Methods of Generating Synthetic Acoustic Logs

from Resistivity Logs for Gas-Hydrate-Bearing

Sediments

By Myung W. Lee

U.S. Geological Survey Bulletin 2170

U.S. Department of the Interior

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

Bruce Babbitt, Secretary

U.S. Geological Survey

Charles G. Groat, Director

This publication is only available on line at:

http://greenwood.cr.usgs.gov/pub/bulletins/b2170/index.html

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

Published in the Central Region, Denver, Colorado

Manuscript approved for publication July 19, 1999

Graphics by Norma J. Maes

Photocomposition by Norma J. Maes

Edited by Lorna Carter
III

Contents

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

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

Acknowledgments ............................................................................................................................... 1

Theoretical Relationship ..................................................................................................................... 1

Logs and Parameters........................................................................................................................... 3

Modified Time Average Equations .................................................................................................... 3

Predicting Velocity from Resistivity .................................................................................................. 3

Without Gas Hydrate Concentration........................................................................................ 3

With Gas Hydrate Concentration.............................................................................................. 4

Discussion ............................................................................................................................................. 4

Conclusions........................................................................................................................................... 9

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

Figures

1–8. Graphs showing:

1. Electrical resistivity measured at Mallik 2L-38 gas hydrate research 

 well with respect to density porosity for depth range 897 m to 1,109 m ......... 4

2. Comparison of P-wave velocities computed from weighted equation 

with W=1.44 and modified time average equations

 where α=1.3 and β =1.68 ........................................................................................ 5

3. Prediction of acoustic values from resistivity log at Mallik 2L-38 well 

using least squares method.................................................................................... 6

4. Prediction of acoustic values from resistivity log at Mallik 2L-38 well 

using modified time average equations without gas hydrate concentration. 7

5. Acoustic values predicted from resistivity log at Mallik 2L-38 well using 

modified time average equations with gas hydrate concentration ................. 8

6. Comparison of acoustic values from resistivity log at Mallik 2L-38 well 

using WE and MTAE2 with gas hydrate concentration...................................... 9

7. True (measured) acoustic values versus predicted acoustic values using 

LSM, MTAE1, and MTAE2 ........................................................................................ 10

8. Theoretical unconsolidation constants with respect to porosity and

weights of WE ........................................................................................................... 10
1

Methods of Generating Synthetic Acoustic Logs from

Resistivity Logs for Gas-Hydrate-Bearing Sediments

By Myung W. Lee

Abstract

Methods of predicting acoustic logs from resistivity logs 

for hydrate-bearing sediments are presented. Modified time 

average equations derived from the weighted equation provide a 

means of relating the velocity of the sediment to the resistivity of 

the sediment. These methods can be used to transform resistivity logs into acoustic logs with or without using the gas hydrate 

concentration in the pore space. All the parameters except the 

unconsolidation constants, necessary for the prediction of acoustic log from resistivity log, can be estimated from a cross plot of 

resistivity versus porosity values. Unconsolidation constants in 

equations may be assumed without rendering significant errors 

in the prediction. These methods were applied to the acoustic 

and resistivity logs acquired at the Mallik 2L-38 gas hydrate 

research well drilled at the Mackenzie Delta, northern Canada. 

The results indicate that the proposed method is simple and 

accurate.

Introduction

One of the problems in the seismic study of hydrate-bearing 

sediments is the lack of available acoustic logs. Older wells usually have resistivity logs, but not acoustic logs. Even in cases 

where acoustic logs exist, most of the time the acoustic values in 

the hydrate-bearing sediment intervals are useless because of 

well bore problems associated with the dissociation of gas 

hydrate. Therefore being able to predict acoustic values from 

resistivity logs or to transform pseudo-velocity logs from resistivity logs is useful in gas hydrate research.

Many methods for predicting acoustic logs from resistivity 

logs have been proposed (Faust, 1951; Kim, 1964; Rudman and 

others, 1975; Brito Dos Santos and others, 1988; Worthington, 

1991). Most of these methods utilize the time average equation 

of Wyllie and others (1958) and Archie’s law (Archie, 1942). 

The difference among the methods is the way to link the time 

average equation to Archie’s law.

