Download Ordinary Kriging

Geomathematical and geostatistical characters of
some clastic Neogene hydrocarbon reservoirs in the
Croatia
(“Stochastic simulations and geostatistics”, J. Geiger)
Tomislav Malvić
Szeged, December 2011
Introduction
0
A T
I A
A
D
HUNGARY
R
dep
Beničanci Field
Okoli Field
Sava depression
0
Osijek
Slavonski
Brod
BIA
Galovac-Pavljani
Field
SER
re s
sio
n
Kloštar Field
Ivanić Field
Stari Gradac-Barcs Nyugat F.
IC
Zagreb
T
IA A
E
S
Dr
ava
C
Mura depress.
R
O
SLOVENIA
100
km
Slavonia-Srijem
depression
50 km BOSNIA and
HERZEGOVINA
Croatian part of the Pannonian Basin System and
locations with the most comprehensive geomathematical analyses
Deterministical
geostatistics
Stochastical
geostatistics
GEOMATH
Neural network
Descriptive
statistics
➲
Such analyses had been done in the largest Croatian
depressions: Sava and Drava.
➲
The most of them were geostatistical, but also some of
them included neural networks and advanced statistics.
➲
Geostatistical analyses had been based on 10-25 data
points.
➲
Neural and statistical analyses was performed on several
dozens data points.
➲
The Sava Depression:
➲
The Drava Depression:
➲
Kloštar,
➲
Stari Gradac-Barcs Nyugat,
➲
Ivanić,
➲
Molve,
➲
Okoli Fields.
➲
Beničanci,
➲
Galovac-Pavljani,
➲
The Bjelovar Subdepression with set
of vertical variograms.
The Sava Depression
➲
The Kloštar Field (the most geomathematical analyzed field):
➲
“Selection of the most appropriate interpolation method for sandstone reservoirs in the
Kloštar oil and gas field” (2008);
➲
“Linearity and Lagrange linear multiplicator in Ordinary Kriging equation” (2009);
➲
“Mapping of Upper Miocene sandstone facies by Indicator Kriging” (2010);
➲
“Using of Ordinary Kriging for indicator variable mapping (example of sandstone/marl
border)” (2010);
➲
“Ordinary Kriging as the most Appropriate Interpolation Method for Porosity in the Sava
Depression Neogene Sandstone” (2010);
➲
“Stochastic simulations of dependent geological variables in sandstone reservoirs of
Neogene age: A case study of Kloštar Field, Sava Depression” (2011).
➲
The Ivanić Field:
➲
“Construction of porosity map by Kriging in the sandstone reservoir, Case study from the
Sava Depression” (2008).
The Drava Depression
➲
The Beničanci Field:
➲
“Application of methods: Inverse distance weighting, ordinary kriging and collocated
cokriging in porosity evaluation, and comparison of results on the Beničanci and Stari
Gradac fields in Croatia” (2003);
➲
“Benefits of application of neural network in porosity estimation (example the Beničanci
Field)” (2007);
➲
“Significance of the amplitude attribute in porosity prediction, Drava Depression Case
Study” (2008);
➲
“Cokriging geostatistical mapping and importance of quality of seismic attribute(s)”
(2009).
➲
The Molve Field:
➲
“Relation between Effective Thickness, Gas Production and Porosity in Heterogeneous
Reservoir: and Example from the Molve Field, Croatian Pannonian Basin” (2010).
➲
The Galovac-Pavljani Field:
➲
“Application of deterministic and stochastic methods in OOIP calculation: Case study of
Galovac-Pavljani field” (2007).
➲
The Stari Gradac-Barcs Nyugat Field:
➲
“Application of methods: Inverse distance weighting, ordinary kriging and collocated
cokriging in porosity evaluation, and comparison of results on the Beničanci and Stari
Gradac fields in Croatia” (2003);
➲
“Improvements in reservoir characterization applying geostatistical modelling (estimation
& stochastic simulations vs. standard interpolation methods)” (2005);
➲
“Reducing variogram uncertainties using the ‘jack-knifing’ method, a case study of the
Stari Gradac-Barcs Nyugat field” (2008).
The 1st period 1999-2006 “Begining”
Recognizing of possible geomathematical
applications
in the analyses of HC reservoirs
in the Croatian part of the Pannonian Basin
System.
