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Bubbles attenuate elastic waves at seismic frequencies: first experimental
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evidence
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Nicola Tisato1*, Beatriz Quintal2, Samuel Chapman2, Yury Podladchikov2, and Jean-Pierre Burg3
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Supporting Information. Methods:
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Laboratory experiments
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The Broad Band Attenuation Vessel (BBAV) was employed to measure seismic attenuation, in
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extensional-mode, in water- (Fig. 2a) and water-glycerin- (Fig. 2b) saturated Berea sandstone
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(BS) samples. The solid frame of the samples can be considered homogeneous at the sample
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scale and isotropic [Madonna et al., 2013]. The specimens were named: BS4, BS5 and BS3, with
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a permeability of 200, 100 and 1000 mD, and porosity: 0.2, 0.18 and 0.21, respectively. The
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samples were sealed with an aluminum foil glued onto their curved surface and a plastic shrink-
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tube (jacket). Such device impeded a free-flow boundary condition that might bias the
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measurements [Dunn, 1986; Yin et al, 1992]. The sample was confined with oil up to 23.7 MPa,
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and an additional vertical stress, ranging between 1 and 2 MPa, was applied on the top of the
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specimen to ensure the good coupling between the specimen and the sample holders.
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Attenuation measurements were performed utilizing the force-oscillation (i.e sub-resonance)
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method [McKavanagh and Stacey, 1974]. A piezoelectric motor served to cyclically vary the
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vertical stress causing a strain of ~1.5×10-6. The analyzed frequencies were logarithmically
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spaced between 0.1 and 100 Hz. The attenuation ( 1 Q E ) was calculated as:
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1 QE  tan  
(7)
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where  is the phase shift between vertical stress and strain. The Young’s modulus (E) equals:
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E
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where  pp and  pp are the vertical peak-to-peak stress and strain, respectively.
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Stress-induced fluid pressure measurements were performed with the sample BS4 saturated with
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water and air (Fig. 1b). Confining pressure was varied with a rate of 0.82±0.2 MPa s-1 and fluid
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pressure was measured by means of a manometer connected to the top of the specimen. Both
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confining and fluid pressure were recorder at 1000 Hz sampling rate with an accuracy of
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±0.01 MPa.
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The BBAV was also equipped to measure transient fluid pressure caused by the oscillatory
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differential stress applied on top of the sample (Fig. 1a) [Tisato and Quintal, 2013]. Up to 5
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pressure sensors, able to sense pressure variations in less than 50 µs, were laterally introduced
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into the sample BS3 saturated with 97% water. Sensors were 200 kPa full scale and vertically
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spaced of ~4.2 cm. The 5 mm diameter sensor holes were sealed to reduce the free space
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between the sensor and the inner wall of the hole limiting the artificial porosity introduced by the
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presence of the sensors. More details on the methods employed to measure attenuation and
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transient fluid pressure are reported in the literature [Tisato and Madonna, 2012; Tisato and
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Quintal, 2013].
 pp
cos 
 pp
(8)
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Numerical experiments
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The deformation caused by an external sinusoidal hydrostatic pressure applied on a water
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volume (198×198×198 μm) containing a micrometric spherical CO2 bubble (18 μm diameter)
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(i.e. water-bubble-volume) was calculated solving three differential equations with an explicit
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finite difference Eulerian-time-marching scheme. The equations consider i) the elastic
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deformation of the bubble, ii) the dynamic dissolution of the gas in the surrounding liquid and
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iii) the diffusion of the gas in the surrounding water. The bubble dissolution rate ( dn dt ) and the
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rate of change of the bubble radius ( dr dt ) are calculated according eqs. 1 and 2. The diffusion
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of the gas in the surrounding water is calculated according to the Fick’s law [Cussler, 1997],
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which was solved in spherical coordinates.
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The external pressure exerted on the system was given as Pf :
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Pf  Pfi  Pf sin(2 f t )  cW k Hpc  2 / r  Pf sin(2 f t )
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where Pfi is the initial fluid pressure (i.e. equilibrium or injection pressure), ΔPf the amplitude of
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the sinusoidal fluid pressure (Fig. 1a) and f the frequency. To numerically solve eq. 1, 2 and the
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diffusion of the gas in the liquid we considered time (t) between 0 and 2/f and frequency (f)
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between 0.1 and 100 Hz. Time and f were discretized in 1024 and 32 steps, respectively.
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At t=0 the code calculated the equilibrium conditions and in particular the initial concentration of
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gas in the water (cW(t=0)) imposing dn dt  0 . At each subsequent time step the code calculates
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the bubble radius, the new concentration of gas in the water (cW(t>0)), and the volume of the
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water whose deformation is considered proportional to its bulk modulus (KH2O). The total
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volume (V) is given by the sum of the bubble and the water volume, and the volumetric strain
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(εV) is calculated as:
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V  
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where Vi is the initial volume (Vi=V(t=0)). Then the real and imaginary part of the bulk modulus
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of the water-bubble-volume are calculated as:
V  Vi
Vi
(9)
(10)
Pˆf
cos  V
ˆV
(11)
Pˆf
sin  V
ˆV
(12)
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Re  K f  
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Im  K f  
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where P̂f and ˆV are the peak-to-peak amplitude of Pf and εV, respectively. V is the phase shift
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between  V and Pf, and it is calculated with a best fit procedure [Bourbié et al., 1987]
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The complex effective Young’s modulus of the saturated Berea sandstone (E) and the related
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attenuation (1/QE) are obtained using Kf and some petrophysical properties of the dry Berea
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sandstone. In particular, the P-wave modulus was calculated as:
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H
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and the P-wave velocity as:
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VP 
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where  is the density of the saturated rock [Bourbié et al., 1987].
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The code was validated by fitting some results presented by Holocher et al., [2003]. We
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simulated the “stagnant flow regime” where an air-gas-bubble is trapped in the subsurface
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groundwater. Suddenly, the water column height and, as a consequence the pressure increases
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causing the partial dissolution of the gas-bubble. The pressure increase is:
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Pfval   gh
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where ρ is the water density (1000 kg m-3), h the water column increase [m], and g is the
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gravitational acceleration (9.81 m s-2).
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Holocher et al. [2003] considered instantaneous gas diffusion in the water surrounding the
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bubble, thus we simulated two cases:
3K 1  
1 
H

