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Ordinary Kriging as the most appropriate interpolation method for porosity in the Sava depression Neogene sandstones Obični kriging kao najprimjerenija interpolacijska metoda za poroznost u neogenskim pješčenjacima Savske depresije Davorin BALIĆ1 and Tomislav MALVIĆ1,2 1INA-Oil industry, Oil & Gas Exploration and Production, Reservoir Engineering & Field Development, Šubićeva 29, 10000 Zagreb, e-mail: davorin.balic@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: Porosity can be mapped by many computer-based interpolated methods. All of them are defined by less and more advanced mathematical equations. There is several works performed on data from the Sava depression that tested appropriation of several such mapping methods, available in almost all computer mapping software packages. Mathematically simpler methods are quicker, but often less precise in cases based on smaller input dataset (approximately 10-15 hard data or lesser). There is another option to use geostatistical deterministical methods like kriging. In kriging methods the most often and very suitable is the Ordinary kriging technique, developed from equations of the Simple Kriging as the basic kriging approach. It can be proven, using matrix equations, that kriging variance (as the measure of error) is decreased in the Ordinary than in the Simple Kriging. The presented analysis was directed in the porosity dataset as the observed input derived from the reservoir. Three simpler methods had been tested (Inverse Distance, Nearest Neighborhood, Moving Average) to obtain porosity maps. Furthermore is proven that variance minimizing in better in the Ordinary Kriging than in the Simple Kriging. Finally, the map interpolated by the Ordinary Kriging and belonging numerical result had been point out as the most appropriate tool (and the interpolation technique) that could be applied for porosity distribution description in reservoir sandstones of Neogene age (from Early Pannonian to Early Pontian). Due to lithological, depositional and tectonic properties similarities in all hydrocarbon reservoirs located in the Sava depression, presented result and given recommendations can be applied for entire depression. Key words: Ordinary Kriging, Simple Kriging, porosity, sandstones, Sava depression, Neogene Sažetak: Poroznost se može kartirati upotrebom brojnih računalnih interpolacijskih metoda. Sve one su temeljene (manje ili više) na naprednim matematičkim jednadžbama. Postoji nekoliko primjera načinjenim na podatcima iz Savske depresije kojima je isprobana upotrebljivost takvih metoda, a sve su bile dostupne u gotovo svim računalnim paketima za kartiranje. Matematički jednostavnije metode su brže, no često i manje precizne u slučajevima kada ulazni skup sadrži mali broj podataka (približno 10-15 ili manje). Postoji i druga mogućnost koja obuhvaća upotrebu geostatističkih determinističkih metoda poput kriginga. Kod te metode vrlo je česta i primjerena upotreba tehnike običnoga kriginga, razvijene iz jednadžbi jednostavnoga kriginga (koji je osnovna tehnika kriginga). Moguće je dokazati, upotrebom matričnih jednadžbi, da je varijanca kriginga (kao mjera pogrješke) manja u tehnici običnoga nego li jednostavnoga kriginga. Prikazan analiza bila je usmjerena na skup vrijednosti poroznosti kao ulazne podatke iz ležišta. Testirane su tri jednostavnije metode (inverzne udaljenosti, najbližega susjedstva, pokretne sredine) kako bi se načinile karte poroznosti. Nadalje, dokazano je i kako je varijanca pogrješke manja kod običnoga nego li jednostavnoga kriginga. Na kraju, karta interpolirana običnim krigingom te numerički rezultat su istaknuti kao najprimjereniji alat (i interpolacijska tehnika) kojom se može opisati raspodjela poroznosti u pješčenjačkim ležištima neogenske starosti (od donjega panona do donjega ponta). Zahvaljujući sličnim litološkim, taložnim i tektonskim svojstvima u svim ležištima ugljikovodika Savske depresije, prikazani rezultat te preporuke mogu se primijeniti na području cijele depresije. Ključne riječi: obični kriging, jednostavni kriging, poroznost, pješčenjaci, savska depresija, neogen 1. INTRODUCTION Porosity can be mapped by many computer-based interpolated methods. All of them are defined by less and more advanced mathematical equations. There is several works performed on data from the Sava depression that tested appropriation of several such mapping methods, available in almost all computer mapping software packages. Mathematically simpler methods are quicker, but often less precise in cases based on smaller input