The very first variogram analyses in CPBS (2002-2003)
Set of vertical variograms in the Bjelovar Subdepression
Porosity behaviour is analysed in vertical dimension regarding potential reservoir units of Mosti,
Poljana and Pepelana lithostratigraphic members (Badenian, Pannonian, Pontian stages).
The analytical tool were variograms.
In the Badenian the largest corrected porosity and range values are calculated on the GalovacPavljani field (7.99 % and 0.64 m) for the Mosti member.
Pannonian sandstones are characterised very often by poor permeable or impermeable
sediments. The highest corrected values are again calculated on the Galovac-Pavljani field (23.3
% and 0.57 m), and the lowest very close on the Velika Ciglena (8.95 % and 0.21 m).
Pontian sandstones have more uniform lithology than Poljana sandstones. The most favourable
reservoir properties are described in the Šandrovac field, where are documented generally the
highest corrected values of 29.99 % and 0.95 m.
Locations/fields with vertical variograms
0,2
0
0
0,5
range (m)
1
semivariogram ()
Figure 3: well Dež-1 / Mosti mb.
semivariogram ()
semivariogram ()
sill=0.44
0,4
40
20
0
sill=22.75
1
sill=4.05
2
0
0,5
range (m)
3
range (m)
Figure 4: well Pav-4 / Mosti mb.
6
4
2
1
Figure 5: well VC-1 / Mosti mb.
Variogram ranges in Badenian clastics
4
5
Variogram ranges in Lower Pontian clastics (Poljana Sandstones)
10
semivariogram ()
semivariogram ()
15
sill=10.56
5
0
1
range (m)
2
2
range (m)
semivariogram ()
semivariogram ()
0
5
3
range (m)
Figure 16: well Ša-35 / Pepelana ss.
semivariogram ()
sill=0.517
0,4
3
range (m)
3
80
sill=54.26
40
0
1
2
3
4
Figure 18: well VC-1 (II) / Pepelana ss.
8
sill=4.86
4
0
0.5
range (m)
1.0
1.5
Figure 17: well VC-1 (I) / Pepelana ss.
0,8
2
2
range (m)
Figure 15: well Ša-5 (II) / Pepelana ss.
sill=18.98
1
semivariogram ()
1
range (m)
40
1
0
3
Figure 14: well Ša-5 (I) / Pepelana ss.
0,0
50
semivariogram ()
semivariogram ()
sill=2.32
1
sill=74.9
Figure 13: well Rov-1 / Pepelana ss.
4
20
100
3
2
Figure 12: well Pav-1 / Pepelana ss.
0
150
5
6
40
20
0
sill=20.2
0.5
1.0
1.5
range (m)
Figure 19: well VC-1 (III) / Pepelana ss.
2.0
Variogram ranges in Lower
Pontian clastics (Poljana
Sandstones)
The very first variogram analyses in CPBS (2003)
Different interpolator algorithms
Results have been compared of the porosity evaluation for:
-Inverse Distance Weighting,
-Ordinary Kriging and
-Collocated Cokriging.
The comparison had been made in the:
-Beničanci field,
-Stari Gradac-Barcs Nyugat field.
The accuracy is determined by:
-the geological evaluation of the isoporosity line shapes and
-calculation of the mean square error (MSE).
The best solution was acquired by the Collocated Cokriging method.
Porosity distribution Beničanci field (IDW
method). Mean square
error was MSE=2.778
Porosity distribution Beničanci field (OK
method).
MSE=2.969.
Experimental variogram
curves – Beničanci field
Porosity distribution – Beničanci field (CC
method). MSE=2.185.
The 2nd period 2007-2010
“Mature geomathematical explorations”
Numerous methods had been applied
in the most of the large
HC reservoirs in the CPBS
The Kloštar Field (Sava Depression)
the best geomathematical analysed hydrocarbon field in
Croatia
The analysed reservoir (sandstone)
Inverse Distance Weighting
Moving Average
Ordinary Kriging
Nearest Neighbourhood
Numerical estimation of maps is performed using a cross validation equation. The following
values were obtained for the different methods (starting with the lowest error):
1. Kriging 366.93 (exact interpolator)
2. Moving average 369.26 (simple matrix smoothing)
3. Inverse distance weighting 371.97 (exact interpolator)
4. Nearest neighbourhood 389.00 (zonal assignment)
1. Differences are relatively small, but the minimum is the kriging results.
2. It is partly surprising that errors obtained by the moving average and nearest
neighbourhood were not higher, especially when comparing by map graphics.
