An analytical model for solute transport through a water-saturated single fracture and permeable rock matrix

The problem of solute transport through a water-saturated single fracture in a permeable rock matrix is examined using an analytical modeling approach. A closed-form analytical solution is obtained that accounts for transverse and longitudinal advective transport in the fracture and matrix and transverse diffusion in the matrix. The solution also accounts for both diffusive and advective solute exchange between the fracture and matrix and a general solute source position in either the fracture or matrix. The novel features are the incorporation of advective transport in the matrix and a general source position into a closed-form solution for the solute-transport problem. Examples of the solution behavior are presented, which demonstrate the effects of matrix advection in combination with advection along the fracture, transverse diffusion in the matrix for solute release in the fracture and matrix. A semianalytical solution in the form of a superposition integral is also derived that includes these transport features, plus independent levels of longitudinal diffusion and dispersion in the matrix and fracture, respectively. Examples are presented that include advective transport in the fracture and matrix, longitudinal and transverse diffusion in the matrix, longitudinal dispersion in the fracture, as well as solute release from the fracture and matrix. An approximate criterion is proposed to evaluate the significance of longitudinal diffusion and dispersion relative to longitudinal spreading caused by fracture-matrix interaction.