U.S. Department of the Interior

U.S. Geological Survey

Scientific Investigations Report 2007–5264

Velocities and Attenuations of Gas

Hydrate-Bearing Sediments
Velocities and Attenuations of Gas 

Hydrate-Bearing Sediments

By Myung W. Lee

Scientific Investigations Report 2007–5264

U.S. Department of the Interior

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

DIRK KEMPTHORNE, Secretary

U.S. Geological Survey

Mark D. Myers, Director

U.S. Geological Survey, Reston, Virginia: 2007

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Suggested citation:

Lee, Myung W., 2007, Velocities and attenuations of gas hydrate-bearing sediments: U.S. Gelogical Survey Scientific 

Investigations Report 2007–5264, 11p.
iii

Contents

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

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

Acknowledgments.........................................................................................................................................2

Theory of Three-Phase Equation ................................................................................................................2

Derivation of equation..........................................................................................................................2

Moduli of dry frame .............................................................................................................................3

Computation of moduli formed by grain and gas hydrate..............................................................3

Dissipation potential.............................................................................................................................4

Velocity and Attenuation Modeling ............................................................................................................4

Real Data Example.........................................................................................................................................4

Discussion.......................................................................................................................................................6

Velocity ...................................................................................................................................................6

Estimation of gas hydrate amounts using velocity..........................................................................7

Attenuation.............................................................................................................................................7

Comparison with other theories.........................................................................................................8

Conclusions...................................................................................................................................................10

References Cited..........................................................................................................................................10

Figures

1–6 Graphs showing—

 1. Modeled velocities and attenuations using LCAM-13 without b13 ................................5

 2. Modeled velocities and attenuations using LCAM-13 with b13......................................5

 3. Measured well log velocities and attenuation at the Mallik 5L-38 well,

Mackenzie Delta,Canada, and calculated velocities and attenuations using

the LCAM-13 ..................................................................................................................................6

 4. Estimated gas hydrate concentrations from the P-wave using the

LCAM-13 and from the nuclear magnetic resonance porosities at the Mallik

5L-38 well,Mackenzie Delta, Canada ........................................................................................7

 5. Calculated attenuation without b13 term at 12 and 2.5 kHz and calculated 

attenuation at 12 and 2.5 kHz......................................................................................................8

 6. Velocities and attenuations calculated from the squirt flow.........................................9

Table

 Table 1. Constants used for the modeling. .......................................................................................5
iv

Selected Abbreviations Used In This Report

GHBS gas hydrate-bearing sediment

NMR nuclear magnetic resonance

VSP vertical seismic profile

mm millimeter

km/s kilometers per second

Hz hertz

kHz kilohertz

MHz megahertz

kg/m3 kilogram per cubic meter

MPa megapascal

GPa gigapascal

cP centipoise 

mD millidarcy
Abstract

Monopole and dipole logging data at the Mallik 5L-38, 

Mackenzie Delta, Canada, provide a challenge for sonic 

velocity and attenuation models used to remotely estimate 

pore-space gas hydrate content. Velocity and attenuation are 

linked, with velocity dispersion causing increased attenuation. 

Sonic waveforms for Mallik 5L-38, however, show no velocity dispersion in gas hydrate-bearing layers, yet are highly 

attenuated. Attenuation models applied to Mallik 5L-38 data 

are shown to be inconsistent with the observed velocity measurements, and therefore are suspect in their ability to predict 

gas hydrate content. A model explicitly linking velocity and 

attenuation data is presented, accurately predicting gas hydrate 

content from velocity data alone while demonstrating that the 

attenuation mechanisms at the Mallik 5L-38 site have not yet 

been identified. 

Introduction

Gas hydrate is an ice-like compound formed by the 

inclusion of gas within cavities in a crystal lattice formed from 

water molecules. Natural gas hydrate occurs in a relatively 

narrow depth range called the hydrate stability zone, which is 

delineated in large part by pressure and temperature, although 

other factors, such as salinity, also contribute. Methane 

hydrate, the most common naturally occurring gas hydrate, 

accumulates in permafrost and in shallow marine sediments 

worldwide (Pellenbarg and Max, 2000). 

