SPE-193483-MS

Geostatistics – Kriging and Co-Kriging Methods in ReservoirCharacterization of Hydrocarbon Rock Deposits

Karen Ifeoma Ochie and Oluwatosin John Rotimi, Department of Petroleum Engineering, Covenant University

Copyright 2018, Society of Petroleum Engineers

This paper was prepared for presentation at the Nigeria Annual International Conference and Exhibition held in Lagos, Nigeria, 6–8 August 2018.

This paper was selected for presentation by an SPE program committee following review of information contained in an abstract submitted by the author(s). Contentsof the paper have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material does not necessarily reflectany position of the Society of Petroleum Engineers, its officers, or members. Electronic reproduction, distribution, or storage of any part of this paper without the writtenconsent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations maynot be copied. The abstract must contain conspicuous acknowledgment of SPE copyright.

AbstractUnderstanding the spatial distribution of the physical and chemical properties of rocks and the fluids theycontain is very essential in determining hydrocarbon deposits, source, seals and aquifers. Population ofpetrophysical data for reservoir analysis has always been challenging in hydrocarbon exploration withlittle or no knowledge about the reservoir or with few pilot wells. Several basic interpolation methods,and geostatistical interpolation methods such as kriging and co-kriging have been used to populate thepetrophysical parameters for better analysis but without much consideration for removal of inherent dataerror. The need then arises for the data to be populated to be error free because an abnormal or anomalousdata set would not be representative, irrespective of the accuracy of the model. In this study, the Gamma test(GT) which is a non-parametric technique is used to assess the quality of the data since it is independent onthe geostatistical model. Five workflow that took cognizance of GT filtered and unfiltered data assemblagewere used. The data is then populated in the models by using geostatistical methods of kriging and co-kriging to interpolate the petrophysical data for further analysis thereby reducing the uncertainties facedduring reservoir characterization and reserves estimation of in-place hydrocarbon and understanding ofintrinsic reservoir heterogeneities. Results were compared to the results obtained when using both a basicinterpolation method: inverse distance method, and geostatistical methods without proper data assessment.Compared to the other interpolation methods, this technique achieved a lower estimation variance errorhence can be said to create a better representation of the reservoir petrophysical parameters.

IntroductionPetrophysics studies the lateral change in content of fluids by presuming the lateral continuity or extent ofthe reservoir when seismic data is unavailable (Adeoye & Enikanselu, 2009) hence determining the accurateestimates of lithology, fluid content and porosity are indispensable since they are the basic petrophysicalparameters (Rotimi, Adeoye, & Oluwatoyin, 2013). Well logs also called hard data are measurementsfrom the wellbore from which petrophysical parameters such as porosity, saturation, volume of shale,permeability, etc., can be obtained. Porosity and permeability are very essential petrophysical parametersof the reservoir (Aghajari & Sasaninia, 2017). Most times the data acquired is not representative ofthe entire hydrocarbon deposit as wells are only drilled in specific locations of interest. Population of

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petrophysical data for reservoir analysis has always been challenging in hydrocarbon exploration withlittle or no knowledge about the reservoir or with few pilot wells. Earth systems are inherently complex,dynamic and contain various characteristics that can make reservoir characterization a very burdensome task(Caumon, 2010) hence the need for geostatistics. Capturing the reservoirs heterogeneity when analyzingwell logs is a major priority in reservoir characterization hence the need for an appropriate interpolationmethod (Rotimi, Ako, & Zhenli, 2014). Geostatistical methods such as kriging and co-kriging are used toimprove reservoir characterization as it is capable of populating petrophysical parameters to characterizethe reservoir's discrete and implicit properties. With more data points which are accurate, the heterogeneityof the reservoir can be captured thus creating a representative geological model. Representative data is amajor prerequisite for the model irrespective of the accuracy of the latter. For this study, the Gamma testwhich is a non-parametric technique is used to assess the quality of the data since it is independent on thegeostatistical model. Several simple mathematical methods exist but with geostatistical methods such askriging and co-kriging the accuracy of the populated data improves.

