Tímarit Verkfræðingafélags Íslands - 01.12.1983, Page 17

On the basis of simple M2-tidal
theory, the volume strain at the surface
is b = -0.25 Ag/g, and we can then
define an apparent formation bulk
modulus
ka ■ 4g(Ap/ag)0 , (103)
where (Ap/Ag)0 is the observed tidal
admittance. Using Hanson’s (1979a) ad-
mittance data, we obtain for the
M2-tides at the seven boreholes.
Since the RR holes open at depths of
the order of 1 to 1.5 km into a valley fill
consisting of tuffs, siltstone and gravel,
we expect that the local wet rock bulk
be exactly in phase with the negative
tidal potential. Hanson (1979) has,
however observed a phase advance of
about 25° at RRGE-1 and a phase delay
of 15° at RRGE-2. These deviations
may be statistical and/or of a structural
nature as pointed out by Hanson
(1979a).
The SS-Elmore-3 well displays an ab-
normally low value for the apparent
bulk modulus. This may result from a
very low matrix coefficient and/or an
open box situation. Furhter data from
the Salton Sea system are needed.
(C) Related theoretical development.
Sneddon and Lowengrub (1969) give
the following formulas for the penny-
shaped cavity of radius R. The displace-
ment of the cavity wall at the center due
to an internal pressure p is given by
U0 = 4(1-v2)RP/tiE (105)
where v is Poisson’s ratio and E the
Young’s modulus. The strain energy is
r ■ 8(1 -v2)p2R3/3E (106)
and the cavity volume elastance is then
simply
Ev ■ (l/pldr/dp (107)
Table III
Apparent bulk tnoduli for M2-tides at Raft River and the Salton Sea
Borehole RRGE-l RRGE-2 RRGE-3 RRGE-4
2.5 1.2 1.2 2.0 x l010Pa
RRGI-6 RRGI-7 SS-Elmore-3
1.2 0.5 0.1 x l010Pa
modulus is of the order of 2 to 3 X 1010
Pa. The apparent bulk moduli values
obtained for RRGE-1 and RRGE-4 fall
within this range. Moreover, RRGE-2,
RRGE-3 and RRGI-6 yield values that
are only slightly below the estimated wet
rock modulus. We can thus conclude
that the matrix coefficient e as defined
by equation (56) above is close to unity.
Moreover, this indicates that the Raft
River geothermal system is consistent
with the closed box model set forth in
section (10) above. Neither the recharge
channels, cap rock nor the upflow chan-
nels provide a sufficient fluid conduc-
tivity to attenuate the tidally induced
pressure oscillations in the main reser-
voir and heating zone. RRGI-7, never-
theless, displays somewhat lower values
which could be the result of either a
lower local matrix coefficient or a local
opening to the surface that can cause an
enhanced attenuation.
Because of the large T-factor, we
would expect the borehole pressure to
12. ELASTOMECH ANICS
(12.i) Symmetric cavity parameters.
The parameters displayed in Table I are
obtained on the basis of formulas given
by Love (1927, pages 142 and 144) for
radially and axially symmetric elastic
systems. To indicate the procedure, we
will turn to the spherically symmetric
case.
Consider a spherical shell in rn<r<r2
with Lame’s parameters k = p
(Poisson’s relation) that is acted on by
in internal pressure p, and an external
pressure p2. The resulting displacement
u(r) is purely radial and is obtained on
the basis of the formula
u(r) - (r/5M)(piri3-p2r23)/(r23-ri3) +
(l/4r)(r23r13/r2)(p1-p2)/(r23-r13) (104)
A similar formula is available in the axi-
symmetric case. We make then the
assumption of no axial strain.
1. REFERENCES
Bodvarsson, G., 1970. Confined fluids as strain
meters, J. Geophys. Res., 75 (14):2711-2718.
Bodvarsson, G., 1978. Convection and Thermo-
elastic Effects in Narrow Vertical Fracture Spaces
with Emphasis on Analytical Techniques, Final
Report for U.S.G.S., pp. 1-111.
Bodvarsson, G. and J. Hanson, 1978. Geother-
mal Reservoir Testing Based on Signals of Tidal
Origin. Workshop on Geothermal Reservoir
Engineering, December, 1978, Stanford University,
Stanford, CA, J. Eng. Ass. lceland, 66, (2), pp.
28—29.
Hanson, J., 1979a. Tidal Pressure Response Well
Testing at the Salton Sea Geothermal Field,
California, and Raft River, Idaho. Report to
Lawrence Livermore Laboratory.
Hanson, J., 1979b. Tidal Pressure Response as a
Reservoir Engineering Tool, Geothermal Resources
Council, TRANSACTIONS, Vol. 3, September,
p. 291—293.
Hanson, J., 1980. Reservoir Response to Tidal
and Barometric Effects. Report to Lawrence Liver-
more Laboratory.
Love, A. E. H., 1927. A Treatise of the
Mathematical Theory of Elasticity, New York,
Dover Publications, 4th ed., 643 pp.
Sneddon, I.N., and M. Lowengrub, 1969. Crack
Problems in the C/assical Theorv of Elasticity, John
Wiley & Sons, Inc., 221 pp.
Sunde, E. D., 1968. Earth Conduction Effects in
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Takeuchi, H., 1950. On the Earth Tide of the
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Acknoledgement
This work was supported by the Lawrence
Livermore Laboratory of the University of
California, CA, U.S.A. under DOE Con-
tract W-7405-ENG-48.
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