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

Abstract

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), σY2 (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 σY2, 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 σY2 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.

Mechanisms of anomalous dispersion in flow through heterogeneous porous media

“…The developed CTRW model, however, can be straightforwardly generalized to d = 3 dimensional media based on similar analytical expressions for the flow velocity inside the inclusions [41]. Furthermore, the present study considers purely advective transport.…”

“…Eames and Bush [40] studied solute dispersion in such media and derived expressions for the dispersion coefficients. Fiori et al [41,42,43] studied anomalous transport in media consisting of inclusions with lognormal distributions of hydraulic conductivity and derived semi-analytical expressions for solute travel times. These solutions have been implemented into a time-domain random walk approach, which is similar to the CTRW [44].…”

Effective retardation factor for transport of reactive solutes in highly heterogeneous porous formations

“…The approach led to accurate solutions when compared with numerical simulations [see e.g., Jankovic et al, 2003;Suribhatla et al, 2011] and was recently used for prediction of MADE transport experiment with favorable results . We recapitulate here the main lines of the procedure; more details can be found in our past publications [e.g., Fiori et al, 2006Fiori et al, , 2007.…”

“…However, in the averaging procedure we have to account that the probability for the generic particle to enter the inclusion j is given by A w /A 5 V(K j )/U, relative area of the upstream capture zone of the block j, as discussed in Fiori et al [2006]. Hence, each of the terms in the summation (3) is multiplied by V(K j )/U before averaging, and the final expression for <s> is…”

Comparison of Fickian and temporally nonlocal transport theories over many scales in an exhaustively sampled sandstone slab

“…[3] It is not a new idea that spatial heterogeneity of a velocity field generates some sort of enhanced ''tailing'' in the breakthrough of a conservative solute past some point (the breakthrough curve, or BTC) [Silliman and Simpson, 1987;Tsang et al, 1988;Moreno et al, 1988;Becker and Shapiro, 2003;Berkowitz et al, 2000Berkowitz et al, , 2006Fiori et al, 2006;Janković et al, 2006;Fernàndez-Garcia and Gómez-Hernández, 2007;Riva et al, 2008;Fernàndez-Garcia et al, 2009;Dentz and Bolster, 2010]. To date, all studies have followed a similar logic chain: (1) assume that the ADE is valid at a small scale; (2) apply the ADE in a heterogeneous velocity field; and (3) observe some delayed BTC tailing.…”