Geophysical Journal InternationalGeophys. J. Int. (2015) 203, 2182–2188 doi: 10.1093/gji/ggv434

GJI Marine geosciences and applied geophysics

Does wettability influence seismic wave propagationin liquid-saturated porous rocks?

Zizhen Wang,1,2 Douglas R. Schmitt2 and Ruihe Wang1

1School of Petroleum Engineering, China University of Petroleum (Huadong), Qingdao, China. E-mail: wzzhprince@126.com2Institute for Geophysical Research, University of Alberta, Edmonton, Canada

Accepted 2015 October 5. Received 2015 October 1; in original form 2015 July 09

S U M M A R YIn order to evaluate the effects of different fluids, we measured the P and S wave speedsthrough a series of synthetic epoxy-bonded carbonate composites with porosities from 21.9 to31.5 per cent and saturated with air, with kerosene and with brine. These observed speeds wereconverted to the saturated bulk and shear moduli and compared to predictions made using Biotand Gassmann formulations. The observed bulk moduli agreed with those calculated for bothkerosene and brine saturation as did the high frequency shear modulus under kerosene satu-ration. The observed shear modulus under water saturation, however, was significantly lowerthan the prediction. After excluding the currently known mechanisms of shear weakening, wesuspect this disparity may be due to variations in the wetting of the epoxy that coats the poressurfaces. The kerosene completely wets this surface while the brine is only weakly hydrophilicwith a wetting angle of 73.6◦. At the molecular scale, this means that Stoke’s no-slip boundarycondition may not always apply as has been more recently demonstrated by many researchersin other disciplines such as microfluidic engineering but the implications for wave propagationin liquid-saturated rocks have not been considered.

Key words: Microstructures; Elasticity and anelasticity; Wave propagation; Acoustic prop-erties.

I N T RO D U C T I O N

Seismic waves propagating and scattering within the earth’s crustcontain substantial information on in situ conditions, but theirproper interpretation necessitates a full understanding of the phys-ical phenomenon affecting them along their travel path. In highlyporous rocks, the models of Gassmann (1951) and Biot (1956a,b)have been widely used for predicting the saturated bulk Ksat andshear μsat elastic moduli of porous rocks saturated with differentfluids. As the simplest example, in the zero-frequency limit of theBiot–Gassmann model, these are given by Gassmann’s (1951) equa-tions

Ksat = Kdry + α2

φ

Kf+ α−φ

Km

(1)

μsat = μdry, (2)

where Kdry, Km and Kf are the bulk moduli of the rock’s dryframe, its mineral constituents, and its saturating fluid, Kdry andKsat are also often referred to as the drained and undrained bulkmoduli, 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

developed eqs (1) and (2) to study seismic wave propagation withthe compressional (VP) and shear (VS) wave speeds:

VP =√

Ksat + 4μsat/3

ρsat(3)

and

VS =√

μsat

ρsat, (4)

where the saturated rock composed of solid minerals and fluid sat-urant of densities ρm and ρf, 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 consideringthe effects of viscosity and internal drag and necessitates additionalconsiderations of viscosity, permeability, pore dimensions, and tor-tuosity. These Gassmann’s and Biot’s models are purely mechanical,they does not account for changes that could result from variationsin chemistry. To be fair, both Gassmann (1951) and Biot (1962) rec-ognized 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-mechanicaleffects have largely been ignored in seismological studies.

2182C© The Authors 2015. Published by Oxford University Press on behalf of The Royal Astronomical Society.

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A key canon of the Gassmann’s model at low frequency is thatthe rock’s frame shear modulus (μsat) should remain constant re-gardless of the saturant (Berryman & Milton 1991). However, thereare 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 de-pending on the saturant, the effective pressure, the pore structure,the existence of clays, and the frequency. Within the literature thesechanges have respectively been called shear ‘stiffening’ or ‘harden-ing’ and ‘softening’ or ‘weakening’. But the survey of this literaturedoes not appear to show any definitive trend towards either weaken-ing or stiffening; more work is necessary. Certainly, water is knownto strongly weaken the apparent strength of rock when taken tofailure (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 areexactly those responsible for the material’s elastic moduli, which ishere the focus of this study. Many of these observations are likelycomplicated by sample containing cracks where local flow effectswill dominate, but it cannot account for all of the variations. Thereason for this discrepancy between Gassmann’s theory and experi-mental findings are still not fully clear (Adam et al. 2006; Fabriciuset al. 2010).

