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.), Head of the Reservoir Physics Laboratory, Corporate Center for the Study of Reservoir Systems (Core and Fluids), Gazprom VNIIGAZ (Moscow); Associate Professor, Department of Fundamental Mathematics, Institute of Physics and Technology, University of Tyumen; d.e.igoshin@utmn.ru

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.

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