Velocity Ratio and its Application to 

Predicting Velocities 

By Myung W. Lee 

U.S. Geological Survey Bulletin 2197 

U.S. Department of the Interior 

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

Gale A. Norton, Secretary 

U.S. Geological Survey 

Charles G. Groat, Director 

Posted online February 2003, version 1.0 

For more information about the USGS and its products:


Telephone: 1-888-ASK-USGS


World Wide Web: http://www.usgs.gov/


This publication is only available online at: 

http://geology.cr.usgs.gov/pub/bulletins/b2197/ 

Any use of trade, product, or firm names in this publication 

is for descriptive purposes only and does not 

imply endorsement by the U.S. Government 
III


Contents 

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


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


Biot-Gassmann Theory (BGT)..................................................................................................................... 2


Velocity Ratio................................................................................................................................................. 2


Modeling the Velocity Ratio ........................................................................................................................ 4


Modeling Velocities...................................................................................................................................... 5


Discussion...................................................................................................................................................... 5


Fluid Effect on Shear Velocity............................................................................................................ 5


Scale G .................................................................................................................................................. 7


Analysis of Exponent n........................................................................................................................ 8


Application to Velocity Prediction..................................................................................................... 9


Velocity Dispersion and Velocity Ratio........................................................................................... 11


Error Analysis ..................................................................................................................................... 11


Conclusion ................................................................................................................................................... 12


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


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


Appendix A................................................................................................................................................... 14


Biot Coefficient................................................................................................................................... 14


Appendix B................................................................................................................................................... 15


Prediction of S-Wave Velocity Based on BGT .............................................................................. 15


Figures 

1. 	 Measured and predicted velocity ratios for dry and wet rock with 


respect to differential pressure ................................................................................................. 3


2. 	 Measured velocity ratios with respect to porosity for data complied by 


Castagna and others (1985) and gas samples from Domenico (1977) ................................. 4


3. 	 Measured velocity ratios with respect to porosity and clay volume content 


for the data of Han and others (1986) at the differential pressure of 30 MPa .................... 5


4. 	 Measured and modeled velocities based on BGT and BGTL with 


respect to differential pressure ................................................................................................. 6


5. Graph showing the effect of scale G on velocities................................................................. 8


6. 	 Measured velocities by Han and others (1986) at the differential pressure 


of 30 MPa and predicted velocities from BGT and BGTL using elastic moduli 


shown in table 1 ............................................................................................................................ 9


7. 	 Predicted and measured velocities for data compiled by 


Castagna and others (1985) ...................................................................................................... 10


A-1 Computed Biot coefficients from velocities of dry samples and 


least-squares fits to calculated coefficients ......................................................................... 14

IV


Table 

1. Elastic constants used for velocity model................................................................................ 4

Velocity Ratio and its Application to 


Predicting Velocities 

By Myung W. Lee 

Abstract 

The velocity ratio of water-saturated sediment derived 

from the Biot-Gassmann theory depends mainly on the Biot 

coefficient—a property of dry rock—for consolidated sediments with porosity less than the critical porosity. With this 

theory, the shear moduli of dry sediments are the same 

as the shear moduli of water-saturated sediments. Because 

the velocity ratio depends on the Biot coefficient explicitly, 

Biot-Gassmann theory accurately predicts velocity ratios with 

respect to differential pressure for a given porosity. However, 

because the velocity ratio is weakly related to porosity, it is 

not appropriate to investigate the velocity ratio with respect to 

porosity (φ). 

A new formulation based on the assumption that the 

velocity ratio is a function of (1–φ)

n yields a velocity ratio that 

depends on porosity, but not on the Biot coefficient explicitly. 

Unlike the Biot-Gassmann theory, the shear moduli of watersaturated sediments depend not only on the Biot coefficient 

but also on the pore fluid. This nonclassical behavior of the 

shear modulus of water-saturated sediment is speculated to be 

an effect of interaction between fluid and the solid matrix, 

resulting in softening or hardening of the rock frame and an 

effect of velocity dispersion owing to local fluid flow. The 

exponent n controls the degree of softening/hardening of the 

formation. Based on laboratory data measured near 1 MHz, 

this theory is extended to include the effect of differential pressure on the velocity ratio by making n a function of differential 

pressure and consolidation. However, the velocity dispersion 

and anisotropy are not included in the formulation. 

Introduction 

The velocity ratio has been used for many purposes, such 

as a lithology indicator, determining degree of consolidation, 

identifying pore fluid, and predicting velocities. The velocity 

ratio usually depends on porosity, degree of consolidation, clay 

content, differential pressure, pore geometry, and other factors. 

The velocity ratio for dry rock or gas-saturated rock is almost 

a constant irrespective of porosity and differential pressure, 

whereas the velocity ratio of wet rock depends significantly on 

porosity and differential pressure. The purpose of this paper 

is to accurately predict elastic velocities of water-saturated 

clastic sediments by utilizing the dependence of the velocity 

ratio on porosity. 

Pickett’s (1963) cross plot shows that P-wave to S-wave 

velocity ratio (Vp/Vs) for sandstone is about 1.6 in lowporosity rocks, drifting to 1.8 in relatively higher porosity 

rocks. His observation implies the dependence of Vp/Vs on 

porosity for sandstone. Gardner and Harris (1968) showed 

that Vp/Vs values > 2.0 are characteristic of water-saturated 

unconsolidated rocks, and values < 2.0 indicate either wellconsolidated rock or the presence of gas in unconsolidated 

sands. Gregory (1976) confirmed this relationship between 

velocity ratio and consolidation and suggested the dependence 

of velocity ratio on porosity. 

Hornby and Murphy (1987) and Murphy and others 

(1992) showed that the velocity ratio increases as clay content 

increases and that Biot-Gassmann theory accurately predicts 

the velocity ratio of unconsolidated water-saturated sand with 

respect to effective pressure. Castagna and others (1985) 

and Han and others (1986) empirically derived relationships 

between velocity ratio, porosity, and clay content. Han and 

others (1986) showed that the velocity ratio increases linearly 

with clay content and porosity. The equation by Castagna and 

others (1985) also implies an increase in velocity ratio with 

increasing porosity. 

The prediction of S-wave velocities for water-saturated 

rocks based on the velocity ratio with a first-order application 

of Biot-Gassmann theory is given by Greenberg and Castagna 

(1992). Empirical relationships by Castagna and others (1985) 

and Han and others (1986) can be used to predict S-wave 

velocity either from porosity and clay content or velocity ratio 

and porosity. Xu and White (1996) investigated the S-wave 

velocity prediction based on bulk and shear moduli of the dry 

rock frame by a combination of Kuster and Toksöz theory 

(1974) and differential effective medium theory, using pore 

aspect ratio. 

