Identification of parameters in an inhomogeneous aquifer by use of the maximum principle of optimal control and quasi‐linearization

Abstract

Optimal identification of aquifer parameters in a distributed system is formulated as an optimal control problem. The dynamics of the head is governed by a second‐order nonlinear partial differential equation. The numerical example presented considers that the parameters to be identified are functions of the space variable. Observations on head variations are available at several observation wells distributed within the system. Spatial discretization is first used to transform the distributed system to a lumped system. The least squares criterion function is then established. After introducing the Lagrange multipliers, the maximum principle is applied to obtain the set of necessary conditions that is optimal. These conditions are expressed in terms of a set of canonic equations of two‐point boundary value type that is easily solved by the technique of quasi‐linearization. Thus aquifer parameters are directly identified on the basis of observational data taken at observation stations. The maximum principle formulation is inherently more accurate and stable, since it minimizes the least squares error over the whole time and space domains. Computationally, it is extremely efficient. The numerical example presented demonstrates simultaneous identification of 11 parameters defined at discretized points along the space variable. Quadratic convergence is also demonstrated by numerical experimentation.