Connecting the dots: Semi-analytical and random walk numerical solutions of the diffusion–reaction equation with stochastic initial conditions

Paster, Amir; Bolster, Diogo; Benson, David A.

doi:10.1016/j.jcp.2014.01.020

Cited by 78 publications

(108 citation statements)

References 56 publications

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“…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.…”

Section: Introductionmentioning

confidence: 99%

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

Henri

Fernàndez-García

2015

Advances in Water Resources

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The interplay between the spatial variability of aquifer properties, mass transfer and chemical reactions often complicates reactive transport simulations. It is well documented that hydro-biochemical properties are ubiquitously heterogeneous and that rate-limited mass transfer typically leads to the conceptualization of an aquifer as a multi-porosity system. Within this context, chemical reactions taking place in mobile/immobile water regions can be substantially different between each other. This paper presents a particle-based method that can efficiently simulate heterogeneity, network reactions and multi-rate mass transfer. The approach is based on the development of transition probabilities that describe the likelihood that particles belonging to a given species and mobile/immobile domain at a given time will be transformed into another species and mobile/immobile domain afterwards. The joint effect of mass transfer and sequential degradation is shown to be non-trivial. A characteristic double peak of daughter products occurs when the degradation capacity in the immobile domain is relatively small. This late rebound of concentrations is not driven by any change in the flow regime (e.g., pumping ceases in the pump-and-treat remediation strategy) but due to the natural interplay between mass transfer and chemical reactions. To illustrate that the method can simultaneously represent mass transfer, spatially varying properties and network reactions without numerical problems, we have simulated the degradation of tetrachloroethylene (PCE) in a three-dimensional fully heterogeneous aquifer subjected to rate-limited mass transfer. Two types of degradation modes were considered to compare the effect of an active biofilm with that of clay pods present in the aquifer. Results of the two scenarios display significantly differences. Biofilms that promote the degradation of compounds in an immobile region are shown to significantly enhance degradation, rapidly producing daughter products and less tailing.

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Section: Introductionmentioning

confidence: 99%

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

Henri

Fernàndez-García

2015

Advances in Water Resources

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“…These authors represented incomplete mixing by setting up a low number of particles into the reactive system; the lower the number of particles the earlier reaction became limited. This was shown later on to represent a continuum-scale model based on the ADRE with stochastic noisy initial conditions [23,24]. While this is an intelligent approach, we would like to stress that caution should be taken when using the statistical fluctuations of a histogram, produced by sub-sampling the concentrations, to represent a true physical phenomenon.…”

Section: Introductionmentioning

confidence: 99%

Do we really need a large number of particles to simulate bimolecular reactive transport with random walk methods? A kernel density estimation approach

Rahbaralam

Fernàndez-García

Sánchez-Vila

2015

Journal of Computational Physics

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Groundwater reactive transport models are an essential tool for the analysis of coupled geochemical processes in Earth Systems. On a molecular level, chemical reactions are the result of collision and subsequent recombination of the molecules present in the system. One way to deal with reactive transport is using a continuum mechanics approach; for example, in a well-mixed system, reactions are a function of reactant concentrations. An alternative is to use a Lagrangian approach, by tracking all molecules and specifying reactions in terms of collision theory. A computationally efficient family of methods to solve the reactive transport problem is based on Particle Tracking (PT). While the number of particles in most realistic applications is in the order of 10^6 í¯-10^9, the number of molecules even in diluted systems might be in the order of fractions of the Avogadro number. Thus, each particle actually represents a group of potentially reactive molecules. The use of a low number of particles may result not only in loss of accuracy, but also may lead to an improper reproduction of the mixing process, limited by diffusion. Recent works have used this effect as a proxy to model incomplete mixing in porous media. In this work, we propose using a Kernel Density Estimation (KDE) of the concentrations that allows getting the expected results for a well-mixed solution with a limited number of particles. The idea consists of treating each particle as a sample drawn from the pool of molecules that it represents; this way, the actual location of a tracked particle is seen as a sample drawn from the density function of the location of molecules represented by that given particle, rigorously represented by a kernel density function. The probability of reaction can be obtained by combining the kernels associated to two potentially reactive particles. We demonstrate that the observed deviation in the reaction vs time curves in numerical experiments reported in the literature could be attributed to a deficient estimation of the concentrations based on particles reconstruction, and not to the occurrence of true incomplete mixing. We further explore the evolution of the kernel size with time, linking it to the diffusion process. Our results show that KDEs are powerful tools to improve reactive transport simulations, and indicates that incomplete mixing in diluted systems should be modeled based on alternative mechanistic models and not on a limited number of particles.

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“…Of particular note are reactive transport processes in which reaction rates are not limited by fundamental molecular interactions, but instead are controlled by mixing processes (primarily diffusion) at interfaces between fluid and/or solid phases containing the various reactants [44][45][46]. These mixing-controlled reactions are often characterized by sharp local gradients and localized reaction zones; example problems include biologically-mediated natural attenuation of dissolved contaminant plumes [47,48], precipitation/dissolution reactions [10,49,50], incomplete mixing effects on reaction rates [51][52][53][54][55][56][57] and biofilm dynamics [11,58]. There have been many representative pore-scale modeling works on transverse mixing-controlled reactive transport [58][59][60][61][62][63][64][65][66].…”

Section: Introductionmentioning

confidence: 99%

Hybrid multiscale simulation of a mixing-controlled reaction

Scheibe

Schuchardt

Agarwal

et al. 2015

Advances in Water Resources

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a b s t r a c tContinuum-scale models, which employ a porous medium conceptualization to represent properties and processes averaged over a large number of solid grains and pore spaces, are widely used to study subsurface flow and reactive transport. Recently, pore-scale models, which explicitly resolve individual soil grains and pores, have been developed to more accurately model and study pore-scale phenomena, such as mineral precipitation and dissolution reactions, microbially-mediated surface reactions, and other complex processes. However, these highly-resolved models are prohibitively expensive for modeling domains of sizes relevant to practical problems. To broaden the utility of pore-scale models for larger domains, we developed a hybrid multiscale model that initially simulates the full domain at the continuum scale and applies a pore-scale model only to areas of high reactivity. Since the location and number of pore-scale model regions in the model varies as the reactions proceed, an adaptive script defines the number and location of pore regions within each continuum iteration and initializes pore-scale simulations from macroscale information. Another script communicates information from the pore-scale simulation results back to the continuum scale. These components provide loose coupling between the pore-and continuum-scale codes into a single hybrid multiscale model implemented within the SWIFT workflow environment. In this paper, we consider an irreversible homogeneous bimolecular reaction (two solutes reacting to form a third solute) in a 2D test problem. This paper is focused on the approach used for multiscale coupling between pore-and continuumscale models, application to a realistic test problem, and implications of the results for predictive simulation of mixing-controlled reactions in porous media. Our results and analysis demonstrate that the hybrid multiscale method provides a feasible approach for increasing the accuracy of subsurface reactive transport simulations.

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Connecting the dots: Semi-analytical and random walk numerical solutions of the diffusion–reaction equation with stochastic initial conditions

Cited by 78 publications

References 56 publications

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

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

Do we really need a large number of particles to simulate bimolecular reactive transport with random walk methods? A kernel density estimation approach

Hybrid multiscale simulation of a mixing-controlled reaction

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