hydraulic flow units: a bayesian approach

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

RESERVOIR ZONATION AND PERMEABILITY ESTIMATION: A BAYESIAN APPROACH

Adolfo D’Windt. PDVSA E&P

Copyright 2007, held jointly by the Society of Petrophysicists and Well Log Analysts (SPWLA) and the submitting authors. This paper was prepared for presentation at the SPWLA 48th Annual Logging Symposium held in Austin, Texas, United States, June 3-6, 2007.

1

ABSTRACT We propose a new hybrid approach to compute unbiased permeability estimates in uncored wells using the theory of Hydraulic Flow Units (HFU) based on the Carman-Kozeny equation. First, a linear regression scheme is applied to obtain the optimal number of HFU present in core data. Next, the results obtained are used as input for the nonlinear optimization scheme based on the probability plot from which statistical parameters of each population are obtained. Subsequently, Bayes’ rule is applied for clustering core data into its respective HFU. Finally, an algorithm based on Bayesian inference is applied to predict permeability in uncored wells. The methodology is applied to a Venezuelan sandstone reservoir and to a Middle East sandstone reservoir. Application of the methodology allows permeability prediction in cored wells with correlation coefficients above 0.95 for the field cases under analysis. Permeability profiles in uncored wells compare well with pressure transient test results. Among primary applications are better productivity index assessments, enhanced petrophysical evaluations, and improved reservoir simulation models. Coupling of Nonlinear optimization with Bayesian inference proves a robust way for performing data clustering providing unbiased estimations. INTRODUCTION Any reservoir description program should address the problem of describing the pore space geometry by subdividing the reservoir into units and assign to them values for those rock parameters being described (Haldoresen, 1986). Core analysis provides a fundamental source of reservoir information because it is the only physical specimen recovered from the reservoir suited for comprehensive rock description at a pore level (microscopic level). Unfortunately, core measurements are both expensive and scarce. On the

other hand, wireline logs, representing a larger volume of investigation (macroscopic level), is one of the most abundant and economical sources of reservoir information being the primary tool for analysis and reservoir description. Permeability is one of the most important petrophysical parameter and it is difficult to estimate in the absence of core measurements. Therefore, relating pore throat attributes (obtained only form core measurements) to wireline log measurements is always a challenge. Amaefule et al. (1996) proposed the hydraulic flow unit concept to be used as a principle for subdividing reservoir in different rock types reflecting different pore-throat attributes. In this regard, the FZI (flow zone indicator) represents the primary parameter for identifying those rock types constituting the foundation of this reservoir characterization tool. Many techniques have successfully been applied in order to both identify the number of clusters present in core data and to properly assign data into its respective cluster. Among these techniques can be mentioned the following: cluster analysis, probability plots (Abbaszadeh et al., 1996), neural networks (Aminian et al., 2003), multivariable regression (Guo et al., 2005), fuzzy logic (Cuddy et al., 2000), and multi-linear graphical clustering (Al-Ajmi et al., 2000). Abbaszadeh et al. (1996) suggested the use of non-linear optimization. However, it has not been applied before to determine the number of HFU present in core data and their statistical properties. Kapur et al., (2000) combined wireline logs and petrologic description via Bayes Theorem in order to produce probability logs for facies identification. A similar approach is applied in this paper. The objective of this paper is to determine HFU by applying non-linear optimization coupled with the Bayes’ rule to perform data clustering. Subsequently HFU is inferred in uncored wells via a bayesian inversion scheme. HYDRAULIC FLOW UNIT CONCEPT A flow unit is defined as a volume of rock where pore throat properties of the porous media that govern hydraulic character of the rock are consistently predictable and significantly different from those of other rocks (Abbaszadeh et al., 1996). A reservoir


ought to be divided into flow units to properly describe its performance when it is subject to different production schemes. Two approaches have been developed in the industry for performing this subdivision: a geological point of view and engineering point of view. Here it will be used the engineering approach based on dynamic definitions. Mohammed (2006) suggested the use of the concept of “engineering facies” or dynamic rock type in order to avoid any confusion with the geological facies definition used in the geological approach. Based on fundamental theory, assuming a bundle of straight capillary tubes, and introducing the concept of mean hydraulic radius (Bird et al., 1960), permeability can be estimated by:

