PROCEEDINGS of the
23rd International Congress on Acoustics
9 to 13 September 2019 in Aachen, Germany

A study on the relationship between rock microstructure and wave
dispersion in carbonates and sandstones
Wei CHENG1; Jing BA*2; José M. CARCIONE31


School of Earth Sciences and Engineering, Hohai University, Nanjing 211100, China

School of Earth Sciences and Engineering, Hohai University, Nanjing 211100, China

Istituto Nazionale di Oceanografia e di Geofisica Sperimentale (OGS), Borgo Grotta Gigante 42c, Sgonico,
Trieste I-34010, Italy.

ABSTRACTThe dependence of the wave properties of common rocks on their microstructure is not fully understood.
Porosity, fluid type and rock texture significantly affect acoustic wave propagation. The P-wave velocities inwater-saturated rocks are predicted from measurements in gas-saturated rocks, using the Gassmann theory.
The dispersion is estimated from the difference between this predicted velocity and the measured one, where
the latter corresponds to the unrelaxed state. In this work, ultrasonic compressional and shear wave velocities
are measured in full-water and full-gas saturated tight carbonates and sandstones. By adding experimental
data from the literature, we evaluate the compressional wave velocity dispersion as a function of porosity for
all these lithologies (240 samples). It shows that dispersion increases with porosity in the low porosity range,
but decreases in the high porosity range. For each lithology, the dispersion peak occurs at a porosity of
approximately 15 %. The new findings here reflect the intrinsic natures of shallow earth rocks, which willcontribute to better understandings on earth rocks for researchers of rock physics and exploration geophysics
areas.
Keywords: Compressional wave velocity dispersion, Porosity, Ultrasonic measurement

1. INTRODUCTION
Rock anelasticity is affected by texture, porosity and pore fluids. Studies on attenuation anddispersion are essential in estimating the properties of rocks and guiding seismic inversion. Jeongand Hsu (1995) showed that ultrasonic attenuation increases with void content in carbon composites.
Klimentos et al. (1990) established a correlation between attenuation, porosity and clay content insandstones based on measurements. Since it has been considered that if the dispersion is completely
characterized for all frequencies then attenuation is known for all frequencies and vice versa(Carcione et al., 2018; Mavko et al., 2009). Dispersion is closely related to porosity and reflects theintrinsic properties of rocks.
The two different lithologies are considered in this work. Carbonates cover a range ofdepositional facies, with complex porous media and considerable textural variability, showing adiversity of pore types, a wide range of pore sizes and fluid distribution heterogeneity (Lopes et al.,2014). Some deep sandstone reservoirs contain high porosity values, however, the tight sandstones
have geological characteristics of low permeability, low porosity and developments of micro-cracks(Guo et al., 2018).
Laboratory measurements of ultrasonic waves in rocks have been frequently used to investigate
the relations between the rock properties and wave properties. Moreover, Gassmann equation can beused to predict the compressional wave velocity of a water-saturated rock based on the properties ofthe dry-rock skeleton and fluid (Gassmann, 1951). However, it is only valid at the low frequency
limit, where the wave-induced pore fluid pressures are equilibrated throughout the pore space (King1chwei@hhu.edu.cn
jingba@188.com (corresponding author)
3
jose.carcione@gmail.com
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and Marsden, 2002). At ultrasonic frequencies, the wave-induced fluid pressure gradient between
stiff and soft pores does not equilibrate. This results in a stiffening effect, and the compressional
wave velocity will be underestimated by the Gassmann equation. The velocity dispersion can be
estimated by calculating the difference between the measurements and Gassmann prediction at fullliquid saturation (Regnet et al., 2015).
In this work, we measure ultrasonic compressional and shear wave velocities in 18 carbonates and17 sandstones at full-gas saturation states and full-water saturation. Together with the experimental
data of the two lithologies in literature (Regnet et al., 2015; Han et al., 1986; Wang, 2016a; Wang,
2016b; Guo et al., 2018; King et al., 2000), dispersion is estimated by using Gassmann theory andthe two measured velocities at full water and full-gas saturations. The relation between dispersion
and porosity is analyzed.

