Relative Permeability Calculation Methods in Multiphase Filtration Problems

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


Release:

2018, Vol. 4. №1

Title: 
Relative Permeability Calculation Methods in Multiphase Filtration Problems


For citation: Shabarov A. B., Shatalov A. V., Markov P. V., Shatalova N. V. 2018. “Relative Permeability Calculation Methods in Multiphase Filtration Problems”. Tyumen State University Herald. Physical and Mathematical Modeling. Oil, Gas, Energy, vol. 4, no 1, pp. 79-109. DOI: 10.21684/2411-7978-2018-4-1-79-109

About the authors:

Aleksandr B. Shabarov, Dr. Sci. (Tech.), Professor, Honored Scientist of the Russian Federation, Professor of the Department of Applied and Technical Physics, Institute of Physics and Technology, University of Tyumen; a.b.shabarov@utmn.ru; eLibrary AuthorID, ORCID, ResearcherID, ScopusID

Alexander V. Shatalov, Postgraduate Student, Engineer, Department of Multiphase Systems Mechanics, University of Tyumen; sashatl@yandex.ru

Pavel V. Markov, Deputy Director General, UNI-CONCORD (Tyumen); Director General, “MicroModel” (Moscow); markov.pv@mail.ru

Natalya V. Shatalova, Teaching Assistant, Department of Electric Power Industry, Industrial University of Tyumen; natalyashatalova@yandex.ru

Abstract:

Computer simulators in oilfield hydrodynamic modelling are used at the stage of engineering design and operation. Their operation requires the knowledge of invading and displacing fluid relative permeabilities, which determine filtration process and, consequently, well production in accordance with generalized Darcy’s low.

This paper presents the relative permeability mathematical definition, the multiphase flow equation system, a literature review, devoted to native and foreign authors relative permeabilities calculation methods; a diagram of methods classification is included. The diagram consists of four large groups: 1) experimental (core sample tests), 2) combined computational and experimental, 3) empirical, field data, and 4) inverse problems solving based. Different ways of sample geometrical model construction, pore space topologies, calculation techniques of multiphase flows in this space structures are described. An analytical research of accuracy and computational cost of the relative permeability curves plotting methods has been performed. Advantages and disadvantages of different approaches are pointed out, potential of the combined computational and experimental method is showed.

References:

  1. Doroginitskaya L. M., Enikeev B. N., Efimov V. A., Isaev G. D., Kostenevich K. A., Malshakov A. V., I. B. Ratnikov, Semenov V. V., Sokova K. I., Fedortsov I. V., Shnurman I. G. 2010. Aktual’nye voprosy petrofiziki slozhno postroennykh kollektorov [Actual Problems of Petrophysics of Complicated Collectors]. Krasnodar: Prosveshchenie-Yug.
  2. Basniev K. S., Kochina I. N., Maksimov V. M. 1993. Podzemnaya gidromekhanika: uchebnik dlya vuzov [Underground Hydromechanics: A Textbook for Universities]. Moscow: Nedra.
  3. Gorinov A. A. 2014. “Postroenie razryvnykh resheniy uravneniya Bakleya-Leveretta v Maple” [Construction of Discontinuous Solutions of the Buckley-Leverett Equation in Maple]. XII Vserossiyskoe soveshchanie po problemam upravleniya [13th All-Russian Convention of the Management Problems] VSPU-2014 (Moscow, 16-19 June), pp. 1536-1540. Moscow: IPU RAN.
  4. Gubaydullin A. A., Igoshin D. E., Khromova N. A. 2016. “Obobshchenie podkhoda Kozeni k opredeleniyu pronitsaemosti model’nykh 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. Demyanov A. Yu., Dinariev O. Yu. 2004. “Primenenie metoda funktsionala plotnosti dlya chislennogo modelirovaniya techeniy mnogokomponentnykh mnogofaznykh smesey” [Application of Density Functional Method for Numerical Simulation of Multicomponent Multiphase Mixture Flows]. Zhurnal prikladnoy mekhaniki i tekhnicheskoy fiziki (PMTF), vol. 45, no 5, pp. 68-78.
  6. Demyanov A. Yu., Dinariev O. Yu., Evseev N. V. 2009. Osnovy metoda funktsionala plotnosti v gidrodinamike [Foundations of the Density Functional Method in Hydrodynamics]. Moscow: Fizmatlit.
  7. Zhizhimontov I. N., Malshakov A. V. 2016. “Metod rascheta koeffitsientov poristosti i pronitsaemosti gornoy porody na osnove krivykh kapillyarnogo davleniya” [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
  8. Igoshin D. E., Maksimov A. Yu. 2015. “Chislennye i analiticheskie otsenki pronitsaemosti poristoy sredy, obrazovannoy kanalami, imeyushchimi vrashchatel’nuyu simmetriyu” [Numerical and Analytical Estimates of Permeability of 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.
  9. Igoshin D. E., Saburov R. S. 2015. “Chislennoe issledovanie zavisimosti pronitsaemosti ot poristosti sredy, obrazovannoy kanalami regulyarnoy struktury” [Numerical Research of Permeability Dependence of Porosity in the Media Formed by Regular Structure Channels]. Tyumen State University Herald. Physical and Mathematical Modeling. Oil, Gas, Energy, vol. 1, no 1, pp. 84-90.
  10. Stepanov S. V., Shabarov A. B., Bembel G. S., Shatalov A. V. 2015. “Issledovanie dinamicheskikh fazovykh pronitsaemostey na osnove chislennogo modelirovaniya dvukhfaznogo techeniya v porovykh kanalakh” [Investigation of Dynamic Phase Permeabilities On The Basis Of Numerical Modeling of Two-Phase Flow in Pore Channels]. Materialy XI Vserossiyskogo s”ezda po fundamental’nym problemam teoreticheskoy i prikladnoy mekhanik [Proceedings of the 11th All-Russian Convention on the Basic Problems of Theoretical and Applied Mechanics], pp. 3600-3601. Kazan.
  11. Shabarov A. B., Saranchin N. V., Chistyakova N. F., Shirshova A. V., Puldas L. A., Stupnikov A. A., Vetrov I. M., Shatalov A. V., Bembel’G. S., Vakulin A. A., Varyukhin S. E., Berdyugin S. V., Medvedev D. N., Molchanov D. A., Vorobyov V. V. 2011. Itogovyy otchet po teme “Chislennoe issledovanie protsessa vytesneniya v masshtabakh kerna dlya polucheniya soglasovannykh krivykh kapillyarnogo davleniya i otnositel’nykh fazovykh pronitsaemostey” [Final report on the topic “Numerical Study of the Process of Displacement in the Core Scale to Obtain Consistent Curves of Capillary Pressure and Relative Permeability”]. University of Tyumen. Framework agreement TNNTs — University of Tyumen of 16 June 2011.
