Simulation of mineral dissolution at the pore scale with evolving fluid-solid interfaces: review of approaches and benchmark problem set


This manuscript presents a benchmark problem for the simulation of single-phase flow, reactive transport, and solid geometry evolution at the pore scale. The problem is organized in three parts that focus on specific aspects: flow and reactive transport (part I), dissolution-driven geometry evolution in two dimensions (part II), and an experimental validation of three-dimensional dissolution-driven geometry evolution (part III). Five codes are used to obtain the solution to this benchmark problem, including Chombo-Crunch, OpenFOAM-DBS, a lattice Boltzman code, Vortex, and dissolFoam. These codes cover a good portion of the wide range of approaches typically employed for solving pore-scale problems in the literature, including discretization methods, characterization of the fluid-solid interfaces, and methods to move these interfaces as a result of fluid-solid reactions. A short review of these approaches is given in relation to selected published studies. Results from the simulations performed by the five codes show remarkable agreement both quantitatively—based on upscaled parameters such as surface area, solid volume, and effective reaction rate—and qualitatively—based on comparisons of shape evolution. This outcome is especially notable given the disparity of approaches used by the codes. Therefore, these results establish a strong benchmark for the validation and testing of pore-scale codes developed for the simulation of flow and reactive transport with evolving geometries. They also underscore the significant advances seen in the last decade in tools and approaches for simulating this type of problem.


  1. 1.

    Adams, J. C.: MUDPACK: Multigrid Portable fortran software for the efficient solution of linear elliptic partial differential equations. Appl. Math. Comput. 34(2), 113–146 (1989).

    Google Scholar 

  2. 2.

    Adams, J.C., Swarztrauber, P.N., Sweet, R.: FISHPACK: efficient fortran subprograms for the solution of separable elliptic partial differential equations. Astrophysics Source Code Library (2016)

  3. 3.

    Adams, M., Colella, P., Graves, D. T., Johnson, J., Keen, N., Ligocki, T. J., Martin, D. F., McCorquodale, P., Modiano, D., Schwartz, P., Sternberg, T., Straalen, B. V.: Chombo software package for AMR applications, design document Lawrence Berkeley National Laboratory Technical Report LBNL-6616E (2015)

  4. 4.

    Anderson, C., Greengard, C.: On vortex methods. SIAM J. Numer. Anal. 22(3), 413–440 (1985).

    Article  Google Scholar 

  5. 5.

    Angot, P., Bruneau, C. H., Fabrie, P.: A penalization method to take into account obstacles in incompressible viscous flows. Numer. Math. 81(4), 497–520 (1999).

    Article  Google Scholar 

  6. 6.

    Archie, G.: The electrical resistivity log as an aid in determining some reservoir characteristics. Trans. AIME 146(01), 54–62 (1942).

    Article  Google Scholar 

  7. 7.

    Balay, S., Abhyankar, S., Adams, M.F., Brown, J., Brune, P., Buschelman, K., Dalcin, L., Eijkhout, V., Gropp, W.D., Kaushik, D., Knepley, M.G., May, D.A., McInnes, L.C., Mills, R.T., Munson, T., Rupp, K., Sanan, P., Smith, B.F., Zampini, S., Zhang, H., Zhang, H.: PETSc Web page. (2018)

  8. 8.

    Battiato, I., Tartakovsky, D. M., Tartakovsky, A. M., Scheibe, T. D.: Hybrid models of reactive transport in porous and fractured media. Adv. Water Resour. 34(9), 1140–1150 (2011).

    Article  Google Scholar 

  9. 9.

    Beale, J. T., Majda, A.: Vortex methods. I: convergence in three dimensions. Math. Comput. 39(159), 1–27 (1982).

    Google Scholar 

  10. 10.

    Beale, J. T., Majda, A.: Vortex methods. II: higher order accuracy in two and three dimensions. Math. Comput. 39(159), 29–52 (1982).

    Google Scholar 

  11. 11.

    Bear, J.: Dynamics of Fluids in Porous Media, vol. 120. Elsevier, New York (1972).

    Google Scholar 

  12. 12.

    Beckingham, L. E., Steefel, C. I., Swift, A. M., Voltolini, M., Yang, L., Anovitz, L. M., Sheets, J. M., Cole, D. R., Kneafsey, T. J., Mitnick, E. H., Zhang, S., Landrot, G., Ajo-Franklin, J. B., DePaolo, D. J., Mito, S., Xue, Z.: Evaluation of accessible mineral surface areas for improved prediction of mineral reaction rates in porous media. Geochim. Cosmochim. Acta 205, 31–49 (2017).

    Article  Google Scholar 

  13. 13.

    Békri, S., Thovert, J., Adler, P.: Dissolution of porous media. Chem. Eng. Sci. 50(17), 2765–2791 (1995).

    Article  Google Scholar 

  14. 14.

    Benioug, M., Golfier, F., Oltean, C., Bues, M. A., Bahar, T., Cuny, J.: An immersed boundary-lattice Boltzmann model for biofilm growth in porous media. Adv. Water Res. 107, 65–82 (2017).

    Article  Google Scholar 

  15. 15.

    Boek, E. S., Zacharoudiou, I., Gray, F., Shah, S. M., Crawshaw, J. P., Yang, J.: Multiphase-flow and reactive-transport validation studies at the pore scale by use of lattice boltzmann computer simulations. SPE J. 22(03), 940–949 (2017).

    Article  Google Scholar 

  16. 16.

    Carman, P.: Fluid flow through granular beds. Chem. Eng. Res. Des. 75, S32–S48 (1997).

    Article  Google Scholar 

  17. 17.

    Chagneau, A., Claret, F., Enzmann, F., Kersten, M., Heck, S., Madė, B., Schäfer, T.: Mineral precipitation-induced porosity reduction and its effect on transport parameters in diffusion-controlled porous media. Geochem. T 16(1). (2015)

  18. 18.

