A Method of Shaly Sand Correction for 

Estimating Gas Hydrate Saturations Using 

Downhole Electrical Resistivity Log Data

U.S. Department of the Interior

U.S. Geological Survey

Scientific Investigations Report 2006-5121
A Method of Shaly Sand Correction for 

Estimating Gas Hydrate Saturations Using 

Downhole Electrical Resistivity Log Data

By Myung W. Lee and Timothy S. Collett

Scientific Investigations Report 2006–5121

U.S. Department of the Interior

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

Dirk Kempthorne, Secretary

U.S. Geological Survey

P. Patrick Leahy, Acting Director

U.S. Geological Survey, Reston, Virginia: 2006

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Suggested citation:

Lee, M.W., and Collett, T.S., 2006, A method of shaly sand correction for estimating gas hydrate saturations using 

downhole electrical resistivity log data: U.S. Geological Survey Scientific Investigations Report 2006-5121, 10 p.
iii

Contents

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

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

Theory..............................................................................................................................................................1

Modeling..........................................................................................................................................................3

Example of Saturation Computation ...........................................................................................................5

Real Data Example.........................................................................................................................................7

Analysis of the Method.................................................................................................................................7

Condition for the Linear Approximation ............................................................................................7

Constraint on Cementation Constant m ............................................................................................8

Error Analysis ........................................................................................................................................8

Summary of the Proposed Method....................................................................................................9

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

Acknowledgments.......................................................................................................................................10

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

Figures

1. Electrical resistivity versus porosity for various clay contents ............................................3

2. Logarithm of porosity versus logarithm of resistivity to show linear relationship.............4

3. Relationship between resistivity of clay and product of connate-water resistivity

and clay volume ............................................................................................................................5

4. Relationship between formation resistivity and porosity.......................................................6

5. Estimated gas hydrate saturation versus true gas hydrate saturation ...............................6

6. Clay-corrected resistivity versus observed resistivity at Mallik 2L-38 well,

Mackenzie Delta, Canada ...........................................................................................................7

7. Relative errors in estimating conductivity contributed by clay versus porosity................9

8. Ratio of water saturation versus formation resistivity times clay term for different relative 

errors in clay term ........................................................................................................................9

Tables

1. Archie constants a* and m* for shaly sands ...........................................................................3
Abstract

Estimation of the amount of nonconductive and conductive constituents in the pore space of sediments, using 

electrical resistivity logs, generally loses accuracy when clays 

are present in the reservoir. Many different methods and clay 

models have been proposed to account for the conductivity of 

clay (for example, the shaly sand correction). In this study, the 

Simandoux model is employed to correct for the clay effect in 

order to more accurately estimate gas hydrate saturations.

This study utilizes the fact that the effect of clay on the 

resistivity of a sediment is manifested in the Archie constants a

and m, values of which are generally a = 1 and m = 2 for cleansand reservoirs. Results of the study indicate that as the clay 

content increases, a also increases whereas m decreases. On the 

basis of the relationship between the Archie constants a and m

with respect to the clay amount, a method of correcting for the 

clay effect on the estimation of water saturation is proposed. 

This method works well if the relationship between porosity 

and resistivity on a log-log plot is approximately linear and if 

accurate Archie constants a and m for clean sand are known. 

However, because of the linearity condition, it is difficult to 

apply the method to low-porosity reservoirs. Gas-hydratebearing sediments generally have high porosities because of 

their shallow depth of occurrence, so the method can be effectively applied in estimating gas hydrate saturations.

Introduction

Most methods of electrical resistivity interpretation are 

based on Archie’s empirical law (Archie, 1942), which works 

well for the estimation of water and nonconducting-constituent 

saturations in the pore space of clean sands, where formation 

matrices are poor conductors. However, Archie’s law is not 

accurate in estimating water saturation for shaly sands, where 

clay minerals are present in the formation matrices. (Note: 

“Shaly sand” is a commonly used term for a clay-bearing sand; 

see Worthington, 1985.) Because clay minerals have high conductivities, in some cases higher than the conductivity of water 

in the formation, the effect of clay on electrical resistivity can 

be significant. Thus, the estimation of water saturation in the 

pore space using electrical resistivity log data is generally inaccurate because the resistivity from the conducting clay material 

was not accounted for.

