Seismic stimulation for enhanced oil recovery Page: 4 of 13
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Pride et al.
Seismic forces driving relative flow
A typical seismic wavelength is always much larger than the size
of the flow cells under consideration. The seismic band of frequen
cies over which it is relatively easy to propagate seismic energy is
approximately 10 to 100 Hz. This corresponds to wavelengths
ranging from approximately 100 to 10 m. The flow cells modeled
here are smaller than a centimeter and are thus far smaller than a seis
mic wavelength.
As a wave traverses such a small cell, it creates an effective force
Fa uniformly acting on the fluid that is moving in relative motion to
the grains. This force consists of both a wavelengthscale pressure
gradient and an apparent force arising from the fact that the reference
frame for the flow (the framework of grains) is accelerating. These
two contributions can be written as
d2u
Fa  Pf Pf 2, (4)
where Vpf is the macroscopic pressure gradient acting across the
flow cell, pf is the fluid density, and u is the average displacement of
the solid grains in the cell. The pressure gradient is present only in
the case of a compressional wave. In the apparent force of the second
term, the acceleration of the frame of reference is equivalent to a
gravitational acceleration and occurs for both compressional waves
and shear waves. This apparent force corresponds to what one feels
when standing on a bus that accelerates.
To model the two force contributions in equation 4, we note that a
seismic wavefront is effectively planar relative to the size of a flow
cell when the distance r from the source to the cell is much larger
than the size of the cell. For a timeharmonic plane wave moving in
the x direction, the seismic strain 6 acting on the bulk material goes
as 6(x, t) = oe"x 0, where c is the wave speed and 6o is the strain
amplitude (taken here to be independent of frequency over the range
of frequencies considered).
For a compressional wave at 10 to 100 Hz, Pride (2005) has
shown that the poroelastic response is effectively undrained, i.e., as
much fluid enters the flow cell as leaves, so there is no net change in
the fluid mass. In this case, the fluidpressure increment pf is related
to the wave strain 6 as pf =  BK,6, where K, is the undrained
bulk modulus and B is called Skempton's (1954) coefficient and is
the ratio of porepressure change to confiningpressure change for
undrained conditions. For liquidsaturated materials found in a sedi
mentary basin, one typically has 0.3 <B <0.7, with large values cor
responding to softer materials. Expressions that detail how the und
rained moduli K, and B depend on the underlying fluid and solid
moduli are given in Appendix A. Further, the seismic velocities at
such frequencies are given asKu + 4G/3
cp = and c
PG P

p(5)
for compressional waves and shear waves, respectively, where G is
the shear modulus (which is independent of the fluid properties at
seismic frequencies) and p is the average density of all the material
in a flow cell.
For a compressional wave, the amplitude of the force driving rela
tive flow is
S=  pB (eqwhere the pf term in brackets corresponds to the apparent force from
the acceleration of the frame of reference and the pB term corre
sponds to the fluidpressure gradient force. This expression is the
principal result of this section. As seen in Appendix A, when some
gas is present in the system, B becomes negligibly small, and we are
left with only the forcing from the acceleration. For liquid systems,
the two terms are of comparable importance. For a shear wave, there
are no compressional changes, and the effective force is simply Fa
  iwps0, arising from acceleration of the framework of grains
alone.
At the pore scale, these seismic forces can be taken as uniform
body forces in the NavierStokes equation for the local relative flow
in the pores. If the oil and water have significantly different densities
pf, one should formally require the apparent inertial force caused by
acceleration of the grains  iwpfc,,6 to be different in the oil and
water phases. However, for most crude oils and pore waters, the den
sity difference is on the order of 20% (with oil being lighter), which
might reasonably be neglected in a first modeling of seismic stimula
tion. The macroscopic fluidpressure gradient given by the pB term
in equation 6 is the same for oil and water phases.
Last, in addition to creating a pressure gradient across a flow cell,
a compressional wave also tries to change the average pressure of
each fluid in the flow cell. Because water is slightly less compress
ible than oil, the associated fluidpressure equilibration will cause a
displacement of the menisci from water patches toward oil patches.
This effect is modeled in Appendix B, where it is determined to be
negligible relative to the menisci displacements driven by the seis
mic forcing Fa of equation 6.
DIMENSIONLESS NUMBERS AND CONDITIONS
NECESSARY FOR STIMULATION TO WORK
To perform numerical simulations pertinent to field experiments,
the dimensionless groups that characterize seismic stimulation must
be identified. As always, two flows will be similar if the dimension
less numbers are the same, even though some of the material proper
ties, force amplitudes, or length scales are different in the two situa
tions.
Oil and water have different viscosities, so their ratio m'a/ 7haier
must be one of the dimensionless groups. For crude oils, one com
monly has 27oi/Y 7ater% 10.
The ratio Fa/F, of seismic forcing to the background fluidpres
sure gradient is another important dimensionless group. The pres
sure gradients used in oilreservoir production are commonly on the
order of a few kPa/m (e.g., tens of bars over 500 m). Combining this
orderofmagnitude appropriate value for F, with FJ  w pfc 6 and
the properties pf = 103 kg/m3 and c, = 3000 m/s gives the rough
field estimate ofFa f
F 104 Hz,(7)
where f = w/2r is frequency in Hertz. Therefore, a 10Hz source
delivering a strain of 6 = 106 (a large but common seismicstrain
level) to some flow cell corresponds to Fa/FO 10', although a 100
Hz source delivering the same strain would yield Fa/Fo 1.
However, it should be kept in mind that Fa falls with distance as
1 /r. If the wave strain is 105 at 10 m from the source position (stim
ulation well), it will be 106 at 100 m and 10' at 1 km. How Fo var
ies spatially is a function of how the production wells are distributed.4
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Pride, S. R.; Flekkoy, E. G. & Aursjo, O. Seismic stimulation for enhanced oil recovery, article, July 22, 2008; Berkeley, California. (digital.library.unt.edu/ark:/67531/metadc895565/m1/4/: accessed January 22, 2019), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.