In this paper, a new method of modeling velocities for 

consolidated and unconsolidated sediments is presented using 

Biot-Gassmann theory with the assumption that the velocity 

ratio is related to porosity. This is an extension of Lee’s (2002) 

theory for unconsolidated sediments. Pickett (1963), Gregory 

(1976), Castagna and others (1985), and Han and others (1986) 

each suggested a relationship between porosity and velocity 

1

2 Velocity Ratio and its Application to Predicting Velocities 

ratio, but their functional relationships between porosity and 

velocity ratio are different from the one proposed in this paper. 

Using measured laboratory data, this new theory is extended 

to include the effect of differential (or effective) pressure on 

velocity ratio or velocities. This new approach is applied to the 

laboratory data from Gregory (1976), Domenico (1977), Han 

and others (1986), and that compiled by Castagna and others 

(1985) with good results. 

Biot-Gassmann Theory (BGT) 

Elastic velocities (i.e., compressional velocity (Vp) and 

shear velocity (Vs)) of water-saturated sediments can be computed from the elastic moduli by the following formulas: 

Vp = ρ

k + 4μ / 3

and Vs = ρ

μ (1) 

where 

k, μ, and, ρ are bulk modulus, shear modulus, and density 

of the formation, respectively. 

The formation density is given by 

ρ = (1−φ )ρ ma +φρ fl (2) 

where 

φ, ρma and ρfl are the porosity, matrix density, and pore fluid 

density, respectively. 

The bulk and shear moduli based on the Biot-Gassmann 

theory (Biot, 1941, 1956; Gassmann, 1951; Domenico, 1977; 

Murphy, 1984; Krief and others, 1990) are given by the following formulas using the Biot coefficient, β: 

k = kma (1− β ) + β2M (3) 

μ = μ ma (1− β ) (4) 

where 

1 (β −φ) + φ = and

M kma k fl

kma, μma, and kfl are the bulk modulus of the matrix, the 

shear modulus of the matrix, and the bulk modulus of 

the fluid, respectively. 

The moduli of a composite matrix including clay volume are 

calculated using Hill’s (1952) averaging method. The Biot 

coefficient describes the ratio of pore volume change to total 

bulk volume change under dry or drained conditions (Mavko 

and others, 1998) and relates the moduli of a dry frame to the 

moduli of the matrix material. 

Lee (2002) derived an alternative equation for the shear 

modulus 

μ maG2

(1−φ)

2n

k

μ = kma + 4μ ma [1− G2

(1−φ)

2n

]/ 3

(5) 

Equation 5 was derived with the assumption that the 

S-wave velocity is related to the P-wave velocity by the following equation: 

Vs =VpGα(1−φ)

n (6) 

where 

α is the Vs / Vp ratio for the matrix material, and 

G is a scale determined to fit the observation. 

As the solidity (complement of porosity) approaches a value of 

one (porosity of zero), the shear velocity of the rock becomes 

equal to the shear velocity of the matrix material. As the 

solidity decreases, the shear velocity in a fluid-saturated rock 

decreases. The exponent n will be used to incorporate changes 

in differential pressure, which is equal to the confining pressure minus pore pressure;. exponent n is also dependent on the 

state of consolidation. For clean sandstone, G is close to 1, 

but G becomes smaller as clay content increases in the matrix. 

Note that Lee (2002) used n = 1 and G = 1 in equation 5 for 

unconsolidated sediments. 

The difference between the classical Biot-Gassmann 

theory (BGT) and the theory proposed by Lee (2002) is in the 

computation of shear modulus for water-saturated sediment. 

The Biot theory assumes that pore fluid does not affect the 

shear modulus of sediment; therefore, the shear moduli of dry 

rock and wet rock are the same. In other words, irrespective 

of the fluid type, the shear modulus of the formation is given 

by equation 4. 

In this paper, when equation 5 is used to calculate the 

shear modulus, it is called BGT by Lee or BGTL; when equation 4 is used to calculate the shear modulus, it is called BGT. 

It is emphasized that there is no difference between BGT and 

BGTL for gas-saturated or dry rocks, and equation 4 should 

be used for both cases. 

Velocity Ratio 

The validity of equation 5 depends on the accuracy of 

the assumption of velocity ratio shown in equation 6. Figure 1 

shows velocity ratios with respect to the differential pressure 

calculated from the Gregory (1976) and Domenico (1977) 

data. From this figure, the following observations can be made. 
For dry sediments:

(a) Velocity ratio is insensitive to porosity if the differential pressure is greater than about 10 MPa. The velocity 

ratio of consolidated sediment that has a porosity of 

0.217 is almost identical to that of unconsolidated sediment having a porosity of 0.382.

(b) The effect of differential pressure on velocity ratio is 

insignificant if the differential pressure is greater than 

about 10 MPa. There is a slight decrease of velocity 

ratio as the differential pressure decreases when the 

differential pressure is greater than 10 MPa; data by 

Winkler (1985) show the same trend. Scattering of the 

velocity ratio at less than 10 MPa may be caused by 

inaccuracy of measurement.

(c) Velocity ratio is insensitive to the degree of consolidation.

For wet sediments:

(a) Velocity ratio depends on porosity.

(b) For consolidated sediment, the velocity ratio increases 

slightly as the differential pressure decreases if the 

differential pressure is greater than 20 MPa; data by 

Winkler (1985) also indicate this trend. However, the 

effect of differential pressure on velocity ratio for consolidated sediment is much smaller than for unconsolidated sediment.

(c) If sediments are unconsolidated, the velocity ratio is 

highly dependent on differential pressure.

In general, the velocity ratio depends on matrix material, 

state of saturation, degree of consolidation, differential pressure, porosity, and other factors such as pore geometry. 

The expected velocity ratios based on BGT or BGTL 

can be written as follows. Based on equations 3 and 4, BGT 

predicts

n

n

DIFFERENTIAL PRESSURE, IN MEGAPASCALS

Vp / Vs

Figure 1. Measured and predicted velocity ratios for dry (open 

symbol) and wet rock (solid symbol) with respect to differential pressure. For consolidated sediments, data by Gregory (1976) measured 

at 1 MHz are used with the assumption that the uniform pressure is 

the same as the differential pressure—this is denoted by circles. For 

unconsolidated sediments, data by Domenico (1977) measured using 

a pulse width of 2 μs are used and denoted as stars. The predicted 

ratios from BGT are represented as solid lines; ratios from BGTL are 

shown as dotted lines. Equations 7 and 8 and the values shown in 

table 1 are used for predicted velocity ratios.

