Contact line friction and surface tension effects on seismic attenuation and effective bulk moduli in rock with a partially saturated crack

Rozhko, Alexander Y.; Bauer, Andreas

doi:10.1111/1365-2478.12653

Cited by 6 publications

(23 citation statements)

References 66 publications

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“…Because the initial far‐field normal stress, the initial fluid pressure and initial contact angles are identical in two identical cracks, the initial saturation degree is also identical in two cracks. Rozhko (), Rozhko and Bauer () and Rozhko () derived equations, describing the initial equilibrium saturation and the capillary pressure of partially saturated crack. The capillary pressure inside the equilibrium crack is calculated as follows (Rozhko ):

\begin{matrix} true \\ right p_{normalcap} & = & left \frac{π}{4} ((), p_{c l} + σ_{n} + p_{we}) \\ left \times \frac{1 - \sqrt{1 - 8 p_{c l} \frac{γ \cos ((), θ)}{π b} \frac{(β + \cot (β) \ln [\cos (β)] - π / 2)}{{(p_{c l} + σ_{n} + p_{we})}^{2} \sin ((), β)}}}{β + \cot ((), β) \ln ([], \cos ((), β)) - π / 2}, \end{matrix}

where

p_{c l}

is the crack closure pressure, calculated as (Rozhko ):

p_{c l} = \frac{b μ}{a (1 - v)},

where μ and v are the Shear modulus and Poisson's ratio of the rock mineral.…”

Section: Mathematical Formulationmentioning

confidence: 99%

“…Although the volume of the wetting fluid phase inside the partially saturated crack

(V_{we})

is calculated as (Rozhko and Bauer ; Rozhko ):

V_{we} = {\overset{V}{true}}_{we} + δ V_{we}

where

\begin{matrix} true \\ right \frac{{\tilde{V}}_{we}}{π a b} & = & left (1 + \frac{σ_{n} + p_{we} + p_{normalcap} (1 + \frac{4}{π} [\frac{β \sin ((), 2 β) - β^{2} + 2 \cos^{2} ((), β) \ln ([], \cos ((), β))}{2 β - \sin ((), 2 β)}])}{p_{c l}}) \\ left \times \frac{(2 β - \sin (2 β))}{π} . \end{matrix}

And

\begin{matrix} true \\ right \frac{δ V_{we}}{b^{2}} & = & left - \frac{(π - 2 θ - \sin (2 θ))}{\cos^{2} ((), θ)} \sin^{2} ((), β) \\ left \times {(1 + \frac{σ_{n} + p_{we} + (\frac{π - 2 β - 2 \cot ((), β) \ln ([], \cos ((), β))}{π}) p_{normalcap}}{p_{c l}})}^{2} . \end{matrix}

…”

Section: Mathematical Formulationmentioning

confidence: 99%

“…Above equations are applicable for the following saturations range of the crack (see details in Rozhko and Bauer ):

\frac{b}{a} \leq S_{we} \leq 1 - \frac{b}{a} .

…”

Section: Mathematical Formulationmentioning

confidence: 99%

“…Because the hysteresis of liquid bridges is a highly non‐linear effect, small stress perturbations of amplitude

Δ τ \sim 10^{2} - 10^{4} Pa

are enough to cause this non‐linearity due to deformation of highly compliant cracks. For example, recent publication of Rozhko and Bauer () have shown that a small stress perturbation

(\sim 10^{2} - 10^{4} Pa)

are sufficient to cause large changes in the contact angle

Δ θ

of the order of a few degrees or even larger. Thus, we cannot use linear (Taylor's) expansions of equations , and , because perturbations of

Δ θ_{i}

are large.…”

Section: Mathematical Formulationmentioning

confidence: 99%

“…Because the initial far-field normal stress, the initial fluid pressure and initial contact angles are identical in two identical cracks, the initial saturation degree is also identical in two cracks. Rozhko (2016), Rozhko and Bauer (2018) and Rozhko (2019) derived equations, describing the initial equilibrium saturation and the capillary pressure of partially saturated crack. The capillary pressure inside the equilibrium crack is calculated as follows (Rozhko 2016):…”

Section: A T H E M a T I C A L F O R M U L A T I O Nmentioning

confidence: 99%

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Effects of pore fluids on quasi‐static shear modulus caused by pore‐scale interfacial phenomena