2. EXPERIMENTAL DATA
2.1 Rock Specimens
The experimental set-up of Guo et al. (2009) is used for the ultrasonic-wave measurements on 18
carbonates (5 limestones and 13 dolomites) and 17 sandstones. The carbonates are collected from
Ordovician and Cambrian formations (> 4.0 km depth), West China, with low -moderate porosity,
dissolved pores and rare clay. The sandstones, collected from the Paleogene formation, in the
Dongying sag, East China (around 4.0 km depth), are composed of feldspar, quartz, and rare clay.

Figure 1. Crossplots of VP and VS for (a) 18 carbonates and (b) 17 sandstones at full gas and full water
saturations. Linear fits are shown.
Ultrasonic P- and S-wave velocity (V P and VS) were measured for each sample at full water and fullgas (nitrogen) saturations, respectively. Measurement frequency is 1 MHz. Figure 1 shows crossplots
of VP and VS for 18 carbonates and 17 sandstones at full gas and full water saturations. The correlationof a linear fitting between VP and VS in carbonates is better than sandstones. The slope of VP versus VS
is 1.9 in carbonates, higher than those in sandstones. For each lithology, the slope of VP versus VS atfull water saturation is lower than that at full gas saturation. In addition to our measurements on the628two lithologies, experimental data by Regnet et al. (2015), Han et al. (1986), Wang (2016a), Wang
(2016b), Guo et al. (2018) and King et al. (2000) are analyzed.
2.2

Determination of The Bulk Modulus of The Mineral (K0)
If the mineral contents, components and geometrical parameters are known, we can use the effective
medium theories to estimate the bulk modulus of the solid (Ba et al., 2016). However, for most rocks, theseproperties can hardly be obtained. In another method the rocks are classified into the same lithology andsub-phase, and by analyzing the set of sorted specimens with similar lithological characteristics and minerals,
a linear fitting of the velocity-porosity relations in low porosity range can give an estimation of the effective
solid modulus for the whole set (Yan et al., 2011). In this work, we taken the porosity less than 8 %. For eachof the two sets, VP and VS at zero porosity are computed, and thus, K0 is obtained.
Modulus K0 is obtained from the limestones, sandstones (our data), and from the results by Han et al.
(1986), Wang (2016a), Guo et al. (2018). K0 is 89.2 GPa for the carbonate set of Wang (2016b) and 40.0 GPa
for the sandstone set of King and Marsden (2002). For our dolomites and the limestones of Regnet et al.
(2015), K0 is taken from Mavko et al. (2009), since the lack of data in the low porosity range.

3. RESULTS AND DISCUSSION
3.1

Gassmann Fluid Substitution
The Gassmann theory (Gassmann, 1951) theory is used to estimate the bulk modulus of water-saturated
rock (K(sat)) based on the velocity measurements of gas-saturated/dry rocks. Gas is light and assumed to cause
no stiffening effect at ultrasonic frequencies, therefore the gas-saturated ultrasonic measurements reflect arelaxed state and give the dry-rock bulk modulus, Kb. The Gassmann theory is based on the assumption thatwave-induced fluid pressures are equilibrated throughout pore space in rock, i. e., the low-frequency (LF)
limit. The Gassmann equation is(LF)
K (sat)K0  K

(LF)
(sat)

Kf
Kb

K0  Kb I ( K0  K f )(1)

where Kf and K0 are the bulk moduli of fluid and mineral mixture, respectively, ϕ is the rock porosity, and Kb
is the bulk modulus of the rock skeleton without pore fluids (e.g., Mavko et al., 2009; Carcione, 2014).
(LF)
is computed from equation (1) for each specimen, where the fluid properties are obtained at theK(sat)

measurement conditions according to Batzle and Wang (1992).
The P-wave velocity dispersion is then estimated as the difference between the measured P-wave velocity(HF)

at ultrasonic frequencies (high-frequency, HF), VP(sat ) , and the velocity obtained from the Gassmann(LF)

equation, VP(sat ) , as follows

Dispersion=

(HF)
(LF)
VP(sat)  VP(sat )(LF)
VP(sat)(2)

