Specifying Scale‐dependent Dispersivity in Numerical Solutions of the Convection–Dispersion Equation

Abstract

Many studies have incorporated various scale‐dependent dispersivity functions into the convection–dispersion equation (CDE). For a given function, there are different ways of specifying the dispersivity values in the discretized subdomains of a numerical solution. These ways would depend on how the function's time or space variable is built into the numerical scheme and on how the different ways may be interrelated. No study has addressed how these different ways may affect resulting breakthrough curves (BTCs) or their applicability to solute transport problems. In this study, five ways were specified and designated as local time, average time, apparent time, local distance, or apparent distance dependent dispersivity in the numerical scheme. The main objective was to demonstrate relationships among these ways by assuming that dispersion in the scale‐dependent CDE was quasi‐Fickian and implicitly based on an estimation of the first moment of the BTC at the given location. How these ways affected calculated BTCs was numerically tested using generalized linear and power scale–dependent dispersivity functions for a one‐dimensional problem with initial solute distribution represented as a Dirac delta function. For either type of function, differences were obvious in BTCs obtained for solute transport conditions with small apparent Peclet numbers using the alternative ways of specifying scale‐dependent dispersivity in the numerical scheme. The differences decreased with increasing apparent Peclet number. The numerical test results also showed that the derived relationships were not applicable for numerically calculating the spatial solute distribution at a given time. These test results were expected to be generally applicable because any other initial‐value solute transport problem can be solved as a superposition of this test problem.