Towards Bayesian Quantification of Permeability in Micro-scale Porous Structures – The Database of Micro Networks

Babak Fazelabdolabadi, Mohammad Hossein Golestan


This article develops a Bayesian framework to quantify the absolute permeability of water in a porous structure from the geometry and clustering parameters of its underlying pore-throat network. These parameters include the network's diameter, transivity, degree, centrality, assortativity, edge density, K-core decomposition, Kleinberg’s hub centrality scores, Kleinberg's authority centrality scores, length, and porosity. In addition, the incorporated clustering aspects of the networks have been determined with respect to several clustering criteria: edge betweenness, greedy optimization of modularity, multi-level optimization of modularity, and short random walks. As such, the article takes the first steps towards creating a database of micro-networks for micro-scale porous structures, to be used as the main input stream for the proposed Bayesian scheme.


Doi: 10.28991/HIJ-2020-01-04-02

Full Text: PDF


Absolute Permeability; Bayesian Network; Database of Micro Networks; Porous Structures.


Andrä, H., Combaret, N., Dvorkin, J., Glatt, E., Han, J., Kabel, M., Keehm, Y., Krzikalla, F., Lee, M., Madonna, C., Marsh, M., Mukerji, T., Saenger, E.H., Sain, R., Saxena, N., Ricker, S., Wiegmann, A., Zhan, X. (2013a). Digital rock physics benchmarks-part II: Computing effective properties. Comput. Geosci. 50, 33–43. doi:10.1016/j.cageo.2012.09.008.

Arns, C.H.; Bauget, F., Limaye, A., Sakellariou, A., Senden, T., Sheppard, A., Sok, R.M., Pinczewski, V., Bakke, S., Berge, L.I., Oren, P., Knackstedt, M. (2005). Pore-Scale Characterization of Carbonates Using X-Ray Microtomography. SPE J. 10, 26–29. doi:10.2118/90368-PA.

Arns, C.H., Knackstedt, M. a., Pinczewski, W.V., Garboczi, E.J. (2002). Computation of linear elastic properties from microtomographic images: Methodology and agreement between theory and experiment. Geophysics 67, 1396. doi:10.1190/1.1512785.

Berg, S., van Wunnik, J. (2017). Shear Rate Determination from Pore-Scale Flow Fields. Transp. Porous Media, 1–18. doi:10.1007/s11242-017-0830-3.

Blunt, M.J., Bijeljic, B., Dong, H., Gharbi, O., Iglauer, S., Mostaghimi, P., Paluszny, A., Pentland, C. (2013). Pore-scale imaging and modelling. Adv. Water Resour. 51, 197–216. doi:10.1016/j.advwatres.2012.03.003.

Hübner, W. (2014). Studying the pore space of cuttings by NMR and μCT. J. Appl. Geophys. 104, 97–105. doi:10.1016/j.jappgeo.2014.02.016.

Knackstedt, M. A., Arns, C., Madadi, M., Sheppard, A. P., Latham, S., Sok, R., … Eberli, G. (2008). Elastic and flow properties of carbonate core derived from 3D X ray-CT images. SEG Technical Program Expanded Abstracts 2008. doi:10.1190/1.3059394.

Knackstedt, M., Latham, S., Madadi, M., Sheppard, A., Varslot, T., Arns, C. (2009). Digital rock physics: 3D imaging of core material and correlations to acoustic and flow properties. Leas. Edge 28, 28–33. doi:10.1190/1.3064143.

Umana, U. S., Ebong, M. S., & Godwin, E. O. (2020). Biomass Production from Oil Palm and Its Value Chain. Journal of Human, Earth, and Future, 1(1), 30–38. doi:10.28991/hef-2020-01-01-04.

Mostaghimi, P.; Blunt, M.J.; Bijeljic, B. (2013) Computations of Absolute Permeability on Micro-CT Images. Mathematical Geosciences 45(1), 103-125. doi:10.1007/s11004-012-9431-4.

