Using of Ordinary Kriging for indicator variable mapping
(example of sandstone/marl border)
Upotreba običnoga kriginga u kartiranju indikatorske varijable
(na primjeru granice pješčenjaka i lapora)
Kristina NOVAK ZELENIKA1 and Tomislav MALVIĆ1,2
1INA-Oil
industry, Oil & Gas Exploration and Production, Reservoir Engineering & Field Development, Šubićeva 29,
10000 Zagreb, e-mail: kristina.novakzelenika@ina.hr (Reservoir Geologist); tomislav.malvic@ina.hr (Adviser)
2University
of Zagreb, Faculty of Mining, Geology and Petroleum Engineering, Pierottijeva 6, 10000 Zagreb, visiting
lecturer
Abstract: There is plenty of opportunity to work with indicator variable in petroleum geology. It
means that each task where we can define two possible results (binary approach), the most often
expressed as 1 (true) and 0 (false), can be finally presented as indicator value. Furthermore, the
reservoir facies analysis is just one of such problem that could be very good analyzed with such
tool. In Croatian part of Pannonian basin system, the majority of reservoirs are presented by
sandstones, that gradually have been changed in marls laterally as well as toward the top and the
base of the reservoir. The analytical goal in this paper was lateral facies change, i.e. the mapping
contours of permeable reservoir part. We used the well data to define indicator variable presented
as 1 for reservoir described by sandstone, and with 0 for the reservoir in dominantly marl type or
pure, basinal marl. Interpolation by hand (very popular in the past) located transition line on the half
distance between 1 and 0 values. But, present-day approach is obligatory based on computer
mapping. It is why we tried to use advanced technique of Ordinary Kriging to interpolate indicator
values, trying to reach approximately ‘half-distanced’ facies line. The successfully results are
presented in this paper.
Key words: Ordinary Kriging, indicator variable, sandstone, marl, facies transition
Sažetak: Postoji veliki broj mogućnosti kako upotrijebiti indikatorsku varijablu u geologiji ležišta
nafte i plina. To podrazumijeva da svaki zadatak gdje možemo odrediti dva moguća rješenja
(binarni pristup), koja su najčešće izražena kao 1 (istina) i 0 (laž), u konačnici može biti
predstavljen kao indikatorska vrijednost. Nadalje, analiza ležišnih facijesa je upravo jedan od takvih
problema koji može uspješno biti riješen upotrebom takvoga alata. U hrvatskom dijelu Panonskoga
bazenskoga sustava, glavnina ležišta je predstavljena pješčenjacima, koji postupno prelaze u
lapore kako bočno tako i u krovini te podini. Cilj analize u ovome radu je praćenje bočne promjene
facijesa, tj. kartiranje propusnih dijelova ležišta. Koristili smo bušotinske podatke za određivanje
indikatorske varijable s vrijednošću 1 (za dio ležišta opisan pješčenjakom) te vrijednošću 0 (za dio
ležišta u kojemu dominira lapor ili je cijeli predstavljen čistim, bazenskim laporom). Ručno
kartiranje (vrlo popularno u prošlosti) stavljalo je liniju promjene facijesa na polovicu udaljenosti
između vrijednosti 1 i 0. No, današnji pristup obavezno uključuje kartiranje računalom. Upravo
stoga pokušali smo upotrijebiti naprednu tehniku običnoga kriginga za interpolaciju indikatorskih
vrijednosti, pokušavajući postići da se ona pruža na polovici udaljenosti između vrijednosti 1 i 0.
Postignuti su uspješni rezultati koji su prikazani u ovome radu.
Ključne riječi: obični kriging, indikatorska varijabla, pješčenjak, lapor, promjena facijesa
1. INTRODUCTION
Mapping of Upper Miocene facies in the Croatian part of the Pannonian basin is one of the most
important and the best explored task in the Croatian petroleum geology. Most of the oil and gas
reservoir in Sava, Drava, Slavonia-Srijem and Mura Depression are in Late Pannonian and Early
Pontian sandstones. Sedimentary model of those sandstones are well known (Vrbanac, 1996,
2002b; Vrbanac et al., 2008) and they represent result of periodically working of turbidit currents
in calm, brackish, lake environment, where mostly marls were settling down. It is possible to notice
structural inversion between time of the sedimentation and structural forming of traps. Older
reservoirs (Paleozoic, Mesozoic and Badenian ages) are mostly results of buried hills or their
heritage structures that was formed in Middle Miocene.
