Multiscale reacting flows in complex microstructures

Abstract

To provide a multiscale reactive flow in complex microstructures.SOLUTION: Embodiments determine behavior of reactive flow systems. One such embodiment defines a plurality of models of the reactive flow system, wherein each defined model represents the reactive flow system at a respective scale. A velocity field for the reactive flow system is determined using a first model, at a first respective scale, of the defined plurality of models and a diffusivity for the reactive flow system is determined using a second model, at a second respective scale, of the defined plurality of models. In turn, a plurality of reaction parameters for the reactive flow system are defined. Then, behavior of the reactive flow system is automatically determined by using the determined velocity field, the determined diffusivity, and the defined plurality of reaction parameters as inputs to a reactive transport solver.SELECTED DRAWING: Figure 1

Description

[Technical Field]

[0001] This application relates to multi-scale reacting flows in complex microstructures. [Background technology]

[0002] (Related Applications) This application claims the benefit of U.S. Provisional Patent Application No. 63 / 487,509, filed February 28, 2023, the entire teachings of which are incorporated herein by reference.

[0003] Reactive flow modeling is a tool for analyzing processes involving fluid flow and chemical reactions, including subsurface carbon capture, groundwater remediation, enhanced oil recovery (EOR) through CO2 injection, in-situ mining through leaching, corrosion, and fouling of structural materials, recycling, and electrochemical systems (batteries, fuel cells, electrolyzers, etc.) for sustainable energy applications, among other examples. [Prior art documents] [Non-patent literature]

[0004] [Non-Patent Document 1] Abbott, JW, Hanke, F., 2022. Kinetically Corrected Monte Carlo-Molecular Dynamics Simulations of Solid Electrolyte Interphase Growth. J. Chem. Theory Comput. 18, 925-934. https: / / doi.org / 10.1021 / acs.jctc.1c00921 [Non-patent document 2] Abd, AS, Abushaikha, AS, 2021 Reactive transport in porous media: a review of recent mathematical efforts in modeling geochemical reactions in petroleum subsurface reservoirs. SN Appl. Sci. 3, 1-28. https: / / doi.org / 10.1007 / s42452-021-04396-9 [Non-patent document 3] Agmon, N., 1995 CHEMICAL PHYSICS The Grotthuss mechanism. Chem. Phys. Lett. 50, 456-462. [Non-patent document 4] Akkermans, RLC, Spenley, NA, Robertson, SH 2021. COMPASS III: automated fitting workflows and extension to ionic liquids. Molecular Simulation, 47, 540-551 [Non-patent document 5] Al-Khulaifi, Y., Lin, Q., Blunt, MJ, Bijeljic, B., 2017. Reaction Rates in Chemically Heterogeneous Rock: Coupled Impact of Structure and Flow Properties Studied by X-ray Microtomography. Environ. Sci. Technol. 51, 4108-4116. https: / / doi.org / 10.1021 / acs.est.6b06224 [Non-patent document 6] Allen, MP, Tildesley, DJ 1987. Computer simulation of Liquids;Clarendon Press,Oxford Science Publications [Non-Patent Document 7] Beckstein, O., Sansom, MSP, 2003. Liquid-vapor oscillations of water in hydrophobic nanopores. Proc. Natl. Acad. Sci. USA 100, 7063-7068. https: / / doi.org / 10.1073 / pnas.1136844100 [Non-patent document 8] Boek, ES, 2014. Molecular dynamics simulations of interlayer structure and mobility in hydrated Li-, Na- and K-montmorillonite clays. Mol. Phys. 112, 1472-1483. https: / / doi.org / 10.1080 / 00268976.2014.907630 [Non-Patent Document 9] Casewit, CJ, Colwell, KS, Rappe, AK, 1992. Application of a Universal Force Field to Main Group Compounds. J. Am. Chem. Soc. 114, 10046-10053. https: / / doi.org / 10.1021 / ja00051a042 [Non-Patent Document 10] Chatterjee A.,Dance T.,Mizukami F.,2005. Effects of Water on the Structure and Bonding of Resorcinol in the Interlayer of Montmorillonite Nanocomposite: A Periodic First Principle Study. Phys. Chem. B, 109, 15, 7306–7313 https: / / doi.org / 10.1021 / jp045775z

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[0005] Although reactive flow modeling tools exist, these tools can benefit from improvements, and embodiments provide such improved capabilities.

[0006] While reactive flow modeling has applications in many industries, much of the progress in this field has been made through the development of geochemical modeling in subsurface systems. Therefore, the exemplary embodiments disclosed herein illustrate a selected representative application in this class: reactive flow modeling of in-situ copper leaching. It should be noted, however, that the embodiments are not limited to simulating / modeling in-situ copper leaching (metals in general); rather, they can be used to analyze any reactive flow system. In-situ leaching involves the underground circulation of acid to dissolve target minerals and extract or produce the metal of interest (copper in this example). Modeling real-time changes in the composition of fluids and fluid-rock interfaces through reactive flow modeling facilitates process optimization by optimizing copper production from in-situ leaching and predicting the potential outcomes of alternative designs and operating conditions.

[0007] Depending on the scale of physics of interest, reactive flow modeling can involve (i) continuum-level models; 42,47 , and (ii) pore-scale model 28,29Two broad categories of models exist: pore-scale and non-pore-scale. At the pore scale, the microstructure is fully resolved, and effective species transport coupled with volumetric surface reactions is simulated directly within the microstructure, with material properties as input. At the continuum scale, the microstructure is not resolved, and the model requires input of specific properties such as effective transport, microstructure-limited reaction rates, material porosity, or the surface-to-volume ratio of the material. The use of three-dimensional (3D) microscopic imaging techniques, such as X-ray microtomography (microCT), as input to the actual pore-scale structure is possible but is not mainstream in reacting flow modeling. For real-world applications, both aspects, scales, and 3D imaging must be included in the model to generate accurate results without requiring experimental measurements or limiting simplified model assumptions. This lack of multiscale aspects in current solutions for reacting flow also involves smaller molecular-level modeling of material, reaction, and diffusion parameters. Another limitation of current reacting flow solvers relates to simultaneously describing multiphase flow (multiple fluid flows, such as oil, water, and gas).

[0008] In summary, existing pore-scale models do not consider flow through unresolved pores (e.g., multiscale flow simulations) and cannot accurately model multiphase flow. Existing continuum-scale models consider simplified flow and transport equations based on permeability and porosity. Therefore, there are no solutions for accurately resolving multiphase and multiscale reacting flow at the pore scale in conjunction with upscaling to simulate field-scale problems. Both pore-scale and continuum-scale models in the literature rely on open-source databases for material, reaction, and diffusion parameters. Finally, accurately considering secondary speciation of flow solutions after primary reactions is often common in continuum models, but is somewhat limited in pore-scale models. Note that "accounting for secondary speciation" can also refer to accurately modeling water chemistry or including homogeneous reactions in reacting flow models.

[0009] To accurately model reacting flow applications as described herein, embodiments implement a reacting flow model with a robust multiphase (coupled liquid and gas flow) and multiscale (flow through resolved and unresolved pores) flow simulator, and a multiscale (pore-scale and continuum-scale) reaction simulator. 3D imaging-based microstructural models are also used in embodiments to avoid simple approximations to property correlations. Furthermore, it is noted that reacting flow modeling utilizes many inputs from material, reaction, and diffusion parameters, which are not easily found in the literature and data repositories for all applications. To solve this problem, embodiments obtain chemical properties through the use of molecular modeling components that calculate chemical properties. Still further, the reacting flow modeling embodiments described herein include homogeneous reactions in addition to heterogeneous reactions in the reacting flow model.

[0010] Embodiments are directed to computer-implemented methods and systems for determining the behavior of a reacting flow system. One such embodiment defines multiple models of the reacting flow system, each defined model representing the reacting flow system at a respective scale. A velocity field of the reacting flow system is then determined using a first model (at a first respective scale) of the defined multiple models, and a diffusivity of the reacting flow system is determined using a second model (at a second respective scale) of the defined multiple models. In one embodiment, determining the velocity field and determining the diffusivity are performed automatically by one or more digital processors. A plurality of reaction parameters for the reacting flow system are then defined. The behavior of the reacting flow system is then automatically determined by using the determined velocity field, the determined diffusivity, and the defined multiple reaction parameters as inputs to a reactive transport solver.

[0011] Another embodiment of the method provides a computer-implemented method and system for determining component concentrations in a reacting flow system.

[0012] In some embodiments, at least one model of the defined plurality of models represents a reacting flow system at a microscale, a molecular scale, or a subsurface scale.

[0013] In some embodiments, at least one model of the defined plurality of models is a geometric model that characterizes the reacting flow system.

[0014] In some aspects, defining a given model of a plurality of models of a reacting flow system includes defining a model of one or more heterogeneous surface reactions as a function of mineral dissolution and precipitation, and then modeling kinetic laws of the defined model of the one or more heterogeneous surface reactions. To proceed, one such embodiment defines the given model based on the modeled kinetic laws and a model of one or more homogeneous bulk reactions.

[0015] In some embodiments, determining a velocity field of a reacting flow system using a first model includes receiving an image of a material within the reacting flow system and segmenting (i.e., bisecting, classifying, etc.) the image into multiple phases, each phase representing a material, solid, or fluid. The velocity field of the reacting flow system is then determined based on the multiple phases in the image using a first model, where the first model is a single-phase fluid flow model.

[0016] In some aspects, the determined velocity field is a multiphase velocity field.

[0017] In some embodiments, the material is porous. In some embodiments, the material further comprises one or more fractures. In some embodiments, the material is a nanoporous clay material. In some embodiments, the material comprises Cu. 0.66 [Al 3.33 Mg 0.66 ][Si8]O 20 It is a Wyoming montmorillonite with the formula [OH]4.