Here, we present observations suggesting that the degree of wet-ting between the fluid and the solid may also affect wave prop-agation. Somewhat unexpectedly, the only study we are aware ofthat explicitly mentions the influence of wetting is that of Wyllieet al. (1958), the same contribution from which originates the time-average equation. They detailed an extensive series of laboratory

measurements of the acoustic wave speeds in samples of porousBerea sandstone, in porous alundum, and in porous TeflonTM (poly-tetrafluoroethylene). Their samples were specifically saturated witha variety of liquids that were either wetting or nonwetting (e.g. wa-ter in porous Teflon). We are unaware of any subsequent studiesthat have examined the influence of wetting on wave propagationin fully saturated materials, although there are numerous studies ofthe related problem of adsorption (e.g. Pimienta et al. 2014).

We first describe the results of laboratory wave speed measure-ments in synthetic carbonates under different full saturations. Weanalyse the results under the Biot–Gassmann paradigm and showthat the sample saturated with a fully wetting fluid is well described,while that saturated with a weakly wetting fluid is not. We suggestthat these different behaviours may be related to the ability of theliquid molecules to ‘slip’ along the liquid–solid interface in oppo-sition to the classic no-slip conditions that would be implicit to theBiot–Gassmann model.

M E T H O D

Following Wang et al. (2015) we manufacture synthetic porousand permeable carbonate cylinders (Fig. 1a) using a mixture ofnatural carbonate rock powder and a two-component epoxy (seeTable S1 for the constituent properties). The mineral compositesof the carbonate powder include both calcite and dolomite, andthe mean grain size is about 200 μm (Fig. 1b). After curing, theepoxy cement is chemically resistant to water or kerosene. Porosityis controlled by randomly adding NaCl grains (∼3 mm width) tothe carbonate–epoxy mixture while it is being placed into remov-able steel cylindrical molds. The mixture is uniaxially compressedat 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 theepoxy. 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 interparticlepores and large vugs. It can also be seen the carbonate cuttings are coated with the epoxy.

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Table 1. Petrophysical characteristics and measured velocities at differentsaturation conditions of all synthetic samples.

Dry Graindensity density NaCl Vuggy Total Permeability

Index (g cm–3) (g cm–3) grains (g) porosity porosity (mD)

V1 2.13 2.76 1.45 0.025 0.227 567V2 2.13 2.73 1.66 0.028 0.219 402V3 2.05 2.75 2.90 0.052 0.256 744V4 1.85 2.70 5.80 0.097 0.315 1497

Brine-saturated Kerosene-saturatedIndex Dry (m s–1) (m s–1) (m s–1)

Vp Vs Vp Vs Vp VsV1 4089 2451 4142 2298 4112 2380V2 4187 2531 4241 2375 4207 2474V3 3831 2329 3880 2132 3896 2265V4 3241 1994 3352 1774 3378 1918

proper curing. Water is then passed through the cylinder to dissolveout the NaCl grains leaving large open voids (Fig. 1c) that supple-ments the porosity of the mixture. An SEM image at smaller scale(Fig. 1d) shows the grain assembly, inter-particle pores, and vuggypores in the synthetic samples. It can be seen from Fig. 1(b) and(d) that the carbonate cutting surfaces are coated with consolidatedepoxy.