Based on the attenuation decrease observed in sandstones 

as a result of decreasing porosity (Shatilo and others, 1998), 

attenuation could be expected to decrease in hydrate-bearing 

sediments as gas hydrate content increases, reducing the 

water-filled porosity. This intuitive relation between attenuation and gas hydrate content runs counter to findings from well 

log data in permafrost (Guerin and Goldberg, 2001; Guerin 

and others, 2005) and marine (Guerin and others, 1999) 

hydrate-bearing sediments. The nonintuitive relation between 

sonic velocity, attenuation, and gas hydrate content is investigated here using a particularly detailed suite of geophysical 

well logs, including monopole and dipole measurement (Collett and others, 2005), obtained in 2000, from JAPEX/JNOC/

GSC Mallik 5L-38 gas hydrate research well. This well was 

drilled to a depth of 1,150 m through sands and sandstones 

interbedded with silt/clay at a locality on the Makenzie Delta, 

Canada (Dallimore and Collett, 1999). 

Monopole and dipole waveform analysis at the Mallik 

2L-38 well indicates that sonic waveforms show strong amplitude losses of compressional and shear waves in the interval 

where gas hydrate is observed (Guerin and Goldberg, 2001). 

The observed attenuation is also confirmed by monopole and 

dipole waveforms measured at the nearby Mallik 5L-38 well. 

Detailed waveform modeling indicates attenuation is linearly 

proportional to the amount of gas hydrate in the sediments 

(Guerin and others, 2005).

The observed attenuation at the Mallik wells appears to 

be opposite to the general behavior of water-saturated sediments, and vertical seismic profile (VSP) data at the Mallik 

2L-38 site indicate attenuation decreases in the presence of 

hydrate (Lee and others, in press). To explain the nonintuitive 

attenuation observed in the Mallik 2L-38 well logs, Guerin 

and Goldberg (2001) used a wave propagation model in frozen 

porous media based on the percolation theory of Leclaire and 

others (1994). Although the predicted and observed S-wave 

attenuation are of the same magnitude, attenuation with 

respect to the gas hydrate concentration differs significantly 

from the observations, and the percolation model predicts 

much lower P- and S-wave velocities than those observed.

Guerin and Goldberg (2005) proposed a model linking 

velocity and attenuation based on the modified percolation 

theory by Carcione and Tinivella (2000) and included the 

squirt flow by Diallo and Appel (2000) in their formulation. 

The major improvement of Guerin and Goldberg (2005) over 

Guerin and Goldberg (2001) is (1) the inclusion of interactions 

between gas hydrate and host sediments through cementation, 

and (2) a frictional coefficient between the solid grain and 

gas hydrate. Guerin and Goldberg (2005) demonstrated that 

reasonable velocities and attenuations comparable to those 

observed at the Mallik 5L-38 well can be predicted.

A model was derived in which velocity and attenuation 

results are linked by their causal relation, demonstrating that 

although the causes of attenuation in well logs are not yet fully 

understood, gas hydrate content can be accurately estimated 

using velocity information. 

Velocities and Attenuations of Gas Hydrate-Bearing 

Sediments

By Myung W. Lee
2 Velocities and Attenuations of Gas Hydrate-Bearing Sediments

Acknowledgments

Well logs were acquired by Schlumberger Ltd. at the 

Mallik 5L-38 gas hydrate research well, which was drilled 

in 2002 to investigate gas hydrate in a collaborative research 

project among the Japan National Oil Company, Geological 

Survey of Canada, U.S. Geological Survey, U.S. Department 

of Energy, GeoForschungZentrum Potsdam, India Ministry 

of Petroleum and Natural Gas / Gas Authority of India, and 

International Scientific Continental Drilling Program. I thank 

W.F. Agena and W.F. Waite, USGS, for many helpful comments and suggestions.

Theory of Three-Phase Equation

The sonic velocities and attenuations discussed here are 

calculated from a set of equations accounting for the three 

phases in the system: sediment grains, gas hydrate, and liquid 

water. The three-phase equations are derived first, followed by 

derivations of the stiffness and frictional loss terms contained 

in the three-phase equations for compressional, P, and shear, S, 

wave speed, and attenuation.

Derivation of equation

Leclaire and others (1994) extended the Biot (1956) 

theory of fluid-saturated media to frozen porous media where 

ice and unfrozen water can coexist. Based on percolation 

theory, which is able to describe the ice matrix transition 

from a continuous to a discontinuous state during a freezing 

process, moduli of a matrix formed from sediment grains and 

ice are derived assuming there is no direct mechanical contact 

between grain and ice. 

