Nonlinear Reactive Transport Research Articles

Understanding the nonlinear reactive transport model in porous catalysts

This paper discusses the steady-state nonlinear reaction-diffusion processes in porous catalysts. This model is based on a nonlinear second-order differential equation that includes a nonlinear term associated with the Michaelis-Menten and non-Michaelis-Menten kinetics of the reaction. The nonlinear equations can be approximately solved using the Taylors series, modified Taylors series and Akabri-Ganji technique to obtain the concentration of dissolved species. The influence of the half-saturation parameter and the characteristic reaction rate on concentration is explored. Sensitivity analysis of parameters is discussed. Our analytical findings were compared with numerical solutions.

The nonlinear reactive transport model: An efficient approximation method based on quasilinearization and Bessel matrix method

The nonlinear reactive transport model: An efficient approximation method based on quasilinearization and Bessel matrix method

Pore-scale study of coke combustion in a matrix-fracture system based on the micro-continuum approach

In situ combustion is an advanced recovery technique used to exploit heavy oil in the fractured reservoirs that make up approximately one-third of global heavy-oil resources. However, the mesoscopic mechanisms of coke combustion in the multiscale matrix-fracture system are not well understood because of the difficulty of performing pore-resolved simulations. In the present study, a pore-resolved micro-continuum approach was used to investigate fully coupled thermal and reactive flows through fractured media that contain nanometer-range coke pores, micrometer-range matrix pores, and sub-millimeter range natural fractures. Image-based simulations were implemented using synthetic geological models to mimic coke deposition patterns based on tomography images. The combustion regime diagram for the fractured media was mapped based on the ignition temperature and the air flux to exhibit three combustion regimes. The regime diagram was compared with that for unfractured media to address the impact of natural fractures on oxygen transport and the burning temperature. The oxygen diffusion mechanism dominated oxygen transport from the fracture into the matrix and led to a desirable smoldering combustion temperature regardless of the air injection rate. Effects of fracture geometries were quantified to demonstrate tortuous and discrete fractures, and matching air injection rates with fracture apertures can suppress air-channeling risk effectively. Possible discrepancies between lab measurements and field operations were demonstrated, and their potential to drive misinterpretation of experimental results was considered. The present pathway from tomography images to synthetic images and numerical simulations extends the “image and compute” technique to resolution of multiscale and nonlinear reactive transport.

Solution of Nonlinear Reaction-Diffusion Model in Porous Catalysts Arising in Micro-Vessel and Soft Tissue Using a Metaheuristic

The steady-state reactive transport model (RTM) is a generalization of the nonlinear reaction-diffusion model in porous catalysts. The RTM is expressed as a non-linear ordinary differential equation of second-order with boundary conditions. Artificial neural network (ANN), Particle swarm optimization (PSO), and hybrid of PSO-SQP (Sequential Quadratic Programming) are used to obtain accurate, approximate solutions to the non-linear RTM. The proposed technique is applied to three different cases of non-linear RTM. The properties of the nonlinear reactive transport model in porous catalysts are investigated by considering various cases based on variation in the half-saturation concentration “ <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula> ” and the characteristic reaction rate “ <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\beta $ </tex-math></inline-formula> .” The stability, reliability, and exactness of the proposed technique are established through comparison with the outcomes of the standard numerical procedure with the RK4 method and along with the different performance indices, which are Root-Mean-Square Error (RMSE), (TIC), Absolute Error (AE), and Mean Absolute Deviation (MAD).

Image-based modelling of coke combustion in a multiscale porous medium using a micro-continuum framework

Non-isothermal reactive transport in complicated porous media is diverse in nature and industrial applications. There are challenges in the modelling of multiple physicochemical processes in multiscale pore structures with various length scales ranging from nanometres to micrometres. This study focuses on coke combustion during in situ crude oil combustion techniques. A micro-continuum model was developed to perform an image-based simulation of coke combustion through a multiscale porous medium. The simulation coupled weakly compressible gas flow, species transport, conjugate heat transfer, heterogeneous coke oxidation kinetics and structural evolution. The unresolved nanoporous coke region was treated as a continuum, for which the random pore model, permeability model and species diffusivity model were integrated as sub-grid models to account for the sub-resolution reactive surface area, Darcy flow and Knudsen diffusion, respectively. A Pe–Da diagram was provided to present five characteristic combustion regimes covering the ignition temperature and air flux in realistic field operations and laboratory measurements. The present model proved to achieve more accurate predictions of the feasible ignition temperature than previous models. Compared with the air flux of $\phi \sim O\textrm{(1) s}{\textrm{m}^\textrm{3}}(\textrm{air})\;{({\textrm{m}^\textrm{2}}\ \textrm{h})^{ - 1}}$ in the field, the increasing air flux in the laboratory transformed the combustion regime from diffusion-limited to convection-limited, which led to an overpredicted burning temperature. Reactive fingering combustion was analysed to understand the potential risks in some experimental measurements. The findings provide a better understanding of coke combustion and can help engineers design sustainable combustion methods. The developed image-based model allows other types of multiscale and nonlinear reactive transport to be simulated.

