On the identification of a spatially varying coefficient appearing in a parabolic partial differential equation

The unknown is the position-dependent storage coefficient s[.] which appears in a linear, partial differential equation of parabolic type with respect to hydraulic potential p[..,.] and multiplies the time derivative of p[.,.]. The spatial derivatives of p.,.] appear in the form div ( a grad p ), where the position-dependent conductivity a[.] is known. The identification of s[.] from knowledge of the measured hydraulic potential z[.,.] relies on the minimisation of the output error functional. The minimisation algorithm is derived from the variational formulation of the inverse problem [J. L. Lions, 1968; G. Chavent, 1971]. A discrete gradient algorithm and the corresponding computer code are described. A stopping ctiterion based on data noise ``energy'' is defined. Numerical and graphical results obtained from (computationally) exact and noisy data are presented. The methods and results apply to the modeling of aquifers and of petroleum reservoirs.