Transport in Porous Media

, Volume 127, Issue 1, pp 85–112 | Cite as

Micro-continuum Framework for Pore-Scale Multiphase Fluid Transport in Shale Formations

  • Cyprien SoulaineEmail author
  • Patrice Creux
  • Hamdi A. Tchelepi


A micro-continuum simulation framework is proposed to study the complex pore-scale dynamics associated with hydrocarbon recovery from shale gas. The model accounts for the presence of immiscible fluid phases and for transport mechanisms in the nanoporous structures including slip flow, adsorption, surface and Knudsen diffusion. We employ the concept of sub-grid models to simulate the transport phenomena in shale gas. Specifically, we use high-resolution FIB–SEM images that provide information on the spatial distribution of the minerals, resolved pore space, and sub-resolution porous regions. The model is used to investigate several production scenarios at the pore-scale. In one setting, the organic matter is in direct contact with a micro-crack; in the other setting, clay regions are sandwiched between the organic matter and the “open” crack. The simulations show that it is important to account for the presence of multiple immiscible fluid phases because they can play a critical role in hydrocarbon production from shale-gas formations both in terms of production rate and in terms of residual mass of hydrocarbon. Moreover, we show that, because of wettability conditions, the rate of hydrocarbon recovery, as well as the ultimate recovery, depends strongly on the spatial distribution of the kerogen and clay in the vicinity of the micro-cracks.


Micro-continuum Source rocks Two-phase Pore-scale Adsorption 


\(\bar{\mathsf {v}}\)

Single-field velocity (\(\hbox {m/s}\))

\(\bar{\mathsf {v}}_r\)

Compression velocity of the gas/liquid interface (\(\hbox {m/s}\))


Single-field concentration for species A (\(\hbox {kg/m}^{3}\))


Single-field pressure (\(\hbox {Pa}\))


Normal vector to the solid surface


Tangent vector to the solid surface

\(\chi _s\)

Indicator function for phase s

\(\Delta P\)

Depletion pressure, \(\Delta P = P^* - P_0\) (\(\hbox {Pa}\))


Normal vector to water/gas interface

\(\lambda \)

Mean free path (\(\hbox {m}\))

\(\varvec{\Phi }_{A}\)

Continuous species transfer function (\(\hbox {kg/m}^2\))

\(\mathsf {F}_A\)

Mass flux of species A (\(\hbox {kg/m}^2/\hbox {s}\))

\(\mathsf {F}_c\)

Surface tension forces (\(\hbox {kg/m}^2/\hbox {s}^2\))

\(\mu \)

Single-field viscosity (\(\hbox {kg/m/s}\))

\(\mu _\mathrm{g}\)

Gas viscosity (\(\hbox {kg/m/s}\))

\(\mu _\mathrm{l}\)

Water viscosity (\(\hbox {kg/m/s}\))

\(\phi \)


\(\phi _s\)

Porosity of phase s

\(\rho \)

Single-field fluid density (\(\hbox {kg/m}^{3}\))

\(\rho _\mathrm{g}\)

Gas density (\(\hbox {kg/m}^{3}\))

\(\rho _\mathrm{l}\)

Water density (\(\hbox {kg/m}^{3}\))

\(\sigma \)

Surface tension (\(\hbox {kg/s}^{2}\))

\(\theta \)

Contact angle


Specific surface area of phase s (\(\hbox {m}^{-1}\))


Cumulative mass of gas produced (\(\hbox {kg}\))


Single-field diffusivity of species A (\(\hbox {m}^2/\hbox {s}\))


Diameter of molecules A (\(\hbox {m}\))


Diffusivity of species A in gas (\(\hbox {m}^2/\hbox {s}\))


Knudsen diffusion coefficient (\(\hbox {m}^2/\hbox {s}\))


Diffusivity of species A in water (\(\hbox {m}^2/\hbox {s}\))


Surface diffusion coefficient (\(\hbox {m}^2/\hbox {s}\))


