ANALYSIS OF WAVE-INDUCED PORE PRESSURE CHANGES

Bulletin of the Seismological Society of America, Vol. 74, No. 4, pp. 1395-1407, August 1984

ANALYSIS OF WAVE-INDUCED PORE PRESSURE CHANGES RECORDED DURING THE 1980 MAMMOTH LAKES, CALIFORNIA,

EARTHQUAKE SEQUENCE

BY GERALD M. MAVKO AND ED HARP

ABSTRACT

Acceleration and wave-induced pore pressure were recorded in a saturated sand during the 1980 Mammoth Lakes, California, earthquake sequence. For the largest event recorded, the pore pressure was observed to be proportional to vertical surface acceleration during the P-wave arrivals and proportional to horizontal surface velocity during the S-wave arrivals. The results can be quantitatively explained with a linear elastic model of a porous saturated medium, such that pore pressure depends on dilatation and is independent of shear strain. A slight frequency dependence in the ratio of pore pressure to dilatation indicates local fluid flow on the scale of individual pores. The good agreement between observations and theory indicates that the deformation was primarily linear, even though maximum shear strains were close.to the typical thresholds for liquefaction.

INTRODUCTION

The mechanical properties of rocks and soils under wave excitation are strongly affected by pore fluids. It is well known, for example, that a water-saturated rock usually has a higher seismic P-wave velocity than the same rock has when dry (Nur and Simmons, 1969). Attenuation of both P and S waves in saturated rocks appears to be dominated by fluid-related losses (Born, 1941, Tit tmann et al., 1972; Winkler and Nut, 1982). In addition, in saturated sands, large amplitude shear waves can lead to a dramatic loss of strength due to liquefaction.

All of these effects are related to wave-induced changes in pore pressure. Several authors (Ikehara, 1970; Seed et al., 1975; Ishihara, 1981) have reported in si tu observations of wave-induced pore pressure fluctuations, and first-order models have been developed (Dobry et al., 1981a) to estimate the strain threshold for pore pressure build-up (liquefaction) in sands. In contrast, many small-strain models for seismic velocity and attenuation (Biot, 1956; Walsh, 1965, 1969; O'Connell and Budiansky, 1974, 1977) do not emphasize pore pressure explicitly, and measurements of seismic velocity and Q are seldom, if ever, accompanied by measurements of wave-induced pore pressure.

In this paper, we analyze recordings of wave-induced pore pressure in saturated sands, measured by Harp et al. (1984) during the 1980 Mammoth Lakes, California, earthquake sequence. The data indicate near-surface deformation primarily in the linear range, and therefore provide a unique test of the role of pore compressibility and fluid motion in models of wave propagation in porous media. The analysis also provides insights into the processes near the threshold of excess pore pressure accumulation.

OBSERVATIONS

The measurements were made in saturated sediments at the east shore of Convict Lake, about 10 km southeast of Mammoth Lakes, California. The lake sediments here are a mixture of gravels in a matrix of medium to coarse sand. Three components of acceleration were measured essentially at the surface using a force-

1395

1396 GERALD M. MAVKO AND ED HARP

balance accelerometer and were recorded digitally at 200 samples per sec. Pore pressure was measured at a depth of -130 cm, essentially below the accelerometer and was also recorded digitally at 200 samples per sec. P and S velocities of the shallow sediments were measured in situ to be 1.6 and 0.13 km/sec, respectively, corresponding to a Poisson's ratio of 0.50. A more complete description of the experiment is given by Harp et al. (1984).

