U.S. Department of the Interior

U.S. Geological Survey

Scientific Investigations Report 2010–5138

Predicting S-Wave Velocities for Unconsolidated Sediments 

at Low Effective Pressure

Predicting S-Wave Velocities for 

Unconsolidated Sediments at Low Effective 

Pressure

By Myung W. Lee

Scientific Investigations Report 2010–5138

U.S. Department of the Interior

U.S. Geological Survey
U.S. Department of the Interior

KEN SALAZAR, Secretary

U.S. Geological Survey

Marcia K. McNutt, Director

U.S. Geological Survey, Reston, Virginia: 2010

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Suggested citation:

Lee, M.W., 2010, Predicting S-wave velocities for unconsolidated sediments at low effective pressure: U.S. Geological 

Survey Scientific Investigations Report 2010–5138, 13 p.
iii

Contents

Abstract ...........................................................................................................................................................1

Introduction.....................................................................................................................................................1

Descriptions of Data ......................................................................................................................................2

Theory ..............................................................................................................................................................3

Biot-Gassmann Theory with Consolidation Parameter ..................................................................3

Parameters  and  ...........................................................................................................................3

Measured S-wave Velocity ..........................................................................................................................5

Predicting S-wave Velocity at Low Effective Pressure ...........................................................................5

Wet Sands Measured at High Frequency .........................................................................................6

Dry Sands Measured at High Frequency ..........................................................................................6

Wet Shale Measured at Low Frequency ..........................................................................................6

Assessment of Prediction Methods ...........................................................................................................7

Comparison with Walton’s Theory .....................................................................................................7

Parameter m ..........................................................................................................................................8

Velocity Dispersion .............................................................................................................................10

Conclusions...................................................................................................................................................11

Acknowledgments .......................................................................................................................................12

References Cited..........................................................................................................................................12

Figures

1. Location map of sites drilled during the Joint Industry Project Leg II Expedition.............2

2. Measured velocities for the Gulf of Mexico sands ................................................................3

3. Estimated parameters from velocities and porosities............................................................4

4. Relation between P-wave and S-wave velocities from various sources ..........................5

5. Measured and predicted S-wave velocities for Gulf of Mexico and 

Galveston sands............................................................................................................................7

6. Measured and predicted S-wave velocities for dry sands ...................................................8

7. Well-log measured and predicted S-wave velocities at the Gulf of Mexico......................9

8. Measured velocities of Gulf of Mexico sands and predicted velocities 

using the Walton’s equation......................................................................................................10

9. Difference between measured and predicted velocities of Gulf of Mexico sands.........11

Abstract

Accurate S-wave velocities for shallow sediments are 

important in performing a reliable elastic inversion for gas 

hydrate-bearing sediments and in evaluating velocity models 

for predicting S-wave velocities, but few S-wave velocities 

are measured at low effective pressure. Predicting S-wave 

velocities by using conventional methods based on the BiotGassmann theory appears to be inaccurate for laboratorymeasured velocities at effective pressures less than about 

4–5 megapascals (MPa). Measured laboratory and well log 

velocities show two distinct trends for S-wave velocities with 

respect to P-wave velocity: one for the S-wave velocity less 

than about 0.6 kilometer per second (km/s) which approximately corresponds to effective pressure of about 4–5 MPa, 

and the other for S-wave velocities greater than 0.6 km/s. To 

accurately predict S-wave velocities at low effective pressure 

less than about 4–5 MPa, a pressure-dependent parameter that 

relates the consolidation parameter to shear modulus of the 

sediments at low effective pressure is proposed. The proposed 

method in predicting S-wave velocity at low effective pressure 

worked well for velocities of water-saturated sands measured 

in the laboratory. However, this method underestimates the 

well-log S-wave velocities measured in the Gulf of Mexico, 

whereas the conventional method performs well for the well 

log velocities. The P-wave velocity dispersion due to fluid in 

the pore spaces, which is more pronounced at high frequency 

with low effective pressures less than about 4 MPa, is probably a cause for this discrepancy. 

Introduction

Gas hydrate, an icelike compound of natural gas and 

water, is widespread in the Gulf of Mexico (GOM) (Milkov 

and Sassen, 2001), particularly on the continental slope with 

water depths in the range of 540–2,000 meters (m). One of 

the objectives of the GOM first Joint Industry Project (JIP Leg 

I) is the development of seismic analysis methods pertinent 

to gas hydrate-bearing sediments (GHBS). For the inversion 

of elastic impedances to estimate gas hydrate saturations, the 

use of predicted S-wave velocity in the absence of measured 

S-wave velocity increases the accuracy of the elastic inversion (Lee, 2006a). However, most theories predicting S-wave 

velocities work well for sediments at high effective pressure, 

possibly greater than 4–5 megapascals (MPa) (for example, 

Greenberg and Castagna, 1992; Xu and White, 1996; Jørstad 

and others, 1999; Lee, 2006b, 2006c). Although there are 

existing theories applicable to shallow sediments (for example, 

Walton, 1987; Dvorkin and others, 1999; Lee, 2002), measurements of S-wave velocities at low effective pressure and at 

the seismic and logging frequencies are lacking to test their 

effectiveness. 