Permeability field, \(k=\sum _s \chi _s k_s\) (\(\hbox {m}^2\))


Permeability related to the nanopores geometry (\(\hbox {m}^2\))


Boltzmann constant (\(k_\mathrm{B}=1.38\times 10^{-23}\hbox {J/K}\))


Single-field relative permeability


Permeability of phase s (\(\hbox {m}^2\))


Permeability correction due to slip and nanoscale effects


Relative permeability to gas


Relative permeability to water


Knudsen number


Molar mass of species A (\(\hbox {g/mol}\))


Mass of gas adsorbed on the surface of the pores (\(\hbox {kg}\))


Total mass of gas in the system (\(\hbox {kg}\))


Mass of gas contained in the volume of the pores (\(\hbox {kg}\))


Initial pressure in the domain (\(\hbox {Pa}\))


Pressure at the outlet (\(\hbox {Pa}\))


Capillary pressure (\(\hbox {Pa}\))


Pressure at which half of the adsorbed molecules is released (\(\hbox {Pa}\))


Quantity of molecules A adsorbed per unit of surface (\(\hbox {kg/m}^{2}\))


Adsorption capacity (\(\hbox {kg/m}^{2}\))


Gas mass flow rate at the outlet (\(\hbox {kg/s}\))


Pore radius (\(\hbox {m}\))


Desorption rate of molecules A (\(\hbox {kg/m}^{2}/\hbox {s}\))


Ideal gas constant (\(R_\mathrm{g}=8.314\hbox {J/mol/K}\))


Water saturation


Temperature (\(\hbox {K}\))


Percentage of the mass of methane adsorbed



We acknowledge the TOTAL STEMS project for financial support and Isabelle Jolivet and Regis Lasnel from TOTAL BGM for the FIB–SEM dataset. We thank the Stanford Center for Computational Earth & Environmental Sciences (CEES) for computational support.


