Stochastic modeling analysis of sequential first-order degradation reactions and non-Fickian transport in steady state plumes

Abstract

Stochastic analyses were performed to examine sequential first-order monomolecular reactions at the microscopic scale and both Fickian and non-Fickian plume reactive transport at the macroscopic scale. An analytical solution was derived for the chemical master equation (CME) for a closed system of irreversible first-order monomolecular reactions. Taking a Lagrangian reference frame of particles migrating from a source, analyses show that the relative concentration of each species in the deterministic analytical solution for 1-D steady state plug flow with first-order sequential degradation is mathematically equivalent to the mean of a multinomial distribution of plume particles moving at constant velocity with sequential transformations described by transition probabilities of a discrete state, continuous-time Markov chain. In order to examine the coupling of reaction and transport terms in subdiffusive-reactive transport equations, a closed-form multispecies analytical solution also was derived for steady state advection, dispersion, and sequential first-order reaction. Using a 1-D continuous-time random walk (CTRW) embedded in Markov chains, computationally efficient Monte Carlo simulations of particle movement were performed to more fully examine effects of subdiffusive-reactive transport with an application to steady state, sequentially degrading multispecies plumes at a site in Palm, Bay, FL. The simulation results indicated that non-Fickian steady state plumes can resemble Fickian plumes because linear reactions truncate the waiting time between particle jumps, which removes lower velocity particles from the broad spectrum of velocities in highly heterogeneous media. Results show that fitting of Fickian models to plume concentration data can lead to inaccurate estimates of rate constants because of the wide distribution of travel times in highly heterogeneous media.

Probabilistic human health risk assessment of degradation‐related chemical mixtures in heterogeneous aquifers: Risk statistics, hot spots, and preferential channels

“…The reductive dechlorination of PCE due to biodegradation can be approximated by a sequential first-order reaction kinetic model [e.g., Clement, 2001;Burnell et al, 2014]. This model assumes that contaminant concentrations are relatively low (below the Michaelis half-saturation constant) [Cunningham and Mendoza-Sanchez, 2006].…”

From stochastic processes to numerical methods: A new scheme for solving reaction subdiffusion fractional partial differential equations

“…Reaction subdiffusion fractional partial differential equations have been widely used in recent years as mathematical models of systems of particles subject to, trapping, obstacles and reactions [1, 2,3,4,5,6,7,8]. Subdiffusion, characterised by a mean squared displacement of diffusing particles that grows slower than linear with time, has been observed in hydrogeology [9,8], physics [10], biology [11], finance [12] and chemistry [13].…”

“…Subdiffusion, characterised by a mean squared displacement of diffusing particles that grows slower than linear with time, has been observed in hydrogeology [9,8], physics [10], biology [11], finance [12] and chemistry [13].…”

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

“…This includes the degradation of chlorinated solvents (e.g., Clement, 1997Clement, , 2001, the decay of radioactive species (e.g., Painter et al, 2007), and the transformation of pesticides, organic phosphates and nitrogen in the environment (e.g., van Genuchten, 1985;Mishra and Mishra, 1991;Vishwanathan et al, 1998). When contaminant concentrations are small, i.e., less than the Michaelis half-saturation constant in the Monod or MichaelisMenten enzyme kinetic model, the microbial biotransformation rates can be described by pseudo-first-order reaction rates (e.g., Bouwer et al, 1981;Vogel et al, 1987;Haston and McCarty, 1999;Burnell et al, 2014). In this context, organic chlorinated solvents are often described by first-order reaction chains schematically described by A → B → C → D, meaning that species A is transformed into species B, B into C and so on.…”

“…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.…”