Tímarit Verkfræðingafélags Íslands - 01.12.1981, Blaðsíða 18

the water level graph given in Fig. 3, we
derive an initial rate of drawdown in
response to the increased production of
u = 1.1 x 10'5 m/s. Assuming that the
production depth is about 10Jm we find
on the basis of equation (1) an estimate
of the porosity of about = 2 x 10'3.
Since equilibrium was not approach-
ed during the present event, we are
unable to apply equation (2) to obtain a
numerical estimate of the fluid conduc-
tivity c. However, since the graph in
Fig. 3 furnishes us with a lower bound
hm for the stationary drawdown hs, we
can convert equation (2) to an inequali-
c^V/2rcghmd (3)
and obtain an upper bound for c. Tak-
ing on the basis of the graph hm = 60
m, we obtain the upper bound of
4x10"8s. Assuming the kinematic
viscosity to be 3 X 10'7 mVs (100°C) we
obtain that the permeability k£í
1.2xl0'14m2 = 12 millidarcy.
The above estimates of the global
permeability turn out to be two to three
orders of magnitude lower than the
values quoted above as the results of the
short-term well interference tests. There
appears to be an inverse relation bet-
ween the time scale of the test signal and
the magnitude of the estimate. The
longer the time scale, the lower the
estimate.
Although a much more elaborate
analysis of the Laugarnes data is in-
dicated, the above discrepancies may be
quite real and reflect the very con-
siderable heterogeneity and fracturing
of the reservoir. Due to local intercon-
nection by fractures, the short-term in-
terference tests performed on adjacent
wells give much higer permeability
estimates than the more integrated
global values obtained with the help of
the total production rate and well data.
On the other hand, we have to em-
phasize that the free surface test has a
strong bias toward the uppermost sec-
tion of the reservoir, and will preferably
reflect average conditions in the reser-
voir cap.
Moreover, we wish to point out that
our identification of the piezometric
surface with the true free surface level is
subject to doubt. The validity of the
assumption depends strongly on the
distribution of flow conductivity of
openings in the observational wells.
The result indicates that considerable
caution is called for in the interpretation
of relatively short-term well interference
tests on complex reservoirs.
SUPPLEMENT
In processing the periodic data, we
can also base our estimates on a lumped
model as shown in Fig. 4. The model is
characterized by a single input conduc-
tance K and a single capacitance S. In
the physical sense, the capacitance
simply represents the effective pore area
of the container shown in Fig. 4. Let f
be the volume rate produced from the
container, h be the average liquid level
in the container counted positive down
and assuming that the ambient water
level is zero, we arrive at the following
equation governing the lumped system
S(dh/dt) + Kh = f (4)
In the case of periodic flow f =
Fexp(iwt) where F is the amplitude and
to is the angular frequency. Let the
response of the liquid level be h =
Hexp (i(tot-a)), and inserting in equa-
tion (4) the resulting output-input
amplitude ratio is found to be
H/F = (K2 + S2 co2)-‘/2 (5)
and the phase angle
a = tan''(Sco/K) (6)
From the graphs in Fig. 3, we find that
we can on the average take F =
0.07 mVs, H = 19 m and a = 0.78 ra-
dians. Solving equations (2) and (3) for
the system parameters we obtain
K = 2.7 x 10'3m2/s = 2.5 x lO^Kg/sPa,
S=1.4xl04m2 (7)
To translate these results into estimates
of the average permeability k and
average porosity <p we observe that the
ground water level depression in Fig. 2
has the shape of a slightly elongated flat
disk with an area of approximately A =
4 km2. For the present purpose we
replace this disk by a circular one with a
radius R = 1.13 km. On the basis of
simple potential theoretical relations
(Sunde, 1968), we find that the contact
conductance of a flat circular disk of
radius R immersed in a porous medium
of fluid conductivity c is simply 8cR. In
the present case, where the disk is plac-
Fig. 3. Hydrographs of welts G7 and G16 and monthly withdrawals of water from 1965 to
1969.
From: Thorsteinsson and Eliasson (1970).
PRODUCTION
Fig. 4. Lumped model of input conductance K and capacitance S.
94 — TÍMARIT VFÍ 1981