Speeding-up Simulation of Multiphase Flow in Digital Images of Heterogeneous Porous Media by Curvelet Transformation

Abstract

Multiphase fluid flow in porous media is important to a wide variety of processes of fundamental scientific and practical importance. Developing a model for the pore space of porous media represents the first step for simulating such flows. With rapid increase in the computation power and advances in instrumentation and imaging processes, it has become feasible to carry out simulation of multiphase flow in two- and three-dimensional images of porous media, hence dispensing with development of models of pore space that are based on approximating their morphology. Image-based simulations are, however, very time consuming. We describe an approach for speeding-up image-based simulation of multiphase flow in porous media based on curvelet transformations, which are specifically designed for processing of images that contain complex curved surfaces. Most porous media contain correlations in their morphology and, therefore, their images carry redundant information that, in the curvelet transform space, can be removed efficiently and accurately in order to obtain a coarser image with which the computations are far less intensive. We utilize the methodology to simulate two-phase flow of oil and water in two-dimensional digital images of sandstone and carbonate samples, and demonstrate that while the results with the curvelet-processed images are as accurate as those with the original ones, the computations are speeded up by a factor of 110–150. Thus, the methodology opens the way toward achieving the ultimate goal of simulation of multiphase flow in porous media, namely, making image-based computations a standard practice.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

References

  1. Aljasmi, A., Sahimi, M.: Efficient image-based simulation of flow and transport in heterogeneous porous Media: Application of curvelet transforms. Geophys. Res. Lett. 47, e2019GL085671 (2020)

    Google Scholar 

  2. Al-Zubi, S., Islam, N., Abbod, M.: Multiresolution analysis using wavelet, ridgelet, and curvelet transforms for medical image segmentation. Int. J. Biomed. Imaging 2011, 136034 (2011)

    Google Scholar 

  3. Arns, C.H., Knackstedt, M.A., Pinczewski, W.V., Garboczi, E.: Computation of linear elastic properties from microtomographic images: methodology and agreement between theory and experiment. Geophysics 67, 1348 (2002)

    Google Scholar 

  4. Arns, C.H., Knackstedt, M.A., Pinczewski, W.V., Lindquist, W.B.: Accurate computation of transport properties from microtomographic images. Geophys. Res. Lett. 28, 3361 (2001)

    Google Scholar 

  5. Aslannejad, H., Hassanizadeh, S.M.: Study of hydraulic properties of uncoated paper: image analysis and pore-scale modeling. Transp. Porous Media 120, 67 (2017)

    Google Scholar 

  6. Aslannejad, H., Hassanizadeh, S.M., Celia, M.A.: Characterization of the interface between coating and fibrous layers of paper. Transp. Porous Media 127, 143 (2019)

    Google Scholar 

  7. Babaei, M., King, P.R.: A comparison between wavelet and renormalization upscaling methods and iterative upscaling-downscaling scheme. SPE Reservoir Simul. Symp. 1, 469 (2011)

    Google Scholar 

  8. Bakhshian, S., Shi, Z., Sahimi, M., Tsotsis, T.T., Jessen, K.: Image-based modeling of gas adsorption and swelling in high-pressure porous formations. Sci. Rep. 8, 8249 (2018)

    Google Scholar 

  9. Bear, J.: Dynamics of Fluids in Porous Media. Dover, Mineola (1972)

    Google Scholar 

  10. Berg, S., Ott, H., Klapp, S.A., Schwing, A., Neiteler, R., Brussee, N., Makurat, A., Leu, L., Enzmann, F., Schwarz, J.-O., Kersten, M., Irvine, S., Stampanoni, M.: Real-time 3D imaging of Haines jumps in porous media flow. Proc. Natl. Acad. Sci. USA 110, 3755 (2013)

    Google Scholar 

  11. Blunt, M.J.: Effects of heterogeneity and wetting on relative permeability using pore level modeling. SPE J. 2, 70 (1997)

    Google Scholar 

  12. Blunt, M.J.: Multiphase Flow in Permeable Media: A Pore-Scale Perspective. Cambridge University Press, Cambridge (2017)

    Google Scholar 

  13. Blunt, M.J., King, M.J., Scher, H.: Simulation and theory of two-phase flow in porous media. Phys. Rev. A 46, 7680 (1992)

    Google Scholar 

  14. Blunt, M.J., King, P.R.: Relative permeabilities from two- and three-dimensional pore-scale network modelling. Transp. Porous Media 6, 407 (1991)

    Google Scholar 

  15. Blunt, M.J., Scher, H.: Pore-level modeling of wetting. Phys. Rev. E 52, 6387 (1995)

    Google Scholar 

  16. Candés, E., Demanent, L., Donoho, D.L., Ying, L.: Fast discrete curvelet transforms. Multiscale Model. Simul. 5, 861 (2005)

