Most correlation equations of rock permeability are usually based on the Euclidean geometry concept. Pore geometry and structure of most porous rocks are very complex, therefore non-Euclidean geometry concept, e.g. fractal theory, is needed to handle such a complexity. This paper presents a new equation for sandstone permeability involving other properties and fractal dimensions of pore space and surface. The equation is derived by combining Newton’s Law of viscosity, Darcy equation, and fractal geometry concept. It is shown that parameters such as tortuosity, internal surface area, and shape factor can be replaced by fractal dimensions. As natural porous media are mostly anisotropic, this study enables us to identify factors that affect the anisotropy. Eighteen sandstone samples with porosity and permeability range from 21 to 37% and 2.76 to 3,644 millidarcies, were employed in this study. The pore space and surface fractal dimensions for each orthogonal direction for each sample was determined by box counting method. The results of this study demonstrate that calculated directional permeability of the high permeability samples is very close to the measured one after corrections were made for pore sizes of less than one micron. This finding suggests that micropores of the samples may be a major factor not contributing to fluid flow. For the low and medium permeability samples, however, an additional pore geometrical correction is needed. The additional correction factor is considerably different for different directions of fluid flow, indicating that the anisotropy is due to the difference in directional pore structural characteristics.
| Published in | American Journal of Science, Engineering and Technology (Volume 3, Issue 2) |
| DOI | 10.11648/j.ajset.20180302.12 |
| Page(s) | 34-45 |
| Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
| Copyright |
Copyright © The Author(s), 2018. Published by Science Publishing Group |
Permeability, Tortuosity, Hydraulic Diameter, Fractal Dimension, Anisotropy
| [1] | Kozeny, J., Uber Kapillare Leitung des Wassers im Boden, Sitzungsberichte der Wiener Akademie der Wissenschaften, Wien, 1927, pp. 137, 271-306. |
| [2] | Carman, P. C., Fluid Flow Through Granular Beds, Trans. Inst. Chem. Eng., London, England, 1937, pp. 15, 150-167. |
| [3] | Scheidegger, A. E., The Physics of Flow through Porous Media, University of Toronto Press, Canada, 1959, pp. 112-133. |
| [4] | Dullien, F. A. L., Porous Media Fluid Transport and Pore Structure, Academic Press, New York, USA, 1979, pp. 75-137. |
| [5] | Bear, J., Dynamics of Fluids in Porous Media, Dover Publication, Inc, New York, USA, 1988, pp. 161-176. |
| [6] | Srisutthiyakorn, N., and Mavko, G., An Improved Kozeny-Carman for Irregular Pore Geometries, Society of Exploration Geophysicists, 2015, pp. 3015-3019. |
| [7] | Haro, C. F., Permeability Modelling in Porous Media: Setting Archie and Carman-Kozeny Right, Paper SPE 100201, Proceedings of SPE International Conference and Exhibition, Veracruz, Mexico, 27 – 30 June 2007. |
| [8] | Krauss, E. D. and Mays, D. C., Modification of the Kozeny-Carman Equation to Quantify Formation Damage by Fines in Clean Unconsolidated Sandstones, Paper SPE 165148, Proceedings of the SPE European Formation Damage Conference and Exhibition, Noordwijk, Netherlands, 5 – 7 June 2013. |
| [9] | Civan, F., Relating Permeability to Pore Connectivity using a Power-Law Flow Unit Equation, Petrophysics, vol. 43, No. 6, 2002, pp. 457-476. |
| [10] | Civan, F., Fractal Formulation of the Porosity and Permeability Relationship Resulting in Power-Law Units Equation – A Leaky Tube Model, Paper SPE 73785, Proceedings of the SPE International Symposium on Formation Damage, Lafayette, LA, USA, 23 – 24 February 2002. |
| [11] | Civan, F., Characterization of Reservoir Flow Units Based on a Power-Law Equation of Permeability Obtained from an Interacting Bundle Leaky Tubes Model, Paper SPE 187289, Proceedings of SPE Annual Technical Conference and Exhibition, San Antonio, TX, USA, 9 – 11 October 2017. |
| [12] | Mandelbrot, B. B., The Fractal Geometry of Nature, Freeman, San Francisco, 1982. |
| [13] | Falconer, K., Fractal Geometry, Mathematical Foundations and Applications, John Wiley & Sons Ltd, Baffins Lane, Chichester, West Sussex, England, 1990, pp. 3 – 53, 133-136. |
| [14] | Katz, A. J. and Thompson, A. H., Fractal Sandstone Pores: Implications for Conductivity and Pore Formation, Phys. Rev. Lett., vol. 54, No. 12, 1985, pp. 1325-1328. |
| [15] | Pape, H., Riepe, L., and Schopper, J. R., Theory of Self-Similar Network Structures in Sedimentary and Igneous Rocks and Their Investigation with Microscopical Methods, Journal of Microscopy, vol. 148, 1987, pp. 121-147. |
