History Matching by Use of Optimal Theory
Authors: Chavent G. , Dupuy M. , Lemmonier P. 
Published: December 2007

Abstract
The optimal control theory has been applied to the problem of determining permeability distribution by matching the history of pressure in a single-phase field, given flow production data. pressure in a single-phase field, given flow production data. The method consists of minimizing a nonquadratic criteria by The steepest descent method; use of an adjoint equation enables the gradient to be obtained numerically in minimum computing time. The method has been tested on a semirealistic example of a field model included in a 9 x 19 grid, with 10 producing wells, using a 5-year pressure and production history. Since the storage capacities are assumed to be known, the transmissivities have been backed up at each grid block (no zonation being needed) in 20 iterations and 100 seconds of CDC 7600 computer time, giving an over-all pressure fit of less than 1 kg/cm2 (equivalent to 14 psi) with the pressures to be adjusted ranging from 482 to 300 kg/cm2 (equivalent to 6,850 to 4,350). As usual, the fitting procedure was continued, for investigation purposes, well beyond the point where satisfactory results from the engineering standpoint have been obtained, which is here about eight iterations. The stability of the procedure with respect to the choice of initial values has been established by numerical experimentation. Moreover, due to the use of the gradient method, no unrealistic value of transmissivities has been generated at any point of the computation. The method is very flexible and is able to take directly into account other types of boundary conditions in monophasic production situations. Extension of the method is currently being tested on the case of multiphase flow problems.

Introduction
As a corollary to the current progress in numerical simulations and also as a necessity by itself, development of history-matching techniques has been news-worthy and internationally widespread in recent years. A number of significant papers on the subject, both review papers and ones expounding new techniques, have been published in papers and ones expounding new techniques, have been published in the literature of the last decade. In this respect, let us cite Dougherty's review paper that provides valuable guides on the matter. Up to 1972, most of the work done had followed the lines of the perturbation method (according to Dougherty's classification) and had been referred to some arbitrary zonation of the reservoir model transmissivity, or storage coefficient, grid. In such an approach, the solution method pertains to the realm of multiple regression by least squares and the problem is treated as a nonlinear form of classical adjustment by same; alternatively, it may be attacked as a nonlinear programming problem, when a maximum absolute deviation norm is substituted to problem, when a maximum absolute deviation norm is substituted to a squared average deviation norm to control the adjustment of the model. In this group of work, the required sensitivity coefficients are obtained by multiple simulation, varying the parameters one at a time, which clearly enough precludes consideration of somewhat refined zonations; a dozen zones appears to be an accepted limit in this regard. In the course of the iterative procedure implementing the nonlinear least-squares adjustment process, oscilltations and other convergence difficulties have been frequently reported, accountable, to a large extent, by a latent quasisingularity of the solution matrix; description of this appears in Jahns. The final explanation behind this resides presumably in the inadequacy of the zonation to the problem (here again, see Jahns). The work of Jacquard was revolutionary in the domain, inasmuch as it gave access within a reasonable amount of computer time to the full set of sensitivity coefficients respective to each node of the transmissivity, or storage coefficient grid. The possibility of removing the zonation constraints was thus basically opened. It was not used, however, due to adherence to the general least-squares reduction concept. Although substantial progress was achieved in this manner, the stability difficulties progress was achieved in this manner, the stability difficulties were not mastered in every case.
SPEJ
P. 74