dataset (approximately 10-15 hard data or lesser). There is another option to use geostatistical deterministical methods like kriging. In kriging methods the most often and very suitable is the Ordinary Kriging technique, developed from equations of the Simple Kriging as the basic kriging approach. It can be proven, using matrix equations, that kriging variance (as the measure of error) is smaller in the Ordinary than in the Simple Kriging. The presented analysis was directed in the porosity dataset as the observed input derived from the reservoir. Three simpler methods had been tested (Inverse Distance, Nearest Neighbourhood and Moving Average) to obtain porosity maps. Furthermore is proven that variance minimizing in better in the Ordinary Kriging than in the Simple Kriging. Finally, the map interpolated by the Ordinary Kriging and belonging numerical result had been point out as the most appropriate tool (and the interpolation technique) that could be applied for porosity distribution description in reservoir sandstones of Neogene age (from Early Pannonian to Early Pontian). Due to lithological, depositional and tectonic properties similarities in all hydrocarbon reservoirs located in the Sava depression, presented result and given recommendations can be applied for entire depression. 2. SIMPLE INTERPOLATION The Inverse Distance Weighting method (Figure 1) estimates values from a relatively simple mathematical expression. The influence of each point is inversely proportional to the distance from an estimated location. The number of points included in the estimation, is defined by circle radii drawn around a selected location. Figure 1: Porosity map interpolated by Inverse Distance Weighting (from: Balić et al., 2008) The Nearest Neighbourhood method (Figure 2) assigns the value of the closest point to each grid node. It is not so much an interpolation method as zonal assignment technique. This method is useful in case of relative large zones without data (blind areas), which need to be schematically mapped. Figure 2: Porosity map interpolated by Nearest Neighbourhood (from: Balić et al., 2008) The Moving Average method (Figure 3) calculates the values of grid nodes from averaged data measured in a particular ellipsoid or circle that surrounds each grid node. There is also a need to define the lowest number of data that can be averaged. This is not exact interpolation but rather a kind of simple method characterised by its trivial smoothing character. Figure 3: Porosity map interpolated by Moving Average (from: Balić et al., 2008) Results are presented on different maps (Balić et al., 2008). Almost all can be recognised by their very different morphologies. The result of Inverse Distance Weighting method (Figure 1) indicates the real porosity distribution in the reservoir and is also similar to the kriging map (Figure 4) which is considered as the most authentic algorithm in the sense of geology and geomathematics. Figure 4: Porosity map interpolated by kriging with isotropic variogram model and range 1100 m (from: Balić et al., 2008) However, the maps obtained by other two methods, which are not exact interpolators, are inappropriate. These are Nearest Neighbourhood (Figure 2) and Moving Average (Figure 3) maps, where the presented isoporosity cannot be meaningfully interpreted. Even if the radii of the searching ellipsoid is changed (moving average), the gradual transition among lines has not been achieved. The Nearest Neighbourhood result strongly emphasises the polygonal presentation. Therefore, these two methods cannot be applied to porosity interpolation in Neogene sandstone reservoirs of the Sava depression. Numerical estimation of maps is performed a cross-validation equation. The following values were obtained for the different methods (starting with the lowest error, from Balić et al., 2008): Ordinary Kriging (366.93), Moving Average (369.26), Inverse Distance Weighting (371.97) and eventually Nearest Neighbourhood (389.00). 