Asymptotic behavior of a random oscillating nonlinear reactive transport in thin turbulent layers

We consider a nonlinear reactive transport of a scalar field in a domain composed of a bounded region lined by turbulent thin layers. We suppose that these layers display a cellular microstructure with thin varying thickness. We suppose that, within the layers, the diffusion, reaction, and velocity coefficients admit random large‐scale structures. Under mixing properties of the diffusion, reaction, and velocity processes, we study the asymptotic behavior of the problem with respect to a vanishing parameter describing the thickness of the layers. We derive the effective flux boundary condition on the surface of the fixed region which coincides with the lower surface of the layers. This flux boundary condition consists in a generalized nonlinear Robin condition involving some nonlocal random effects which emerge asymptotically due to the random fluctuations of the reactive transport parameters in the layers.

Upscaling non-linear reactive transport in correlated velocity fields

While commonly non-local transport models have been shown to reproduce breakthrough curves resulting from transport in heterogeneous aquifers successfully, open questions include the formal link between the upscaled governing equations and the sub-scale heterogeneity, and the ability to account for the effect of heterogeneity on effective chemical reaction rates in the presence of non-linear multi-component reactions. Time-domain random walk approaches based on velocity Markov models provide a framework to resolve these issues by incorporating the spatial correlation of velocity of a solute particle in consecutive locations along its trajectory. These approaches often rely on particle tracking approaches, which, however, can be computationally burdensome especially when non-linear reactions are sought to be modeled. In this paper, an integro-differential equation is proposed for upscaling multi-component reactive transport in heterogeneous media that relies on copulas for representing the velocity correlation structure. For this purpose, we express concentration or flux of solutes as a distribution over their velocity. We then derive the integro-differential equation that governs the evolution of concentration distribution over a quantity defined as velocity-rank. In this way, the spatial evolution of breakthrough curves away from the source is predicted based on ergodic cross-sectional velocity distributions and a parameterized copula function which expresses the correlation between velocity-ranks of a solute particle along its trajectory. We demonstrate the validity of the proposed model by comparing breakthrough curves for conservative and non-linearly reacting solutes based on realizations of hydraulic conductivity fields to the results of the upscaled model.

Novel applications of intelligent computing paradigms for the analysis of nonlinear reactive transport model of the fluid in soft tissues and microvessels

This article presents a methodology to solve a one-dimensional steady-state nonlinear reactive transport model (RTM) that is meant for fluid and solute transport model of soft tissues and microvessels. The methodology integrates the artificial neural network (ANN), genetic algorithms (GAs), and pattern search (PS) aided by active-set technique (AST) and interior-point technique (IPT). The RTM is represented with nonlinear second-order system based on the boundary value problem of ordinary differential equation. The ANN modeling is used for governing expression of RTM to form a fitness function in mean square sense, and optimization solvers based on the GA, PS, GA-AST, GA-IPT, PS-AST, PS-IPT are used for viable learning of weights. Proposed techniques are applied to different nonlinear RTMs based on variation in the characteristic reaction rate and half-saturation concentration. The proposed stochastic numerical solutions are compared with state-of-the-art solvers in order to check the accuracy and convergence based on sufficient large multiple runs of the algorithms.

Multiple solutions of a nonlinear reactive transport model using least square pseudo-spectral collocation method

The recognition and the calculation of all branches of solutions of the nonlinear boundary value problems is difficult obviously. The complexity of this issue goes back to the being nonlinearity of the problem. Regarding this matter, this paper considers steady state reactive transport model which does not have exact closed-form solution and discovers existence of dual or triple solutions in some cases using a new hybrid method based on pseudo-spectral collocation in the sense of least square method. Furthermore, the method usages Picard iteration and Newton method to treat nonlinear term in order to obtain unique and multiple solutions of the problem, respectively.

