Reservoir Zonation and Permeability Estimation

8/3/2019 Reservoir Zonation and Permeability Estimation

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SPWLA 48th Annual Logging Symposium, June 3-6, 2007

RESERVOIR ZONATION AND PERMEABILITY ESTIMATION:

A BAYESIAN APPROACH

Adolfo DWindt. PDVSA E&P

Copyright 2007, held jointly by the Society of Petrophysicists and Well LogAnalysts (SPWLA) and the submitting authors.

This paper was prepared for presentation at the SPWLA 48th Annual LoggingSymposium held in Austin, Texas, United States, June 3-6, 2007.

1

ABSTRACT

We propose a new hybrid approach to computeunbiased permeability estimates in uncored wells usingthe theory of Hydraulic Flow Units (HFU) based on the

Carman-Kozeny equation. First, a linear regressionscheme is applied to obtain the optimal number of HFUpresent in core data. Next, the results obtained are usedas input for the nonlinear optimization scheme based onthe probability plot from which statistical parameters ofeach population are obtained. Subsequently, Bayes ruleis applied for clustering core data into its respectiveHFU. Finally, an algorithm based on Bayesianinference is applied to predict permeability in uncoredwells.

The methodology is applied to a Venezuelan sandstonereservoir and to a Middle East sandstone reservoir.Application of the methodology allows permeability prediction in cored wells with correlation coefficientsabove 0.95 for the field cases under analysis.Permeability profiles in uncored wells compare wellwith pressure transient test results.

Among primary applications are better productivityindex assessments, enhanced petrophysical evaluations,and improved reservoir simulation models. Coupling ofNonlinear optimization with Bayesian inference provesa robust way for performing data clustering providingunbiased estimations.

INTRODUCTION

Any reservoir description program should address the problem of describing the pore space geometry bysubdividing the reservoir into units and assign to themvalues for those rock parameters being described(Haldoresen, 1986). Core analysis provides afundamental source of reservoir information because itis the only physical specimen recovered from thereservoir suited for comprehensive rock description at a pore level (microscopic level). Unfortunately, coremeasurements are both expensive and scarce. On the

other hand, wireline logs, representing a larger volumeof investigation (macroscopic level), is one of the mostabundant and economical sources of reservoirinformation being the primary tool for analysis andreservoir description. Permeability is one of the mostimportant petrophysical parameter and it is difficult toestimate in the absence of core measurements.Therefore, relating pore throat attributes (obtained onlyform core measurements) to wireline log measurementsis always a challenge.

Amaefule et al. (1996) proposed the hydraulic flow unitconcept to be used as a principle for subdividingreservoir in different rock types reflecting differentpore-throat attributes. In this regard, the FZI (flow zoneindicator) represents the primary parameter foridentifying those rock types constituting the foundationof this reservoir characterization tool.

Many techniques have successfully been applied inorder to both identify the number of clusters present incore data and to properly assign data into its respectivecluster. Among these techniques can be mentioned thefollowing: cluster analysis, probability plots

(Abbaszadeh et al., 1996), neural networks (Aminian etal., 2003), multivariable regression (Guo et al., 2005),fuzzy logic (Cuddy et al., 2000), and multi-lineargraphical clustering (Al-Ajmi et al., 2000). Abbaszadehet al. (1996) suggested the use of non-linearoptimization. However, it has not been applied beforeto determine the number of HFU present in core dataand their statistical properties. Kapur et al., (2000)combined wireline logs and petrologic description viaBayes Theorem in order to produce probability logs forfacies identification. A similar approach is applied inthis paper. The objective of this paper is to determineHFU by applying non-linear optimization coupled with

the Bayes rule to perform data clustering.Subsequently HFU is inferred in uncored wells via abayesian inversion scheme.

HYDRAULIC FLOW UNIT CONCEPT

A flow unit is defined as a volume of rock where porethroat properties of the porous media that governhydraulic character of the rock are consistently predictable and significantly different from those ofother rocks (Abbaszadeh et al., 1996). A reservoir

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ought to be divided into flow units to properly describeits performance when it is subject to different production schemes. Two approaches have beendeveloped in the industry for performing thissubdivision: a geological point of view and engineering

point of view. Here it will be used the engineeringapproach based on dynamic definitions. Mohammed(2006) suggested the use of the concept of engineeringfacies or dynamic rock type in order to avoid anyconfusion with the geological facies definition used inthe geological approach.