    Google Scholar 

  17. Chandler, R., Koplik, J., Lerman, K., Willemsen, J.: Capillary displacement and percolation in porous media. J. Fluid Mech. 119, 249 (1982)

    Google Scholar 

  18. Dashtian, H., Sahimi, M.: Coherence index and curvelet transformation for denoising geophysical data. Phys. Rev. E 90, 042810 (2014)

    Google Scholar 

  19. Daubechies, I.: Orthonormal basis of compactly supported wavelets. Commun. Pure Appl. Math. 41, 901 (1988)

    Google Scholar 

  20. Daubechies, I.: Ten Lecture on Wavelets. SIAM, Philadelphia (1992)

    Google Scholar 

  21. Donoho, D.L.: Wedgelets: nearly minimax estimation of edges. Ann. Statist. 27, 859 (1999)

    Google Scholar 

  22. Ebrahimi, F.: Invasion percolation: A computational algorithm for complex phenomena. Comput. Sci. Eng. 12(2), 84 (2010)

    Google Scholar 

  23. Ebrahimi, F., Sahimi, M.: Multiresolution wavelet coarsening and analysis of transport in heterogeneous porous media. Phys. A 316, 160 (2002)

    Google Scholar 

  24. Ebrahimi, F., Sahimi, M.: Multiresolution wavelet scale up of unstable miscible displacements in flow through porous media. Transp. Porous Media 57, 75 (2004)

    Google Scholar 

  25. Francois, M.M., Cummins, S.J., Dendy, E.D., Kothe, D.B., Sicilian, J.M., Williams, M.M.: A balanced-force algorithm for continuous and sharp interfacial surface tension models within a volume tracking framework. J. Comput. Phys. 213, 141 (2006)

    Google Scholar 

  26. Friedlingstein, P., Solomon, S.: Contributions of past and present human generations to committed warming caused by carbon dioxide. Proc. Natl. Acad. Sci. USA 102, 10832 (2005)

    Google Scholar 

  27. Ghanbarian, B., Sahimi, M., Daigle, H.: Modeling relative permeability of water in soil: Application of effective-medium approximation and percolation theory. Water Resour. Res. 52, 5025 (2016)

    Google Scholar 

  28. Ghassemzadeh, J., Hashemi, M., Sartor, L., Sahimi, M.: Pore network simulation of fluid imbibition into paper during coating processes: I. Model development. AIChE J. 47, 519 (2001)

    Google Scholar 

  29. Ghassemzadeh, J., Sahimi, M.: Pore network simulation of fluid imbibition into paper during coating III: Modeling of the two-phase flow. Chem. Eng. Sci. 59, 2281 (2004)

    Google Scholar 

  30. Gueyffier, D., Li, J., Nadim, A., Scardovelli, R., Zaleski, S.: Volume-of-fluid interface tracking with smoothed surface stress methods for three-dimensional flows. J. Comput. Phys. 152, 423 (1999)

    Google Scholar 

  31. Heiba, A.A., Sahimi, M., Scriven, L.E., Davis, H.T.: Percolation theory of two-phase relative permeability. SPE Reservoir Eng. 7, 123 (1992)

    Google Scholar 

  32. Helmig, R., Schulz, P.: Multiphase Flow and Transport Processes in the Subsurface. Springer, Berlin (1997)

    Google Scholar 

  33. Herrmann, F.J., Wang, D., Hennenfent, G., Moghaddam, P.P.: Curvelet-based seismic data processing: A multiscale and nonlinear approach. Geophysics 73, A1 (2007)

    Google Scholar 

  34. Hunt, A.G., Sahimi, M.: Flow, transport, and reaction in porous media: Percolation scaling, critical-path analysis, and Effective-Medium Approximation. Rev. Geophys. 55, 993 (2017)

    Google Scholar 

  35. Iglauer, S., Favretto, S., Spinelli, G., Schena, G., Blunt, M.J.: X-ray tomography measurements of power-law cluster size distributions for the nonwetting phase in sandstones. Phys. Rev. E. 82, 056315 (2010)

    Google Scholar 

  36. Kantzas, A., Chatzis, I.: Network simulation of relative permeability curves using a bond correlated-site percolation model of pore structure. Chem. Eng. Commun. 69, 191 (1988)

    Google Scholar 

  37. Knackstedt, M.A., Sheppard, A.P., Sahimi, M.: Pore network modeling of two-phase flow in porous rock: The effect of correlated heterogeneity. Adv. Water Resour. 24, 257 (2001)

    Google Scholar 

  38. Kohanpur, A.H., Rahromostaqim, M., Valocchi, A.J., Sahimi, M.: Two-phase flow of CO\(_2\)-brine in a heterogeneous sandstone: characterization of the rock and comparison of the lattice-Boltzmann, pore-network, and direct numerical simulation methods. Adv. Water Resour. 135, 103439 (2020)