Late Miocene sandstones were deposited in pull-aparts or structural depressions, located between
regional strike-slip faults. Later they become structural uplifts (anticlines) or traps for hydrocarbons,
as a result of the inversion due to compression resulted from N-S stress. Hydrocarbons started to
migrate at the end of Pliocene and in Quaternary.
Sedimentation in small, structural depressions had great influence on locations of facies, as well as
on boundaries of the reservoir and distribution of porosity and permeability. Fault boundaries of the
structural depressions are also boundaries between turbidity sandstone and basin marl. It is
important to remember that sandstones had been deposited periodically, and in their top or base is
always marl. Facies boundaries are not sharp and there is wide transition zone in the basin plain
as well as in the direction of turbidite current palaeotransport (Malvić, 2006). In the palaeotransport
direction, there is transition from the coarse grained to fine grained sandstone, and decreasing of
the porosity. Laterally there is also transition from coarse grained to fine grained sandstone, siltite
and finally marl. Boundaries of all these transitions are not sharp and there is several transitional
lithotypes.
Mapping of facies is very important task for creating the scheme of sedimentary environment,
shapes and boundaries of hydrocarbon reservoirs. It is impossible to show all variety on one
clearly map and that is the reason that several facies mapping methods were developed. One of
the well known methods is mapping based on SP curve (in Croatia it was represented e.g. in
papers of e.g. Saftić, 1998; Steiner, 1999; Vrbanac, 2002a).
Deterministic mapping of facies as inputs use categorical variables, shown with numbers of
discrete values. Every facies should be represented with one categorical value (e.g. 1 for coarse
grained sandstone, 2 for fine grained sandstone, 3 for silt and 4 for marl) and their borders should
be mapped. There is well known geostatistical method of Indicator Kriging which is used for
categorical variable mapping. Indicator variable is showed through two categorical values (1 and
0). Also analyzed Upper Miocene sediments could be described with 1 (sandstones) and 0 (marls).
Of course, 1 could not present only pure sandstones and 0 pure marls, because we could lost all
transitional facies. This is why it must be adjust cutoff or threshold for variable used in facies
description. The permeable reservoir saturated by hydrocarbons is described as 1 and water
saturated parts or pure marl with 0. The porosity cutoff value between 1 and 0 values is 12%.
After this indicator transformation of data (porosity is normal distributed) we can get a new nonparameter group of value, with cumulative distribution function or abbr. CDF (Journel, 1983;
Deutsch & Journel, 1997; Richmond, 2002). Spatial dependency of new indicator values was
defined with variogram model.
2. INDICATOR TRANSFORMATION
Almost all geostatistics methods interpolate numerical values in different way. Otherwise, Indicator
Kriging is used for creating the maps with only two values, so called „two-type map“, i.e. for binary
mapping, where the conditional probability is evaluated so that the mapped variable could be
described with one of two possible values (Journel, 1983; Solow, 1986; Deutsch & Journel,
1997). The basic of Indicator Kriging is indicator transformation.
2.1. Short indicator theory
As it has been mentioned, measured data are transformed in indicator values (using limiting
values) into variable 0 and 1. It is the most frequently used in the geology for facies mapping, i.e.
for distinguishing of two lithotypes.
On the Figure 1 there are two facies, in which every of 35 points is defined by its place in 2D
coordinate system.
These data are put in some continuous, random field (group). Indicator variable in every measured
location will be calculated using limiting value by the term:
1 if Z ( x )  v cutoff
I ( x)  
0 if Z ( x )  v cutoff
Where are:
I(x)
- Indicator variable
Z(x)
- originally measured value
cutoff
- limiting value
(Equation 1)
Figure 1: Grouping of measured values in 2 classes (in coordinate system). Presented facies analysis was
made on the base of reservoir porosity value, and the limiting value was 12% (smaller were defined watersaturated facies 0, and equal or greater oil-saturated sandstone facies 1)
Total cumulative probability that the measured value will be less than limiting value (and that
indicator variable will be 0) could be calculated by the equation:
N
P( Z ( x)  vcutoff )    j  I ( x j )
(Equation 2)
j 1
Where are:
P
- probability
Z(x)
- measured value
vcutoff
- limiting value
j
- weighting coefficient
I(xj)
- indicator value on the 'x' location (in j class)
In special cases, when the variogram analysis shows that all measured values 'Z' on locations 'xj'
(j=1, 2…n) are independent (without spatial dependency), the weighting coefficient will be
calculated by mean value (  j 
1
). Then kriging can be replaced by simpler, e.g. IDW, method
N
for indicator mapping. Otherwise, if data distributed in continuous space showed autocorrelation
property, indicator variogram should be made. Modeling of such variogram is the same as
calculating any other experimental variogram, and its approximation can be made with any
adequate theoretical model.
