Skip to main content Accessibility help
  • Print publication year: 2020
  • Online publication date: April 2020

10 - Understanding Unrest and Forecasting Eruptions


One of the main aims of the science of volcanology, and that of volcanotectonics in particular, is to understand volcanic unrest periods. By ‘understanding’ I mean that the signals coming from the volcano during the unrest can be interpreted in terms of plausible physical and chemical processes occurring inside the volcano. By volcanic ‘unrest’ we mean an increase in various physical and chemical signals, suggesting that associated processes within the volcano operate at different rates, intensities, or both. By interpreting the unrest period in terms of correct physical processes, there is a chance of assessing the volcanic hazard, namely the probability that the unrest period results in an eruption. Furthermore, when the understanding of the processes giving rise to the signals is accurate, not only the location of the eruption site but also the likely size (volume) of the eruption can be forecasted.

Agustsdottir, T., Woods, J., Greenfield, T., 2016. Strike-slip faulting during the 2014 Bardarbunga–Holuhraun dike intrusion, central Iceland. Geophysical Research Letters, 43, 14951503, doi:10.1002/2015GL067423.
Allmann, B. P., Shearer, P. M., 2009. Global variations of stress drop for moderate to large earthquakes. Journal of Geophysical Research, 114, doi:10.1029/2008JB005821.
Al Shehri, A., Gudmundsson, A., 2018. Modelling of surface stresses and fracturing during dyke emplacement: application to the 2009 episode at Harrat Lunayyir, Saudi Arabia. Journal of Volcanology and Geothermal Research, 356, 278303.
Anderson, T. L., 2005. Fracture Mechanics: Fundamentals and Applications, 3rd edn. London: Taylor & Francis.
Arioldi, G., Muirhead, J. D., Zanella, E., White, J. D. L., 2012. Emplacement process of Ferrar Dolerite sheets at Allan Hills (South Victoria Land, Antarctica) inferred from magnetic fabric. Geophysical Journal International, 188, 10461060.
Bachmann, C. E., Wiemer, S., Woessner, J., Hainzl, S., 2011. Statistical analysis of the induced Basel 2006 earthquake sequence: introducing a probability-based monitoring approach for enhanced geothermal systems. Geophysical Journal International, 186, 793807.
Barton, C. A., Zoback, M. D., Moos, D., 2010. Fluid flow along potentially active faults in crystalline rock. Geology, 23, 683686.
Bazargan, M., Gudmundsson, A., 2019. Dike-induced stresses and displacements in layered volcanic zones. Journal of Volcanology and Geothermal Research, 384, 189205.
Becerril, L., Galindo, I., Gudmundsson, A., Morales, J. M., 2013. Depth of origin of magma in eruptions. Scientific Reports, 3, 2762, doi:10.1038/srep02762.
Bedford, A., 1985. Hamilton’s Principle in Continuum Mechanics. London: Pitman Publishing.
Bergerat, F., Angelier, J., 1998. Fault systems and paleostresses in the Vestfirdir Peninsula. Relationship with the Tertiary paleo-rifts of Skagi and Snaefells (northwest Iceland). Geodinamica Acta, 11, 105118.
Bonaccorso, A., Aoki, Y., Rivalta, E., 2017. Dike propagation energy balance from deformation modeling and seismic release. Geophysical Research Letters, 44, 54865494.
Bonafede, M., Rivalta, E., 1999a. The tensile dislocation problem in a layered elastic medium. Geophysical Journal International, 136, 341356.
Bonafede, M., Rivalta, E., 1999b. On tensile cracks close to and across the interface between two welded elastic half-spaces. Geophysical Journal International, 138, 410434.
Bostrom, N., Cirkovic, M. M., 2008. Global Catastrophic Risk. Oxford: Oxford University Press.
Boudreau, A., Simon, A., 2007. Crystallization and degassing in the basement sill, McMurdo DryValleys, Antarctica. Journal of Petrology, 48, 13691386.
