# Multi-scale Image-Based Pore Space Characterisation and Pore Network Generation: Case Study of a North Sea Sandstone Reservoir

## Abstract

In this paper, we examine the pore space geometry and topology of a North Sea sandstone reservoir rock based on multi-scale scanning electron microscopy. The reservoir was subjected to extensive diagenesis which has resulted in a complex pore space with a wide range in pore sizes. We quantify the pore size and pore coordination number distributions, and we find that the mean and standard deviation of the coordination number are power law functions of pore radius, where the scaling exponent varies from 0.3 to 0.5. We present a 2D stochastic algorithm to generate a pore network based on statistical information. The algorithm incorporates the concept of a weighted planar stochastic lattice which is a construction that naturally leads to scale-free character with power law behaviour. We validate the algorithm against SEM imaging by showing that it can reproduce the observed clustering and a realistic spatial distribution of pore space elements. We also try to explore the relationship between fluid flow properties of reservoir rock and 2D pore image features.

## Keywords

Porous media Pore network SEM imaging Multi-scale## List of symbols

- \(A_\mathrm{mean}\)
Proportionality factor in the relationship between \(Z_\mathrm{mean}(r)\) and \(r\)

- \(A_\mathrm{stdev}\)
Proportionality factor in the relationship between \(Z_\mathrm{stdev}(r)\) and \(r\)

- \(E\)
Euclidean distance

- \(E_{i}\)
Euclidean distance of pixel \(i\)

- \(n\)
Exponent of \(\frac{r}{r_{1}}\) in the pore size probability distribution

- \(N_\mathrm{e}\)
Number of elements

- \(N_\mathrm{e}(r)\)
Number of elements with radius greater than \(r\)

- \(N_{\mathrm{e,total}}\)
Total number of elements

- \(N_\mathrm{l}\)
Number of links

- \(N_\mathrm{l}(r)\)
Number of links with radius greater than \(r\)

- \(N_{\mathrm{l,total}}\)
Total number of links

- \(r\)
Radius of element

- \(r_{i}\)
Radius of element \(i\)

- \(r_\mathrm{min}\)
Minimum element radius

- \(r_\mathrm{max}\)
Maximum element radius

- \(r_{0}\)
Pore size distribution parameter, near the minimum radius

- \(r_{1}\)
Pore size distribution parameter, near the maximum radius

- \(r_{x}\)
Characteristic radius of largest pores which control permeability

- \(Z\)
Coordination number

- \(Z_{i}\)
Coordination number of element \(i\)

- \(Z_\mathrm{mean}\)
Mean coordination number

- \(Z_\mathrm{stdev}\)
Standard deviation of the coordination number

- \(\alpha \)
Scaling index in the pore size probability distribution

- \(\beta \)
Scaling index in the relationship between \(Z_\mathrm{mean}(r)\) and \(r\)

- \(\phi \)
Porosity

- \(\chi \)
Connectivity function

## Notes

### Acknowledgements

The authors would like to thank Anasuria Operating Company Ltd (AOC) for making available the rock samples for analysis. SEM imaging was performed at the University of Aberdeen Centre for Electron Microscopy, Analysis and Characterisation (ACEMAC). We would like to thank John Still for invaluable assistance in the SEM imaging work. We would like to acknowledge financial support from Royal Society through the International cost share programme with reference IE151092. The authors acknowledge the very helpful comments from three anonymous reviewers.