We made four samples (Fig. 1a) with total porosities rangingfrom 21.9 to 31.5 per cent as determined using a Boyle’s Law Hepycnometer (Table 1). Permeability to water was measured usinga standard Darcy’s law flow through method. It should be notedthat these four samples have similar matrix composites and com-paction process, so they should have similar grain density (around2.70 g cm–3) and interparticle porosity (around 20 per cent), whichis confirmed by our measurements and also indicates that the mix-ing is reproducible. Further, we do not expect these samples tocontain 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 mod-ulus and viscosity) and the densities of the brine and kerosene donot vary from each other significantly (Table S1). Both VP and VS

were measured using standard ultrasonic pulse-transmission mea-surements under full saturations of air, brine, and kerosene (see Fig.S1 for the experimental protocol). The central frequencies of thetransducers are 0.5 MHz for P wave and 0.25 MHz for S wave. Thereceived signals are acquired at 50 MHz for 2000–5000 samples(Fig. S2). Estimates of the cut-off frequency (Table S2) fall in therange of 30–150 kHz (Johnson 1984; Sarout 2012) lying just belowthe useful bandwidth of the signals. High-pass filters (>0.1 MHzfor P wave and >0.05 MHz for S wave) are applied to remove lowfrequency bias. The travel time is picked at the first peak of thereceiving signal, and then used to calculate the velocity after cal-ibration; this transit time determination criterion provides the bestdetermination of the wave’s phase speed over the useful frequen-cies of the pulse (Molyneux & Schmitt 2000). The complete set ofobserved VP and VS are provided in Table 1.

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 saturationrelative to the dry measures. These observations all follow expected

trends but wave speeds on their own are difficult to interpret becausethey depend on both the constituent moduli and the bulk density. Themoduli are more revealing and they are calculated from the observedwave speeds and densities using directly eqs (3) and (4) (see TableS3). The normalized bulk moduli Kn = Ksat/Kdry all increase uponsaturation as expected (Fig. 2b) according to Gassmann’s eq. (1)with the larger values for the brine saturated Kn entirely consistentwith the greater Kf of brine. However, the normalized shear moduliμn = μsat/μdry indicates that they may decrease upon brine satu-ration and weakly increase upon kerosene saturation, respectively(Fig. 2b) in disagreement with Gassmann’s eq. (2).

D I S C U S S I O N

Predictions versus observations

We are confident that these materials do not contain microcracksdue to their construction and that, excepting additional effects notanticipated, we expect the saturated behaviour to be described underthe Biot–Gassmann paradigm. This approach has been successful innumerous other studies in highly porous materials such as sinteredglass beads (see review in Bouzidi & Schmitt 2009) or sinteredcorundum (Schmitt 2015) where application of Biot’s full equationshas 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) wavespeeds according to the Biot’s model (Biot 1956a,b; Berryman1980) using the algorithms provided in Mavko et al. (2009), detailsof the calculations are included in the supplementary information.These are converted to the frequency dependent Ksat and μsat witheqs (3)–(5). The Ksat predicted at higher frequencies show littledispersion from the zero-frequency limit Gassmann-predicted Ksat

(Fig. 2c). That is, the Biot–Gassmann model over the range offrequencies calculated predicts well Ksat for both brine and kerosenesaturations.

The situation differs somewhat for μsat. The predicted μsat aremore dispersive as illustrated by the height of the symbols inFig. 2(d). The predicted kerosene-saturated μsat agree well withthe observations. This together with the good agreement for thecorresponding Ksat in Fig. 2(c) suggests that the Biot–Gassmanntheory adequately models the behaviour under kerosene saturation.In contrast, the predicted brine-saturated μsat significantly exceedthose observed (Fig. 2d).

The carbonate cuttings in our synthetic samples are coated withconsolidated epoxy (Fig. 1d) that is chemically resistant to brineand kerosene. Therefore, both brine and kerosene cannot change thegrain-grain contacts in our synthetic samples. This means that thecurrently known mechanisms of water weakening are not suitable toexplain modulus variations here. As already noted, from the purelymechanical perspective inherent to the Biot–Gassmann model thetwo liquids are not significantly different, and it is important to askwhy the model predicts the kerosene saturated values well but failsto predict the brine saturated values.

