Accelerated reactive transport simulations in heterogeneous porous media using Reaktoro and Firedrake. (English) Zbl 1485.86005

Summary: This work investigates the performance of the on-demand machine learning (ODML) algorithm introduced in [the fourth author et al., “Accelerating reactive transport modeling: on-demand machine learning algorithm for chemical equilibrium calculations”, Transp. Porous Media 133, No. 2, 161–204 (2020; doi:10.1007/s11242-020-01412-1)] when applied to different reactive transport problems in heterogeneous porous media. This approach was devised to accelerate the computationally expensive geochemical reaction calculations in reactive transport simulations. We demonstrate that even with a strong heterogeneity present, the ODML algorithm speeds up these calculations by one to three orders of magnitude. Such acceleration, in turn, significantly advances the entire reactive transport simulation. The performed numerical experiments are enabled by the novel coupling of two open-source software packages: Reaktoro [the fourth author, “Reaktoro: An open-source unified framework for modeling chemically reactive systems” (2015), https://reaktoro.org] and Firedrake [F. Rathgeber et al., ACM Trans. Math. Softw. 43, No. 3, Article No. 24, 27 p. (2017; Zbl 1396.65144)]. The first library provides the most recent version of the ODML approach for the chemical equilibrium calculations, whereas, the second framework includes the newly implemented conservative Discontinuous Galerkin finite element scheme for the Darcy problem, i.e., the Stabilized Dual Hybrid Mixed(SDHM) method [Y. Núñez et al., “A Mixed-Hybrid Finite Element Method Applied to Tracer Injection Processes”, Int. J. Model. Simul. Petroleum Ind., 6, 51–59 (2012)].


86A05 Hydrology, hydrography, oceanography
68T05 Learning and adaptive systems in artificial intelligence
76S05 Flows in porous media; filtration; seepage
76V05 Reaction effects in flows


Zbl 1396.65144
Full Text: DOI arXiv


[1] Ahusborde, E.; El Ossmani, M.; Id Moulay, M., A fully implicit finite volume scheme for single phase flow with reactive transport in porous media, Math. Comput. Simul., 164, 3-23 (2019) · Zbl 07316717
[2] Alnæs, MS; Logg, A.; Ølgaard, KB; Rognes, ME; Wells, GN, Unified form language: a domain-specific language for weak formulations and partial differential equations, ACM Trans. Math. Softw., 40, 2, Art. 9, 37 (2014) · Zbl 1308.65175
[3] Amir, L.; Kern, M., Preconditioning a coupled model for reactive transport in porous media, Int. J. Numer. Anal. Model., 16, 1, 18-48 (2019) · Zbl 1412.65126
[4] Arnold, DN; Brezzi, F., Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates, RAIRO Modél. Math. Anal. Numér., 19, 1, 7-32 (1985) · Zbl 0567.65078
[5] Arnold, D. N., Brezzi, F., Cockburn, B., Marini, L. D.: Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39(5), 1749-1779 (2001/02) · Zbl 1008.65080
[6] Azad, VJ; Li, C.; Verba, C.; Ideker, JH; Isgor, OB, A COMSOL-GEMS interface for modeling coupled reactive-transport geochemical processes, Comput. Geosci., 92, 79-89 (2016)
[7] Bȧchler, D.; Kohl, T., Coupled thermal-hydraulic-chemical modelling of enhanced geothermal systems, Geophys. J. Int., 161, 2, 533-548 (2005)
[8] Berrone, S., Della Santa, F., Pieraccini, S., Vaccarino, F.: Machine learning for flux regression in discrete fracture networks. GEM - Int. J. Geomath. 12(9) (2021) · Zbl 1476.65291
[9] Bethke, CM, Geochemical and Biogeochemical Reaction Modeling (2007), New York: Cambridge University Press, New York
[10] Bilke, L.; Flemisch, B.; Kalbacher, T.; Kolditz, O.; Helmig, R.; Nagel, T., Development of Open-Source porous media simulators: Principles and experiences, Transp. Porous Media, 130, 337-361 (2019)
