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Geophysical Journal International
Geophys. J. Int. (2015) 203, 2182–2188
doi: 10.1093/gji/ggv434
GJI Marine geosciences and applied geophysics
Does wettability influence seismic wave propagation
in liquid-saturated porous rocks?
Zizhen Wang,1,2 Douglas R. Schmitt2 and Ruihe Wang1
1 School
of Petroleum Engineering, China University of Petroleum (Huadong), Qingdao, China. E-mail: [email protected]
for Geophysical Research, University of Alberta, Edmonton, Canada
2 Institute
Accepted 2015 October 5. Received 2015 October 1; in original form 2015 July 09
Key words: Microstructures; Elasticity and anelasticity; Wave propagation; Acoustic properties.
I N T RO D U C T I O N
Seismic waves propagating and scattering within the earth’s crust
contain substantial information on in situ conditions, but their
proper interpretation necessitates a full understanding of the physical phenomenon affecting them along their travel path. In highly
porous rocks, the models of Gassmann (1951) and Biot (1956a,b)
have been widely used for predicting the saturated bulk Ksat and
shear μsat elastic moduli of porous rocks saturated with different
fluids. As the simplest example, in the zero-frequency limit of the
Biot–Gassmann model, these are given by Gassmann’s (1951) equations
K sat = K dry +
φ
Kf
α2
+ α−φ
Km
μsat = μdry ,
(1)
(2)
where Kdry , Km and Kf are the bulk moduli of the rock’s dry
frame, its mineral constituents, and its saturating fluid, Kdry and
Ksat are also often referred to as the drained and undrained bulk
moduli, respectively. μdry is the shear modulus of the dry frame, α =
1-Kdry /Km is the Biot’s coefficient, and φ is the porosity. Gassmann
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C
developed eqs (1) and (2) to study seismic wave propagation with
the compressional (VP ) and shear (VS ) wave speeds:
K sat + 4μsat /3
(3)
VP =
ρsat
and
VS =
μsat
,
ρsat
(4)
where the saturated rock composed of solid minerals and fluid saturant of densities ρ m and ρ f , respectively, will have a bulk density
ρsat = (1 − ϕ)ρm + ϕρf .
(5)
Gassmann’s eqs (1) and (2) are the low frequency limit of Biot’s
(1956a,b) model that includes frequency dispersion by considering
the effects of viscosity and internal drag and necessitates additional
considerations of viscosity, permeability, pore dimensions, and tortuosity. These Gassmann’s and Biot’s models are purely mechanical,
they does not account for changes that could result from variations
in chemistry. To be fair, both Gassmann (1951) and Biot (1962) recognized that their models did not incorporate such effects. Indeed,
Gassmann’s caveats are almost always repeated when his eqs (1) and
(2) are provided in the literature, but despite this extra-mechanical
effects have largely been ignored in seismological studies.
The Authors 2015. Published by Oxford University Press on behalf of The Royal Astronomical Society.
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SUMMARY
In order to evaluate the effects of different fluids, we measured the P and S wave speeds
through a series of synthetic epoxy-bonded carbonate composites with porosities from 21.9 to
31.5 per cent and saturated with air, with kerosene and with brine. These observed speeds were
converted to the saturated bulk and shear moduli and compared to predictions made using Biot
and Gassmann formulations. The observed bulk moduli agreed with those calculated for both
kerosene and brine saturation as did the high frequency shear modulus under kerosene saturation. The observed shear modulus under water saturation, however, was significantly lower
than the prediction. After excluding the currently known mechanisms of shear weakening, we
suspect this disparity may be due to variations in the wetting of the epoxy that coats the pores
surfaces. The kerosene completely wets this surface while the brine is only weakly hydrophilic
with a wetting angle of 73.6◦ . At the molecular scale, this means that Stoke’s no-slip boundary
condition may not always apply as has been more recently demonstrated by many researchers
in other disciplines such as microfluidic engineering but the implications for wave propagation
in liquid-saturated rocks have not been considered.
Wettability influences seismic wave propagation
measurements of the acoustic wave speeds in samples of porous
Berea sandstone, in porous alundum, and in porous TeflonTM (polytetrafluoroethylene). Their samples were specifically saturated with
a variety of liquids that were either wetting or nonwetting (e.g. water in porous Teflon). We are unaware of any subsequent studies
that have examined the influence of wetting on wave propagation
in fully saturated materials, although there are numerous studies of
the related problem of adsorption (e.g. Pimienta et al. 2014).
