GEOSTATISTICAL TOOLS IN RESERVOIR CHARACTERISATION

GEOSTATISTIČKI ALATI U KARAKTERIZACIJI LEŽIŠTA

Tomislav MALVIĆ1

1INA-Oil Industry Plc., E&P of Oil and Gas, Reservoir Engineering & Field Development

Department, Šubićeva 29, 10000 Zagreb, DSc., BSc. in Geology

SAŽETAK

Geostatistika je vrlo snažan i standardan alat kod izrade geoloških modela ležišta

ugljikovodika. Takvo značenje poprimila je 80-tih godina 20. stoljeća te ga ima još i danas.

Geostatistika uključuje matematičku teoriju razvijenu za opisivanje ponašanja

regionalizirane varijable. Danas se geostatistika upotrebljava u mnogih područjima naftne

geologije ali i drugim geoznanostima koje se bave opisivanjem prostornog ponašanja

regionaliziranih varijabli. Temelji se na variogramskoj analizi te uključuje različite

interpolacijske i simulacijske tehnike. Prednosti geostatističke interpolacije (različite tehnike

kriginga) uočljive su na gotovo svakom skupu s 15 ili više podataka. Skupina metoda

kokriginga dopušta uključivanje jedne ili više sekundarnih varijabli kojima se poboljšava

proces kartiranja primarne varijable. Najčešće primjer je upotreba seizmičkih atributa kao

dodatnog izvora informacija kod kartiranja poroznosti. Druga skupina geostatističkih

metoda obuhvaća stohastičke simulacije. To je vrlo koristan alat koji se upotrebljava za

kreiranje niza jednakovrijednih realizacija kartirane varijable. Pojedinačne realizacije

odabiru se na temelju nekoliko geoloških kriterija.

Ključne riječi: geostatistika, kriging, kokriging, stohastičke simulacije, poroznost

Predložena kategorija rada: stručni članak

ABSTRACT

Geostatistics is very powerful and standard tool in geological modelling of hydrocarbon

reservoirs, starting from eighties in 20th century and last to date. It includes mathematical

theory developed for describing of regional variable behaviour. Today geostatistics is used

widely in petroleum geology as well as other geosciences that handle by spatial distribution

of regional variable. It is based on variogram analysis and different interpolation and

simulation techniques. The advantages of geostatistics interpolations (different Kriging

techniques) are obviously for almost every datasets that include 15 or more point data. Set

of Cokriging methods allow to include one or more secondary variables to improve

mapping process of primary variable. The most often example is porosity mapping

supported by seismic attributes. Second group geostatistical methods encompass

stochastical simulations. This very useful tool allows creating a set of equiprobable

realizations of mapped variable. Particular realization can be selected based on several

geological criteria.

Key words: geostatistics, Kriging, Cokriging, stochastic simulations, porosity

Proposed paper category: Professional paper

1. INTRODUCTION

Exploration and production of oil and gas is large field of activity. Geological modelling and

reservoir characterisation play one of the major roles in petroleum business. New and improved

techniques come very fast. Geostatistics is one of the main tools in geological modelling,

recognised and global applied approx. from eighties in the 20th century to date. It can be

applied in all phases of geological modelling, i.e. in exploration as well as in development phase

of hydrocarbons (Figure 1).

Figure 1: Application of geostatistics in different stages of geological modelling

Number of input is key-value for applying mostly deterministic or stochastical geostatistical

methods. After1 reservoirs can be classified as:

Deterministic are reservoirs where inter-well area is excellent known, correlated and

internal reservoir architecture is solved. Such type is very rare and can be observed on

small or very mature fields, where is well’s density high. Geostatistics deterministic

methods (Kriging/Cokriging) could be applied.

Stochastic reservoirs includes mostly deterministic parameters, but also are

characterised with some degree of un-predictability, i.e. inter-well areas include some

stochastic. This type is the main target for geostatistics whether deterministic or

stochastic methods.

Mostly un-predictable reservoirs are very rare type, always connected with

exploration localities, few wells or potential plays. Estimation could be done only with

analogy or Monte Carlo method.

In the following chapters several (deterministic and stochastic) examples are selected from

Development Department practice in geostatistical modelling. Geostatistics analyses are

performed in several oil and gas reservoirs and lithofacies in the Drava depression from

Palaeozoic to Badenian age. Names of wells and fields are stayed hidden as company

business information, but all examples are transparent and clearly describe benefit of

geostatistics in geological modelling.

2. BASICS OF GEOSTATISTICS THEORY

Geostatistics assumes spatial data analysis. The most common spatial tool is variogram

defined by formula (Eq. 1).