2emhk r

2φ

=τ

(1)

where is effective porosity, τ tortuosity, and reφ mh is the mean hydraulic radius. Equation (1) provides a relationship between permeability and mean hydraulic radius showing its strong relationship with pore geometry. By combining porosity, rmh, and surface area per unit grain volume (Sgv) with equation (1), the Carman-Kozeny model for a generalized geometry is obtained (Amaefule et al., 1993)

( )3e

22 2gv e

1kF S 1

φ=

τ − φ (2)

where k is given in µm2 and Sgv is in µm-1. The effective porosity is obtained either from well logs or core measurements. From (1) follows that

2mhe

kr = τ

φ (3)

The mean hydraulic radius has a strong correlation with different petrophysical parameters such as (Amaefule, et al., 1988): stress corrected porosity and permeability, capillary pressure derived pore throat radius, formation factor, cation exchange capacity, saturation exponent, and relative permeability among others. Additionally, the mean hydraulic radius can be correlated with the characteristic length used in the definition of Reynolds number for porous media (Jones, 1987). Thus, the selection of ek/φ as a predictor of pore space

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

10 03141

e

e e s gv

k.

F Sφ

=φ − φ τ

(4)

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

e

kRQI 0.0314=φ

(5)

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

s gv

1FZIF S

=τ

(6)

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

zRQI FZI= φ (7)

and

1e

ze

φφ =

− φ (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 φz =1 equals to FZI. Having obtained FZI we are in capability of determining intrinsic petrophysical properties of a given hydraulic unit and, by doing so, a reservoir can be divided into a discrete number of hydraulic units. Once a HFU or engineering facie is identified, permeability is calculated by (Amaefule et al., 1993):

( )

32

210141

e

e

k FZI φ=

−φi (9)

The Carman-Kozeny equation provides good estimates for well-sorted samples from which the average particle size diameter is known (Wu, 2004). However, knowledge of grain diameter and specific surface area is critical in the Carman-Kozeny model. The latter represents a major limitation of such model. Also, applicability of the Carman-Kozeny model is questionable in the presence of diagenesis (Abbaszadeh et al., 2000).

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One of the advantages of the engineering facies definition over the geological definition of facies is that the FZI has a strong correlation with irreducible water saturation, specific surface, grain size, and mineral content (Svirsky et al., 2004). Moreover, J-Function derived water saturation values can be obtained using the FZI concept (Desouky, 2003) and be successfully used 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 in carbonates with consistent results. Nevertheless, when reservoir rock deviates from the capillary bundle model the Carman-Kozeny model fails in predicting permeability. Therefore it is important to introduce a modification. A variable exponent in the porosity group is introduced as well as a correction for the degree of cementation (Civan, 2002-2003). This modification leads to obtain straight lines with slopes different than one in the RQI-φz plot. NON-LINEAR OPTIMIZATION Because of random errors and minor fluctuations of geological factors controlling petrophysical attributes, data will cluster around the straight lines showing some scatter (Abbaszadeh et al., 1996), consequently the FZI will be distributed around an expected value. Since the FZI can be represented as the product of several factors, according to the Central Limit Theorem, its probability distribution will be log-normal (Jensen, 1997). This observation has profound implications because the properties -mean and standard deviation- of the log-normal distribution or, more broadly speaking, the Gaussian distribution, are well known, being given this probability distribution by:

( )21 x

21f x, , e2

−µ⎛ ⎞− ⎜ ⎟σ⎝ ⎠µ σ =σ π

(10)