During field testing of phase II of the JIP Leg I, drilling, 

coring, and logging were conducted to test, validate, and 

adjust various models developed during the phase I program. 

Two field sites were selected (fig. 1): one at Atwater Valley 

(AT13) and the other at Keathley Canyon (KC151). Lee and 

others (2008) measured mechanical and electromagnetic properties of recovered sediments at the two sites in order to gain 

insight into processes accompanying nucleation and growth of 

gas hydrate, and also measured S-wave velocities at the effective pressures less than about 2.5 MPa.

In 2009, the JIP conducted its Leg II (JIP Leg II) loggingwhile-drilling (LWD) operations at three sites (Walker Ridge, 

Green Canyon, and Alaminos Canyon) in the northern GOM 

(fig. 1). These locations were identified, primarily from the 

3-D seismic data, to test geological and geophysical interpretation methods to prospect for gas hydrate-bearing sand 

reservoirs (Hutchinson and others, 2009; Shedd and others, 

2009). A primary scientific objective of the drilling was to 

collect high-quality LWD data to further refine estimates of 

gas hydrate saturations (Boswell and others, 2009). During 

this expedition, a suite of LWD well logs, including S-wave 

velocities at shallow depths, were acquired.

The purpose of this investigation is to determine whether 

the overestimation of S-wave velocity is due to (1) the inaccuracy of the rock physics model based on the Biot-Gassmann 

theory (BGT) at low effective pressure, or (2) other causes 

such as velocity dispersion, which is more pronounced at 

low effective pressure with high-frequency measurement. If 

the overestimation of S-wave velocity is due to the velocity 

dispersion, conventional BGT-based methods can be used 

Predicting S-Wave Velocities for Unconsolidated 

Sediment at Low Effective Pressure

By Myung W. Lee
2 Predicting S-Wave Velocities for Unconsolidated Sediment at Low Effective Pressure

AC21 GC955

WR313

KC151

Houston New 

Orleans

AT13

Figure 1. Location map of sites drilled during the Joint Industry Project Leg II Expedition and 

KC151 site, Gulf of Mexico (modified from Boswell and others, 2009).

by either accounting for the velocity dispersion or by using 

velocities measured at low frequencies such as well log 

(1–60 kilohertz [Hz]) and seismic (less than 100 Hz) velocities. If the overestimation is due to inaccuracy of BGT-based 

methods at low effective pressure, a new rock physics model 

applicable at low effective pressure is desirable and would be 

an improvement. 

This report uses the measured S-wave velocities for 

recovered core samples by Lee and others (2008) at the 

KC151 site, reconstituted sands by Zimmer (2003), and 

well-log data at the Walker Ridge and Alaminos Canyon sites 

acquired during JIP Leg II (Boswell and others, 2009). For 

velocities, BGT with dry moduli of the sediments proposed 

by Pride (2005) and Lee (2005) is used. In order to model the 

effect of effective pressure on velocity, a pressure-dependent 

consolidation parameter is incorporated into BGT.

Descriptions of Data

S-wave velocities (Vs) of reconstituted sediment at the 

KC151 site measured in the frequency range of 5–70 kHz 

(Lee and others, 2008) and reconstituted GOM and Galveston sands measured by Zimmer (2003) with a frequency of 

150 kHz along with well log velocities were analyzed to study 

the effect of the effective pressure on Vs. Effective pressure 

used by Lee and others (2008) ranged from 0 to 2.31 MPa 

with a single loading and unloading cycle and 0.1 to 20 MPa 

with many loading and unloading cycles for measurements by 

Zimmer (2003). 

The S-wave velocities at the Walker Ridge site 

(WR313–H well located at the water depth of 6,450 ft 

[1,953 m]) and at the Alaminos Canyon site (AC21–A well, 

located at the water depth of 4,889 ft or [1,490 m]) were 

acquired using MP3 (Schlumberger’s newer generation 

acoustic tool). The tool, located 35.45 ft (10.8 m) from the 

drill bit, was the closest tool in the LWD logging assembly. Because of the short distance from the drill bit, adverse 

effects of the borehole such as washouts were minimized. The 

P-wave velocities (Vp) were estimated from the waveform in 

the frequency range of 2.5–4.5 kHz and S-wave velocities 

were processed using frequencies less than 3 kHz. Although 

high-quality P-wave velocities were acquired throughout the 

logged interval (Guerin and others, 2009), the S-wave velocities were interpretable only for limited depth ranges due to 

the low signal-to-noise ratio of the shear arrivals. In spite of 

this limitation of the S-wave velocity, the LWD S-wave data 

provide a critical dataset to test rock physics models at low 

effective pressure, as presented in this report. 