(1)

Where are:

N(h) - number of data pairs at distance “h” (inside searching neighbourhood area)

z(un) - value at location un

z(un+h) - value at location un+h

Calculation of experimental variogram is necessary input for different geostatistical interpolation

or simulation techniques, like Kriging (Eq. 2), Cokriging (Eq. 3) and Sequential Gaussian

Simulations.

(2)

(3)

Where are:

zk / zc - points estimated by Kriging / Cokriging

i / j - weight coefficient for Kriging / Cokriging calculated from matrix equation

zi - hard data of primary variable (inside searching neighbourhood area)

sj - hard data of secondary variable (inside searching neighbourhood area)

The Kriging includes several interpolation techniques (see e.g. ref.2,3) like Simple, Ordinary,

Block and other Kriging interpolations. Stochastical simulations are geostatistical tool with

different purpose than interpolation techniques (see e.g. ref.4,5). Simulation algorithm transforms

input data in normal distribution with known mean and standard deviation. Well data are

respected as hard data (conditional simulation). Variogram models and Kriging map are used

as additional input data. Estimated points are sequentially used as new “hard-data”, and

procedure has been repeating until all grid points are not simulated, defining one realization.

The simulation aim is make statistically representative set of equally probable realizations,

describing reservoir heterogeneity.

3. SOME EXAMPLES OF VARIOGRAMS AND KRIGED MAPS

3.1. Experimental variograms in Drava depression

Experimental variogram modelling, with porosity as primary variable, is performed at several

reservoirs in Drava depression6,7. These models included 10-30 point values. Several porosity

variograms are shown on Figures 2 and 3. Each variogram is combined with corresponding

Ordinary Kriging map (using local mean as substitution for global mean), to reflect influence of

variogram ranges to porosity mapping. Quality of each map was validated using cross-

validation8. Variograms can be distinguished by reservoir age, lithology and number of data. All

figures represent geological (and variogram) structures with primary axis strike 30-210o.

The Figure 2 shows variogram and Kriging estimation in Lower Triassic heterogeneous

lithology (quartzites, filites, dolomites...), obtained at the Kalinovac field. Dataset includes 11

point data, and variogram range is based on “first sill crossing” for primary axis. Kriging map is

very smoothed, without “bull-eyes” effect. Cross-validation result was 1.13 (compared with 6.15

as maximum obtained for this lithofacies at other field).

The next example (Figure 3) presents variogram models and Kriging interpolation made in

mostly carbonate lithofacies (dolomites, carbonate breccias). Such lithofacies is characterised

with mostly stochastical porosity distribution, what means that any interpolation technique have

to include significant error. Also, such lithology is most problematic for geostatistics using, often

stress “bull-eyes” effect. But this example shows very successful variogram and Kriging

estimation, where isolines are elongated along structure (porosity distribution follows structural

style). Cross-validation result was 2.80 (12.10 was result at field with maximum interpolation

error in this lithofacies).

Figure 2: Experimental variogram from Lower Triassic lithofacies (ranges 2500x2000 m)

and corresponding Kriging porosity map.

Figure 3: Experimental variogram from Upper Triassic lithofacies (ranges 2000x900)

and corresponding Kriging porosity map

3.2. Cokriging improvement – example from Drava depression

Kriging maps are not unique solution, i.e. sometimes they need to be compared with other

types of solutions. Generally it could be done in case of:

a) Small-in-number input (e.g.<12 hard-data values) when Kriging would need to be

compared with Inverse Distance Weighting, Nearest Neighbourhood or other non-

geostatistical methods,

b) Seismic attribute(s) that could considered as relevant secondary variable, when Kriging

is compared with Cokriging solution.

Seismic attribute as secondary variable is often way how to improve geostatistical results. In

such case Cokriging techniques need to be tested (Eq. 3). Such connection need to be proved

as objective as it is possible, and it is usual done by statistical tests like:

Standard Pearson’s correlation coefficient,

Statistical t-test or

Tables with probability indicating on true or false correlation9,10.

Such testing was performed on the Beničanci field in the Drava depression6. Significant rank

correlation is found for pair porosity-reflection strength. Collocated Cokriging (CC) map shows

significantly more reservoir heterogeneity, due to seismic attribute, in inter-well area than

Ordinary Kriging and Inverse Distance Weighting maps (Figure 4). More important, this

heterogeneity better reflects true geological picture of breccia reservoir. Also, Cokriging was

confirmed as favourable mapping technique also by cross-validation

(CC<IDW<OK=2.19<2.78<2.97). Seismic could be very useful additional source of information,

especially when number of wells is small (14 in this example).

Figure 4: Comparing of IDW (left), OK (centre) and CC (right) porosity maps.