    Chaniotis, A., Poulikakos, D.: High order interpolation and differentiation using b-splines. J. Comput. Phys. 197(1), 253–274 (2004).

    Article  Google Scholar 

  19. 19.

    Chatelain, P., Curioni, A., Bergdorf, M., Rossinelli, D., Andreoni, W., Koumoutsakos, P.: Billion vortex particle direct numerical simulations of aircraft wakes. Comput. Methods Appl. Mech. Eng. 197(13-16), 1296–1304 (2008).

    Article  Google Scholar 

  20. 20.

    Chatelin, R., Poncet, P.: A hybrid grid-particle method for moving bodies in 3D, stokes flow with variable viscosity. SIAM J. Sci. Comput. 35(4), B925–B949 (2013).

    Article  Google Scholar 

  21. 21.

    Chatelin, R., Sanchez, D., Poncet, P.: Analysis of the penalized 3D variable viscosity stokes equations coupled to diffusion and transport. ESAIM: Math. Modell. Numer. Anal. 50(2), 565–591 (2016).

    Article  Google Scholar 

  22. 22.

    Chen, L., Kang, Q., Carey, B., Tao, W. Q.: Pore-scale study of diffusion–reaction processes involving dissolution and precipitation using the lattice Boltzmann method. Int. J. Heat Mass Transf. 75, 483–496 (2014).

    Article  Google Scholar 

  23. 23.

    Chou, L., Garrels, R. M., Wollast, R.: Comparative study of the kinetics and mechanisms of dissolution of carbonate minerals. Chem. Geol. 78(3-4), 269–282 (1989).

    Article  Google Scholar 

  24. 24.

    Cocle, R., Winckelmans, G., Daeninck, G.: Combining the vortex-in-cell and parallel fast multipole methods for efficient domain decomposition simulations. J. Comput. Phys. 227(21), 9091–9120 (2008).

    Article  Google Scholar 

  25. 25.

    Colella, P., Graves, D., Ligocki, T., Modiano, D., Straalen, B. V.: EBChombo software package for Cartesian grid, embedded boundary applications. Tech. Rep., Applied Numerical Algorithms Group, Lawrence Berkeley National Laboratory. Unpublished. Available at (2003)

  26. 26.

    Cottet, G., Koumoutsakos, P.: Vortex methods: theory and practice. IOP Publishing (2001)

  27. 27.

    Cottet, G. H., Etancelin, J. M., Perignon, F., Picard, C.: High order semi-Lagrangian particle methods for transport equations: numerical analysis and implementation issues. ESAIM: Math. Modell. Numer. Anal. 48(4), 1029–1060 (2014).

    Article  Google Scholar 

  28. 28.

    Cottet, G. H., Poncet, P.: Advances in direct numerical simulations of 3D wall-bounded flows by vortex-in-cell methods. J. Comput. Phys. 193(1), 136–158 (2004).

    Article  Google Scholar 

  29. 29.

    Curti, E., Xto, J., Borca, C., Henzler, K., Huthwelker, T., Prasianakis, N.: Modelling ra-baryte nucleation/precipitation kinetics at the pore scale : application to radioactive waste disposal. Eur. J. Mineral. In press (2019)

  30. 30.

    Deng, H., Molins, S., Trebotich, D., Steefel, C., DePaolo, D.: Pore-scale numerical investigation of the impacts of surface roughness: Upscaling of reaction rates in rough fractures. Geochim. Cosmochim. Acta 239, 374–389 (2018).

    Article  Google Scholar 

  31. 31.

    Deng, H., Steefel, C., Molins, S., DePaolo, D.: Fracture evolution in multimineral systems: the role of mineral composition, flow rate, and fracture aperture heterogeneity. ACS Earth Space Chem. 2(2), 112–124 (2018).

    Article  Google Scholar 

  32. 32.

    El Ossmani, M., Poncet, P.: Efficiency of multiscale hybrid grid-particle vortex methods. Multiscale Model. Simul. 8(5), 1671–1690 (2010).

    Article  Google Scholar 

  33. 33.

    Ellis, B., Peters, C., Fitts, J., Bromhal, G., McIntyre, D., Warzinski, R., Rosenbaum, E.: Deterioration of a fractured carbonate caprock exposed to CO2-acidified brine flow. Greenhouse Gas. Sci. Technol. 1(3), 248–260 (2011).

    Google Scholar 

  34. 34.

    Gazzola, M., Chatelain, P., van Rees, W.M., Koumoutsakos, P.: Simulations of single and multiple swimmers with non-divergence free deforming geometries. J. Comput. Phys. 230(19), 7093–7114 (2011).

    Article  Google Scholar 

  35. 35.

    Golfier, F., Zarcone, C., Bazin, B., Lenormand, R., Lasseux, D., Quintard, M.: On the ability of a darcy-scale model to capture wormhole formation during the dissolution of a porous medium. J. Fluid Mech. 457, 213–254 (2002).

    Article  Google Scholar 

  36. 36.

    Gray, F., Cen, J., Shah, S., Crawshaw, J., Boek, E.: Simulating dispersion in porous media and the influence of segmentation on stagnancy in carbonates. Adv. Water Res. 97, 1–10 (2016).

    Article  Google Scholar 

  37. 37.

    Hirt, C., Amsden, A., Cook, J.: An arbitrary lagrangian-eulerian computing method for all flow speeds. J. Comput. Phys. 14(3), 227–253 (1974).

    Article  Google Scholar 

  38. 38.

    Hirt, C., Nichols, B.: Volume of fluid (vof) method for the dynamics of free boundaries. J. Comput. Phys. 39 (1), 201–225 (1981).

    Article  Google Scholar 

  39. 39.

