New Considerations on Analytical Solutions Employed in Tracer Flow Modeling

A methodology commonly used to obtain analytical and semi-analytical solutions to describe spike and finite-step tracer injection tests is discussed. In these cases, solutions to the diffusion–convection equation are derived from the solution of a different problem, namely the continuous injection of a tracer. Within this procedure, spike injection results from the time derivative of this solution, and finite-step injection from the superposition of two solutions shifted in time. In this paper we show that although this methodology is mathematically correct, attention should be paid to the properties of the solutions. Their boundary conditions may not represent physically acceptable situations, since these conditions are inherited from a different problem. The application of the methodology to a simple one-dimensional case of a tracer pulse diffusing in a homogeneous, semi-infinite reservoir shows serious problems regarding boundary conditions and mass conservation. These problems has not probably been found before since tracer breakthrough curves are not very sensitive to them. However, the problems clearly show up when the tracer distribution in space is analyzed. We conclude that the traditional methodology should not be employed. Equations should be solved imposing the specific boundary and initial conditions corresponding to the original system under consideration.