  12. Kadet V. V., Khurgin Ya. I. 2006. Sovremennye veroyatnostnye podkhody pri reshenii zadach mikro- i makrourovnya v neftegazovoy otrasli [Modern Probabilistic Approaches to Solving Micro- and Macro-Level Problems in the Oil and Gas Industry]. Moscow; Izhevsk: Regulyarnaya i khaoticheskaya dinamika, Institut komp’yuternykh issledovaniy.
  13. Kadet V. V. 2008. Metody teorii perkolyatsii v podzemnoy gidromekhanike [Methods of Percolation Theory in Underground Hydromechanics]. Moscow: TsentrLitNefteGaz.
  14. Doroginitskaya L. M., Dergacheva T. N., Anashkin A. R., Kolyvanov A. I., Kushnarev S. V., Khudyakova L. D., Romanov E. A., Golikov N. A., Melkozerova S. N. 2007. Kolichestvennaya otsenka dobyvnykh kharakteristik kollektorov nefti i gaza po petrofizicheskim dannym i materialam GIS [Quantitative Estimation of Oil and Gas Reservoirs Production Characteristics on Petrophysical Data and GIS Materials], p. 163. Tomsk: STT.
  15. Kollinz R. 1964. Techeniya zhidkostey cherez poristye materialy [Flows of Liquids through Porous Materials]. Edited by G. I. Barenblatt; translated from English by R. L. Salganik. Moscow: MIR.
  16. Kupershtokh A. L. 2012. “Trekhmernoe modelirovanie dvukhfaznykh sistem tipa zhidkost’–par metodom reshetochnykh uravneniy Bol’tsmana na GPU” [Three-Dimensional Modeling of Two-Phase Systems of Liquid-Vapor Type by the Boltzmann Lattice Equations on the GPU]. Vychislitel’nye metody i programmirovanie, vol. 13, pp. 130-138.
  17. Leybenzon L. S. 1947. Dvizhenie prirodnykh zhidkostey i gazov v poristoy srede [The Motion of Natural Liquids and Gases in a Porous Medium]. Moscow-Leningrad: Gosudarstvennoe izdatel’stvo tekhniko-teoreticheskoy literatury.
  18. Lysenko V. D. 2004. “O fazovykh pronitsaemostyakh” [On Phase Permeabilities]. Neftepromyslovoe delo, no 12, pp. 4-9.
  19. Malshakov A. V., Efimov V. A. 1991. “Pronitsaemost’ i perkolyatsionnye svoystva porovogo prostranstva osadochnykh gornykh porod” [Permeability and Percolation Properties of the Pore Space of Sedimentary Rocks]. Inzhenerno-fizicheskiy zhurnal, vol. 61, no 4, pp. 635-640.
  20. Markov P. V., Rodionov S. P. 2015. “Ispol’zovanie modeley mikrostruktury poristoy sredy pri raschete fil’tratsionnykh kharakteristik dlya gidrodinamicheskikh modeley” [Use of Models of the Microstructure of a Porous Medium in the Calculation of Filtration Characteristics for Hydrodynamic Models]. Neftepromyslovoe delo, no 11, pp. 64-75.
  21. Markov P. V., Rodionov S. P. 2016. “Metod stokhasticheskoy generatsii modeley porovykh setey po raspredeleniyam ikh parametrov” [The Method of Stochastic Generation of Models of Pore Networks Based on the Distribution of Their Parameters]. Vestnik kibernetiki, no 3 (23), pp. 18-15.
  22. Masket M. 1953. Fizicheskie osnovy tekhnologii dobychi nefti [Physical Foundations of Oil Production Technology]. Translated from English by M. A. Geyman. Moscow-Leningrad: Gostoptekhizdat.
  23. Shandrygin A., Shelepov V., Ramazanov R., Andrianov N., Klemin D., Nadeev A., Safonov S., Yakimchuk I. 2016. “Mekhanizm vytesneniya malovyazkoy nefti iz mikroneodnorodnoy poristoy sredy polimernymi rastvorami” [The Mechanism of Displacement of Low-Viscosity Oil from a Microinhomogeneous Porous Medium by Polymer Solutions]. Proceedings of the International Conference “Rossiyskaya neftegazovaya tekhnicheskaya konferentsiya i vystavka SPE” [Russian Oil and Gas Technical Conference and the SPE Exhibition] (Moscow, Russia, 24-26 October). Moscow. SPE-182037-RU.
  24. Ovcharov V. V. 2014. “Obzor metodov rascheta i protsedur korrektirovki krivykh otnositel’nykh fazovykh pronitsaemostey dlya gidrodinamicheskogo modelirovaniya zalezhey uglevodorodov” [Review of Calculation Methods and Procedures for Correcting the Relative Permeability Curves for Hydrodynamic Modeling of Hydrocarbon Deposits]. Vestnik kibernetiki, no 1 (13), pp. 10-16.
