Application of iterated Laplace transformation to tracer transients in heterogeneous porous media

Abstract

The Laplace transformation technique has been widely applied to modeling of tracer transport in oil and geothermal reservoirs, and in groundwater aquifers. However, mathematical models of many flow and transport problems could only be obtained as Laplace space solutions, and hence, their computations had to involve a numerical inversion technique. In this work, we employ the iterated Laplace transformation technique to develop novel closed form solutions to the tracer transport models in heterogeneous media. Two types of configurations have been considered: tracer transport in single fracture located in low-permeability matrix and tracer transport in a double porosity medium consisting of flowing and dead-end pore systems. In addition, both linear and radial flow geometries have been considered for both configurations. Applications of iterated Laplace transform technique to these four types of models are presented as fundamental examples and their numerical results were used as benchmarking for the numerical inversion results from Stehfest and Dubner and Abate algorithms. As the technique is quite versatile, we expect that the method should gain widespread acceptance to develop solutions to a wide range of problems in flow and transport in porous media and improve the application of nonlinear regression technique to these solutions. This work has achieved four important objectives: first, two novel Laplace transform relations that are useful in tracer studies are presented. Second, the present work serves to verify/invalidate the results of numerical inversion algorithms. In addition, it provides better insight into tracer transport mechanisms. Finally, it serves as a powerful tool of design and interpretation of tracer tests. All four objectives are illustrated in this work.