The Numerical Calculation of Permeability in Two-Dimensional Porous Media with Skeleton of Randomly Arranged Overlapping Disks

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


Release:

2016, Vol. 2. №4

Title: 
The Numerical Calculation of Permeability in Two-Dimensional Porous Media with Skeleton of Randomly Arranged Overlapping Disks


About the authors:

Aleksey S. Gubkin, Junior Researcher, Tyumen Branch of Khristianovich Institute of Theoretical and Applied Mechanics of the Siberian Branch of the RAS; Senior Lecturer, Department of Mechanics of Multiphase Systems, Tyumen State University; alexshtil@gmail.com

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

Dmitry V. Trapeznikov, Mathematical Modeling Engineer, Automation and Digital Modeling Department, University of Tyumen; d.v.trapeznikov@utmn.ru

Abstract:

The article presents the two-dimensional model of a porous medium with random micro-inhomogeneities and the results of numerical calculation of single-phase flow of a Newtonian fluid through a model porous medium. The results of calculations based on Darcy’s equation allowed to find the absolute permeability coefficients in the longitudinal and transverse directions. The principle of the skeleton constructing the medium is that the region in the form of a rectangle with sides Lx and Ly is randomly cast with disks of random radius in the range from Rmin to Rmax. The algorithm includes two model parameters: δin and δout, giving the minimum imposition of overlapping disks and the minimum distance between the non-overlapping discs. Thus, the skeleton of the porous medium was formed until achieving the preassigned value of porosity. Then the pore space is obtained by subtracting the skeleton from the treated area. The geometry of the problem and the computational grid were built in open package Salome, the numerical solution of the Navier-Stokes equations for a given pressure drop at the boundaries of the area was considered in the open package OpenFOAM. The difference founded values of the absolute permeability in the longitudinal and transverse directions is no more than half which speaks of the volume of the calculated area being close to the representative.

References:

  1. Altunina L. K., Kuvshinov V. A. 2007. “Fiziko-khimicheskie metody uvelicheniya nefteotdachi plastov neftyanykh mestorozhdeniy” [Physico-Chemical Methods of Enhanced Oil Recovery of Oil Deposits]. Russian Chemical Reviews, vol. 76, no 10, pp. 1034-1052.
  2. Anderson J. D. 1995. Computational Fluid Dynamics: The Basics with Applications. McGraw-Hill Science.
  3. Aziz Kh., Settari E. 2004. “Matematicheskoe modelirovanie plastovykh system” [Mathematical Modeling of Reservoir Systems].
  4. Gubaidullin A. A., Igoshin D. Ye., Khromova N. A. 2016. “Obobshchenie podkhoda Kozeni k opredeleniyu pronitsaemosti modelnykh poristykh sred iz tverdykh sharovykh segmentov” [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, vol. 2, no 2, pp. 105-120. DOI: 10.21684/2411-7978-2016-2-2-105-120
  5. Gubajdullin A. A., Maksimov A. Ju. 2015. “Sobstvennye chastoty prodolnyh kolebanij kapli v suzhenii kapilljara” [Natural Frequencies of Longitudinal Oscillations of a Droplet in the Constriction of the Capillary Tube]. Tyumen State University Herald. Physical and Mathematical Modeling. Oil, Gas, Energy, vol. 1, no 2, pp. 85-91.
  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. Ye., Khromova N. A. 2016. “Fil'tratsionno-emkostnye svoystva periodicheskoy poristoy sredy romboedricheskoy struktury so skeletom iz sharovykh segmentov” [Filtration-Capacitive Properties of the Periodic Porous Medium Rhombohedral Structure of the Skeleton of the Ball Segments]. Tyumen State University Herald. Physical and Mathematical Modeling. Oil, Gas, Energy, vol. 2, no 3, pp. 107-127. DOI: 10.21684/2411-7978-2016-2-3-107-127
  10. Igoshin D. E., Khromova N. A. 2016. “Gidravlicheskoe soprotivlenie izvilistykh kanalov” [Hydraulic Resistance of Tortuous Channels]. Proceedings in Cybernetics, no 3 (23), pp. 8-17.
  11. 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.
  12. 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.
  13. 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.
  14. 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.
  15. Ishkova Z. A., Kolunin V. S. 2016. “Opredelenie kapillyarnykh svoystv melkoporistoy sredy metodom nachala kristallizatsii vody” [The Determination of Capillary Properties of Finely Porous Medium by the Onset of Water Crystallization]. Tyumen State University Herald. Physical and Mathematical Modeling. Oil, Gas, Energy, vol. 2, no 1, pp. 19–25. DOI: 10.21684/2411-7978-2016-2-1-19-25
  16. Kadet V. V. 2008. “Metody teorii perkolyatsii v podzemnoy gidromekhanike” [percolation theory methods in underground fluid mechanics]. Moscow: TsentrLitNefteGaz.
  17. Kislitsyn A. A., Potapov A. G. 2015. “Issledovanie raspredeleniya por po razmeram v poristoy srede s pomoshch'yu yadernogo magnitnogo rezonansa” [The Study of Pores Distribution According to Their Sizes in Porous Media by Nuclear Magnetic Resonance]. Tyumen State University Herald. Physical and Mathematical Modeling. Oil, Gas, Energy, vol. 1, no 3(3), pp. 52–59.
  18. Kurchatov I.M., Laguntsov N. I., Tsodikov M. V., Fedotov A. S., Moiseev I. I. 2008. “Priroda anizotropii pronitsaemosti i kataliticheskoy aktivnosti” [Nature of permeability anisotropy and catalytic activity]. Kinetics and Catalysis, vol. 49, no 1, pp. 129-134.9
  19. 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.
  20. Masket M. 2006. “Techenie odnorodnykh zhidkostey v poristoy srede” [The flow of homogeneous fluids in porous media]. Moscow-Izhevsk: Institut komp'yuternykh issledovaniy.
  21. Ministry of Energy of Russia. 2015. “Energeticheskaja strategija Rossii na period do 2035 goda” [Energy Strategy of Russia for the period to 2035].
  22. Roldugin V. I. 2003. “Svoystva fraktal'nykh dispersnykh system” [Fractal Properties of Disperse Systems]. Russian Chemical Reviews, vol. 72, no 11, pp. 1027-1054. 
  23. Romm E. S. 1985. “Strukturnye modeli porovogo prostranstva gornykh porod” [Structural Model of the Pore Space of Rocks]. Leningrad: Nedra.
  24. Shabarov A. B., Shatalov A. V. 2016. “Poteri davlenija pri techenii vodoneftjanoj smesi v porovyh kanalah” [Pressure Drops in Water-Oil Mixture Flow in Porous Channels]. Tyumen State University Herald. Physical and Mathematical Modeling. Oil, Gas, Energy, vol. 2, no 2, pp. 50-72. DOI: 10.21684/2411-7978-2016-2-2-50-72
  25. Shvidler M. I. 1985. “Statisticheskaya gidrodinamika poristykh sred” [Statistical Hydrodynamics of Porous Media]. Moscw: Nedra.
  26. Sidnyaev N. I., 2004. “Chislennoe modelirovanie polucheniya pronitsaemykh poroshkovykh materialov formiruyushchikhsya pri spekanii” [Numerical Simulation of Producing Permeable Powder Material Formed by Sintering]. Journal on Composite Mechanics and Design, vol. 10, no 1, pp. 93-107.
  27. Sologaev V. I. 2002. “Fil'tratsionnye raschety i komp'yuternoe modelirovanie pri zashchite ot podtopleniya v gorodskom stroitel'stve” [Filtration Calculations and Computer Modeling in the Protection against Flooding in Urban Construction]. Omsk.
  28. Stepanov S. V., Shabarov A. B., Bembel G. S. 2016. “Vychislitelnaja tehnologija dlja opredelenija funkcii mezhfaznogo vzaimodejstvija na osnove modelirovanija techenija v kapilljarnom klastere” [Computer Technology for Determination of Interphase Interaction Function Based on Flow Simulation in Capillary Cluster]. Tyumen State University Herald. Physical and Mathematical Modeling. Oil, Gas, Energy, vol. 2, no 1, pp. 63-71. DOI: 10.21684/2411-7978-2016-2-1-63-71
  29. Succi S. 2001. The Lattice Boltzmann Equation for Fluid Dynamics and Beyond. Oxford University Press.
  30. Uryev N. B., Kuchin I. V. 2006. “Modelirovanie dinamicheskogo sostoyaniya dispersnykh system” [Modeling of the Dynamic State of Disperse Systems]. Russian Chemical Reviews, vol. 75, no 1, pp. 36-63.
  31. Yampol'skiy Yu. 2007. P. “Metody izucheniya svobodnogo ob"ema v polimerakh” [Methods of Study of the Free Volume in Polymers]. Russian Chemical Reviews, vol. 76, no 1, pp. 66-87.
  32. Zhizhimontov I. N., Malshakov A. V. 2016. “Metod rascheta kojefficientov poristosti i pronicaemosti gornoj porody na osnove krivyh kapilljarnogo davlenija” [The Method of Determining the Coefficients of Porosity and Permeability of the Rock on the Basis of Capillary Pressure Curves]. Tyumen State University Herald. Physical and Mathematical Modeling. Oil, Gas, Energy, vol. 2, no 1, pp. 72-81. DOI: 10.21684/2411-7978-2016-2-1-72-81
  33. Zhuravlev A. S., Zhuravlev E. S. 2016. “Vlijanie neodnorodnostej filtracionno-emkostnyh parametrov na processy migracii i akkumuljacii uglevodorodov v estestvennyh geologicheskih sistemah” [The Heterogeneity Effect of Reservoir Properties on Migration and Accumulation of Hydrocarbons in Natural Geological Systems]. Tyumen State University Herald. Physical and Mathematical Modeling. Oil, Gas, Energy, vol. 2, no 1, pp. 101-109. DOI: 10.21684/2411-7978-2016-2-1-101-109