  1. Abu-Al-Saud, M.O., Riaz, A., Tchelepi, H.A.: Multiscale level-set method for accurate modeling of immiscible two-phase flow with deposited thin films on solid surfaces. J. Comput. Phys. 333, 297–320 (2017)CrossRefGoogle Scholar
  2. Al Hinai, A., Rezaee, R., Esteban, L., Labani, M.: Comparisons of pore size distribution: a case from the Western Australian gas shale formations. J. Unconv. Oil Gas Resour. 8(C), 1–13 (2014)CrossRefGoogle Scholar
  3. Barenblatt, G., Zheltov, I., Kochina, I.: Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks. J. Appl. Math. Mech. 24(5), 1286–1303 (1960)CrossRefGoogle Scholar
  4. Blunt, M.J., Bijeljic, B., Dong, H., Gharbi, O., Iglauer, S., Mostaghimi, P., Paluszny, A., Pentland, C.: Pore-scale imaging and modelling. Adv. Water Resour. 51, 197–216 (2013)CrossRefGoogle Scholar
  5. Bousige, C., Ghimbeu, C.M., Vix-Guterl, C., Pomerantz, A.E., Suleimenova, A., Vaughan, G., Garbarino, G., Feygenson, M., Wildgruber, C., Ulm, F.-J., Pellenq, R.J.-M., Coasne, B.: Realistic molecular model of kerogen’s nanostructure. Nat. Mater. 15(5), 576–82 (2016)CrossRefGoogle Scholar
  6. Brace, W.F., Walsh, J., Frangos, W.T.: Permeability of Granit under high pressure. J. Geophys. Res. 73(6), 2225–2236 (1968)CrossRefGoogle Scholar
  7. Brackbill, J., Kothe, D., Zemach, C.: A continuum method for modeling surface tension. J. Comput. Phys. 100(2), 335–354 (1992)CrossRefGoogle Scholar
  8. Brooks, R., Corey, A.: Hydraulic properties of porous media. Colo. State Univ. Hydro Pap. 3, 27 (1964)Google Scholar
  9. Bultreys, T., Van Hoorebeke, L., Cnudde, V.: Multi-scale, micro-computed tomography-based pore network models to simulate drainage in heterogeneous rocks. Adv. Water Resour. 78, 36–49 (2015)CrossRefGoogle Scholar
  10. Békri, S., Laroche, C., Vizika, O.: Pore network models to calculate transport and electrical properties of single or dual-porosity rocks. In: International Symposium of the Society of Core Analysts, Toronto (2005)Google Scholar
  11. Cheng, P., Tian, H., Xiao, X., Gai, H., Li, T., Wang, X.: Water distribution in overmature organic-rich shales: implications from water adsorption experiments. Energy Fuels 31(12), 13120–13132 (2017)CrossRefGoogle Scholar
  12. Cui, X., Bustin, A.M.M., Bustin, R.M.: Measurements of gas permeability and diffusivity of tight reservoir rocks: different approaches and their applications. Geofluids 9(3), 208–223 (2009)CrossRefGoogle Scholar
  13. Damian, S.M.: An extended mixture model for the simultaneous treatment of short and long scale interfaces. Ph.D. thesis, FACULTAD DE INGENIERÍA Y CIENCIAS HÍDRICAS UNIVERSIDAD NACIONAL DEL LITORAL (2013)Google Scholar
  14. Dehghanpour, H., Lan, Q., Saeed, Y., Fei, H., Qi, Z.: Spontaneous imbibition of brine and oil in gas shales: effect of water adsorption and resulting microfractures. Energy Fuels 27(6), 3039–3049 (2013)CrossRefGoogle Scholar
  15. Dong, X., Liu, H., Hou, J., Wu, K., Chen, Z.: Phase equilibria of confined fluids in nanopores of tight and shale rocks considering the effect of capillary pressure and adsorption film. Ind. Eng. Chem. Res. 55(3), 798–811 (2016)CrossRefGoogle Scholar
  16. Duan, Z., Møller, N., Greenberg, J., Weare, J.H.: The prediction of methane solubility in natural waters to high ionic strength from 0 to 250 C and from 0 to 1600 bar. Geochim. Cosmochim. Acta 56(4), 1451–1460 (1992)CrossRefGoogle Scholar
  17. Egermann, P., Lenormand, R., Longeron, D., Zarcone, C.: A fast and direct method of permeability measurement on drill cutting. SPE Reserv. Eval. Eng. 8(04), 269–275 (2005)CrossRefGoogle Scholar
