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Proposed Moduli of Dry Rock and

Their Application to Predicting

Elastic Velocities of Sandstones

By Myung W. Lee

Scientific Investigations Report 2005-5119

U.S. Department of the Interior

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

Gale A. Norton, Secretary

U.S. Geological Survey

P. Patrick Leahy, Acting Director

U.S. Geological Survey, Reston, Virginia: 2005

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Suggested citation:

Lee, Myung W., 2005, Proposed moduli of dry rock and their application to predicting elastic velocities of sandstones: 

U.S. Geological Survey Scientific Investigations Report 2005-5119, 14 p.
Contents

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

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

Theory ...................................................................................................... 2

Velocity Equation .................................................................................... 2

Moduli of Dry Frame .............................................................................. 2

Comparison with MacBeth Equation ............................................................ 4

Modeling ................................................................................................... 6

Velocities with Respect to Porosity ............................................................... 6

Velocities of Gas-Hydrate-Bearing Sediments ................................................ 7

Results and Applications ................................................................................. 9

Comparison with Other Moduli .................................................................. 9

Predicting S-Wave Velocity from P-Wave Velocity and Porosity ........................... 9

Consolidation Parameter α ........................................................................ 10

Conclusions ................................................................................................ 12

References Cited .......................................................................................... 13

Figures

 1–11. Graphs showing:

 1. Relation between consolidation parameter and velocity ratio ........................... 3

 2. Relation between consolidation parameter and Biot coefficient ........................ 4

 3. Relations between dry-rock properties with respect to porosity ........................ 5

 4. Comparison between predicted velocities of water-saturated sandstones ............ 6

 5. Comparison between predicted and measured velocities .............................. 7

 6. Calculated velocities for gas-hydrate-bearing sediments .............................. 7

 7. Velocities and gas-hydrate concentrations estimated at the Mallik 5L-38 well ...... 8

 8. Comparison between the S-wave velocities from P-wave velocity and

porosity for data from Han and others ...................................................... 10

 9. Comparison between the S-wave velocities from P-wave velocity and

porosity for well-log data at the Alpine-1 well ............................................. 11

 10. Measured and predicted S-wave velocities for dry rocks .............................. 11

 11. Estimated consolidation parameters from P-wave velocity and porosity ............ 12

Table

 1. Elastic constants used for this study ...................................................... 4
Abstract

Velocities of water-saturated isotropic sandstones under 

low frequency can be modeled using the Biot-Gassmann 

theory if the moduli of dry rocks are known. On the basis 

of effective medium theory by Kuster and Toksöz, bulk and 

shear moduli of dry sandstone are proposed. These moduli are 

related to each other through a consolidation parameter and 

provide a new way to calculate elastic velocities. Because this 

parameter depends on differential pressure and the degree of 

consolidation, the proposed moduli can be used to calculate 

elastic velocities of sedimentary rocks under different in-place 

conditions by varying the consolidation parameter. This theory 

predicts that the ratio of P-wave to S-wave velocity (Vp /Vs) 

of a dry rock decreases as differential pressure increases and 

porosity decreases. This pattern of behavior is similar to that 

of water-saturated sedimentary rocks. If microcracks are 

present in sandstones, the velocity ratio usually increases as 

differential pressure increases. This implies that this theory is 

optimal for sandstones having intergranular porosities. Even 

though the accurate behavior of the consolidation parameter 

with respect to differential pressure or the degree of consolidation is not known, this theory presents a new way to predict 

S-wave velocity from P-wave velocity and porosity and to calculate elastic velocities of gas-hydrate-bearing sediments. For 

given properties of sandstones such as bulk and shear moduli 

of matrix, only the consolidation parameter affects velocities, 

and this parameter can be estimated directly from the measurements; thus, the prediction of S-wave velocity is accurate, 

reflecting in-place conditions.

Introduction

In order to relate elastic velocities (P-wave velocity, Vp, 

and shear-wave velocity, Vs) to physical properties of sedimentary rocks, an accurate forward modeling is required. Elastic 

velocities are generally dispersive and depend on many factors 

such as porosity, differential pressure, and degree of consolidation. Biot (1956) developed theoretical formulas for predicting 

the frequency-dependent elastic velocities for water-saturated 

rocks for all ranges of frequency, and Geertsma and Smit 

(1961) analyzed the Biot equation for dilatational waves in 

the low-frequency range. Gassmann (1951) derived equations for elastic velocities of the fluid-saturated porous media 

at zero-frequency. The Gassmann (1951) theory predicts the 

bulk modulus of the fluid-saturated porous medium from the 

known bulk moduli of the solid matrix, the frame, and the pore 

fluid and assumes that the shear modulus of sedimentary rocks 

is not affected by fluid saturation. The essential input for the 

application of the Gassmann equation is the bulk and shear 

moduli of the dry frame, which are usually measured in the 

laboratory or calculated using theories such as a contact theory 

(Digby, 1981; Murphy and others, 1993; Winkler, 1983).

