Seismoelectric Phenomena in Fluid-Saturated Sediments

Transcription

1 UCRL-JRNL Seismoelectric Phenomena in Fluid-Saturated Sediments G. I. Block, J. G. Harris April 25, 2005 Journal o Geophysical Research

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3 Seismoelectric Phenomena in Fluid-Saturated Sediments Gareth I. Block Lawrence Livermore National Laboratory 7000 East Avenue, L-206, Livermore, CA John G. Harris Ctr. QEFP, Northwestern University 2137 N. Sheridan Rd., Evanston, IL Abstract: Seismoelectric phenomena in sediments arise rom acoustic wave-induced luid motion in the pore space, which perturbs the electrostatic equilibrium o the electric double layer on the grain suraces. Experimental techniques and the apparatus built to study this electrokinetic (EK) eect are described and outcomes or studies o seismoelectric phenomena in loose glass microspheres and medium-grain sand are presented. By varying the NaCl concentration in the pore luid, we measured the conductivity dependence o two kinds o EK behavior: (1) the electric ields generated within the samples by the passage o transmitted acoustic waves, and (2) the electromagnetic wave produced at the luid-sediment interace by the incident acoustic wave. Both phenomena are caused by relative luid motion in the sediment pores this eature is characteristic o poroelastic (Biot) media, but not predicted by either viscoelastic luid or solid models. A model o plane-wave relection rom a luidsediment interace using EK-Biot theory leads to theoretical predictions that compare well to the experimental data or both sand and glass microspheres. 1

4 1 Introduction We developed electrokinetic techniques in the laboratory to monitor acoustic wave propagation in electrolyte-saturated, unconsolidated sediments. Because most underwater imaging and naval operations require predicting the acoustical properties o the seabed, one application o our work is to the question: Are ocean seabed sediments best described by viscoelastic luid or solid models, or poroelastic ones? Experimentally derived, ad hoc viscoelastic luid and solid models commonly used in ocean acoustics [Hamilton, 1972, 1974] are oten unable to predict how requency-dependent behavior varies with sediment type because they lack a direct connection between micro- and macroscale properties. To remedy this situation, a number o researchers (e.g., Stoll [1983, 2002], Chotiros [1995], Williams [2001], and Williams et al. [2002]) have studied models based on the Biot theory o poroelasticity [Biot, 1956a, 1956b]. And extensions have been made to these models to account or grain contact [Biot, 1962; Tutuncu and Sharma, 1992], stick-slip phenomena [Buckingham, 1997, 1998, 2005], and squirt low [Murphy et al., 1986; Dvorkin and Nur, 1993; Chotiros and Isakson, 2004]. While traditional acoustical methods have had diiculty distinguishing experimentally among the predictions o these competing theories, only poroelastic media are capable o generating a macroscopic, EK response. Techniques that measure seismoelectric phenomena oer a unique means o assessing how successully Biot theory (and its extensions) model sediments. The coupled EK-Biot theory developed by Pride [1994] describes how acoustic waves generate electromagnetic (EM) waves (the seismoelectric eect), and reciprocally, how electromagnetic ields generate acoustic waves (the electroseismic eect) in electrolyte-saturated, porous media. Both seismoelectric and electroseismic imaging 2

5 techniques have been developed or ield research [Thompson and Gist, 1991; Mikhailov et al., 1997; Zhu et al., 1999], well-logging [Chandler, 1981], and modeled numerically [Haartsen and Toksöz, 1996; Pride and Haartsen, 1996; Haartsen and Pride, 1997]; Beamish [1999] provides a good review o these techniques in seismology. In the context o material characterization, a simpliied orm o EK-Biot theory was used to study the permeability and pore eatures o consolidated rock and sandstones at requencies below 100 Hz [Pengra and Wong, 1995; Pengra et al., 1999], and constant low-rate EK measurements have been perormed in unconsolidated sand [Ahmed, 1964]. We measured high-requency ( khz) seismoelectric potentials in laboratory experiments using unconsolidated medium-grain sand and loose glass microspheres, which were saturated by NaCl electrolytes with a range o electrical conductivities. In the ollowing sections we describe these measurements and compare them to predictions derived rom coupled EK-Biot theory. Section 2 introduces EK phenomena and the coupled EK-Biot theory. The apparatus and typical data sets in medium-grain sand and loose glass microspheres are described in Section 3. In Section 4, we derive expressions or plane-wave relection and transmission rom a luid-sediment interace to analyze how seismoelectric-wave phenomena depend on requency, angle o incidence, and poreluid conductivity. We compare our experimental results with the predictions o the EK- Biot theory in Section 5. Section 6 summarizes and concludes with a discussion o the impact o EK measurement techniques in sediment acoustics. Material properties and analyses o the samples tested are given in Appendices A and B, respectively. 3

6 2 Electrokinetics 2.1 Electric Double Layer Electrokinetic phenomena arise because an electric double layer orms near the grain surace, as shown in Figure 1. The bare surace o silicon dioxide (SiO, the prime constituent o glass and sand) oten carries a small negative charge due to naturally deprotonated silanol groups (Si-O 2 ). When in contact with an electrolyte say, NaCl in water this surace charge creates an electric potential that aects the charge distribution in the surrounding luid. A physical model or this structure was developed by Gouy [1910], Debye and Huckel [1923], and Stern [1924] who developed the double layer concept. In the simplest case, counter ions (Na + ) in the pore luid are attracted to and adsorbed by the negatively charged grains; they are bound chemically in an atomically thin, immobile layer. Further rom the surace is a distribution o mobile counter ions in a diuse layer [Hiemenz and Rajagopalan, 1997]. The potential at the interace between the immobile and diuse layers is called the ζ potential. It is sensitive to the available binding sites at the grain surace, as well as to the electrolyte concentration and ph o the pore luid. Because the electric potential decays exponentially away rom the grain wall, the eective thickness o the electric double layer is oten less than 10 nm. A simple example o EK phenomena arises in the case o luid low in a silica capillary [Rice and Whitehead, 1965]. An electric ield aligned parallel to the grain wall causes the ions in the diuse layer to move, dragging the pore luid along with it because o the presence o viscous stresses. The reciprocal eect also exists: an applied pressure gradient will create both luid low and ionic convection current, which in turn, produces 4

