Large time asymptotics in contaminant transport in porous media

In this paper we derive large time solutions of the partial differential equations modelling contaminant transport in porous media for initial data with bounded support. While the main emphasis is on two space dimensions, for the sake of completeness we give a brief summary of the corresponding results for one space dimension. The philosophy behind the paper is to compare the results of a formal asymptotic analysis of the governing equations as $t \to \infty $ with numerical solutions of the complete initial value problem. The analytic results are obtained using the method of dominant balance which identifies the dominant terms in the model equations determining the behaviour of the solution in the large time limit. These are found in terms of time scaled space similarity variables and the procedure results in a reduction of the number of independent variables in the original partial differential equation. This generates what we call a reduced equation, the solution of which depends crucially on the value of a parameter appearing in the problem. In some cases the reduced equation can be solved explicitly, while others have a particularly intractable structure which inhibits any analytic or numerical progress. However, we can extract a number of global and local properties of the solution, which enables us to form a reasonably complete picture of what the profiles look like. Extensive comparison with numerical solution of the original initial value problem provides convincing confirmation of our analytic solutions. In the final section of the paper, by way of motivation for the work, we give some results concerning the temporal behaviour of certain moments of the two-dimensional profiles commonly used to compute physical parameter characteristics for contaminant transport in porous media.