Some Free Boundary Problems Arising in Rock Mechanics

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Abstract

В этой статье мы рассматриваем несколько физических процессов в механике горных пород, которые описываются задачами со свободной границей. Некоторые из них известны (задачи Муската), другие совершенно новые (подземное выщелачивание и динамика трещин в подземных горных породах).

About the authors

A M Meirmanov

Yachay Tech University; Belgorod State University

Email: anvarbek@list.ru
Yachay, Ecuador; Belgorod, Russia

O V Galtsev

Belgorod State University

Email: galtsev_o@bsu.edu.ru
Belgorod, Russia

O A Galtseva

Belgorod State University

Email: galtseva@bsu.edu.ru
Belgorod, Russia

References

  1. Ладыженская О. А. Математические вопросы динамики вязкой несжимаемой жидкости. - М.: Наука, 1970.
  2. Ладыженская О. А., Солонников В. А., Уральцева Н. Н. Линейные и квазилинейные уравнения параболического типа. - М.: Наука, 1967.
  3. Рубинштейн Л. И. Проблема Стефана. - Рига: Звайгзне, 1967.
  4. Жиков В. В., Козлов С. М., Олейник О. А. Усреднение дифференциальных операторов. - М.: Физматлит, 1993.
  5. Anderson T. L. Fracture mechanics. Fundamentals and applications. - Boca Raton: CRC Press, 1995.
  6. Antontsev S. N., Kazhikhov A. V., Monakhov V. N. Boundary Value Problems in Mechanics of Nonhomogeneous Fluids. - Amsterdam-NewYork-Oxford-Tokyo: North-Holland, 1990.
  7. Antontsev S., Meirmanov A., Yurinsky V. A free boundary problem for Stokes equations: classical solutions// Interfaces Free Bound. - 2000. - 2. - C. 413-424.
  8. Bo¨hm M. On a nonhomogeneous Bingham fluid// J. Differ. Equ. - 1985. - 60. - C. 259-284.
  9. Brady P. V., House W. A. Surface-controlled dissolution and growth of minerals// В сб.: «Physics and chemistry of mineral surfaces». - Boca Raton: CRC Press, 1996. - C. 225-306.
  10. Burridge R., Keller J. B. Poroelasticity equations derived from microstructure// J. Acoustic Soc. Amer. - 1981. - 70. - C. 1140-1146.
  11. Cohen C. E., Ding D., Quintard M., Bazin B. From pore scale to wellbore scale: Impact of geometry on wormhole growth in carbonate acidization// Chem. Eng. Sci. - 2008. - 63. - C. 3088-3099.
  12. Colding T. N., Minicozzi II W. P., Pedersen E. K. Mean Curvature Flow// Bull. Am. Math. Soc. (N.S.). - 2015. - 52. - C. 297-333.
  13. Ferna´ ndez-Cara E., Guille´n F., Ortega R. R. Some theoretical results for visco-plastic and dilatant fluids with variable density// Nonlinear Methods Appl. - 1997. - 28. - C. 1079-1100.
  14. Freidman A. Variational principles and free-boundary problems. - New York: John Wiley & Sons Inc., 1982.
  15. Giga Y., Takahashi S. On global weak solutions of the nonstationary two-phase Stokes flow// SIAM J. Math. Anal. - 1994. - 25. - C. 876-893.
  16. Golfier F., Zarcone C., Bazin B., Lenormand R., Lasseux D., Quintard M. On the ability of a Darcyscale model to capture wormhole formation during the dissolution of a porous medium// J. Fluid Mech. - 2002. - 457. - C. 213-254.
  17. Kalia N., Balakotaiah V. Effect of medium heterogeneities on reactive dissolution of carbonates// Chem. Eng. Sci. - 2009. - 64. - C. 376-390.
  18. Kasahara K. Earthquake mechanics. - Cambridge: Cambridge University Press, 1981.
  19. Kenneth W. W., Raymond E. D., Larry M. P., Stanley G. G. Chemistry. - Belmont: Brooks, 2014.
  20. Lukkassen D., Nguetseng G., Wall P. Two-scale convergence// Int. J. Pure Appl. Math. - 2002. - 2.- C. 35-86.
  21. Malvern L. E. Introduction to Mechanics of a Continuum Medium. - Englewood Cliffs: Prentice-Hall Inc., 1969.
  22. Meirmanov A. The Stefan problem. - Berlin-New York: Walter de Gruyter, 1992.
  23. Meirmanov A. The Muskat problem for a viscoelastic filtration// Interfaces Free Bound. - 2011. - 13.- C. 463-484.
  24. Meirmanov A. M. Mathematical models for poroelastic flows. - Paris: Atlantis Press, 2013.
  25. Meirmanov A., Galtsev O. Displacement of oil by water in a single elastic capillary// Boundary Value Problems. - 2017. - 2017. - C. 1-26.
  26. Meirmanov A., Galtsev O. Dynamics of cracks in rocks// Int. J. Evol. Equ. - 2017. - 10. - C. 214-227.
  27. Meirmanov A., Zimin R., Shiyapov K. The Muskat problem at the microscopic level for a single capillary// Bound. Value Probl. - 2015. - 71.- C. 1-8.
  28. Monakhov V. N. Boundary-value problems with free boundaries for elliptic systems of equations. - Providence: AMS, 1983.
  29. Nguetseng G. A general convergence result for a functional related to the theory of homogenization// SIAM J. Math. Anal. - 1989. - 20. - C. 608-623.
  30. Nguetseng G. Asymptotic analysis for a stiff variational problem arising in mechanics// SIAM J. Math. Anal.- 1990.- 21. - C. 1394-1414.
  31. Nouri A., Poupaud F. An existence theorem for the multifluid Navier-Stokes problem// J. Differ. Equ. - 1995. - 13. - C. 463-484.
  32. Panga M. K. R., Ziauddin M., Balakotaiah V. Two-scale continuum model for simulation of wormholes incarbonate acidization// A.I.Ch.E. Journal. - 2005. - 51. - C. 3231-3248.
  33. Sanchez-Palencia E. Non-homogeneous media and vibration theory. - Berlin: Springer, 1980.
  34. Solonnikov V. A., Ladyzhenskaya O. A. On unique solvability of an initial-boundary value problem for viscous nonhomogeneous fluids// J. Soviet Math. - 1978. - 9. - C. 697-749.
  35. Solonnikov V. A., Padula M. On the local solvability of free boundary problem for the Navier-Stokes equations// J. Math. Sci. - 2010. - 170. - doi: 10.1007/s10958-010-0099-3.
  36. Solonnikov V. A., Tani A. Free boundary problem for the Navier-Stokes equations for a compressible fluid with a surface tension// J. Soviet Math. - 1992. - 62. - C. 2814-2818.
  37. Whitham G. B. Linear and nonlinear waves. - New York: Willey, 1999.

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