[11] Bochev, PB; Gunzburger, MD, Finite element methods of least-squares type, SIAM Rev., 40, 4, 789-837 (1998) · Zbl 0914.65108
[12] Brezzi, F.; Fortin, M., A minimal stabilisation procedure for mixed finite element methods, Numer. Math., 89, 3, 457-491 (2001) · Zbl 1009.65067
[13] Brooks, A. N., Hughes, T. J. R.: Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations. Comput. Methods Appl. Mech. Engrg. 32(1-3), 199-259. FENOMECH ”81, Part I (Stuttgart 1981) (1982) · Zbl 0497.76041
[14] Carrayrou, J., Hoffmann, J., Knabner, P., Kräutle, S., de Dieuleveult, C., Erhel, J., Van der Lee, J., Lagneau, V., Mayer, K. U., MacQuarrie, K. T. B.: Comparison of numerical methods for simulating strongly nonlinear and heterogeneous reactive transport problems – the momas benchmark case. Comput. Geosci. (14), 483-502 (2010a) · Zbl 1426.76723
[15] Carrayrou, J., Kern, M., Knabner, P.: Reactive transport benchmark of MoMaS. Comput. Geosci. (14) (2010b) · Zbl 1425.76236
[16] Carrayrou, J.; Mosé, R.; Behra, P., Operator-splitting procedures for reactive transport and comparison of mass balance errors, J. Contam. Hydrol., 68, 3-4, 239-268 (2004)
[17] Centler, F.; Shao, H.; De Biase, C.; Park, CH; Regnier, P.; Kolditz, O.; Thullner, M., GeosysBRNS-a flexible multidimensional reactive transport model for simulating biogeochemical subsurface processes, Comput. Geosci., 36, 3, 397-405 (2010)
[18] Cockburn, B.; Gopalakrishnan, J., A characterization of hybridized mixed methods for second order elliptic problems, SIAM J. Numer. Anal., 42, 1, 283-301 (2004) · Zbl 1084.65113
[19] Cockburn, B.; Gopalakrishnan, J.; Lazarov, R., Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems. SIAM, J. Numer. Anal., 47, 2, 1319-1365 (2009) · Zbl 1205.65312
[20] Correa, MR; Loula, AFD, Stabilized velocity post-processings for Darcy flow in heterogeneous porous media, Comm. Numer. Methods Engrg., 23, 6, 461-489 (2007) · Zbl 1259.76018
[21] Correa, MR; Loula, AFD, Unconditionally stable mixed finite element methods for Darcy flow, Comput. Methods Appl. Mech. Engrg., 197, 17-18, 1525-1540 (2008) · Zbl 1194.76109
[22] Damiani, LH; Kosakowski, G.; Glaus, MA; Churakov, SV, A framework for reactive transport modeling using FEniCS-reaktoro: governing equations and benchmarking results, Comput. Geosci., 24, 3, 1071-1085 (2020) · Zbl 1439.86004
[23] de Dieuleveult, C., Erhel, J.: A global approach to reactive transport: application to the MoMas benchmark. Comput. Geosci. (14), 451-464 (2010) · Zbl 1425.76294
[24] Debye, P.; Hückel, E., The theory of electrolytes. 1. lowering of freezing point and related phenomena, Phys. Zeitsch., 24, 85-206 (1923)
[25] Drummond, S.: Boiling and Mixing of Hydrothermal Fluids: Chemical Effects on Mineral Precipitation. Ph.d, Pennsylvania State University (1981)
[26] 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)
[27] Fuks, O.; Tchelepi, HA, Limitations of physics informed machine learning for nonlinear two-phase transport in porous media, J. Mach. Learn. Model. Comput., 1, 1, 19-37 (2020)
[28] Gamazo, P.; Slooten, LJ; Carrera, J.; Saaltink, MW; Bea, S.; Soler, J., PROOST: Object-Oriented approach to multiphase reactive transport modeling in porous media, J. Hydroinf., 18, 2, 310-328 (2016)
[29] Georget, F.; Prėvost, JH; Huet, B., A reactive transport simulator for variable porosity problems, Comput. Geosci., 21, 1, 95-116 (2017)
[30] Georget, F.; Prévost, JH; Huet, B., A reactive transport simulator for variable porosity problems, Comput. Geosci., 21, 1, 95-116 (2017)
[31] Guo, B.; Hong, Y.; Qiao, G.; Ou, J., A COMSOL-PHREEQC interface for modeling the multi-species transport of saturated cement-based materials, Construct. Build Mater., 187, 839-853 (2018)