We first describe the results of laboratory wave speed measurements in synthetic carbonates under different full saturations. We
analyse the results under the Biot–Gassmann paradigm and show
that the sample saturated with a fully wetting fluid is well described,
while that saturated with a weakly wetting fluid is not. We suggest
that these different behaviours may be related to the ability of the
liquid molecules to ‘slip’ along the liquid–solid interface in opposition to the classic no-slip conditions that would be implicit to the
Biot–Gassmann model.
METHOD
Following Wang et al. (2015) we manufacture synthetic porous
and permeable carbonate cylinders (Fig. 1a) using a mixture of
natural carbonate rock powder and a two-component epoxy (see
Table S1 for the constituent properties). The mineral composites
of the carbonate powder include both calcite and dolomite, and
the mean grain size is about 200 μm (Fig. 1b). After curing, the
epoxy cement is chemically resistant to water or kerosene. Porosity
is controlled by randomly adding NaCl grains (∼3 mm width) to
the carbonate–epoxy mixture while it is being placed into removable steel cylindrical molds. The mixture is uniaxially compressed
at 20 MPa stress and then placed at a temperature of 75◦ C to ensure
Figure 1. (a) The four synthetic carbonate samples. The diameter of these samples is 25.4 ± 0.02 mm. (b) The carbonate cuttings before being mixed with the
epoxy. The mean grain size is about 200 μm. (c) CT images of sample V4 at different section planes. (d) The SEM image of sample V4 with both interparticle
pores and large vugs. It can also be seen the carbonate cuttings are coated with the epoxy.
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A key canon of the Gassmann’s model at low frequency is that
the rock’s frame shear modulus (μsat ) should remain constant regardless of the saturant (Berryman & Milton 1991). However, there
are a number of experimental observation in which both sandstones
(Wyllie et al. 1958; Khazanehdari & Sothcott 2003) and carbonates
(Peselnik 1962; Assefa et al. 2003; Røgen et al. 2005; Adam et al.
2006; Baechle et al. 2009; Bakhorji 2010; Fabricius et al. 2010;
Verwer et al. 2010; Njiekak et al. 2013) can become more
(μsat >μdry ) and less (μsat < μdry ) rigid after fluid saturation depending on the saturant, the effective pressure, the pore structure,
the existence of clays, and the frequency. Within the literature these
changes have respectively been called shear ‘stiffening’ or ‘hardening’ and ‘softening’ or ‘weakening’. But the survey of this literature
does not appear to show any definitive trend towards either weakening or stiffening; more work is necessary. Certainly, water is known
to strongly weaken the apparent strength of rock when taken to
failure (e.g. Lajtai et al. 1987; Baud et al. 2000; David et al. 2015)
but it is not clear that the effects leading to failure of the rock are
exactly those responsible for the material’s elastic moduli, which is
here the focus of this study. Many of these observations are likely
complicated by sample containing cracks where local flow effects
will dominate, but it cannot account for all of the variations. The
reason for this discrepancy between Gassmann’s theory and experimental findings are still not fully clear (Adam et al. 2006; Fabricius
et al. 2010).
Here, we present observations suggesting that the degree of wetting between the fluid and the solid may also affect wave propagation. Somewhat unexpectedly, the only study we are aware of
that explicitly mentions the influence of wetting is that of Wyllie
et al. (1958), the same contribution from which originates the timeaverage equation. They detailed an extensive series of laboratory
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Z. Wang, D.R. Schmitt and R. Wang
Table 1. Petrophysical characteristics and measured velocities at different
saturation conditions of all synthetic samples.