Based on fundamental theory, assuming a bundle ofstraight capillary tubes, and introducing the concept ofmean hydraulic radius (Bird et al., 1960), permeabilitycan be estimated by:

2e

mhk r

2

(1)

where is effective porosity, tortuosity, and re mh isthe mean hydraulic radius.

Equation (1) provides a relationship between permeability and mean hydraulic radius showing itsstrong relationship with pore geometry. By combining

porosity, rmh, and surface area per unit grain volume(Sgv) with equation (1), the Carman-Kozeny model for ageneralized geometry is obtained (Amaefule et al.,1993)

3e

22 2

gv e

1k

F S 1

(2)

where k is given in m2 and Sgv is in m-1. The

effective porosity is obtained either from well logs orcore measurements.

From (1) follows that

2mhe

kr

(3)

The mean hydraulic radius has a strong correlation withdifferent petrophysical parameters such as (Amaefule,et al., 1988): stress corrected porosity and permeability,capillary pressure derived pore throat radius, formationfactor, cation exchange capacity, saturation exponent,and relative permeability among others. Additionally,the mean hydraulic radius can be correlated with thecharacteristic length used in the definition of Reynoldsnumber for porous media (Jones, 1987). Thus, the

selection ofek/ as a predictor of pore space

attributes is both useful and physically sound. Based onthese observations and from equation (3), equation (2)is rearranged to give the following:

1

0 0314 1e

e e s gv

k

. F S

(4)

The units of k are md. Now, a reservoir quality index(RQI) is defined by

e

kRQI 0.0314

(5)

Also, the Flow Zone Indicator (FZI) is defined as

s gv

1FZI

F S

(6)

RQI and FZI are given in m. According to equations(5) and (6), (4) is rewritten as:

zRQI FZI (7)

and

1e

z

e

(8)

Thus, in a log-log plot, core data corresponding to a particular hydraulic flow unit will plot as a unit slope

straight line with intersect at z =1 equals to FZI.Having obtained FZI we are in capability ofdetermining intrinsic petrophysical properties of a givenhydraulic unit and, by doing so, a reservoir can bedivided into a discrete number of hydraulic units. Oncea HFU or engineering facie is identified, permeability iscalculated by (Amaefule et al., 1993):

32

21014

1

e

e

k FZI

(9)

The Carman-Kozeny equation provides good estimatesfor well-sorted samples from which the average particlesize diameter is known (Wu, 2004). However,knowledge of grain diameter and specific surface areais critical in the Carman-Kozeny model. The latterrepresents a major limitation of such model. Also,applicability of the Carman-Kozeny model isquestionable in the presence of diagenesis (Abbaszadehet al., 2000).

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One of the advantages of the engineering faciesdefinition over the geological definition of facies is thatthe FZI has a strong correlation with irreducible watersaturation, specific surface, grain size, and mineralcontent (Svirsky et al., 2004). Moreover, J-Function

derived water saturation values can be obtained usingthe FZI concept (Desouky, 2003) and be successfullyused for allowing accurate 3D reservoir modeling(Obeida et al., 2005).

Despite its limitations, the Carman-Kozeny model has been widely applied both in sandstone and incarbonates with consistent results. Nevertheless, whenreservoir rock deviates from the capillary bundle modelthe Carman-Kozeny model fails in predicting

permeability. Therefore it is important to introduce amodification. A variable exponent in the porosity groupis introduced as well as a correction for the degree of

cementation (Civan, 2002-2003). This modificationleads to obtain straight lines with slopes different than

one in the RQI-z plot.

NON-LINEAR OPTIMIZATION

Because of random errors and minor fluctuations ofgeological factors controlling petrophysical attributes,data will cluster around the straight lines showing somescatter (Abbaszadeh et al., 1996), consequently the FZIwill be distributed around an expected value.

Since the FZI can be represented as the product of

several factors, according to the Central LimitTheorem, its probability distribution will be log-normal(Jensen, 1997). This observation has profoundimplications because the properties -mean and standarddeviation- of the log-normal distribution or, more

broadly speaking, the Gaussian distribution, are wellknown, being given this probability distribution by:

21 x

21f x, , e2

(10)

In equation (10) and are the expected value and the

standard deviation.

In a probability plot the logarithm of the FZI will plotas a straight line. Unfortunately, when several

populations (flow units) are present in the data, it iscommon to observe superposition of severaldistributions.