  18. Fathi, E., Akkutlu, I.Y.: Lattice boltzmann method for simulation of shale gas transport in kerogen. SPE J. 18(01), 27–37 (2013)CrossRefGoogle Scholar
  19. Graveleau, M., Soulaine, C., Tchelepi, H.A.: Pore-scale simulation of interphase multicomponent mass transfer for subsurface flow. Transp. Porous Media 120(2), 287–308 (2017)CrossRefGoogle Scholar
  20. Gu, X., Cole, D.R., Rother, G., Mildner, D.F.R., Brantley, S.L.: Pores in marcellus shale: a neutron scattering and FIB-SEM study. Energy Fuels 29(3), 1295–1308 (2015)CrossRefGoogle Scholar
  21. Guibert, R., Nazarova, M., Horgue, P., Hamon, G., Creux, P., Debenest, G.: Computational permeability determination from pore-scale imaging: sample size, mesh and method sensitivities. Transp. Porous Media 107(3), 641–656 (2015)CrossRefGoogle Scholar
  22. Guo, Z., Zhao, T.: Lattice boltzmann model for incompressible flows through porous media. Phys. Rev. E 66(3), 036304 (2002)CrossRefGoogle Scholar
  23. Hao, S., Dengen, Z., Adwait, C., Meilin, D.: Quantifying shale oil production mechanisms by integrating a delaware basin well data from fracturing to production. Unconventional Resources Technology Conference (URTEC) (2016)Google Scholar
  24. Haroun, Y., Legendre, D., Raynal, L.: Volume of fluid method for interfacial reactive mass transfer: application to stable liquid film. Chem. Eng. Sci. 65(10), 2896–2909 (2010)CrossRefGoogle Scholar
  25. Heller, R., Zoback, M.: Adsorption of methane and carbon dioxide on gas shale and pure mineral samples. J. Unconv. Oil Gas Resour. 8, 14–24 (2014)CrossRefGoogle Scholar
  26. Horgue, P., Prat, M., Quintard, M.: A penalization technique applied to the volume-of-fluid method: wettability condition on immersed boundaries. Comput. Fluids 100, 255–266 (2014)CrossRefGoogle Scholar
  27. Horgue, P., Soulaine, C., Franc, J., Guibert, R., Debenest, G.: An open-source toolbox for multiphase flow in porous media. Comput. Phys. Commun. 187, 217–226 (2015)CrossRefGoogle Scholar
  28. Hu, Y., Devegowda, D., Sigal, R.: A microscopic characterization of wettability in shale kerogen with varying maturity levels. J. Nat. Gas Sci. Eng. 33, 1078–1086 (2016)CrossRefGoogle Scholar
  29. Issa, R.I.: Solution of the implicitly discretised fluid flow equations by operator-splitting. J. Comput. Phys. 62, 40–65 (1985)CrossRefGoogle Scholar
  30. Jasak, H.: Error analysis and estimation for the finite volume method with applications to fluid flows. Ph.D. thesis, Department of Mechanical Engineering Imperial College of Science, Technology and Medicine (1996)Google Scholar
  31. Karniadakis, G., Beskok, A., Aluru, N.: Microflows and Nanoflows: Fundamentals and Simulation. Volume 29 of Interdisciplinary Applied Mathematics. Springer, New York (2005)Google Scholar
  32. Klinkenberg, L.: The permeability of porous media to liquids and gases. In: Drilling and Production Practice. American Petroleum Institute, pp. 200-2013 (1941)Google Scholar
  33. Lee, T., Bocquet, L., Coasne, B.: Activated desorption at heterogeneous interfaces and long-time kinetics of hydrocarbon recovery from nanoporous media. Nat. Commun. 7, 11890 (2016)CrossRefGoogle Scholar
  34. Loucks, R.G., Reed, R.M., Ruppel, S.C., Jarvie, D.M.: Morphology, genesis, and distribution of nanometer-scale pores in siliceous mudstones of the mississippian barnett shale. J. Sediment. Res. 79(12), 848–861 (2009)CrossRefGoogle Scholar
  35. Luffel, D.L., Guidry, F.K.: New core analysis methods for measuring reservoir rock properties of devonian shale. J. Pet. Technol. 44(11), 269–275 (1992)Google Scholar