Sediments obtained at the KC151 site, at 4,369 ft 

(1,323 m) water depth in the northern GOM (fig. 1) are classified as inorganic clays with high plasticity (Yun and others, 2006). Sediments at the KC151 site are similar to those 

measured by MP3 during JIP Leg II. 
Theory 3

Theory

Biot-Gassmann Theory with Consolidation 

Parameter

Vp and Vs of fluid (water)-saturated sediments can be 

written as: 

 V k p

b

= + 4 3 



/ and Vs

b

= 



 (1)

where k and  are bulk and shear moduli of the sediment, b

is the bulk density of sediment given by      b s = − + w ( ) , 1

and subscripts s and w refer to sediment grain and fluid, 

respectively.

According to BGT, the bulk modulus of the sediments is 

given by

 k K = − + s p p av ( ) 1 K

2   (2)

with 1

K K K av

p

s w

= − +

( )    ,

and the shear modulus given by (Lee, 2005) is

    = − s s ( ) 1 (3)

where  p and s are appropriate Biot coefficients for bulk and 

shear moduli.

The Biot coefficient shown in equations 2 and 3 can be 

written as (Lee, 2005):

   

 p = ++

( )

( )

1

1

and   

 s = ++

( )

( )

1

1 ,

 with  

 = ++

1 2

1 (4)

where is the consolidation parameter (Pride, 2005; Lee, 

2005). Pride (2005) used a constant value for  , whereas Lee 

(2005) used  as a function of the consolidation parameter. 

Lee (2005) showed that using  as a function of yields more 

accurate velocities, particularly for unconsolidated sediments. 

Parameters α and γ

The consolidation parameter depends on the effective 

pressure and degree of consolidation and should be a function 

of depth or effective pressure. Depth (or effective pressure) 

dependent is proposed by the following equation:

   i o o i  n

o o i

n = ( / ) ( / ) p p d d ≈ (5)

whereo is the consolidation parameter at the effective 

pressure po or depth do, i is the consolidation parameter at 

the effective pressure pi or depth di, and n is an appropriate 

constant. 

As mentioned previously, Pride (2005) used a constant 

 irrespective of the consolidation parameter. Lee (2005) 

used  as a function of (equation 4) in such a way that

 increases as increases. Equation 5 indicates that the 

logarithm of with respect to the effective pressure is a linear 

function of pressure. The slope of the linear function controls 

how rapidly the velocity changes with respect to effective 

pressure. 

Figure 2 shows measured velocities of GOM sands 

(Zimmer, 2003), and figure 3 shows various calculated  ’s. Figure 3A shows the calculated, assuming 

   = + + ( ) / ( ) 1 2 1 from the P-wave velocity and porosity 

data for GOM reconstituted sands using the method proposed 

by Lee (2006c). As shown in figure 3A,  is virtually constant 

irrespective of whereas increases as the effective pressure decreases. However, the increase of logarithmic is 

not a linear function of the logarithmic effective pressure as 

assumed in equation 5. In other words, the exponent n shown 

in equation 5 is not a constant, but rather a function of effective pressure. 

To a first-order approximation, there are two linear 

functions of as a function of effective pressure, with the 

transition occurring at approximately 4 to 5 MPa (fig. 3A). 

EFFECTIVE PRESSURE, IN MEGAPASCALS

VELOCITY, IN KILOMETERS PER SECOND

0

0.1 1 10

0.5

1.0

1.5

2.0

2.5

P-wave

S-wave

γ = (1+mα)/(1+α)

Measured

Predicted assuming m = 2

Predicted assuming m = 2 for p > 5 MPa

m = 6/p0.6 for p < 4 MPa

Figure 2. Measured velocities for the Gulf of Mexico 

(GOM) sands by Zimmer (2003) with various predicted 

S-wave velocities.
4 Predicting S-Wave Velocities for Unconsolidated Sediment at Low Effective Pressure

A B

EFFECTIVE PRESSURE, IN MEGAPASCALS

PARAMETERS α AND γPARAMETERS α AND γ

1

0.1 1 10 0.1 1 10

10

100

1

10

100

Transition

Transition

Assuming γ is a function of α

Assuming α and γ are independent

α γ

α γ

Figure 3. Estimated parameters from velocities and porosities of Gulf of Mexico sands measured by Zimmer (2003). 