    Google Scholar 

  39. Larson, R.G., Scriven, L.E., Davis, H.T.: Percolation theory of residual phases in porous media. Nature 268, 409 (1977)

    Google Scholar 

  40. Lemmens, H.J., Butcher, R., Botha, P.W.S.K.: FIB/SEM and SEM/EDX: a new dawn for the SEM in the core lab? Petrophysics 52, 452 (2011)

    Google Scholar 

  41. Ma, J., Plonka, G.: Computing with curvelets: From image processing to turbulent flows. Comput. Sci. Eng. 11(2), 72 (2009)

    Google Scholar 

  42. Mallat, S.G.: A theory for multiresolution signal decomposition: the wavelet representation. IEEE Trans. Pattern Recog. Machine Intell. 11, 674 (1989)

    Google Scholar 

  43. Mallat, S.G.: Multiresolution approximations and wavelet orthonormal bases of \(L^2(R)\). Trans. Am. Math. Soc. 315, 69 (1989)

    Google Scholar 

  44. Mehrabi, A.R., Sahimi, M.: Coarsening of heterogeneous media: application of wavelets. Phys. Rev. Lett. 79, 4385 (1997)

    Google Scholar 

  45. Neelamani, R., Baumstein, A.I., Gillard, D.G., Hadidi, M.T., Soroka, W.L.: Coherent and random noise attenuation using the curvelet transform. The Leading Edge 27, 129 (2008)

    Google Scholar 

  46. Nordbotten, J.M., Celia, M.A.: Geological Storage of CO\(_2\): Modeling Approaches for Large-Scale Simulation. Wiley, New York (2011)

    Google Scholar 

  47. Oak, M., Baker, L., Thomas, D.: Three-phase relative permeability of Berea sandstone. J. Pet. Technol. 42, 1054 (1990)

    Google Scholar 

  48. Olhede, S., Walden, A.T.: The Hilbert spectrum via wavelet projections. Proc. R. Soc. Lond. A 460, 955 (2004)

    Google Scholar 

  49. Pancaldi, V., Christensen, K., King, P.R.: Permeability up-scaling using Haar wavelets. Transp. Porous Media 67, 395 (2007)

    Google Scholar 

  50. Piri, M., Blunt, M.J.: Three-dimensional mixed-wet random pore-scale network modeling of two- and three-phase flow in porous media. I. Model description. Phys. Rev. E 71, 026301 (2005)

    Google Scholar 

  51. Piri, M., Blunt, M.J.: Three-dimensional mixed-wet random pore-scale network modeling of two- and three-phase flow in porous media. II. Results. Phys. Rev. E 71, 026302 (2005)

    Google Scholar 

  52. Porter, M.L., Wildenschild, D., Grant, G., Gerhard, J.I.: Measurement and prediction of the relationship between capillary pressure, saturation, and interfacial area in a NAPL-water-glass bead system. Water Resour. Res. 46, W08512 (2010)

    Google Scholar 

  53. Raeini, A.Q.: Modelling Multiphase Flow Through Micro-CT Images of the Pore Space, Ph.D. Thesis, Imperial College of London (2013)

  54. Raeini, A.Q., Blunt, M.J., Bijeljic, B.: Modelling two-phase flow in porous media at the pore scale using the volume-of-fluid method. J. Comput. Phys. 231, 5653 (2012)

    Google Scholar 

  55. Raeini, A.Q., Blunt, M.J., Bijeljic, B.: Direct simulations of two-phase flow on micro-CT images of porous media and upscaling of pore-scale forces. Adv. Water Resour. 74, 116 (2014)

    Google Scholar 

  56. Rasaei, M.R., Sahimi, M.: Upscaling and simulation of waterflooding in heterogeneous reservoirs using wavelet transformations: Application to the SPE-10 model. Transp. Porous Media 72, 311 (2008)

    Google Scholar 

  57. Rasaei, M.R., Sahimi, M.: Upscaling of the permeability by multiscale wavelet transformations and simulation of multiphase flows in heterogeneous porous media. Comput. Geosci. 13, 187 (2009)

    Google Scholar 

  58. Rezapour, A., Ortega, A., Sahimi, M.: Upscaling of geological models of oil reservoirs with unstructured grids using lifting-based graph wavelet transforms. Transp. Porous Media 127, 661 (2019)

    Google Scholar 

  59. Sahimi, M.: Flow and Transport in Porous Media and Fractured Rock, 2nd edn. Wiley, Weinheim (2011)

    Google Scholar 

  60. Sahimi, M., Heiba, A.A., Davis, H.T., Scriven, L.E.: Dispersion in flow through porous media: II. Two-phase flow. Chem. Eng. Sci. 41, 2123 (1986)

