An evaluation of boundary conditions for one‐dimensional solute transport: 1. Mathematical development

Abstract

The Laplace transform method is employed to obtain the solution to several boundary value problems in which mixing occurs in reservoirs attached to a porous medium. Flow is assumed to be uniformly one‐dimensional throughout the porous medium. Solutions are obtained for both continuous and discontinuous concentrations at the reservoir‐medium boundary and for resident and flux‐averaged concentrations. A mass balance conducted on each solution shows that neither continuity condition can be proved generally superior. An evaluation of the parametric sensitivity of the solutions is also conducted and shows that the distinction between the continuity conditions and the flux and resident concentrations is only important at small Peclet numbers. The only exception to this is the distinction between flux‐averaged versus resident continuous concentrations in a finite solution domain. In this case, the flux concentrations were found to be insensitive to the volume of the downstream reservoir and, consequently, the flux and resident concentrations can differ substantially for larger reservoirs. In consideration of the general behavior of the solutions, three conceptual problems were identified which must be investigated by physical modeling: (1) determination of the appropriate solution at small Peclet number, (2) the possibility that dispersive mass flux occurs from the upstream reservoir under conditions of continuous concentration, and (3) the use of the flux transformation for continuous concentration in a finite solution domain. The physical modeling is discussed in a companion paper.