The velocity ratio, Vp/Vs , is a useful parameter to evaluate the characteristics of velocity models of dry rocks. Pickett 

(1963) demonstrated that Vp/Vs of the partially gas-saturated 

sediment are almost constant, irrespective of porosity. Based 

on this observation, Krief and others (1990) derived the shear 

modulus of dry rock, which is a simple function of the Biot 

coefficient similar to the bulk modulus predicted from the Biot 

theory. Krief’s theory predicts a constant Vp/Vs for a dry rock 

irrespective of porosity and differential pressure. The contact 

theory predicts that Vp/Vs decreases as differential pressure 

increases and is independent of porosity. The Kuster and 

Toksöz (1974) theory (KTT), which was developed based on 

wave scattering theory, provides moduli of the dry frame and 

predicts that Vp/Vs increases as porosity increases. 

Recently, Pride (2005) presented bulk and shear moduli 

(which are functions of porosity and a consolidation parameter) of dry sandstones. The consolidation parameter depends 

on differential pressure and the degree of consolidation. This 

theory predicts that the velocity ratio, Vp/Vs, decreases as differential pressure increases and porosity decreases. MacBeth 

(2004) derived an empirical relation of moduli with respect 

to differential pressure based on the theory by Sayers and 

Kachanov (1995) and demonstrated that, in general, Vp/Vs

decreases as differential pressure increases, mainly because of 

microcracks present in sandstones. 

An advantage of using moduli derived by Pride (2005) is 

that there is a consolidation parameter both in bulk and shear 

moduli, and this parameter can be treated as a free parameter in predicting velocities or matching observed velocities. 

Therefore, moduli recommended by Pride (2005) provide an 

accurate method of predicting S-wave velocities from P-wave 

Proposed Moduli of Dry Rock and Their Application to 

Predicting Elastic Velocities of Sandstones

By Myung W. Lee
velocity and porosity. Many approaches exist for predicting 

S-wave velocity. For example, (1) Greenberg and Castagna 

(1992) proposed a semiphysical model based on the BiotGassmann theory (BGT); (2) Xu and White (1996) predicted 

S-wave velocity by a combination of Kuster and Toksöz 

(1974) theory and differential effective medium theory (Cheng 

and Toksöz, 1979), using pore aspect ratios to characterize 

compliance of sand and clay components; (3) Lee (2003) 

predicted S-wave velocity by using the modified BGT under 

the assumption that Vp/Vs is a function of porosity. Also, many 

empirical methods are used (for example, Castagna and others, 

1985; Han and others, 1986; Wang, 2000). Because there are 

no other input parameters except P-wave velocity and porosity when using the moduli by Pride, the prediction of S-wave 

velocity is unbiased and simple. 

Elastic velocities of gas-hydrate-bearing sediments have 

been investigated by many researchers (for example, Carcione 

and Tinivella, 2000; Helgerud, 2001; Jakobsen and others, 

2002; Lee, 2002). Assuming a pore-filling model of gas 

hydrate in the pore, the proposed equation by Pride (2005) 

provides a new way of calculating elastic velocities of gashydrate-bearing sediments.

In this paper, moduli of dry rocks similar to those recommended by Pride (2005) are derived to predict elastic velocities of water-saturated sedimentary rocks having different 

in-place conditions under the assumption that velocity dispersion, attenuation, and anisotropy can be ignored. This theory is 

applied to consolidated sediments measured by Han and others 

(1986), semiconsolidated sediments by Gregory (1976), and 

unconsolidated sediments by Hamilton (1971) and Domenico 

(1977). This theory is also applied to well-log velocities 

for unconsolidated sediments acquired at the Alpine-1 well 

on the North Slope of Alaska and at the Mallik 5L-38 well, 

Mackenzie Delta, in Canada. The applied theory results in 

good agreement between measured and predicted velocities.

Theory

Velocity Equation

Elastic velocities (that is, compressional [P]-wave velocity [Vp] and shear [S]-wave velocity [Vs]) of water-saturated 

sedimentary rocks can be computed from the elastic moduli by 

the following formulas:

V k p = + 4 3 m

r

/ and Vs = m

r

 (1)

where k, m , and r are bulk modulus, shear modulus, and 

density of the water-saturated sedimentary rocks, respectively. 

The formation density is given by 

r f = - ( ) 1 r f ma + r fl , (2)

where f , rma , and rfl are porosity, matrix density, and 

pore-fluid density, respectively. The bulk and shear moduli 

of a composite matrix —for example, quartz and clay— are 

calculated using Hill’s averaging method (1952). 

Under the low-frequency approximation, the classical 

Biot-Gassmann theory (BGT) (Biot, 1941, 1956; Gassmann, 

1951) predicts bulk modulus of water-saturated rocks from 

the following equations, if the Biot coefficient ( b ) of the dry 

frame is known. 

k k = - ma ( ) 1 + M

2 b b , (3)

where

1

M k k ma fl

= - + ( ) b f f

and k is the bulk modulus of water-saturated sediments, and 

kma and kfl are bulk modulus of matrix (constitutes the skeleton of the formation) and bulk modulus of fluid, respectively. 

