© Author(s) 2022. CC Atribution 4.0 License Comparison of mapping efficiency for small datasets using inverse distance weighting vs. moving average, Northern Croatia Miocene hydrocarbon reservoir Primerjava učinkovitosti kartiranja za majhne podatkovne nize z uporabo metode inverzne utežene razdalje in metode drsečega povprečja v miocenskem rezervoarju ogljikovodikov, Severna Hrvaška Josip IVŠINOVIĆ1 & Tomislav MALVIĆ2 1Vodnikova 17, HR-10000 Zagreb, Croatia; e-mail: jivsinovic@gmail.com 2Faculty of Mining, Geology and Petroleum Engineering, University of Zagreb, Zagreb, Croatia; e-mail: tomislav. malvic@rgn.unizg.hr Prejeto / Received 7. 1. 2022; Sprejeto / Accepted 15. 7. 2022; Objavljeno na spletu / Published online 22. 07. 2022 Key words: Sava depression, Croatia, interpolation, hydrocarbon reservoirs, mapping spatial comparison Abstract Mapping of geological variables in the Croatian part of the Pannonian Basin System (CPBS) is mostly based on small input datasets. In the case of the analyzed hydrocarbon field "B", reservoir "K", due to the complex geological structure and pronounced tectonics, the interpretations are restricted on several blocks, where each has very limited dataset. The porosity (19 data) and permeability (18 data) variables were analyzed. The applied interpolation methods are the Inverse Distance Weighting (IDW) and the Moving Average (MA). They were compared and analyzed by visual inspection of the obtained maps, comparison of mathematical background and by calculation of cross-validation (CV). The cross-validation value for the porosity of the "K" reservoir in the case of IDW application is 0.0011, and in the case of MA 0.0010; while in the case of permeability the IDW is 480.84, and in the case of MA 1346.41. According to the visual review of maps, the values of descriptive statistics of estimated values and the results of cross-validation, the IDW method is recommended for mapping the porosity and permeability of reservoirs blocks in the Sava Depression. Izvleček Kartiranje geoloških spremenljivk v hrvaškem delu Panonskega bazena temelji večinoma na majhnih vhodnih podatkovnih nizih. V primeru preučevanega polja ogljikovodikov »B«, rezervoarja »K« je zaradi kompleksne geološke zgradbe in močno izražene tektonike, interpretacija omejena na nekaj blokov, od katerih ima vsak zelo omejen nabor podatkov. Analizirane spremenljivke so bile poroznost (19 podatkov) in prepustnost (18 podatkov). Kot interpolacijski metodi sta bili uporabljeni metoda inverzne utežene razdalje (IUR) in metoda drsečega povprečja (DP). Metodi smo primerjali in analizirali s pomočjo vizualnega pregleda dobljenih kart, primerjavo matematičnega ozadja in z izračunom navzkrižne validacije. Vrednost navzkrižne validacije za poroznost rezervoarja »K« pri uporabi IDW je 0,0011, v primeru uporabe DP pa 0,0010. Vrednost navzkrižne validacije v primeru prepustnosti pa je bila pri uporabi IDW 480,84 in pri uporabi DP 1346,41. Glede na vizualni pregled kart, vrednosti opisne statistike ocenjenih vrednosti in rezultate navzkrižne validacije, se je metoda IDW izkazala za priporočljivo metodo pri kartiranju poroznosti in prepustnosti rezervoarskih blokov v Savski depresiji. GEOLOGIJA 65/1, 47-57, Ljubljana 2022 https://doi.org/10.5474/geologija.2022.003 Introduction Complex geological structures result in a rel- atively small volumes and consequently small datasets. The tectonics causes the fragmentation in several block, often separate hydrodynamic units, what negatively affected production. In such complex conditions, it is necessary to set up a reliable spatial model of selected variables. Any successful application of the recommended inter- polation method is always welcome because can be repeated in similar geological environments as the best approach. Here are analyzed the po- rosity and permeability of the Neogene reservoir “K” using Moving Average (MA) and Inverse Distance Weighting (IDW) methods. 