3 (1 )

4 2 2

μ β

β

μ − = + + 















ma ma

ma

s

p k M

V

V (7)

This indicates that the velocity ratio is a function of the 

Biot coefficient at a given porosity and depends on the porosity through the modulus M and β. Because the quantity 

β2M/(μma(1–β)), shown in equation 7, is small for porosity 

less than the critical porosity, which is 36–40 percent for wellrounded granular packs (Nur and others, 1998), the effect of 

porosity on velocity ratio for consolidated sediment is insignificant. Murphy and others (1992) also showed that the pore 

fluid contribution to velocity ratio is insignificant at low porosity. Because the Biot coefficient is a function of differential 

pressure at a given porosity, equation 7 could accurately model 

the velocity ratio with respect to differential pressure. However, it is not suitable for the prediction of velocity ratio with 

respect to porosity changes. 

On the other hand, BGTL predicts the following velocity 

ratio:

[ n

n

n

ma

ma

s

p

G

G

G

k

V

V

2 2

2 2

2 2

2

3 (1 )

41 (1 )

3 (1 )

4

φ

φ

μ φ −

− − + − = + 













 (8)

This equation indicates that the velocity ratio does not depend 

on the Biot coefficient directly. This implies that, in this formulation, whether the sediment is consolidated or not, velocity 

ratio is only a function of porosity. Therefore, it is not suitable 

to model velocity ratio with respect to differential pressure, 

but this equation accurately predicts velocity ratio with respect 

to porosity.

Velocity Ratio 

]

3
Modeling the Velocity Ratio

In order to model the velocity ratio with respect to the 

differential pressure and porosity, Biot coefficients shown in 

Appendix A are used with the parameters shown in table 

1. Appendix A shows two examples in which the Biot coefficient depends on differential pressure and presents general 

Biot coefficients applicable to consolidated and unconsolidated 

sediments. The modeled velocity ratios are shown in fi gure 1 

as solid and dotted lines. As mentioned previously, there is no 

difference between BGT and BGTL for a dry rock. Therefore, 

both theories accurately predict velocity ratios for dry rocks 

irrespective of the degree of consolidation, and velocity ratios 

depend only on the matrix material because M is negligible 

in equation 7.

the trend of dry rock as BGT predicts. For unconsolidated 

sediments with porosities greater than 40 percent, BGTL with 

n = 1 agrees well with the measured values. For porosities 

less than 40 percent, BGTL with n = 1⁄2 and n = 1 performs 

equally well. The scattering of velocity ratios shown in fi gure 2 

is partly due to the clay content of the samples. Figure 3 shows 

the measured velocity ratios using Han’s data (Han and others, 

1986) at the differential pressure of 30 MPa with respect to 

porosity and clay volume content. Predicted velocity ratios 

from BGT and BGTL are also shown in figure 3. It appears 

that, for clay content of less than 10 percent, BGTL with 

n = 1⁄2 is applicable; for high clay content, BGTL with n = 1 

performs better. 

Table 1. Elastic constants used for velocity model.

[From Helgerud and others (1999). Shear modulus and bulk modulus reported 

in Gpa; density reported in g/cm3] 

Value used 

Shear modulus of quartz 45 

Bulk modulus of quartz 36 

Shear modulus of clay 6.85 

Bulk modulus of clay 20.9 

Bulk modulus of water 2.29 

 Density of quartz 2.65 

 Density of clay 2.58 

For wet sediments, BGTL with n = 0.5 accurately predicts 

the magnitude of velocity ratio for consolidated sediments 

(Gregory data), but it fails to predict an increase in velocity 

ratio with decreasing differential pressure. Note that G = 1 

is used unless it is explicitly mentioned in this paper. BGT 

predicts the trend of velocity ratio with respect to differential 

pressure, but its magnitude is much higher than observed. 

On the other hand, BGT accurately predicts velocity ratios 

for unconsolidated sediments with respect to differential pressure, whereas BGTL with n = 1 fails to predict the velocity 

ratio. Constant values of velocity ratios calculated from BGTL 

irrespective of differential pressure are due to the fact that 

the variation of porosity changes with respect to differential 

pressure is ignored. Figure 1 also indicates that the velocity 

ratio for wet sediment depends on porosity. However, it is 

difficult to establish a relationship, partly because of the difference in consolidation and partly because only two porosities 

are shown. 

Figure 2 shows velocity ratios calculated from published 

data, mostly complied by Castagna and others (1985), with 

predicted ratios from BGT and BGTL. Figure 2 suggests that, 

for all porosity ranges, the velocity ratio increases as porosity 

increases except for all nine gas-sand samples, which follow 

POROSITY

n

n

Vp / Vs

-

Figure 2. Measured velocity ratios with respect to porosity for 

data complied by Castagna and others (1985) and gas samples from 

Domenico (1977). The predicted velocity ratio from BGT is represented 

by a solid line, BGTL with n = 0.5 by a dashed line, and BGTL with 

n = 1 by a dotted line. Equations 7 and 8 and the values shown in table 

1 are used for predicted velocity ratios. Data complied by Castagna 

and others (1985) are measured using frequencies ranging from 200 

kHz to 2 MHz.

4 Velocity Ratio and its Application to Predicting Velocities
Based on figures 1, 2, and 3, the following general observations can be made.

1. For water-saturated and consolidated sediments, BGTL 

with n = 1⁄2 is appropriate for modeling the velocity 

ratio and is preferred to BGT.

2. For water-saturated unconsolidated sediments, BGTL 

with n = 1 is appropriate and is preferred to BGT.

3. BGT predicts accurate velocity ratios for dry rock and 

models well the effect of the differential pressure for 

unconsolidated sediments.

 4. The exponent n appears to increase as clay content 

increases. However, it is shown that the effect of clay 

can be treated better by G.

