Approximations for transport parameters and self-averaging properties for point-like injections in heterogeneous media

Numerical investigations on ergodicity of solute transport in heterogeneous aquifers

“…Since the asymptotic effective coefficients in given realizations do not tend to their expectation (a necessary condition for ergodicity), the transport is not ergodic. Quantitative numerical estimations for the deviation from the macrodispersion model of the transport in realizations of a Gaussian velocity field with finite correlation lengths (in both transverse and longitudinal directions) were recently obtained, for the case of point‐like initial plumes [ Eberhard , 2004]. The numerical computations of Eberhard [2004], based on approximate solutions of the transport equation, show that the fluctuations of the longitudinal effective coefficient tend to zero, indicating that a necessary condition for ergodicity is fulfilled.…”

“…Quantitative numerical estimations for the deviation from the macrodispersion model of the transport in realizations of a Gaussian velocity field with finite correlation lengths (in both transverse and longitudinal directions) were recently obtained, for the case of point‐like initial plumes [ Eberhard , 2004]. The numerical computations of Eberhard [2004], based on approximate solutions of the transport equation, show that the fluctuations of the longitudinal effective coefficient tend to zero, indicating that a necessary condition for ergodicity is fulfilled. To prove the ergodicity of transport in velocity fields with finite correlation lengths it is necessary to check whether the sufficient conditions, formulated for actual concentrations, are verified.…”

“…Condition e2 for space averaged concentration was used by Dagan [1984, 1987] to investigate the ergodicity under the assumption that the ensemble averaged concentration is already Gaussian at finite times. Condition e2 also formulates the self‐averaging property of the longitudinal effective coefficient, investigated by Clincy and Kinzelbach [2001] and Eberhard [2004]. The condition e1 alone, applied to effective coefficients or to second moments of the plume, is often referred to as “ergodicity condition” [ Fiori , 1998; Naff et al , 1998b; Zhang and Seo , 2004; Dagan , 2004].…”

“…Even though abundant literature has been produced in the last two decades, the investigations on ergodicity were in most cases limited to the study of the second moments of the solute plume and the conclusions often disagree. For instance, it is accepted that the length scale to reach the asymptotic limit predicted by the stochastic model is impractical for real contamination problems [ Berkowitz , 2001; Schwarze et al , 2001; Dentz et al , 2002, 2003; Eberhard , 2004]. However, recent numerical investigations conclude that, for extended initial plumes, the dispersivities behave ergodically, at relatively small distances from the injection domain [ Janković et al , 2003].…”

“…The presumption that the hydraulic conductivity has a lognormal distribution and small variance is an accepted simplification leading to Gaussian velocity fields which can capture the essential features of the stochastic model [ Cortis et al , 2004; Eberhard , 2004]. In the present context, this choice is not a limitation since the existence of the upscaled Gaussian distribution for the ensemble averaged concentration, used as reference in our investigations, is ensured for Gaussian velocity fields when the velocity variance goes to zero.…”