Jökull - 01.06.2000, Page 24

Fiona S. Tweed
problem noted by other workers (e.g. Hooke et al,
1990, p. 70).
The closure rate according to Nye’s theory is gi-
ven by:
Í=A(±?) (1)
where:
n = flow law constant, 3 (Paterson, 1994)
A = flow law constant, 6.8-10_16s_1 (kPa)-3 (Pater-
son, 1994)
AP = pressure difference between the water in the
tunnel and the surrounding ice
Assuming that the water flow is at atmospheric
pressure, AP is equal to the ice overburden pressure,
P, which is given by:
P = tg piCos2/3 (2)
where:
t = vertical thickness of ice above the tunnel
g = gravitational acceleration, 9.82 ms-2
pi = ice density, 900 kg/m3
/3 = slope angle
The conduit slope angle at Sólheimajökull (/3 =
1.65°) was derived by topographically surveying the
entrance and exit portal outlets at ground level and,
in the absence of subglacial topographic informati-
on, assuming a constant slope between the two points.
The maximum ice thickness (71.7 m) was calculated
by means of topographic surveying of the ice above
the tunnel over the length of the glacier and construct-
ing a profile of the ice surface. A projected subglacial
bedrock surface profile was also constructed (in the
absence of data concerning the bedrock topography)
and the maximum glacier thickness was obtained by
subtracting the constructed bedrock elevation from
the measured elevation of the ice surface.
Under overburden pressure of ice alone a tunnel at
atmospheric pressure at Sólheimajökull would reach
63% closure (1 — e-1) in approximately 182 days. Us-
ing a tunnel radius of 2.5 m, the initial closure rate, (r)
predicted by Nye’s model is equivalent to approxima-
tely 14 mm per day. It should be noted that the closure
rate defined by Nye’s model (equation 1) refers to a
relative radius reduction in an exponential decay of
the tunnel radius due to ice pressure; the rate of tunnel
closure therefore decreases as the radius of the tunnel
becomes smaller.
B) TUNNEL EXPANSION BY MELT
WIDENING
Offsetting tunnel closure by deformation of ice is the
process of tunnel expansion due to melt-widening.
Measurements taken during the summer of 1989,
1990 and 1996 and during a winter visit to the site
in 1991 indicate that the maximum likely discharge
from the river draining Jökulsárgil is approximately
12 m3s-1 and that the minimum is roughly 2 m3s-1.
Hooke’s (1984) model of melt-widening is given by
the following equation (neglecting a small term due
to the depression of the melting point of ice due to the
ice overburden pressure).
where:
rh = melt rate
g = gravitational acceleration, 9.82 ms-2
Ds = diameter of tunnel
H = heat of fusion, 3.34x 105 Jkg-1
pw = density of water, 1000 kg/m3
Pi = density of ice, 900 kg/m3
Q = discharge
/3 = slope of conduit
Hooke’s (1984) model was applied to the tunn-
el at Sólheimajökull. The model predicts a melt rate
of 21 mm per day at the minimum water flow rate of
2 m3s-1 and up to 124 mm per day at the maximum
discharge of 12 m3s-1.
C) IMPLICATIONS FOR TUNNEL DYNAMICS
AT SÓLHEIMAJÖKULL
Even under conditions of low river discharge
(2 m3s-1) the rate of melting of ice from the tunnel
walls is sufficient to offset the closure of the tunn-
el due to ice overburden pressure (a melt rate of
21 mm against a tunnel closure rate of 14 mm).
Thus the models predict that the tunnel at Sólheima-
jökull will not seal. This is not in agreement with
geomorphological observations, which indicate than
an ice-dammed lake is occasionally formed in Jök-
ulsárgil at present (as further discussed below). The
22 JÖKULL No. 48