Faust (1951) developed an empirical formula to relate 

velocities to depth and age of the rocks. Using Archie’s law with 

his empirical formula, Faust developed a relationship between 

the velocity and apparent resistivity, which is thought to be 

applicable primarily to permeable formations.

Kim (1964) demonstrated that a mathematical relation 

could be developed between apparent resistivity and travel time 

(acoustic value). The mathematical formula indicates that three 

parameters, related to the physical properties of the sediments, 

are sufficient to predict acoustic values from resistivity values. 

Instead of using physically significant parameters to derive the 

constants, Kim solved nonlinear simultaneous equations using 

three different resistivity-acoustic values. In essence, this is similar to the least squares method of nonlinear curve fitting. 

Because Kim selected good and consistent resistivity-acoustic 

values, it can be said that this method is based on the weighted 

least squares method.

Rudman and others (1975) slightly modified Kim’s 

approach and used the average scale function (a scale function is 

a predictive function that specifies a corresponding transit time 

for any resistivity value) to predict acoustic values from resistivity measurements; they suggested the application, with caution. 

They recommended that the resistivity range recorded be examined and those logs with anomalous values be discarded.

Brito Dos Santos and others (1988) presented a method utilizing the resistivity of mud, resistivity of mud-filtrate-containing rock, and resistivity of clay with Bussian’s (1983) equation, 

which is more general than Archie’s equation for the sediment’s 

resistivity. In essence, this method provides a better prediction 

of acoustic values by accounting for the shaly sand effect on the 

resistivity logs. To account for the clay effect on acoustic logs as 

well as resistivity logs, Worthington (1991) presented equations 

containing explicit clay terms in the time average and resistivity 

equations.

In this report, Archie’s equation for dirty shaly sands and 

modified time average equations using unconsolidation constants are combined to link the resistivity values to the velocities 

through the common parameter of porosity. The proposed methods implicitly account for the clay effect by using modified 

matrix velocity (Lee and others, 1996) and Archie’s parameter 

for dirty sands. These methods were applied to the log data 

acquired at the Mallik 2L-38 gas hydrate research well with 

good agreement.

Acknowledgments

Well logs were acquired by Schlumberger Ltd. at the Mallik 

2L-38 gas hydrate research well, which was drilled to investigate 

gas hydrate in a collaborative research project among Japan 

National Oil Company, Japan Petroleum Exploration Company, 

Geological Survey of Canada, and U.S. Geological Survey. I 

thank Tim Collett and Dave Taylor for many helpful comments. 

Theoretical Relationship

The simplest equation relating the acoustic value to sediment’s porosity is the time-average equation by Wyllie and others (1958). This equation works well for consolidated 

sediments, but it is problematic when applied to unconsolidated 

sediments. In order to overcome the unconsolidation problem of 
2 Methods of Generating Synthetic Acoustic Logs from Resistivity Logs for Gas-Hydrate-Bearing Sediments

the sediments using the time average equation, particularly for 

the hydrate-bearing sediments, Lee and others (1996) introduced 

a weighted equation (WE). This WE works well for hydratebearing sediments or permafrost samples (Lee and others, 1996; 

Lee and Collett, 1999). However, predicting velocities from 

resistivities using WE is not simple, mainly because the velocity 

from WE equation is a complex function of porosity. Thus, 

equations similar to the time average equation are useful to formulate the relationship between the resistivity of sediments and 

the transit time of the compessional wave. To account for the 

unconsolidation of sediments and to relate the porosity to the 

velocity, WE is approximated in the following way.

WE for non-hydrate-bearing sediments is defined by (Lee 

and others, 1996):

 (1)

where Lwe is the slowness computed using the weighted equation, Lwd is the slowness computed using the Wood equation, Lta

is the slowness computed using the time-average equation, W is 

the weight and φ is the porosity of the sediment.

The time average equation is defined by

 (2)

where Sf is the slowness of the pore fluid, and Sm is the slowness 

of the modified matrix, which accounts for the clay content in 

sediment, as explained in Lee and others (1996).

