Reactive Transport Equations Research Articles

Application of Analytical Solutions of the Reactive Transport Equation for Underground Methanation Reactors

Application of Analytical Solutions of the Reactive Transport Equation for Underground Methanation Reactors

Porous-DeepONet: Learning the Solution Operators of Parametric Reactive Transport Equations in Porous Media

Reactive transport equations in porous media are critical in various scientific and engineering disciplines, but solving these equations can be computationally expensive when exploring different scenarios, such as varying porous structures and initial or boundary conditions. The deep operator network (DeepONet) has emerged as a popular deep learning framework for solving parametric partial differential equations. However, applying the DeepONet to porous media presents significant challenges due to its limited capability to extract representative features from intricate structures. To address this issue, we propose the Porous-DeepONet, a simple yet highly effective extension of the DeepONet framework that leverages convolutional neural networks (CNNs) to learn the solution operators of parametric reactive transport equations in porous media. By incorporating CNNs, we can effectively capture the intricate features of porous media, enabling accurate and efficient learning of the solution operators. We demonstrate the effectiveness of the Porous-DeepONet in accurately and rapidly learning the solution operators of parametric reactive transport equations with various boundary conditions, multiple phases, and multi-physical fields through five examples. This approach offers significant computational savings, potentially reducing the computation time by 50–1000 times compared with the finite-element method. Our work may provide a robust alternative for solving parametric reactive transport equations in porous media, paving the way for exploring complex phenomena in porous media.

On the Flow of CO2-Saturated Water in a Cement Fracture

Cement fractures represent preferential leakage pathways in abandoned wells upon exposure to a CO2-rich fluid. Understanding fracture alteration resulting from geochemical reactions is critical for assessing well integrity in CO2 storage. This paper describes a mathematical model used to investigate the physical and the chemical changes in cement properties when CO2-saturated water is injected into a wellbore. This study examines the flow of a solution of CO2-saturated water in a two-dimensional fractured cement. In this approach, a micro-continuum equation based on the Darcy–Brinkman–Stokes (DBS) equation is used as the momentum balance equation; in addition, reactive transport equations are used to study the coupled processes of reactant transport and geochemical reactions, and the model for cement porosity alteration and fracture enhancement. This paper focuses on the effects of cement porosity, fracture aperture size, and surface roughness. Mineral dissolution and precipitation mechanisms are also considered. Our simulations show that smaller initial fracture apertures tend to a high mineral precipitation self-sealing. However, a complete sealing of the fracture is not observed due to the continuous flow of CO2-saturated water. The calcite precipitation mechanism of a rough fracture (random zigzag shape) differs from that of a smooth/flat fracture surface.

A computational framework based on explicit local chemical equilibrium for coupled chemo-hydro-mechanical effects on fluid-infiltrating porous media

Conventionally, a finite element framework for coupled hydro-mechanical-chemical processes in porous media adopts simplified chemical reactions, which consider superficial chemical effects of complicated multiphysics mechanisms. This paper presents a novel computational framework for geochemistry of reactive multi-component systems to explicitly capture the mineral dissolution/precipitation of saturated porous media. The resultant of this framework can be applied in oil recovery engineering and used to simulate the production scenarios. The model is also fundamental for studying on creep behaviors from the view of the coupling effects between hydro-geochemistry. In particular, a complete chemistry system of ▪ is considered, including intra-aqueous chemical reactions and kinetic rate-limiting chemical reactions. Besides, an algorithm is proposed such that the complete chemistry system is scalable for various chemical reactions and is easily incorporated into existing finite element frameworks for hydro-mechanical processes of porous media. This complete chemistry system is implemented to establish a local chemical equilibrium. The geochemical reactive transport equations are then sequentially coupled with the balance laws such that coupling among fluid advection-diffusion, solid deformation, chemical concentration advection-diffusion, intra-aqueous chemical reaction, and kinetic rate-limiting chemical reaction can be simulated. Phenomenological relations for porosity and permeability changes with mineral saturation in solutions are employed. We first verify the proposed framework to capture the advection-diffusion of different chemical species under steady-state and transient conditions. Then a consolidation problem under the chemical reactions is presented to further demonstrate the capabilities of the proposed computational framework.

