Multivariate statistical log log-facies classification on a shallow marine reservoir

Journal of Petroleum Science and Engineering 61 (2008) 88–93

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Journal of Petroleum Science and Engineering j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / p e t r o l

Research paper

Multivariate statistical log log-facies classification on a shallow marine reservoir Hong Tang a,⁎, Christopher D. White b a b

Chevron Energy Technology Company, 14141 Southwest Freeway, Sugar Land, TX 77584, United States Craft & Hawkins Department of Petroleum Engineering, Louisiana State University, Baton Rouge, LA 70803, United States

a r t i c l e

i n f o

Article history: Received 8 April 2007 Accepted 19 May 2008 Keywords: beta distributions Bayesian multinomial logistic regression discriminant analysis facies classification analysis of variance

a b s t r a c t Sedimentary facies are important in reservoir characterization because flow properties are commonly assigned using facies-specific correlations. Multivariate statistical methods provide a powerful vehicle to extract facies responses from different well logs, to predict facies in uncored wells and evaluate uncertainty. Previous study shows a difficulty in bin selection to accurately reproduce the samples' conditional probability distribution. In this paper, a new method uses empirical beta distributions to model the distribution of petrophysical properties conditional to facies. Petrophysical property distributions are assumed conditionally independent, simplifying the use of Bayes rule. Three multivariate statistical methods (beta-Bayesian, multinomial logistic regression, and discriminant analysis) are examined in this paper using log and facies data from a western African sandstone reservoir. Three derived probability logs compare the prediction performance of the statistical methods as well as illustrate influences of log combinations and sample size. Two-way analysis of variance compares prediction accuracy of the models. For a given dataset, there are no significant differences (with 90% confidence) in predictions by the three methods. Additional logs improve prediction accuracy from 30 to above 80%. Final prediction accuracy is 82 to 90% for these three algorithms. Including 20 to 25% of the complete core and facies data in model construction provides accurate predictions; models were validated against the data not used in model construction. The fitted classification models can generate three-dimensional log-derived facies distributions for geologic modeling and reservoir simulation. The three methods are straightforward, efficient, and have quantifiable prediction errors. © 2008 Elsevier B.V. All rights reserved.

1. Introduction Sedimentary facies classifications can support reservoir characterization because flow properties are commonly assigned using facies-specific correlations. In uncored wells, sedimentary facies cannot be observed directly, and facies are inferred from petrophysical data. Some well log data, which are sensitive to lithology such as gamma spectrum, spontaneous potential, density and neutron logs, are good indicators of facies. Traditionally, histograms and cross plots are used to visualize and quantify the cutoffs for wireline logs based on the log responses on different lithofacies. They are straightforward and then are widely used. However, the determination of cutoffs is subjective. It becomes more difficult when more logs are involved. With increase of computer power, many algorithms have been applied in automatic well log log-facies classification. Two major directions are artificial intelligence methods and multivariate statistic methods. Typical artificial intelligence methods are artificial neural network and fuzzy logic. Multivariate statistics method includes discriminant analysis, cluster analysis, regression analysis and Bayese analysis and other multivariate methods.

⁎ Corresponding author. E-mail addresses: [email protected] (H. Tang), [email protected] (C.D. White). 0920-4105/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.petrol.2008.05.004

Numerous successful studies of application of artificial neural networks in facies classification have been reported. Baldwin et al. (1990) used self-organizing maps. Rogers et al. (1992), and Karpur et al. (2000) introduced back-propagation feedforward neural networks for the estimation of lithologies from logs. Tang and Ji (2006) used probability based neural network in volcanic facies classification. Saggaf and Nebrija (2000) used a neural network that performs vector quantization of input data by competitive learning. Qi and Carr (2006), reported a successful application of a single hidden layer neural network in examine carbonate lithofacies. Although neural network method is flexible and robust, their convergence is often unreliable and slow; training the network requires considerable parameter tweaking, which is strongly based on interpreters' experience (Saggaf and Nebrija, 2000). On the other hand, multivariate statistics methods are flexible and provide clear mathematical expression of log-facies relationship. Anderberg (1973) and Rao (1973) provide a pioneer extensive analysis of multidimensional statistical analysis methods. Wolff and Pelissier-Combescure (1982) used principal component analysis to cluster the log values into separate facies that can be regarded as indicator of lithology. Kapur et al. (2000) applied a bin-dependent Bayes method in facies classification. It's successful when sample data are abundant. However, unfortunately, the samples (or core) are usually limited because of the expensive cost and long coring time.