[32] Harvie, CE; Møller, N.; Weare, JH, The prediction of mineral solubilities in natural waters: The Na-K-Mg-Ca-H-Cl-SO4-OH-HCO3-CO3-CO2-H2O system to high ionic strengths at 25∘,C, Geoch. Cosmoch. Acta, 48, 4, 723-751 (1984)
[33] He, W.; Beyer, C.; Fleckenstein, JH; Jang, E.; Kolditz, O.; Naumov, D.; Kalbacher, T., A parallelization scheme to simulate reactive transport in the subsurface environment with OGS#IP hreeqc 5.5.7-3.1.2, Geosci. Model Dev., 8, 10, 3333-3348 (2015)
[34] Helgeson, H. C., Delany, J. M., Nesbitt, H. W., Bird, D. K.: Summary and critique of the thermodynamic properties of rock-forming minerals. Amer. J. Sci. 278 A(1), 229 (1978)
[35] Helgeson, HC; Kirkham, DH, Theoretical prediction of the thermodynamic behavior of aqueous electrolytes at high pressures and temperatures: I. Summary of the thermodynamic/electrostatic properties of the solvent, Am. J. Sci., 274, 10, 1089-1198 (1974)
[36] Helgeson, HC; Kirkham, DH, Theoretical prediction of the thermodynamic behavior of aqueous electrolytes at high pressures and temperatures: II. Debye-huckel parameters for activity coefficients and relative partial molal properties, Am. J. Sci., 274, 10, 1199-1261 (1974)
[37] Helgeson, HC; Kirkham, DH, Theoretical prediction of the thermodynamic properties of aqueous electrolytes at high pressures and temperatures: III. Equation of state for aqueous species at infinite dilution, Am. J. Sci., 276, 2, 97-240 (1976)
[38] Helgeson, HC; Kirkham, DH; Flowers, GC, Theoretical prediction of the thermodynamic behavior of aqueous electrolytes at high pressures and temperatures: IV. Calculation of activity coefficients, osmotic coefficients, and apparent molal and standard and relative partial molal properties to 600 C, Am. J. Sci., 281, 10, 1249-1516 (1981)
[39] 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. Comput. Geosci. (14), 421-433 (2010) · Zbl 1425.76240
[40] Jacques, D., Simunek, J.: User manual of the multicompenent variably-saturated flow and transport model hp1 (2005)
[41] Jara, D.; de Dreuzy, JR; Cochepin, B., TREaclab: An object-oriented implementation of non-intrusive splitting methods to couple independent transport and geochemical software, Comput. Geosci., 109, 281-294 (2017)
[42] Jasakh, H.: OpenFOAM: Open source CFD in research and industry. International Journal of Naval Architecture and Ocean Engineering (2012)
[43] 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)
[44] Kolditz, O., Bauer, S., Bilke, L., Böttcher, N., Delfs, J., Fischer, T., Görke, U., Kalbacher, T., Kosakowski, G., McDermott, C., Park, C., Rad, U. F., Rink, K., Shao, H., Sun, F., Sun, Y., Singh, A., Taron, J., Walther, M., Wang, W., Watanabe, N., Wu, N., Xie, M., Xu, W., Zehner, B.: Opengeosys. OpenGeoSys (OGS) is a scientific open source project for the development of numerical methods for the simulation of thermo-hydro-mechanical-chemical (THMC) processes in porous and fractured media (2019)
[45] Kolditz, O., Görke, U.-J., Shao, H., Wang, W.: Thermo-Hydro-Mechanical-Chemical Processes in Porous Media - Benchmarks and Examples, vol. 86. Springer (2012)
[46] Kolditz, O., Nagel, T., Shao, H., Wang, W., Sebastian, B.: Thermo-Hydro-Mechanical-Chemical Processes in Fractured Porous Media. Modelling and Benchmarking - From Benchmarking to Tutoring. Springer International Publishing (2018)
[47] 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)
[48] Kulik, D., Berner, U. R., Curti, E.: Modeling Chemical Equilibrium Partitioning with the GEMS-PSI Code. Technical Report March Paul Scherrer Institut, Villigen (2004)
[49] Kulik, DA; Wagner, T.; Dmytrieva, SV; Kosakowski, G.; Hingerl, FF; Chudnenko, KV; Berner, UR, GEM-Selektor geochemical modeling package: revised algorithm and GEMS3k numerical kernel for coupled simulation codes, Comput. Geosci., 17, 1, 1-24 (2013) · Zbl 1356.86022
