Tímarit Verkfræðingafélags Íslands - 01.12.1983, Side 13

A = As + aQS (69)
It is important that when R/ds« 1, the
term As dominates in (68) and (69). In
this limit, the cavity admittance is the
same as for the steady state case.
Equations (62) and (65) show that an
oscillatory driving point test of the half-
space when the amplitudes p0 and q0
can be observed yields the oscillatory
surface admittivity «0 and since u will
be known, we are then able to derive the
formation characteristic admittivity on
the basis of (64). The situation is thus
different from the steady state cavity
case where the fluid conductivity c is be-
ing obtained. As already emphasized,
an oscillatory driving point spherical
cavity test is equivalent to the steady
state test when (R/ds)«l.*
Since the oscillatory surface driving
point flow/pressure test of the Darcy
half-space yields only the formation ad-
mittivity a that depends on the product
(cs), further test data of a different
nature are required to separate the two
factors. The most obvious way of ob-
taining additional data is to extend the
oscillatory driving point test to a trans-
fer test by observing the pressure
amplitude p(z) at a depth z in the solid.
An integration of the one-dimensional
form of (59) shows that
p(z) = pQexp[-(l+i)z/ds] (70)
An evaluation of either amplitude or
phase relations of the transfer test will
thus yield data on the skin-depth
ds = (2c/ p s u )1/2 and thereby on the
ratio (c/s). On the basis of (60) and (61)
we find that
c * s.Afw/2)^ (71)
and
s = (a/pds)(2/u,)J (72)
The separation of the parameters c and
s can thus be obtained on a combined
driving point and transfer test.
(6.iv) Oscillatory strain tests. Another
way of testing the Darcy half-space is to
subject it to transitory, in particular,
oscillating strain that pumps fluid
in/out of the surface. The surface
pressure is maintained at a constant
level that can be taken to be zero. In this
case we assume that the imposed effec-
tive volume strain is known and that the
amplitude of the resulting surface llow
* Phase shifts are observed wlien (R/d^) is
not small compared to unity.
density can be observed. Let b be the
uniform imposed volume strain and <5
be the dilaticity number as defined by
equation (52) above such that the effec-
tive pore strain is <5b. The equation for
the fluid pressure in the solid is then ob-
tained from (21) above
3tp ' a3zzp = -(s/s)3tb (73)
where z is the coordinate into the solid.
Since ðb is assumed to be uniform, this
equation is solved by the substitution
u = p - (öb/ s) (74)
such that
8tu ■ a8zzu = 0 (75)
and the surface boundary condition is
thus
u = -(6 b/s), z = o (76)
It is thus evident that the transitory or
oscillatory strain test where the surface
flow is the field observable is completely
analog to the above driving port test.
The amount of information is the same.
Obviously, by observing the pressure at
a point inside the solid, the strain test
can be extended to a transfer type of
test. The same applies in the case of the
cavity.
7. THE TIDAL TEST IN POROUS
FORMATIONS
(7.i) The simple field model. The
theory of Darcy flow reservoir testing
on the basis of volume strain imposed
by the solid earth tides has been discuss-
ed in a number of papers. Here we will
refer to the papers by Bodvarsson
(1970) and by Bodvarsson and Hanson
(1978). Omitting details, we will below
very briefly state the main results.
The field situation discussed involves
a vertical borehole (see Fig. 3) of cross
section f that has been drilled into a
fluid saturated porous half-space where
Darcy type flow can be assumed. The
borehole is filled with liquid up to a cer-
tain level and is connected at the depth
H to the formation by a spherical cavity
of radius R. The hole is cased down to
the cavity.
A uniform oscillatory strain of tidal
origin is imposed on the system causing
oscillatory movement of the borehole li-
quid level that can be observed. Let h be
the amplitude of this movement. Apply-
ing the same notation as above, and
assuming (R/ds)«l, but (ds/H)«l,
we find that to the first order,
h = -(sb/Pgs)T/(l+T). (77)
where g is the acceleration of gravity
and T is the tidal factor
T - -iAsS/o. (78)
The quantity As = 47tcR is the steady
state admittance of the cavity and
S = g/f (79)
is the stiffness of the borehole with a
free liquid level. It is to be noted that
(78) holds for more general cavity
shapes provided the linear dimensions
are small compared to the skin depths
ds. Moreover, (78) holds also for a dif-
ferent type of stiffness. In the case of a
closed liquid filled borehole with a total
liquid mass M
S « dp/dm = l/cfM (80)
where Cf is the compressibility of the li-
quid.
Assuming that (R/ds)«l, the
observation of h at two different fre-
quencies will, in principle, yield data on
the effective pore pressure amplitude
(<5b/s) and the tidal factor T and
thereby on the admittance As. Knowing
the cavity form, the admittance can
then furnish data on the formation con-
ductivity c.
Analog to the case discussed in sec-
tion (4) above, the resolution of the
tidal factor breaks down when T7Í>1
and the system is then a pure presso-
meter. As a matter of fact, the resolu-
tion of T in open hole tidal tests in
porous formations is limited to forma-
tion permeabilities below a few hundred
millidarcys.
(7.ii) The simple lumped model. It is of
some conceptual importance to show
that the mechanism of the above tidal
Figure 3. Simple field inodel, borehole open
ing into a splierical cavily.
TÍMARIT VFÍ 1983 - 93