Decay of the initial oil concentration discontinuity in the Buckley--Leverett model
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DOI:
https://doi.org/10.32523/3007-0155/bulmathenu.2024/2.1Abstract
We consider a free boundary problem for a one-dimensional system of Buckley-Leverett equations, describing the displacement of oil by a suspension. For this problem we formulated conditions for the strong decay of the discontinuity of the initial oil concentration. We will prove that the phenomenological Buckley-Leverett model does not adequately describe the physical process under consideration. To do this, we will study the problem of the decay of a discontinuity in the initial concentration of oil, when at rest in one half of the domain there is oil, and in the other half of the domain there is a suspension, and these domains are separated by an impenetrable partition. At the initial moment of time, the partition is removed and a non-negative suspension velocity is maintained at the injection wells. An accurate analysis of the unique solution to the Buckley-Leverett model shows that at the initial moment of time, oil begins to displace the suspension, resulting in the formation of a zone of mixing of oil and suspension. If the velocity of the suspension at the injection wells is high enough, then at some point in time the natural option of displacing oil by the suspension begins to be realized.}
\keywords{Free boundary problems, transport equations, displacement of oil by suspension, strong discontinuity conditions.
References
Friedman A., Kinderlehrer D. A one phase Stefan problem, Indiana University Mathematics Journal. 1975. Vol. 24. No 11. P. 1005-1035. DOI: https://doi.org/10.1512/iumj.1975.24.24086
Meirmanov A. On the classical solution of the multidimensional Stefan problem for quasilenear parabolic equations, Mathematics of the USSR-Sbornik. 1981. Vol. 40. Issue 2. P. 157–178. DOI: https://doi.org/10.1070/SM1981v040n02ABEH001795
Meirmanov A. The Stefan problem. De Gruyter, 1992. 244 p. DOI: https://doi.org/10.1515/9783110846720
Meirmanov A., Galtsev O., Zimin R. Free boundaries in Rock Mechanics. De Gruyter, 2017. 220 p. DOI: https://doi.org/10.1515/9783110546163
Meirmanov A. Mathematical models for poroelastic flows. Atlantis Press, 2013. 417 p. DOI: https://doi.org/10.2991/978-94-6239-015-7
Solonnikov V.A. Solvability of a problem on the evolution of a viscous incompressible fluid, bounded by a free surface, on a finite time interval, Algebra i Analiz. 1991. Vol. 3. No 1. P. 222-257; St. Petersburg Math. J. 1992. Vol. 3. No 1. P. 189-220.
Solonnikov V. A. Lectures on Evolution Free Boundary Problems: Classical Solutions [in book: Mathematical Aspects of Evolving Interfaces]. 2003. Vol. 1812. P. 123-175. DOI: https://doi.org/10.1007/978-3-540-39189-0_4
Friedman A. A free boundary problem associated with multiscale tumor models, Math. Mod. Nat. Phenom. 2009. Vol. 4. No 3. P. 134-155. DOI: https://doi.org/10.1051/mmnp/20094306
Ovsyannikov L.V., Makarenko N.I., Nalimov V.I. and others Nelinejnye problemy teorii poverhnostnyh i vnutrennih voln [Nonlinear problems in the theory of surface waves, Nonlinear problems in the theory of surface waves]. Novosibirsk, Nauka, Sibirskoe otdelenie, 1985. 318 p. [in Russian]
Monakhov V.N. Boundary value problems with free boundaries for elliptic systems of equations. Providence, Rhode Island: American Mathematical Society, 1983. Vol. 57. 522 p. DOI: https://doi.org/10.1090/mmono/057
Lavrentev M. A., Shabat B. V. Problemy gidrodinamiki i ih matematicheskie modeli [Hydrodynamics problems and their mathematical models]. Moscow, Izdatel'stvo Nauka, 1973. 416 p. [in Russian]
Buckley S.E., and Leverett M.C. Mechanism of fluid displacements in sands, Transactions of the AIME. 1942. Vol. 146. P. 107-116. DOI: https://doi.org/10.2118/942107-G
Antontsev S.N., Kazhikhov A.V., Monakhov V.N. Boundary value problems in mechanics of nonhomogeneous fluids [in book series: Studies in Mathematics and its Applications]. Amsterdam: North-Holland Publishing Co. 1990. Vol. 22. P. ii-vii, 1-309.
Ovsyannikov L. V. Vvedenie v mehaniku sploshnyh sred [Introduction to continuum mechanics]. Novosibirsk , Novosibirsk State University, parts I, II. 1976. [in Russian]
Barenblatt G.I., Entov V.M., Ryzhik V.M. Teorija nestacionarnoj fil'tracii zhidkosti i gaza [Theory of non-stationary filtration of oil and gas]. Moscow, Nedra, 1972. 288 p. [in Russian].
Rozneberg M.D. Mnogofaznaja mnogokomponentnaja fil'tracija pri dobyche nefti i gaza [Multiphase and multicomponent filtration during oil and gas production]. Moscow, Nedra, 1976. 335 p. [in Russian].
Alishaev M.G., Rosenberg M.D., Teslyuk E.V. Neizotermicheskaja fil'tracija pri razrabotke neftjanyh mestorozhdenij [Non-isothermal filtration in the development of oil fields]. Moscow, Nedra, 1985. 271 [in Russian].
Burridge R., Keller J. B., Poroelasticity equations derived from microstructure, J. Acoust. Soc. Am. 1981. Vol. 70. Is. 4. P. 1140-1146. DOI: https://doi.org/10.1121/1.386945
Sanchez-Palencia E. Non-homogeneous media and vibration theory: Lecture Notes in Physics 127. Berlin–New York: Springer-Verlag, 1980.
Ladyzhenskaja O.A., Solonnikov V.A., Ural'seva N.N. Linear and Quasi-linear equations of parabolic type. Providence R.I.: American Mathematical Society, 1968. Vol. 23. 648 p. DOI: https://doi.org/10.1090/mmono/023
Lions J.L. Quelques metodes de resolution des problemes aux limites non lineaire. Paris: Dunon Gauthier-Villars, 1969.
Vol'pert A.I., Hudjaev, S.I. Analysis in classes of discontinuous functions and equations of mathematical physics. Mechanics: Analysis 8, Springer, 1985. 696 p.
Natanson I. Theory of function of real variable. Courier Dover Publications, 2014. 544 p.
Kolmogorov A.N., Fomin S.V. Introductory real analysis. New York: Dover Publications, INC, 1975. 403 p.