  25. OST 39-235-89 Neft’. Metod opredeleniya fazovykh pronitsaemostey v laboratornykh usloviyakh pri sovmestnoy statsionarnoy fil’tratsii [Oil. Method for Determination of Phase Permeabilities in Laboratory Conditions for Joint Stationary Filtration].
  26. Altunin A. E., Sokolov S. V., Stepanov S. V., Cheremisin N. A., Shabarov A. B. 2013. “Raschetnyy metod polucheniya OFP na osnove resheniya obobshchennykh uravneniy Bernulli dlya sistemy porovykh kanalov” [Calculation Method for Obtaining OFP Based on the Solution of the Generalized Bernoulli Equations for a Pore Channel System]. Neftepromyslovoe delo, no 8, pp. 40-46.
  27. Sotnikov O. S. 2009. “Sovershenstvovanie metodov opredeleniya otnositel’nykh fazovykh pronitsaemostey i ikh primeneniya pri gidrodinamicheskom modelirovanii razrabotki neftyanykh mestorozhdeniy” [Improvement of Methods for Determining the Relative Phase Permeabilities and Their Application in the Hydrodynamic Modeling of the Development of Oil Deposits]. Cand. Sci. (Tech.) diss. abstract. Bugulma.
  28. Sotnikov O. S. 2009. “Sovershenstvovanie metodov opredeleniya otnositel’nykh fazovykh pronitsaemostey i ikh primeneniya pri gidrodinamicheskom modelirovanii razrabotki neftyanykh mestorozhdeniy” [Improvement of Methods for Determining Relative Phase Permeabilities and Their Application in Hydrodynamic Modeling of Oilfield Development]. Cand. Sci. (Tech.) diss. Bugulma.
  29. Stepanov S. V. 2006. “Ispol’zovanie dannykh razrabotki mestorozhdeniy nefti dlya polucheniya krivykh fazovykh pronitsaemostey” [Use of Oil Development Data to Produce Curves of Phase Permeabilities]. Neftyanoe khozyaystvo, no 4, pp. 112-114.
  30. Stepanov S. V. 2016. “Kompleks vychislitel’nykh tekhnologiy dlya povysheniya kachestva modelirovaniya razrabotki neftyanykh i gazoneftyanykh mestorozhdeniy” [A Complex Of Computational Technologies for Improving the Modeling Quality of the Development of Oil And Gas-Oil Fields]. Dr. Sci. (Tech.) diss. Tyumen.
  31. Stepanov S. V., Shabarov A. B., Bembel G. S. 2016. “Vychislitel’naya tekhnologiya dlya opredeleniya funktsii mezhfaznogo vzaimodeystviya na osnove modelirovaniya techeniya v kapillyarnom 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
  32. Tarasevich Yu. Yu. 2002. Perkolyatsiya: teoriya, prilozheniya, algoritmy: uch. posobie [Percolation: Theory, Applications, Algorithms]. Moscow: Editorial URSS.
  33. Tiab D., Donaldson E. C. 2009. Petrofizika: teoriya i praktika izucheniya kollektorskikh svoystv gornykh porod i dvizheniya plastovykh flyuidov [Petrophysics: Theory and Practice of Measuring Reservoir Rock and Fluid Transport Properties]. Translated from English. Moscow: Premium Inzhiniring.
  34. Timoshenko A. A. 2008. “Opredelenie otnositel’nykh fazovykh pronitsaemostey pri dvukhfaznoy fil’tratsii zhidkostey po promyslovym dannym” [Determination of Relative Phase Permeabilities in Two-Phase Filtration of Liquids from Field Data]. Optimizatsiya tekhnologii razrabotki neftyanykh mestorozhdeniy: sb. nauch. tr., pp. 145-154. Tyumen: TsESSIYa.
  35. Wolcott D. 2001.Razrabotka i upravlenie mestorozhdeniyami pri zavodnenii [Design, Performance and Surveillance Methods to Optimize Rate and Reserves]. Moscow: Yukos. 
  36. Vorobyov V. V. 2016. RF Patent 158561 “Ustroystvo dlya opredeleniya fazovykh pronitsaemostey” [Device for Determining Phase Permeabilities]. MPK G01N 15/08, E21B 47/00. No 2015111395/15; applied 30 March 2015; published 10 January 2016, patent bulletin no 1.
  37. A Shabarov. B., Vorobyov V. V. 2016. RF Patent 166252 “Ustroystvo dlya opredeleniya fazovykh pronitsaemostey” [Device for Determining Phase Permeabilities]. MPK G01N 15/08. No 2016118428/28; applied 11 May 2016; published 20 November 2016, patent bulletin no 32.