H. Tang, C.D. White / Journal of Petroleum Science and Engineering 61 (2008) 88–93

In this paper, a new method is discussed using beta function and Bayes rule to overcome the limitation of sample size. It also provides a vehicle to integrate geological experience, termed as geological prior probabilities. Furthermore a regression based classification method– multivariate logistic regression is introduced. This method is used in production risk mitigation, but rarely applied in facies classification. These two methods are compared with a discriminant analysis method. We first briefly discuss the problem. Three methods are proposed. Classification procedure is discussed. It follows by an application on a West African field. Several practical issues are discussed later. 2. Statement of problem

for specific facies to the probability of log reading for all facies, which is called likelihood. The left side of this Eq. (1) P(F|X) is the posterior probability, or probability of a facies for given log data. The posterior probability and likelihood guide facies classification. If the log reading is independent from facies, or uninformative for facies prediction, then P(X|F) = P(X). Eq. (1) changes to P(F|X) = P(F), which means posterior equals to prior and is not informative. Otherwise, if P(X|F) is larger (or smaller) than P(X), then P(F|X) should be larger or smaller than the P(F). Assuming multiple logs' probability distribution are independent, Eq. (1) changes to following equation: P ðFjX; Y; Z Þ ¼

Multivariate facies classification requires data to train the model. There are properties of typical samples: the cores are always sampled in a biased way; the reservoir facies usually have more samples than the nonreservoir facies. There is always a limitation of sample sizes, because the subsurface coring is usually expensive and time consuming. Furthermore, traditional bin-dependent classification methods may introduce errors in selection of bin size. “If too few bins are selected, the FOP (Facies Occurrence Probability, here referred to as the conditional probability) lacks the ability to discriminate between the adjacent log readings; if there are too many bins, the FOP will not be estimated precisely” (Kapur et al., 2000). The key objectives of this study are to answer the practical questions: how much samples are enough? How to use geological information to overcome the bias of limited core data and difficulties of bin selection?

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P ð X; Y; ZjF Þ P ðF Þ~P ð XjF ÞP ðYjF ÞP ðZjF ÞP ðF Þ P ð X; Y; Z Þ

ð2Þ

in which P(F|X,Y,Z) is the probability of facies F, given readings of log X, Y, and Z. P(X,Y,Z|F) is the conditional probability of log readings range with in X, Y, and Z, if in facies F. From Eq. (2), including new logs may or may not affect the posterior P(F|X,Y,Z). The posterior will differ from the prior only if the conditional probabilities of log responses given facies differ from the unconditional distributions. The beta function (Fig. 1) is chosen because it is simple to understand and flexible to manipulate. The shape of the beta function is controlled by only two factors: shape factor α and position factor β. For example, when α and β are equal, the PDF curves are symmetric. When α and β are both equal to one, the beta function becomes a uniform function. In this paper, the beta function is used to combine Bayes theorem to predict lithofacies.

3. Method introduction 3.2. Multinomial logistic regression Three multivariate statistical methods (beta-Bayesian, multinomial logistic regression, and discriminant analysis) are examined in this study. The techniques are illustrated using log and facies data from a western African sandstone reservoir. 3.1. Beta-Bayesian method (BBM) A Bayesian method provides an approach to combine the prior information on fraction of facies from core observation with the wireline log data. In our case, Bayes formula is: P ðFjX Þ ¼

P ð XjF Þ P ðF Þ: PðX Þ

ð1Þ

Eq. (1) is used to classify facies F from given log data X. X is any log data such as gamma ray, neutron, or density. P(F) is the proportion (or probability) of different facies in training set, which is also called the Þ prior or from analog data. PPððXjF X Þ is the ratio of probability a log reading