    Huang, H., Li, X.: Pore-scale simulation of coupled reactive transport and dissolution in fractures and porous media using the level set interface tracking method. J. Nanjing Univ. (Nat. Sci.) 47(3), 235–251 (2011).

    Google Scholar 

  40. 40.

    Huber, C., Shafei, B., Parmigiani, A.: A new pore-scale model for linear and non-linear heterogeneous dissolution and precipitation. Geochim. Cosmochim. Acta 124(0), 109–130 (2014).

    Article  Google Scholar 

  41. 41.

    Issa, R. I.: Solution of the implicitly discretised fluid flow equations by operator-splitting. J. Comp. Phys. 62(1), 40–65 (1986).

    Article  Google Scholar 

  42. 42.

    Jasak, H.: Error analysis and estimation for the finite volume method with applications to fluid flows. Ph.D. thesis, University of London (1996)

  43. 43.

    Kang, J., Prasianakis, N., Mantzaras, J.: Thermal multicomponent lattice boltzmann model for catalytic reactive flows Physical Review E - Statistical, Nonlinear and Soft Matter Physics 89(6). (2014)

  44. 44.

    Kang, Q., Lichtner, P. C., Viswanathan, H. S., Abdel-Fattah, A. I.: Pore scale modeling of reactive transport involved in geologic co2 sequestration. Transp. Porous Med. 82(1), 197–213 (2010)

    Article  Google Scholar 

  45. 45.

    Kang, Q., Zhang, D., Chen, S.: Simulation of dissolution and precipitation in porous media. J. Geophys. Res. Solid Earth 108(B10), 1–5 (2003).

    Article  Google Scholar 

  46. 46.

    Kang, Q., Zhang, D., Chen, S., He, X.: Lattice Boltzmann simulation of chemical dissolution in porous media. Phys. Rev. E 65(3), 036,318 (2002).

    Article  Google Scholar 

  47. 47.

    Kang, Q., Zhang, D., Chen, S., He, X.: Lattice Boltzmann simulation of chemical dissolution in porous media. Phys. Rev. E 036(3), 318 (2002).

    Google Scholar 

  48. 48.

    Kozeny, J.: Ueber kapillare Leitung des Wassers im Boden. Sitzungsber Akad. Wiss. Wien (1927)

  49. 49.

    Lai, P., Moulton, K., Krevor, S.: Pore-scale heterogeneity in the mineral distribution and reactive surface area of porous rocks. Chem. Geol. 411, 260–273 (2015).

    Article  Google Scholar 

  50. 50.

    Landrot, G., Ajo-Franklin, J. B., Yang, L., Cabrini, S., Steefel, C. I.: Measurement of accessible reactive surface area in a sandstone, with application to CO2 mineralization. Chem. Geol. 318, 113–125 (2012).

    Article  Google Scholar 

  51. 51.

    Li, L., Peters, C. A., Celia, M. A.: Effects of mineral spatial distribution on reaction rates in porous media. Water Resour. Res. 43(1), W01,419 (2007).

    Article  Google Scholar 

  52. 52.

    Li, X., Huang, H., Meakin, P.: Level set simulation of coupled advection-diffusion and pore structure evolution due to mineral precipitation in porous media. Water Resour. Res 44(12). (2008)

  53. 53.

    Li, X., Huang, H., Meakin, P.: A three-dimensional level set simulation of coupled reactive transport and precipitation/dissolution. Int. J. Heat Mass Tran. 53(13), 2908–2923 (2010).

    Article  Google Scholar 

  54. 54.

    Lichtner, P.C.: The quasi-stationary state approximation to coupled mass transport and fluid-rock interaction in a porous medium. Geochim. Cosmochim. Acta 52(1), 143–165 (1988).

    Article  Google Scholar 

  55. 55.

    Liu, M., Shabaninejad, M., Mostaghimi, P.: Impact of mineralogical heterogeneity on reactive transport modelling. Comput. Geosci. 104, 12–19 (2017).

    Article  Google Scholar 

  56. 56.

    Liu, M., Shabaninejad, M., Mostaghimi, P.: Predictions of permeability, surface area and average dissolution rate during reactive transport in multi-mineral rocks. J. Petrol. Sci. Eng. 170, 130–138 (2018).

    Article  Google Scholar 

  57. 57.

    Luhmann, A. J., Tutolo, B. M., Bagley, B. C., Mildner, D. F. R., Seyfried, W. E., Saar, M. O.: Permeability, porosity, and mineral surface area changes in basalt cores induced by reactive transport of CO2-rich brine. Water Resour. Res. 53(3), 1908–1927 (2017).

    Article  Google Scholar 

  58. 58.

    Luquot, L., Gouze, P.: Experimental determination of porosity and permeability changes induced by injection of CO2 into carbonate rocks. Chem. Geol. 265(1-2), 148–159 (2009).

    Article  Google Scholar 

  59. 59.

    Maes, J., Geiger, S.: Direct pore-scale reactive transport modelling of dynamic wettability changes induced by surface complexation. Adv. Water Resour. 111, 6–19 (2018).

    Article  Google Scholar 

  60. 60.

    Magni, A., Cottet, G. H.: Accurate, non-oscillatory, remeshing schemes for particle methods. J. Comp. Phys. 231(1), 152–172 (2012).

    Article  Google Scholar 

  61. 61.

    Menke, H., Bijeljic, B., Blunt, M.: Dynamic reservoir-condition microtomography of reactive transport in complex carbonates: effect of initial pore structure and initial brine pH. Geochim. Cosmochim. Acta 204, 267–285 (2017).

    Article  Google Scholar 

  62. 62.

    Miller, K., Vanorio, T., Keehm, Y.: Evolution of permeability and microstructure of tight carbonates due to numerical simulation of calcite dissolution. J. Geophys. Res. Solid Earth 122(6), 4460–4474 (2017).

