An eigenvalue numerical technique for solving unsteady linear groundwater models continuously in time

Abstract

A procedure for solving the differential groundwater flow equation is presented herein. Using a finite difference or finite element discretization scheme, a set of simultaneous linear equations is obtained. The eigenvalues and eigenvectors of a matrix, which is a function of the coefficients of the set, are the key to the solution. A vector L is obtained straightforwardly by combining the eigenvector matrix A, the eigenvalue vector α, the pumping vector P, and the initial head vector H. Vector L, which depends on time, can be expressed simply and explicitly as a function of the eigenvalues. Piezometric heads can be obtained by combining A and L. L is the only vector that needs to be computed as P changes with time. In this way, influence functions of a piezometric head, flow velocity and flow depletion of a stream connected with the aquifer under a unit stress, can be obtained explicitly and continuously in time. The method can be applied to confined as well as to leaky aquifers and to one‐, two‐, or three‐ dimensional linear models. Its main advantage lies in the fact that it is unnecessary to repeatedly solve a matrix for every time increment. The method is particularly useful for groundwater management problems in which a large number of alternatives have to be evaluated.