3.2

Dispersion Versus Porosity in Tight Rocks
The dispersion is estimated by using equation (2) in the two sets of tight rocks (carbonate and sandstone)
with three sub-phases, limestone, dolomite, sandstone, as is shown in Figure 2. The dispersion is generally
positive in the tight rocks, indicating the predicted velocity by Gassmann theory is lower than the measured
ultrasonic P-wave velocity. Similar phenomena were reported by King et al. (2002), and attributed to thepresence of compliant pores or open microcracks, and the related squirt-flow mechanism. However, in thesetight specimens, the dispersion is mostly less than 5 %, and apparently increases with porosity. It is also
shown the sub-phases of dolomite and limestone in carbonates share the same trend of linear fitting, each629

phase cannot be distinct from the others by dispersion. However, by the dispersion in these tight rocks, thetwo lithologies can be distinguished, with sandstone having the higher dispersion and fitted slope (0.21), andcarbonates having the lower slope (0.14).

Figure 2. Crossplot of P-wave velocity dispersion and porosity for the tight carbonates (including dolomites
and limestones) and sandstones in our measurements. The dashed linear represent the regressions.
3.3

Relation Between Porosity and Dispersion
Porosities of specimens in the tests of this study are mostly less than 12 % (tight rocks). The measured
conditions of these data reflect the actual in-situ conditions of typical shallow geological formations. Figure 3
presents the crossplots of porosity and dispersion for the two lithologies. The most interesting point is that
although rocks exhibit a trend of dispersion increasing with porosity in low porosity range (<15 %), but
decreases for moderate to high porosities (> 15 %). A Gaussian fitting on the different lithologies shows thatthe dispersion peaks at a porosity approximately 15% (the porosities in carbonate and sandstone are 15.7 %and 14.9 %, respectively).
In addition, some negative values of dispersion in Figure 3a and 3b indicate that the fitted/given K0 ineach rock set may be not appropriate for each specimen, however there is no effective approach to precisely
determine K0 for each sample, in lack of the detailed lithology and mineral information. The analysis here
provides the general statistical trend for the considered rocks.

Figure 3. Crossplots of P-wave dispersion and porosity for the whole datasets of (a) carbonates and (b)sandstones. Gaussian fits are shown.

4. CONCLUSIONS
Ultrasonic measurements are performed on the two sets of tight rocks, including 18 carbonates

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and 17 sandstones, at the full-water and full-gas saturation states. By assuming the rock matrix is
relaxed at full-gas saturation, the Gassmann theory is used to predict the P-wave velocity at fullwater saturation at the low frequency limit. Wave dispersion is then estimated by the difference
between the Gassmann prediction and ultrasonic measurements. In the tight rock specimens of thisstudy, the dispersion increases with porosity, and the different litholog ies can be distinguished withthe different linear fitting slopes. However, regarding each lithology, the sub -phases (e. g., dolomite
and limestone in carbonates) cannot be distinct from each other. By incorporating the experimental
data in literature, a statistical analysis on the 9 data sets with 240 specimens shows the P-wave
dispersion peaks at a moderate porosity around 15 % (carbonate 15.7 %; sandstone 14.9 %) for
different lithologies, which indicates the dispersion decreases with porosity in the hig h porosity
range, instead of increases. The characteristics of other lithologies or rock types remain to beinvestigated. The combination research of geology and geophysics on this topic will help
highlighting people’s understanding on shallow earth rocks.

ACKNOWLEDGEMENTS
Zhaobing Hao helped in the ultrasonic measurements on rock specimens. This work is supportedby the Distinguished Professor Program of Jiangsu Province, China, the Cultivation Program of the“111 Plan”, China (BC2018019), and the Fundamental Research Funds for the Central Universities(2016B13114), China.

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