# Method of numerical analysis of fluid flows in porous media under a cas-cade of hydraulic structures

## DOI:

https://doi.org/10.26577/JMMCS-2019-1-616## Keywords:

fluid flow in porous media, stream function, superposition principle, R-function method, Ritz method## Abstract

Fluid flows in porous media are widespread in nature and they often come to the need for consideration in the course of economic activity. The paper deals with the problem of the theory of stationary fluid flows in porous media in the ground under the construction of hydraulic structures under the assumption that Darcy’s law is fulfilled. The mathematical model of this problem is the elliptic equations for the stream function with boundary conditions of the second kind on sections of the boundary of the reservoir and boundary conditions of the first kind on sections of the boundary that are impermeable to liquid. At the same time, the formulation of the problem includes the unknown values of the total fluid flow rates under each of the hydraulic structures of the cascade, for the determination of which additional integral relations are formulated. For the numerical analysis of the problem, it is proposed to use the structural-variational method (the R-functions method), which will make it possible to fully take into account in the computational algorithm all the geometric and analytical information that is included in the formulation of the problem. In accordance with the principle of superposition from the original problem, a transition was made to a set of boundary-value problems with known boundary conditions. For each of these problems, according to the method of R-functions, the structures of the solution are constructed that accurately take into account all the boundary conditions, and the use of the Ritz variational method for approximation of the uncertain component is justified. After that, of the additional integral relations, the approximate values of the unknown flow rates of the fluid, and hence the approximate solution of the original problem, are found. A computational experiment was carried out for the case of a constant filtration coefficient in an area that has the form of the lower half of a ring with two semicircular burials located symmetrically. The proposed method of numerical analysis has shown its effectiveness in solving a test problem and can be used to solve applied problems. The advantages of the developed numerical method are the possibility of obtaining the solution of the boundary value problem in the form of a single analytical expression and the exact satisfaction of all the boundary conditions.

## References

[2] Podgornij O.R. “Chisel’nij analіz metodom R-funkcіj fіl'tracіjnih techіj u neodnorіdnomu gruntі” [Numerical analysis by the R-functions method of flow in po-rous inhomogeneous soils], Matematichne ta komp’yuterne modelyuvannya. (ser. fіz.-mat. nauki), Vol. 18 (2018) : 147-162.

[3] Khan W. et al. “Exact Solutions of Navier Stokes Equations in Porous Media”, International Journal of Pure and Applied Mathematics, Vol. 96, № 2 (2014) : 235-247.

[4] Daly E., Basser H., Rudman M. “Exact solutions of the Navier-Stokes equa-tions generalized for flow in porous media”, The European Physical Journal Plus, Vol. 133, № 5 (2018) : 173.

[5] Bomba A.Ja., Bulavac’kij V.M., Skopec’kij V.V. Nelіnіjnі matematichnі modelі procesіv geogіdrodinamіki [Nonlinear mathematical models of processes of geo-hydrodynamics], (K.: Nauk. dumka, 2007) : 292.

[6] Brebbia C.A., Connor J.J. Finite element techniques for fluid flow, (Newness-Butterworths, London, 1976) : 310.

[7] Ferziger J.H., Peric M. Computational Methods for Fluid Dynamics, (Berlin: Springer, 2002) : 423 p.

[8] Chung T.J. Computational Fluid Dynamics, (United Kingdom: CUP, 2002) : 1022.

[9] Wessiling P. Principles of computational Fluid Dynamics (Berlin: Springer, 2001) : 644.

[10] Ljashko I.I., Velikoivanenko I.M., Lavrik V.I., Mistec'kij G.E. Metod ma-zhorantnyh oblastej v teorii fil'tracii [The method of majorant domains in filtration theo-ry], (K.: Nauk. dumka, 1974) : 202.

[11] Ljashko N.I., Velikoivanenko N.M. Chislenno-analiticheskoe reshenie kraevyh zadach teorii fil’tracii [Numerical-analytical solution of boundary value prob-lems of filtration theory], (K.: Nauk. dumka, 1973) : 264.

[12] Vabishhevich P.N. Metod fiktivnyh oblastej v matematicheskoj fizike [The method of fictitious areas in mathematical physics], (M.: Izd-vo MGU, 1991) : 156.

[13] Zhang D. Stochastic methods for flow in porous media: coping with uncer-tainties, (San Diego: Academic Press, 2001) : 368.