    Article  Google Scholar 

  63. 63.

    Molins, S., Trebotich, D., Arora, B., Steefel, C. I., Deng, H.: Multi-scale model of reactive transport in fractured media: diffusion limitations on rates transport in porous media.

  64. 64.

    Molins, S., Trebotich, D., Miller, G. H., Steefel, C. I.: Mineralogical and transport controls on the evolution of porous media texture using direct numerical simulation. Water Resour. Res. 53(5), 3645–3661 (2017).

    Article  Google Scholar 

  65. 65.

    Molins, S., Trebotich, D., Steefel, C. I., Shen, C.: An investigation of the effect of pore scale flow on average geochemical reaction rates using direct numerical simulation. Water Resour. Res 48(3). (2012)

  66. 66.

    Molins, S., Trebotich, D., Yang, L., Ajo-Franklin, J. B., Ligocki, T. J., Shen, C., Steefel, C. I.: Pore-scale controls on calcite dissolution rates from flow-through laboratory and numerical experiments. Environ. Sci. Technol. 48(13), 7453–7460 (2014).

    Article  Google Scholar 

  67. 67.

    Monaghan, J.: Extrapolating b splines for interpolation. J. Comput. Phys. 60(2), 253–262 (1985).

    Article  Google Scholar 

  68. 68.

    Monaghan, J. J.: Extrapolating B splines for interpolation. J. Comp. Phys. 60(2), 253–262 (1985).

    Article  Google Scholar 

  69. 69.

    Noiriel, C., Daval, D.: Pore-Scale geochemical reactivity associated with CO2 Storage: New frontiers at the Fluid–solid interface. Account. Chem. Res. 50(4), 759–768 (2017).

    Article  Google Scholar 

  70. 70.

    Noiriel, C., Luquot, L., Made, B., Raimbault, L., Gouze, P., van der Lee, J.: Changes in reactive surface area during limestone dissolution: an experimental and modelling study. Chem. Geol. 265(1), 160–170 (2009).

    Article  Google Scholar 

  71. 71.

    Oltėan, C., Golfier, F., Buės, M. A.: Numerical and experimental investigation of buoyancy-driven dissolution in vertical fracture. J. Geophys. Res. Solid Earth 118(5), 2038–2048 (2013).

    Article  Google Scholar 

  72. 72.

    Oostrom, M., Mehmani, Y., Romero-Gomez, P., Tang, Y., Liu, H., Yoon, H., Kang, Q., Joekar-Niasar, V., Balhoff, M. T., Dewers, T., Tartakovsky, G. D., Leist, E. A., Hess, N. J., Perkins, W. A., Rakowski, C. L., Richmond, M. C., Serkowski, J. A., Werth, C. J., Valocchi, A. J., Wietsma, T. W., Zhang, C.: Pore-scale and continuum simulations of solute transport micromodel benchmark experiments. Computat. Geosci. 20(4), 857–879 (2016).

    Article  Google Scholar 

  73. 73.

    Ovaysi, S., Piri, M.: Pore-scale dissolution of CO2+SO2 in deep saline aquifers. Int J. Greenh. Gas Con. 15, 119–133 (2013).

    Article  Google Scholar 

  74. 74.

    Parmigiani, A., Huber, C., Bachmann, O., Chopard, B.: Pore-scale mass and reactant transport in multiphase porous media flows. J. Fluid Mech. 686, 40–76 (2011).

    Article  Google Scholar 

  75. 75.

    Pereira-Nunes, J. P., Blunt, M. J., Bijeljic, B.: Pore-scale simulation of carbonate dissolution in micro-CT images. J. Geophys. Res. Solid Earth 121(2), 558–576 (2016).

    Article  Google Scholar 

  76. 76.

    Poncet, P.: Topological aspects of three-dimensional wakes behind rotary oscillating cylinders. J. Fluid Mech. 517, 27–53 (2004).

    Article  Google Scholar 

  77. 77.

    Poncet, P.: Finite difference stencils based on particle strength exchange schemes for improvement of vortex methods. J. Turbul. 7, N23 (2006).

    Article  Google Scholar 

  78. 78.

    Poncet, P.: Analysis of direct three-dimensional parabolic panel methods. SIAM J. Numer. Anal. 45(6), 2259–2297 (2007).

    Article  Google Scholar 

  79. 79.

    Poncet, P.: Analysis of an immersed boundary method for three-dimensional flows in vorticity formulation. J. Comput. Phys. 228(19), 7268–7288 (2009).

    Article  Google Scholar 

  80. 80.

    Poncet, P., Hildebrand, R., Cottet, G. H., Koumoutsakos, P.: Spatially distributed control for optimal drag reduction of the flow past a circular cylinder. J. Fluid Mech. 599, 111–120 (2008).

    Article  Google Scholar 

  81. 81.

    Prasianakis, N., Ansumali, S.: Microflow simulations via the lattice boltzmann method. Commun. Comput. Phys. 9(5), 1128–1136 (2011).

    Article  Google Scholar 

  82. 82.

    Prasianakis, N., Karlin, I., Mantzaras, J., Boulouchos, K.: Lattice boltzmann method with restored galilean invariance. Physical Review E - Statistical, Nonlinear and Soft Matter Physics 79(6). (2009)

  83. 83.

    Prasianakis, N., Rosén, T., Kang, J., Eller, J., Mantzaras, J., Büchi, F.: Simulation of 3d porous media flows with application to polymer electrolyte fuel cells. Commun. Comput. Phys. 13(3), 851–866 (2013).

    Article  Google Scholar 

  84. 84.

    Prasianakis, N.i., Curti, E., Kosakowski, G., Poonoosamy, J., Churakov, S.V.: Deciphering pore-level precipitation mechanisms. Sci. Rep. 7(1), 13,765 (2017).

