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A discontinuous Galerkin method for a diffuse-interface model of immiscible two-phase flows with soluble surfactant

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Abstract

A numerical method using discontinuous polynomial approximations is formulated for solving a phase-field model of two immiscible fluids with a soluble surfactant. The proposed scheme is shown to decay the total free Helmholtz energy at the discrete level, which is consistent with the continuous model dynamics. The scheme recovers the Langmuir adsorption isotherms at equilibrium. Simulations of spinodal decomposition, flow through a cylinder and flow through a sequence of pore throats show the dynamics of the flow with and without surfactant. Finally the numerical method is used to simulate fluid flows in the pore space of Berea sandstone obtained by micro-CT imaging.

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References

  1. Neugebauer, J.M.: Detergents: An overview. In: Deutscher, M.P. (ed.) Guide to Protein Purification, Methods in Enzymology, vol. 182, pp 239–253. Academic Press (1990). https://doi.org/10.1016/0076-6879(90)82020-3

  2. Hasenhuettl, G.L., Hartel, R.W.: Food Emulsifiers and Their Applications, 3rd edn. Springer International Publishing, New York (2019). https://doi.org/10.1007/978-3-030-29187-7

    Book  Google Scholar 

  3. Chen, X., Feng, Q., Liu, W., Sepehrnoori, K.: Modeling preformed particle gel surfactant combined flooding for enhanced oil recovery after polymer flooding. Fuel 194, 42–49 (2017). https://doi.org/10.1016/j.fuel.2016.12.075

    Article  Google Scholar 

  4. Halpern, D., Jensen, O., Grotberg, J.: A theoretical study of surfactant and liquid delivery into the lung. J. Appl. Physiol. (Bethesda Md. : 1985) 85(1), 333–352 (1998). https://doi.org/10.1152/jappl.1998.85.1.333

    Article  Google Scholar 

  5. Teigen, K.E., Song, P., Lowengrub, J., Voigt, A.: A diffuse-interface method for two-phase flows with soluble surfactants. J. Comput. Phys. 230(2), 375–393 (2011). https://doi.org/10.1016/j.jcp.2010.09.020

    Article  Google Scholar 

  6. Stone, H.A., Leal, L.G.: The effects of surfactants on drop deformation and breakup. J. Fluid Mech. 220, 161–186 (1990). https://doi.org/10.1017/S0022112090003226

    Article  Google Scholar 

  7. Milliken, W.J., Stone, H.A., Leal, L.G.: The effect of surfactant on the transient motion of Newtonian drops. Phys. Fluids A Fluid Dyn. 5(1), 69–79 (1993). https://doi.org/10.1063/1.858790

    Article  Google Scholar 

  8. Li, X., Pozrikidis, C.: The effect of surfactants on drop deformation and on the rheology of dilute emulsions in Stokes flow. J. Fluid Mech. 341, 165–194 (1997). https://doi.org/10.1017/S0022112097005508

    Article  Google Scholar 

  9. Zhang, J., Eckmann, D., Ayyaswamy, P.: A front tracking method for a deformable intravascular bubble in a tube with soluble surfactant transport. J. Comput. Phys. 214(1), 366–396 (2006). https://doi.org/10.1016/j.jcp.2005.09.016

    Article  Google Scholar 

  10. Muradoglu, M., Tryggvason, G.: A front-tracking method for computation of interfacial flows with soluble surfactants. J. Comput. Phys. 227(4), 2238–2262 (2008). https://doi.org/10.1016/j.jcp.2007.10.003

    Article  Google Scholar 

  11. Lai, M.C., Tseng, Y.H., Huang, H.: An immersed boundary method for interfacial flows with insoluble surfactant. J. Comput. Phys. 227(15), 7279–7293 (2008). https://doi.org/10.1016/j.jcp.2008.04.014

    Article  Google Scholar 

  12. Xu, J.J., Zhao, H.K.: An Eulerian formulation for solving partial differential equations along a moving interface. J. Sci. Comput. 19(1), 573–594 (2003). https://doi.org/10.1023/A:1025336916176

    Article  Google Scholar 

  13. Renardy, Y.Y., Renardy, M., Cristini, V.: A new volume-of-fluid formulation for surfactants and simulations of drop deformation under shear at a low viscosity ratio. Eur. J. Mech. B/Fluids 21(1), 49–59 (2002). https://doi.org/10.1016/S0997-7546(01)01159-1

    Article  Google Scholar 

  14. James, A.J., Lowengrub, J.: A surfactant-conserving volume-of-fluid method for interfacial flows with insoluble surfactant. J. Comput. Phys. 201(2), 685–722 (2004). https://doi.org/10.1016/j.jcp.2004.06.013

