determination of multivariate relationships between ... · determination of multivariate...

Paper 111, CCG Annual Report 13, 2011 (© 2011)

111-1

Determination of Multivariate Relationships between

Petrophysical and Rock Mechanical Properties

Mehdi Khajeh and Jeff Boisvert

A geological model is one of the main inputs for the simulation process and geostatistical techniques are

used widely to produce these models. Structural, facies and property models are included in each

geostatistical realization. Petrophysical properties, i.e. porosity, permeability and saturations, are the only

properties required for conventional flow simulation but in the case of coupled geomechanical-flow

simulation process rock mechanical properties should be modeled stochastically as well. Spatial modeling

of multiple variable is one of the most common challenges in the field of geostatistics. Conventional

Gaussian techniques is the most common approach for cosimulation of multiple variable which is

applicable only when multi-variate gaussianity assumption is confirmed. Alternative techniques such as

stepwise conditional transformation should be considered if variables don’t show gaussianity very well.

Statistical analysis, i.e. univariate and bivariate analysis, is the primary step of each geostatistical

modeling process to select the best work flow for geostatistical modeling in the case of dealing with

multiple variables. In this paper, bivariate relationship between petrophysical, i.e. porosity, attributes used

for determination of rock mechanical properties, i.e. dipole sonic logs and density log, and rock mechanical

properties, i.e. poison ratio and young modulus, is investigated. Data set related to one of western

Canadian oilsands basins is used for that purpose.

Introduction

Almost all natural soils are highly variable and rarely homogeneous. Lithological and inherent

spatial variability of soils, are two categories of soil heterogeneity. To improve the accuracy of recovery

performance predictions, detailed high-resolution geological models are built geostatistically, which are

applied in numerical simulation process. Structural, facies and property modeling are three main parts of

each geological models. Petrophysical and rock mechanical properties are two groups of properties which

could be modeled stochastically. In the case of conventional flow simulation process, in which soil

deformation has not significant effect on recovery performance, petrophysical properties, i.e. porosity,

permeability and saturations, are the only properties which are modeled stochastically however,

considering heterogeneous rock mechanical properties has a significant effect on predicted reservoir

performance for the cases in which soil deformation could not be ignored and coupled geomechanical

flow simulation process should be considered instead of flow simulation alone. A comprehensive

geological model consisting of petrophysical and rock mechanical properties as well as the in-situ stress

state is termed a Mechanical Earth Model (MEM).

Like petrophysical properties, log data is one of the main sources of data used for determination

of rock mechanical properties. Dipole Sonic logs, i.e. compression and shear velocity logs, and density log

are the main logs from which elastic properties could be determined. Equation 1 and 2 show the

equations from which poison ratio (ν) and young modulus (E) are determined respectively.

1

15.0

2

2

−

∆∆

−

∆∆

=

c

s

c

s

t

t

t

t

υ (1)

( )( )( )υ

υυρ

−

+−

∆=

1

121

2

c

b

tE (2)

where Δts is shear wave transformation time, Δtc is compression wave transformation time and ρb is bulk

density. The cosimulation of multiple properties remains a significant outstanding problem in reservoir

characterization. Gaussian simulation techniques are the most common and simple simulation

approaches used in reservoir property modeling. The use of Gaussian techniques require variables to be


111-2

multivariate Gaussian; however, earth science phenomena are rarely Gaussian (Leuangthong and

Deutsch, 2000). Transformation techniques are applied to make the model variables Gaussian. The

conventional technique is the normal score transformation. This technique generates univariate Gaussian

distributions but do not ensure bivariate or multivariate Gaussianity. Instead, the multivariate

distributions may show signs of non-linearity, mineralogical constraints, and heteroscedasticity as shown

in Figure 1.

