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: (Reservoir Geologist); (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 non- parameter 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:
(Equation 1)
Where are:
I(x) - Indicator variable
Z(x) - originally measured value
ncutoff - limiting value
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 water-saturated 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:
(Equation 2)
Where are:
P - probability
Z(x) - measured value
vcutoff - limiting value
lj - 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 (). Then kriging can be replaced by simpler, e.g. IDW, method 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:
(Equation 3)
Where are:
Cj - 'j' class
w(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 (f>16%), 2 fine grained sandstone (f=12-16%), and 3 marl (f<12%).
Cumulative probability could be defined as:
(Equation 4)
Where are:
w - observed facies
C - facies class
li - 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:
(Equation 5)
then data 'w(x)' really belongs to the 'Cj' class.
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 / Notes /Nuggeteffect / / 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 nuggeteffect 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)
Figure 8: Facies distribution created with Gaussian model (25 data)
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).