Hydraulic unit prediction using support vector machine

Journal of Petroleum Science and Engineering ∎ (∎∎∎∎) ∎∎∎–∎∎∎

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Hydraulic unit prediction using support vector machine Syed Shujath Ali, Syed Nizamuddin, Abdulazeez Abdulraheem, Md. Rafiul Hassan, M. Enamul Hossain n King Fahd University of Petroleum and Minerals, Dhahran 31260, KSA

art ic l e i nf o

a b s t r a c t

Article history: Received 29 April 2012 Accepted 6 September 2013

Hydraulic Flow Units or hydraulic units (HUs) concept is becoming popular for grouping reservoir rocks of similar petrophysical properties with the main goal of having a better estimate of permeability. HU approach has an advantage that it addresses the development of permeability in reservoir rocks from fundamentals of geology and physics of flow at pore network scale. The aim of the present study is to predict HUs for the un-cored sections of the wells in a carbonate reservoir using Support Vector Machines (SVMs). HUs for un-cored sections were predicted using wire line logs as input and the associated conventional core values as guides to SVMs. HUs for the core data were identified using three popular correlations such as Kozeny–Carmen (KC) equation, Nooruddin and Hossain equation, and the power law flow unit equation. The experimental results on a Middle East field data show that a better HU prediction accuracy is achieved using Nooruddin and Hossain correlation in comparison with using KC and the power law flow unit correlation. A further analysis to the predicted value reveals that better prediction accuracy is achieved if the granularity of HU class boundary is enlarged to the neighboring classes. Although Nooruddin and Hossain correlation could better relate wire line logs to HUs, however the permeability calculated from the predicted HUs showed less error with the power law flow unit correlation. Considering this, we achieved a maximum of 97% accuracy which encourages the application of SVMs in HU unit prediction using well log data for the un-cored sections of well or for the wells which do not have core data. & 2013 Elsevier B.V. All rights reserved.

Keywords: Kozeny–Carmen equation porosity–permeability wire line logs Nooruddin and Hossain equation reservoir characterization

1. Introduction Reservoir characterization and simulation is a challenging task due to its inherent chaotic behavior. Rock permeability has been used to simulate reservoirs due to its influence on the hydrocarbon rate of production, ultimate recovery, optimal placement of wells, pressure and fluid contact evaluation. The importance of permeability is reflected in the number of studies (Leverett, 1941; Tixier, 1949; Wyllie and Rose, 1950; Timur, 1968; Coates and Dumanoir, 1974; Nooruddin and Hossain, 2011) that have established permeability prediction techniques based on correlations with well log data. However, the accuracy and reliability of the reported experimental results based on the correlations are limited mainly due to the lack of or a week establishment of correlation between permeability and well logs. HU approach has an advantage that it addresses the development of permeability in reservoir rocks from fundamentals of geology and physics of flow at pore network scale. A HU is defined as the representative volume of total reservoir rock within which geological properties that control fluid flow are internally n

Corresponding author. Tel.: þ 966 13 860 2305; fax: þ966 13 860 4447. E-mail addresses: [email protected], [email protected] (M.E. Hossain).

consistent and predictably different from properties of other rocks (Bear, 1972). The concept of HU has widely been used in reservoir characterization (Amaefule et al., 1993; Abbaszadeh et al., 1996; Elgaghah et al., 1998; Nooruddin and Hossain, 2011). Existing literature have shown an improved reservoir characterization by classifying reservoir rock into hydraulic flow units. Gardner and Albrechtsons (1995) observed a significant improvement in the reservoir description through the refinement of the permeability model using hydraulic flow unit concept. Guo et al. (2007) showed that hydraulic flow concept proved to be an effective technique for rock-typing in clastic reservoirs in South America. Shahvar et al. (2010) observed an enhanced prediction of relative permeability by discretizing reservoir rock based on hydraulic flow units for a carbonate reservoir in Iran. Shenawi et al. (2009) developed generalized porosity–permeability transforms based on hydraulic unit technique with excellent accuracy for carbonate reservoir in Saudi Arabia. Orodu et al. (2009) expressed a satisfactory estimation of permeability from HUs, considering high reservoir heterogeneity, availability of less number of cored wells and poor well log response correlation to permeability. Svirsky et al. (2004) were able to resolve the challenges in Siberian Oil field using the concept of hydraulic flow units. Usually a generalized KC equation (Wyllie and Gardner, 1958) approximating the fluid flow in a porous medium is used to relate