3. It is probably a result of the relatively limited input dataset, which can not reflect the true
advantage of using exact interpolators.
The Kloštar Field (Sava Depression)
Neural analysis of e-logs
Well log representing the interval of the 1st sandstone “series”
saturated with oil
When determining the lithological component in wells Klo–A and Klo–B with RBF and
MLP neural networks, achieved is excellent correspondence between true and predicted
values.
Prediction of hydrocarbon saturation in well Klo–B with a neural network trained in well
Klo–A gave excellent correspondence between true and predicted values.
Results show the great potential of neural networks’ application in petroleum geology
research, where they could be used to quickly acquire results from well logs, to obtain
vertical and lateral correlation of such logs, and to solve other petroleum geology
problems.
The Ivanić Field (Sava Depression)
example of the large input dataset
The applicability of kriging interpolation is tested by averages of porosity dataset of 82 values.
Original porosity values are calculated from e-logs for sandstone reservoirs of the Pannonian
age. It is saturated with oil.
Semivariogram surface map
Primary axis
Secondary axis
The Beničanci Field (Drava Depression)
The correlation between seismic amplitude and reservoir
porosity (2008)
The analysed reservoir (breccia)
The physical meaning of seismic amplitude
The seismic wave reflection process occurs at boundaries between rock layers with different
acoustic impedances (products of seismic velocities and densities).
The reflectivity function on boundary is defined with the amplitude ratio of input and output
seismic waves, R.
In the “soft” materials (low acoustic impedance, lower density and greater porosity) the
seismic wave arrival causes longer particle movements and a little pressure increase.
In the “hard” rocks (higher acoustic impedance, lower porosity and greater density) is the shorter
particle movements and pressure increase.
The reflected seismic wave amplitude changes are a good indicator for elastic properties.
Correlation between amplitude and porosity
3D seismic data were interpreted on a grid of 50 x 50 nodes. Each contained the average of
absolute amplitude, instantaneous frequency, instantaneous phase, reflection strength, the
highest amplitude and RMS amplitude (“Root Mean Square”).
Lately, the 14 well locations had mean porosities.
Correlation could be done for 14 pairs between attributes and porosity values. This number of 14
inputs was not large enough to approximate these datasets by normal (Gaussian) curve,
what is a precondition for Pearson’s correlation coefficient calculation.
This encouraged the use of non-parametric Spearman ranking correlation coefficient, which
used median value instead of mean and standard deviation:
The highest correlation was reached between porosity and reflection strength values, which are
ranked in Table 1.
Spearman rank correlation was r’=-0.64
The reflection strength being accepted as a secondary variable.
The experimental variograms (porosity, 14 data)
The Cokriging porosity maps (14 data, 2 variables)
Theoretical work on Ordinary Kriging equations
(2009)
Linearity and Lagrange Linear Multiplicator in the Equations of Ordinary Kriging
The equations of Simple and Ordinary Kriging are compared to outline their differences in the
estimation procedure. Emphasis is given to the Lagrange multiplicator as a variable that allows
the minimization of variance in Ordinary Kriging.
A detailed presentation of equation sets provides a better understanding of the Simple and
Ordinary Kriging algorithms for geological engineers, as the two most-used geostatistical
techniques (included Indicator Kriging as the third).
The conclusion includes proposals for the determination of the Lagrange multiplicator value
in any Ordinary Kriging equation.
Dataset no. 1 – manual calculation of weighting coefficient
Simple Kriging
Ordinary Kriging
Difference in the estimation variance with Simple and Ordinary Kriging
techniques (same dataset)
Ordinary Kriging
Matrix B can be calculated as:
0x0.3805 + 12.65x0.4964 + 21.54x0.1232 + 1x(0.9319)=8.001 8in the 1st row)
In other rows the same procedure is applied.
Simple Kriging
Variance=7.63 m2
Standard deviation=2.76 m.
The variance of Ordinary Kriging can be calculated
from:
2 = 1x1(Z1-Z)+2x2(Z2-Z)+...+m
Variance=6.70 m2
Standard deviation=2.59 m.
