Jökull - 01.12.1982, Page 26

B. The outward conduction flow of heat in the
region is gk where g is the vertical temperature grad-
ient such that the undisturbed temperature at the
depth D is Tg = gD. We place thex-coordinate in the
direction of the flow. Let T (x) be the temperature of
the water and as'sume T(o) = 0. Since the water
takes up heat from below, and loses heat to the
surface, the differential equation for the tempera-
tures is
cm(dT/dx) = k[g-(T/D)] (5)
where c is the specific heat of the water. It is con-
venient to introduce the coefficient b = kL/cmD
such that (5) can be restated as
(dT/dx)+(b/L)T= (b/L)T0 (6)
This equation has the solution
T = T()[I-exp( -bx/L) ] (7)
The temperature at the end x = L is thus
T = eT0 (8)
where
e = l-exp(-b) (9)
which can be inverted
b = -ln(l-e) (10)
We note that
b = kL/cmD = kLB/cm BD = k/cqD (11)
where q is the mass flow of water per unit sheet area.
The factor e can be interpreted as the end tempera-
ture ratio and
e =-e/ln(l-e) (12)
is then the efíiciency of heat recovery.
In the Icelandic environment k = 2 W/m °C
approximately and since c = 4 X lCf3 J/kg °C we
find that b = 5 X 10_4/qD. Equation (10) can then
be restated
q = -5 X 10'4/Dln(l-e) (13)
The honzontal sheet in a transient situation. This case
has been discussed at some length by Bodmrsson
(1974), and we will therefore only restate the main
results.
We assume now that the sheet was embedded at t
= 0 in an infinite homogeneous conducting space
and that q is a constant mass flow per unit sheet
area. Using the same notation as above and assum-
ing the same values for c and k as above and a =
10~6m2/s Bodvarsson (1974) finds that
q = l/2t1/2erf_1(e) (14)
where erf“1 () is the inverse error-function. Since the
outflow temperature is a function of time, e is the
present value of the temperature ratio. In the case of
a half-space, equation (14) is valid for t<D2/a
where D is again the depth of the sheet.
Equations (13) and (14) both furnish expressions
for the specific thermal water productivity per unit
sheet area. To obtain the total sheet area required
for a given mass flow at given conditions, we have
only todivide the mass flowby q.
Convective downward migration of vertical fracture
spaces. Because of the coupling of fluid convection
and thermoelastic fields, the theory of CDM is very
complex and we have, as ofnow, little quantitative
understanding of the phenomenon. Bodvarsson
(1982a) has, however, arrived at an estimate of the
rate of CDM and of the uptake of heat by the
convecting water. The rateofdownward migration
v is estimated by
v~a/d (15)
where a is again the thermal diffusivity of the for-
mation and d is an appropriate length that is of the
same order as the penetration depth of the down-
ward migrating thermal conduction front. This
quantity enters into the equation for the tempera-
ture front in the following way. Let the migrating
front temperaturediflerential be AT, and the equa-
tion for the front is then expressed by AT exp(-z/d)
where z is the depth coordinate. It is likely that d is
of the order of KTmeters resulting in the estimate v
~ 10“6/102 = 10"8m/s = 0.3 m/year. This figure
appears consistent with observations in the Reykja-
vik geothermal system in southwestern Iceland
(Bodvarsson 1982b).
Based on the above concepts Bodvarsson (1982a)
estimates the rate of heat uptake in CDM by the
expression
H = 2C(avM)1/2 (16)
where H is now the rate of heat uptake per unit
horizontal fracture length, C is the average sensible
volumetric heat capacity of the formation relative to
the average temperature of the convecting water
and M is the depth ofmigration that has taken place
at the time H is being estimated by (16).
THE REYKHOLTSDALUR
GEOTHERMAL SYSTEM
The thermal area in Reykholtsdalur in Western
Iceland is one of the most active LT fields in Ice-
land. An overview of the locations of the hot springs
and their relation to local geological features is giv-
24 JÖKULL 32. ÁR