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Please cite this article as: Ali, S.S., et al., Hydraulic unit prediction using support vector machine. J. Petrol. Sci. Eng. (2013), http://dx.doi. org/10.1016/j.petrol.2013.09.005i

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permeability with porosity based on pore characteristics. However, various studies (Reis and Acock, 1994; Lucia, 1995; Civan, 2001, 2002) have also indicated that the KC equation is inherently over simplified and does not satisfactorily represent the porosity– permeability relationship for natural porous media. Civan (2001) theoretically derived and verified a power-law flow unit equation by incorporating the fractal attributes of pores in irregular porous media into a bundle of tortuous leaky hydraulic tubes as a model of porous media. The power-law flow unit equation is reported to practically circumvent the limitations of the Kozeny–Carmen (KC) equations with minimum modifications and more accurately describe the porosity– permeability relationship for an actual geological porous formation. Civan (2003) extensively verified their power-law flow unit equation using experimental data. Recently Nooruddin and Hossain (2011) modified the KC permeability–porosity model to predict permeability using hydraulic units (HUs) concept. In this study the HUs were identified based on the above discussed three correlations: Kozeny–Carmen (KC) equation, the power law flow unit equation and the Nooruddin and Hossain equation. A number of studies have also employed various methods in order to predict HUs for the un-cored sections of the wells. As hydraulic units are delineated based on flow zone index (FZI) variable, most of the researchers opted to predict FZI from well log data and then group the predicted FZI as hydraulic units (Al-Ajmi and Holditch, 2000; Soto et al., 2001; Soto and Garcia, 2001; Svirsky et al., 2004). FZI has been predicted either using regression technique (Soto et al., 2001) or artificial intelligence technique such as artificial neural networks (ANNs) (Al-Ajmi and Holditch, 2000; Soto and Garcia, 2001; Shahvar et al., 2009). Some studies aimed to relate HUs directly to well log data (Abbaszadeh et al., 1996; Aminian et al., 2003). In addition, different mapping techniques have been used to correlate wire logs to HUs. Most of the researchers relied on probabilistic approach to tie well logs to HUs (Altunbay et al., 1997; Orodu et al., 2009). There is only one study by Aminian et al. (2003) that used soft computing technique where ANN has been used to predict HUs for un-cored wells with well log data as input to the model. Although ANN is a popular predictive algorithm for regression problems, it is not well known as a classification technique. Moreover, it is a research challenge to choose the best network structure and there is risk to get stuck at local minima. In contrast, SVM has been popular in exhibiting an excellent performance for classification problems since its development by Vapnik (1995). SVM is built based on the statistical learning theory and structural risk minimization principle. These theories do not suffer from stucking at local minima since the resultant solution is the only optimal solution given its parameters. One of the objectives of this study is to utilize the capabilities of the SVM model to predict hydraulic units for uncored sections of wells for carbonate reservoirs in Saudi Arabia, based upon well log data and thereby to predict the permeability. In summary this study estimates the permeability for the uncored wells from the predicted hydraulic units by applying SVMs for Saudi Arabian carbonate reservoir. Finally, a comparative analysis is done to prove empirically the applicability and effectiveness of Nooruddin and Hossain, and the power-law flow unit models over the KC model for deriving permeability from the predicted HUs.

2. Preliminaries In this section we introduce the SVM and hydraulic flow unit theory which have been used to predict HUs.

