Parameter Identification in an Inhomogeneous Medium With the Finite-Element Method
Authors: Yoon Y.S. , Yeh W.W-G. 
Published: December 2007

Abstract
This paper studies the development of a systematic procedure for solving the problem of parameter identification. The parameter function to parameter identification. The parameter function to be identified is the permeability embedded in a nonlinear, partial differential equation of parabolic type. Finite elements are used to represent the unknown function parametrically in terms of nodal values over a suitable discretization of a flow region. These nodal values are then determined by a least-squares fitting between the observed and calculated flow potentials, subject to linear inequality constraints imposed upon the parameters to be identified. To handle such constraints, Rosen's gradient projection technique is combined with the Gauss-Newton method. Numerical procedures are presented for the solution of the procedures are presented for the solution of the partial differential equation using the Galerkin partial differential equation using the Galerkin method and a predictor-corrector approach. Simultaneous identification of 25 parameters embedded in a two-dimensional, nonlinear, diffusion-type equation is demonstrated by two examples.

Introduction
Mathematical models have been used as powerful tools in analyzing groundwater or oil reservoir systems. The application of such models to real field problems requires the determination of parameters embedded in partial differential parameters embedded in partial differential equations. These parameters are not simply measurable from die physical point of view.
Recently, considerable attention has been directed toward developing analytical procedures for solving the parameter identification problem. Proposed methods may be classified into two Proposed methods may be classified into two groups: the direct approach and the indirect approach. The direct approach treats the parameters as dependent variables in the form of a formal boundary-value problem. If the flow potentials are known over the entire flow region, the original governing equation becomes a linear, first-order, partial differential equation of the hyperbolic type partial differential equation of the hyperbolic type in terms of the unknown permeability function. With the aid of boundary conditions and flow data, a unique solution of the equation can be obtained. Nelson and Nelson and McCollum proposed the energy dissipation method for solving the unknown parameters. Frind and Pinder used a Galerkin parameters. Frind and Pinder used a Galerkin method for steady flow only. Another direct-method approach that specially takes account of unsteady flow problems is the minimization of the residual error involved in the differential equation. When the error functional is defined as the sum or maximum norm, the minimization problem can be solved by the linear programming technique. The time component in a linear differential equation may be also eliminated by an integration with respect to time. This was studied by Nutbrown. When the flow data and necessary boundary conditions are lacking, the inverse problem yields an infinite number of possible solutions. The inverse problem, even in the case where its solution is guaranteed to be unique, is not generally well posed. In order to obtain a meaningful solution to the inverse problem, it is necessary to impose certain prior problem, it is necessary to impose certain prior restrictions on the admissible function space of parameters. The methods that have been proposed parameters. The methods that have been proposed in the literature include the use of a flatness criterion by Emsellem and de Marsilly and a multiple objective decision process by Neuman. For the direct approach a difficulty arises in the data requirement. In field practice, observation wells are sparsely distributed in an arbitrary fashion rather than regularly distributed; and only a limited number of wells are available. An attempt to obtain space and/or time derivative approximations of flow potentials from the insufficient data by finite-difference or finite-element methods would be a multisource of errors in the results of parameter identification. Sagar et al. suggested the method of spline interpolation to fit the discrete data of observations; but the method still requires a sufficient number of the observation points to adequately approximate the whole flow potential surface in the region under consideration.
The indirect approach is an optimization procedure in which the algorithm starts from a set of initial estimates of the parameters and improves it in an iterative manner until the model response is sufficiently close to that of the field observation.
SPEJ
P. 217