[0018] In some aspects, determining the diffusivity of the reacting flow system using the second model includes providing parameters of the bulk salt solution and parameters of the salt solution in the clay interlayer nanopores as inputs to a molecular dynamics simulation. These inputs are then used to perform the molecular dynamics simulation as a function of temperature and salt concentration. In one such embodiment, the results of performing the molecular dynamics simulation indicate the diffusivity of the reacting flow system.

[0019] In some embodiments, the diffusivity is an ionic diffusivity. In some embodiments, the ionic diffusivity is a copper (Cu) 2+ is the ionic diffusivity of

[0020] In some embodiments, a plurality of reaction parameters for a reacting flow system are defined using input data, hi some embodiments, the input data is obtained from at least one of simulation results and a database.

[0021] In some aspects, determining the behavior of the reacting flow system by using the determined velocity field, the determined diffusivity, and the defined plurality of reaction parameters as inputs to a reactive transport solver includes solving an advection-diffusion-reaction equation using the reactive transport solver with the inputs. In one such embodiment, the results of the solution are indicative of the behavior of the reacting flow system.

[0022] In some embodiments, the behavior of the reacting flow system is a concentration profile. In some embodiments, the concentration profile is a copper (Cu) 2+ 1 is a concentration profile of

[0023] In some aspects, the method further includes updating the first model and the second model based on the determined behavior of the reactive slow system, determining an updated velocity field of the reacting flow system using the updated first model, determining an updated diffusivity of the reacting flow system using the updated second model, and determining an updated behavior of the reacting flow system by using the updated determined velocity field, the updated determined diffusivity, and the defined plurality of reaction parameters as inputs to a reactive transport solver. In some aspects, the method further includes iterating (i) updating, (ii) determining an updated velocity field, (iii) determining an updated diffusivity, and (iv) determining an updated behavior of the reacting flow system until the determined updated behavior of the reacting flow system reaches a steady state.

[0024] The embodiments described herein also provide computer-implemented methods and systems for determining component concentrations in a reactive flow system.

[0025] Yet another embodiment is directed to a system including a processor and a memory having computer code instructions stored thereon, in one such embodiment, the processor and memory are configured to cause the computer code instructions to cause the system to perform any embodiment or combination of embodiments described herein.

[0026] Another embodiment is directed to a cloud computing implementation for determining the behavior of a reactive flow system. One such embodiment is directed to a computer program product executed by a server in communication with one or more clients over a network. The computer program product includes program instructions that, when executed by a processor, cause the processor to perform any embodiment or combination of embodiments described herein.

[0027] It is noted that the method, system, and computer program product embodiments may be configured to implement any embodiment or combination of embodiments described herein.

[0028] Yet another embodiment provides a multiscale reactive flow model for simulating in situ copper leaching in heterogeneous porous microstructures. In one such embodiment, a workflow is utilized that combines fluid flow simulation with advection-diffusion-reaction simulation, both of which are used to model the reactive flow. Such a workflow may include flow through resolved and unresolved pore structures and may utilize parameters from molecular simulations (ion diffusivities) and reaction databases (reaction rate parameters). The embodiment has also been validated by comparing the determined results with other open-source codes for modeling calcite dissolution during acid injection. A molecular mechanics model of clay is also presented to estimate ion diffusivities in nanoporous media, and a dissolved salt in water model is implemented to estimate ion diffusivities within open fractures. The model is applied to copper mining by leaching to analyze reactive flow through a fractured digital rock model of an underground sample. Results were analyzed by tracking the concentration distribution along the pore space structure and calculating the copper outlet concentration to confirm the leaching pathway. Several sensitivity studies described herein below demonstrate the robustness of the embodiment and demonstrate the importance of acid inlet flow conditions and different reaction types and scales on copper production. One embodiment systematically increases the complexity of the model from a single-scale surface reaction model to also include competing bulk solution reactions, flowing through porous media to model multi-scale reacting flow. Results from the embodiment demonstrate that a multi-scale flow model with homogeneous bulk reactions and heterogeneous surface reactions accurately models copper leaching. [Brief explanation of the drawings]

[0029] The foregoing will become apparent from the following more particular description of example embodiments, as illustrated in the accompanying drawings, in which like reference characters refer to the same parts throughout the different views. The drawings are not necessarily to scale, emphasis instead being placed upon illustrating the embodiments.

[0030] [Figure 1] FIG. 1 is a flow chart illustrating a method for determining the behavior of a reacting flow system, according to one embodiment. [Figure 2] Figure 2 is a schematic diagram showing copper leaching by acid injection at the pore scale. [Figure 3] FIG. 3 is an exemplary embodiment of a workflow for determining characteristics of a reacting flow system, according to one embodiment. [Figure 4] Figure 4 shows the steps of an exemplary sequential workflow used to obtain velocity fields from Lattice Boltzmann Method (LBM) simulations with and without the multiscale option, including flow in clay nanopores. [Figure 5A] FIG. 5A illustrates the surface reactions of dissolution and precipitation on a mineral particle in an exemplary embodiment. [Figure 5B] FIG. 5B illustrates an exemplary mechanism for combining surface and bulk reactions to obtain the concentration of an exemplary species. [Figure 5C] FIG. 5C is an exemplary reactive flow method for updating the concentration of an exemplary species within a voxel of a microstructural model according to one embodiment. [Figure 6A] FIG. 6A is a plot of H+ concentration as a function of time in a two-dimensional (2D) geometry of an exemplary calcite pellet placed in a rectangular channel. [Figure 6B] Figure 6B is a comparative plot of H+ concentration (M) as a function of geometric length between the exemplary simulation and GeoChemFoam. [Figure 6C] FIG. 6C is an exemplary simulation of spatial H+ concentration using GeoChemFoam. [Figure 7A]FIG. 7A is an exemplary digital rock microstructure used to model copper leaching by acid injection, according to one embodiment. [Figure 7B] FIG. 7B is a plot of steady-state copper concentration as a function of time as determined according to one embodiment. [Figure 8A] Figure 8A shows an example of a molecular dynamics (MD) simulation structure of copper chloride in bulk water. [Figure 8B] FIG. 8B shows an exemplary MD structure of clay containing copper chloride in hydrated nanopores, according to one embodiment. [Figure 8C] FIG. 8C shows plots of ionic diffusivities of copper and chloride in bulk water presented at different salt concentrations and temperatures, respectively. [Figure 8D] FIG. 8D shows plots of ionic diffusivities of copper and chloride in bulk water presented at different salt concentrations and temperatures, respectively. [Figure 8E] FIG. 8E is a plot of the ionic diffusivities of copper and chloride in the nanoporous clay model presented at different temperatures and 0.1 M chloride ions. [Figure 9] Figure 9 shows the three-dimensional (3D) microstructure of an exemplary rock having copper ferrite minerals (white areas) with fractures. [Figure 10A] FIG. 10A is a surface plot of copper outlet concentration as a function of velocity (Peclet number) and reaction rate (Dankeler number). [Figure 10B] FIG. 10B is a contour plot of the copper outlet concentration as a function of rate (Peclet number) and reaction rate (Dankeler number). [Figure 11A] FIG. 11A is a plot of the outlet concentration of copper as a function of time at a Peclet number of 10 and various Dankeler numbers. [Figure 11B] FIG. 11B is a plot of the outlet concentration of copper as a function of time at a Dankeler number of 2.5 and various Peclet numbers. [Figure 11C] FIG. 11C is a plot of steady-state copper concentration as a function of Dankeler number. [Figure 11D]FIG. 11D is a plot of steady-state copper concentration as a function of Peclet number. [Figure 12A] FIG. 12A is a plot of copper species concentration as a function of inlet pH. [Figure 12B] FIG. 12B is a plot of iron species concentration as a function of inlet pH. [Figure 13A] FIG. 13A is a plot of the outlet Cu 2+ concentration as a function of time for an exemplary model using only heterogeneous surface reactions. [Figure 13B] FIG. 13B is a plot of the outlet Cu 2+ concentration as a function of time for an exemplary model using both a homogeneous bulk reaction and a heterogeneous surface reaction. [Figure 13C] FIG. 13C is a plot of steady-state Cu2+ outlet concentration as a function of inlet acid concentration. [Figure 14A] FIG. 14A is an exemplary digital rock cross-sectional image visualizing copper concentration through a fracture in a single-scale reacting flow system. [Figure 14B] FIG. 14B is a cross-sectional image of an exemplary digital rock visualization of copper concentration through a fracture in a multi-scale reacting flow system. [Figure 14C] FIG. 14C is a plot of the copper outlet concentration as a function of time for each of the reactive flow systems shown in FIGS. 14A-B. [Figure 15] FIG. 15 is an exemplary embodiment of a multi-scale reactive flow simulation of a metal leached through the pore space. [Figure 16] FIG. 16 is a simplified block diagram of a computer system for determining the behavior of a reactive flow system, according to one embodiment. [Figure 17] FIG. 17 is a simplified block diagram of a computer network environment in which an embodiment of the present invention may be implemented. DETAILED DESCRIPTION OF THE INVENTION

[0031] A description of an exemplary embodiment follows.