  36. Ma, J., Sanchez, J.P., Wu, K., Couples, G.D., Jiang, Z.: A pore network model for simulating non-ideal gas flow in micro-and nano-porous materials. Fuel 116, 498–508 (2014)CrossRefGoogle Scholar
  37. 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)CrossRefGoogle Scholar
  38. Maes, J., Soulaine, C.: A new compressive scheme to simulate species transfer across fluid interfaces using the volume-of-fluid method. Chem. Eng. Sci. 190, 405–418 (2018)CrossRefGoogle Scholar
  39. Maxwell, J.C.: VII. On stresses in rarified gases arising from inequalities of temperature. Philos. Trans. R. Soc. Lond. 170, 231–256 (1879)CrossRefGoogle Scholar
  40. Meakin, P., Huang, H., Malthe-Sørenssen, A., Thøgersen, K.: Shale gas: opportunities and challenges. Environ. Geosci. 20(4), 151–164 (2013)CrossRefGoogle Scholar
  41. Mehmani, A., Prodanović, M., Javadpour, F.: Multiscale, multiphysics network modeling of shale matrix gas flows. Transp. Porous Media 99(2), 377–390 (2013)CrossRefGoogle Scholar
  42. Ochoa-Tapia, J.A., Del Rio, J.A., Whitaker, S.: Bulk and surface diffusion in porous media: an application of the surface-averaging theorem. Chem. Eng. Sci. 48(11), 2061–2082 (1993)CrossRefGoogle Scholar
  43. Orgogozo, L., Renon, N., Soulaine, C., Hénon, F., Tomer, S., Labat, D., Pokrovsky, O., Sekhar, M., Ababou, R., Quintard, M.: An open source massively parallel solver for richards equation: mechanistic modelling of water fluxes at the watershed scale. Comput. Phys. Commun. 185(12), 3358–3371 (2014)CrossRefGoogle Scholar
  44. Ougier-Simonin, A., Renard, F., Boehm, C., Vidal-Gilbert, S.: Microfracturing and microporosity in shales. Earth Sci. Rev. 162, 198–226 (2016)CrossRefGoogle Scholar
  45. Roman, S., Abu-Al-Saud, M.O., Tokunaga, T., Wan, J., Kovscek, A.R., Tchelepi, H.A.: Measurements and simulation of liquid films during drainage displacements and snap-off in constricted capillary tubes. J. Colloid Interface Sci. 507, 279–289 (2017)CrossRefGoogle Scholar
  46. Rudman, M.: Volume-tracking methods for interfacial flow calculations. Int. J. Numer. Methods Fluids 24(7), 671–691 (1997)CrossRefGoogle Scholar
  47. Rusche, H.: Computational fluid dynamics of dispersed two-phase flows at high phase fractions. Ph.D. thesis, Imperial College London, University of London (2003)Google Scholar
  48. Scheibe, T.D., Perkins, W.A., Richmond, M.C., McKinley, M.I., Romero-Gomez, P.D.J., Oostrom, M., Wietsma, T.W., Serkowski, J.A., Zachara, J.M.: Pore-scale and multiscale numerical simulation of flow and transport in a laboratory-scale column. Water Resour. Res. 51(2), 1023–1035 (2015)CrossRefGoogle Scholar
  49. Shabro, V., Torres-Verdín, C., Javadpour, F.: Numerical simulation of shale-gas production: from pore-scale modeling of slip-flow, knudsen diffusion, and langmuir desorption to reservoir modeling of compressible fluid. In: North American Unconventional Gas Conference and Exhibition (2011)Google Scholar
  50. Sheng, M., Li, G., Huang, Z., Tian, S., Shah, S., Geng, L.: Pore-scale modeling and analysis of surface diffusion effects on shale-gas flow in kerogen pores. J. Nat. Gas Sci. Eng. 27, 979–985 (2015)CrossRefGoogle Scholar
  51. Song, W., Yao, J., Ma, J., Couples, G., Li, Y.: Assessing relative contributions of transport mechanisms and real gas properties to gas flow in nanoscale organic pores in shales by pore network modelling. Int. J. Heat Mass Transf. 113, 524–537 (2017)CrossRefGoogle Scholar
  52. 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. Transp. Porous Media 113(1), 227–243 (2016)CrossRefGoogle Scholar
  53. 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)CrossRefGoogle Scholar