Acceleration and pore pressure for the largest earthquake recorded by Harp et al. (1984) are shown in Figure 1. The event had a coda magnitude of 3.3, a depth of ~1.7 kin, and an epicentral distance from the recording site of - 5 km (R. Lester, personal communication, 1982). The epicenter-to-station azimuth was N41°E. The peak horizontal surface acceleration (vector sum of north and east components) is -180 cm/sec 2, and the peak recorded pore pressure fluctuation is -0.008 bar (0.8 kN/m 2). The surface angles of incidence were estimated by tracing rays from the hypocenter, assuming a P velocity of 5.0 km/sec and an S velocity of 2.8 km/sec (Mueller et al., 1981) below the lake sediments, giving N18 ° for the first P arrivals and ~2 ° for the first S arrivals (Figure 2). Estimates of angle of incidence from the ratio of vertical to horizontal acceleration amplitudes appear to be unreliable, because mode conversions along the ray path cause some S arrivals during the main P-wave train and some P arrivals during the main S-wave train (Figure 2).

The induced pressure fluctuations are visibly related to the ground motion (Figure 1). During the P-wave arrivals (0.1 to 0.9 sec in Figure la), the pore pressure is approximately proportional to the vertical component of surface acceleration; the peaks and troughs are in phase, and the largest pressure peaks generally coincide with the largest acceleration peaks. The pore pressure appears to be slightly smoother (i.e., rounder peaks, fewer high frequencies) than the acceleration.

During the S-wave arrivals (0.9 to 3.0 sec in Figure lb, c), the pore pressure appears to be approximately proportional to the ground velocity, because the pressure lags 1 cycle behind the surface horizontal acceleration, and it is significantly smoother. There is no evidence of development of a step-like pore pressure excess typical of liquefaction, as reported for example by Ishihara (1981).

LINEAR ELASTIC ANALYSIS

Elastic Wave Propagation

Many of the first-order features in the relations between pore pressure and ground motion in Figure 1 can be understood in terms of purely elastic wave propagation. We begin by deriving the relations between acceleration at the surface and acceleration, strain, and pore pressure slightly below the surface. We assume that the saturated sediments behave as a linear elastic solid. That is, we assume that the grain packing in the sediments does not change irreversibly by sliding Or compaction during wave excitation and that the frequency is low enough that viscous fluid effects are negligible. A consequence of these assumptions is that the induced pore pressure is predicted to be proportional to the solid dilatational strain. With the elastic effects identified, observed deviations from elastic behavior can be discussed in terms of fluid motion.

P arrivals. Consider first the ground motion from incident plane P waves. Near the free surface, the vector ground displacement is the sum of displacements from the upgoing (compressional) P wave, Pu, the reflected downgoing P wave, P~, and the downgoing converted P-to-SV-wave, PSd, as shown in Figure 2a. Each frequency

ANALYSIS OF WAVE-INDUCED PORE PRESSURE CHANGES 1397

ACCEL. PRES. crn/s 2 kN/m 2

50.11.O (.01 bar)

i 25.-0.5

[ :!, [l I:

,'~i~ ,I/:, 'L i .

-25. t-0.5

-50.1 El o ~ ~

ACCEL. PRES. cm/s 2 kN/m 2

15011.0 (.01 bar)

TIME (s)

-7 -0.5

-150L b 1 2 3

TIME (s)

ACCEL. PRES. cm/s 2 kN/m 2

1.0 (.01 bar)

7510"5

= i i,

-75 -0.5

_15(3[ C 0 1 2 3

TiME (s)

Fro. 1. Acceleration (dotted) measured at the surface and pore pressure (solid) measured 1.3 m below the surface. (a) Vertical component of acceleration compared with pressure. (b) East-west component. (c) North-south component.

1 3 9 8 GERALD M. MAVKO AND ED HARP

component has the form:

9" [" / s i n 0 -c°s-----~0 >] Pu = Po(o:)(sin 8, 0, - cos ) e x p [ ~ w [ ~ x V~ z - t

cos )] = 0, x + - - C z

[. s i n e c o s ¢ t)] PSa = Po(~O)rv~(cos 4, O, - s i n ~b)exp toJ(---~- x + ~ z - (1)

where P0(w) is the Fourier transform of the P-wave amplitude at the origin, Vp is the near-surface P velocity, Vs is the near-surface S velocity, rp, is the free surface P-wave reflection coefficient, and rp~ is the free surface reflection coefficient for P- to -SV mode conversion (Aki and Richards, 1980, p. 141). The coordinate axes are

