Block Iterative Strategies for Multiaquifer Flow Models

Abstract

When groundwater flow takes place in aquifer-aquitard systems characterized by a high contrast of hydrogeological parameters and non-linear aquitard behavior, quasi — three-dimensional models of flow must be solved by a fully numerical approach. The numerical implementation with finite difference or finite element methods involves large systems whose solution requires much CPU time and computer storage. A new solution strategy of the resulting algebraic equations is suggested to overcome non-linearity in the aquitards. The global nonlinear system is decoupled into a number of smaller subsystems, which are consistent with the geologic structure of the multiaquifer system and are ideally suited for updating the nonlinear hydrologic parameters in the aquitards. The aquifer and the aquitard equations are solved separately with the modified conjugate gradient (MCG) and the Thomas algorithms, respectively, while the final coupled solution is obtained with an iterative procedure. The procedure, as is naturally suggested by the special block tridiagonal pattern of the coefficient matrix, can be shown to be equivalent to a block Gauss-Seidel strategy and can therefore be generalized into a block SOR (successive over-relaxation) strategy, the blocks corresponding to the aquifer and aquitard equations. The convergence properties of the new SOR scheme are initially analyzed with linear porous systems for which the optimum over-relaxation factor ω _opt can be theoretically computed, and the asymptotic convergence rate is studied in relation to the geometry, the hydrogeological parameters of the multiaquifer system and the size of the numerical model. The results obtained with steady state sample problems show that if flow in the various aquifers is weakly interconnected (i.e. the hydraulic exchange between the aquifer units is quite limited) ω _opt is close to 1, while in strongly coupled systems ω _opt falls into the upper range (1.5 <ω _opt <2) and the SOR asymptotic convergence rate can be as much as one order of magnitude larger than the Gauss-Seidel rate. Subsequently the block iterative method has been extended to non-linear porous media and early results indicate that the relaxation procedure with ω = ω _opt provides a significant acceleration of convergence as well. The linear theory, however, does no more apply and ω _opt must be assessed empirically.