##
**A framework for reactive transport modeling using FEniCS-Reaktoro: governing equations and benchmarking results.**
*(English)*
Zbl 1439.86004

Summary: Reactive transport codes are widely applied in geoscience to predict or reconstruct spatial and temporal evolution of geochemical systems. To provide an accurate description of natural systems at different spatial and temporal scales, the reactive transport code has to deal with coupling of different physical and chemical phenomena. Many reactive transport codes have been developed in the past and each of these codes has specific strengths and limitations. Here, we present a new versatile reactive transport framework based on the FEniCS equations solver and the chemical solver Reaktoro. This development was motivated by the need for an advanced open-source tool allowing user-friendly modeling environment and, at the same time, full control over the numerical methods. Unlike most of the currently available codes, the developed FEniCS-Reaktoro framework offers full flexibility in setting up the reactive transport simulations of arbitrary complexity in terms of process couplings, simulation domain geometry and the boundary conditions applied. The simulations are setup using a simple high-level scripting language intuitively linked to the equation based model definition without the need of advanced programming skills. The chemical solver Reaktoro allows thermodynamic modeling of multicomponent multiphase system with several fluids and solid phases, including highly non-ideal solid solutions. The coupling of transport and chemistry is implemented using the sequential non-iterative approach (SNIA) in which the transport of the aqueous components and the chemical reactions are solved in two consequent steps. The flexibility and results of the FEniCS-Reaktoro framework are demonstrated against several widely accepted reactive transport benchmarks.

### MSC:

86-08 | Computational methods for problems pertaining to geophysics |

65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |

65Y15 | Packaged methods for numerical algorithms |

### Keywords:

reactive transport; electrochemical transport; multicomponent diffusion; finite element method; porous media; Gibbs energy minimization; operator splitting approach### Software:

PROOST; ParCrunchFlow; TReacLab; CrunchFlow; PETSc; SUPCRT92; HYDROGEOCHEM; COMSOL-GEMS; FEniCS; MIN3P-THCm; PHREEQC; IPython; NumPy; GeoSysBRNS; MoMaS; FEniCS-Reaktoro; ORCHESTRA; PFLOTRAN; OpenGeoSys; FEniCS-HPC; TOUGHREACT; PhreeqcRM; Matplotlib
PDF
BibTeX
XML
Cite

\textit{L. H. Damiani} et al., Comput. Geosci. 24, No. 3, 1071--1085 (2020; Zbl 1439.86004)

Full Text:
DOI

### References:

[1] | Steefel, CI; DePaolo, DJ; Lichtner, PC, Reactive transport modeling: an essential tool and a new research approach for the Earth sciences, Earth Planet Sci. Lett., 240.3-4, 539-558 (2005) |

[2] | Bächler, D.; Kohl, T., Coupled thermal-hydraulic-chemical modelling of enhanced geothermal systems, Geophys. J Int., 161.2, 533-548 (2005) |

[3] | Darland, J.E., Inskeep, W.P.: Effects of pore water velocity on the transport of arsenate. J. Am. Chem. Soc. (1997) |

[4] | Lichtner, PC, Continuum model for simultaneous chemical reactions and mass transport in hydrothermal systems, Geochim. Cosmochim. Acta, 49.3, 779-800 (1985) |

[5] | Tournassat, C.; Steefel, CI, Ionic transport in nano-porous clays with consideration of electrostatic effects, Rev. Min. Geochem., 80, 287-329 (2015) |

[6] | Muniruzzaman, M.; Rolle, M., Modeling multicomponent ionic transport in groundwater with IPhreeqc coupling Electrostatic interactions and geochemical reactions in homogeneous and heterogeneous domains, Adv. Water Res., 98, 1-15 (2016) |

[7] | Konda, SSM, Chemical reactions modulated by mechanical stress: extended Bell theory, J. Chem. Phys., 135.16, 164103 (2011) |

