Stochastic evaluation of mixing-controlled steady-state plume lengths in two-dimensional heterogeneous domains

We study plumes originating from continuous sources that require a dissolved reaction partner for their degradation. The length of such plumes is typically controlled by transverse mixing. While analytical expressions have been derived for homogeneous flow fields, incomplete characterization of the hydraulic conductivity field causes uncertainty in predicting plume lengths in heterogeneous domains. In this context, we analyze the effects of three sources of uncertainty: (i) The uncertainty of the effective mixing rate along the plume fringes due to spatially varying flow focusing, (ii) the uncertainty of the volumetric discharge through (and thus total mass flux leaving) the source area, and (iii) different parameterizations of the Darcy-scale transverse dispersion coefficient. The first two are directly related to heterogeneity of hydraulic conductivity. In this paper, we derive semi-analytical expressions for the probability distribution of plume lengths at different levels of complexity. The results are compared to numerical Monte Carlo simulations. Uncertainties in mixing and in the source strength result in a statistical distribution of possible plume lengths. For unconditional random hydraulic conductivity fields, plume lengths may vary by more than one order of magnitude even for moderate degrees of heterogeneity. Our results show that the uncertainty of volumetric flux through the source is the most relevant contribution to the variance of the plume length. The choice of different parameterizations for the local dispersion coefficient leads to differences in the mean estimated plume length.