The Biot coefficient is defined as b = -1 k k d m / a , 

where kd is the dry-rock bulk modulus. The Biot coefficient 

is only defined for the bulk modulus because it is the ratio of 

pore-volume change to total bulk-volume change under dry 

or drained conditions. However, the BGT does not provide a 

relation between the shear modulus and matrix material using 

the Biot coefficient.

Moduli of Dry Frame

Within the poroelastic framework, dry moduli of the 

frame are undetermined and must be specified a priori. Generally, moduli are measured in the laboratory (Murphy, 1984), 

are predicted by theory —for example, contact theory by 

Digby (1981)— or can be derived under a specific assumption. 

If it is assumed that the velocity ratio, Vp/Vs, is constant irrespective of porosity, which is approximately true for partially 

gas saturated sedimentary rocks (Pickett, 1963), and is equal 

to the velocity ratio of matrix, the shear modulus of dry rock 

( β φ φ = − − − 1 1 3 1 ( ) ) can be derived by the following formula (Krief and oth ( ) -

ers, 1990):

m m d m = - a ( ) 1 b (4)

Conventionally, it is assumed that fluid in the pore space 

does not change the shear modulus of sedimentary rocks. In 

other words, the shear modulus of dry rock is the same as 

that of water-saturated sedimentary rock or m m = d , where 

m is the shear modulus of the water-saturated rock. Therefore, under this assumption, equation 4 can also be used for 

water-saturated rock. As shown in equations 1−4, by using the 

Gassmann theory, velocities of water-saturated sedimentary 

rocks can be obtained if moduli of the dry frame are known.

Pride (2005) provided the following dry-frame moduli for 

consolidated sandstones:

2 Proposed Moduli of Dry Rock and Their Application to Predicting Elastic Velocities of Sandstones
k k d

ma = -

+

( )

( )

1

1

f

af 

(5)

m m m f

af d

ma = = -

+

( )

( . )

1

1 1 5 

(6)

where a is a consolidation parameter that represents the degree 

of consolidation between grains. Effective medium theories 

can be approximately manipulated into expressions of this 

form and predict thata depends on both the shape of the cavity 

and the ratio mma ma /k (Pride, 2005). The factor 1.5 in equation 6 (2 or 5/3 are also reasonable) is somewhat arbitrary but 

yields an accurate Vp/Vs ratio for sandstones.

 In order to generalize the Pride equation, the shear 

modulus of dry rock is proposed in the following equation:

m m m f

gaf d

ma = = -

+

( )

( )

1

1 (7)

where

g

a

a = ++

1 2

1 .

 (8)

Note that when a = 1, g = 1 5. , which is identical to 

equation 6. When a = 2, then g = 5 3 / , and as a increases, 

g approaches 2. Therefore, equation 8 covers all reasonable 

values suggested by Pride (2005).

In order to relate equations 5 and 7 to equations 3 and 4, 

the following coefficients are defined using equations 5 and 7: 

b f a

af p

d

ma

k

k o - = ++

1 1

1

( )

( )

Þ k k d m = - a p ( ) 1 b (9)

b m

m

f ga

gaf s

d

ma

o - = +

+

1 1

1

( )

( ) Þ m m = - ma bs ( ) 1 . (10)

Parameter bp , equation 9, has the real meaning of the 

Biot coefficient, but bs , equation 10, is just a convenient 

parameter to put the shear modulus in the same form as the 

bulk modulus. It is noted that bp is always less than or equal 

(when a = 0) to bs for a given degree of consolidation and 

porosity and is the same as bs at f = 0 and f = 1 for all values 

of a.

In the framework of Pride (2005), velocities of sandstones 

for a given porosity are determined only by the consolidation 

parameter a . For sandstones, a lies in the approximate range 

2< a < 20 (Pride, 2005). As the Biot coefficient decreases, 

moduli of the dry frame increase. Therefore, as a decreases, 

moduli (or velocities) increase. In this new formulation, velocities of sandstones for a given porosity are determined only by 

the parameter a, which could vary between 0 and infinity.

Figure 1 shows relation between parameters a and g . 

At a = 0, g = 1, and at a =100, g » 1. 99. As parameter a

increases from 0 to 10, parameter g increases rapidly and 

approaches asymptotically to g = 2 asa further increases. 

The dotted line in figure 1 shows the velocity ratio of dry 

rock (Vp/Vs) having f = 0.25 with elastic constants shown in 

table 1. For all ranges of a, the velocity ratio varies between 

1.48 and 1.74. The velocity ratio is insensitive to variation in 

porosity. The velocity ratios determined by Domenico (1977) 

for unconsolidated sands with f= 0.382, and by Gregory 

(1976) for semiconsolidated sandstones with f = 0.217, fall 

within this range for differential pressure greater than 10 

megapascals (MPa).

