Forecasting the propagation paths of fluid-driven fractures, particularly dikes and inclined sheets

&amp;lt;p&amp;gt;It is of great importance in many fields to be able to forecast the likely propagation paths of fluid-driven factures. These include mineral veins, human-made hydraulic fractures, and dikes/inclined sheets. The physical principles that control the propagation of all fluid-driven fractures are the same. Here the focus is on dikes and inclined sheets where the selected path determines whether, where, and when a particular dike/sheet reaches the surface to erupt. Here we provide analytical and numerical models on dike/sheet paths in crustal segments (including volcanoes) that include layers of various types (lava flows, pyroclastic flows, tuff layers, soil layers, etc) as well as mechanically weak contacts and faults. The modelling results are then compared with, and tested on, actual data of two types. (a) Seismic data on the paths of dikes/sheets as well as human-made hydraulic fractures, and (b) field data on the actual propagation paths of dikes/sheets in layered and faulted rocks&amp;lt;/p&amp;gt;&amp;lt;p&amp;gt;The numerical results show that, particularly in stratovolcanoes, the paths are likely to be complex with common deflections along layer contacts, in agreement with field observations. &amp;amp;#160;Also, some dikes/sheets may use existing faults as parts of their paths, primarily steeply dipping and recently active normal faults. The propagation path is thus not entirely in pure mode I but rather partly in a mixed mode. The energy required to propagate the dike/sheet is mainly the surface energy needed to rupture the rock, to form two new surfaces and move them apart as the fracture propagates. The energy available to drive the fracture is the stored elastic energy in the hosting crustal segment.&amp;lt;/p&amp;gt;&amp;lt;p&amp;gt;From its point of initiation in the magma-chamber roof, a dike/sheet can, theoretically, select any one of an infinite number of paths to follow to its point of arrest or eruption. It is shown that the eventual path selected is the one of least action, that is, the path along which the time integral of the difference between the kinetic and potential energies is an extremum (normally a minimum) relative to all other possible paths with the same endpoints. If the kinetic energy is omitted, and there are no constraints, then least action becomes the minimum potential energy, which was postulated as a basis for understanding dike propagation by Gudmundsson (1986). Here it is shown how this theoretical framework can help us make reliable forecasts of dike/sheet paths and associated volcanic eruptions.&amp;lt;/p&amp;gt;&amp;lt;p&amp;gt;Gudmundsson, A., 1986. Formation of dykes, feeder-dykes, and the intrusion of dykes from magma chambers. Bulletin of Volcanology, 47, 537-550.&amp;lt;/p&amp;gt;&amp;lt;p&amp;gt;Gudmundsson, A., 2020. Volcanotectonics: Understanding the Structure, Deformation, and Dynamics of Volcanoes. Cambridge University Press, Cambridge.&amp;lt;/p&amp;gt;&amp;lt;p&amp;gt;Drymoni, K., Browning, J. Gudmundsson, A., 2020. Dyke-arrest scenarios in extensional regimes: insights from field observations and numerical models, Santorini, Greece. Journal of Volcanology and Geothermal Research (in press).&amp;lt;/p&amp;gt;&amp;lt;p&amp;gt;&amp;amp;#160;&amp;lt;/p&amp;gt;