Methodology

Geostatistical MethodsGeostatistics is the statistical study of spatial, regionalized variables. It was first created by Danie G. Krigeto predict gold mines, and later developed by Georges Matheron who built its theoretical basis in 1960(Dhaher & Lee, 2013). It usually has the lowest variance of estimation in comparison to other interpolationmethods (Adhikary, Muttil, & Yilmaz, 2017).

Kriging is the most known geostatistical method (Husanovic & Malvic, 2014) used to interpolate asurface from a scattered set of known points (Crystal, 2015). It is appropriate because it shows a particulartrend in the distribution of thickness, which is not observed when using simpler mathematical interpolationalgorithms. The basic kriging relation is presented in equation 1.

(1)

Where Zk is value estimated by kriging, λi is weighting coefficient of the primary variable for each location‘i', and zi is measured values in points.

Co-kriging (equation 2), is an upgrade of kriging because it works on the same principle as krigingand additionally needs auxiliary data, more variography (covariance between the primary and secondaryvariables) and time in order to capture the covariance between two or more related regionalized variables(Yalcin, 2005). It is basically a multivariate equivalent to kriging (Hooshmand, Delghandi, Izadi, & Aali,2011) which employs the secondary variable to estimate the first variable (Zhao, Zhou, Wang, Xin, &Chen, 2014). Availability and inclusion of a number of secondary variable data in simulation is the mainreason for its introduction and computation in correlation with primary variable for data population (Malvic,Barisic, & Futvic, 2009) which makes it more accurate than traditional kriging since it has a smallervariance of estimation error (Xu, Sun, Russell, & Innanen, 2015), (Carter, et al., 2011). This variance isbecause prediction of rock properties is done with more than one attribute since every attribute has a uniqueinformation about the reservoir it represents (Guerrero, Vargas, & Montes, 1996).

(2)

Where Zc is the value estimated by co-kriging, xj is weighting coefficient of the secondary variable foreach location ‘j', and sj is known value of the secondary variable, control point (hard data).

Gamma TestWhen characterizing a reservoir, it is important to develop a systematic feature selection scheme that wouldguide the choice of the most representative features to estimate the petrophysical parameters. The Gamma

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test (GT) is a non-linear modelling and data analysis tool that allows examination of the nature of an input/output in a numerical data-set selection scheme (Dunn & Everitt, 2001) which helps in saving time andin choosing the optimum inputs and avoids over-fitting of data sets. It helps in choosing the best, inputcombination before calibrating and testing models, therefore reducing the uncertainty encountered duringinput selection (Ursula & Jorge, 2014). It can be used to find the best embedding dimension and lag timesfor time series analysis (Kemp S. E., 2006) e.g. climate modelling, predicting the Earth's future temperaturebased on past temperatures, carbon dioxide levels, solar radiation at regular or non-regular intervals (Takens,1981). With an efficient implementation of the Gamma test, relevant input factors could conceivably befound using a combination of correlation (Durrant, 2002) and a random or exhaustive embedding searchapproach. Gamma test can be considered a non-linear equivalent of the sum squared error used in regression(Kemp, Wilson, & Ware, 2006) which removes errors from data by estimating the variance of that part ofthe output that cannot be accounted for by a smooth model, which corresponds to noise in the data set. Asmooth model is one in which the transformation from input to output is continuous and has bounded firstpartial derivatives over the input space.

Suppose we have equation 3; a set of M input output observations of the form:

(3)

Where the inputs xi ∈ Rm are vectors confined to some closed bounded set C ⊂ Rm, and the correspondingoutputs yi ∈ R, are scalars. The relationship between an input xi and its corresponding output yi can beexpressed as equation 4.

(4)

Where f is some smooth unknown function representing the system, f:C → R, C ⊂ Rm, ri is a randomvariable having expectation zero representing noise.

The Gamma test allows the variance of the noise variable r, Var(r) to be estimated, despite that f is anunknown smooth equation with bound derivatives (Jones, Evans, & Kemp, 2007). The noise in the datacan be generated when there is inaccuracy of measurement and the underlying relationship between inputand output is not smooth.