In order to correct for clay conductivity’s effects on 

formation resistivity, a number of clay models have been proposed; of these, Worthington (1985) summarized all available 

shaly sand models. One of the earlier models was based on the 

assumption that the conductivity of an aggregate of conductive 

particles saturated with conducting fluid can be represented by 

resistors in parallel (Wyllie and Southwick, 1954). Simandoux 

(1963) used this concept and proposed a shaly sand model that 

shows the conductivity of the formation to be the sum of the 

conductivity through the water and the conductivity through the 

clay minerals (hereafter referred to as the Simandoux model). 

Other models include (1) a shaly sand model by Waxman and 

Smits (1968), based on the fact that clay particles contribute 

exchange cations to the electrolyte, thereby increasing the 

conductivity of the formation; and (2) a dual-water model by 

Clavier and others (1977), based on the assumption that the 

exchange cations contribute to the conductivity of clay-bound 

water that is spatially separated from the bulk water.

All models have essentially two components—one for the 

conductivity of the water in the formation and the other for the 

conductivity of the clay. The difference among the models is 

the way the clay conductivity is computed.

In our study, the Simandoux model was used to account 

for the effects of the clay conductivity on the formation 

resistivity. On the basis of Archie’s law and the Simandoux 

equation, a relationship between the amount of clay conductivity and the Archie constants is established, and a method 

of correcting for the clay effect on the estimation of water 

saturations is proposed.

Theory

The electrical resistivity of clean sands, saturated with 

100 percent water (R0), can be expressed through the use of the 

Archie equation (Archie, 1942) in the following way:

A Method of Shaly Sand Correction for Estimating Gas 

Hydrate Saturations Using Downhole Electrical

Resistivity Log Data

By Myung W. Lee and Timothy S. Collett
1 1

R a 0 R = -m f w , (1)

where R0 is the formation resistivity, Rw is the resistivity of 

the connate water, a and m are Archie constants, and φ is the 

porosity. Archie constants a and m can be derived empirically; 

m is related to the cementation of the sediment and is commonly called the cementation factor. Equation 1 indicates that 

a plot of log φ versus log R0 is linear and the slope is given 

by m. The linear relationship between log φ and log R0 was 

considered to be the most important property for the purposes 

of our study, and the relationship is widely used in well-log 

analyses (Pickett, 1966).

The electrical conductivity of an aggregate of conductive particles saturated with a conducting electrolyte can be 

represented by resistivity elements in parallel (Wyllie and 

Southwick, 1954). Therefore, the effect of clay in sediments 

(Simandoux, 1963) can be formulated as

1 1 1 1 1 1

R a 0 R

V

R FR Q FR

= + m FR Q - - o + = +

f

f

w

c

c w

c

w

w c

( ) ( )

, (2)

where F is a formation factor for clean sands (F = aφ–m), Vc is 

the volume fraction of clay in solid matrix, Rc is the resistivity 

of the clay, and Qc is the effective clay conductivity:

Q V c c = - Rc ( ) 1 f / . (3)

The effective clay conductivity (Qc) is defined in this 

report as the clay effect on the resistivity measurement. From 

equation 2, the porosity of the sediments using clean-sand 

Archie parameters (a, m) is given by

f- = -

m R

aR Q R

0

w c 1 0 ( ) . (4)

Because the porosity is a positive number, (1 – QcR0) should 

be greater than zero. Furthermore, because (1 – QcR0) is less 

than 1, porosities estimated by using equation 4 are greater 

than true porosities; thus, for shaly sands, the porosity calculation from a resistivity measurement is higher than the true 

porosity, if the clay correction is not applied.

As demonstrated in the next section, the cross plot of log 

φ versus log R0 is approximately linear even for shaly sands 

under certain conditions. If a* and m* are defined for the 

Archie parameters derived from shaly sands by linearly fitting 

the logarithm of resistivities and logarithm of porosities, equation 2 can be written as

1 1 1

R a 0 R F R = = * * -m * f w w , (5)

where F* is the formation factor for shaly sands and is given by 

F* = a*φ–m*. Combining equations 2 and 5 yields the following:

1 1

a R aR * * m m Q

w w

c f f - = + -

. (6)

From equation 6, a relationship between parameters for clean 

and shaly sands can be written as

a

a

aQ R

m

m

m

* *

( ) f ff

-

-

= - 1 + c w . (7)

Therefore, the conductive term can be written as

Q a a

aa R

m m m

c

w

= - - f f * * *

*

( )

. (8a)