X X

z'~ ~ ~\,~ ~ Vp =1.6 km/s z ~ " II \r" / ~\ i~/ j \ V s= .13

/Pu ~Sd Pd Vp=5.o

P SV Vs= 2.8

(a) (b) Fro. 2. Ray geometry and coordinates used in modeling for (a) P and (b) S waves incident on the

bottom of the low-velocity lake sediments. Waves represented by solid rays were modeled to estimate acceleration and strain. Dashed rays show examples of other phases tha t were ignored, but may be important in some cases.

chosen with the origin at the surface; z is positive downward; y is perpendicular to the direction of propagation; and the horizontal component of propagation is toward increasing x. The axes and angles 0 and ~ are illustrated in Figure 2a.

The near-surface acceleration is obtained by twice differentiating and summing the upgoing and downgoing waves in equation (1). In particular, the vertical component of acceleration at the origin is given by

Ap(z = 0, co) = -002Po(~)(-cos 0 + rpp cos 0 - rpssin ¢)exp(-i00t). (2)

Therefore, the vertical acceleration at the surface is a measurement of the upgoing P-wave amplitude Po (w).

As shown in Figure 2a, we expect that upgoing S waves are generated by the incident P waves at the base of the low-velocity lake sediments, as well as at other discontinuities along the ray path. These mode-converted S waves, ignored in the above derivation, are near-vertically incident in the low-velocity surface sediments for the event shown in Figure 1. They do not, therefore, significantly affect our


interpretation of vertical acceleration using equation (2), or the discussion of pore pressure that follows. The reader is warned, however, that cases may arise where these phases must be accounted for.

The dilatation, ep, associated with the P-wave train at a point below the origin (x = 0, y = 0, z) is the sum of the dilatations of the upgoing and downgoing P waves

iwPo [ [-iwcos O ) [io~cos Oz_)] e p - Vp [exp~ z + rpp e xp[- ~ e -i'~t. (3)

Combining equations (2) and (3), the dilatation at depth z can be written in terms of the vertical surface acceleration as

[ /-iwcos 0 z] /io~cos 0 1 [exp[- ~-~p / +rppeXp~----~p z)]

ep(Z, w) = iooVp [-cos 0 + rppCOS 0 - rp~sin ¢] Ap(z = O, w). (4)

For small angles of incidence (cos 0 ~ 1, sin ¢ << 1), this simplifies to

sin( ) ep(z, ~) - - - Ap(z = 0, w). (5)

Furthermore, for depths small compared to the seismic wavelength, equation (5) simplifies to

z-L Ap(z = 0, w). eAz , " V, 2 (6)

With model parameters appropriate for the data shown in Figure 1 (w < 100~r; z = 130 cm, Vp = 1600 m/sec, V~ - 130 m/sec, P-wave angles of incidence < 40°), the expression (6) predicts values within ~1 per cent of those given by expression (4).

It is clear from equation (6) that the dilatation ep at depth z associated with incident P waves is proportional to the surface vertical acceleration. The ratio of ep and A, is predicted to be approximately independent of frequency and to depend linearly on depth.

S arrivals. The near-surface ground displacement from incident plane SV waves is the sum of vector displacements from the upgoing S V waves, S Vu, the downgoing reflected SV wave, SVa, and the downgoing converted P wave S-Pd, as shown in Figure 2b. Each frequency component has the form

. sin ¢ cos ¢ SVu = So(w)(cos ¢, 0, sin ¢)exp tw(---~- x V~ _ _ z t)]

[. s ine c o s ¢ ) ] SVd = So(o~)r~(cos ¢, - s i n ¢)exp ~¢o(---V-~ x + ~ z - t

r.,"sinO co8o )] SPa = So(w)rw(sin O, O, cos vlexp[,w[---~p x + --V~-p z - t (7)


where So(co) is the Fourier transform of the incident SV-wave amplitude, r~ is the free surface S V reflection coefficient, and rsp is the free surface reflection coefficient for the S V - t o - P mode conversion (Aki and Richards, 1980, p. 141). The coordinates and angles 0, ¢ are illustrated in Figure 2b.