## References

- Akpokodje, M., Melvin, A., Churchill, J., Burns, S., Morris, J.S.K., Wakefield, M., Ameerali, R.: Regional study of controls on reservoir quality in the Triassic Skagerrak Formation of the Central North Sea. Proceedings of the 8th Petroleum Geology Conference, pp. 125–146 (2017). https://doi.org/10.1144/PGC8.29
- Barabasi, A., Albert, R.: Emergence of scaling in random networks. Science
**286**, 509–512 (1999)CrossRefGoogle Scholar - Bashtani, F., Maini, B., Kantzas, A.: Single-phase and two-phase flow properties of mesaverde tight sandstone formation; random-network modeling approach. Adv. Water Resour.
**94**, 174–184 (2016). https://doi.org/10.1016/j.advwatres.2016.05.006 CrossRefGoogle Scholar - Békri, S., Laroche, C., Vizika, O.: SCA 2005—Pore Network Models to Calculate Transport and Electrical Properties of Single or Dual-porosity Rocks. International Symposium of the Society of Core Analysts, vol. 35, Toronto (2005)Google Scholar
- Blunt, M.J., Bijeljic, B., Dong, H., Gharbi, O., Iglauer, S., Mostaghimi, P., Paluszny, A., Pentland, C.: Pore-scale imaging and modelling. Adv. Water Resour.
**51**, 197–216 (2013). https://doi.org/10.1016/j.advwatres.2012.03.003 CrossRefGoogle Scholar - Borgefors, G.: Distance transformations in arbitrary dimensions. Comput. Vis. Graph. Image Process.
**27**(3), 321–345 (1984). https://doi.org/10.1016/0734-189X(84)90035-5 CrossRefGoogle Scholar - Bultreys, T., Van Hoorebeke, L., Cnudde, V.: Multi-scale, micro-computed tomography-based pore network models to simulate drainage in heterogeneous rocks. Adv. Water Resour.
**78**, 36–49 (2015). https://doi.org/10.1016/J.ADVWATRES.2015.02.003 CrossRefGoogle Scholar - Bultreys, T., De Boever, W., Cnudde, V.: Imaging and image-based fluid transport modeling at the pore scale in geological materials: a practical introduction to the current state-of-the-art. Earth Sci. Rev.
**155**, 93–128 (2016). https://doi.org/10.1016/J.EARSCIREV.2016.02.001 CrossRefGoogle Scholar - Cannon, S.J.C., Gowland, S.: Facies controls on reservoir quality in the Late Jurassic Fulmar Formation, Quadrant 21, UKCS. Geol. Soc. Lond. Spec. Publ.
**114**(114), 215–233 (1996). https://doi.org/10.1144/GSL.SP.1996.114.01.10 CrossRefGoogle Scholar - Cook, J., Goodwin, L., Boutt, D.: Systematic diagenetic changes in the grain-scale morphology and permeability of a quartz-cemented quartz arenite. AAPG Bull.
**95**(6), 1067–1088 (2011)CrossRefGoogle Scholar - De Boever, E., Varloteaux, C., Nader, F.H., Foubert, A., Békri, S., Youssef, S., Rosenberg, E.: Quantification and prediction of the 3D pore network evolution in carbonate reservoir rocks. Oil Gas Sci. Technol.
**67**(1), 161–178 (2012). https://doi.org/10.2516/ogst/2011170 CrossRefGoogle Scholar - Freire-Gormaly, M., Ellis, J., MacLean, H.L., Bazylak, A.: Pore structure characterization of Indiana limestone and pink dolomite from pore network reconstructions. Oil Gas Sci. Technol. (2015). https://doi.org/10.2516/ogst/2015004 Google Scholar
- Gowland, S.: Facies characteristics and depositional models of highly bioturbated shallow marine siliciclastic strata: an example from the Fulmar Formation (Late Jurassic), UK Central Graben. Geology of the Humber Group; Central Graben and Moray Firth. Geol. Soc. Lond. Spec. Publ.
**114**, 185–214 (1996)CrossRefGoogle Scholar - Hassan, M., Hassan, M., Pavel, N.I.: Scale-free network topology and multifractality in a weighted planar stochastic lattice. New J. Phys. (2010). https://doi.org/10.1088/1367-2630/12/9/093045 Google Scholar
- Idowu, N., Blunt, M.: IPTC 12292 Pore-scale modelling of rate effects in waterflooding. Presented at the International Petroleum Technology Conference, Kuala Lumpur, Malaysia (2008)Google Scholar
- Jiang, Z., van Dijke, M.I.J., Wu, K., Couples, G.D., Sorbie, K.S., Ma, J.: Stochastic pore network generation from 3D rock images. Transp. Porous Media