Generally, the Archie parameters for a clean sand are 

chosen to be a = 1 and m = 2. The deviations of Archie parameters a* and m* from a = 1 and m = 2 for shaly sands represent 

the effect of clay on the formation resistivity measurement. If 

the values a = 1 and m = 2 are assumed for the clean sand, the 

conductive term can be written as

Q a R a m m

c = - w

- f f * * * * ( ) 1 / 2

. (8b)

Equations 8a and 8b indicate that the effect of clay on the 

resistivity can be computed if Archie parameters a* and m* 

can be estimated from the resistivity of shaly sands. Because 

Qc should be positive, (a – a*φm–m*) should be positive; thus, 

there is a constraint for Archie parameters a* and m* when 

equations 8a and 8b are used for the clay correction. To have 

a positive value of Qc, the following relationship should be 

satisfied:

m m > + a a * * ln( / )/ lnf . (9)

The formation’s water saturation (Sw) calculated from the 

resistivity log value (Rt) for shaly sand without explicit clay 

correction has been given by Archie (1942) as

S a R

R m

n

w

w

t

= æ

è

ç

ç

çç

ö

ø

÷

÷

÷

÷

*

*

/

f

1

, (10a)

where n is an empirically derived parameter close to 2 and Rt

is the formation resistivity with gas hydrate. The parameter n

varies between 1.715 (unconsolidated sediment) and 2.1661 

(sandstone); although somewhat dependent on the lithology of 

the reservoir, n is typically 1.9386 (Pearson and others, 1983).

On the basis of equation 10a, the water saturation for 

clean sand with a = 1, m = 2, and n = 2 can be calculated as

S

R R w

w t = /

f . (10b)

 A Method of Shaly Sand Correction for Estimating Gas Hydrate Saturations
For shaly sands with explicit clay correction using the 

term Qc, the water saturation can be written from equations 2 

and 10b as

S

R RQ R w

w t c t = ( ) 1- /

f . (11a)

In summary, equations 2, 8, and 11a indicate that the water 

saturation for shaly sands can be computed by estimating the 

effective clay conductivity (Qc) using equation 8, subtracting 

the Qc term from the observed conductivity of the formation 

(inverse of formation resistivity) using equation 2, and determining water saturation using equation 11a.

In the general case, when a ≠ 1, m ≠ 2, and n ≠ 2, water 

saturation can be written as

S aR RQ R m

n

w

w t c t = æ -

è

ç

ç

çç

ö

ø

÷

÷

÷÷

( )/

/

1 1

f . (11b)

Modeling

Figure 1 shows graphs of formation resistivity versus 

porosity, as computed with equation 2; the parameters for the 

model are also shown. In this model, Archie constants for 

clean sands are a = 1 and m = 2. When the resistivity of the 

clay in the formation is much less than the resistivity of the 

connate water (fig. 1A), the log-log plot of resistivity versus 

porosity is approximately linear if the porosity is greater than 

about 30 percent. However, when the resistivity of the clay is 

greater than the resistivity of the connate water (fig. 1B), there 

is no apparent linear relationship between log R0 and log φ.

When porosity is more than 30 percent, the accuracy of 

the linearity between log R0 and log φ is demonstrated in figure 

2 where plots of linear equations computed with least-squares 

methods are shown for resistivity data modeled as in figure 1A. 

Table 1 shows the a* and m* values estimated from the linear 

approximations. As the clay content increases, the data indicate 

that (1) a* increases but m* decreases in this model and (2) the 

linearity between log R0 and log φ degrades but is still good at a 

clay content as high as 75 percent (fig. 2D).

Under what conditions does log R0 versus log φ show 

the linear relationship? As indicated in equation 2, the Archie 

equation for clean sand (Qc = 0) is perfectly linear in the log R0

versus log φ plot. Therefore, it can be inferred from equation 2 

that when FRwQc is smaller than 1, the linear approximation of 

log R0 versus log φ is better for shaly sand. Choosing the maximum value of FRwQc

, one that provides an approximate linear 

relationship between log R0 and log φ, is subjective. In our 

study, the value for the maximum allowable FRwQc was set at 

0.4, corresponding to a value calculated from parameters used 

for the data shown in figure 2D. So the degree of linearity with 

FRwQc < 0.4 is similar or better than that shown in figure 2D.