48 Josip IVŠINOVIĆ & Tomislav MALVIĆ The MA method was applied in different re- search areas: economics (Fan & Wang, 2020; Rau- dys & Pabarškaitė, 2018), food industry (Kolko- va, 2018), medicine (Mustapa et al., 2019), geology (Balić et al., 2008), environment protection (Ku- mar et al., 2020), agriculture (Hatchett et al., 2009), transportation (Adeniran et al., 2018; Len- kutis et. al., 2021), energy (Alsharif et al., 2019; Xin et al., 2020) etc. The IDW method is also widely used in various research areas (Achilleos, 2011; Al-Hassan & Ad- jei, 2015; Moeletsi et al., 2016; Tunçay et al., 2016; Ikechukwu et al., 2017; Maleika, 2020; Liu et al., 2021). In petroleum geology, IDW is used to map a small set of geological variables in the area of the Croatian Pannonian Basin System (CPBS), which has proven to be a reliable interpolation method (Ivšinović, 2018a; Malvić et al., 2019; Malvić et al., 2020; Ivšinović & Malvić, 2020). Also, this could be often applied method in mapping of hydrocar- bon reservoirs worldwide (Wenli et al., 2021; Liu et al., 2020; Otchere et al., 2021) or subsurface re- source in general (Busygin et al., 2019). The paper analyzes the possibility of apply- ing the MA on particular subsurface structure in the CPBS, the reservoir “K” of the field “B” located in the western part of the Sava Depres- sion (Fig. 1). The analyzed input data set is less than 20 points, which according to the classifica- tion (Malvić et al., 2019) belongs to a small data- set. The mapping can be also done with different hybrid algorithms, no using pure interpolation, but connecting points from the very dense seis- mic, gravimetric or similar grids (e.g., Vrdoljak et al., 2021; Lemenkova, 2021). As quality check the MA results are compared with the IDW method, which is previously proven as very reliable map- ping method in the research area (Malvić et al., 2019; Ivšinović & Malvić, 2020). This research is continuation of analyses of different mathematical interpolation algorithms application in the Miocene reservoirs of the CPBS. This task is important, not only for better know- ing of the geological subsurface of this part of the PBS, but also because such algorithm is mostly applied in hydrocarbon reservoirs, still in pro- duction, and results have also economic value. The testing of different mathematical interpolation algorithms extensively started in this part more than decade ago (e.g., Balić, et al., 2008; Malvić, 2008), where some of the knowing algorithms had been compared regarding their efficiency in map- ping of geological variables collected in Miocene of the CPBS. So, Balić et al., (2008) compared also in the Sava Depression, in the Kloštar Field, four interpolations, namely: Inverse Distance Weight- ing, Nearest Neighborhood, Moving Average and Kriging, using 20 porosity values from Upper Miocene reservoir. Interestingly, cross validation/ mean square errors formula resulted in very sim- ilar values, orderly: Kriging 366.93, Moving Av- erage 369.26, Inverse Distance Weighting 371.97, Nearest Neighborhood 389.00. It was concluded that limited dataset influenced variogram model as well as successfulness of exact interpolators like (Ordinary Kriging or IDW, especially be- cause distribution could not be concluded. Au- thors pointed out that calculation of residual map can help in the interpretation of zones with higher uncertainty. Simultaneously, Malvić (2008) per- formed comparison between three geostatistical approaches – Kriging, CoKriging and stochasti- cal simulation (i.e., deterministic vs. stochastic) for datasets taken in the Drava Depression (also in the CPBS). The analysis was done with only 14 data points, so the application of geostatistics strongly depended on assumption of normal dis- tribution (the 2nd order stationarity assumption for Kriging and related methods). The simulations are described as the most descriptive approach, where zones of errors are the easiest recognizable. In analysis presented here, the input datasets are not characterized with normal distribution as well as the entire dataset can be set up in only three classes, so distribution cannot be examined with any statistical formal test for normality like Shapiro-Wilk, Kolmogorov-Smirnov, Lilliefors and Anderson-Darling (Razali & Wah, 2011). To bypass the normality condition, we selected two methods where such distribution is not strong condition, namely Inverse Distance Weighting and Moving Average, to improve conclusions ob- tained in earlier publications, especially in (e.g., Balić et al., 2008). Geological settings of analyzed area Analyzed oil reservoir is part of the typical hydrocarbon field in the CPBS. The geographi- cal position of the analyzed field “B” is shown in Figure 1. The reservoir is composed of medium to fine- grained sandstone, altered with marls. The age is Upper Miocene. A typical geological column of the western part of the Sava Depression is shown in Figure 2. The reservoir had been formed by the depo- sition of turbidities originating from the eastern part of the Alps. A schematic representation of the sedimentary environment during the Upper Pan- nonian and Lower Pontian is shown in Figure 3. 