A, consolidation parameter  from the P-wave velocity and porosity assuming    = + + ( ) / ( ) 1 2 1 . B, parameters

and  calculated from the P-wave, S-wave velocities, and porosity assuming and  are independent variables.

For effective pressure less than 5 MPa, the linear function is 

given by  = − 1 72 0 15 . . p and for p > 5 MPa, it is given by 

 = − 1 81 0 35 . . p . The calculated S-wave velocities using parameters shown in figure 3A are shown in figure 2 as red circles. 

The calculated S-wave velocities are accurate for p > 5MPa, 

whereas the calculated S-velocities are much higher than those 

measured for p < 4 MPa. This implies that equation 4 is accurate to model velocities of sediments for an effective pressure 

greater than about 4–5 MPa, but overestimates S-wave 

velocity when p < 4 MPa. 

Figure 3B shows calculated and  assuming these 

parameters are independent of each other. These parameters 

are calculated by fitting both P- and S-wave velocities by 

adjusting and  independently. Figure 3B indicates, when 

p > 5 MPa, that and  , as calculated under the assumption 

that these parameters are independent, are similar to those calculated assuming    = + + ( ) / ( ) 1 2 1 . In other words, equation 

4 is accurate for p > 4–5 MPa. However, the calculated and 

in figure 3B differ from those shown in figure 3A for 

p < 4 MPa. To accurately model both P- and S-wave velocities 

for p < 4–5 MPa, equation 4 is not adequate and new parameters are required. 

To model velocities at low effective pressure, the following equations are proposed:

   = + + ( ) / ( ) 1 1 m with m p = − 6 0 6. for p < 4 MPa and 

m = 2 for p>5Mpa, (6)

and a linear interpolation between 2 and m for 4 <p < 5 MPa. 

The large open circles in figure 2 show the calculated S-wave 

velocities using equation 6. The calculated S-wave velocities 

are accurate for all ranges of effective pressure. It is noted that 

the parameter transition near p ≈4–5 MPa shown in figure 3 

is based on velocities for clean sands. Transition pressure 

for shale or shaly sand, therefore, may be different from that 

shown in figure 3, and it could be updated when such data are 

available. 

Equation 6 is valid only for fluid-saturated sediment. In 

the case that the pore space is filled with air (dry rock) or sediment is partially gas saturated,  = 1 should be used irrespective of the effective pressure (Lee, 2005).
Predicting S-wave Velocity at Low Effective Pressure 5

Measured S-wave Velocity

Figure 4 shows the relation between P- and S-wave 

velocities from various sources. Data by Han and others 

(1986) are for consolidated sediments and other data are for 

unconsolidated sediments. There are two distinct trends: one is 

for S-wave velocity greater than about 1 km/s 

(V V p = s 2 286 0 672 . . ) and the other is for velocity less than about 

0.6 km/s (V V p = s 2 042 0 103 . . ). Modeling results using pressuredependent consolidation parameter = 3 5 0 6 ( / ) . p with a 

constant m = 2 for consolidated sediment having = 0 15 . (red 

solid line), and for unconsolidated sediment (red dashed line) 

having = 0 38 . with = 40 5 0 6 ( / ) . p are shown for comparison. 

Although the modeled relation for unconsolidated sediments 

mimics the measured relation for S-wave velocity up to 1 

km/s, it deviates significantly from the measured relation for 

S-wave velocities less than about 0.6 km/s. Figure 4 indicates 

that two different velocity trends exist and suggests that two 

different approaches would be required to predict S-wave 

velocity, similar to that suggested in figure 2. 

Predicting S-wave Velocity at Low 

Effective Pressure

One important factor to be considered for the elastic 

inversion of unconsolidated shallow sediment is the S-wave 

velocity at low effective pressure (Lee, 2006a). It is desirable to accurately predict the S-wave velocity less than about 

0.6 km/s or at low effective pressure from the P-wave velocity 

to effectively implement the seismic inversion. S-wave velocities less than 0.6 km/s are common. Kumar and others (2006) 

reported that S-wave velocities estimated from multicomponent ocean-bottom seismograph data at the Hydrate Ridge, 

offshore Oregon, are in the range of 0.15 to 0.35 km/s. Also, 

S-wave velocities analyzed from the multicomponent oceanbottom cable data in the GOM by Hardage and others (2006) 

are in the range of 0.3 km/s. The well log S-wave velocities 

at Hydrate Ridge that were acquired during Ocean Drilling 

Program Leg 204 vary from 0.25 km/s to 0.65 km/s (Lee and 

Collett, 2006). 