2.1. Support vector machine Support Vector Machines are learning systems, that are based on Statistical Learning Theory (SLT) and the principle of Structural Risk Minimization (SRM). In principal, the SVM is a supervised learning system where the training data are mapped into a high dimensional feature space in which an optimal separating hyperplane between the two classes of the labeled data is obtained using the quadratic programming. The hyper-plane is then pulled back to the input space via inverse algebra and thus it becomes a non-linear decision system to separate/classify the input data (Cortes and Vapnik, 1995). In fact, the input space in SVM is mapped into the high dimensional feature space using a Kernel function K which is one of the main building blocks of SVMs. The Kernel trick is a computational short-cut to avoid the implicit definition of feature space mapping (Scholkopf et al., 1997). In order to use the Kernel approach we need to create a complicated feature space, then the inner product in that space is to be produced and following this a method to compute that value using the original inputs is to be designed. In other words the Kernel function implicitly uses the feature space where the mapping from input space to feature space can be avoided. ! ! ! ! Let us consider we have n vectors x 0 ; x 1 ; x 2 ; :::; x n from m the vector space R which belongs to either of the two classes {þ1,  1}. The data are labeled as follows: ! ð1Þ ð x j ; yj Þ : j ¼ 0; ::::; n  1 where yj A þ1;  1 Our goal is to find a hyper-plane as in Eq. (2) which would classify the input vector effectively: !T ! w x j þb ¼ 0 ð2Þ ! where, w ¼a weight vector and b¼a bias. Thus it is assumed that the two classes can be separated by considering the two margins parallel to the hyper-plane. !T ! w x j þ b Z1; for yj ¼ þ 1 !T ! w x j þ b r  1;

for yj ¼  1

ð3Þ

where, 0 rj r ðn  1Þ.By combining the above two equations we get !T ! yj ð w x j þbÞ Z 1;

for 0 r jr ðn  1Þ

ð4Þ

Eq. (4) is valid for a dataset which is linearly separable. For such a linearly separable data the goal for SVM is to find the optimal weight and bias values such that the obtained hyper-plane separates the two classes of training data with maximum margin. For a dataset which is not completely separable (i.e., data instances are badly scattered), a classification violation is allowed in the SVM formulation. For this case, Eq. (4) is modified as in Eq. (5), where we introduce n non-negative (slack) variables ξj . !T ! yj ð w x j þbÞ Z 1  ξj

for 0 rj r ðn  1Þ ð5Þ ! In Eq. (5) those vectors x j , for which the value of ξj is greater than zero, does not satisfy Eq. (4). Thus, the sum of ξj is used to measure the misclassification due to using the above mentioned hyperplane. To find an optimal hyper-plane after having identified the embedded constraints the following Lagrange function is defined: 1!T ! n  1 ! !T ! LP ðw ; b; αÞ ¼ w w  ∑ αj ðyj ðw x j þ bÞ  1Þ 2 j¼0

ð6Þ

where, αj Z0 are Lagrange multiplier. ! ! By computing the gradient of LP ð w ; b; αÞ with respect to w and !n b the optimized values of weights w are achieved. By putting the values of optimal weights in Eq. (6) the following equation is

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where

achieved: n1

LD ðαÞ ¼ ∑ αj  j¼0

1 n1 n1

!T

∑ ∑ ααyy x 2i¼0j¼0 i j i j

j

! xj

ð7Þ

ϕz ¼

ϕe 1  ϕe

ð16Þ

The above mentioned solution aims to separate linearly separable dataset. For the nonlinear decision system, we extend the method by mapping the data points into a high dimensional space, called feature space, as in Eq. (8).

FZI can be calculated from Eq. (15) for each of the measurements of permeability and porosity. If Irq vs ϕz is plotted on log–log coordinates, data samples with similar FZI values will be located around a single unit slope straight line with a mean FZI value. The mean FZI value is the intercept of a unit-slope line with the coordinate ϕz ¼1. Samples with significantly different FZIs will lie on other parallel unit-slope lines. The basic idea is to identify groupings of data that form the unit-slope straight lines representing similar flow characteristics and hence a distinct hydraulic unit. Permeability of a sample point is then calculated from the corresponding HU using its mean FZI value and the matching sample porosity using Eq. (17) which is obtained by rearranging Eq. (15).