B

1 10 100 1,000 10,000

1

0.1

0.01

RESISTIVITY, IN OHM-METERS

A

0.1 1 10 100 1,000

RESISTIVITY, IN OHM-METERS

POROSITY, IN FRACTIONS

1

0.1

0.01

POROSITY, IN FRACTIONS

Model parameters

a = 1, m = 2

RW = 10 Ω⋅m

RC = 5 Ω⋅m

Model parameters

a = 1, m = 2

RW = 0.35 Ω⋅m

RC = 5 Ω⋅m

Clay = 50%

Clay = 25%

Clay = 0%

Clay = 75%

Clay = 50%

Clay = 0%

Clay = 75%

Clay = 25%

Figure 1. Graphs showing electrical resistivity versus porosity 

for various clay contents that were calculated from the Simandoux 

(1963) equation. Archie parameters for clean sand are a = 1 and 

m = 2. A, Results when resistivity of the connate water (Rw) is 

lower than resistivity of the clay (Rc) (Rw = 0.35 Ω·m, Rc = 5 Ω·m). 

B, Results when resistivity of the connate water is higher than 

resistivity of the clay (Rw = 10 Ω·m, Rc = 5 Ω·m).

Table 1. Archie constants a* and m* for shaly sands.

[These constants were derived by least-squares fitting of log-porosity and 

log-resistivity shown in figure 2]

Lithology a* m*

Clean sand 1.00 2.00

25 percent clay content 1.03 1.89

50 percent clay content 1.06 1.79

75 percent clay content 1.09 1.70

Modeling 
RESISTIVITY, IN OHM-METERS

POROSITY, IN FRACTIONS

Clay content = 0%

log φ = −0.500 log R0 – 0.228

0.1

1

0.1 1 10

A

RESISTIVITY, IN OHM-METERS

POROSITY, IN FRACTIONS

0.1

1

0.1 1 10

B RESISTIVITY, IN OHM-METERS POROSITY, IN FRACTIONS

0.1

1

0.1 1 10

D

RESISTIVITY, IN OHM-METERS

POROSITY, IN FRACTIONS

C

0.1 1 10

0.1

1

Clay content = 25 %

log φ = −0.529 log R0 – 0.234

Clay content = 75 %

log φ = −0.588 log R0 – 0.247

Clay content = 50%

log φ = −0.558 log R0 – 0.240

 A Method of Shaly Sand Correction for Estimating Gas Hydrate Saturations
Figure 3 shows the ranges of connate-water resistivities, 

clay resistivities, and clay contents (termed clay volumes, Cv), 

with respect to the minimum porosity of interest, that result in 

an approximate linear relationship between log R0 and log φ. 

The values under the curves approximately satisfy the linearity 

condition. As indicated in figure 3, as the porosity of a sediment 

increases, the log R0 versus log φ relationship is linear under 

many different combinations of resistivity and clay content.

Example of Saturation Computation

In order to evaluate the feasibility and the accuracy of the 

proposed method of computing water saturations with equation 11, the model we used consists of 500 random porosities 

uniformly distributed over a range of 20–40 percent and random gas hydrate saturations uniformly distributed over a range 

of 0–80 percent. The resistivity of gas hydrate is assumed to 

be infinite (similar to hydrocarbons), the resistivity of connate 

water is 0.4 Ω·m, the resistivity of clay is 20 Ω·m, and the 

clay content is 25 percent of the total solids. Figure 4A shows 

the computed formation resistivities with respect to porosities with a = 1 and m = 2. The solid dots, appearing as a thick 

line in figure 4A, denote the resistivities for 100 percent water 

saturation (base line), and open circles indicate resistivities 

with variable hydrate saturations.

The Archie parameter for shaly sands can be estimated by 

applying the least-squares method to the base-line data (solid 

dots shown in fig. 4A). The linear approximation of the baseline values gives a* = 1.08 and m* = 1.90. Note that the effect 

of clay on the resistivity manifests itself as a* being more than 

1 and m* being less than 2 (table 1). Through the use of equation 11, the water saturations are computed from the resistivity values shown in figure 4A. To evaluate the efficiency and 

accuracy of the method, the corrected resistivities of sediments 

with clay are shown as solid dots in figure 4B. This evaluation 

used water-filled porosity (φw), defined by

f f w = - h ( ) 1 S , (12)

where Sh is the gas hydrate saturation. Because the solid dots 

in figure 4B represent resistivities of sediments without clay 

contribution (the contribution of clay on the formation resistivity was eliminated), the Archie constants should be a* = a = 

1 and m* = m = 2, which are the Archie parameters for clean 

sands. The values of Archie constants from the least-squares 

fitting of solid dots in figure 4B agree with the prediction.

The purpose of a clay correction for the electrical resistivity is to accurately estimate the amounts of nonconducting 

constituents in the pore space, such as gas, oil, or gas hydrate. 