| [16] | Pape, H., Clauser, C., and Iffland, J., Permeability Prediction for reservoir Sandstones and Basement Rocks Based on Fractal Pore Space Geometry, SEG Expanded Abstract, 1998. |
| [17] | Pape, H., Clausser, C., and Iffland, J., Permeabiity Prediction Based on Fractal Pore-Space Geometry, Geophysics, vol. 64, 1999, pp. 1447-1460. |
| [18] | Pape, H., Arnold, J., Pechnig, R., Clausser, C., Talnishnikh, E., Anferova, S., and Iffland, Blümlich, B., Permeability Prediction for Low Porosity Rocks by Mobile NMR, Pure and Applied Geophysics, vol. 166, 2009, pp. 1125-1163. |
| [19] | Hansen, J. P. and Skjeltorp, A. T., Fractal Pore Space and Rock Permeability Implications, Physical Review B, vol. 378, No. 4, 1988, pp. 2635-2638. |
| [20] | Shen, P. and Li, K., A New Method for Determining the Fractal Dimension of Pore Structures and Its Application, Proceedings of the 10th Offshore South East Asia Conference, Singapore, 6 – 9 December 1994. |
| [21] | Abdassah, D., Permadi, P., Sumantri, Y., and Sumantri, R, Saturation Exponent at Various Wetting Conditions: Fractal Modelling of Thin Sections, J. Petroleum Sci. & Eng., 1998, pp. 20, 147-154. |
| [22] | Moulu, J-C., Vizika, O., Kalaydjian, F., and Duquerroix, J-P., A New Model for Three-Phase Relative Permeabilities Based on a Fractal Representation of the Porous Media, SPE 38891, Proceedings of SPE Annual Technical Conference and Exhibition, San Antonio, TX, USA, 5 – 8 October 1997. |
| [23] | Civan, F., Improved Permeability Equation from the Bundle-of-Leaky-Capillary-Tubes Model, Paper SPE 94271, Proceedings of SPE Production Operations Symposium, Oklahoma City, OK, USA, 17 – 19 April 2005. |
| [24] | Li, K., Characterization of Rock Heterogeneity Using Fractal Geometry, Paper SPE 86975, SPE International Thermal Operations and Heavy Oil Symposium and Western Regional Meeting, Bakersfield, CA, USA, 16 – 18 March 2004. |
| [25] | Xu, P. and Yu, B. M., Developing a New Form of Permeability and Kozeny-Carman Constant for Homogeneous Porous Media by Means of Fractal Geometry, Advanced Water Resources, vol. 31, No. 1, 2008, pp. 74-81. |
| [26] | Zhang, Z. and Weller, A., Factal Dimension of Pore Space Geometry of an Eocene Sandstone Formation, Geophysics, vol. 79, No. 6, 2014, pp. 377-387. |
| [27] | Lei, G., Dong, P., Wu, Z., Mo, S., Gai, S., Zhao, C., & Liu, Z. K., A Fractal Model for the Stress-Dependent Permeability and Relative Permeability in Tight Sandstones, Journal of Canadian Petroleum Society, January 2015, pp. 36-48. |
| [28] | Tiab, D. and Donaldson, E. C., Petrophysics: Theory and Practice of MeasuringReservoir Rock and Fluid Transporting Properties, Second Edition, Elsevier Inc., Gulf Professional Publishing, Massachusetts, USA, 2004. |
| [29] | Widarsono, B., Muladi, A., and Jaya, I., Permeability Vertical-to-Horizontal Anisotropy in Indonesian Oil and Gas Reservoirs: A General Review, Paper SPE 103315, Proceedings of the First International Conference and Exhibition in Mexico, Cancun, Mexico, 31 August – 2 September 2006. |
| [30] | Pirson, S. J., Oil Reservoir Engineering, McGraw-Hill Book Company Inc., New York, USA, Second Edition, 1958, pp. 97-103. |
| [31] | Sumantri, Y., Permeability Anisotropy Study of Sandstone Through the Pore Size Distribution and Fractal of Thin Sections Analysis, Dissertation, Department of Petroleum Engineering, Graduate Program, Bandung Institute of Technology, 2007. |
| [32] | Sumantri, Y., Permadi, P., and Putra, J. C. E., Estimating Permeability for Sandstones Having Multi-Modal Pore Size Distribution, Proceedings of International Geosciences Conference and Exhibition, Jakarta, Indonesia, 14 – 16 August 2006. |
| [33] | Bear, J., Dynamics of Fluids in Porous Media, Dover Publication, Inc, New York, 1988. |
| [34] | Perez-Rosales, C., On the Relationship between Formation Resistivity Factor and Porosity, SPE Journal, Society of Petroleum Engineers, Richardson, TX, USA, 1982, pp. 531-536. |
| [35] | Ishutov, S. and Hasiuk, F. J., 3D Printing Berea sandstone: Testing a New Tool for Petrophysical Anaysis of Reservoirs, Petrophysics, vol. 58, No. 6, 2017, pp. 592-602. |
| [36] | Meng, B., Determination and Interpretation of Fractal Properties of the Sandstone Pore System, Materials and Structures/Materiaux et Constructions, Vol. 29, May 1996, pp. 195-205. |
| [37] | Winsauer, W. O., Shearin, Jr., H. M., Masson, P. H., and Williams, M., Resistivity of Brine-Saturated Sands in Relation to Pore Geometry, Bull. AAPG, vol. 36, No. 2, 1952, pp. 253-277. |
APA Style
Yosaphat Sumantri, Pudji Permadi. (2018). A Study of Sandstone Permeability Anisotropy Through Fractal Concept. American Journal of Science, Engineering and Technology, 3(2), 34-45. https://doi.org/10.11648/j.ajset.20180302.12
ACS Style
