estimation of saturation exponent from nuclear ... · estimation of saturation exponent from...

Estimation of Saturation Exponent from NuclearMagnetic Resonance (NMR) Logs in LowPermeability Reservoirs

Liang Xiao • Zhi-qiang Mao • Gao-ren Li • Yan Jin

Received: 26 November 2011 / Revised: 3 May 2012 / Published online: 3 June 2012

� Springer-Verlag 2012

Abstract The resistivity experimental measurements of 36 core samples, which

were drilled from low permeability reservoirs of southwest China, illustrate that the

saturation exponents are not agminate, but vary from 1.627 to 3.48; this leads to a

challenge for water saturation estimation in low permeability formations. Based on the

analysis of resistivity experiments, laboratory nuclear magnetic resonance (NMR)

measurements for all 36 core samples, and mercury injection measurements for 20 of

them, it was observed that the saturation exponent is proportional to the proportion of

small pore components and inversely proportional to the logarithmic mean of NMR T2

spectrum (T2lm). For rocks with high proportion of small pore components and low

T2lm, there will be high saturation exponents, and vice versa. The proportion of small

pore components is characterized by three different kinds of irreducible water satu-

rations, which are estimated by defining 30, 40 and 50 ms as T2 cutoffs separately. By

integrating these three different kinds of irreducible water saturations and using T2lm, a

technique of calculating the saturation exponent from NMR logs is proposed and the

corresponding model is established. The credibility of this technique is confirmed by

L. Xiao (&)

Key Laboratory of Geo-detection, China University of Geosciences, Beijing, Ministry of Education,

No. 29, Xueyuan Road, Haidian, Beijing 100083, People’s Republic of China

e-mail: [email protected]

Z. Mao

College of Geophysics and Information Engineering, China University of Petroleum,

Beijing, People’s Republic of China

G. Li

Research Institute of Exploration and Development, Changqing Oilfield Company,

PetroChina, Shaanxi, People’s Republic of China

Y. Jin

Southwest Oil and Gas Field Branch Company, PetroChina,

Sichuan, People’s Republic of China

123

Appl Magn Reson (2013) 44:333–347

DOI 10.1007/s00723-012-0366-1

Applied

Magnetic Resonance

comparing the predicted saturation exponents with the results from the core analysis.

For more than 85 % of core samples, the absolute errors between the predicted satu-

ration exponents from NMR logs and the experimental results are lower than 0.25.

Once this technique is extended to field application, the accuracy of water saturation

estimation in low permeability reservoirs will be improved significantly.

1 Introduction

Water saturation (thus related to hydrocarbon saturation) is an indispensable input

parameter in formation evaluation, and it also plays a very important role in

reservoir development program formulation. Generally, water saturation is calcu-

lated using Archie’s equations after the necessary parameters have been obtained

[1]. Archie’s equations can be expressed as Eqs. (1) and (2):

F ¼ R0

Rw

¼ a

/m ð1Þ

Ir ¼Rt

Ro

¼ b

Snw

ð2Þ

where R0 is the rock resistivity at full water saturation, Rt the true formation

resistivity, Rw the formation water resistivity, the units of which are X m, F the

formation factor, Ir the resistivity index, / the porosity in fraction, a and b the

lithology factors, m the cementation exponent, Sw the water saturation in fraction,

and n is the saturation exponent.

Combining with Eqs. (1) and (2), a derivative expression can be written as follows:

Sw ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

a� b� Rw

/m � Rt

n

s

: ð3Þ

From Eq. (3), it can be observed that the values of a, b, m, n, Rw, / and Rt must

be obtained first for the water saturation calculation, / and Rt can be acquired from

conventional logs [2–4], and Rw can be checked from the formation water salinity

using Schlumberger’s log interpretation charts [5].

