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DOI 10.1007/s11004-011-9326-9

Validation Techniques for Geological Patterns

Simulations Based on Variogram and Multiple-PointStatistics

S. De Iaco S. Maggio

Received: 8 January 2009 / Accepted: 24 September 2010 / Published online: 31 March 2011

International Association for Mathematical Geosciences 2011

Abstract Traditional simulation methods that are based on some form of kriging are

not sensitive to the presence of strings of connectivity of low or high values. They are

particularly inappropriate in many earth sciences applications, where the geological

structures to be simulated are curvilinear. In such cases, techniques allowing the re-

production of multiple-point statistics are required. The aim of this paper is to point

out the advantages of integrating such multiple-statistics in a model in order to al-

low shape reproduction, as well as heterogeneity structures, of complex geologicalpatterns to emerge. A comparison between a traditional variogram-based simulation

algorithm, such as the sequential indicator simulation, and a multiple-point statistics

algorithm (e.g., the single normal equation simulation) is presented. In particular, it is

shown that the spatial distribution of limestone with meandering channels in Lecce,

Italy is better reproduced by using the latter algorithm. The strengths of this study

are, first, the use of a training image that is not a fluvial system and, more impor-

tantly, the quantitative comparison between the two algorithms. The paper focuses

on different metrics that facilitate the comparison of the methods used for limestone

spatial distribution simulation: both objective measures of similarity of facies real-

izations and high-order spatial cumulants based on different third- and fourth-order

spatial templates are considered.

Keywords Validation methods Variogram Multiple-point statistics Training

image Curvilinear structures Snesim High order cumulants

S. De Iaco () S. MaggioDipartimento di Scienze Economiche e Matematico-Statistiche, Universit del Salento,

Complesso Ecotekne, Via per Monteroni, Lecce, Italy

e-mail:sandra.deiaco@unisalento.it

S. Maggio

e-mail:sabrina.maggio@unisalento.it

mailto:sandra.deiaco@unisalento.itmailto:sabrina.maggio@unisalento.itmailto:sabrina.maggio@unisalento.itmailto:sandra.deiaco@unisalento.it

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1 Introduction

In most cases of spatial soil analysis, curvilinear geological structures need modeling.

However, traditional simulation methods based on some form of kriging are unable to

reproduce the presence of this kind of geological heterogeneities (non-elliptical pat-terns). Although second-order statistics can adequately provide a complete statistical

description of Gaussian processes, they are inappropriate for modeling geological

phenomena that are typically far away from Gaussianity and are characterized by

complex non-linear spatial patterns (Guardiano and Srivastava1993; Journel1997;

Tjelmeland1998; and others). The limitations of variogram-based techniques for rep-

resenting realistic geological continuity can be overcome by using multiple-point

statistics techniques (Strbelle and Journel2000; Strbelle2002; Krishnan and Jour-

nel 2003). Multiple-point statistics involve three or more points at a time and are

much more informative than the traditional variogram, which is a two-point statistic;

hence, when integrated in a model, multiple-statistics allow shape reproduction, as

well as long-range connectivity patterns, in the subsurface. However, data at sparse

sample localizations are often not sufficient to infer such multiple statistics. For this

reason, multiple-point geostatistics relies on the concept of training images (concep-

tual geological models), from which higher order statistics can be borrowed.

A fair number of techniques dealing with spatial complexity have been proposed

recently, such as the Markov random field based approaches (Daly 2004; Tjelme-

land and Eidsvik2004), a multiple-point method based on filters (Zhang et al. 2006;

Wu et al.2008), the sequential simulation method simpat based on the calculation of

similarity between patterns (Arpat and Caers 2007) and other related developments(Boucher2009; Chugunova and Ly2008; Mirowski et al. 2009; Remy et al. 2009;

Scheidt and Caers2009; Gloaguen and Dimitrakopoulos2009; Honarkhah and Caers

2010; and others). Moreover, Dimitrakopoulos et al. (2010) used spatial cumulants

(combinations of moment statistical parameters) to characterize non-Gaussian ran-

dom variables, focusing on the understanding of the interrelation of cumulant char-

acteristics and in situ behavior of geological entities or processes. Mustapha and Dim-

itrakopoulos (2010b) introduced a new simulation framework for complex non-linear

geological and other spatial patterns that is based on the use of spatial cumulants in

the high-dimensional space of Legendre polynomials. In all of these recent contri-butions, the two-point variogram is replaced with a training image in order to repro-

duce complex geological patterns. Although these are all considerable advances, it

is very important to have a well-defined spatial stochastic modeling framework as it

contributes to the geostatistical capacity to describe non-Gaussian and/or non-linear

geological phenomena.