The horizontal acceleration at the observation point at the origin on the free surface is obtained by twice differentiating and summing the horizontal components in equation (7)

A~,(z = 0, co) = -o~2S0(co)(cos ~b + r~cos O + r~psin 0)exp(- icot). (8)

Analogous to equation (2), we see here that the x component of surface acceleration is a measure of the incident SV-wave amplitude.

For small angles of incidence (cos ¢ ~- 1, sin 0 << 1) the horizontal acceleration at depth z in terms of the surface acceleration is

A ~ ( z , ~) ~- cos A ~ ( z = O, co). (9)

That is, the horizontal acceleration at depth is in-phase with the surface acceleration and varies as the cosine of depth.

The dilatation e~v at a depth z associated with the S V arrivals results only from the mode-converted P-wave generated at the free surface and is given by the divergence of SPd in equation (7)

ico //cocos 0 e~,(z, w ) = r~pSo(w ) --~v exp~----~p zjexp(-icot) . (10)

Neither Sv nor S H waves have dilatational strain, and there is no converted P- wave generated at the free surface by the S H arrivals.

Combining equations (8) and (10), the dilatation at depth z is written in terms of the surface acceleration as

• //cocos 0 z ~ ]

e~v(z, co) = A ~ ( z = O, co). (11) ~Vp(cos ¢ + r~cos ¢ + rwsin 0)

For very small angles of incidence, equation (11) simplifies to

r cos( ) e~(z, co) - n~v(z = O, co)

2co Vp 2co V~ A~v(z = O, co). (12)

Furthermore, for depths small compared to a seismic wavelength (z << Vp/co), equation (12) simplifies to

rsp___~z es,(z, co) ~- 2Vp2 Asv(z = O, w) +

rsp A~v(z = O, co)

2vp ico (13)


With the model parameters appropriate for the data in Figure 1 (S V-wave angle of incidence -2°), the imaginary part of expression (13) is within about 3 per cent of the imaginary part of expression (11), while the real part of (13) overestimates the real part of (11) by about 10 per cent. For low frequencies ( f < 50 Hz), the imaginary part of equation (11) is a factor of 4 or more larger than the real part, so that the imaginary part of (13) describes the dominant behavior quite well.

Equation (13) indicates that, during the SV-wave arrivals, the dilatation at shallow depths is dominated by an out-of-phase term proportional to surface acceleration divided by (i~), i.e., the integral of acceleration. There is, in addition, a much smaller term proportional to acceleration similar to equation (6) derived for incident P waves•

The largest component of SV-wave shear strain for very small angles of incidence is ex,, given by one-half the z derivative of the horizontal components of displacement in equation (7)

- S o ( o ~ ) o ~ s m \ s~

Cxz(Z, o~) ~- exp(- iwt). (14) Vs

In terms of the surface acceleration, the strain can be written

• ~ o z slnt-- \ys/

exz(Z, ~) - - - A ~ v ( z = 0, o~). (15) 2wV~

The maximum shear strain at depth is in-phase with the surface acceleration and depends on the sine of depth.

As shown in Figure 2b, we expect that upgoing P waves are generated by the incident S V waves at the base of the low-velocity lake sediments. These P waves, ignored in the derivation of expressions (8) to (15), will add to the surface vertical acceleration and will cause a dilatation that is in-phase with acceleration, exactly as derived in equations (4) to (6). Because the pore pressure to acceleration relationship during S V arrivals in Figure 1 is primarily out-of-phase, we conclude that the secondary P waves are not important. However, cases may arise where their contribution is significant. In addition, S H waves, ignored in the above, can affect the interpretation of horizontal acceleration and maximum shear strain. The data in Figure 1 indicate primarily S V motion in this case (north and east motion in phase, approximately same amplitude; epicenter to station azimuth ~N41°E), but in general both S V and S H must be considered.

