. Therefore, it is of utmost importancethat an automatic history matching algorithm be asefficient as possible. Setting aside for the moment the issue of code complexity, the efficiency of analgorithm depends on two factors, the computationtime needed per iteration and the number ofiterations needed for convergence (whereconvergence is usually defined in terms of reaching acertain level of incremental change in either theparameters themselves or the objective function). Formost iterative optimization methods, the speed ofconvergence increases with the complexity of thealgorithm.
SPEJ
P. 551^