!T ! ! ! x i x j -〈ϕð x i Þ:ϕð x j Þ〉

k ¼ 1014ðFZI mean Þ2

The solution is found by minimizing LP and maximizing LD. The points in between the two margins depicted by LP and LD exhibiting nonzero values for αj are called the support vectors. Once, the support vectors and values of bias bn (bn is the optimal bias) are determined the two classes are separated by investigating the signs of Eq. (7). 2.2. Kernel function

ð8Þ

To map the input space to a suitable feature space is a tedious task and hence Kernel function is introduced where the mapping is ! done implicitly. The Kernel function between two vectors x i and ! x j is defined as follows: !! ! ! Kð x i x j Þ ¼ 〈ϕð x i Þ:ϕð x j Þ〉 ð9Þ Using Eqs. (8) and (7) we get n1

LD ðαÞ ¼ ∑ αj  j¼0

1 n1 n1 !! ∑ ∑ α α y y Kð x i x j Þ 2i¼0j¼0 i j i j

ð10Þ

Our goal is to maximize the value of LD(α) to optimize the SVM for non-linear dataset. Solutions are available via quadratic programming.

ϕ3 ð1  ϕe Þ2

ð17Þ

Theoretically, there should exist one FZI value for each HU. However, it is found that there is a distribution of FZI values around its true mean results due to random measurement errors in core samples. When multiple HU groups exist, identification of each mean FZI would require decomposition of the overall FZI distribution into its constituting units or delineation of HUs. The techniques usually employed in delineating HUs are histogram analysis, probability plot and cluster analysis (Abbaszadeh et al., 1996). In the present study the HUs were classified based on a popular correlation by Guo et al. (2007) as   HU ¼ Round 2 lnðFZIÞ þ 10:6 ð18Þ

2.3. Hydraulic flow unit theory The present section briefly discusses the three correlations used in the present study to identify hydraulic units. 2.3.1. Kozeny–Carmen equation A generalized Kozeny–Carmen (KC) equation approximating the fluid flow in a porous medium is given by Eq. (11) as k¼

ϕ3e

1

ð1  ϕe Þ2 F S τ2 S2gv

ð11Þ

where k is the permeability in μm2, ϕe is the effective porosity, Fs is the shape factor, τ is the tortuosity and Sgv is the surface area per unit grain volume in μm  1. The effective porosity can be obtained either from core data or appropriate log data. The term F 2s S2gv is a function of geological characteristics of porous media and varies with changes in pore geometry. Thus, this term captures the geological aspect of HUs. Eq. (11) can be written as follows: sffiffiffiffiffi k ϕ 1 pffiffiffiffiffi 0:0314  ð12Þ ¼ ϕe ð1  ϕe Þ F S τSgv where, the constant 0.0314 is the conversion factor from μm2 to mD. Flow Zone Indicator (FZI) and Reservoir Quality Index (RQI) are defined as follows: 1 FZI ¼ pffiffiffiffiffi F S τSgv sffiffiffiffiffi k I rq ¼ 0:031  ϕe I rq ¼ ϕz FZI

ð13Þ

ð14Þ ð15Þ

2.3.2. Power-law flow unit equation The power-law flow unit equation given by Civan (2001) as shown in Eq. (19) is reported to practically circumvent the limitations of the Kozeny–Carmen equations with minimum modifications. sffiffiffiffi  β K ϕ ð19Þ ¼Γ ϕ αϕ The permeability can be calculated by re-arranging Eq. (19) as shown in Eq. (20) K ¼ 1014:24ðΓ 2 Þ

ϕ2β þ 1 ðα  ϕÞ2β

ð20Þ

where Γ, α, and β are the FZI or pore-connectivity functions, the cement exclusion factor and the power-law exponent, respectively. The cement exclusion factor α is given by α ¼ 1  αc , where αc is the volume fraction of the cementation and grain consolidation in bulk porous media. Ordinarily, the cement exclusion factor α is 1, when the cementation effect, which is a measure of grain consolidation by cementing, fusing and other means, is zero and hence αc ¼0. The values of β and Γ may vary significantly if fluid flow through pore throats is restricted. The appropriate values of α and β were obtained by correlating experimental data of permeability–porosity by rearranging Eq. (19) and plotting ð2βÞ  1 log ðK=ϕΓ 2 Þ on Y-axis and log ðϕ=ðα  ϕÞÞ on X-axis and determining the best estimates of parameters α and β of flow units by strategically designed straight line plotting till we get the highest coefficient of regression R2 value (Civan, 2003). For the permeability–porosity data used in the present study the values of α and β obeying the above method were obtained as 0.6 and 1 respectively.

