GEOSTATISTICAL TOOLS IN RESERVOIR CHARACTERISATION
GEOSTATISTIČKI ALATI U KARAKTERIZACIJI LEŽIŠTA
Author:
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
Key words: geostatistika, kriging, kokriging, stohastičke simulacije, poroznost
Ključne riječi: geostatistics, Kriging, Cokriging, stochastic simulations, porosity
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.
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.
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. After Jensen et al. (2000) 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.
BASICS OF GEOSTATISTICS THEORY
Geostatistics assumes spatial data analysis. The most common spatial tool is variogram
defined by formula (Eq. 1).
N (h)
1
2
2 (h) 
  z (un )  z (un  h)
N (h) n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.
n
z k   i  z i
(2)
i 1
n
m
i 1
j 1
zC   i  zi    j  s j
(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. Hohn, 1988; Isaaks &
Srivastava, 1989) like Simple, Ordinary, Block and other Kriging interpolations. Stochastical
simulations are geostatistical tool with different purpose than interpolation techniques (see e.g.
Dubrule, 1998; Kelkar and Perez, 2002). 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.
SOME EXAMPLES OF VARIOGRAMS AND KRIGED MAPS
Experimental variograms in Drava depression
Experimental variogram modelling, with porosity as primary variable, is performed at several
reservoirs in Drava depression (Malvić and Đureković, 2003; Malvić and Đureković, 2004).
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-validation (Davi, 1987). 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
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 nongeostatistical 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 correlation (Kalkomey, 1997;
Chambers and Yarus, 2002).
Such testing was performed on the Beničanci field in the Drava depression (Malvić and
Đureković, 2003). 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 crossvalidation (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.
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 field (Malvić
and Smoljanović, 2004; Smoljanović and Malvić, 2005). 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
DISCUSSION
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.
REFERENCES
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.
Davi, B., 1987, Uses and Abuses of Cross Validation in Geostatistics: Mathematical Geology, v.
19 (3), p. 241-248.
Dubrule, O., 1998, Geostatistics in Petroleum Geology: AAPG, Tulsa.
Hohn, M. E., 1988, Geostatistics and Petroleum Geology: Van Nostrand Reinhold, p. 264, New
York.
Isaaks, E. and R. Srivastava, 1989, An Introduction to Applied Geostatistics: Oxford University
Press, p. 561, New York.
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.
Kalkomey, C. T., 1997, Potential risks when using seismic attributes as predictors of reservoir
properties: The Leading Edge, March, p. 247-251.
Kelkar, M. and G. Perez, 2002, Applied Geostatistics for Reservoir Characterization: Society of
Petroleum Engineers, 264 p., Richardson.
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.
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.
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.
ACKNOWLEDGMENT
The presented researching is done through activity of Reservoir Engineering & Field
Development Department (INA-Exploration and Production of Oil and Gas). Thanks all
colleagues who participate in some parts of researching and application of geostatistics.