3. ADVANCED MAPPING 3.1. Mathematical fundamentals of kriging The principle of kriging is shown most simply with sets of equations that define the method. Kriging is applied to the estimation of the values of a regionalised variable at a selected location (Zk), based on surrounding existing values (Zi). Each such location is assigned a relevant weighting coefficient (i), and the calculation of this is the most demanding part of the kriging algorithm. The value of a regionalised variable can be defined as: Z i Z xi (Equation 1) Where is: xi - is the value at the known location. Moreover, the value of a regionalised variable estimated by kriging based on n points is: n Z k i Z i (Equation 2) i 1 Where are: i - is the weighting coefficient for a particular location “i”; Zi - are known values, the so-called “control points” (hard data); Zk - is the value estimated by kriging. These equations represent the system of linear kriging equations. Previous equation also can be written as the matrix: A B (Equation 3) In matrices A and B the values are expressed as variogram values, i.e., these values depend only on the distances and orientations between the control points and not on their values. The third matrix includes weighting coefficients, which are simply estimated from a system with “n” equations with “n” unknown variables. 3.2. Simple Kriging theory Simple Kriging, as its name implies, is the simplest kriging technique. The full matrix equation is: ( Z1 Z1 ) ( Z1 Z 2 ) ... ( Z1 Z n ) 1 ( Z1 Z ) ( Z Z ) ( Z Z ) ... ( Z Z ) ( Z Z ) 2 1 2 2 2 n 2 2 ( Z n Z1 ) ( Z n Z 2 ) ... ( Z n Z n ) n ( Z n Z ) (Equation 4) Where are: - are the variogram values; Z1…Zn - are known measured values at points; Z - is the point at which new values are estimated from known (hard) data (Z1…Zn). Kriging uses dimensionless point data that represent the values of the regionalized variable. In Simple Kriging, it is assumed that the regionalized variable has second order stationary, the excepted value is everywhere constant and known, and the covariance function is known [c(x,y)=Cov(Z(x),Z(y))]. Furthermore, when this estimation is performed at the control point, the error can also be calculated at the point as: Z real Z estimated (Equation 5) If there is no external drift in the variable and the sum of all weighting coefficients is 1, unbiasedness is achieved. The difference between all the measured and estimated values is called the estimation error or kriging variance and it is expressed as: n 2 (Z i 1 real Z estimated ) i2 n (Equation 6) In an ideal case, kriging tries to calculate the optimal weighting coefficients that will lead to the minimal estimation error. Such coefficients, which lead to an estimation of unbiasedness with minimal variance, are calculated by solving of the matrix equations system. If the matrices in previous equation are represented by linear equations, it can be written as: (Z1-Z1)x1 + g(Z1-Z2)x2 + … + (Z1-Zn)xn = (Z1-Z) (Z2-Z1)x1 + g(Z2-Z2)x2 + … + (Z2-Zn)xn = (Z2-Z) …………… (Equation 7) (Zn-Z1)x1 + g(Zn-Z2)x2 + … + (Zn-Zn)xn = (Zn-Z) Moreover, for previous linear equation to be considered unbiased, an additional condition must be fulfilled, i.e., the sum of all weighting coefficients is 1. This condition is achieved by adding new conditions to the kriging matrices. Technique of Ordinary Kriging is the most-often used unbiasedness estimation. The estimation is the selected point that can be represented by the following equation: 1 ( Z 1 Z 0 ) ... (Z Z ) n 0 n Z ( Z Z ) ... ( Z 1 1 1 Zn ) ... ... ... ( Z Z ) ... ( Z Z ) n 1 n n (Equation 8) Moreover, for previous linear equation to be considered unbiased, an additional condition must be fulfilled, i.e. the sum of all weighting coefficients is 1. This condition is achieved by adding new conditions to the kriging matrices. Technique of Ordinary Kriging is the most-often used unbiasedness kriging technique. 3.3. Ordinary Kriging theory All kriging techniques (except Simple Kriging) have added some constraints to the matrices, to minimize the error k2 ( x ) , and these techniques are unbiasedness estimations. Generally, these factors would describe some external limit (restriction) on the input data, which cannot simply be observed in the measured values. The most-used kriging technique is probably Ordinary Kriging, and we therefore analyse the constraint factor in Ordinary Kriging equations, called the Lagrange multiplicator. As discussed above, if the sum of all weighting coefficient is 1, kriging expression can be written as: (Z1-Z1)x1 + g(Z1-Z2)x2 + … + (Z1-Zn)xn + m = (Z1-Z) (Z2-Z1)x1 + g(Z2-Z2)x2 + … + (Z2-Zn)xn + m = (Z2-Z) …………… (Equation 9) (Zn-Z1)x1 + g(Zn-Z2)x2 + … + (Zn-Zn)xn + m = (Zn-Z) 1 + 2 + … + n + 0 = 1 If such a system of linear equations is shown as kriging matrices it can be written as: (Z1 Z1 ) (Z1 Z 2 ) ( Z Z ) ( Z Z ) 2 1 2 2 ( Z n Z 1 ) ( Z n Z 2 ) 1 1 ... ( Z 1 Z n ) 1 1 ( Z 1 Z ) ... ( Z 2 Z n ) 1 2 ( Z 2 Z ) 1 ... ( Z n Z n ) 1 n ( Z n Z ) ... 