Bradley, J., 1965. Intrusion of major dolerite sills. Transactions of the Royal Society of New Zealand, 3, 2755.
Cashman, K. V., Sparks, R. S. J., 2013. How volcanoes work: a 25 year perspective. Geological Society of America Bulletin, 125, 664690.
Cashman, K. V., Sparks, R. S. J., Blundy, J. D., 2017. Vertically extensive and unstable magmatic systems: a unified view of igneous processes. Science, 355, 6331, doi:10.1126/science.aag3055.
Cayol, V., Cornet, F. H., 1998. Three-dimensional modelling of the 1983–1984 eruption of Piton de la Fournaise volcano, Reunion Island. Journal of Geophysical Research, 103, 1802518037.
Chaussard, E., Amelung, F., 2014. Regional controls on magma ascent and storage in volcanic arcs. Geochemistry, Geophysics, Geosystems, 15, doi:10.1002/2013GC005216.
Chesner, C. A., Rose, W. I., Deino, A., Drake, R., Westgate, J. A., 1991. Eruptive history of Earth’s largest Quaternary caldera (Toba, Indonesia) clarified. Geology, 19, 200203.
Chevallier, L., Woodford, A., 1999. Morpho-tectonics and mechanism of emplacement of the dolerite rings and sills of the western Karoo, South Africa. South African Journal of Geology, 102, 4354.
Crosweller, H. S., Arora, B., Brown, S. K, Cottrell, , et al., 2012. Global database on large magnitude explosive volcanic eruptions (LaMEVE). Journal of Applied Volcanology, 1, doi:10.1186/2191–5040-1–4.
Davis, P. M., 1983. Surface deformation associated with a dipping hydrofracture. Journal of Geophysical Research, 88, 58265834.
Davis, R. J., Mathias, S. A., Moss, J., Hustoft, S., Newport, L., 2012. Hydraulic fractures: how far can they go? Marine and Petroleum Geology, 37, 16.
Davis, R. J., Foulger, G. R., Mathias, S., 2013. Reply: Davis et al. (2012). Hydraulic fractures: how far will they go? Marine and Petroleum Geology, 43, 519521.
Dym, C. L., Shames, I. H., 2013. Solid Mechanics: A Variational Approach. Berlin: Springer Verlag.
Dzurisin, D., 2006. Volcano Deformation: New Geodetic Monitoring Techniques. Berlin: Springer Verlag.
Fedotov, S. A., 1985. Estimates of heat and pyroclast discharge by volcanic eruptions based upon the eruption cloud and steady plume observations. Journal of Geodynamics, 3, 275302.
Fedotov, S. A., Chirkov, A. M., Gusev, N. A., Kovalev, G. N., Slezin, Yu. B., 1980. The large fissure eruption in the region of Plosky Tolbachik Volcano in Kamchatka, 1975–1976. Bulletin of Volcanology, 43, 4760.
Fialko, Y., Khazan, Y., Simons, M., 2001. Deformation due to a pressurized horizontal circular crack in an elastic half-space, with applications to volcano geodesy. Geophysical Journal International, 146, 181190.
Fisher, K., 2014. Hydraulic fracture growth: real data. Presentation given at GTW-AAPG/STGS Eagle Ford plus Adjacent Plays and Extensions Workshop, San Antonio, Texas, February 24–26.
Fisher, K., Warpinski, N., 2011. Hydraulic fracture-height growth: real data. Society of Petroleum Engineers Annual Technical Conference and Exhibition, SPE 145949.
Fleming, T. H., Heimann, A., Foland, K. A., Elliot, D. H., 1997. 40Ar/39Ar geochronology of Ferrar Dolerite sills from the Transantarctic Mountains, Antarctica: implications for the age and origin of the Ferrar magmatic province. Geological Society of America Bulletin, 109, 533546.
Flewelling, S. A., Tymchak, M. P., Warpinski, N., 2013. Hydraulic fracture height limits and fault interactions in tight oil and gas formations. Geophysical Research Letters, 40, 36023606.
