Fluid Movement in Earth Materials

An analytical method of determining the ionic diffusion transverse to the direction of flow in granular media is presented. The basic assumption that the frontal zone created by longitudinal dispersion is fully developed and stationary before the transverse diffusion can cause spreading is made to simplify the mathematical analysis. Solutions were obtained in terms of both a series of Bessel's functions and a hypergeometric series. Numerical computations were made and results are presented in graphical form. INTRODUCTION The necessity for disposal of radioactive and other wastes and for more quantitative knowledge of microscopic flow in porous media has focused interest on dispersion phenomena occurring in saturated flow through porous media. Various laboratory techniques have been developed for determining the dispersion coefficient appearing in the analytical solutions. Emphasis, however, has been placed on studies of longitudinal dispersion for example, those of Beran (1957) and Day (1956) and by the writer (Dispersion in porous media, doctoral dissertation, Northwestern Univ., 1958). The data available on transverse diffusion are still extremely meager, owing to the difficulty in modeling transverse diffusion and measuring the concentration distribution once a model is constructed. The Geological Survey in its ground-water research office at Phoenix, Ariz., has developed methods for measuring transverse diffusion, by means of radioactive isotopes and commercially available counters (Skibitzke and others, 1960). However, the studies to date have been directed toward feasibility of the methods, and quantitative data are not available. This paper will consider the mathematical aspects of an approximate description of the phenomena by dealing with a simplified mathematical model. ACKNOWLEDGMENTS The problem treated in this report was brought to the attention of the writer by H. E. Skibitzke, mathematician in charge of the Phoenix ground-water research office. The writer is greatly indebted to Mr. Skibitzke, whose initial development and ideas greatly facilitated the preparation of this report. The author would like to express appreciation also to Prof. Richard Skalak, of Columbia University, for his thorough discussion and review of the paper. Professor Skalak points out that Goldstein, in his paper "Some Two-Dimensional Diffusion Problems with Circular Symmetry" (1932), obtained equation 27. However, his paper was not available to the writer at the time of writing. Professor Skalak points out also that there is a shorter method of derivation of equation 9 by use of the Hankel transform. The author utilized the Laplace transformation because this method has a wider range of applicability and is a more familiar method of approach. TWO-DIMENSIONAL DISPERSION The differential equation that describes the twodimensional dispersion of a contaminant hi saturated granular media, where radial symmetry exists, is