Task 1: Magma upwelling as driving mechanism for the stress build-up in the elastic lithosphere

Tensile cracks are often employed to model magma migration in rift zones or within volcanic edifices, through lateral or feeding dykes. In a crack model, the overpressure of magma $\Delta p$ with respect to the horizontal stress in the host rock, is assumed to be responsible for dyke opening and propagation. Most crack models of dykes have been developed so far in homogeneous media. The most simple heterogeneous medium has been considered, made up of two welded half-spaces, characterized by different elastic parameters. The analytical solutions available for the elementary dislocation problem in such a medium (Bonafede and Rivalta 1999) have been employed to set up an integral equation with generalized Cauchy kernel, representing the condition for static equilibrium. The unknown in such a problem is the dislocation density distribution, whose singular behaviour has been studied near the crack tips and near the intersection with the interface between the two media. When the crack is in half-space 1 but touches the interface, the order of singularity of the dislocation density distribution at the interface changes from the classical behaviour $\sim r^{-1/2}$ to $\sim r^{-b}$ (where r is the distance from the interface) and the order of infinity b is obtained solving a trascendental compatibility equation; some results are shown in Table 6.

 

Table 6: Crack touching the interface.

$m=\mu_1/\mu_2$	$\infty$	10	5	2	1	0.5	0.2	10-1
b	0.255	0.312	0.352	0.430	0.500	0.576	0.678	0.752

 

A crack crossing the interface z=0 between the two half-spaces with rigidities $\mu_1$ (in z>0) and $\mu_2$ (in z<0) has been considered in detail. A system of generalized Cauchy equations is obtained, which is solved for the dislocation density distributions of each crack section. An internal singularity in the dislocation density distribution appears at the intersection between the crack plane and the interface. This singularity is again of the type r^-b on both sides of the interface and its order b depends only upon the elastic parameters of the media in welded contact and the ratio between the crack lengths in the two half-spaces (see Table 7). More specifically, the order of singularity b does not depend on the stress drop.

 

Table 7: Crack crossing the interface.

$m=\mu_2/\mu_1$	1	0.5	0.1	0.05	0.001
b	0	0.030	0.170	0.208	0.245

 

The horizontal stress component induced by crack opening is plotted in Figure 45, assuming 5 MPa overpressure within the crack. From a comparison with solutions pertinent to a homogeneous medium, it appears that layering can be responsible of stress changes, localized along the the interface, which may be considerably higher than the overpressure within the dyke. These results provide useful hints for the interpretation of induced seismicity in rift zones and in volcanic areas. The detailed results of this research are reported in Bonafede and Rivalta (1999).

  

Figure: Horizontal normal stress induced by rifting in proximity of a structural discontinuity (horizontal dashed line). The harder side of the interface is affected by strong compressive stresses. Other stress components (not shown) are also affected by layering, but to a lesser extent. Rifting is modelled as a tensile crack with overpressure $\Delta p$.