Modeling Velocities

Figure 4 shows theoretical predictions of velocities with 

respect to differential pressure using the data given by Gregory 

(1976) and Domenico (1977). For dry rock, there is no difference between BGT and BGTL, as mentioned earlier. To

compute velocities, the Biot coefficient shown in Appendix 

A (equation A-3 for Domenico data and equation A-4 for 

Gregory data) with parameters shown in table 1 is used. Figure 

4A shows the result for the unconsolidated sediment investigated by Domenico (1977). For the dry rock, P-wave velocities by BGT are almost identical to measured velocities and 

computed S-wave velocities are slightly higher than measured 

velocities. For the wet rock, BGT (dotted line) slightly underestimates both P- and S-wave velocities, but it predicts the 

velocity trend with respect to differential pressure very accurately. On the other hand, BGTL with n = 1 (dash-dot-dash 

line) predicts accurate P-wave velocities for a differential pressure of less than 10 MPa, but it underestimates the P-wave 

velocity at higher differential pressure—the underestimation 

of velocity increases as the differential pressure increases. For 

S-wave velocity, BGTL accurately predicts the velocity only 

at a differential pressure of about 15 MPa; it overestimates 

S-wave velocity for differential pressure less than 15 MPa; 

and it underestimates S-wave velocity for differential pressure 

greater than 15 MPa. 

Figure 4B shows similar results for the consolidated sediment investigated by Gregory (1976). For dry rock, BGT predicts accurate P-wave velocity at high differential pressure 

(greater than 30 MPa), but it overestimates S-wave velocity for 

differential pressure greater than 10 MPa. BGTL with n = 0.5 

underestimates both P- and S-wave velocities.

Results shown in figure 4 confirm that BGT is effective

in predicting velocity changes with respect to differential pressure (equation 7). This result agrees with Hornby and Murphy 

(1987), who demonstrated that BGT accurately predicts the 

effect of effective pressure on the velocity of fully water saturated unconsolidated sand. However, BGTL is not suitable to 

predict the effect of differential pressure, as shown in equation 

8. Results denoted by heavy dashed lines in figure 4 will be 

discussed later.

Discussion

Fluid Effect on Shear Velocity

In poroelasticity, it is usually assumed that the fl uid does 

not affect the shear modulus of rock unless the viscosity is 

high. Therefore the shear modulus of dry rock and watersaturated rock are the same, and BTG uses the same shear 

modulus for dry and wet rocks. However, BGTL incorporates 

a dependence of fluid property in the shear modulus through 

modulus M, which measures the variation in hydraulic pressure needed to force an amount of water into the formation 

without any change in formation volume (Krief and others, 

1990). This dependence of fluid on shear velocity comes from 

the assumption shown in equation 6. Equation 5 indicates that 

the shear modulus depends on the bulk modulus, which, in 

turn, depends on the properties of the pore fluid in the sedin Cv

n Cv

n Cv

n

POROSITY

Vp / Vs

Cv

Figure 3. Measured velocity ratios with respect to porosity and clay 

volume content (Cv) for the data of Han and others (1986) at the differential pressure of 30 MPa. The central frequencies of transducers 

used for P- and S-waves are 1.0 and 0.6 MHz, respectively. Equations 

7 and 8 and the values shown in table 1 are used for the predicted 

velocity ratios.

Discussion 5
ment. On the other hand, BGT indicates that bulk modulus 

is related to the shear modulus through the Biot coeffi cient, 

which is independent of pore fl uid. 

Equation 5 indicates that, when n is greater than about 

0.25 and the porosity is less than the critical porosity, the shear 

modulus computed from BGTL is less than that from BGT. 

In other words, the formation becomes softer. The degree of 

softness increases with increasing n. If n is near 0.25, the 

difference between BGT and BGTL is negligible. 

According to Biot (1956), the fluid actually affects 

S-wave velocity through coupling between the pore fl uid and 

rock frame. The coupling constant, χ, appears in the following 

equation:

The usual S-wave velocity or velocity of BGT is given 

by equation 9 at zero frequency. This kind of dependence 

of S-wave velocity on the presence of fluid is different from 

the result of BGTL. Note that the S-wave velocity is always 

greater than the S-wave velocity at zero frequency; the pore 

fluid increases the S-wave velocity as the coupling constant 

decreases. 

One of the assumptions in Gassmann’s equation and 

implicit in the BGT is that the pore fluid does not interact with 

the solid in a way that would soften or harden the frame. In 

reality, the pore fluid will inevitably interact with the rock’s

solid matrix to change the surface energy. When a rock is 

saturated by a fluid, the fluid may either soften or harden 

the matrix—a shaly sandstone is often softened because of 

clay swelling (Wang, 2000). Therefore, the exponent n could 

be interpreted as a parameter controlling the degree of the 

softening/hardening effect of fluid on the formation. One problem with this interpretation is that the softening/hardening 

occurs only for the shear modulus, not for the bulk modulus. 

Therefore, this may not be a problem for the velocity ratio 

(i.e., equation 8), but it may produce asymmetrical error in 

predicting P- and S-wave velocity from the porosity.

Another interpretation of fluid effect on S-wave velocity 

n n

n n

DIFFERENTIAL PRESSURE, IN MEGAPASCALS

A B VELOCITY, IN KILOMETERS PER SECOND VELOCITY, IN KILOMETERS PER SECOND

Figure 4. Measured and modeled velocities based on BGT and BGTL with respect to differential pressure. For elastic moduli, the values shown 

in table 1 are used and the Biot coefficient is computed using Appendix A, equations A-3 and A-4. A, Unconsolidated sediments measured by 

Domenico (1977). B, Consolidated sediments measured by Gregory (1976).

1/ 2

[1 /( )]















− = ρ ρ φ ρχ

μ

fl

Vs (9)

where the coupling constant, χ, ranges from 1 for no coupling 

at infinite frequency to ∞ for perfect coupling at zero frequency.

6 Velocity Ratio and its Application to Predicting Velocities
2

is velocity dispersion. BGT is a theory for low frequency, 

so there is no interaction between fluid and solid in rocks 

because fluid and solid in rocks moves in phase. However, at 

high frequencies, fluid interacts with the solid in such a way 

that velocity increases as the frequency increases (i.e., velocity 

dispersion) (Dvorkin and Nur, 1992; Dvorkin and others, 1993, 

1994; Mavko and Jizba, 1993). In Biot’s model, pore fluid 

participates in a solid’s motion by viscous friction and inertia 

coupling (Biot’s flow), and a small amount of velocity dispersion occurs for frequencies of less than 1 MHz (Dvorkin and 

others, 1993). A different mechanism of fluid flow (the squirt 

flow), perpendicular to wave propagation, is associated with 

the squirting of pore fluid out of cracks and results in high 

velocity dispersion for frequencies greater than sonic frequencies (kHz range). As apposed to BGT, BGTL is based on the 

velocity ratio (measured at any frequency), so its formulation 

implicitly includes the effect of velocity dispersion. 