Figure 2 shows the calculated Biot coefficients with 

respect to parameter a for a clean sandstone having porosity of 0.25 using the elastic constants shown in table 1. As 

indicated in figure 2, as a increases, both bp and bs increase, 

which means that velocities decrease. Also note that bs is 

always greater than bp. The behavior of the Biot coefficients 

indicates that the dependence of velocity on differential pressure can be modeled by decreasing parameter a as differential 

pressure increases. Figure 2 also demonstrates that velocities 

Figure 1. Relation between consolidation parameter (α) and 

velocity ratio (solid line) and relation between consolidation 

parameter (α) and parameter γ. φ, porosity; Vp, P-wave velocity; 

Vs , S-wave velocity.

Theory 3
4 Proposed Moduli of Dry Rock and Their Application to Predicting Elastic Velocities of Sandstones

Table 1. Elastic constants used for this study.

[GPa, gigapascals; kg/m3, kilograms per cubic meter]

Parameters Values used Sources

Shear modulus of quartz 44 GPa Carmichael (1989) 

Bulk modulus of quartz 38 GPa Carmichael (1989)

Shear modulus of clay 6.85 GPa Helgerud and others 

(1999)

Bulk modulus of clay 20.9 GPa Helgerud and others 

(1999)

Bulk modulus of water 2.29 GPa

Density of quartz 2,650 kg/m3 Helgerud and others 

(1999)

Density of clay 2,580 kg/m3 Helgerud and others 

(1999)

for consolidated rocks and unconsolidated sediments can be 

modeled using different values of a, where a for the consolidated rock is less than that of the unconsolidated sediment. 

For comparison, the Biot coefficient using the Pride 

equation (equation 6) is shown as dotted line in figure 2. The 

Biot coefficient of S-wave using the Pride equation is greater 

than that from the proposed equation (equation 7) when a < 

1 and is less than that from the proposed equation when a > 

1. This implies that equation 7 predicts lower velocities than 

those predicted from the Pride equation for most rocks for a 

given porosity. 

Comparison with MacBeth Equation

The moduli defined in equations 5 and 7 are similar to 

those defined by MacBeth (2004): 

k p k

E e d

d

k

p pk

( ) / = *

+

¥

- 1 

(11)

where p is differential pressure and kd

¥ is the bulk modulus of 

dry rock at infinite differential pressure. Let us assume that the 

consolidation parameter is given bya a = + a -

o

p p e k

1

/ *

. Note 

that ao is the consolidation parameter at infinite pressure and 

( ) a a o + 1 is the consolidation parameter at p = 0. Then, the 

bulk modulus given in equation 5 can be written as

k p k

E e d

ma

o k

p pk

( ) ( )

( )( ) / = *

-

+ + -

1

1 1

f

fa 

(12)

where 

Ek

o

= +

fa

fa

1

1 . (13)

Comparing equation 11 with equation 12, 

k k d

ma

o

¥ = -

+

( ) 1

1

f

fa .

 

(14)

Equation 14 indicates that the bulk modulus at high differential pressure decreases as porosity and a increase. Likewise, 

the shear modulus defined by MacBeth (2004), which is 

m m

m

m d

d

p p p

E e

( ) / = *

+

¥

- 1 ,

 

(15)

can be related to equation 7. The relation is

m m f

gfa d

ma

o

¥ = -

+

( ) 1

1

E

o

m

gfa

gfa = +

1

1 . (16)

Figure 2. Relation between consolidation parameter (α) and 

the Biot coefficient for a rock having porosity (φ ) = 0.25 and clay 

volume content (Cv ) = 0. β p and β s are the Biot coefficients for 

the P-wave velocity and S-wave velocity, respectively.
In the framework of this model, pk

*

= pm

*

 and E E m 3 k . The 

velocity ratio, Vp/Vs, is given by

V

V

p k

s

= +

4

3 m .

Therefore, the velocity ratio can be deduced from the ratio of 

moduli (k /m), which is given by the following equation for a 

dry rock: 

k E e

E e

d

d

o

p p

o k

p p m

gfa

fa

m = + +

+ +

-

-

( )( )

( )( )

/

/

*

*

1 1

1 1

(17)

where p*= pk

*

= pm

*

. Equation 17 implies that Vp/Vs ratio 

increases as porosity and the consolidation parameter increase, 

and the ratio decreases as differential pressure increases. This 

characteristic is different from the general behavior of sandstones investigated by MacBeth (2004). The Vp/Vs shown in 

equation 17 agrees well with that of water-saturated sedimentary rocks (Lee, 2003). 