[8] | Dmitrii, A., Kulik GEM-selektor geochemical modeling package: revised algorithm and GEMS3k numerical kernel for coupled simulation codes, Computat. Geosci., 17.1, 1-24 (2013) · Zbl 1356.86022 |

[9] | Centler, F., GeosysBRNS-a exible multidimensional reactive transport model for simulating biogeochemical subsurface processes, Comput. Geosci.-UK., 36.3, 397-405 (2010) |

[10] | Steefel, C., Molins, S.: CrunchFlow software for Modeling Multicomponent Reactive Flow and Transport (2016) |

[11] | Nardi, A., Interface COMSOL-PHREEQC (iCP), an efficient numerical framework for the solution of coupled multiphysics and geochemistry, Comput. Geosci.-UK., 69, 10-21 (2014) |

[12] | Guo, B., A COMSOL-PHREEQC interface for modeling the multi-species transport of saturated cement-based materials, Construct. Build Mater., 187, 839-853 (2018) |

[13] | Azad, VJ, A COMSOL-GEMS interface for modeling coupled reactive-transport geochemical processes, Comput. Geosci., 92, 79-89 (2016) |

[14] | Samper, J., et al.: Core^2D. A code for non-isothermal water flow and reactive solute transport. Users Manual version 2 (2000) |

[15] | Yapparova, A., Reactive transport modelling of dolomitisation using the new CSMP++GEM coupled code: governing equations, solution method and benchmarking results, Transport Porous Med., 117.3, 385-413 (2017) |

[16] | Elakneswaran, Y.; Ishida, T., Development and verification of an integrated physicochemical and geochemical modelling framework for performance assessment of cement-based materials, J. Adv. Concr. Technol., 12.4, 111-126 (2014) |

[17] | Jacques, D., Simunek, J.: User manual of the multicompenent variably-saturated ow and transport model hp1 (2005) |

[18] | Mayer, K.U.: A numerical model for multicomponent reactive transport in variably saturated porous media. PhD thesis (1999) |

[19] | Bethke, C.M.: Geochemical Reaction Modeling: Concepts and Applications. Oxford University Press, Oxford, pp. 416 (1996) |

[20] | Li, D., OpenGeoSys-ChemApp: a coupled simulator for reactive transport in multiphase systems and application to CO2 storage formation in Northern Germany, Acta Geotech., 9.1, 67-79 (2014) |

[21] | Kosakowski, G.; Watanabe, N., OpenGeoSys-Gem: a numerical tool for calculating geochemical and porosity changes in saturated and partially saturated media, Phys. Chem. Earth, 70-71, 138-149 (2014) |

[22] | He, W., A parallelization scheme to simulate reactive transport in the subsurface environment with OGS#IPhreeqc 5.5.7-3.1.2, Geosci. Model Dev., 8.10, 3333-3348 (2015) |

[23] | Meeussen, JCL, ORCHESTRA: An object-oriented framework for implementing chemical equilibrium models, Environ. Sci. Tech., 37.6, PMID: 12680672, 1175-1182 (2003) |

[24] | Lichtner, P.C., et al.: PFLOTRAN User Manual A Massively Parallel Reactive Flow and Transport Model for Describing Surface and Subsurface Processes (2015) |

[25] | Parkhurst, D.L., Appelo, C.A.J.: Description of Input and Examples for PHREEQC Version 3 - A Computer Program for Speciation, Batch-Reaction, One-Dimensional Transport, and Inverse Geochemical Calculations. U.S. Geological Survey Techniques and Methods, book 6, chapter A43, 497 p. U.S. Geological Survey Techniques and Methods, book 6, chapter A43, 6-43A (2013) |

[26] | Gamazo, P., PROOST: object-oriented approach to multiphase reactive transport modeling in porous media, J. Hydroinform., 18, 2, 310-328 (2016) |

[27] | Georget, F.; Prévost, JH; Huet, B., A reactive transport simulator for variable porosity problems, Computat. Geosci., 21.1, 95-116 (2017) |