Folch, A., Marti, J., 2004. Geometrical and mechanical constraints on the formation of ring-fault calderas. Earth and Planetary Science Letters, 221, 215255.
Francis, E. H., 1982. Magma and sediment – I: emplacement mechanism of late Carboniferous tholeiite sills in northern Britain. Journal of the Geological Society, 139, 120.
Fung, Y. C., Tong, P., 2001. Classical and Computational Solid Mechanics. Singapore: World Scientific Publishing.
Galindo, I., Gudmundsson, A., 2012. Basaltic feeder dykes in rift zones: geometry, emplacement, and effusion rates. Natural Hazards and Earth System Sciences, 12, 36833700.
Galland, O., Scheibert, J., 2013. Analytical model of surface uplift above axisymmetric flat-lying magma intrusions: implications for sill emplacement and geodesy. Journal of Volcanology and Geothermal Research, 253, 114130.
Gautneb, H., Gudmundsson, A., 1992. Effect of local and regional stress fields on sheet emplacement in West Iceland. Journal of Volcanology and Geothermal Research, 51, 339356.
Gelman, S. E., Gutierrez, F. J., Bachmann, O., 2013. On the longevity of large upper crustal silicic magma reservoirs, Geology, 41, 759762, doi:10.1130/g34241.1.
Geshi, N., Neri, M., 2014. Dynamic feeder dyke systems in basaltic volcanoes: the exceptional example of the 1809 Etna eruption (Italy). Frontiers in Earth Science, 2, doi:10.3389/feart.2014.00013.
Geshi, N., Shimano, T., Chiba, T., Nakada, S., 2002. Caldera collapse during the 2000 eruption of Miyakejima volcano, Japan. Bulletin of Volcanology, 64, 5568.
Geshi, N., Kusumoto, S., Gudmundsson, A., 2010. Geometric difference between non-feeder and feeder dikes. Geology, 38, 195198.
Geshi, N., Kusumoto, S., Gudmundsson, A., 2012. Effects of mechanical layering of host rocks on dike growth and arrest. Journal of Volcanology and Geothermal Research, 223–224, 7482.
Geyer, A., Marti, J., 2008. The new worldwide collapse caldera database (CCDB): a tool for studying and understanding caldera processes. Journal of Volcanology and Geothermal Research, 175, 334354.
Geyer, A., Marti, J., 2014. A short review of our current understanding of the development of ring faults during collapse caldera formation. Frontiers in Earth Science, 2, doi:10.3389/feart.2014.00022.
Goldstein, H., Poole, C. P., Safko, J. L., 2013. Classical Mechanics. New York, NY: Pearson.
Gonnermann, H. M., Manga, M., 2013. Dynamics of magma ascent in the volcanic conduit. In Fagents, S. A., Gregg, T. K. P., Lopes, R. M. C. (eds.), Modeling Volcanic Processes. Cambridge: Cambridge University Press, pp. 5584.
Greenland, L. P., Rose, W. I., Stokes, J. B., 1985. An estimate of gas emissions and magmatic gas content from Kilauea volcano. Geochimica et Cosmochimica Acta, 49, 125129.
Greenland, L. P., Okamura, A. T., Stokes, J. B., 1988. Constraints on the mechanics of the eruption. In Wolfe, E. W (ed.), The Puu Oo Eruption of Kilauea Volcano, Hawaii: Episodes through 20, January 3, 1983 through June 8, 1984. US Geological Survey Professional Paper, 1463. Denver, CO: US Geological Survey, pp. 155164.
Gretener, P. E., 1969. On the mechanics of the intrusion of sills. Canadian Journal of Earth Sciences, 6, 14151419.
Gudmundsson, A., 1983. Form and dimensions of dykes in eastern Iceland. Tectonophysics, 95, 295307.
Gudmundsson, A., 1986. Formation of dykes, feeder-dykes and the intrusion of dykes from magma chambers. Bulletin of Volcanology, 47, 537550.