**94**(2), 571–593 (2012). https://doi.org/10.1007/s11242-011-9792-z CrossRefGoogle Scholar - Jiang, Z., Van Dijke, M.I.J., Sorbie, K.S., Couples, G.D.: Representation of multiscale heterogeneity via multiscale pore networks. Water Resour. Res.
**49**(9), 5437–5449 (2013). https://doi.org/10.1002/wrcr.20304 CrossRefGoogle Scholar - Katz, A., Thompson, A.: A quantitative prediction of permeability in porous rock. Phys. Rev. B
**34**, 8179–8181 (1987). https://doi.org/10.1103/PhysRevB.34.8179 CrossRefGoogle Scholar - Keehm, Y., Mukerji, T., Nur, A.: Permeability prediction from thin sections: 3D reconstruction and lattice-Boltzmann flow simulation. Geophys. Res. Lett. (2004). https://doi.org/10.1029/2003GL018761 Google Scholar
- Lai, J., Wang, G., Cao, J., Xiao, C., Wang, S., Pang, X., Dai, Q., He, Z., Fan, X., Yang, L., Qin, Z.: Investigation of pore structure and petrophysical property in tight sandstones. Mar. Pet. Geol.
**91**, 179–189 (2018). https://doi.org/10.1016/j.marpetgeo.2017.12.024 CrossRefGoogle Scholar - Lindquist, W.B., Lee, S.M., Coker, D.A., Jones, K.W., Spanne, P.: Medial axis analysis of void structure in three-dimensional tomographic images of porous media. J. Geophys. Res. Solid Earth
**101**(B4), 8297–8310 (1996). https://doi.org/10.1029/95JB03039 CrossRefGoogle Scholar - Manwart, C., Torquato, S., Hilfer, R.: Stochastic reconstruction of sandstones. Phys. Rev. E
**62**, 893–899 (2000). https://doi.org/10.1103/PhysRevE.62.893 CrossRefGoogle Scholar - Naraghi, M.E., Spikes, K., Srinivasan, S.: 3D reconstruction of porous media from a 2D section and comparisons of transport and elastic properties. SPE Reserv. Eval. Eng.
**20**(02), 342–352 (2017). https://doi.org/10.2118/180489-PA CrossRefGoogle Scholar - Okabe, H., Blunt, M.J.: Pore space reconstruction of vuggy carbonates using microtomography and multiple-point statistics. Water Resour. Res.
**43**(12), 3–7 (2007). https://doi.org/10.1029/2006WR005680 CrossRefGoogle Scholar - Øren, P., Bakke, S., Arntzen, O.: SPE52052 Extending predictive capabilities to network models. SPE J.
**3**, 324–336 (1998)CrossRefGoogle Scholar - Otsu, N.: A threshold selection method from gray-level histograms. IEEE Trans. Syst. Man Cybern.
**9**(1), 62–66 (1979). https://doi.org/10.1109/TSMC.1979.4310076 CrossRefGoogle Scholar - Patzek, T.W.: Verification of a complete pore network simulator of drainage and imbibition. SPE J.
**6**(02), 144–156 (2001). https://doi.org/10.2118/71310-PA CrossRefGoogle Scholar - Rabbani, A., Jamshidi, S., Salehi, S.: An automated simple algorithm for realistic pore network extraction from micro-tomography images. J. Pet. Sci. Eng.
**123**, 164–171 (2014). https://doi.org/10.1016/j.petrol.2014.08.020 CrossRefGoogle Scholar - Rabbani, A., Ayatollahi, S., Kharrat, R., Dashti, N.: Estimation of 3-D pore network coordination number of rocks from watershed segmentation of a single 2-D image. Adv. Water Resour.
**94**, 264–277 (2016). https://doi.org/10.1016/J.ADVWATRES.2016.05.020 CrossRefGoogle Scholar - Santiago, A.: Multiscaling of porous soils as heterogeneous complex networks. Nonlinear Process. Geophys.
**15**, 893–902 (2008)CrossRefGoogle Scholar - Silin, D., Patzek, T.: Pore space morphology analysis using maximal inscribed spheres. Physica A
**371**, 336–360 (2006). https://doi.org/10.1016/j.physa.2006.04.048 CrossRefGoogle Scholar - Tahmasebi, P., Sahimi, M.: Reconstruction of three-dimensional porous media using a single thin section. Phys. Rev. E (2012). https://doi.org/10.1103/PhysRevE.85.066709 Google Scholar
- Verscheure, M., Fourno, A., Chiles, J.: Joint inversion of fracture model properties for \(\text{ CO }_2\) storage monitoring or oil recovery history matching. Oil Gas Sci. Technol.
**67**, 221–235 (2012). https://doi.org/10.2516/ogst/2011176 CrossRefGoogle Scholar - Vogel, H.J., Roth, K.: Quantitative morphology and network representation of soil pore structure. Adv. Water Resour.
**24**(3–4), 233–242 (2001). https://doi.org/10.1016/S0309-1708(00)00055-5 CrossRefGoogle Scholar - Wu, K., Nunan, N., Crawford, J.W., Young, I.M., Ritz, K.: An efficient Markov chain model for the simulation of heterogeneous soil structure. Soil Sci. Soc. Am. J.
**68**(2), 346–351 (2004). https://doi.org/10.2136/sssaj2004.0346 CrossRefGoogle Scholar - Yakubo, K., Koroak, D.: Scale-free networks embedded in fractal space. Phys. Rev. E Stat. Nonlinear Soft Matter. Phys. (2011). https://doi.org/10.1103/PhysRevE.83.066111
- Youssef, S., Rosenberg, E., Gland, N., Kenter, J., Skalinski, M., Vizika, O.: SPE 111427 high resolution CT and pore-network models to assess petrophysical properties of homogeneous and heterogeneous carbonates. In: SPE/EAGE Reservoir Characterization Conference, Abu Dhabi, UAE (2007)Google Scholar