Commun. Comput. Phys., 6 (2009), pp. 131-161. Laplace-Transform Finite Element Solution of Nonlocal and Localized Stochastic Moment Equations of Transport Eric Morales-Casique 1*, Shlomo P.Neuman 11 Department of Hydrology and Water Resources, University of Arizona, Tucson, Arizona 85721, USA. Received 26 November 2007; Accepted (in revised version) 25 October 2008 Available online 18 November 2008 Abstract Morales-Casique et al. (Adv. Water Res., 29 (2006), pp. 1238-1255) developed exact first and second nonlocal moment equations for advective-dispersive transport in finite, randomly heterogeneous geologic media. The velocity and concentration in these equations are generally nonstationary due to trends in heterogeneity, conditioning on site data and the influence of forcing terms. Morales-Casique et al. (Adv. Water Res., 29 (2006), pp. 1399-1418) solved the Laplace transformed versions of these equations recursively to second order in the standard deviation $\sigma_Y$ of (natural) log hydraulic conductivity, and iteratively to higher-order, by finite elements followed by numerical inversion of the Laplace transform. They did the same for a space-localized version of the mean transport equation. Here we recount briefly their theory and algorithms; compare the numerical performance of the Laplace-transform finite element scheme with that of a high-accuracy ULTIMATE-QUICKEST algorithm coupled with an alternating split operator approach; and review some computational results due to Morales-Casique et al. (Adv. Water Res., 29 (2006), pp. 1399-1418) to shed light on the accuracy and computational efficiency of their recursive and iterative solutions in comparison to conditional Monte Carlo simulations in two spatial dimensions. AMS subject classifications: 65Z05, 65Z05, 68U20, 76S05 Notice: Undefined variable: pac in /var/www/html/issue/abstract/readabs.php on line 164 Key words: Geologic media, porous media, random heterogeneity, conditioning, stochastic transport, moment equations, nonlocality, localization, Laplace transform, finite elements, ultimate-quickest. *Corresponding author. Email: emorales@hwr.arizona.edu (E. Morales-Casique), neuman@hwr.arizona.edu (S. P. Neuman)