7 an electric potential. To study these phenomena in poroelasticity at a macroscopic scale, the average acoustic and electromagnetic ields in the presence o a complex network o capillaries must be determined. Previous models o seismoelectric phenomena in geophysics [Frenkel, 1944; Fitterman, 1978; Auriault and Strzekecki, 1981; Neev and Yeats, 1989] and colloidal chemistry [O'Brien, 1988] did not use the ull set o Maxwell s equations and/or limited their scope to the low-requency case. They thereore ailed to predict key theoretical and experimental behaviors such as EM-wave generation at a luid-sediment interace caused by a high-requency, incident acoustic wave that are a robust eature in our experimental data. It is or this reason that we use the ull coupled EK-Biot theory. 2.2 Coupled Electrokinetic-Biot Theory Poroelasticity depends (oten implicitly) on the notion o a hierarchy o scales. Transducers ix the scale o our observations. Consider, or example: (1) the size o their active suraces, which orm a portion o the macroscopic boundary, (2) their requency response and bandwidth, which together deine a range o resolvable time scales, and (3) the strength o their interactions with the material, which determines just how ar rom equilibrium the material will be driven. We ocus on small amplitude disturbances with operating requencies less than 1 MHz, so that the acoustic wavelength is always much larger than typical grain-scale heterogeneities. The sediments are also assumed to be homogeneous and isotropic on the macroscale in what ollows. Because poroelasticity depends on the kinematics o both the luid and solid, it requires two macroscale balance laws. To derive these laws, Pride [1994] applied the 5

8 techniques o volume averaging [Slattery, 1967] to locally average the microscale acoustic and electromagnetic ield equations. Coupling along the interaces between the luid and solid phases results in nontrivial (and oten requency-dependent) poroelastic coeicients. The irst law describes the balance o linear momentum: ( s ) u + w = τ (1) 2 ω ρbulk ρ bulk. The second law describes macroscale luid low; it is a orm o Darcy's law, which we give shortly. A time dependence o exp( iωt), with angular requency ω, is assumed, and overbars denote locally averaged quantities. Here, bulk ( 1 ) ( 1 ) τ = φ τ φpi ρ = φ ρ + φρ bulk bulk w = φ( u u ). s s (2) The variables u s, u, w, p, and τ s are the average luid displacement, solid displacement, relative luid displacement, luid pressure, and solid stress tensor, respectively. The luid density ρ and solid density ρ s combine to orm a bulk density ρ bulk based on the luid volume raction φ. Constitutive relations are given in (A.1) (A.3) in Appendix A. Volume-averaged versions o Maxwell s equations are also determined or the porous medium: E= iωµ H H = iωε E+ J. 0 bulk (3) Here, E and H are the average electric and magnetic ields and J is the average current density. The magnetic permeability µ 0 is assumed constant or both the luid and solid. The bulk permittivity o the porous medium, 6

9 φ ε ε ( κ κ ) κ bulk = 0 s + s, α (4) is deined in terms o the porosity φ, sediment tortuosity α, vacuum permittivity ε 0, and dielectric constants or the luid and solid phases, κ and κ, respectively. The primary result o averaging is that Darcy s and Ohm s laws are coupled when the presence o the electric double layer is taken into account. Following Pride [1994], these lux-orce relations can be written in a symmetric orm: s ( ω) ( 2 ) ( ) k iωw = p+ ρ ω us + L ω E η 2 ( ω)( p ρ ω ) σ ( ω) J = L + u + E. s bulk (5) The irst line in (5) is an augmented orm o Darcy s law based on the viscosity o the luid η and dynamic permeability k ( ω ) (deined in (A.4) (A.6)), which captures the eect o sound-speed dispersion and attenuation in sediments by modeling the transition rom viscous pore low at low requencies to a type o boundary-layer low near the grain suraces at high requencies [Pride et al., 1992]. The electrokinetic-coupling coeicient L( ω ) (deined in (A.7) (A.9)) depends explicitly on the electric double-layer properties. When L( ω ) is set to zero, Darcy's and Ohm's laws are uncoupled and we obtain the original Biot theory and Maxwell s equations. The second line o (5) is a generalization o Ohm s law: it consists o contributions rom bulk electromigration and ionic convection currents generated by luid low in the pore space. The bulk electrical conductivity σ ( ) in EK-Biot theory and is discussed at length in Section 3.3. bulk ω plays an important role 7

10 3 Laboratory Experiment 3.1 Apparatus A diagram o our apparatus is given in Figure 2. Seismoelectric phenomena are excited by an acoustic, sine-wave burst injected at the top o the apparatus; this wave propagates down the luid-illed tube and impacts the saturated sediment about 750 µs later. The incident wave generates relected and transmitted acoustic waves, as well as EM waves at the interace; the transmitted acoustic wave is accompanied by a quasistatic electric ield in the sediment. By varying the NaCl concentration in the pore luid between S/m and 0.12 S/m, we determined the conductivity dependence o (1) the electric potential localized within the support o the transmitted acoustic wave, and (2) the EM waves generated at the interace. The various electric ields are measured by laboratory grade, sintered Ag/AgCl electrodes (A-M Systems) that are ixed in a vertical array both above and below the luid-sediment interace at nine vertical positions. The apparatus consists o a PVC tube, hal-illed with de-ionized (DI) water in which NaCl is dissolved, and hal-illed with glass microspheres or medium-grain sand saturated by this electrolyte. The saturating luid is drained rom below and simultaneously replenished rom above with a dierent NaCl solution ater each measurement sequence, and the sediment is kept ully saturated throughout this process. A 100 khz (center requency) submersible acoustic transducer at the top o the apparatus is controlled by a data acquisition PC and driven with a 50 khz sine-wave burst or approximately 60 µs. While the amplitude o the transmitted wave is not a maximum 8