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2.3.3. Nooruddin and Hossain equation Recently, Nooruddin and Hossain (2011) proposed a modified K–C relation in which they transformed the tortuosity term to a measurable petrophysical property of cementation factor as shown in Eq. (21) as k¼

1

ϕ2m þ 1

f g a2 S2gv

ð1 ϕÞ2

ð21Þ

Rearranging and taking the square root of Eq. (21) results in the following form: 1 sffiffiffiffiffi 0 0:031 

k B 1 C ϕ ϕm  1 ¼ @qffiffiffiffiffi A ϕe ð1  ϕÞ f aS g

ð22Þ

gv

The left hand side of Eq. (22) is the reservoir quality index (Irq) where permeability (k) is in mD. The first and second terms of RHS qffiffiffiffiffi ð1= f g aSgv Þðϕ=ð1  ϕÞÞ are the modified flow zone indicator FZIm. Since the normalized porosity index ϕz equals to ðϕ=ð1  ϕÞÞ, rearranging Eq. (22) gives I rq ¼ FZI m ϕm  1 ϕz 0

ð23Þ 1

1 B C FZI m ¼ @qffiffiffiffiffi A f g aSgv

ð24Þ

Taking the logarithm of both sides of Eq. (23) results in the following relationship: log ðI rq Þ ¼ log ðFZI m Þ þ log ðϕz Þ þ ðm  1Þlog ðϕÞ

ð25Þ

It can be noticed that if the cementation exponent (m) equals to one, then Eq. (25) becomes identical to Amaefule's et al. (1993) model. With this modified KC equation, each group of rocks having similar FZIm will constitute an HFU. The permeability of a sample point belonging to certain HU is then calculated using its mean FZIm as given in Eq. (26). k ¼ 1014ðFZI mðmeanÞ Þ2

ϕ2m þ 1 ð1  ϕe Þ2

ð26Þ

3. Hydraulic flow units identification and analysis of the data A case for one of the fields in Saudi Arabia is presented in this study. Wire line log data along with core permeability and porosity for five wells were available for the study. The wells are named consecutively from well 1 to well 5. Log data at reservoir depth interval was extracted and interpolated at depth for which core data is available. First, FZI values for the KC approach, power-law model, and Nooruddin and Hossain model were calculated from core permeability and porosity using Eq. (15), Eq. (19) and Eq. (23), respectively. HU numbers for the core data were then determined using the concept of Eq. (18), which is a function of FZI alone. At similar depth intervals, we now have three sets of HUs. One set being identified using KC approach, second by the power-law model and third with the modified KC model (i.e. the Nooruddin and Hossain model). Hydraulic flow units delineated for all the five wells combined is shown in Fig. 1. Fig. 1a–c shows the distribution of HUs with depth, for standard Kozeny–Carmen (KC) correlation (Eq. (11)), power-law correlation (Eq. (19)) and modified KC correlation (Eq. (21)). HU numbers for the three approaches were obtained using the concept of Eq. (18). HU numbers range between 7 and 15 for KC approach, between 7 and 18 for power law correlation and between 10 and 20 for modified KC approach. Although the HU distribution shows similar trend for the all three cases, the actual values seem to be shifted. For example, at similar depths for which KC approach indicates HU number 7, the power law equation labels them as 8 and the modified KC approach labels them as HU number 11. This could be attributed to the difference in the approach by which FZI has been characterized. For the power law equation and modified KC approach, a modified flow zone index (FZIm & Γ) takes into account the effect of cementation exponent apart from pore throat properties, as shown in Eq. (19) and Eq. (23). Although the ranges of FZI for the two approaches are different, HU numbers were calculated by the same correlation (Eq. (18)) which is generally used to delineate HUs for KC approach. The role of the constant term in Eq. (18) is only to shift the numbering of the HUs. Increasing the constant results in higher HU numbers and decreasing it results in lower HU numbers, but numbering is relative to each other. The constant

Fig. 1. Hydraulic Unit distribution for all five wells combined using (a) KC correlation, (b) power-law correlation, and (c) modified KC correlation.

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Fig. 2. shows a comparison of the HU distribution for the three approaches, with HU for power law correlation and MKC correlation being delineated using new correlation constant as shown in Eqs. (26) and (27).