49Comparison of mapping efficiency for small datasets using inverse distance weighting vs. moving average, Northern Croatia SL OV EN IA Zagreb Osijek 0 100 km HUNGARY BOSNIA AND HERZEGOVINA SERBIA N Legend: Border of Depression State border Analysed area (field "B") Mura Depr. Sava Depression Drava Depression Slavonia-Srijem Depression Western part Bjelovar Fig. 1. Geographical positi- on of the western part of the Sava Depression and field “B” (Ivšinović et al., 2020). Fig. 2. Typical geological column of the Sava Depression (e.g., Novak Zelenika, 2013; Novak Zelenika et al., 2018). 50 Josip IVŠINOVIĆ & Tomislav MALVIĆ Fig. 3. Sedimentary environment during the Upper Pannonian and Lower Pontian western part of the Sava Depression (Ivšinović et al., 2021). -900 -1000 -1100 -1000 -900 -950 5 029 000 5 028 000 6 42 2 00 0 6 4 21 00 0 6 42 0 00 0 Legend: well deviation of well 1300 isoline normal fault 158 170 171 175 120 167 25a 173 25 168 162 161 169 166 102 101 174 149 148 N reverse fault correlation profile 172 area of interpolation Fig. 4. Structural map of reservoir “K” (Malvić et al., 2020). 51Comparison of mapping efficiency for small datasets using inverse distance weighting vs. moving average, Northern Croatia Reservoir hydrodynamics is defined by faults, which created several blocks with different per- meabilities. In the beginning the contact oil-wa- ter was unique for entire reservoir, i.e., reservoir was single hydrodynamic unit. However, after some recovery period, parts are characterized with larger permeability caused that that fluids faster moving upward as well as contact, and consequently larger portion of water in produced fluid, until the production was not ceased. The structural map of the analyzed deposit “K” of the field “B” is shown in Figure 4. The reservoir is classified as layer type or with structural-strati- graphic trap. Simultaneously, reservoir pressure was dropped, but also minor communication of fluids had happened through fault zones. Methodology The applied interpolations were Moving Aver- age (MA), and Inverse Distance Weighting (IDW). Their results had been compared visually, nu- merically and theoretically. Moving Average (MA) The MA method (Eq. 1) assigns values to spa- tial points by determining the mean value of measured data located within a particular area around the grid node (un-known value that is es- timated by MA). The minimum (sometimes also maximum) amount of analyzed data should be defined. The value calculated in each grid node is equal to the arithmetic mean of the measured data located within the defined range of spatial dependence. The mathematical equation for cal- culating the moving average value is (e.g., John- ston et al., 1999; Ekhosuehi & Omorogbe Dick- son, 2016; Rusdiana et al., 2020): (1) where: yi,j-interpolated value of moving average method; n- number of data set; xi,j-k- input data. Inverse Distance Weighting (IDW) The IDW method is based on the distance (ponder with exponent, mostly second power) be- tween the measured data and the location where unknown value is estimated. Such values are in grid nodes. The measured data included in calcu- lation are values inside the searching radius/ra- dii, i.e. inside circle/ellipsoid of spatial depend- ence. The mathematical expression for inverse distance estimation (e.g., Setianto & Triandini, 2013; Ivšinović, 2018b) is: (2) where: ziu-estimated value; di-distance to “i-th” location; p-power of distance; zi-measured values at “i-th” location. The result of IDW interpolation depends on the value of the exponent p, which is obtained ex- perimentally and has a different value in differ- ent fields of science. In subsurface mapping, for the CPBS, the proposed value exponent “p” is 2. Cross-validation (CV) Cross-validation (CV) or out-of-sample test- ing is a method of assessing the quality of a map obtained by implemented an interpolation