[0032] Reactive flow modeling is a tool for analyzing any process involving fluid flow and chemical reactions, such as geological carbon capture. 14,23 , groundwater purification 43 , Enhanced Oil Recovery (EOR) with CO2 injection 26 , corrosion and fouling of structural materials 41 , and redox flow batteries 11 One such application of interest in this study is the in-situ leaching of copper. 38 In-situ leaching involves the circulation of acid to dissolve the target mineral to produce or extract the metal of interest. As global demand for copper increases, in-situ leaching has proven to be a low-cost method of extracting copper. 32,38 Many challenges remain for efficiently extracting copper from in situ leaching, such as determining the optimal composition of the lixiviant, achieving uniform contact of the injection fluid with the minerals, and varying the permeability and porosity of the rock due to mineral dissolution and precipitation. Therefore, modeling real-time changes in the composition of the fluid and fluid-rock interface through reactive flow modeling is beneficial for optimizing copper production from in situ leaching and for predicting the possible outcomes of alternative designs and operating conditions. Therefore, reactive flow modeling can be used to optimize real-world in situ copper leaching. Furthermore, while embodiments are described herein as being used to simulate and optimize in situ copper leaching, it should be noted that the embodiments are not limited to such systems and can be used to model and optimize any reactive flow system.

[0033] Although reacting flow has applications in many industries, most of the advances in this field have been related to geochemical modeling in subsurface systems. 2,27,28,31,49 In reacting flow modeling, the continuum level model 42 There are two broad categories of models: microscale and pore-scale models. 5,21The use of continuum-level and pore-scale models typically depends on the scale of physics of interest. At the pore scale, the microstructure is fully resolved, and effective species transport coupled with surface reactions per volume is simulated directly within the microstructure, with the inputs being material properties. At the continuum scale, the microstructure is unresolved, and effective transport and reaction rates, limited by microstructural connectivity, must be provided as inputs. Inputs are often provided using simplified microstructural property relationships, such as permeability and surface-to-volume ratio porosity correlations for a spherical pack of parallel tubes. The strategy used by existing methods is to first model reactive transport at the pore scale for a given microstructure, and then upscale the property relationships and models to the continuum scale to accommodate larger scales in the study. 25 Laboratory experiments 30 Advances in both pore-scale and reactive flow modeling have demonstrated the importance of this multiscale strategy for better understanding the pore-scale physics of reacting flows over the past few decades. 2 .

[0034] Two important properties of interest in reacting flows are the reaction rate and the diffusion coefficient. Reaction parameters such as rate constants, equilibrium constants, and thermodynamic parameters are available in open source thermodynamic databases such as the LLNL database, the PHREEQC database, and the MINTEQ database. 35 , Reaktoro 13 , TOUGHREACT 47 Several open-source reacting flow simulators, such as , utilize these databases to perform chemical reaction and thermodynamic calculations and update the composition of fluid phases. Each database targets a specific application and depends on the application being modeled, so choosing an appropriate database is recommended.

[0035] The diffusion of solutes in water in nanoporous materials such as clays and in bulk fluids is often modeled using molecular dynamics simulations. 8,22,39,40,45 Clays have unique properties, including swelling in the presence of water, which is an important reason for facilitating ion migration within clays.10 Some researchers have addressed these issues in the past, developing robust force fields for ion and clay atomic interactions (molecular models). 12 However, these studies are limited to simple salts of alkali and alkaline earth metals at room temperature. Many reaction stream applications involve the use of Na + and Ca 2+ For simple species such as Cu, it is possible to find diffusion parameters from the literature, and therefore researchers generally do not focus explicitly on diffusion modeling. However, there is limited literature and data available for less common ionic species, such as Cu. 2+ is one such case. Thus, it is shown in this paper that incorporating molecular simulations into a reacting flow workflow provides a general framework for any application where diffusivity is one of the parameters of choice.

[0036] The numerical simulation method for fluid flow described herein is the Lattice Boltzmann Method (LBM). LBM is widely used as a direct simulation method for dissolvable pore-scale events to determine physical properties such as wettability, capillary effects, and viscous fingering phenomena. 16,20,44,48 LBM is an explicit method, often performed on a cubic lattice, that solves a discrete version of the Boltzmann equation and has its origins in kinetic theory. While multiphase capabilities are available to simulate immiscible multiphase scenarios such as oil and water, the examples described herein use single-phase flow LBM. Here, single-phase flow simulations extracted from direct X-ray microtomography imaging (microCT) of representative samples are performed on 3D microstructure models, with the goal being to calculate the flow velocity field within the interconnected pore space of the 3D microstructure. Furthermore, LBM 17,18Multiscale extension to allow partial flow through some semi-permeable materials or porous media (PM) regions, where pore connectivity is not explicitly resolved at the model resolution of the 3D microstructure. Nanoporous clays are examples of such materials, and the multiscale extension described herein can calculate the effective flow velocity field in regions filled with these types of semi-permeable materials.

[0037] This paper presents a model that combines the above-mentioned components of multiscale reacting flow, including microCT imaging into the flow velocity field, homogeneous and heterogeneous reactions, ion diffusion, resolved pore structure, and unresolved pore material regions. The embodiment includes a multiscale, multispecies reacting flow workflow that has been implemented and validated. Below, the application of one embodiment to mineral dissolution in 3D digital rock images is presented to illustrate the effects of reaction rates, velocity, reaction type, pH, and multiscale flow through porous media. The methodology used to model in situ copper leaching is also discussed. The reacting flow workflow is introduced, and the different components of reacting flow are described. The exemplary embodiment begins with a validation case of water diffusivity in the bulk using molecular simulations. Next, validation of the reacting flow methodology is performed by modeling a test case of calcite particle dissolution and comparing the results with published results of other pore-scale reacting flow simulations. Following this validation, a model of copper leaching is presented. To obtain the diffusivity for the reactive flow model, a molecular model of bulk copper ions in aqueous solution was used for diffusion in micropores and clay nanopores for diffusivity in porous media. Using the reactive flow model, a series of sensitivity tests was performed to investigate the effects of pH, fluid velocity, and surface reactivity on copper formation. A homogeneous bulk reaction was then incorporated into the model to understand its effect on copper formation. The model was extended to include transport through both open fractures and nanoporous clay and to demonstrate its effect on copper formation.

[0038] Models such as those described above can be utilized in the methods described herein to determine the behavior of a reacting flow system. FIG. 1 illustrates one such exemplary method 110 for determining the behavior of a reacting flow system. Method 110 is computer-implemented and may be performed by any combination of hardware and software as known in the art. For example, method 110 may be implemented via one or more processors having associated memory storing computer code that causes the processor to implement steps 111, 112, 113, 114, and 115 of method 110. Furthermore, while embodiments are described herein as being capable of being implemented in software provided by the applicant, it should be noted that embodiments are not limited to being implemented in existing software; instead, embodiments may be practiced using any combination of hardware and software as known in the art.

[0039] Returning to FIG. 1 , method 110 begins in step 111 by defining, e.g., in computer memory, multiple models of a reacting flow system, each defined model representing the reacting flow system at a respective scale. According to one embodiment of method 110, exemplary scales include the microscale, the molecular scale, and the subsurface scale, among others. In an exemplary embodiment, models for different scales are defined in step 111 using different techniques. For example, 3D imaging by X-ray tomography, 3D molecular modeling, and / or 3D seismic surveying, among other examples, may be used in step 111 to generate computer models at each scale. According to one embodiment, at least one model of the multiple models defined in step 111 is a geometric model that characterizes the reacting flow system. In another embodiment of method 110, a given model is defined in step 111 as a model of one or more heterogeneous surface reactions. In one such embodiment, reaction rate laws are modeled for the defined models of one or more heterogeneous surface reactions as a function of mineral dissolution and precipitation. A given model is then defined based on the modeled reaction rate laws and one or more homogeneous bulk reaction models. According to one embodiment, heterogeneous surface reactions refer to reactions between species dissolved in the fluid and mineral surfaces in contact with the fluid. Furthermore, in one embodiment, homogeneous surface reactions refer to reactions between species dissolved in the fluid. Reaction rate laws are provided for both types of reactions in embodiments.

[0040] Returning to FIG. 1 , in step 112, method 110 continues by determining a velocity field of the reacting flow system using a first model (at a first respective scale) of the defined plurality of models. According to one embodiment, the determined velocity field is a multiphase velocity field. In an exemplary embodiment, the velocity field of the reacting flow system is determined in step 112 using the first model by receiving an image of the material in the reacting flow system and segmenting (i.e., bisecting, classifying, etc.) the image into phases or segments representing multiple phases, e.g., material, solid, and fluid. A velocity field is then determined based on the multiple phases in the image using the first model, the first model being a single-phase fluid flow model. According to one embodiment, a computational fluid dynamics (CFD) simulation is implemented in step 112 to determine the velocity field. In an exemplary embodiment, the first model is a microCT model, which is utilized in step 112 for a Lattice Boltzmann Method (LBM)-based flow simulation to determine the velocity field. According to one such embodiment, the LBM-based flow simulation technique implemented in step 112 enables sub-resolution and accurate multiphase flows. For example, in one such embodiment, functionality described in U.S. Patent Publication No. 2022 / 0207219 A1 may be used in step 112.

[0041] According to one embodiment, the material (e.g., the material shown in the received image) is porous. According to another embodiment, the material may further comprise one or more fractures. In another embodiment, the material is a nanoporous clay material. According to yet another embodiment, the material is Cu. 0.66 [Al 3.33 Mg 0.66 ][Si8]O 20 It is a Wyoming montmorillonite with the formula [OH]4.