    Article  Google Scholar 

  85. 85.

    Prasianakis, N.i., Gatschet, M., Abbasi, A., Churakov, S.V.: Upscaling strategies of porosity-permeability correlations in reacting environments from pore-scale simulations. Geofluids 2018, 1–8 (2018).

    Article  Google Scholar 

  86. 86.

    Qian, Y. H., D’Humiėres, D., Lallemand, P.: Lattice BGK models for navier-stokes equation. Europhys. Lett. 17(6), 479–484 (1992).

    Article  Google Scholar 

  87. 87.

    Rosén, T., Eller, J., Kang, J., Prasianakis, N., Mantzaras, J., Büchi, F.: Saturation dependent effective transport properties of pefc gas diffusion layers. J. Electrochem. Soc. 159(9), F536–F544 (2012).

    Article  Google Scholar 

  88. 88.

    Sadhukhan, S., Gouze, P., Dutta, T.: Porosity and permeability changes in sedimentary rocks induced by injection of reactive fluid: a simulation model. J. Hydrol. 450-451, 134–139 (2012).

    Article  Google Scholar 

  89. 89.

    Sallès, J., Thovert, J. F., Adler, P. M.: Deposition in porous media and clogging. Chem. Eng. Sci. 48 (16), 2839–2858 (1993).

    Article  Google Scholar 

  90. 90.

    Sanchez, D., Hume, L., Chatelin, R., Poncet, P.: Analysis of the 3d non-linear stokes problem coupled to transport-diffusion for shear-thinning heterogeneous microscale flows, applications to digital rock physics and mucociliary clearance. ESAIM: Mathematical Modelling and Numerical Analysis (Under revision)

  91. 91.

    Saxena, N., Hofmann, R., Alpak, F. O., Berg, S., Dietderich, J., Agarwal, U., Tandon, K., Hunter, S., Freeman, J., Wilson, O. B.: References and benchmarks for pore-scale flow simulated using micro-CT images of porous media and digital rocks. Adv. Water Res. 109, 211–235 (2017).

    Article  Google Scholar 

  92. 92.

    Schneider, C.A., Rasband, W.S., Eliceiri, K.W.: NIH image to ImageJ: 25 years of image analysis. Nat. Methods 9(7), 671–675 (2012).

    Article  Google Scholar 

  93. 93.

    von der Schulenburg, D. A. G., Pintelon, T. R. R., Picioreanu, C., Loosdrecht, M. C. M. V., Johns, M. L.: Three-dimensional simulations of biofilm growth in porous media. AIChE J. 55(2), 494–504 (2009).

    Article  Google Scholar 

  94. 94.

    Sanchez, D., Hume, L., Chatelin, R., Poncet, P.: Analysis of non-linear Stokes problem coupled to transport-diffusion for shear-thinning heterogeneous microscale flows, applications to digital rock physics and mucociliary clearance. Math. Model. Numer. Anal. 53, 1083–1124 (2019)

    Article  Google Scholar 

  95. 95.

    Soulaine, C., Gjetvaj, F., Garing, C., Roman, S., Russian, A., Gouze, P., Tchelepi, H.: The impact of sub-resolution porosity of x-ray microtomography images on the permeability. Transport Porous Med. 113(1), 227–243 (2016).

    Article  Google Scholar 

  96. 96.

    Soulaine, C., Roman, S., Kovscek, A., Tchelepi, H. A.: Mineral dissolution and wormholing from a pore-scale perspective. J. Fluid Mech. 827, 457–483 (2017).

    Article  Google Scholar 

  97. 97.

    Soulaine, C., Roman, S., Kovscek, A., Tchelepi, H. A.: Pore-scale modelling of multiphase reactive flow. Application to mineral dissolution with production of CO2. J. Fluid Mech. 855, 616–645 (2018).

    Article  Google Scholar 

  98. 98.

    Soulaine, C., Tchelepi, H. A.: Micro-continuum approach for pore-scale simulation of subsurface processes. Transport Porous Med. 113(3), 431–456 (2016).

    Article  Google Scholar 

  99. 99.

    Starchenko, V., Ladd, A. J. C.: The development of wormholes in laboratory-scale fractures: perspectives from three-dimensional simulations. Water Resour. Res. 54(10), 7946–7959 (2018).

    Article  Google Scholar 

  100. 100.

    Starchenko, V., Marra, C. J., Ladd, A. J. C.: Three-dimensional simulations of fracture dissolution. J. Geophys. Res. Solid Earth 121, 6421–6444 (2016).

    Article  Google Scholar 

  101. 101.

    Steefel, C. I., Appelo, C. A. J., Arora, B., Jacques, D., Kalbacher, T., Kolditz, O., Lagneau, V., Lichtner, P. C., Mayer, K. U., Meeussen, J. C. L., Molins, S., Moulton, D., Shao, H., VSimu̇nek, J., Spycher, N., Yabusaki, S.B., Yeh, G.T.: Reactive transport codes for subsurface environmental simulation. Computat. Geosci. 19(3), 445–478 (2014).

    Article  Google Scholar 

  102. 102.

    Steefel, C. I., Beckingham, L. E., Landrot, G.: Micro-continuum approaches for modeling pore-scale geochemical processes. Rev. Mineral. Geochem. 80(1), 217–246 (2015).

    Article  Google Scholar 

  103. 103.

    Steefel, C. I., Molins, S., Trebotich, D.: Pore scale processes associated with subsurface CO2 injection and sequestration. Rev. Mineral. Geochem. 77(1), 259–303 (2013).

    Article  Google Scholar 

  104. 104.

    Succi, S.: The lattice Boltzmann equation: for fluid dynamics and beyond. Oxford University Press (2001)

  105. 105.

    Succi, S., Foti, E., Higuera, F.: Three-dimensional flows in complex geometries with the lattice Boltzmann method. EPL 10(5), 433–438 (1989).

