History Matching by Spline Approximation and Regularization in Single-Phase Areal Reservoirs
- Tai-youn Leo (California Inst. of Tech.) | Costas Kravaria (U. of Michigan) | John H. Seinfeld (California Inst. of Tech.)
- Document ID
- Society of Petroleum Engineers
- SPE Reservoir Engineering
- Publication Date
- September 1986
- Document Type
- Journal Paper
- 521 - 534
- 1986. Society of Petroleum Engineers
- 5.5 Reservoir Simulation, 7.2.2 Risk Management Systems, 5.5.8 History Matching, 1.6.9 Coring, Fishing, 4.3.4 Scale
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Summary. An automatic history-matching algorithm is developed from bicubic spline approximations of permeability and porosity distributions and from the theory of regularization to estimate permeability or porosity in a single-phase, two-dimensional (2D) areal reservoir from well pressure data. The regularization feature of the algorithm, the theoretical details of which are described by Kravaris and Seinfeld, is used to convert the ill-posed history-matching problem into a well-posed problem. The algorithm uses Nazareth's conjugate gradient method as its core minimization method. A number of numerical experiments are carried out to evaluate the performance of the algorithm. Comparisons with conventional (nonregularized) automatic history-matching algorithms indicate the superiority of the new algorithm with respect to the parameter estimates obtained. A quasioptimal regularization parameter is determined without requiring a priori information on the statistical properties of the observations.
The process of estimating unknown properties, such as permeability and porosity, in a mathematical reservoir permeability and porosity, in a mathematical reservoir model to give the best fit to measured well pressure and production data is commonly called history matching. production data is commonly called history matching. Because the properties in an inhomogeneous reservoir vary with location, an infinite number of parameters is required conceptually for a full description of the reservoir. From a computational point of view, a reservoir simulator contains only a finite number of parameters, corresponding to each gridblock or element in the spatial domain. In field-scale simulations, it is not unusual for the reservoir domain to consist of about 10,000 gridblocks, and consequently 20,000 or more parameters may need to be estimated simultaneously. This potential large dimensionality of the unknown parameters distinguishes the reservoir history-matching problem from other parameter-estimation problems in science and engineering. parameter-estimation problems in science and engineering. Moreover, the standard reservoir history-matching problem is mathematically ill-posed, and this ill-posed nature, problem is mathematically ill-posed, and this ill-posed nature, coupled with such a large number of unknown parameters, lies at the root of the difficulties in its solution. parameters, lies at the root of the difficulties in its solution. The ill-posed nature of the history-matching problem is manifested by numerical instabilities in the estimated parameters. Such instabilities are well documented in the parameters. Such instabilities are well documented in the petroleum engineering and hydrology literature. petroleum engineering and hydrology literature. The principal approach that has been used to alleviate ill-conditioning in the parameter estimates is to decrease the number of unknown parameters and, if possible, to utilize any available information to constrain the space of unknown parameters. One commonly used approach for reducing the number of unknown parameters is to divide the reservoir into a relatively small number of zones and to assume uniform properties within each zone. While this approach is effective in reducing the number of unknowns, sufficient a priori information usually is not available to enable specification of the zones on any physical basis. Moreover, although it limits the dimension of physical basis. Moreover, although it limits the dimension of the parameter space, zonation does not alleviate the fundamental ill-posed nature of the problem. An alternative to zonation is to use prior information expressed in the form of an assumed probability distribution for the reservoir properties. If certain a priori knowledge is assumed about the parameter mean values and correlations, the history-matching objective function can be modified to include a term that penalizes the weighted deviations of the parameters from their assumed mean values. A form of Bayesian estimation can then be used to determine the unknown parameters. While it has been shown that better-conditioned estimates may be obtained when a priori statistical information is used, sufficient knowledge of the nature of the unknown parameters generally is not available to specify the parameters needed to carry out a Bayesian estimation.
The critical problems in generating an effective algorithm for history matching are two-fold: (1) the original problem must be defined in a manner that alleviates the ill-posed nature of the problem; and (2) an efficient computational algorithm must be developed for solving the large, constrained, nonlinear minimization problem that results from any history-matching problem.
With respect to the inherent ill-posed nature of the history-matching problem, Kravaris and Seinfeld have shown that the concept of regularization can be applied rigorously to the estimation of spatially varying parameters in partial-differential equations of parabolic type. parameters in partial-differential equations of parabolic type. The regularization idea, first advanced by Tikhonov and Arsenin, has been widely used in the solution of ill-posed integral equations, but had not been developed for the estimation of parameters in partial-differential equations. partial-differential equations. SPERE
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