Low-Frequency Pore Pressure

The wave-induced dilatational strain given by equations (4) and (11) are the average strains over a volume much larger than a typical grain size but much smaller than a seismic wavelength. The induced pore pressure in the saturated medium corresponding to the average strain depends on frequency, making the medium approximately viscoelastic. However, in the low-frequency limit, the re- sponse is elastic and the pore pressure can be predicted using the static form of the Betti-Rayleigh reciprocity theorem (Fung, 1965; Jaeger and Cook, 1969). The


theorem states that for an elastic body acted upon separately by two sets of tractions, the work done by the first set of tractions acting through the displacements produced by the second set of tractions is equal to the work done by the second set of tractions acting through the displacements produced by the first set of tractions.

To apply the reciprocity theorem, consider the two sets of tractions shown in Figure 3. The sample has total volume V and pore volume vp which is a subset of V. The system on the left is loaded by an externally applied uniform hydrostatic stress a and an internal uniform hydrostatic fluid stress P resulting in pore volume change AVp. The average volumetric strain of the rock on the left is e = a/Ks where Ks is the bulk modulus of the saturated composite. The assumption of uniform fluid pressure is only reasonable when the frequency is low enough for viscous fluid flow to approximately equilibrate the pore pressure over distances larger than a grain size but much smaller than a seismic wavelength. Deviations for this at slightly higher frequencies are discussed later.

The system on the right has the same hydrostatic stress a applied to both the

$ :7

0-

f

--->

V

FIO. 3. Soil sample under two sets of loads.

internal and external surfaces. Consequently, the sample deforms as though it is solid (no pore space). The volumetric strain of the sample and also of the pore space is ,~/Ko where Ko is the bulk modulus of the solid mineral grains.

Applying the reciprocity theorem, we write

a ~ - , ~ A V p = a - P . (16)

The fluid stress is related to the fluid strain through the fluid bulk modulus K t

,z [ AVp~ p = ~,\--~-]. (17)

Combining equations (16) and (17) and rearranging gives the induced fluid stress


in terms of the composite dilatation

[Kt(go- Ks)] e

= R e (18)

where n is the porosity. The quanti ty/~ has the dimensions of an elastic modulus, as expected for the ratio of stress to strain. It is related to Skempton's (1954) empirical pore pressure coefficient B, except that equation (18) predicts that the induced pore pressure is always less than the applied hydrostatic stress as long as the saturated composite shear modulus is nonzero. (A nonzero shear modulus allows the pore stiffness to be nonzero, and therefore not all the applied load is transmitted to the pore fluid.) In the range of linear elastic deformation, the pore pressure is independent of applied shear strain. In the limit of poorly consolidated soils in which (Poisson's ratio) v ~ 0.5, Ko >> Ks, Ko >> K I, and K i ~ npVp 2, equation (18) simplifies to the result more recognizable to soil engineers

P "" Kse = a (19)

TABLE 1

MODEL PARAMETERS

K! 2.0 101° dyne/cm 2 K0 3.6-5.0 1011 dyne/cm 2 Ks 4.9-5.4 101° dyne/cm ~ rs~ 0.011-0.017 Vp 1.6 105 cm/sec V, 1.3 104 cm/sec Z 1.3 10 ~ cm n 0.35-0.45 p 1,9-2.1 gm/cm a

where a is the average or hydrostatic stress in the saturated composite. For the material parameters (Table 1) appropriate for the data shown in Figure 1, the pore pressure predicted by (19) is about 10 per cent larger than predicted by (18).