11 at this operating requency, the seismoelectric response levels, which decrease with increasing requency, are largest or the 50 khz burst. Short-time bursts or the transducer input signals ensure that the electrode responses are not due to electrical crosstalk. Electrode data is ampliied 60 db, averaged 1000 times or each measurement, and band-pass iltered between 2 khz and 500 khz to remove unwanted noise. The entire apparatus is set inside a copper-mesh Faraday cage, which acts as common ground or one port o the 60 db pre-ampliier, as well as or the other laboratory instruments (the transducer, PC, power ampliier, etc.). The copper-mesh cage increases signal-to-noise levels more than ten-old, and thereore plays an important role in the experiment. See the igure caption or urther details. In a separate calibration measurement, we determined the pressure waveorm o the incident acoustic wave at the position o a (virtual) luid-sediment interace by ixing a hydrophone at the end o a onemeter portion o the PVC tube. 3.2 Discussion o a Typical Data Set Figures 3(a) and 3(b) depict seismoelectric potentials measured or the 50 khz sinewave burst in loose glass microspheres saturated by a NaCl solution with a pore-luid conductivity o S/m. A number o separate arrivals can be distinguished. Arrivals occur near 750 µs in both the water and sediment, almost simultaneously at each o the electrodes along the vertical array. These signals correspond to EM waves in the luid and sediment they are generated at the luid-sediment interace and propagate at the speed o light in each medium. Similar observations o EM waves in consolidated porous rock and sandstone are well documented [Beamish, 1999], and are similar in magnitude 9

12 to the EM waves observed in our glass and sand data. Figure 3(a) also shows that electrodes above the luid-sediment interace measure a small voltage that is coincident with the irst passage o the incident acoustic wave, perhaps caused by a pressure-wave disturbance o the double layers ormed on each electrode. Saturated medium-grain sand exhibits the same qualitative eatures; Figures 4(a) and 4(b) depict the seismoelectricresponse levels in sand using a pore-luid conductivity o S/m. The second arrivals in Figures 3(b) and 4(b) are seen to move out with increasing depth in the model sediments. Comparing arrival times or electrodes #7 and #9 leads to estimated wavespeeds o 1730 m/s and 1690 m/s (at 50 khz) or the second arrival in glass microspheres and medium-grain sand, respectively. We also applied a matched ilter technique with a broad-band chirp (in the khz band, using electrodes #3 and #8) to calculate group velocities o approximately 1770 m/s or both sand and glass microspheres. These values are typical o the wavespeeds o the Biot ast wave in unconsolidated sediments within this requency range (see Stoll [1983], or example). Accordingly, we believe the second arrival is essentially a plane, Biot ast wave. We are unable to discern any Biot slow waves or shear waves in the data (but slowwave motion is essentially diusive in unconsolidated media [Johnson and Plona, 1982]); separate arrivals rom higher-order modes in the sediment tube are also diicult to detect. The seismoelectric potentials coincident with the ast-wave arrival are generally less than 1 mv and decrease monotonically with increasing NaCl concentration, or the range o conductivities tested. Figure 5 depicts this behavior, using electrode #8, or the 50 khz sine-wave burst in glass microspheres over the ull range o conductivities. 10

13 3.3 Bulk and Pore-Fluid Conductivity The coupled EK-Biot theory described in Section 2 yields an expression o the bulk sediment conductivity, φ 2Σ σbulk ( ω) = σ + α Λ ( ω), (6) in terms o the pore-luid conductivity σ, pore-throat dimension Λ (oten a ixed raction o the grain diameter), and surace conductance o the diuse layer Σ. Coupled EK-Biot theory provides an explicit relationship between Σ and two orms o surace conduction (requency-dependent electroosmosis and electromigration; see Eqn. (242) in Pride [1994]). Because the σ bulk σ relation plays a critical role in determining seismoelectric behavior, each conduction mechanism must be accurately accounted or in order to compare numerical predictions to experimental data. Another issue is chemical equilibration in the pore space. The conductivity o the solution used to saturate the sediments (which we denote by σ w ) increases rapidly until it plateaus at the value o the equilibrium in situ pore-luid conductivity σ ; equilibration oten occurs within a ew hours and is only apparent or σ 0.01 S/m. The ph o the pore-luid showed a similar increase rom ph 7 to ph during this time. Append B details a treatment process that helped stabilize the grain surace chemistry o our samples. A separate study o the equilibration behavior provided a means o determining σ rom known values o σ w ; our results also showed that (1) chemical equilibration was substantially reduced or the treated samples, and (2) allowing or suicient time w 11

14 between runs (greater than our hours) would minimize the eect o this variability on our experimental data [Block, 2004]. Beore each run, an HP 4192A impedance analyzer is used to determine the electrical impedance (at 1 khz) o the luid above the water-sediment interace (between electrodes #1 and #2) and within the sediment (between electrodes #7 and #8). The impedances are then converted to conductivities ollowing a straightorward calibration procedure [Block, 2004]; the resulting data is summarized in Figure 6. The σ bulk σ relation that aects seismoelectric response ollows an almost linear trend or both medium-grain sand ( * ) and loose glass microspheres ( + ). However, there is a noticeable shit away rom linearity or the weakest electrolytes. As expected, the σ bulk σ w relation (indicated by symbols with holes in their centers) is strongly nonlinear. Figure 6 also includes two glass data points that correspond to the same input luid conductivity, σ w : although DI water was used in both cases, one point corresponds to 16 hours o equilibration (diamond), while the other corresponds to an equilibration time o only 4 hours. Error ellipses represent two standard deviations rom the mean. Although Pride s [1994] theory predicts a negligible contribution rom the surace conductance Σ or the range o pore-luid conductivities that we tested, other researchers ound experimental evidence that σ bulk is signiicantly aected by surace conduction in sandstones [Glover et al., 1994; Nettelblad et al., 1995] and clay-bearing sands [Wildenschild et al., 2000] containing just 1 3 % clay. A small amount o clay and silt was present in our sand, even ater numerous runs. For reasons that we explain in Section 5, we chose a itted value o 8 Σ= S to match both the bulk conductivity data and seismoelectric data or our medium-grain sand (dashed curve in Figure 6). This value 12