[28] | Soetaert, K.; Meysman, F., Reactive transport in aquatic ecosystems Rapid model prototyping in the open source software R, Environ. Model. Softw., 32, 49-60 (2012) |

[29] | Xu, T., TOUGHREACT-a simulation program for non-isothermal multiphase reactive geochemical transport in variably saturated geologic media: applications to geothermal injectivity and CO2 geological sequestration, Comput. Geosci.-UK., 32.2, 145-165 (2006) |

[30] | Jara, D.; de Dreuzy, J-R; Cochepin, B., TReacLab: an object-oriented implementation of non-intrusive splitting methods to couple independent transport and geochemical software, Comput. Geosci-UK., 109, 281-294 (2017) |

[31] | Su, D.; Ulrich Mayer, K.; MacQuarrie, KTB, Parallelization of MIN3p-THCm: a high performance computational framework for subsurface flow and reactive transport simulation, Environ. Model. Softw., 95, 271-289 (2017) |

[32] | Trebotich, D., High-resolution simulation of pore-scale reactive transport processes associated with carbon sequestration, Comput. Sci. Eng., 16.6, 22-31 (2014) |

[33] | James, J., Beisman ParCrunchFlow: an efficient, parallel reactive transport simulation tool for physically and chemically heterogeneous saturated subsurface environments, Computat. Geosci., 19.2, 403-422 (2015) · Zbl 1392.86005 |

[34] | Smith, W.R., Missen, R.W.: Chemical reaction equilibrium analysis : theory and algorithms. Wiley, New York, pp. 364 (1982) |

[35] | Leal, MM, Computational methods for reactive transport modeling: an extended law of mass-action, xLMA, method for multiphase equilibrium calculations, Adv. Water Res., 96, 405-422 (2016) |

[36] | Comsol Multiphysics, The Platform for Physics-Based Mod- eling and Simulation. Comsol Inc., Burlington, (2013) |

[37] | Bell, LSJ; Binning, PJ, A split operator approach to reactive transport with the forward particle tracking Eulerian Lagrangian localized adjoint method, Adv Water Res., 27.4, 323-334 (2004) |

[38] | Saaltink, MW; Carrera, J.; Ayora, C., On the behavior of approaches to simulate reactive transport, J. Contam. Hydrol., 48.3-4, 213-35 (2001) |

[39] | Carrayrou, J., Comparison of numerical methods for simulating strongly nonlinear and heterogeneous reactive transport problems-the MoMaS benchmark case, Computat. Geosci., 14.3, 483-502 (2010) · Zbl 1426.76723 |

[40] | Hoffmann, J.; Kräutle, S.; Knabner, P., A parallel global-implicit 2-D solver for reactive transport problems in porous media based on a reduction scheme and its application to the MoMaS benchmark problem, Computat. Geosci., 14.3, 421-433 (2010) · Zbl 1425.76240 |

[41] | Yeh, GT; Tripathi, VS, A critical evaluation of recent developments in hydrogeochemical transport models of reactive multichemical components, Water Resour. Res., 25.1, 93-108 (1989) |

[42] | Walter, AL, Modeling of multicomponent reactive transport in groundwater: 1. Model development and evaluation, Water Resour. Res., 30.11, 3137-3148 (1994) |

[43] | Samper, J.; Xu, T.; Yang, C., A sequential partly iterative approach for multicomponent reactive transport with CORE2D, Computat. Geosci., 13.3, 301-316 (2009) · Zbl 1338.76066 |

[44] | Li, L., Expanding the role of reactive transport models in critical zone processes, Earth-Sci. Rev., 165, 280-301 (2017) |

[45] | Ravi, A., Patel A three-dimensional lattice Boltzmann method based reactive transport model to simulate changes in cement paste microstructure due to calcium leaching, Con- Str. Build. Mater., 166, 158-170 (2018) |

