geostatistical applications in petroleum reservoir modelling

IntroductionAlthough geostatistics was initially developedfor mining resource evaluation (Krige, 1951;Matheron, 1963; Journel and Huijbregts,1978), it has been used quite extensively inthe petroleum industry in the past two to threedecades. Geostatistical analysis and geologicalmodelling have been commonly employed forhydrocarbon characterization. The success ofgeostatistics in geoscience applications restsupon the spatial descriptions of geologicalphenomena, from characterizing rockcontinuities using variograms (Jones and Ma,2001) to three-dimensional (3D) represen-tations of natural phenomena (Deutsch andJournel 1992; Ma et al., 2009).

Geostatistical methods for buildinggeological models all make use of thevariogram. The variogram describes the degreeof discontinuity of a variable as a function oflag distance and direction. For a categoricalvariable such as facies, the variogramdescribes the size of geological objects (Jonesand Ma, 2001). In hydrocarbon resourceevaluation, facies are an intermediate-scalerock property as they govern petrophysicalproperties and fluid flow. This paper discussesthe main characteristics of a lithofacies

indicator variogram, and presents methods formodelling lithofacies, including sequentialindicator simulation that integrates propensityanalysis of descriptive geology, hierarchicalfacies, or lithofacies modelling workflows.

Reservoir models of continuous propertiesin hydrocarbon resource evaluation generallyinclude porosity, fluid saturations, andpermeability. This paper briefly discussesporosity modelling by using kriging andsequential Gaussian simulation, andpermeability modelling by using collocated co-simulation. Based on the scale of subsurfaceheterogeneities, hierarchical modellingframeworks are presented not only for a two-step facies-lithofacies modelling workflow, butalso for dealing with the more generalmultilevel modelling methodology, from thecategorical variable of facies to continuousvariables of porosity and permeability.

Facies and lithofacies modellingFacies and lithofacies are discrete variables inwhich the rock property describes categories ofthe rock quality (Ma, 2011). An indicatorvariable represents a binary state with twopossible outcomes: presence or absence. Forthree or more lithofacies, the indicator variablemay be defined in terms of one lithofacies andall the others combined to indicate the absenceof that selected lithofacies.

Indicator variogramMost indicator variograms show the second-order stationarity with a definable plateau(Jones and Ma, 2001). The lithofaciesvariogram observed across stratigraphicformations is commonly strongly cyclical withlag distance. This has been termed a hole-effect variogram in geostatistics (Ma andJones, 2001). Cyclicity and amplitudes in hole-

Geostatistical applications in petroleumreservoir modellingby R. Cao*, Y. Zee Ma† and E. Gomez†

SynopsisGeostatistics was initially developed in the mining sector, but has beenextended to other geoscience applications, including forestry, environ-mental science, soil science, and petroleum science and engineering. Thispaper presents methods, workflows, and pitfalls in using geostatisticsfor hydrocarbon resource modelling and evaluation. Examples arepresented of indicator variogram analysis of categorical variables,lithofacies modelling by sequential indicator simulation and hierarchicalworkflow, porosity modelling by kriging and stochastic simulation,collocated cokriging for integrating seismic data, and collocated cosimu-lation for modelling porosity and permeability relationships. Thesemethods together form a systematic approach that can be effectivelyused for modelling natural resources.

Keywordsfacies modelling, propensity, multilevel or hierarchical modeling, object-based modeling, collocated cosimulation, porosity, permeability.

* China University of Petroleum, Beijing, China.† Schlumberger, Denver, Colorado, USA.© The Southern African Institute of Mining and

Metallurgy, 2014. ISSN 2225-6253.

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Geostatistical applications in petroleum reservoir modelling

effect variograms are strongly affected by the relativeabundance of each lithofacies and by the means and standarddeviations of the sizes of lithofacies objects (e.g. sandstoneand shale bodies).

Sample density is very important for accurately describingan experimental variogram (Ma et al., 2009). In a reservoirwith a low sandstone fraction, if individual sandstone orother lithofacies bodies are sampled densely (e.g. several datasamples for each sand body), the experimental variogram willlikely show spatial correlations and possibly a certain degreeof cyclicity. However, if individual lithofacies bodies aresampled with only one observation each, the indicatorvariogram will appear with a strong nugget effect. This isconsistent with the notion that geology is by no meansrandom, but that sparse sampling can make it appearrandom.