A Compositional Global Implicit Approach for Modeling Coupled Multicomponent Reactive Transport

AbstractReactive transport modeling has become widely used to help improve understanding of hydrogeochemical processes from the pore scale to the watershed scale. In recent years, the scope of reactive transport applications has increased toward a higher level of complexity and process coupling. For example, the production and consumption of water as well as porosity evolution associated with the dissolution and precipitation of hydrated minerals can impact system evolution. Waste rock weathering, carbon sequestration, or the degradation of engineered barriers in radioactive waste repositories all constitute applications in which geochemistry and hydrodynamics can strongly influence each other. For these purposes, the traditional formulation of reactive transport simulators, which decouples groundwater flow and reactive transport processes, is limited. We present a global implicit compositional approach, which integrates the flow processes directly into the reactive transport and geochemical framework. This approach solves the flow field implicitly with the reactive transport equations, simultaneously accounting for water consumption and production due to geochemical reactions. Applications show that the model allows tackling complex reactive transport problems while accounting for intra‐aqueous reactions, redox reactions, and reactions involving mass transfer with the gas and solid phases. The presented simulations also demonstrate that the compositional and traditional approaches yield similar results for complex geochemical systems with relatively low reactivity.

A staggered projection scheme for viscoelastic flows

We develop a numerical scheme for the flow of viscoelastic fluids, including the Oldroyd-B and FENE-CR constitutive models. The space discretization is staggered, using either the Marker-And-Cell (MAC) scheme for structured nonuniform grids, or the Rannacher and Turek (RT) nonconforming low-order finite element approximation for general quadrangular or hexahedral meshes. The time discretization uses a fractional-step algorithm where the solution of the Navier–Stokes equations is first obtained by a projection method and then the transport-reaction equation for the conformation tensor is solved by a finite volume scheme. In order to obtain consistency, the space discretization of the divergence of the elastic part of the stress tensor in the momentum balance equation is derived using a weak form of the MAC scheme. For stability and accuracy purposes, the solution of the transport-reaction equation for the conformation tensor is split into pure convection steps, with a change of variable to the log-conformation tensor, and a reaction step, which consists in solving one ODE per cell via an Euler scheme with local sub-cycling. Numerical computations for the flow in the lid-driven cavity at Weissenberg numbers above one and the flow around a confined cylinder confirm the efficiency of the scheme.

The effects of mechanical instability on PDGF mediated inflammatory response at early stage of fracture healing under diabetic condition

Background and ObjectiveMechanical stability plays an important role in fracture healing process. Excessive interfragmentary movement will continuously damage the tissue and newly formed capillaries at the fracture site, which leads to overproduction of platelet-derived growth factor (PDGF) that attracts more macrophages into fracture callus, ultimately persistent and enhanced inflammatory response happens. For diabetic condition, the impact of mechanical instability of fracture site on inflammatory response could be further compliciated and the relevant research in this field is relatively limited. MethodsBuilding on previous experimental studies, this study presents a numerical model consisting of a system of reactive-transport equations representing the transport as well as interactions of different cells and cytokines within the fracture callus. The model is initially validated by available experimental data, and then implemented to investigate the role of mechanical stability of fracture site in inflammatory response during early stage of healing. It is assumed that there is an increased release of PDGF due to the rupture of blood vessels resulting from mechanical instability, which leads to increased production of inflammatory cytokines (i.e., TNF-α). The bone healing process under three different conditions were investigated, i.e., mechanically stable condition with normal inflammatory response (Control, Case 1), mechanically unstable condition with normal inflammatory response (Case 2) and mechanically unstable condition with diabetes (Case 3). ResultsMechanical instability can promote the macrophage infiltration and thus induce an enhanced and prolonged inflammatory response, which could impede the MSCs proliferation during the early fracture healing stage (e.g., compared with the control condition, the MSCs concentration in unstable fracture with normal inflammatory response can be reduced by 3.2% and 5.2% on day 2 and day 10 post-fracture, respectively). Under diabetic condition, the mechanical instability of fracture site could lead to a significant increase of TNF-α concentration in fracture callus (Case 3) in comparison to control (Case 1) (e.g., three-fold increase in TNF-α concentration compared to control). In addition, the results show that the mechanical instability affects the cell differentiation and proliferation in fracture callus in a spatially dependent manner, e.g., for diabetic fracture patients, the mechanical instability could potentially decrease the concentration of MSCs, osteoblasts and chondrocytes by around 39%, 30% and 29% in cortical callus, respectively, in comparison to control. ConclusionThe mechanical instability together with diabetic condition can significantly affect the natural resolution of inflammation during early stage of healing by turning acute inflammation into chronic inflammation which is characterized by a continuously upregulated TNF-α pathway.