Dry
Grain
density density
NaCl
Vuggy
Total
Permeability
(mD)
Index (g cm–3 ) (g cm–3 ) grains (g) porosity porosity
V1
V2
V3
V4
2.13
2.13
2.05
1.85
Index
Dry (m s–1 )
Vp
Vs
4089
2451
4187
2531
3831
2329
3241
1994
V1
V2
V3
V4
2.76
2.73
2.75
2.70
1.45
1.66
2.90
5.80
0.025
0.028
0.052
0.097
Brine-saturated
(m s–1 )
Vp
Vs
4142
2298
4241
2375
3880
2132
3352
1774
0.227
0.219
0.256
0.315
567
402
744
1497
Kerosene-saturated
(m s–1 )
Vp
Vs
4112
2380
4207
2474
3896
2265
3378
1918
DISCUSSION
Predictions versus observations
We are confident that these materials do not contain microcracks
due to their construction and that, excepting additional effects not
anticipated, we expect the saturated behaviour to be described under
the Biot–Gassmann paradigm. This approach has been successful in
numerous other studies in highly porous materials such as sintered
glass beads (see review in Bouzidi & Schmitt 2009) or sintered
corundum (Schmitt 2015) where application of Biot’s full equations
has satisfactorily explained laboratory observations.
We first calculated the frequency-dependent (from 1 to 1010 Hz,
the latter far exceeding the ∼105 Hz of the measurements) wave
speeds according to the Biot’s model (Biot 1956a,b; Berryman
1980) using the algorithms provided in Mavko et al. (2009), details
of the calculations are included in the supplementary information.
These are converted to the frequency dependent Ksat and μsat with
eqs (3)–(5). The Ksat predicted at higher frequencies show little
dispersion from the zero-frequency limit Gassmann-predicted Ksat
(Fig. 2c). That is, the Biot–Gassmann model over the range of
frequencies calculated predicts well Ksat for both brine and kerosene
saturations.
The situation differs somewhat for μsat . The predicted μsat are
more dispersive as illustrated by the height of the symbols in
Fig. 2(d). The predicted kerosene-saturated μsat agree well with
the observations. This together with the good agreement for the
corresponding Ksat in Fig. 2(c) suggests that the Biot–Gassmann
theory adequately models the behaviour under kerosene saturation.
In contrast, the predicted brine-saturated μsat significantly exceed
those observed (Fig. 2d).
The carbonate cuttings in our synthetic samples are coated with
consolidated epoxy (Fig. 1d) that is chemically resistant to brine
and kerosene. Therefore, both brine and kerosene cannot change the
grain-grain contacts in our synthetic samples. This means that the
currently known mechanisms of water weakening are not suitable to
explain modulus variations here. As already noted, from the purely
mechanical perspective inherent to the Biot–Gassmann model the
two liquids are not significantly different, and it is important to ask
why the model predicts the kerosene saturated values well but fails
to predict the brine saturated values.
Wetting behaviour
R E S U LT S
As expected, VP and VS generally decrease with total porosity
(Fig. 2a). Further, VP increases and VS decreases upon saturation
relative to the dry measures. These observations all follow expected
‘Wettability’ refers here to the ability of a fluid to adhere to a solid
and microscopically relies on the degree of molecular attraction
between the fluid and the solid (e.g. Guéguen & Palciauskas 1994).
The ability of a fluid to ‘wet’ a solid surface is determined experimentally by placing a drop of the liquid onto a flat surface of the
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proper curing. Water is then passed through the cylinder to dissolve
out the NaCl grains leaving large open voids (Fig. 1c) that supplements the porosity of the mixture. An SEM image at smaller scale
(Fig. 1d) shows the grain assembly, inter-particle pores, and vuggy
pores in the synthetic samples. It can be seen from Fig. 1(b) and
(d) that the carbonate cutting surfaces are coated with consolidated
epoxy.
We made four samples (Fig. 1a) with total porosities ranging
from 21.9 to 31.5 per cent as determined using a Boyle’s Law He
pycnometer (Table 1). Permeability to water was measured using
a standard Darcy’s law flow through method. It should be noted
that these four samples have similar matrix composites and compaction process, so they should have similar grain density (around
2.70 g cm–3 ) and interparticle porosity (around 20 per cent), which
is confirmed by our measurements and also indicates that the mixing is reproducible. Further, we do not expect these samples to
contain crack porosity that would disallow application of the Biot–
Gassmann model in the analysis (e.g. Johnson et al. 1994b; Bouzidi
& Schmitt 2009).
The samples were saturated successively with air, 1.4 mol l–1
NaCl brine, and kerosene. The mechanical properties (bulk modulus and viscosity) and the densities of the brine and kerosene do
not vary from each other significantly (Table S1). Both VP and VS
were measured using standard ultrasonic pulse-transmission measurements under full saturations of air, brine, and kerosene (see Fig.