Lower variance
Calculation of Ordinary Kriging with variation of the Lagrange multiplicators
Dataset no. 2 and variogram parameters
(a) Lagrange multiplicator 0.06
(b) Lagrange multiplicator 0.9
(b) Lagrange multiplicator -0.9
Covariance matrix of Ordinary Kriging
Methodology how to estimate Lagrange valid for minimum kriging variance
Cokriging in the Molve Field (2009)
The maximal correlation was calculated between
porosity and reflection strength. Correlation
significance was checked using t-test. Calculated
value is t=2.22, and tcritical=1.76 (for =5%).
So, the calculated correlation is statistically
significant.
The secondary variable is sampled at much more
grid nodes than primary (2 500 vs. 16 nodes). It is
why anisotropic experimental variogram is modelled
from secondary variable data. This model is defined
by:
- Azimuth of primary axis 120º;
- Lag-spacing about 350 m;
- Primary range 4 000 m (spherical theoretical
model without nugget);
- Secondary range 2 900 m (spherical theoretical
model without nugget).
The 3rd period 2010-2011
“Advanced and specific applications”
Application of Indicator Kriging and
using of simulation as regular tool
The Kloštar Field (Sava Depression)
application of Indicator Kriging
Marlitic lithofacies
Sandy lithofacies
Experimental and theoretical variograms for
cutoffs 14, 18, 19, 20, 22%.
Variogram cutoffs
Resulting probability maps for cutoffs 14, 18,
19, 20, 22%.
1. Most of the kriging techniques are linear, but some of them are not. In fact, these are linear
techniques applied on some non-linear transformation of the data.
2. Indicator transformation presented in analysis is one of such non-linear transformation and
Indicator Kriging is non-linear technique as well.
3. Such application in this analysis resulted in indicator variograms for different porosity cutoffs
and in set of probability maps for such cutoffs.
4. Using of these maps revealed the areal extension of porosity probability below defined cutoff.
The Kloštar Field (Sava Depression)
SGS, SIS and Indicator Kriging
Thickness distribution (left, scale 0-25 m), porosity (middle, 0-25%), and depth (right, 600-1100 m)
hard data
In deterministic solution (that is also “zero” realization for simulation) the following values are
always known:
a) Mean value, variance (µ, σ2);
b) Kriging variance (σ2);
c) The interval allowed for simulated values (determined from the relationship between the mean
and the variance);
d) If the allowed interval encompasses three standard deviations (±3σ) around a mean (µ) each
cell, then 99% of all possible solutions are included.
When all previous values are known and the type of simulation is defined, then the values of all
model “blank” cells can be estimated using the SGS method.
The 1st (left) and 100th (right) realizations for porosity (scale 0-25%) - SGS
POROSITY
The 1st (left) and 100th (right) realizations for thickness (scale 0-30m) - SGS
Histogram of porosity - SGS
DEPTH
The 1st (left) and 100th (right) realizations for depth (scale 600-1100m) - SGS
Histogram of depth - SGS
THICKNESS
The 1st (left) and 100th (right) realizations for thickness (scale 600-1100m) - SIS
Probability map for thickness,
cutoff more than 13 m - SIS
Probability map for thickness,
cutoff less than 13 m - IK
Probability map for thickness,
cutoff more than 9 m - SIS
Probability map for thickness,
cutoff less than 9 m - IK
Histogram of thickness – SGS (left) and SIS (right)
Conclusions
Practical development of geostatistics can be done in several ways.
In presented examples it can be easily followed through simple three phases:
1. The simple and comprehensive application of variogram analysis (modelling);
2. The first using of different, usual (linear) kriging techniques (Simple and Ordinary Kriging);
3. The next “jump” on “non-linear” Indicator Kriging;
4. Theoretical analysis of kriging equations – i.e. the “basic” techniques Simple and Ordinary K.;
5. The using of simulation, mostly the simplest – Sequential Gaussian Simulations;
6. Combination of Indicator Kriging and (Sequential) Indicator Simulations and mutual
interpretation;
7. Using of simulation for obtaining new (hopefully) more descriptive histograms.
Selection of geological variable:
1. Theoretically it can be any reservoir variable;
2. Preferable is selection of variable with (theoretically) normal distribution (like porosity) or lognormal (permeability, but rarely).
Regarding other part of geomathematics:
1. The “clasicall” descriptive statistics or t-test or F-test can be useful in introductory analysis of
raw data and reveal some distribution characteristics or data group connections;
2. The application of neural networks is different tool that can be connected with geostatistics
only occasionally or locally, e.g., in some mapping application with several algorithms or
Neural Kriging;
3. The neural algorithms are proven good tool for vertical analysis of e-logs.