Figure 1: Examples of problematic bivariate distributions for Gaussian simulation: non-linear

relations (left), constraints (centre) and heteroscedasticity (right) (Leuangthong 2003)

In the case of appearance of these features which confirm that data don’t follow multi-variate

Gaussianity assumption, alternative techniques such as stepwise conditional transformation technique

should be used and applied instead of conventional Gaussian techniques for cosimulation of multiple

variables. Univariate and multivariate statistical analysis are preliminary steps which should be performed

before geostatistical modeling process. Investigating the results obtained from multivariate statistical

analysis helps in selecting the most precise multivariate geostatistical modeling work flow.

In this work, univariate and bivariate statistical analysis is performed for petrophysical, i.e.

porosity, attributes used for determination of rock mechanical properties, i.e. dipole sonic logs and

density log, and also rock mechanical properties, i.e. poison ratio and young modulus, is investigated.

Data set related to one of western Canadian oilsand reserves is used for that purpose. By defining simple

cut-off on porosity, available data for this study is divided in two facies which is facies 1 (Ф<0.28) and

facies 2 (Ф>0.28).

Results

Figure 2 and Figure 3 shows univariate statistical analysis of facies 1 and facies 2 respectively. Porosity,

poison ratio, young modulus, compression wave velocity, shear wave velocity, compression to shear wave

velocity ratio and density are 7 attributes which are considered for analysis in each facies. Differences in

statistical information of each attribute could be seen by comparing corresponding histogram of each

attribute in Figure 2 (facies 1) and Figure 3 (facies 2). To investigate bivariate statistical analysis, first data

is transformed from original unit to normal score unit system and then scatter plot of each two variables

has been plotted. Just as an example, in Figure 4 and 5 the scatter plot of porosity respect to other

properties are shown. From this analysis the correlation coefficients existed between variables could be

determined. In figure 6 and figure 7 correlation coefficient matrices of attributes are shown respectively

for facies 1 and facies 2. As could be seen although for some bivariate correlation coefficients are almost

the same in facies 1 and facies 2 but for some others, there is significant difference in correlation

coefficients. The correlation coefficients existed between porosity and compression velocity could be

mentioned as one set of bivariate in which there is considerable difference in correlation coefficient.

Discussion and Conclusions

In this study, multivariate relationships between petrophysical and rock mechanical properties was

investigated. For that purpose the data set related to one of Canadian oilsand basin was considered and

seven attributes, i.e. porosity, young modulus, poison ratio, compression velocity, shear velocity,

compression/shear velocity and density, were considered. The correlation coefficient matrix obtained for


111-3

each facies and it has been shown that for some set of bivariate there is considerable difference in the

correlation coefficient existed between two attributes.

References

Leuangthong, O. and Deutsch, C. V. (2000). Stepwise Conditional Transformation for Simplified

Cosimulation of Reservoir Properties. CCG Annual Report-Report No.2 .

Leuangthong, O. (2003). Stepwise Conditional Transformation, Ph.D. Thesis, University of Alberta.

Figure 2: Histograms of attributes (facies1)

Fre

quency

Porosity (%)

5 10 15 20 25

0.00

0.04

0.08

0.12

0.16Porosity_Facies1

Number of Data 1298

mean 20.45std. dev. 3.46

coef. of var 0.17

maximum 27.99upper quartile 22.83

median 20.03lower quartile 18.31

minimum 6.81

Fre

quency

Poison Ratio

0.30 0.35 0.40 0.45 0.50

0.00

0.04

0.08

0.12

0.16

Poison Ratio_Facies1Number of Data 1298


coef. of var 0.06



minimum 0.32

Fre

quency

Young Modulus (GPa)

