Jökull - 01.12.1976, Page 24

Fig. 2. Maximum entropy power spectrum o£
the record in Fig. 1. Prediction error filter
length =10 and D.C. component removed.
Mynd 2. Aflróf hámarksentropi límiritsins í
Mynd 1.
THE BOREHOLE-CAVITY RESONATOR
The simple model to be investigated here
and sketched in Fig. 3 consists of a homogene-
ous and isotropic linearly (Hookean) elastic
half-space with a cavity at depth which is con-
SURFACET
Fig. 3. Borehole-cavity resonator with a flat
circular cavity.
Mynd 3. Sveiflukerfi borholu, sem tengd er
flötu hringlaga holrými.
22 JÖKULL26. ÁR
nected with the free surface by a vertical pipe
or borehole. For the present purpose we will
assume that the cavity is a flat horizontal cir-
cular fracture of radius R and having a very
small width b. The system contains a fluid
such that the cavity is filled and there is a
fluid column of height h in the borehole. The
cross section of the borehole is a.
In order to avoid mathematical complexities
of little relevance for the main physical pheno-
mena of interest, we will make the following
simplifying assumptions:
(1) The depth of the borehole is substantial-
ly greater than the radius R such that we can
assume that the cavity is embedded in an in-
finite elastic space. The assumption of a hori-
zontal cavity is then theoretically strictly not
necessary. Our final results will hold for any
other position provided the cavity is connected
with the borehole.
(2) On the other hand, the dimensions h and
R are limited and b so small (b« R/100) that
the compressibility of the fluid can be neglect-
ed. Hence, during oscillations, the differential
fluid pressure p is assumed to be constant
throughout the cavity and equal to the differ-
ential pressure due to the fluid column in the
borehole.
Fig. 4. Coordinate system placed in the walls
of the cavity.
Mynd 4. Hnitakerfi sett i vegg holrýmis.
(3) Let the volume of the cavity be V. In
order to obtain the volume elastance e = dV/dp
we place as shown in Fig. 4 a cylindrical co-
ordinate system with the radial coordinate r in
each wall of the cavity such that the z-axis is
vertical into the solid. We assume that because
of the flatness of the cavity, the elastic displace-
rnent vector in the adjacent rock is of the form
u = (0,0,w(P,t)) where P = (r,z). In other words,
the displacement in both walls of the cavity
has only a vertical component w(P,t). More-
over, we assume symmetry about the center
plane and that