[46] | Fowler, D., Atmospheric composition change: ecosystems- atmosphere interactions, Atmos. Environ., 43.33, 5193-5267 (2009) |

[47] | Dentz, M., Mixing, spreading and reaction in heterogeneous media: a brief review, J. Contam. Hydrol., 120-121, 1-17 (2011) |

[48] | Ian, G., Microbiology in nuclear waste disposal: interfaces and reaction fronts, FEMS Microbiol. Rev., 20.3-4, 545-556 (1997) |

[49] | Kosakowski, G.; Berner, U., The evolution of clay rock/cement interfaces in a cementitious repository for low and intermediate level radioactive waste, Phys. Chem. Earth Parts A/B/C, 64, 65-86 (2013) |

[50] | Kolditz, O., OpenGeoSys: an open-source initiative for numerical simulation of thermo - hydro - mechanical / chemical (THM/c) processes in porous media, Environ. Earth Sci., 67.2, 589-599 (2012) |

[51] | Choo, J.; Lee, S., Enriched Galerkin finite elements for coupled poromechanics with local mass conservation, Comput. Methods Appl. Mech Eng., 341, 311-332 (2018) · Zbl 1440.74120 |

[52] | Lehmann, C.; Kolditz, O.; Nagel, T., The FEM Simulation Software OpenGeoSys, vol. 6, 29-48 (2018), Cham: Springer, Cham |

[53] | Sharma, PK; Joshi, N.; Ojha, CP, Reactive transport through porous media using finite-difference and finite-volume methods, J. Hydraul. Eng., 18.1, 11-19 (2012) |

[54] | Prasianakis, NI, Deciphering pore-level precipitation mechanisms, Sci. Rep., 7.1, 1-9 (2017) |

[55] | Zhao, C-b; Schaubs, P.; Hobbs, B., Effects of porosity heterogeneity on chemical dissolution-front instability in fluid-saturated rocks, J. Cent South Univ., 24.3, 720-725 (2017) |

[56] | Hatanaka, A., The impact of tortuosity on chloride ion diffusion in slag-blended cementitious materials, J. Adv. Concr. Technol., 15.8, 426-439 (2017) |

[57] | Navarre-Sitchler, A., Evolution of porosity and diffusivity associated with chemical weathering of a basalt clast, J. Geophys. Res., 114.F2, F02016 (2009) |

[58] | Ma, R., Assessment of controlling processes for field-scale uranium reactive transport under highly transient flow conditions, Water Resour. Res., 50.2, 1006-1024 (2014) |

[59] | Van Loon, L.R., et al.: Anisotropic diffusion in layered Argillaceous rocks: a case study with Opalinus clay (2004) |

[60] | Hommel, J.; Coltman, E.; Class, H., Porosity-permeability relations for evolving pore space: a review with a focus on (bio-)geochemically altered porous media, Transport Porous Med., 124.2, 589-629 (2018) |

[61] | Trinchero, P., Implications of grain-scale mineralogical heterogeneity for radionuclide transport in fractured media, Transport Porous Med., 116.1, 73-90 (2017) |

[62] | Steefel, CI; Yabusaki, SB; Ulrich Mayer, K., Reactive transport benchmarks for subsurface environmental simulation, Computat. Geosci., 19.3, 439-443 (2015) |

[63] | Hunter, JD, Matplotlib: a 2D graphics environment, Comput. Sci. Eng., 9.3, 90-95 (2007) |

[64] | Perez, F.; Granger, BE, IPython: A system for interactive scientific computing, Comput. Sci. Eng., 9.3, 21-29 (2007) |

[65] | McKinney, W.: Data structures for statistical computing in Python (2010) |

[66] | Oliphant, T.E.: Guide to NumPy. 2nd USA: Createspace Independent Publishing Platform (2015) |

[67] | Martin, S., Alnaes The FEniCS Project Version 1.5, Arch. Numer. Softw., 3.100, 9-23 (2015) |