Carcione and Tinivella (2000) extended the percolation 

theory by Leclaire and others (1994) to include the potential and kinetic energy contributions because of the contact 

between sediment grains and ice. They referred to this theory 

as the modified LCA model or LCAM. However, the frictional 

loss caused by the relative motion between sediment grains 

and ice was ignored. They applied this theory for elastic wave 

velocities in gas hydrate-bearing sediments by replacing ice 

with gas hydrate.

In this report, the LCAM is extended to include the frictional loss caused by the interaction between sediment grains, 

referred to here simply as “grains,” and gas hydrate. Moduli 

of a matrix formed by grains and gas hydrate were computed 

using dry frame moduli, similar to those recommended by 

Pride and others (2004), instead of the percolation theory. This 

report describes only the essential part of the theory of the 

LCAM to model velocities and attenuations of gas hydratebearing sediment (GHBS). Details of the LCAM are left to 

Carcione and Tinivella (2000). This report closely follows 

the notations of Carcione and Tinivella (2000) except that 

subscript i, which stands for ice, is changed into the subscript 

h, for gas hydrate. Note the author of this report assumes the 

intrinsic attenuations of gas hydrate and grains are negligible, 

so moduli of grains and gas hydrate are all real. 

The equation of motion of the LCAM can be written as 

follows:

where u is the displacement field, and R is the stiffness matrix, 

 is the shear matrix, r* is the mass matrix, and A is the friction matrix. Elements of matrices are given by

Note that the present theory, referred to as LCAM-13 in this 

report, differs from the LCAM only in elements of A matrix, 

where b13 = 0 in the LCAM formulation. In addition to the 

difference in attenuation formulation, the major difference 

between LCAM and this report is the way of computing those 

elements, which will be discussed later. 

P-wave velocities (V) and attenuations (inverse of quality 

factor Q) are given by

where Λi are called the eignevalues of  relative to R (Leclaire 

and others, 1994) and the effective density  is defined as 

S-wave velocities and attenuations are given by

where Ωi are the eignevalues of  relative to . Therefore, all 

theories based on the Leclaire and others (1994) predict three 

P-wave propagations and two S-wave propagations in GHBS. 

If the attenuations are ignored, fast velocities of gas 

hydrate-bearing sediments at low frequencies can be calculated from the stiffness matrix alone and are given as follows:

R11

R12

R13

R12

R22

R23

R13

R23

R33

11

0

13

0

0

0

13

0

33

b11 b13

–b11

–b13

–b11

b11 b33

–b33

–b13

–b33

b33 b13



11



12



13



12



22



23



13



23



33

R  ,

, and

,

 . 

 

 

A 

(1)

i = 1,2,3 (2)

(3)

(5)

i = 1,2 (4)
Theory of Three-Phase Equation 3

where the bulk density of gas hydrate-bearing sediments 

(rb) is given by rb = (1 ‐ φ)rs + (1 – Ch)rw + Ch rh where 

is porosity, Ch is the pore space gas hydrate saturation, and 

subscripts s, w, and h refer to sediment grains, water and gas 

hydrate, respectively.

Moduli of dry frame 

The key component for deriving velocities from the 

LCAM-13 is computing the bulk and shear moduli for a frame 

formed by grains and gas hydrate. Carcione and Tinivella 

(2000) used the Kuster-Toksöz theory (1974) to drive these 

moduli. Here, however, moduli similar to those recommended 

by Pride and others (2004) and Lee (2005) are used. Lee 

(2005) proposed the following moduli for dry rocks:

where Kd , Ks , d , and s are bulk modulus of dry frame, bulk 

modulus of the grain, shear modulus of dry frame, and shear 

modulus of the grain, respectively, and a is a consolidation 

parameter. As a increases, the dry frame moduli decrease, 

resulting in a velocity decrease. If a = 1, equation 7 is identical to that recommended by Pride and others (2004).