Wetting behaviour

‘Wettability’ refers here to the ability of a fluid to adhere to a solidand microscopically relies on the degree of molecular attractionbetween the fluid and the solid (e.g. Gueguen & Palciauskas 1994).The ability of a fluid to ‘wet’ a solid surface is determined exper-imentally by placing a drop of the liquid onto a flat surface of the

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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 normalizedby 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 rangeof Biot–Gassmann modulus dispersion while the horizontal error bar is that estimated by propagation of observational uncertainties.

solid with all of this done in air. In this geometry, the surface ten-sions between the solid–liquid, the liquid–air and the liquid–solidinterfaces compete against each other and determine the drop’s finalshape (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. Ifthe liquid and solid are strongly attracted to one another, the liquidspreads completely over the solid surface with � = 0◦, and theliquid is said to completely wet the solid in this case. Conversely,should they repel one another (or the liquid has a very strong self-attraction) then the liquid makes a ball shape with � approaching180◦, this is the completely non wetting case. In general, 0◦ ≤ � ≤180◦ and as such � is a measure of the degree of molecular at-traction between the liquid and the solid as illustrated in Fig. 3awhich shows the evolution of the shape of the drop from completelywetting (strong attraction) to completely non wetting (repelled).Wettability is an important topic in many fields such as petroleumengineering and physical chemistry and the reader is directed totexts such as Adamason & Gast (1997) for additional details.

We measured the wetting angle of brine on a flat surface ofthis epoxy to be 73.6◦ (Fig. 3b) indicating that it is only weakly

hydrophilic as is expected for the epoxy employed. For the sakeof comparison, water strongly wets quartz with � = 4.1◦ (Askviket al. 2005). In contrast, once the kerosene drop falls on the sameepoxy 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 signifi-cant differences in these liquids’ physio-chemical behaviour.Water’s molecular structure is highly polarized with distinct pos-itively and negatively charged ends. In contrast, kerosene istypically a mixture of a number of saturated hydrocarbons con-taining 6 to 16 carbon atoms in each molecule. Major com-pounds typically include straight- and branched-chain alkanes(paraffins) and cycloalkanes (napthenes) with lesser fractions ofalkylbenzenes, alkylnapthealenes and olefins. The H–C bondsare nonpolar and consequently so are these compounds. Thisleads to substantial differences in how they wet an epoxy-coatedsurface.

The degree of wetting is symptomatic of the way in which thesolid and the liquid interact with one another across their bound-ary. Classical hydrodynamics presumes the liquid molecules im-mediately at the surface are essentially permanently bound there

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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 consolidatedepoxy surface at room temperature and pressure. (d) Schematic of no-slip, slip and complete slip boundary conditions at the liquid–solid interface [adaptedfrom 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 lami-nar 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 greatdeal of evidence for slip (Lauga et al. 2007) from, for example,the measurement of permeability (Steenkamer et al. 1995; Blunt1997), in effects in microfluidic devices (e.g. Ralston et al. 2008),

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Wettability influences seismic wave propagation 2187

and in resonances of transverse-mode ultrasonic sensors (e.g. Ellis& Hayward 2003).

Although this remains under active study, the amount of slipallowed appears to correlate with the wetting angle (Lauga et al.2007). That is, the less hydrophilic a surface is the greater themolecular slip that is allowed (Fig. 3d). This suggests that the dif-ferences here in the wetting of the brine and the kerosene in thecurrent experiments may be symptomatic of the propensity for slipto occur along the liquid–solid interface. For example, Vinogradova& Yakubov (2003) observed slip at the Polystyrene–NaCl solutioninterface with wetting angle of 90◦. There are observations of slipeven when the wetting angle is only 35◦ (Cho et al. 2004). Thecompletely wetting kerosene conforms with the no-slip boundarycondition while the weakly hydrophilic brine may not (e.g. Ho et al.2011).

Implications for Biot—Gassmann models

The no-slip boundary condition is implicit to the Biot–Gassmannmodels by dint of Biot’s (1956b) words ‘the condition U1 = 0at the wall’. As the kerosene completely wets the epoxy surface,this assumption is likely valid for the kerosene-saturated mediumand, consequently, may explain the ability of the Biot–Gassmannmodel to explain those observations. In contrast, if slip does occuracross the brine-epoxy surface then this boundary condition maynot hold perfectly resulting in the divergence of the theory from theobservations.

