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_Y² (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 z 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_Y², 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_Y² 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.