WATER RESOURCES RESEARCH, VOL. 31, NO. 1, PAGES 237-243, JANUARY 1995 Comment on "Stochasticanalysisof the transport of kinetically sorbing solutes in aquifers with randomly heterogeneous hydraulic conductivity" by H. A.M. Quinodoz and A. J. Valocchi Roko Andri•evid Desert ResearchInstitute, Water Resources Center, University of Nevada Systems,Las Vegas 1. Introduction and Scope In a recent paper, Quinodoz and Valocchi [1993] pre- sented a spatial moments analysis of kinetically sorbing solutes undergoing linear reversibleadsorption in randomly heterogeneous porous formations. The objective of this comment is to introducean alternativeapproach to obtaining the spatial momentsof a particle displacement in a closed form and to present a few constructive comments on both approaches, with particular emphasison their predictive capabilities in actual field settings. The comments are struc- tured as follows: First, the alternative approach in deriving the closed-form expressionfor the first two displacement momentsis presentedand comparedwith the results of Quinodoz and Valocchi [ 1993]. Second, theuse of developed spatial moments for predicting the displacement of kineti- cally sorbing solutes in actual field realizations is discussed. 2. Spatial Moments The solute kinetics are modeled as a linear reversible reactionOs/Ot= kr(KdC - $), as in Quinodozand Valoc- chi's equation (2) followingthe same notation,namely,Kd - kf/k r as theequilibrium distribution coefficient, with kf and kr as forward and reverse rate coefficients, andR - 1 + Kd as the retardation factor. Then, following the Lagrangian formulation for the nonreactive solute [Dagan, 1984], the random displacement vector X(t) of the kinetically sorbing solutes can be expressed with a modified kinematic relation X(t) = ••* V[X(r,)] dr, + Xd = Ut, + fj* v[X(r,)] dr, + Xa t, -< t (1) where X(t) is a three-dimensional displacement vector of the reactivesolute plume, ¾(x) is the steady velocityfield which possesses a statistical homogeneity resulting in decomposi- tion V(x) = U + v(x), where (V(x)) = U denotes the ensemble average vector of the velocity field and v(x) is a random fluctuation with zero mean and covariance matrix Rv(x i - xj) depending only ontheseparation between two points, Xe is the pore-scale movements (molecular diffusion Copyright 1995by the American Geophysical Union. Paper number 94WR02377. 0043-1397/95/94 WR-02377502.00 is neglected), and r, is the integration variable going from 0 to t,, where t, -• t. Thus the right-hand side of (1) is expressed in the new time variablet,, which is a fractionof the actual time t that a particle stays in the liquid phase and relates to the work of Quinodoz and Valocchi [1993] by t,(t) = 13(t)t = tm(t). Sincethe particleis sorbing with the soil, the random fluid residence time appears as an upper limit in the integrationof the velocity field. The difficulties and possible approximative solutionof the kinematicrela- tion for the nonreactive case (equation (1) replacing t, with t) were extensively discussed by Dagan [1989]. In the case of a kinetically sorbing solute, the additional random vari- able t, appears in the kinematic relation. We start our solution process with formal first-orderapproximation of the Lagrangian displacement by expanding the right-hand side of (1) in the Taylor series up to the first order around the ensemblemean fluid residence time, (t,(t)), and the ensem- ble mean displacement,(X(t)), and by neglecting the pore- scale dispersion,which yields X(t) = Ut, + f•* v[X(r,)] dr, = U(t,)+ v[(X(r,))] dr, + t,={t,} X(t)={X(t)} + X'(t)V x X(t)=(X(t)) where t[ = t, - (t,) denotesthe fluctuationsaround the mean residencetime, X' (t) is the residual displacement, and V,., and Vx denote thegradient of (1) with respect to thefluid residence time and particle displacement, respectively. The neglect of the pore-scale dispersion coefficient compared to other terms in (1) implies that the case with large Peclet numbers is considered. Neuman et al. [1987] have consid- ered other classes of problems with small Peclet numbers, but these specialcasesare not consideredhere. After differentiating,t, and X(t) are replaced with corre- sponding mean values such that the first displacement mo- ment, (X(t)) = U(t,), is obtained by taking the ensemble average of (2) and assuming independence betweenfluctua- tions of the random velocity field and the fluid residence time. Keepingonly first-order terms, the displacement resid- ual is, then, given as 237