In this paper, after a brief review of multiple-point geostatistical concepts includ-

ing the idea of training image and the general definition of multiple-point statis-

tics, the single normal equation methodology is introduced in order to point out

how multiple-point statistics can be incorporated in a model for simulation pur-

poses. Next, the empirical framework concerning limestone spatial distribution is

presented and simulated realizations obtained by using a traditional variogram-based

algorithm (sequential indicator simulation, sisim) (Journel and Huijbregts 1978;

Deutsch and Journel 1998) and a multiple-point statistics algorithm (single nor-

mal equation simulation, snesim) (Strbelle2000; Remy 2001) are compared. The

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idea is to compare the conditional simulations produced by using two indicator-

based algorithms when the patterns characterizing the phenomenon under study

are complex. Objective measures of the similarity of facies realizations are con-

sidered in order to determine if the sets of realizations generated by sisim and

snesim algorithms have the properties required, in terms of geological shape, ho-mogeneity and connectivity. A facies realization analyzer program (faprop), pro-

vided by Soleng et al. (2006), is used first. The following are then computed: the

global volume fractions of each facies, the number of bodies, the surface areas, and

the volumes and extensions in x and y directions for the bodies. Secondly, an al-

gorithm for calculating high-order spatial cumulants (hosc), recently proposed by

Mustapha and Dimitrakopoulos (2010a), is applied; cumulants based on different

third- and fourth-order spatial templates are computed. Hence, a statistical analy-

sis of realizations, as well as a quantitative comparison between the realizations

properties and the training image, is presented. At the end, we will highlight howlimestone spatial distribution with meandering channels are better reproduced by us-

ing a multiple-point statistics algorithm. This further confirms that variogram-based

methods are not sensitive to the presence of curvilinear patterns (Soleng et al. 2006;

Dimitrakopoulos et al.2010).

2 Multiple-Point Geostatistics

Multiple-point statistics are often used in geostatistical simulation because of theirability to reproduce complex geological patterns (e.g., undulating channels). Some

interesting applications can be found in the field of petroleum engineering (Strbelle

and Journel2000; Strbelle2002). A key justification for the use of multiple-point

statistics is that they contain a wealth of information compared to second-order mea-

sures (Pan and Szapudi 2005). These statistics can be borrowed from a training image,

which contains information about geological structures prevailing in the subsurface,

as well as about the spatial distribution of the phenomenon under study in general.

Note that geologists usually have a prior conceptual idea of the heterogeneity pat-

terns present in the spatial domain. As a result, training images can sometimes be

hand-drawn by an expert; that is, they need not be conditioned to any local data. The

training images are only bound by the principles of stationarity (the patterns do not

change over the training image) and repetitiveness. Repetitiveness dictates the mini-

mum size of the training. Accordingly, the reproduction of large scale patterns such

as channels would require a large training image, since multiple-point statisticslike

any statisticstend to fluctuate, and that fluctuation becomes larger as the distances

over which these statistics are calculated become larger (Caers and Zhang2002).

2.1 Multiple-Point Statistics

Unlike the variogram, which is a two-point statistic based on a two-point configura-

tion (Fig.1(b)), the multiple-point statistics involve three or more points at a time (Kr-

ishnan and Journel2003) and they are based on three-point configuration (Fig. 1(c))

or configuration of more than three points (Fig.1(d) and (e)).

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Fig. 1 Geometric

configurations of variogram and

multiple-point statistics

2.1.1 Formalism of Multiple-Point Statistics

LetZ be a random field, defined on the domain D Rd (d 3), such that any ran-

dom variable Z(u) at location u has K mutually exclusive possible states {zk, k=

1,2, . . . ,K}. In general, under the usual stationarity hypothesis, a multiple-point

statistic is a measure of the joint variability at more than two locations at a time.

It has been defined by Strbelle and Journel (2000) as the probability that n vari-

ablesZ(u+h1),Z(u+h2), . . . ,Z(u+hn)are jointly in the stateszk1, zk2, . . . , zkn ,

respectively; hence, it is obtained by the expected value of the product of the n cor-

responding indicator variables

(h1,h2, . . . ,hn;k1, k2, . . . , kn)=E n=1

I (u+ h;k), (1)

where I (u+ h;k) is an indicator variable at location u+h , such that I (u+

h;k )= 1, ifZ(u+ h)= zk andI (u+ h;k )= 0 otherwise.

