Jökull - 01.12.1976, Síða 25

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Fig. 5. Resonance period for the borehole-
cavity system according to equation (16).
Mynd 5. Eiginsveiflutími borholukerfis sam-
kvæmt jöfnu (16).
w(P,t) = 0 for r §: R (1)
Hence, the displacement is assumed to be limit-
ed to the region r < R.
(4) In computing the resonant frequency of
the Helmholtz cavity mode we neglect the kine-
tic energy of the fluid in the cavity.
The basic differential equation for the dis-
placement vector u(P, t) in a Hookean solid
with no impressed volume forces is (Love, 1927)
/xV-u + (X + /r)VV • u = pdnu (2)
where p0 is the density and X and p, the Lamé
constants of the solid. Based on the assumptions
(1) to (4) above, the equation for w(P,t) is con-
sequently,
p03ttW + Lw = 0, P in (0,R;0,“=) (3)
where L is the operator
L = -[p(arr+(l/r)3r) + (X + 2p)3J (4)
and the boundary conditions are
w = 0, r = R and p = -(X + 2p)3zw ,
for r < R (5)
Assuming purely harmonic motion with the
angular frequency a),
w(P,t) = w(P)exp(i(yt) (6)
we obtain the following eigenvalue equation for
the amplitude w(P)
Lw = p0o)2w, (7)
with the above boundary conditions.
In the first approximation we can indentify
the Helmholtz mode with the lowest eigenmode
of (7) and assume for this case that
w(P) = AJ0(kr)exp[-zV(pX2 - p0co2)/(X + 2p)] (8)
where k = 2.4/R and A is a constant. A brief
investigation shows that at the assumed condi-
tions
Poft)2« pX~, (9)
that is, we can neglect the inertia of the rock.
Moreover, assuming Poisson’s relation X = p.,
we obtain then with the help of (5) approxi-
mately
w(r,0) = (PR/4p)J0(2.4r/R), (10)
and by an integration over 0 ^ r ^ R approxi-
mately (including both halfs)
B
e = (47r/p) fwrdr = 0.69R3/p (11)
J O
Having obtained the elastance of the cavity we
can now write the equation for the dynamics
of the fluid column in the borehole. Let x be
the vertical displacement of the column (posi-
tive downward), then in a force free case
ahpD2x + pa = 0 (12)
where D = d/dt and p is the density of the
fluid. Since for small x and p
e = dV/dp = ax/p (13)
or
x = pe/a
we have the simple equation for the pressure
oscillations
D2P + (a/ehp)P = 0 (14)
and hence the resonant frequency
m = Va/ehp = Vpa/0.69hpR3 (15)
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