Laplace-Transform Finite Element Solution of Nonlocal and Localized Stochastic Moment Equations of Transport

Abstract

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 σ 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.

Transient water age distributions in environmental flow systems: The time‐marching Laplace transform solution technique

“…To return into the real age‐domain, the fields or need to be numerically inverted. Various authors [e.g., see Sudicky , 1989; Cornaton , 2003; Cornaton and Perrochet , 2006a; Morales‐Casique and Neuman , 2009] have shown that the numerical inversion of Laplace transform finite element solutions are generally accurate when using the algorithm of Crump [1976] and when further combining it to the quotient‐difference algorithm, as proposed by de Hoog et al [1982]. The Laplace transform schemes can suffer from oscillations near discontinuities during the inversion, in relation to the nonuniform convergence of the series in the inversion formula.…”

“…A control of these oscillations can be done by increasing the number of discrete Laplace variables and decreasing the mesh/grid size near discontinuities. It can be reported that a value for n ranging between 10 and 20 generally yields accurate solutions in the presence of sharp gradients [ Cornaton , 2003; Cornaton and Perrochet , 2006a; Morales‐Casique and Neuman , 2009].…”

Finite Element Modeling of Diffusion in Fractured Porous Media by Using Hierarchical Material Properties

“…The time independency of the transformed diffusion problem (Equation ) renders its numerical solution possible without time stepping, and therefore avoids both the necessity to satisfy the stability condition which dictates smaller time steps for finer mesh and the requirement to compute the solutions for each time step regardless of the interest at a single time (e.g., Cai & Costache, 1994; Morales‐Casique & Neuman, 2009; Sudicky & McLaren, 1992).…”

Review of inverse Laplace transform algorithms for Laplace-space numerical approaches

“…Numerical methods can similarly benefit from the Laplace transform, converting the time-dependence of a differential equation to parameter dependence. Laplace-space finite-element approaches have seen application to groundwater flow and solute transport (e.g., [Sudicky and McLaren(1992), Morales-Casique and Neuman(2009)]), and Laplace-space BEM has also been used in groundwater applications (e.g., [Kythe(1995), §10.3] or [Liggett and Liu(1982), §10.1]). The Laplace transform analytic element method [Kuhlman and Neuman(2009)] is a transient extension of the analytic element method.…”