A modified Darcy’s law for viscoelastic flows of highly dilute polymer solutions through porous media

Viscoelastic flows of polymer solutions in complex geometries can generate a strong localization of stress within small regions of the fluid and the formation of birefringent strands. In porous media , these localized structures of stress drive preferential flow paths and increase global dissipation. Modeling the impact of such effects at Darcy or larger scales is a daunting task—one of the reasons being the lack of approaches using homogenization theories to help figure out both the correct form of the averaged transport equations and the relevant set of effective parameters. Here we homogenize the incompressible Oldroyd-B equations at zero Reynolds number to obtain a Darcy scale model that captures the effect of localized polymeric stress. This model consists of an advection—reaction transport equation for the average conformation tensor along with a form of Darcy’s law that contains an additional forcing term associated with structures of localized stress. The derivation is based upon a limit of high dilution, a regime where the Oldroyd-B model can be transformed into a sequence of linear problems using asymptotic developments. We validate our approach in test cases corresponding to flows in a channel and through arrays of circles. Besides providing a new model for viscoelastic flows in porous media , our work also shows that modeling viscoelastic flows through porous media is not simply a matter of determining an apparent permeability tensor—the homogenized model cannot be easily reduced to a simple form of Darcy’s law—but rather requires the development of specific homogenized models that capture the coupling between the transport of the polymeric stress and momentum. • The Oldroyd-B equations are homogenized using asymptotics and volume averaging. • We derive a Darcy-scale model for viscoelastic flow in porous media. • Our model couples a form of Darcy’s law with transport of the conformation tensor. • Our model captures the effect of stress localization at pore-scale on the average flow. • The notion of effective permeability alone seems to be insufficient for viscoelastic flows.

Novel local fluctuation-preserving upscaling techniques for heterogeneous reaction-transport equation in porous media

A body of research in the field of lithium-ion batteries is dedicated to the investigation of the influence on their behavior by the complex microstructure of their porous electrodes. Experimental and numerical studies reported the distinctly localized spatial fluctuations of overpotential and lithium ion concentration which are not observed in the simulation results based on a widely used lithium-ion cell’s DFN (Doyle-Fuller-Newman) model. Using a combination of perturbation technique and asymptotic analysis, we rigorously derived the analytical results that explain the fluctuation dynamics that was reported earlier in the microstructure-resolving simulations; importantly, it was theoretically demonstrated that the localized fluctuations generally do not disappear in the homogenization limit which is usually used to justify the use of DFN. We reported new numerical results that proves that the approximations made in our theoretical analysis are indeed applicable to the physical and chemical parameters corresponding to the real lithium-ion cells. Our ansatz may be of use for accurate derivation of DFN-like electrochemistry mathematical models beyond the narrow scope of this report, in particular for the models that accurately account for the cell degradation processes which are sensitive to the particle-scale local environment in the electrode.

