Transport in Porous Media

, Volume 94, Issue 1, pp 19–46 | Cite as

On the Initiation of a Hydrothermal Eruption Using the Shock-Tube Model



In this work, the authors introduce the shock-tube model for a hydrothermal eruption in a geothermal reservoir. The governing equations, based on the multiphase Euler equations and a Darcy-type law, are solved using a three-phase weighted sub-system numerical solver. Results are then presented which show the importance of the geometry of the geothermal reservoir in predicting the initiation of a hydrothermal eruption. In particular, the porosity, permeability, and cohesion of the reservoir are shown to significantly affect the pressure difference required to initiate an eruption. Finally, the authors show the importance of the initial liquid water/water vapour volume fractions in determining the size of an eruption, and further show boiling to be of major importance.


Hydrothermal eruption Porous medium Boiling Euler equations Shock-tube 

List of Symbols


Density (kg/m3)


Fluid velocity (m/s)


Pressure (Pa)


Entropy (J/K)


Internal energy (J)


Temperature (K)


Dynamic viscosity (Pa s)


Specific gas constant (J/kg/K)


Interface velocity (m/s)


Interface pressure (Pa)

\({\phi_{\rm v,l}}\)

Volume fraction of phase of water


Volume fraction of fluid


Porosity of the porous medium


Permeability of the porous medium


Cohesion of porous medium (Pa/m)


Ergun coefficient


Specific volume (m3/kg)


Time (s)


Vertical coordinate (m)


Particle diameter (m)


Acceleration due to gravity (m/s2)


Interaction weight


Border length


Wave speed (m/s)


