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.
Similar content being viewed by others
References
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)
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)
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)
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)
Aslannejad, H., Hassanizadeh, S.M.: Study of hydraulic properties of uncoated paper: image analysis and pore-scale modeling. Transp. Porous Media 120, 67 (2017)
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)
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)
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)
Bear, J.: Dynamics of Fluids in Porous Media. Dover, Mineola (1972)
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)
Blunt, M.J.: Effects of heterogeneity and wetting on relative permeability using pore level modeling. SPE J. 2, 70 (1997)
Blunt, M.J.: Multiphase Flow in Permeable Media: A Pore-Scale Perspective. Cambridge University Press, Cambridge (2017)
Blunt, M.J., King, M.J., Scher, H.: Simulation and theory of two-phase flow in porous media. Phys. Rev. A 46, 7680 (1992)
Blunt, M.J., King, P.R.: Relative permeabilities from two- and three-dimensional pore-scale network modelling. Transp. Porous Media 6, 407 (1991)
Blunt, M.J., Scher, H.: Pore-level modeling of wetting. Phys. Rev. E 52, 6387 (1995)
Candés, E., Demanent, L., Donoho, D.L., Ying, L.: Fast discrete curvelet transforms. Multiscale Model. Simul. 5, 861 (2005)
Chandler, R., Koplik, J., Lerman, K., Willemsen, J.: Capillary displacement and percolation in porous media. J. Fluid Mech. 119, 249 (1982)
Dashtian, H., Sahimi, M.: Coherence index and curvelet transformation for denoising geophysical data. Phys. Rev. E 90, 042810 (2014)
Daubechies, I.: Orthonormal basis of compactly supported wavelets. Commun. Pure Appl. Math. 41, 901 (1988)
Daubechies, I.: Ten Lecture on Wavelets. SIAM, Philadelphia (1992)
Donoho, D.L.: Wedgelets: nearly minimax estimation of edges. Ann. Statist. 27, 859 (1999)
Ebrahimi, F.: Invasion percolation: A computational algorithm for complex phenomena. Comput. Sci. Eng. 12(2), 84 (2010)
Ebrahimi, F., Sahimi, M.: Multiresolution wavelet coarsening and analysis of transport in heterogeneous porous media. Phys. A 316, 160 (2002)
Ebrahimi, F., Sahimi, M.: Multiresolution wavelet scale up of unstable miscible displacements in flow through porous media. Transp. Porous Media 57, 75 (2004)
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)
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)
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)
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)
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)
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)
Heiba, A.A., Sahimi, M., Scriven, L.E., Davis, H.T.: Percolation theory of two-phase relative permeability. SPE Reservoir Eng. 7, 123 (1992)
Helmig, R., Schulz, P.: Multiphase Flow and Transport Processes in the Subsurface. Springer, Berlin (1997)
Herrmann, F.J., Wang, D., Hennenfent, G., Moghaddam, P.P.: Curvelet-based seismic data processing: A multiscale and nonlinear approach. Geophysics 73, A1 (2007)
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)
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)
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)
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)
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)
Larson, R.G., Scriven, L.E., Davis, H.T.: Percolation theory of residual phases in porous media. Nature 268, 409 (1977)
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)
Ma, J., Plonka, G.: Computing with curvelets: From image processing to turbulent flows. Comput. Sci. Eng. 11(2), 72 (2009)
Mallat, S.G.: A theory for multiresolution signal decomposition: the wavelet representation. IEEE Trans. Pattern Recog. Machine Intell. 11, 674 (1989)
Mallat, S.G.: Multiresolution approximations and wavelet orthonormal bases of \(L^2(R)\). Trans. Am. Math. Soc. 315, 69 (1989)
Mehrabi, A.R., Sahimi, M.: Coarsening of heterogeneous media: application of wavelets. Phys. Rev. Lett. 79, 4385 (1997)
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)
Nordbotten, J.M., Celia, M.A.: Geological Storage of CO\(_2\): Modeling Approaches for Large-Scale Simulation. Wiley, New York (2011)
Oak, M., Baker, L., Thomas, D.: Three-phase relative permeability of Berea sandstone. J. Pet. Technol. 42, 1054 (1990)
Olhede, S., Walden, A.T.: The Hilbert spectrum via wavelet projections. Proc. R. Soc. Lond. A 460, 955 (2004)
Pancaldi, V., Christensen, K., King, P.R.: Permeability up-scaling using Haar wavelets. Transp. Porous Media 67, 395 (2007)
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)
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)
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)
Raeini, A.Q.: Modelling Multiphase Flow Through Micro-CT Images of the Pore Space, Ph.D. Thesis, Imperial College of London (2013)
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)
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)
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)
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)
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)
Sahimi, M.: Flow and Transport in Porous Media and Fractured Rock, 2nd edn. Wiley, Weinheim (2011)
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)
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)
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)
Shokri, N.: Pore-scale dynamics of salt transport and distribution in drying porous media. Phys. Fluids 26, 012106 (2014)
Shokri, N., Lehmann, P., Or, D.: Characteristics of evaporation from partially-wettable porous media. Water Resour. Res. 45, W02415 (2009)
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)
Shokri, N., Sahimi, M., Or, D.: Morphology, propagation dynamics and scaling characteristics of drying fronts in porous media. Geophys. Res. Lett. 39, L09401 (2012)
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)
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)
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)
Starck, J.-L., Candés, E.J., Donoho, D.L.: The curvelet transform for image denoising. IEEE Trans. Image Process. 11(6), 670 (2002)
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)
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)
Ubink, O.: Numerical Prediction of Two Fluid Systems with Sharp Interfaces, Ph.D. Thesis, Imperial College of London (1997)
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)
Wilkinson, D., Willimsen, J.F.: Invasion percolation: a new form of percolation theory. J. Phys. A 16, 3365 (1983)
Woiselle, A., Starck, J.-L., Fadili, J.: 3D curvelet transforms and astronomical data restoration. Appl. Comput. Harmonic 28, 171 (2010)
Ying, L., Demanet, L., Candés, E.: 3D discrete curvelet transform. Proceedings of SPIE5914, Wavelets XI, 591413 (2005)
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
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11242-021-01559-5