meth- od. It is based on the re-placement of the any measured (original) data(s) from an entire meas- ured dataset and is replaced by a new value(s) estimating from the existing data set. In geolo- gy the most often is applied “p=1”, i.e., leaving- one-out CV. Differences between measured and estimated value on the same location is error. The very often such error are calculated using mean square error algorithm (MSE). The mathematical formula for calculating MSE is (e.g., Malvić & Novak Zelenika, 2013): (3) where: MSE- mean square error; measured- known value at location “i”; estimated- unknown value (deleted) at location “i”; n - number of lo- cations. The quality of the map obtained by the inter- polation method is based on the numerical value of the MSE, the lower the value of the MSE the better the applied interpolation method. Mathematical differences in MA and IDW and reflections of searching radius Basically, both methods are considered as mathematically simpler interpolation methods, that highly depend on number of input data. However, both use so called searching radius or radii. If anisotropy could be recognized and quantitatively described, two axes/ellipsoid of dependency are defined, if not – the circle out- lined area with hard data included in calculation. The small datasets (n<20) hardly allow unambig- uous recognition of anisotropy, even with lot of , j , 1 1 n i i j k k y x n −= = ∑ 1 2 1 2 1 2 ... 1 1 1... n p p p n IU p p p n zz z d d dz d d d + + = + + 2 1 1 (measured estimated) n i MSE n = = −∑ 52 Josip IVŠINOVIĆ & Tomislav MALVIĆ qualitative geological data, so the most often MA or IDW works with searching radius or circle (of spatial dependance), without any further sepa- ration of circle in quadrants or octants, because most of them would be without single hard value, i.e., would be empty. Definition of searching radius is much more important in MA then IDW, because MA does not honor distances among unknown value/location and known, measured, hard data. However, one of the geological axioms is that if what two meas- urements are more distant in space, the differ- ence must be larger. It is valid for any geological variable, like thickness, granulometry, porosity, permeability, depths etc., simply because any ge- ological value is result of particular geological environment, limited in space and time. Conse- quently, the equally weighting of each measured value (MA) inside searching radius surely does not honor such rule (Equation 1), especially be- cause area of spatial dependance has to cover large part or entire analyzed area to include at least one measured value. Oppositely, the IDW honor the distances among estimated and measured values, equally for any direction (Equation 2). Even for single reservoir this could be crucial advantage, be- cause reservoir’s lithofacies is not homogeneous and will gradually change from the center of structure, especially if depocenter remained in the current structure top, toward the reservoir margins. This presumption has been tested in this work and elaborated with results, discussion and conclusions. Results and discussion The analyzed variables were obtained from laboratory measurements of well cores and log- ging measurements. Data from reservoir “K” are shown in Figure 5. The analyzed porosity (19 data) and permea- bility (18 data) belong to a small datasets. They are characterized by intervallic grouped data - in the case of permeability two, and in the case of porosity three groups, which is results of an- alytical approximation and reservoir heteroge- neity. However, the problem is what the formal normality tests cannot be applied, and empirical Q-Q plot cannot be calculated. It is why those two interpolation algorithms, where normal distribu- tion is not strong condition for their application, had been selected. Based on selected area and structural axes, the searching radius was in both algorithms set on 628 m. It is done using the “rule of the thumb” that if data do not allow create reliable spatial model, omnidirectional or anisotropic, for search- ing radius is recommended to use structural or depositional axes and their ration (if exists). So, 628 m is about half of the quadrat used to border researched area and is used for omnidirection- al/circle searching radius. The research area is Well Surface X Surface Y Porosity (part of units) J-101 6421096 5028877 0.217 J-120 6420658 5029068 0.272 J-161 6420957 5028870 0.217 J-162 6421034 5028593 0.217 J-167 6420529 5028674 0.217 J-168 6420699 5028475 0.315 J-169 6420724 5028825 0.217 J-170 6420349 5028926 0.223 J-174 6421298 5028863 0.217 J-175 6420475 5029136 0.223 J-158 6420303 5028910 0.223 J-171 6420576 5028970 0.223 J-172 6420928 5029147 0.223 J-102 6421208 5028926 0.217 J-148 6421126 5028437 0.217 J-149 6420959 5028501 0.217 J-166 6420771 5028650 0.217 J-25 6420546 5028460 0.315 J-173 6420539 5028382 0.217 Fig. 5. Raw data set of po- rosity and permeability of reservoir “K” (Malvić et al., 2020). 