[0042] Method 110 continues at step 113 by determining the diffusivity of the reacting flow system using a second model (at a second respective scale) of the defined plurality of models. In one embodiment, the diffusivity is determined at step 113 using known techniques. In one such embodiment, the diffusivity is determined using the Forcite module of the BIOVIA Material Studio® software application. According to one embodiment of method 100, the diffusivity of the reacting flow system is determined using the second model by first providing parameters of the bulk salt solution and parameters of the salt solution in the clay interlayer nanopores as inputs for a molecular dynamics simulation. Second, a molecular dynamics simulation is performed using these inputs as a function of temperature and salt concentration. The results of performing the molecular dynamics simulation indicate the diffusivity of the reacting flow system. In one exemplary embodiment, the second model, i.e., the model used at step 113, is a chemical composition model. In one such embodiment, the chemical composition model is used in the molecular dynamics simulation to determine the diffusivity at step 113. According to one embodiment, the molecular dynamics simulation method may be implemented in BIOVIA Materials Studio® using the Forcite module. In another embodiment, the diffusivity is an ionic diffusivity. According to one embodiment, the ionic diffusivity is 2+ is the ionic diffusivity of

[0043] At step 114, the method 110 continues by defining a plurality of reaction parameters for the reacting flow system. According to one embodiment, the plurality of reaction parameters for the reacting flow system are defined using input data. According to one embodiment, the input data is obtained from at least one of simulation results and a database.

[0044] Continuing, at step 115, method 110 automatically determines the behavior of the reacting flow system by using the determined velocity field, the determined diffusivities, and the defined reaction parameters as inputs to a reactive transport solver. According to one embodiment, the behavior is determined at step 115 using Equation 5, described below. In an exemplary embodiment, the reactive transport equation is solved at step 115 to determine the behavior of the reacting flow system. According to one embodiment, determining the behavior of the reacting flow system by using the determined velocity field, the determined diffusivities, and the defined reaction parameters as inputs to a reactive transport solver includes solving an advection-diffusion-reaction equation using the reactive transport solver with the inputs. In one such embodiment, the solution results indicate the behavior of the reacting flow system. In an embodiment of method 110, the determined behavior of the reacting flow system is a concentration profile, e.g., a species concentration profile. According to another embodiment, the concentration profile is a concentration profile of copper (Cu) 2+ In another exemplary embodiment, the behavior determined in step 115 is the overall production rate of the species (product) of interest.

[0045] The method 110 may further include updating the first model and the second model based on the determined behavior of the reacting flow system, determining an updated velocity field of the reacting flow system using the updated first model, determining an updated diffusivity of the reacting flow system using the updated second model, and determining the updated behavior of the reacting flow system by using the updated velocity field, the updated determined diffusivity, and the defined plurality of reaction parameters as inputs to a reactive transport solver. Furthermore, one such embodiment of the method 110 may further include iterating (i) updating, (ii) determining the updated velocity field, (iii) determining the updated diffusivity, and (iv) determining the updated behavior of the reacting flow system until the determined updated behavior of the reacting flow system reaches a steady state.

[0046] [Illustration] An exemplary embodiment provides a workflow for solving multi-scale reacting flow in porous microstructures. To model reacting flow, an embodiment models chemical interactions (involving molecular models, heterogeneous reactions, and homogeneous reactions) combined with flow through the microstructure. The resulting learnings can be extended from microscale simulations to field-scale simulations. Reacting flow has traditionally been studied in either microscale or field-scale simulations. The exemplary workflow embodiment described herein bridges the gap to simulating reacting flow at multiple scales.

[0047] According to one embodiment, a reacting flow modeling workflow combines various components described herein (e.g., multiphase, multiscale reacting flow), such as microCT imaging into the velocity field, multiphase flow, homogeneous and heterogeneous reactions, ion diffusion, resolved pore structure, and multiscale transport modeling including unresolved porous material regions. A demonstration of this multiphase, multiscale, multispecies reacting flow workflow is presented, using mineral dissolution in 3D digital rock images to understand the effects of reaction rates, kinetics, reaction type, pH, and multiscale flow through porous media.

[0048] Embodiments have multiple components that enable unique opportunities for global optimization in design and operation by collaborating in a simulation ecosystem such as Dassult Systemes 3DS Experience Platform, which enables multi-scale and multi-physics modeling, simulation, and process optimization.

[0049] [Example Methodology] [Example: Copper Leaching] An exemplary application of the method 110 of FIG. 1 is provided below.

[0050] At the field scale, in situ leaching is performed using a series of wells to inject acid into underground formations, where fluid flows through interconnected fractured rock containing target minerals and product solution after mineral dissolution reactions are extracted through the wells and subsequent hydrometallurgical processing to extract copper from the solution. Figure 2 is a schematic diagram illustrating copper leaching by acid injection at the pore scale of a fractured rock model 225. At the field scale, acid is injected into fractured underground rock through a parent well 221, and the copper-bearing solution 224 is collected for further processing through a child well. At the pore scale, acid is injected into the fractured rock to produce copper at the outlet. The digital rock model 225 (i.e., a computer-based model) has four segmented phases: reactive minerals 226a, unreactive minerals 226b, nanoporous minerals 226c, and pores 226d. Figure 2 illustrates copper production at the pore scale, which can be modeled using embodiments by solving for flow, mass transport, and chemical reactions. In the model 225 illustrated in Figure 2, acid is injected at one end of the rock (inlet) 221, and then the acid moves due to advection 222a-b through a fracture network (black region 226d). In the process, the acid diffuses 223a-b through nanoporous clay (dark gray region 226c), reacts with minerals (white region 226a), and copper is collected at the far end of the rock (outlet) 224. The different physics of copper leaching illustrated in Figure 2 can be systematically and separately addressed in a multiscale reactive flow workflow according to one embodiment.

[0051] [Example: Reactive Flow Workflow] An exemplary implementation of the application of the method 110 of FIG. 1 to copper leaching is described below.

[0052] An embodiment can connect different parts of a reacting flow process using workflow 330, as illustrated in Figure 3. In workflow 330 of Figure 3, in step 331, a microCT image (e.g., 337) is segmented into different regions and fed to an LBM single-phase fluid flow simulator to determine the velocity field (e.g., 338) in step 332. In step 333, a reaction database such as PHREEQC is used to obtain kinetic parameters for reaction calculations, such as those shown in Table 339. Additionally, in step 334, molecular simulations are performed using a solver such as BIOVIA Materials Studio® version 2021 to determine ion diffusion coefficients (as illustrated in model 340). Next, in step 335, the velocity field (from step 332), the rate parameters (from step 333), and the diffusivity (from step 334) are used as inputs to a reactive transport simulation solver, which determines advection-diffusion-reaction equations that can be used in step 336 to determine different species concentration profiles (such as those illustrated in plot 341) within the 3D pore space of the digital rock model (e.g., model 342).

[0053] [Example: Fluid flow modeling] An exemplary implementation of step 112 of method 110 in the copper leaching example is described below.

[0054] Digital rock (e.g., computer-based models, computer-aided design (CAD) models, etc.) applications of LBM can be used in embodiments to predict the fluid flow permeability of porous rock (represented by the model). 16.20,44,483D imaging methods, such as microCT imaging, may be employed, which are suitable for pore-scale description of fluid dynamics. LBM is based on the equations of motion of fluid particles and represents a statistical description of the molecular behavior of fluid particles. LBM can be used to simulate the dynamic behavior of fluid flow without directly solving the continuum fluid dynamics equations. Furthermore, LBM-based fluid solvers are considered a competitive alternative to traditional Navier-Stokes PDE-based numerical methods, especially in applications involving complex geometries, such as porous media flow in digital rock models.

[0055] The computer-based model geometry used in an LBM solver in one embodiment can include 3D volume elements corresponding to voxels from a microCT scan image. In addition, the computer-based model used in an embodiment can include pore / solid surface elements called surfels. The use of surfels allows for a high-fidelity representation of the pore / solid geometry interface with effective sub-voxel resolution accuracy. Slip boundary conditions cannot be applied between the fluid and surfels. The use of surfels is an inherent feature of the particular LBM applied by one such embodiment and can make computational costs manageable by allowing a practical number of grid cells to be used in the rock volume.

[0056] Here, a multiscale framework can also be considered. Simulations can be performed directly in the resolved pore space, while partial flow is possible through the unresolved porous medium (PM). For the rock sample used in one such embodiment, the unresolved PM regions can be associated with clay nanoporous materials. Transport through the PM regions can be calculated by applying LBM multiscale extension to regions in the 3D model identified as clay nanoporous materials during segmentation. 17,18 This can be achieved by applying the following equation: The effective PM permeability can be estimated and input independently to control the transport intensity within the PM domain.

[0057] FIG. 4 shows a sequence of images, from left to right, illustrating the steps of an exemplary sequential workflow used to determine a velocity field from an LBM simulation with and without the multiscale option of including flow in clay nanopores. Image 441 is an input microCT image, with gray values ​​representing X-ray absorption coefficients. Image 442 illustrates a segmented rock model (generated based on microCT image 441), where phases in the rock model are indicated by shading. Specifically, shading 445a depicts reactive minerals, shading 445b depicts unreactive minerals, shading 445c depicts microporous minerals, and shading 445d depicts porosity. Model 442 can then be used in an LBM simulation to determine the velocity field, according to one embodiment. Image 443 illustrates the resulting velocity field from an LBM simulation that considers only flow through the resolved pore space. Image 444 shows the resulting velocity field from an LBM multiscale simulation that considers flow through both the resolved pore space and the PM and marked regions. Note that the calculated fluid flow permeability values ​​are the same for both flow simulations, indicating that the contribution of PM regions to flow is negligible compared to the flow contribution from the larger fracture network. While this is true, it is expected that a non-negligible contribution to the copper production rate is still achievable as more areas are marked as reactive minerals when PM regions are added to the contacted flow paths.

[0058] [Example: Reaction Modeling] An exemplary implementation of step 114 of method 110 for an exemplary copper leaching example is provided below.

[0059] In one embodiment, the reactive transport model has two reaction categories: a heterogeneous surface reaction category and a homogeneous bulk reaction category. First, in one embodiment, the kinetics of surface reactions are modeled as a function of mineral dissolution and precipitation on the concentrations of aqueous species participating in the reaction (based on transition state theory).