In equation (10) µ and σ are the expected value and the standard deviation. In a probability plot the logarithm of the FZI will plot as a straight line. Unfortunately, when several populations (flow units) are present in the data, it is common to observe superposition of several distributions. For the case of a multi modal distribution the probability plot is not linear, rather a smooth curve is obtained, thus, attempting to identify straight line in such a plot is though and inaccurate. Moreover, because

of superposition effects, the number of clusters present will be masked introducing bias to interpretations extracted from the plot. Hence, data clustering using the probability plot requires a rigorous approach. Non-linear optimization provides a robust way to decompose a superposition of a multi-modal Gaussian distribution into its component parents. The goal is to minimize the cost function given by the least squares criteria

( )( ) (([ ]∑=

−=m

1i

2i

calcii

measi FZIlnFFZIlnFχ )) (11)

where ( )( )i

calci FZIlnF and are the calculated

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

(( imeasi FZIlnF ))

i). The term ( )( )i

measi FZIlnF is obtained from measured data.

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

( )( ) (∑=

=N

1jijji

calci zFfFZIlnF ) (12)

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

( ) ⎥⎦

⎤⎢⎣

⎡⎟⎠

⎞⎜⎝

⎛⋅+= ∑

=

N

1ii 2

erff121zF z (13)

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

lnFZI

lnFZI

lnFZI-z µ=

σ (14)

The standard variable z may be estimated from rational approximations (Jensen, 1997), µ ln(fzi), and σln(fzi) are the mean and standard deviation of ln(fzi) One of the major advantages of this approach is that the mean and standard deviation of each population are calculated values, not approximations. On the other, it is critical to provide the algorithm with initial guess values close to the solution because non-physical solutions may arise. Also, convergence problems might take place. Another problem that has to be dealt with is how to figure out the optimum number of HFU present in the core data.

3


Because the optimum number of clusters is not known in advance, an iterative procedure is required to determine the number of hydraulic flow units. In order to 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-φz 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 a plateau.

Once the optimum number of HFU has been determined, the mean, standard deviation and the fraction of the sample corresponding to each HFU can be determined from the minimization of equation (11). An alternative linear optimization algorithm (Al-Ajmi, et al., 2000) using the RQI-φ

z plot can be applied with equivalent results. BAYES THEOREM Once the statistics (µ and σ) of each distribution have been obtained, the next step is to properly assign core measurements to their respective HFU. The basis of this approach is the fact that a particular FZI may correspond to any HFU, but it will take place with different values of probabilities for each HFU. This may be visualized according to the probability tree shown in Figure 1.

FZIi

FZIi

FZIi

.

.

.

.

ff HFU1

HFU1

ffHFU2HFU2

ffHFHNHFHN

( )i 1P lnFZI lnfzi HFU∈


( )i nP lnFZI lnfzi HFU∈

FZIi

FZIi

FZIi

.

.

.

.

ff HFU1

HFU1

ffHFU2HFU2

ffHFHNHFHN



( )i nP lnFZI lnfzi HFU∈

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

Each branch in the probability tree leads to the same FZI but with different probability. Then, from the Bayes’ Theorem (D’Windt, 2005):

4

( ) ( )( )∑

=

∈

∈=∈ N

1iii

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 term P(HFUi│ln(FZIj)∈ ln(fzi)) is the probability of obtaining a particular HFU given that a particular ln(FZI) is “in the neighborhood of” ln(fzi). The term fi is an a priori estimate of the probability occurrence of a given HFU. The term P(ln(FZIj)∈ ln(fzi)│HFUi) is the probability that a particular ln(FZI) is within certain interval given that it belongs to a particular HFU, it is calculated using equation (10). The application of equation (15) permits the determination of the boundaries of each HFU in a simple way. Probability logs can be generated for each HFU so the boundaries are easily identified. The procedure is simple and intuitive; the decision rule is based on a probability value. INVERSE PROBLEM: BAYESIAN INVERSION Once core measurements have been clustered into their respective parents, the inverse problem must be addressed. That is, predicting hydraulic flow units on wells without core measurements based only wireline logs. Figure 2 shows a probability tree with the different alternatives or paths available leading to a particular set of wireline logs. It is assumed that different HFU may result in the same set of log values, but this takes place at a different probability values.