Case StudyThe data used for comparison was gotten from the Teapot Dome field or Naval Petroleum reserve field No.3(NPR3), a federally owned oilfield, located 35 miles north of Casper, Wyoming (Doll, et al., 1995). Ninewells were available and their logs which were provided in the .LAS format contained the caliper, gammaray, density, neutron porosity, density porosity, SP, resistivity and acoustic logs for each of the nine wells.Five different workflow were used in other to achieve the objectives, compare estimation variance andvalidate significance of research. The first two techniques involved populating permeability using krigingand for co-kriging, the secondary variable was porosity after conducting the Gamma test and selectingrepresentative attributes. The other techniques involved populating the data without a Gamma test. For thelast technique, the inverse distance method which is a basic mathematical method, was used to populatethe data.

Results and DiscussionThe gamma test assisted in selection of the best combination of attributes for the reservoir model. A maskcan be defined as a string of binary digits (0s and 1s) with length of the total number of inputs that onewants to analyze using the gamma test (Ursula & Jorge, 2014). In this study, each well log had nine differentattributes shown in Table 1, hence the mask length is nine. A ‘1’ in the mask means the attribute is to beincluded for accurate modelling while a ‘0’ means the log is not needed (Jones, Evans, & Kemp, 2007).

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Generally, the higher the number of 1's, the higher the gamma static. When the combination has a highgamma static i.e. closer to one, it simply indicates that there is noise in the data therefore you would want tominimize the use of such data or eliminate its use completely. This would require reliance on other availabledata hence incurring more 1's until the best combination is achieved for the lowest possible gamma static.Of all the masks, the best combination for each well was used for to train a neural net to create a facieslog. This is very important because the horizons used in further petrophysical analysis is dependent on thefacies log hence is very important to train the data properly using representative data. From the GammaTest, the best combination for each well are as presented in Table 1. A new facie log was created with thecombinations to supervise the training of the neural net. This new log was then used to create new horizonsfor the reservoir grid which captured the uncertainties in characterization especially the shaly sands.

Table 1—Results from the Gamma Test

Well Depth Calliper GammaRay

Density log NeutronPorosity

DensityPorosity

SP log Resistivitylog

Acousticlog

Γ

Well 1 0 0 1 0 0 1 0 1 0 3.046e-5

Well 2 0 0 1 0 1 1 0 1 0 0.0001849

Well 3 0 0 1 0 1 1 1 1 0 0.0004939

Well 4 0 0 1 0 1 1 0 1 1 0.0001932

Well 5 0 0 1 0 0 1 1 1 0 0.0002129

Well 6 0 0 1 0 0 1 1 1 0 0.0002999

Well 7 0 0 1 0 1 1 0 1 0 0.0002255

Well 8 0 1 1 0 1 1 0 1 0 0.0006156

Well 9 0 0 1 1 0 0 1 1 0 0.0001674

Due to the improvement in the characterization process, there were several alterations in the surfacesearlier built with untested data as heterogeneities that were earlier by-passed are now captured. A scatterplotof the two variables studied which are permeability and porosity is shown in Figure 1.

Figure 1—scatterplot of primary and secondary variables

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Permeability from the well logs was upscaled using the arithmetic average method for the createdreservoir simulation case. Prior to property modelling, the estimated value for the upscaled log and thesimulated permeability log had the same value as shown in Figure 2. The upscaled log had values mostlyhigher than the observed data from the hard data.

Figure 2—Histogram plot of permeability distribution before property modelling

To effectively establish the significance of this study, a mathematical interpolation method which wasthe inverse distance method was first used to populate the permeability values. From Figure 3, the modelhad an estimation variance of 1.6299. The estimation variance was that high due to the data obtained forpermeability from the hard data. Generally, the lower the estimation variance (closer to zero), the moreaccurate the interpolation technique.