The first approximation of WE, which is called the modified time average equation 1 (MTAE1), is defined as follows, 

from equation (1): 

 (3)

where α is assumed to be a constant, whereas in fact it is a function of porosity and physical properties of sediments. The accuracy of this approximation will be discussed later. Since W is 

positive and Lwd is greater than Lta, α is always greater than or 

equal to 1.0 and called the α-unconsolidation constant.

The second approximation of WE, which is called the modified time average equation 2 (MTAE2), is defined as follows, 

using equations (1) and (2):

 

(4)

where β is defined as a constant and called the β-unconsolidation 

constant. As in MTAE1, β is always greater than or equal to 1.0. 

When the sediment pore spaces are occupied by the gas 

hydrates, the modified time average equations can be written as 

(Lee and others, 1996):

Lwe Lta + Wφ Lwd Lta = ( ) –

Lta (S f Sm – )φ Sm = +

MTAE1 Lwe Lta 1

Wφ(Lwd Lta – )

Lta

+ ------------------------------------- αLta ≈ = ≈

MTAE2 Lwe (S f Sm = – )φ 1 Wφ

W L( wd Sm – )

S f Sm – – + ------------------------------------ Sm ≈ +

(S f Sm – )βφ Sm ≈ +

 (5a)

 (5b)

where C is the gas hydrate concentration and Sh is the slowness 

of the gas hydrates.

The electrical property of sediment can be described by 

Archie’s formula (Archie, 1942), which is given by

 (6)

where Rt is the resistivity of the sediments, Rw is the resistivity 

of pore fluid, Cw is the fluid saturation, and a, n, and m are 

Archie’s empirically derived parameters. Usually a ranges from 

0.55 to 2.26, n varies between 1.7 and 2.2 (Pearson and others, 

1983), and m can have values between 1 and 3 (Labo, 1987). If 

the gas hydrate in the pore space is considered as part of the 

matrix, then the water saturation Cw is equal to 1 and the porosity in equations (1) and (6) can be considered as water-filled (φf ) 

porosity, which is defined φf = (1–Ch)φ, where Ch is the gas 

hydrate concentration. 

For clean sands, a = 1.0 and m =2.0 are proper parameters 

to use. The deviation of the empirically derived a and m from 

those of the clean sands is partly due to the clay content in the 

sediments in addition to the complex pore geometry. In this 

report, we assume that the clay in the formation resistivity is 

manifested in Archie’s parameters a and m.

With 100 percent water saturation or using the water-filled 

porosity in the Archie’s equation, the water saturation can be set 

to 1 for hydrate-bearing sediments. So, defining Q=(1/aRw) and 

substituting the porosity in equation (6) into the modified time 

average equations, the relationship between the acoustic and 

resistivity can be written as 

(7)

Or

 (8)

where S ́ is defined as a normalized acoustic, η1 and η2 are α ’s 

for MTAE1, and η1 = β and η2 =1 for MTAE2. Equation (8) 

shows a linear relationship between a normalized acoustic and 

resistivity in a Log-Log plot. Equation (6) also indicates a linear 

relationship between porosity and the resistivity in a Log-Log 

plot. If data quality is good, the slope from equation (8) should 

be close to that estimated from the resistivity log using equation 

(6).

Most of the parameters necessary for the use of equation (7) 

to compute the P-wave acoustic values from resistivity log values can be estimated directly from the resistivity versus density 

data. For 100 percent water saturation (baseline data without gas 

hydrate concentration), the slope of the Log-Log plot of porosity 

MTAE1 α CSh ( ) 1 – C S f Sm [ ] + – φ αSm = +

MTAE2 β CSh ( ) 1 – C S f Sm [ ] + – φ Sm = +

φ

–m Cn

w Rt

aRw

---------------

      =

Log10

S η2Sm –

η1Sw η1Sm – -------------------------------- Log10 ≡ S′ Log10Q

m – ------------------- Log10Rt

m = – --------------------

S Sw–Sm ( )η1φ η2Sm + Sw–Sm ( )η1Q–1 ⁄ mRt

–1 ⁄ m η2Sm = = +
Logs and Parameters 3

and resistivity provides the parameter m, and the intercept at Rt = 

1 ohm-m provides the value for Q = 1/(aRw).

