Jökull - 01.12.1984, Side 44

Mgv = (eMa-a(Mt-Ma) + (Lj+e+a)Mi) / Lv (6)
Mgv = Mt - Ma - M, - Mgv (3)
where
a = c2(T3-T2) and e = c^T^-Ti).
The equations above have been used to calcu-
late the unknown masses, the steam fraction x =
Mgv / (Mgv+Mgw), and the terms in the energy
equation. Fig. 9 shows how the results vary with
the geothermal mass fraction k. These computa-
tions were done for the temperature at the bot-
tom of the lake equal to the boiling temperature
at 30 bar overburden pressure, that is T3 = 235
°C. The average temperature of the lake was
assumed to be T2 = 6°C. (The choice of T2 is not
critical for the results, in the possible range from
0°C to, say 10°C). Other values used in the com-
putations were: Mt = 6.6-1011 kg/yr, Ma =
1.5-1011 kg/yr, T, = 0°C, L, = 335 kJ/kg, c, =
4.218 kJ/kg K, c2 = 4.245 kJ/kg and Mr = 0.M011
kg/yr (NEA data, Sigurjón Rist pers. comm.),
and, Lv = 1770 kj/kg.
The results in Fig. 9 show that the energy
balance limits k to the range from 0.09 to 0.21.
For k=0.09 the entire discharge to the lake must
be steam if the required energy is to be provided.
For k = 0.21 no steam could be present as heat
from the geothermal water provides all the
required energy. As k increases the mass of
geothermal water is increased but the mass of
steam is decreased. But when the mass of steam
is reduced by a certain amount the mass of water
is increased by (1+Lv/a) times (typically three
times) this amount. Therefore, the mass of ice
(Mj) required for melting is reduced as k
increases and the total thermal power of the
geothermal area is reduced from 5300 MW to
4300 MW (see Fig. 9).
Estimates of geothermal mass fraction.
Further interpretation of the calculations, illus-
trated in Fig. 9, depends on the estimate of the
mass fraction k for the Grímsvötn geothermal
system. This estimate can be done with the aid of
equation (1) which applies for a substance whose
concentratíon is neither changed during storage
in the Grímsvötn lake nor when the water runs
subglacially to Skeidarársandur. The geothermal
mass fraction could be estimated if we knew the
original concentration of all five components.
Consider a non-volatile substance whose con-
centration in the geothermal vapour (Cgv) is
negligible compared to that of the water phase.
Further the concentration in the meltwater com-
ponent at Grímsvötn is equal to the normal con-
centration in the rivers on Skeidarársandur.
From equations (1), (2), (5), (6) and (7) we
obtain for Cgv=0 and Ca=Ci=Cr the estimate
k=(A+S)/(R+B) (8)
where
A=(e Ma + (L( + e)(Mt - Ma))/Lv (9)
B = (Li + e + a)(Mt + Mr)/Lv (10)
R=(Mt + Mr)(Cgw - Cr)/Cgw (11)
S=(Mt + Mr)(Cj - Cr)/Cgw (12)
and e=Ci(T2-T|) and a=c2(T3-T2). All parameters
and numerical values are the same as in the
section above.
We believe this model applies for the poorly
soluble silica. Ice melted in the geothermal area
is originally precipitation and contains small
amounts of dissolved silica (<1 mg/kg). How-
ever, meltwater from the surface as well as the
bed may have reacted with rocks on the way to
the lake in a similar way as has water discharged
into the glacier rivers. For Skeidará the normal
concentration of silica is Cr=10-20 mg/kg (see
Fig.5) when the river is not influenced by jökul-
hlaups. The mean concentration of silica is 13±
-5 mg/kg for rivers in Iceland (NEA data). The
measured concentration of silica is Cj=44-60 mg/
kg in water from jökulhlaups (Table 2). In most
jökulhlaups the silica concentration has been
near 60 mg/kg; this value Cj=60 mg/kg and Cr=
13 mg/kg is used in our calculations. The com-
putations show that the geothermal mass fraction
k increases from 0.12 to 0.18 as the concentration
Cgw in the geothermal water entering the lake
decreases from 800 to 300 mg/kg.
The concentration of silica in the geothermal
fluid can be estimated on the basis of assumptions
about the likely reservoir properties at Gríms-
vötn. The silica concentration in high-tempera-
ture areas is controlled by the reservoir tempera-
42 JÖKULL 34. ÁR