Abstract
Transport of solutes in porous media at the laboratory scale is governed by an Advection Dispersion Equation (ADE). The advection is by the fluid velocity $U$ and dispersion by $D_{dL}=U\alpha _{dL}$, where the longitudinal dispersivity $\alpha _{dL}$ is of the order of the pore size. Numerous data revealed that the longitudinal spreading of plumes at field scale is characterized by macrodispersivity $\alpha _{L}$, larger than $% \alpha _{dL}$ by orders of magnitude. This effect is attributed to heterogeneity of aquifers manifesting in the spatial variability of the logconductivity $Y$. Modeling $Y$ as a stationary random field and for mean uniform flow (natural gradient), $\alpha _{L}$ could be determined in an analytical form by a first order approximation in $\sigma _{Y}^{2}$ (variance of $Y$) of the flow and transport equations. Recently, models and numerical simulations for solving transport in highly heterogeneous aquifers ($\sigma _{Y}^{2}>1$), primarily in terms of the mass arrival (the breakthrough curve BTC), were advanced. In all cases ergodicity, which allows to exchange the unknown BTC with the ensemble mean, was assumed to prevail for large plumes, compared to the logconductivity integral scale. Besides, the various statistical parameters characterizing the logconductivity structure as well as the mean flow were assumed to be known deterministically. The present paper investigates the uncertainty of the non-ergodic BTC due to the finiteness of the plume size as well as due to the uncertainty of the various parameters on which the BTC depends. By the use of a simplified transport model we developed in the past (which led to accurate results for ergodic plumes), we were able to get simple results for the variance of the BTC. It depends in an analytical manner on the flow parameters as well as on the dimension of the initial plume relative to the integral scale of logconductivity covariance. The results were applied to the analysis of the uncertainty of the plume spatial distribution of the MADE transport experiment. This was achieved by using the latest, recent, analysis of the MADE aquifer conductivity data.
Highlights
Aquifers pollution by various contaminants constitutes a major threat to fresh water resources all over the world
As previously stated our study focuses on the BTC M which is relevant to many applications and is quite robust and less prone to uncertainty than point concentration
The aim of the present study is to provide a discussion of uncertainty in modeling transport in three-dimensional heterogeneous aquifers, with application to the MAcro Disperion Experiment (MADE) transport experiment (Zheng et al, 2011) as a platform for discussion; we summarize what we have learned in the last two decades or so, in view of applications, with a particular focus on uncertainty due to plume sampling and incomplete knowledge of parameters that are both important for MADE
Summary
Aquifers pollution by various contaminants constitutes a major threat to fresh water resources all over the world. Low K, (iii) like the K field (see Figure 1) the plume is seemingly erratic and defies a representation by smooth functions but, at the same time, it makes the identification of point concentration at a given location an elusive goal, (iv) in contrast, global measures like the mass arrival at vertical planes over the entire domain (the BTC M) smooth out the variations and the extent of spreading can be quantified for instance by αL, (v) it was found that the presence of the pore scale dispersion (primarily the vertical one) causes mixing which affects the local C, but has a minor effect on M (Fiori and Dagan, 2000) and (vi) space averages like M are the ones of interest in many applications, e.g., those in which the goal is to determine the mass of solute pumped by wells which intercept the plume (Fiori et al, 2016). The plan of the paper is as follows: section 2 provides an overview of concepts development and paper aims, recapitulating some of our recent developments in transport of ergodic plumes; section 3 addresses the modeling of uncertainty in the prediction of the BTC, the main topic of the paper; section 4 presents the application of the uncertainty analysis to the MADE-1 experiment, relying on the latest published data; section 5 summarizes and concludes the study
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