Logs and Parameters

Resistivity and acoustic logs were acquired at the Mallik 

2L-38 gas hydrate research well at the Mackenzie Delta, 

northern Canada, in 1998 (Collett and others, 1999). The quality 

of data is excellent, and porosities derived from the density log 

were used in this study. Previous studies (Lee and Collett, 1999) 

estimated some relevant parameters for this study. A WE 

(weighted equation) with the weight of W= 1.44 and the exponent of n = 1 with other parameters such as Sf = 0.667 s/m, Sh = 

0.303 s/m, and Sm = 0.2024 s/m was used in the previous study 

(Lee and Collett, 1999) to estimate gas hydrate concentration 

from acoustic logs. The slowness of modified matrix (Sm) is calculated from Han and others’ relation (1986) using the volume 

clay content of 30 percent. These parameters were used in this 

study.

Figure 1 shows a cross plot of Log10 (φ) versus Log10 (Rt)

for the depth range of 897 m to 1,109 m. Using the slope of the 

linear approximation for the non-hydrate-bearing sediment data 

shown in figure 1, m is estimated as m = 1.95. The intercept of 

the linear approximation at Rt = 1 ohm-m is about 0.63. 

Substituting this value into equation (8), it can be shown that 

Q = 0.406.

Modified Time Average Equations

To examine the behavior of modified time average equations with respect to WE for the velocities of unconsolidated 

sediments, velocities from WE with W=1.44 and Sm = 0.2024 

s/m are compared with those from the modified time average 

equations. Figure 2 shows the theoretical velocities using WE 

with those from the modified time average equations. The computation of unconsolidation parameters can be done in two ways: 

1. Match the observed velocity at a given porosity with the 

modified time average equations shown in equations (3) and (4). 

2. When the parameters for WE are known, the unconsolidation parameters can be calculated using W in the definition of 

unconsolidation parameters. 

In this example, unconsolidation constants were computed by 

matching the velocities for the sediment having a porosity of 33 

percent, and they are given by α =1.3 and β =1.68. Without 

unconsolidation correction, the time average equation predicts a 

velocity of 2.81 km/s for 33 percent sediment porosity with 30 

percent clay volume content, while the MTAE2 predicts a velocity of 2.16 km/s. The velocities predicted from WE are very 

close to those calculated from MTAE; however, the theoretical 

velocities from MTAE1 are less than those from WE or MTAE2 

for porosity less than 33 percent and greater than those from WE 

and MTAE2 for porosity greater than 33 percent. As is shown 

later, each equation, MTAE1 or MTAE2, has its own advantage 

in predicting acoustic velocities from resistivities. 

Predicting Velocity from Resistivity

Velocities of hydrate-bearing sediments can be predicted 

from resistivity logs with or without knowing gas concentrations 

in the pore space. 

Without Gas Hydrate Concentration

The basic equation predicting the acoustic values from the 

resistivities is equation (7). Because the gas hydrate concentrations are not explicitly utilized in the method, the porosity in 

equation (7) can be considered as the water-filled porosity. One 

well-known method is fitting acoustic values to resistivity values 

using the least squares method (Kim, 1964; Rudman and others, 

1975). As indicated in equation (7), the acoustic values can be 

written in the following way:

 (9)

where parameters A, B, and D can be estimated by the least 

squares method. Notice that in applying equation (9), any 

explicit information such as Archie’s parameters or slowness of 

the constituents of sediments is not required, but acoustic and 

resistivity logs should be available for analysis. Originally Kim 

(1964) proposed a method of obtaining three parameters by solving three nonlinear simultaneous equations from three resistivity-acoustic values. This is similar to the least squares method 

mentioned here, where all available resistivity-acoustic values 

are used in the least squares method. Because the selection of 

three pairs of resistivity-acoustic values is subjective, Kim’s 

approach can be considered as a weighted least squares method.

An example of the application of this method is shown in 

figure 3. Figure 3A shows the least squares fitting curve with 

acoustic and resistivity data, and figure 3B shows the comparison of measured versus predicted acoustic values with depth. 

Because the prediction is done using the least squares method, 

the difference of the average acoustic values between the measured and predicted is zero. 

