Tímarit Verkfræðingafélags Íslands - 01.12.1983, Blaðsíða 14

test can be easily explained on the basis
of the simple lumped linear two-
capacitor model portrayed in Figure 4
below. The large capacitor represents
the formation and the small one the
borehole that acts as a pressometer.
Both are filled with a compressible li-
quid connected by a linear conductor
that represents the cavity at the bottom
of the hole. The solid earth tides impose
oscillations of the volume of the large
capacitor, causing pressure oscillations
within the system that are observed in
the pressometer.
Let p, be the pressure and C, the
capacitance of the large capacitor.
Moreover, let p2 and C2 be the same
parameters for the pressometer, A be
thc conductance of the conductor con-
necting the two elements and V be the
amplitude of the volume oscillation of
thc large capacitor. We have then the
simple cquations for the mass balance
C^Dp^ = A(p2 - pj) + Vexp(iut), (81)
C2Dp2 = A (p, - P2) (82)
D = d/dT (83)
In terms of amplitudes we find on the
basis of (81) and (82) that
p2 = AV/MuC, + A)(iwC2 + A) - A2] (84)
Assuming now that the capacitance of
the large capacitor C, and the conduc-
tancc A satisfy thc inequality
uC^ >> A (85)
we obtain the pressure amplitude in the
pressometer
P2 = AV/iuC^iuC, + A) (86)
that can be rewritten
P2 = P0T/(HT) (g7)
where
Po-ÍV/luC,) (88)
is the unperturbed pressure amplitude in
the large capacitor and
T = -iAS/w (89)
where
s = 1/c, (90)
is the stiffness of the large capacitor.
Equation (89) is of the same form as
(78) above.
The 2-capacitor lumped models is
relevant only to porous formation cases
of a large ds and dsJ is about dimension
of large capacitor.
More complicated systems are easily
approximated in terms of networks.
8. STRESS/STRAIN
ENHANCEMENT
Elastic inhomogeneities lead to varia-
tions in the stress and strain fields. The
most prominent stress enhancement ef-
fects are found at various types of high
bulk modulus inclusions where stress
focusing such as bridge and notch ef-
fects can be prominent. Enhanced strain
is associated with inclusions of high
compressibility. A few details will be
discussed below.
(8.i) The compacl inclusions. The case
of stress enhancement at an im-
permeable spherical inclusion has
already been discussed in section (3.iv)
above. Equation (17) shows that a stress
enhancement by a factor of 1.8 can be
obtained at and in a spherical inclusion
that has a considerably higher bulk
modulus than the surrounding rock.
Moreover, considering a spherical inclu-
sion of high compressibility we find on
the basis of Table I that the dilaticity
may reach values of 1.35/// where p is
the shear modulus of the surrounding
rock. The dilaticity in terms of the bulk
modulus k is 2.2/k. Assuming Poisson’s
relations, the strain enhancement factor
for the inclusion can tlius reach a value
of about 2. These are fairly prominent
effects that result from stress focusing.
(8.ii) Fault zones. The tidal dilatance of
major fault zones is a matter of par-
ticular interest. Obviously, the fault
gouge is a flat slab-like inclusion thal
represents an inclusion with elastic pro-
perties that usually will be different
frorn those of the surrounding rock.
Moreover, the gouge is frequently
permeable to water and the tidally in-
duced fluid movements may affect the
stress field. We are thus confronted
with a very complex situation with the
possibility of significant fluid
pressure/rock stress coupling. More-
over, it is obvious that the orientation
of the fault to the imposed stress field is
of great importance. Some aspects of
the mechanics involved are discussed in
section (13) below. As a main result the
following estimate is given for the frac-
ture zone dilatance under a periodic (in
time) stress field that is perpendicular to
the plane of the fracture zone.
Stated very briefly, the fracture zone
acts as a slit that has an active depth
that probably can be taken to be of the
same order as the hydraulic skin depth
ds for the fracture gouge at the frequen-
cy of the imposed stress. Correcting
(133) by a factor of 1.5 the dilatance of
the fracture width w as measured at the
surface is then
Dw = 1-3ds/u (91)
where p is the shear modulus of the sur-
rounding rock. At a stress amplitude of
ct, the total width increment is thus
Aw = 1.3dsa/y (92)
and the dilativity thus
dw = l.3ds/wu (93)
Consider now a tidal stress field of an
amplitude 10’ Pa that is perpendicular
to the fracture zone. Since (ds/w) may
frequently be of the order of 10, and
// = 2x 1010 Pa can be assumed, equa-
tion (93) gives a strain of I0"6. The
strain enhancement across a fauit zone
as measured at the surface may thus
reach values of 102. The figure depends,
of course, critically on the skin depth of
the fault gouge that is a not too well
defined and obviously poorly known
parameter.
(8.iii) Bridge and notch effects. Closely
associated with the above phenomena
are various types of stress focusing by
bridge and notch effects. For example,
a small section of the gouge in a fault
zone may have a relatively high Young’s
modulus. This leads to a stress focusing
by a bridge across the fault zone. More-
over, there obviously is the possibility
for a substantial stress notch effect at
the active depth (about equal to skin
depth) of the fault zone.
The bridge and notch effects are
notoriously difficult topics, in par-
94 — TIMARIT VFI 1983