Jökull - 01.12.1980, Page 59

either side it is broken by extension cracks and
small normal faults bounding a central
graben” (Pollard and Holzhausen 1979). This
model, originally proposed for the fissure
swarms on Hawaii, is therefore very similar to
the single dyke model; the main difference
being that each small graben, inside the fissure
swarm, is formed by a separate dyke. This
mo'del appears to be able to explain the Vogar
fissure swarm; the only difficulty I see at the
moment is how it can account for the reverse
inclination and the closed faults.
A horizonlal sill (or sills)
General comments. The horizontal intrusion is
supposed to have given rise to the fractures by
lifting the layers above it. The intrusion is
assumed to have the form of a sill. This is a
very simple model, hence it is easy to handle
mathematically and test the results by data
from the Vogar fissure swarm. The model is of
course only a hypothesis, but there is though
some evidence supporting it. Borehole data
indicates that there are in fact some sills below
the Reykjanes Peninsula; and some of the
boreholes are very near the Vogar area
(Arnórsson el al. 1975).
The model. The sill is assumed to have an
elliptical cross section. This appears to be the
most common shape of such intrusions; as in-
dicated by experiments and field observations
(Pollard 1973, Pollard and Johnson 1973). A
simplified outline of the model is given in
Fig. 14. As usual in mathematical analysis, I
assume that the rock behaves as homogeneous,
isotropic, elastic material. As every geologist
knows, this assumption is not strictly valid.
However, its validity has been discussed so
often before (e.g. Farmer 1968, Fyfe et al. 1978),
that I see no point in discussing it further. I
will first consider the intrusion itself in detail,
and later discuss its effect on the surface, i.e.
the fracture formation.
The details are given by Sneddon (1946) and
Sneddon and Lowengrub (1969). A mathematical
crack is opened under an internal magma
overpressure. Only the upper half of the ellipse
is considered, as shown in Fig. 15. The magma
overpressure is considered constant over the
surface of the crack. The value of the normal
component of displacement w is given by:
w = 2(l— v2) P„ E'1 (a2—x2)1/2 (2)
where v is Poisson’s ratio, Po is the magma
overpressure, i.e. the pressure of the magma
minus the lithostatic pressure, E is Young’s
modulus, and a is the half-length of the major
axis of an ellipse with centre at the origin of
the coordinate system.
Clearly, when x = 0, i.e. at the centre of the
ellipse, w is maximum and equal to the half-
length of the minor axis (i.e. equal to the
semi-minor axis). Hence,
wmil = 2(1- v-’)P0E>a (3)
Fig. 14. A simplified picture of
the model for a single horizontal
sill. The intrusion is supposed to
take place at a contact between
different rock types. U.A. means
uplifted area. — Mynd 14. Ein-
földuð mynd af líkani fynr láre'tla
sillu. Innskotið verður á mótum ólíkra
berggerða. U.A. táknar upplyft svceði.
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JÖKULL 30. ÁR 57