Page 353 - Caldera Volcanism Analysis, Modelling and Response

328 Agust Gudmundsson

At shallow crustal depths in tectonically active areas, the field Young’s modulus
of a rock unit depends strongly on the fracture frequency of that unit (Priest, 1993).
It is well known that Young’s modulus of a rock mass is normally less than that of a
laboratory sample of the same type of rock. This difference is mainly attributed to
fractures and pores in the rock mass, which do not occur in small laboratory samples
(Farmer, 1983; Priest, 1993). With increasing number of fractures, in particular in a
direction perpendicular to loading, the ratio E is /E la (E in situ/E laboratory) shows a
rapid decay. Similar results are obtained for elastic materials in general (Sadd, 2005).
Thus, as the fracture frequency and porosity increase in a rock unit, its Young’s
modulus normally decreases.
Another parameter of great importance for collapse caldera formation is rock
strength. Usually, one distinguishes between three types of rock strengths: tensile
strength, shear strength and compressive strength. Theoretically, the shear strength
should be about twice the tensile strength, and the compressive strength about 10
times the tensile strength ( Jaeger and Cook, 1979). These theoretical predictions
are generally supported by observations. Laboratory tensile strengths range up to a
few tens of megapascals ( Jumikis, 1979; Myrvang, 2001), compressive strengths up
to a few hundred megapascals and shear strengths are somewhere in between these
extremes, mostly close to twice the tensile strengths ( Jumikis, 1979; Bell, 2000;
Nilsen and Palmstro ¨m, 2000; Myrvang, 2001). For ring-fault formation, tensile and
shear strengths are the most important.
The field or in situ values of these strengths, however, are normally much lower
than the laboratory values. Perhaps, the best-studied field strength is the tensile
strength. It has been estimated from numerous hydraulic fracturing experiments in
solid rocks worldwide. This method of testing is very suitable for magma chambers
and dyke emplacement since the in situ tensile strength is estimated from the fluid
pressure (in excess of the minimum compressive stress) that is needed to fracture
open the rock (Amadei and Stephansson, 1997). The results indicate that the tensile
strength of solid rocks has a comparatively narrow range, that is, 0.5–6 MPa and
most commonly 2–3 MPa (Haimson and Rummel, 1982; Schultz, 1995; Amadei
and Stephansson, 1997). For comparison, the driving shear stress (estimated from
the stress drop) of earthquakes generally ranges between 1 and 10 MPa and is most
commonly 3–6 MPa (Kanamori and Anderson, 1975; Scholz, 1990), or roughly
twice the tensile strength. At the high temperatures close to the margin of a fluid
magma chamber, the tensile (and thus the shear) strength is likely to decrease
somewhat, but still be within the ranges indicated above.
In some numerical models in this paper, I use the laboratory rock stiffnesses but
scale them down to reasonable field values using the information above. The
highest stiffness used in the models, 100 GPa, may occur in nature, but is here used
mainly to emphasise the stress-field effects of the contrast with the soft layers. In
none of the models, however, do I use the most extreme stiffnesses one might
encounter in composite volcanoes. For example, laboratory measurements of
volcanic tuffs yield stiffnesses as low as 0.05–0.1 GPa (Afrouz, 1992; Bell, 2000), in
which case the field stiffnesses could be even lower. Similarly, laboratory
measurements of some rocks yield stiffnesses as high as 150–200 GPa (Myrvang,
2001). In the models, all the stiffness values used are within the range of 1–100 GPa.