Computation of moduli formed by grain and

gas hydrate

To compute stiffness and shear matrices, three different 

volume fractions are defined. The proportions of sediment 

grains (s),water (w), and gas hydrate (h) are given by the following formulas:

 s=1–, w=(1–Ch), and h=Ch . (8)

For computing velocities, key matrix elements are R11, 

R33, 11, and 33. These matrix elements are given in the following equation by the volume fractions shown in equation 8 

(Leclaire and others, 1994; Carcione and Tinivella, 2000):

 R11=[(1– c1)s]

2

Kav + Ksm + 411/3 

 R33=[(1– c3)h]

2

Kav + Khm + 433/3 

 11=[(1– g1)s]2av + sm

 33=[(1– g3)h]

2

av + hm (9)

where

 is an angular frequency, and  is the viscosity of pore fluid.

To compute those four elements, the bulk and shear 

moduli of the matrix formed by grains (Ksm and sm) and the 

bulk and shear moduli of the matrix formed by gas hydrate 

(Khm and hm ) should be known. These moduli are difficult 

to evaluate. Leclaire and others (1994) used the percolation 

theory to derive these moduli. Carcione and Tinivella (2000) 

used a different equation for sm , but used the same form for 

Ksm. For example, Carcione and Tinivella (2000) used the following formula for sm:

sm = [smKT – smo][h /(1 – s)]3.8 + smo (10)

where smKT is the shear modulus of sediment matrix calculated using the Kuster-Toksöz theory (1974) and smo is the 

shear modulus of matrix at full water saturation. Carcione and 

Tinivella (2000) used smo =1.19 gigapascals (GPa) in their 

example, whereas the equation by Leclaire and others (1994) 

yields smo= 0. 

In this paper, the moduli formed by grains and gas 

hydrate are computed from equations 6 and 7 using the following apparent water-filled porosity (as ). For the matrix formed 

by the grains (that is, Ksm and sm ), as = w + h is used for the 

porosity in equations 6 and 7. Therefore, 

The term h in the apparent porosity is a small number when 

 is small and accounts for the smaller impact of gas hydrate 

accumulation relative to compaction on velocity. The logic 

behind using the apparent porosity instead of the water-filled 

porosity (w) is that the velocity increase owing to accumulating gas hydrate is less than that expected from porosity 

reduction due to compaction. When  = 0, the apparent porosity is the same as the water-filled porosity and the calculated 

velocities using LCAM-13 with  = 0 are higher than those 

measured, as shown later in this report. 

For the matrix formed by the gas hydrate phase (that is, 

Khm and hm), ah = 1 – h is used as the apparent porosity. In 

other words,

(12)

(6)

(7)

with

(11)
4 Velocities and Attenuations of Gas Hydrate-Bearing Sediments

Note that the major differences among Leclaire and others 

(1994), Carcione and Tinivella (2000), and this report are in 

the calculation of the moduli of these matrices. For example, 

Leclaire and others (1994) used constant Ksm and sm, Carcione 

and Tinivella (2000) used a constant Ksm but a variable sm, 

depending on the gas hydrate concentration as shown in equation 10, and this report uses variable Ksm and sm, depending on 

the amount of gas hydrate concentration. 

The dominant contribution of gas hydrate to velocities comes from equation 11, which indicates that the frame 

moduli increase as the water-filled porosity decreases, similar 

to the effect of compaction on velocity. The cementation effect 

on shear modulus is ignored, meaning the aforementioned 

approach treats the gas hydrate as a load-bearing component 

of sediments. 

Dissipation potential

The dissipation potential (D) used by Leclaire and others 

(1994) has the following form:

where u1, u2, and u3 are displacements in the solid grain, water, 

and gas hydrate, respectively, and b11 and b33 are the friction 

coefficients between solid and water and between gas hydrate 

and water. As indicated in equation 13, dissipation caused by 

relative velocity between grains and gas hydrate is ignored 

under the assumption that there is no direct contact between 

grains and gas hydrate. In this paper, the friction between solid 

and gas hydrate is formally included by adding

to equation 13. Because the term b13 is included in the friction 

matrix, the elements of A, shown in equation 1, are different 

from Leclaire and others (1994). The theoretical derivation 

of b13 in the friction matrix is beyond the scope of the current 

study. However, using the approach of Guerin and Goldberg 

(2005), b13 is estimated in such a way that the calculated 

attenuations using LCAM-13 fit the observed attenuations. 

