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 two- dimensional 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 mathemat- ical 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. I. I NTRODUCTION L IQUID filtering processes occur in porous environment, which, depending on its physical, chemical and mechan- ical properties, belongs to the group of anisotropic (filtering properties at each point of environment are equal in all direc- tions, i.e. the corresponding functions in mathematical models are scalar functions) or ananisotropic (filtering properties at each point of environment are different in each direction, i.e. the corresponding functions in mathematical models are vector functions) materials. Likewise, porous environment layers can be divided into two types - ”active/productive” and ”passive/non-productive” layers. In addition, in real processes (objects, phenomena) ”active/productive” layers do not only show anisotropic or ananisotropic, homogeneous or inhomo- geneous filtering properties, but are always flexuous and with variable thickness. Similar problems appear in the processes of acquiring liquid energy feedstock (oil, gas; for instance, see [1], [2]); when op- erating hydro-technical and hydro-ameliorative constructions (for instance, see [3]-[7]), as well as when designing and Sh.E. Guseynov is with the Liepaja University of Latvia and the Institute mail: sh.e.guseinov@inbox.lv; Home-Page: http://www.guseynov.lv) Jekaterina V. Aleksejeva is with the Institute of Mathematical Sciences and Janis S. Rimshans is with the Institute of Mathematical Sciences and constructing them; in the fight against the problem of pollution and salification of agricultural areas by ground waters (for instance, see [8], and in other dynamic processes, described by 2D elliptic equations. Solution of such problems requires elaboration of filtering process theory in those models of porous medium, which are most adequate to the natural conditions. II. FORMULATION OF THE DIRECT LINEAR FILTERING PROBLEM IN THE ANISOTROPIC POROUS ENVIRONMENT Suppose that our studied inhomogeneous porous domain is an anisotropic structure having a periodic volume (not compulsory with a constant period), and the main element of the structure is a rectangular prism. The coefficient of permeability written as a product k (x, y, z)= K (α,β,γ )= K {1} (α) K {2} (β) K {3} (γ ) , (1) where α = α (x, y, z) ,β = β (x, y, z) ,γ = γ (x, y, z) are auxiliary functions of arguments, which, firstly, define the geometry of the periodic structure of the porous environment, and periods by α, β and γ are dimensions of periodic structure elements (rectangular prisms), which form a porous region; and secondly, satisfies these conditions: 1. ∇α, ∇β = ∇α, ∇γ  = ∇β, ∇γ ≡ 0, (2) where ·, · denotes a scalar derivative; 2. max  α(x,y,z)+Tper. (α(x,y,z))  α(x,y,z)    ∂ ∂α(x,y,z)  x (α,β,γ )  i 1 +y (α,β,γ )  i 2 + z (α,β,γ )  i 3    dα, β(x,y,z)+Tper. (β(x,y,z))  β(x,y,z)    ∂ ∂β(x,y,z)  x (α,β,γ )  i 1 +y (α,β,γ )  i 2 + z (α,β,γ )  i 3    dβ, γ(x,y,z)+Tper. (γ(x,y,z))  γ(x,y,z)    ∂ ∂γ(x,y,z)  x (α,β,γ )  i 1 +y (α,β,γ )  i 2 + z (α,β,γ )  i 3    dγ   L. (3) 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 Information Technologies, Liepaja, Latvia (e-mail: janis.rimsans@liepu.lv) Information Technologies, Liepaja, Latvia (e-mail: j.v.aleksejeva@gmail.com) of Mathematical Sciences and Information Technologies, Liepaja, Latvia (e- Keywords—Porous media, filtering, permeability, elliptic PDE. World Academy of Science, Engineering and Technology International Journal of Mathematical and Computational Sciences Vol:7, No:10, 2013 1510 International Scholarly and Scientific Research & Innovation 7(10) 2013 scholar.waset.org/1307-6892/17031 International Science Index, Mathematical and Computational Sciences Vol:7, No:10, 2013 waset.org/Publication/17031