A similarity solution for reaction front propagation in a fracture-matrix system.

We propose a new composite similarity variable, based on which a similarity solution is derived for reaction front propagation in fracture-matrix systems. The similarity solution neglects diffusion/dispersion within the fracture and assumes the existence of a sharp reaction front in the rock matrix. The reaction front location in the rock matrix is shown to follow a linear decrease with distance along the fracture. The reaction front propagation along the fracture is shown to scale like diffusion (i.e. as the square root of time). The similarity solution using the composite similarity variable appears to be applicable to a broad class of reactive transport problems involving mineral reactions in fracture-matrix systems. It also reproduces the solutions for non-reactive solute and heat transport when diffusion/dispersion/conduction are neglected in the fracture. We compared our similarity solution against numerical simulations for nonlinear reactive transport of an aqueous species with a mineral in the rock matrix. The similarity solutions agree very well with the numerical solutions, especially at later times when diffusion limitations are more pronounced.This article is part of the themed issue 'Energy and the subsurface'.

Exposure-time based modeling of nonlinear reactive transport in porous media subject to physical and geochemical heterogeneity

Transport of reactive solutes in groundwater is affected by physical and chemical heterogeneity of the porous medium, leading to complex spatio-temporal patterns of concentrations and reaction rates. For certain cases of bioreactive transport, it could be shown that the concentrations of reactive constituents in multi-dimensional domains are approximately aligned with isochrones, that is, lines of identical travel time, provided that the chemical properties of the matrix are uniform. We extend this concept to combined physical and chemical heterogeneity by additionally considering the time that a water parcel has been exposed to reactive materials, the so-called exposure time. We simulate bioreactive transport in a one-dimensional domain as function of time and exposure time, rather than space. Subsequently, we map the concentrations to multi-dimensional heterogeneous domains by means of the mean exposure time at each location in the multi-dimensional domain. Differences in travel and exposure time at a given location are accounted for as time difference. This approximation simplifies reactive-transport simulations significantly under conditions of steady-state flow when reactions are restricted to specific locations. It is not expected to be exact in realistic applications because the underlying assumption, such as neglecting transverse mixing altogether, may not hold. We quantify the error introduced by the approximation for the hypothetical case of a two-dimensional, binary aquifer made of highly-permeable, non-reactive and low-permeable, reactive materials releasing dissolved organic matter acting as electron donor for aerobic respiration and denitrification. The kinetically controlled reactions are catalyzed by two non-competitive bacteria populations, enabling microbial growth. Even though the initial biomass concentrations were uniform, the interplay between transport, non-uniform electron-donor supply, and bio-reactions led to distinct spatial patterns of the two types of biomass at late times. Results obtained by mapping the exposure-time based results to the two-dimensional domain are compared with simulations based on the two-dimensional, spatially explicit advection-dispersion-reaction equation. Once quasi-steady state has been reached, we find a good agreement in terms of the chemical-compound concentrations between the two approaches inside the reactive zones, whereas the exposure-time based model is not able to capture reactions occurring in the zones with zero electron-donor release. We conclude that exposure-time models provide good approximations of nonlinear bio-reactive transport when transverse mixing is not the overall controlling process and all reactions are essentially restricted to distinct reactive zones.

Modeling and analysis of reactive solute transport in deformable channels with wall adsorption–desorption

We show well posedness for a model of nonlinear reactive transport of chemical in a deformable channel. The channel walls deform due to fluid–structure interaction between an unsteady flow of an incompressible, viscous fluid inside the channel and elastic channel walls. Chemical solutes, which are dissolved in the viscous, incompressible fluid, satisfy a convection–diffusion equation in the bulk fluid, while on the deforming walls, the solutes undergo nonlinear adsorption–desorption physico‐chemical reactions. The problem addresses scenarios that arise, for example, in studies of drug transport in blood vessels. We show the existence of a unique weak solution with solute concentrations that are non‐negative for all times. The analysis of the problem is carried out in the context of semi‐linear parabolic PDEs on moving domains. The arbitrary Lagrangian–Eulerian approach is used to address the domain movement, and the Galerkin method with the Picard–Lindelöf theorem is used to prove existence and uniqueness of approximate solutions. Energy estimates combined with the compactness arguments based on the Aubin–Lions lemma are used to prove convergence of the approximating sequences to the unique weak solution of the problem. It is shown that the solution satisfies the positivity property, that is, that the density of the solute remains non‐negative at all times, as long as the prescribed fluid domain motion is ‘reasonable’. This is the first well‐posedness result for reactive transport problems defined on moving domains of this type. Copyright © 2015 John Wiley &amp; Sons, Ltd.