Mostaghimi, P.; Liu, M.; Arns, C.H. (2016) Numerical Simulation of Reactive Transport on Micro-CT Images. Mathematical Geosciences 48(8) 963-983. doi:10.1007/s11004-016-9640-3.

Oostrom, M.; Mehmani, Y.; Romero-Gomez, P.; Tang, Y.; Liu, H.; Yoon, H.; Kang, Q.; Joekar-Niasar, V.; Balhoff, M. T.; Dewers, T.; Tartakovsky, G. D.; Leist, E. A.; Hess, N. J.; Perkins, W. A.; Rakowski, C. L.; Richmond, M. C.; Serkowski, J. A.; Werth, C. J.; Valocchi, A. J.; Wietsma, T. W.; Zhang, C. (2016). Pore-scale and continuum simulations of solute transport micromodel benchmark experiments. Computational Geosciences 20 (4), 857-879. doi:10.1007/s10596-014-9424-0.

Øren, P., Bakke, S., Rueslåtten, H. (2006). Digital core laboratory: Rock and flow properties derived from computer generated rocks. Int. Symp. Soc. Core Anal. 1–12.

Saenger, E.H., Enzmann, F., Keehm, Y., Steeb, H., (2011). Digital rock physics: Effect of fluid viscosity on effective elastic properties. J. Appl. Geophys. 74, 236–241. doi:10.1016/j.jappgeo.2011.06.001.

Sain, R., Mukerji, T., Mavko, G. (2014). How computational rock-physics tools can be used to simulate geologic processes, understand pore-scale heterogeneity, and refine theoretical models. Lead. Edge 33, 324–334. doi:10.1190/tle33030324.1.

Saxena, N., Mavko, G. (2016). Estimating elastic moduli of rocks from thin sections: Digital rock study of 3D properties from 2D images. Comput. Geosci. 88, 9–21. doi:10.1016/j.cageo.2015.12.008.

Saxena, N., Hofmann, R., Alpak, F. O., Berg, S., Dietderich, J., Agarwal, U., … Wilson, O. B. (2017). References and benchmarks for pore-scale flow simulated using micro-CT images of porous media and digital rocks. Advances in Water Resources, 109, 211–235. doi:10.1016/j.advwatres.2017.09.007.

Saxena, N., Mavko, G., Hofmann, R., Srisutthiyakorn, N. (2017b). Estimating permeability from thin sections without reconstruction: Digital rock study of 3D properties from 2D images. Comput. Geosci. 102, 79–99. doi:10.1016/j.cageo.2017.02.014.

Yang, X., Mehmani, Y., Perkins, W. A., Pasquali, A., Schönherr, M., Kim, K., … Scheibe, T. D. (2016). Intercomparison of 3D pore-scale flow and solute transport simulation methods. Advances in Water Resources, 95, 176–189. doi:10.1016/j.advwatres.2015.09.015.

Eugene, Y.L.; LeBoeuf, J.; Basu, P.K.; Mahadevan S. (2005). Stochastic modeling of the permeability of randomly generated porous media. Advances in Water Resources 28(8), 835-844. doi:10.1016/j.advwatres.2005.01.007.

Ahmadi, M. A., & Chen, Z. (2019). Comparison of machine learning methods for estimating permeability and porosity of oil reservoirs via petro-physical logs. Petroleum, 5(3), 271–284. doi:10.1016/j.petlm.2018.06.002.

Alqahtani, N.; Armstrong, R.T.; Mostaghimi, P. (2018). Deep Learning Convolutional Neural Networks to Predict Porous Media Properties. Society of Petroleum Engineers SPE Asia Pacific Oil and Gas Conference and Exhibition - Brisbane, Australia. doi:10.2118/191906-MS.

Arigbe, O.D.; Oyeneyin, M.B.; Ghazi, M.D. (2019). Real-time relative permeability prediction using deep learning. Journal of Petroleum Exploration and Production Technology 9(2), 1271-1284. doi:10.1007/s13202-018-0578-5.