[14] Jenny P., Lee S.H., Tchelepi H.A. “Adaptive multiscale finite-volume meth-od for multiphase flow and transport in porous media”, Multiscale Modeling & Simula-tion, Vol. 3, № 1 (2005) : 50 – 64.

[15] Wang J.G., Leung C.F., Chow Y.K. “Numerical solutions for flow in porous media”, International Journal for numerical and analytical methods in geomechanics, Vol. 27, № 7 (2003) : P. 565-583.

[16] Bastian P. “Higher order discontinuous Galerkin methods for flow and transport in porous media”, Challenges in Scientific Computing-CISC 2002 (Springer, Berlin, Heidelberg, 2003) : 1 – 22.

[17] Lee H.K.H. et al. “Markov random field models for high-dimensional param-eters in simulations of fluid flow in porous media”, Technometrics, Vol. 44, № 3 (2002). : 230 – 241.

[18] Gray W. G., Miller C. T. “Examination of Darcy's law for flow in porous media with variable porosity”, Environmental science & technology, Vol. 38, № 22 (2004) : 5895 – 5901.

[19] Hoteit H. et al. “Numerical reliability for mixed methods applied to flow problems in porous media”, Computational geosciences, Vol. 6, № 2 (2002) : 161 – 194.

[20] Rvachev V.L., Sheiko T.I. “R-functions in boundary value problems in me-chanics”, Appl. Mech. Rev., Vol. 48, № 4 (1995) : 151 – 188.

[21] Kravchenko V.F., Rvachev V.L. Algebra logiki, atomarnye funkcii i vejvlety v fizicheskih prilozhenijah [Logic algebra, atomic functions and wavelets in physical applications], (M.: Fizmatlit, 2006) : 416.

[22] Sidorov M.V., Storozhenko A.V. “Matematicheskoe komp’juternoe modelirovanie nekotoryh fil’tracionnyh techenij” [Mathematical and computer modeling of some fluid flows in porous media], Radiojelektronika i informatika, № 4 (2004) : 58-61.

[23] Blishun A.P., Sidorov M.V., Jalovega I.G. “Matematicheskoe modelirovanie i chislennyj analiz fil’tracionnyh techenij pod gidrotehnicheskimi sooruzhenijami s pomoshh’ju” [Mathematical modeling and numerical analysis of fluid flows in porous media under hydraulic structures using the R-function method], Radiojelektronika i informatika, № 2 (2010) : 40 – 46.

[24] Blishun A.P., Sidorov M.V., Jalovega I.G. “Primenenie metoda R-funkcij k chislennomu analizu fil’tracionnyh techenij pod gidrotehnicheskimi sooruzhenijami” [Application of the method of R-functions to the numerical analysis of fluid flows in po-rous media under hydraulic structures], Vіsnik Zaporіz’kogo nacіonal’nogo unіversitetu (ser. fіz.-mat. nauki), № 1 (2012) : 50 – 56.

[25] Aleksidze M.A. Fundamental'nye funkcii v priblizhennyh resheniyah gra-nichnyh zadach [Fundamental functions in approximate solutions of boundary value problems], (M.: Nauka, 1991) : 352.

[26] Rvachev V.L. Teorija R-funkcij i nekotorye ejo prilozhenija [The R-functions theory and some of its applications] (K.: Nauk. dumka, 1982) : 552.

[27] Maksimenko-Shejko K.V. R-funkcii v matematicheskom modelirovanii geometricheskih ob"ektov i fizicheskih polej [R-functions in mathematical modeling of geometric objects and physical fields], (Harkіv, ІPMash NAN Ukrayiny, 2009) : 306.

[28] Podgornij O.R. “Matematichnі modelі fіl’tratsіjnikh techіj ta zastosuvannya metodu R-funktsіj dlya ikh chisel’nogo analіzu” [Mathematical modeling of flow in porous media and application of R-function’s method for their numerical analysis], Radіoelektronіka ta іnformatika, № 1 (2018) : 40-47.

[29] Mikhlin S.G. Variatsionnye metody v matematicheskoj fizike [Variational methods in mathematical physics] (M.: Nauka, 1970) : 511.

[30] Rektorys K. Variational methods in mathematics, science and engineering (Springer Science & Business Media, 2012) : 571.

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## How to Cite

*Journal of Mathematics, Mechanics and Computer Science*,

*101*(1), 115–131. https://doi.org/10.26577/JMMCS-2019-1-616