Scale G

The accuracy of BGTL depends on how accurately the 

observed Vp/Vs of sediments agrees with the velocity ratio of 

matrix material at zero porosity. Many factors, such as the 

aspect ratio of pore space, clay volume content, degree of 

compaction, and differential pressure, affect the velocity ratio 

and cause discrepancy between predicted and observed velocities. In BGTL formulation, the exponent n accounts for the 

effect of differential pressure and the degree of consolidation/ 

compaction and scale G attempts to account for the effect 

of clay (or gas hydrate) on velocity. The effect of clay on 

velocity is partially corrected by using Hill’s (1952) averaging 

method of elastic moduli. The other effect of clay on velocity 

by BGTL comes from the fact that the shear modulus of the 

matrix used for the derivation of the Biot coefficient (which 

used a clean sandstone with the shear modulus shown in table 

1) is different from the shear modulus of matrix of the sediment, such as a shaly sandstone. Because the Biot coefficient 

depends on the shear modulus of the matrix, a different Biot 

coefficient should be used for different matrix material. Also, 

the presence of clay in the matrix reduces the overall aspect 

ratio of the pore space (Xu and White, 1996) and affects 

the velocity ratio. Introducing a scale, G, in the formulation 

of BGTL is an attempt to correct the discrepancy between 

observed and calculated velocity ratios. There are two ways 

of implementing G in BGTL formulation. One approach is to 

use the same Biot coefficient that was derived from a clean 

sandstone (such as in equation A-6) for shaly or gas-hydratebearing sediments. Numerically this is the same as treating 

αG as a modified velocity ratio of matrix material. The other 

method is to use a different Biot coefficient depending on G. 

A different Biot coefficient for shaly sandstone can be 

derived as follows. 

The equation for β can be written, at zero porosity, as 

μ ≈ μ ma (1 − β )G2 (10) 

Discussion 7 

If β1 is used as β computed with G = 1 and βG is the β

computed with G other than 1, the Biot coefficient with G 

other than 1 is given by 

β G ≈ 1− (

1− β1 ) (11)

G2

Introducing different Biot coefficients depending on G 

in BGTL results in computed velocities that are closer to 

measured velocities, but it also limits the application of the 

theory. Because the frame shear modulus cannot be greater 

than the shear modulus of the matrix, the lower limit of β

is zero. Usually G is less than 1, so equation 11 yields β

values less than 0, which is not physically possible. The porosity that yields negative β using equation 11 is defined as a 

cut-off porosity, φco. Therefore, the use of equations 3 and 5 

to compute the moduli of sediments is limited to porosities 

greater than φco. For example, when G = 0.8, a negative βG 

results when β is less than (1–G2) or 0.36. Therefore, from 

equation A-6, the cut-off porosity is about 5 percent. In other 

words, BGTL given in equations 3 and 5 is correct only for 

porosities greater than about 5 percent. At the cut-off porosity, 

the Biot coefficient is zero and the following relation exists. 

k = kma = ksk

kmaμ maG2(1−φco ) 2n 

μ = kma + 4μ ma [1− G2

(1−φco )

2n

]/ 3 ≈

(12) 

μ maG2(1−φco )

2n when G ≈ 1 

It is shown that, at the cut-off porosity, the bulk modulus of the 

formation is the same as the bulk modulus of the matrix or the 

skeleton, but the shear modulus of the formation is less than 

the shear modulus of the matrix. 

At φ = 0, the modulus, M, is given by M = kma / β. 

This modulus is negative when β is negative, so there is no 

physical meaning to M. However, when M is inserted into 

equation 5, the shear modulus is given by μ ≈ G2μma at φ = 0 

and G ≈ 1.0. This relationship with equation 12 indicates 

that the shear modulus reasonably increases when the porosity 

decreases from the cut-off porosity to zero porosity. Therefore, 

for this porosity range (zero to cut-off porosity), the shear 

modulus of the formation has a physically realizable value 

and S-wave velocity can be predicted from the BGTL. It is 

proposed that the shear-wave velocity can be obtained beyond 

the cut-off porosity by interpolating the shear modulus at φ

= 0 and φ = φco. It is noted that the bulk modulus cannot 

be interpolated between φ = 0 and φ = φco because, at both 

porosities, the bulk modulus is given by k = kma. 

The difference between the two approaches can be examined from figure 5. Figure 5 indicates that both velocities, Vp 

and Vs, predicted from the first approach are lower than veloci-
ties using G = 1. However, velocities predicted from second 

approach show that P-wave velocity with G = 0.9 is almost 

the same as that computed with G = 1 and Vs is lower than 

that with G = 1. Both approaches yield the same velocity 

ratios, as shown in figure 5. The first approach compensates 

for the discrepancy between measured and calculated velocity 

ratio by lowering both Vp and Vs, whereas the second approach 

compensates primarily by lowering Vs. Both approaches can 

be applied to analyze observed data. In the case that BGTL 

overestimates both Vp and Vs, the first approach is preferable. 

On the other hand, in the case that BGTL predicts accurate 

P-wave velocity but overestimates S-wave velocity, the second 

approach is suitable. All examples in this paper are calculated 

using the first approach of implementing G in BGTL.

The magnitude of G depends on the matrix. The more the 

shear modulus of the matrix deviates from the shear modulus 

of clean sandstone, the smaller G becomes. For a clean sandstone, G is close to 1 because the shear moduli of various kinds 

of quartz are similar to each other and to the value given in 

table 1. Based on first-approach numerical experiments on the 

data of Han and others (1986), it is suggested that

for consolidated sediments having an average clay volume 

content of 15 percent. For practical applications of BGTL, 

G may be treated as a free parameter to fit the observation. 

For unconsolidated sediments, data at the Mallik 2L-38 well 

indicate that G becomes small as the gas-hydrate concentration 

increases; G is close to 0.85 for sediments with high gashydrate concentration (more than 80 percent).

Analysis of Exponent n

The exponent n is a free parameter in BGTL that can be 

chosen to fit the measured data. In the case that BGTL is used 

to predict S-wave velocity from porosity and P-wave velocity, 

an optimum n can be estimated from the P-wave data by 

fitting the observation to the BGTL. However, if the purpose is 

modeling of P- and S-wave velocities from porosity, n should 

be known beforehand.

General trends of the exponent n can be inferred from 

figures 1, 2, and 3 and from data presented by Han and 

others (1986). Analysis of Han’s data and figures 1, 2, and 3 

suggest that (1) as the consolidation increases, the exponent n

decreases. (2) as the clay content increases, n increases. and 

(3) as the differential pressure decreases, n increases. 