Gudmundsson, A., 1988. Effect of tensile-stress concentration around magma chambers on intrusion and extrusion frequencies. Journal of Volcanology and Geothermal Research, 35, 179194.
Gudmundsson, A., 2002. Emplacement and arrest of sheets and dykes in central volcanoes. Journal of Volcanology and Geothermal Research, 116, 279298.
Gudmundsson, A., 2003. Surface stresses associated with arrested dykes in rift zones. Bulletin of Volcanology, 65, 606619.
Gudmundsson, A., 2006. How local stresses control magma-chamber ruptures, dyke injections, and eruptions in composite volcanoes. Earth-Science Reviews, 79, 131.
Gudmundsson, A., 2009. Toughness and failure of volcanic edifices. Tectonophysics, 471, 2735.
Gudmundsson, A., 2011a. Rock Fractures in Geological Processes. Cambridge: Cambridge University Press.
Gudmundsson, A., 2011b. Deflection of dykes into sills at discontinuities and magma-chamber formation. Tectonophysics, 500, 5064.
Gudmundsson, A., 2012. Strengths and strain energies of volcanic edifices: implications for eruptions, collapse calderas, and landslides. Natural Hazards and Earth System Sciences, 12, 22412258.
Gudmundsson, A., 2014. Energy release in great earthquakes and eruptions. Frontiers in Earth Science, 2, doi:10.3389/feart.2014.00010.
Gudmundsson, A., 2016. The mechanics of large volcanic eruptions. Earth-Science Reviews, 163, 7293.
Gudmundsson, A., 2017. The Glorious Geology of Iceland’s Golden Circle. Berlin: Springer Verlag.
Gudmundsson, A., Brenner, S. L., 2001. How hydrofractures become arrested. Terra Nova, 13, 456462.
Gudmundsson, A., Brenner, S. L., 2004. How mechanical layering affects local stresses, unrests, and eruptions of volcanoes. Geophysical Research Letters, 31, doi:10.1029/2004GL020083.
Gudmundsson, A., Homberg, C., 1999. Evolution of stress fields and faulting in seismic zones. Pure and Applied Geophysics, 154, 257280.
Gudmundsson, A., Loetveit, I. F., 2005. Dyke emplacement in layered and faulted rift zone. Journal of Volcanology and Geothermal Research, 144, 311327.
Gudmundsson, A., Philipp, S. L., 2006. How local stress fields prevent volcanic eruptions. Journal of Volcanology and Geothermal Research, 158, 257268.
Gudmundsson, A., Bergerat, F., Angelier, J., Villemin, T. 1992. Extensional tectonics of Southwest Iceland. Bulletin of the Geological Society of France, 163, 561570.
Gudmundsson, A., Friese, N., Galindo, I., Philipp, S. L., 2008. Dike-induced reverse faulting in a graben. Geology, 36, 123126.
Gupta, A. B., 2015. Classical Mechanics and Properties of Matter. Kolkata: Books & Allied.
Hamill, P., 2014. A Student’s Guide to Lagrangians and Hamiltonians. Cambridge: Cambridge University Press.
Holmes, A., 1965. Principles of Physical Geology. London: Thomas Nelson.
Isida, M., 1955. On the tension of a semi-infinite plate with an elliptic hole. Scientific Papers of the Faculty of Engineering, Tokushima University, 5, 7595.
Jaeger, J. C., 1961. The cooling of irregularly shaped igneous bodies. American Journal of Science, 259, 721734.
Janssen, V., 2008. GPS-Based Volcano Deformation. Saarbrücken: VDM Verlag.
Kanamori, H., Anderson, D.L., 1975. Theoretical basis of some empirical relations in seismology. Bulletin of the Seismological Society of America, 65, 10741095.
Karson, J. A., 2017. The Iceland plate boundary zone: propagating rifts, migrating transforms, and rift-parallel strike-slip faults. Geochemistry, Geophysics, Geosystems, 18, 40434054.
Kavanagh, J. L., Sparks, R. S. J., 2011. Insights of dyke emplacement mechanics from detailed 3D dyke thickness datasets. Journal of the Geological Society of London, 168, 965978.