Yosaphat Sumantri; Pudji Permadi. A Study of Sandstone Permeability Anisotropy Through Fractal Concept. Am. J. Sci. Eng. Technol. 2018, 3(2), 34-45. doi: 10.11648/j.ajset.20180302.12
AMA Style
Yosaphat Sumantri, Pudji Permadi. A Study of Sandstone Permeability Anisotropy Through Fractal Concept. Am J Sci Eng Technol. 2018;3(2):34-45. doi: 10.11648/j.ajset.20180302.12
Share This Article
Twitter
Linked In
Facebook
Copy
Download@article{10.11648/j.ajset.20180302.12,
author = {Yosaphat Sumantri and Pudji Permadi},
title = {A Study of Sandstone Permeability Anisotropy Through Fractal Concept},
journal = {American Journal of Science, Engineering and Technology},
volume = {3},
number = {2},
pages = {34-45},
doi = {10.11648/j.ajset.20180302.12},
url = {https://doi.org/10.11648/j.ajset.20180302.12},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajset.20180302.12},
abstract = {Most correlation equations of rock permeability are usually based on the Euclidean geometry concept. Pore geometry and structure of most porous rocks are very complex, therefore non-Euclidean geometry concept, e.g. fractal theory, is needed to handle such a complexity. This paper presents a new equation for sandstone permeability involving other properties and fractal dimensions of pore space and surface. The equation is derived by combining Newton’s Law of viscosity, Darcy equation, and fractal geometry concept. It is shown that parameters such as tortuosity, internal surface area, and shape factor can be replaced by fractal dimensions. As natural porous media are mostly anisotropic, this study enables us to identify factors that affect the anisotropy. Eighteen sandstone samples with porosity and permeability range from 21 to 37% and 2.76 to 3,644 millidarcies, were employed in this study. The pore space and surface fractal dimensions for each orthogonal direction for each sample was determined by box counting method. The results of this study demonstrate that calculated directional permeability of the high permeability samples is very close to the measured one after corrections were made for pore sizes of less than one micron. This finding suggests that micropores of the samples may be a major factor not contributing to fluid flow. For the low and medium permeability samples, however, an additional pore geometrical correction is needed. The additional correction factor is considerably different for different directions of fluid flow, indicating that the anisotropy is due to the difference in directional pore structural characteristics.},
year = {2018}
}
TY - JOUR T1 - A Study of Sandstone Permeability Anisotropy Through Fractal Concept AU - Yosaphat Sumantri AU - Pudji Permadi Y1 - 2018/10/12 PY - 2018 N1 - https://doi.org/10.11648/j.ajset.20180302.12 DO - 10.11648/j.ajset.20180302.12 T2 - American Journal of Science, Engineering and Technology JF - American Journal of Science, Engineering and Technology JO - American Journal of Science, Engineering and Technology SP - 34 EP - 45 PB - Science Publishing Group SN - 2578-8353 UR - https://doi.org/10.11648/j.ajset.20180302.12 AB - Most correlation equations of rock permeability are usually based on the Euclidean geometry concept. Pore geometry and structure of most porous rocks are very complex, therefore non-Euclidean geometry concept, e.g. fractal theory, is needed to handle such a complexity. This paper presents a new equation for sandstone permeability involving other properties and fractal dimensions of pore space and surface. The equation is derived by combining Newton’s Law of viscosity, Darcy equation, and fractal geometry concept. It is shown that parameters such as tortuosity, internal surface area, and shape factor can be replaced by fractal dimensions. As natural porous media are mostly anisotropic, this study enables us to identify factors that affect the anisotropy. Eighteen sandstone samples with porosity and permeability range from 21 to 37% and 2.76 to 3,644 millidarcies, were employed in this study. The pore space and surface fractal dimensions for each orthogonal direction for each sample was determined by box counting method. The results of this study demonstrate that calculated directional permeability of the high permeability samples is very close to the measured one after corrections were made for pore sizes of less than one micron. This finding suggests that micropores of the samples may be a major factor not contributing to fluid flow. For the low and medium permeability samples, however, an additional pore geometrical correction is needed. The additional correction factor is considerably different for different directions of fluid flow, indicating that the anisotropy is due to the difference in directional pore structural characteristics. VL - 3 IS - 2 ER -
Information
Email: