Journal of Petroleum Science and Engineering 82-83 (2012) 113-119

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Journal of Petroleum Scienceand Engineering

ELSEVIER jou rnalhomepage: www.elsevier.com/locate/petrol

A robust Krigingmodel for predicting accumulative outflow from a mature reservoirconsidering a new horizontal well

Hossein Mohammadi a, Abbas Seifi a.*, Toomaj Foroud b. Department of Industrial Engineering, Amirkabir Universiry of Technology, Tehran, Iran

b Department of Petroleum Engineering, Amirkabir Universiry of Technology, Tehran, Iran

ARTICLE INFO ABSTRACT

The objective of this research is to build a response surface based on Kriging method to predict accumulativeoutflow from a mature reservoir as a function of location, direction and length of a new horizontal well over a

planning horizon. Investigating the impacts of horizontal well parameters on the outflow using repetitiveruns of reservoir flow simulation would involve enormous computation time for evaluating possible well

placement scenarios. Instead, we construct a Kriging model to build a good approximation of accumulatedoutflow from a reservoir when a new horizontal well is to be drilled. Kriging method is a geostatistical ap-

proach in which the spatial correlation of samples can be estimated via a variogram or a covariance matrix.The objective of this research is to find a more accurate estimate of the covariance matrix used in the con-struction of Kriging model. The method indudes an optimization model to smoothen the noisy experimentalcovariance matrix which in turn leads to enhancing the fitting process. We have observed a very good fit ofthe resulting Kriging model in a case study on a mature oil reservoir in Iran for which a dynamic simulationmodel was set up previously. However, it is necessary to add the nugget effect to the model in order to yieldbetter predictions.

Article history:Received 17 May 2011

Accepted 7 January 2012

Available online 15 January 2012

Keywords:krigingrobust estimationhorizontal wellaccumulative outflow

~ 2012 Elsevier B.V.All rights reserved.

1. Introduction

Exploitationof mature oil reservoirs has forced researchers to findways for enhancing production from such reservoirs.An effectivewayto do so isdrillinghorizontalwells.Horizontalwells can improverecov-ery factordue to enlargingcontact area with the oil reservoirand possi-bility of reaching trapped oil packs in remote areas. However,this kindof wells requires extra investment for drilling and completion.There-fore, it is necessary to find a reliable relation between the parametersof a horizontal well and the resulting accumulativeoil production.

Metamodel is a surrogate mathematical equation which is used tomimic input-output relation of a complicated system. Metamodelingcan be used to create a fast analysis module by approximating theexisting computer simulation model in order to achieve more effi-cient analysis.This efficiencyusually comes at a price of loosing accu-racy and one has to be concerned about this trade-off when using ametamodel. Metamodeling techniques also shed lights on the func-tional relationship between input and output parameters. A commonfeature of metamodeling approach is to identify an efficient set ofcomputer runs to be performed and used by a regression analysis tocreate an approximating model of the computer simulation.

.Corresponding author at: Amirkabir University ofTechnology, 424 Hafez Ave.. P.O.Box 15875-4413. Tehran, Iran. Tel.: +982164545377; fax: +982166954569.

E-mail addresses:mohammadiindus@autac.ir(H.Mohammadi).aseifi@aulac.ir(A. Seifi), toomaLforoud@autac.ir(T.Foroud).

0920-4105/$ - see front matter ~ 2012 Elsevier B.V.All rights reserved.doi: I0.1016/j.petroI.2012.01.004

There has been an increasing trend of research on this subject duringlast two decades. Guo and Evans (1993) developed an economic modelto assess feasibility of drilling horizontal wells in naturally fractured car-bonate reservoirs. They also derived an analytical method to forecast fu-ture production performance in naturally fractured reservoirs.Aanonsen et at. (1995) suggested a method for well placement optimi-zation while considering geological uncertainties. They employed a re-sponse surface method and a Kriging model to reduce the requiredsimulation runs. Wagenhofer and Hatzignatiou (1996) tried to optimizehorizontal well depth using water and gas coning concepts. They built adynamic simulation model of a reservoir first and then developed a sur-rogate model to be used instead of the original simulator. Pan andHorne (1998) investigated least squares and Kriging methods to beused as metamodels for reservoir simulations in several cases, includingfield development optimization. Dejean and Blanc (1999) applied re-sponse surface methodology (RSM) and a quadratic model form to op-timize the well location. Guyaguler (2003) created a hybrid geneticalgorithm by combining neural networks and Kriging model with a dy-namic simulator to optimize the location of a production and injectionwell. Yeten et at. (2003) applied Artificial Neural Networks (ANN) to es-timate the value of the objective function representing total well lengthand average contact permeability. Centilmen et at. (1999) introduced aneuro-simulation technique to bridge between a reservoir simulatorand a predictive ANN.Nakajima and Schoizer (2003) proposed a perfor-mance analysis method to identify the effective parameters of a hori-zontal well. Foroud and Seifi (2010) extended the concept of

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114 H.Mohammadiet al.1Journalof PetroleumSdenceandEngineering82-83 (2012) 113-119

surrogate based optimization for horizontal wells. They used threefunctions, polynomial,multiplicative and RBFas surrogate models tofind the best one in terms of fitness and prediction ability. Finally,they found the properties of an optimal horizontal well using a geneticalgorithm on the best surrogate model.

In recent years, support vector machine (SVM)has gained moreattention in geosciences. As it can be seen in Li and Castagna(2004), Youxiand Jun (2007), and Zhao and Zhou (2005), this tech-nique is mostly used to classifyseismic data patterns for explorationand reservoir characterization.

Krigingis a geostatistical method which approximate the responsefunction based on a spatial correlation among sample points. Sackset al. (1989) applied a Krigingapproximation to the design of com-puter experiments. Simpson et al. (1998) reported the optimizationof an aerospike nozzle problem using Krigingapproximation. Giuntaet al. (1998) compared Kriging models and polynomial regressionmodels, while Jin et al. (2000) compared some metamodeling tech-niques for mathematical test problems and one engineering problem.

The aim of this paper is investigation of applicability of Krigingmethod as a surrogate model to predict the performance of horizontalwells in a mature fractured reservoir. The objective is finding an ap-propriate Krigingmodel to be applicable as a metamodel in case ofhorizontal well performance in a hydrocarbon reservoir. The casestudied herein is a mature oil reservoir in Iran for which a dynamicsimulation model was set up previously by Foroud and Seifi (2010).In this paper, we study the application of a Krigingmodel and its en-hancement to this case study.

2. Problem definition

The purpose of this paper is to build a Krigingmodel to predict ahorizontal well performance in a mature oil field.The objective func-tion herein is selected as the accumulativeoutflow from a mature res-ervoir over 20 years. Parameters considered herein are location,direction and length of the horizontal well. The only available toolto calculate the accumulated outflow as a function of horizontalwell features is a dynamic simulation model. Todo so,a fractured ma-ture reservoir was selected to be used as a case study. This reservoir isa heterogeneous and fractured reservoir which is located in west ofIran and is considered as the second most important potential reser-voirs rock. It is a saturated reservoir with existing phases of oil, gasand water in an initial pressure equal to 4100 psi. Depths of water-oil contact and gas-oil contact are 900 and 195 ft respectively. Inte-gration of petrography, lithofacies and petrophysical studies shows,shaly limestone and dolomitic limestone interbedded with thinlayer of shales. Crossplot K-Th and K-Pe,reveals that Chlorite is thedominant clay minerals in that area. The reservoir also contains 5faults that were used as inner boundary lines to create grid blocksin order to improve stream line simulation. The reservoir fluid, rockand fracture properties are summarized in Table 1. A grid structure

Table1ReselVoir fluid and rock properties.

Parameter Unit

psiagleegleeSCFISIBRbbl/SbblcpRbbl/Sbbl1/psi1/psifractionmdfractionft

Quantity

30570.86240.70488501.45340.86391.012.10x 10-61.2x 10-60.005500110

Bubble point pressure

Density of stock tank at S.C

Density of reselVoir fluid at PbGOR at S.C

Bo at Pb

Oil viscosity at Pb

Bw at 4100 psiaCw at 4100 psia

Rock compressibility at 4100 psia

Fracture porosityFracture permeabilityFracture NTG

DZ matrix

Fig. 1. Grid cell distribution of the reselVoir model.

with dimensions of 40 x 32x 6 was created which can be observedin Fig.1.

Production from this field has started since 1999 by drilling twovertical wells and a horizontal well which are illustrated in Fig.1. Pro-duction continued with these three wells until 2006 when a major re-duction was observed in oil production rate. So after work over onwells and infill drilling, it was decided to define a new horizontalwell to enhance the production ofthe field.

Parameters of the horizontal well selected in a way to be usedboth as inputs of the Eclipsesimulator and as variables in metamodelbuilding process. For the well location, we choose I,J and Kvariableswhich are the indices of the horizontal well heel block. Heel of a hor-izontal well is the point in which the well turns from vertical to hor-izontal direction. 1is block index in X direction, J is block index in Ydirection and K represents block index in Z direction. Length of thewell is the other parameter must be defined. We considered the num-ber of blocks in which the well penetrates horizontally as the lengthof the horizontal well and it was represented by L.The last variableto be defined is the direction of the horizontal well in the reservoirthat was denoted by N. Hence direction is a continuous parameterand cannot be introduced to Eclipsesimulator directly; we had to in-vent a coding system for direction delineation. The coding system isdepicted in Fig.2.

After defining all variables we suggest the following function

Q =f(l,J,K,L,N),

in which Q is the accumulated oil production (bbl) over 20 years ofoperation and is defined as a function of horizontal well parameters.In this paper, the function 'f is fitted by a Kriging model which isexplained next.

3. The Kriging model

A Krigingmodel provides a basis for function approximation as-suming that some form of spatial correlation exists among datapoints. It takes into account the fact that a system property varies in

Fig. 2. Direction encoding.

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H.Mohammadiet01.(Journalof PetroleumScienceandEngineering82-83 (2012) 113-119 115

space according to the spatial correlation and uses a weighted aver-age of existing data points for estimation at an unsampled location.The weights are allocated to the sampled data in such a way thatthe variance of estimation is minimized and the estimates are unbi-ased (Webster and Oliver,2007). Hence,Krigingis the best linear un-biased predictor (BLUP).The theory behind Krigingmodels is basedon the concept of regionalized variables. A regionalized variable re-fers to a phenomenon which is dependent on space or time. Kriginghas been used widely in earth science, particularly in geostatistics.Geostatistical applications are usually limited to three dimensions:however the model can be extended to any dimension. The ordinaryKrigingmodel used in this study is briefly explained next.

Suppose a property at an unsampled point xo,denoted by Z(xo) isto be estimated from a set of n sampled data points in the parameterspace. Then, the Krigingestimation is a convex combination of func-tion values at the given data points. i.e.,

n

i(xo) = L AjZ(Xj),j~1

where the weights Ajmust sum to one. Inordinary Kriging,the con-n

straint 2:Ai = 1 is imposed using Lagrangemethod with the multipli-j~1

er J.LThegivendata are somerealizationsofa randomfunctionZ(x)with the variogramy(h). Variogramis a plot of the average dissimilar-ities between the sample data points versus their lag distance of h.The variogram takes large values where there is little correlation be-tween the data points and takes small values where the correlation isstrong.

The weights of ordinary Krigingmodel are obtained by minimiz-ing the estimation variance given by (Webster and Oliver,2007):

cJ1= E[(Z(Xo)-Z(Xo)r]n n n

= -y(xo,xo)- LLAjAjY(Xj,Xj) + 2LA;Y(X"Xo),i~1 j~1 j=1

where Y(Xj,Xj) is the variogram of Z between the data points Xi and Xj'Then, the Kriging equations can be represented in matrix form as:

A1I.=b,

where

A =

[

Y(XI:,XtJ

y(xn, XI)1

Y(XI:,Xn) 1

] [

~I

] [

Y(XI:,XO)

]

. 1 11.= . b= .y(xn,xn) l' An' Y(Xn'XO)'

1 0 f..L 1

The weights and the Lagrangemultiplier ~ are obtained by solvingthe system in Eq. (3). Then, the Krigingvariance is estimated by:

a2(xO) = bT}...

In geostatistical studies, the variogram which has the best fit tothe data would be chosen.The correlation structure of the underlyingfunction can be extracted by fitting a theoretical variogram model tothe data, although the variogram is not reproduced by its Kriginges-timates. However, ifthe variogram used has a very narrow correlationrange. the ordinary Kriging estimates will be close to the mean re-sponse over all existing points. This would be unsatisfactory since itwould never estimate a value higher than the values at the knowndata points. Therefore, the variogram used for optimization musthave a very long correlation range. This is why space fillingsamplinghas been recommended in the literature (Guayules.2003).

It is also possible to use covariance function instead of variogramin a Krigingmodel. The relationship between variogram and covari-ance is as follows:

C(h) = a2 -y(h), (5)

(1)

where C(h) is the covariance function of Euclideandistance hand dis the variance of responses at sample points.

Building a Krigingmodel consists of the following steps that arediscussed in more details for the case studied herein:

(1) Defining the model variables including dependent and inde-pendent variables.

(2) Choosing initial points (seed points), and determining re-sponses at these points (which is in our case the cumulativeoil production).

(3) Calculatingexperimental variogram or covariance estimates.(4) Fittingthe theoretical variogram over the experimental values.(5) Extractingthe robust estimate of the covariance function.(6) Setting up the Krigingsystem of equations to determine the

weights according to (3) and predicting the responses at thedesired test points.

3.1. Model variables

Asdescribed in Section 2. the dependent variable is considered tobe the accumulative oil production of the reservoir over 20 yearswhich is herein referred to as the response, and the independent vari-ables are location coordinates, direction and length of the horizontalwell.

3.2. The initial points and the responses

(2)

To build the variogram and covariance functions and set up theKrigingsystem in Eq. (3), it is necessary to have some initial samplepoints. These points have to be distributed uniformly in the variablesspace to cover more area. We have chosen 125 points selectively byconsidering possible values in the five dimensional variable space.The number of sample points has been recommended in the literatureto be more than one hundred in order to get smoother and more re-liable experimental variogram and covariance curves. In addition,our experience shows that less number of initial points may lead toa noisy variogram which is not suitable for curve fitting.The responseat each sample point has been obtained using Eclipsesoftware.

(3)

3.3. Experimental variogram and covariance

The experimental variogram and covariance estimates have beencomputed using the responses at the 125 initial points according toEqs. (6) and (7), respectively. The variogram is a graph of semivar-iance yaccording to a lag distance h that is given by:

(4) y(h) = L [z(Xj)-z(Xj+ h)]22N(h) ,

(6)

wherez(Xj)is the responseof samplepointxjandN(h) denotesthetotal number of sample pairs for the lag distance h. However,asmall tolerance is placed on the lag h because our reservoir has an ir-regular grid as it is depicted in Fig.1.

The semivariance and variogram estimates for different values of hare shown in Table2 and depicted in Fig.3.

Fig.3 shows that the estimated variogram is too noisy after lagdis-tance 30. This is due to having less number of sample points in thatrange.

Asshown in Eq. (5). semivariance and covariance have a linear re-lationship and can be used interchangeably in the Kriging model.

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Therefore, we can calculate the unbiased experimental covarianceusing the method in Xiong et al. (2007) as follows:

C (h) = L:[Z(Xj)Z(Xj +h)] - L:Z(Xj)L:Z(Xj +h)ov N(h)-1 N(h)(N(h)-1)'

The numerical results and the graph of experimental covariance

are shown in Table 3 and Fig. 4.

Fig. 4 shows that the variation in the estimated covariance is high.

However, it approaches zero as the lag distance increase which is in

line with our expectation. We apply some corrections in the negativecovariance values to make it smoother.

3.4. Fitting the theoretical variogram

Two methods have been employed herein to find suitable theoret-

ical functions over the experimental variogram and covariance. Thefirst one is to use regression analysis in order to find the unknown pa-

rameters of a predefined variogram function such as linear, Gaussian,exponential and spherical functions. Another approach ensures that

4.5 X 1013

4

3.5

40

CD 30c:

.g! 2.5~'E 2CDen 1.5

0.5

00 5 10 15 20 25

Lag distance (h)

30 35

Fig. 3. Experimental variogram of 125 initial points.

(7)

the positive semi-definiteness of the resulting correlation matrix ismaintained.

We have chosen a spherical function to fit the theoretical vario-gram as shown below:

y(h) = -2.6 x 1012+ (4.1 x 1014)[(~3~~4)-0.5(63~.4)1(8)

The fitted variogram is depicted against the experimental values

in Fig. 5. This figure shows that we do not get a good fit using this the-oretical variogram. Therefore, we need another method to get a betterfit which is discussed in the next section.

3.5. Robust extraction of covariance matrix

The correlation between any two locations with distance of h, isgiven by:

p(h) = Cov(h)0-2(9)

1.4

1.2

c:

20.6ca

~ 0.40

() 0.2

0

-0.2

-0.40 35 4015 20 25

Lag distance (h)

305 10

Fig. 4. Experimental covariance of 125 initial points.

116 H.Mohammadiet aL/ journal of PetroleumSdenceandEngineering82-83 (2012) 113-119

Table 2 Table 3Semivariances of the initial points. Experimental covariance for 125 initial points.

Lag distance Semivariance Lag distance Semivariance Lag distance Covariance Lag distance Covariance(h) (''I) (h) ('Y) (h) (h)

1.049311 2.88x 10" 16.16242 1.15 x 1013 1.049311 1.52 x 1013 16.16242 5.18x 10122.069335 1.58 x 1012 16.55295 1.05 x 1013 2.069335 1.44 x 1013 16.55295 8.87 x 10 122.828427 2.81 x 1012 20.20371 1.68 x 1013 2.828427 1.38 x 1013 20.20371 2.14x 10123.046924 2.16x 1012 20.74417 1.97x 1013 3.046924 1.44 x 1013 20.74417 1.08x 10123.605551 3.64 x 1012 21.17975 1.79x 1013 3.605551 1.47 x 1013 21.17975 -6.1 xl0"4.147324 3.02 x 1012 21.76235 1.24 x 1013 4.147324 1.1x 1013 21.76235 -3.5xl0'25.144189 2.87x 1012 22.43328 2.18x 1013 5.144189 1.33 x 1013 22.43328 7.45 x 10"5.830952 4.45 x 10'2 22.72832 2.1 x 1013 5.830952 2.13 x 1013 22.72832 4.71 x 10"6.224072 3.66 x 1012 2321656 1.68xl013 6.224072 1.31 x 1013 23.21656 1.42 x 10126.708204 8.97xl012 23.72834 2.03 x 1013 6.708204 1.38 x 1013 23.72834 -l.5xlO"7.211103 5.26x 1012 28.37085 1.36 x 1013 7.211103 1.14 x 1013 28.37085 3.5xl0"7.81025 4.74 x 1012 28.69585 l.4x 1013 7.81025 1.56 x 1013 28.69585 6.7xl0"8.485281 8.18x 1012 29.13615 1.59x 1013 8.485281 -1.1 x 10" 29.13615 6.01 x 10"10.24966 6.81 x 1012 29.55221 1.54xl013 10.24966 1.04 x 1013 29.55221 8.46 x 101010.76784 7.05 x 1012 30.20228 3.98 x 1013 10.76784 9.68 x 1012 30.20228 -2.7xlOlO11.2479 7.01 x 1012 30.74838 3.81 x 1013 11.2479 9.16x 1012 30.74838 3.27 x 10"11.76124 9.94 x 1012 31.15847 3.08 x 1013 11.76124 7.56 x 1012 31.15847 4.88 x 10"12.25924 6.29 x 1012 31.74628 3.85 x 1013 12.25924 1.09 x 1013 31.74628 1.79 x 10"12.60733 8.58 x 1012 32.16427 3.92 x 1013 12.60733 6.7 x 10'2 32.16427 -4.5x 10"13.17749 7.07 x 1012 32.62913 1.14 x 1013 13.17749 4.46 x 1012 32.62913 -4.9x 10"14.31522 9.1xl012 36.22549 4.46 x 1013 14.31522 3.94 x 1012 36.22549 -2x 101014.68156 9.34 x 1012 36.70504 4.35 x 1013 14.68156 4.02 x 1012 36.70504 -5.1 x 10915.17532 9.6 x 10'2 37.06751 3.17x 1013 15.17532 327x 1012 37.06751 4.41 x 1010

15.69682 1.04x 1013 15.69682 5.04 x 10'2

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H.Mohammadi et al./ Journalof Petroleum Science and Engineering 82-83 (2012) 113-119

X 10134.5

4

3.5

~ 3

.ffi 2.5

~ 2'E 1.5OJC/) 1

0.5

0

-0.50

II'01/11

Yv

5 10 15 20 25

Lag distance (h)

30 35 40

Fig. 5. Fitting experimental variogram (in blue) using a spherical model (green line).

According to Xiong et aI. (2007), a general form of spatial correlationfunction between any two points with distance of h is given as follows:

(bh

)S-1

p(h)=22 Ks-dbh)r(s-l)-I,

where Kis the modified Bessel function of the second kind, lis thegamma function, and band s are two real numbers which define theshape of the correlation function.

Giventhe sample variance62and pair (h, Cov(h)),where Cov(h) iscomputed from Eq. (6), the spatial covariance extraction problem canbe formulated as the followingoptimization problem:

[ (bh

)S-1

]

2

~~nL 262 2 KS-I(bh)r(s-l)-I-Cov(h).

Once the optimization problem in Eq. (11) is solved and band sare obtained, these parameters are plugged in Eq.(10) to get the spa-tial correlation function.

It should be noted that the experimental covariancevalues used asdata in the optimization problem Eq. (11) are noisy and thus the so-lutions may not be reliable. As mentioned in Section 3.3, we use an-other technique to extract a spatial correlation matrix to be fed intoEq. (11). This helps us to significantly reduce the noise of experimen-tal covariance and make the resulting correlation matrix positivesemi-definite with the diagonal elements being equal to one. Wesolve the following optimization problem in Eq. (12) to find the near-est correlation matrix to the given sample correlation matrix Awhichmay not be positive semi-definite:

m~n~ IV\-XIIFs.t.dig(X) = 1X~O.

This problem is known as nearest correlation matrix used in othercontext as reported in Higham (2002). The distance is measured inFerobenius norm defined as:

IIXIIF= (~;:X/).'/'

Fig. 6 shows how such a matrix can be obtained by solving Eq. (12)

in our case study. It can also reduce the variation in the original sam-ple correlation matrix.

Estimating the values of parameters band s using a positive semi-definite correlation matrix obtained by solvingEq. (11) has led to ac-curate estimation these values:

b = 0.1720,s = 2.5249.

117

1.4

1.2 .." '" , sample correlatio

,- robustcorrelation

c: 0.8

g 0.6a:!~ 0.4(;() 0.2

0

-0.2

-0.40 5 10 15 20 25

Lag distance (h)

30 35 40

Fig. 6. Correlation values based on a valid correlation matrix (green dotted line) andbased on a sample correlation matrix (red dotted line).

Fig.7 shows the shape of the resulting covariance function wellfitted over the valid correlation values.

(10)3.6. The Kriging predictions

(11)

At this stage, the Krigingsystem of equations in Eq.(3) was set upand solved for each test point in our case study. The coefficientmatrixin the left hand side ofthe Krigingsystem Eq.(3) was estimated oncebased on the variogram and the covariance functions obtained in theprevious stages. The right hand side vector, however, was estimatedforeachnewtest pointusingCov(h)or y(h)inwhichh is the lagdis-tance between the new test point and each of those 125 samplepoints.

We have then calculated the response (cumulative oilproduction)using the weights obtained by solving Krigingequations. It should benoted that Krigingis an exact estimator which means the estimationsof Krigingmodel for the initial sample points must be exact. We havetested our implementation for this condition and observed negligibleerrors in the order of 10-12. Thisobservationallowsus to use thefitted Krigingmodel for prediction.

Table4 shows the predictions of the Krigingmodel based on bothfitted covarianceand variogram. The column under Q.imcontains thecumulative oil production estimated by the numerical flow simulatorused in this research and serves as a basis for comparison with theKriging predictions. Contrary to our expectation, initial predictionsof the Krigingmodel based on a robust estimate of covariance matrixwere relatively weaker which could be due to the nugget effect. Thenugget effect describes the discontinuity of the correlation func-tionp(h)as the distance h approaches zero. In our spatial correlation

(12)18 X 10'2

(13)

16

f

" "14 "

12 ~\OJ 'g10a:!.;:: 8~8 6

4

2

0

-20 5 10 15 20 25

Lag distance (h)

30 35 40

Fig. 7. The theoretical covariance function (green line) fitted over the covariance values(blue line).

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model (Fig.6), we can see a sudden drop from one to 0.9 at lag dis-tance zero.Therefore, a nugget effectof 0.1 has been used to describethe discontinuity of the correlation function at the origin accordingtothe following equation (Sasena, 2002):

Pnew(h)= (l-nugget)p(h),

in whichOsnugget<l. Using the nugget effect led to a relative im-provement in the Kriging model prediction. Such an adjustment isnot needed for Krigingprediction based on variogram since the vario-gram accounts for the nugget effect by its predefined parameters.

Fig.8 depicts the same values as reported in Table4 graphically.Itcan be observed that the trends in both Kriging model predictionsagree with those of simulated cumulative oil production (Q.jrn)andthe error is below 10%except at two points 5 and 10. We have im-proved the predictions at those two points using a localizingmethodas described in Section 4.

4. Localizing the Kriging model

As mentioned in some literature like Haas (1990), the localization

of the weights in Kriging system may improve its prediction. There-fore. in this section we employ a localized Kriging model using covari-ance tapering at points 5 and 10 to reduce their predictions error.

Comparison of Kriging outputs50000000a

~ 400000000ti 30000000"-g20000000!i'6 10000000..~ 0..:;E"v

,"- -...-

2 345 6 7 8 9 10 11 12 13 14 15

Test points

--Qsim --QKriging (cov) -iit--QKriging (variogram)

Fig. 8. Outputs of two Krigingmodels as compared to simulated values for the 15 testpoints.

Table 5

Error of Kriging predictions at sample points 5 and 10.