Upscaling Diffusion and Nonlinear Reactive Mass Transport in Homogeneous Porous Media

In this work, we revisit the upscaling process of diffusive mass transfer of a solute undergoing a homogeneous reaction in porous media using the method of volume averaging. For linear reaction rate kinetics, the upscaled model exhibits a vis-a-vis correspondence with the mass transfer governing equation at the microscale. When nonlinear reactions are present, other methods must be adopted to upscale the nonlinear term. In this work, we explore a linearization approach for the purpose of solving the associated closure problem. For large rates of nonlinear reaction relative to diffusion, the effective diffusion tensor is shown to be a function of the reaction rate, and this dependence is illustrated by both numerical and analytical means. This approach leads to a macroscale model that also has a similar structure as the microscale counterpart. The necessary conditions for the vis-a-vis correspondence are clearly identified. The validation of the macroscale model is carried out by comparison with pore-scale simulations of the microscale transport process. The predictions of both concentration profiles and effectiveness factors were found to be in acceptable agreement. In an appendix, we also briefly discuss an integral formulation of the nonlinear problem that may be useful in developing more accurate results for the upscaled transport and reaction equations; this approach requires computing the Green function corresponding to the linear transport problem.

On the multiplicity of solutions of the nonlinear reactive transport model

The generalization of the nonlinear reaction–diffusion model in porous catalysts the so called one dimensional steady state reactive transport model is revisited. This model, which originates also in fluid and solute transport in soft tissues and microvessels, has been recently given analytical solution in terms of Taylor’s series for different families of reaction terms. This article considers the mentioned model without advective transport in the case of including Michaelis–Menten reaction term and shows that it is exactly solvable and furthermore, gives analytical exact solution in the implicit form for further physical interpretation. It is also revealed that the problem may admit unique or dual or even more triple solutions in some domains for the parameters of the model.

A new approach for a class of nonlinear boundary value problems with multiple solutions

In this paper, an approach based on the variational iteration method (VIM) is proposed with an auxiliary parameter to predict the multiplicity of the solutions of homogeneous nonlinear ordinary differential equations with boundary conditions. The proposed approach is capable to predict and calculate all branches of the solutions simultaneously. Four practical problems are chosen to show the efficiency and importance of the proposed method. The proposed approach successfully detects multiple solutions to Bratu’s problem, the model of mixed convection flows in a vertical channel, the nonlinear model of diffusion and reaction in porous catalysts and the nonlinear reactive transport model.

Reactive transport of uranium with bacteria in fractured rock: Model development and sensitivity analysis

A numerical model for the reactive transport of uranium and bacteria in fractured rock was newly developed. The conceptual model consists of four phases (fracture, fracture surface, matrix pore, and matrix solid) and eight constituents (solutes in the fracture, on the fracture surface, on mobile bacteria, on immobile bacteria, in the rock matrix pores and on the rock matrix solids, and bacteria in the fracture and on the fracture surface). In addition to the kinetic sorption/desorption of uranium and bacteria, uranium reduction reaction accompanying with bacteria growth was considered in the reactive transport. The non-linear reactive transport equations were numerically solved using the symmetric sequential iterative scheme of the operator-splitting method. The transport and kinetic reaction modules in the developed model were separately verified, and the results were reasonably acceptable. From the sensitivity analysis, the uranium transport was generally more sensitive to the sorption rate rather than desorption rate of U(VI). Considering a uranium reduction reaction, bacteria could considerably retard the uranium transport no matter the uranium sorption/desorption rates. As the affinity of U(VI) onto the bacteria becomes higher than that onto a rock fracture surface, a biofilm effect, rather than a colloidal effect, of the bacteria becomes more influential on the uranium transport.

Unique and multiple PHAM series solutions of a class of nonlinear reactive transport model

The purpose of this paper is to visit a class of nonlinear reactive transport model in the case including advective and diffusive transport with the Michaelis-Menten reaction term. We apply the method so-called predictor homotopy analysis method (PHAM) which has been recently proposed to predict multiplicity of solutions of nonlinear BVPs. Consequently two consequential matters are indicated which confirms the authority of PHAM to identify multiple solutions: (i) The Talylor series solutions are improved by the so-called convergence-controller parameter (ii) The possibility of existence of multiple solutions is discovered in some cases for the model.