2 Determination of the Values of a, b, m and n

To calculate water saturation from conventional logs using Archie’s equation, the

determinations of the values of a, b, m and n are crucial. Generally, the

determinations of a, b, m and n rely on the resistivity experimental measurements of

the target core samples. To obtain the necessary resistivity experimental data, the

needed procedures should be applied as follows: (1) every waterless core sample is

saturated using the used saline water, and the rock resistivity R0 at full water

saturation is measured; in this study, the salinity of the used saline water is

13.00 mg/l. (2) The oil is used as the displacing medium, and the centrifugal method

is used to vary the water saturation (Sw) of core samples, and the corresponding rock

334 L. Xiao et al.

123

resistivity Rt of every core sample under different water saturations are measured.

(3) R0, Rt and Sw are collected as a data set to obtained the value a, b, m and n.

For conventional reservoirs, after the representative core samples were drilled

from the intended intervals for the resistivity experiment, the fixed values of a, b,

m and n can separately be obtained from the cross plots of the porosity with the

formation factor, and the water saturation with the resistivity index using the power

function. However, for low permeability sands, not rigorous power function exists

between the porosity and the formation factor, the water saturation and the resistivity

index due to the complicated pore structure and the strong heterogeneity [6]. Wang

and Sharma [7] and Mao et al. [8, 9] had proposed that the tendency of porosity and

formation factor would be changed when the porosities of core samples are lower

than 9.0 %, and they had demonstrated that this change is caused by the poor pore

structure of low permeability plug samples. Mao et al. [8] had developed a novel

method to obtain the accurate values of a and m from the porosity using binary

regression. This method has been confirmed to be effective and is used widely [6]. In

the low permeability formations mentioned in this study, a and m can be determined

using Mao’s method precisely. Thus, the technique of determining a and m from the

porosity that has been proposed by Mao et al. [8] is not introduced in this paper.

It is really a challenge to determine the saturation exponent in low permeability

reservoirs, as the cross plot of the water saturation with the resistivity index is

divergent and a fixed saturation exponent is difficult to acquire. Figure 1 shows the

cross plot of the water saturation with the resistivity index of 36 core samples,

which were drilled from low permeability reservoirs of southwest China. It can be

observed that the relationship between the water saturation and the resistivity index

for all core samples is not consistent. The saturation exponent for single core sample

varies from 1.627 to 3.48. In this case, water saturation calculated using the

regressed fixed saturation exponent from all 36 core samples would be inaccurate.

y = 1.0415x-2.0797

R2 = 0.9054

1

10

0.1 1

Water saturation, fraction

Res

istiv

ity e

xpon

ent,

Ir

Fig. 1 Relationship of water saturation and resistivity index for 36 core samples in low permeabilitysands of southwest China

Estimation of Saturation Exponent from NMR Logs 335

123

The best method is to estimate the water saturation using various saturation

exponents along with the target intervals.

3 Influencing Factors of Saturation Exponent in Low Permeability Sandstones

To acquire accurate saturation exponents for water saturation estimation at low

permeability, it is necessary to understand the influencing factors and the variation

of the saturation exponent. Based on the qualitative analysis of the core thin section

and mercury injection capillary pressure experimental data, Mao et al. [8] had point

out that the saturation exponents were related to rock pore structure. However, the

quantitative relationship between them was not established, and an applicable

technique was not proposed. Nuclear magnetic resonance (NMR) logs have a unique

advantage in indicating reservoir pore structure. From the measured NMR T2

distribution, the information of pore size and distribution can be obtained [10–14].

Rocks with macropore and good pore structure will display long transversal

relaxation time, and wide T2 distribution due to the contribution of surface

relaxation. On the contrary, short transversal relaxation time and narrow NMR T2

distribution mean poor pore structure for rocks (Fig. 2). Mercury injection capillary

pressure curves can be used to obtain the pore throat radius distribution, which is

useful in evaluating the pore throat size and the connectivity [15, 16].

To quantitatively display the relationship between the saturation exponents with the

pore structure, all 36 core samples, shown in Fig. 1, have been chosen for rock

resistivity and laboratory NMR experimental measurements; 20 of them were studied

in mercury injection experiments. The experimental parameters of NMR measure-

ments are designed as follows: polarization time (TW): 6.0 s; inter-echo spacing (TE):

0.2 ms; the number of echoes per echo train (NE): 4096; scanning number: 128.