    Article  Google Scholar 

  106. 106.

    Sweet, R. A.: A parallel and vector variant of the cyclic reduction algorithm. J. Sci. Stat. Comput. 9(4), 761–765 (1988)

    Article  Google Scholar 

  107. 107.

    Szymczak, P., Ladd, A. J. C.: Microscopic simulations of fracture dissolution. Geophys. Res. Lett. 31(23), 1–4 (2004).

    Article  Google Scholar 

  108. 108.

    Szymczak, P., Ladd, A. J. C.: Wormhole formation in dissolving fractures. J. Geophys. Res. Solid Earth 114(B6), 1–22 (2009).

    Article  Google Scholar 

  109. 109.

    Tang, Y., Valocchi, A. J., Werth, C. J., Liu, H.: An improved pore-scale biofilm model and comparison with a microfluidic flow cell experiment. Water Resour. Res. 49(12), 8370–8382 (2013).

    Google Scholar 

  110. 110.

    Tartakovsky, A. M., Meakin, P., Scheibe, T. D., West, R. M. E.: Simulations of reactive transport and precipitation with smoothed particle hydrodynamics. J. Comp. Phys. 222(2), 654–672 (2007).

    Article  Google Scholar 

  111. 111.

    Trebotich, D., Adams, M. F., Molins, S., Steefel, C. I., Shen, C.: High-resolution simulation of pore-scale reactive transport processes associated with carbon sequestration. Comput. Sci. Eng. 16(6), 22–31 (2014).

    Article  Google Scholar 

  112. 112.

    Trebotich, D., Graves, D.: An adaptive finite volume method for the incompressible Navier-Stokes equations in complex geometries. Comm. App. Math. Com. Sc. 10(1), 43–82 (2015).

    Article  Google Scholar 

  113. 113.

    Tukoviċ, ž., Jasak, H.: A moving mesh finite volume interface tracking method for surface tension dominated interfacial fluid flow. Comput. Fluids 55, 70–84 (2012).

    Article  Google Scholar 

  114. 114.

    Vilcȧez, J., Morad, S., Shikazono, N.: Pore-scale simulation of transport properties of carbonate rocks using FIB-SEM 3d microstructure: implications for field scale solute transport simulations. J. Nat. Gas Sci. Eng. 42, 13–22 (2017).

    Article  Google Scholar 

  115. 115.

    Xu, Z., Huang, H., Li, X., Meakin, P.: Phase field and level set methods for modeling solute precipitation and/or dissolution. Comput. Phys. Commun. 183(1), 15–19 (2012).

    Article  Google Scholar 

  116. 116.

    Xu, Z., Meakin, P.: Phase-field modeling of solute precipitation and dissolution. J. Chem. Phys. 014(1), 705 (2008).

    Google Scholar 

  117. 117.

    Xu, Z., Meakin, P.: Phase-field modeling of two-dimensional solute precipitation/dissolution: Solid fingers and diffusion-limited precipitation. J. Chem. Phys. 044(4), 137 (2011).

    Google Scholar 

  118. 118.

    Xu, Z., Meakin, P., Tartakovsky, A. M.: Diffuse-interface model for smoothed particle hydrodynamics. Phys. Rev E 79(3). (2009)

  119. 119.

    Yang, Y., Bruns, S., Stipp, S., Sørensen, H.: Impact of microstructure evolution on the difference between geometric and reactive surface areas in natural chalk. Adv. Water Res. 115, 151–159 (2018).

    Article  Google Scholar 

  120. 120.

    Yoon, H., Valocchi, A.J., Werth, C.J., Dewers, T.: Pore-scale simulation of mixing-induced calcium carbonate precipitation and dissolution in a microfluidic pore network. Water Resour. Res. 48(2). W02524 (2012)

  121. 121.

    Zhao, B., MacMinn, C.W., Juanes, R.: Wettability control on multiphase flow in porous media: A benchmark study on current pore-scale modeling approaches. 71st Annual Meeting of the APS Division of Fluid Dynamics. In: Bull. Am. Phys. Soc. American Physical Society. (2018)

Download references


This material is based upon work supported as part of the Center for Nanoscale Control of Geologic CO2, an Energy Frontier Research Center funded by the US Department of Energy, Office of Science, Office of Basic Energy Sciences under award DE-AC02-05CH11231. Chombo-Crunch simulations used resources of the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility supported by the Office of Science of the US Department of Energy under contract DE-AC02-05CH11231. Development of dissolFoam was supported by the US Department of Energy, Office of Science, Office of Basic Energy Sciences, Chemical Sciences, Geosciences, and Biosciences Division under Award Number DE-FG02-98ER14853 and DE-SC0018676. Development of the advanced mesh relaxation in dissolFoam was sponsored by the Laboratory Directed Research and Development Program of Oak Ridge National Laboratory, managed by UT-Battelle, LLC, for the US Department of Energy. Vortex method simulations acknowledge the HPC resources of cluster Pyrene (UPPA-E2S, Pau, France) and the support of the Carnot Institute ISIFoR under contract RugoRX. N.I.P. acknowledges support from Swiss National Science Foundation, SNSF project No: 200021_172618, and the Swiss National Supercomputing Centre (CSCS). The benchmark problem set was proposed and developed by S.M. and C.S. Manuscript preparation was led by S.M. Chombo-Crunch simulations were conducted by S.M., OpenFOAM-DBS by C.S., lattice Boltzmann by A.A. and N.I.P., vortex by P.P., and dissolFoam by A.L. and V.S. These authors are listed according to when they joined the benchmark effort. Chombo-Crunch was developed by D.T. and S.M., CrunchFlow by C.I.S., OpenFOAM-DBS by C.S., the lattice Boltzmann code by N.I.P., the vortex code by P.P., and dissolFoam by A.L. and V.S. Part III experiments were conducted by S.R.