Coreflooding and Steady-State Principles Determine Reaction Kinetics

This article, written by JPT Technology Editor Chris Carpenter, contains highlights of paper SPE 209380, “Reaction Kinetics Determined From Coreflooding and Steady-State Principles for Stevns Klint and Kansas Chalk Injected With MgCl2 Brine at Reservoir Temperature,” by Pål Østebø Andersen, Reidar Inge Korsnes, and Andre Tvedt Olsen, University of Stavanger, et al. The paper has not been peer reviewed. A methodology is presented for determining reaction kinetics from coreflooding. A core is flooded with reactive brine at different compositions with injection rates varied systematically. Each combination is performed until steady state, when effluent concentrations no longer change significantly with time. The authors also propose shut-in tests where brine reacts statically with the core for a defined period and then is flushed out. The residence time and produced brine composition is compared with the flooding experiments. This design allows characterization of the reaction kinetics from a single core. The steady-state method allows computationally efficient matching even with complex reaction kinetics. Introduction The complete paper presents a methodology to interpret reactive flow experiments. The main principle is to focus on the steady-state data collected when effluent concentrations do not change further with time for given injected composition, injection rate, core properties, temperature, and pressure. Such measurements depend only on the dynamic equilibrium between reactions, advection, and to some extent dispersion, while transient events and mechanisms can be ignored. The authors hypothesize that, by measuring such data under different injected compositions, injected rates, and temperatures, characterization of reaction kinetics is possible with great detail. This is demonstrated by performance of reactive flow experiments on chalk cores at varied conditions with the purpose of collecting steady-state data. A key advantage from the interpretation perspective is that complicated reactive transport equations can be simplified by ignoring the changes in time and only simulating and matching the steady state. The experimental setup is detailed in the complete paper, as is a general description of advection-dispersion-reaction systems, a description of steady states in an advection-dispersion-reaction system and an advection-reaction system, and a theoretical description of a magnesium chloride (MgCl2)/calcite system.

Total Lagrange implementation of a finite-deformation continuum dislocation dynamics model of mesoscale plasticity

We present a computational algorithm for solving the recently developed finite-deformation continuum dislocation dynamics theory of mesoscale plastic deformation of single crystals (Starkey et al., 2020). This CDD theory is based on a vector density representation of dislocations governed by curl-type transport-reaction equations subjected to the divergence-free constraint of the appropriate dislocation density. These density evolution equations are to be solved simultaneously with the finite-deformation crystal mechanics. Specifically, our algorithm aims to solve the referential form of the governing equations for a representative volume element (RVE) subject to remote uniform loading. The mechanical fields at the mesoscale are thus split into RVE-averages plus fluctuating components and treated using a strain-driven homogenization scheme. A virtual work-based total Lagrange formulation was used to discretize the governing mechanics equations. A first-order system least squares finite element formulation was used to solve the transport equations. The two schemes are coupled in a staggered fashion. As a part of the crystal mechanics discretization, we derive a consistent tangent modulus and show that the stress update for this model is both linear and global. This linear stress update comes at the cost of solving the dislocation transport equations at every time step to update the plastic distortion caused by dislocation motion. Several test problems are given, demonstrating the ability of the discretization scheme to solve the problem, including the expansion of dislocation loop-like bundles under constant velocity and driven by a mean deformation gradient, dynamic recovery of two oppositely oriented tilt boundaries in a single crystal, and a uniaxial tension test of a single crystal with one slip system activated. In most of these examples, the evolution behavior of the dislocations in the finite deformation regime is demonstrated.

Novel Local Fluctuation-Preserving Upscaling Techniques for Heterogeneous Reaction-Transport Equation in Porous Media

Novel Local Fluctuation-Preserving Upscaling Techniques for Heterogeneous Reaction-Transport Equation in Porous Media

GeoChemFoam: Direct Modelling of Multiphase Reactive Transport in Real Pore Geometries with Equilibrium Reactions