Figure 3A shows the relation between porosity and moduli of dry frame at high differential pressure (that is, kd

¥), and 

figure 3B shows the derived Biot coefficients using equations 

9 and 10. The large dots and circles in figure 3A are moduli 

at infinite differential pressure shown in table 2 of MacBeth 

(2004). Thin solid and dotted lines represent the least-squares 

fit (LSF) curves of the data. Thick solid and dashed lines are 

calculated moduli for a clean sandstone using equations 5 

and 7 and elastic constant shown in table 1 with a = 4. The 

difference between the moduli at zero porosity derived from 

this study and that of MacBeth is partly caused by clay content 

in the measured samples. Figure 3A indicates that the moduli 

predicted by the proposed equation deviated more from that by 

MacBeth as porosity increases beyond about 0.3.

Figure 3B shows the calculated Biot coefficients using 

equations 9 and 10 for the result shown in figure 3A. The Biot 

coefficients for the MacBeth data are calculated from the LSF 

curves, and the bulk and shear moduli of matrix used for MacBeth data are those at zero porosity shown in figure 3B. The 

Biot coefficient derived from dry bulk moduli (bp ) is always 

less than that derived from the shear moduli (bs ), even though 

Figure 3. Relations between dry-rock properties with respect to porosity. A, Relation between dry-rock moduli and porosity. LSF, 

least-squares fit. B, Relation between the Biot coefficient and porosity. The Biot coefficient derived by Krief and others (1990) is given by

β φ φ = − − − 1 1 3 1 ( ) ( ) . φ , porosity; β p and β s are the Biot coefficients for the P-wave velocity and S-wave velocity, respectively.

Theory 5
6 Proposed Moduli of Dry Rock and Their Application to Predicting Elastic Velocities of Sandstones

the difference derived from MacBeth is much smaller than that 

from the proposed equation. For consolidated sedimentary rocks 

with porosity less than a critical porosity, which is about 0.4 for 

sandstones (Nur and others, 1998), the difference between bp

and bs approximately increases as porosity increases. 

Note that the application of equations 3 and 4 for bulk 

and shear moduli implies that the same Biot coefficients are 

used for bulk and shear moduli; that is, bp = bs =b. Krief and 

others (1990) proposed a Biot coefficient, b, applicable to 

equations 3 and 4, and this Biot coefficient is also shown in 

figure 3B. For porosities less than about 0.2, the Biot coefficient, b , derived from Krief and others (1990) is similar to bp

derived from equation 9; but as porosity increases, b becomes 

much larger than bp . However, the Biot coefficient based on 

the moduli analyzed by MacBeth (2004) agrees better with 

that from Krief and others (1990). Figure 3B indicates that at 

porosities less than about 0.2, velocities predicted using the 

Biot coefficient from Krief and others (1990) are larger than 

those using the Biot coefficients derived from equations 9 

and 10 with a= 4, and at porosities greater than about 0.25, 

the opposite is true. Figure 3 implies that, in order to fit the 

observed data or to accurately predict velocities for a wide 

range of porosities using the proposed equations (equations 

5 and 7), the consolidation parameter should be a function of 

porosity as well as the degree of consolidation and differential 

pressure. 

Modeling

Velocities with Respect to Porosity

As indicated previously, for a given rock, the consolidation parameter, a , controls velocities when using equations 5 

and 7. Figure 4 shows predicted velocities of water-saturated 

sandstones from porosity and dry moduli by equations 5 and 7 

and comparisons with measured velocities by Han and others 

(1986). Figures 4A and 4B show the results of measurements 

at 5 MPa and 40 MPa, respectively. The fractional errors using 

a = 5.4 for 5-MPa data are 0.00± 0.05, and 0.01± 0.08 for 

P-wave and S-wave velocities, respectively. If a = 4 is used 

for the prediction, the fractional errors would be 0.05± 0.05 

and 0.09± 0.09 for P- and S-wave velocities, respectively. This 

implies that the predicted velocities are highly sensitive to a. 

The fractional errors for predicted velocities using 

a= 3.2 for 40-MPa data are 0.00± 0.04 and 0.01± 0.05 for Pand S-wave velocities, respectively. As expected, the optimal 

value of a for 40-MPa data is less than that for 5-MPa data, 

Figure 4. Comparison between predicted velocities of water-saturated sandstones, which are computed by applying the Gassmann 

theory to the dry moduli derived from equations 5 and 7, and measured velocities by Han and others (1986). A, Velocities measured at 5 

megapascals (MPa). B, Velocities measured at 40 MPa. Vp , P-wave velocity; Vs , S-wave velocity.
and the predicted velocities for 40-MPa data are somewhat 

more accurate than those for 5-MPa data. 