[0060]

number

[0061] JPEG0007766725000002.jpg22158

[0062]

number

[0063] JPEG0007766725000004.jpg27159

[0064] From the above two equations, we can observe that the mineral surface reactions are driven by multiple parallel mechanisms. To illustrate, we can take copper ferrite (the reactive mineral in the model) as an example to see the final rate equations and parameters. The reaction of copper ferrite with acid is as follows:

[0065]

number

[0066] In this paper, the reaction rate is assumed to depend on the proton activity. Therefore, the reaction rate equation for the copper ferrite surface reaction can be written as follows:

[0067]

number

[0068] The parameters used in equation (4) are listed in Table 1.

[0069] [Table 1]

[0070] JPEG0007766725000008.jpg11155

[0071] For bulk homogeneous species, the reaction rate is fast and in one embodiment, equilibrium is assumed to be reached immediately. 34 Therefore, relationships between aqueous species are found using the law of mass action. For example, the bulk reactions considering the species in equation (3) are listed in Table 2.

[0072] [Table 2]

[0073] [Example: Reactive transport modeling] An exemplary implementation of step 115 of method 110 for an exemplary copper leaching example is provided below.

[0074] In one embodiment, advection-diffusion-reaction equations are solved to model the transport of each species.

[0075]

number

[0076] JPEG0007766725000011.jpg43164

[0077] Figure 5A shows the acid (H + )562 reaction flows on the mineral surface553, and Cu 2+ 563a and / or Fe 3+ 563b and indicates a surface reaction that can modify the mineral geometry based on the Saturation Index (SI). In one embodiment, the Saturation Index SI is used to define whether the reaction is a dissolution 551 or precipitation 552 reaction.

[0078] In one embodiment, the surface reaction illustrated in FIG. 5A is combined with a bulk homogeneous reaction. FIG. 5B shows this combined chemical reaction at the voxel level in a digital rock 3D model 554, with shading indicating particles 555a, pores 555b, and reactive minerals 555c. In one such embodiment, when acid encounters a reactive mineral 555c voxel instead of a particle 555a voxel (visualized in FIG. 5B), the acid produces a constant amount of total copper (a mathematical construct used to facilitate concentration calculations) in the adjacent pore 555b voxel. In one such embodiment, the bulk reaction is assumed to be at equilibrium; therefore, the bulk reaction can be assumed to be a well-mixed batch reactor in which total copper is instantaneously converted to copper species in different amounts based on an equilibrium constant. To solve the transport and reaction equations, in one embodiment, a global implicit scheme is used by converting the nonlinear equations into simultaneous linear equations using the total concentration 556, including primary and secondary species. The set of partial differential equations (PDEs) for mass transport (advection-diffusion-reaction) and algebraic relationships used in such an embodiment are listed herein in a further example.

[0079] FIG. 5C is a flowchart of a method 557 summarizing the numerical implementation of the reactive flow modeling described herein above in connection with FIGS. 5A-B, according to one embodiment. The workflow 557 of FIG. 5C begins with step 558 receiving a segmented image, i.e., a segmentation, of a 3D microCT image of a material of interest, e.g., porous rock. In step 559, the received image segmentation is used to determine a velocity field. To proceed, in step 560, the advection-diffusion-reaction equation is solved (e.g., as described above in connection with FIG. 5B) for pore voxels adjacent to reactive minerals, and the advection-diffusion equation is solved for the remainder of the pore voxels, i.e., pore voxels not adjacent to reactive minerals. Geometric changes due to dissolution and precipitation are not considered in this embodiment; instead, a short-time profile is simulated, allowing for the reaction rate of the dissolution portion of the mineral reaction to be considered. Furthermore, in the workflow 557 of FIG. 5C, as the product concentration begins to increase, precipitation is suppressed by equilibrating the precipitation rate to zero. After the transport equation is solved in step 560, method 557 moves to step 561, where the total concentration is obtained and the concentrations of the secondary species are obtained using a linear transformation as shown in the algebraic speciation equations (a4-a9). Further, method 557 may iterate 563 (i.e., repeat steps 560 and 561) until the copper concentration reaches a steady state.

[0080] According to one embodiment of method 557, the transport equation is solved in step 560 using a finite difference method. One embodiment uses a structured uniform grid obtained from a voxelized microCT image of the pore space, where the time derivative of equation (5) is discretized using a first-order implicit method and the advection term is discretized using an upwind difference scheme.

[0081] [Example: Diffusion Modeling] In one embodiment, MD simulations are performed using the Forcite module of Material Studio, based on theory as detailed in Reference 6. In one such embodiment, the force field parameters of the atoms on the clay are modeled using the CLAYFF force field (Reference 12) and the COMPASIII force field (updated with a flexible water model) for water and ions (References 4 and 37). The clay model in this embodiment is based on Al in the octahedral sheets. 3+ and Mg 2+ With isomorphous substitution of Cu 0.66 [Al 3.33 Mg 0.66 ][Si8]O 20 Typical Wyoming montmorillonite with a unit cell formula of [OH]. The partial charges on the clay surface are taken from previous literature. 19,22 According to one embodiment, the clay simulation model includes two regularly replicated simulation cells with parallel surfaces of montmorillonite, each containing Cu 2+ The nanopores consist of 45 unit cells with counterions and water, similar to the 2W water from previous studies. 19 Furthermore, in one such embodiment, bulk copper chloride simulations are performed at different salt concentrations in a cubic box filled with ions and water at different concentrations and temperatures.

[0082] The clay simulation was equilibrated with NVT for 1 ns, followed by NVT at 1 bar for 1 ns to relax the montmorillonite layers. A generation run of ion diffusion in the nanoporous clay simulation can be carried out using a Nose barostat with a 1 fs time step in the NVT ensemble at 298, 398, and 498 K for 5 ns simulations. 33 A leapfrog algorithm can be used as an integrator. One embodiment uses the particle mesh Ewald method to calculate long-range electrostatic interactions. 15 According to one embodiment, the cutoff for Lennard-Jones (LJ) interactions and electrostatic interactions is set to 12 Å.

[0083] One embodiment calculates simple ionic self-diffusion coefficients for reactive transport simulations from the gradient of the mean square displacement (MSD). 19 A few water molecules and protons (H) are known to have a more complex mechanism involving quantum tunneling called the Grotthuss mechanism. + ) to obtain the diffusivity 3 Dynamically corrected MC-MD simulations were performed in Materials Studio 1 , or other such simulation tools known to those skilled in the art.

[0084] [Example: Partial Differential Equations (PDEs) Used in the Embodiments]

[0085]

number

[0086] On the other hand, the set of algebraic speciation equations is

[0087]

number

[0088] [Verification and application of the embodiment] In this section, the reacting flow method (described herein) is applied and discussed in a basic validation case, followed by a discussion of the application of the method to a flow through fractured rock case. Each of the following sections is based on a different component of the reacting flow workflow studied, with results ranging from nanoscale molecular simulations to microscale fluid flow simulations. Results from these different sections can be combined to model copper leaching through fractured rock or other such reacting flow processes / systems.

[0089] [Basic verification] Water Diffusivity We first validated the water diffusivities obtained from molecular simulations according to the present embodiment. These diffusivities are listed in Table 3. Water diffusivities were calculated using the Forcite module in BIOVIA Materials Studio® for different force-field water models and compared with experimental water diffusivities. The four-point water model (TIP4P / 2005) and the flexible three-point water model (SPC / Fw, now incorporated into the COMPASIII force-field model) were found to be more accurate than other water force-field models. Furthermore, SPC / Fw was shown to better predict the hydrogen-bonding structure and thermodynamic properties necessary to accurately model clay interfaces. Therefore, the flexible three-point water force-field model was used to simulate ion diffusion in clay nanopores herein.

[0090] [Table 3]

[0091] [Calcite dissolution] The work described herein was validated with a benchmark model of single-phase reacting flow at the pore scale. 29 This benchmark case did not involve any multicomponent reactions or geometrical evolution. It was intended to validate the implementation of the reacting flow and its boundary conditions. A simple 2D geometry of a calcite pellet placed in a rectangular channel was simulated as presented in Reference 29, with simulation parameters taken from that reference. A concentration profile 662 with a diffusion boundary layer around the calcite mineral is shown in FIG. 6A. In the concentration profile 662, concentrations are plotted according to key 669. Plot 661 shows the average outlet acid concentration (mol / cc) 666b as a function of time (sec) 666a for several reacting flow simulation methods 665a-f. Plot 661 also shows the H from simulation results 665f of one embodiment. +The concentration profiles are compared with results obtained on the same geometry using other reacting flow pore-scale simulations665a-e, namely Vortex665a, Dissolvefoam665b, Lattice Boltzmann665c, OpenFOAM DBS665d, and Chombo-crunch665e. In this study, the geometry did not evolve with the reaction, and therefore short simulation times (around 3 seconds to reach steady state) were used. 29 ) results were compared. The short-time averaged outlet concentration profiles from the simulations were in good agreement with the results from Ref. 29, which validates the reactive flow embodiment used herein. As shown in Figure 6A, the steady-state concentration profile 662 around the calcite pellet exhibits lower H at the surface. + The lines for Dissolvefoam 665b and OpenFOAM DBS 665d almost overlap.

[0092] FIG. 6B is a plot 663 showing the boundary effect of the response on a surface. Plot 663 shows the H + This effect is illustrated by showing the average spatial concentration 667a-c. Plot 663 shows the results of the GeoChemFoam code 668a and the results of the embodiment 668b. The spatial concentration profile 664 from the simulation was compared to reactiveTransportALEfoam (as shown in FIG. 6C), which shares the openFOAM library as part of the GeChemFoam code, and concentration differences 670a-c were observed. 28。 Agreement between codes, especially on the thickness of the diffusion layer and H + The rate of consumption was good.