Rozhko

2019

Geophysical Prospecting

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It is evident from the laboratory experiments that shear moduli of different porous isotropic rocks may show softening behaviour upon saturation. The shear softening means that the shear modulus of dry samples is higher than of saturated samples. Shear softening was observed both at low (seismic) frequencies and high (ultrasonic) frequencies. Shear softening is stronger at seismic frequencies than at ultrasonic frequencies, where the softening is compensated by hardening due to unrelaxed squirt flow. It contradicts to Gassmann's theory suggesting that the relaxed shear modulus of isotropic rock should not depend upon fluid saturation, provided that no chemical reaction between the solid frame and the pore fluid. Several researchers demonstrated that the shear softening effect is reversible during re‐saturation of rock samples, suggesting no permanent chemical reaction between the solid frame and the pore fluid. Therefore, it is extremely difficult to explain this fluid–rock interaction mechanism theoretically, because it does not contradict to the assumptions of Gassmann's theory, but contradicts to its conclusions. We argue that the observed shear softening of partially saturated rocks by different pore fluids is related to pore‐scale interfacial phenomena effects, typically neglected by the rock physics models. These interface phenomena effects are dependent on surface tension between immiscible fluids, rock wettability, aperture distribution of microcracks, compressibility of microcracks, porosity of microcracks, elastic properties of rock mineral, fluid saturation, effective stress and wave amplitude. Derived equations allow to estimate effects of pore fluids and saturation on the shear modulus and mechanical strength of rocks.

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\begin{matrix} true \\ right p_{normalcap} & = & left \frac{π}{4} ((), p_{c l} + σ_{n} + p_{we}) \\ left \times \frac{1 - \sqrt{1 - 8 p_{c l} \frac{γ \cos ((), θ)}{π b} \frac{(β + \cot (β) \ln [\cos (β)] - π / 2)}{{(p_{c l} + σ_{n} + p_{we})}^{2} \sin ((), β)}}}{β + \cot ((), β) \ln ([], \cos ((), β)) - π / 2}, \end{matrix}

where

p_{c l}

is the crack closure pressure, calculated as (Rozhko ):

p_{c l} = \frac{b μ}{a (1 - v)},

where μ and v are the Shear modulus and Poisson's ratio of the rock mineral.…”

Section: Mathematical Formulationmentioning

confidence: 99%

“…Although the volume of the wetting fluid phase inside the partially saturated crack

(V_{we})

is calculated as (Rozhko and Bauer ; Rozhko ):

V_{we} = {\overset{V}{true}}_{we} + δ V_{we}

where

\begin{matrix} true \\ right \frac{{\tilde{V}}_{we}}{π a b} & = & left (1 + \frac{σ_{n} + p_{we} + p_{normalcap} (1 + \frac{4}{π} [\frac{β \sin ((), 2 β) - β^{2} + 2 \cos^{2} ((), β) \ln ([], \cos ((), β))}{2 β - \sin ((), 2 β)}])}{p_{c l}}) \\ left \times \frac{(2 β - \sin (2 β))}{π} . \end{matrix}

And

\begin{matrix} true \\ right \frac{δ V_{we}}{b^{2}} & = & left - \frac{(π - 2 θ - \sin (2 θ))}{\cos^{2} ((), θ)} \sin^{2} ((), β) \\ left \times {(1 + \frac{σ_{n} + p_{we} + (\frac{π - 2 β - 2 \cot ((), β) \ln ([], \cos ((), β))}{π}) p_{normalcap}}{p_{c l}})}^{2} . \end{matrix}

…”

Section: Mathematical Formulationmentioning

confidence: 99%

“…Above equations are applicable for the following saturations range of the crack (see details in Rozhko and Bauer ):

\frac{b}{a} \leq S_{we} \leq 1 - \frac{b}{a} .