GeoChemFoam is an open-source OpenFOAM-based toolbox that includes a range of additional packages that solve various flow processes from multiphase transport with interface transfer, to single-phase flow in multiscale porous media, to reactive transport with mineral dissolution. In this paper, we present a novel multiphase reactive transport solver for simulations on complex pore geometries, including microfluidic devices and micro-CT images, and its implementation in GeoChemFoam. The geochemical model includes bulk and surface equilibrium reactions. Multiphase flow is solved using the Volume-Of-Fluid method, and the transport of species is solved using the continuous species transfer method. The reactive transport equations are solved using a sequential operator splitting method, with the transport step solved using GeoChemFoam, and the reaction step solved using Phreeqc, the US geological survey’s geochemical software. The model and its implementation are validated by comparison with analytical solutions in 1D and 2D geometries. We then simulate multiphase reactive transport in two test pore geometries: a 3D pore cavity and a 3D micro-CT image of Bentheimer sandstone. In each case, we show the pore-scale simulation results can be used to develop upscaled models that are significantly more accurate than standard macro-scale equilibrium models.

On-the-fly reduced order modeling of passive and reactive species via time-dependent manifolds

One of the principal barriers in developing accurate and tractable predictive models in turbulent flows with a large number of species is to track every species by solving a separate transport equation, which can be computationally impracticable. In this paper, we present an on-the-fly reduced order modeling of reactive as well as passive transport equations to reduce the computational cost. The presented approach seeks a low-rank decomposition of the species to three time-dependent components: (i) a set of orthonormal spatial modes, (ii) a low-rank factorization of the instantaneous species correlation matrix, and (iii) a set of orthonormal species modes, which represents a low-dimensional time-dependent manifold. Our approach bypasses the need to solve the full-dimensional species to generate high-fidelity data — as it is commonly performed in data-driven dimension reduction techniques such as the principle component analysis. Instead, the low-rank components are directly extracted from the species transport equation. The evolution equations for the three components are obtained from optimality conditions of a variational principle. The time-dependence of the three components enables an on-the-fly adaptation of the low-rank decomposition to transient changes in the species. Several demonstration cases of reduced order modeling of passive and reactive transport equations are presented.

On the computational solution of vector-density based continuum dislocation dynamics models: A comparison of two plastic distortion and stress update algorithms

Continuum dislocation dynamics models of mesoscale plasticity consist of dislocation transport-reaction equations coupled with crystal mechanics equations. The coupling between these two sets of equations is such that dislocation transport gives rise to the evolution of plastic distortion (strain), while the evolution of the latter fixes the stress from which the dislocation velocity field is found via a mobility law. Earlier solutions of these equations employed a staggered solution scheme for the two sets of equations in which the plastic distortion was updated via time integration of its rate, as found from Orowan's law. In this work, we show that such a direct time integration scheme can suffer from accumulation of numerical errors. We introduce an alternative scheme based on field dislocation mechanics that ensures consistency between the plastic distortion and the dislocation content in the crystal. The new scheme is based on calculating the compatible and incompatible parts of the plastic distortion separately, and the incompatible part is calculated from the current dislocation density field. Stress field and dislocation transport calculations were implemented within a finite element based discretization of the governing equations, with the crystal mechanics part solved by a conventional Galerkin method and the dislocation transport equations by the least squares method. A simple test was first performed to show the accuracy of the two schemes for updating the plastic distortion, which shows that the solution method based on field dislocation mechanics is more accurate. This method then was used to simulate an austenitic steel crystal under uniaxial loading and multiple slip conditions. By considering dislocation interactions caused by junctions, a hardening rate similar to discrete dislocation dynamics simulation results was obtained. The simulations show that dislocations exhibit some self-organized structures as the strain is increased.

Reaction-diffusion and reaction-subdiffusion equations on arbitrarily evolving domains.

Reaction-diffusion equations are widely used as the governing evolution equations for modeling many physical, chemical, and biological processes. Here we derive reaction-diffusion equations to model transport with reactions on a one-dimensional domain that is evolving. The model equations, which have been derived from generalized continuous time random walks, can incorporate complexities such as subdiffusive transport and inhomogeneous domain stretching and shrinking. Inhomogeneously growing domains are frequently encountered in biological phenomena involving stochastic transport, such as tumor growth and morphogen gradient formation. A method for constructing analytic expressions for short-time moments of the position of the particles is developed and moments calculated from this approach are shown to compare favorably with results from random walk simulations and numerical integration of the reaction transport equation. The results show the important role played by the initial condition. In particular, it strongly affects the time dependence of the moments in the short-time regime by introducing additional drift and diffusion terms. We also discuss how our reaction transport equation could be applied to study the spreading of a population on an evolving interface. From a more general perspective, our findings help to mitigate the scarcity of analytic results for reaction-diffusion problems in geometries displaying nonuniform growth. They are also expected to pave the way for further results, including the treatment of first-passage problems associated with encounter-controlled reactions in such domains.