[0093] [Application to copper leaching]

[0013] Embodiments have been validated for modeling copper leaching in fractured rock microstructures, where the rock is modeled as a cube with a side length of 300 μm. An exemplary model 770 is shown in FIG. 7A. In model 770, the white regions 771 in the rock are reactive minerals, the light gray regions 772 in the rock are unreactive minerals, the dark gray regions 773 in the rock are microporous minerals, and the black regions 774 in the rock are pores. To validate copper leaching, a simulation was performed using model 770, where acid 775 was injected into one end of rock 770 and copper solution 776 was extracted at the exit of rock 770. Rock 770 was divided into four regions, where copper ferrite was used as the reactive mineral 771, smectite was considered for nanoporous minerals 773, solid rock was considered for unreactive minerals 772, and voids were considered for pores 774. Molecular simulations are further described herein to obtain diffusivities within the pores (fractures) 774 of the clay nanoporous mineral 773 and digital rock 770, followed by the results of a reactive flow simulation to illustrate the effect of inlet flow 775 conditions and reaction type on copper production / extraction 776. Figure 7B depicts a comparison of the outlet concentrations 778a of copper production / extraction 776 from Figure 7A over time (seconds) 778b. The outlet concentrations 778a were calculated for a given value of pH=2 779 and plotted 777 as shown in Figure 7B, where the concentrations plateaued after approximately 0.2 seconds.

[0094] [Ion Diffusivity in Clay Nanopores] To obtain the diffusivity values ​​required as part of the reactive flow calculations through open fractured and nanoporous clay materials, we used MD simulations of bulk salt solutions (represented by model 880 shown in Figure 8A) as well as salt solutions in clay interlayer nanopores (represented by model 884 shown in Figure 8B). The bulk salt solution model 880 includes copper ions 881, chloride ions 882, and water molecules 883, represented by close-packed spheres in the bulk salt solution. The clay nanopores 885a-b shown in Figure 8B are from a generic Wyoming montmorillonite, taken from reference 1. 19In Figure 8B, copper ions 886, chloride ions 887, and water molecules 888 are all represented by spheres between the montmorillonite layers 885a-b. Figure 8B also includes atomic density profiles 889 along the clay pores for water oxygen 893a, copper 893c, and chloride 893b. The densities of water oxygen 893a, copper 893c, and chloride 893b within the clay nanopores as shown in plot 889 oscillate. The density oscillations are a standard feature resulting from finite size effects near the nanopore walls. 7 Simulations were carried out as a function of temperature and salt concentration to understand the dependence of the diffusivity on these external factors, as was done in many previous studies on clay nanopores. 19,22,36 .

[0095] Figure 8C is a plot 890 showing diffusivity values ​​896a of copper versus concentration 895a as a function of temperature: 498 K 894a, 398 K 894b, and 298 K 894c. Figure 8D is a plot 891 showing diffusivity values ​​896b of chloride versus salt concentration 895b in bulk liquid water as a function of temperature: 498 K 894d, 398 K 894e, and 298 K 894f. Plots 890 and 891 show that the diffusivities 896a-b of both ions increased with temperature (which was expected) but decreased with increasing salt concentration 895a-b. This decrease in diffusivity with increasing concentration change is attributed to increased ion-ion clustering at high salt concentrations, which reduced the mobility of each ion.

[0096] Similarly, Figure 8E is a plot 892 showing the diffusivity values ​​897 of copper 899a and chloride 899b inside a clay nanopore model at different temperatures 898. Some previous studies have investigated the diffusivity of Na in clay nanopores. + have reported diffusivities of simple ions such as copper, and the work described here extends that previous work to other ions important for copper leaching. Figure 8E shows that the diffusivity of copper ions in clay is approximately two orders of magnitude slower than the expected bulk water value due to increased interactions with the charged clay surface.

[0097] [Copper leaching by acid injection in a fractured rock model] FIG. 9 depicts views 990a-c of a 3D microstructural model of an exemplary rock having copper ferrite minerals and fractures. FIG. 990a depicts the entire 3D rock model 994, while FIG. 990b illustrates an upper portion 995 of the model 994, and FIG. 990c illustrates a lower portion 996 of the model 994. FIG. 9 also includes a scale illustrating copper concentrations 993a-c as depicted in FIGS. 990a-c. Model 994 was used to simulate acid injection in a rock microstructure to demonstrate the ability of embodiments to handle multi-species reactive transport in complex 3D microstructures. In this section, examples of reactive flow simulations are described, along with several sensitivity studies utilizing different flow and reaction conditions using dimensionless numbers. The effects of having only heterogeneous surface reactions versus adding multi-species homogeneous reactions are also considered. Finally, flow through porous media is introduced to model multi-scale reactive flow through both fractures and porous media and compare it to flow through fractures alone. The simulation results are discussed by examining the 3D rock concentration profile and presenting the acid outlet concentration and the composition of the outlet solution.

[0098] A 3D steady-state concentration profile of copper in fracture 992 is utilized to visualize flow patterns and reactions. Copper plumes form above the white copper ferrite minerals 991a (represented by the white minerals in the 3D model) due to the acid reacting with them. The copper then advects and diffuses into the remainder of the fracture 992 due to copper flow and concentration gradients. High concentrations 993a are shown near the reactive minerals, while low concentrations 993c are shown in the rest of the geometry flowing over the light gray 991c and dark gray 991b minerals.

[0099] 10A-B illustrate plots 1010 and 1011, respectively, showing the results of a sensitivity study. Specifically, plot 1010 illustrates a sensitivity study utilizing different Peclet numbers 1012a (Pe, shown on the x-axis) and Dankeler numbers 1012b (Da, shown on the y-axis) to understand the effect of mineral inlet velocity and surface reactivity on copper production 1012c (outlet concentration, shown on the z-axis). Plot 1010 in FIG. 10A shows that the effect of the Dankeler number 1012b on copper production 1012c was greater than the Peclet number 1012a because the Dankeler number 1012b represents the effect of increasing the rate of forward reactions to produce more copper, as illustrated by the slope 1013 of plot 1010.

[0100] Contour plot 1011 in FIG. 10B was used to determine the values ​​of the Dankeler number 1014b and the Peclet number 1014a that would be kept constant in future experiments as shown in FIGS. 11A and 11B. Contour plot 1011 plots the Dankeler number 1014b versus the Peclet number 1014a, with each series corresponding to a different concentration of copper. For the results described in the following sections, a constant of 10 for the Peclet number 1014a and 2.5 for the Dankeler number 1014b were used. Dashed lines 1015a-b are values ​​selected for reaction rate and velocity for further analysis. Dashed lines 1015a-b shown in FIG. 10B were used to further visualize the steady-state copper concentration profiles in FIGS. 11A-D while varying only one parameter at a time.

[0101] 11A-D are plots 1110, 1111, 1112, and 1113, respectively, showing the results of a sensitivity study to capture transient behavior to steady state as a function of time.

[0102] FIG. 11A is a plot 1110 of outlet concentration 1114a versus time 1115a. Plot 1110 includes profiles 1117a-g for Dankeler numbers of 10, 2.5, 0.5, 0.125, 0.025, 0.005, and 0.001. Plot 1110 illustrates that each concentration 1114a profile 1117a-g reached steady state simultaneously based on the dashed line 1116 representing a Peclet number of 10, but the steady-state value of the copper 1114a concentration increased with increasing Dankeler number 1117a-g. The lines representing 0.005 Da 1117f and 0.001 Da 1117g nearly overlap.

[0103] 11B is a plot 1111 of outlet concentration 1114b versus time 1115b. Plot 1111 includes series 1118a-e for Peclet numbers of 500, 250, 50, 10, and 2.5. The lines representing Pe of 500 1118a and Pe of 250 1118b nearly overlap.

[0104] FIG. 11C is a plot 1112 of steady-state concentration 1120 versus Dankeler number 1121. Plot 1112 shows that, at a given inlet velocity, more copper was produced when surface reactivity increased. This can be seen in FIG. 11C, which shows the steady-state concentration of copper 1120 at the outlet compared to the Dankeler number. When the Peclet number 1118a-e was varied at a Dankeler number of 2.5, as shown by the dashed line 1119 in FIG. 11B, the time for the concentration profile to reach a steady state changed, as shown in FIG. 11B. When the Peclet number 1118a-e was varied at a Dankeler number of 2.5, as shown by the dashed line 1119 in FIG. 11B, the time for the concentration profile to reach a steady state changed, as shown in the comparison plot 1111 in FIG. 11B.

[0105] Figure 11D is a plot 1113 of steady-state concentration 1122 versus Peclet number 1123. Plot 1113 shows that the steady-state value 1122 of copper at the outlet appears to have a dependence on the Peclet number 1123, such that low rates correspond to low concentrations, while high rates approach an asymptotic value for concentration. Beyond the sensitivity of each individual parameter shown in Figures 11A-D, corresponding to dashed lines 1015a and 1015b in Figure 10B, it can also be observed that while changing the reactivity increased overall copper production, if there were still high copper concentrations around the copper ferrite locations, there appeared to be some limitation imposed by the microstructure that limited additional copper production. This meant that increasing the reactivity was not beneficial unless the rate was also increased to remove copper from the system. This interesting observation indicates the role of microstructural connectivity in the balance between reactivity and transport.

[0106] [Incorporation of homogeneous bulk reactions] In this section, we investigate the interplay between competing homogeneous bulk and surface reactions on mineral surfaces in the 3D pore space of fractured rock. The homogeneous bulk reactions used in this model are listed in Table 2. When these bulk reactions are added to the system of equations, the effects of speciation and pH on outlet copper production can be understood.