Velocity and Attenuation Modeling

Figures 1 and 2 show velocities and attenuations calculated for a sediment having  = 0.32, Cv = 0.1 (fractional 

volume clay content), and Ch = 0.3 using the LCAM-13 with 

constants listed in table 1 with  = 0.12. The result of the 

LCAM-13 without b13 is shown in figure 1, and the result with 

b13, which is given by 109h

2

kg m–3s–1, is shown in figure 2. 

Figure 1 indicates that fast P- and S-waves are nearly 

constant for frequencies to 1 MHz, meaning they are practically nondispersive (fig. 1A: Vp1 is about 2.5 km/s and Vs1

 is 

about 1.0 km/s). The slow P-waves are dispersive, but the slow 

S-wave is non-dispersive. The calculated attenuation is small 

for the fast P- and S-waves, as expected from their nondispersive behavior (fig.1B). Comparing figures 1B and 1C for frequencies higher than 10 kHz, the shear wave attenuation Qs1

–1 

is orders of magnitude higher than the P-wave attenuation Qp

–1.

In figure 2, the effects of including friction between 

grains and gas hydrate can be seen. The fast P- and S-waves 

are still almost nondispersive, but all slow P- and S-waves are 

highly dispersive (fig. 2A). The biggest difference between 

figures 1 and 2 is in the attenuation of slow waves. Figure 2 

indicates that Qp2

–1 and Qs2–1 with b13 are orders of magnitude 

higher than those without b13. The results shown in figures 1 

and 2 indicate that the effect of b13 in the LCAM formulation 

has only a minor change on the fast P- and S-waves.

Real Data Example

The Mallik 2L-38 and Mallik 5L-38 wells drilled in the 

Mackenzie Delta, Canada, provided data for evaluating the 

three-phase equation. The following summarizes the observed 

seismic properties of GHBS at these well locations.

(1) Velocities derived from the vertical seismic profiles 

(VSP) are similar to those from well logging. Well log P-wave 

velocities were measured using a monopole with a dominant 

frequency of 12 kHz, and S-wave velocities were measured 

using a dipole tool with the frequency of 2.5 kHz. The average of P-wave velocities derived from the surface VSP data 

is about 4 percent slower than that derived from the well log 

(Lee and others, in press).

(2) P-wave attenuation derived from the well logs indicates that attenuation increases almost linearly as gas-hydrate 

concentration increases and is represented by Guerin and 

others (2005) as:

 Qp

–1= 0.0342 + 0.076Ch

. (14)

The average gas hydrate concentration at the Mallik 2L-38 

well or Mallik 5L-38 well is about 30 percent, with concentrations reaching 90 percent (Lee and Collett, 1999; 2005). The 

P-wave attenuation of GHBS at 30 percent saturation predicted 

by equation 14 is about Qp

–1 0.057. 

Using cross-hole seismic data, Pratt and others (2005) 

estimated Qp

–1, in the absence of gas hydrate, is on the order 

of 0.01, but the presence of gas hydrate increases the apparent values of Qp

–1 to more than 0.1 in places. Assuming that 

Qp

–1= 0.1 is the attenuation of gas hydrate-bearing sediments 

with Ch = 0.9, and two-thirds of the apparent attenuation is 

because of gas hydrate, similar to monopole attenuation shown 

in equation 14, the attenuation at Ch = 0.3 is Qp

–1 0.05 at a 

dominant frequency of about 500 Hz for the cross-hole data. 

VSP data indicate that non-GHBS is on the order of 

Qp

–1 0.013, whereas the average attenuation of GHBS is 

(13)
Real Data Example 5

Table 1. Constants used for the modeling.

[GPa, gigapascal; kg/m3, kilogram per cubic meter; mD, millidarcy; cP, centipoise]

Ks = 38 GPa s = 44 GPa s= 2,650 kg/m3 so = 100 md r13 = 0.5

Kc = 20.9 GPa c = 6.85 GPa c = 2,580 kg/m3 ho = so × 106md r23 = 0.5

Kh = 6.41 GPa h = 2.54 GPa h = 910 kg/m3 = 1.8 cP r31 = 0.5

Kw =2.29 GPa w = 0 w = 1,000 kg/m3 Cv = 0.1 r12 = 0.5

 = 0.32 = 25

Subscript c stands for clay. Unless parameters are specifically specified, these parameters were used in the modeling. 

