Estimation of parameters in partial differential equations -- with applications to petroleum reservoir description

Citation

Chen, Wen Hsiung (1974) Estimation of parameters in partial differential equations -- with applications to petroleum reservoir description. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/6XF8-F790. https://resolver.caltech.edu/CaltechETD:etd-10252005-143807

Abstract

The determination of parameters in dynamical systems, on the basis of noisy experimental data, is called the parameter estimation problem or inverse problem. In this dissertation, several methods for parameter estimation are derived for systems governed by partial differential equations, so-called distributed parameter systems.

The first class of problems, investigated in Chapter II, is that in which the parameters to be estimated are constants. This class of problems is important for it includes most cases of practical interest. Techniques based on gradient optimization, quasilinearization, and collocation methods are developed. A method of determining confidence intervals for parameter estimates is presented, a method which enables one to design experiments (and measurements) that lead to the best estimates of the parameters. The effectiveness of these methods for estimating constant parameters is illustrated through the estimation of the diffusivity in the heat equation, the estimation of the activation energy for a single reaction from dynamic plug flow reactor data, and the estimation of the permeabilities in a two-region reservoir model. The numerical results also show the advantage of using data taken at optimally chosen measurement locations to estimate the parameters.

Many physical systems contain spatially varying parameters, for example, the permeability distribution in a petroleum reservoir model. In Chapter III, two approaches are presented for the estimation of spatially varying parameters. The first is a method of steepest descent based on consideration of the unknown parameter vector as a control vector. The second is based on treating the parameter as an additional state vector and employing least square filtering. The key feature of the former method is that the parameters are considered as continuous functions of position rather than as constant in a certain number of spatial regions. This technique may offer significant savings in computing time over conventional gradient optimization methods, such as steepest descent and Gauss-Newton in which the parameters are considered as uniform in a certain number of zones. Two examples are presented to illustrate the use of the method and its comparison to other algorithms.

In certain cases, the location of the boundary of a system may not be known, such as the boundary of a petroleum reservoir. In the case of oil reservoirs it is very important to be able to estimate the area and shape (or the location of the boundary) of a reservoir so that the production policies can be optimized. A method based on the variation of a functional defined on a variable region is developed in Chapter IV. The computational applications of this method are illustrated in determining the locations of the boundaries of a one-dimensional and a two-dimensional petroleum reservoir.

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