Finally, the low-frequency induced pore pressure in terms of acceleration during P arrivals is found by combining equation (6) with equation (18), giving

zg1(Ko - Ks) P(z, o~) ~- nVp2(K ° _ K[) Ap(z = 0, ~0) (20)

which predicts that pressure is proportional to acceleration for linear deformation. For poorly consolidated soils, the corresponding expression obtained from equations (6) and (19) is

P(z , ~o) "~ pzAp(z = 0, ~o). (21)

Similarly, combining the imaginary part of equation (13) with equation (18) gives

P(z , ~ ) ~- rs~KI(K° - Ks) Asv(z = O, ~) 2Ypn(Ko - K t) iw (22)


which predicts that the pressure during the S V arrivals is approximately proportional to surface ground velocity, the first integral of acceleration. The corresponding expression for poorly consolidated soils, obtained from equations (13) and (19) is

P(z, ~) - r~K~A~v (z = 0, ~). (23) 2

DISCUSSION

The largest wave-induced pore pressures are usually observed during S-wave arrivals. As derived above, for linear elastic propagation of incident S V waves, the dilatational strain (and therefore the pore pressure) is expected to be out-of-phase with the surface acceleration, and both the maximum shear strain and acceleration at depth are in-phase with the surface acceleration. The observed pore pressure during the S arrivals in Figure 1 is clearly out-of-phase with the horizontal acceleration, suggesting that the measurements were made in the linear elastic range and that dilatational strain (average or hydrostatic stress) is the important parameter controlling pore pressure.

Quantitative comparison between observations and the linear theory can be made using equations (20) and (22) or the approximate forms (21) and (23). Ranges of values for the various model parameters are shown in Table 1. The fluid modulus Ks, porosity n, the mineral modulus K0 were estimated from tables of typical values (Clark, 1966; Lambe and Whitman, 1969) and are the least well-determined. The composite modulus Ks is obtained from the measured values of Vs and Vp and depends weakly on n through the composite density p. The SV-to-P reflection coefficient r.~p is computed assuming the S V angle of incidence (2 ° to 3 °) estimated in the "Observations" section and the measured Vp and Vs. Substituting these into equation (18) gives a range of/~ = 4.2 × 10 +1° to 5.2 × 10 +1° dyne/cm 2. The ratio of the amplitudes of pore pressure to acceleration from equation (20) is then P/A 210 to 260 dyne-sec 2/cm 3 for P waves and from equation (22) is P/A ~- (1400/~o) to (2800/~) dyne-sec 2/cm 3 for S waves.

The amplitude spectra and spectral ratios of the pore pressure and acceleration records in Figure 1 are shown in Figure 4. The spectra during the P-wave arrivals (Figure 4a) were computed for a window extending from 0.1 to 0.9 sec, cosine smoothed over the first and last 10 per cent of the window and padded with zeros outside the window. The spectra during the S-wave arrivals (Figure 4b) were similarly computed for a window extending from 0.95 to 2.05 sec. Antialiasing filters in the recording system eliminated most of the signal at frequencies higher than 50 Hz. Theoretical values for the spectral ratios, based on equations (20) and (22), are superimposed. The observed pressure-to-acceleration spectral ratio during the P- wave arrivals (Figure 4a) is lower than the predicted ratio by a factor of 2 or 3. The fit would be better if, for example, the in situ porosity n or bulk modulus Ks were larger than estimated in Table 1. During the S-wave arrivals, the agreement between observed and predicted ratios is quite good (Figure 4b).

The observed ratio during P arrivals (Figure 4a) shows a negative slope, which we attribute to fluid motion. Two primary modes of fluid motion in porous media have been proposed in the literature: Blot (1956) considered coupled solid-fluid motions averaged over a scale much larger than a pore size. Details of local flow and pore geometry are neglected and lumped into global parameters. Physically there are two sources of flow in the Blot model (White, 1965). The first results from


relative accelerations of the solid and liquid. Both P and S waves cause acceleration of the solid matrix, while inertia causes the liquid to lag. The second Biot mechanisms occurs only with P waves. Differential compression between peaks and troughs of the waves produces pressure gradients in the pore fluid. The fluid, in turn, diffuses through the solid according to Darcy's law. The second mode of fluid motion, neglected in the early Biot (1956) model, is on a more local scale. When a section of rock or soil is stressed by a passing wave, heterogeneities in pore shape and orientation produce large pressure gradients and flow (squirt) on the scale of individual pores (Biot, 1962; Mavko and Nur, 1975; O'Connell and Budiansky, 1977). In particular, under the compression of a P wave, thinner pores have a higher

