Page 354 - Caldera Volcanism Analysis, Modelling and Response

Magma-Chamber Geometry, Fluid Transport, Local Stresses and Rock Behaviour 329

Many authors have suggested that surrounding the magma chamber there is a
shell of rock that behaves as elastic-plastic or viscoelastic (e.g., Bonafede et al., 1986;
Chery et al., 1991; Burov and Guillou-Frottier, 1999; Newman et al., 2001;
Jellinek and DePaolo, 2003). It is of course quite possible that the rocks in such a
shell behave as, for example, elastic–plastic, and in the numerical models below the
von Mises shear stress ( Jaeger and Cook, 1979) is used to indicate the likely location
of ring faults. Most dykes in the roofs and envelopes of shallow magma chambers
studied in Iceland, however, are extension fractures that seem to propagate
primarily as elastic cracks during volcanic unrest periods (Gudmundsson, 2006). In
this paper, the rock properties prior to failure are assumed elastic.

5. Magma-Chamber Rupture and Fluid Transport
Along a Dyke

For magma to be transported out of a magma chamber, a sheet or a dyke must
be initiated at the margin of the chamber and be able to propagate into the host
rock. For an eruption to occur, the sheet or dyke must be able to propagate through
all the rock layers and contacts between the point of rupture at the margin of the
chamber and the free surface of the associated volcano. Based on the Griffith crack
theory, and supported by analogy with the results of numerous hydraulic-fracture
experiments worldwide (Valko and Economides, 1995; Yew, 1997), a magma-filled
chamber ruptures and initiates a sheet or dyke when the following condition is
satisfied:

(1)
p þ p ¼ s 3 þ T 0
1 e
Here p 1 denotes the lithostatic stress (or pressure) at the depth of the chamber;
p e ¼ p t p l , the excess magmatic pressure, is the difference between the total magma
pressure, P t , in the chamber at the time of its rupture and the lithostatic stress or
pressure; and s 3 and T 0 denote the minimum principal stress and the in situ tensile
strength, respectively, at the site of rupture of the chamber. Here and elsewhere in
this paper, compressive stress is considered positive. Therefore, when there is an
absolute tension, s 3 is negative (but is here given by its absolute value), whereas the
maximum compressive principal stress, s 1 , is always positive.
The local s 3 and T 0 are the relevant parameters in Equation (1). It follows that
sheet injection occurs when the condition of Equation (1) is reached at any point at
the margin of the chamber, irrespective of the chamber shape and depth below the
surface. The condition of Equation (1) is reached by increasing p e , decreasing s 3 or
both (Gudmundsson, 2002, 2006). Since Equation (1) refers to the local s 3 stress
concentration, effects due to the shape of the magma chamber, including
irregularities at its boundary, are automatically taken into account. Thus, Equation
(1) does not represent the condition for the rupture of a fluid-filled cavity in terms
of maximum and minimum compressive regional stresses (s H ; s h ), as is normal
when dealing with hydraulic fractures initiated from boreholes (Valko and
Economides, 1995; Amadei and Stephansson, 1997; Yew, 1997; Charlez, 1997).