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Transition to instability of the interface in geothermal systems. (English) Zbl 1065.76091
Summary: High-temperature geothermal reservoir in porous media is under consideration, consisting of two high-permeability layers, which are separated by a low-permeability stratum. The thermodynamic conditions are assumed to imply that the upper and lower high-permeability layers are filled in by water and by vapour, respectively. In these circumstances the low-permeability stratum possesses the phase transition interface, separating domains occupied by water and vapour. The stable stationary regimes of vertical phase flow between water and vapour layers in the low-permeability stratum may exist. Stability of such regimes, where the heavier fluid is located over the lighter one, is supported by a heat transfer caused by temperature gradient in Earth’s interior. We give the classification of possible types of transition to instability of vertical flows in such a system under the condition of smallness of advective heat transfer in comparison with the conductive one. It is found that, in the non-degenerate case, there exist three different scenarios of the onset of instability of stationary vertical phase transition flows. Two of them are accompanied by the bifurcations of the destabilizing vertical flow, leading to appearance of horizontally non-homogeneous regimes with non-constant shape of the interface. The bifurcations correspond to the simple resonance and to the 1:1-resonance, which typically arise in reversible systems.

76E20 Stability and instability of geophysical and astrophysical flows
76S05 Flows in porous media; filtration; seepage
86A05 Hydrology, hydrography, oceanography
76T30 Three or more component flows
80A20 Heat and mass transfer, heat flow (MSC2010)
Full Text: DOI
[1] White, D.E.; Muffler, L.J.P.; Truesdell, A.H., Vapor-dominated hydrothermal systems compared with hot water systems, Econ. geol., 66, 75-97, (1971)
[2] Grant, M.A., Geothermal reservoir modeling, Geothermics, 12, 251-263, (1983)
[3] Schubert, G.; Straus, J.M., Gravitational stability of water over steam in vapour-dominated geothermal system, J. geoph. res., 85, 6505-6512, (1980)
[4] Tsypkin, G.; Il’ichev, A., Gravitational stability of the interface in water over steam geothermal reservoirs, Transport in porous media, 55, 183-199, (2004)
[5] O’Sullivan, M.J., Geothermal reservoir simulation, Int. J. energy res., 9, 319-332, (1985)
[6] Fitzgerald, S.D.; Woods, A.W., The instability of a vapourization front in hot porous rock, Nature, 367, 450-453, (1994)
[7] Tsypkin, G.G., On water-steam phase transition front in geothermal reservoirs, (), 483-490
[8] Iooss, G.; Adelmeyer, M., Topics in bifurcation theory and applications, (1999), World Scientific Singapore · Zbl 0968.34027
[9] Chandrasekhar, S., Hydrodynamic and hydromagnetic stability, (1967), Oxford University Press New York · Zbl 0142.44103
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