On the Optimal Identification of Parameters in a Parabolic System
Authors: Yeh William W-G. 
Published: December 2007

This paper was prepared for the 48th Annual Fall Meeting of the Society of Petroleum Engineers of AIME, to be held in Las Vegas, Nev., Sept. 30-Oct. 3, 1973. Permission to copy is restricted to an abstract of not more than 300 words. Illustrations may not be copied. The abstract should contain conspicuous acknowledgement of where and by whom the paper is presented. Publication elsewhere after publication in the JOURNAL OF PETROLEUM TECHNOLOGY or the SOCIETY OF PETROLEUM ENGINEERS JOURNAL is usually granted upon request to the Editor of the appropriate journal provided agreement to give proper credit is made.
Discussion of this paper is invited. Three copies of any discussion should be sent to the Society of Petroleum Engineers office. Such discussion may be presented at the above meeting and, with the paper, may be considered for publication in one of the two SPE magazines.

Abstract
This paper presents an analysis of the problem of parameter identification in unsteady problem of parameter identification in unsteady fluid flow through a porous medium. The response of the system is governed by a second order partial differential equation of parabolic type. Identification is an inverse process whereby the parameters embedded in a differential equation is parameters embedded in a differential equation is determined from observations of systems input and output along with appropriate initial and boundary conditions. These parameters are usually not physically measurable. In general, the governing physically measurable. In general, the governing equation is non-linear with no closed-form solution.
In this paper, this inverse problem is solve by quasilinearization, maximum principle, gradient method, influence coefficient method and linear programming. A comparison is made between these programming. A comparison is made between these methods. The problem of convergence and stability is discussed and demonstrated by numerical experimentation.

Introduction
The literature dealing with parameter identification in fluid flow through a porous medium in a distributed system has been sparse. Yeh and Tauxe and Marino and Yeh showed the application of the technique of quasilinearization allied with finite difference scheme in aquifer parameter identification under unsteady state parameter identification under unsteady state conditions. However, Seinfeld described a gradient method, Becker and Yeh developed the influence coefficient method, Yeh and Becker demonstrated the use of linear programming and Lin and Yeh presented the maximum principle in the identification presented the maximum principle in the identification of parameters in partial differential equations of different types.
The problem under consideration is that of an unsteady flow in an unconfined aquifer and stream interaction system. Figure 1 shows the flow configurations. The governing equation can be expressed as:
(1)
subject to the following initial and boundary conditions:
h = h (x),
h = h (t), (2)o
h—— = 0,x
where
h = head in the aquiferx = space variablet = time variable D = diffusivity of the aquifer