Another method of predicting acoustic values from resistivity logs is to use MTAE1 or MTAE2 with explicit information about the resistivity log and physical properties of 

corresponding sediments. The required parameters are m and Q

from the resistivity log and slowness (inverse of velocity) of 

modified matrix and unconsolidation constants for the acoustic 

properties of sediments. In principle, this method is similar to 

Kim’s method, whereby the parameters were derived from the 

physical properties of sediments rather than estimated using 

least squares method. Kim’s parameters A, B, and D can be 

identified as

 (10)

Figure 4 shows the predicted acoustic values using MTAE1 

and MTAE2 with parameters shown in the previous section. 

S ARt

B = + D

A Sw–Sm ( )η1Q–1 ⁄ m, B= –1 ⁄ m D, η2Sm = =
4 Methods of Generating Synthetic Acoustic Logs from Resistivity Logs for Gas-Hydrate-Bearing Sediments

0.1 1 10 100 1000

0.1

1

Mallik 2L-38

Gas-hydrate-bearing

sediments

Archie's equation with

a=0.62, m =2.15, Rw = 0.4 ohm-m

Non-gas-hydrate-bearing

sediments

RESISTIVITY, IN OHM-METERS

POROSITY

Overall the prediction using MTAE1 is better than that using 

MTAE2. The agreement between the prediction of MTAE1 and 

the measured acoustic values for the hydrate-bearing sediments 

(acoustic values less than about 0.4 s/m) is good. Both equations, MTAE1 and MTAE2, overestimate acoustic values for 

high acoustic values greater than about 0.45 s/m. 

With Gas Hydrate Concentration

Gas hydrate concentration from the resistivity log can be 

obtained from Archie’s equation and is given by:

 (11)

where Ch is the gas hydrate concentration and Cw is the water 

saturation. Using MTAE1 or MTAE2 given in equation (5), 

acoustic values can be predicted explicitly plugging the gas 

hydrate concentration into MTAE’s. Figure 5 shows the result of 

predicted acoustic utilizing MTAE1 and MTAE2 with gas 

hydrate concentrations estimated from the resistivity log using 

a = 1.02, m =1.95, n = 1.9386, and Rw = 0.4 ohm-m. As opposed 

to the results shown in figure 4, the acoustic log predicted from 

MTAE2 is better than that from MTAE1. 

Ch= 1 Cw – 1

aRw

φ

mRt

-------------

1 ⁄ n

= 1 – 1

Qφ

mRt

------------------

1 ⁄ n

= –

WE can also be used to predict acoustic velocity if gas concentrations are available. Because the P-wave behavior is similar to that in MTAE2, the predicted values from WE are close to 

those from MTAE2. Figure 6 shows a cross plot of predicted 

acoustic values from WE versus those from MTAE2. Except at 

low slowness less than 0.3 s/m, the two predictions are almost 

identical. 

Discussion

The least squares method or Kim’s approach is well known 

for predicting acoustic logs from resistivity logs (Kim, 1964; 

Rudman and others, 1975) and works well, as indicated in figure 

3. However, the problem with this method is that it provides for 

no control over the particular characteristics of the well site, 

such as the salinity of connate water, cementation factor in the 

Archie’s constant, or the unconsolidation factor of the sediments. As shown in equation (10), the least squares parameters 

A, B, and D are related to the physical properties of sediments—

but this information was not included in the least squares 

method. 

When predicting acoustic from resistivity data without 

explicit use of gas hydrate concentration, using MTAE1 seems 

to give better results than using MTAE2. MTAE1 is simple to 

use, and most of the constants, except the unconsolidation factor 

α, necessary for the prediction, can be estimated from resistivity 

Figure 1. Electrical resistivity measured at the Mallik 2L-38 gas hydrate research well with respect to density porosity for depth range of 897 m to 1,109 m. Open circles, resistivity values for non-hydrate-bearing sediments; solid dots, 

hydrate-bearing sediments. Straight line is the Archie’s equation with a =1.02, m =1.95, and Rw = 0.4 ohm-m.
Discussion 5

