History Matching in Two-Phase Petroleum Reservoirs

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Abstract
An automatic history-matching algorithm based on an optimal control approach has been formulated for joint estimation of spatially varying permeability and porosity and coefficients of relative permeability functions in two-phase reservoirs. The algorithm uses pressure and production rate data simultaneously. The performance of the algorithm for the waterflooding of one- and two-dimensional hypothetical reservoirs is examined, and properties associated with the parameter estimation problem are discussed. Introduction There has been considerable interest in the development of automatic history-matching algorithms. Most of the published work to date on automatic history matching has been devoted to single-phase reservoirs in which the unknown parameters to be estimated are often the reservoir porosity (or storage) and absolute permeability (or transmissibility). In the single-phase problem, the objective function usually consists of the deviations between the predicted and measured reservoir pressures at the wells. Parameter estimation, or history matching, in multiphase reservoirs is fundamentally more difficult than in single-phase reservoirs. The multiphase equations are nonlinear, and in addition to the porosity and absolute permeability, the relative permeabilities of each phase may be unknown and subject to estimation. Measurements of the relative rates of flow of oil, water, and gas at the wells also may be available for the objective function. The aspect of the reservoir history-matching problem that distinguishes it from other parameter estimation problems in science and engineering is the large dimensionality of both the system state and the unknown parameters. As a result of this large dimensionality, computational efficiency becomes a prime consideration in the implementation of an automatic history-matching method. In all parameter estimation methods, a trade-off exists between the amount of computation performed per iteration and the speed of convergence of the method. An important saving in computing time was realized in single-phase automatic history matching through the introduction of optimal control theory as a method for calculating the gradient of the objective function with respect to the unknown parameters. This technique currently is limited to first-order gradient methods. First-order gradient methods generally converge more slowly than those of higher order. Nevertheless, the amount of computation required per iteration is significantly less than that required for higher-order optimization methods; thus, first-order methods are attractive for automatic history matching. The optimal control algorithm for automatic history matching has been shown to produce excellent results when applied to field problems. Therefore, the first approach to the development of a general automatic history-matching algorithm for multiphase reservoirs would seem to proceed through the development of an optimal control approach for calculating the gradient of the objective function with respect to the parameters for use in a first-order method. SPEJ P. 521^

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