53Comparison of mapping efficiency for small datasets using inverse distance weighting vs. moving average, Northern Croatia 42.3 km2. Obtained permeability and porosity in- terpolations by IDW and MA methods are shown in Figure 6, while the results of cross-validation and descriptive statistics are shown in Table 1. The results given on Figure 6 are influenced with, and reflected in, the complexity of the geo- logical structure, and especially depositional en- vironments that are responsible for clastic types and their petrophysics. Here is also included the problem of measurement errors and consequently fitting of measured values into classes. It is well known that laboratory and logging results can differ, even for an order of magnitude, so some- times they are not directly comparable or have to be used jointly only with caution. The step of caution had been selection of values in group, de- creasing influence of measurement error, but also decreasing data spectrum, what made interpola- tion significantly less representative, especially on smaller scales of changes. The IDW porosity map (Fig. 6, a) shown many outliners of individual points resulted in so- called bull-eyes and even on NW part butterfly effects. That was expected due to the distances of measured data, applied power exponent and values of minimum and maximum. According to Table 1 IDW porosity values have standard devi- ation of 0.016 and MSE 0.0011. The MA porosity map (Fig. 6, b) is more uni- formly shaped at first glance. No butterfly or bull- Fig. 6. Maps of reservoir “K” obtained by interpolation methods: (a) Porosity (IDW); (b) Porosity (MA); (c) Permeability (IDW); (d) Permeability (MA). Table 1. Cross-validation values and descriptive statistics of estimated values for IDW and MA methods for geological vari- ables porosity and permeability of reservoir “K”. Variable Method Cross-validation (MSE) Min Max Mean Standard deviation Porosity IDW 0.0011 0.219 0.293 0.232 0.016 Porosity MA 0.0010 0.217 0.243 0.231 0.006 Permeability IDW 480.84 36.33 115.97 90.97 28.97 Permeability MA 1346.41 63.95 121.20 93.06 13.34 54 Josip IVŠINOVIĆ & Tomislav MALVIĆ eyes are pronounced, which is understandable because it is a local calculation of the mean val- ue around the estimated data, i.e., in the search- ing radius. So, if averaging is dominant, and to- tal number of data is small, the consequence is that many estimations will be averaged with the same values. Furthermore, it means that complex shapes like butterfly of bull-eye effects will not be developed. Moreover, the transition area be- tween the data is not emphasized, but island-like irregularities are formed in this area (between wells J-170 and J-175 or J-162 and J-174. Due to the data averaging, the usual butterfly or bull- eyes effects cannot occur, as in the IDW method. That is clearly seen by the circular appearance of the “green” surface in the center of Figures 6, b. Comparing the data of descriptive statistics (Ta- ble 1) the values of standard deviation (0.006) and MSE (0.0010) are slightly smaller than in the IDW. The IDW permeability map (Fig. 6, c) empha- sized the butterfly or bull-eyes effects. In the northwestern part of the reservoir, it is less pro- nounced, while in the southeastern more. Fig. 6, c clearly shows the zone of abrupt changes be- tween wells J-169 and J-171 or J-167 and J-158. The transition zone between the zone of higher reservoir permeability and lower reservoir per- meability can be follow along entire map and correspond to fault zone shown on structural map (Fig. 4). The data of the descriptive statistics from Table 1 showed relatively small standard deviation (28.97) and high MSE (480.84). On