Variability of travel time of solute particles in single fractures with spatially variable aperture

The arrival time of solutes from the source to a prescribed compliance boundary is a global transport quantity that is of practical interest in many applications, such as risk assessment for environmentally sensitive sites in regulatory analyses. Estimates of uncertainty are necessary and can play an important role in the decision-making process. The objective of this work is therefore to quantify the variability of solute travel time in individual fractures with spatially variable apertures. The variability of solute transport in fractures is largely determined by the heterogeneity of the apertures of naturally occurring rock fractures. To model solute transport, the fracture apertures and flow fields are considered as spatial random functions. Using the relationship between the two-dimensional depth-averaged solute conservation equation and the Fokker-Planck equation, it is possible to formulate the solute convection velocity in such a way that a general expression for the solute travel time variance can be developed by taking into account the effects of the variation in fracture aperture. A closed-form expression for the variance of travel time in the mean flow direction is derived for the case of advection-dominated solute transport. This derived expression allows the analysis of the effects of the variation of the fracture aperture on the variability of the solute travel time.