1 6 11 16 21

0.00

0.05

0.10

0.15

0.20

0.25Young Modulus_Facies1

Number of Data 1298


coef. of var 0.65



minimum 1.22

Fre

quency

Vp

200 300 400 500 600

0.00

0.04

0.08

0.12

0.16

Compression Velocity_Facies1Number of Data 1298

mean 414.44std. dev. 53.22

coef. of var 0.13



minimum 211.94

Fre

quency

Vs

500 1000 1500 2000 2500

0.00

0.04

0.08

0.12

0.16 Shear Velocity_Facies1Number of Data 1298

mean 1425.70std. dev. 356.14

coef. of var 0.25



minimum 568.36

Fre

quency

Vp/Vs

0.15 0.25 0.35 0.45 0.55

0.00

0.04

0.08

0.12

Vp/Vs_Facies1Number of Data 1298


coef. of var 0.20



minimum 0.15

Fre

quency

RHOB (Kg/m3)

2000 2100 2200 2300 2400 2500 2600

0.00

0.05

0.10

0.15

0.20

Bulk Density_Facies1Number of Data 1298

mean 2359.34std. dev. 72.62

coef. of var 0.03



minimum 2040.18


111-4

Figure 2: Histograms of attributes (facies 2)

Fre

quency

Porosity (%)

25 35 45 55

0.00

0.05

0.10

0.15

0.20

Porosity_Facies2Number of Data 1308


coef. of var 0.09



minimum 29.03

Fre

quency

Poison Ratio

0.2 0.3 0.4 0.5

0.00

0.05

0.10

0.15

0.20

Poison Ratio_Facies2Number of Data 1308


coef. of var 0.10



minimum 0.23

Fre

quency

Young Modulus (GPa)

0 5 10 15 20

0.00

0.05

0.10

0.15

0.20

Young Modulus_Facies2Number of Data 1308


coef. of var 0.59



minimum 0.89

Fre

quency

Vp

200 300 400 500 600

0.00

0.05

0.10

0.15

0.20

Compression Velocity_Facies2Number of Data 1308

mean 418.13std. dev. 75.20

coef. of var 0.18



minimum 209.49

Fre

quency

Vs

500 1000 1500 2000 2500

0.00

0.05

0.10

0.15

0.20

Shear Velocity_Facies2Number of Data 1308

mean 1209.44std. dev. 433.84

coef. of var 0.36



minimum 620.17

Fre

quency

Vp/Vs

0.15 0.25 0.35 0.45 0.55

0.00

0.04

0.08

0.12

Vp/Vs_Facies2Number of Data 1308


coef. of var 0.21



minimum 0.16

Fre

quency

RHOB (Kg/m3)

1700 1900 2100 2300 2500

0.0

0.1

0.2

0.3 Bulk Density_Facies2Number of Data 1308

mean 2109.41std. dev. 66.00

coef. of var 0.03



minimum 1765.87


111-5

Figure 3: Scatter plots of porosity respect to other attributes (facies 1)

NS

_P

ois

on

Ra

tio

NS_Porosity (%)

Porosity vs Poison Ratio

-4 -3 -2 -1 0 1 2 3 4

-4

-3

-2

-1

0

1

2

3

4Number of data 1298Number plotted 1298

X Variable: mean 0.000std. dev. 0.999

min. -3.363max. 3.363

Y Variable: mean 0.000std. dev. 0.999

min. -3.363max. 3.363

correlation 0.025rank correlation 0.009

NS

_Y

ou

ng

Mo

du

lus (

GP

a)

NS_Porosity (%)

Porosity vs Young Modulus

-4 -3 -2 -1 0 1 2 3 4

-4

-3

-2

-1

0

1

2

3



min. -3.363max. 3.363


min. -3.363max. 3.363

correlation -0.282rank correlation -0.272

NS

_V

p

NS_Porosity (%)

Porosity vs Vp

-4 -3 -2 -1 0 1 2 3 4

-4

-3

-2

-1

0

1

2

3



min. -3.363max. 3.363


min. -3.363max. 3.363


NS

_V

s

NS_Porosity (%)

Porosity vs Vs

-4 -3 -2 -1 0 1 2 3 4

-4

-3

-2

-1

0

1

2

3



min. -3.363max. 3.363


min. -3.363max. 3.363


NS

_V

p/V

s

NS_Porosity (%)