Erofeev, A., Orlov, D., Ryzhov, A., & Koroteev, D. (2019). Prediction of Porosity and Permeability Alteration Based on Machine Learning Algorithms. Transport in Porous Media, 128(2), 677–700. doi:10.1007/s11242-019-01265-3.

Wu, J., Yin, X., & Xiao, H. (2018). Seeing permeability from images: fast prediction with convolutional neural networks. Science Bulletin, 63(18), 1215–1222. doi:10.1016/j.scib.2018.08.006.

Srisutthiyakorn, N. (2016) Deep Learning Methods for Predicting Permeability from 2-D/3-D Binary Segmented Images. Society of Exploration Geophysicists SEG Technical Program Expanded Abstracts - Dallas, Texas. doi:10.1190/segam2016-13972613.1.

Sun, H., Al-Marzouqi, H., & Vega, S. (2019). EPCI: A new tool for predicting absolute permeability from computed tomography images. Geophysics, 84(3), F97–F102. doi:10.1190/geo2018-0653.1.

Van der Linden, J. H., Narsilio, G. A., & Tordesillas, A. (2016). Machine learning framework for analysis of transport through complex networks in porous, granular media: A focus on permeability. Physical Review E, 94(2). doi:10.1103/physreve.94.022904.

Mehmani, Y., Oostrom, M., & Balhoff, M. T. (2014). A streamline splitting pore-network approach for computationally inexpensive and accurate simulation of transport in porous media. Water Resources Research, 50(3), 2488–2517. doi:10.1002/2013wr014984.

Bhattacharya, S., & Mishra, S. (2018). Applications of machine learning for facies and fracture prediction using Bayesian Network Theory and Random Forest: Case studies from the Appalachian basin, USA. Journal of Petroleum Science and Engineering, 170, 1005–1017. doi:10.1016/j.petrol.2018.06.075.

Martinelli, G., Eidsvik, J., Sinding-Larsen, R., Rekstad, S., & Mukerji, T. (2013). Building Bayesian networks from basin-modelling scenarios for improved geological decision making. Petroleum Geoscience, 19(3), 289–304. doi:10.1144/petgeo2012-057.

Lindberg, D. V., Rimstad, E., & Omre, H. (2015). Inversion of well logs into facies accounting for spatial dependencies and convolution effects. Journal of Petroleum Science and Engineering, 134, 237–246. doi:10.1016/j.petrol.2015.09.027.

Csardi, G., & Nepusz, T. (2006). The igraph software package for complex network research. InterJournal, complex systems, 1695(5), 1-9.

Clauset, A., Newman, M. E. J., & Moore, C. (2004). Finding community structure in very large networks. Physical Review E, 70(6). doi:10.1103/physreve.70.066111.

Blondel, V. D., Guillaume, J.-L., Lambiotte, R., & Lefebvre, E. (2008). Fast unfolding of communities in large networks. Journal of Statistical Mechanics: Theory and Experiment, 2008(10), P10008. doi:10.1088/1742-5468/2008/10/p10008.

Pons, P., & Latapy, M. (2005). Computing Communities in Large Networks Using Random Walks. Lecture Notes in Computer Science, 284–293. doi:10.1007/11569596_31.

Dong, H., & Blunt, M. J. (2009). Pore-network extraction from micro-computerized-tomography images. Physical Review E, 80(3). doi:10.1103/physreve.80.036307.

Korb, K., Nicholson, A. (2004). Bayesian Artificial Intelligence. Chapman and Hall. CRC Press, Florida, United States.

Scutari, M. (2010). Learning Bayesian Networks with the bnlearn R Package. Journal of Statistical Software, 35(3), 1-22. Available online: (accessed on December 2020).

Nagarajan R., Scutari M., Lèbre S. (2013) Bayesian Networks in R with Applications in Systems Biology. Springer-Verlag, New York, United States.

S1 Sandstone (2009). Micro-CT image of S1 sandstone and extracted networks (Dong and Blunt, 2009, doi:10.1103/PhysRevE.80.036307). doi:10.6084/m9.figshare.1189274.