2.2 Single Normal Equation Methodology

Multiple-point statistics simulation is one of the most recent spatial modeling tech-

niques to have been put into use, especially for the reproduction of curvilinear

and long-range connectivity patterns. The single normal equation methodology is

frequently applied in this context (Journel and Alabert 1989; Srivastava 1992;

Guardiano and Srivastava1993; Strbelle and Journel2000; Strbelle2002), in or-

der to integrate such multiple-point statistics in a soil characterization model. A short

presentation of this methodology is given here and a multiple statistics inference from

a training image is also discussed. Let n= {h,= 1, . . . , n} be a geometric tem-

plate and dn= {Z(u+ h)= zk ,= 1, . . . , n} be an associated data event of size

n centered at an unsampled location u to be modeled. The goal is to evaluate the

conditional probability distribution of variable Z(u), given the nearestn sample val-

ues Z(u)= zk , =1, . . . , n. Let A0 be an indicator random variable, such that

A0= 1, ifZ(u)= zk andA0= 0, otherwise, andD be the indicator random variable

associated with the case that dn occurs. It is easy to show that D can be written as

the product of the binary random variablesA associated with each conditioning data

valuezk , = 1, . . . , n.

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Hence, the generalized indicator kriging formalism can be used and the required

conditional probability can be given by the following simple indicator kriging expres-

sion

P{A0= 1 |D = 1} =E{A0} +

1 E{D}, (2)

where E{A0} is the prior probability that Z(u) at the unsampled location u is in

the state zk (i.e., prior to the knowledge of the observed data event D =1), and

E{D}is the probability that the conditioning data event occurs. The single resulting

extended normal equation (kriging system) provides the single weight . One need

not be surprised if the single weightobtained from the kriging system leads to the

notion of conditional probability, provided by Bayes and given as follows

P{A0= 1 |D = 1} =

E{A0 D}

E{D} =

P{A0= 1,D = 1}

P{D = 1} . (3)

Further specifications can be found in Strbelle (2000).

Hence, as an alternative to a form of kriging based on a variogram model, the con-

ditional probability can be estimated by using the corresponding proportions obtained

from the training image. Invoking a form of stationarity, the conditional probability

in (3) can be inferred using the training ratio between the number of replicates of the

data event dn associated with the occurrence of the state zk at the central location u

and the number of all replicates of the data event dn.

Unlike in the variogram-based algorithm, the conditional probability is not ex-

pressed as a linear combination of the n single data indicators. As a result, it is not

necessary to solve a kriging system in order to obtain the conditioning data weights,

and even less to estimate and model the two-point covariance or variogram: all one

needs is to find a suitable training image. Of course, finding an appropriate training

image might be hard in a three-dimensional space.

2.2.1 Single Normal Equation Simulation Algorithm

Thesnesimalgorithm is an efficient pixel-based sequential simulation algorithm that

obtains multiple-point statistics from the training images, exports them to the geo-

statistical model and anchors them to the actual subsurface data. For each location u

along a random path, the set of local data values and their spatial configuration

called data eventis recorded. The training image is scanned for replicates that

match this event. The central node values corresponding to the replicates are used

to calculate the conditional probability of the central value given to the data event.

The main steps of the snesim algorithm are described in detail in Strbelle (2000.

2002).

The snesim algorithm has been implemented in sgems software (Remy et al. 2009),

which is built by using the Geostatistical Template Library (Remy 2001). This al-

gorithm is essentially the first practical algorithm proposed for the application of

multiple-point geostatistics.

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3 Empirical Framework and Simulations

In many earth sciences applications, a stochastic model is needed to capture the pat-

terns of geological heterogeneities in the subsurface in question. This may be difficult

or impossible within a particular modeling framework. The aim of the following sec-tion is to illustrate how a model that includes multiple-point statistics can improve

traditional variogram-based inference, especially in terms of shape reproduction. For

this reason, a thorough quantitative validation of the results of the two-point and

multiple-point simulation methods is provided and their performance is assessed.

Different metrics that facilitate the comparison of results, in terms of general fea-

tures of the bodies and the degree of homogeneity and connectivity, are calculated.