1 0 m 1 (Equation 10) The number of weighting coefficients and control points can be very large, but contemporary computers can successfully solve numerically demanding tasks. The estimation can be performed simply by calculating the influence of all control points weighted by their associated coefficients according to: Z 1 Z1 2 Z 2 ... n Z n (Equation 11) The calculation of the estimation variance includes adding the Lagrange coefficient: 2 1 (Z1 Z ) 2 (Z 2 Z ) ... n (Z n Z ) m (Equation 12) Here are shown the two probably most-used kriging techniques. In general, in all linear kriging techniques (since they are constrained optimization problems), the associated equations can be divided into two parts: a) In one part of the equations, the spatial dependence (spatial correlation) of the measured data is calculated, usually using a variogram; b) The other part of the equations includes different constraints, resulting in the sum of all weighting coefficient being equal to 1. 4. CONCLUSION The Ordinary Kriging technique is more appropriate for the interpolation of point data (compared with Simple Kriging). In lot of papers it was proven that geostatistical interpolation (as well as stochastic estimation) is the best approach to mapping geological variables. It is concluded that kriging techniques can be successfully applied with minimal of 10 to 15 data points. Any representative statistical dataset of geological variables must include at least 30 data points. This means that we cannot conclude the value of the area population mean (expected value) based only on the usually available datasets. For this reason, the so-called “local mean” is most often applied, which is calculated only from the hard data encompassed by the searching radii. This favours the Ordinary Kriging technique. Therefore, we must be very careful to analyse the variables with Ordinary Kriging equations. Most of the variables are the standard elements of all kriging techniques, but there is a unique variable, the Lagrange (linear) multiplicator. It has been shown (Figure 5) that this value can be negative, but the kriging variance cannot be negative (mathematically impossible). If the value is randomly sampled many times from some interval (e.g. -1,1), one value will produce the least possible kriging variance. The user only needs to select the width of sampling interval and number of samples. This procedure can be described through four steps (similar like in paper Malvić & Balić, 2009): test stopping direction of changing “m” value 2 minimum of variance the case when the Lagrange multiplicator is decreased in negative values m -0.3 -0.2 -0.1 0.01 0.1 0.2 negative kriging variance negative kriging variance m -0.3 -0.2 -0.1 0.01 0.1 0.2 direction of changing “m” value test stopping 2 minimum of variance the case when the Lagrange multiplicator is increased in positive values Figure 5: Graphical representation of how to select the most appropriate value for Lagrange multiplicator using random sampling (from Malvić & Balić, 2009) 1. Based on experience, it can be assumed that value of the Lagrange multiplicator should occur in the interval (-1,1). Some values can be very close to 0. For these reasons, we consider that the starting point in the random sampling should be set at 0.01. 2. In the next step, the starting value must be decreased (in steps of e.g. 0.05), i.e. it becomes negative. Again, the appropriate kriging variance can be calculated and registered. Such a procedure should be repeated until the kriging variance is positive. Using this procedure, the value of m with the least kriging variance can be selected (Figure 5 left). 3. The positive side of the Lagrange value should then be checked, again increasing the value in steps of +0.05. Reduction in the variance will be observed, and the procedure should be continued until this reduction vanishes. The first time when the variance is increased, the calculation is stopped (Figure 5 right). 4. It is important to know that if, for the first positive ‘m’ (m>0), the variance immediately started increasing, then it can be considered that the minimum kriging variance is found in the negative interval (less than 0) and random sampling can be ceased (Figure 5 left). We believe that these four rules completely describe the correct procedure for selecting the Lagrange multiplicator in Ordinary Kriging equations. Consequently, this kriging technique is one of the best interpolation algorithms for mapping geological variables. 5. REFERENCES 1. Balić, D., Velić, J. & Malvić, T.: Selection of the most appropriate interpolation method for sandstone reservoirs in the Kloštar oil and gas field. Geologia Croatica, 61, 1, 27–35 (2008). 2. Malvić, T. & Balić, D. (2009): Linearity and Lagrange Linear Multiplicator in the Equations of Ordinary Kriging. Nafta, 59, 1, 31-37.