5

Fig. 3. Permeability distribution for each HU identified using KC approach for five wells combined.

term in Eq. (18) was iterated such that the HUs for power law equation and modified KC equation fall in the similar range as the HUs for KC approach and is shown in Eqs. (26) and (27) respectively. HU ¼ Round½2 lnðΓÞ þ 9:6

ð27Þ

HU ¼ Round½2 lnðFZI m Þ þ 6:6

ð28Þ

Fig. 2 shows a comparison of the HU distribution for the three approaches, with HU for power law equation and MKC equation being delineated using new correlation constant as shown in Eqs. (26) and (27). Now, the dataset consists of well log data and core derived HUs for the three approaches for the five wells available. In Section 4, log data of each well are related to the respective HUs through the model of SVM and HUs for the un-cored sections are predicted. In order to have a better understanding of rock properties characterized by each HU, knowledge of distribution of permeability and porosity across each HU is necessary. Figs. 3 and 4 show the range of permeability and porosity respectively for each HU. Permeability for the least HU number 7 ranges from 0.06 mD to 0.6 mD. As the HU number increases from 7 to 13 upper limit of permeability range increases and so does the concentration of higher permeability values. Only a couple of points belonging to HU 14 and HU 15 were observed. This indicates that, in a reservoir rock with a higher HU, an increased flow of reservoir fluid is expected which is widely suggested by the previous studies. Distribution of porosity across each HU was observed to be highly scattered. As Fig. 3 indicated that permeability range increases with the increase of HU number, it seemed a good idea to match the profile of HU with that of permeability profile as shown in Fig. 5. The profiles of HU and permeability were a match for most of the depth intervals, indicating a strong relationship between HUs and permeability. The contribution of permeability in characterizing HUs seems to be more when compared to porosity. The conflict between the two profiles at some depth intervals can be attributed

Fig. 4. Porosity distribution for each HU identified using KC approach for five wells combined.

to the scatter in the porosity values because the HU is a function of both permeability and porosity. Hydraulic units delineated using KC, power law equation and MKC equation are shown in Figs. 6, 7 and 8 respectively. It can be observed that some points in each HU defined by KC approach have been promoted by at least one unit higher by the power law and MKC approach. This interprets that, for some rock points, power law equation and modified KC approach suggests a better hydraulic fluid flow when compared to the flow characterized by KC approach. Although, in order to validate this, mineralogical and textural characteristics of each HU determined from

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Permeability (mD) 0.01

0.1

1

10

100

1000

800 HU Permeability

Depth (ft)

600

400

200

0 6

7

8

9

10

11

12

13

14

15

HU Fig. 5. Comparison of permeability and hydraulic units for five wells combined. Fig. 7. Hydraulic Unit delineated using power law equation.

Fig. 6. Hydraulic Unit delineated using KC approach.

Fig. 8. Hydraulic Unit delineated using MKC correlation.

petro-graphical study is needed. In this study, we expect that a better delineation of HUs will reflect in a better prediction of HUs from well log data, thus reflecting the effectiveness of KC, power law equation and modified KC correlation in grouping HUs.

logs to the core derived HUs. In our experiment we aim to utilize the capabilities of SVM to predict HUs using log data. Each well consists of log data and three sets of HUs identified using the three approaches mentioned earlier. We, assume that the accuracy of SVM in predicting HUs (which are identified using the three approaches mentioned) from log data, directly relates to the efficiency of these three approaches in grouping rock with similar fluid flow characteristics. Better accuracy in prediction of HUs indicates that well logs could identify a better relation with the respective HU model. Most common and easily available wire line logs such as neutron porosity (NPHI), density log (RHOB), gamma ray (GR)

4. Hydraulic unit prediction from well log data using SVM Existing literature indicate that predictions of hydraulic units for the uncored sections were usually made using statistical or probabilistic methods. In contrast, in this study we use a novel artificial intelligence technique called SVM in order to relate well

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and water saturation (SWT) were chosen as input to the model. For each of the five wells, classified HUs along with the corresponding wire line log data were trained with multiple classification SVM algorithms with 70% of the data. The validation check was performed on 15% of the data points and the testing of the model was carried out by feeding the input log values for the remaining 15% unseen data into the model. The testing phase represents the case of predicting HUs for the uncored section of the wells. The resulting HUs were compared with the originally classified HUs. The performance of the model was evaluated by calculating the percentage of correct classification (PCC) given by the equation below: PCC ¼

