From analytical solutions of solute transport equations to multidimensional time‐domain random walk (TDRW) algorithms

Abstract

In this study, new multi-dimensional time-domain random walk (TDRW) algorithms are derived from approximate one-dimensional (1-D), two-dimensional (2-D), and three-dimensional (3-D) analytical solutions of the advection-dispersion equation and from exact 1-D, 2-D, and 3-D analytical solutions of the pure-diffusion equation. These algorithms enable the calculation of both the time required for a particle to travel a specified distance in a homogeneous medium and the mass recovery at the observation point, which may be incomplete due to 2-D or 3-D transverse dispersion or diffusion. The method is extended to heterogeneous media, represented as a piecewise collection of homogeneous media. The particle motion is then decomposed along a series of intermediate checkpoints located on the medium interface boundaries. The accuracy of the multi-dimensional TDRW method is verified against (i) exact analytical solutions of solute transport in homogeneous media and (ii) finite-difference simulations in a synthetic 2-D heterogeneous medium of simple geometry. The results demonstrate that the method is ideally suited to purely diffusive transport and to advection-dispersion transport problems dominated by advection. Conversely, the method is not recommended for highly dispersive transport problems because the accuracy of the advectiondispersion TDRW algorithms degrades rapidly for a low P eclet number, consistent with the accuracy limit of the approximate analytical solutions. The proposed approach provides a unified methodology for deriving multi-dimensional time-domain particle equations and may be applicable to other mathematical transport models, provided that appropriate analytical solutions are available.

Time domain random walks for hydrodynamic transport in heterogeneous media

“…The efficiency of particle tracking can be noticeably enhanced by moving a particle over a fixed distance imposed by the computational mesh, for example, in a time that corresponds to the transit time over this distance [e.g., Pollock , ; McCarthy , ; Banton et al ., ; Noetinger and Estebenet , ]. Variants of this general methodology have been known in the literature under the terms time domain random walk (TDRW) [ Banton et al ., ; Delay et al ., ; Cvetkovic et al ., ; Bodin , ] and CTRW [ McCarthy , ; Noetinger and Estebenet , ]. As a general concept, the TDRW moves particles on a lattice with jumps of fixed distance whose direction and transition times are determined by the local advection and dispersion (or diffusion).…”

Surface water and groundwater: unifying conceptualization and quantification of the two “water worlds”

“…This is particularly the case for biopores such as worm burrows and root channels. Worm burrows provide a high amount of organic carbon and worms "catalyse" microbiological activity due to their enzymatic activity (Bundt et al, 2001;Binet et al, 2006;Bolduan and Zehe, 2006;. Similarly, plant roots provide litter and exude carbon substrates to facilitate nutrient uptake.…”

Random Walk Methods for Modeling Hydrodynamic Transport in Porous and Fractured Media from Pore to Reservoir Scale

“…The time of flight to the interface is drawn from exact or approximate analytical solutions to transport in homogeneous media. Developed first in one-dimensional media for simulating solute transport in fracture segments Bodin et al [2003], Painter and Cvetkovic [2005], TDRW have been extended to multidimensional media Bodin [2015], Delay et al [2002]. When local analytical solutions of the travel time distribution can be obtained, TDRW methods are highly efficient to couple the advective and diffusive/dispersive transport processes.…”