[68] | Allan, MM, An overview of computational methods for chemical equilibrium and kinetic calculations for geochemical and reactive transport modeling, Pure Appl. Chem., 89.5, 597-643 (2017) |

[69] | Alnæs, MS, Unified form language: a domain speci fic language for weak formulations of partial differential equations, ACM Trans. Math. Softw., 40.2, 9:1-9:37 (2014) |

[70] | Hoffman, JJJ; Jansson, N., FEniCS-HPC automated predictive high-performance finite element computing with applications in aerodynamics. Parallel Process, Appl. Math., 9573, 356-365 (2016) |

[71] | Abhyankar, S., et al.: PETSc/TS: A modern scalable ODE/DAE Solver Library. arXiv:1806.01437 (2018) |

[72] | Spycher, N.; Pruess, K.; Ennis-King, J., CO2-H2O mixtures in the geological sequestration of CO2. I. Assessment and calculation of mutual solubilities from 12 to 100 ∘C and up to 600 bar, Geochim. Cosmochim. Acta, 67.16, 3015-3031 (2003) |

[73] | Spycher, NF; Reed, MH, Fugacity coefficients of H2, CO2, CH4, H2O and of H2O- CO2-CH4 mixtures: a virial equation treatment for moderate pressures and temperatures applicable to calculations of hydrothermal boiling, Geochim. Cosmochim. Acta, 52.3, 739-749 (1988) |

[74] | Matschei, T.; Lothenbach, B.; Glasser, FP, Thermodynamic data for hydrated solids in Portland cement system CaO-Al2o3-Sio2-CaSO4-CaCO3- Fe2O3-MgO-H2O, Cem. Concr. Res., 37, 1379-1410 (2007) |

[75] | Thoenen, T., et al.: The PSI/Nagra Chemical Thermodynamic Database 12/07 Nuclear Energy and Safety Research Department Laboratory forWaste Management (LES) (2014) |

[76] | Johnson, JW; Oelkers, EH; Helgeson, HC, SUPCRT92: a software package for calculating the standard molal thermodynamic properties of minerals, gases, aqueous species, and reactions from 1 to 5000 bar and 0 to 1000 ∘c, Comput. Geosci., 18.7, 899-947 (1992) |

[77] | Wigger, C.; Van Loon, LR, Importance of interlayer equivalent pores for anion diffusion in clay-rich sedimentary rocks, Environ. Sci. Tech., 51.4, 1998-2006 (2017) |

[78] | Van Loon, LR; Glaus, MA; Müller, W., Anion exclusion effects in compacted bentonites: towards a better understanding of anion diffusion, Appl. Geochem., 22.11, 2536-2552 (2007) |

[79] | Altmann, S., Diffusion-driven transport in clayrock formations, Appl. Geochem., 27.2, 463-478 (2012) |

[80] | Bear, J., Bachmat, Y.: Introduction to Modeling of Transport Phenomena in Porous Media. Springer, Dordrecht (1990) · Zbl 0743.76003 |

[81] | Samson, E.; Marchand, J., Numerical solution of the extended Nernst-Planck model, J Colloid Interface Sci., 215, 1-8 (1999) |

[82] | Daus, AD; Frind, EO, An alternating direction Galerkin technique for simulation of contaminant transport in complex groundwater systems, Water Resour. Res., 21.5, 653-664 (1985) |

[83] | Courant, R.; Friedrichs, K.; Lewy, H., ÜBer die partiellen Differenzengleichungen der mathematischen Physik, Mathematische Annalen, 100.1, 32-74 (1928) · JFM 54.0486.01 |

[84] | Isaacson, E., Keller, H.B.: Analysis of numerical methods dover books on mathematics. Dover Publications (1994) |

[85] | Murdoch, JR, What is the rate-limiting step of a multistep reaction?, J. Chem. Educ., 58.1, 32 (1981) |

[86] | Zhang, Y.: Geochemical kinetics. Princeton University Press, pp. 631 (2008) |

