Indicator KrigingCase study; Geological Models of Upper Miocene Sandstone Reservoirs at the Kloštar Oil and Gas Field Kristina Novak Zelenika Zagreb, November 2013
Introduction • Application of mathematics in geology is relatively new approach in interpretation of underground geological relations. • Two great scientists are founders of this discipline: Prof. Dr. Daniel Krige and Prof. Dr. George Matheron. • Geostatistical methods can be divided into deterministical and stochastical methods.
Introduction – determinism • In deterministical methods, all the conditions which can influence to estimation, have to be completely known (mustn't have randomness of any kind in variables description). • Deterministical results can be unambiguously described by the completely known finite conditions. • It is clear that geological underground is only one, but since the description of the underground is based on well data (point data) it is not possible to be absolutely sure that the solution obtained with geostatistical methods is absolutely correct (all geostatistical methods contain some uncertainty). • Deterministical methods give only one solution. • It is more correct to call them deterministical interpolation methods.
Introduction – stochastics • Stochastical realizations provide different number of solution for the same input data set. • The solutions can be very similar, but never identical, and all obtained solutions or results are equally probable. There are conditional and unconditional simulations. • In stochastical processes number of realizations can be any number we want. • It is very clear that more realizations will cover more uncertainty area, i.e. the more realizations there are, the lower uncertainty is.
Indicator Kriging Where: I(x) - indicator variable; z(x) - measured value; cutoff - cutoff value. Location map of 38 data: 1 represents sandstone, 0 represents other lithofacies Recommended no. of cutoffs: 5-11 Results: probabilities
What are the principles of indicator formalism in Indicator Kriging? • Indicatorformalism: • Indicatortransformationcanbeinterpreted as follows:
If v is continuous variable • Inthiscaseweshouldcreatecumulativeprobabilitydistributionof v fromthe data values: • Sincewegeneralyhavefinitenumberof data, thecumulativeprobabilitydistributionfunctionmay change withtheincreasing or decreasingnumberofavailable data. • That is whythecumulativeprobabilitydistributionfunctioniscalledconditionalprobabilitydistributionfunction (ccdf) • It is conditionedbynumberofavailable data
Nextstep: Introducetheindicatorformalism for thisccdfin a way to subdividethe total rangeusing k cut-offvalues
According to ccdfwecandefinethecorrespondingprobabilities for all thesecut-offs:
Wecanchoose a particularcut-off, say 2m • Allthelocationscanbecategorizedintwo groups: The first one is the set oflocationswheretheactualthickness is smallerthan 2 m Thenext group is locationswheretheactualthicknessislargerthan 2 m Usingthiscut-offwecandefineanindicatorvariable, whichtakes 1 for all locationswherethethickness is smallerthan 2, and takes 0 for all otherlocations
Inthiswaywecandefine all otherindicatorvariables • Actually, thelargernumberofcut-offs, the more precisethecontinousccdfderived are and this is theprincipleofIndicatorKriging
Withrespectof 5 indicatorcut-offs (2, 4, 6, 8, 10 m), wecancreate 5 pointmapsshowingtheactualvalues (0 or 1). • Thatmeanswehave 5 pointmaps – one for eachcut-off • Eachmapcontainsonly 0 and 1 values • Unfortunately, wecannotperformanymeaningfulestimationwiththesevalues
But, theycanhold some othermeaning: • Wesupposethat at anyparticularwelllocationtheprobabilityofthethicknesssmallerthan a particularcut-offcanbederivedfromthe global probabilitydistributionofthickness • Wecanconcludethataftermakingindicatortransformation, theprobabilitiesoftheirvalueequals 1 canbeestimated
Thisestimationcanbeperformed for eachindividualcut-offseparately • As a resultwe got gridsshowingtheprobabilitiesthattheindicatorvariabletake 1 value
Output of Indicator Kriging • Ineachrowtheprobabilitiesincreasebyincreasingofthecut-offvalues • Alloftheseprobabilitiesbelong to a particular grid point • UsingIndicatorKrigingtheccdf at a grid pointcanbeestimated • Thefinalresultwecanget is ccdf for each grid point
If v is a categorical variable • Rocktype • TheIndicatorKrigingofthatvariablegivestheprobabilitythatthisrocktypeappears at a particularlocation
Conclusion • The Indicator Kriging is a specific geostatistical technique for spatial phenomena with weak stationarity. • In fact, this kriging technique is weaker than any other kriging approximation. • However, this technique is designed for estimating lateral uncertainty. • This approach estimates the local probability distributions on grid cells.