0.0 0.2 0.4 0.6

2

4

5

3

1

S= (Sf-Sm ) +Sm

S= α

β

(Sf-Sm ) +αSm

POROSITY

VELOCITY, IN KILOMETERS PER SECOND

 Modified time-average equation 1 with α=1.3

 Modified time-average equation 2 with β=1.68

 Weighted equation (WE) 

φ

φ

and density logs. The advantage of MTAE1 over MTAE2 when 

estimating acoustic logs without using the gas hydrate concentration comes from the fact that MTAE1 is a better approximation of the velocities of hydrate-bearing sediments with respect 

to the water-filled porosity as shown by results from data 

obtained at the Mallik 2L-38 well site. The P-wave velocities of 

hydrate-bearing sediments are less than the P-wave velocities of 

non-hydrate-bearing sediments at the same water-filled porosities. Because of this, MTAE1 is a better approximation of the 

velocities of hydrate-bearing sediments.

However, when gas hydrate concentrations are explicitly 

used in the prediction, MTAE2 is better than MTAE1, because 

MTAE2 is a better approximation to WE than MTAE1. As indicated in Lee and others (1996), WE describes the behavior of 

non-hydrate-bearing sediments accurately, and the prediction of 

velocities from WE for hydrate-bearing sediments is accurate. 

Therefore, using MTAE2 with explicit hydrate concentration is a 

better method than using MTAE1. As shown in figure 5, using 

MTAE1 with explicit gas hydrate concentrations overestimates 

the acoustic values for hydrate-bearing sediments.

Comparison of three methods, least squares (LSM), 

MTAE1 without gas hydrate concentrations, and MTAE2 with 

gas hydrate concentrations is shown in figure 7, where measured 

acoustic values are plotted against the predicted acoustic values 

with the least squares linear fitting curves. The slope and intercept of the linear equation contain the information of the overall 

performance of the various methods. Based on figure 7, we can 

say that overall, MTAE1 without gas hydrate concentrations 

works best for the Mallik 2L-38 well. However, when S>0.4 

s/m, LSM works best, and when S<0.4 s/m, MTAE2 with gas 

hydrate concentrations works best for these data. 

The only unknown parameter not able to be estimated from 

the resistivity log in applying MTAE1 or MTAE2 is the unconsolidation parameter. If a few acoustic values for non-hydratebearing sediments are available close to the hydrate-bearing 

zone, we can estimate the unconsolidation constants by matching the velocities at the particular porosity (the average porosity 

is optimum) as explained in the previous section.

The fractional error of predicted acoustic values caused by 

the error in the unconsolidation constant in MTAE1 can be 

derived from equation (7), and it is given by

 (12)

For MTAE2, the error function can be written as 

 (13)

The fractional error in the predicted acoustic value is proportional to the fractional error in unconsolidation constants. In 

∆S

S

------ ∆α

α = -------

∆S

S

------ ∆β

β

------ 1

Sm

S – ------     = =

Figure 2. Comparison of P-wave velocities computed from weighted equation with W = 1.44 and modified time average 

equations with α = 1.3 and β = 1.68. Unconsolidation parameters were computed at 33 percent porosity.
6 Methods of Generating Synthetic Acoustic Logs from Resistivity Logs for Gas-Hydrate-Bearing Sediments

900 950 1000 1050 1100

0.2

0.3

0.4

0.5

0.6

B

Mallik 2L-38

DEPTH, IN METERS

SONIC, IN SECONDS PER METER

 Observed

 Predicted from resistivity using LSM

1 10 100 1000

0.1

1

A

S=1.225 R-0.03878 -0.743

Mallik 2L-38

RESISTIVITY, IN OHM-METERS

SONIC, IN SECONDS PER METER

 Observed

 Least squares fitting 

order to evaluate the range of unconsolidation error introduced 

in the prediction, unconsolidation constants are computed from 

the definitions shown in equations (3) and (4) and are plotted 

with respect to the porosity and weights of WE (W = 0.75, W = 

1.0, W = 1.25, and W = 1.5) in figure 8. As W increases, the 

unconsolidation constants increase. For a given weight, the 

variation of unconsolidation constant β is in the range of 0.1 

and is between 0.1 (W = 0.75) and 0.2 (W = 1.5) for α. This 

Figure 3. Prediction of acoustic values from resistivity log at Mallik 2L-38 well using least squares method. A, Open circles, measured resistivity-acoustic values; solid circles, least squares curve. B, Measured 

