Tímarit Verkfræðingafélags Íslands - 01.12.1983, Blaðsíða 10

(20)
P * (1/Ec)(m/D) - Ya
It is convenient to express a in terms
of the volume strain of the surrounding
rock b such that
P = (1/EC)(m/o) - ykrb (21)
and hence is the case of pure strain
where m = 0
P ■ -Ykrb (22)
4. THE ULTRASIMPLE TIDAL
TEST ON A BOREHOLE-CAVITY
SYSTEM
Upon the results obtained above, we
are now in the position of considering a
tidal test on the simple system shown in
Figure 1 consisting of a borehole con-
nected to a single cavity at some depth
in a formation of neglibible permeabili-
Figure I. Borehole-cavity system.
Tidal forces cause a periodic straining
of the formation resulting in a periodic
oscillation of the cavity volume and
thereby an oscillation of the free liquid
level in the borehole that can be observ-
ed.
Let a be the imposed (isotropic) for-
mation stress, p the liquid pressure in
the cavity, p the density of the liquid, V
the cavity volume at p = 0, h the liquid
level in the borehole as measured
relative to the average level and f be the
cross section of the hole. Neglecting in-
ertia forces and flow resistances, we
have then on the basis of the above
notation
p ■ gph (23)
and the conservation of liquid volume
oDy + p(Ev + cfV)+ fh = 0 (24)
We abbreviate the expression in the
parenthesis by the cavity capacitance
Ec(Ev + cfV) and insert p from (23) such
that
h ■ -oDv/(qPEc + f) (25)
which can be rewritten
h * -h0/(l+T) (26)
where
h0 - oDv/f (27)
is the cavity volume increment
measured as a water level and T is the
tidal factor
T = gPEc/f (28)
It is sometimes convenient to restate
(26) as
h - -h-|T/(l + T) (29)
where h, is now to be taken as a
pressure amplitude measured in terms
of a water level defined by
h-| = oDv/ppEc = oy/pa (30)
Equations (26) and (29) are the main
results. Two extreme cases can now be
considered. Let T»l, that is, the
cavity be very soft, and we obtain then
h = h,, that is, the water level furnishes
a measure of the formation stress (if ev
known). In the case of flat cavities of a
very small aperture, Dv is about equal
to Ef and then h = h, = a/gp. The
borehole-cavity system with T>$>1
therefore acts as a stress-or pressometer.
Considering, on the other hand, the
very stiff cavity where T<SC1, we find
that h = -erDv/f and the water level is
then directly proportional to the
negative volume increment of the cavi-
ty. The borehole-cavity system with
T «1 acts the as a dilatometer. The
difference between these two cases is
important and indicates the type of data
that can be obtained on the basis of
tidal testing.
It is of interest to obtain some picture
of the cavity dimensions involved in the
case T = 1 that is intermittent between
the above extremes. Consider the case
of the penny-shaped cavity of radius R
with the volume elastance given in Table
I. We assume that the borehole is of
standard dimensions with a diameter of
0.25 m and hence f = 0.05 m2. On the
basis of (28) the case T = 1 then implies
Ec = 5xl0‘6 kg/s2. Assuming a small
cavity aperture such that fluid com-
pressibility can be neglected,
Ec = EV2RJ/p. Let p = 2xl010 Pa and
we find then that R = 37 m. Because of
the third power of the radius, cavities
with RS 25 m will be stiff whereas
cavities with RS50 m will be soft.
It is frequently convenient to rewrite
equation (28)
T = MS (31)
where M = pEc is the total cavity mass
elastance and
S = g/f (32)
is the borehole stiffness (see Bodvarsson
and Hanson, 1978).
In this form the above equations are
valid for more general cases of borehole
stiffness.
Finally, it should be pointed out that
on the above non-resistive model, the
borehole water level oscillates in phase
with the imposed stress. Introducing a
flow resistance, for example, at the
borehole cavity connection, leads to a
phase lag. In the case equation (23)
would have to be restated as
p = gph + (eq/f) (33)
where q is the mass flow from the cavity
to the borehole and /? is an appropriate
friction factor that has the dimensions
of a velocity. Since q/pf=dh/dt, we
obtain for (23)
P = gph + BpDh (34)
where D = d/dt. Assuming all variables
oc exp (i u t) we obtain in amplitude
form an equation corresponding to (26)
above
h = -hQ/(l + T + iT,), (35)
where the resistive part of the tidal fac-
tor is
T-| = poiBEc/f (36)
The phase lag caused by the flow
resistance is
< = tg'1(T^/(1+T)1 = tq'1[uB/(g+(f/pEc))] (37)
5. POROUS FORMATION
CAPACITIVITY AND PORE
PRESSURE REACTANCE
In the present context, we are in-
terested in the elastic response of porous
rock to both external strain and internal
pore pressure. The parameters of par-
ticular importance are
90 — TÍMARIT VFÍ 1983