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Using of Ordinary Kriging for indicator variable mapping (example of sandstone/marl border)

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Using of Ordinary Kriging for indicator variable mapping(example of sandstone/marl border)

Kristina NOVAK ZELENIKA1 and Tomislav MALVIĆ1,2

1INA-Oil industry, Oil & Gas Exploration and Production, Reservoir Engineering & Field Development, Šubićeva 29, 10000 Zagreb

2University of Zagreb, Faculty of Mining, Geology and Petroleum Engineering, Pierottijeva 6, 10000 Zagreb, Assistant Professor

e-mails:

[email protected] (Reservoir Geologist)

[email protected] (Adviser)

Visiting lecture for

IAMG student chapter in Szeged, Hungary

October 2009

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 of the structural depressions are also often boundaries between turbidity sandstone and basin marl.

Turbiditic sandstones had been deposited periodically, and in their top or base is always marl.

Facies boundaries are not sharp. There is wide transition zone in toward basinal plain as well as in the direction of turbidite current palaeotransport (Malvić, 2006).

Boundaries of transitional zones are not sharp and there are several transitional lithotypes.

Deterministic mapping of facies is based on categorical variables, expressed with numbers of discrete values.

Every facies should be represented with one discrete value (e.g. 1 for coarse grained sandstone, 2 for fine grained sandstone, 3 for silt and 4 for marl).

There is well known geostatistical method of Indicator Kriging which is used for categorical variable mapping when Indicator variable is showed through two categorical values (1 and 0). But, we used OK is some modified way for mapping facies line.

Analyzed are 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.

It was used two cutoff (threshold) for selection (using of Boolean operator AND)

1)Porosity of 12% = all lesser value are facies =0, otherwise=1

2)Permeable reservoir saturated by hydrocarbons is described as 1 and water saturated parts or pure marl with 0.

SHORT INDICATOR THEORY

Measured data are transformed in indicator values 0 and 1. It is the most frequently used in the geology 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.

Indicator variable in every measured location will be calculated using limiting value by the term:

Figure 1: Grouping of measured values

in 2 facies classes (theoretical example)

Where are:

I(x)- Indicator variable

Z(x)- originallymeasured value

cutoff- limiting value

Mapping of indicator variables - approach

- Distributuion of sandstone and marl in analyzed model was matched with our generally knowledge about Neogene geology in the Croatian structural depressions.
- It was assumed that results could be extended and applied on all oil/gas field in the Croatian part of Pannonian basin.
- We based our analysis on two facies distribution as it was schematically presented on Figure 1.
- Experimental variograms and theoretical model approximation have been made in program Variowin 2.21.
- Eventually, indicator variables in 2D space had been interpolated by the Ordinary Kriging technique.

Case 1: mapping with 25 input data

- Variogram surface map is defined as Figure 2.
- It helped to define principal and subordinate axis.
- These two axes are:
- principal with direction 135º - 315º and
- subordinate with direction 45º -225º.

Figure 2: Variogram surface map for two facies

Experimental variograms

(angular tolerance 45o)

Figure 3: Principal axis, direction 135º - 315º

Figure 4: Subordinate axis, direction 45º - 225º

Theoretical variograms

Figure 5: Principal axis,

Gaussian model,

range=8.02

Figure 6: Subordinate axis,

Gaussian model,

range=4.26

Interpolation of indicator values by Ordinary Kriging

Ordinary Kriging interpolation were performed in program Surfer8.0TM, using ranges from indicator variograms.

Facies maps had been drawn applying two theoretical variogram models: Gaussian and Spherical.

Gaussian model should match better experimental variogram points.

Spherical model led to better geological map (the facies line was smoother and closer to half-distance between 0 and 1).

Lithological legend on Figures 7 and 8 is coloured:

marl = light yellow, sandy marl = yellow, marly sand = green and sand = grey blue.