15 o Σ is approximately one ith the value that Wildenschild et al. [2000] ound or their cleaned Ottawa sand samples. To compare and contrast these results with the unmodiied orm o EK-Biot theory, we used the explicit version o Σ derived by Pride [1994] to interpret our measurements using loose glass microspheres (solid curve). Equation (6) also allows us to determine the ormation actor, F α / φ 4, rom the inverse o the slope o the solid curve in the large-conductivity limit. This approximate value o F is within the expected range o 3.5 to 5 that is observed in sediments with similar grain sizes. Following Williams et al. [2002], who measured porosities o between 0.36 and 0.40 or medium grain sand, we assumed φ 0.38 or both media. Then the tortuosity is estimated to be: α 1.52, which is also within the expected range o 1.35 to 2.25 or unconsolidated ocean sediments. 4 Numerical Modeling To predict the signal received at each electrode, we determine all o the electromagnetic disturbances that result rom an acoustic plane wave scattering rom an EK-Biot hal-space. Assuming that the incident wave is plane and that the wall does not aect the incident and scattered waves is a reasonable approximation because the group velocities are very close to their plane-wave counterparts, and little dispersive behavior was seen in Figures 3 and 4. Accordingly, we expect that only the lowest-order mode is signiicant in our data and that it approximates a plane wave. Another simpliication results rom the act that the electrokinetic coupling in our experiment is predominantly one way: While the incident pressure wave generates pore- 13

16 luid motion and hence a measurable electric ield, the secondary pore-luid motion that would result rom this electric ield is negligible. This statement within EK-Biot theory is equivalent to the approximation that ηl k 2 ( ω) ( ω) ( ) σ ω iωε, (7) bulk bulk which we assume here. Note that bulk are indicated by (5), and σ ( ) bulk ε is given by (4), the role o L( ) k( ) ω, ω, and η ω is deined in (6). We calculate the relative luid motion vector w independently o the EM ields by irst solving the simpler problem o relection rom a Biot hal-space. At the luid-sediment interace, relative luid motion in the EM boundary conditions acts as a source (a layer o dipoles) or an electromagnetic wave; this is the origin o the EM wave noted in our experimental results. The transmitted acoustic wave in the sediment also generates a quasi-static electric ield within its support, as indicated previously. 4.1 Relection rom a Fluid-Sediment Interace The Biot relection problem is solved by Stoll [1981] and Stern et al. [1985]. This analysis results in vector and scalar potentials within the luid and sediment (Eqns. (63) (67) in Stern et al. [1985]), rom which it is straightorward to calculate the relative luid displacement at the interace, namely, ( ) p, 0 w = w (8) θ w w z= 14

17 or a given time-harmonic component o the acoustic pressure p w incident at angle θ w (additional dependencies on angular requencyω and the luid and sediment properties are implicit). We set p w 1 z = = or the moment. The total relative luid displacement is 0 where w s is the shear-wave contribution, and w = ws + wp + w ps, (9) w and w are contributions rom the ast and slow compressional waves, respectively. These terms act as sources or the electromagnetic waves in what ollows. As shown in Figure 7, unit wavevectors or the plane waves (or propagation in the x- z plane) are deined as: p ps kˆ = cosθ xˆ + sinθ zˆ w w w kˆ = cosθ xˆ sinθ zˆ rw rw rw kˆ = cosθ xˆ sin θ zˆ, ew ew ew (10) or the incident, relected, and EM waves in water, respectively. The EM wavenumber k w = ω / cw, where cw is the phase velocity in water. In the sediment, kˆ = cosθ xˆ + sinθ zˆ eb eb eb kˆ = cosθ xˆ + sin θ zˆ, l l l (11) where l = s, p, and ps or the shear, ast, and slow waves, respectively. Wavenumbers or the EM modes, k and k, are deined shortly, while k, k, and k can be ound in ew eb s p ps Eqns. (87) and (92) in Pride and Haartsen [1996]. Note that phase matching is enorced at the interace, so that an exp( ik cos x) mechanical waveields. Maxwell s equations in water are solved by setting l θ dependence is ound or each o the EM and l 15

18 ( ik ˆ ˆ ) H = h exp k x (12) ew ew ew ew and ( ik ˆ ˆ ) E = e exp k x, (13) ew ew ew ew where ( ) k = iωµ σ iωε κ (14) ew 0 w 0 w is the EM wavenumber in water. In the sediment, (3) and (5) lead to where 2 2 iωη L Hb + kebhb = w, (15) k ( ) k = iωµ σ iωε (16) eb 0 bulk bulk is the so-called bare EM wavenumber in sediment. Because the EM ields do not generate mechanical motion (using the approximation in (7)), the EM wavenumber is determined by the sediment's electrical properties alone (the exact value is ound by solving an eigenvalue problem derived rom the ully coupled EK-Biot equations in Pride and Haartsen [1996]). The magnetic ield in the sediment is split into two parts, H = H + H. The irst term is the transverse EM-wave, ( ik ˆ ˆ ) b eb mb H = h exp k x. (17) eb eb eb eb The second term is a mechanically induced part, (15), namely, 2 2 ( ks keb) H mb, which is the particular solution to ks ωηl H ˆ mb = ks w s. (18) k 16

19 It is generated only by shear-wave motion, w s. The electric ield solution in the sediment can also be separated into two parts, Eb = Eeb + E mb, where ( ik ˆ ˆ ) E = e exp k x (19) eb eb eb eb travels at the (bare) EM wavespeed in the sediment. It is then straightorward to determine that using (3), (5), and (18). While 2 0 L keb Emb = ωµ ωη w, 2 p + wps w 2 2 s (20) keb k ( ks keb) E mb has components along each o the polarizations o the transmitted waves, the shear-wave contribution is negligible since k 2 2 s keb. The mechanically induced electric and magnetic ields act as source terms in the EM boundary conditions at the water-sediment interace; thus, ( ) z ( ) zˆ E E = zˆ E eb ew mb = 0 z = 0 zˆ H H = zˆ H. eb ew mb z = 0 z = 0 (21) We assume that the interace is uncharged prior to any disturbance, so that there are no additional charge or current sources in (21). Following Haartsen and Pride [1997], we treat only the case where the particle displacement is in the x-z plane: the PSVTM (pressure, shear-vertical, transverse magnetic) case. The electric ields e ew and e are eb determined by enorcing (21). A complete list o all the electric ields, both those carried along by the acoustic waves and those excited at the interace, are given in Eqns. (4.46) (4.53) in Block [2004]. 17

