Fully-coupled 3D modelling of magmatic dike propagation - finite pulse release from a point source

<p>Magmatic dikes are a naturally occurring type of fluid-driven fractures [1] propagating in the lithosphere driven by buoyancy (more precisely by the difference between the in-situ minimum horizontal stress gradient and the magma weight). Fully-coupled modelling of these 3D fractures is very challenging and most contributions until today have been restricted to 2D plane-strain. These 2D investigations have highlighted the importance of the head-tail structure, notably the fact that lubrication flow in the tail is driving the growth of the hydrostatic head [2, 3]. We investigate the 3D development of a buoyant dike from a point source, focusing on the case of a finite volume release under homogeneous conditions (homogeneous material properties and buoyancy contrast). We use the fully coupled planar 3D hydraulic fracture growth solver PyFrac based on the implicit level set algorithm [4].</p><p>This configuration shows an early time behaviour heavily dominated by the effects of the pulse release. The initially radial hydraulic fracture transitions toward a large time buoyant dike solution. At large time our simulations tends to the finger-like/constant breadth solution [5] albeit extremely slowly. Our results confirm the 3D toughness dominated head structure and the importance of the viscous tail as the driving mechanism for the dynamics of such a 3D Weertman’s pulse (form of the head). Depending on the initial phase of the pulse release, we observe an overshoot of the dike breadth when it is initially strongly dominated by viscous dissipation. Using a scaling analysis, we characterize the transition from the early time radial finite pulse fracture to the late dike constant breadth solution. Our simulations show, that the time when the buoyant force takes its full dominance is crucial and governs the existence (or not) of an overshoot. Mainly we show that the overshoot depends on a transitional time/lengthscale. A detailed understanding of the fracture propagation after the end of the finite volume release (yet without buoyancy) is key to quantify this lengthscale. We thus present scalings and semi-analytical solutions for this case and discuss its relevance for the transition toward a buoyancy driven dike propagation.</p><p>[1]  E. Rivalta, B. Taisne, A.P. Bunger, and R.F. Katz. Tectonophysics, 638:1–42, 2015.</p><p>[2]  J. R. Lister and R. C. Kerr. J. Geohpys. Res. Solid Earth, 96(B6):10049–10077, 1991.</p><p>[3]  S. M. Roper and J. R. Lister. J. Fluid Mech., 536:79–98, 2005.</p><p>[4]  A. P. Peirce and E. Detournay. Comput. Methods in Appl. Mech. Eng., 197(33-40):2858–2885, 2008.</p><p>[5]  L.N. Germanovich, D. I. Garagash, Murdoch, L., and Robinowitz M. AGU Fall meeting, 2014.</p>

CURVES OF CONSTANT BREADTH ACCORDING TO DARBOUX FRAME IN GALILEAN SPACE G3>

In this work, the curves of constant breadth according to Darboux frame in the 3-dimensional Galilean Space are investigated. Firstly the curves of constant breadth according to Darboux frame are determined then the differential equation of the constant breadth curve with this frame is found. After that some special cases of this differential equation are researched.

The construction of the space-like surface of constant breadth

The objective of this study is to define an ovaloid surface on the convex closed space-like surfaces of constant breadth when principal curvatures of these surfaces are continuous, non-vanishing functions, and to obtain some special geometrical properties of this ovaloid surface by using the radius of curvature, diameter of the surface in [Formula: see text].

A special interpretation of the concept constant breadth for a space curve

The definition of curve of constant breadth in the literature is made by using tangent vectors, which are parallel and opposite directions, at opposite points of the curve. In this study, normal vectors of the curve, which are parallel and opposite directions are placed at the exit point of the concept of curve of constant breadth. In this study, on the concept of curve of constant breadth according to normal vector is worked. At the conclusion of the study, is obtained a system of linear differential equations with variable coefficients characterizing space curves of constant breadth according to normal vector. The coefficients of this system of equations are functions depend on the curvature and torsion of the curve. Then is obtained an approximate solution of this system by using the Taylor matrix collocation method. In summary, in this study, a different interpretation is made for the concept of space curve of constant breadth, the first time. Then this interpretation is used to obtain a characterization. As a result, this characterization we?ve obtained is solved.