Page 356 - Caldera Volcanism Analysis, Modelling and Response

Magma-Chamber Geometry, Fluid Transport, Local Stresses and Rock Behaviour 331

solidification), W the fracture width in a direction perpendicular to the flow
direction (assuming WcDu, the fracture cross-sectional area perpendicular to the
flow is A ¼ DuW ), m the dynamic (absolute) viscosity of the magma, r r the host-
rock density, r the magma density (assumed constant), g the acceleration due to
m
gravity, and @p =@L the pressure gradient in the direction of the flow. This equation
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ignores possible stress gradients in the host rock, for example, due to variation in
mechanical properties and thus local stresses, topography of the volcano or both.
When the flow is vertical, that is, along a vertical dyke, the dip a ¼ 901 so that
sin a ¼ 1, and substituting z for L (Figure 13), we get:

3
Du W @p e
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Q ¼ ðr r Þg (3)
z r m
12m @z
Similarly, when the magma flow is horizontal, either along a sill or a laterally
propagating dyke, then the dip a ¼ 01 and sin a ¼ 0. Consequently, the first term
in the brackets in Equation (2) drops out. For a sill in the horizontal xy-plane,
the width W is measured along the y-axis; for or a laterally propagating dyke in the
vertical xz-plane, the width W is measured along the z-axis. In either case, if the
magma flow is assumed to be along a mechanical layer with a density equal to that
of the magma (the assumption of a ‘neutral buoyancy’) and the length L is measured
along the x-axis, then we may substitute x for L in Equation (2) to obtain
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volumetric magma flow rate Q :
x
3
Du W @p e
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Q ¼ (4)
x
12m @x
Thus, in the absence of a stress gradient, the only pressure gradient for driving
magma flow through a dyke (or a sill) emplaced laterally along a neutral buoyancy
layer is due to the excess pressure p in the magma chamber.
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In terms of mechanics of collapse caldera formation, Equation (4) indicates that
for fluid to be driven out of a chamber there must be excess pressure p 40 in the
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chamber. If this excess pressure falls to zero, there is no pressure gradient available to
drive the magma out of the chamber, and the magma flow should stop. It follows
that for the underpressure models or ‘withdrawal of magmatic support’ for
explaining ring-fault formation, which commonly assume that underpressure is
generated because of flow of magma out of the chamber along a laterally
propagating dyke, it must be shown that somehow Equation (4) is not valid during
caldera collapse.
One suggested reason for Equation (4) not being valid is that once a conduit has
opened to the surface during an eruption, the static pressure in a magma column
may be less than that in a similar and adjacent column of host rock (Folch, personal
communication, 2007). The magmastatic pressure is p ¼ r gh, where h is depth
m m
and all the symbols are as defined above. For comparison, the lithostatic pressure in
a rock column would be p ¼ r gh, where all the symbols are defined above. When
l
r
r 4r , it follows that p op and the possibility may exist that p þ p op ¼ s 3
m
l
e
r
l
m
m
even if p 40. If this were correct, there could be a positive excess pressure so that
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