Consider an experimental horizontal variogram generatedwhile only a few wells have been drilled in the area. Theclosest spacing of the wells may be so great that a variogramcannot be determined for short lag distances. For instance, ifthe object size varies substantially for one lithofacies, thevariability may override any signal that the periodicstratigraphy may carry. If the experimental variogram iscalculated isotropically or with a wide tolerance around aspecified direction, non-isotropic bodies (e.g. channels) willproduce variograms with a mixture of spatial continuities.

For applications, an experimental variogram is typicallyfitted to a theoretical model. The spherical and exponentialmodels, possibly with a partial nugget effect, are mostcommonly used in practice. These models imply monoton-ically increasing variance with increasing lag distance.

Sequential indicator simulationSequential indicator simulation (SIS) is one of the mostcommonly used methods to model lithofacies because of itscapabilities in integrating various data. In particular,lithofacies probability maps or volumes can be used toconstrain the positioning of the geological objects in a SISmodel. As a matter of fact, because of the complexity ofdepositional history, a geological process is simultaneouslyindeterministic and causal. Such a phenomenon can bedescribed by the propensity, which enables quantitativeintegration of descriptive geology with lithofacies frequencydata at wells (Ma, 2009). Such integration can producelithofacies probabilities that convey the descriptive geology,and at the same time honour the available data.Subsequently, these lithofacies probabilities can be used toconstrain stochastic modelling for building realistic geologicalmodels. The boundaries in the propensity zoning are ofteninterpretive, in which case the geologist can incorporate theconceptual depositional model.

When no probability is used for conditioning, thelithofacies model is driven only by the indictor variogram anddata from the wells. As a result of the scarcity of the data, thefacies objects are often distributed unrealistically randomly,especially in the areas of no control points. In the exampleshown in Figure 1A, reef facies should occur only in theeastern rim of the shelf, but the model generates the reeffacies everywhere (Figure 1B).

By integrating the propensity analysis (Figure 2A) basedon the conceptual depositional model, the facies probability

maps were generated (Figures 2B to 2E). The facies modelconstrained by these probability maps shows most reef faciesin the east of the area (Figure 2F). The propensity influenceis obvious in the model.

Hierarchical facies modellingApart from SIS, other techniques for modelling facies includeobject-based modelling (OBM, see e.g., Ma et al. 2011),Boolean, and truncated Gaussian methods (Deutsch andJournel, 1992; Falivene et al., 2006). All these categoricalmodelling techniques can be hierarchized in two or morelevels for multilevel facies modelling. For example, facies may

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Figure 2—(A) Propensity zoning generated by considering the faciesfrequency data at the wells. (B)-(E) Facies probability maps combiningthe propensities in (A) and facies frequency data at the wells: (B)foreslope, (C) reef, (D) tidal flat, and (E) lagoon. (F) Facies model builtusing SIS with the probability maps (B-E)

Figure 1—(A) A carbonate-rimmed shelf with eight wells. The reservoirarea extends about 7200 m in the east-west direction, 2700 m north-south, and the thickness is about 17 m (interval with facies display).Four facies are present out of nine depositional facies in Wilson’sgeneralized facies belts (Wilson, 1975). (B) Facies model built using SISwithout using a conceptual depositional model

be first modelled using fluvial OBM or the truncated Gaussianmethod, and then modelled again using SIS based on themodel already constructed. This is because the OBM andtruncated Gaussian methods generally produce larger faciesobjects in the model, and SIS can be further used to modelsmall-scale heterogeneities. OBM is suitable for clear shapedefinitions of geological features such as channels and bars.The truncated Gaussian method is suitable when the order ofthe facies is clearly definable.

The workflow of modelling lithofacies by SIS following atruncated Gaussian modelling or OBM has been applied to anumber of hydrocarbon field development case studies,including carbonate ramps and shallow marine depositionalenvironments. Here, an example of constructing a faciesmodel using OBM based on the model constructed by SIS ispresented. A facies model of sand-shale (Figure 3A) was firstbuilt using SIS with facies probability maps. An OBM methodwas then used to generate channels with splays (Figure 3B).Splays occurs on the edges of the channels, but are notinvariably present. Notice that channels erode bothpreviously deposited shale and sand.

