Jökull - 01.12.1966, Side 52

to the sqtiare root of the recovery time, if the
flow is kept constant.
It is obvious that these conditions will again
confront us with an optimization problem, that
is, the problem of recovering the optimum
amount of energy from a given source. A study
of this problem is as yet premature, mainly
because of the almost complete lack of field
data. In order to obtain the long-term tempera-
ture behavior of producing geothermal areas,
very precise temperature measurements must
be made at the well-heads. No suc.h data are
available as yet.
Moreover, the model presented in section
(2.2) is probably too simple for practical pur-
poses. One of the main complications is the
possibility of short-cuts of cold water into the
producing zones during the pumping of wells.
This phenomenon has not been observed so
far, but it is a real possibility and mav limit
the permissible draw-down in wells.
But in order to obtain a qualitative feeling
for the conditions involved, we may take an
elementary look at the problem, and derive the
optimum flow policy in the most simple cases.
Suppose that we have a source which, at a
constant production during a time interval T,
will produce a total volume of thermal water
of QT11 with a temperature above a certain
minimum. The figures Q and n are constants.
The tliermal water will be used for heating
purposes. After the tirne T the temperature
drops below the minimum, and we will assume
that the water cannot be utilized at the lower
temperature. In the case of the model present-
ecl in section (2.2) we have n = í/ó. In the
case of a given fixed volume of water we have
n = 0.
The flow from the source is QTn-1 ancl
the discounted total profit, or fuel savings,
during the interval T is
SQT”-i / exp(— rt)dt =
‘° (6)
SQTn-l (1 — exp(— rt))/r,
where S is the profit per unit volume, and r
is the interest rate.
We will assume that the capital requirements
of tlie heating plant are proportional to the
power, that is, proportional to the flow. In the
case of geothermal heating plants this is a
fairly goocl assumption. Thus, the total invest-
ment will be KQTn-i where K is the invest-
ment per unit flow.
The time interval T defines the optimaf flow
policy, and we have to maximize the following
quantity:
SQT--1 (1 exp( rT))/r -KQTn-i
The maximum is obtained for a T satisfving
the equation
(1 + Tr/(1 - n)) exp(— rT) = 1 - rK/S.
In the case of geothermal heating plants in
Iceland the ratio K/S can be assumed to be
around 5. This will be the case for thermal
water with a temperature of 100° C. In the
case of n = i/9 we find that T 1,9/r. At an
interest rate of 7% per annum we have an
optimum recovery time of 27 years. In the case
of a fixecl amount of water, that is. n = 0, we
would have obtained T = 16 years.
REFERENCE:
Bodvarsson, G. 1963. An Appraisal of the
Potentialities of Geothermal Resources in
Iceland. Tímarit Verkfraedingafél. Islands.
Vol. 48, nr. 5, 1-7.
— 1966. Direct Interpretation Methods in Ap-
plied Geophysics. Geoexploration, Vol. 4,
113-138.
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