Page 357 - Caldera Volcanism Analysis, Modelling and Response

332 Agust Gudmundsson

p 40 which could drive out the magma even if the magmastatic pressure p m plus
e
the excess pressure were less than the minimum principal compressive stress s 3 .
This possibility, as well as the dynamic effects, certainly needs to be explored.
Ring-fault slips may also result in explosive build up of gas pressure in the associated
chamber, by which means the excess pressure is maintained as positive, that is,
p 40 during the collapse (Gudmundsson, 1998a). The present suggestion that
e
p þ p op ¼ s 3 and p 40, however, seems to imply a rigid rather than elastic
e
l
e
m
crust, that is, one with an infinite Young’s modulus. Since the shallow crust is
known to behave, to a first approximation, as elastic, the proposal in its present form
does not seem very likely. While further work is definitely needed on this
important topic, current understanding indicates that Equation (4) is approximately
valid for magma flow from a chamber during ring-fault formation or slip.
6. Stress Fields Triggering Ring-Fault Initiation

Field observations show that most ring faults and ring dykes originate near the
lateral ends of the associated magma chambers (Figures 1, 2 and 4). A necessary
condition for a ring fault to form, or an existing one to slip, is thus that the shear
stress and the near-surface tensile stress favouring dip-slip faults must peak above the
lateral ends of the associated magma chamber. Therefore, the stresses must peak
above the ‘equator’ of a spherical chamber and the ‘vertices’ of an oblate ellipsoidal
(sill-like) chamber. This is because ring faults are primarily shear fractures and thus
cannot form or slip unless the local shear stress satisfies the condition for failure.
This condition is normally represented by the Navier–Coulomb criterion, von
Mises criterion and other similar criteria ( Jaeger and Cook, 1979).
Unrest periods and eruptions are much more common in collapse calderas than
slips on existing ring faults (Newhall and Dzurisin, 1988). The local stress field
around a shallow magma chamber may trigger tens of thousands of sheet injections
during its lifetime, resulting in hundreds or thousands of eruptions, while caldera
collapses remain very infrequent. The local stress field, in turn, depends primarily
on the shape of the magma chamber, the loading conditions and the mechanical
properties of the host rock (Gudmundsson, 1998a, 1998b; Gudmundsson and
Brenner, 2005). As indicated above, there are three ideal shapes that a magma
chamber can have: prolate ellipsoidal, oblate ellipsoidal and spherical (Figure 10). A
prolate ellipsoidal magma chamber has a vertical, long axis and is normally unlikely
to generate local stresses suitable for ring-fault formation (Tsuchida and Nakahara,
1970; Tsuchida et al., 1982; Gudmundsson, 1998a, b). The focus here is therefore
on circular (spherical) and sill-like (oblate ellipsoidal) magma chambers.
Based on the magma-chamber geometries and earlier results as to likely stress
states for initiating ring faults (Gudmundsson, 1998a, 2007; Gudmundsson and
Nilsen, 2006), four types of loading conditions are used in the present numerical
models. The loadings are: (1) internal magmatic excess pressure in the chamber
(that is, pressure in excess of the lithostatic stress or overburden pressure at the
margin of the chamber); (2) external horizontal tensile stress applied to the crustal
segment hosting the chamber; (3) magma accumulation and excess pressure at the