To illustrate the factors that heavily affect the saturation exponent, the resistivity

and laboratory NMR experimental results for 36 core samples and mercury injection

measurements for 20 core samples have been analyzed. Four representative core

samples with saturation exponent increasing from 1.627 to 3.48 are compared and

displayed through Figs. 3, 4, 5 and 6. In these figures, the correlation of the water

saturation and the resistivity index, the corresponding laboratory NMR T2

distribution and the pore throat radius distribution that acquired from mercury

injection capillary pressure curve are displayed in (a), (b) and (c), respectively. For

the core sample no. 2, no mercury injection data have been obtained.

From a comparison of the data displayed in Figs. 3, 4, 5 and 6, several

regularities can be observed:

1. The saturation exponent is heavily affected by the rock pore structure. For core

samples with good pore structure (with wide NMR T2 distribution), the

proportion of large pore components is dominated and the corresponding

saturation exponent is low, like for the core samples 1 and 2 shown in Figs. 3

and 4. On the other hand, when the rocks are dominated by micro porosity, the

proportion of small pore components is high, and the corresponding saturation

exponent increase, like for the core samples 3 and 4 shown in Figs. 5 and 6.

336 L. Xiao et al.

123

2. The saturation exponent is hardly affected by the pore throat radius, as for the

core samples 1, 3, and 4. Their saturation exponents vary strongly, but their

distributions of pore throat radii are not different, especially for the core

samples 1 and 3. Their saturation exponents and NMR T2 distributions are

significantly different, while the morphologies of the pore throat radius

distributions are almost the same.

3. The saturation exponent is not relevant to rock porosity and permeability, but it

is inversely proportional to T2lm. This is because high T2lm means wide NMR T2

distributions and thus leads to low saturation exponents.

4 A Novel Model for Estimating the Saturation Exponent from NMR Logs

4.1 Estimating the Saturation Exponent Parameters from NMR Logs

From Figs. 3, 4, 5 and 6, we can conclude that the saturation exponent is

proportional to the proportion of small pore components and inversely proportional

0.1 1 10 100 1000

T 2

T2

-

0.1 1 10 100 1000

Rel

ativ

e am

plitu

de

Relaxation time , ms T 2Relaxation time , ms

Rel

ativ

e am

plitu

de

H1 nuclei

Equivalent rock pore space

Spin-echo train

NMR distribution

Multi- exponential

inversion

0 300 600 900

Decay time, ms

Am

plitu

de

0 300 600 900

Decay time, ms

Am

plitu

de

Fig. 2 Relationship of rock pore size with the corresponding NMR T2 distribution


123

to the T2lm. These parameters must be obtained first to estimate the saturation

exponent precisely. In this aspect, NMR logs have unique advantages [17–21]. T2lm

can be obtained from the NMR logs directly, but the proportion of small pore

components needs to be characterized.

In this study, different kinds of irreducible water saturations, which are calculated

by defining 10, 20, 30, 40, 50, 60, 70 and 100 ms as T2 cutoffs separately, are

y = 1.0252x-1.627

R2 = 0.998

1

10(a)

(b)

(c)

0.1 1


Res

istiv

ity in

dex

Core No. 1

por.=14.0%

perm.=0.49 mD

0

0.05

0.1

0.15

0.2

0.25

0.3

0.1 10 1000 100000

T 2, ms

Am

plitu

de

Core No. 1

T 2lm =30.6 ms

0

10

20

30

0.01 1 100 10000

R c, um

Am

plitu

de

Core No. 1

Fig. 3 Experimental results ofcore sample no. 1

338 L. Xiao et al.

123

chosen to characterize the proportion of small pore components. The irreducible

water saturation can be estimated using Eq. (4),

Swirr ¼R T2cutoff

T2 minSðTÞdt

R T2 max

T2 minSðTÞdt

ð4Þ

where Swirr is the estimated irreducible water saturation from NMR logs using the

defined T2 cutoff, T2min the minimum transverse relaxation time, T2max the

maximum transverse relaxation time, T2cutoff the defined T2 cutoff, which is used to

estimate the irreducible water saturation; the units of them are ms and S(T) is the

porosity distribution function, which is associated with the T2 relaxation time.