Author information



Corresponding author

Correspondence to Sergi Molins.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Electronic supplementary material

Below is the link to the electronic supplementary material.

(ZIP 1.48 GB)


Appendix A: Additional/alternative equations

Flow at the pore scale may be described by the incompressible Navier-Stokes (33) and (34):

$$ \nabla\cdot \textbf{u} = 0, $$
$$ \frac{\partial \textbf{u}}{\partial t}+\left( \textbf{u}\cdot\nabla \right)\textbf{u}+\frac{1}{\rho}\nabla p=\nu\nabla^{2}\textbf{u}, $$

as well as the Stokes (1) and (2). In these benchmarks, the Reynolds number is sufficiently small that fluid inertia can be neglected; thus, these two approaches are equivalent.

In the dissolution benchmarks (parts II and III), codes may take advantage of the large time scale separation between boundary motion and transport processes to solve the steady-state transport equation directly,

$$ \boldmath{\nabla} \cdot (\mathbf{u} c) = D \nabla^{2}c. $$

Time-dependent solutions of transport and reaction (part I) are more tightly coupled than dissolution (parts II and III), because tA and tR are often of the same order, especially for relatively fast reacting minerals such as carbonates. Both global implicit and operator splitting approaches have been used for time-dependent transport, with the time stepping in the operator splitting constrained by the Courant-Friedrichs-Levy (CFL) criterion

$$ {\Delta} t<\frac{\Delta x}{\max{(u)}}. $$

Appendix B: Analysis and comparison of results

B.1. Upscaled parameters

Simulation results are compared in terms of the evolution with time of upscaled parameters. These upscaled parameters include the volume (V ) and surface area of all reacting reacting mineral surfaces (A) and the average reaction rate (R). The average rate is calculated as follows:

$$ R=\frac {Q (c_{out}-c_{in})} {\xi A}, $$

where ξ is the stoichiometric coefficient, cin is the (uniform) concentration at the inlet, given by the boundary conditions, and cout is the flux-weighted-average outlet concentration,

$$ c_{out}= \frac{{\int}_{\delta S} c {\textbf{u}} \cdot d \boldsymbol{s}} {Q}. $$

The volumetric flux at the outlet Q is found by integrating over the outlet area

$$ Q={\int}_{\delta S} \mathbf{u} \cdot ds. $$

In addition to these upscaled parameters, simulation results are compared on the basis of the geometry of the grain at different time points and the concentration contours are prescribed times.

B.2. Grid convergence

As methods for simulating of moving boundary problems vary greatly, we want to investigate the impact of grid resolution on results for each method separately. For this purpose, the simulations were run at different resolutions (Figs. 1415, and 9) in the main text. Results for the grain volume and surface area were analyzed to ensure grid convergence of the methods, and choose a resolution for which results will be assumed to have converged within a reasonable accuracy.

Fig. 14

Grid convergence tests results for the time evolution of the grain volume (part II) from a OpenFOAM-DBS, b lattice Boltzmann, c vortex, d Chombo-Crunch, and e dissolFoam simulations

Fig. 15

Grid convergence tests results for the time evolution of the grain surface area (Part II) from a OpenFOAM-DBS, b lattice Boltzmann, c vortex, d Chombo-Crunch, and e dissolFoam simulations

Appendix C: Notes on unit conversion for concentrations and rates

The conversions from the parameters reported by [96] to the units used in Part III are presented. Mass fraction is converted to molar concentration using

$$ c = \frac{\rho f} {M}, $$

where c is the molar concentration of protons (molcm− 3), M is the molar weight of acid (gmol− 1), ρ is the fluid density (gcm− 3), and f is the mass fraction of acid. The inlet concentration (0.05%) is converted to mol cm− 3 as follows:

$$ c_{in} = \frac{0.92 \text{g} \text{cm}^{-3} \times 0.0005} {36.5 \text{g} \text{mol}^{-1}} = 1.26 \cdot 10^{-5} \text{mol} \text{cm}^{-3}. $$

In the formulation used in this manuscript (Section 2), the first-order reaction is expressed as a function of the activity coefficient and the molar concentration of H+ (26). Assuming that \(\gamma _{\text {H}^{+}}=1000 \text {cm}^{3} \text {mol}^{-1}\), the proton concentration \(c_{\text {H}^{+}}\) must be in mol cm− 3 so that the product \(k_{\text {H}^{+}} \gamma _{\text {H}^{+}}\) has units of cm s− 1. The conversion from the rate constant used in [96] (\(k_{\text {H}^{+}} \gamma _{\text {H}^{+}} = 0.5 \text {cm\ s} ^{-1}\)) is

$$ k_{\text{H}^{+}} = \frac{0.5 \text{cm} \text{s} ^{-1}}{1000 \text{cm}^{3} \text{mol}^{-1}} = 5 \times 10^{-4} \text{mol} \text{cm}^{-2} \text{s} ^{-1}. $$

However, in [96], this rate is applicable to the rate of HCl consumption when reacting with calcite according to the following stoichiometry

$$ \text{CaCO}3_{(\text{s})} + \text{2HCl} -> \text{CaCl}_{2} + \text{H}_{2}\text{CO}_{3}. $$