The weak Galerkin finite element method for the transport-reaction equation

We present and analyze a weak Galerkin finite element method for solving the transport-reaction equation in d space dimensions. This method is highly flexible by allowing the use of discontinuous finite element on general meshes consisting of arbitrary polygon/polyhedra. We derive the L2-error estimate of O(hk+12)-order for the discrete solution when the kth-order polynomials are used for k≥0. Moreover, for a special class of meshes, we also obtain the optimal error estimate of O(hk+1)-order in the L2-norm. A derivative recovery formula is presented to approximate the convection directional derivative and the corresponding superconvergence estimate is given. Numerical examples on compatible and non-compatible meshes are provided to show the effectiveness of this weak Galerkin method.

A Numerical Simulator for Modeling the Coupling Processes of Subsurface Fluid Flow and Reactive Transport Processes in Fractured Carbonate Rocks

Water–rock interactions can alter rock properties through chemical reactions during subsurface transport processes like geological CO2 sequestration (GCS), matrix acidizing, and waterflooding in carbonate formations. Dynamic changes in rock properties cause a failure of waterflooding and GCS and could also dramatically affect the efficiency of the acidizing. Efficient numerical simulations are thus essential to the optimized design of those subsurface processes. In this paper, we develop a three-dimensional (3D) numerical model for simulating the coupled processes of fluid flow and chemical reactions in fractured carbonate formations. In the proposed model, we employ the Stokes–Brinkman equation for momentum balance, which is a single-domain formulation for modeling fluid flow in fractured porous media. We then couple the Stokes–Brinkman equation with reactive-transport equations. The model can be formulated to describe linear as well as radial flow. We employ a decoupling procedure that sequentially solves the Stokes–Brinkman equation and the reactive transport equations. Numerical experiments show that the proposed method can model the coupled processes of fluid flow, solute transport, chemical reactions, and alterations of rock properties in both linear and radial flow scenarios. The rock heterogeneity and the mineral volume fractions are two important factors that significantly affect the structure of conductive channels.

A sequential non-iterative approach for modeling multi-ionic species reactive transport during localized corrosion

Multi-ionic reactive transport modeling can provide a better understanding of localized corrosion in iron and steel alloys. However, one-step numerical methods used to solve reactive transport equations in a fully-coupled manner may suffer from poor conditioning and numerical convergences issues. In this paper, a sequential non-iterative approach (SNIA) is developed to enable robust numerical simulation of multi-ionic reactive transport models for localized corrosion. The electro-diffusive mass transport equations are decoupled from the homogeneous equilibrium chemical reactions equations, and are solved sequentially without iteration, leading to a two-step numerical method. The nonlinear mass transport equations are discretized using the standard finite element method, linearized and solved using the Newton-Raphson method. To ensure the non-negativity of ionic concentrations, the nonlinear chemical reactions equations are solved using the Newton-Raphson method together with the under-relaxation technique. Numerical studies, including model comparison, parametric and experimental validation studies, are conducted to simulate localized crevice/cavity corrosion of generic carbon steel and iron-chromium binary alloys and to demonstrate the efficacy of the proposed approach.

Richardson diffusion in neurons.

The dynamics of an initial wave packet affected by random noise is considered in the framework of a comb model. The model is relevant to a diffusion problem in neurons where the transport of ions can be accelerated by an external random field due to synapse fluctuations. In the present specific case, it acts as boundary conditions, which lead to a reaction transport equation with multiplicative noise. The temporal behavior of the mean squared displacement is estimated analytically, and it is shown that the spreading of the initial wave packet corresponds to Richardson diffusion.