Therefore, it is desirable to analyze the proposed method relative to gas hydrate saturation. Figure 5A shows the estimated 

gas hydrate saturation (1 – Sw) without the clay correction relative to true saturation based on the Archie equation for shaly 

sand using equation 10a with a* = 1.08, m* = 1.9, and n = 

1.9386. As indicated, the estimated gas hydrate saturations are 

less than the true hydrate saturation (about 7 percent less). Figure 5B shows the comparison of gas hydrate saturation after 

applying a clay correction to the resistivity and reveals that 

the estimated gas hydrate saturations are accurate. Thus, this 

model study indicates that the method of using equation 11 for 

the correction of clay on the electrical resistivity is feasible.

Figure 2 (previous page). Plots of logarithm of the porosity 

versus logarithm of the resistivity to show the linear relationship. 

The model parameters are identical to those for figure 1A, which 

are a = 1, m = 2, water resistivity (Rw) = 0. 35 Ω·m, and clay 

resistivity (Rc) = 5 Ω·m. Plots are for A, clean sand; B, 25 percent 

clay; C, 50 percent clay; and D, 75 percent clay.

0 10 20 30 40 50

RESISTIVITY OF CLAY (RC),

IN OHM-METERS

2

4

6

8

10

-2

0

RESISTIVITY OF CONNATE WATER × CLAY VOLUME (

RW

CV),

IN OHM-METERS

φ = 50%

φ = 30%

φ = 10%

(1 – φ)φ−2 RWCV < 0.4RC

Figure 3. Graph showing relationship between the resistivity 

of clay (Rc) and the product of connate-water resistivity (Rw) 

and clay volume (Cv) required to yield an approximate linear 

relationship between log resistivity and log porosity. The 

values under each curve satisfy the linearity condition.

Example of Saturation Computation 
B

1 10 100 1,000

1

0.1

0.01

FORMATION RESISTIVITY, IN OHM-METERS

A

0.1 10 100

FORMATION RESISTIVITY, IN OHM-METERS

WATER-FILLED POROSITY, IN FRACTIONS

0.7

0.1

POROSITY, IN FRACTIONS

Modeled with

RW = 0.4 Ω⋅m, a = 1, m = 2

Porosity 20%–40%

Hydrate concentration 0–80%

Resisivity of sediment

with clay correction

(a* = 1.0, m* = 2.0 & RW = 0.4 Ω⋅m

Gas-hydrate-bearing sediment

Non-hydrate-bearing

sediment

Figure 4. Plots showing relationship between formation 

resistivity and porosity. A, Resistivities versus porosities 

calculated by using a = 1, m = 2, water resistivity (Rw) = 0.4 Ω·m, 

clay volume (Cv) = 25 percent, and clay resistivity (Rc) = 20 Ω·m. 

Porosity is uniformly distributed over a range of 20–40 percent. 

Solid dots (mainly merged into a line) represent resistivities 

for the base line (no gas hydrate saturation), and open circles 

are for the resistivities for gas-hydrate-bearing sediments with 

random hydrate saturation distributed over a range of 0–80 

percent. B, The resistivity versus water-filled porosity after 

application of a clay correction.

B

0.0 0.5 1.0

1.0

0.5

0.0

A

0.0 0.5 1.0

TRUE GAS HYDRATE SATURATION (Sh),

IN FRACTIONS

TRUE GAS HYDRATE SATURATION (Sh),

IN FRACTIONS

ESTIMATED GAS HYDRATE

SATURATION (

S h,est), IN FRACTIONS

ESTIMATED GAS HYDRATE

SATURATION (

S h,est), IN FRACTIONS

0.5

1.0

0.0

With clay correction

Sh,est = 0.979Sh + 0.01

No clay correction

Sh,est = 0.93Sh + 0.02

Figure 5. Plots showing estimated gas hydrate saturation (Sh,est) 

versus true gas hydrate saturation (Sh) for the model shown in 

figure 4A. A, Without clay correction. B, With clay correction.

 A Method of Shaly Sand Correction for Estimating Gas Hydrate Saturations
Real Data Example

Figure 6 shows the clay-corrected resistivity versus the 

observed resistivity from the well log acquired at the Mallik 

2L-38 gas hydrate research well, Mackenzie Delta, Canada. 