XXii

XXii

XXii

..

....

..

..

..

P(HFU

1

P(H

FU1))

P(HFU2P(HFU2))

P(HFUnP(HFUn))

( )i 1P X HU∈ξ

( )i 2P X HU∈ξ

( )i nP X HU∈ξ

XXii

XXii

XXii

..

....

..

..

..

P(HFU

1

P(H

FU1))

P(HFU2P(HFU2))

P(HFUnP(HFUn))

( )i 1P X HU∈ξ

( )i 2P X HU∈ξ

( )i nP X HU∈ξ

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

In the case of multiple wells logs, equation (15) is modified so that the probability of occurrence of HFU given a wireline log data set may be calculated, the required expression is the following (D’Windt, 2005):


( ) ( )( )

i j ii j n

i j ii 1

f P X HFUP HFU X

f P X HFU=

∈ξ∈ξ =

∈ξ∑

(16)

where fi the a priori estimate of a particular HFU obtained from core data, Xj represents a group of log values which are near a set of given log values ξ. The term P(HUi│Xj∈ξ) is the probability of obtaining a HFU given that the wireline log readings are “in the neighborhood of” ξ and P(Xj│HUi) is the probability of obtaining a given set of log readings given a particular HFU. When using log data we are dealing with discrete data, thus probability given by equation (16) can be estimated by:

( ) ( )( )j j

i jj

n X HFUP HFU X

n X

∈ξ∈ξ =

∈ξ (17)

The term n(X j∈ξ│HFUj) is the number of data points belonging to HFUj in a given interval or bin and n(X

j∈ξ) is the number of all data points falling in the

bin. For bins without data, probabilities are interpolated using an inverse distance method (Isaaks, 1989). FIELD CASES I – Sandstone Reservoir: This formation is a Eocene clastic reservoir located at the 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 is up to 900 ft. A total of 21 wells have been drilled with core data acquired only on three of them. Permeability ranges from less than 0.1 md up to 2000 md. Net pay porosity is between 12% and 25%, non-pay rocks are considered to have less than 10% porosity. Data Corrections. Porosity and permeability core data has to be both Klinkenberg and stress-corrected to simulate reservoir-confining conditions. Stress corrections are made according to Jones (1986). Figure 3 shows a permeability-porosity cross plot the sandstone reservoir at net overburden (NOB) conditions. Variation of 2 orders of magnitude for a given porosity indicates that other factors –rather than porosity itself- are governing formation transmissibility. An exponential or potential model for the k-φ 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

Perm

eabi

lity

(mD

) @ N

OB

Figure 3: Permeability-Porosity relationship for sandstone reservoir

Hydraulic Flow Unit Identification. Preliminary analysis of the probability RQI-φz plot (Figure 4) indicates that there is not clear boundary among the different flow units. Furthermore, it is challenging to determine the number of flow units present in the core measurements

0.001

0.01

0.1

1

10

0.01 0.1 1PHIz @ NOB

RQ

I @ N

OB

(mic

rons

)

Figure 4: RQI-φz relationship for sandstone reservoir; the plot does not show clear boundaries between HFU

From the minimization of the cost function given by equation (11) it is determined that 7 clusters or HFU are required to properly model core data (Figure 5). For 7 clusters the cost function flattens indicating that a further increase of the number of clusters does not significantly reduce the objective function.

5


0.001

0.01

0.1

1

0 1 2 3 4 5 6 7 8Number of Clusters

Cos

t Fun

ctio

n

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

Once the optimum number of cluster is determined, the minimization of equation (11) allows the calculation of the statistics of each parent or HFU. Figure 6 shows the match given by non-linear optimization. A close approximation on the probability plot is obtained by using 7 HFU.