Figure 1. Modeled velocities and attenuations using LCAM-13 without b13 . (A) Velocities. (B) P-wave attenuation. (C) S-wave 

attenuation.

Figure 2. Modeled velocities and attenuations using LCAM-13 with b13 which is given by 109n

2

kg m–3s–1. The gas hydrate 

saturation is 0.3. (A) Velocities. (B) P-wave attenuation. (C) S-wave attenuation. 
6 Velocities and Attenuations of Gas Hydrate-Bearing Sediments

Qp

–1 0.006 (Lee and others, in press). Unlike the monopole or 

cross-hole data, the attenuation of GHBS of VSP data with a 

dominant frequency of 60 Hz is less than that of non-GHBS. 

 (3) S-wave attenuation derived from well logs also indicates that attenuation increases almost linearly as gas hydrate 

concentration increases and is represented by Guerin and 

others (2005) as:

 Qs

–1= 0.0615 + 0.134Ch (15)

Therefore, Qs

–1 0.1 at Ch

 = 0.3. There are no other available 

S-wave attenuation data measured at other frequencies.

Figure 3 shows calculated and measured velocities and 

attenuations for well log data acquired at Mallik 5L-38. The 

gas hydrate concentrations shown in figure 3A are derived 

from the nuclear magnetic resonance (NMR) porosity, as 

shown in Kleinberg and others (2005). The parameters for 

three-phase equation are identical to those for figure 2. 

Calculated velocities agree well with those measured using 

monopole and dipole waveforms. The calculated attenuations, 

however, are much smaller than observed. 

Discussion

Velocity

Velocities calculated using the present method, with 

 = 25, and  = 0.12, agree well with the observed velocities. 

The consolidation parameter  can be estimated from the 

non-GHBS as shown in Lee (2005). Figure 3A also shows the 

calculated velocities using different values of . When  = 1, 

which is similar to the Leclaire and others (1994) approach, 

the calculated velocities are much smaller than observed 

velocities, and shear velocities are almost independent of gas 

hydrate concentration. The result with  = 1 is the same as the 

result of the model that assumes gas hydrate floats in the pore 

fluid and increases seismic velocities primarily by increasing 

the moduli of the pore fluid (Helgerud, 2001). On the other 

hand, when  = 0 the calculated velocities are much higher 

than the measured velocities, but the relation between Vp and 

Vs is similar to the measured relation. The result with  = 0.12 

agrees closely with measured velocities and calculated velocities using the modified Biot-Gassmann theory by Lee (Lee 

and Collett, 2005) for gas hydrate saturations less than about 

80 percent. 

Figure 3 Measured well log velocities and attenuation at the Mallik 5L-38 well, Mackenzie Delta, Canada, and calculated 

velocities and attenuations using the LCAM-13 with b13 =109n

2

kg m–3s–1. (A) Velocities. (B) Attenuations.
Discussion 7

Estimation of gas hydrate amounts

using velocity

Figure 4 shows gas hydrate concentrations estimated 

from the P-wave velocity using the LCAM-13 compared with 

those estimated from the NMR porosity. The parameters for 

the LCAM-13 are identical to the previous examples except in 

porosity and in clay volume content, which are taken directly 

from the well logs (Collett and others, 2005).

Velocity-based gas hydrate saturation estimates depend 

on how acoustic effects of the pore-scale interaction between 

gas hydrate and the host sediment are modeled, and therefore 

vary depending on the velocity model. Gas hydrate saturations 

based on the NMR log, however, depend only on the accuracy of the log and on the bulk volume of gas hydrate present, 

irrespective of pore-scale interactions between gas hydrate and 

the host sediments. Therefore, gas hydrate estimations from 

the NMR log can serve as a reference. The excellent agreement between the saturations shown in figure 4 implies that 

the employed velocity model, which assumes gas hydrate is a 

load-bearing component of the system, is accurate for the well 

log velocities measured at the Mallik 5L-38 well site.

Attenuation

Comparing figures 1C and 2C, accounting for frictional 

motion between grains and gas hydrate by including the b13

term increases the predicted attenuation. As figure 5 shows, 

dependence of attenuation on hydrate content varies with the 

b13, the attenuation changing more noticeably for the S-wave 

than for the P-wave.

