Numerical modelling of magma transport in dykes
Research highlights
► We developed a new mechanical code to model the magmatic processes. ► Results explain why observation of frozen dykes show crystals aligned parallel to flow direction. ► Presence of crystals modifies the velocity profile from a typical Poiseuille flow shape to a Bingham-type shape. ► The segregation of granitic melt from an ascending crystal-rich magma is physically possible.
Introduction
The processes related to magma ascent from source zones towards emplacement sites of magmatic intrusions in the upper crust constitute a major subject of interest in Earth sciences, especially in terms of understanding intra-crustal differentiation. In particular, the mechanism leading to granitic melt migration towards the upper continental crust (represented by the “diapirism” and “dyking” end-members) has been controversially discussed throughout the 20th century (see reviews by Clemens, 1998, Petford, 2003). Nevertheless, since the study of Clemens and Mawer (1992), it is now largely agreed that the most viable mechanism for the migration of magma, from the deep partial melting zone where they form, to the upper crust where they emplace, is dyking (e.g., Clemens, 1998, Clemens, 2003, Clemens and Petford, 1999, Petford et al., 1994, Scaillet et al., 1998).
The rheology and the behaviour of a multiphase magma (i.e., composed of suspended crystals carried by a viscous medium) subject to a given pressure gradient are governed by the amount of crystals and their geometry (e.g., Bagdassarov and Dorfman, 1998). Here we do not consider the effect of gas bubbles because the processes that we study occur at a depth where volatiles are dissolved in the melt phase. When the amount of crystals is small, silicate melts are considered as Newtonian fluids and their behaviour follows the Einstein–Roscoe relations (Einstein, 1906, Roscoe, 1953). Above a critical solid fraction of suspended crystals, depending on their size, shape and distribution in the magma, the suspension can form a rigid skeleton (Philpotts et al., 1998), which introduces a yield stress in the magmatic suspension and thus results in an effective non-Newtonian rheology (e.g. Kerr and Lister, 1991). The volume of crystals at which the transition from a Newtonian to a non-Newtonian rheology occurs has been estimated somewhere between 15 and 50 vol.% (Champallier et al., 2008, Hallot et al., 1996, Kerr and Lister, 1991, Petford, 2003, Petford and Koenders, 1998, Walsh and Saar, 2008). When the magma contains ca. 55% of solid particles, only residual liquids can escape the rigid skeleton formed by crystals, an effect known as the “rigid percolation threshold” (Vigneresse et al., 1996). According to these authors, when ca. 75% of crystallisation occurs, the entire system becomes totally locked, preventing further mechanical melt percolation from occurring.
Field observations of crystal arrangement in frozen dykes reveal that both their repartitions and their orientations are not random (e.g., Chistyakova and Latypov, 2010, Paterson, 2009, Smith, 2002). In many instances, the crystals are found to be orientated with their major axis parallel, or at a low angle, to the edges of the dyke following the magma flow direction. However, the processes of crystal re-orientation and of their spatial organisation acting during magma transport cannot be directly observed and thus have to be modelled. The rotation of solids immersed in a deforming viscous medium has been addressed in numerous analogue experiments (e.g., Arbaret et al., 1996, Arbaret et al., 2001, Marques and Burlini, 2008, Marques and Coelho, 2001, Marques and Coelho, 2003, Van den Driessche and Brun, 1987, Willis, 1977) and numerical studies (Bons et al., 1996, Mandal et al., 2001, Marques et al., 2005a, Marques et al., 2005b, Samanta et al., 2002, Schmid, 2005). Also, observations of the crystal size distribution within dykes often show that the crystals are sorted by their size, increasing from the edges of the dyke to the centre (e.g., Nkono et al., 2006 and references therein). This can be attributed to their mechanical segregation during the magmatic transport, a phenomenon known as the “Bagnold effect” (e.g., Bagnold, 1954, Barrière, 1976, Bhattacharji, 1967, Komar, 1972a, Komar, 1972b). Besides the fact that these two phenomena should occur coevally during crystal-bearing magma transport in dykes, this complex mechanism remains poorly constrained. Numerous experimental (e.g., Bagdassarov and Dorfman, 1998) and numerical (e.g., Deubelbeiss et al., 2010) studies have been undertaken on magmatic suspension containing particles. However, they mainly addressed the issue of quantifying the effective viscosity of the crystal-melt system and the related rheological consequences, with an emphasis on volcanism (Caricchi et al., 2007, Costa et al., 2007, Dingwell, 1996, Melnik and Sparks, 1999, Papale, 1999, Taisne and Jaupart, 2011).