6

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

0.1

1

10

-3 -2 -1 0 1 2 3Z

FZI

MatchData

Figure 6: Probability plot of core data showing non-Linear regression match obtained from minimization of misfit function

It has to be noted that the parameters (µ and σ) obtained from minimization of the misfit function correspond to the lognormal distribution. Therefore, proper transformations are needed to obtain the actual values of FZI for each HFU. To complete such a task (Jensen, 1997) the following expressions are used:

( )2ln FZIi lnFZI+0.5

FZI =eµ σ

µ (18)

( )2lnFZI2 2

FZI ln FZI= eσ 1σ µ − (19)

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

Table 1. Non-linear optimization results. Statistics for each HFU

HFU fi µFZIi σFZIi1 0.0425 0.1496 0.1863 2 0.1593 0.6489 0.0277 3 0.2116 1.3032 0.0040 4 0.1575 2.0360 0.0098 5 0.1873 3.4079 0.0740 6 0.2044 5.8613 0.0952 7 0.0373 8.5924 0.1409

For the sake of comparison, alternative choices for clustering data such as: K-means, Ward’s algorithm, and Kohonen’s self-organizing maps techniques were applied in order to evaluate the consistency of the proposed method. Table 2 shows a comparison of the expected value of FZI for each HFU obtained from different methods Table 2. Comparison of expected value of FZI from different clustering results

HFU Non-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 that the four different methods provide values for the FZI in a close range indicating consistency in results. For the case of the Kohonen’s self-organizing maps technique, the absolute differences with the non-linear results are less than 5% in average (excluding HFU 1). Further more, hypothesis test for the mean between Kohonen’s and non-linear results show that, with the exception of the HFU1 and HFU7, differences are not statistically significant. Once the optimal number of clusters is determined it is necessary to establish the boundaries of each HFU in order to classify the core data. In this paper a Bayesian approach is applied to perform this task. A probability


of occurrence for each HFU given a particular value of FZI is determined by means of the Bayes´ rule. This is accomplished by applying of equation (15). The corresponding HFU for a given FZI will be that one with 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

Prob

abili

ty o

f Ocu

rren

ce HFU1

HFU2

HFU3

HFU4

HFU5

HFU6

HFU7

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

It is important to remark that boundaries for each HFU are easily determined by a probability value. There is no need for clustering algorithms such as Ward or K-means algorithms for clustering core data into its respective parent, a simple probability criteria is sufficient. The RQI-φz plot (Figure 8) shows unit slope lines passing through each cluster or HFU following the classification made based on the Bayes’ Theorem; here the intersection at φz=1 provides the expected value FZI for each HFU.

0.001

0.01

0.1

1

10

0.01 0.1 1PHIz @ NOB

RQ

I @ N

OB

(mic

rons

)

HU1 HU2 HU3 HU4 HU5 HU6 HU7

Figure 8: RQI-φz plot presenting core data grouped by Hydraulic Flow Unit

Once core measurements are assigned to their respective HFU, k-φ cross-plot (Figure 9) shows a variation of less than a half of logarithmic cycle for each HFU. Variability within a particular HFU is small. Variation in permeability, for a given porosity, is dramatically reduced when reservoir is subdivided into flow 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 @

NO

B (m

D)

HU1 HU2 HU3 HU4 HU5 HU6 HU7

Figure 9: k-φ cross plot for Field Case I

Hydraulic flow unit inference from log data. Solution of the inverse problem requires dealing with wireline log data but, prior to any kind of analysis, log data must be depth-matched to core measurements and corrected for environmental effects. Also, wireline log responses have to be normalized; this is a mandatory and sensitive task (Hunt et al., 1996). Environmentally-corrected wireline logs are correlated with FZI via Spearman’s rank correlation method (Amaefule et al., 1988). Among the different logs available, it was determined that gamma-ray, Neutron porosity, and density porosity logs gave the highest correlation coefficient with FZI.

In order to apply equation (17), a Bayesian algorithm was implemented with the log measurements and with conventional core laboratory measurements. Application of equation (17) allowed the determination of probabilities of occurrence for each HFU for a given set of wireline logs, being selected that HFU with the highest probability. Once a HFU was determined, permeability was calculated by using equation (9).

Predicted and measured permeability are shown in a log-log cross plot in Figure 10. Correlation coefficient is equal to 0.97, indicating a high accuracy in the results provided by the Bayesian inversion method.

7


0.01

0.1

1

10

100

1000

10000

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

K ca

lcul

ated

(mD

)

Figure 10: Calculated-Measured permeability cross plot for core 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 with core data. Calculated permeability profile reproduces the general tendency shown by core measurements

10550

10650

10750

10850

10950

1 10 100 1000 10000

K(mD)

Dep

th (f

eet)

K coreK Calculated

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

11000

11100

11200

11300

11400

11500

0.1 1 10 100 1000 10000

K(mD)

Dep

th (f

eet)

K coreK Calculated

Figure 12: Permeability profile for Well B; comparison between Bayesian generated probability and actual core data These figures illustrate the ability of the inversion scheme to reproduce permeability by using only well logs assuming that no core data had been taken at all.

Dynamic validation. Proposed HFU scheme is tested in a well without core data completed in the reservoir. Production data available includes RFT, BUP and PLT measurements. The pressure transient test indicated a kh of 10400 mD-ft and a skin of 2. From PLT/RFT results it was calculated a PI of 25.65 STB/D/psi. Bayesian inversion was applied using available log data in this well determining a kh of 10950 md-ft (relative permeability data was used to correct for liquid saturation effects; kro@Swi was estimated at 0.73). A synthetic PI of 24.05 STB/D/psi was calculated by using pseudo-steady Darcy´s equation. Calculated PI and kh values differed only in 5% and 6% respectively from actual data.

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

8


response can be simulated with a high degree of confidence.

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 applied methodology. Figure 13 is a permeability-porosity cross-plot showing variation of to 4 orders of magnitude 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 @

NO

B (m

D)

v

Figure 13: Permeability-Porosity relationship for Middle East Reservoir.

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

0.1

1

10

-3 -2 -1 0 1 2 3Z

FZI

MatchData

Figure 14: Probability plot of core data showing non-linear regression match obtained from minimization of cost function – field case II

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

ble 3 HFU statistica pertie ase II HFU fi µFZIi σFZIi

1 0.3645 0.5848 0.0328 2 0.1754 1.1432 0.0006 3 0.1738 1.9595 0.0164 4 0.1705 3.1936 0.0321 5 0.0492 4.6912 0.1023 6 0.0667 7.4872 0.2858

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

0.00

0.20

0.40

0.60

0.80

1.00

0.1 1 10FZI

Pro

babi

lity

of o

curr

ence

HU1HU2HU3HU4HU5HU6

as a function of the

ow Zone Indicator (FZI). Field case II

forming clusters with small variability within them.

Figure 15: Probability log for each HFUFl

RQI-φz cross plot (Figure 16) shows 6 different HFU represented. Despite data being tightly clustered, proposed algorithm is capable of separating one from other

0.001

0.01

0.1

1

10

0.01 0.1 1PHIz @ NOB

RQ

I @ N

OB

(mic

rons

)

HU1 HU2 HU3 HU4 HU5 HU6

C

Figure 16: RQI-φz plot presenting core data grouped by Hydraulic Flow Unit – Middle East Reservoir


10

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

Inspection of Figure 17 indicates little variability within a given HFU in the k-φ 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 1000K (mD)

Dep

th (f

t)

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 @

NO

B (m

D)

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-φ cross plot – Field case II

For this field-case several well logs were available: gamma-ray, neutron density, FDC, shallow and deep resistivity, microlog, and sonic among others. Rank correlation determined that only three well logs were required: GR, FDC, and Neutron porosity. Inversion scheme permitted to determine permeability with a correlation coe 2

th

0.01

0.1

1

10

100

1000

10000

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

K c

alcu

late

d (m

D)

ls with core data drilled in the reservoir. The inversion process accurately 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 the reservoir; comparison between calculated permeability and actual core data.