S1 for the experimental protocol). The central frequencies of the
transducers are 0.5 MHz for P wave and 0.25 MHz for S wave. The
received signals are acquired at 50 MHz for 2000–5000 samples
(Fig. S2). Estimates of the cut-off frequency (Table S2) fall in the
range of 30–150 kHz (Johnson 1984; Sarout 2012) lying just below
the useful bandwidth of the signals. High-pass filters (>0.1 MHz
for P wave and >0.05 MHz for S wave) are applied to remove low
frequency bias. The travel time is picked at the first peak of the
receiving signal, and then used to calculate the velocity after calibration; this transit time determination criterion provides the best
determination of the wave’s phase speed over the useful frequencies of the pulse (Molyneux & Schmitt 2000). The complete set of
observed VP and VS are provided in Table 1.
trends but wave speeds on their own are difficult to interpret because
they depend on both the constituent moduli and the bulk density. The
moduli are more revealing and they are calculated from the observed
wave speeds and densities using directly eqs (3) and (4) (see Table
S3). The normalized bulk moduli Kn = Ksat /Kdry all increase upon
saturation as expected (Fig. 2b) according to Gassmann’s eq. (1)
with the larger values for the brine saturated Kn entirely consistent
with the greater Kf of brine. However, the normalized shear moduli
μn = μsat /μdry indicates that they may decrease upon brine saturation and weakly increase upon kerosene saturation, respectively
(Fig. 2b) in disagreement with Gassmann’s eq. (2).
Wettability influences seismic wave propagation
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solid with all of this done in air. In this geometry, the surface tensions between the solid–liquid, the liquid–air and the liquid–solid
interfaces compete against each other and determine the drop’s final
shape (Fig. 3a) which is described by the equilibrium, or wetting,
angle () between the air–liquid interface relative to the liquid–
solid interface from the point of intersection of all three phases. If
the liquid and solid are strongly attracted to one another, the liquid
spreads completely over the solid surface with = 0◦ , and the
liquid is said to completely wet the solid in this case. Conversely,
should they repel one another (or the liquid has a very strong selfattraction) then the liquid makes a ball shape with approaching
180◦ , this is the completely non wetting case. In general, 0◦ ≤ ≤
180◦ and as such is a measure of the degree of molecular attraction between the liquid and the solid as illustrated in Fig. 3a
which shows the evolution of the shape of the drop from completely
wetting (strong attraction) to completely non wetting (repelled).
Wettability is an important topic in many fields such as petroleum
engineering and physical chemistry and the reader is directed to
texts such as Adamason & Gast (1997) for additional details.
We measured the wetting angle of brine on a flat surface of
this epoxy to be 73.6◦ (Fig. 3b) indicating that it is only weakly
hydrophilic as is expected for the epoxy employed. For the sake
of comparison, water strongly wets quartz with = 4.1◦ (Askvik
et al. 2005). In contrast, once the kerosene drop falls on the same
epoxy it rapidly spreads and completely wets the same surface
( = 0◦ ; Fig. 3c) indicating that it is highly oleophilic.
These variations in attraction are likely due to significant differences in these liquids’ physio-chemical behaviour.
Water’s molecular structure is highly polarized with distinct positively and negatively charged ends. In contrast, kerosene is
typically a mixture of a number of saturated hydrocarbons containing 6 to 16 carbon atoms in each molecule. Major compounds typically include straight- and branched-chain alkanes
(paraffins) and cycloalkanes (napthenes) with lesser fractions of
alkylbenzenes, alkylnapthealenes and olefins. The H–C bonds
are nonpolar and consequently so are these compounds. This
leads to substantial differences in how they wet an epoxy-coated
surface.
The degree of wetting is symptomatic of the way in which the
solid and the liquid interact with one another across their boundary. Classical hydrodynamics presumes the liquid molecules immediately at the surface are essentially permanently bound there
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Figure 2. (a) Observed wave speeds for dry, brine and kerosene saturation. The relative error is within the symbol. (b) Saturated bulk modulus Ksat normalized
by the dry modulus Kdry and shear modulus μsat normalized by the dry modulus μdry . (c) Ksat predicted using the Biot–Gassmann model versus Ksat observed.
(d) μsat predicted using the Biot–Gassmann model versus μsat observed. In (c) and (d) the vertical extension of the data symbol represents the calculated range
of Biot–Gassmann modulus dispersion while the horizontal error bar is that estimated by propagation of observational uncertainties.