Porosity vs Vp/Vs

-4 -3 -2 -1 0 1 2 3 4

-4

-3

-2

-1

0

1

2

3



min. -3.363max. 3.363


min. -3.363max. 3.363


NS

_R

HO

B (

Kg

/m3

)

NS_Porosity (%)

Porosity vs Density

-4 -3 -2 -1 0 1 2 3 4

-4

-3

-2

-1

0

1

2

3



min. -3.363max. 3.363


min. -3.363max. 3.363



111-6

Figure 4: Scatter plots of porosity respect to other attributes (facies 2)

Figure 5: Correlation Coefficients (facies1) Figure 6: Correlation Coefficient (facies2)

NS

_P

ois

on R

atio

NS_Porosity (%)

Porosity vs Poison Ratio

-4 -3 -2 -1 0 1 2 3 4

-4

-3

-2

-1

0

1

2

3



min. -3.365max. 3.365


min. -3.365max. 3.365


NS

_Y

oung M

odulu

s (

GP

a)

NS_Porosity (%)

Porosity vs Young Modulus

-4 -3 -2 -1 0 1 2 3 4

-4

-3

-2

-1

0

1

2

3



min. -3.365max. 3.365


min. -3.365max. 3.365

correlation -0.010rank correlation 0.012

NS

_V

p

NS_Porosity (%)

Porosity vs Vp

-4 -3 -2 -1 0 1 2 3 4

-4

-3

-2

-1

0

1

2

3



min. -3.365max. 3.365


min. -3.365max. 3.365

correlation 0.003rank correlation -0.032

NS

_V

s

NS_Porosity (%)

Porosity vs Vs

-4 -3 -2 -1 0 1 2 3 4

-4

-3

-2

-1

0

1

2

3



min. -3.365max. 3.365


min. -3.365max. 3.365


NS

_V

p/V

s

NS_Porosity (%)

Porosity vs Vp/Vs

-4 -3 -2 -1 0 1 2 3 4

-4

-3

-2

-1

0

1

2

3



min. -3.365max. 3.365


min. -3.365max. 3.365


NS

_R

HO

B (

Kg/m

3)

NS_Porosity (%)

Porosity vs Density

-4 -3 -2 -1 0 1 2 3 4

-4

-3

-2

-1

0

1

2

3



min. -3.365max. 3.365


min. -3.365max. 3.365


Correlation Matrix (Facies1)

Porosity

Poison Ratio

Young Modulus

Vp

Vs

Vp/Vs

Density

Po

rosity

Po

iso

n R

atio

Yo

un

g M

od

ulu

s

Vp

Vs

Vp

/Vs

De

nsity

1.00 0.02 -0.28 0.36 0.23 -0.07 -0.75

0.02 1.00 -0.71 0.17 0.81 -0.97 -0.01

-0.28 -0.71 1.00 -0.49 -0.80 0.71 0.21

0.36 0.17 -0.49 1.00 0.65 -0.23 -0.36

0.23 0.81 -0.80 0.65 1.00 -0.87 -0.18

-0.07 -0.97 0.71 -0.23 -0.87 1.00 0.02

-0.75 -0.01 0.21 -0.36 -0.18 0.02 1.00

Correlation Matrix (Facies2)

Porosity

Poison Ratio

Young Modulus

Vp

Vs

Vp/Vs

Density

Poro

sity

Pois

on R

atio

Young M

odulu

s

Vp

Vs

Vp/V

s

Density

1.00 -0.09 -0.01 0.00 -0.07 0.09 -0.91

-0.09 1.00 -0.65 0.44 0.88 -1.00 0.13

-0.01 -0.65 1.00 -0.51 -0.68 0.65 0.01

0.00 0.44 -0.51 1.00 0.75 -0.44 -0.01

-0.07 0.88 -0.68 0.75 1.00 -0.88 0.08

0.09 -1.00 0.65 -0.44 -0.88 1.00 -0.13

-0.91 0.13 0.01 -0.01 0.08 -0.13 1.00

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