A facies realization analyzer program (faprop), generated by Soleng et al. (2006),

is applied to the training image as well as to the realizations produced by using the

sisimand snesimalgorithms. Hence, the comparison criteria are based on observablephysical measures such as the number of bodies, the direction of maximum variabil-

ity, the global volume fraction (as a measure of abundance), the maximal directional

extension of bodies (as a measure of connectivity), the ratio of surface area to volume

(as a measure of ruggedness and shape). Moreover, cumulative distribution functions

of the volume of bodies in the realizations are further analyzed. Secondly, an algo-

rithm for calculating high-order spatial cumulants, recently proposed by Mustapha

and Dimitrakopoulos (2010a), is used. The comparison criteria then have involved

high-order spatial cumulants based on specific spatial templates up to order four. As

it is explained in Sect.4.2, they are referred to cumulants of orders three and four,

which are enabled to describe the complex geological configuration of the character-

istic under study.

In particular, the empirical framework of interest concerns the spatial distribution

of limestone and dolomitic limestone in Lecce, Italy. Conditional simulations of the

occurrence of limestone and dolomitic limestone (LDL) in Lecce are performed by

using both sisim and snesim. Multiple-point statistics are scanned directly from an

available geological map, used as a training image. Hence, the following steps are

considered:

1. Description of available information about the phenomenon to examine (data setand training image).

2. Conditional simulations made by using the traditional geostatistical tools.

3. Conditional simulations made by using the multiple-point geostatistics tools.

3.1 Description of Available Information

In this analysis, the geological structure of LDL spatial distribution in Lecce, Italy

has been examined. Let

I (u,LDL)=

1 in case of LDL occurrence,

0 otherwise.

In order to generate conditional simulations of the spatial random field {I (u,LDL),

u D R2}, a sample data set of 140 indicator values i(ui,LDL),i = 1,2, . . . ,140,

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Fig. 2 Posting map of the

sample indicator values

has been selected, by using a patterned sampling, from an exhaustive limestone map

of 24,642 points (Fig.2).

Moreover, an available geolithologic map of Lecce district with eight possible

states (Fig.3(a)) has been used as a training image (Fig.3(b)). In particular, the entire

body of geolithologic information has been coded in bits 0/1 (1 stands for limestone

and dolomitic limestone occurrence and 0 stands for other types of geolithologic

components occurrence) and a training image with a size of 135 185 pixels has

been obtained.

Note that the overall proportion of limestone and dolomitic limestone in the train-

ing image is 24%.

It is evident that the spatial distribution of limestone presents specific curvilinear

patterns of heterogeneity. Accordingly, the modeling of these patterns should require

the use of multiple-point statistics in order to allow long range continuous channels

reproduction.

3.2 Conditional Simulations Made by Using Traditional Geostatistical Tools

In order to apply the sisim algorithm based on an appropriate variogram model, we

computed the sample directional indicator variograms of the binary variable, ob-

served at 140 points. Through a review of the directional sample indicator variograms,

a geometric anisotropy with direction of maximum continuity elongated at 135 was

detected.

An exponential model with the parameters (i) nugget effect= 0.00415, (ii) effec-

tive range = 14.7 km, and (iii) sill value = 0.1815 was fitted to the directional sample

variogram at 135.

Figure4shows the sample indicator variogram together with the fitted model.

Hence, thesisimalgorithm, based on the variogram model, has been applied and

six simulations over an 86 121 grid have been obtained, as shown in Fig. 5. In

order to improve spatial continuity, elliptical neighborhoods have been considered

with major axis along 135 (N45W).

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(a)

(b)

Fig. 3 aGeo-lithologic map and b training image

3.3 Conditional Simulations Made by Using Multiple-Point Statistics

In order to use the multiple-point statistics simulation algorithm, we began by fixing

a template of size 10, 10= {h,= 1,2, . . . ,10}, elongated at 135

(N45W), as il-

lustrated in Fig.6. A multiple grid approach (Gmez-Hernndez1991; Remy2001)

was adopted by simulating three nested and increasingly finer grids, and a servosys-

tem correction set to 0.5 was applied. Figure7 shows the rescaled data templates of

size 10; that is, the data template rescaled proportionally to the spacing of the nodes

within the grid to be simulated. This is done to capture the large scale structures of

the training images. It is well known that this approach improves the reproduction of

long-range continuous channels (Yuhong2006).