Figure 5 shows modeled velocities for unconsolidated 

sediments. The large dots and circles represent velocities for 

deep marine sediments measured by Hamilton (1971), and 

small circles and dots are well-log velocities acquired at the 

Alpine-1 well, North Slope of Alaska. Equations 5 and 7 

with a = 40 and a volume clay content (Cv ) of 0.3 predict 

accurate P- and S-wave velocities for the Hamilton data but 

underestimate velocities for the well-log data. For comparison, 

modeled velocities using the modified Biot-Gassmann theory 

(BGTL) by Lee (2002) with n = 1 and Cv = 0.3 are shown as 

dashed lines. The predicted velocities for Hamilton data using 

the BGTL are inferior to those predicted from the proposed 

equation; but for well-log data at the Alpine-1 well, the BGTL 

performs better. In order to predict more accurate well-log 

velocities using equations 5 and 7, a different a, smaller than 

40, could have been used. Results shown in figures 4 and 5 

indicate that velocities can be predicted accurately by using 

equations 5 and 7, if an accurate consolidation parameter is 

known. However, the parameter a is difficult to estimate and 

is highly variable, as discussed herein. 

Velocities of Gas-Hydrate-Bearing Sediments

Velocities of gas-hydrate-bearing sediments are strongly 

affected by how gas hydrates interact with porous media, and 

Helgerud (2001) shows four different cases of gas-hydrate 

deposits in the pore space. Among four different gas-hydrateaccumulation models, the pore-filling model accurately 

predicts in-place velocities (Helgerud, 2001; Lee, 2002; Kleinberg and others, 2003). 

In the context of the proposed equations (equations 5 and 

7), the consolidation parameter a can be estimated by fitting 

the measured velocities of non-gas-hydrate-bearing sediment. 

For a givena, the velocities of gas-hydrate-bearing sediments 

can be calculated using the pore-filling model, which only 

affects the dry-frame bulk and shear moduli of matrix material, which consists of sand, clay, and gas hydrate, as shown in 

Helgerud (2001) and Helgerud and others (1999).

Figure 5. Comparison between predicted and measured velocities. Solid lines show velocities predicted by applying Gassmann 

theory to the moduli of dry frame by equations 5 and 7 with α= 40 

and clay volume content of 0.3. Dashed lines represent velocities 

predicted by the modified Biot-Gassmann theory by Lee (2003; 

BGTL) with n =1 and clay volume content (Cv) of 0.3. Velocities of 

the Alpine-1 well are from depths of 4,000 to 4,500 feet.

Figure 6. Calculated velocities for gas-hydrate-bearing sediment 

using various methods. BGTL, modified Biot-Gassmann theory by 

Lee (2002).

Modeling 7
8 Proposed Moduli of Dry Rock and Their Application to Predicting Elastic Velocities of Sandstones

Figure 6 shows calculated elastic velocities with respect 

to gas-hydrate concentrations for a sediment having f = 0.32 

and Cv = 0.1. Velocities calculated from the BGTL (Lee, 2003), 

with n =1 and G e C C

h = + v - - 0 9552 0 0448 0 18 0 06714 2 . . . / . (Ch

is the gas-hydrate concentration), are shown as dotted lines and 

serve as reference velocities for comparison. Velocities computed using a = 20 are shown as line-circle-line and are less 

than reference velocities where gas-hydrate saturations are less 

than about 0.9. As gas hydrate fills the pore space, the waterfilled porosity decreases. Therefore, the consolidation parameter a should decrease (higher velocities) to fit the reference 

velocities. The solid lines in figure 6 are computed velocities 

using the following equation for a with ao = 20:

a a = + -

o

C e h ( . . ) ( / . ) 0 59 0 41 0 376

 (18)

where ao is the consolidation parameter at Ch = 0. The computed velocities using equation 18 are close to the reference 

velocities for gas-hydrate concentrations as much as about 0.8 

and become higher than the reference velocities as concentrations increase further. Because the gas-hydrate concentrations are limited by the free water in the pore, the maximum 

hydrate concentration in sediments is usually less than about 

0.9. Therefore, equation 18, with equations 5 and 7, provides 

accurate velocities for gas-hydrate-bearing sediments.

Figure 7A shows well-log velocities of non-gas-hydratebearing sediments recorded at the Mallik 5L-38 well, Mackenzie Delta, in Canada. Modeled velocities with a = 20 agree 

well with the velocities of non-gas-hydrate-bearing sediments. 

Figure 7B compares gas-hydrate concentrations estimated from 

the P-wave velocities and from the S-wave velocities using 

a= 20. Except for gas-hydrate concentrations higher than 

about 0.8, the gas-hydrate concentrations estimated from the 

P-wave velocities are much less than those from the S-wave 

velocities, as the modeled velocities shown in figure 6 indicate. Figure 7C shows gas-hydrate concentrations estimated 

Figure 7 (above and facing page). Velocities and gas hydrate concentrations estimated at the Mallik 5L-38 well, Mackenzie Delta, 

in Canada. A, Measured velocities and predicted velocities from the proposed equations (equations 5 and 7) with α = 20 for nongas-hydrate-bearing sediments. B, Gas-hydrate concentrations estimated using the proposed equation. C, Gas-hydrate concentrations estimated using the proposed equation with a variable consolidation parameter, which decreases with increasing gas-hydrate 

concentration.
using equation 18. Gas-hydrate concentrations estimated from 

P-wave velocities are similar to those estimated from S-wave 

velocities.

