Solute transport along a single fracture with a finite extent of matrix: A new simple solution and temporal moment analysis

A new simple and robust solution, based on uniform steady-state flow velocity, is developed for the problem of solute transport in a fracture-matrix system with a finite penetration depth of a radioactive contaminant into the rock matrix. The solution is an extension of Liu et al. (2017) to finite penetration depth and an alternative solution strategy to the problem solved by Sudicky et al. (1982). The solution takes the form of a convolution of two functions. The first function describes the probability density function of the residence time distribution of a conservative solute resulting merely from advection and Fickian dispersion. The second function is actually the impulse response of the fracture-matrix system in the case of a plug flow without any hydrodynamic dispersion. As a result, the effects of Fickian dispersion and matrix diffusion on solute transport are decoupled, and thus the resulting breakthrough curve can be analyzed in terms of those two functions. In addition to this, the derived Péclet numbers of those two functions, based on temporal moments, are also found to be associated with the derived Péclet number of the resulting breakthrough curve. By comparing the Péclet numbers of those two functions, the contribution of Fickian dispersion and matrix diffusion to solute spreading is determined in a straightforward way. This can aid to find out the dominating mechanism on solute transport, and therefore the performance of breakthrough curve can be approximated by only one function in some specific cases.