Line 0.5 represent border between facies 0 and 1 (black bold lines on Figures 7 and 8).

That line is very similar to the ‘hand-made’ facies line, which would be drown on the half distance between points 0 and 1 (red line on Figures 7 and 8).

Figure 7: Facies distribution created with Gaussian model

Figure 8: Facies distribution created with Spherical model

Case 2: mapping with 37 input data

Number of input data was increased on 37.

- Figure 9 indicates on two axes:
- principal in direction 135º - 315ºand
- subordinate in direction 45º -225º.

Figure 9: Variogram map of the facies distribution

Experimental variograms

(angular tolerance 45o)

Figure 10:Principal axis,

direction 135º - 315º

Figure 11:Subordinate axis,

direction 45º - 225º

Theoretical variograms

Figure 12:Principal axis (135º - 315º),

Gaussian model,

range=8.85

Figure 13:Subordinate axis (45º - 225º), Gaussian model,

range=3.50

Lithological legend on Figures 7 and 8 is coloured:

marl = light yellow, sandy marl = yellow, marly sand = green and sand = grey blue.

Line 0.5 represent border between facies 0 and 1 (black bold lines on Figures 14 and 15).

That line is very similar to the ‘hand-made’ facies line, which would be drown on the half distance between points 0 and 1 (red line on Figures 14 and 15).

Figure 14: Facies distribution created with Gaussian model

Figure 15: Facies distribution created with Spherical model

CONCLUSIONS

- Mapping of the facies 0 and 1 (marl and sandstone) was made in Surfer8.0TM program, using the geostatistical technique of Ordinary Kriging, interpolated isolines 0, 0.5 and 1.
- Number of input data in the first case was 25 and this number was increased in the second case on 37.
- Line 0.5 is representing border of facies 0 and 1, i.e. line of lateral facies changing in Upper Miocene sediments (it also can be considered as line between sandstone and marl).
- The 0.5 line is very similar in the both cases as the borders ‘interpolated by hand’, i.e. with line on half distance between data 0 and 1.
- Interpolated 0.5 line can be applied for any modelling purpose as (depending on selected cutoff variable):
- border between oil-saturated and water-saturated sandstone facies or
- border between sandstone and marl in the same depositional environment.

- BOOKS:
- Deutsch, C.V. & Journel, A.G. (1997) GSLIB: Geostatistical Software Library and User's Guide, 2nd edn. Oxford University Press, New York, 369 pp.
- De Smith, M.J., Goodchild, M.F. & Longley, P.A. (2007) Geospatial Analysis, 2nd edn. Troubador Publishing, 516 pp.
- Pannatier, Y. (1996) Software for Spatial Data Analysis in 2D. Springer-Verlag, New York, 91 p.
- PAPERS:
- Journel, A.G. (1983) Non-parametric estimation of spatial distributions. Mathematical Geology, 15, 3, 445-468.
- Malvić, T. (2006) Middle Miocene Depositional Model in the Drava Depression Described by Geostatistical Porosity and Thickness Maps (Case study: Stari Gradac-Barcs Nyugat Field). Rudarsko-geološko-naftni zbornik, 18, 63-70.
- Richmond, A. (2002) An alternative implementation of indicator kriging. Computers & Geosciences. 28, 4, 555-565.
- Solow, R. (1986) Mapping by simple indicator kriging. Mathematical Geology, 18, 3, 335-352.
- Vrbanac, B. (2002b) Facies and Facies Architecture of the Ivanić Grad Formation (Upper Pannonian) – Sava Depression, NW Croatia.Geologia Croatica, 55, 1, 57-78.
- Vrbanac, B.; Velić, J. and Malvić, T. (2008) Sedimentation of Late Pannonian clastic deposits in main and marginal basins (Sava depression vs. Bjelovar subdepression).Geophysical Research Abstracts, Vol. 10, sessions committee (ed.). Vienna, Copernicus Group, 2 p.