the MA permeability map (Fig. 6, d) is clearly seen no uniform transition zone between the different permeability values. This can be seen in Figure 6, d through the peninsula-like shape between wells J-166 and J-167 or J-101 and J-102. The values are averaged as seen from the descriptive statistics in Table 1 and although standard deviation is smaller than by IDW (13.34) the MSE is extremely high (1346.41). The differ- ence between the original and estimated data (MSE) is twice as large in the case of the min val- ues of the set. This can be seen from the interpo- lated map (Fig. 6, d) that the permeability values of wells J-167 and J-169 differ significantly, and the measured values are the same (121.2·10-9 m2). Regarding large difference between MSE(IDW) and MSE (MA), here 480.84 vs. 1346.41, the range of permeability is much higher than of porosity, so consequently any estimation will varies sig- nificantly depending on applied mathematical al- gorithm, like it is shown in results. As can be seen from Table 1, the cross-vali- dation/MSE value has a lower value of MA than IDW in the case of porosity, while in the case of throughput it is the reverse case. The difference in cross-validation in bandwidth between the MA and IDW methods is 0.0001. This difference can be ignored due to the other results of the de- scriptive statistics in Table 1. It can be observed that the coverage of the estimated values in the IDW method is much more realistic than in the case of the MA method. In such cases, the rule of accepting the interpolation method to the amount of the lower CV value can be deviated from. Most importantly, the estimated values obtained by interpolation methods must be an approximate reflection of the measured data. Conclusions Here is solved one local problem of selection the appropriate interpolation algorithm for small datasets of petrophysics measured in the sub- surface Miocene hydrocarbon reservoirs in the Northern Croatia (CPBS). So, the results are of interest for researchers engaged in the studying of such reservoirs in the Sava Depression as well as in all other depressions in the CPBS. Howev- er, all researching that included interpolation of small geological datasets (n<=20 points) could find those achievements and conclusions worth of testing in their own explorations, whatever the data are collected from surface or subsurface. The specific outcomes obtained with this analy- sis are: • Visual inspection of the obtained interpolat- ed maps revealed that the maps obtained by the IDW method are more acceptable for interpre- tation of reservoir petrophysics compared to the MA method. • The permeability map of the “K” reservoir ob- tained by the MA method can be described with mosaic effect, with sharply wavy, peninsular, and island shapes, which is not an unusual case in mapping of small datasets, but cannot be in- terpreted with sense. This is characterized by a high value of cross-validation. • Descriptive statistics, histograms and maps showed that the values obtained by the IDW method are closer to the ranges of original data- sets. The differences between IDW and MA are about 25 % for porosity and more than 200 % for permeability. • The calculated cross-validation/MSE for the MA method in the case of porosity is 0.0010, and permeability is 1346.41 for reservoir “K”. In the case of the IDW method the MSE for porosity is 0.0011, while for permeability 480.84. The dif- ference between the MSE values for porosity can 55Comparison of mapping efficiency for small datasets using inverse distance weighting vs. moving average, Northern Croatia be neglected with respect to the results of de- scriptive statistics and visual inspection of the obtained maps. • This researching showed that mapping the porosity and permeability of Neogene reservoir in the CPBS highly depends on the experience of the interpreter in the application of differ- ent mathematical methods and geological un- der-standing of spatial distribution of selected variables. The MA showed one large disadvan- tages – in the case of small datasets there is not enough measured values inside searching ra- dius for reliable calculation of average. So, the “distance weighting” approach of the IDW is far better approach for mapping of reservoirs in such case. 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