Vector of conserved variables


Vector of fluxes





Vapour phase


Liquid phase








Pertaining to time t = 0


Flashing front


Eruption jet








Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Abgrall R., Perrier V.: Asymptotic expansion of a multiscale numerical scheme for compressible multiphase flows. Multiscale. Model. Simul. 5, 84–115 (2006)CrossRefGoogle Scholar
  2. Abgrall R., Saurel R.: Discrete equations for physical and numerical compressible multiphase mixtures. J. Comp. Phys. 186, 361–396 (2003)CrossRefGoogle Scholar
  3. Alidibirov M., Dingwell D.B.: Three fragmentation mechanisms for highly viscous magma under rapid decompression. J. Volcanol. Geotherm. Res. 100, 413–421 (2000)CrossRefGoogle Scholar
  4. Andrianov N., Saurel R., Warnecke G.: A simple method for compressible multiphase mixtures and interfaces. Int. J. Numer. Methods Fluids 41, 109–131 (2003)CrossRefGoogle Scholar
  5. Barker, H. R.: Natural hazards 2006. Technical Report, NIWA and GNS (2006)Google Scholar
  6. Bdzil J.B., Menikoff R., Son S.F., Kapila A.K., Stewart D.S.: Two-phase modeling of deflagration-to-detonation transition in granular materials: a critical examination of modeling issues. Phys. Fluids 11(2), 378–402 (1999)CrossRefGoogle Scholar
  7. Bercich, B.J.: Mathematical modelling of natural hydrothermal eruptions. In: Report, Department of Engineering Science, School of Engineering, University of Auckland, Auckland, Sept 1992Google Scholar
  8. Brennen C.E.: Fundamentals of Multiphase Flow. Cambridge University Press, Cambridge (2005)Google Scholar
  9. Browne P.R.L., Lawless J.V.: Characteristics of hydrothermal eruptions, with examples from New Zealand and elsewhere. Earth-Sci. Rev. 52, 299–331 (2001)CrossRefGoogle Scholar
  10. Castro, C.E., Toro, E.F.: Godunov schemes for compressible multiphase flows. In: European Conference on Computational Fluid Dynamics, Egmond aan Zee (2006)Google Scholar
  11. Chang C.H., Liou M.S.: A robust and accurate approach to computing compressible multiphase flow: stratified flow model and AUSM+ -up scheme. J. Comp. Phys. 225, 840–873 (2007)CrossRefGoogle Scholar
  12. Chapman C.J.: High Speed Flow. Cambridge University Press, Cambridge (2000)Google Scholar
  13. Christiansen, R.L.: Preliminary assessment of volcanic and hydrothermal hazards in yellowstone national park and vicinity. Technical Report, USGS (2007)Google Scholar
  14. Drew D.A.: Mathematical modeling of two-phase flow. Annu. Rev. Fluid Mech. 15, 261–291 (1983)CrossRefGoogle Scholar
  15. Fullard, L.A., Lynch, T.A.: An introduction to the shock tube model for hydrothermal eruptions. In: Conference Proceedings: ISTP-21, Kaohsiung, 2–5 Nov 2010Google Scholar
  16. Ghangir F.G., Nowakowski A.F., Nicolleau F.C.G.A., Michelitsch T.M.: The application of HLLC numerical solver to the reduced multiphase model. Proc. World Acad. Sci. Eng. Technol. 30, 302–308 (2008)Google Scholar
  17. Koyaguchi, T., Mitani, N.K.: An analysis of volcanic explosions on the basis of the shock-tube model. In: Technical report, Kyoto University, Kyoto (2004)Google Scholar
  18. Koyaguchi T., Mitani N.K.: A theoretical model for fragmentation of viscous bubbly magmas in shock tubes. J. Geophys. Res. 110, 1–21 (2005)CrossRefGoogle Scholar
  19. Lee W.T., Fowler A.C., Scheu B., McGuinness M.J.: A theoretical model of the explosive fragmentation of vesicular magma. Proc. R. Soc. A 466, 731–752 (2009)Google Scholar
  20. Liou M.S., Nguyen L., Theofanous T.G., Chang C.H.: How to solve compressible multifluid equations: a simple, robust and accurate method. AIAA J. 46(9), 2345–2356 (2008)CrossRefGoogle Scholar
  21. McKibbin, R.: An attempt at modelling hydrothermal eruptions. In: Proceedings of the 11th New Zealand Geothermal Workshop 1989. University of Auckland, Auckland, pp. 267–273 (1989)Google Scholar
  22. McKibbin, R.: Could non-condensible gases affect hydrothermal eruptions? In: Conference Proceedings: (18th) New Zealand Geothermal Workshop. University of Auckland, Auckland, Nov 1996Google Scholar
  23. Miyoshi T., Kusano K.: A multi-state HLL approximate riemann solver for ideal magnetohydrodynamics. J. Comp. Phys. 208, 315–344 (2005)CrossRefGoogle Scholar
  24. Munkejord S.T., Papin M.: The effect of interfacial pressure in the discrete-equation multiphase model. Comput. Fluids 36, 742–757 (2007)CrossRefGoogle Scholar
  25. Namiki A., Manga M.: Response of a bubble bearing viscoelastic fluid to rapid decompression: implications for explosive volcanic eruptions. Earth Planet. Sci. Lett. 236, 269–284 (2005)CrossRefGoogle Scholar
  26. Powers J.M.: Two-phase viscous modeling of compaction of granular materials. Phys. Fluids 16(8), 2975–2990 (2004)CrossRefGoogle Scholar
  27. Qiang L., Jian-hu F., Ti-min C., Chun-bo H.: Difference scheme for two-phase flow. Appl. Math. Mech. 25, 536–545 (2004)CrossRefGoogle Scholar
  28. Rogers, G.F.C., Mayhew, Y.R.: Thermodynamic and transport properties of fluids. In: Technical report, Basil Blackwell Publisher, Oxford (1983)Google Scholar
  29. Saurel R., Abgrall R.: A multiphase Godunov method for compressible multifluid and multiphase flows. J. Comput. Phys. 150, 425–467 (1999)CrossRefGoogle Scholar
  30. Saurel R., Lemetayer O.: A multiphase model for compressible flows with interfaces, shocks, detonation waves and cavitation. J. Fluid Mech. 431, 239–271 (2001)CrossRefGoogle Scholar
  31. Smith, T.A., McKibbin, R.: An investigation of boiling processes in hydrothermal eruptions. In: Proceedings of the 21st New Zealand Geothermal Workshop 1999. University of Auckland, Auckland, pp. 699–704 (1999)Google Scholar
  32. Smith, T.A.: Mathematical modelling of underground flow processes in hydrothermal eruptions. PhD thesis, Massey University, Palmerston North (2000)Google Scholar
  33. Toro E.F.: Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction. Springer-Verlag Telos, New York (1997)Google Scholar
  34. Toro E.F., Spruce M., Speares W.: Restoration of the contact surface in the HLL-riemann solver. Shock Waves 4, 25–34 (1994)CrossRefGoogle Scholar
  35. Weir G.J., Young R.M., McGavin P.N.: A simple model for geyser flat, whakarewarewa. Geothermics 21, 281–304 (1992)CrossRefGoogle Scholar
  36. Wohletz, K.H., Valentine, G.A.: Computer simulations of explosive volcanic eruptions. In: Technical report, Earth and Environmental Sciences Division, Los Alamos National Laboratory, Los Alamos (1990)Google Scholar
  37. Wohletz K.H., McGetchin T.R., Sandford M.T. II, Jones E.M.: Hydrodynamic aspects of caldera-forming eruptions: numerical models. J. Geophys. Res. 89, 8269–8285 (1984)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.Institute of Fundamental SciencesMassey UniversityPalmerston NorthNew Zealand

Personalised recommendations