    Article  Google Scholar 

  15. Hameed, M., Siegel, M., Young, Y.N., Li, J., Booty, M.R., Papageorgiou, D.T.: Influence of insoluble surfactant on the deformation and breakup of a bubble or thread in a viscous fluid. J. Fluid Mech. 594, 307–340 (2008). https://doi.org/10.1017/S0022112007009032

    Article  Google Scholar 

  16. Cahn, J.W., Hilliard, J.E.: Free energy of a nonuniform system. I. Interfacial free energy. J. Chem. Phys. 28(2), 258–267 (1958). https://doi.org/10.1063/1.1744102

    Article  Google Scholar 

  17. Laradji, M., Guo, H., Grant, M., Zuckermann, M.J.: The effect of surfactants on the dynamics of phase separation. J. Phys. Condens. Matter 4(32), 6715–6728 (1992). https://doi.org/10.1088/0953-8984/4/32/006

    Article  Google Scholar 

  18. Pätzold, G., Dawson, K.: Numerical simulation of phase separation in the presence of surfactants and hydrodynamics. Phys. Rev. E 52, 6908–6911 (1995). https://doi.org/10.1103/PhysRevE.52.6908

    Article  Google Scholar 

  19. Diamant, H., Andelman, D.: Kinetics of surfactant adsorption at fluid-fluid interfaces. J. Phys. Chem. 100(32), 13732–13742 (1996). https://doi.org/10.1021/jp960377k

    Article  Google Scholar 

  20. Komura, S., Kodama, H.: Two-order-parameter model for an oil-water-surfactant system. Phys. Rev. E 55, 1722–1727 (1997). https://doi.org/10.1103/PhysRevE.55.1722

    Article  Google Scholar 

  21. Diamant, H., Ariel, G., Andelman, D.: Kinetics of surfactant adsorption: the free energy approach. Colloids Surf. A Physicochem. Eng. Asp. 183-185, 259–276 (2001). https://doi.org/10.1016/S0927-7757(01)00553-2

    Article  Google Scholar 

  22. van der Sman, R.G.M., van der Graaf, S.: Diffuse interface model of surfactant adsorption onto flat and droplet interfaces. Rheol. Acta 46(1), 3–11 (2006). https://doi.org/10.1007/s00397-005-0081-z

    Article  Google Scholar 

  23. Liu, H., Zhang, Y.: Phase-field modeling droplet dynamics with soluble surfactants. J. Comput. Phys. 229(24), 9166–9187 (2010). https://doi.org/10.1016/j.jcp.2010.08.031

    Article  Google Scholar 

  24. Engblom, S., Do-Quang, M., Amberg, G., Tornberg, A.K.: On diffuse interface modeling and simulation of surfactants in two-phase fluid flow. Commun. Comput. Phys. 14(4), 879–915 (2013). https://doi.org/10.4208/cicp.120712.281212a

    Article  Google Scholar 

  25. Yang, X.: Numerical approximations for the Cahn–Hilliard phase field model of the binary fluid-surfactant system. J. Sci. Comput. 74(3), 1533–1553 (2018). https://doi.org/10.1007/s10915-017-0508-6

    Article  Google Scholar 

  26. Yang, X., Ju, L.: Linear and unconditionally energy stable schemes for the binary fluid-surfactant phase field model. Comput. Methods Appl. Mech. Eng. 318, 1005–1029 (2017). https://doi.org/10.1016/j.cma.2017.02.011

    Article  Google Scholar 

  27. Zhu, G., Kou, J., Sun, S., Yao, J., Li, A.: Decoupled, energy stable schemes for a phase-field surfactant model. Comput. Phys. Commun. 233, 67–77 (2018). https://doi.org/10.1016/j.cpc.2018.07.003

    Article  Google Scholar 

  28. Zhu, G., Kou, J., Sun, S., Yao, J., Li, A.: Numerical approximation of a phase-field surfactant model with fluid flow. J. Sci. Comput. 80(1), 223–247 (2019). https://doi.org/10.1007/s10915-019-00934-1

    Article  Google Scholar 

  29. Zhu, G., Kou, J., Yao, J., Li, A., Sun, S.: A phase-field moving contact line model with soluble surfactants. J. Comput. Phys. 405, 109170 (2020). https://doi.org/10.1016/j.jcp.2019.109170

    Article  Google Scholar 

  30. Wang, K., Luo, J., Wei, Y., Wu, K., Li, J., Chen, Z.: Practical application of machine learning on fast phase equilibrium calculations in compositional reservoir simulations. J. Comput. Phys. 401, 109013 (2020). https://doi.org/10.1016/j.jcp.2019.109013