(line) and predicted (open circle) acoustic values with respect to depth.
Discussion 7

900 950 1000 1050 1100

0.2

0.3

0.4

0.5

0.6

B

Mallik 2L-38

DEPTH, IN METERS

SONIC, IN SECONDS PER METER

 Observed P-wave sonic

 Predicted from resistivity using MTAE2

 without gas hydrate concentration

900 950 1000 1050 1100

0.2

0.3

0.4

0.5

0.6

A

Mallik 2L-38

 Observed P-wave sonic

 Predicted from resistivity using MTAE1

 without gas hydrate concentration 

DEPTH, IN METERS

SONIC, IN SECONDS PER METER

figure indicates that the assumption that unconsolidation parameters are constants irrespective of porosity is reasonable. If we 

assume that the error in α is 0.1 and α varies 1.2 to 1.4, then the 

possible fractional error in the acoustic value is 7 percent to 8.5 

percent. 

The predicted acoustic error owing to the error in β is less 

than that from the error in α. Let’s assume that the minimum 

value of Sm/S is about 0.5. Then the fractional error ∆S/S 

caused by ∆β/β is about half the error caused by ∆α/α . The 

hatched regions in figure 8 show values of the weight-porosity 

pairs which yield fractional errors in acoustic values less than 

about 7 percent if α =1.3 and β =1.7 are used for unconsolidation constants for the prediction. 

Pressure-temperature conditions control the occurrence of 

hydrate-bearing sediments. Because of the limited conditions of 

gas hydrate stability, hydrate-bearing sediments occur in shallow depths, usually 500 to 1,500 m in permafrost regions and 

200-1,000 m sub-bottom depth in deep marine environments. 

Figure 4. Prediction of acoustic values from resistivity log at Mallik 2L-38 well using modified 

time average equations without gas hydrate concentration. A, using MTAE1; B, using MTAE2.
8 Methods of Generating Synthetic Acoustic Logs from Resistivity Logs for Gas-Hydrate-Bearing Sediments

900 950 1000 1050 1100

0.2

0.3

0.4

0.5

0.6

B

Mallik 2L-38

DEPTH, IN METERS

SONIC, IN SECONDS PER METER

 Observed P-wave sonic

 Predicted from resistivity using MTAE2

 with gas hydrate concentration 

900 950 1000 1050 1100

0.2

0.3

0.4

0.5

0.6

A

Mallik 2L-38

 Observed P-wave sonic

 Predicted from resistivity using MTAE1

 with gas hydrate concentration 

SONIC, IN SECONDS PER METER

DEPTH, IN METERS

Therefore the range of porosity and the weight of WE applicable 

to hydrate-bearing sediments could be small; for example, at the 

Mallik 2L-38 well, porosity varies between 20 percent and 40 

percent, and weight of WE varies between 1.44 and 1.6 depending on the clay content (Lee and Collett, 1999). 

Because of the shallow occurrence of hydrate-bearing 

sediments in the permafrost area, porosities and weights of 

WE may fall within the hatched region of figure 8. Therefore it is reasonable to assume that the unconsolidation constants estimated from the data at the Mallik 2L-38 well can be 

Figure 5. Acoustic values predicted from resistivity log at Mallik 2L-38 well using modified time average equations with gas hydrate concentration. A, using MTAE1; B, using MTAE2.
Conclusions 9

0.2 0.3 0.4 0.5 0.6

0.2

0.3

0.4

0.5

0.6

Mallik 2L-38

St =1.05Sw-0.01

PREDICTED SONIC USING WE, IN SECONDS PER METER

PREDICTED SONIC USING MTAE2, IN SECONDS PER METER

applied to other permafrost areas in the Arctic region without 

rendering significant errors to the prediction of acoustic 

values.

