Solute Transport in a Porous Medium: A Mass-Conserving Solution for the   Convection-Dispersion Equation in a Finite Domain
  This dissertation resolves a longstanding discussion of a mathematical problem important in contaminant hydrogeology and chemical-reaction engineering, the proper mathematical description for a miscible solute undergoing longitudinal convective-dispersive transport with production, decay, and sorption in a porous medium. Initial and input concentrations may be any continuously differentiable functions and the problem is stated for a finite domain. This domain yields a mass balance which requires Robin (i.e., third-type) boundaries, which describe a continuous flux but a discontinuous resident-concentration. The discontinuity in the resident concentration at the outflow boundary yields an underdetermined system when the exit concentration is not experimentally measured. This is resolved by defining the unknown effluent concentration from a semi-infinite problem which satisfies a Dirichlet (i.e., first-type) condition at the origin. The solution is represented in a uniformly convergent series of real variables.   The large volume of antecedent literature on finite solutions for convective-dispersive transport equations grew out of the historical precedents set by Danckwerts (1953) and Wehner and Wilhelm (1956) whom made simplifying assumptions of continuous boundary concentrations. This dissertation includes the demonstration that continuous-concentration hypotheses, whether rendered as Dirichlet or homogeneous Neumann (i.e., second-type) conditions, satisfy external mass conservation yet fail to provide solutions that are internally consistent with the governing equation. 
W. J. Golz