Time-periodic transport in heterogeneous porous media

Abstract

Analytical solutions for the one-dimensional convection-diffusion equation in a semi-infinite, heterogeneous domain are obtained for periodic boundary conditions. The kinematic dispersion of the porous medium is taken to be either a linear function of distance, or an exponential function of distance that asymptotically becomes constant. The analytic solutions are given in terms of Kelvin functions and hypergeometric functions, respectively, which are computed using Mathematica. The solutions are compared with each other and with the constant dispersivity case, and the amplitudes and phase shifts are studied with respect to the frequency of the input wave at the boundary. All three models respond differently to different single frequency inputs, showing different decay rates at lower frequencies and more pronounced phase shifts at higher frequencies. The use of single frequency, periodic inputs appears to give more distinguishable solutions than traditional pulse-type data which contains multiple spectral frequencies, and the resulting solutions can be used as a verification for numerical schemes for convection-diffusion equations.