[50] Langevin, C., Hughes, J., Banta, E., Provost, A., Niswonger, R., Panday, S.: MODFLOW 6 Modular Hydrologic Model version 6.1.0: U.S. Geological Survey Software Release (2019)
[51] Leal, A. M.: Reaktoro: A unified framework for modeling chemically reactive systems (2015)
[52] Leal, AM; Blunt, MJ; LaForce, TC, Efficient chemical equilibrium calculations for geochemical speciation and reactive transport modelling, Geochim. Cosmochim. Acta, 131, 301-322 (2014)
[53] Leal, AM; Blunt, MJ; LaForce, TC, A chemical kinetics algorithm for geochemical modelling, Appl. Geochem., 55, 46-61 (2015)
[54] Leal, AM; Kulik, DA; Kosakowski, G., Computational methods for reactive transport modeling: A Gibbs energy minimization approach for multiphase equilibrium calculations, Adv. Water Resour., 88, 231-240 (2016)
[55] Leal, AM; Kulik, DA; Kosakowski, G.; Saar, MO, Computational methods for reactive transport modeling: An extended law of mass-action, xLMA, method for multiphase equilibrium calculations, Adv. Water Resour., 96, 405-422 (2016)
[56] Leal, A. M. M., et al.: autodiff, a modern, fast and expressive C++ library for automatic differentiation. https://autodiff.github.io (2018)
[57] Leal, AMM; Kulik, DA; Smith, WR; Saar, MO, 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)
[58] Leal, AMM; Kyas, S.; Kulik, DA; Saar, MO, Accelerating reactive transport modeling: on-demand machine learning algorithm for chemical equilibrium calculations, Transp. Porous Media, 133, 2, 161-204 (2020)
[59] Li, D.; Bauer, S.; Benisch, K.; Graupner, B.; Beyer, C., 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)
[60] Lichtner, PC, Continuum model for simultaneous chemical reactions and mass transport in hydrothermal systems, Geochim. Cosmochim. Acta, 49, 3, 779-800 (1985)
[61] Lichtner, P. C., Hammond, G. E., Lu, C., Karra, S., Bisht, G., Andre, B., Mills, R. T., Kumar, J., Frederick, J. M.: PFLOTRAN Web page. http://www.pflotran.org (2019)
[62] Logg, A.; Wells, GN, DOLFIN, ACM Trans. Math. Softw., 37, 2, 1-28 (2010) · Zbl 1364.65254
[63] Logg, A., Wells, G. N.: Automated Solution of Differential Equations by the Finite Element Method, volume 84 of Lecture Notes in Computational Science and Engineering. Springer, Berlin (2012) · Zbl 1247.65105
[64] Loula, AFD; Correa, MR; Guerreiro, JNC; Toledo, EM, On finite element methods for heterogeneous elliptic problems, Internat. J. Solids Struct., 45, 25-26, 6436-6450 (2008) · Zbl 1168.74459
[65] Malta, SMC; Loula, AFD, Numerical analysis of finite element methods for miscible displacements in porous media, Numer. Methods Partial Differ. Equ., 14, 4, 519-548 (1998) · Zbl 0932.76035
[66] Malta, SMC; Loula, AFD; Garcia, ELM, Numerical analysis of a stabilized finite element method for tracer injection simulations, Comput. Methods Appl. Mech. Engrg., 187, 1-2, 119-136 (2000) · Zbl 0958.76044
[67] Marcato, A., Boccardo, G., Marchisio, D.: A computational workflow to study particle transport and filtration in porous media: Coupling CFD and deep learning. Chem. Eng. J. 417(128936) (2021)
[68] Masud, A.; Hughes, TJR, A stabilized mixed finite element method for Darcy flow, Comput. Methods Appl. Mech. Engrg., 191, 39-40, 4341-4370 (2002) · Zbl 1015.76047
[69] Mayer, K., Frind, E., Blowes, D.: Multicomponent reactive transport modeling in variably saturated porous media using a generalized formulation for kinetically controlled reactions. Water Resour. Res. 38(1174) (2002)
[70] Meeussen, J.: ORCHESTRA: An object-oriented frame- work for implementing chemical equilibrium models. Technical Report 37 (2003)
[71] Mo, S.; Zhu, Y.; Zabaras, N.; Shi, X.; Wu, J., Deep convolutional Encoder-Decoder networks for uncertainty quantification of dynamic multiphase flow in heterogeneous media, J. Comput. Phys., 5, 1, 703-728 (2019)
