Geodesic
Investigation of the transition to instability of the water boiling front during injection into a geothermal reservoir
Teoretičeskaâ i matematičeskaâ fizika, Tome 211 (2022) no. 2, pp. 347-357 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We discuss the stability problem for a boiling front moving at a constant speed in a porous permeable geothermal reservoir. We study the dispersion equation obtained by the method of normal modes. The decay of unstable small perturbations corresponding to large values of the dimensionless wave number is shown analytically. We construct neutral stability curves in the plane of the main parameters. The evolution of the neutral curves with changing parameters shows that an increase in the permeability and initial temperature, as well as a decrease in porosity and initial pressure, leads to an expansion of the instability region. We investigate the dependence of the critical dimensionless wave number on the permeability of a porous medium at which the transition to instability occurs. The obtained critical values give an estimate of the characteristic size of the most unstable perturbation, which varies depending on the process parameters. This size ranges from half a meter to several meters at characteristic values of the geothermal reservoir parameters. Possible types of transitions to instability of the interfaces in filtration problems are discussed.
Mots-clés : filtration, dispersion equation
Keywords: Darcy's law, boiling surface, stability, curves of neutral stability. 
@article{TMF_2022_211_2_a12,
     author = {G. G. Tsypkin},
     title = {Investigation of the~transition to instability of the~water boiling front during injection into a~geothermal reservoir},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {347--357},
     year = {2022},
     volume = {211},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2022_211_2_a12/}
}
TY  - JOUR
AU  - G. G. Tsypkin
TI  - Investigation of the transition to instability of the water boiling front during injection into a geothermal reservoir
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2022
SP  - 347
EP  - 357
VL  - 211
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/TMF_2022_211_2_a12/
LA  - ru
ID  - TMF_2022_211_2_a12
ER  - 
%0 Journal Article
%A G. G. Tsypkin
%T Investigation of the transition to instability of the water boiling front during injection into a geothermal reservoir
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2022
%P 347-357
%V 211
%N 2
%U http://geodesic.mathdoc.fr/item/TMF_2022_211_2_a12/
%G ru
%F TMF_2022_211_2_a12
G. G. Tsypkin. Investigation of the transition to instability of the water boiling front during injection into a geothermal reservoir. Teoretičeskaâ i matematičeskaâ fizika, Tome 211 (2022) no. 2, pp. 347-357. http://geodesic.mathdoc.fr/item/TMF_2022_211_2_a12/

[1] P. G. Saffman, G. I. Taylor, “The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid”, Proc. Roy. Soc. London Ser. A, 245:1242 (1958), 312–329 | DOI | MR

[2] G. H. de Rooij, “Modeling fingered flow of water in soils owing to wetting front instability: a review”, J. Hydrology, 231–232:1–4 (2000), 277–294 | DOI

[3] S. M. S. Shokri-Kuehni, B. Raaijmakers, T. Kurz, D. Or, R. Helmig, N. Shokri, “Water table depth and soil salinization: from pore-scale processes to field-scale responses”, Water Resour. Res., 56:2 (2020), e2019WR026707, 13 pp. | DOI

[4] G. G. Tsypkin, “Ob ustoichivosti poverkhnostei ispareniya i kondensatsii v poristoi srede”, Izv. RAN. MZhG, 2017, no. 6, 70–78 | DOI | DOI | MR

[5] A. Riaz, M. Hesse, H. A. Tchelepi, F. M. Orr, “Onset of convection in a gravitationally unstable diffusive boundary layer in porous media”, J. Fluid Mech., 548 (2006), 87–111 | DOI | MR

[6] S. H. Hejazi, J. Azaiez, “Stability of reactive interfaces in saturated porous media under gravity in the presence of transverse flows”, J. Fluid Mech., 695 (2012), 439–466 | DOI | MR

[7] G. G. Tsypkin, “Neustoichivost legkoi zhidkosti nad tyazheloi pri dvizhenii poverkhnosti razdela v poristoi srede”, Izv. RAN. MZhG, 2020, no. 2, 70–76 | DOI | DOI

[8] S. D. Fitzgerald, A. W. Woods, “The instability of a vaporization front in hot porous rock”, Nature, 367:6462 (1994), 450–453 | DOI

[9] G. Schubert, J. M. Straus, “Gravitational stability of water over steam in vapor-dominated geothermal systems”, J. Geophys. Res., 85:B11 (1980), 6505–6512 | DOI

[10] G. G. Tsypkin, A. T. Il'ichev, “Gravitational stability of the interface in water over steam geothermal reservoirs”, Transp. Porous Media, 55:2 (2004), 183–199 | DOI | MR

[11] A. T. Ilichev, G. G. Tsypkin, “Neustoichivosti odnorodnykh filtratsionnykh techenii s fazovym perekhodom”, ZhETF, 134:4(10) (2008), 815–830 | DOI

[12] Z. H. Khan, D. Pritchard, “Liquid–vapour fronts in porous media: Multiplicity and stability of front positions”, Internat. J. Heat Mass Transfer, 61 (2013), 1–17 | DOI

[13] Z. H. Khan, “Transition to instability of liquid–vapour front in a porous medium cooled from above”, Internat. J. Heat Mass Transfer, 70 (2014), 610–620 | DOI

[14] A. T. Il'ichev, G. G. Tsypkin, “Transition to instability of the interface in geothermal systems”, Eur. J. Mech. B Fluids, 24:4 (2005), 491–501 | DOI | MR

[15] P. S. Ramesh, K. E. Torrance, “Stability of boiling in porous media”, Internat. J. Heat Mass Transfer, 33:9 (1990), 1895–1908 | DOI

[16] P. Amili, Y. C. Yortsos, “Stability of heat pipes in vapor-dominated systems”, Internat. J. Heat Mass Transfer, 47:6–7 (2004), 1233–1246 | DOI

[17] V. A. Shargatov, A. T. Il'ichev, G. G. Tsypkin, “Dynamics and stability of moving fronts of water evaporation in a porous medium”, Internat. J. Heat Mass Transf., 83 (2015), 552–561 | DOI

[18] V. A. Shargatov, G. G. Tsypkin, “Influence of capillary pressure gradient on connectivity of flow through a porous medium”, Internat. J. Heat Mass Transf., 127, Part C (2018), 1053–1063 | DOI

[19] A. T. Il'ichev, V. A. Shargatov, “Dynamics of water evaporation fronts”, Comput. Math. Math. Phys., 53:9 (2013), 1350–1370 | DOI | MR

[20] V. A. Shargatov, S. V. Gorkunov, A. T. Il'ichev, “Dynamics of front-like water evaporation phase transition interfaces”, Commun. Nonlinear Sci. Numer. Simul., 67 (2019), 223–236 | DOI | MR

[21] G. G. Tsypkin, “Ob ustoichivosti fronta inzhektsii v vysokotemperaturnye porody”, Izv. RAN. MZhG, 2021, no. 6, 66–73 | DOI | DOI