Abstract
Stability of vertical flows in geothermal systems is investigated in the case when the domain occupied by water (heavy fluid) is located over the domain occupied by vapor. It is found that under the transition to an unstable regime in a neighborhood of the existing solution, a pair of new solutions appears as a result of the turning point bifurcation. We consider the dynamics of a narrow band of weakly unstable and weakly nonlinear perturbations of the plane surface of the water-to-vapor phase transition. It is shown that such perturbations obey the generalized Ginzburg-Landau-Kolmogorov-Petrovsky-Piscounov equation.
Similar content being viewed by others
References
D. E. White, L. J. P. Muffler, and A. H. Truesdell, “Vapor-dominated hydrothermal systems compared with hot-water systems,” Econ. Geol. 66, 75–97 (1971).
M. A. Grant, “Geothermal reservoir modeling,” Geothermics 12(4), 251–263 (1983).
S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability (Clarendon Press, Oxford, 1961).
G. Schubert and J. M. Straus, “Gravitational stability of water over steam in vapor-dominated geothermal systems,” J. Geophys. Res. 85(B11), 6505–6512 (1980).
P. G. Drazin, Introduction to Hydrodynamic Stability (Cambridge Univ. Press, Cambridge, 2002).
G. G. Tsypkin and A. T. Il’ichev, “Stability of a steady front for the water-vapor phase transition in geothermal reservoirs,” Dokl. Akad. Nauk 378(2), 197–200 (2001) [Dokl. Phys. 46 (5), 359–362 (2001)].
G. Tsypkin and A. Il’ichev, “Gravitational stability of the interface in water over steam geothermal reservoirs,” Transp. Porous Media 55, 183–199 (2004).
A. T. Il’ichev and G. G. Tsypkin, “Transition to instability of the interface in geothermal systems,” Eur. J. Mech. B: Fluids 24, 491–501 (2005).
V. E. Odintsova, “Transition to instability of a phase interface in a porous medium in the isothermal approximation,” Izv. Ross. Akad. Nauk, Mekh. Zhidk. Gaza, No. 1, 123–133 (2009) [Fluid Dyn. 44, 103–111 (2009)].
A. T. Il’ichev and G. G. Tsypkin, “Weakly nonlinear theory for the instability of long-wave perturbations,” Dokl. Akad. Nauk 416(2), 192–194 (2007) [Dokl. Phys. 52 (9), 499–501 (2007)].
A. T. Il’ichev and G. G. Tsypkin, “Catastrophic transition to instability of evaporation front in a porous medium,” Eur. J. Mech. B: Fluids 27, 665–677 (2008).
A. T. Il’ichev and G. G. Tsypkin, “Instabilities of uniform filtration flows with phase transition,” Zh. Eksp. Teor. Fiz. 134(4), 815–830 (2008) [J. Exp. Theor. Phys. 107, 699–711 (2008)].
A. Kolmogoroff, I. Pretrovsky, and N. Piscounoff, “Étude de l’équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique,” Bull. Univ. État Moscou, Math. Méc. 1(6), 1–25 (1937).
Author information
Authors and Affiliations
Additional information
Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2013, Vol. 281, pp. 188–198.
Rights and permissions
About this article
Cite this article
Il’ichev, A.T., Tsypkin, G.G. Classification of the types of instability of vertical flows in geothermal systems. Proc. Steklov Inst. Math. 281, 179–188 (2013). https://doi.org/10.1134/S0081543813040159
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0081543813040159