Analysis of concentration under non-ergodic transport as sampled in natural aquifers
Creators: name: Fiorotto, Virgilio
name: Caroni, Elpidio
name: Nicolini, Matteo
name: Nicolini, Matteo
Publisher: name: XVI International Conference on Computational Methods in Water Resources
Year: 2006

description: 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 Péclet 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 Péclet 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).