    Article  Google Scholar 

  31. Zhang, T., Li, Y., Li, Y., Sun, S., Gao, X.: A self-adaptive deep learning algorithm for accelerating multi-component flash calculation. Comput. Methods Appl. Mech. Eng. 369, 113207 (2020). https://doi.org/10.1016/j.cma.2020.113207

    Article  Google Scholar 

  32. Frank, F., Liu, C., Alpak, F.O., Riviere, B.: A finite volume / discontinuous Galerkin method for the advective Cahn–Hilliard equation with degenerate mobility on porous domains stemming from micro-CT imaging. Comput. Geosci. 22(2), 543–563 (2018). https://doi.org/10.1007/s10596-017-9709-1

    Article  Google Scholar 

  33. Frank, F., Liu, C., Alpak, F.O., Berg, S., Riviere, B.: Direct numerical simulation of flow on pore-scale images using the phase-field method. SPE J. 23(05), 1833–1850 (2018). https://doi.org/10.2118/182607-PA

    Article  Google Scholar 

  34. Liu, C., Frank, F., Thiele, C., Alpak, F.O., Berg, S., Chapman, W., Riviere, B.: An efficient numerical algorithm for solving viscosity contrast Cahn–Hilliard–Navier–Stokes system in porous media. J. Comput. Phys. 400, 108948 (2020). https://doi.org/10.1016/j.jcp.2019.108948

    Article  Google Scholar 

  35. Zhang, T., Li, Y., Li, C., Sun, S: Effect of salinity on oil production: review on low salinity waterflooding mechanisms and exploratory study on pipeline scaling. Oil Gas Sci. Technol. Rev. IFP Energ. Nouvelles 75, 50 (2020). https://doi.org/10.2516/ogst/2020045

    Article  Google Scholar 

  36. Feng, X.: Fully discrete finite element approximations of the Navier–Stokes–Cahn–Hilliard diffuse interface model for two-phase fluid flows. SIAM J. Numer. Anal. 44(3), 1049–1072 (2006)

    Article  Google Scholar 

  37. Bao, K., Shi, Y., Sun, S., Wang, X.P.: A finite element method for the numerical solution of the coupled Cahn-Hilliard and Navier-Stokes system for moving contact line problems. J. Comput. Phys. 231, 8083–8099 (2012)

    Article  Google Scholar 

  38. Guo, A., Lin, P., Lowengrub, J.: A numerical method for the quasi-incompressible Cahn–Hilliard–Navier–Stokes equations for variable density flows with a discrete energy law. J. Comput. Phys. 276, 486–507 (2014)

    Article  Google Scholar 

  39. Kou, J., Sun, S., Wang, X.: Linearly decoupled energy-stable numerical methods for multicomponent two-phase compressible flow. SIAM J. Numer. Anal. 56, 3219–3248 (2018)

    Article  Google Scholar 

  40. Giesselmann, J., Pryer, T.: Energy consistent discontinuous Galerkin methods for a quasi-incompressible diffuse two-phase flow model. ESAIM Math. Model. Numer. Anal. 49, 275–301 (2015)

    Article  Google Scholar 

  41. Riviere, B.: Discontinuous Galerkin methods for solving elliptic and parabolic equations: theory and implementation. Frontiers in applied mathematics Society for industrial and applied mathematics (2008)

  42. Yue, P., Zhou, C., Feng, J.J.: Spontaneous shrinkage of drops and mass conservation in phase-field simulations. J. Comput. Phys. 223(1), 1–9 (2007). https://doi.org/10.1016/j.jcp.2006.11.020

    Article  Google Scholar 

  43. Andrä, H., Combaret, N., Dvorkin, J., Glatt, E., Han, J., Kabel, M., et al.: Digital rock physics benchmarks. Part I: Imaging and segmentation. Comput. Geosci. 50, 25–32 (2013). https://doi.org/10.1016/j.cageo.2012.09.005

    Article  Google Scholar 

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Acknowledgements

The authors thank Dr. Steffen Berg for useful discussions on surfactant models. Ray and Riviere acknowledge funding from a Shell-Rice collaboration. Riviere is also partially funded by NSF-DMS 1913291.

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Authors and Affiliations

  1. Department of Aerospace and Mechanical Engineering, University of Southern California, Los Angeles, TX, USA

    Deep Ray

  2. Department of Computational and Applied Mathematics, Rice University, Houston, TX, USA

    Chen Liu & Beatrice Riviere

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  1. Deep Ray
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  2. Chen Liu
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Correspondence to Deep Ray.

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Ray, D., Liu, C. & Riviere, B. A discontinuous Galerkin method for a diffuse-interface model of immiscible two-phase flows with soluble surfactant. Comput Geosci 25, 1775–1792 (2021). https://doi.org/10.1007/s10596-021-10073-y

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