All the procedures previously shown are easily implemented in a spreadsheet environment; no special commercial software is required for performing these tasks. CONCLUSIONS A novel technique is presented and successfully applied for clustering core data. Non-linear optimization is coupled with Bayes’ Theorem for grouping core data into its respective parent. Bayesian inversion proves to be a powerful tool for predicting rock properties based only on log data.


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Application of the methodology allows the determination of the true (not estimates) mean and standard deviation of each HFU. Accurate permeability prediction is achieved by means of the application of the Carman-Kozeny model. Reproduction of well performance behavior indicates that outcomes of the inversion process can be used as input for numerical simulation purposes. Consistency in results when compared to other techniques validates the application of the proposed methodology rendering it suitable for data analysis for predicting pore-throat attributes. Productivity assessments can be performed in an easy and accurate way reducing the need for acquisition of expensive testing data. REFERENCES Abbaszadeh, M., Fuji, H., Fujimoto, F., 1996, “Permeability Prediction by Hydraulic Flow Units – Theory and Applications”. SPE Formation Evaluation. December. 263-271 Abbaszadeh, M., Koide, N., Murahashi, Y., 2000, “Integrated Characterization and Flow Modeling of a Heterogeneous Carbonate Reservoir in Daleel Field, Oman”. SPE Formation Evaluation and Engineering. Vol 3. 150-159. Amaefule, J., Kersey, D., Marshall, D., Poewl, J., Valencia, L., Keelan, 1988, “Reservoir Description: A Practical synergistic engineering and geological approach based on analysis of core data”. SPE paper 18167 presented at the 63rd Annual Technical Conference and Exhibition of the Society of Petroleum Engineers, Houston, Tx. Amaefule, J., Altumbay, M., Tiab, D., Kersey, D., Keelan., D., 1993, “Enhanced Reservoir Description: Using Core and Log Data to Identify Hydraulic Flow Units and Predict Permeability in Uncored Intervals/Wells”. SPE paper 26436. 68th Annual Technical Conference and Exhibition of the Society of Petroleum Engineers, Houston, Tx. Aminian, K., Ameri, S., Oyerokun, A., Thomas, B., 2003, “Prediction of Flow Units and Permeability Using Artificial Neural Networks” SPE paper 83586 presented at the SPE Wester Regional/AAPG Pacific Section Joint Meeting. Long Beach, California