Finally, another fundamental aspect of magma dynamics is the capacity of crystal-melt segregation to occur during magma ascent in dykes, which has important consequences for magmatic differentiation processes. Crystal-melt fractionation is controlled by factors such as the density difference between the solid and liquid phases, the viscosity of the melt phase, the crystal size and the dynamics of the system. In granitic rocks, such a process is considered to be difficult to initiate, because of (i) the common belief that granitic melts are highly viscous and (ii) the lack of a sufficiently high density difference between the minerals and the residual liquid. Granitic melt viscosities are in the range 104–106 Pa.s (e.g., Clemens, 1998, Scaillet et al., 1998), and the density difference between the melt and common crystals is typically in the range of 200–400 kg.m− 3.
To our knowledge, numerical modelling has not been used to study crystal-melt segregation processes that might take place in dykes at depth. In order to address this issue, we propose to use a fluid dynamic description of creeping flow (Stokes) to represent both the crystal and melt phases in one coupled system. We first verify that the numerical method developed can reproduce the known behaviour of highly viscous, or rigid, inclusion subject to both simple and pure shear boundary conditions. In these tests, we quantitatively compared our numerical results to analytical solutions. Then, we describe a model setup that can be used to understand field observations such as crystal orientation in dykes. Finally, we show that crystal-melt segregation is actually a viable mechanism during granitic magma ascent. In this paper, we present examples where all of the crystals introduced possess identical material properties and geometry.
Section snippets
Numerical model
In order to study the dynamics of a crystal suspension in an ascending magmatic flow, we define the system to be composed of highly viscous fluids. The equations governing creeping flow in two-dimensions are given by the Stokes equations (Eqs. (1), (2)) subject to the incompressibility constraint (Eq. (3)):where P, τij, ρ and g correspond to the pressure, the deviatoric stress tensor, the density, and the gravitational acceleration,
Verification of the numerical scheme
Before using our code to study flows with randomly distributed crystals, we performed several experiments involving a viscous inclusion for which we had an analytic solution for the velocity and pressure field. These tests were conducted in order to understand the discretisation errors associated with the method, and to verify that these discretisation errors decreased at the appropriate rate as the numerical resolution in the model was increased. In the crystal-free case, the flow induced by a
Model setup
Our model setup is constructed in a manner to produce an effective pressure gradient between the base and the top of a channel (simulating the dyke), by using a rigid piston pushed in a fluid perforated by a hole (Fig. 5). The fluid corresponds to the melt phase of the magmatic material that can be filled with crystals. The term magmatic, as in a mineralogical point of view, means a mixture composed of both melt and crystals (as in Fig. 5). The two main advantages of such a model are (1) that a
Conclusion and perspectives
In this study we have presented a numerical technique to model magmatic flows within a dyke. The main results can be summarised as follow:
- (1)
Our code has been verified against several analytical solutions that possess characteristics similar to those found in crystal-melt systems. These tests indicate that the marker to node interpolation using 1-cell area is more accurate for problems that include rotating, non-circular rigid-bodies.
- (2)
The simulations presented here clearly illustrate that crystals
Acknowledgments
We first thank F.O. Marques and the GeoMod2010 team for the workshop organisation where multiple discussions and debates have initiated this work. Discussions with B. Kaus and M. Dabrowski were greatly appreciated. A special “cпacибo” is addressed to T. Gerya for his constant support and enthusiasm during coding using the M-I-C method. Multiple stimulating discussions with E. Hallot, K. Gallagher, P. Boulvais, M. Poujol, W. Husson and B. Cordonnier were also appreciated. Finally, we also thank
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