The Generalization of the Kozeny Approach to Determining the Permeability of the Model Porous Media Made of Solid Spherical Segments

Tyumen State University Herald. Physical and Mathematical Modeling. Oil, Gas, Energy


Release:

2016, Vol. 2. №2

Title: 
The Generalization of the Kozeny Approach to Determining the Permeability of the Model Porous Media Made of Solid Spherical Segments


About the authors:

Amir A. Gubaidullin, Dr. Sci. (Phys.-Math.), Professor, Сhief Researcher, Tyumen Branch of the Khristianovich Institute of Theoretical and Applied Mechanics of the Siberian Branch of the Russian Academy of Sciences; eLibrary AuthorID, ORCID, Web of Science ResearcherID, Scopus AuthorID, a.a.gubaidullin@yandex.ru

Dmitry E. Igoshin, Cand. Sci. (Phys-Math.), Senior Researcher, Tyumen Branch of the Khristianovich Institute of Theoretical and Applied Mechanics of the Siberian Branch of the Russian Academy of Sciences; Associate Professor, Department of Fundamental Mathematics and Mechanics, Department of Applied and Technical Physics, University of Tyumen; igoshinde@gmail.com

Nadezhda A. Khromova, Research Engineer, Tyumen Branch of Khristianovich Institute of Theoretical and Applied Mechanics of the Siberian Branch of the Russian Academy of Sciences; khromova.n.a@gmail.com

Abstract:

To establish the connection between the porosity, permeability, and pore or grains size of the porous medium, Kozeny considered fictitious soil-pile as some kind of a filling with balls. However, in real earth material the shape of the particles, which make up the skeleton, may differ substantially from the spherical one.

The aim of this paper is to generalize the Kozeny approach to take into consideration the case of the porous system, the skeleton of which is formed with spherical segment adjacent to each other. As an example, the authors consider the model periodic structure, the permeability values of which have been previously defined on the basis of numerical solution of the Navier–Stokes equations. The model periodic structure patterns of four types are presented: simple cubic, hexagonal simple, body-centered cubic, and face-centered cubic. The sphere intersection degree is a dimensionless modeling parameter that determines the environment porosity and voidage.

The generalized approach allowed to obtain the permeability values for the four types of the considered structures, and to compare them with the corresponding numerical solutions. The results show that the proposed approach suggests good results in the case of body-centered cubic structure in a wide range of porosity (0.32 ≥ m ≥ 0.04). For the face-centered cubic structure the result is satisfactory in the porosity range of 0.26 ≥ m ≥ 0.14. In the case of the simple cubic and hexagonal structures the method of minimal voidage more preferred to assess the permeability.

References:

  1. Atyutskaya L. Yu., Bebiya A. G., Milyukova I. V. 2013. “Kontrol udelnoy poverkhnosti tseolita metodom Karmana-Kozeni v protsesse mekhanicheskoy aktivatsii” [Control of the Specific Surface Area of the Zeolite by Karman-Kozeny Method in the Process of Mechanical Activation]. Polzunovskiy almanakh, no 1, pp. 95–97.
  2. Dedov A. V. 2013. “Ispolzovanie modeli Kozeni dlya prognozirovaniya pronitsaemosti netkanykh igloprobivnykh materialov” [Using Kozeny Models to Predict the Permeability of Needle-Punched Nonwoven Materials]. Inorganic Materials: Applied Research, no 5, pp. 15–17.
  3. Dmitriev M. N., Dmitriev N. M. 2005. “K opredeleniyu filtratsionnogo chisla Reynoldsa i kharakternogo lineynogo razmera dlya idealnykh i fiktivnykh poristykh sred” [Determination of the Filtration of the Reynolds Number and the Characteristic Linear Dimension of the Ideal and Fictitious Porous Media]. Fluid Dynamics, no 4, pp. 97–104.
  4. Dolzhik K., Khmelevska I. 2014. “Raschetnaya otsenka filtratsii nesvyaznykh gruntov” [The Estimated Filtering Loose Soils]. Soil Mechanics and Foundation Engineering, no 5, pp. 2–5.
  5. Filippova K. E., Lukina Yu. Yu. 2015. “Kontrol udelnoy poverkhnosti tseolitsoderzhashchikh gornykh porod Suntarskogo mestorozhdeniya Khonguruu metodom Karmana-Kozeni v protsesse mekhanicheskoy aktivatsii” [Control the Specific Surface Area of Zeolite Rocks Suntarsky Field Khonguruu by Karman-Kozeny in the Process of Mechanical Activation]. Aktualnye napravleniya nauchnykh issledovaniy: ot teorii k praktike, no 1 (3), pp. 260–262.
  6. Gubaydullin A. A., Maksimov A. Yu. 2013. “Modelirovanie dinamiki kapli nefti v kapillyare s suzheniem” [Modeling the Dynamics of the Oil Droplets in the Capillary with the Restriction]. Tyumen State University Herald, no 7, pp. 71–77.
  7. Igoshin D. E. 2015. “Chislennoe opredelenie pronitsaemosti v srede periodicheskoy struktury, obrazovannoy razvetvlyayushchimisya kanalami” [Numerical Determination of Permeability in the Medium with a Periodic Structure Formed by Branching Channels]. Avtomatizatsiya, telemekhanizatsiya i svyaz v neftyanoi promyshlennosti, no 12, pp. 30–33.
  8. Igoshin D. E., Khromova N. A. 2015. “Osnovnye filtratsionnye svoystva poristoy sredy, obrazovannoy soobshchayushchimisya osesimmetrichnymi kanalami” [Main Filtration Properties of the Porous Medium Formed Communicating Axially Symmetric Channels]. Tyumen State University Herald. Physical and Mathematical Modeling. Oil, Gas, Energy, vol. 1, no 4, pp. 69–79.
  9. Igoshin D. E., Maksimov A. Yu. 2015. “Chislennye i analiticheskie otsenki pronitsaemosti poristoy sredy, obrazovannoy kanalami, imeyushchimi vrashchatelnuyu simmetriyu” [Numerical and Analytical Assessment of the Permeability of a Porous Medium Formed by Channels Having Rotational Symmetry]. Tyumen State University Herald. Physical and Mathematical Modeling. Oil, Gas, Energy, vol. 1, no 3, pp. 112–121.
  10. Igoshin D. E., Nikonova O. A. 2015. “Pronitsaemost poristoy sredy periodicheskoy struktury s razvetvlyayushchimisya kanalami” [Permeability of the Porous Medium Periodic Structure with Branching Channels]. Tyumen State University Herald. Physical and Mathematical Modeling. Oil, Gas, Energy, vol. 1, no 2, pp. 131–141.
  11. Igoshin D. E., Nikonova O. A., Mostovoy P. Ya. 2014. “Modelirovanie poristoy sredy regulyarnymi upakovkami peresekayushchikhsya sfer” [Modeling Porous Medium Regular Packages Intersecting Spheres]. Tyumen State University Herald, no 7, pp. 34–42.
  12. Igoshin D. E., Saburov R. S. 2015. “Chislennoe issledovanie zavisimosti pronitsaemosti ot poristoy sredy, obrazovannoy kanalami regulyarnoy struktury” [Numerical Study Depending on the Permeability of the Porous Medium, Formed a Regular Structure of Channels]. Tyumen State University Herald. Physical and Mathematical Modeling. Oil, Gas, Energy, vol. 1, no 1, pp. 84–90.
  13. Leybenzon L. S. 1947. Dvizhenie prirodnykh zhidkostey i gazov v poristoy srede [Movement of Natural Fluids in Porous Media]. Moscow: Gosudarstvennoe izdatelstvo tekhniko-teoreticheskoy literatury.
  14. Loytsyanskiy L. G. 2003. Mekhanika zhidkosti i gaza [Mechanics of Liquid and Gas]. Moscow: Drofa.
  15. Muskat 2004 (1946). Techeniye odnorodnykh zhidkostey v poristoy srede [The Flow of Homogeneous Fluids Through Porous Media]. Moscow, Izhevsk: Institute of Computer Science.
  16. Ren X., Zhao Y., Deng Q., Kang J., Li D., Wang D.. 2016. A Relation of Hydraulic Conductivity — Void Ratio for Soils Based on Kozeny-Carman equation. Engineering Geology, vol. 213, pp. 89–97. DOI: 10.1016/j.enggeo.2016.08.017
  17. Sorokin A. G. 2012. “Teoreticheskoe modelirovanie koeffitsienta pronitsaemosti pri filtratsii neszhimaemykh zhidkostey” [Theoretical Modeling of the Permeability Coefficient of Filtration of Incompressible Liquids]. Izvestiya vysshikh uchebnykh zavedeniy. Geologiya i razvedka, no 6, pp. 47–54.
  18. Stepanov S. V., Shabarov A. B., Bembel G. S. 2016. “Vychislitelnaya tekhnologiya dlya opredeleniya funktsii mezhfaznogo vzaimodeystviya na osnove modelirovaniya techeniya v kapillyarnom klastere” [Computing Technology to Determine the Function of Interfacial Interaction on the Basis of Flow Simulation in a Capillary Cluster]. Tyumen State University Herald. Physical and Mathematical Modeling. Oil, Gas, Energy, vol. 2, no 1, pp. 63–71.
  19. Tang T., J.M. McDonough. 2016. A Theoretical Model For The Porosity–Permeability Relationship. International Journal of Heat and Mass Transfer, vol. 103, pp. 984–996. DOI: 10.1016/j.ijheatmasstransfer.2016.07.095
  20. Vasilevskiy M. V., Romandin V. I., Zykov E. G. 2013. “Kharakteristiki sostoyaniya dispersnoy sredy na filtruyushchey podlozhke obespylivayushchego ustroystva” [The Features of the Dispersed State of the Environment on the Filter Substrate of De-Dusting Devices]. Russian Physics Journal, no 9-3, pp. 40–42.
  21. Yevseev F. A., Aliev A. E., Bogdanova E. V. 2015. “Eksperimentalnaya proverka tochnosti metoda Karmana-Kozeni dlya izmereniya udelnoy poverkhnosti chastits” [Experimental Verification of the Accuracy of the Karman-Kozeny Method for Measuring the Specific Surface Area of Particles]. Novyy universitet. Seriya: Tekhnicheskie nauki, no 9–10 (43–44), pp. 53–58.