For the case of a multi modal distribution the probability plot is not linear, rather a smooth curve isobtained, thus, attempting to identify straight line insuch a plot is though and inaccurate. Moreover, because

of superposition effects, the number of clusters presentwill be masked introducing bias to interpretationsextracted from the plot. Hence, data clustering using the

probability plot requires a rigorous approach.

Non-linear optimization provides a robust way todecompose a superposition of a multi-modal Gaussiandistribution into its component parents. The goal is tominimize the cost function given by the least squarescriteria

m

1i

2

icalcii

measi FZIlnFFZIlnF (11)

where icalci FZIlnF and are the calculated

and measured cumulative probability of obtaining avalue less than or equal to ln(FZI

imeasi FZIlnF

i). The term

imeasi FZIlnF is obtained from measured data.

The cumulative probability distribution for amultimodal distribution is given by (Sinclair, 1976):

N

1j

ijjicalci zFfFZIlnF (12)

where fi is the fraction of the data belonging to aparticular population, N is the number of HFU, and F(z)is given by (Jensen, 1997):

N

1i

i2

erff12

1zF

z(13)

In equation (13) erf() is the error function and z is thestandard normal variable defined by

lnFZI

lnFZI

lnFZI-z

(14)

The standard variable z may be estimated from rationalapproximations (Jensen, 1997), ln(fzi), andln(fzi) are themean and standard deviation of ln(fzi)

One of the major advantages of this approach is that themean and standard deviation of each population arecalculated values, not approximations. On the other, itis critical to provide the algorithm with initial guessvalues close to the solution because non-physicalsolutions may arise. Also, convergence problems mighttake place. Another problem that has to be dealt with ishow to figure out the optimum number of HFU presentin the core data.

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Because the optimum number of clusters is not knownin advance, an iterative procedure is required todetermine the number of hydraulic flow units. In orderto overcome these two adverse situations, it is

performed a preliminary graphical clustering analysis as

follows:

1. Assume the number of HFU in dataset

2. Estimate FZI values from the RQI-z plot.3. Perform non-linear optimization in order to obtain

the minimum of the misfit function given by (11)for the assumed number of HFU

4. Increase the number of HFU and go back to step 2;perform this step until the misfit function reaches aplateau.

Once the optimum number of HFU has beendetermined, the mean, standard deviation and the

fraction of the sample corresponding to each HFU canbe determined from the minimization of equation (11).An alternative linear optimization algorithm (Al-Ajmi,

et al., 2000) using the RQI-z plot can be applied withequivalent results.

BAYES THEOREM

Once the statistics ( and) of each distribution havebeen obtained, the next step is to properly assign coremeasurements to their respective HFU. The basis ofthis approach is the fact that a particular FZI maycorrespond to any HFU, but it will take place with

different values of probabilities for each HFU. Thismay be visualized according to the probability treeshown in Figure 1.

FZIi

FZIi

FZIi

.

.

.

.

ff HFU1

HFU

1

ffHFU2HFU2

ffHFH

N

HFHN

i 1P lnFZI lnfzi HFU


i nP lnFZI lnfzi HFU

FZIi

FZIi

FZIi

.

.

.

.

ff HFU1

HFU

1

ffHFU2HFU2

ffHFH

N

HFHN



i nP lnFZI lnfzi HFU

Figure 1: Probability Tree showing alternative branches foreach HFU leading to the same FZI

Each branch in the probability tree leads to the sameFZI but with different probability. Then, from theBayes Theorem (DWindt, 2005):

4

N

1i

ii

iii

HFU)fziln()FZIln(Pf

HFU)fziln()FZIln(Pf)fziln()FZIln(HFUP

(15)

The symbol means in the neighborhood of (Kapur,2000). N is the number of HFU. The termP(HFUiln(FZIj)ln(fzi)) is the probability ofobtaining a particular HFU given that a particular

ln(FZI) is in the neighborhood of ln(fzi). The term fiis an a priori estimate of the probability occurrence of agiven HFU. The term P(ln(FZIj) ln(fzi)HFUi) is the

probability that a particular ln(FZI) is within certaininterval given that it belongs to a particular HFU, it iscalculated using equation (10).

The application of equation (15) permits thedetermination of the boundaries of each HFU in asimple way. Probability logs can be generated for eachHFU so the boundaries are easily identified. The

procedure is simple and intuitive; the decision rule isbased on a probability value.

INVERSE PROBLEM: BAYESIAN INVERSION

Once core measurements have been clustered into theirrespective parents, the inverse problem must beaddressed. That is, predicting hydraulic flow units onwells without core measurements based only wirelinelogs.

Figure 2 shows a probability tree with the differentalternatives or paths available leading to a particular setof wireline logs. It is assumed that different HFU mayresult in the same set of log values, but this takes placeat a different probability values.

XXii

XXii

XXii

..

....

..

..

..

P(HFU

1

P(HFU

1))

P(HFU

2P(H

FU2))

P(HFUn

P(HFUn))

i 1P X HU

i 2P X HU

i nP X HU

XXii

XXii

XXii

..

....

..

..

..

P(HFU

1

P(HFU

1))

P(HFU

2P(H

FU2))

P(HFUn

P(HFUn))

i 1P X HU

i 2P X HU

i nP X HU

Figure 2: Probability Tree showing alternative branches foreach HFU leading to the same set of wireline log readings

In the case of multiple wells logs, equation (15) ismodified so that the probability of occurrence of HFUgiven a wireline log data set may be calculated, therequired expression is the following (DWindt, 2005):


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i j i

i j n

i j ii 1

f P X HFUP HFU X

f P X HFU

(16)

where fi the a priori estimate of a particular HFUobtained from core data, Xj represents a group of logvalues which are near a set of given log values . Theterm P(HUiXj) is the probability of obtaining aHFU given that the wireline log readings are in theneighborhood of and P(XjHUi) is the probability ofobtaining a given set of log readings given a particularHFU. When using log data we are dealing with discretedata, thus probability given by equation (16) can beestimated by:

j j

i j

j

n X HFUP HFU X

n X

(17)

The term n(X jHFUj) is the number of data points belonging to HFUj in a given interval or bin and n(X j) is the number of all data points falling in the

bin. For bins without data, probabilities are interpolatedusing an inverse distance method (Isaaks, 1989).

FIELD CASES

I Sandstone Reservoir:

This formation is a Eocene clastic reservoir located atthe center of the basin of Lake Maracaibo (Venezuela);

at approximately 11,000 ft (TVD). Sand deposits are primarily of channel-type corresponding to a fluvial-deltaic deposition system. Total reservoir thickness isup to 900 ft. A total of 21 wells have been drilled withcore data acquired only on three of them. Permeabilityranges from less than 0.1 md up to 2000 md. Net pay

porosity is between 12% and 25%, non-pay rocks areconsidered to have less than 10% porosity.

Data Corrections. Porosity and permeability core datahas to be both Klinkenberg and stress-corrected tosimulate reservoir-confining conditions. Stresscorrections are made according to Jones (1986).

Figure 3 shows a permeability-porosity cross plot thesandstone reservoir at net overburden (NOB)conditions. Variation of 2 orders of magnitude for agiven porosity indicates that other factors rather than

porosity itself- are governing formation transmissibility.

An exponential or potential model for the k-relationship will not properly reproduce formation

permeability (Jennings et al., 2001).

0.0001

0.001

0.01

0.1

1

10

100

1000

10000

0 0.05 0.1 0.15 0.2 0.25Porosity (fraction) @ NOB

Permeability(mD)@

NOB

Figure 3: Permeability-Porosity relationship for sandstonereservoir

Hydraulic Flow Unit Identification. Preliminary

analysis of the probability RQI-z plot (Figure 4)indicates that there is not clear boundary among thedifferent flow units. Furthermore, it is challenging todetermine the number of flow units present in the coremeasurements

0.001

0.01

0.1

1

10

0.01 0.1 1PHIz @ NOB

RQI@N

OB(microns)

Figure 4: RQI-z relationship for sandstone reservoir; the plotdoes not show clear boundaries between HFU

From the minimization of the cost function given byequation (11) it is determined that 7 clusters or HFU arerequired to properly model core data (Figure 5).For 7 clusters the cost function flattens indicating that afurther increase of the number of clusters does notsignificantly reduce the objective function.

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0.001

0.01

0.1

1

0 1 2 3 4 5 6 7 8

Number of Clusters

CostFunction

Figure 5: Cost function (least-square criteria) as a function ofnumber of clusters

Once the optimum number of cluster is determined, theminimization of equation (11) allows the calculation ofthe statistics of each parent or HFU. Figure 6 shows thematch given by non-linear optimization. A closeapproximation on the probability plot is obtained byusing 7 HFU.

6

It is important to state that any attempt of fittingstraight lines directly in the plot lead to a wrong numberof HFU. In this particular case maybe 4 or 5 straightlines might be fitted leading to wrong statistics.

0.1

1

10

-3 -2 -1 0 1 2 3

Z

FZI

Match

Data

Figure 6: Probability plot of core data showing non-Linearregression match obtained from minimization of misfitfunction

It has to be noted that the parameters ( and) obtainedfrom minimization of the misfit function correspond tothe lognormal distribution. Therefore, propertransformations are needed to obtain the actual valuesof FZI for each HFU. To complete such a task (Jensen,1997) the following expressions are used:

2ln FZIi lnFZI+0.5FZI =e

(18)

2lnFZI2 2

FZI ln FZI= e 1 (19)

Table 1 shows the statistics obtained from theoptimization scheme for each HFU.

Table 1. Non-linear optimization results. Statisticsfor each HFU

HFU fi FZIi FZIi1 0.0425 0.1496 0.18632 0.1593 0.6489 0.02773 0.2116 1.3032 0.00404 0.1575 2.0360 0.00985 0.1873 3.4079 0.07406 0.2044 5.8613 0.0952

7 0.0373 8.5924 0.1409

For the sake of comparison, alternative choices forclustering data such as: K-means, Wards algorithm,and Kohonens self-organizing maps techniques wereapplied in order to evaluate the consistency of the

proposed method. Table 2 shows a comparison of theexpected value of FZI for each HFU obtained fromdifferent methods

Table 2. Comparison of expected value of FZI fromdifferent clustering results

HFUNon-

Linear

Ward

Algorithm

K-Means Kohonen

1 0.150 0.196 0.203 0.226

2 0.649 0.564 0.627 0.675

3 1.303 0.969 1.046 1.219

4 2.036 1.672 1.647 2.038

5 3.408 2.578 2.513 3.327

6 5.861 4.582 4.457 5.393

7 8.592 7.409 7.297 7.890

From the results presented in table 2 it is observed thatthe four different methods provide values for the FZI ina close range indicating consistency in results. For thecase of the Kohonens self-organizing maps technique,

the absolute differences with the non-linear results areless than 5% in average (excluding HFU 1). Furthermore, hypothesis test for the mean between Kohonensand non-linear results show that, with the exception ofthe HFU1 and HFU7, differences are not statisticallysignificant.

Once the optimal number of clusters is determined it isnecessary to establish the boundaries of each HFU inorder to classify the core data. In this paper a Bayesianapproach is applied to perform this task. A probability


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of occurrence for each HFU given a particular value ofFZI is determined by means of the Bayes rule. This isaccomplished by applying of equation (15). Thecorresponding HFU for a given FZI will be that onewith the highest probability of occurrence. Results of

the application of equation (15) are shown in Figure 7.

0.00

0.20

0.40

0.60

0.80

1.00

0.1 1 10FZI

ProbabilityofOcurrence

HFU1

HFU2

HFU3

HFU4

HFU5

HFU6

HFU7

Figure 7: Probability log for each HFU as a function of theFlow Zone Indicator (FZI). Venezuelan reservoir

It is important to remark that boundaries for each HFUare easily determined by a probability value. There isno need for clustering algorithms such as Ward or K-means algorithms for clustering core data into itsrespective parent, a simple probability criteria is

sufficient. The RQI-z plot (Figure 8) shows unit slope

lines passing through each cluster or HFU following theclassification made based on the Bayes Theorem; here

the intersection at z=1 provides the expected value FZIfor each HFU.

0.001

0.01

0.1

1

10

0.01 0.1 1PHIz @ NOB

RQI@N

OB(microns)

HU1 HU2 HU3 HU4 HU5 HU6 HU7

Figure 8: RQI-z plot presenting core data grouped byHydraulic Flow Unit

Once core measurements are assigned to their

respective HFU, k- cross-plot (Figure 9) shows avariation of less than a half of logarithmic cycle foreach HFU. Variability within a particular HFU is small.Variation in permeability, for a given porosity, is

dramatically reduced when reservoir is subdivided intoflow units with distinctive pore-throat attributes.

0.001

0.01

0.1

1

10

100

1000

10000

0.05 0.10 0.15 0.20 0.25Porosidad (fraccion) @ NOB

K@N

OB(mD)

HU1 HU2 HU3 HU4 HU5 HU6 HU7

Figure 9: k- cross plot for Field Case I

Hydraulic flow unit inference from log data. Solutionof the inverse problem requires dealing with wirelinelog data but, prior to any kind of analysis, log data must

be depth-matched to core measurements and correctedfor environmental effects. Also, wireline log responses

have to be normalized; this is a mandatory and sensitivetask (Hunt et al., 1996).

Environmentally-corrected wireline logs are correlatedwith FZI via Spearmans rank correlation method(Amaefule et al., 1988). Among the different logsavailable, it was determined that gamma-ray, Neutron

porosity, and density porosity logs gave the highestcorrelation coefficient with FZI.

In order to apply equation (17), a Bayesian algorithmwas implemented with the log measurements and withconventional core laboratory measurements.Application of equation (17) allowed the determination

of probabilities of occurrence for each HFU for a givenset of wireline logs, being selected that HFU with thehighest probability. Once a HFU was determined,

permeability was calculated by using equation (9).

Predicted and measured permeability are shown in alog-log cross plot in Figure 10. Correlation coefficientis equal to 0.97, indicating a high accuracy in the results

provided by the Bayesian inversion method.

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0.01

0.1

1

10

100

1000

10000

0.01 0.1 1 10 100 1000 10000K core (mD)

Kcalculated(mD)

Figure 10: Calculated-Measured permeability cross plot forcore data of three wells; r2 = 0.97

Figure 11 and Figure 12 show a comparison between

actual and calculated values for two of the wells withcore data. Calculated permeability profile reproducesthe general tendency shown by core measurements

10550

10650

10750

10850

10950

1 10 100 1000 10000

K(mD)

Depth(feet)

K core

K Calculated

Figure 11: Permeability profile for Well A; comparison between Bayesian generated permeability and actual coredata. Venezuelan sandstone reservoir

11000

11100

11200

11300

11400

11500

0.1 1 10 100 1000 10000

K(mD)

Depth(feet)

K core

K Calculated

Figure 12: Permeability profile for Well B; comparisonbetween Bayesian generated probability and actual core data

These figures illustrate the ability of the inversionscheme to reproduce permeability by using only welllogs assuming that no core data had been taken at all.

Dynamic validation. Proposed HFU scheme is tested ina well without core data completed in the reservoir.Production data available includes RFT, BUP and PLTmeasurements. The pressure transient test indicated akh of 10400 mD-ft and a skin of 2. From PLT/RFTresults it was calculated a PI of 25.65 STB/D/psi.

Bayesian inversion was applied using available log datain this well determining a kh of 10950 md-ft (relative permeability data was used to correct for liquidsaturation effects; kro@Swi was estimated at 0.73).A synthetic PI of 24.05 STB/D/psi was calculated byusing pseudo-steady Darcys equation. Calculated PIand kh values differed only in 5% and 6% respectivelyfrom actual data.

These results indicate that not only permeability is properly reproduced but also that dynamic well

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response can be simulated with a high degree ofconfidence.

Ta l pro s Field c

9

II - Middle East Reservoir:

Core data available from a Middle East sandstone

reservoir was analyzed using the previously appliedmethodology. Figure 13 is a permeability-porositycross-plot showing variation of to 4 orders ofmagnitude in permeability.

0.1

1

10

100

1000

10000

0.05 0.10 0.15 0.20 0.25 0.30

Porosity (fraction) @ NOB

K@

NOB(mD)

v

Figure 13: Permeability-Porosity relationship for Middle EastReservoir.

Non-linear optimization on the probability plot allowsthe determination of 6 HFU present in core data(Figure 14). Observe the smoothness of the curve,which cause to be challenging attempting to drawstraight lines directly in the plot.

0.1

1

10

-3 -2 -1 0 1 2 3Z

FZI

Match

Data

Figure 14: Probability plot of core data showing non-linearregression match obtained from minimization of cost function

field case II

Table 3 presents the statistical properties of each HFUobtained from the non-linear optimization

ble 3 HFU statistica pertie ase II

HFU fi FZIi FZIi1 0.3645 0.5848 0.03282 0.1754 1.1432 0.00063 0.1738 1.9595 0.01644 0.1705 3.1936 0.03215 0.0492 4.6912 0.10236 0.0667 7.4872 0.2858

Application of equation (15) allows the determinationof probability logs (Figure 15) for each HFU permittingthe identification of each HFU boundaries.

0.00

0.20

0.40

0.60

0.80

1.00

0.1 1 10

FZI

Probabilityofocurrence

HU1

HU2

HU3

HU4HU5

HU6

as a function of theow Zone Indicator (FZI). Field case II

forming clusters with small variability withinthem.

Figure 15: Probability log for each HFUFl

RQI-z cross plot (Figure 16) shows 6 different HFUrepresented. Despite data being tightly clustered,

proposed algorithm is capable of separating one fromother

0.001

0.01

0.1

1

10

0.01 0.1 1PHIz @ NOB

RQI@

NOB(microns)

HU1 HU2 HU3 HU4 HU5 HU6

C

Figure 16: RQI-z plot presenting core data grouped byHydraulic Flow Unit Middle East Reservoir

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10

t; less than an order inmagnitude is observed when previously 4 orders ofmagnitude was the order of day.

Inspection of Figure 17 indicates little variability within

a given HFU in the k- cross plo

0.01

0.1

1

10

100

1000

10000

8200

8250

8300

8350

8400

8450

8500

8550

8600

0.1 1 10 100 1000

K (mD)

Depth(ft)

K core (mD)

K Calculated (mD)

0.00 0.05 0.10 0.15 0.2Porosidad @ N

0 0.25 0.30 0.35OB

K@N

OB(mD)

HU1 HU2 HU3 HU4 HU5 HU6

C

fficient of r = 0.96 (Figure 18). Again,e Bayesian inversion showed accuracy in predicting

permeability.

Figure 17: k- cross plot Field case II

For this field-case several well logs were available:gamma-ray, neutron density, FDC, shallow and deepresistivity, microlog, and sonic among others. Rankcorrelation determined that only three well logs wererequired: GR, FDC, and Neutron porosity. Inversionscheme permitted to determine permeability with acorrelation coe 2

th

0.01

0.1

1

10

100

1000

10000

0.01 0.1 1 10 100 1000 10000

K core (mD)

Kcalculated(mD)

ls with core datadrilled in the reservoir. The inversion processaccurately reproduces permeability.

Figure 18: Calculated-Measured permeability cross plot; r2 =0.96. Middle East reservoir

Figure 19 presents one of the wel

Figure 19: Permeability profile for a well drilled in thereservoir; comparison between calculated permeability andactual core data.

All the procedures previously shown are easilyimplemented in a spreadsheet environment; no specialcommercial software is required for performing thesetasks.

CONCLUSIONS

A novel technique is presented and successfully appliedfor clustering core data. Non-linear optimization iscoupled with Bayes Theorem for grouping core datainto its respective parent.

Bayesian inversion proves to be a powerful tool forpredicting rock properties based only on log data.


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Application of the methodology allows thedetermination of the true (not estimates) mean andstandard deviation of each HFU.

Accurate permeability prediction is achieved by means

of the application of the Carman-Kozeny model.

Reproduction of well performance behavior indicatesthat outcomes of the inversion process can be used asinput for numerical simulation purposes.

Consistency in results when compared to othertechniques validates the application of the proposedmethodology rendering it suitable for data analysis for

predicting pore-throat attributes.

Productivity assessments can be performed in an easyand accurate way reducing the need for acquisition of

expensive testing data.

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ACKNOWLEDGMENTS

The author thanks PDVSA for permission to publishthis work and for permitting the use of field data.

A note of special gratitude goes to Onaida Pereira forher assistance and Dr. Rodolfo Soto for his advice, andfor providing data for this research. Specialappreciation is granted to Jesus Salazar for hiscorrections.

ABOUT THE AUTHOR

Adolfo DWindt has worked for eight years asreservoir engineer for PDVSA ,Maracaibo, Venezuela,dealing with sandstone reservoirs under primary

depletion and waterflooding projects. His researchinterests include well transient analysis, formationevaluation, productivity assessment, and uncertaintyanalysis. He received a B.Sc degree in 1997 fromUniversidad del Zulia, and M.Sc. degrees from the

University of Texas at Austin, in 2004 and fromUniversidad del Zulia in 2006, all in PetroleumEngineering.

Reservoir Zonation and Permeability Estimation

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