Test point <1<rig;ng (cov) <1<rig;ng (variog,am) <1<'ig;ng (local)

5

10

-14.2-16.0

-13-18.0

-9.4-12.9

(14)

The basic idea of covariance tapering is that we taper the spatialcorrelation values beyond a certain lag distance to zero (Furrer etaI., 2006). Tapering the correlation function has to be done using acompactly supported correlation function with a finite support. Inter-ested readers may refer to Gneiting (2002) for more detail on com-pactly supported correlation function.

Suppose thatpis a correlation function and pois a compactly sup-ported correlation function with support of O.The tapered correlationfunction is the product of pandpo:

Prap(h) = p(h)Pe(h). (15)

For covariance tapering in our model, we use the Bohman's com-pactly supported correlation function which has the followingdefini-tion (Kaufmanet aI.,2011):

{(1- ~)cos(

nh

) sin (q!!) h<O

p(h;0) = 0 0 + rr- h~O0(16)

As we can see in Table 5, using covariance tapering in our casestudy with support 0=12, the prediction error at points 5 and 10have been reduced to - 9.4 and - 12.9, respectively.

5. Conclusions

In this paper, we construct a Krigingmodel to be used as a surro-gate for predicting the cumulative outflow from a reservoir when anew horizontal well is to be drilled. The Kriging models developedherein are in a five dimensional space and use both variogram and co-variance functions.The objective of this research is to find a more ac-curate estimate of the covariance matrix used in the construction ofKrigingmodel. The findings of this study are summarized as follows:

(1) The robust extraction of covariancematrix leads to a significantnoise reduction in the experimental covariance. We have ob-served a very good fit of the resulting Krigingmodel on a ma-ture oil reservoir in Iran for which a dynamic simulationmodel was set up previously.

(2) The predictions of Krigingmodel based on estimates of covari-ance matrix and variogram function are similar in our casestudy. However, we find it very useful and necessary to addthe nugget effect and covariance tapering to the model inorder to yield better predictions.

The Krigingmodel could be further enhanced if it is constructedon the high potential region in the system domain. Furthermore,

118 H.Mohammadiet aL/JournalofPetroleumSdence and Engineering82-83 (2012) 113-119

Table4Krigingpredictions using the variogram and robust covariance estimates with nugget effect.

Test point I J K N L Q,im <1<riging (cov) :tError <1<riging(vanag,am) :tError

1 10 30 6 5 4 43,546,008 40,794,811 6.3 40,848,216 6.22 33 22 6 3 3 37,880,912 34,132,544 9.9 34,256,978 9.63 5 26 6 7 4 44,787,352 43,529,934 2.8 43,361,751 3.24 20 8 6 2 7 33,832,208 34,071,176 -0.7 34,813,872 -2.95 11 11 6 5 3 34,650,684 39,575,190 -14.2 39,171,145 -13.06 39 16 6 7 3 33,876,700 35,002,705 -3.3 34,572,899 -2.17 21 13 6 4 2 33,888,828 34090857 -0.6 34,135,173 -0.78 17 4 6 4 3 37,051,192 34,407,932 7.1 34,662,312 6.49 36 14 6 5 5 34,672,492 35,649,994 -2.8 35,668,481 -2.910 16 15 6 1 8 32,730,312 37,952,983 -16.0 38,620,670 -18.011 23 24 6 6 3 33,821,852 34,268,460 -1.3 34,289,444 -1.412 38 28 6 8 4 33,908,356 34,568,294 -1.9 34,026,378 -0.313 32 25 6 3 8 33,908,356 34,654,400 -2.2 34,883,920 -2.914 15 17 6 8 8 42,847,304 39,805,931 7.1 39,902,680 6.915 18 27 6 6 3 38,326,156 36,450,680 4.9 36,469,545 4.8RMSE 7.10 7.20

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H.Mohammad;et al./ Journalof PetroleumScienceandEngineering82-83 (2012) 113-119

data balancing in the initial sampling could improve the model ro-bustness and is suggested for future research.

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