determination of multivariate relationships between ... · determination of multivariate...
TRANSCRIPT
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
Paper 111, CCG Annual Report 13, 2011 (© 2011)
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
Paper 111, CCG Annual Report 13, 2011 (© 2011)
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
mean 0.45std. dev. 0.02
coef. of var 0.06
maximum 0.49upper quartile 0.46
median 0.45lower quartile 0.44
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
mean 3.86std. dev. 2.51
coef. of var 0.65
maximum 20.66upper quartile 4.57
median 3.56lower quartile 2.28
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
maximum 590.13upper quartile 442.06
median 414.90lower quartile 389.31
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
maximum 2366.52upper quartile 1701.40
median 1349.92lower quartile 1197.17
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
mean 0.30std. dev. 0.06
coef. of var 0.20
maximum 0.51upper quartile 0.34
median 0.30lower quartile 0.26
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
maximum 2692.04upper quartile 2400.01
median 2367.47lower quartile 2318.25
minimum 2040.18
Paper 111, CCG Annual Report 13, 2011 (© 2011)
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
mean 35.38std. dev. 3.16
coef. of var 0.09
maximum 55.21upper quartile 37.83
median 35.30lower quartile 33.14
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
mean 0.41std. dev. 0.04
coef. of var 0.10
maximum 0.49upper quartile 0.45
median 0.42lower quartile 0.38
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
mean 4.77std. dev. 2.82
coef. of var 0.59
maximum 16.57upper quartile 7.00
median 4.20lower quartile 2.10
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
maximum 627.13upper quartile 443.33
median 392.41lower quartile 372.46
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
maximum 2589.18upper quartile 1502.84
median 1063.54lower quartile 850.23
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
mean 0.37std. dev. 0.08
coef. of var 0.21
maximum 0.59upper quartile 0.44
median 0.37lower quartile 0.30
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
maximum 2481.33upper quartile 2145.90
median 2108.12lower quartile 2062.92
minimum 1765.87
Paper 111, CCG Annual Report 13, 2011 (© 2011)
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
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 -0.282rank 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
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 0.364rank correlation 0.343
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
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 0.226rank correlation 0.200
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
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 -0.071rank correlation -0.048
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
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 -0.747rank correlation -0.808
Paper 111, CCG Annual Report 13, 2011 (© 2011)
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
4Number of data 1308Number plotted 1308
X Variable: mean 0.000std. dev. 1.000
min. -3.365max. 3.365
Y Variable: mean 0.000std. dev. 1.000
min. -3.365max. 3.365
correlation -0.09rank correlation -0.0
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
4Number of data 1308Number plotted 1308
X Variable: mean 0.000std. dev. 1.000
min. -3.365max. 3.365
Y Variable: mean 0.000std. dev. 1.000
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
4Number of data 1308Number plotted 1308
X Variable: mean 0.000std. dev. 1.000
min. -3.365max. 3.365
Y Variable: mean 0.000std. dev. 1.000
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
4Number of data 1308Number plotted 1308
X Variable: mean 0.000std. dev. 1.000
min. -3.365max. 3.365
Y Variable: mean 0.000std. dev. 1.000
min. -3.365max. 3.365
correlation -0.068rank correlation -0.082
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
4Number of data 1308Number plotted 1308
X Variable: mean 0.000std. dev. 1.000
min. -3.365max. 3.365
Y Variable: mean 0.000std. dev. 1.000
min. -3.365max. 3.365
correlation 0.09rank correlation 0.0
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
4Number of data 1308Number plotted 1308
X Variable: mean 0.000std. dev. 1.000
min. -3.365max. 3.365
Y Variable: mean 0.000std. dev. 1.000
min. -3.365max. 3.365
correlation -0.911rank correlation -0.927
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