Application of a moving grid method to a class of 1D brine transport problems in porous media

SUMMARY The background of this paper is the study of transport of pollutants by groundwater flow when released from a repository in a rock salt formation. Flow in regions surrounding such formations may be strongly influenced by variations in salt concentrations, a factor requiring special attention in the development of realistic mathematical models for predicting transport of pollutants. Indispensable for this development are advanced numerical methods. The aim of this paper is to illustrate the application of such a method to a class of non-linear brine transport problems in one space dimension. Our method is based on the method of lines for solving time-dependent partial differential equations. The method is of the finite difference type, implicit and thus applicable to wide classes of (one-space-dimensional) partial differential equation systems. The main feature of the method, however, is that it can automatically move the spatial grid for evolving time and thus is able to refine the grid in regions with large, special transitions. The grid refinement has proven to be a very valuable facility in the numerical modelling of brine transport problems involving low and high salt concentrations. From the user’s point of view an additional advantage of the moving grid method is that it can be implemented in advanced, user-oriented method-of-lines software packages based on implicit stiff ODE solvers. In the brine transport application discussed here we have used the package SPRINT.

[1]  J. M. Sanz-Serna,et al.  Stability and convergence at the PDE/stiff ODE interface , 1989 .

[2]  J. Verwer,et al.  Stability of Runge-Kutta Methods for Stiff Nonlinear Differential Equations , 1984 .

[3]  J. Verwer,et al.  A numerical study of three moving-grid methods for one-dimensional partial differential equations which are based on the method of lines , 1990 .

[4]  E. Dorfi,et al.  Simple adaptive grids for 1-d initial value problems , 1987 .

[5]  S. Majid Hassanizadeh,et al.  On the modeling of brine transport in porous media , 1988 .

[6]  Linda R. Petzold,et al.  Numerical solution of initial-value problems in differential-algebraic equations , 1996, Classics in applied mathematics.

[7]  A moving-grid interface for systems of one-dimensional time-dependent partial differential equations , 1989 .

[8]  S. Hassanizadeh,et al.  Experimental study of coupled flow and mass transport: a model validation exercise. , 1990 .

[9]  Joke Blom,et al.  A moving grid method for one-dimensional PDEs based on the method of lines , 1988 .

[10]  Joke Blom,et al.  On the use of the arclength and curvature monitor in a moving-grid method which is based on the method of lines , 1989 .

[11]  Martin Berzins,et al.  A Method for the Spatial Discretization of Parabolic Equations in One Space Variable , 1990, SIAM J. Sci. Comput..

[12]  Martin Berzins,et al.  Developing software for time-dependent problems using the method of lines and differential-algebraic integrators , 1989 .