To illustrate the correlation of all the experimental parameters obtained from

laboratory NMR and mercury injection measurements with the saturation exponent,

the correlations of them are analyzed and listed in Table 1.

Table 1 illustrates that the saturation exponents are strongly correlated with

Swirr_30, Swirr_40, Swirr_50, but the correlation with Swirr_10, Swirr_20, Swirr_60, Swirr_70

and Swirr_100 was reduced. This is because that for majority of core samples, the

y = 1.0132x-1.885

R2 = 0.9982

1

10(a)

(b)

0.1 1

Water saturation, fractionR

esis

tivity

inde

x

Core No. 2

por.=15.9%

perm.=1.05 mD

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.1 10 1000 100000

T 2, ms

Am

plitu

de

Core No. 2

T 2lm=75.182 ms



123

NMR T2 distribution mainly ranges from 20 to 60 ms. When the T2 relaxation time

is lower than 30 ms and higher than 60 ms, nearly no T2 spectrum exists. T2lm is the

overall signature of NMR T2. Hence, it is associated with the pore structure.

y = 0.9828x-2.234

R2 = 0.9955

1

10(a)

(b)

(c)

0.1 1

Water saturation, fractionR

esis

tivity

inde

x

Core No. 3

por.=8.2%

perm.=0.27 mD

0

0.05

0.1

0.15

0.2

0.1 10 1000 100000

T 2, ms

Am

plitu

de Core No. 3

T 2lm =28.2 ms

0

10

20

0.01 1 100 10000

R c, um

Am

plitu

de

Core No. 3


340 L. Xiao et al.

123

Core porosity, permeability, T2 cutoff and parameters obtained from the

mercury injection measurements are weakly correlated with the saturation

exponent.

y = 0.9609x-3.4801

R2 = 0.9969

1

10(a)

(b)

(c)