Multinomial logistic regression (MLR) assumes all samples are from one of n populations (in our study, n facies). MLR uses the logit transform of category probability as response to regress regression; the logit is the log of probability ratios (Eq. (3)). This method then estimates odds ratio of one outcome to a reference outcome (here, facies 8). MLR estimates n − 1 probabilities; the nth probability is determined from ∑ni¼1 pi ¼ 1. Using a linear model,

 n−1 P P Logit i ¼ ln i ¼ β0 þ ∑ βi Xi þ e ð3Þ Pn Pn i ¼1 in which Pi is the probability of occurrence of facies i, given log data X1,…Xi. 0,… i are the coefficients of regression and e: is the error between model and prediction (SAS Institute, 1989). A maximum likelihood method computes the regression coefficients, β . The regression coefficients quantify the effects of unit changes in independent variables (i.e., the log data) on probability of each facies. These effects on category

Fig. 1. Fitting conditional probability with beta function. (a) Beta function to fit conditional probability; b, c) the cumulative probability of gamma ray and neutron log are fitted by beta function using nonlinear solver and probability density function are inferred and used in BBM.

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Table 1 Facies classification and petrophysical properties Facies number

Facies name

P(F)

Average porosity (%)

Average permeability (md)

1 2 3 5 7 8 9 10

Turbidite Debris Lagoon/marsh Abandoned channel Sand flat Shallow marine Shoreface Lower shoreface

0.14 0.01 0.07 0.13 0.01 0.22 0.21 0.21

31.61 17.86 23.48 27.53 22.81 30.94 26.78 21.12

2334.68 338.55 1098.51 5381.35 1796.44 3419.11 908.89 782.77

P ðFjX Þ ¼

After Karpur et al. (2000).

probability are, by the nature of logistic function, nonlinear. The estimated category probabilities can be computed from a matrix equation: 2 6 6 6 6 4

1 1 N 1

N

1

1

1

32 3 2 P1 −C1 6 7 6 −C2 7 76 P2 7 6 6 7 6 N 7 76 N 7 ¼ 6 1 −Cn−1 54 Pn−1 5 4 1 1 Pn

3

0 0 N 0 1

7 7 7 7 5

ð4Þ

n−1

β0 þ∑ βi Xi

in which Ci ¼ e Pi ¼

Ci n

1þ ∑

Ck

i¼1

one of n populations (D = (D1,… Dn)). Probability density functions (PDF's) for groups are expressed f1(x),…fn(x). The groups are assumed to be normally distributed; however, DA is robust with respect to the normality assumption. For linear discriminant functions, a common variance is used for all facies. Fortunately, DA is also robust to this assumption (so long as variances and covariances of all variables are within a factor of four of one another). Normalized data (range from 0–1) satisfy this requirement. The DA classification boundaries between groups are linear in form. According to Bayes theorem and maximum likelihood theory (Hand, 1981):

ði ¼ 1; N n ¼ 1ÞPi ¼

k¼1

n

Ck

ð5Þ

1þ ∑ k¼1

The summary of probability at each depth equals to 1. At each depth, the rock is assigned to the facies with the maximum probability. A confidence measurement can evaluate the selection. 3.3. Discriminant analysis Discriminant analysis (DA) is a multivariate method to group samples into categories. Like MLR, DA assumes that all samples are from

ð6Þ

That is, observations should be assigned to the facies with maximum likelihood.    l ¼ 1; N k ð7Þ D4l ¼ X : Pl fl ð X Þ ¼ max Pj fj ðxÞ The maximum likelihood method (Eq. (7)) is same as minimizing the pairwise generalized squared distances within groups, which can be expressed by following Eq. (8), from observation to the allocated group. A datum x is allocated to facies j,  V   D2 ðXÞ ¼ X−X C −1 X−X þ lnjTrðC Þj−2lnP