…”

Section: Mathematical Formulationmentioning

confidence: 99%

“…Because the hysteresis of liquid bridges is a highly non‐linear effect, small stress perturbations of amplitude

Δ τ \sim 10^{2} - 10^{4} Pa

are enough to cause this non‐linearity due to deformation of highly compliant cracks. For example, recent publication of Rozhko and Bauer () have shown that a small stress perturbation

(\sim 10^{2} - 10^{4} Pa)

are sufficient to cause large changes in the contact angle

Δ θ

of the order of a few degrees or even larger. Thus, we cannot use linear (Taylor's) expansions of equations , and , because perturbations of

Δ θ_{i}

are large.…”

Section: Mathematical Formulationmentioning

confidence: 99%

Section: A T H E M a T I C A L F O R M U L A T I O Nmentioning

confidence: 99%

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Effects of pore fluids on quasi‐static shear modulus caused by pore‐scale interfacial phenomena

Rozhko

2019

Geophysical Prospecting

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Effective Fluid Bulk Modulus in the Partially Saturated Rock and the Amplitude Dispersion Effects

Rozhko

2020

JGR Solid Earth

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Frequency dispersion is a well‐known effect in geophysics, which means that waves of different wavelengths propagate at different velocities. Amplitude dispersion is a less‐known effect, which means that waves of different amplitudes propagate at different velocities. Herewith, we consider the alteration of the interfacial energy during wave‐induced two‐phase fluid flow in a partially saturated rock and demonstrate that this leads to a nonlinear amplitude dispersion effect. When the wave amplitude is small, seismic waves cause bending of the interface menisci between immiscible fluids at the pore scale. However, when the wave amplitude is sufficiently large, the interface menisci will slip at the pore scale, causing attenuation of the elastic energy by the contact line friction mechanism. At the zero frequency limit, all viscous dissipation models predict zero attenuation of the elastic wave energy, while this approach predicts a nonzero attenuation due to a static contact angle hysteresis effect. Herein, we extend the Gassmann's theory with three extra terms, which can be obtained from standard laboratory tests: pore‐size distribution and interfacial tension between immiscible fluids and rock wettability (advancing and receding contact angles). We derive closed‐form analytical expressions predicting the effective fluid modulus in partially saturated rock, which falls between Voigt and Reuss averages. Next, we demonstrate that the nonlinear amplitude dispersion effect leads to energy transfer between different frequencies. This may explain the low‐frequency microtremor anomalies, frequently observed above hydrocarbon reservoirs, when the low‐frequency energy of ocean waves (0.1–1 Hz) is converted to higher frequencies (2–6 Hz) by partially saturated reservoirs.

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Seismic Wave Attenuation and Dispersion Due to Partial Fluid Saturation: Direct Measurements and Numerical Simulations Based on X‐Ray CT

et al. 2021

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Quantitatively assessing seismic attenuation caused by fluid pressure diffusion (FPD) in partially saturated rocks is challenging because of its sensitivity to the spatial fluid distribution. To address this challenge we performed depressurization experiments to induce the exsolution of carbon dioxide from water in a Berea sandstone sample. In a first set of experiments we used medical X‐ray computed tomography (CT) to characterize the fluid distribution. At an equilibrium pressure of approximately 1 MPa and applying a fluid pressure decline rate of approximately 0.6 MPa per minute, we allowed a change in saturation of less than 1%. The gas was heterogeneously distributed along the length of the sample, with most of the gas exsolving near the sample outlet. In a second set of experiments, at the same pressure and temperature, following a very similar exsolution protocol, we measured the frequency dependent attenuation and modulus dispersion between 0.1 and 1,000 Hz using the forced oscillation method. We observed significant attenuation and dispersion in the extensional and bulk deformation modes, however, not in the shear mode. Lastly, we use the fluid distribution derived from the X‐ray CT as an input for numerical simulations of FPD to compute the attenuation and modulus dispersion. The numerical solutions are in close agreement with the attenuation and modulus dispersion measured in the laboratory. Our approach allows for accurately relating attenuation and dispersion to the fluid distribution, which can be applied to improving the seismic monitoring of the subsurface.

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Contact line friction and surface tension effects on seismic attenuation and effective bulk moduli in rock with a partially saturated crack

Cited by 6 publications

References 66 publications

Effects of pore fluids on quasi‐static shear modulus caused by pore‐scale interfacial phenomena

Effects of pore fluids on quasi‐static shear modulus caused by pore‐scale interfacial phenomena

Effective Fluid Bulk Modulus in the Partially Saturated Rock and the Amplitude Dispersion Effects

Seismic Wave Attenuation and Dispersion Due to Partial Fluid Saturation: Direct Measurements and Numerical Simulations Based on X‐Ray CT

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