Results and Applications

Comparison with Other Moduli 

The contact theory (Digby, 1981; Murphy and others, 

1993; Winkler, 1983) predicts that the ratio of bulk to shear 

moduli for dry rock is independent of porosity but dependent on differential pressure. Contact theory predicts that 

the velocity ratio (Vp /Vs ) decreases as differential pressure 

increases. Application of equations 5 and 7 predicts the same 

relation. Equations 5 and 7 also predict that the Vp /Vs ratio 

increases as porosity increases, but this differs from the prediction of the contact theory. 

Murphy and others (1993) suggested the following 

approximate linear relation between the frame and matrix 

moduli for sandstones at high differential pressure: 

k k d m = - a ( . 1 2 5f) and m m d m = - a ( . 1 2 5f). (19)

According to Murphy and others (1993), the velocity 

ratio of dry rock at high differential pressure is independent of 

porosity, which is the same as the prediction of Krief and others (1990). The magnitudes of Biot coefficient estimated from 

the least-squares fit (LSF) to data analyzed by MacBeth (2004), 

which are shown in figure 3, are similar to those from Murphy 

and others (1993). Even though both the proposed method 

and the MacBeth (2004) data indicate b b p s < , the difference 

between bp and bs for the MacBeth data is negligible. 

The Kuster and Toksöz (1974) theory predicts that the 

velocity ratio increases as porosity increases. Therefore, the 

prediction based on equations 5 and 7 is similar to the Kuster 

and Toksöz theory for the relation between porosity and velocity ratio and is similar to the contact theory for the relation 

between velocity ratio and differential pressure. The MacBeth 

(2004) formula predicts increasing, constant, decreasing, 

or alternate velocity ratios with respect to differential pressure. However, for most rocks, the velocity ratio increases 

with pressure. The discrepancy between the observation by 

MacBeth (2004) and the prediction of the proposed equation 

comes from differences in pore geometry. Sandstones containing microcracks behave differently from rocks containing 

intergranular porosities. As differential pressure increases, 

small microcracks close rapidly. Consequently, the P-wave 

velocity increases rapidly as differential pressure increases, 

resulting in increasing Vp /Vs with increasing differential pressure. Therefore, equations 5 and 7 work well for sedimentary 

rocks having intergranular porosities and less accurately for 

those having microcracks. 

Predicting S-Wave Velocity from

P-Wave Velocity and Porosity

Use of equations 5 and 7 provides a means for accurately 

predicting S-wave velocities of water-saturated sandstones 

from P-wave velocity and porosity because a single parameter 

a relates both bulk and shear moduli of dry frame. Let us 

define the predicted P-wave velocity using the BGT with the 

dry moduli derived from equations 5 and 7 as Vp

*

 and as Vp

m

for measured P-wave velocity. The consolidation parameter 

can be calculated by solving the following equation:

V V p p

* m ( ) a - = 0 . (20)

Figure 8 shows the predicted S-wave velocities for 

sandstones measured at 5 MPa and 40 MPa by Han and others 

(1986) by solving equation 20 using the Newton-Raphson 

method (Press and others, 1986). The fractional errors for 

predicted S-wave velocities are 0.01± 0.04 and 0.00± 0.04 

for 5-MPa and 40-MPa data, respectively. Figure 9 shows the 

predicted S-wave velocities for unconsolidated sediments at 

the Alpine-1 well, North Slope of Alaska. The fractional error 

for S-wave velocities is – 0.01± 0.04. 

Results and Applications 9
10 Proposed Moduli of Dry Rock and Their Application to Predicting Elastic Velocities of Sandstones

The accuracy of predicted S-wave velocities for dry 

rock is different from that of water-saturated rock. Figure 10 

shows the predicted S-wave velocities of dry rock measured 

by Domenico (1977) and Gregory (1976), using equation 7 

for the shear moduli. The fractional error of the predicted 

S-wave velocity is – 0.12 ± 0.01 for the Domenico data and 

– 0.08 ± 0.04 for the Gregory data. Both results indicate a 

large underestimation of S-wave velocities. The line and 

star in figure 10 shows the predicted S-wave velocities using 

g = 1 in equation 7. The fractional errors using g = 1 are 

0.00 ± 0.01 and 0.00 ± 0.04 for the Domenico data and the 

Gregory data, respectively. Using g = 1 is the same as using 

identical Biot coefficients for P- and S-wave velocities, which 

is predicted by Murphy and others (1993) and Krief and others (1990). The result shown in figure 10 indicates that the 

proposed moduli are accurate when calculating velocities of 

water-filled sedimentary rocks. However, use of g = 1 for 

dry rock appears to be more accurate. The Biot coefficients 

based on the LSF to moduli at infinite pressure derived from 

179 sets of data by MacBeth (2004), as shown in figure 3B, 

indicate that g = 1 is a close approximation for shear modulus 

of sandstones at high differential pressure.