The Archie constants for shaly sand used for figure 6 are a* 

= 1.02 and m* = 1.95, and Rw is 0.25 Ω·m; for clean sands, 

the constants are a = 1 and m = 2. Figure 6A shows the result 

when using a constant clay content of 30 percent. As indicated 

in equation 8, the effective clay conductivity (Qc) is a function 

of porosity if Rw, a*, and m* are constant. Therefore, when 

using a constant clay content, the estimated Qc provides an 

average electrical conductivity for the clay. The average Qc

for the result shown in figure 6A is 0.021±0.005 mho·m. By 

using an average porosity of 0.31 and an average clay content 

of 0.324, the average resistivity of clay of about 10 Ω·m can 

be calculated from equation 3. The resistivity of clay estimated 

in our study is comparable to the value that Miyairi and others 

(1999) used in their well-log analysis at Mallik 2L-38 (that 

is, 5 Ω·m). The clay-corrected resistivity was greater than 

the observed resistivity, and the difference between the two 

increased as the measured resistivity increased (overcompensation). These results occurred because a constant clay content 

was used even though sediments having high resistivities are 

cleaner than sediments having low resistivities at this well site.

Figure 6B shows the clay-corrected resistivity when 

using a variable clay content based on the well logs. Clay 

contents were calculated from the gamma log by using the 

formula pertinent to Tertiary clastic sediments (Western Atlas 

International Inc., 1995). Archie constants depend on the clay 

content in sediment, as demonstrated in table 1. In cases where 

the clay content varies, the Archie constants a* and m* are 

linearly interpolated with respect to the clay content. When 

using variable clay contents, the clay-corrected resistivity is 

almost linearly related to the observed resistivity (fig. 6B), but 

is about 20 percent higher.

Analysis of the Method

Reliable estimates of the clay contribution to formation 

resistivity are dependent on a number of important factors, 

such as clay types, the nature of pore space (connected or isolated), the degree of lithologic heterogeneity of the reservoir, 

and the accuracy of resistivity measurements. In this section, 

elements relevant to using equation 11 for determining the 

clay correction are discussed.

Condition for the Linear Approximation

There are many models accounting for the clay effect on 

the resistivity of sediments (for example, Simandoux, 1963; 

Waxman and Smits, 1968; Poupon and Leveaux, 1971; Clavier 

and others, 1977). In our study, we used the Simandoux model, 

which considers the conductivity of the formation to be the sum 

of conductivity of water in clean sand and the conductivity of 

clay. The contribution of clay (that is, the effective clay conductivity, Qc

) to the total conductivity is proportional to the total 

clay content in the matrix and the conductivity of clay. Although 

the explicit expressions of other models are different from the 

Simandoux model, the essence of the expressions is similar. 

For example, Qc in the model of Waxman and Smits (1968) is 

expressed by the product of cation exchange capacity per unit 

volume and equivalence of conductance of the clay, which is 

equivalent to the conductivity of clay material in the formation.

B

1 10 100 1,000

A

1 10 100 1,000

MEASURED FORMATION RESISTIVITY,

IN OHM-METERS

MEASURED FORMATION RESISTIVITY,

IN OHM-METERS

1

100

10

1,000

1

100

10

1,000

CALCULATED FORMATION RESISTIVITY, IN

OHM-METERS (WITH CLAY CORRECTION)

CALCULATED FORMATION RESISTIVITY, IN

OHM-METERS (WITH CLAY CORRECTION)

Variable clay

contents determined

from well-log analysis

Clay = 30%

Least-squares fit

Least-squares fit

from figure 6B

Figure 6. Plots showing clay-corrected resistivity versus 

observed resistivity at the Mallik 2L-38 well, Mackenzie Delta, 

Canada. Archie constants for shaly sand are a* = 1.02 and m* = 

1.95, and for a clean sand they are a = 1 and m = 2. The resistivity 

of connate water was set to 0.25 Ω·m. A, Using constant clay 

contents (average of 30 percent). B, Using variable clay contents 

from well-log analysis.

Analysis of the Method 
The only constraint in using equation 11 for the estimation of clay contribution is the linear nature of log R0

 versus 

log φ. As long as the plot of log R0 versus log φ can be approximated by a linear equation, this clay-correction method can 

be applied irrespective of the clay model. As indicated in the 

previous section, the plot of log R0 versus log φ for 100 percent water-saturated sediments is approximately linear, if the 

quantity FRwQc—which is the ratio of the formation conductivity contributed by the clay to the conductivity contributed 

by the fluid—is less than approximately 0.4. The ranges of 

water conductivity, clay conductivity, and volume of clay that 

satisfy the linear condition for various porosities are shown in 

figure 3. The values under the curves satisfy the linear condition. As shown in figure 3, it is difficult to satisfy the linear 

requirement for low-porosity sediments, particularly those 

with less than 10 percent porosity. Bussian (1983) showed 

that, on the basis of real resistivity measurements, the linear 

condition breaks down when porosity is less than 4 percent. 