20 To determine the seismoelectric potentials along the electrode array, we set l l l E = V = ik V, with l = p and ps, where mb mb l mb 2 p ωµη 0 L i V exp 2 ( ˆ ˆ mb = wp ikp k p x ) (22) k kp k eb is the potential (or voltage) generated by a ast wave propagating in the sediment. Similarly, 2 ps ωµη 0 L i V exp 2 ( ˆ ˆ mb = wps ikpsk ps x ) (23) k kpsk eb is the potential (or voltage) generated by a slow wave propagating in the sediment. We plot ps V mb in the subsequent igures to compare it with the ast-wave potential, although we were unable to identiy slow waves in our experimental data. Transverse EM waves measured in water and sediment are also measured by the l l Ag/AgCl electrodes. We set E xˆ = ik V, with l = p and ps, where eb l eb V V p eb p ew 2 ωµη cos exp 0 Lik θ = wp k k k k ( ik kˆ xˆ ) ew p eb eb 3 eb ew eb eb ew ( sinθ + sinθ ) 2 ωµη cos exp 0 L i θ = wp k k k k k ( ik kˆ xˆ ) p ew ew ( sinθ + sinθ ) ew eb ew eb eb ew. (24) Note that the potentials in (22), (23), and (24) are ound directly in terms o the ast- and slow-wave components o the relative luid displacements, w and w, respectively, which depend implicitly on the incident angle θ w, angular requency ω, and material properties o the luid and bulk sediment. p ps 18

21 4.2 Numerical Predictions The predicted potentials, evaluated at z = 0, are plotted or various situations in the ollowing igures. We have assumed the conductivities o the pore luid and water above sediment to be equal or simplicity. The potentials have been scaled by the input pressure magnitude (at the interace), so that the y axes have units o nanovolts per Pascal. Unless otherwise stated, the material properties are those ound to best it the experimental data shown in the next section and are those summarized in Appendix A. Figures 8(a) and 8(b) depict how the magnitudes o the potentials vary with pore-luid conductivity in medium-grain sand (using a constant surace conductance Σ ) and loose glass microspheres (using the Σ derived by Pride [1994]), respectively. Both igures assume a ixed requency = 50 khz and incident angle θ = π / 2. A peak is predicted or EM-wave potentials in sand when the contributions rom surace and pore conduction w balance, i.e., when σ S/m in (6). While the seismoelectric potentials are predicted to decay as 1/ σ or large conductivities in both media, using the itted surace conductance Σ lessens the seismoelectric response or weak electrolytes. Figures 8(c) and 8(d) depict the behavior in medium-grain sand; the requency and angle dependence o the potentials are qualitatively similar in both media. Figure 8(c) predicts that the ast-wave potential decays as 1/ above the transition requency deined in (A.6), but remains essentially constant or requencies below this value. Because the EM-wave potentials peak near this requency, broad-band measurements o EM phenomena might be used to determine pore-scale eatures, which are intimately connected with the transition requency. 19

22 Figure 8(d) depicts the magnitudes o the potentials versus angle o incidence, where a strong increase is seen at the ast-wave critical angle (about 30 ) or p V mb, p V eb, and V. One aspect o the plane-wave problem is that the EM-wave potentials exhibit a p ew rapid decay to zero near normal incidence (within one degree or these sediments). This phenomenon is caused by the uniorm dipole layer at the interace, but ceases to be an issue when there is a deviation rom planarity (e.g., due to a slight angle in the sediment surace) or when the dipole layer exists only over a inite region (as in our apparatus). 5 Comparing Theory to Data We use the peak magnitude o the second arrivals (as in Figures 3(b) and 4(b)) to compare our laboratory data to numerical predictions. To simulate ast-wave voltages at the position o electrode #8 in the sediment, we deine p π Vpredict = Vmb θ w =, z = m pw, z= 0 2 (25) using the Fourier transorm o the measured 50 khz signal, p w. Note that w p is calculated per unit applied pressure: or a known p w and sediment type, we use the Biot relection problem to generate w p at normal incidence. A time series or the predicted voltage is then computed using (22) and (25) or the range o pore-luid conductivities tested in the laboratory. One o the main diiculties o comparing EK-Biot predictions to data is the number o parameters involved. The only parameters that we varied to it the data, by estimation 20

23 and trial and error, were the DC permeability k 0 and the ζ potential. The Kozeny- Carman relation [Johnson et al., 1987], 8k α, φ 0 Λ= (26) was used to relate k 0 to the pore-throat dimension Λ, tortuosity α, and volume raction φ. This is an experimentally determined relationship that is approximately valid or these sediments. Sand and glass microspheres were assumed to have the same values or the other EK-Biot parameters. Also, while the ζ potential is known to depend on pore-luid conductivity and ph [Ishido and Mizutani, 1981], best-it values or both sand and glass microspheres were ζ 40 mv, which is near the lower range or silica over this range o conductivities (and at ph 9). Best-it values or the DC permeabilities were k m 12 2 or medium-grain sand and k m 12 2 or loose glass microspheres. Equation (26) predicts eective pore radii or sand and glass to be approximately Λ 16 µ m and Λ 19 µ m, respectively. Figure 9 shows the results o comparing the peaks o the ast-wave potential at electrode #8 or both data and theory. Error ellipses represent two standard deviations rom the mean. The it or glass microspheres (solid curve) is based on the unmodiied orm o EK-Biot theory, while the predictions or medium-grain sand (dashed curve) rely on a itted value o the surace conductance Σ to simultaneously match both the astwave voltage and bulk conductivity data. Individual data points are accurately predicted and there is a clear similarity between theoretical and experimental trends or both curves. The unmodiied orm o EK-Biot theory predicts that the magnitude o the seismoelectric potential increases as the conductivity is lowered, and this trend is 21

