Steady plumes in heterogeneous porous formations: A stochastic Lagrangian approach

Abstract

We study three-dimensional steady plumes in heterogeneous porous formation by a stochastic analytical approach. Flow is uniform in the mean and the hydraulic conductivity K is a space random function, resulting in a random local concentration C. The primary concern of our work is to characterize C in terms of its mean value and its fluctuation; this is achieved, for low/medium heterogeneity, by a rigorous Lagrangian model based on the first-order approximation. Large-scale advection and pore-scale dispersion (PSD) govern the movement of solute particles through the medium of log conductivity YlnK. Transport is mainly governed by vertical and lateral spreading, different from the extensively studied transient transport, which is in turn governed by longitudinal dispersion. The concentration field depends on the Peclet number Pe, which controls dilution; the latter results from the intertwining of large-scale advection and PSD. The mean concentration 〉C〈 recovers the well-known solution for homogeneous media with the obvious differences in the dispersion/diffusion parameters. The standard deviation σC, evaluated here for the first time, fully analytically within the Lagrangian approach for steady state plumes, exhibits a maximum far from the plume centerline; the coefficient of variation can be very large in the plume fringe. The rigorous analytical evaluation of σC requires a huge computational effort in order to calculate the trajectory moments and perform the numerical integrations needed. We propose two simplified solutions with different degrees of approximation, that can be used in applications and preliminary studies: (i) the first one is obtained neglecting the longitudinal component of the trajectory fluctuations; the solution brings considerable simplifications although numerical integrations are still required in order to evaluate σC; this approximation is quite accurate for the mean concentration while slightly overestimates the concentration variance; (ii) the second one leads to a closed-form expression for σC; although somewhat conservative, this simple approximated solution works very well in a broad range of distances from the source, and may be used for a preliminary assessment of C in applications. © 2013. American Geophysical Union. All Rights Reserved.