Equation 2 shows that at 4 percent porosity, RwCv should be 

less than (6.67 × 10–4)Rc to meet the linearity condition. On the 

basis of the assumption that Rc is 20 Ω·m and Rw is 1 Ω·m, Cv

should be less than about 1.3 percent, which demonstrates that 

the proposed method using Archie constants a* and m* may 

not work well for sediments with less than 10 percent porosity.

Constraint on Cementation Constant m

Because clay increases the conductivity of composite 

material irrespective of m values, the resistivity of shaly sand is 

always less than the resistivity of clean sand at the same porosity. Because of the contribution from clay to the resistivity, the 

slope of the log R0 versus log φ plot decreases as the clay effect 

increases, as shown in figure 1. Therefore, the Archie constant 

m, which is given by the slope of the log R0 versus log φ plot, 

for shaly sand is less than m for clean sand as long as the clay 

content is constant. Consequently, m – m* in equation 8a is 

always positive in this clay model, and equation 9 predicts 

that when a* is greater than 1, m is greater than m*. Equation 

9 also predicts that when a* is less than 1, m is less than m*. 

Although the clay model we propose cannot predict m* greater 

than m or a* less than 1.0, it is generally true that a* is less 

than 1.0 for real data. For example, a* for the Humble equation 

(Labo, 1987) is 0.62, so m should be less than m*.

Equation 9 provides the lower bound for m. The upper 

bound for m can be derived from equation 11, expressed as 

1 – RtQc ≥ 0 ⇒ RtQc ≤ 1. This relationship can be written as

m m

aR a R R a m m

< + [ - ] * ln

ln

( )/( ) * * * *

t

f w tf

f , (13a)

or, alternatively, by using the water saturation shown in equation 11a:

m m

a a S a n

< +* -* * ln( / / )

ln

w

f . (13b)

Where uncertainty exists, a bound for m can be calculated 

from equations 9 and 13. In the case that a = 1 and Sw = 1 

(water-saturated sediments) or high water saturation, the 

upper bound is large, and only the lower bound is important. 

Because both upper and lower bounds of m depend on porosity as well as Archie’s parameters, an accurate estimation 

of m is not practical. However, equations 9 and 13 provide 

some guidelines for choosing m. For example, if a = 1.0, 

a* = 1.6, m* = 1.6, φ = 0.4, and Sw = 0.5, equations 9 and 13 

yield 2.11 < m < 2.42. This inequality indicates that if m = 2 

is used for the clay correction, Qc is negative. Therefore, m

should be greater than 2.11 to apply the proposed method for 

the clay correction.

Error Analysis

Archie constants a = 1 and m = 2 were assumed for 

clean-sand reservoirs in our study. Theoretically, a is always 

1 for the clean sand, but m can vary from 1 to infinity according to the shape of the sediment’s particles (Bussian, 1983). 

The cementation factor m = 2 corresponds to the electrical 

resistivity of the infinitely long rod perpendicular to the 

electric field. One drawback in applying a clay correction 

using Qc is uncertainty in the value for m, when it differs 

from 2 for clean sands. The relative error associated with this 

uncertainty can be written by the following formula, derived 

from equation 8a:

D D Q

Q

a

a a

m

m m

m m

c

c

= -

-

-

-

* *

* *

(ln )

( )

f f

f . (14)

Because (a – a*φm–m*) is a small number, the relative error 

of ∆Qc/Qc for a given error in m (that is, ∆m) is large. Figure 7 

shows relative errors, Qc/Qc, associated with an error in m so 

that ∆m = 0.1 for the clean sand with respect to the porosity 

for the three clay models (clay content = 25, 50, and 75 percent) shown in figure 1A. For most cases, the error is greater 

than 100 percent for a 25 percent clay model. To reduce this 

error to less than 10 percent for a sediment with 30 percent 

porosity, ∆m should be less than about 0.01.