As can be inferred from equation 8, as n increases, 

Vp/Vs increases. An increase of Vp/Vs is related to the consolidation of the sediment and also to clay content. Therefore, 

a decrease of n with increasing consolidation agrees with 

the observation of Gregory (1976) that Vp/Vs increases as 

consolidation decreases. The effect of clay can be modeled by 

using a clay-dependent exponent, but it is more effective to use 

a scale G, as discussed in the previous section. 

In order to derive a functional relation between the exponent and the differential pressure for clean sandstone, results 

shown in figure 4 were carefully examined. By fi tting the 

predictions of BGTL with different exponents to the velocities 

of unconsolidated wet sediment at different differential pressures (fi gure 4A), the following equation is derived for the 

exponent:

G

Cv

G g

g g G2

POROSITY

VELOCITY, IN KILOMETERS PER SECOND

VELOCITY RATIO, 

Vp / 

V

V s

p / Vs

Figure 5. Graph showing the effect of scale G on velocities. Note 

that the second approach of implementing G in BGTL changes P-wave 

velocity slightly, but S-wave velocity changes significantly.

/ 0.06714 0.9552 0.0448 Cv G e− = + (13)

/17.78 0.67 0.77 p n e− = + (14)

where 

p is the differential pressure in MPa.

Based on equation 14 and the above observations, a general 

formula for exponent n as a function of differential pressure 

and degree of consolidation is proposed as follows:

n [ e m p 0.67 0.77 / − /17.78 = + (15)

where 

m is a constant incorporating the effect of consolidation. 

8 Velocity Ratio and its Application to Predicting Velocities

]
Because equation 14 is derived from data on unconsolidated 

sediment (Domenico, 1977), m = 1 is appropriate for unconsolidated sediment and m = 3 is appropriate for consolidated 

sediment at high differential pressure, as demonstrated in the 

following section. 

The result of using n as a function of differential pressure 

is shown in fi gure 4A as heavy dashed lines. As indicated 

in fi gure 4A, BGTL with a variable n, which is computed 

from equation 15 with m = 1, accurately predicts both P- and 

S-wave velocities. In order to model the effect of differential 

pressure on the velocity of consolidated sediment (Gregory,

1976), an exponent calculated with m = 2 is tried and the result 

is shown in fi gure 4B. (Note that m = 2 instead of m = 3 

is used for the Gregory data. The reason is that the Biot coefficient calculated for the Gregory data is higher than the one 

calculated from the Raymer and others (1980) data (Appendix 

A). This implies that the sample used by Gregory (1976) is 

less consolidated than the sample used by Raymer and others 

(1980)). Figure 4B indicates that BGTL with the variable 

exponent, shown as heavy dashed lines, accurately predicts 

the effect of differential pressure on velocities, albeit predicted 

P-wave velocities are slightly lower than measured velocities.

The above estimation of n is based on sediment having a 

porosity of 0.382 (Domenico data), and it also appears to be 

reasonable for sediment having a porosity of 0.217 (Gregory 

data). Does this hold for sediments with other porosities? The 

next section investigates this question. 

Application to Velocity Prediction

In order to see the effectiveness of a variable n defi ned 

in equation 15, data by Han and others (1986) are used. Han’s 

data includes porosities that range from 2 to 30 percent and Cv

that ranges from 0 to 51 percent. Both BGT and BGTL can be 

used to predict velocities from porosity, and either can be used 

to predict S-wave velocity from porosity and P-wave velocity. 

With BGTL, the S-wave velocity can be predicted from porosity and P-wave velocity using the following formula, which is 

derived from equations 1 and 4: 

4 / 3

(1 )

2 2

ma ma

p

n

ma

k

V

μ

μ φ ρ

μ

+

− = (16)

MEASURED VELOCITY, 

IN KILOMETERS PER SECOND

MEASURED S-WAVE VELOCITY, 

IN KILOMETERS PER SECOND

A B

PREDICTED S-WAVE VELOCITY, 

IN KILOMETERS PER SOCOND

MEASURED VELOCITY, IN KILOMETERS PER SECOND

Figure 6. Measured velocities by Han and others (1986) at the differential pressure of 30 MPa and predicted velocities from BGT and BGTL 

using elastic moduli shown in table 1. A, S-wave velocity predicted from measured P-wave velocity and porosity. For BGT, the Biot coefficient 

is calculated from Appendix B. B, P- and S-wave velocities predicted from porosity. Biot coefficient by Raymer and others (1980)—equation 

A-6 is used.

Discussion 9
In BGT, the S-wave velocity can be derived from the 

P-wave velocity cient. Therefore, in 

order to predict S-wave velocity from P-wave velocity and 

porosity using BGT, equation B-1, shown in Appendix B, is 

solved for the Biot coefficient using the measured P-wave 

velocity. In the following examples, when BGT is used to 

predict S-wave velocity from P-wave velocity and porosity, the 

Biot coefficient calculated from Appendix B is used. When 

predicting velocities from porosity only, the Biot coeffi cient 

shown in equation A-6 (Appendix A) is used because the Biot 

coefficient cannot be calculated without velocity data.

Figure 6A shows S-wave velocities predicted from 

P-wave velocities and porosities using Han’s data at 30 MPa 

based on BGTL and BGT. A variable exponent n from equation 15 with m = 3, p = 30 MPa, scale G given in equation 

13, and the values shown in table 1 are used for BGTL. The 

fractional S-wave velocity error (∆Vs / Vs) from BGTL is 

0.01±0.03, whereas it is 0.06±0.03 from BGT.

Figure 6B shows P- and S-wave velocities predicted using 

only porosities. Figure 6B indicates that both BGTL and 

BGT predict reasonable velocities, but velocities predicted 

from BGTL are more accurate. Overall, BGT slightly overestimates the velocities and BGTL slightly underestimates velocities. The fractional errors for P-wave velocities predicted from 

BGTL and BGT are –0.01±0.02 and 0.03±0.03, respectively, 

indicating the same amount of error for both methods. However, the fractional errors for S-wave velocities predicted from 

BGT and BGTL are 0.09±0.05 and 0.00±0.05, respectively, 

indicating more accurate S-wave velocities from BGTL. 