Advantages and disadvantages • Advantages: • It does not need normality of the input data set • It can be inplemented in case of bimodal distribution • Since it estimates probabilities, it may show the connectivity of the largest values (very important in production plans or EOR projects) • Disadvantages: • Success of IK strongly depends on the correct selection of the cut-offs values. The fewer the numbers of cut-offs are, the fewer details of the distribution can be got.
Introduction – research location • There are many reservoirs in Croatian part of Pannonian Basin interpreted with deterministical and stochastical methods (like reservoirs of the fields Ivanić, Molve, Kalinovac, Stari Gradac-Barcs Nyugat, Beničanci, Ladislavci, Galovac-Pavljani, Velika Ciglena). • Kloštar Field was very detail analyzed in the joint study of INA and RGNF, led by Prof. Dr. J. Velić and Prof. Dr. T. Malvić. • Kloštar Field was chosen as research location i.e. its sandstone reservoirs as objects with high and accurate base of the measured data and many geostatistical results and interpretations.
Introduction – used methods and analyzed variables Used methods Deterministic Stochastic OK IK SGS SIS Analyzed variables Porosity Depth Thickness
Introduction - goals • Goals: • (1) Construction of geostatistical model of the Kloštar field (reservoirs T and Beta); using of geostatistics as tool for improving of mapping accuracy • (2) Geostatistical models will represent upgrade for previously available deterministic models from field study.
Location of the Kloštar Field Kloštar Field location (CVETKOVIĆ et al., 2008)
About the Kloštar Field wells • Total no. of wells: 197 • Measured wells: 57 • Technically abandoned: 73 • Water injection wells: 5 • Production wells: 62
Location of the Beta and T reservoirs Location of the Beta reservoir Location of the T reservoir
Lithology and log curves of Klo-62 well Lithology and log curves of Klo-145 well
Core data – cores from INA laboratory Klo – 57(788.9 – 793.3 m, III m) Rocks top section of T+U+V reservoir Determination: Lithoarenite(VELIĆ & MALVIĆ, 2008) Klo – 82 (1404.6 – 1411.7 m, II m) Beta Reservoir Determination: Lithoarenite (VELIĆ & MALVIĆ, 2008)
Structural modeling of the Kloštar Field • Kloštar Field is anticline with direction northwest-southeast • Normal fault (Kloštar fault) divides structure into two parts, northeastern and southwestern • Conceptual models were constructed based on structural maps of the Upper Pannonian and Lower Pontian reservoirs, well data and structural maps and palaeotectonic profiles from the paper VELIĆ et al. (2011)
Structural modeling of the Kloštar Field • During Badenian to Late Pannonian new accommodation space opened • Sandstone reservoirs were deposited Evolution of the Kloštar Field during Late Pannonian
Structural modeling of the Kloštar Field • At the transition from Late Pannonian to Early Pontian normal fault appeared, which caused downlifting of the NE part • NE of the fault and SW of the Moslavačkagora Mt. new deeper area for sedimentation was created • It is very possible that two source of material were active: • (1) Eastern Alps and • (2) Moslavačkagora Mt. Evolution of the Kloštar Field during Early Pontian
Structural modeling of the Kloštar Field • During Late Pontiantranspression began, which is active still today • Main normal faults changed to reverse. • Smaller faults in the field are normal because of the local extension at the top of the Kloštar structure Evolution of the Kloštar Field during Late Pontian Evolution of the Kloštar Field during Pliocene and Quaternary
Deterministical geostatistical mapping of the reservoir variables Analyzed variables of the Beta reservoir Analyzed variables of the T reservoir
Indicator Kriging mapping of the Beta reservoir porosity – data transformation Indicator transformation of the porosity input data
Indicator Kriging mapping of the Beta reservoir porosity – variograms Experimental variograms (left) and their approximation with theoretical curves (right) of the Beta reservoir porosity for cutoffs: a-15%, b-16%, c-18% and d-19%
