On One Mathematical Model for Filtration of Weakly Compressible Chemical Compound in the Porous Heterogeneous 3D Medium. Part I: Model Construction with the Aid of the Ollendorff Approach

Authors: Sharif E. Guseynov, Jekaterina V. Aleksejeva, Janis S. Rimshans

Abstract:

A filtering problem of almost incompressible liquid chemical compound in the porous inhomogeneous 3D domain is studied. In this work general approaches to the solution of twodimensional filtering problems in ananisotropic, inhomogeneous and multilayered medium are developed, and on the basis of the obtained results mathematical models are constructed (according to Ollendorff method) for studying the certain engineering and technical problem of filtering the almost incompressible liquid chemical compound in the porous inhomogeneous 3D domain. For some of the formulated mathematical problems with additional requirements for the structure of the porous inhomogeneous medium, namely, its isotropy, spatial periodicity of its permeability coefficient, solution algorithms are proposed. Continuation of the current work titled ”On one mathematical model for filtration of weakly compressible chemical compound in the porous heterogeneous 3D medium. Part II: Determination of the reference directions of anisotropy and permeabilities on these directions” will be prepared in the shortest terms by the authors.

Keywords: Porous media, filtering, permeability, elliptic PDE.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1088256

References:

[1] P. Y. Kochina, Theory of ground water motion, Moscow, Russia: Science, 1977, 664 p.
[2] F. Ollendorff, Currents in the ground, M.L. GNTI, 1932, 215 p.
[3] L. S. Leibenzon, Motion of natural liquids and gasses in the porous environment, M.L. GNTI, 1947, 244 p.
[4] R. Collins, Motion of liquids through porous materials, Moscow, Russia: World, 1964, 350 p.
[5] G. I. Barenblatt, V. M. Yentov, V. M. Ryzhik, Motion of liquids and gasses in natural layers, Moscow, Russia: Earth interior, 1984, 208 p.
[6] Y. Ber, D. Zaslavsky, S. Irmey, Physical and mathematical principles of water filtering, Moscow, Russia: World, 1971, 452 p.
[7] N. N. Kochina, P. Y. Kochina, V. N. Nilokayevsky, The world of underground liquids, Moscow, Russia: Institute of Physics, Russian Academy of Sciences, 1994, 112 p.
[8] P. Y. Kochina, N. N. Kochina, Problems of motion with free surface in underground hydrodynamics, Journal ”March of physical science”, 1994, 176 p.
[9] V. N. Emikh, Hydrodynamical analysis of two dimensional filtering streams with free boundaries, Thesis in physics and mathematics, Novosibirsk, Russia: Novosibirsk State University, 1984, 64 p.
[10] V. A. Tolpaev, Mathematical models of two dimensional filtering in anisotropic non-homogeneous multilayer environments, Doctoral (Dr.Sc.Math.) Thesis in physics and mathematics, Moscow, Russia: Institute for Problems in Mechanics, Russian Academy of Sciences, 2004, 40 p.
[11] G. V. Golubev, G. G. Tumashev, Filtering of incompressible liquid in non-homogeneous porous environment, Kazan, Russia: Kazan State University, 1972, 195 p.
[12] V. P. Pilatovsky, Fundamentals of thin layer hydromechanics, Moscow, Russia: Earth interior, 1966, 317 p.
[13] V. A. Tolpaev, V. V. Paliev, Equation of continuity in two dimensional filtering models of liquids and gasses in distorted layers of finite thickness, Saratov, Russia: Journal of Saratov University, Series: Mathematics. Mechanics. Computer Science, Vol. 7, Issue 2, 2007, pp. 49-53.
[14] V. I. Aravin, Calculation of flat filtering in the ground with curvilinear anisotropy, Journal VNIIG in the name of B. E. Bedeneev, Vol. 104, pp. 3-9.
[15] V. I. Aravin, S. N. Aravin, Theory of motion of iquids and gasses in undistorted porous environment, Moscow, Russia: ”Gostehizdat”, 1953, 616 p.
[16] A. E. Scheidegger, The physics of flow through porous media, Toronto, Canada: University of Toronto Press, 1974, 353 p.
[17] K. S. Basniev, N. M. Dmitriev, R. D. Kanevskaya, V. M. Maksimov, Underground hydrodynamics, Moscow, Russia: Institure of Computer Research, 2006, 488 p.