S-WAVE VELOCITY, IN KILOMETERS PER SECOND

P-WAVE VELOCITY, IN KILOMETERS PER SECOND

0.1

1

2

3

4

5

6

1 4

Prasad (2002),

Alpine-1 well,

Modeled using αO = 40(5/p)0.6 with φ = 0.38

Modeled using αO = 3(40/p)0.6 with φ = 0.15

Mount Elbert well

Han and others (1986) at 5 MPa

Han and others (1986) at 40 MPa

Zimmer (2003)

Vp= 2.042Vs

 0.103

Vp= 2.286Vs

 0.672

Figure 4. Relation between P-wave and S-wave 

velocities from various sources; laboratory data from 

Han and others (1986), Prasad (2002), and Zimmer 

(2003), and well log data from the Mount Elbert and 

Alpine–1 wells, North Slope of Alaska.
6 Predicting S-Wave Velocities for Unconsolidated Sediment at Low Effective Pressure

The use of equations 1 and 4 provides a means of 

predicting S-wave velocities based on P-wave velocities and 

porosities of sediments because the parameter relates both 

bulk and shear moduli of a dry frame. Defining the predicted 

P-wave velocity using the BGT, with the moduli derived from

equations 1 and 4Vp

* as and asVpm for measured P-wave 

velocity, the consolidation parameter can be calculated by 

solving the following equation (Lee, 2006c):

 V V p p

* m ( ) .  − = 0 (7)

Therefore, the shear modulus can be calculated using equation 

3 with equation 4 by substituting the consolidation parameter 

estimated from equation 7 into equation 4. 

If m shown in equation 6 is used to solve equation 7, it 

is the proposed method of predicting S-wave velocity in this 

report. If m=2 for all effective pressure shown in equation 6 

is used to solve equation 7, it is the conventional method and 

identical to that shown in Lee (2006c). 

Wet Sands Measured at High Frequency

Using m=2 in equation 6 (conventional method) resulted 

in overestimating S-wave velocities that are predicted on the 

basis of the P-wave velocities measured at the high frequency 

of 150 kHz for GOM sands by Zimmer (2003), as shown 

in figure 2. Currently available rock physics models (for 

example, Walton, 1987; Greenberg and Castagna, 1992; Xu 

and White, 1996; Lee, 2006b) consistently predict a slightly 

higher S-wave velocity from the porosity and P-wave velocity 

for GOM sands. Figure 5A shows predicted S-wave velocities for GOM and Galveston sands (Zimmer, 2003) with m=2 

irrespective of the effective pressure (conventional method), 

whereas figure 5B shows the predicted S-wave velocities using 

m p = − 6 0 6. for effective pressures less than 4 MPa and m =2 for 

pressures greater than 5 MPa (proposed method). Figure 5A

indicates that using m = 2 results in accurate prediction of 

S-wave velocities for effective pressures greater than about 

5 MPa, which corresponds to an S-wave velocity of about 

0.6 km/s but overestimates S-wave velocities for effective 

pressures less than about 5 MPa. The overestimation increases 

as the effective pressure decreases. 

Predicted S-wave velocities for GOM sands using 

the proposed method are more accurate than those for the 

Galveston sands because the parameter m was derived specifically from the GOM sands. However, the use of m as a function of effective pressure shown in equation 6 also worked 

fairly well for the Galveston sands. 

The S-wave prediction methods suggested by Greenberg 

and Castagna (1992), Xu and White (1996), and Lee (2006b, 

2006c) are based on the Gassmann type equation, which is 

valid at low frequency and ignores the velocity dispersion. 

Thus, one interpretation for the cause of the overestimation 

of S-wave velocity for GOM sands at low effective pressure 

is that P-wave velocity dispersion was not accounted for. An 

alternative interpretation is that the conventional method based 

on the BGT is not accurate at low effective pressure.

Dry Sands Measured at High Frequency

If overestimation of S-wave velocities for wet GOM or 

Galveston sands is caused by velocity dispersion due to fluid 

in the pore space, the prediction of S-wave velocities for dry 

rock (air filled in the pore spaces) should be accurate using the 

same rock physics model shown in this report. Figures 6A and 

6B show the measured and predicted S-wave velocities for dry 

samples of GOM and Galveston sands measured by Zimmer 

(2003) using  = 1 irrespective of the effective pressure. As 

opposed to the results shown in figure 5, the predicted S-wave 

velocities are accurate both for the GOM and Galveston sands. 

The results shown in figure 6 indicate that the pore fluid is 

probably the primary cause for the overestimation of S-wave 

velocities shown in figure 5A.

Wet Shale Measured at Low Frequency

The MP3 P-wave data were processed with a 4.5-kHz 

low-pass filter, and the S-wave data were processed with a 

3.0-kHz low-pass filter. Figure 7A shows the measured MP3 

shear-wave velocities with two predicted S-wave velocities 

from the measured P-wave velocity and porosity at WR313–H 

well, GOM. S-wave velocities using a constant m =2 (conventional method) are higher than those predicted using m p = − 6 0 6.