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Fig. 4 Sample indicator

variogram and its model

Fig. 5 Simulated maps obtained by usingsisimalgorithm

Multiple-point statistics involving ten points have been estimated by scanning thetraining image and a search tree (Strbelle 2000, 2002) has been created for each

nested simulation grid. Hence, the snesim algorithm, based on multiple-point statis-

tics, has been applied and six other simulations over the same 86 121 grid have

been obtained, as illustrated in Fig.8.

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Fig. 6 Template of size 10,10,

elongated at 135

Fig. 7 Rescaled data templates

of size 10

Fig. 8 Simulated maps obtained by usingsnesimalgorithm

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4 Validation and Comparison of Results

At the end, a quantitative analysis of the results was performed in order to check if

the two geostatistical simulation algorithms are able to generate realizations where

the essential characteristics of the training image are captured and represented.The validation and comparison are based on general features of bodies and high-

order cumulants. Note that systematic definitions of high-order spatial cumulants can

be found in Dimitrakopoulos et al. (2010). It is essential to underline that cumulants

are an extension of the well-known covariance function, which is capable of capturing

complex spatial configurations and connectivity patterns in a geological sense. They

allow for large possible combinations of random point variables and they completely

characterize non-linear and non-Gaussian stationary and ergodic random fields.

4.1 Comparison Based on General Features of Realization Bodies

Firstly, we present the comparison criteria of facies realizations based on physically

measurable quantities: the number of bodies,n; the global volume fraction,P (mea-

sure of abundance); the direction of maximum variability,D; the maximal extensions

in x and y directions, Max(X), Max(Y) (measure of connectivity); the mean vol-

ume, V; the mean area, A; and the mean value of the area to volume ratio, A/V

(measure of ruggedness and shape). The program faprop, proposed by Soleng et al.

(2006), was applied to the training image in Fig. 3(b) and the above-mentioned cri-

teria were evaluated. Then, the same computations were considered for realizationsfrom snesim and sisim algorithms. For each criterion, the realizations values have

been normalized with respect to the corresponding training image values, so that the

closer the comparison criteria are to unity, the more similar the simulated maps are

to the training image.

The snesim algorithm was more accurate than the sisim algorithm, in the sense

that the resulting realizations were more similar to the training image. The failure of

the variogram-based algorithm is clear from the box and whisker plots in Fig.9. The

sisim algorithm produced too many tiny objects; as a result, the number of bodies

was too large and the mean volume of the objects was too small. Moreover, the bod-ies were too rugged; this is reflected in the box and whisker plots of Fig.9, where the

mean ratio of area to volume was high. On the contrary, the multi-point method re-

produced the features of the training image quite well. For all criteria, the realizations

values were close to unity and their variability was very low. The same conclusion

can be drawn when looking at Fig.10, where the cumulative distribution functions

of the bodies volume of each of the two sets of realizations were compared to the

corresponding training image distribution.

4.2 Comparison Based on High-Order Cumulants

In this section, objective measures of similarity of facies realizations such as high-

order cumulants (Dimitrakopoulos et al.2010) are considered in order to compare the

spatial architecture of the training image with respect to the two sets of realizations

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Fig. 9 Box and whisker plots of comparison criteria obtained from simulated maps

(generated bysisimand snesim). Taking into account that limestone spatial distribu-

tion is characterized by a geometric anisotropy with a direction of maximum conti-

nuity elongated at 135, anisotropic experimental high-order cumulants were based

on different spatial templates. They were referred to the following: (i) cumulants of

order two, at 135; (ii) cumulants of order three, whose templates directions are:

{315,45}, {135,135}, {315,135}, as shown in Fig.11, (a)(c); (iii) cumulants

of order four, whose templates directions are {135,135,135}, {315,135,45}, as

shown in Fig.11(d) and (e). The above experimental cumulants have been computed

usinghoscfor both the training image and the simulated realizations. For each crite-

rion, mean square errors (between experimental cumulants of simulated realizations

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Fig. 10 Cdf of the bodies volume of training image compared toacdfs of the bodies volume of realiza-

tions fromsisimand b cdfs of the bodies volume of realizations from snesim

Fig. 11 Third-order and

fourth-order cumulant templates

used for calculations

and cumulants of the training for each lag) have been calculated so that the closer the

comparison criteria are to zero, the more similar the spatial characteristics of simu-

lated maps are to the training image.