Kriging and stochastic simulation of porosityPorosity is one of the most important petrophysical variablesin hydrocarbon resource characterization, as it describes thesubsurface pore space available for fluid storage.Geostatistical methods for modelling porosity include krigingand sequential Gaussian simulation (SGS). Kriging generallyproduces smoother results, as the variance of the krigingmodel is commonly smaller than the variance of the dataused in the kriging. SGS can be considered to be a two-stepmodelling workflow that performs a stochastic simulationbased on kriging results. Sometimes, cokriging or cosimu-lation can be used when more densely sampled seismic datais available and can be calibrated with porosity.

In addition, the lithofacies model is often used toconstrain the spatial distribution of porosity using SGS,because in the hierarchy of subsurface heterogeneities,

depositional facies govern spatial and frequency character-istics of porosity to a large extent. Even though porosity canstill be quite variable within each facies, the porositystatistics by facies generally exhibit less variation (Ma et al.,2008). Figure 4 compares two porosity models constructedwith the two different lithofacies models presented earlier(Figure 2).

Collocated cokriging and collocated cosimulationCollocated cokriging is a simplified version of cokriging (Xuet al., 1992). Instead of using a number of data points fromthe secondary variables, collocated cokriging uses only onedata point that is collocated (i.e. at the same position) withthe estimation point of the primary variable. Thus theestimator by collocated cokriging is such that:

[1]

where m is the mean of the primary variableλj are the weights of the data points of the primary

variableλ0 is the weight of the collocated data point of the

secondary variablemy is the mean of the secondary variableY(x0) is the collocated secondary variable.

The simple kriging solution to Equation [1] is a linearsystem of equations obtained by the least-square method thatminimizes the estimation error. That is, in a block matrixform:

[2]

where Czz represents the sample covariance matrix

Czy, or its transpose, cyz is the vector of covariancesbetween each of the data points in the primaryvariable and the collocated data point of thesecondary variable

C00 is the variance of the secondary variable (equal to 1after normalization)


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Figure 3—(A) Facies model built using SIS with facies probability maps.(B) Facies model constructed hierarchically using OBM utilizing thefacies model in (A) as the first-step model

Figure 4—Comparison of two porosity models constructed using SGS.(A) Porosity model constrained to the unconstrained lithofacies model(Figure 1B). (B) Porosity model constrained to the constrainedlithofacies model (Figure 2F)

(B)

(A)


Λsk is the vector of the simple kriging weights for theprimary variable

λ0 is the weight for the collocated data point of thesecondary variable

cz is the vector of covariances between each of thedata points and the estimation point of the primaryvariable

Czy is the covariance between the primary andsecondary variables.

Special cases:(1) When the collocated secondary variable is not used, the

method is reduced to simple kriging(2) When there is no sample data in the kriging

neighborhood, collocated cokriging uses the sole datapoint of the secondary variable at the location, andbecomes the linear regression. Indeed, λ0 is obtained asthe correlation coefficient between the two variables(scaled by their ratio of standard deviations), which isthe solution of the linear regression

(3) An advantage of collocated cokriging compared to linearregression is that it has the exactitude property, inheritedfrom all the kriging methods, while linear regressiondoes not. That is, it honours all the sample data. Theproof is quite straightforward. Making one of the knownpoints in Equations [1] and [2] the estimation point willlead to that point having a weight equal to 1 while all theother weights become zero.

Collocated cosimulation is somewhat similar to sequentialGaussian simulation, except that their kriging estimationequations are slightly different (see Equations [1] and [2]). Itcan also be formulated as the summation of an estimationand a simulated error.