Kavanagh, J., Menand, T., Sparks, R. S. J., 2006. An experimental investigation of sill formation and propagation in layered elastic media. Earth and Planetary Science Letters, 245, 799813.
Kavanagh, J., Boutelier, D., Cruden, A. R., 2015. The mechanics of sill inception, propagation and growth: experimental evidence for rapid reduction in magmatic overpressure. Earth and Planetary Science Letters, 421, 117128.
Kumagai, H., Ohminato, T., Nakano, M., et al., 2001. Very-long-period seismic signals and caldera formation at Miyake Island, Japan. Science, 293, 687690.
Kusumoto, S., Gudmundsson, A., 2014. Displacement and stress fields around rock fractures opened by irregular overpressure variations. Frontiers in Earth Science, 2, doi:10.3389/feart.2014.00007.
Kusumoto, S., Geshi, N., Gudmundsson, A., 2013. Inverse modeling for estimating fluid-overpressure distributions and stress intensity factors from arbitrary open-fracture geometry. Journal of Structural Geology, 46, 9298.
Lacazette, A., Geiser, P., 2013. Comment on Davis et al., 2012 – Hydraulic fractures: how far will they go? Marine and Petroleum Geology, 43, 517519.
Lauthold, J., Muntener, O., Baumgartener, L. P., et al., 2014. A detailed geochemical study of a shallow arc-related laccolith: the Torres del Paine Mafic Complex (Patagonia). Journal of Petrology, 54, 273303.
Lu, Z., Dzurisin, D., 2014. InSAR Imaging of Aleutian Volcanoes: Monitoring a Volcanic Arc from Space. Berlin: Springer Verlag.
Manconi, A., Walter, T R., Amelung, F., 2007. Effects of mechanical layering on volcano deformation. Geophysical Journal International, 170, 952958.
Marinoni, L. B., Gudmundsson, A., 1999. Geometry, emplacement, and arrest of dykes. Annales Tectonicæ, 13, 7192.
Marti, J., Geyer, A., Folch, A., Gottsmann, J., 2008. A review on collapse caldera modelling. In Gottsmann, J., Marti, J. (eds.), Caldera Volcanism: Analysis, Modelling and Response. Amsterdam: Elsevier, pp. 233283.
Marti, J., Villasenor, A., Geyer, A., Lopez, C., Tryggvason, A., 2017. Stress barriers controlling lateral migration of magma revealed by seismic tomography. Scientific Reports, 7, doi:10.1038/srep40757.
Mason, B. G., Pyle, D. M., Oppenheimer, C., 2004. The size and frequency of the largest explosive eruptions on Earth. Bulletin of Volcanology, 66, 735748, doi:10.1007/s00445-004-0355-9.
Masterlark, T., 2007. Magma intrusion and deformation predictions: sensitivities to the Mogi assumptions. Journal of Geophysical Research, 112, doi:10.1029/2006JB004860.
Meirovitch, L., 2003. Methods of Analytical Dynamics. New York, NY: Dover.
Menand, T., Daniels, K. A., Benghiat, P., 2010. Dyke propagation and sill formation in a compressive tectonic environment. Journal of Geophysical Research, 115, doi:10.1029/2009JB006791.
Michel, J., Baumgartner, L., Putlitz, B., Schaltegger, U., Ovtcharova, M., 2008. Incremental growth of the Patagonian Torres del Paine laccolith over 90 k.y. Geology, 36, 459462, doi:10.1130/G24546A.1.
Mindlin, R. D., 1936. Force at a point in the interior of a semi-infinite solid. Physics, 7, 195202.
Mogi, K., 1958. Relations between eruptions of various volcanoes and the deformations of the ground surfaces around them. Bulletin of the Earthquake Research Institute University of Tokyo, 36, 99134.
Murase, T., McBirney, A. R., 1973. Properties of some common igneous rocks and their melts at high temperatures. Geological Society of America Bulletin, 84, 35633592.
