https://doi.org/10.26089/NumMet.v27r108

Estimation of three-phase relative permeabilities from the results of two-phase pore-scale numerical simulation

Authors

  • Dmitry I. Prokhorov
  • Veronica D. Chepelenkova
  • Vadim V. Lisitsa

Keywords:

digital rock physics
domain decomposition
pore-scale simulation
two-phase flows
three-phase relative permeability

Abstract

The article presents a parallel algorithm for numerical simulation of two-phase flows in digital rock images. The algorithm is based on the phase-field method and the finite-difference approach, which includes solving the Poisson equation using conjugate gradients with a spectral preconditioner. The domain decomposition method and CUDA and MPI technologies are used to parallelize the algorithm. The algorithm is applied to calculate the relative phase permeabilities using the steady-state method for oil and water, oil and CO2 in a supercritical form. Based on the calculated two-phase permeabilities, ternary relative permeability diagrams are constructed for a three-phase system using the Stone I, Stone II, and Baker models. The justification of the choice of liquid parameters for two-phase modeling is carried out.


Downloads

Published

2026-03-19

Issue

Vol. 27 (2026): Issue 1.

Section

Methods and algorithms of computational mathematics and their applications

Authors

Dmitry I. Prokhorov

Trofimuk Institute of Petroleum Geology and Geophysics of SB RAS

• Researcher

Veronica D. Chepelenkova

Trofimuk Institute of Petroleum Geology and Geophysics of SB RAS

• Junior Researcher

Vadim V. Lisitsa

Sobolev Institute of Mathematics of SB RAS

• Head of Department

References

  1. N. Alyafei, Fundamentals of Reservoir Rock Properties(Hamad Bin Khalifa University Press, Doha, 2021).
    doi 10.5339/Fundamentals_of_Reservoir_Rock_Properties_2ndEdition
  2. M. J. Oak, L. E. Baker, and D. C. Thomas, “Three-Phase Relative Permeability of Berea Sandstone,” Journal of Petroleum Technology 42 (08), 1054–1061 (1990).
    doi 10.2118/17370-pa
  3. H. L. Stone, “Probability Model for Estimating Three-Phase Relative Permeability,” Journal of Petroleum Technology 22 (02), 214–218 (1970).
    doi 10.2118/2116-PA
  4. H. L. Stone, “Estimation of Three-Phase Relative Permeability And Residual Oil Data,” Journal of Canadian Petroleum Technology 12 (04), 53–61 (1973).
    doi 10.2118/73-04-06
  5. F. Jiang and T. Tsuji, “Estimation of three-phase relative permeability by simulating fluid dynamics directly on rock-microstructure images,” Water Resources Research 53 (1), 11–32 (2017).
    doi 10.1002/2016WR019098
  6. O. Axelsson, X. He, and M. Neytcheva, “Numerical Solution of theTime-Dependent Navier–Stokes Equation for Variable Density–Variable Viscosity. Part I,” Mathematical Modelling and Analysis 20 (2), 232–260 (2015).
    doi 10.3846/13926292.2015.1021395
  7. V. Balashov and A. Zlotnik, “On a New Spatial Discretization for a Regularized3D Compressible Isothermal Navier–Stokes–Cahn–Hilliard System ofEquations with Boundary Conditions,” Journal of Scientific Computing 86 (3), Article Number 33 (2021).
    doi 10.1007/s10915-020-01388-6
  8. J. Kim, “Phase-Field Models for Multi-Component Fluid Flows,” Communications in Computational Physics 12 (3), 613–661 (2012).
    doi 10.4208/cicp.301110.040811a
  9. L. Zhang, C. Xu, Y. Guo, et al., “The Effect of Surface Roughness on Immiscible Displacement Using Pore Scale Simulation,” Transport in Porous Media 140 (3), 713–725 (2021).
    doi 10.1007/s11242-020-01526-6
  10. G. Zhu, J. Kou, J. Yao, et al., “A phase-field moving contact line model with soluble surfactants,” Journal of Computational Physics 405, Article Number 109170 (2020).
    doi 10.1016/j.jcp.2019.109170
    http://hdl.handle.net/10754/660524 Cited March 12, 2026.
  11. H. Andra, N. Combaret, J. Dvorkin, et al., “Digital rock physics benchmarks - Part I: Imaging and segmentation,” Computers and Geosciences 50 (4), 25–32 (2013).
    doi 10.1016/j.cageo.2012.09.005
  12. C. Madonna, B. Quintal, M. Frehner, et al., “Synchrotron-based X-ray tomographic microscopy for rock physics investigations,” Geophysics 78 (1), D53–D64 (2013).
    doi 10.1190/GEO2012-0113.1
  13. H. Yu and X. Yang, “Numerical approximations for a phase-field moving contact line model with variable densities and viscosities,” Journal of Computational Physics 334 (3), 665–686 (2017).
    doi 10.1016/j.jcp.2017.01.026
  14. F. J. Carrillo, I. C. Bourg, and C. Soulaine, “Multiphase flow modeling in multiscale porous media: An open-source micro-continuum approach,” Journal of Computational Physics: X 8, Article Number 100073 (2020).
    doi 10.1016/j.jcpx.2020.100073
  15. G. Fu, “A divergence-free HDG scheme for the Cahn-Hilliard phase-field model for two-phase incompressible flow,” Journal of Computational Physics 419, Article Number 109671 (2020).