S2 Sandstone (2009). Micro-CT image of S2 sandstone and extracted networks (Dong and Blunt, 2009, doi:10.1103/PhysRevE.80.036307). doi:10.6084/m9.figshare.1189275.

S3 Sandstone (2009). Micro-CT image of S3 sandstone and extracted networks (Dong and Blunt, 2009, doi:10.1103/PhysRevE.80.036307). doi:10.6084/m9.figshare.1189276.

Gostick, J.; Aghighi, M.; Hinebaugh, J.; Tranter, T.; Hoeh, M.; Day, H.; Sharqawy, M.; Spellacy, B.; Bazylak, A.; Burns, A.; Lehnert, W.; Putz, A. (2016). OpenPNM: A Pore Network Modeling Package. Computing in Science & Engineering. 18(4), 60-74. doi:10.1109/mcse.2016.49.

Ladd, A.J.C.; Verberg, R. (2001). Lattice-Boltzmann Simulations of Particle-Fluid Suspensions. Journal of Statistical Physics 104(4-5) 1191-1251. doi:10.1023/a:1010414013942.

He, S.; Ariyaratne, C.; Vardy, A.E. (2008). A computational study of wall friction and turbulence dynamics in accelerating pipe flows. Computers & Fluids 37(6) 674-689. doi:10.1016/j.compfluid.2007.09.001.

Latt, J. (2009). Palabos, Parallel Lattice Boltzmann Solver. Geneva , Switzerland. Available online: (accessed 01 February 2018).

OpenFOAM. The open source CFD toolbox. London, United Kingdom Available online: (accessed 01 February 2018).

Raeini, A.Q.; Blunt, M.J.; Bijeljic, B. (2014), Direct simulations of two-phase flow on micro-CT images of porous media and upscaling of pore-scale forces. Advances in Water Resources 74, 116-126. doi:10.1016/j.advwatres.2014.08.012.

Mohammadmoradi, P.; Bashtani, F.; Goudarzi, B.; Taheri, S.; Kantzas, A. (2017). Pore Network and Morphological Characterization of Pore-Level Structures. SPE184964 in SPE Canada Heavy Oil Technical Conference 2017, Calgary, Alberta, Canada.

Scutari, M. (2017). Bayesian Network Constraint-Based Structure Learning Algorithms: Parallel and Optimized Implementations in the bnlearn R Package. Journal of Statistical Software, 77(2), 1-20. doi:10.18637/jss.v077.i02.

Kursa, M. B., Rudnicki, W.R. (2010). Feature Selection with the Boruta Package. Journal of Statistical Software, 36(11), 1-13. Available online:

Romanski, P.; Kotthoff, L. (2018). FSelector: Selecting Attributes. R package Version 0.31. Available online: https://CRAN.R (accessed 01 February 2018).

Costa, A. (2006). Permeability-porosity relationship: A reexamination of the Kozeny‐Carman equation based on a fractal pore-space geometry assumption. Geophysical Research Letters 33(2). doi:10.1029/2005GL025134.

Newman, M.; Girvan, M. (2004). Finding and evaluating community structure in networks. Phys. Rev. E 69, 026113. doi:10.1103/physreve.69.026113.

Bijeljic, B. Raeini, A.; Mostaghimi, P.; Blunt, M.J. (2013). Predictions of non-Fickian solute transport in different classes of porous media using direct simulation on pore-scale images. Physical Review E 87(1), 013011. doi:10.1103/PhysRevE.87.013011.

Yi, Z., Wu, X., & Li, F. (2018). Ranking Spreaders in Complex Networks Based on the Most Influential Neighbors. Discrete Dynamics in Nature and Society, 2018, 1–6. doi:10.1155/2018/3649079.

Full Text: PDF

DOI: 10.28991/HIJ-2020-01-04-02


  • There are currently no refbacks.

Copyright (c) 2020 Babak Fazelabdolabadi, Mohammad Hossein Golestan