[0107] Figure 12A is a bar graph 1220 showing the outlet concentration 1222 of copper species 1224 and 1225 as a function of inlet acidity 1223a-f, specifically at pH 0, 1, 2, 3, 4, and 8. Figure 12B is a bar graph 1221 showing the outlet concentration 1226 of iron species 1228a-d as a function of inlet acidity 1227a-f, specifically at pH 0, 1, 2, 3, 4, and 8. One observation from Figures 12A-B is that the product concentration at the outlet decreased with increasing pH, which is attributed to the fact that the production of copper 1224 and 1225 and iron 1228a-d is primarily controlled by surface reactions that require high acid concentrations (low pH) for good reactivity. Competing reactions in the bulk produce secondary copper and iron species, preferential species for copper and iron species, e.g., variations in inlet pH conditions. This is an effect that needs to be properly captured in any reactive flow simulator; otherwise, biased estimates of primary species concentrations may result. Although this embodiment involves a simple subset of homogeneous bulk reactions, in real-world applications several competing surface and bulk reactions will be combined with large porous rock sizes, and this speciation issue will have a significant impact on product concentrations. Therefore, the selected values ​​for the inlet acid concentration and species considered in the aqueous model are very important for optimizing any leaching application.

[0108] To proceed, copper ions (Cu 2+ The effect of competing bulk reactions on the primary species produced was investigated, and the results are shown in Figures 13A-C. Figure 13A is a plot 1330 showing outlet concentrations 1331 for a heterogeneous reaction compared to time 1332 for several different pHs 1333a-f (specifically, at pH 0, 1, 2, 3, 4, and 8). Figure 13B is a plot 1334 showing outlet concentrations 1335 for homogeneous and heterogeneous reactions compared to time 1336 for several different pHs 1337a-f (specifically, at pH 0, 1, 2, 3, 4, and 8). Figure 13C is a plot 1340 of steady-state concentration 1341 compared to acid concentration 1342. Plot 1340 includes a series of heterogeneous and homogeneous reactions 1343 and a series of heterogeneous reactions 1344 only.

[0109] Comparing plots 1330 (FIG. 13A) and 1334 (FIG. 13B) shows that the concentration of copper ions at the outlet (1331 in FIG. 13A and 1335 in FIG. 13B) decreased as the inlet pH conditions (1333a-f in FIG. 13A and 1337a-f in FIG. 13B) increased due to the decreased surface reactivity of only the heterogeneous reaction 1330 compared to the homogeneous and heterogeneous reactions 1331. Interestingly, when a competing bulk reaction was added, copper ion production was higher than when only the surface reaction was present (as shown in plot 1340 in FIG. 13C). Plot 1340 comparing the heterogeneous and homogeneous reactions 1343 with only the heterogeneous reaction 1344 based on steady-state concentration 1341 and acid concentration 1342 indicates that the competing bulk reaction outpaced the surface reaction (Cu 2+ ) and thus move the reaction forward. Therefore, having multiple bulk reactions has a significant effect on the composition and quantity of the products at the outlet, which helps to accurately model the copper leaching problem.

[0110] [Acid injection copper leaching in fractured rock and clay - multi-scale] The possibility of flow through fractured and nanoporous clay (e.g., a multiscale reactive transport problem) was also investigated. Transport through the nanoporous material (in this embodiment, clay) occurred primarily due to the diffusion of species through nanoporous regions segmented in the 3D microstructure model. These diffusivities were estimated using MD simulations as presented herein.

[0111] Figure 14A shows a cross section of a digital rock fracture model 1440 with flow only through fracture 1444. The cross section 1440 shows material phases using shading, with shading 1442a, 1442b, and 1442c representing reactive, unreactive, and microporous minerals, respectively. Figure 14A also includes legend 1443, whereby copper concentrations are indicated based on the shading. Figure 14A shows that copper concentrations are higher adjacent to reactive minerals 1442a. Copper concentrations are lower adjacent to unreactive minerals 1442b and microporous minerals 1442c.

[0112] Figure 14B shows a cross section of a digital rock fracture model 1441 with flow through fracture 1445 and nanoporous clay 1446c. Cross section 1441 illustrates material phases using shading, with shading 1446a, 1446b, and 1446c representing reactive minerals, unreactive minerals, and nanoporous clay, respectively. Figure 14B also includes legend 1448, whereby copper concentrations are indicated based on the shading. Multiscale flow through fracture 1445 and nanoporous clay region 1446c indicates that copper begins to penetrate through dark gray region 1446c, e.g., at location 1447, which represents nanoporous clay in the model. Incorporating flow through nanoporous clay helped reach other reactive minerals 1446a that were not in direct contact with fracture 1445, resulting in an increase in the estimated copper production. The copper concentration around the unreacted mineral 1446b remains relatively low compared to the nanoporous clay and reactive minerals.

[0113] FIG. 14C shows plot 1449 of outlet copper concentration 1450 as a function of time 1451. Plot 1449 includes series 1452 for single-scale reacting flow (e.g., as shown in FIG. 14A ) and series 1453 for multi-scale reacting flow (e.g., as shown in FIG. 14B ). It can be observed in plot 1449 that multi-scale flow 1453 takes a relatively longer time to reach steady state because transport through porous media is relatively slower. An interesting observation from FIG. 14C was that more copper was produced over time than when using the single-scale reacting flow 1452 model. This means that incorporating a multi-scale flow model through a nanoporous material, as in an embodiment, plays an important role in improving the accuracy of the reacting flow production model's estimation of copper leaching.

[0114] An exemplary digital rock model 1550 that may be utilized in embodiments is shown in Figure 15. Figure 15 includes a close-up view 1551 showing copper leaching 1551 of a fractured middle portion (fractures in clay material) of rock 1552 through pore space surrounded by reactive minerals 1553a, microporous minerals 1552c, and unreactive minerals 1553b. Figure 15 also includes a close-up view 1554 of clay, modeled as Wyoming montmorillonite, above and below the digital rock as two layers 1557a-b, with copper ions 1558, chloride ions 1559, and water molecules 1560 between them. Still further, Figure 15 includes a view 1555 of the fluid in the fracture, modeled as a bulk molecular fluid containing copper ions 1561, chloride ions 1562, and water molecules 1563.

[0115] [Example Method Workflow] Described herein is a computer-implemented method for determining the behavior of a reacting flow system.

[0116] An embodiment provides a workflow for reactive transport simulations using single-scale and multiscale flows with multispecies reactions. In one embodiment, the velocity fields for both single-scale and multiscale flows are taken from LBM single-phase flow simulations and combined with molecular dynamics simulations for ion diffusivities and reactions from available databases for multiscale reactive flow simulations. The reactive transport model in the embodiment was validated with short-time simulations of calcite dissolution, and the results were compared with published results from other reactive flow codes. The molecular simulation portion of the embodiment was also validated using estimates of water diffusivity and compared with other available experimental values.

[0117] Using a systematic multiscale reactive flow workflow, we investigated the importance of surface reactions, competing bulk reactions, pH, and flow through porous media, combined with their sensitivity to both transport and reaction parameters. In all of these results, we analyzed the outlet copper ion concentration, the product of copper leaching, and investigated the composition of the product stream when homogeneous bulk reactions are included in addition to heterogeneous surface dissolution reactions. In conjunction with the outlet concentration, we examined 3D concentration profiles in digital rock models to illustrate the influence of both transport and reactivity, as well as microstructural connectivity.

[0118] Through these simulation results, it was observed that including homogeneous bulk reactions along with surface reactions (because they have a significant effect on the composition of the outlet stream) is important to accurately model the concentration of copper ions produced from copper leaching. It was also observed that pH significantly affects the chemical equilibrium of the aqueous solution, resulting in different relative fractions in the species composition of the outlet stream. Finally, multiscale flow through porous materials was included, which modeled nanoporous clay within the rock. This multiscale extension had a significant effect on the outlet copper ion concentration, as it could reach more reactive minerals that were not initially in direct contact with the fractures. Incorporating multiscale flow through fractures and nanoporous clay was shown to have a significant effect on predicting copper ion production from copper leaching.

[0119] Embodiments provide a robust workflow for multiscale reactive flow in 3D microstructured models, where reactions and fluid transport are coupled using molecular dynamics simulations, LBM fluid flow simulations, and reactive transport simulations. Furthermore, embodiments provide a simple and effective method for including bulk homogeneous reactions in reactive flow simulations, in addition to surface heterogeneous reactions. This has a significant effect on copper ion production in the outlet flow. Still further, embodiments can simulate multiscale flow through fractured and nanoporous materials combined with homogeneous and heterogeneous reactions to accurately model reactive flow through digital rock models.

[0120] The accurate results of the embodiments can be used in various real-world reactive flow applications. For example, the embodiments can be used to operate / control reactive flow systems, such as copper leaching applications, among other examples. In one such exemplary implementation, the embodiments can be utilized to determine optimized operating parameters for a copper leaching application, and then the real-world system can be controlled to operate according to the determined optimized parameters. Another example is the behavior of a reactive flow of CO2 dissolved in water flowing through concrete, a porous material, and reacting with mineral cement solid components.

[0121] The reactive flow modeling workflow described herein combines the following capabilities to accurately model reactive flow simulations: (1) accurate microstructural models based on 3D imaging (microCT), (2) multiscale flow through resolvable and non-resolvable pore(s), (3) a multiphase flow simulator to solve coupled gas and liquid simulations, (4) heterogeneous and homogeneous reactions to model reactive flow, (5) upscaling from pore-scale flow simulations to model field-scale reactive flow simulations, (6) molecular simulations to estimate the material, reaction, and diffusion properties necessary to model reactive flow, (7) a robust workflow in 3D microstructural models for multiphase, multiscale, and multispecies reacting flows (here, we combine reactions and fluid transport using molecular dynamics simulations, LBM fluid flow simulations, and reactive transport simulations), and (8) an effective method to include bulk homogeneous reactions in reactive flow simulations in addition to surface heterogeneous reactions. This has a significant effect on determining species generation in the outlet flow, as shown in the embodiments. (9) Multiphase flow through resolvable and non-resolvable porous materials combined with homogeneous and heterogeneous reactions to accurately model reacting flow through digital rock models. (10) Implementation of embodiments on Dassault Systemes' 3DEXPERIENCE® platform enables the optimization of multiphase, multiscale, multispecies reacting transport models, which can then be scaled up for field-scale simulations in industries such as mining, oil and gas, energy storage technology, chemical industry, and materials corrosion.