(proposed method) and agree better with the measured MP3 

shear-wave velocity data. The effective pressure was estimated 

assuming a normal pressure regime in this area. For example, 

the effective pressure at 500 ft below sea floor (fbsf) is about 

1.5 MPa. 

The S-wave velocities by Lee and others (2008) were 

measured from reconstituted samples collected at the KC151 

site, between the WR and AC sites (fig. 1) and lithologies at 

both sites are shale. The MP3 data at different wells acquired 

during JIP Leg II show a similar P-wave velocity trend with 

depth, particularly at depths less than about 500 fbsf. Therefore, it is assumed that the P-wave velocities at the KC151–2 

well are similar to those measured P-waves at the Walker 

Ridge and Alaminos Canyon areas. The measured S-wave 

velocities by Lee and others (2008) were also plotted in figure 

7A for WR313–H as stars. Although the locations of the two 

sites are not the same, the S-wave velocities with depth (effective pressure) at the KC151 site agree better with the predicted 

S-wave velocities using m = 2 rather than m as a function of 

the effective pressure. Figure 7B shows predicted and measured S-wave velocities at the AC21–A well and supports the 

previous observation at the WR313–H well. 

In summary, the conventional method accurately predicts 

S-wave velocities at all ranges of effective pressure from the 

P-wave velocities measured at low frequencies less than 70 

kHz, whereas it overestimates S-wave velocities from P-wave 

velocities measured at low effective pressure less than 4 MPa 
Assessment of Prediction Methods 7

0

0 0.25 0.50 0.75 1.00 0 0.25 0.50 0.75 1.00

0.25

0.50

0.75

1.00

0

0.25

0.50

0.75

1.00

MEASURED S-WAVE VELOCITY,

IN KILOMETERS PER SECOND

PREDICTED S-WAVE VELOCITY, IN KILOMETERS PER SECONDPREDICTED S-WAVE VELOCITY, IN KILOMETERS PER SECOND

MEASURED S-WAVE VELOCITY,

IN KILOMETERS PER SECOND

Using a constant m = 2 Using varying m

GOM sand (Zimmer, 2003)

Galveston sand (Zimmer, 2003)

Predicted = measured

GOM sand (Zimmer, 2003)

Galveston sand (Zimmer, 2003)

Predicted = measured

A B

Figure 5. Measured and predicted S-wave velocities for Gulf of Mexico (GOM) and Galveston sands by Zimmer (2003). 

A, using the conventional method (equation 4). B, using the proposed method (equation 6).

at frequencies greater than 150 kHz. On the other hand, the 

proposed method accurately predicts S-wave velocities at 

all ranges of effective pressure from the P-wave velocities 

measured at high frequencies, but it underestimates S-wave 

velocities from P-wave velocities measured at low frequencies 

such as well log velocities. 

Assessment of Prediction Methods

Comparison with Walton’s Theory 

To test whether the overestimation of S-wave velocities for wet GOM sands is caused by the inaccuracy of the 

rock physics model used here (equations 1–5), the theory by 

Walton (1987), which is well established for the determination of velocities of unconsolidated sediments, is investigated. 

For unconsolidated sands and soils, bulk and shear moduli 

of the frame are well modeled using the following variant of 

Walton’s theory (see Pride [2005] for details):

k q p

C

p p

p p d

o

s

i o

i o

= ⎡ −

⎣

⎢ ⎤

⎦

⎥

{ + [ ] }

1

6

4 1

1 16 9

2 2

4 2

1 3

4 1 24

( ) /

/ ( )

/

/



 (8)

   = =d

d 3k

5

where q is the coordination number (average number of 

grain contacts per grain) and increases as effective pressure 

increases (reference effective pressure po is commonly on the 

order of 10 MPa) and Cs is a compliance parameter given by

 C

K s

s s s

= +

+

⎛

⎝

⎜

⎞

⎠

⎟

1

4

1 1

   / 3 .

A sediment with unimodal grain-size distributions and random 

grain packs, which is similar to reconstituted GOM sands by 

Zimmer (2003), typically has 0 32 0 36 . . < <  and 8 11 < < q .

Figure 8 shows the calculated P- and S-wave velocities 

using Walton’s (1987) equation with  = 0 4. , q = 9, and po

=18 MPa as solid red lines. The predicted S-wave velocities 

from Walton’s equation agree well with measured S-wave 

velocities, and S-wave velocities are proportional to about 
8 Predicting S-Wave Velocities for Unconsolidated Sediment at Low Effective Pressure

1/4 power of effective pressure. Also shown in figure 8 as 

blue dashed lines are the modeled velocities using the BGT 

with = 45 5 0 6 ( / ) . p with = 0 4. and m = 2. The results using 

the BGT are almost identical to those of the Walton (1987) 

for effective pressure less than 5 MPa but are more accurate for effective pressures greater than 5 MPa. By using a 

pressure-dependent coordination number or other adjustment 

of parameters for Walton’s equation, the performance of the 

Walton model could be improved. However, figure 8 indicates 

that the result using BGT with m=2 is as good as the result by 

Walton’s equation. 

