Modeling flow and transport in highly heterogeneous three‐dimensional aquifers: Ergodicity, Gaussianity, and anomalous behavior—2. Approximate semianalytical solution

Flow and transport take place in a heterogeneous medium of lognormal distribution of the conductivity K . Flow is uniform in the mean, and the system is defined by U (mean velocity), σ Y 2 (log conductivity variance), and integral scale I . Transport is analyzed in terms of the breakthrough curve of the solute, identical to the traveltime distribution, at control planes at distance x from the source. The “self‐consistent” approximation is used, where the traveltime τ is approximated by the sum of τ pertinent to the different separate inclusions, and the neglected interaction between inclusions is accounted for by using the effective conductivity. The pdf f (τ, x ), where x is the control plane distance, is derived by a simple convolution. It is found that f (τ, x ) has an early arrival time portion that captures most of the mass and a long tail, which is related to the slow solute particles that are trapped in blocks of low K . The macrodispersivity is very large and is independent of x . The tail f (τ, x ) is highly skewed, and only for extremely large x / I , depending on σ Y 2 , the plume becomes Gaussian. Comparison with numerical simulations shows very good agreement in spite of the absence of parameter fitting. It is found that finite plumes are not ergodic, and a cutoff of f (τ, x ) is needed in order to fit the mass flux of a finite plume, depending on σ Y 2 and x / I . The bulk of f (τ, x ) can be approximated by a Gaussian shape, with fitted equivalent parameters. The issue of anomalous behavior is examined with the aid of the model.