Block iterative strategies for multiaquifer flow models
G. Gambolati, P. Teatini
Dept. Mathematical Methods and Models for Scientific
Applications, University of Padova, Padova, Italy
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 non-linear 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 non-linear
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 ωopt 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)
ωopt is close to 1, while in
strongly coupled systems ωopt
falls into the upper range
(1.5<ωopt<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 = ωopt
provides a significant acceleration of convergence as well.
The linear theory, however, does no more apply and ωopt must
be assessed empirically.