[72] Müller, S., Schüler, L.: Geostat-framework/gstools: Reverberating red (version v1.1.0). doi:10.5281/zenodo.3468230 (2019)
[73] 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 Resour., 98, 1-15 (2016)
[74] Nardi, A.; Idiart, A.; Trinchero, P.; De Vries, LM; Molinero, J., Interface COMSOL-PHREEQC (iCP), an efficient numerical framework for the solution of coupled multiphysics and geochemistry, Comput. Geosci., 69, 10-21 (2014)
[75] Nguyen, N. C., Peraire, J., Cockburn, B.: Hybridizable discontinuous Galerkin methods. In: Spectral and High Order Methods for Partial Differential Equations, pp. 63-84. Springer (2011) · Zbl 1216.65160
[76] Núñez, Y., Faria, C., Loula, A., Malta, S.: A mixed-hybrid finite element method applied to tracer injection processes. Int. J. Model. Simul. Petroleum Industry, 6 (2012)
[77] Núñez, YR; Faria, CO; Loula, AFD; Malta, SMC, A hybrid finite element method applied to miscible displacements in heterogeneous porous media, Rev. Int. Mé,tod. Numér. Cálc. Diseño Ing., 33, 1-2, 45-51 (2017)
[78] Núñez, YR; Faria, CO; Malta, SMC; Loula, AFD, The influence of velocity field approximations in tracer injection processes, TEMA Tend. Mat. Apl. Comput., 19, 2, 347-367 (2018)
[79] Oliveira, TD; Blunt, MJ; Bijeljic, B., Modelling of multispecies reactive transport on pore-space images, Adv. Water Resour., 127, 192-208 (2019)
[80] Parkhurst, D.; Appelo, C., User’s guide to PHREEQC (Version 2)—A computer program for speciation, batch-reaction, one-dimensional transport, and inverse geochemical calculations, USGS Water-Resour. Invest. Report, 99, 4259, 326 (1999)
[81] Parkhurst, D., Appelo, C.: Description of input and examples for PHREEQC version 3—A computer program for speciation, batch-reaction, one-dimensional transport, and inverse geochemical calculations. In Groundwater Book 6, Modeling Techniques, chapter A43, pp. 497. U.S. Geological Survey Techniques and Methods (2013)
[82] Peaceman, DW, Fundamental of Numerical Reservoir Simulation (1977), Amsterdam: Elsevier, Amsterdam
[83] Pitzer, KS, Thermodynamics of electrolytes. I. Theoretical basis and general equations, J. Phys. Chem., 77, 2, 268-277 (1973)
[84] Raissi, M.; Perdikaris, P.; Karniadakis, GE, Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations, J. Comput. Phys., 378, 1, 686-707 (2019) · Zbl 1415.68175
[85] Rathgeber, F., Ham, D. A., Mitchell, L., Lange, M., Luporini, F., McRae, A. T., Bercea, G. T., Markall, G. R., Kelly, P. H.: Firedrake: Automating the finite element method by composing abstractions. ACM Trans. Math. Softw. 43(3) (2016) · Zbl 1396.65144
[86] Samper, J., Juncosa, R., Delgado, J., Montenegro, L.: CORE. A code for non-isothermal water flow and reactive solute transport. Users manual version 2 (2000)
[87] Shock, E.; Helgeson, HC, Calculation of the thermodynamic and transport properties of aqueous species at high pressures and temperatures: Correlation algorithms for ionic species and equation of state predictions to 5 kb and 1000∘c, Geochim. Cosmochim. Acta, 52, 8, 2009-2036 (1988)
[88] Shock, EL; Oelkers, EH; Johnson, JW; Sverjensky, DA; Helgeson, HC, Calculation of the thermodynamic properties of aqueous species at high pressures and temperatures. Effective electrostatic radii, dissociation constants and standard partial molal properties to 1000 ∘c and 5 kbar, J. Chem. Soc. Faraday Trans., 88, 6, 803 (1992)
[89] Simunek, J.; van Genuchten, M., Modeling non-equilibrium flow and transport processes using HYDRUS, Vadose Zone J., 7, 2, 782-797 (2008)
[90] Smith, W.; Missen, R., Chemical Reaction Equilibrium Analysis: Theory and Algorithms (1982), New York: Wiley-Interscience, New York
