Concentration and temperature transients in heterogeneous porous media: Part I: Linear transport

The transport of solute/heat in porous media is modeled by the convection–dispersion equation. In the study of such transport processes, while the resident concentration has always been used to develop the governing differential equations, the flux concentration has been the most commonly measured quantity in experiments. Hence, earlier studies have emphasized that care must be exercised in the quantitative interpretation of experiments to avoid inconsistent use of solutions with the actual conditions of experiments. They have further demonstrated that performing an actual variable transformation in the homogeneous medium differential equation of resident concentrations shows that the flux concentration also satisfies the convection dispersion equation including those with linear reaction. Hence, the physical meanings of solutions with respect to these two concentrations are inferred from boundary conditions. The earlier studies have also provided a classification of solutions of homogeneous medium models based on these two concentration variables. This work first generalizes the theory of dependent variable transformation for dispersive transport and further extends it to include the heterogeneous medium models that assume a diffusive transport between the fracture and matrix phases. Then, two new solutions of the heterogeneous medium solutions are derived. Hence, the experimentalist is furnished with the heterogeneous medium model solutions in which at least one of them is consistent with the actual conditions of an experiment. Secondly, in this work, the concept of block geometry functions (BGFs) is extended to include frequency distributions of multiple block sizes more likely to exist in heterogeneous media than a single block. It was found that BGFs of various distributions with λmin less than 0.1 differ only slightly, and hence, may be represented by the BGF of the mean block size. Otherwise care must be exercised to include the block size distribution effect. In addition, numerical Laplace inversion of complete solutions having those slightly differing BGFs is found to lead to significant differences for cases where mobile and immobile phase fractions are close. Finally, interpretation of tracer return profiles in a heterogeneous system by employing a nonlinear regression technique is illustrated. Simulated field data are matched by a four-parameter theoretical model and sensitivity of results to parameter values is investigated.