Indicator Kriging mapping of the Beta reservoir porosity Probability map for porosity less than cutoff 16% Probability map for porosity less than cutoff 15% Probability map for porosity less than cutoff 19% Probability map for porosity less than cutoff 18%
Indicator Kriging mapping of the Beta reservoir thickness – data transformation Indicator transformation of the thickness input data
Indicator Kriging mapping of the Beta reservoir thickness – variograms Experimental variograms (left) and their approximation with theoretical curves (right) of the Beta reservoir thickness for cutoffs: a-7m, b-9m, c-15m and d-21m
Indicator Kriging mapping of the Beta reservoir thickness Probability map for thickness less than cutoff 9m Probability map for thickness less than cutoff 7m Probability map for thickness less than cutoff 15m Probability map for thickness less than cutoff 21m
Indicator Kriging mapping of theT reservoir porosity – data transformation Indicator transformation of the porosity input data
Indicator Kriging mapping of the T reservoir porosity – variograms Experimental variograms (left) and their approximation with theoretical curves (right) of the T reservoir porosity for cutoffs: a-14%, b-18%, c-19% , 20% and d-22%
Indicator Kriging mapping of the T reservoir porosity Probability map for porosity less than 14% Probability map for porosity less than 18% Probability map for porosity less than 19% Probability map for porosity less than 20% Probability map for porosity less than 22%
Indicator Kriging mapping of the T reservoir thickness – data transformation Indicator transformation of the thickness input data
Indicator Kriging mapping of the T reservoir thickness – variograms Experimental variograms (left) and their approximation with theoretical curves (right) of the T reservoir thickness for cutoffs: a-5m, b-9m, c-13m, 17m and d-21m
Indicator Kriging mapping of the T reservoir thickness Probability map for thickness less than 9m Probability map for thickness less than 5m Probability map for thickness less than 17m Probability map for thickness less than 13m
Discussion and conclusion • 1st assumption - higher porosity represents sandy lithofacies and lower marly lithofacies. • In this way it was possible to distinguish sandstones, marly sandstones, sandy marls and pure marls. • 2nd assumption - higher thicknesses should point to central part of depositional channel, where the coarsest material was deposited. • In Upper Pannonian reservoir Beta higher porosity locations matched higher thickness locations. • In Lower Pontian reservoir highest thicknesses were only partly matched higher porosities. • In the deepest parts of the depositional channel sandstones were deposited and toward the channel margins more and more marly component could be expected.
Main material transport direction in Upper Pannonian was NW-SE. • Lateral thickness changes points to transition into marls and sandy marls. • The coarsest material was deposited in local synclines and today they can be recognized with the highest thicknesses of the sandy layers. • Thin marls and clayey marls were deposited in the N and NE direction, i.e. in the direction of the Moslavačkagora Mt. Material transport direction during Late Pannonian interpreted on the probability map for the porosity higher than 18% (left) and thickness higher than 15 m (right)
The coarsest material in this part of the Sava Depression mostly came from north. • Part of material was transported parallel with the fault toward SE. • Locations of the highest probabilities for the highest thicknesses does not match location of the highest probabilities for the highest porosity. • The highest thicknesses match sandstone and marl intercalations, so it could not represent depositional channel. • Probability map for porosity more accurate shows depositional channel than the probability map for thickness. Material transport direction during Early Pontian interpreted on the probability map for the porosity higher than 19% (left) thickness higher than 13 m (right)