    Google Scholar 

  61. Sankey, M.H., Holland, D.J., Sederman, A.J., Gladden, L.F.: Magnetic resonance velocity imaging of liquid and gas two-phase flow in packed beds. J. Magn. Reson. 196, 142 (2009)

    Google Scholar 

  62. Sheppard, S., Mantle, M.D., Sederman, A.J., Johns, M.L., Gladden, L.F.: Magnetic resonance imaging study of complex fluid flow in porous media: flow patterns and quantitative saturation profiling of amphiphilic fracturing fluid displacement in sandstone cores. Magn. Reson. Imaging. 21, 365 (2003)

    Google Scholar 

  63. Shokri, N.: Pore-scale dynamics of salt transport and distribution in drying porous media. Phys. Fluids 26, 012106 (2014)

    Google Scholar 

  64. Shokri, N., Lehmann, P., Or, D.: Characteristics of evaporation from partially-wettable porous media. Water Resour. Res. 45, W02415 (2009)

    Google Scholar 

  65. Shokri, N., Lehmann, P., Or, D.: Liquid phase continuity and solute concentration dynamics during evaporation from porous media—pore scale processes near vaporization surface. Phys. Rev. E 81, 046308 (2010)

    Google Scholar 

  66. Shokri, N., Sahimi, M., Or, D.: Morphology, propagation dynamics and scaling characteristics of drying fronts in porous media. Geophys. Res. Lett. 39, L09401 (2012)

    Google Scholar 

  67. Shokri-Kuehni, S.M.S., Vetter, T., Webb, C., Shokri, N.: New insights into saline water evaporation from porous media: complex interaction between evaporation rates, precipitation and surface temperature. Geophys. Res. Lett. 44, 5504 (2017)

    Google Scholar 

  68. Shokri-Kuehni, S.M.S., Norouzirad, M., Webb, C., Shokri, N.: Impact of type of salt and ambient conditions on saline water evaporation from porous media. Adv. Water Resour. 105, 154 (2017)

    Google Scholar 

  69. Shokri-Kuehni, S.M.S., Bergstad, M., Sahimi, M., Webb, C., Shokri, N.: Iodine k-edge dual energy imaging reveals the influence of particle size distribution on solute transport in drying porous media. Sci. Rep. 10, 10731 (2018)

    Google Scholar 

  70. Starck, J.-L., Candés, E.J., Donoho, D.L.: The curvelet transform for image denoising. IEEE Trans. Image Process. 11(6), 670 (2002)

    Google Scholar 

  71. Swerin, A.: Dimensional scaling of aqueous ink imbibition and inkjet printability on porous pigment coated paper A revisit. Ind. Eng. Chem. Res. 57, 49 (2018)

    Google Scholar 

  72. Tahmasebi, P., Sahimi, M., Kohanpur, A.H., Valocchi, A.J.: Pore-scale simulation of flow of CO\(_2\) and brine in reconstructed and actual 3D rock cores. J. Pet. Sci. Eng. 155, 21 (2017)

    Google Scholar 

  73. Ubink, O.: Numerical Prediction of Two Fluid Systems with Sharp Interfaces, Ph.D. Thesis, Imperial College of London (1997)

  74. Wildenschild, D., Armstrong, R.T., Herring, A.L., Young, I., Young, I.M., Carey, J.W.: Exploring capillary trapping efficiency as a function of interfacial tension, viscosity, and flow rate. Energy Procedia 4, 4945 (2014)

    Google Scholar 

  75. Wilkinson, D., Willimsen, J.F.: Invasion percolation: a new form of percolation theory. J. Phys. A 16, 3365 (1983)

    Google Scholar 

  76. Woiselle, A., Starck, J.-L., Fadili, J.: 3D curvelet transforms and astronomical data restoration. Appl. Comput. Harmonic 28, 171 (2010)

    Google Scholar 

  77. Ying, L., Demanet, L., Candés, E.: 3D discrete curvelet transform. Proceedings of SPIE5914, Wavelets XI, 591413 (2005)

Download references

Acknowledgements

A.A. is grateful to the Public Authority for Applied Education and Training of Kuwait for a Ph.D. scholarship. This work was also supported in part by the National Science Foundation.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Muhammad Sahimi.

Additional information

Publisher's Note

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

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Aljasmi, A., Sahimi, M. Speeding-up Simulation of Multiphase Flow in Digital Images of Heterogeneous Porous Media by Curvelet Transformation. Transp Porous Med 137, 215–232 (2021). https://doi.org/10.1007/s11242-021-01559-5

Download citation

Keywords

  • Two-phase fluid flow
  • Porous media
  • Curvelet transformation