It is noteworthy that an (auto-) covariance function issymmetrical about the origin (zero lag distance), but a cross-covariance function is not necessarily symmetrical. In such a

case, cokriging cannot be simplified to collocated cokrigingbecause the delay effect (see e.g. Papoulis, 1965) incorrelation between the primary and secondary variablescannot be easily conveyed into the auto-covariance function.Moreover, even if the cross-covariance is symmetrical, thereare some important approximations in reducing cokriging to acollocated cokriging (Deutsch and Journel, 1992).Theoretically, only some special cases verify the assumptions(Rivoirard, 2001). In practice, the simplification in collocatedcokriging and co-simulation make these methods very usefulin honouring the relationships between two variables.

Collocated cokriging can be quite effective in integratingseismic data into petrophysical property models (Xu et al.,1992). Collocated cosimulation is generally more useful formodeling permeability as it can honour the porosity-permeability relationship, often termed the phi-Krelationship. An example of modelling the phi-K relationshipis discussed below in comparison to linear regression.

The phi-K relationship in the data is often a cloudednonlinear correlation (Figure 5A). Imposing a lineartransform between the porosity and logarithmeticpermeability can significantly distort the frequency statisticsof the permeability. In fact, the difference in frequency distri-bution between the permeability model generated using astandard linear regression and the well log data is commonlystriking, as shown by the example in Figure 5B. The phi-Ktransform, in which the logarithm of permeability isestimated from the porosity using a linear transform, reducesthe permeability because the exponential of the mean issmaller than the mean of the exponential (Delfiner, 2007).This effect of reduction can be easily seen by comparing thefrequency distribution of the permeability from the linearregression to the original frequency distribution of the data.Indeed, a linear transform creates more low permeabilityvalues and skews the histogram toward the lowerpermeability values (Figure 5B).

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Figure 5 (A) Cross-plot between the horizontal permeability (millidarcy) and porosity (fractional). Green data points are based on samples, and red on themodel using cocosim. The solid line is the linear regression between porosity and the logarithm of permeability. (B) Histograms comparison of the well logpermeability (green) and permeability model (blue) from the phi-K linear transform (logarithmic scale). Note the large discrepancies between the twohistograms. (C) Comparison of the well log permeability and permeability model generated by cocosim (logarithmic scale)

(A)

(B) (C)

Instead of using a phi-K transform or cloud transform,collocated cosimulation or ’cocosim’ (Ma et al., 2008) can beused to honour the relationship between the permeability andporosity. The histogram of the permeability model usingcocosim closely matches the histogram of the well logpermeability data (Figure 5C). Besides avoiding the biasintroduced in a phi-K transform (Ma, 2010), there are otheradvantages in using cocosim, including the 3D porositymodel constraining the 3D permeability model, andhonouring the data at the well locations.

ConclusionIn 3D modelling of subsurface processes, there is commonly asignificant lack of hard data. To generalize the data to a 3Dmodel, critical inferences must be made. The quality andaccuracy of the model depend on not only the quality andquantity of data, but also how the inference is drawn andmade. It is often said that ‘garbage in, garbage out’.However, it should be noted that the ’data in’ does notnecessarily mean a good model will result; it can still result in’garbage out’ due to erroneous inference from the data to themodel. Geostatistical methods provide tools for betterinference from limited data in constructing a 3D reservoirmodel. These include incorporation of depositional interpre-tation using propensity analysis, variogram analysis, and thehierarchical modelling framework.

Propensity analysis can help the transition fromqualitative description to quantitative analysis, bridge the gapbetween the descriptive geology and quantitative modelling,and provides useful constraints to condition the facies modelto be geologically realistic. Variogram analysis can helpcharacterize the continuity of rock properties, includinggeological object size and anisotropy. A broad hierarchicalmodelling workflow is an efficient way of modelling multi-scale subsurface heterogeneities, from large-scale structuraland stratigraphic heterogeneities, to intermediate-scale faciesheterogeneities, to smaller-scale petrophysical properties.Furthermore, some discrete variables in one category of abroad hierarchical workflow, such as facies, can be hierar-chically modelled with two or more levels of modelling bydifferent methods. An example of combining object-basedmodeling and SIS was given in this paper.

Geostatistical models of discrete lithofacies variables areimportant because of their use in constraining porosity andpermeability models. Collocated cosimulation is an effectiveway to model porosity-permeability relationship whilehonouring the known permeability data and constraining thepermeability model to the previously generated porositymodel.

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