Traveltime-based descriptions of transport and mixing in heterogeneous domains

Modeling mixing-controlled reactive transport using traditional spatial discretization of the domain requires identifying the spatial distributions of hydraulic and reactive parameters including mixing-related quantities such as dispersivities and kinetic mass transfer coefficients. In most applications, breakthrough curves (BTCs) of conservative and reactive compounds are measured at only a few locations and spatially explicit models are calibrated by matching these BTCs. A common difficulty in such applications is that the individual BTCs differ too strongly to justify the assumption of spatial homogeneity, whereas the number of observation points is too small to identify the spatial distribution of the decisive parameters. The key objective of the current study is to characterize physical transport by the analysis of conservative tracer BTCs and predict the macroscopic BTCs of compounds that react upon mixing from the interpretation of conservative tracer BTCs and reactive parameters determined in the laboratory. We do this in the framework of traveltime-based transport models which do not require spatially explicit, costly aquifer characterization. By considering BTCs of a conservative tracer measured on different scales, one can distinguish between mixing, which is a prerequisite for reactions, and spreading, which per se does not foster reactions. In the traveltime-based framework, the BTC of a solute crossing an observation plane, or ending in a well, is interpreted as the weighted average of concentrations in an ensemble of non-interacting streamtubes, each of which is characterized by a distinct traveltime value. Mixing is described by longitudinal dispersion and/or kinetic mass transfer along individual streamtubes, whereas spreading is characterized by the distribution of traveltimes, which also determines the weights associated with each stream tube. Key issues in using the traveltime-based framework include the description of mixing mechanisms and the estimation of the traveltime distribution. In this work, we account for both apparent longitudinal dispersion and kinetic mass transfer as mixing mechanisms, thus generalizing the stochastic-convective model with or without inter-phase mass transfer and the advective-dispersive streamtube model. We present a nonparametric approach of determining the traveltime distribution, given a BTC integrated over an observation plane and estimated mixing parameters. The latter approach is superior to fitting parametric models in cases wherein the true traveltime distribution exhibits multiple peaks or long tails. It is demonstrated that there is freedom for the combinations of mixing parameters and traveltime distributions to fit conservative BTCs and describe the tailing. A reactive transport case of a dual Michaelis-Menten problem demonstrates that the reactive mixing introduced by local dispersion and mass transfer may be described by apparent mean mass transfer with coefficients evaluated by local BTCs.