Indicator Kriging

Case 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.

Introduction – determinism and stochastics

Indicator Kriging theory

Indicator Kriging

cutoff

cutoff

vxz

vxzxI

)( if 0

)( if 1)(

Where: I(x) - indicator variable;z(x) - measured value;cutoff - cutoff value.

Location map of 38 data: 1 represents sandstone,

0 represents other lithofaciesRecommended no. of cutoffs: 5-11 Results: probabilities

What are the principles of indicator formalism in Indicator Kriging?

Indicator formalism:

Indicator transformation can be interpreted as follows:

If v is continuous variable

In this case we should create cumulative probability distribution of v from the data values:

Since we generaly have finite number of data, the cumulative probability distribution function may change with the increasing or decreasing number of available data.

That is why the cumulative probability distribution function is called conditional probability distribution function (ccdf)

It is conditioned by number of available data

Next step: Introduce the indicator formalism for this ccdf in a way to subdivide the total range using k cut-off values

According to ccdf we can define the corresponding probabilities for all these cut-offs:

We can choose a particular cut-off, say 2m All the locations can be categorized in two groups:

The first one is the set of locations where the actual thickness is smaller than 2 m

The next group is locations where the actual thickness is larger than 2 m

Using this cut-off we can define an indicator variable, which takes 1 for all locations where the thickness is smaller than 2, and takes 0 for all other locations

In this way we can define all other indicator variables Actually, the larger number of cut-offs, the more precise the

continous ccdf derived are and this is the principle of Indicator Kriging

With respect of 5 indicator cut-offs (2, 4, 6, 8, 10 m), we can create 5 point maps showing the actual values (0 or 1).

That means we have 5 point maps – one for each cut-off Each map contains only 0 and 1 values Unfortunately, we cannot perform any meaningful estimation with

these values

But, they can hold some other meaning: We suppose that at any particular well location the probability

of the thickness smaller than a particular cut-off can be derived from the global probability distribution of thickness

We can conclude that after making indicator transformation, the probabilities of their value equals 1 can be estimated

This estimation can be performed for each individual cut-off separately

As a result we got grids showing the probabilities that the indicator variable take 1 value

Output of Indicator Kriging

In each row the probabilities increase by increasing of the cut-off values

All of these probabilities belong to a particular grid point Using Indicator Kriging the ccdf at a grid point can be estimated The final result we can get is ccdf for each grid point

If v is a categorical variable

Rock type

The Indicator Kriging of that variable gives the probability that this rock type appears at a particular location

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.

Conclusion

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.

Case study; Kloštar Field

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

Stochastic

Used methods

Deterministic

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

Našice

Požega

Virovitica

Kutina

Križevci

Varaždin

ZAGREB

Vukovar

Vinkovci

Slavonski Brod

OsijekKarlovac

Sava river

Sava river

Drava river

Mura river

Dun av ri ver

S a v a d e p r e s s i o n

0 100 200 km

C R O A T I A

S L O V E N I A

H U N G A R Y

S E R B I A

D r a v a d e p r e s s i o n

M u r adepression

Slavonsko-srijemskadepression

KLOŠTAR FIELD

Bj elovarsubdep ression

Il ovasubd epressio n

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

Core data – cores from INA laboratory

Klo – 57 (788.9 – 793.3 m, III m) Rocks top section of T+U+V reservoirDetermination: Lithoarenite (VELIĆ & MALVIĆ, 2008)

Klo – 82 (1404.6 – 1411.7 m, II m) Beta ReservoirDetermination: 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 down lifting of the NE part

NE of the fault and SW of the Moslavačka gora Mt. new deeper area for sedimentation was created

It is very possible that two source of material were active: (1) Eastern Alps and (2) Moslavačka gora Mt.