Results are shown in Figs. 12and13. Note that for the fourth-order cumulants

maps, three cross sections have been chosen and analyzed, in order to help to clarify

the results. Looking at the box and whisker plots (Figs. 12 and13), it is evident that

the geological patterns of limestone spatial distribution captured by the high-order

cumulants are better reproduced by the snesim realizations. This is because the snesim

realizations are characterized by approximately the same high-order cumulant maps

as the training. This means that high cumulant values which identify the interactions

of blocks associated with limestone and dolomitic limestone occurrences (gray blocks

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Fig. 12 Box and whisker plots of comparison criteria based on second-order and third-order cumulants

in Figs.3(b),5, and8) are often located at the same lags for both the training and the

snesimrealizations. It is clearly understandable why the comparison of experimental

cumulants is so important in validating simulations of complex patterns. Hence, the

box and whisker plots of Figs.12and13give a further quantitative confirmation that

the complex spatial configurations and patterns of the snesim realizations are closer

to the training ones than the geological properties of the sisimrealizations. The box

and whisker plots associated with the snesim realizations in Fig. 12 are all closer

to zero than the ones related to sisim realizations. They are also characterized by

less variability. Note that the variability of the errors is greater along the direction of

maximum continuity for bothsisimand snesimrealizations. A significant difference

between thesnesimbox and whisker plots, and thesisimbox and whisker plots can be

noted for the template direction {315, 135}. This is what we should expect when

the template (Fig.6) used in this paper forsnesimsimulation is taken into account.

Looking at the box and whisker plots of Fig.13, it is clear that the experimental

fourth-order cumulants, which are capable of capturing more complex spatial char-

acteristics and interactions among gray blocks (Figs.3(b),5, and8), have produced

for snesim lower mean square errors, for all directions and especially for non-zero

lags.

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Fig. 13 Box and whisker plots of comparison criteria based on fourth-order cumulants

4.3 Visual Inspection

Six images (three from the six sisimmaps and three from the six snesim maps) that

have best reproduced the proportion of limestone and dolomitic limestone and the

channel-like objects have been chosen for a graphic comparison (Fig. 14). Hence,

the subjective impression of similarity or dissimilarity obtained through visual in-

spection of the simulated maps has merely confirmed the results obtained from the

above quantitative analysis. It is evident that the limestone spatial distribution with

meandering channels, as well as the most important characteristics of the geolog-

ical structure, is better reproduced by using a multiple-point simulation algorithm

(snesim).

5 Conclusions

In this paper, after a brief review of multiple-point geostatistics, a comparison be-

tween a traditional variogram-based simulation algorithm and a multiple-point statis-

tics algorithm was presented by using a specific empirical framework. Information

about geological structures prevailing in the subsurface was crucial, since multiple-

point statistics have been inferred from the training image reflecting the expected

patterns of geological heterogeneities under study.

A quantitative analysis was proposed in order to point out how the two sets of real-

izations reproduce the complex geological structures of the characteristic of interest.

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Fig. 14 Selected maps obtained by usinga the sisimalgorithm andb thesnesimalgorithm

Thefapropprogram and thehoscalgorithm were used in order to provide a complete

comparison and validation between thesnesim-derived and thesisim-derived simula-

tions. It was highlighted how limestone spatial distribution with meandering channels

and other statistics are better reproduced by using a multiple-point statistics algorithm

(snesim). This has further confirmed that traditional methods (sisim) based on someform of kriging are not sensitive to the presence of non elliptical patterns (Soleng

et al.2006). Results shown in this paper confirm that the use of multiple-point geo-

statistics is a powerful tool to characterize subsurface heterogeneity for applications

in a wide variety of complex geological settings. This could be further studied in

detail for both binary and continuous images, for two- and three-dimensional fields,

and using other spatial simulation techniques for comparisons, such as simulation

by simulated annealing, boolean simulation of ellipses, truncated Gaussian simula-

tion, as well as the most recent simulation algorithms based on the use of a training

image.

Acknowledgements The authors would like to thank Prof. Posa who stimulated their interest in this

field, the reviewers for their useful comments and suggestions, Petter Abrahamsen, the Research Director

SAND of the Norwegian Computing Center, for providing the faprop (Facies Properties Analyzer) pro-

gram, and Hussein Mustapha of the Department of Mining and Materials Engineering of McGill University

(Canada), for providing thehosc(High-order spatial cumulants) program as well as for his precious hints.

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