0.1 1


Res

istiv

ity in

dex

Core No. 4

por.=10.58%

perm.=0.62 mD

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.1 10 1000 100000

T 2, ms

Am

plitu

de

Core No. 4

T 2lm =20.41 ms

0

10

20

0.01 1 100 10000

R c, um

Am

plitu

de

Core No. 4



123

Ta

ble

1C

orr

elat

ion

so

fth

esa

tura

tion

exp

onen

tan

dth

eex

per

imen

tal

par

amet

ers

ob

tain

edfr

om

lab

ora

tory

NM

Ran

dm

ercu

ryin

ject

ion

mea

sure

men

ts

Sat

ura

tio

nex

po

nen

tP

oro

sity

Per

mea

bil

ity

T2cuto

fflo

g(T

2lm

)S

wir

r_10

Sw

irr_

20

Sw

irr_

30

Sw

irr_

40

Sw

irr_

50

Sat

ura

tio

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Po

rosi

ty-

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91

.00

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mea

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.00

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ff-

0.4

80

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0.6

11

.00

log

(T2lm

)-

0.5

80

.66

0.4

90

.67

1.0

0

Sw

irr_

10

0.5

2-

0.6

5-

0.3

2-

0.6

2-

0.9

61

.00

Sw

irr_

20

0.6

1-

0.6

6-

0.4

2-

0.6

6-

0.9

80

.97

1.0

0

Sw

irr_

30

0.6

5-

0.6

6-

0.4

9-

0.6

4-

0.9

70

.92

0.9

81

.00

Sw

irr_

40

0.6

5-

0.6

5-

0.5

2-

0.6

1-

0.9

60

.87

0.9

50

.99

1.0

0

Sw

irr_

50

0.6

4-

0.6

4-

0.5

4-

0.5

8-

0.9

50

.84

0.9

20

.98

1.0

01

.00

Sw

irr_

60

0.6

2-

0.6

2-

0.5

6-

0.5

7-

0.9

40

.81

0.9

00

.96

0.9

91

.00

Sw

irr_

70

0.6

1-

0.6

0-

0.5

7-

0.5

7-

0.9

30

.80

0.8

80

.95

0.9

80

.99

Sw

irr_

100

0.5

6-

0.5

5-

0.6

1-

0.5

9-

0.9

10

.76

0.8

40

.91

0.9

40

.96

So

rtin

gco

effi

cien

t-

0.1

50

.64

0.4

40

.40

0.6

6-

0.6

1-

0.6

0-

0.6

3-

0.6

5-

0.6

7

Var

iati

on

coef

fici

ent

-0

.30

0.7

30

.79

0.6

20

.76

-0

.64

-0

.69

-0

.74

-0

.78

-0

.80

P50

0.5

1-

0.7

1-

0.2

9-

0.5

0-

0.7

80

.78

0.7

70

.78

0.7

80

.77

R50

-0

.29

0.5

51

.00

0.7

20

.60

-0

.42

-0

.51

-0

.58

-0

.62

-0

.64

Pd

0.1

9-

0.5

8-

0.2

5-

0.4

1-

0.6

60

.64

0.6

00

.60

0.6

00

.61

Rm

ax

-0

.33

0.6

90

.89

0.6

60

.70

-0

.54

-0

.63

-0

.70

-0

.75

-0

.77

Rm

-0

.31

0.6

10

.99

0.7

10

.65

-0

.4-

0.5

7-

0.6

4-

0.6

8-

0.7

1

342 L. Xiao et al.

123

Ta

ble

1co

nti

nu

ed

Sw

irr_

60

Sw

irr_

70

Sw

irr_

100

So

rtin

gco

effi

cien

tV

aria

tio

nco

effi

cien

tP

50

R50

Pd

Rm

ax

Rm

Sat

ura

tio

nex

po

nen

t

Po

rosi

ty

Per

mea

bil

ity

T2cuto

ff

log

(T2lm

)

Sw

irr_

10

Sw

irr_

20

Sw

irr_

30

Sw

irr_

40

Sw

irr_

50

Sw

irr_

60

1.0

0

Sw

irr_

70

1.0

01

.00

Sw

irr_

100

0.9

80

.99

1.0

0

So

rtin

gco

effi

cien

t-

0.6

8-

0.6

8-

0.6

71

.00

Var

iati

on

coef

fici

ent

-0

.82

-0

.83

-0

.85

0.8

91

.00

P50

0.7

60

.75

0.7

0-

0.7

4-

0.6

91

.00

R50

-0

.67

-0

.68

-0

.75

0.4

50

.80

-0

.32

1.0

0

Pd

0.6

10

.61

0.5

9-

0.8

9-

0.7

10

.83

-0

.27

1.0

0

Rm

ax

-0

.80

-0

.81

-0

.84

0.7

40

.95

-0

.52

0.8

9-

0.5

11

.00

Rm

-0

.73

-0

.74

-0

.80

0.5

70

.88

-0

.40

0.9

9-

0.3

60

.95

1.0

0

Inth

ista

ble

,S

wir

r_10,

Sw

irr_

20,

Sw

irr_

30,

Sw

irr_

40,

Sw

irr_

50

,S

wir

r_60,

Sw

irr_

70

and

Sw

irr_

100

are

the

irre

du

cib

lew

ater

satu

rati

on

sca

lcu

late

dfr

om

the

NM

RT

2d

istr

ibu

tio

nu

sin

g

10

,2

0,

30

,4

0,

50

,6

0,

70

and

10

0m

sas

T2

cuto

ffs

P50

isth

em

ercu

ryin

ject

ion

pre

ssu

reco

rres

po

ndin

gto

50

.0%

mer

cury

inje

ctio

nsa

tura

tio

n,

R50

isth

ep

ore

thro

atra

diu

sco

rres

po

nd

ing

to5

0.0

%m

ercu

ryin

ject

ion

satu

rati

on

,P

dis

the

thre

sho

ldp

ress

ure

,R

max

isth

em

axim

um

po

reth

roat

rad

ius,

Rm

isth

eav

erag

ep

ore

thro

atra

diu

s


123

4.2 A Novel Model of Estimating Saturation Exponent from NMR Logs

Based on the analysis described above, Swirr_30, Swirr_40, Swirr_50 and T2lm are chosen

