Modelling the shape of a growing fluid-filled crack and computing its propagation velocity: application to magmatic dykes.
Authors: Maccaferri Francesco , Furst Severine , Pinel Virginie 
Published: March 2022

&lt;p&gt;&lt;span&gt;The physics describing fluid-filled fracture growth is simple to describe, but extremely challenging to implement in an analytical, and even in a numerical modelling scheme. The fracturing process is governed by the equations for a brittle-elastic medium, while the internal flow is described by fluid dynamics equations. The pressure profile within the fluid-filled crack, the crack shape, and the velocity of crack growth, results from the solution of the coupled elastic and fluid-dynamic problem, that is far from been trivial. Magmatic dykes can be seen as a sub-set of the larger family of fluid-filled fractures. So far, two main schools have been established for modelling magmatic dykes: they have been named &amp;#8220;fracture dominated&amp;#8221; and &amp;#8220;viscous dominated&amp;#8221;, according to the fracture propagation regime that they target. Fracture dominated models are used when the fluid viscosity contributes with a negligible forcing to the total budget of the problem. They can describe complex crack shapes, account for heterogeneous stress fields and crustal heterogeneity, and compute the direction of crack growth. However they give no information about the crack propagation velocity. On the other hand, the viscous dominated school, drastically simplifies the crack geometry and the crustal structures, but can account for the interaction between elastic and viscous forces, hence it can compute the crack propagation velocity along a prescribed trajectory.&lt;/span&gt;&lt;/p&gt;&lt;p&gt;&lt;span&gt;A few years ago, we teamed up, coming from these two different modelling schools, with the aim of merging our approaches in a single modelling scheme. Here we present a&lt;/span&gt; &lt;span&gt;new modelling scheme, which computes&lt;/span&gt; &lt;span&gt;the dynamic shape of a moving fluid-filled crack, built with the BE technique, in plane strain approximation (2D). Our model account for heterogeneous crustal stress and complex fracture propagation paths, and compute the crack shape considering&lt;/span&gt; &lt;span&gt;the fluid viscosity and the crack propagation velocity. The crack velocity can be given as input to&lt;/span&gt; &lt;span&gt;our model, or&lt;/span&gt; &lt;span&gt;computed as output in the assumption&lt;/span&gt; &lt;span&gt;that the main sources of energy dissipation are the brittle fracturing and the laminar viscous flow.&lt;/span&gt; &lt;span&gt;We compare our model results with previous numerical models from the fracture dominated and viscous dominated schools, and present &lt;/span&gt;&lt;span&gt;the&lt;/span&gt;&lt;span&gt; implications of our findings with regards to some of the most important parameters characterising a magmatic intrusion, such as its volume, buoyancy and viscosity of magma, and rock fracture toughness. &lt;/span&gt;&lt;span&gt;E&lt;/span&gt;&lt;span&gt;ventually &lt;/span&gt;&lt;span&gt;we&lt;/span&gt; &lt;span&gt;show an application of the model to the rising&lt;/span&gt; &lt;span&gt;of the dyke that fed the 1998 Piton de la Fournaise eruption&lt;/span&gt; &lt;span&gt;(La R&amp;#233;union Island).&lt;/span&gt;&lt;/p&gt;