[87] | Arnold, D.N., et al.: Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems (2002) · Zbl 1008.65080 |

[88] | Riviére, B.: Discontinuous Galerkin methods for solving elliptic and parabolic equations : theory and implementation. SIAM, Society for Industrial and Applied Mathematics, pp. 190 (2008) · Zbl 1153.65112 |

[89] | Zhang, C.; Zarrouk, SJ, Rosalind Archer. A mixed finite element solver for natural convection in porous media using automated solution techniques, Comput. Geosci.-UK., 96, 181-192 (2016) |

[90] | Houston, P.; Sime, N., Automatic symbolic computation for discontinuous Galerkin finite element methods, SIAM J. Sci Comput., 40.3, C327-C357 (2018) · Zbl 1397.65268 |

[91] | Berner, U.; Kulik, DA; Kosakowski, G., Geochemical impact of a low-pH cement liner on the near field of a repository for spent fuel and high-level radioactive waste, Phys. Chem. Earth, 64, 46-56 (2013) |

[92] | Steefel, CI; Maher, K., Fluid-rock interaction: a reactive transport approach, Rev. Mineral. Geochem., 70.1, 485-532 (2009) |

[93] | Quintard, M.; Whitaker, S., Transport in ordered and disordered porous media: volume - averaged equations, closure problems, and comparison with experiment, Chem. Eng. Sci., 48.14, 2537-2564 (1993) |

[94] | Lasaga, A.C.: Kinetic Theory in the Earth Sciences, pp. 811. arXiv:1011.1669v3 (1998) |

[95] | Rumbaugh, J., Jacobson, I., Booch, G.: The unified modeling language reference manual. Addison-Wesley, pp. 550 (1999) |

[96] | Rasouli, P., Benchmarks for multicomponent diffusion and electrochemical migration, Computat. Geosci., 19.3, 523-533 (2015) |

[97] | Lichtner, P.C.: Principles and Practice of Reactive Transport Modeling (1994) |

[98] | Martin, A., Seeming steady-state uphill diffusion of 22Na+ in compacted montmorillonite, Environ. Sci. Technol., 47.20, 11522-11527 (2013) |

[99] | Glaus, MA, Cation diffusion in the electrical double layer enhances the mass transfer rates for Sr2+, Co2+ and Zn2+ in compacted illite, Geochim. Cosmochim. Acta, 165, 376-388 (2015) |

[100] | Massimo, R., Nernst-Planck-based description of transport, coulombic interactions, and geochemical reactions in porous media: modeling approach and benchmark experiments, Water Resour. Res., 54.4, 3176-3195 (2018) |

[101] | Maes, N., Determination of the diffusion coefficient of ionic species in boom clay by electromigration: feasibility study, Radiochim. Acta, 82.s1, 183-190 (1998) |

[102] | Berner, U.: Radionuclide concentration limits in the cementitious near-field of an ILW repository (2003) |

[103] | Jenni, A., In situ interaction between different concretes and Opalinus clay, Phys. Chem. Earth, 70-71, 71-83 (2014) |

[104] | Engesgaard, P.; Kipp, LK, A geochemical transport model for redox-controlled movement of mineral fronts in groundwater ow systems: a case of nitrate removal by oxidation of pyrite, Water Resour. Res., 28.10, 2829-2843 (1992) |

[105] | Prommer, H.; Barry, DA; Zheng, C., MODFLOW/ MT3DMS - Based reactive multicomponent transport modeling, Ground Water, 41.2, 247-257 (2002) |

[106] | Shao, H., Modeling reactive transport in non-ideal aqueous-solid solution system, Appl. Geochem., 24.7, 1287-1300 (2009) |

[107] | He, W.: Code verification: Engesgaard benchmark. Open- GeoSys Tutorial: Computational Hydrology III: OGS # IPhreeqc Coupled Reactive Transport Modeling. Springer International Publishing, Berlin, pp. 31-35 (2018) |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.