as the input parameters to establish a model to estimate the saturation exponent.

With the 36 studied core samples, multivariate regression is used. The regression

model is established and expressed as Eq. (5).

n ¼ 0:546þ 0:292� logðT2lmÞ þ 0:009� Swirr 30 þ 0:061� Swirr 40

� 0:044� Swirr 50; correlation coefficient: 0:776ð5Þ

Equation (5) illustrates that the precision of the saturation exponent estimation

model is improved when the parameters Swirr_30, Swirr_40, Swirr_50 and T2lm are

introduced. If these three fixed T2 cutoffs of 30, 40 and 50 ms are determined, the

proportions of small pore components could be characterized and the consecutive

saturation exponents can be estimated from the NMR field logs after this technique

is extended to field application.

0 1 2 3 4 5

Saturation exponent

Measured n

Predicted n

Fig. 7 Comparison ofsaturation exponents acquiredfrom experimental resistivitymeasurements of core samplesand calculated from NMR fieldlogs

344 L. Xiao et al.

123

5 Case Studies

To confirm the reliability of the mentioned technique in this study, saturation

exponents acquired from the experimental resistivity measurements of core samples

and calculated from NMR field logs are compared in Fig. 7. This comparison shows

that for the vast majority of core samples, the predicted saturation exponents are

close to the experimental results. To quantitatively evaluate the absolute errors of

the predicted saturation exponents and the core results, the cross plot of these two

kinds of saturation exponents is made and shown in Fig. 8. These two figures

illustrate that the estimated saturation exponents from NMR field logs using the

proposed technique are credible and the absolute errors for more than 85 % of core

samples are lower than 0.25. In reservoirs with consecutive NMR field logs, this

technique can be applied for saturation exponent estimation and this will be

valuable for water saturation calculation in low permeability sands.

6 Conclusions

In low permeability reservoirs, the saturation exponents are divergent and a fixed

value cannot be regressed from the cross plot of the water saturation with the

resistivity index to estimate water saturation accurately.

1

2

3

4

1 2 3 4

Experimental saturation exponents from core samples

Pred

icte

d sa

tura

tion

expo

nent

s fr

om N

MR

fie

ld lo

gs

+0.25

-0.25

Fig. 8 Cross plot of the predicted saturation exponents and the core results


123

The rock resistivity, laboratory NMR and mercury injection measurements of

core samples illustrate that the saturation exponent is heavily affected by the rock

pore structure. Thus, it is proportional to the proportion of small pore components

and inversely proportional to T2lm. The saturation exponent is not relevant to rock

porosity, permeability and the rock pore throat radius distribution.

The proportion of small pore components can be characterized by the irreducible

water saturations predicted from the NMR T2 distribution after defining 30, 40 and

50 ms as the fixed T2 cutoffs. An estimation model for the saturation exponent can

be established based on the corresponding irreducible water saturations and T2lm.

The saturation exponents predicted from NMR field logs using the proposed

model in this paper are credible, and they are close to the measured core results. For

more than 85 % of the core samples, the absolute errors of these two kinds of

saturation exponents are lower than 0.25. This ensures that the proposed technique

and model are reliable and can be extended to field application to estimate the

saturation exponents from NMR field logs. This is valuable for water saturation

calculation in low permeability sandstones.

Acknowledgments The authors thanks for the supporting of the Fundamental Research Funds for the

Central Universities, China (No. 2011YXL009) to this research work.

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