The predicted P-wave velocities using the Walton equation and BGT with consolidation parameter with equation 

4 behave similarly. For effective pressure less than about 3 

MPa, both equations predict similar P-wave velocities that are 

slightly less than the measured velocities. On the other hand, 

for effective pressure greater than about 3 MPa, BGT predicts 

accurate P-wave velocities, whereas the Walton equation 

underestimates the P-wave velocity. The key observation of 

figure 8 is that both BGT and Walton’s theory underestimate 

the P-wave velocities for GOM wet sands for a given S-wave 

velocity, and the underestimation increases as the effective 

pressure decreases, which is similar to the behavior of the 

overestimation of S-wave velocity shown in figure 2. In other 

words, the underestimation of the P-wave velocity or overestimation of S-wave velocity is not likely caused by the inaccuracy of the rock physics model shown in equations 1–5.

The parameter m, shown in equation 6, determines 

whether or not the conventional BGT-based rock physics 

models are accurate at low effective pressure. If using m as a 

function of the effective pressure is more accurate than using 

m = 2 irrespective of the effective pressure, the conventional 

method is inaccurate. Figure 9A shows the predicted P-wave 

velocities using m = 2 irrespective of the effective pressure 

for GOM sands by Zimmer (2003). If a pressure dependent 

m —for example, m p = 6 0 6 / . —is used, the predicted P-wave 

velocities would be almost identical to the measured P-wave 

velocities indicated as solid dark green dots in figure 9A. 

Figure 9B shows the difference between predicted P-wave 

velocities with m = 2 and measured P-wave velocities for 

GOM and Galveston sands. The difference increases as the 

0.1

1

0.1

1

0.01 0.1 1 10 0.01 0.1 1 10

EFFECTIVE PRESSURE, IN MEGAPASCALS EFFECTIVE PRESSURE, IN MEGAPASCALS

VELOCITY, IN KILOMETERS PER SECONDVELOCITY, IN KILOMETERS PER SECOND

Gulf of Mexico dry sand Galveston dry sand

Measured P-wave velocity (Zimmer, 2003)

Measured S-wave velocity (Zimmer, 2003)

Predicted S-wave velocity

Measured P-wave velocity (Zimmer, 2003)

Measured S-wave velocity (Zimmer, 2003)

Predicted S-wave velocity

A B

Figure 6. Measured and predicted S-wave velocities for dry sands by Zimmer (2003). A, Gulf of Mexico (GOM) sands. 

B, Galveston sands.

Parameter m
Assessment of Prediction Methods 9

Figure 7. Well-log measured and predicted S-wave velocities at the 

Gulf of Mexico with S-wave velocities measured by Lee and others 

(2008). A, WR313–H well. B, AC21–A well.

0

0.2

0.4

0.6

0.8

1.0

0

0.2

0.4

0.6

0.8

1.0

0 500 1,000

0 500 1,000

DEPTH BELOW SEA FLOOR, IN FEET

DEPTH BELOW SEA FLOOR, IN FEET

S-WAVE VELOCITY, IN KILOMETERS PER SECOND S-WAVE VELOCITY, IN KILOMETERS PER SECOND

A

B

Measured MP3 S-wave velocity at WR313–H

Measured S-wave velocity by Lee and others (2008)

Predicted S-wave velocity using m = 2

Predicted S-wave velocity using m =6/p0.6

Measured MP3 S-wave velocity at AC21–A

Measured S-wave velocity by Lee and others (2008)

Predicted S-wave velocity using m = 2

Predicted S-wave velocity using m =6/p0.6
10 Predicting S-Wave Velocities for Unconsolidated Sediment at Low Effective Pressure

effective pressure decreases, similar to the behavior of the 

P-wave velocity dispersion (Zimmer, 2003). 

This result implies that the velocity dispersion can be 

accounted for by using m p = 6 0 6 / . , at least for the GOM sands. 

If the overestimation of the S-wave is caused by the inaccuracy of the BGT-based method, m p = 6 0 6 / . provides more 

accurate S-wave velocities at low effective pressure irrespective of the frequency. However, the results shown for the well 

log velocities and those measured by Lee and others (2008) 

indicate that m = 2, the conventional method, is more accurate. 

Therefore, the overestimation of S-wave velocities for GOM 

sands is likely due to P-wave velocity dispersion rather than 

the inaccuracy of the conventional method. 