When the fluid is completely wetting, the fluid at the solid–fluidinterface is static (Fig. 3d) and tends to maintain immobility, whichwould apply a shear force to the solid to resist mobility when thestress wave propagates through the saturated rock. However, whenthe fluid is less wetting, it would allow slip at the solid–fluid interface(Fig. 3d). The solid–fluid interface becomes less shear-resistant thanexpected. Such decrease in shear resistance accumulates up at allfluid–solid interfaces, and the effective shear moduli of the saturatedrock become weakened at the macroscopic scale.

However, if interfacial slip is responsible for decreasing the rigid-ity under brine saturation as shown in Figs 2(b) and (d), it is notimmediately obvious how to include it in the Biot’s model. Biot’smodel incorporates numerous phenomena at the pore scale thatalready relate to interactions between the liquid and the solid. Forexample, one implication of slip is that the rate of fluid flow throughthe medium increases with consequent modifications to the effectivepermeability or fluid viscosity. In a similar vein, slip might affectboth the viscous and inertial drag forces experienced as the fluid isdisplaced relative to the solid and hence require modification of themass coupling coefficients. Geometric factors such as the viscousskin depth or the dynamic connected porosity (see Johnson et al.1994a) might also be influenced. Many of these factors are cou-pled to one another, and resolution of this problem awaits suitablemodification of the Biot–Gassmann equations to accommodate slip.We note investigations of the relationships between acoustics andinterfacial slip are still nascent (Huang & Szlufarska 2012) but weare as yet unaware of any studies through porous media.

C O N C LU S I O N S

Our observations indicate that wettability affects the wave speedsand elastic moduli of fully liquid-saturated synthetic carbonates.Further, the observations obtained on the porous samples when sat-urated with completely wetting kerosene are predicted well with ex-isting Biot–Gassmann theory. This same theory significantly over-

predicts the shear moduli, however, when the same porous materialsare saturated with weakly wetting brine. These observations sug-gest that extra-mechanical factors can be important to understandingwave propagation in liquid-saturated porous media.

Wettability is known to correlate with the propensity for molecu-lar slip. As such, the fully wetting kerosene is expected to conformto the no-slip assumption while the weakly wetting brine may not.This is in opposition to the conventional ‘no-slip’ boundary con-dition of hydrodynamics; an assumption that is implicit within theBiot–Gassmann models and their subsequent adaptations (e.g. John-son et al. 1994b). The success and failure of the Biot–Gassmannequations to in respectively predicting the kerosene- and the brine-saturated observations may be due to differences in the molecularslip allowed for the two liquids. Regardless of whether molecularslip is responsible, once mechanical effects are accounted for theonly obvious difference between two liquids is their wetting be-haviour. Additional experimental and theoretical work is necessaryto both examine the face value effects of wetting on wave propaga-tion and to go further to test the underlying physical mechanisms.

A C K N OW L E D G E M E N T S

This research was supported by the National Natural ScienceFoundation of China (No.51274230), China Scholarship Coun-cil (201406450011) and the Excellent Doctoral Dissertation Cul-tivation Program of China University of Petroleum (Huadong)(UPC201403011). DRS is supported by the Canada Research Chairsprogram.

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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 ver-sion of this paper:

Figure S1. The ultrasonic measurement procedures for differentsaturation conditions.Figure S2. Typical received P and S waveforms (sample V4) atdifferent saturation conditions. The lower three waveforms are theP waveforms at different saturation conditions. And the upper threewaveforms are the S waveforms at different saturation conditions.Table S1. The physical properties of the solid constituents and porefluids.Table S2. Gassmann- and Biot-predicted bulk and shear moduli atbrine- and kerosene saturations based on dry observations.Table S3. Observed bulk and shear moduli at different saturationconditions.Table S4. The relative error of observed bulk and shearmoduli. (http://gji.oxfordjournals.org/lookup/suppl/doi:10.1093/gji/ggv434/-/DC1).

Please note: Oxford University Press is not responsible for the con-tent or functionality of any supporting materials supplied by theauthors. Any queries (other than missing material) should be di-rected to the corresponding author for the paper.

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