Figure 7 shows predictions using a different set of data 

compiled by Castagna and others (1985). As pointed out by 

Koesoemadinata and McMechan (2001), the data were missing 

some explicit auxiliary information. It is assumed that the data 

were acquired at a differential pressure of 50 MPa and the 

average clay volume content is 15 percent. Equation 15 with 

m = 3 and p = 50 MPa yields n = 0.25. Using table 1, 

n = 0.25, and G given by equation 13, S-wave velocities 

predicted from BGT and BGTL, with predictions by Castagna 

and others (1985), are shown in fi gure 7A. All three predictions are similar, but the predictions by Castagna and others 

(1985) are the most accurate. However, their accuracy was 

achieved by using the wet bulk modulus calculated from 

both P- and S-wave velocities. Predicted P- and S-wave velocities from BGTL, shown in fi gure 7B, are more accurate 

than those predicted from BGT. The fractional errors for Pand S-wave velocities predicted from BGTL are 0.03±0.11 

A B

PREDICTED S-WAVE VELOCITY, 

IN KILOMETERS PER SECOND

MEASURED S-WAVE VELOCITY, 

IN KILOMETERS PER SECOND

MEASURED VELOCITY, 

IN KILOMETERS PER SECOND

PREDICTED VELOCITY, IN KILOMETERS PER SECOND

Figure 7. Predicted and measured velocities for data compiled by Castagna and others (1985). For the prediction, a differential pressure of 50 

MPa and a clay volume content of 15 percent are assumed using elastic moduli shown in table 1. A, S-wave velocity predicted from measured 

P-wave velocity and porosity. For BGT, the Biot coefficient is calculated using Appendix B. B, P- and S-wave velocities predicted from porosity. 

Biot coefficient by Raymer and others (1980)—equation A-6 is used.

10 Velocity Ratio and its Application to Predicting Velocities

through the Biot coeffi
and 0.06±0.24, respectively, whereas they are 0.07±0.11 and 

0.17±0.25, respectively, for velocities predicted from BGT. 

More scattering in figure 7B compared to figure 6B may be 

caused by using average values for differential pressure and 

clay volume content. 

Velocity Dispersion and Velocity Ratio 

The velocity ratio mainly depends on porosity, differential 

pressure, clay content, and frequencies of measurements. It 

has been known that elastic wave velocities in water-saturated 

sediments are dispersive (cf. references in Mavko and others, 

1998), so the velocity ratio would be dependent on the frequency of measurement. In the BGTL formulation, the exponent n incorporates the effect of differential pressure and the 

scale G compensates for the effect of clay on the velocity ratio. 

However, the proposed BGTL does not include the effect of 

velocity dispersion. The exponent n and scale G were based on 

the data by Gregory (1976) measured at 1 MHz, by Domenico 

(1977) measured at about 0.5 MHz (pulse width of 2 μs), 

and by Han and others (1986) measured at 1 MHz for the 

P-wave velocity and at 0.6 MHz for the S-wave velocity. The 

question is how much error is introduced by ignoring velocity 

dispersion in BGTL. 

In order to examine the magnitude of error in BGTL 

by ignoring the velocity dispersion, an empirical formula by 

Koesoemadinata and McMechan (2001), who used a variety 

of data measured with frequency ranges from 380 Hz to 1 

MHz, is used. Their analysis indicates that the dispersive part 

of P-wave velocity is 0.338f, where f is the frequency in MHz; 

it is 0.368f for the S-wave velocity. The P-wave velocity of a 

sandstone having a porosity of 15 percent is 4.237 km/s, and it 

is 2.532 km/s for the S-wave at 0 Hz and p = 40 MPa. These 

velocities are 4.575 km/s and 2.9 km/s for P- and S-wave 

velocity, respectively, at 1 MHz. Therefore, Vp/Vs at 0 Hz is 

1.673 and is 1.578 at 1 MHz, which is about a 6 percent 

decrease in Vp/Vs going from 0 Hz to 1 MHz. However, velocity itself has a larger error, namely 8 percent for the P-wave 

velocity and 15 percent for the S-wave velocity. 

Velocity dispersion requires that the exponent n should 

be dependent on frequency as well as differential pressure to 

accurately predict velocities. If the exact amount of velocity 

dispersion is known, the exponent n can be adjusted to fit 

the measurements. However, quantifying velocity dispersion 

is difficult. The velocity dispersion in sediments is very important and yet poorly understood (Wang and Nur, 1990). Note 

that Wang and Nur (1990) stated “We still do not know exactly 

how much velocity dispersion occurs in fluid-saturated rock 

from seismic to laboratory ultrasonic frequencies.” McDonal 

and others (1958) observed that there appears to be no detectable dispersion of velocity with frequency in consolidated 

sedimentary rocks at frequencies of less than 1 MHz. Blangy 

and others (1993) analyzed Troll sandstone and concluded that 

ultrasonic measurements can be used directly for detailed seismic work without correction for frequency dispersion, probDiscussion 11 

ably because of high permeability of Troll sands and the lack 

of grain-scale local flow effect (Mavko and Jizba, 1991). 

Therefore, the proposed BGTL is optimum for velocities 

measured at ultrasonic frequencies and for sandstones with 

high permeability without grain-scale local flow; it is also optimum for consolidated sediments with no appreciable velocity 

dispersion. Because the exponent n is based on ultrasonic frequency, the appropriate n for velocities measured at frequencies less than 1 MHz would be a little larger than n calculated 

from equation 15. 

Error Analysis 

For a given Biot coefficient, predicted errors in BGT 

come from the error in the calculation of matrix material. In 

BGTL, in addition to error in the calculation of matrix material, errors depend on exponent n and scale G . The error in 

the predicted S-wave velocity from BGTL can be written in the 

following form using equation 6: 

∆Vs ∆V ∆α ∆n ∆G = p +

α

+ n ln(1−φ) + (17)

Vs Vp n G

When the measured P-wave velocity is used for S-wave prediction, the first term can be ignored. Therefore, the fractional 

error in S-wave velocity is proportional to the fractional error 

in the matrix material, to the fractional error in scale G, and 

is linearly proportional to the fractional error in the exponent. 

The error caused by n depends on the magnitude of n and 

porosity and increases as the porosity and exponent increase. 

For consolidated sediments, n is usually 0.5. Assume that a 

fractional error in n is ±20 percent. Then the fractional error in 

S-wave velocity using BGTL is about ±4 percent for a porosity 

of 30 percent and ±1.6 percent for a porosity of 10 percent. 

The average porosity reported in the data of Han and others 

(1986) is 15 percent, and the fractional S-wave velocity error 

from BGTL is less than 1 percent (figure 6A). Therefore, the 

exponent used for the data of Han and others (1986) at 30 MPa 

has an error of about 12 percent, assuming no other errors. 

The error in the matrix can be reduced by a proper use of 

elastic moduli of the matrix material based on well-log analysis, laboratory measurements, or published data. However, the 

error associated with n and G is more complicated and difficult 

to estimate because optimum n and G are difficult to compute 

without velocity data. Therefore, a judicious choice of G and 

n, guided from equations 13 and 15, is important to accurately 

predict velocities. 
12 Velocity Ratio and its Application to Predicting Velocities 

Conclusion 

The classical Biot-Gassmann theory (BGT) accurately 

predicts the effect of differential pressure on velocities or 

the velocity ratio. However, explicit Biot coefficients as a function of differential pressure are required to predict velocities. 

