A perturbation solution for dyke propagation in an elastic medium with graded density

Abstract

SUMMARY We present a perturbation method to investigate the steady‐state propagation of a dyke from an over pressured source (e.g. a magma chamber) into a semi‐infinite elastic solid with graded mass density. The non‐linear dyke propagation/magma transport problem is reduced to a series of linear problems using a perturbation technique with the small non‐dimensional parameter ɛ= 12ηV/(H2Δρ0g), where η is the magma viscosity, V the propagation velocity, Δρ0 the difference between the densities of the host rock and magma at the dyke tail, g the gravitational acceleration and H a parameter on the order of maximum dyke thickness. In general, the perturbation method is applicable to mafic dyke propagation at a relatively low propagation velocity wherein ɛ remains small (e.g. less than 0.1). We describe an integral equation approach to obtain the stress intensity factor at the dyke tip and the separation displacement of the two dyke surfaces. Numerical examples are presented to examine the effects of various physical parameters, for example, buoyancy, density gradation, propagation velocity and overpressure on the dyke propagation behaviour. It is found that dyke propagation could reach a steady‐state in some specific ranges of growth for a given density gradient of the host rock. The propagation tends to decelerate when the dyke tip approaches the level of neutral buoyancy.

A volcanic origin for the outflow channels of Mars: Key evidence and major implications

“…Though lithostatic pressure is a major control on the level to which magma can flow, magma will only rise beyond a stalled chamber if a path already exists and can be opened by magma pressure or if fluid pressures are sufficient to open a new path by overcoming the tensional strength of overlying rock (e.g., Rubin and Pollard, 1987;Rubin, 1995). In the absence of excess pressures such as those related to dynamic inflow into a magma chamber (e.g., Chen et al, 2007) or to favorable gradients of tectonic stress (e.g., Rubin, 1995), magma intrusions will generally reach no shallower than the level given by Eq.…”

Magma‐driven multiple dike propagation and fracture toughness of crustal rocks

“…Equation (12) is a nonlinear integral equation for the dislocation density function ϕ( r ). Here we employ a perturbation method [ Chen et al , 2007] to attack the nonlinear problem. In the perturbation method, the nonlinear equation (12) reduces to the following series of linear integral equations with the density function ϕ( r ) given by where ɛ is a perturbation parameter defined as in which D is a parameter with an order of the dike base thickness.…”

“…This problem involves nonlinear coupling of solid mechanics, fracture mechanics and fluid mechanics, necessitating a relatively simple geometrical configuration in order to obtain a semi‐analytical solution. Using a perturbation method that we recently developed [ Chen et al , 2007] for the low velocity (e.g., less than 0.5 m/s) propagation of a single dike filled with low viscosity magma (e.g., less than 100 Pa‐s), we analyze an array of parallel, periodically spaced dikes that grow simultaneously from an overpressured source into a semi‐infinite host rock. Although the parallel dikes observed in many localities on Earth and other planets may be injected separately at different times, the possibility of simultaneous injection cannot be excluded, as discussed above.…”

Subcritical dyke propagation in a host rock with temperature-dependent viscoelastic properties

“…Spence & Turcotte 1985; Spence et al . 1987; Pollard & Segall 1987; Lister & Kerr 1991; Clemens & Mawer 1992; Rubin, 1995a,b, 1998; Meriaux & Jaupart 1998; Bonafede & Rivalta 1999; Dahm 2000; Ito & Martel 2002; Rivalta & Dahm 2006; Chen et al . 2007; Jin & Johnson 2008).…”

“…Weertman 1971; Spence & Sharp 1985; Spence et al . 1987; Lister 1990, 1991, 1994a,b; Rubin 1995a,b, 1998; Bonafede & Rivalta 1999; Meriaux & Jaupart 1998; Bolchover & Lister 1999; Dahm 2000; Menand & Tait 2002; Kuhn & Dahm 2004; Roper & Lister 2005; Rivalta & Dahm 2006; Chen et al . 2007; Taisne & Jaupart 2009; Traversa et al .…”