Macroscopic behavior and random‐walk particle tracking of kinetically sorbing solutes

Abstract

Abstract. Analytical expressions are derived for the zeroth, first, and second spatial moments of sorbing solutes that follow a linear reversible kinetic mass transfer model. We determine phase-transition probabilities and closed-form expressions for the spatial moments of a plume in both the sorbed and aqueous phases resulting from an arbitrary initial distribution of solute between the phases. This allows for the evaluation of the effective velocity and dispersion coefficient for a homogeneous domain without resorting to numerical modeling. The equations for the spatial moments and the phase-transition probabilities are used for the development of a new random-walk particle-tracking method. The method is tested against three alternate formulations and is found to be computationally efficient without sacrificing accuracy. We apply the new random-walk method to investigate the possibility of a double peak in the aqueous solute concentration resulting from kinetic sorption. The occurrence of a double peak is found to be dependent on the value of the Damk6hler number, and the timing of its appearance is controlled by the mass transfer rate and the retardation factor. Two ranges of the Damk6hler number leading to double peaking are identified. In the first range (Da• -< 1), double peaking occurs for all retardation factors, while in the second range (1 _< Da• -< 3), this behavior is most significant for R > 12. IntroductionA thorough understanding of sorption processes is necessary for accurate groundwater transport predictions. Sorption in porous media is often described by assuming local equilibrium. The local-equilibrium assumption is not always valid, however, and sorption rates must be considered in some cases, especially where the adsorption and desorption reactions are slow relative to the seepage velocity [ where D is the dispersion coefficient, k is a mass transfer coefficient, Tim is the mobile phase porosity, and v is the seepage velocity. In other words, most of the desorption had to occur after the transport effects ceased to affect the breakthrough curve. Furthermore, double peaking occurred only when (kL/Tiv) was "not too large," where TI is the total porosity and L is the length of the column. Sample runs were performed only for a Peclet number Pe = vL/D = 10 and a retardation coefficient R = 10, and only point sources were examined. Previous Particle-Tracking MethodsA simple and efficient numerical method is needed to investigate the behavior of kinetically sorbing solutes. The particletracking method [Ahlstrom and Foote, 1976; Prickett et al., 1981] is a Lagrangian approach in which a large number of particles is tracked in order to simulate conservative-tracer transport. In contrast to Eulerian simulation techniques, negative concentrations cannot occur, concentration profile artificial smoothing and oscillations are greatly reduced, and dispersive fluxes are approximated accurately, provided that the number of particles is high enough. The random-walk method is therefore particularly applicable...

Toward efficiency in heterogeneous multispecies reactive transport modeling: A particle‐tracking solution for first‐order network reactions

“…The method extends the concept of transition probabilities [e.g., Michalak and Kitanidis, 2000;Salamon et al, 2006aSalamon et al, , 2006b] to first-order reaction networks and develops analytical solutions of the transition matrices associated with serial reactions.…”

“…Some of them are based on transition probabilities [Kinzelbach, 1987;Andričević and Foufoula-Georgiou, 1991;Michalak and Kitanidis, 2000] and others on the probability distribution of the particle residence time in the liquid and solid phase [Valocchi and Quinodoz, 1989;Painter et al, 2008]. In general, the efficiency of these methods depends on the parameters adopted to simulate transport.…”

A generalized Damköhler number for classifying material processing in hydrological systems

“…Thus, Da can be considered an important non-dimensional number within the dual lens approach. Michalak and Kitanidis (2000) used Da to characterise the appearance of chemical sorption peaks in simulation experiments. Wehrer and Totsche (2003) used Da to characterise when breakthrough of contaminants took place under nonequilibrium conditions, Ocampo et al (2006) explored the removal of nitrate by riparian zones and demonstrated that 50 % nitrate removal occurred when Da < 1 and increased to nearly 100 % at Da = 2-20.…”

A random walk solution for modeling solute transport with network reactions and multi-rate mass transfer in heterogeneous systems: Impact of biofilms

“…In subsurface hydrology, this concept has been used in the past to simulate solute transport with sorption/desorption processes (Valocchi and Quinodoz , 1989;Michalak and Kitanidis, 2000), rate-limited mass transfer processes , and kinetic network reactions (Henri and Fernàndez-Garcia, 2014). Following Michalak and Kitanidis (2000) and Henri and Fernàndez-Garcia (2014), transition probabilities can be determined from the evolution of the zeroth spatial moments of the solute plume. This is the procedure employed here.…”

“…In this context, Particle Tracking Methods (PTMs) constitute an efficient numerical alternative to simulate reactive transport (Kitanidis, 1994;Henri and Fernàndez-Garcia, 2014). Even though a large variety of methods exist to simulate rate-limited mass transfer processes with particle tracking (Benson and Meerschaert, 2009;Delay and Bodin, 2001;Dentz and Berkowitz , 2003;Tsang and Tsang, 2001), this method is still limited in the type of chemical reactions available, which include sorption (Tompson, 1993;Valocchi and Quinodoz , 1989;Michalak and Kitanidis, 2000), radioactive decay (Wen and Gómez-Hernández , 1996;Painter et al, 2007), first-order network reactions (Burnell et al, 2014;Henri and Fernàndez-Garcia, 2014), and simple bimolecular reactions (Benson and Meerschaert, 2008;Ding et al, 2013;Edery et al, 2009Edery et al, , 2010Paster et al, 2014) among others. None of the methods available nowadays supports multi-porosity systems with network reactions in three-dimensional randomly heterogeneous porous media.…”