Newhall, C. G., Dzurisin, D., 1988. Historical Unrest of Large Calderas of the World. Reston, VA: US Geological Survey.
Newhall, C. G., Self, S., 1982. The volcanic explosivity index (VEI): an estimate of explosive magnitude for historical volcanism. Journal of Geophysical Research, 87, 12311238.
Okada, Y., 1985. Surface deformation due to shear and tensile faults in a half-space. Bulletin of the Seismological Society of America, 75, 11351154.
Okada, Y., 1992. Internal deformation due to shear and tensile faults in half space. Bulletin of the Seismological Society of America, 82, 10181040.
Passarelli, L., Rivalta, E., Cesca, S., Aoki, Y., 2015. Stress changes, focal mechanisms, and earthquake scaling laws for the 2000 dike at Miyakejima (Japan). Journal of Geophysical Research, 120, 41304145.
Paterson, M. S., Wong, T. W., 2005. Experimental Rock Deformation: The Brittle Field, 2nd edn. Berlin: Springer Verlag.
Philipp, S., Philipp, S. L., Afsar, F., Gudmundsson, A., 2013. Effects of mechanical layering on the emplacement of hydrofractures and fluid transport in reservoirs. Frontiers of Earth Science, 1, doi:10.3389/feart.2013.00004.
Pollard, D. D., Delaney, P. T., Duffield, W. A., Endo, E. T., Okamura, A. T., 1983. Surface deformation in volcanic rift zones. Tectonophysics, 94, 541584.
Pyle, D. M., 2000. Sizes of volcanic eruptions. In Sigurdsson, H. (ed.), Encyclopedia of Volcanoes. New York, NY: Academic Press, pp. 263269.
Reddy, J. N., 2002. Energy Principles and Variational Methods in Applied Mechanics, 2nd edn. Hoboken, New Jersey: Wiley.
Richards, T. H., 1977. Energy Methods in Stress Analysis. Chichester: Ellis Horwood.
Rivalta, E., Taisne, B., Bunger, A. P., Katz, R. F., 2015. A review of mechanical models of dike propagation: schools of thought, results and future directions. Tectonophysics, 638, 142.
Roth, F., 1993. Deformations in a layered crust due to a system of cracks: modeling the effect of dike injections or dilatancy. Journal of Geophysical Research, 98, 45434551.
Rougier, J., Sparks, S., Cashman, K., Brown, S., 2018. The global magnitude–frequency relationship for large explosive eruptions. Earth and Planetary Science Letters, 482, 621629.
Rubin, A. M., 1995. Propagation of magma-filled cracks. Annual Reviews of Earth and Planetary Sciences, 23, 287336.
Rubin, A. M., Pollard, D. D., 1988. Dike-induced faulting in rift zones of Iceland and Afar. Geology, 16, 413417.
Ryan, M. P., 1993. Neutral buoyancy and the structure of mid-ocean ridge magma reservoirs. Journal of Geophysical Research, 98, 2232122338.
Sanford, R. J., 2003. Principles of Fracture Mechanics. Upper Saddle River, NJ: Prentice-Hall.
Scholz, C. H., 1990. The Mechanics of Earthquakes and Faulting. Cambridge: Cambridge University Press.
Segall, P., 2010. Earthquake and Volcano Deformation. Princeton, NJ: Princeton University Press.
Segall, P., Llenos, A. L., Yun, S. H., Bradley, A. M., Syracuse, E. M., 2013. Time-dependent dike propagation from joint inversion of seismicity and deformation data. Journal of Geophysical Research, 118, doi:10.1002/2013JB010251.
Self, S., 2006. The effects and consequences of very large explosive volcanic eruptions. Philosophical Transactions of the Royal Society A, 364, 20732097.
Shapiro, S. A., 2018. Fluid-Induced Seismicity. Cambridge: Cambridge University Press.
Sigmundsson, F., Hreinsdottir, S., Hooper, A., et al., 2010. Intrusion triggering of the 2010 Eyjafjallajökull explosive eruption. Nature, 468, 426430.