  38. Vorobyov V. V., Grigoryev B. V. 2015. RF Patent 2572476 “Ustroystvo dlya opredeleniya fazovykh pronitsaemostey” [Device for Determining Phase Permeabilities]. MPK E21B 49/00. No 2014122094/03; applied 30 May 2014; published 10 December 2015, patent bulletin no 34.
  39. Vorobyov V. V., Grigoryev B. V. 2016. RF Patent 159112 “Ustroystvo dlya smesheniya zhidkikh faz” [Device for Mixing Liquid Phases]. MPK B01F 5/06, B01F 13/00. No 2015128327/05; applied 13 July 2015; published 27 January 2016, patent bulletin no 3.
  40. Fatikhov S. Z. 2012. “K voprosu vychisleniya otnositel’nykh fazovykh pronitsaemostey” [To the Problem of Calculating the Relative Phase Permeabilities]. Neftegazovoe delo: elektron. nauchn. zhurnal, no 1. Accessed on 7 July 17. http://www.ogbus.ru
  41. Fatikhov S. Z. 2012. “Remasshtabirovanie setochnykh modeley neftyanykh mestorozhdeniy s uchetom mikroneodnorodnosti poristoy sredy” [Rescaling Net Models of Oil Fields with Allowance for the Microinhomogeneity of the Porous Medium]. Cand. Sci. (Phys.-Math.) diss. Tyumen.
  42. Chugaev R. R. 1982. Gidravlika: uchebnik dlya vuzov [Hydraulics: A University Textbook]. 4th edition, revised. Leningrad: Energoizdat. Leningr. otd-nie.
  43. Shabarov A. B. 2013. Gidrogazodinamika: ucheb. posobie [Hydro and Gas Dynamics]. 2nd edition, revised; p. 156. Tyumen: UTMN Publishing House.
  44. Shabarov A. B., Shatalov A. V. 2016. “Geometricheskaya model’ porovogo prostranstva dlya rascheta fil’tratsii nefti i vody” [Geometric Model of Porous Space for Calculation of Oil and Water Filtration]. Proceedings of the 9th workshop for young researchers “Teplofizika, teplotekhnika, gidrogazodinamika. Innovatsionnye tekhnologii” [Thermophysics, Heat Engineering, Hydrodynamics. Innovative Technologies], pp. 25-36. Tyumen: UTMN Publishing House.
  45. Shabarov A. B., Shatalov A. V. 2016. “Poteri davleniya pri techenii vodoneftyanoy smesi v porovykh kanalakh” [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
  46. Shabarov A. B., Shatalov A. V. 2017. “Klasternaya model’ rascheta otnositel’nykh fazovykh pronitsaemostey pri dvukhfaznoy fil’tratsii” [A Cluster Model for Calculating the Relative Phase Permeabilities in Two-Phase Filtration]. Proceedings of the All-Russian Conference for Students and Young Researchers “Molodezh’. Nauka. Tekhnologii” [Youth. Science. Technologies] in 4 parts. Edited by E. G. Gurova and S. V. Makarov; pp. 74-79. Novosibirsk: NGTU.
  47. Shvidler M. I. 1985. Statisticheskaya gidrodinamika poristykh sred [Statistical Hydrodynamics of Porous Media]. Moscow: Nedra.
  48. Ellanskiy M. M. 2001. “Edinaya teoreticheskaya model’ pronitsaemosti produktivnykh otlozheniy s mezhgranulyarnym tipom pustot” [A Unified Theoretical Model for the Permeability of Productive Deposits with an Intergranular Type of Void]. Geofizika, no 6, pp. 1-14. Accessed on 6 July 17. http://mmell.narod.ru/Index.htm
  49. Efros D. A. 1963. Issledovaniya fil’tratsii neodnorodnykh sistem [Studying the Filtration of Inhomogeneous Systems]. Leningrad: Gostoptekhizdat.
  50. Akanji L. T., Matthai S. K. 2010. “Finite Element-Based Characterization of Pore-Scale Geometry and Its Impact on Fluid Flow”. Transport in Porous Media, vol. 81, no 2, pp. 241-259. DOI: 10.1007/s11242-009-9400-7
  51. Al-Gharbi M. S. 2004. “Dynamic Pore-Scale Modelling of Two-Phase Flow”. PhD Thesis. June. London: Imperial College of the University of London.
  52. Al-Kharusi A. S., Blunt M. J. 2007. “Network Extraction from Sandstone and Carbonate Pore Space Images”. Journal of Petroleum Science and Engineering, vol. 56, pp. 219-231. DOI: 10.1016/j.petrol.2006.09.003
  53. Allen M. P., Tildesley D. J. 1987. Computer Simulation of Liquids. Oxford Science Publications.