2.2. Indicator transformation of the greater number of facies in two categorical variables
Previous case shows simple transformation measured discontinuous numerical data (e.g. porosity)
in two indicator facieses. Still, it is possible to apply similar procedure even when the input map
contains several facies (Figure 2). Then every facies could be defined as individual class or
category (C), and depending on its value, it could be made indicator transformation in analyzed
space (or random continuous field) from:
1 if  ( x)  C j
I j ( x)  
0 if  (x)  C j
j  1,2,3
(Equation 3)
Where are:
Cj
- 'j' class
(x)
- facies type (on 'x' location)
Ij(x)
- indicator variable (on 'x' location)
Figure 2: Three facies derived from the same data as on the Figure 1. In analysed sedimentary environment
facies 1 would be defined as course grained sandstone (>16%), 2 fine grained sandstone (=12-16%), and
3 marl (<12%).
Cumulative probability could be defined as:
N
P( ( x)  C j )  Pj ( x)   i  I j ( xi )
i 1
(Equation 4)
Where are:

- observed facies
C
- facies class
i
- weighting coefficient on 'i' location.
xi
- 'i' data location
Ij
- indicator variable value for 'j' class
Simplified, if the probability of particular class is equal to the term:
Pj ( x)  max( Pi ( x), i  1,2,3)
then data '(x)' really belongs to the 'Cj' class.
(Equation 5)
3. INDICATOR DATA MAPPING
Analyzed (imaginary) model was matched with our generally accepted one that describes
conditions in Croatian depressions, therefore evaluation of used methods and results can be
transferred on every existed oil/gas field in Croatian part of Pannonian basin. We based our
analysis on two facies distribution as it was schematic given in Figure 1. Experimental variograms
and theoretical model approximation have been made in program Variowin 2.21 (Pannatier, 1996).
Experimental variograms can be approximated by one of many theoretical models, however it is
important to mention that calculation of experimental variogram and therefore choosing the most
appropriate theoretical model is rather an art rather than a science. Every variogram model ends
with choosing one of many possible theoretical models. De Smith et al. (2007) presented 14 of
them, and here are re-published equations of 5 the most popular (Table 2.1).
Model
Formula
Nugget effect
Notes
Simple constant. May be added to all
models. Models with a nugget
will not be exact
Linear
No sill. Often used in combination
with other functions. May be
used as a ramp, with a constant
sill value set at a range, a
Spherical
Useful when the nugget effect is
important but small. Given as
the
default
model
in
some
packages.
Exponential
k is a constant, often k=1 or k=3.
Useful when there is a larger
nugget and slow rise to the sill.
Gaussian
k is a constant, often k=3. Can be
unstable
without
a
nugget.
Provides a more s-shaped curve
Table 2.1: Theoretical, univariate, isotropic variogram models (from de Smtih et al., 2007)
3. 1. Case 1: mapping with 25 input data
Experimentally variogram and approximation with theoretical model was made in program Variowin
2.21 (Pannatier, 1996). Number of input values was 25.
First step was creation of the variogram map. It helped us to define principal and subordinate axis.
Figure 3 shows variogram surfaced map of facies 0 and 1.
Figure 3: Variogram surface map for two facies (25 data)
Observing the variogram map indicate that there are two axes: principal, direction 135º - 315º, and
subordinate, direction 45º -225º. It was followed by experimental variogram for both axes, as it can
be seen on the Figures 4 and 5.
Figure 4: Variogram for the principal axe, direction 135º - 315º, angular tolerance 45 º (25 data)
Figure 5: Variogram for the subordinate axe, direction 45º - 225º, angular tolerance 45 º (25 data)
These variograms were approximated with theoretical curves shown on Figures 6 and 7.