The calculated attenuations without the b13 term (fig. 5A) 

show that P-wave and S-wave attenuations initially decrease as 

gas hydrate concentration increases, then increase for hydrate 

saturations greater than 80 percent. Attenuations calculated 

with b13 = 0.25×109h

2

kg m–3s–1, however, show a more complicated dependence on gas hydrate saturation (fig. 5B). Including b13, the attenuation decrease with respect to gas hydrate 

saturation is slower than that calculated without b13. The 

P-wave attenuation at 12 kHz is orders of magnitude smaller 

than the S-wave attenuation at 2.5 kHz for all ranges of gas 

hydrate saturation.

S-wave attenuation is sensitive to the magnitude of b13 ; figure 

5B shows the result of attenuation with b13 = 0.25×109h

2

kg m–3s–1, 

which is one-fourth of the b13 value used in figure 3B. The 

P-wave attenuation is still small, whereas the S-wave attenuation increases significantly as gas hydrate saturation increases. 

Figure 4. Estimated gas hydrate concentrations from the P-wave using the 

LCAM-13 and from the nuclear magnetic resonance porosities at the Mallik 5L-38 

well, Mackenzie Delta, Canada.
8 Velocities and Attenuations of Gas Hydrate-Bearing Sediments

Numerical results indicate that the S-wave attenuation is 

maximized for b13 = 0.25×109h

2

kg m–3s–1 and that b13 affects 

the S-wave attenuation more than the P-wave attenuation.

As shown in figure 3B, calculated attenuations are far 

lower than observed estimates at the Mallik 5L-38 well. 

Because causality requires a specific relationship between 

velocity and attenuation, a large attenuation is generally 

associated with high velocity dispersion (Dvorkin and Uden, 

2004). If high attenuation occurs at 12 kHz for the P-wave 

and 2.5 kHz for the S-wave, the P-wave and S-wave velocities 

from 60 Hz VSP data would have been much less than the well 

log velocities. However, Sakai (1999) and Lee and others (in 

press) show that Vp and Vs estimated from VSP data are comparable to measured well log velocities at the Mallik 2L-38 

well, indicating negligible velocity dispersion, consistent with 

small calculated attenuations. 

Comparison with other theories

Several proposed attenuation mechanisms can account for 

the counterintuitive attenuation increase with increasing velocity (or gas hydrate concentration) observed at the Mallik sites. 

Figure 5. (A) Calculated attenuation without b13 term at 12 kilohertz (kHz) and 2.5 kHz. (B) Calculated attenuation at 12 kHz and 2.5 kHz 

with b13 = 0.25 ×109h

2

kg m–3s–1 .

One example is attenuation caused by a squirt flow (Dvorkin 

and Nur, 1993; Dvorkin and others, 1994, 1995). Squirt flow 

attenuation is orders of magnitude higher than predicted 

based on Biot flow. Figure 6 shows an example of velocities 

and attenuations calculated using the squirt flow theory by 

Dvorkin and others (1995) using a squirt length of 0.4 mm and 

model parameters identical to the result shown in figures 1 and 

2, such as porosity, clay content, permeability, viscosity and 

gas hydrate concentration. The result indicates high P-wave 

velocity dispersion, increasing from about 2.5 km/s for frequencies less than about 100 kHz to 3.5 km/s for frequencies 

greater than 1 megahertz (MHz). Because significant dispersion occurs above 100 kHz, the predicted velocities are close 

to measured velocities shown in figure 3. 

To match observations at the Mallik wells, squirt length 

and permeability can be adjusted. Decreasing the squirt flow 

length decreases the attenuation while increasing the frequencies of peak attenuation and the onset of velocity dispersion. 

As permeability decreases, attenuation increases. By adjusting permeability and squirt length, it is possible to predict 

P-wave attenuation to the order of magnitude observed at the 

Mallik sites. However, the predicted S-wave attenuation is 

much less than the predicted P-wave attenuation (fig. 6), in 
Discussion 9

assumed that particles having similar density to the fluid 

participate in the acoustic vibration, providing a lower bound 

of the suspension model. These bounds differ by orders of 

magnitude, however, and provide only a loose constraint on 

the measured attenuation. Moreover, Urick’s (1948) theory 

predicts decreases in attenuation for volume concentration of 

the particles exceeding 20 percent, contradicting the behavior 

of the observed attenuation in GHBS. The particle suspension 

models of Sewell (1910) or Urik (1948) do not appear to be 

viable for the Mallik site.