4. STOCHASTICAL SIMULATIONS

Stochastical simulations are very strong tool for different kind of estimation and can be

observed as improvements for deterministic approach. There is several kind of simulation –

sequential and indicator tools are very often types. Here is shown example (Figure 5) of

Sequential Gaussian Simulations (SGS) applied in purpose of:

1. Obtaining new better porosity histogram (simulated + hard data),

2. Simulation of reservoir porosity distribution in several possible realizations.

Mathematically and naturally possible porosity variability (based on inputs) can be observed in

two selected realization for Palaeozoic reservoir (Figure 5) at the Stari Gradac field7,11. Two

“extreme” realization (P1 and P99, where P1 means that 99 % of all realizations describe larger

total reservoir porosity) show huge difference in porosity distribution that could be mapped on

the same field (all realization are equally possible). Set of all 100 realizations is also excellent

base for calculating more precise histogram of analysed variable (Figure 5, right).

Figure 5: Variability of porosity expressed with P1 and P99 realizations

and histogram of simulated data

5. DISCUSSION AND CONCLUSIONS

Geostatistics offers many advantages in deterministic and stochastic approaches to geological

modelling. Kriging techniques are generally the best spatial estimation method, and stochastic

simulations are excellent tool for describing uncertainty in reservoir.

There is no doubt that applying of Kriging and Sequential Gaussian Simulations improved

geological models and especially porosity distribution at several hydrocarbon fields in Croatia.

Limit of successful application of Kriging is defined at approx. 10-15 input porosity values.

There is no need to use more demanding geostatistics for mapping smaller datasets. It means

that geostatistics could be used only on the fields with enough number of wells/data. In such

case, using of geostatistics will lead to better reservoir characterisation (porosity mapping,

OGIP/OOIP calculations) and finally to higher recovery of hydrocarbons.

6. REFERENCES

1. Jensen, J. L., L. W. Lake, P. W. M. Corbett and D.J. Goggin, 2000, Statistics for

Petroleum Engineers and Geoscientists: Elsevier Science B.V., 338 p., Amsterdam.

2. Hohn, M. E., 1988, Geostatistics and Petroleum Geology: Van Nostrand Reinhold, p.

264, New York.

3. Isaaks, E. and R. Srivastava, 1989, An Introduction to Applied Geostatistics: Oxford

University Press, p. 561, New York.

4. Dubrule, O., 1998, Geostatistics in Petroleum Geology: AAPG, Tulsa.

5. Kelkar, M. and G. Perez, 2002, Applied Geostatistics for Reservoir Characterization:

Society of Petroleum Engineers, 264 p., Richardson.

6. Malvić, T. and M. Đureković, 2003, Application of the Methods: Inverse Distance

Weighting, Ordinary Kriging and Collocated Cokriging in the Porosity Evaluation and

Results Comparison in the Beničanci and Stari Gradac Field: Nafta, 54, 9, 331-340,

Zagreb.

7. Malvić, T. and S. Smoljanović, 2004, Geostatistical Estimation and Simulation

Approaches for More Detailed OGIP Calculations (Stari Gradac - Barcs Nyugat Field):

IOR Methods for Economical Oil Recovery from Small Size and/or Marginal Oil Fields /

Steiner, I. (ur.).Zagreb : Petroleum Summer School, RGNF, 119-128.

8. Davis, B., 1987, Uses and Abuses of Cross Validation in Geostatistics: Mathematical

Geology, v. 19 (3), p. 241-248.

9. Kalkomey, C. T., 1997, Potential risks when using seismic attributes as predictors of

reservoir properties: The Leading Edge, March, p. 247-251.

10. Chambers, R. L. and J. M. Yarus, 2002, Quantitative Use of Seismic Attributes for

Reservoir Characterization: RECORDER, Canadian SEG, v. 27, p. 14-25, June.

11. Smoljanović, S. and T. Malvić, 2005, Improvements in reservoir characterisation

applying geostatistical modelling (estimation & stochastic simulations vs. standard

interpolation methods): Nafta, 56, 57-63, Zagreb.

ACKNOWLEDGEMENT

The presented researching is partially done through activity of Reservoir Engineering &

Field Development Department. This is part of INA-Oil Industry Plc.

AUTHOR

Tomislav Malvić, graduate engineer in geology, PhD in Natural Sciences, INA- Industry of

Oil Plc., Exploration and Production of Oil and Gas, Reservoir Engineering & Field

Development Department, expert, Šubićeva 29, 10000 Zagreb, tomislav.malvic@ina.hr

APPENDIX

Figure list:

Figure 2: Application of geostatistics in different stages of geological modelling

Figure 2: Experimental variogram from Lower Triassic lithofacies (ranges 2500x2000 m)

and corresponding Kriging porosity map.

Figure 3: Experimental variogram from Upper Triassic lithofacies (ranges 2000x900) and

corresponding Kriging porosity map

Figure 4: Comparing of IDW (left), OK (centre) and CC (right) porosity maps

Figure 5: Variability of porosity expressed with P1 and P99 realizations and histogram of

simulated data