    doi 10.1016/j.jcp.2020.109671
  16. J. Yang and J. Kim, “A novel Cahn–Hilliard–Navier–Stokes model with a nonstandard variable mobility for two-phase incompressible fluid flow,” Computers & Fluids 213, Article Number 104755 (2020).
    doi 10.1016/j.compfluid.2020.104755
  17. J. Zhao and D. Han, “Second-order decoupled energy-stable schemes for Cahn–Hilliard–Navier–Stokes equations,” Journal of Computational Physics 443 (1), Article Number 110536 (2021).
    doi 10.1016/j.jcp.2021.110536
  18. M. Griebel and P. Zaspel, “A multi-GPU accelerated solver for the three-dimensional two-phase incompressible Navier–Stokes equations,” Computer Science – Research and Development 25 (1–2), 65–73 (2010).
    doi 10.1007/s00450-010-0111-7
  19. Q. Zhang and X.-P. Wang, “Phase field modeling and simulation of three-phase flow on solid surfaces,” Journal of Computational Physics 319, 79–107 (2016).
    doi 10.1016/j.jcp.2016.05.016
  20. T. S. Khachkova, V. V. Lisitsa, G. V. Reshetova, and V. A. Tcheverda, “Numerical estimation of electrical resistivity in digital rocks using GPUs,” Numerical Methods and Programming [Vychislitel’nye Metody i Programmirovanie] 21 (3), 306–318 (2020).
    doi 10.26089/NumMet.v21r326
    https://www.mathnet.ru/eng/vmp1012 Cited March 12, 2026.
  21. K. Luo, Z. Zhuang, J. Fan, and N. E. L. Haugen, “A ghost-cell immersed boundary method for simulations of heat transfer in compressible flows under different boundary conditions,” International Journal of Heat and Mass Transfer 92, 708–717 (2016).
    doi 10.1016/j.ijheatmasstransfer.2015.09.024
  22. Y. Saad, Iterative Methods for Sparse Linear Systems(SIAM, Philadelphia, 2003).
  23. A. A. Manaev and V. V. Lisitsa, “Spectral preconditioner for solving the Poisson equation,” Numerical Methods and Programming [Vychislitel’nye Metody i Programmirovanie] 26 (2), 111–128 (2025).
    doi 10.26089/NumMet.v26r208
    https://www.mathnet.ru/eng/vmp1153 Cited March 12, 2026.
  24. D. I. Prokhorov, “Domain decomposition for the numerical solution of the Cahn–Hilliard equation,” Numerical Methods and Programming [Vychislitel’nye Metody i Programmirovanie] 26 (1), 17–32 (2025).
    doi 10.26089/NumMet.v26r102
  25. P. Chiquet, J.-L. Daridon, D. Broseta, and S. Thibeau, “CO_2/water interfacial tensions under pressure and temperature conditions of CO_2 geological storage,” Energy Conversion and Management 48 (3), 736–744 (2007).
    doi 10.1016/j.enconman.2006.09.011
  26. D. Yang, P. Tontiwachwuthikul, Y. Gu, “Interfacial Tensions of the Crude Oil + Reservoir Brine + CO_2 Systems at Pressures up to 31 MPa and Temperatures of 27°C and 58°C,” Journal of Chemical & Engineering Data50 (4), 1242–1249 (2005).
    doi 10.1021/je0500227
  27. T. Ahmed, Reservoir Engineering Handbook(Gulf Professional Publishing, Houston, 2010).
  28. L. Li, J. Zheng, Y. Shi, et al., “Mechanisms of Fluid Migration and CO_2 Storage in Low Permeability Heavy Oil Reservoirs Using High-Pressure Microfluidic CO_2 Flooding Experiment,” Energy & Fuels 38 (9), 7997–8008 (2024).
    doi 10.1021/acs.energyfuels.4c00869
  29. A. Hemmati-Sarapardeh, S. Ayatollahi, M.-H. Ghazanfari, and M. Masihi, “Experimental Determination of Interfacial Tension and Miscibility of the CO_2–Crude Oil System; Temperature, Pressure, and Composition Effects,” Journal of Chemical & Engineering Data 59 (1), 61–69 (2014).
    doi 10.1021/je400811h
  30. Z. Yang, X. Liu, Z. Hua, et al., “Interfacial tension of CO_2 and crude oils under high pressure and temperature,” Colloids and Surfaces A: Physicochemical and Engineering Aspects 482, 611–616 (2015).
    doi 10.1016/j.colsurfa.2015.05.058
  31. F. Haeri, D. Tapriyal, C. Matranga, et al., “Variation of CO_2-Brine Contact Angles on Natural Rocks of Different Compositions,” Journal of Energy and Power Technology 3 (4), Article Number 046 (2021).
    doi 10.21926/jept.2104046
  32. X. Li and X. Fan, “Effect of CO_2 phase on contact angle in oil-wet and water-wet pores,” International Journal of Greenhouse Gas Control 36, 106–113 (2015).
    doi 10.1016/j.ijggc.2015.02.017
  33. R. Gupta and D. Maloney, “Intercept Method – A Novel Technique To Correct Steady-State Relative Permeability Data for Capillary End Effects,” SPE Reservoir Evaluation & Engineering 19 (02), 316–330 (2015).
    doi 10.2118/171797-pa
  34. M.J. Blunt, A. Alhosani, Q. Lin, et al., “Determination of contact angles for three-phase flow in porous media using an energy balance,” Journal of Colloid and Interface Science 582 (Pt A), 283–290 (2021).
    doi 10.1016/j.jcis.2020.07.152

License

Copyright (c) 2026 Д.И. Прохоров, В.Д. Чепеленкова, В.В. Лисица

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.