Analysis of concentration under non-ergodic transport as sampled in natural aquifers

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Groundwater is probably the major source of water supply in the world, and the predictive ability in describing the fate of chemical contaminants in soils is of great importance when performing risk assessment and designing effective and efficient techniques to mitigate such problems. Most of environmental regulations (e.g., U.S. EPA, 1988; European Union, Directive 80/778/EEC) define water quality standards and acceptability in terms of concentration thresholds, and thus the prediction in natural aquifers must be performed with reference to the concentration probability of excess relative to the relevant threshold value. Natural porous formation are inherently heterogeneous, and solute plumes transported exhibit irregular shapes. Transport of an inert solute in heterogeneous porous formation is determined by large-scale advection and pore-scale dispersion, the relative importance given by the Peclet number. The first is mainly controlled by the spatial variability of hydraulic conductivity while the second, acting at scales lower than the heterogeneity characteristic length, is usually neglected. The prediction of the concentration field, due to the irregular variation of permeability, is affected by uncertainty, which has been set in a theoretical framework by regarding the permeability as a random space function. Several investigations have been conducted, for small values of the log-conductivity variance, in order to define the first and second moment of concentration, both in Eulerian and in Lagrangian framework. These analyses have been carried out under the ergodic hypothesis (satisfied when the solute initial characteristic lengths are much larger than the heterogeneity correlation scale), in which case the position of the barycenter of the plume can be regarded as deterministic. Moreover, these approaches give only an estimate of mean concentration and variance, while no consideration has been made about the underlying pdf. Recently, Fiorotto and Caroni (Trans. Porous Media, 48, 2002), and Caroni and Fiorotto (Trans. Porous Media, 59, 2005), analyzed, under ergodic conditions, the statistical properties of solute concentration in natural aquifers as sampled in observation wells. The aim of the present paper is to extend such previous research, in particular investigating to which extent the ergodic hypothesis may be assumed valid, and analyzing the statistical properties of the position of the barycenters of the solute plume, thus giving an estimate of the uncertainty in the prediction. The calculations, in Lagrangian framework, take advantage of the reverse formulation where, instead of considering the destination of the injected particles, the origin of the particle being sampled is sought. The advantage is that the concentration can be simulated using a reduced number of particles, while the accurate forward computation of the concentration requires a large number of particles, increasing up to prohibitive levels as long as the sampling area tends to shrink into a point. The analyses, have considered different sizes of the solute initial plume, and have been carried out varying the log-conductivity variance and the Peclet number, to quantify the relative role of the macro and the pore scale dispersion processes. In the case of small values of the log-conductivity variance, the methodology allows the derivation of an analytical expression for concentration mean, variance and pdf, while for high values, a Monte Carlo approach in a two-dimensional heterogeneous and statistically isotropic aquifer, characterized by log-normally distributed trasmissivity with an exponential covariance, has been developed. In the last case, the adoption of the Beta function to fit the concentration pdf proves valid for practical application, under the ergodic hypothesis (Caroni and Fiorotto, Trans. Porous Media, 59, 2005). Simulations show that, under non-ergodic transport, the uncertainty in the prediction of the barycenters of the plumes may be described by a multinormal random variate: this allows an estimate of the overall concentration pdf, which may be obtained by the convolution between the two distributions (in this case reducing to the mere product, being the processes uncorrelated).