Conclusions

Methods of predicting acoustic logs from resistivity logs 

using modified time average equations are presented, for 

hydrate-bearing sediments. Unlike some other methods, clay 

terms are not included explicitly in the formulation, because it is 

assumed that the effect of the clay is manifested in the Archie’s 

constants and in the velocity of the modified matrix. All the 

parameters necessary for the transform can be estimated from 

the resistivity log except the unconsolidation constants. 

When gas hydrate concentrations were not explicitly used 

in the prediction, using MTAE1 is better than using MTAE2, but 

using MTAE2 is better than using MTAE1 when gas hydrate 

concentrations were explicitly used in the prediction.

The range of unconsolidation constants to be used for 

hydrate-bearing sediments is small because the depths of 

occurrence of hydrate-bearing sediments are shallow, 

restricted by the gas hydrate stability condition. Therefore 

the unconsolidation constants estimated at the Mallik 2L-38 

well might be appropriate throughout the permafrost region 

in transforming resistivity logs into acoustic logs.

References Cited

Archie, G.E., 1942, The electrical resistivity log as an aid in determining 

some reservoir characteristics: Journal of Petroleum Technology, v. 

5, p. 1–8.

Brito Dos Santos, W.L., Ulrych, T.J., and De Lima, O.A.L., 1988, A new 

approach for deriving pseudovelocity logs from resistivity logs: Geophysical Prospecting, v. 36, p. 83–91.

Bussian, A.W., 1983, Electrical conductance in porous medium: Geophysics, v. 48, p. 1258–1268.

Collett, T.S., Lewis, R., Dallimore, S.R., Lee, M.W., Mroz, T.H., and Uchida, 

T., 1999, Mallik 2L-38 downhole well log display-detailed evaluation 

of gas hydrate reservoir properties, in Dallimore, S.R., and others, 

eds., Scientific results from JAPEX/JNOC/GSC Mallik 2L-38 Gas 

Hydrate Research Well, Mackenzie Delta, Northwest Territories, 

Canada: Geological Survey of Canada Bulletin 544, p. 295–311.

Figure 6. Comparison of predicted acoustic values from resistivity log at Mallik 2L-38 

well using WE and MTAE2 with gas hydrate concentration.
10 Methods of Generating Synthetic Acoustic Logs from Resistivity Logs for Gas-Hydrate-Bearing Sediments

0.2 0.3 0.4 0.5 0.6

0.2

0.3

0.4

0.5

0.6

C

Prediction using MTAE2 

with hydrate concentration

PS=0.78TS+0.07

TRUE SONIC, IN SECONDS PER METER

PREDICTED SONIC, IN SECONDS PER METER

0.2 0.3 0.4 0.5 0.6

0.2

0.3

0.4

0.5

0.6

B

Prediction using MTAE1

without hydrate concentration

PS = 0.93TS +0.04

TRUE SONIC, IN SECONDS PER METER

PREDICTED SONIC, IN SECONDS PER METERE

0.2 0.3 0.4 0.5 0.6

0.2

0.3

0.4

0.5

0.6

A

Prediction using 

least squares method

PS =0.70 TS +0.11

TRUE SONIC, IN SECONDS PER METER

PREDICTED SONIC IN SECONDS PER METER

0.1 0.2 0.3 0.4 0.5 0.6 0.7

1.0

1.2

1.4

1.6

1.8

2.0

Sm = 0.2 s/m

W = 0.75

W = 1.0

W = 1.25

W = 1.5

W = 1.5

W = 1.25

W = 1.0

W = 0.75

POROSITY

UNCONSOLIDATION PARAMETERS

α

β

Figure 7. True (measured) acoustic values versus predicted acoustic 

values using LSM, MTAE1, and MTAE2. Open circles, true and predicted 

acoustic values; straight lines, least square fitting curves. A, LSM without gas hydrate concentration. B, MTAE1 without gas hydrate concentration. C, MTAE2 with gas hydrate concentration.

Figure 8. Theoretical unconsolidation constants with respect to porosity and weights of WE. Slowness of modified matrix of 0.2024 s/m is used. 

Hatched areas indicate weight-porosity pairs, which yield the fractional 

error of the predicted acoustic value to be less than about 7 percent if α

=1.3 and β =1.7 are used for unconsolidation constants.
References Cited 11

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