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Z. Wang, D.R. Schmitt and R. Wang
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Figure 3. (a) Schematic of the wetting angle of a fluid on a flat solid surface. Wetting angle measurements of (b) brine and (c) kerosene on the consolidated
epoxy surface at room temperature and pressure. (d) Schematic of no-slip, slip and complete slip boundary conditions at the liquid–solid interface [adapted
from Lauga et al. (2007) and Mchale et al. (2010)].
(see Blake 1990). This is called the ‘no-slip’ boundary condition
(Fig. 3d) and it has been commonly assumed in describing laminar flow (Landau & Lifshitz 1959; McHale et al. 2010). However,
this assumption cannot be derived from first principles and even
its originator (Stokes 1851) worried about it. There is now a great
deal of evidence for slip (Lauga et al. 2007) from, for example,
the measurement of permeability (Steenkamer et al. 1995; Blunt
1997), in effects in microfluidic devices (e.g. Ralston et al. 2008),
Wettability influences seismic wave propagation
and in resonances of transverse-mode ultrasonic sensors (e.g. Ellis
& Hayward 2003).
Although this remains under active study, the amount of slip
allowed appears to correlate with the wetting angle (Lauga et al.
2007). That is, the less hydrophilic a surface is the greater the
molecular slip that is allowed (Fig. 3d). This suggests that the differences here in the wetting of the brine and the kerosene in the
current experiments may be symptomatic of the propensity for slip
to occur along the liquid–solid interface. For example, Vinogradova
& Yakubov (2003) observed slip at the Polystyrene–NaCl solution
interface with wetting angle of 90◦ . There are observations of slip
even when the wetting angle is only 35◦ (Cho et al. 2004). The
completely wetting kerosene conforms with the no-slip boundary
condition while the weakly hydrophilic brine may not (e.g. Ho et al.
2011).
Implications for Biot—Gassmann models
C O N C LU S I O N S
Our observations indicate that wettability affects the wave speeds
and elastic moduli of fully liquid-saturated synthetic carbonates.
Further, the observations obtained on the porous samples when saturated with completely wetting kerosene are predicted well with existing Biot–Gassmann theory. This same theory significantly over-
predicts the shear moduli, however, when the same porous materials
are saturated with weakly wetting brine. These observations suggest that extra-mechanical factors can be important to understanding
wave propagation in liquid-saturated porous media.
Wettability is known to correlate with the propensity for molecular slip. As such, the fully wetting kerosene is expected to conform
to the no-slip assumption while the weakly wetting brine may not.
This is in opposition to the conventional ‘no-slip’ boundary condition of hydrodynamics; an assumption that is implicit within the
Biot–Gassmann models and their subsequent adaptations (e.g. Johnson et al. 1994b). The success and failure of the Biot–Gassmann
equations to in respectively predicting the kerosene- and the brinesaturated observations may be due to differences in the molecular
slip allowed for the two liquids. Regardless of whether molecular
slip is responsible, once mechanical effects are accounted for the
only obvious difference between two liquids is their wetting behaviour. Additional experimental and theoretical work is necessary
to both examine the face value effects of wetting on wave propagation and to go further to test the underlying physical mechanisms.
AC K N OW L E D G E M E N T S
This research was supported by the National Natural Science
Foundation of China (No.51274230), China Scholarship Council (201406450011) and the Excellent Doctoral Dissertation Cultivation Program of China University of Petroleum (Huadong)
(UPC201403011). DRS is supported by the Canada Research Chairs
program.
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S U P P O RT I N G I N F O R M AT I O N
Additional Supporting Information may be found in the online version of this paper:
Figure S1. The ultrasonic measurement procedures for different
saturation conditions.
Figure S2. Typical received P and S waveforms (sample V4) at
different saturation conditions. The lower three waveforms are the
P waveforms at different saturation conditions. And the upper three
waveforms are the S waveforms at different saturation conditions.
Table S1. The physical properties of the solid constituents and pore
fluids.
Table S2. Gassmann- and Biot-predicted bulk and shear moduli at
brine- and kerosene saturations based on dry observations.
Table S3. Observed bulk and shear moduli at different saturation
conditions.
Table S4. The relative error of observed bulk and shear
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