[0122] [Computer Support] FIG. 16 is a simplified block diagram of a computer-based system 2020 that may be used to determine the behavior of a reactive flow system according to any of the various embodiments described herein. The system 2020 includes a bus 2023. The bus 2023 serves as an interconnect between the various components of the system 2020. Connected to the bus 2023 is an input / output device interface 2026 for connecting various input and output devices, such as a keyboard, mouse, display, and speakers, to the system 2020. A central processing unit (CPU) 2022 is connected to the bus 2023 and provides for the execution of computer instructions to implement the embodiments. The memory 2025 provides volatile storage for data used in the execution of computer instructions to implement the embodiments described herein, such as method 110. The storage 2024 provides non-volatile storage for software instructions, such as an operating system (not shown) and configuration of the embodiments. The system 2020 also includes a network interface 2021 for connecting to any of the various networks known in the art, including wide area networks (WANs) and local area networks (LANs).

[0123] It should be understood that the exemplary embodiments described herein may be implemented in many different ways. In some cases, the various methods and machines described herein may each be implemented by a physical, virtual, or hybrid general-purpose computer, such as computer system 2020, or a computer network environment, such as computer environment 2120, described herein below in connection with FIG. 17. Computer system 2020 may be converted into a machine that performs the methods described herein by loading software instructions into either memory 2025 or non-volatile storage 2024, for example, for execution by CPU 2022. Those skilled in the art should further appreciate that system 2020 and its various components may be configured to perform any embodiment or combination of embodiments described herein. Furthermore, system 2020 may implement the various embodiments described herein utilizing any combination of hardware, software, and firmware modules operatively coupled internally or externally to system 2020.

[0124] 17 illustrates a computer network environment 2120 in which an embodiment of the present invention may be implemented. In the computer network environment 2120, a server 2121 is linked to clients 2123a-n via a communications network 2122. The environment 2120 may be used to enable the clients 2123a-n, alone or in combination with the server 2121, to perform any of the methods described herein. In a non-limiting example, the computer network environment 2120 provides a cloud computing embodiment, a software as a service (SAAS) embodiment, etc.

[0125] The embodiments or aspects thereof may be implemented in hardware, firmware, or software. If implemented in software, the software may be stored on any non-transitory computer-readable medium configured to allow a processor to load the software, or a subset of its instructions. The processor is then configured to execute the instructions and operate or cause an apparatus to operate in the manner described herein.

[0126] Furthermore, firmware, software, routines, or instructions may be described herein as performing certain operations and / or functions of a data processor, although it will be understood that such description contained herein is merely for convenience and that in reality such operations result from a computing device, processor, controller, or other device executing the firmware, software, routines, instructions, etc.

[0127] It should be understood that the flow diagrams, block diagrams, and network diagrams may include more or fewer elements, or may be arranged differently, or may be represented differently, but it should also be understood that the particular implementation may dictate the number of block diagrams and network diagrams that illustrate the execution of an embodiment implemented in a particular way.

[0128] Accordingly, further embodiments may also be implemented in a variety of computer architectures, physical, virtual, cloud computers, and / or any combination thereof, and therefore the data processors described herein are intended for illustrative purposes only and not as limitations of the embodiments.

[0129] While exemplary embodiments have been particularly shown and described, it will be understood by those skilled in the art that various changes in form and details may be made in the embodiments without departing from the scope of the embodiments as encompassed by the appended claims.

[0130] For example, the foregoing description and details of the illustrated embodiments refer to Applicant-Assignee (Dassault Systemes Americas Corporation) and Dassault Systemes tools and platforms for purposes of illustration, but not limitation. Other similar tools and platforms are suitable.

[0131] The teachings of all patents, published applications, and references cited herein are incorporated herein by reference in their entirety.

Claims

1. 1. A computer-implemented method for determining the behavior of a reacting flow system, comprising: defining a plurality of models of a reacting flow system, each defined model representing the reacting flow system at a respective scale; determining a velocity field of the reacting flow system using a first model at a first respective scale of the defined plurality of models; determining a diffusivity of the reacting flow system using a second model at a second respective scale of the defined plurality of models, wherein the determining of the velocity field and the determining of the diffusivity are performed automatically by one or more digital processors; and defining a plurality of reaction parameters for the reactive flow system; and automatically determining the behavior of the reacting flow system by using the determined velocity field, the determined diffusivity, and the defined plurality of reaction parameters as inputs to a reactive transport solver.

2. The method of claim 1 , wherein the given scale is a microscale, a molecular scale, or a subsurface scale.

3. The method of claim 1 , wherein at least one model of the defined plurality of models is a geometric model that characterizes the reacting flow system.

4. defining a given model of the plurality of models of the reacting flow system; defining a model of one or more heterogeneous surface reactions; modeling rate laws as a function of mineral dissolution and precipitation for said defined model of said one or more heterogeneous surface reactions; and defining a given model based on the modeled kinetic laws and a model of one or more homogeneous bulk reactions.

5. determining the velocity field of the reacting flow system using the first model; receiving an image of material within the reactive flow system; segmenting the image into a plurality of phases, each phase representing a material, a solid, or a fluid; and determining the velocity field of the reacting flow system based on the plurality of phases using the first model, wherein the first model is a single-phase fluid flow model.

6. The method of claim 5 wherein the material is porous.

7. The method of claim 5 , wherein the material further comprises one or more fractures.

8. The method of claim 5 , wherein the material is a nanoporous clay material.

9. The material is Cu 0.66 [Al 3.33 Mg 0.66 ][Si 8 ]O 20 [OH] 4 6. The method of claim 5, wherein the Wyoming montmorillonite has the formula:

10. The method of claim 1 , wherein the determined velocity field is a multiphase velocity field.

11. determining the diffusivity of the reacting flow system using the second model; providing parameters of the bulk salt solution and parameters of the salt solution in the clay interlayer nanopores as input for a molecular dynamics simulation; 2. The method of claim 1, comprising: performing a molecular dynamics simulation using the inputs of the molecular dynamics simulation as a function of temperature and salt concentration, wherein results of performing the molecular dynamics simulation indicate the diffusivity of the reacting flow system.

12. The method of claim 11 , wherein the diffusivity is an ionic diffusivity.

13. The ion diffusivity is copper (Cu) 2+ The method of claim 12, wherein the ionic diffusivity is

14. The method of claim 1 , wherein the plurality of reaction parameters for the reacting flow system are defined using input data.

15. The method of claim 14 , wherein the input data is obtained from at least one of a simulation result and a database.

16. determining the behavior of the reacting flow system by using the determined velocity field, the determined diffusivity, and the defined plurality of reaction parameters as inputs to the reactive transport solver; 2. The method of claim 1, comprising: solving an advection-diffusion-reaction equation using the reactive transport solver with the input, the results of the solving being indicative of the behavior of the reacting flow system.

17. The method of claim 1 , wherein the determined behavior of the reactive flow system is a concentration profile.

18. The concentration profile is copper (Cu) 2+ The method of claim 17, wherein the concentration profile is:

19. updating the first model and the second model based on the determined behavior of the reacting flow system; determining an updated velocity field of the reacting flow system using the updated first model; and determining an updated diffusivity for the reacting flow system using the updated second model; and 2. The method of claim 1, further comprising: determining an updated behavior of the reacting flow system by using the updated determined velocity field, the updated determined diffusivity, and the defined plurality of reaction parameters as inputs to the reactive transport solver.

20. 20. The method of claim 19, further comprising repeating (i) the updating, (ii) the determining an updated velocity field, (iii) the determining an updated diffusivity, and (iv) the determining an updated behavior of the reacting flow system until the determined updated behavior of the reacting flow system reaches a steady state.

21. 1. A computer-based system for determining the behavior of a reacting flow system, the computer-based system comprising: a processor; a memory having computer code instructions stored therein, wherein said processor and said memory use said computer code instructions to: defining a plurality of models of the reacting flow system, each defined model representing the reacting flow system at a respective scale; determining a velocity field of the reacting flow system using a first model at a first respective scale of the defined plurality of models; determining a diffusivity of the reacting flow system using a second model at a second respective scale of the defined plurality of models; defining a plurality of reaction parameters for the reactive flow system; and automatically determining the behavior of the reacting flow system by using the determined velocity field, the determined diffusivity, and the defined plurality of reaction parameters as inputs to a reactive transport solver.

22. 1. A non-transitory computer program product for determining the behavior of a reacting flow system, the non-transitory computer program product comprising: A computer-readable medium comprising program instructions that, when executed by a processor, defining a plurality of models of the reacting flow system, each defined model representing the reacting flow system at a respective scale; determining a velocity field of the reacting flow system using a first model at a first respective scale of the defined plurality of models; determining a diffusivity of the reacting flow system using a second model at a second respective scale of the defined plurality of models; defining a plurality of reaction parameters for the reactive flow system; automatically determining the behavior of the reacting flow system by using the determined velocity field, the determined diffusivity, and the defined plurality of reaction parameters as inputs to a reactive transport solver; a computer-readable medium that causes the processor to perform Including, Non-transitory computer program products.

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