Hydraulic conductivity estimation in partially saturated soils using the adjoint method

Abstract

An iterative algorithm based on the adjoint method for the estimation of the saturated hydraulic conductivity k in a partially saturated soil Q is proposed. Groundwater flow in Q is assumed to be described by Richards equation. The optimization problem minimizes the L2-error between the pressure head values p(k,x, t) calculated as the solution of a direct problem and the measured values of the pressure head at discrete points inside the domain Q. The exact gradient of the cost functional is obtained by solving an appropriate adjoint problem, which is derived from the equations of the Gaˆteaux derivatives of the pressure head with respect to the parameter k. A finite element procedure is used to obtain approximate solutions of the direct and adjoint problems and the Gaˆteaux derivatives. A discrete form of the expression of the gradient of the cost functional at the continuous level is used inside a nonlinear conjugate gradient iteration to solve the optimization problem. A numerical example showing the implementation of the algorithm to estimate the saturated hydraulic conductivity k(x) during an hypothetical infiltration experiment in a heterogeneous soil is also presented.