An analytical model for the ascent speed of a viscous fluid batch in three dimensions
Authors: Davis Timothy , Rivalta Eleonora 
Published: March 2022

&lt;p&gt;There are few analytical models of 3D dyke ascent due in part to the algebraic complexity of deriving such solutions but also due to a lack of numerical schemes that can be used to test the validity of their simplifying assumptions. Recent developments in hydro-fracture codes allow for numerical simulation of constant inflow/finite batches of fluid rising towards the ground surface (Zia and Lecampion, 2020). Such schemes allow us to formulate and test some analytical approximations of this process.&lt;/p&gt;&lt;p&gt;Recently, analytical formulations have reproduced in three dimensions the self-sustaining ascent of a batch of fluid, where a fracture ascends upwards once a given &amp;#8220;critical&quot; volume of fluid is injected (Davis et al., 2020; Salimzadeh et al., 2020; Smittarello et al., 2021). The critical volume is dependent on: the rock stiffness, the density contrast between the fluid and rock and the rock toughness. Such formulations have been verified numerically, showing that relatively small batches of fluid are required before these begin to ascend towards the ground surface. In particular, these estimated critical volumes are below observed eruptive volumes and far below typical industrial fluid injection volumes. We investigate how accounting for fluid flow in the model can lead to better estimates of the critical volumes, ascent timescales and the fracture size.&lt;/p&gt;&lt;p&gt;We first detail an approximation of the ascent speed for a given volume of fluid, deriving an approximate maximum ascent speed of a fracture. We show this speed is linearly proportional to the injected volume and inversely proportional to the material stiffness and fluid viscosity. Secondly, we adapt the 2D similarity solution of Spence and Turcotte (1990), showing how to scale this in 3D. This solution describes how the ascent speed decelerates from its initial velocity. We note that in particular the decay in the front velocity is dependent on volume (V) and time (t) with the following scaling V&lt;sup&gt;(1/2)&lt;/sup&gt;/t&lt;sup&gt;(2/3)&lt;/sup&gt;. Our resulting analytical solution matches well to decay speeds from 3D numerical experiments with a finite fluid batch. We discuss the implications this scaling has on the ascent speed of magmatic intrusions and the stability of industrial operations.&lt;/p&gt;&lt;p&gt;Lastly, we briefly discuss formulations describing how density, stress and stiffness interfaces can trap ascending fractures.&lt;/p&gt;&lt;p&gt;Davis, T., Rivalta, E. and Dahm, T., 2020. Critical fluid injection volumes for uncontrolled fracture ascent. &lt;em&gt;Geophysical Research Letters&lt;/em&gt;, &lt;em&gt;47&lt;/em&gt;(14), p.e2020GL087774.&lt;/p&gt;&lt;p&gt;Salimzadeh, S., Zimmerman, R.W. and Khalili, N., 2020. Gravity Hydraulic Fracturing: A Method to Create Self&amp;#8208;Driven Fractures. &lt;em&gt;Geophysical Research Letters&lt;/em&gt;, &lt;em&gt;47&lt;/em&gt;(20), p.e2020GL087563.&lt;/p&gt;&lt;p&gt;Smittarello, D., Pinel, V., Maccaferri, F., Furst, S., Rivalta, E. and Cayol, V., 2021. Characterizing the physical properties of gelatin, a classic analog for the brittle elastic crust, insight from numerical modeling. &lt;em&gt;Tectonophysics&lt;/em&gt;, &lt;em&gt;812&lt;/em&gt;, p.228901.&lt;/p&gt;&lt;p&gt;Spence, D.A. and Turcotte, D.L., 1990. Buoyancy&amp;#8208;driven magma fracture: A mechanism for ascent through the lithosphere and the emplacement of diamonds. &lt;em&gt;Journal of Geophysical Research: Solid Earth&lt;/em&gt;, &lt;em&gt;95&lt;/em&gt;(B4), pp.5133-5139.&lt;/p&gt;&lt;p&gt;Zia, H. and Lecampion, B., 2020. PyFrac: A planar 3D hydraulic fracture simulator. &lt;em&gt;Computer Physics Communications&lt;/em&gt;, &lt;em&gt;255&lt;/em&gt;, p.107368.&lt;/p&gt;