  54. Avraam D. G., Payatakes A. C. 1995. “Generalized Relative Permeability Coefficients during Steady-State Two-Phase Flow in Porous Media, and Correlation with the Flow Mechanisms”. Transport in Porous Media, vol. 20, no 1, pp. 135-168. DOI: 10.1007/BF00616928
  55. Blunt M. J., King M. J., Scher H. 1992. “Simulation and Theory of Two-Phase Flow in Porous Media”. Physical Review A, vol. 46, no 12, pp. 7680-7699. DOI: 10.1103/PhysRevA.46.7680
  56. Brooks R. H., Corey A. T. 1964. “Hydraulic Properties of Porous Media”. Hydrology Papers, no 3. Fort Collins, Colorado: Colorado State U.
  57. Buckley S. E., Leverett M. C. 1942. “Mechanism of Fluid Displacement in Sands”. Transactions of the American Institutetute of Mining, Metallurgical and Petroleum Engineers, vol. 146, no 1, pp. 107-116. DOI: 10.2118/942107-G
  58. Burdine N. T., Gournay L. S., Reichertz P. P. 1950. “Pore Size Distribution of Petroleum Reservoir Rocks”. Journal of Petroleum Technology, vol. 2, no 7, pp. 195-204. DOI: 10.2118/950195-G
  59. Burdine N. T. 1953. “Relative Permeability Calculations from Pore Size Distribution Data”. Journal of Petroleum Technology, vol. 5, no 3, pp. 71-78. DOI: 10.2118/225-G
  60. Chierici G. L. 1984. “Novel Relations for Drainage and Imbibition Relative Permeabilities”. Society of Petroleum Engineers Journal, vol. 24, no 3, pp. 275-276. DOI: 10.2118/10165-PA
  61. Corey A. T. 1954. “The Interrelation between Gas and Oil Relative Permeabilities”. Producers Monthly, November, pp. 38-41. 
  62. Blunt M. J., Jackson M. D., Piri M., Valvatne P. H. 2002. “Detailed Physics, Predictive Capabilities and Macroscopic Consequences for Pore-Network Models of Multiphase Flow”. Advances in Water Resources, vol. 25, no 8, pp. 1069-1089. DOI: 10.1016/S0309-1708(02)00049-0
  63. Dias M. M., Payatakes A. C. 1986. “Network Models for Two-Phase Flow in Porous Media. Part 1. Immiscible Microdisplacement of Non-Wetting Fluids”. Journal of Fluid Mechanics, vol. 164, pp. 305-336. DOI: 10.1017/S0022112086002574
  64. Dias M. M., Payatakes A. C. 1986. “Network Models for Two-Phase Flow in Porous Media. Part 2. Motion of Oil Ganglia”. Journal of Fluid Mechanics, vol. 164, pp. 337-358. DOI: 10.1017/S0022112086002586
  65. Dong H. 2007. “Micro-CT Imaging and Pore Network Extraction”. PhD Thesis. London: Imperial College of London.
  66. Fatt I., Dykstra H. 1951. “Relative Permeability Studies”. Journal of Petroleum Technology, vol. 3, no 9, pp. 249-256. DOI: 10.2118/951249-G
  67. Fatt I. 1956. “The Network Model of Porous Media, II. Dynamic Properties of a Single Size Tube Network”. Transactions of the American Institutetute of Mining, Metallurgical and Petroleum Engineers, vol. 207, pp. 160-181.
  68. Shabro V., Torres-Verdín C., Javadpour F., Sepehrnoori K. 2012. “Finite-Difference Approximation for Fluid-Flow Simulation and Calculation of Permeability in Porous Media”. Transport in Porous Media, vol. 94, no 3, pp. 775-793. DOI: 10.1007/s11242-012-0024-y
  69. Gates J. I., Lietz W. T. 1950. “Relative Permeabilities of California Cores by the Capillary — Pressure Method”. Proceedings of Drilling and Production Practice Conference (1 January), pp. 285-302. New York. API-50-285.
  70. Guibert R., Horgue P., Debenest G., Quintard M. 2016. “A Comparison of Various Methods for the Numerical Evaluation of Porous Media Permeability Tensors from Pore-Scale Geometry”. Mathematical Geosciences, vol. 48, no 3, pp. 329-347. DOI: 10.1007/s11004-015-9587-9
  71. Gunstensen A. K., Rothman D. H. 1993. “Lattice-Boltzmann Studies of Immiscible Two-Phase Flow through Porous Media”. Journal of Geophysical Research, vol. 98, no B4, pp. 6431–6441. DOI: 10.1029/92JB02660
  72. Hirt C. W., Nichols B. D. 1981. “Volume of Fluid (VOF) Method for the Dynamics of Free Boundaries”. Journal of Computational Physics, vol. 39, pp. 201-225. DOI: 10.1016/0021-9991(81)90145-5