Figure 6: Theoretical curve of the principal variogram axe, direction 135º - 315º, Gaussian model,
range=8.02 (25 data)
Ordinary Kriging were made in program Surfer8.0TM, and input data were variogram values. Maps
of facies distribution were made on the base of two variogram models: Gaussian and Spherical.
According to the variogram Gaussian model should match better, but the maps are showing that
Spherical model gives better geological solution. There is facies distribution map, interpolated with
Gaussian model shown on Figure 8, and there is facies distribution map interpolated with
Spherical model on Figure 9.
Figure 7: Theoretical curve of the subordinate variogram axis, direction 45º - 225º, Gaussian model,
range=4.26 (25 data)
0
13
1
0
12
1
0
11
1
1
0
10
1
9
1
0
8
1
1
7
0
1
1
6
5
1
1
4
0
1
1
3
1
2
0
1
1
1
1
2
3
4
5
6
7
8
9
10
11
12
13
Figure 8: Facies distribution created with Gaussian model (25 data)
0
13
1
0
12
1
0
11
1
1
0
10
1
9
1
0
8
1
1
7
0
1
1
6
5
1
1
4
0
1
1
3
1
2
0
1
1
1
1
2
3
4
5
6
7
8
9
10
11
12
13
Figure 9: Facies distribution created with Spherical model (25 data)
Line 0.5 represent border between two facies 0 and 1. As you can see, border line value 0.5 is very
similar to the ‘hand-made’ border, which is drown on the half way distance between points 0 and 1
(red line on the Figures 8 and 9). The colors on Figures 8 and 9 indicate on the following
lithologies: marl (light yellow), sandy marl (yellow), marly sand (green) and sand (grey blue).
3.2. Case 2: mapping with 37 input data
Experimental variogram as well as approximation with theoretical model were also made in
program Variowin 2.21 (Pannatier, 1996). Number of input data was increased on 37. Variogram
surface map of the facies 0 and 1 distribution for 37 input data is given on the Figure 10.
Figure 10: Variogram map of the facies distribution (37 data)
Main and additional axes variograms for increased (37) number of data can be seen on Figures 11
and 12, and the approximations with theoretical curves on Figures 13 and 14.
Figure 11: Variogram for principal axis, direction 135º - 315º, angular tolerance 45º (37 data)
Picture 12: Variogram for subordinate axis, direction 45º - 225º, angular tolerance 45 º (37 data)
Figure 13: Theoretical curve for principal axis (135º - 315º), Gaussian model, range=8.85 (37 data)
Figure 14: Theoretical curve for subordinate axis (45º - 225º) Gaussian model, range=3.50 (37 data)
Ordinary Kriging map of facies distribution with Gaussian model can be seen on Figure 15. Figure
16 shows facies distribution maps. In both cases the line 0.5 is very similar to ‘hand-made’ border
between facies 0 and 1 (red line).
0
0
13
1
1
0
12
1
0
11
1
1
0
0
10
1
1
9
0
1
0
8
1
1
0
7
0
1
1
1
1
6
5
0
1
1
4
0
1
1
1
1
1
3
0
2
0
1
1
1
1
2
3
4
5
6
7
8
9
10
11
12
13
Figure 15: Facies distribution created with Gaussian model (37 data)
0
0
13
1
1
0
12
1
0
11
1
1
0
0
10
1
1
9
0
1
0
8
1
1
0
7
0
1
1
1
1
6
5
0
1
1
4
0
1
1
1
1
1
3
0
2
0
1
1
1
1
2
3
4
5
6
7
8
9
10
11
12
13
Figure 16: Facies distribution created with Spherical model (37 data)
4. CONCLUSION
Mapping of the facies 0 and 1 (marl and sandstone) was made in Surfer8.0 TM program, using the
geostatistical technique of Ordinary Kriging, which we have, with careful using of isolines (0, 0.5
and 1) for indicator mapping. Number of input data in the first case was 25 and this number was
increased in the second case on 37. Facies distribution maps were made for minor and increased
number of data. Line 0.5 is representing facies 0 and 1 border, i.e. in this case of Upper Miocene
sediments, sandstone and marl border. As it could be on the facies distribution map seen, the 0.5
line is very similar in the both cases to the ‘hand-interpolated’ borders, using the half distance
between well data 0 and 1. It could be defined as border line between porous, oil-saturated and
low effective porous, water-saturated facies.
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