24 exhibited by the glass data. However, seismoelectric potentials in medium-grain sand appear to grow less rapidly at lower conductivities. Combining the itted surace conductance Σ with EK-Biot theory allowed us to predict this eature. Because EK-Biot theory is a broad-band model in contrast to the other EK models discussed in Section 2.2, which hold only in the low-requency limit we are able to compare our high-requency results to the data o Ahmed [1964], who measured EK voltages generated by constant low rates in sand and loose glass microspheres, and Pengra and Wong [1999], who studied low-requency EK behavior in consolidated porous media. The results shown here all within the ranges o both datasets (see Block [2004] or details), which provides another argument or the reliability o EK-Biot theory ((5), in particular) and the plane-wave model described in Section 4. 6 Summary and Conclusions Medium-grain sand and loose glass microspheres were studied using pore-luid conductivities that ranged between S/m and 0.12 S/m. Electrodes buried in the sediment measure two types o seismoelectric phenomena: (1) arrivals rom an EM wave generated at the interace, which is recorded at all electrodes simultaneously, and (2) electric potentials carried along with transmitted acoustic waves in the sediment. Electrodes above the water-sediment interace measure the EM-wave arrival in water, as well as a small disturbance o the electrode double layers caused by the incident acoustic wave. Fast-wave potentials in the sediments are oten greater than 500 µv, while the EMwave potentials are usually 100 µv in magnitude. These values correspond to eiciencies greater than 150 nv/pa and 30 nv/pa, at 50 khz, respectively. 22

25 EK-Biot theory is able to predict the trends and magnitudes o the laboratory data with good accuracy or a large range o pore-luid conductivities. But we emphasize that a robust model o σ ( σ ) is critical or ield applications that utilize electrokinetic and bulk conductivity data rom clay-bearing sands and sandstones. The results rom seismoelectric (and electroseismic) imaging o oil and gas reservoirs may also be aected when EM-wave phenomena are used to determine the properties o rocks saturated by resh water and/or hydrocarbons. The implications o our results in the ield o ocean seabed acoustics are twoold. First, a clear distinction can be made between the kinematics o poroelasticity and that o other theories. The viscoelastic luid and solid models that are commonly used rely on a single, macroscopic displacement ield because the dynamics o the separate phases are lost (or ignored) while upscaling. In contrast, poroelasticity allows the luid and solid rame to undergo relative motion and preserves this two-phase behavior on the macroscale. The resulting dissipation mechanism depends on the average relative luid displacement ( w ), which plays a critical role in both acoustic and seismoelectric behavior theories that do not retain this mechanism are incapable o predicting the EK phenomena described here. We can turn this argument around: because sediments are well-modeled by EK-Biot theory, they should also exhibit Biot properties in situations where the pore luid is not an electrolyte (i.e., with L( ω) 0 ). Second, experimentally derived (ad hoc) models o the seabed oer no details on how wave propagation depends on sediment microstructure. Rigorous averaging is not only more powerul by providing a direct connection between eective-medium and porescale properties but it is essential to predicting key experimental behaviors, such as 23

26 EM-wave generation at a luid-sediment interace and the broad-band requency dependence o seismoelectric phenomena, that are a robust eature in our data. Acknowledgements The work o G. Block was supported by the Oice o Naval Research, Department o Ocean, Atmosphere and Space Sciences, under contract N , and under the auspices o the U.S. Department o Energy by UC, LLNL under contract W-7405-ENG -48, supported speciically by the Geosciences Research Program o the DOE Oice o Basic Energy Sciences, Division o Chemical Sciences, Geosciences and Biosciences. The authors also wish to thank the Applied Research Laboratory at the University o Texas at Austin, N.P. Chotiros, and J.G. Berryman. Appendix A: Material Properties The material properties or our sediment samples are given in Table 1. The DC permeability k 0, pore-throat dimension Λ, and ζ potential are determined by parameter its and the Kozeny-Carman relation (26), as described in Section 5. Medium-grain sand (mesh #4 sand-blasting sand) and glass microspheres (lead-ree, borosilicate glass rom Glen Mills, Inc., NJ) have grain diameters o approximately 250 µm and 350 µm, respectively. The constitutive relations or two-phase, isotropic poroelastic media (see Pride et al. [1992]) are as ollows: 24

27 T 2 τbulk = ( KG us + C w) I+ Gr us + us usi 3 p= C u + M w, s (A.1) where ( 1 φ) K r + φk + + Ks KG = 1+ K + Ks C = 1+ 1 K M =, φ 1+ (A.2) and the parameter is K = ( 1 φ ) Ks K r. φk (A.3) s The bulk and shear rame moduli o the sediment, K and G, respectively, are oten assumed complex to model or inelastic behavior that is not accounted or by Biot theory; we assumed that both media had the same rame properties, and chose their values in accordance with the accepted range discussed in the ocean acoustics literature. The dynamic Darcy permeability, r r ( ) 1/2 k ω ω 4 ω = 1 i i, k0 ωt m ω t (A.4) is derived in Pride [1994]. Here, k 0 is the DC permeability, and m φ = Λ 2, (A.5) αη φ η = (A.6) ωt α k0 ρ 25

28 is the Biot transition requency discussed in Sections 2 and 5. orm: The EK coupling coeicient [Pride, 1994] using the Debye approximation takes the ( ω) L m d ωρ = i i d L0 ωt 4 Λ η 2 ω 3/ , (A.7) where L φ ε κ ζ = d α η Λ (A.8) The Debye length in a NaCl solution, εκ kt = +, (A.9) ( ) 0 B d bna bcl 2σ characterizes the thickness o the electric double layer, and is generally less than 10 nm; it depends on Boltzmann s constant k B 23 ( J/K), the ambient temperature T (298 K), the pore-luid conductivity σ, and the ionic mobilities o Na + and Cl, b and b Na Cl, respectively. Table 1: Water and sediment (sand & glass) properties K (GPa) 2.4 α 1.52 K s (GPa) 32 φ 0.38 K r (MPa) 44( i) κ 80 (MPa) i κ 3 G r ( + ) s ρ 2650 ζ (mv) 40 3 s (kg/m ) 3 ρ (kg/m ) 1023 Na (s/kg) b 2.9 x η (kg/ms) 10-3 b Cl (s/kg) 4.4 x k 0 ( µ m ) 8 & 11 µ 0 (H/m) 10-3 Λ ( µ m) 16 & 19 ε 0 (F/m) 8.85 x

