# Seismic stimulation for enhanced oil recovery Page: 3 of 13

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Seismic stimulation

where k is the permeability of the cell, yf is the fluid's viscosity, and

F is the total effective-force driving flow (with units of a pressure

gradient). This total force can be decomposed asF = F0 + AF, + Fa,

(2)

Symbol

A

Bwhere F, is the background (i.e., production) pressure gradient and Bf

AF, and Fa are created by changes in the stimulation well. If the pres-

sure perturbation applied in the stimulation well has a component

that is steady in time and if this well is perforated, there will be a stat-

ic force perturbation AF, that simply adds to F, to create a total static

forcing coming from all wells in the region. By conservation of fluid Ca

mass, this AF, has an amplitude that falls off with distance r as 1 /r; d

specifically, for a homogeneous reservoir, one has D

f

AF = 7fAQ (3) Fo

k4lrr2 Fa

where AQ, is the perturbed volumetric flow rate across the perforat- AF,

ed portion of the stimulation well. G

Force Fa is the amplitude of any time-harmonic forcing created by h

the fluid-pressure perturbations applied in the stimulation borehole.

If this borehole is perforated, the time-harmonic forcing has contri- i

butions both from an oscillating fluid-pressure diffusion that some- k

times is called the Biot slow wave (Biot, 1956) and from a propagat- Kd

ing seismic wave (see Figure 1). The diffusional contribution has a Kf

spatial falloff dominated by a factor e - rd, where the diffusive skin K

depth d is given by d = D/w with D the fluid-pressure diffusivity K,

given by D = kM/y1f. The fluid-storage incompressibility M, to an

excellent approximation (Pride, 2005), is given by M = Kf /o,

where Kf is the fluid's bulk modulus and 0 is porosity. Using values M

for a highly permeable sandstone saturated with water (k N

= 10-12 m2, 0 = 0.2, Kf = 2 X 109 Pa, yr = 10-3 Pa s), one then Pf

obtains D = 10 m2/s as almost an upper bound for the diffusivity in q

rocks. So for all frequencies f = wi27r> 10 Hz, which correspond

to the seismic frequencies of interest here, the skin depth is far less r

than a meter. Thus, the diffusive contribution to Fa is always com- R

pletely negligible for flow cells more than a meter from the stimula- Rao~,

tion borehole. Rip

Thus, the principal contribution to Fa is coming from the seismic S

wave, which (apart from a small loss because of intrinsic attenua-

tion) is conserving energy as it propagates outward spherically. En- Sm

ergy conservation requires the energy density in the wave to fall off t

with distance as 1 /r2, and because the energy density is proportional T

to the strain squared, the strain must fall as 1/r. The wave-induced u

pressure gradient Fa goes as the strain multiplied by an elastic modu- Uc

lus and divided by the wavelength and thus also falls off as 1 /r. We

conclude that because a static (DC) perturbation has a fluid force

AF, falling as 1/r2 (although the seismic perturbations have fluid volt

forces Fa falling as 1/r), it might be more cost-efficient to use a stim- P

ulation well as a source of seismic waves than as a means to perturb 17f

the steady background driving force. Therefore, AF, will be set to A

zero, and Fa will represent exclusively the acoustic (seismic) force

perturbations.

Finally, because it has been shown that seismic-wave forcing P

dominates over the slow wave for all distances r> 1 m from the a

stimulation well, the well need not be perforated, i.e., fluid exchang- 0

es between the stimulation well and the reservoir can be neglected O

entirely. However, because a perforated well lining is less stiff than a

nonperforated lining, the seismic coupling between borehole andMeaning

Lattice-Boltzmann interface parameter

Skempton's coefficient when both fluids are in

the pores

Skempton's coefficient when fluid f is in the

pores

Seismic P-wave velocity

Seismic S-wave velocity

Capillary number

Skin depth of pressure diffusion

Fluid-pressure diffusivity

Frequency in Hertz

Background (steady) pressure gradient

Seismic-force perturbation

Steady-force perturbation

Shear modulus of rock

Downstream length of a stuck oil bubble

Flux of oil

Permeability

Drained bulk modulus

Bulk modulus of fluid f

Bulk modulus of grain material

Undrained bulk modulus

Hydraulic throat radius

Fluid-storage incompressibility

Number of lattice points to each side

Wave-induced fluid-pressure increment

Darcy flow velocity

Distance from seismic source

Reynolds number

Downstream meniscus radius

Upstream meniscus radius

Stimulation number

Surface area of the menisci

Time

Dimensionless frequency number

Displacement of solid grains

Characteristic flow speed

Volume fraction of fluid f

Specific volume of oil produced

Interface angle in the lattice-Boltzmann model

Shear viscosity of fluid f

Lattice-Boltzmann collision parameter

Mass density of fluid f

Mass density of bulk rock

Surface tension

Wave strain

Contact angle

Circular frequency3

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### Reference the current page of this Article.

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/m1/3/: accessed November 19, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.