(13)
(14)
(15)
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1) The gas diffuses instantaneously in the surrounding water;
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2) The gas diffuses according to the Fick’s law as aforementioned.
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Both the cases, calculated for 4 different h values (0.05, 0.35, 0.65 and 0.95 m), reach a steady
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state after ~30 min similarly to what calculated by Holocher et al. [2003]. However, the second
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case shows a slower dynamic (Figure S1), which is in agreement with the model as the diffusion
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of the gas requires time to occur.
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The simulation of wave propagation in the subsurface presenting a domain saturated with water
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and CO2 microbubbles was performed by means of the two-dimensional (2D) numerical code
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called Sofi2D [Bohlen, 2002]. The source function was a Ricker wavelet with center frequency
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of 5 Hz. The viscoelasticity of the GZ was implemented as a third-order generalized standard
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linear solid (GSLS) model with QP=25 at frequencies 2, 4 and 6 Hz. The characteristic
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parameters for the spring stiffness and viscosity of the dashpots were assigned using the least-
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squares optimization [Bohlen, 2002]. A fourth-order finite difference, time-explicit scheme was
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employed to evaluate the displacement field in the nodes of the mesh.
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Approximation of the critical frequency (fc) for WIGED
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We combined equation 1, 4 and 9 obtaining:
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dI  
3RT DW
r
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.
(16)
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We calculated the time derivative of 9 and, combining it with 5, we obtained:
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d R  2 f r Pf cos(2 f t )
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Finally, we equated the amplitudes of the harmonic oscillations in 16 and 17 and we obtained a
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proxy for fc:

Pf  2 / r 
3RT DW
 cw 
  
k Hpc
r



Pfi  Pf sin(2 f t )  2 / r  3RT DW
Pf sin(2 f t )
 cw 
 
k Hpc
k Hpc r


(17)
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f 
3RT DW
 fc .
2 kHpc r 2
(18)
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Considering the properties reported in table S1 fc ≈ 11 Hz which is less than half order of
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magnitude higher than the attenuation peak frequency.
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Supporting Information. Tables:
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Table S1. Fluid and Berea sandstone properties utilized to calculate the attenuation caused by the
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dynamic bubble exsolution-dissolution of CO2 in water [Hart and Wang, 1995; Cussler, 1997;
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Tisato and Madonna, 2012; Tisato and Quintal, 2013].
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Supporting Information. Figures:
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Figure S1. Validation of the numerical model: the radius (r) of the bubble as a function of time
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has been calculated after that a step of pressure ( Pfval ) has been applied at time=0 min. The
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“stagnant flow regime” data, presented by Holocher et al. [2003], are represented by the circles.
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The two cases which have been modelled are shown by the continuum (case 1: no diffusion) and
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the dashed (case 2: diffusion) line.
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Figure S2. 2D model utilized to simulate wave propagation with and without the influence of the
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WIGED mechanism. From the bottom to the top the lithologies are: i) basalt, ii) carbonates, iii)
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anhydrites, and iv) sandstones. Velocities (VP and VS) and densities () are typical for the
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lithologies [Jaeger et al., 2007]. We assumed that the carbonates represent a geological CO2
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storage with injection point in the center of the model. The anhydrites are an imperfect seal and
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the leakage creates an overlying gassy zone (GZ) saturated with water and CO2. For depth
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<700 m we assume that the CO2 is a gas phase creating micrometric bubbles in the pores of the
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sandstone [Span and Wagner, 1996; Cussler, 1997]. Under this circumstances the P-wave
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attenuation of the GZ is 0.05 at 3.6 Hz (ii) and the VP dispersion is 180 m/s between 1 and 10 Hz
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(Fig. 3e). An explosion generates a 5 HZ Ricker elastic signal at the source point and it is
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recorder by the 6 receivers.
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Supporting References
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Dunn, K. J. (1986), Acoustic attenuation in fluid-saturated porous cylinders at low frequencies,
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J. Acoust. Soc. Am., 79(6), 1709–1721.
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Madonna, C., B. Quintal, M. Frehner, B. S. G. Almqvist, N. Tisato, M. Pistone, F. Marone, and
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E. H. Saenger (2013), Synchrotron-based X-ray tomographic microscopy for rock physics
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investigations, Geophysics, 78(1), D53–D64, doi:10.1190/geo2012-0113.1.
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McKavanagh, B., and F. D. Stacey (1974), Mechanical hysteresis in rocks at low strain
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amplitudes and seismic frequencies, Phys. Earth Planet. Inter., 8(3), 246–250.