  73. Honarpour M., Koederitz L., Harvey A. H. 1986. Relative Permeability of Petroleum Reservoirs. Boca Raton, Florida: CRC Press, Inc.
  74. Huet C. C., Rushing J. A., Newsham K. E., Blasingame T. A. 2005. “A Modified Purcell/Burdine Model for Estimating Absolute Permeability from Mercury-Injection Capillary Pressure Data”. Proceedings of the International Petroleum Technology Conference (Doha, Qatar, 21-23 November). IPTC 10994.
  75. Jerauld G. R., Salter S. J. 1990. “The Effect of Pore-Structure on Hysteresis in Relative Permeability and Capillary Pressure — Pore-Level Modeling”. Transport in Porous Media, vol. 5, no 2, pp. 103-151. DOI: 10.1007/BF00144600
  76. Joekar-Niasar V., Hassanizadeh S. M. 2012. “Analysis of Fundamentals of Two-Phase Flow in Porous Media Using Dynamic Pore-Network Models: A Review”. Critical Reviews in Environmental Science and Technology, vol. 42, no 18, pp. 1895-1976. DOI: 10.1080/10643389.2011.574101
  77. Joekar-Niasar V., S Hassanizadeh. M. 2012. “Uniqueness of Specific Interfacial Area–Capillary Pressure–Saturation Relationship under Non-Equilibrium Conditions in Two-Phase Porous Media Flow”. Transport in Porous Media, vol. 94, no 2, pp. 465-486. DOI: 10.1007/s11242-012-9958-3
  78. Joekar-Niasar V., van Dijke R., Hassanizadeh S. M. 2012. “Pore-Scale Modeling of Multiphase Flow and Transport: Achievements and Perspectives”. Transport in Porous Media, vol. 94, no 2, pp. 461-464. DOI: 10.1007/s11242-012-0047-4
  79. Johnson E. F., Bossler D. P., Naumann V. O. 1959. “Calculation of Relative Permeability from Displacement Experiments”. Transactions of the American Institutetute of Mining, Metallurgical and Petroleum Engineers, no 216, pp. 370.
  80. Lomeland F., Ebeltoft E., Wibeke H. T. 2005. “A New Versatile Relative Permeability Correlation”. Proceedings of the International Symposium of the Society of Core Analysts (Toronto, Canada, 21-25 August). Toronto. SCA2005-32.
  81. Meakin P., Tartakovsky A. M. 2009. “Modeling and Simulation of Pore-Scale Multiphase Fluid Flow and Reactive Transport in Fractured and Porous Media”. Reviews of Geophysics, vol. 47, no 3. 47 p. RG3002.
  82. Markov P. V. 2016. “The Relative Permeability Regions Assignment on the Basis of Pore Network Models Parameters Distributions”. Saint Petersburg 2016 International Conference & Exhibition (Saint Petersburg, Russia, 11-14 April). DOI: 10.3997/2214-4609.201600231
  83. Markov P. V., Rodionov S. P. 2017. “Rock Typing on the Basis of Pore-scale Models and Complex Well Log Interpretation Parameters”. International Conference & Exhibition “Tyumen 2017”, EAGE (Tyumen, Russia, 11-14 April).
  84. O’Meara D. J., Lease W. O. 1983. “Multiphase Relative Permeability Measurements Using an Automated Centrifuge”. SPE Conference Paper. 12 p. 12128-MS.
  85. 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, vol. 20, no 4, pp. 857-879. DOI: 10.1007/s10596-014-9424-0
  86. Øren P. E., Bakke S., Avntzen O. J. 1998. “Extending Predictive Capabilities to Network Models”. Society of Petroleum Engineers Journal, vol. 3, no 4, pp. 324-336.
  87. Osher S., Fedkiw R. 2002. Level Set Methods and Dynamic Implicit Surfaces. New York: Springer-Verlag.
  88. Pan C., Hilpert M., Miller C. T. 2004. “Lattice-Boltzmann Simulation of Two-Phase Flow in Porous Media”. Water Resources Research, vol. 40, no 1. 14 p. W01501.
  89. Payatakes A. C., Dias M. M. 1984. “Immiscible Microdisplacement and Ganglion Dynamics in Porous Media”. Reviews in Chemical Engineering, vol. 2, no 2, pp. 85-174. DOI: 10.1515/REVCE.1984.2.2.85
  90. Piri M. 2003. “Pore-Scale Modeling of Three-Phase Flow”. PhD Thesis. London: Imperial College of the University of London, December.
  91. Prodanovic M., Bryant S. L. 2006. “A Level Set Method for Determining Critical Curvatures for Drainage and Imbibition”. Journal of Colloid and Interface Science, vol. 304, pp. 442-458. DOI: 10.1016/j.jcis.2006.08.048