Sobradelo, R., Bartolini, S., Martí, J., 2013. HASSET: a probability event tree tool to evaluate future volcanic scenarios using Bayesian inference. Bulletin of Volcanology, 76, 115.
Sobradelo, R., Martí, J., Kilburn, C., López, C., 2015. Probabilistic approach to decision-making under uncertainty during volcanic crises: retrospective application to the El Hierro (Spain) 2011 volcanic crisis. Natural Hazards, 76, 979998.
Sparks, R. S. J., Aspinall, W. P., Crosweller, H. S., Hincks, T. K., 2013. Risk and uncertainty assessment of volcanic hazards. In Rougier, J., Sparks, R. S. J., Hill, L (eds.), Risk and Uncertainty Assessment for Natural Hazards. Cambridge: Cambridge University Press, pp. 365397.
Spera, F. J., 2000. Physical properties of magmas. In Sigurdsson, H. (ed.), Encyclopedia of Volcanoes. New York, NY: Academic Press, pp. 171190.
Steketee, J. A., 1958. On Volterra’s dislocations in a semi-infinite elastic medium. Canadian Journal of Physics, 36, 192205.
Sun, R. J. 1969. Theoretical size of hydraulically induced horizontal fractures and corresponding surface uplift in an idealized medium. Journal of Geophysical Research, 74, 59956011.
Tauchert, T. R., 1981. Energy Principles in Structural Mechanics. Malabar, FL: Krieger.
Tibaldi, A., 2015. Structure of volcano plumbing systems: a review of multi-parametric effects. Journal of Volcanology and Geothermal Research, 298, 85135.
Townsend, M., Pollard, D. D., Smith, R., 2017. Mechanical models for dikes: a third school of thought. Tectonophysics, 703–704, 98118.
Tsuchida, E., Nakahara, I., 1970. Three-dimensional stress concentration around a spherical cavity in a semi-infinite elastic body. Japan Society of Mechanical Engineers Bulletin, 13, 499508.
Valko, P., Economides, M. J., 1995. Hydraulic Fracture Mechanics. New York, NY: Wiley.
Villemin, T., Bergerat, F., Angelier, J., Lacasse, C., 1994. Brittle deformation and fracture patterns on oceanic rift shoulders: the Esja peninsula, SW Iceland. Journal of Structural Geology, 16, 16411654.
Volterra, V., 1907. On the equilibrium of multiply-connected elastic bodies. Annales scientifiques de l’École Normale Supérieure, 24, 401517 (in French; English translation).
Wadge, G., 1981. The variation of magma discharge during basaltic eruptions. Journal of Volcanology and Geothermal Research, 11, 139168.
Wallerstein, D. V., 2002. A Variational Approach to Structural Analysis. New York, NY: Wiley.
Washizu, K., 1975. Variational Methods in Elasticity and Plasticity. Amsterdam: Elsevier.
Woods, A. W., Huppert, H. E., 2003. On magma chamber evolution during slow effusive eruptions. Journal of Geophysical Research, 108, 2403, doi:10.1029/2002JB002019.
Wu, Y. S. (ed.), 2017. Hydraulic Fracture Modeling. Houston, TX: Gulf Publishing.
Yew, C. H., Weng, X., 2014. Mechanics of Hydraulic Fracturing, 2nd edn. Houston, TX: Gulf Publishing.
Yokoyama, I., 1957. Energies in active volcanoes. Bulletin of the Earthquake Research Institute Tokyo, 35, 7597.
Zang, A., Stephansson, O., 2010. Stress Field of the Earth’s Crust. Berlin: Springer Verlag.
Zoback, M. D., Harjes, H. P., 1997. Injection-induced earthquakes and crustal stress at 9 km depth at the KTB deep drilling site, Germany. Journal of Geophysical Research, 102, 1847718491.
Zobin, V. M., 2003. Introduction to Volcanic Seismology. Amsterdam: Elsevier.