Velocity Dispersion

Velocity dispersion is primarily caused by the interaction 

between the pore fluid and the porous solid. Velocities increase 

as frequency increases because the viscosity of the fluid resists 

shearing of the rock volume and is more pronounced at high 

frequency. As shown in the previous example for the GOM 

sands, BGT predicts accurate S-wave velocities for high effective pressure, whereas it overestimates S-wave velocities at 

low effective pressure. This discrepancy is mainly caused by 

the P-wave velocity dispersion.

The effect of pore fluid is more pronounced on the 

P-wave velocities at low effective pressure (Zimmer, 2003). 

EFFECTIVE PRESSURE, IN MEGAPASCALS

Gulf of Mexico sands by Zimmer (2003)

VELOCITY, IN KILOMETERS PER SECOND

0

0.1 1 10

0.5

1.0

1.5

2.0

2.5

P-wave

S-wave

Measured

Modeled using Lee (2006c)

Modeled using Walton (1987)

Figure 8. Measured velocities of Gulf of Mexico (GOM) sands 

and predicted velocities using the Walton’s equation and the BiotGassmann theory with the consolidation parameter with m = 2 shown 

in this report. 
Conclusions 11

A B

EFFECTIVE PRESSURE, IN MEGAPASCALS EFFECTIVE PRESSURE, IN MEGAPASCALS

VELOCITY, IN KILOMETERS PER SECOND

P-WAVE VELOCITY DIFFERENCE, IN KILOMETERS PER SECOND

0 0

0.05

0.10

0.15

0.20

0.5

1.0

1.5

2.0

2.5

0 5 10 15 20 0.1 1 10

P-wave

S-wave

Measured velocities of 

 GOM sands (Zimmer, 2003)

GOM sand

Galveston sand

Predicted P-wave velocities with m = 2

Figure 9. Difference between measured and predicted velocities of Gulf of Mexico (GOM) sands by Zimmer (2003). 

A, predicted P-wave velocities from S-wave velocities for GOM sands using m = 2. B, difference between the 

predicted P-wave velocity with m = 2 and the measured velocities for GOM and Galveston sands.

S-wave velocities predicted using Gassmann theory for a variety of samples are close to the measured velocities, whereas 

the predicted P-wave velocities are always smaller than the 

measured velocities (Zimmer, 2003). On the basis of this 

observation, Zimmer concluded that the dispersion mechanism 

significantly affects only the bulk modulus of the sediments, 

not the shear modulus. Consequently, the Gassmann type 

equation (such as equations used in this report) predicts higher 

S-wave velocity calculated from the P-wave velocity that was 

measured at low effective pressure with higher frequencies. 

As shown previously, the predicted S-wave velocities 

from the P-wave velocities are accurate for dry sands (fig. 

6) and well-log velocities at low frequencies (fig. 7). Taking 

these results with the velocity dispersion, the overestimation 

of S-wave velocity at low effective pressure from the P-wave 

velocity measured at high frequencies in the range of 150 kHz 

is primarily caused by velocity dispersion.

Conclusions

Measured data indicate that there are two distinct relations between the P-wave velocity and the S-wave velocity. 

One trend is for the S-wave velocity that is less than about 

0.6 km/s, which corresponds approximately to an effective 

pressure less than 4–5 MPa. The other relation is valid for the 

S-wave velocity that is greater than about 0.6 km/s. However, the same rock physics model with m = 2 (conventional 

method) can be applied to relate the P-wave velocity to the 

S-wave velocity, if the P-wave velocity is measured at low 

frequencies. 

Conventional rock physics models, which are based on 

the classical Biot-Gassmann theory, predicted higher S-wave 

velocities from the P-wave velocities, which were measured at 

high frequencies and at low effective pressures less than about 

4–5 MPa, and the overestimation increases as the effective 
12 Predicting S-Wave Velocities for Unconsolidated Sediment at Low Effective Pressure

pressure decreases. The primary reason for the overestimation 

of the S-wave velocity is likely due to the P-wave velocity 

dispersion.

The logging-while-drilling velocity logs acquired in the 

Gulf of Mexico indicate that Biot-Gassmann type equations, 

which are the low-frequency approximation of the Biot equation, appear to be appropriate for predicting S-wave velocities 

for well logs or seismic data. However, because the S-wave 

velocities at low effective pressure investigated in this report 

were limited to shale, more accurate velocity data for sands 

measured both at low effective pressure and low frequency are 

desirable to further refine the S-wave prediction method.

Acknowledgments

I would like to thank Timothy Collett (U.S. Geological 

Survey) for providing the MP3 well-log data for this study 

and for his continuous support and encouragement throughout 

the investigation. I also thank Warren Agena and Lauri Burke 

(U.S. Geological Survey) for their many suggestions and 

comments. 

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