  92. Purcell W. R. 1949. “Capillary Pressures — Their Measurement Using Mercury and the Calculation of Permeability Therefrom”. Journal of Petroleum Technology, vol. 1, no 2, pp. 39-48. DOI: 10.2118/949039-G
  93. Raeini A. Q., Bijeljic B., Blunt M. J. 2014. “Numerical Modelling of Sub-pore Scale Events in Two-Phase Flow through Porous Media”. Transport in Porous Media, vol. 101, no 2, pp. 191-213. DOI: 10.1007/s11242-013-0239-6
  94. Heriot-Watt Institute of Petroleum Engineering. 2010. Reservoir Engineering. Manual. Edinburgh: Heriot-Watt Institute of Petroleum Engineering.
  95. Ryazanov A. V. 2012. “Pore-Scale Network Modelling of Residual Oil Saturation in Mixed-Wet Systems”. PhD Thesis. Edinburgh: Heriot-Watt University, Institute of Petroleum Engineering.
  96. Sahimi M. 2011. Flow and Transport in Porous Media and fractured Rock. From Classical Methods to Modern Approaches. 2nd Edition. Weinheim: WILEY-VCHVerlag GmbH & Co. KGaA. DOI: 10.1002/9783527636693
  97. Shandrygin A. N. 2014. “Digital Core Analysis for Flow Process Evaluation Is Myth or Reality?”. Proceedings of the SPE Russian Oil and Gas Exploration and Production Technical Conference and Exhibition 2014 (Moscow, 14-16 October). Moscow, 171216-MS.
  98. Avraam D. G., Kolonis G. B., Roumeliotis T. C., Constantinides G. N., Payatakes A. C. 1994. “Steady-State Two-Phase Flow through Planar and Nonplanar Model Porous Media”. Transport in Porous Media, vol. 16, no 1, pp. 75-101. DOI: 10.1007/BF01059777
  99. Kunz P., Zarikos I. M., Karadimitriou N. K., Huber M., Nieken U., Hassanizadeh S. M. 2016. “Study of Multi-Phase Flow in Porous Media: Comparison of SPH Simulations with Micro-Model Experiments”. Transport in Porous Media, vol. 114, no 2, pp. 581-600. DOI: 10.1007/s11242-015-0599-1
  100. Stukan M. R., Ligneul P., Boek E. S. 2012. “Molecular Dynamics Simulation of Spontaneous Imbibition in Nanopores and Recovery of Asphaltenic Crude Oils Using Surfactants for EOR Applications”. Oil & Gas Science and Technology. Rev. IFP Energies nouvelles, vol. 67, no 5, pp. 737-742.
  101. Tartakovsky A. M., Meakin P. 2006. “Pore Scale Modeling of Immiscible and Miscible Fluid Flows Using Smoothed Particle Hydrodynamics”. Advances in Water Resources, vol. 29, no 10, pp. 1464-1478. DOI: 10.1016/j.advwatres.2005.11.014
  102. Thomeer J. H. 1983. “Air Permeability as a Function of Three Pore-Network Parameters”. Journal of Petroleum Technology, vol. 35, no 4, pp. 809-814. DOI: 10.2118/10922-PA
  103. Valavanides M. S., Payatakes A. C. 2001. “True-to-Mechanism Model of Steady-State Two-Phase Flow in Porous Media, using Decomposition into Prototype Flows”. Advances in Water Resources, vol. 24, no 3-4, pp. 385-407. DOI: 10.1016/S0309-1708(00)00063-4
  104. Valavanides M. S. 2012. “Steady-State Two-Phase Flow in Porous Media: Review of Progress in the Development of the DeProF Theory Bridging Pore to Statistical Thermodynamics Scales”. Oil & Gas Science and Technology. Rev. IFP Energies nouvelles, vol. 67, no 5, pp. 787-804.
  105. Welge H. J. 1952. “Simplified Method for Computing Oil Recovery by Gas or Water Drive”. Transactions of the American Institutetute of Mining, Metallurgical and Petroleum Engineers, vol. 195, pp. 91-98. DOI: 10.2118/124-G
  106. Wilkinson D., J. F. Willemsen 1983. “Invasion Percolation: A New form of Percolation Theory”. Journal of Physics A: Mathematical and General, vol. 16, no 14, pp. 3365-3376. DOI: 10.1088/0305-4470/16/14/028
  107. Xiong Q., Baychev T. G., Jivkov A. P. 2016. “Review of Pore Network Modelling of Porous Media: Experimental Characterisations, Network Constructions and Applications to Reactive Transport”. Journal of Contaminant Hydrology, vol. 192, pp. 101-117. DOI: 10.1016/j.jconhyd.2016.07.002