29 Appendix B: Sample Characterization Both the surace charge density and ζ potential o naturally occurring silica are oten reduced by adsorption o organic material and/or the recombination o partially-bound oxygen to neighboring silicon atoms. We attempted to stabilize the surace-chemical properties and maximize the surace-charge density o our samples ollowing Hau et al. [2003]. The samples were rinsed with de-ionized water ater each step. A strong suluric acid was used to remove organic impurities (15% by vol. at 100 C or 30 min.). Next, the sample was rinsed with sodium hydroxide (10% by vol. at 100 C or 30 min.) so that each hydroxide ion hydrolyzed SiO 2 to orm silanol and silanol salt groups. Hydrochloric acid (10% by vol. at 100 C or 30 min.) was then applied to displace the Na + ions, yielding mainly silanol groups. Ater interaction with water, ph 7 8, the silanol groups were deprotonated to produce a maximal density o unbound oxygen on the surace (and hence a maximalζ potential). The entire procedure was expected to produce a negatively charged surace with ζ potentials o approximately 65 mv at neutral ph. While chemical treatment stabilizes the surace-chemical properties o the samples and helped to minimize the eects o equilibration discussed in Section 3.3, both untreated and treated samples exhibited the same ast-wave potential levels (and had best-it values o ζ 40 mv ). The seismoelectric datasets discussed in this paper are based entirely on the treated samples. 27

30 Reerences Ahmed, M. U. (1964), A laboratory study o streaming potentials, Geophys. Prosp., 12, Auriault, J. L. and T. Strzekecki (1981), On the electro-osmotic low in a saturated porous medium, Int. J. Eng. Sc., 19, Beamish, D. (1999), Characteristics o near-surace electrokinetic coupling, Geophys. J. Int., 137, Biot, M. A. (1956a), Theory o propagation o elastic waves in a luid saturated porous solid I. Low requency range, J. Acoust. Soc. Am., 28, Biot, M. A. (1956b), Theory o propagation o elastic waves in a luid saturated porous solid II. Higher requency range, J. Acoust. Soc. Am., 28, Biot, M. A. (1962), Mechanics o deormation and acoustic propagation in porous media, J. Appl. Phys., 33, Block, G. (2004), Coupled acoustic and electromagnetic disturbances in a granular material saturated by a luid electrolyte, University o Illinois at Urbana-Champaign, Champaign, Illinois. Buckingham, M. J. (1997), Theory o acoustic attenuation, dispersion, and pulse propagation in unconsolidated granular materials including marine sediments, J. Acoust. Soc. Am., 102, Buckingham, M. J. (1998), Theory o compressional and shear waves in luid-like marine sediments, J. Acoust. Soc. Am., 103, Buckingham, M. J. (2005), Compressional and shear wave properties o marine sediments: Comparisons between theory and data, J. Acoust. Soc. Am., 117, Chandler, R. N. (1981), Transient streaming potential measurements on luid-saturated porous structures: an experimental veriication o Biot's slow wave in the quasi-static limit, J. Acoust. Soc. Am., 70, Chotiros, N. P. (1995), Biot model o sound propagation in water-saturated sand, J. Acoust. Soc. Am., 97, Chotiros, N. P. and M. J. Isakson (2004), A broadband model o sandy ocean sediments: Biot-Stoll with contact squirt low and shear drag, J. Acoust. Soc. Am., 116, Debye, P. and E. Huckel (1923), On the theory o electrolytes. I. Freezing point depression and related phenomena, Physikalische Zeitschrit, 24, Dvorkin, J. and A. Nur (1993), Dynamic poroelasticity: A uniied model with the squirt and the Biot mechanisms, Geophysics, 58, Fitterman, D. V. (1978), Electrokinetic and magnetic anomalies associated with dilatant regions in a layered Earth, J. Geophys. Res., 83, Frenkel, J. (1944), On the theory o seismic and seismoelectric phenomena in moist soil, J. Physics (Soviet), 8, Glover, P. W. J., P. G. Meredith, P. R. Sammonds, and S. A. F. Murrell (1994), Ionic surace electrical conductivity in sandstone, J. Geophys. Res., 99, Gouy, G. (1910), About the electric charge on the surace o an electrolyte, J. Phys. A, 9,

31 Haartsen, M. W. and S. R. Pride (1997), Electroseismic waves rom point sources in layered media, J. Geophys. Res., 102, Haartsen, M. W. and M. N. Toksöz (1996), Dynamic streaming currents rom seismic point sources in homogeneous poro-elastic media, Geophys. J. Int., 132, Hamilton, E. L. (1972), Compressional-wave attenuation in marine sediments, Geophysics, 37, Hamilton, E. L., (1974), Prediction o deep-sea sediment properties: state-o-the-art, in Deep-sea sediments: physical and mechanical properties, Inderbitzen, A. K., pp. 1-44, Plenum Press, New York. Hau, W. J. W., D. W. Trau, N. J. Sucher, M. Wong, and Y. Zohar (2003), Surace-chemistry technology or microluidics, J, Micromech. Microeng., 13, Hiemenz, P. and R. Rajagopalan (1997), Principles o colloid and surace chemistry, Marcel Dekker, Inc. New York. Ishido, T. and H. Mizutani (1981), Experimental and theoretical basis o electrokinetic phenomena in rockwater systems and its applications to geophysics, J. Geophys. Res., 86, Johnson, D. L., J. Koplik, and R. Dashen (1987), Theory o dynamic permeability and tortuosity in luidsaturated porous media, J. Fluid Mech., 176, Johnson, D. L. and T. J. Plona (1982), Acoustic slow waves and the consolidation transition, J. Acoust. Soc. Am., 72, Mikhailov, O. V., M. W. Haartsen, and M. N. Toksöz (1997), Electroseismic investigation o the shallow subsurace: ield measurements and numerical modeling, Geophysics, 62, Murphy, W. F., K. W. Winkler, and R. L. Kleinberg (1986), Acoustic relaxation in sedimentary rocks: dependence on grain contacts and luid saturation, Geophysics, 51, Neev, J. and F. R. Yeats (1989), Electrokinetic eects in luid-saturated poroelastic media, Phys. Rev. B, 40, Nettelblad, B., B. Ahlen, G. A. Niklasson, and R. M. Holt (1995), Approximate determination o surace conductivity in porous media, J. Phys. D: Appl. Phys., 28, O'Brien, R. W. (1988), Electro-acoustic eects in a dilute suspension o spherical particles, J. Fluid Mech., 190, Pengra, D., S. X. Li, and P. Wong (1999), Determination o rock properties by low-requency AC electrokinetics, J. Geophys. Res., 104, Pengra, D. and P. Wong, (1995), Pore size, permeability and electrokinetic phenomena, in Access in nanoporous materials, Pinnavaia, T. J. and Tho, M. F., pp , Plenum Press, New York. Pride, S. R. (1994), Governing equations or the coupled electromagnetics and acoustics o porous media, Phys. Rev. B, 50, Pride, S. R., A. F. Gangi, and F. D. Morgan (1992), Deriving the equations o motion or porous isotropic media, J. Acoust. Soc. Am., 92, Pride, S. R. and M. W. Haartsen (1996), Electroseismic wave properties, J. Acoust. Soc. Am., 100,