[91] Steefel, C.; Depaolo, D.; Lichtner, P., 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)
[92] Steefel, C. I.: Crunchflow: Software for modeling multicomponent reactive flow and transport. Technical report (2009)
[93] Steefel, CI, Reactive transport at the crossroads, Rev. Mineral. Geochem., 85, 1, 1-26 (2019)
[94] Steefel, CI; Appelo, CAJ; Arora, B.; Jacques, D.; Kalbacher, T.; Kolditz, O.; Lagneau, V.; Lichtner, PC; Mayer, KU; Meeussen, JCL; Molins, S.; Moulton, D.; Shao, H.; Šimunek, J.; Spycher, N.; Yabusaki, SB; Yeh, GT, Reactive transport codes for subsurface environmental simulation, Comput. Geosci., 19, 3, 445-478 (2015) · Zbl 1323.86002
[95] Sun, A., Yoon, H., Shih, C.-Y., Zhong, Z.: Applications of physics-informed scientific machine learning in subsurface science: A survey. (arXiv:2104.04764 [physics.geo-ph]) (2021)
[96] Tang, M., Liu, Y., Durlofsky, L. J.: A deep-learning-based surrogate model for data assimilation in dynamic subsurface flow problems. J. Comput. Phys. 130(109456) (2020) · Zbl 1436.76058
[97] Tanger, JC; Helgeson, HC, Calculation of the thermodynamic and transport properties of aqueous species at high pressures and temperatures; revised equations of state for the standard partial molal properties of ions and electrolytes, Am. J. Sci., 288, 1, 19-98 (1988)
[98] Van der Lee, J.; De Windt, L.; Lagneau, V.; Goblet, P., Module-oriented modeling of reactive transport with HYTEC, Comput. Geosci., 29, 3, 265-275 (2003)
[99] Wagner, W.; Pruss, A., The IAPWS formulation 1995 for the thermodynamic properties of ordinary water substance for general and scientific use, J. Phys. Chem. Ref. Data, 31, 2, 387 (2002)
[100] Wen, G., Hay, C., Benson, S. M.: CCSNet: a deep learning modeling suite for CO2 storage. (arXiv:2104.01795 [physics.flu-dyn]) (2021)
[101] White, M., Oostrom, M.: Stomp subsurface transport over multiple phases version 4.0 user’s guide. Technical report (2006)
[102] Xiao, Y., Whitaker, F., Xu, T., Steefel, C.: Reactive transport modeling: Applications in subsurface energy and environmental problems (2018)
[103] Xu, T.; Sonnenthal, E.; Spycher, N.; Pruess, K., 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., 32, 2, 145-165 (2006)
[104] Yan, B., Harp, D. R., Chen, B., Pawar, R.: A Physics-Constrained Deep Learning Model for Simulating Multiphase Flow in 3D Heterogeneous Porous Media. (arXiv:2105.09467 [physics.geo-ph]) (2021)
[105] Yapparova, A.; Gabellone, T.; Whitaker, F.; Kulik, DA; Matthȧi, SK, Reactive transport modelling of dolomitisation using the new CSMP++GEM coupled code: Governing equations, solution method and benchmarking results, Transp. Porous Media, 117, 3, 385-413 (2017)
[106] Yeh, G., Tsai, C., Ni, C.: Hydrogeochem 6.0: A model to couple thermal-hydrology-mechanics-chemical (thmc) processes user guide. Technical report (2013)
[107] Zheng, C., Wang, P.: Mt3dms: A modular three-dimensional multispecies transport model for simulation of advection, dispersion and chemical reactions of contaminants in ground water systems: documentation and user’s guide. Technical report. http://hydro.geo.ua.edu/mt3d (1999)
[108] Zhou, Z.; Shi, L.; Zha, Y., Seeing macro-dispersivity from hydraulic conductivity field with convolutional neural network, Adv. Water Resour., 138, 103545, 421-433 (2020)
[109] Zhu, Y.; Zabaras, N., Bayesian deep convolutional encoder-decoder networks for surrogate modeling and uncertainty quantification, J. Comput. Phys., 366, 1, 415-447 (2018) · Zbl 1407.62091
[110] Zhu, Y.; Zabaras, N.; Koutsourelakis, P-S; Perdikarisc, P., Physics-constrained deep learning for high-dimensional surrogate modeling and uncertainty quantification without labeled dataa, J. Comput. Phys., 394, 1, 56-81 (2019) · Zbl 1452.68172
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.