To maintain consistency with the rate expressed for calcite, one must multiply by the stoichiometric coefficient of HCl in Eq. 43,

$$ k_{\text{H}^{+}} = 10^{-3} \text{mol} \text{cm}^{-2} \text{s} ^{-1}. $$

Appendix D: Additional information on numerical choices and parameters

D.1. Lattice Boltzmann dimensionalization

Dimensionalization of the LB computations is a process that needs special care. Lattice Boltzmann unit conversion to physical units can be done after matching the characteristic non-dimensional Reynolds, Péclet, and Damköhler numbers. For a 256 × 128 discretization grid Ly = 128 (in lattice units), each lattice space unit in parts I and II corresponds to w/128 = 3.91 × 10− 4cm. For the current setup, viscosity is defined as ν = τfρT. The relaxation parameter for the fluid phase, τf, is set to τf = 0.5 in lattice units. By equating Re=ReLB= 0.6, using the aforementioned viscosity, the inlet velocity can be calculated as uinLB = 0.00078125 (in lattice units), which corresponds to \(\textbf {u}_{in}=0.12 \text {cm\ s} ^{-1}\). Once the lattice velocity is set, the duration of the time step δt can be calculated by equating the inlet velocities: δt = 2.54 × 10− 6 s. Note that the time step is dictated by the slow advective flow, and by choosing to keep the same time step for all processes. This leads to a fully coupled advection-diffusion-reaction scheme applicable to all flow and chemical conditions. Separation of time scales is possible by solving for steady-state flow, then steady-state reactive transport, and finally the solid geometry update. Such an approach would be sufficient for these benchmarks and would greatly reduce the number of time steps to reach the solution.

Diffusivity is defined as D = τgT. By equating the Péclet numbers Pe=PeLB= 600, the relaxation parameter τg, which corresponds to the diffusive time scale, is set to τD = 0.0005, for the species that follow the advection-diffusion equation. Finally, by equating the Damköler numbers DaII=DaII-LB= 178.15, the rate constant \(k_{\text {H}^{+} \text {LB}}=10^{-3.2364}\). For this dimensionalization Ma<Kn<< 1, thus recovering the incompresible Navier-Stokes equations.

D.2. Discussion on interpolation kernel for Lagrangian methods

The choice of the kernel Λ used for re-meshing the particle is crucial for the accuracy of vortex and particle methods. Indeed, in order to avoid holes and accumulation of particles that would ruin the convergence, particle information Fp (including vorticity, concentration, ...) in volumes vp located at positions ξp is remeshed on to a new structured mesh (with cell volumes \(\tilde {v}_{q}\)). This mesh defines a new set of particles \(\tilde {F}_{q}\) at locations \(\tilde {\xi }_{q}\) by means of the following convolution:

$$ \begin{array}{@{}rcl@{}} \tilde{F}_{q}&=& F*{\Lambda} (\tilde{x}_{q})=\int F(y){\Lambda}(\tilde{x}_{q}-y)dy\\ &=& \sum\limits_{p} F_{p}{\Lambda}(\tilde{x}_{q}-x_{p})v_{p}, \end{array} $$

since the set of particles is mathematically defined by the generalized function \(\displaystyle F=\sum \limits _{p} F_{p}\delta _{x_{p}}v_{p}\), based of Dirac functions at xp. In practice, when Λ is the “hat” (or “tent”) function, the reaction stays confined on the fluid/solid interface, but exhibits a pH over-estimation close to the stagnation points, thus over-estimating the reaction rate. When this kernel is smoother but positive in order to be sign preserving, such as the first-order cubic spline M4, the fluid/solid boundary becomes fuzzy and requires us to force the reaction on the interface by means of the function ∥∇𝜖∥, as in [96]. When using the second-order kernel \(M_{4}^{\prime }\) from [67], which is non sign preserving since the integral of \(x^{2}M_{4}^{\prime }(x)\) is zero, no negative concentration appears despite the jump of acid concentration at the body but it leads underestimation of reaction rate. However, the hydrodynamic flow is computed with better accuracy using \(M_{4}^{\prime }\), as expected [28]. Consequently, the short-supported function M3, smoother than the hat function with a support smaller than M4, has been chosen for interpolating and remeshing the chemical concentrations, while \(M_{4}^{\prime }\) has been chosen for the interpolation hydrodynamic values (velocity and vorticity).

In practice for the present benchmark, for which the reaction properties (bounds and positivity) have to be strictly satisfied, the choice of the remeshing kernel is mainly driven by the following arguments:

  • The hat function, is good for the estimation of reaction rate but does respect the pH bounds (pH overshoots below 2 can occur),

  • • The kernel M4 is smooth but M4(0) = 2/3 ≠ 1; thus, it is diffusive: pH bounds are good but reaction rate is under-estimated (see formula A.4 of [18] for definition),

  • \(M_{4}^{\prime }\) (formula 4.5 of [20]) is algebraically mass-conservative, smooth, and second order, but its negative values induce oscillations at concentration jumps and over-estimate the reaction rate. Furthermore, it is not mathematically sign preserving, although negative concentrations were never been observed in this benchmark,

  • M3 (formula A.3 of [18]) is smoother than hat, first order and sign preserving, with short support. It is the best choice for reactive flows like the one considered in the present study; the reaction rate is well estimated (a bit higher than the hat function and closer to other curves) and does not go lower than the initial pH= 2 bound, consistent with this purely dissolution process,

  • M6 and \(M_{6}^{\prime }\) supports are too large for this geometry, and cannot handle correctly the final state of the dissolution.

Consequently, the kernel \(M_{4}^{\prime }\) is the best choice for hydrodynamic computations (for particle remeshing and interpolation of velocity and vorticity from and to grids), while M3 is the best choice for interpolation and transfer of concentrations.

D.3. Darcy-Brinkman-Stokes parameter values

Parameters specific to Darcy-Brinkman-Stokes code simulations are presented in Table 6.

Table 6 Parameters for Darcy-Brinkman-Stokes equations

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Molins, S., Soulaine, C., Prasianakis, N.I. et al. Simulation of mineral dissolution at the pore scale with evolving fluid-solid interfaces: review of approaches and benchmark problem set. Comput Geosci (2020).

Download citation


  • Pore scale
  • Reactive transport
  • Moving boundary
  • Benchmark
  • Review of approaches