Tímarit Verkfræðingafélags Íslands - 01.04.1981, Blaðsíða 17
and we can define the mass stiffness of
the well
S = dp/dm = g/f, (7)
This quantity measures the increase in
well bottoni pressure when a unit mass
of liquid is added to the well. Using
these expressions, the tidal factor can be
expressed
T = iAS/'w (8)
which along with (4) is our final result
for the above simple model illustrated in
Fig. 1.
It is to be noted that the above expres-
sion (8) will also hold for a closed well
situation. We have then only to redefine
the mass stiffness
S = dp/dm = 1//?M (9)
where M is the liquid mass in the well
and P is the compressibility of the li-
quid. In the case of a gas cap, (9) will
have to be adjusted accordingly. In the
case of the closed well, equation (4) will
have to be expressed in terms of the
well-head pressure rather than a water
level.
interpretation
OF WELL DATA
In thc relatively simple situation
described above, the interpretation of
well data is based on equations (4) and
(8). Invariably, p and f can be taken to
be known. Since in most practical cases,
the effective dilatation rb and forma-
tion capacitivity s are of less interest
than the formation fluid conductivity c,
the latter quantity is generally the
primary target of any interpretation of
observational tidal well data. Equations
(4), (8) and (6) show that c has to be
derived from the tidal factor T and that
this factor can be separated if we are
able to observe the water level
amplitude at two different tidal fre-
quencies. Since the tidal force field in-
cludes a number of frequencies this will
generally be possible. Being, in princi-
Ple, able to obtain T and thus on the
basis of (8) the admittance A, the fluid
conductivily will have to be derived with
the help of (6). In the rather idealistic
situation with a spherical well-cavity of
know radius rQ, we are therefore, in
principle, able to reach our goal of ob-
taining a numerical estimate of c.
From the practical point of view, the
procedure will, however, break down if
T 1. The factor T/(l +T) is then
aPproximately equal to unity and
the water level amplitude h will be
independent of the fluid conductivty.
In practice, we can expect this
difficulty to become serious when
about T > 3. Since in most cases the
factor 4ng/íu will be of the order of 107
(MKS), we see that the above inequality
implies r0c>3x 10'7 (MKS). For water
at 100°C with v = 3.10'7mJ/s we obtain
then in terms of permeability rQk >
3 x 10'7x 3 x 10'7s“0.1 darcy-meters.
Therefore, taking, for example, r0 =
0.5, we find that the above difficulty
becomes serious for permeabilities in ex-
cess of 200 millidarcy. In other words,
the tidal test based on open well situa-
tions is sensitive only to small to
medium permeabilities. Due to increas-
ed stiffness S, the applicability in the
case of closed wells is more restricted.
DEVIATIONS FROM THE
BASIC MODEL
Non-spherical well-reservoir con-
nection. The assumption of a spherical
cavity is perhaps to most obvious ideal-
ization in the above basic model. Un-
fortunately, the symmetry of the
pressure field will be broken in the case
of a non-spherical cavity, and the above
simple relations may, in principle, not
apply. However, provided the dimen-
sions of a non-spherical cavity are much
smaller than the skin depth d of the
medium, and tliis will mostly be the
case, the difficulties arising are not too
important from the more global point
of view. The global pressure field at
some proper distance from the cavity
will be approximately spherically sym-
metric and the above analysis will large-
ly be valid. The most serius casualty is
that the cavity admittance is not given
by the simple relation (6) and other
analog relations have to be relied on. In
practical cases, there may be difficulties
in establishing the form of the cavity.
Most frequently, however, the well-
reservoir connection consists of an open
section of the well. Let the radius of the
well be Tj and the length of the open
section be L. An elementary potential
theoretical exercise shows that when L
>> r, the admittance can then be
(Sunde, 1968) approximated by
A = 27tcL/ln(L/r|) (10)
It is of interest to point out that an open
section of L = lOm and r, = O.lm has
approximately the same admittance as a
spherical cavity of rQ = 1.1 m.
Multi-well setting. Another impor-
tant deviation from the basic model in-
volves cases where there is ntore than
one well opening into the reservoir. This
situation may lead to a well-well interac-
tion and pressure field scattering. The
practical criterion for interaction is ob-
tained by comparing the well-well
distance to the skin depth d ol' the
medium. In general, two wells will in-
teract noticeably if thc distance between
the well-reservoir cavities is approx-
imately equal or less than the skin depth
at tidal frequencies.
The analysis of multi-well situations
is more complex than the results given
above. In the case of spherical cavities,
the solution for the pressure amplitude
field will then have to be constructed as
a sum over solutions of the type (3), that
*s y
p= j (Bj/r:)exp|-(l +i)r:/d]-(eb/s),
(11)
where the summand is centered at
the j1*1 well-cavity, rj is the distance
from the field point to the centcr of the
j1*1 cavity and the Bjs are integration
constants. A boundary condition of the
type (2) applies at each well-cavity and
the constants Bj are obtained by solving
a set of linear algebraic equations. An
estimate for the fluid conductivity can
then be obtained along similar lines as
indicated obove. We will, however,
refrain from a further discussion. Ob-
viously, neglecting well-well interaction
in multi-well situations leads to an
underestimate of the formation fluid
conductivity c.
REFERENCES
Arditty, P.C., Ramey, H.J., Jr., and A.M. Nur,
1978. Response of a closed well-reservoir
system to stress induced by earth tides. SPE
paper 7484 Houston, Texas.
Bodvarsson, G., 1970. Confined fluids as strain
meters. .1. Geophys. Res. 75(14):2711-2718.
Bodvarsson, G., 1977. lnterpretation of borehole
tides and other elastomechanical oscillatory
phenomena in geothermal systems. 3rd
Workshop on Geothermal Reservoir Enginecr-
ing, Dccember, 1977, Stanford University,
Stanford, California.
Bodvarsson, G., 1978a. Convection and ther-
moelastic effects in narrow vertical fracture
spaces with emphasis on analytical techniques.
Final Report for U.S.G.S. pp. 1-111.
Bodvarsson, G. 1978b. Mechanism of rescrvoir
testing. 4th Workshop on Geothermal Reser-
voir Engineering, December 1978, Stanford
University, Stanford, California.
Bredehoeft, J.D., 1967. Response of well-acquifcr
systems to earth tides. J. Geophys. Rcs.,
72(12), 3075-3087.
Sunde, E.D., 1968. Earth Conduction Effects in
Transmission Systents. Dover Pulications, Inc.,
New York. pp 370.
Acknowlcdgements:
Tliis work was supportcd by the National Sciencc
Foundation of the U.S.A. under Grant No. EAR
77-23938.
TIMARIT VFI 1981 — 29