Number of corrrect HU classification Total number of data points

ð29Þ

SVM with polynomial Kernels was used to train and test the model. For each well the SVMs complexity constant (C) and polynomial exponent (E) was varied in order to tune the SVM to obtain the best possible performance. Table 1 shows the classification performance in terms of PCC during training validation and testing for the HUs identified using KC correlation. During testing, a 66.67% PCC was obtained for both Well-1 and Well-2 and a PCC of approximately 74% could be achieved for Well-3 and Well-4, whereas best performance with a PCC of 83.7% was achieved for Well-5. A study was also conducted to find out how far the HUs were misclassified. To do so, we assumed if a HU is classified in adjacent unit, we count it as a correct classification. For example, if a log data point originally belongs to HU4, the predicted HU will be counted as correct even if it belongs either to HU3 or HU4 or HU5 i.e. the HU itself and the neighboring HU. This can indicate if the misclassifications fall in an immediate HU or in a HU which is far away. Table 1 also lists the PCC considering the prediction of HU in the neighboring class (NC) to be correct, under the column titled NC. It can be seen that the PCC was almost close to 100% indicating that the HUs misclassified by SVM fall in the neighboring HUs and not far away. Tables 2 and 3 list the percentage of correct classification of HU when power law and MKC correlations were used as the governing equations. Comparing the performance of HU prediction during testing for the three cases in which HUs were identified using three different approaches, it can be observed that modified KC approach outperformed power law equation for all the five wells,

Table 1 Classification performance (PCC) during training, validation and testing for five wells using SVM for KC approach. Well name

Training

Validation

Testing

NC

Well Well Well Well Well

43.37 45.16 50.29 71.43 54.39

49.05 24.24 50 57.14 18.37

66.67 66.67 74.29 74.286 83.67

90.74 100 97.14 100 97.96

1 2 3 4 5

Table 2 Classification performance (PCC) during training, validation and testing for five wells using SVM for power law equation. Well name

Training

Validation

Testing

NC

Well Well Well Well Well

47.98 65.16 80.59 41.50 35.09

33.33 17.65 48.65 38.71 10.20

43.40 72.73 47.22 59.38 61.22

86.79 87.88 77.78 93.75 97.96

1 2 3 4 5

7

Table 3 Classification performance (PCC) during training, validation and testing for five wells using SVM for modified KC approach. Well name

Training

Validation

Testing

NC

Well Well Well Well Well

89.65 78.79 76.53 66.14 66.46

42.86 52.94 28.57 59.26 36.36

83.33 81.25 71.43 81.48 64.71

100 100 90.48 100 97.06

1 2 3 4 5

while it was better for three wells (Well-1, Well-2 and Well-4) when compared to the traditional KC approach. Also, the HU predictions using KC approach were better for four wells when compared to power law equation. It has to be made clear that SVM acts as a mapping tool between the well logs and the HUs identified using core data. The performance of SVM in predicting HUs for the three models could reflect the efficiency of these models in grouping rocks based on their flow characteristics. Based on this assumption, it can be stated that modified KC approach proves to be a better model in grouping HUs considering the better accuracy of SVM in predicting HUs identified by this approach.