  54. Soulaine, C., Roman, S., Kovscek, A., Tchelepi, H.A.: Pore-scale modeling of multiphase reactive flow: application to mineral dissolution with production of CO\(_2\). J. Fluid Mech. 855, 616–645 (2018)CrossRefGoogle Scholar
  55. Soulaine, C., Tchelepi, H.A.: Micro-continuum approach for pore-scale simulation of subsurface processes. Transp. Porous Media 113, 431–456 (2016)CrossRefGoogle Scholar
  56. Succi, S.: Mesoscopic modeling of slip motion at fluid-solid interfaces with heterogeneous catalysis. Phys. Rev. Lett. 89(6), 064502 (2002)CrossRefGoogle Scholar
  57. Swami, V., Clarkson, C.R., Settari, A.: Non-darcy flow in shale nanopores: do we have a final answer? In: SPE Canadian Unconventional Resources Conference, Society of Petroleum Engineers (2012)Google Scholar
  58. Tinni, A., Fathi, E., Agarwal, R., Sondergeld, C., Akkutlu, Y., Rai, C.: Shale permeability measurements on plugs and crushed samples. In: SPE Canadian, pp. 1–14 (2012)Google Scholar
  59. Vengosh, A., Kondash, A., Harkness, J., Lauer, N., Warner, N., Darrah, T.H.: The geochemistry of hydraulic fracturing fluids. Proc. Earth Planet. Sci. 17, 21–24 (2017)CrossRefGoogle Scholar
  60. Wang, C.-Y., Beckermann, C.: A two-phase mixture model of liquid–gas flow and heat transfer in capillary porous media-I. Formulation. Int. J. Heat Mass Transf. 36, 2747–2747 (1993)CrossRefGoogle Scholar
  61. Wang, J., Chen, L., Kang, Q., Rahman, S.S.: Apparent permeability prediction of organic shale with generalized lattice Boltzmann model considering surface diffusion effect. Fuel 181, 478–490 (2016a)CrossRefGoogle Scholar
  62. Wang, J., Chen, L., Kang, Q., Rahman, S.S.: The lattice boltzmann method for isothermal micro-gaseous flow and its application in shale gas flow: a review. Int. J. Heat Mass Transf. 95, 94–108 (2016b)CrossRefGoogle Scholar
  63. Waples, D.W.: Organic Geochemistry for Exploration Geologists. Burgess Pub. Co, Minneapolis (1981)Google Scholar
  64. Warren, J.E., Root, P.J.: The behavior of naturally fractured reservoirs. SPE J. 3(3), 245–255 (1963)Google Scholar
  65. Weller, H.G., Tabor, G., Jasak, H., Fureby, C.: A tensorial approach to computational continuum mechanics using object-oriented techniques. Comput. Phys. 12(6), 620–631 (1998)CrossRefGoogle Scholar
  66. Williams-Kovacs, J.D., Clarkson, C.R.: A modified approach for modelling 2-phase flowback from multi-fractured horizontal Shale Gas Wells. Unconv. Resour. Technol. Conf. 2008, 2121–2137 (2015)Google Scholar
  67. Zanganeh, B., Ahmadi, M., Hanks, C., Awoleke, O.: The role of hydraulic fracture geometry and conductivity profile, unpropped zone conductivity and fracturing fluid flowback on production performance of shale oil wells. J. Unconv. Oil Gas Resour. 9, 103–113 (2015)CrossRefGoogle Scholar
  68. Zhang, T., Li, X., Sun, Z., Feng, D., Miao, Y., Li, P., Zhang, Z.: An analytical model for relative permeability in water-wet nanoporous media. Chem. Eng. Sci. 174, 1–12 (2017)CrossRefGoogle Scholar
  69. Zhang, X., Xiao, L., Shan, X., Guo, L.: Lattice boltzmann simulation of shale gas transport in organic nano-pores. Sci. Rep. 4, 4843 (2014)CrossRefGoogle Scholar
  70. Zhou, Z., Teklu, T., Li, X., Hazim, A.: Experimental study of the osmotic effect on shale matrix imbibition process in gas reservoirs. J. Nat. Gas Sci. Eng. 49(November 2017), 1–7 (2018)CrossRefGoogle Scholar

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© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Energy Resources EngineeringStanford UniversityStanfordUSA
  2. 2.CNRS/TOTAL/UNIV PAU & PAYS ADOUR, Laboratoire des Fluides Complexes et leurs Reservoirs-IPRA, UMR5150PauFrance

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