The error in water saturation owing to the error in Qc can 

be derived easily from equation 11 in the case of a = 1 and 

m = 2. By using Sw as the estimated water saturation determined 

from the erroneous Qc and by using Sw as the estimated water 

saturation determined from the correct Qc, then

S

S

R Q

RQ

R Q

RQ

w

w

t c

t c

t c

t c

*

( ) = - - » - - 11

1

2 1

D D

. (15)

Figure 8 shows the ratio of water saturation for two relative errors in Qc

—one is for a relative error of 100 percent, and 

the other is for a relative error of 50 percent. When the relative 

error (∆Qc/Qc) is less than 50 percent, the error in estimating 

water saturation is less than about 20 percent.

 A Method of Shaly Sand Correction for Estimating Gas Hydrate Saturations
Summary of the Proposed Method

Equation 11 is a general formula in the sense that it can 

be applied to any kind of clay model that satisfies the linear condition, because the equation does not depend on the 

explicit expression of the clay model. Rather, it uses only the 

effective clay conductivity term Qc irrespective of the clay 

model. The details of how to calculate the conductivity of clay 

and how to estimate the effect of clay on the resistivity of the 

formation depend on the specific clay model. However, our 

proposed method—which is based on Archie parameters a* 

and m* for estimating the effective clay conductivity in order 

to estimate water saturation—is independent of any explicit 

clay model. This aspect is different from other methods. For 

example, that used by Waxman and Smits (1968) requires 

parameters specific to the clay model such as cation exchange 

capacity and equivalent conductance of the clay.

Archie constants a* and m* for shaly sand, calculated 

from resistivity versus porosity data with 100 percent water 

saturation, represent average values for the sedimentary interval analyzed. As the clay content in sediments changes, the 

dependence of a* and m* with respect to clay content should 

be incorporated in equation 8 to estimate a reliable Qc. Table 

1 indicates that a* and m* are nearly linear with respect to the 

clay content. The difference between a clay correction using 

a constant clay content and that using a variable clay content 

is shown in figure 6. Therefore, it is possible to use variable 

Archie constants for clay corrections by linearly interpolating 

the Archie parameters between clean and shaly sand.

Conclusions

From model studies based on the Simandoux equation, 

we propose a method of correcting the clay contribution to 

resistivity measurements. In this method, Archie constants a* 

and m* for shaly sands are used to compute clay conductivity 

from the resistivity log under the assumption of known Archie 

parameters for clean sands. To apply the method, the plot of 

0.0 0.3 0.6

POROSITY, IN FRACTIONS

2.0

1.0

1.5

0.5

0.0

RELATIVE ERROR IN CLAY TERM (

∆

QC /QC)

Clay = 25%

Clay = 75%

Clay = 50%

Figure 7. Graph showing relative errors in estimating the 

conductivity contributed by clay versus porosity. The parameters 

for various clay contents are identical to those for figure 1A, and 

the error in m, ∆m, is 0.1.

0.0 0.2 0.4

FORMATION RESISTIVITY × CLAY TERM (RtQc)

1.50

1.00

1.25

0.75

0.50

RATIO OF WATER SATURATION (

S*W/SW)

∆Qc /Qc = 1.0

∆Qc /Qc = 0.5

∆Qc /Qc = –0.5

∆Qc /Qc = –1

Figure 8. Graph showing the ratio of water saturation versus the 

formation resistivity times the clay term (RtQc) for different relative 

errors in the clay term (Qc). Sw is the water saturation estimated 

from the resistivity when there is no error in Qc, and Sw* is the 

estimated water saturation when there is a relative error in Qc. 

Solid dots indicate an error of 100 percent, and open circles are 

errors of 50 percent.

Conclusions 
log R0

 versus log φ should be approximately linear, a condition that is satisfied if the conductivity contributed by the clay 

is less than about 40 percent of the conductivity contributed 

by the fluid. Because of the linear condition, however, the 

method is inadequate for low-porosity reservoirs (those with 

approximately less than 10 percent porosity). The linearity 

requirement also implies that with (1) high water salinities and 

high sediment porosities and (2) low clay contents and low 

clay conductivities, the proposed method is more effective in 

estimating gas hydrate saturation.

The proposed method has been tested only in the case 

that a* is greater than a and m* is less than m. Even though 

applicable to a wide range of reservoir conditions, further 

study is required to evaluate the method’s accuracy and effectiveness in all cases of Archie parameters.

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, the Geological Survey of Canada, and 

United States Geological Survey. We thank Phillip Nelson and 

Timothy Hester for many helpful comments.

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10 A Method of Shaly Sand Correction for Estimating Gas Hydrate Saturations