Without an explicit Biot coefficient for a particular data set, the 

Biot coefficient given by Raymer and others (1980) provides 

an accurate P-wave velocity at differential pressures higher 

than about 30 MPa. Because the velocity ratio derived from 

BGT is not sensitive to porosity variation at low porosities, it is 

not suitable to model the velocity ratio with respect to porosity. 

BGT is easy to use, but it usually overestimates velocities, 

particularly the S-wave velocity. 

Biot-Gassmann theory by Lee (BGTL), which is formulated under the assumption that the velocity ratio is a function 

of porosity (φ) such that (1–φ)

n accurately predicts velocity 

ratio or velocities with respect to porosity, but not with respect 

to differential pressure. In order to overcome this problem, a 

variable exponent n as a function of the differential pressure, 

which is estimated from measured velocities of unconsolidated 

sediment, is proposed and applied to the consolidated sediment. However, velocity dispersion is not included in the 

BGTL formulation. This investigation suggests that BGTL is 

preferable to BGT in most cases. 

One of the consequences of the BGTL formulation is the 

dependence of S-wave velocity on pore fluid. This coupling of 

pore fluid to the S-wave velocity is interpreted as a manifestation of the effect of fluid interaction on the solid matrix. 

The prediction of S-wave velocity based on BGTL 

requires properties of matrix material, including clay content, 

porosity (or porosity and P-wave velocity), the Biot coefficient 

such as proposed by Raymer and others (1980) or Lee (2002), 

and differential pressure. The accuracy of prediction of velocity increases as differential pressure increases because the error 

associated with the exponent decreases as n decreases or as 

differential pressure increases. 

Exponent n is based on data measured at ultrasonic frequencies—the proposed BGTL is optimum for velocity data 

measured in the range of 1 MHz. However, the effect of velocity dispersion could be incorporated by using a frequencydependent n if the exact amount of velocity dispersion is known. 

The application of BGTL is more complex than the application of BGT because the exponent n depends on many factors. A judicious choice of n and G is essential to accurately 

predict velocities, and feasible n and G values can be estimated 

from the general guidelines and examples presented in this 

paper. 

Acknowledgments 

I thank Philip Nelson, John Miller, and Carol Molnia for 

many helpful comments for the earlier version of the paper. 

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Appendix A

Biot Coefficient

Wave velocity in a dry rock depends on differential 

pressure as well as porosity. Therefore, the Biot coeffi cient 

depends on porosity and differential pressure. Two examples 

of Biot coefficients depending on differential pressure are calculated from measured velocities.

The Biot coefficient can be calculated from equations 3 

and 4 using the following equation:

For hard formations, the equation by Raymer and others 

(1980), which is written in the following form by Krief and 

others (1990), can be used.

ma

d pd sd

k

(V 4V / 3) 1

2 2 − = − ρ β (A-1)

or

ma

dVsd

μ

ρ β

2

= 1 − (A-2)

In equations A-1 and A-2, ρd, Vpd, Vsd, are the density of dry 

rock, P-wave velocity of dry rock, and S-wave velocity of dry 

rock, respectively.

Figure A-1 shows calculated Biot coeffi cients using 

unconsolidated sands measured by Domenico (1977) with 

respect to differential pressure. Equations A-1 and A-2 yield 

slightly different Biot coefficients, so the average is taken as 

the Biot coefficient in this study. The porosity of the dry rock 

sample is 0.3817 at 2.7 MPa and decreases to 0.3672 at 34 

MPa. A least-squares fitting curve to the calculated Biot coefficient is also indicated and is given by

/ 36.61 0.92767 0.0635 p e− β = + (A-3)

where 

p is the differential pressure in MPa.

Figure A-1 also shows calculated Biot coeffi cients using 

consolidated sediment having a porosity of 0.217, which was 

measured by Gregory (1976). A least-squares fitting curve is 

given by

–p

-

-

DIFFERENTIAL PRESSURE, IN MEGAPASCALS

BIOT COEFFICIENT, 

–p

Figure A-1. Computed Biot coefficients from velocities of dry samples 

and least-squares fits to calculated coefficients. The computed Biot 

coefficient is the average of Biot coefficients computed from P- and 

S-wave velocities. 

/ 27.41 0.65948 0.23256 p e− β = + (A-4)

In the case that there is no measured Biot coeffi cient, 

the following Biot coefficient can be used. For unconsolidated 

sediments, the following Biot coefficient is suggested (Lee, 

2002) 

0.99494

1

184.05

( 0.56468)/ 0.10817 +

+

− = φ + β

e

(A-5)

3.8 β = 1− (1−φ) (A-6)

The Biot coefficient predicted from equation A-5 is 0.966 

at a porosity of 0.382. The computed Biot coeffi cient using 

equation A-5 agrees with the measured Biot coeffi cient from 

the Domenico (1977) data around the differential pressure 

of 18 MPa. Therefore, the Biot coefficient proposed by Lee 

(2002) is adequate for a differential pressure of about 20 MPa. 

For consolidated sediments with a porosity of 0.217, equation 

A-6 yields β = 0.605. The measured Biot coeffi cient from 

Gregory (1976) data is always greater than that predicted by 

equation A-6 for all ranges of differential pressure. This may 

suggest that the sample used by Gregory is less consolidated 

than the sample used by Raymer and others (1980). 

14 Velocity Ratio and its Application to Predicting Velocities
Appendix B 15 

Appendix B 

Prediction of S-Wave Velocity Based on BGT 

P-wave and S-wave velocities are related through the 

Biot coefficient as shown in equations 1 through 3. If these 

equations are solved using measured P-wave velocities, the 

following quadratic equation results: 

aβ2 + bβ + c = 0 (B-1) 

where 

a = 4μ mak fl / 3 , 

b = ρk flVp

2 − kma [k fl(1+φ − kmaφ]− 4μ ma [k fl(1+φ − kmaφ]/ 3 , and 

c = φ(k fl − kma )(−ρVp

2 + kma + 4μ ma / 3) . 

The computed Biot coefficient from equation B-1 with 

equation 4 can be used to calculate the S-wave velocity. 

Manuscript approved for publication January 30, 2003 

Published in the Central Region, Denver, Colorado 

Editing, layout, photocomposition—Richard W. Scott, Jr. 

Graphics—Myung W. Lee