Consolidation Parameter a

When velocities are calculated using equations 5 and 7, 

a is the only parameter to choose. The general behavior of 

a is known. For example, a becomes small as the degree of 

consolidation and differential pressure increases. But the precise value of a , on the other hand, is not known and should 

be estimated from the data or assumed for a given sedimentary rock. In the case that P- and (or) S-wave velocities with 

porosity are known, the parameter a can be estimated by 

solving equation 20. 

Figure 8. Comparison between the predicted S-wave velocity, which is calculated from the P-wave velocity and porosity using the dryrock moduli from equations 5 and 7, and measured S-wave velocity by Han and others (1986). A, Velocities measured at 5 megapascals 

(MPa). B, Velocities measured at 40 MPa. Vs, S-wave velocity.
Figure 9 (left). Comparison between the predicted S-wave velocity, which is calculated from the P-wave velocity and porosity 

using the dry rock moduli from the proposed equation, and the 

measured well-log velocity at the Alpine-1 well, North Slope of 

Alaska. φ , porosity; Vp, P-wave velocity.

Figure 10 (below). Measured and predicted S-wave velocities 

for dry rocks. Measured velocities are shown as line-circle-line. 

Line-dot-line indicates the predicted S-wave velocity (Vs) using 

equation 7, and line-star-line represents for the predicted Swave velocity using equation 7 with γ = 1. A, Unconsolidated dry 

sediment measured by Domenico (1977). B, Semiconsolidated 

dry rock measured by Gregory (1976). φ, porosity; Vp, P-wave 

velocity; β p and β s are the Biot coefficients for the P-wave 

velocity and S-wave velocity, respectively.

Results and Applications 11
12 Proposed Moduli of Dry Rock and Their Application to Predicting Elastic Velocities of Sandstones

Figure 11 shows the estimated a , which is derived from 

the porosity and P-wave velocity when predicting S-wave 

velocities shown in figures 8 and 9. Figure 11A, based on data 

from Han and others (1986), indicates that, for a given set of 

sandstones, the consolidation parameter a varies between 2 

to 10 for 5-MPa data and between 1 and 5 for 40-MPa data. 

The consolidation parameters a of 5.4 and 3.2 used for figure 

4 are the average values of a shown in figure 11A. Figure 

11B, using data from the Alpine-1 well, indicates that a varies between around 10 and 30, even though the depth range 

is only 4,000 to 4,500 ft. Figure 11 implies that, although a

is related to differential pressure and degree of consolidation, determining an accurate a is not simple; thus, it can 

be considered as a convenient free parameter to fit observed 

velocities. Pride (2005) suggested that if a is greater than 20 

or 30, moduli predicted by Walton (1987) are preferred. However, the prediction of P- and (or) S-wave velocities shown in 

figures 5 through 9 indicates that the moduli derived from the 

proposed equations work well for consolidated rocks as well 

as for unconsolidated sediments. 

Conclusions

By applying the BGT to moduli of dry frame proposed 

here, elastic velocities of water-saturated sandstones rocks can 

be accurately predicted. The following conclusions are drawn 

from the present study:

1. The Biot coefficient estimated from the P-wave 

velocity is smaller than that estimated from the S-wave velocity; this behavior is similar to the prediction of MacBeth 

(2004) but different from that of Murphy and others (1993) or 

Krief and others (1990).

2. Contrary to the prediction of MacBeth (2004), 

the velocity ratio, Vp/Vs, decreases as differential pressure 

increases, similar to the prediction of contact theory. This 

discrepancy is caused by microcracks present in the measured 

Figure 11. Estimated consolidation parameters from the P-wave velocity and porosity. A, Estimated from velocities of sandstones 

measured by Han and others (1986). B, Estimated from velocities of well-log data acquired at the Alpine-1 well, North Slope of Alaska. 

MPa, megapascals.
sandstones, and moduli predicted by the proposed equation are 

optimal for sandstones with granular porosities.

3. The Vp/Vs ratio increases as porosity increases, which 

is similar to the prediction of Kuster and Toksöz (1974) but 

different from Krief and others (1990), who predicted that the 

velocity ratio is independent of porosity.

4. Accurate S-wave velocities of water-saturated sandstones can be predicted because the consolidation parameter is 

related to both P- and S-wave velocities. Parameter a can be 

estimated if either P- or S-wave velocity, as well as porosity, is 

known.

5. Although velocities of water-saturated sandstones can 

be predicted accurately, based on the proposed equations, predicting velocities of gas-hydrate-bearing sediments by using 

the proposed equation is not accurate. However, by making 

the consolidation parameter a decrease with increasing gashydrate concentration, accurate gas-hydrate concentrations (or 

velocities) can be estimated.

6. For a dry rock, the proposed equation is not accurate. 

However, by using g = 1, accurate velocities, particularly at 

high differential pressure, can be predicted.

7. Parameter a is related to the degree of consolidation and differential pressure. The exact behavior, however, is 

difficult to predict. In practical applications, parameter a can 

be viewed as a free parameter to fit the observation. 

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