Al-Ajmi, F., Holditch S., 2000, “Permeability Estimation Using Hydraulic Flow Units in a Central Arabia Reservoir”. SPE Paper 63254 presented at the 2000 SPE Annual Technical Conference and Exhibition. Dallas, Texas Bird, R.B., Stewart, W.E and Lightfoot, E.N., 1960, Transport Phenomena. Civan, F., 2002, “Fractal Formulation of the Porosity and Permeability Relationship Resulting in A Power-Law Flow Unit Equation – A leaky Tube Model”. SPE Paper 73785 presented at the SPE International Symposium and Exhibition of Formation Damage Control held in Lafayette, Louisiana Civan, F., 2003, “Leaky-Tube Permeability Model for Identification, Characterization, and Calibration of Reservoir Flow Units”. SPE paper 84603 presented at the SPE Annual Technical Conference and Exhibitiom. Denver. Cuddy, S.J., 2000, “Litho-Facies and Permeability Prediction From Electrical Logs Using Fuzzy Logic”. SPE Reservoir Eval. & Eng Vol. 3, No. 4, 319-324 Desouky, S., 2003, “A new Method for Normalization of Capillary Pressure Curves”. Oil and Gas Technology. Rev. IFP. Vol 58. No. 5. 551-556 D’Windt, A., 2005, "Hydraulic flow unit identification by bayesian inference for permeability prediction”. Msc Thesis. Universidad de Zulia. Guo, G., Diaz, K., Paz, F., Smalley, J., Waninger, E.A., 2005, “Rock Typinc as an Effective Tool for Permeability and Water-Saturation Modeling. A Case Study in a Clastic Reservoir in the Oriente Basin”. SPE Paper 97033 presented at the 2005 SPE Annual Technical Conference and Exhibition, Dallas, Texas Haldoresen, H.H., 1986, “Simulator Parameter Assignment and the Problem of Scale in Reservoir Engineering”. Reservoir Characterization. Academic Press, inc. 293-340 Hunt, E., Ahmed A., Pursell, D., 1996, “Fundamentals of log analysis. Part IV: Normalizing logs with histograms”. World Oil. October. 101-102. Isaaks, E., Srivastava, R., 1989, An introduction to applied geostatistics. Oxford University Press. Jennings, J., Lucia, J., 2001, “Predicting Permeability From Well Logs in Carbonates With a Link to Geology for Interwell Permeability Mapping”. SPE paper 71336.


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SPE Annual Technical Conference and Exhibition. New Orleans, Lousiana. Jensen, J., Lake, L., Corbett, P., Goggin, D, 1997, Statistics For Petroleum Engineers and Geoscientists. Prentince Hall. Jones, S.C., 1987, “Using the inertial Coeffiecient to Characterize Heterogeneity in Reservoir Rocks”. SPE paper 16949 presented at the 62nd Annual Technical Conference and Exhibition of the Society of Petroleum Engineers, Houston, Tx. Kapur, L., Lake, L., Sepehrnoori, K., 2000, “Probability logs for facies classification”. In Situ. Vol. 4. No. 1. 57-78 Mohammed, D., 2006, Personal conversation. Obeida, T.A., Al-Mehairi, Y.S., Suryanarayana, K., 2005, “Calculation of Fluid Saturations From Log-Derived J-Functions in Giant Complex Middle-East Carbonate Reservoir”. SPE Paper 95169 presented at the 2005 SPE Annual Technical Conference And Exhibition. Dallas, Texas. Sinclair, A. J., 1976, “Applications of Probability Graphs in Mineral Exploration”. Association of Exploration Geologist Special Vol. 1. Svirsky, D., Ryazanov, A., Pankov, M., Corbett, P., Posyoesv, A., 2004, “Hydraulic Flow Units Resolve Reservoir Description Challenges in a Siberian Oil Field”. SPE paper 87056 presented at SPE Asia Pacific Conference on Integrated Modeling for Asset Management. Kuala Lumpur, Malaysia. Wu, T., 2004, ”Permeability Prediction and Drainage Capillary Pressure Simulation in Sandstone Reservoirs”. Phd Dissertation. Texas A&M University. ACKNOWLEDGMENTS The author thanks PDVSA for permission to publish this work and for permitting the use of field data. A note of special gratitude goes to Onaida Pereira for her assistance and Dr. Rodolfo Soto for his advice, and for providing data for this research. Special appreciation is granted to Jesus Salazar for his corrections. ABOUT THE AUTHOR Adolfo D’Windt has worked for eight years as reservoir engineer for PDVSA ,Maracaibo, Venezuela, dealing with sandstone reservoirs under primary

depletion and waterflooding projects. His research interests include well transient analysis, formation evaluation, productivity assessment, and uncertainty analysis. He received a B.Sc degree in 1997 from Universidad del Zulia, and M.Sc. degrees from the University of Texas at Austin, in 2004 and from Universidad del Zulia in 2006, all in Petroleum Engineering.

hydraulic flow units: a bayesian approach

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