Guerin and Goldberg (2005) obtained a significant attenuation increase by including cementation in 13, making the 

inertial coupling coefficient, and including squirt flow into R12, 

R22, and R23 (see eq. 1). As mentioned by Guerin and Goldberg 

(2005), the inclusion of squirt flow only affects the P-wave 

attenuation. Figure 5 indicates that the inclusion of squirt 

flow proposed by Guerin and Goldberg (2005) in LCAM-13 

increases the P-wave attenuation by an order of magnitude. 

However, even adding the attenuative elements mentioned by 

Guerin and Goldberg (2005) does not cause the attenuation to 

increase enough to agree with the measured attenuation at the 

Mallik 5L-38; the calculated attenuations are still much less 

than the measured attenuations.

Figure 6. Velocities and attenuations calculated from the squirt flow. (A) Velocities. (B) Attenuations.

contradiction with observations at the Mallik in which S-wave 

attenuation exceeds the P-wave attenuation (fig. 3). Also, the 

squirt flow mechanism does not predict high attenuation at low 

frequencies. Therefore, the squirt flow mechanism alone does 

not explain the attenuation observed at the Mallik 5L-38 well 

site. 

Klimentos and MaCann (1990) presented a relation 

between P-wave attenuation and clay content and porosity 

at 1 MHz and 40 megapascals (MPa). The dominant part 

of P-wave attenuation is attributed to the clay content of 

the sediments, and attenuation is linearly related to the clay 

volume content. They analyzed the observed attenuation using 

particle suspension models of Sewell (1910) and Urick (1948) 

and concluded that there is reasonable order of magnitude 

agreement between the theoretical and experimental results. 

Similarities in the linear dependence of attenuation on clay 

content and gas hydrate concentration indicate that the particle 

suspension theory could be applied to GHBS. 

In the particle suspension theory, attenuation arises from 

the viscous interaction between pore fluid and suspended 

particles. Sewell’s (1910) model assumed that particles are of 

such great density that they do not participate in the acoustic motion of the fluid and gives the upper boundary of the 

suspension model. At the other extreme, Urick’s (1948) model 
10 Velocities and Attenuations of Gas Hydrate-Bearing Sediments

Failure to predict high attenuation for GHBS may result 

from: (1) theoretical inadequacies in the models previously 

discussed, and (2) interpretive inadequacies concerning the 

data itself. Recently, Lee and Waite (2007) questioned the 

magnitude of attenuation analyzed at the Mallik site by Guerin 

and others (2005). They argued that an amplitude loss owing 

to the source coupling to the surrounding medium, rather than 

intrinsic attenuation of the GHBS itself, accounts for at least 

half the observed amplitude loss at the Mallik site. It is unclear 

whether the failure of current theory to predict attenuations 

high enough to compare with observed attenuation levels is 

caused by incompleteness of the theory, by observational inadequacies, or both. Estimating gas hydrate from acoustic data 

using velocity and attenuation is, however, important. Accordingly, attenuation models proposed in the future need to honor 

the causality condition relating sonic velocity and attenuation 

so as to demonstrate not only a match to observed attenuation, 

but to observed velocities as well.

Conclusions

A modified three-phase percolation theory that includes 

frictional interactions between the sediment matrix and gas 

hydrate accurately predicts the velocities measured at the Mallik 5L-38 well. This approach yields accurate P- and S-wave 

velocities that can be used to accurately estimate the gas 

hydrate content of the sediments.

As with attenuation models based on squirt flow and 

frictional motion between sediment grains and gas hydrate, 

however, the model presented here fails to predict attenuations comparable to measured attenuations at the Mallik 5L-38 

well. All attenuation models considered in this study predict 

attenuations that are orders of magnitude less than measured 

attenuations in the frequency range between 60 and 12,000 Hz 

at the Mallik site. Failure of the current state-of-the-art attenuation models when considering the Mallik site indicates that 

the proposed attenuation mechanisms and attenuation data 

interpretation need to be reconsidered. To be viable, future 

attenuation models need to reflect the causal relation between 

velocity and attenuation by accurately predicting both.

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