Evolution of the Kloštar Field during Early Pontian

Structural modeling of the Kloštar Field During Late Pontian transpression 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

WellPorosity

(%)Depth

(m)Thickness

(m)

Klo-5 18,0 1365,0 3,0Klo-19   1502,5 15,5Klo-60 17,9 1447,0 23,0Klo-62 15,3 1400,0 22,5Klo-63 16,6 1437,0 9,0Klo-64 15,0 1397,0 10,0Klo-70 12,2 1387,5 3,5Klo-73 13,3 1373,0 4,0Klo-74   1358,0 20,5Klo-75 17,5 1375,0 20,0Klo-76 18,5 1362,5 14,5Klo-77   1386,0 22,0Klo-78 16,2 1376,5 13,5Klo-79 18,5 1393,0 14,0Klo-81 19,1 1362,0 11,5Klo-82 18,3 1396,5 18,5Klo-83 16,0 1368,5 8,5Klo-84   1406,0 9,0Klo-86   1338,0 8,5Klo-87 17,3 1409,0 10,0Klo-88 15,5 1405,0 8,0Klo-89 17,9 1395,0 7,0Klo-163   1394,0 18,0

WellPorosity

(%)Depth

(m)Thickness

(m)

Klo-1 19,9 940,0 13,0Klo-12 19,5 991,0 12,0Klo-16 19,6 916,0 12,0Klo-20 21,1 1026,0 13,0Klo-22 23,3 966,0 11,5Klo-23 20,5 1014,0 12,0Klo-24 20,1 1020,5 11,0Klo-26 21,2 1016,0 9,5Klo-27 17,9 880,0 20,0Klo-28 19,2 994,0 17,0Klo-35 13,8 790,0 3,0Klo-43 5,5 765,5 4,5Klo-48 19,7 1019,0 13,5Klo-57 18,2 795,0 25,0Klo-58 21,8 803,0 6,0Klo-59 18,1 890,0 9,0Klo-71 18,5 838,0 10,0Klo-72 19,6 785,0 11,0Klo-95 22,0 957,0 8,0Klo-104 18,4 912,5 6,0

Analyzed variables of the T reservoir Analyzed variables of the Beta 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 15%

Probability map for porosity less than

cutoff 18%

Probability map for porosity less than cutoff 16%

Probability map for porosity less than cutoff 19%

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 7m

Probability map for thickness less

than cutoff 9m

Probability map for thickness less than cutoff 15m

Probability map for thickness less than cutoff 21m

Indicator Kriging mapping of the T 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 5m

Probability map for

thickness less than 9m

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 Moslavačka gora 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)

13 Upper Pannonian and Lower Pontian cores were examined. Upper Pannonian reservoirs have more mica. Local material source could not be directly interpreted based on 13 core

data. Local material source should be noticed on the probability maps as direction

NNE-SSW. NE part of the reservoir has high probability that porosity is higher than 18%.

It was concluded that it was possible that Moslavačka gora Mt. was a local source for one part of sandy and silty material

Material transport direction interpreted on the probability map for the porosity lower (left) and higher (right) than 18%

Discussion and conclusion Geostatistical methods were used for detail modeling of the two most

important and significantly different reservoirs of the Kloštar Field.

Every geological model is always stochastical because it contains uncertainty.

It is possible to perform additional geostatistical analysis by increasing number of input data and number of mapped reservoir variables.

Reliability of the model also depends on used software.

Mapped variables were porosity and thickness of the Beta and T reservoirs.

All previous solutions as well as E-logs were taken into the consideration.

Two mentioned reservoirs were chosen as the most widespread, the thickest and typical Upper Miocene reservoirs.

Discussion and conclusion

The Indicator Kriging method have been used in the probability mapping of the certain variable value.

Probability maps for certain cutoff value showed material transport direction and distribution channel location.

The Indicator Kriging maps proved heterogeneity of the reservoirs by existence of different lithofacies starting with sandstones in the central part of the channel to marly sandstones, sandy marls and marls. In this way it is easier to create precise boundary around the reservoirs and to get accurate estimation of the original hydrocarbon in place.

The methodology applied in the Kloštar Field can be used in all Upper Pannonian and Lower Pontian sandstone reservoirs in the Sava Depression, primarily because all depositional conditions, migrations and traps forming were almost the same.

Thank you for your attention!