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The Lagrangian concentration approach for determining dilution in aquifer transport: Theoretical analysis and comparison with field experiments

Transport of nonreactive solutes in aquifers of random permeability is considered. The local concentration C, which is a random function of space and time, can be modeled through 〈C〉, the ensemble mean, and σC2, the variance. Because of the irregular spatial distribution of the concentration, which is a consequence of the spreading and dilution of the solute plume, the uncertainty in the local concentration prediction can be very large, and the mean concentration 〈C〉 may not accurately predict the concentration distribution in the field, in particular, the peak concentration. The aim of the present paper is to quantify dilution through the peak concentration by introducing a quantity denoted as Lagrangian concentration (CL). According to the proposed methodology the combined effects of dispersion, which is ruled by large‐scale advection, and dilution, which is a consequence of pore‐scale dispersion, are analyzed by following each Darcy‐scale solute particle constituting the initial plume by focusing on the mass exchanges between the examined particle and the surrounding ones. After definition and derivation of CL and its statistical moments, illustrative examples are shown for a small solute injection. Then the comparison of the theoretical results with the Borden and Cape Cod experimental findings shows that the Lagrangian concentration gives a reasonable prediction of the peak concentration: conversely, the often employed mean concentration 〈C〉 underestimates the maximum concentration. Furthermore, the fluctuations of CL are also smaller than those for C, as indicated by the analysis of the variance. The results are of some importance for applications in which the environmental regulations are formulated in terms of maximal admissible local concentrations.