32 1315. Rice, C. L. and R. Whitehead (1965), Electrokinetic low in a narrow cylindrical capillary, J. Phys. Chem., 69, Slattery, J. C. (1967), Flow o viscoelastic luids through porous media, Am. Ch. Eng. Journal, 13, Stern, M., A. Bedord, and H. R. Millwater (1985), Wave relection rom a sediment layer with depthdependent properties, J. Acoust. Soc. Am., 77, Stern, O. (1924), The theory o the electrolytic double layer, Zeitschrit ur Elektrochemie, 30, Stoll, R. D. (1981), Relection o acoustic waves at a water-sediment interace, J. Acoust. Soc. Am., 70, Stoll, R. D. (1983), Sediment Acoustics, Lecture Notes in Earth Science, Springer Verlag, Berlin. Stoll, R. D. (2002), Velocity dispersion in water-saturated granular sediment, J. Acoust. Soc. Am., 111, Thompson, A. H. and G. A. Gist (1991), Electroseismic prospecting, SEG Expanded Abstract, EM 2.1, Tutuncu, A. N. and M. M. Sharma (1992), The inluence o luids on grain contact stiness and rame moduli in sedimentary rocks, Geophysics, 57, Wildenschild, D., J. J. Roberts, and E. D. Carlberg (2000), On the relationship between microstructure and electrical and hydraulic properties o sand-clay mixtures, Geophys. Res. Lett., 27, Williams, K. L. (2001), An eective density luid model or acoustic propagation in sediments derived rom Biot theory, J. Acoust. Soc. Am., 110, Williams, K. L., D. R. Jackson, E. I. Thorsos, D. Tang, and S. G. Schock (2002), Comparison o sound speed and attenuation measured in a sandy sediment to predictions based on the Biot theory o porous media, IEEE J. Ocean Eng., 27, Zhu, Z., M. W. Haartsen, and M. N. Toksöz (1999), Experimental studies o electrokinetic conversions in luid-saturated borehole models, Geophysics, 64,

33 Figure Captions Figure 1: Electric double layer near a grain surace. Electrokinetics and luid motion are coupled by the diuse layer, which is ree to move with the pore luid. Figure 2: Seismoelectric apparatus. The apparatus is based on a cylindrical tube geometry that is 7.62 cm in diameter. Ag/AgCl electrodes are buried in a vertical array within the sediment and above the watersediment interace. A copper-mesh Faraday cage was used to isolate the apparatus rom electrical intererence and provide a universal ground. The transducer ace diameter is 2.54 cm. Electrode and interace positions are accurate to 0.5 cm and 1 cm (between sample changes), respectively. Figure 3: Loose glass microspheres at S/m. Electrodes measure (a) a simultaneous EM wave arrival in the luid, as well as an additional response caused by an acoustic-wave disturbance o the electrolyte around each electrode, and (b) two arrivals in sediment: an EM wave arriving simultaneously at all electrodes, and a potential coupled to the transmitted wave, that moves out in time with deeper depths. Figure 4: Medium-grain sand at S/m. Seismoelectric responses depicted in (a) and (b) are qualitatively similar to those observed in Figure 3, although smaller in amplitude. Figure 5: Electrokinetic voltages or a range o conductivities or electrode #8 (buried in loose glass microspheres). Time series or conductivities between S/m and 0.12 S/m have been superposed. Figure 6: Bulk versus pore-luid conductivity or medium-grain sand ( * ) and loose glass microspheres ("+"). The σ bulk σ relation that aects seismoelectric response ollows an almost linear trend or both sample types. As expected, the σ bulk σ w relation (indicated by symbols with holes in their centers) is strongly nonlinear. Predictions using a itted surace conductance or sand (dashed curved) are compared to the unmodiied orm o EK-Biot theory or glass microspheres (solid curve). The diamond corresponds to glass microspheres saturated with DI water ater 16 hours o equilibration. Error ellipses represent two standard deviations rom the mean. Figure 7: Relection rom an EK-Biot hal-space produced by an incident pressure wave in water with wavevector k ˆ w and angle θ w (relative to the horizontal axis). Two waves in the water are generated: a relected acoustic wave ( k ˆ rw ( k ˆ s ), ast ( k ˆ p ), and an EM wave ( k ˆ eb ). ) and an EM wave ( k ˆ ew ), and our waves in the sediment: slow ( k ˆ ps ), shear Figure 8: Numerical predictions o seismoelectric potentials near a water-sediment interace versus poreluid conductivity in (a) medium-grain sand, and (b) loose glass microspheres, with requency = 50 khz and θw = π / 2. (c) Potentials versus requency in medium-grain sand, with θ = π / 2 and σ = 0.01 S/m. (d) Potentials versus angle o incidence in medium-grain sand, w with = 50 khz and σ = 0.01 S/m. Figure 9: Peaks o the ast-wave potentials (measured at electrode #8) versus the bulk conductivity. Data points or medium-grain sand ("*") and loose glass microspheres ("+") are compared to predictions or sand (dashed curved), based on a itted surace conductance, and to glass (solid curve), using the unmodiied orm o EK-Biot theory. The diamond corresponds to glass microspheres saturated with DI water ater 16 hours o equilibration. Error ellipses represent two standard deviations rom the mean. 31

34 Figure 1 32

35 Figure 2 33

36 Figure 3 34

37 Figure 4 35

38 Figure 5 36

39 Figure 6 37

40 Figure 7 38