5. Discussion Identification of HUs in most of the research studies was carried by histogram analysis or cumulative probability distribution of FZI (Aminian et al., 2003). These techniques are biased to the core dataset being used to estimate HUs. The number of HUs that can be characterized for a reservoir and the boundaries or range of FZI belonging to each HU depends on observer's point of view. HUs numbered in one study cannot be compared to HUs in another study. Guo et al. (2007) introduced a standard practice to number HUs based on core data, making it possible to characterize each HU number with a distinct range of permeability and porosity. This allows comparison of HU studies across different fields and formations, possible. Proper delineation of HUs is more critical when relating wire line log data to core data with the aim of predicting HUs for uncored wells. If HUs grouping is not appropriate or if the grouped HUs do not reflect reservoir rock of similar flow patterns, well logs will not be able to capture the flow patterns for un-cored section, resulting in a poor performance of any predictive model. It is difficult to characterize HUs based on the range of permeability and porosity to which they belong. In the case of permeability, as the HU number increases, higher permeability values become the characteristics of that unit, but the lower permeability values are common for all the HUs, as can be seen from Fig. 3. On the other hand, a whole range of porosity exists for each HU as seen in Fig. 4. The range of porosity values belonging to HU7 also exists for rest of the HUs. This makes the prediction of HUs from well logs a challenging task. This is reflected in relatively low percentage of accuracy in predicting HUs for uncored sections (testing) of the three approaches as seen in Tables 1, 2 and 3, respectively. Better prediction of HUs using modified KC approach is obtained in the present study when compared to that using KC and power law correlation. The KC model usually works well for uniform synthetic porous media and homogeneous rocks. It has difficulty in capturing nonlinearity introduced due to tortuosity, cementation exponent and shape and size factors. Especially the non-linear nature of tortuosity is strongly impacted by the cementation exponent. The modified KC correlation proposed by Nooruddin and Hossain (2011) takes into account the effect of

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Table 4 Comparison of error in permeability estimation using KC, power law and MKC correlation. Well name

Well Well Well Well Well

1 2 3 4 5

AAE KC

Power law

MKC

0.17 0.95 0.59 0.30 0.37

3.32 0.83 0.29 0.27 0.33

0.24 0.90 0.35 0.28 0.35

Fig. 10. Comparison of core permeability and permeability estimated using KC, power law and modified KC correlation for Well 2.

Fig. 9. Comparison of core permeability and permeability estimated using KC, power law and modified KC correlation for Well 1.

cementation exponent and thus better prediction of HUs can be attributed to better delineation or grouping of HUs when compared to the groupings obtained using KC approach. In order to further compare the effectiveness of KC, power law and modified KC correlation, permeability was estimated from the respective equations (Eqs. (17), (20), (26)) using mean FZI values of the predicted HUs and corresponding sample porosities. Table 4 compares the Average Absolute Error (AAE) obtained during permeability estimation using KC, power law and modified KC correlation. In addition, Figs. 9–13 show a comparison of the permeability profile obtained using KC, power law and MKC equations with core permeability for all the five wells. Power law equation outperformed traditional KC and Modified KC correlation in estimating permeability for the four wells (Wells 2–5). Both, the power law model and the MKC model seem to follow the trend of the core permeability for all wells, except for Well 1. It is worth noting that, although, MKC equation was more effective in relating well logs to HUs which is reflected in its better accuracy in predicting HUs, power law equation yielded less error in permeability estimated from these predicted HUs. This can be due to the power law model, having an advantage of a better relationship between permeability and porosity which reflects in low error in permeability.

Fig. 11. Comparison of core permeability and permeability estimated using KC, power law and modified KC correlation for Well 3.

6. Conclusion In this work, a novel technique of SVM is presented and successfully applied to predict HUs from well log data for the un-cored sections of the wells. Comparison of the performance efficiency in predicting HUs was done between KC, power law and modified KC correlation. Results show that during HU prediction using SVM, the

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parameters are not generally available. This study proves that characterization of heterogeneity using readily available geophysical well logs is a better option if researchers use either power law or modified KC as the basic model for identifying HUs and support vector machine for predicting them.

Acknowledgments The author(s) would like to acknowledge the support provided by King Abdulaziz City for Science and Technology (KACST) through the Science & Technology Unit at King Fahd University of Petroleum and Minerals (KFUPM) for funding this work through project No. 11-OIL 2144-04 as part of the National Science, Technology and Innovation Plan.

References

Fig. 12. Comparison of core permeability and permeability estimated using KC, power law and modified KC correlation for Well 4.

Fig. 13. Comparison of core permeability and permeability estimated using KC, power law and modified KC correlation for Well 5.

modified KC model outperformed over the power law model for all the five wells. The modified KC model also outperformed for three wells out of a total of five wells when KC and modified KC correlation were used to discretize HUs. Interestingly a better estimate of permeability from the predicted HUs was obtained for most of the wells using the power law model. Although, the power law model performs better in terms of permeability estimation, the MKC model performs reasonably good compared to the KC model. The modified KC model is easy to handle however the power law model requires rigorous tuning of the model parameters (α, β) for the specific core data and experimental values of these model

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