Q Acceleration b

200- E

E lO0-

E

0 - - 0

~model Pressure

Acceleration

20 40 Hz

6'o

Pressure

~_200

ol

Pressure

, , , , . i

20 40 Hz

60

FIG. 4. (a) Amplitude spectra and spectral ratio of pore pressure and vertical acceleration during the P-wave train (0.1 to 0.9 sec in Figure 1). The shaded band on the spectral ratio plot shows the static ratio predicted by the model. The dark sloping line highlights the apparent linear trend that indicates possible fluid flow effects. (b) Amplitude spectra and spectral ratio of pore pressure and horizontal acceleration during the S-wave train (0.95 to 2.05 sec in Figure 1). The acceleration is the square root of the sum of the squares of the spectral amplitudes of the two horizontal components. The shaded band on the spectral ratio indicates the model prediction.

induced pressure than rounder pores. At zero frequency, Biot and squirt mechanisms predict the same ratio of pore presusre to total dilatation, as given by equation (18). At nonzero frequencies, the Biot model predicts a positive slope for the spectral ratio of pore pressure to total dilation, while the local squirt model predicts a negative slope for the pressure in the roundest pores and a positive slope for the pressure in the thinnest pores. Because the measured pore pressure is probably representative of the larger, rounder pores, the negative slope in Figure 4a indicates that the local squirt model of induced fluid motion dominates any contribution by the Biot mode of flow.


In contrast to the data in Figure 1 and the analysis so far, soils often deform nonlinearly at sufficiently large shear strains, so that the induced pore pressure is proportional to applied shear strain as well as applied dilatational strain (e.g., Scott, 1963). For example, the onset of seismically induced liquefaction is observed to occur when shear strains reach a threshold of the order 10 -4 (Drnevich and Richart, 1970; Dobry et al., 1981b). The maximum shear strain for the data in Figure 1 can be estimated using equation (15). At a given frequency, equation (15) predicts that the maximum shear strain occurs at a depth z = (~/2) V,/w and has a magnitude A~v(O, w)/2o~V,. At 10 Hz, the depth of maximum shear strain is ~3.3 m, and the strain is 1.1 × 10 -4. Therefore, it appears that the maximum shear strain is close to the empirical threshold for pore pressure accumulation at a depth of 2.0 m below the pressure transducer.

CONCLUSIONS

Analysis of recorded pore pressure and acceleration resulting from one of the 1980 Mammoth Lakes, California, earthquakes indicates deformation primarily in the linear range. There is no evidence of development of a step-like excess pore pressure typical of liquefaction. Furthermore, the pore pressure appears to be proportional to dilatation and (to first order) independent of shear strain, even though the shear strains were close to typical thresholds for liquefaction. An elastic model of a porous saturated medium, similar to those usually applied to rocks, quantitatively explains the observations quite well.

The analysis suggests that future measurement programs should include recording of three components of ground motion, to allow separation of P-, SV-, and SH- wave motions. Furthermore, pore pressure should be recorded at several depths, because acceleration, strain, and pore pressure are all predicted to depend on depth. Finally, complete characterization of the soil and installation of sensors that minimizes changes to the soil will simplify the analysis.

ACKNOWLEDGMENTS

The authors would like to thank Ralph Archuleta, Bill Joyner, Bill Murphy, and Les Youd for useful discussions on the analysis. Comments by an anonymous reviewer helped to improve the manuscript.

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Geophysics 47, 1-15.

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Manuscript received 12 August 1983

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ANALYSIS OF WAVE-INDUCED PORE PRESSURE CHANGES