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Figure 3—Histogram plot of permeability distribution using inverse distance method

Permeability was then modelled using the geostatistical method of kriging. The estimation variancewas 1.5998 which established this method of population as more accurate than the simpler mathematicaltechnique used. The variogram functions in all directions is shown in Figure 4 with a sill of 1 since thefunction was normalized. Normally, a weighted least squares method is used to fit a linear model to asemivariogram as was done here. The parameters of the variogram is a direct contributor to the final krigingmodel. From the post-kriging permeability histogram shown in Figure 5 it can be observed that the valuesfrom this model were closer to the hard data than that from the upscaled logs.

Figure 4—experimental and theoretical semivariogram for the primary variable

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Figure 5—Histogram plot of permeability distribution using kriging

Co-kriging unlike kriging requires an additional variable also known as the secondary variable. For thisstudy, the secondary variable used to co-krig permeability was porosity. This means that the permeabilityvalue for each point was dependent on the weighting coefficient of porosity and the log value of porosity.The model populated by this method in Figure 6 had a smaller estimation error than kriging.

Figure 6—Histogram plot of permeability distribution using co-kriging

Figure 7 is a new improved theoretical variogram modeled sequel to the gamma test workflow, fittedbetter with the experimental variogram when compared to using only kriging. After conducting the gammatest and characterizing the reservoir again based on the improved facie log, the permeability was then

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modelled using kriging as shown in Figure 8. The estimation variance was lower than that obtained withoutusing a gamma test to train the facie log. The estimation variance obtained using this technique was 1.5776.Generally, the reservoir grid was better defined after the data was trained with the gamma test.

Figure 7—experimental and theoretical variogram for the primary variable after gamma test

Figure 8—Histogram plot of permeability distribution using GT + kriging

A histogram plot (Figure 9) of the co-kriging simulation algorithm was used to populate the trained datashows that GT trained raw logs data had appreciable influence on the model. It had the lowest estimationvariance as shown in Figure 10. From the study, it was observed that after the gamma test, the horizonmap for the reservoir was more defined attributing to the fact that more heterogeneity was captured, thusincreasing the mean and standard deviation for the trained data as shown in Table 2.

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Figure 9—Histogram plot of permeability distribution using GT + co-kriging

Figure 10—Estimation Variance

Table 2—Statistics

Method/workflow Standard deviation Mean Estimation Variance

Inverse Distance 905.2626 555.3921 1.6299

Kriging 856.8638 535.5675 1.5998

Co-kriging 830.5398 555.9527 1.4939

Kriging + GT 6357.0305 4029.5730 1.5776

Co-kriging + GT 3040.1116 2119.1795 1.4346

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ConclusionsThis study has shown that geostatistical methods are generally more beneficial in population ofpetrophysical data given their lower estimation variance in comparison to basic mathematical methods. Theresults has shown that co-kriging was a better technique than kriging since it had a lower estimation variancebecause it relied on the influence of the secondary variable when interpolating for a primary variable. Thegamma test which is a non-parametric technique was an effective tool for validating representative datasince it was used in assessing the quality of the data hence it was independent on the geostatistical model.From the trained facie log, it was observed that more heterogeneities were captured and this reflected whenthe primary petrophysical data was populated using kriging and co-kriging. The estimation variance waslower and the agreement between the experimental and theoretical variogram was more accurate indicatingthat this technique was a better approach. These result shows that although co-kriging is a better techniquethan kriging, it is more accurate when data for modelling is subjected to a non-parametric test such as thegamma test prior to analysis which has been seen to be capable of reducing estimation error variance asthe case in this study.

AcknowledgementThe Rocky Mountain Oilfield Testing Center (RMOTC) and the United States Department of Energy isimmensely acknowledged for the release of data for research and permission to publish.

NomenclatureZk = value estimated by krigingλi = weighting coefficient of the primary variable for each location ‘i'zi = measured values in points

Zc = value estimated by co-krigingxj = weighting coefficient of the secondary variable for each location ‘j'sj = known value of the secondary variable, control point (hard data)

M = a set of input output observationsC⊂Rm = components of closed bounded set

f = some smooth unknown function representing the systemri = random variable having expectation zero representing noisexi = inputs confined to some closed bounded setyi = outputΓ = gamma static

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