Modeling anomalous transport in randomly heterogeneous media with nonlocal moment equations

Representations of transport dynamics based on a Fickian analogy of the kind embedded in the traditional advection-dispersion equation (ADE) fail to capture (a) temporal variations in solute spreading rates or (b) tailing of late arrival times often exhibited by point and section-averaged solute breakthrough curves (BTCs). These two phenomena are considered to signify anomalous transport. Elsewhere we presented exact stochastic moment equations to model tracer transport in randomly heterogeneous aquifers. We also developed closure schemes which allow solving these equations numerically at various orders of approximations. We have previously shown that (a) conditional and unconditional first and second concentration and solute flux moments computed in this manner agree closely with those obtained by Monte Carlo simulation for well-conditioned or mildly heterogeneous media and (b) that mean concentration spreading rates vary with time and degree of conditioning as well as boundaries. Here we demonstrate numerically that, in the unconditional case, our moment solution generates BTCs with long tails at log hydraulic conductivity variances as low as 0.01; as this variance increases, early arrival times and late time tailing become more pronounced. Concentration variance at fixed locations acquires asymmetric shapes with long tails. Our results demonstrate that statistical moments of the standard (non-anomalous) ADE produce behavior similar to that commonly attributed to anomalous transport.