An analytical method to solve a general class of nonlinear reactive transport models

Abstract Three recent papers published in Chemical Engineering Journal studied the solution of a model of diffusion and nonlinear reaction using three different methods. Two of these studies obtained series solutions using specialized mathematical methods, known as the Adomian decomposition method and the homotopy analysis method. Subsequently it was shown that the solution of the same particular model could be written in terms of a transcendental function called Gauss’ hypergeometric function. These three previous approaches focused on one particular reactive transport model. This particular model ignored advective transport and considered one specific reaction term, R( C) = ϕ 2 C n , only. Here we generalize these previous approaches and develop an exact analytical solution for a general class of steady state reactive transport models that incorporate (i) combined advective and diffusive transport and (ii) any sufficiently differentiable reaction term R( C). The new solution is a convergent Maclaurin series. The Maclaurin series solution can be derived without any specialized mathematical methods nor does it necessarily involve the computation of any transcendental function. Applying the Maclaurin series solution to certain case studies shows that the previously published solutions are particular cases of the more general solution outlined here. We also demonstrate the accuracy of the Maclaurin series solution by comparing with numerical solutions for particular cases.

Optimal reconstruction of concentrations, gradients and reaction rates from particle distributions

Random walk particle tracking methodologies to simulate solute transport of conservative species constitute an attractive alternative for their computational efficiency and absence of numerical dispersion. Yet, problems stemming from the reconstruction of concentrations from particle distributions have typically prevented its use in reactive transport problems. The numerical problem mainly arises from the need to first reconstruct the concentrations of species/components from a discrete number of particles, which is an error prone process, and then computing a spatial functional of the concentrations and/or its derivatives (either spatial or temporal). Errors are then propagated, so that common strategies to reconstruct this functional require an unfeasible amount of particles when dealing with nonlinear reactive transport problems. In this context, this article presents a methodology to directly reconstruct this functional based on kernel density estimators. The methodology mitigates the error propagation in the evaluation of the functional by avoiding the prior estimation of the actual concentrations of species. The multivariate kernel associated with the corresponding functional depends on the size of the support volume, which defines the area over which a given particle can influence the functional. The shape of the kernel functions and the size of the support volume determines the degree of smoothing, which is optimized to obtain the best unbiased predictor of the functional using an iterative plug-in support volume selector. We applied the methodology to directly reconstruct the reaction rates of a precipitation/dissolution problem involving the mixing of two different waters carrying two aqueous species in chemical equilibrium and moving through a randomly heterogeneous porous medium.

Approaches to modelling coupled flow and reaction in a 2D cementation experiment

Porosity evolution at reactive interfaces is a key process that governs the evolution and performances of many engineered systems that have important applications in earth and environmental sciences. This is the case, for example, at the interface between cement structures and clays in deep geological nuclear waste disposals. Although in a different transport regime, similar questions arise for permeable reactive barriers used for biogeochemical remediation in surface environments. The COMEDIE project aims at investigating the coupling between transport, hydrodynamics and chemistry when significant variations of porosity occur. The present work focuses on a numerical benchmark used as a design exercise for the future COMEDIE-2D experiment. The use of reactive transport simulation tools like Hytec and Crunch provides predictions of the physico-chemical evolutions that are expected during the future experiments in laboratory. Focus is given in this paper on the evolution during the simulated experiment of precipitate, permeability and porosity fields. A first case is considered in which the porosity is constant. Results obtained with Crunch and Hytec are in relatively good agreement. Differences are attributable to the models of reactive surface area taken into account for dissolution/precipitation processes. Crunch and Hytec simulations taking into account porosity variations are then presented and compared. Results given by the two codes are in qualitative agreement, with differences attributable in part to the models of reactive surface area for dissolution/precipitation processes. As a consequence, the localization of secondary precipitates predicted by Crunch leads to lower local porosities than for predictions obtained by Hytec and thus to a stronger coupling between flow and chemistry. This benchmark highlights the importance of the surface area model employed to describe systems in which strong porosity variations occur as a result of dissolution/precipitation. The simulation of highly non-linear reactive transport systems is also shown to be partly dependent on specific numerical approaches.