. Probability for each facies is then 1

fj ðxÞPj P ð XjF Þ P ðF Þ ¼ ~Pj fj ðxÞ PðXÞ Pð X Þ

ð8Þ

4. Data introduction To compare the effects of the three types of classification, these methods are applied to Kapur et al. (2000) published data from a West Africa sandstone reservoir. Wireline log data, core petrophysics, and core-based facies classifications are available. There are 8 sedimentary facies: turbidite, debris, lagoon/marsh, abandoned channel, sand flat, shallow marine, shoreface, and lower shoreface. Core classified facies and related petrophysical parameters are illustrated in Table 1. The turbidite facies (facies 1) and shallow marine facies (facies 8) are major reservoir facies with high average porosity and permeability. Among

Fig. 2. The confidence measurements of beta-Bayesian method using different log combination (a–c) illustrate the confidence measurements of BBM within 5850–5950 (m) for 3 kinds of log combinations between Gamma, Neutron and Density. (a) Comparison of facies prediction using GR only with observed facies classification; (b) Comparison of facies prediction using gamma and neutron logs; (c) Comparison of facies prediction using all three logs. Decrease of the enclosed area indicates that prediction accuracy increases from left to right.

H. Tang, C.D. White / Journal of Petroleum Science and Engineering 61 (2008) 88–93 Table 2 Accuracy comparison of all methods with different log combinations Log combination

Gamma ray Neutron Density Gamma ray/neutron Gamma ray/density Neutron/density All three logs

% Accuracy of prediction for different methods Beta-Bayesian method

Multinomial logistic reg.

Discriminate analysis

56.50 44.87 72.10 71.00 78.50 79.50 80.63

52.63 39.32 73.02 77.41 78.88 78.14 83.03

55.25 24.63 73.13 62.88 73.38 71.50 77.50

For all methods, additional logs increase prediction accuracy.

the total 800 samples, 22% are shallow marine facies and 14% are turbidite facies. For accurate geological reserve calculation, the facies classification methods need to minimize the prediction error among these facies. Facies 2 and 7 are least common facies with prior facies proportion 1% each which might pose difficulty for facies prediction. 5. Classification procedure Prior distributions must be obtained; these give the expected fractions facies in the section to be analyzed. Prior distribution is computed from the fraction of the subsampled training sets in this research. Alternatively, the geological empirical facies proportion can approximate the prior distribution to correct for bias. The second step in the beta-Bayesian classification is to fit conditional probability models for wireline responses given facies to beta distributions, for each lithology and log. The conditional probability distributions P(X|F) may have different shapes, skewness, mean and variance which are not easily fit by normal or lognormal distributions. The beta distribution (Fig. 1) can fit these conditional distributions. The conditional cumulative distribution functions (ccdf) are computed using a randomly selected subsample. The random sampling procedure assumes the subsample sets include all 8 facies. If any subsample misses any facies, such sample is not qualify for training set and is abandoned Nonlinear regression is used to fit ccdf's for each facies using the least squares criterion. Petrophysical property distributions are assumed conditionally independent to simplify the use of Bayes rule. The conditional probabilities are used with a simplified form of Bayes rule that assumes that the petrophysical property distributions are conditionally independent. This conditional independence can be expressed mathematically by stipulating that P(xi|fj) does not depend on xk, k ≠ i. Finally, the priors, conditional probabilities, and log measurements are combined using Eq. (2). The measurement point is classified as belonging to the facies with the highest posterior probability.

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The beta-Bayesian method is checked by predicting data not in the training data set. Classification errors can then be tabulated. Other prediction methods (MLR and DA) are checked in the same way, using a subsample for model computations and the full sample for checking. Multinomial logistic regression method and discriminant analysis use the CATMOD and DISCRIM procedures of SAS software (SAS Institute Inc., 1989). Using the same training set, MLR uses maximum likelihood method to predict the regression coefficient. A discriminant function was developed from the training set and used to classify all 900 observations. 6. Prediction uncertainty measurements One advantage of statistic classification methods is the ability of quantifying the prediction uncertainty (or confidence measurement); these methods estimate probability logs for all facies. The uncertainty in these predictions can be compared quantitatively using two derived logs to evaluate prediction uncertainty. The probability of the most probable facies is called overall confidence. The higher overall probability represents the higher confidence of such classification, and vise versa. The distinguishing ability is defined as the ability to differentiate one facies from other facies, which is estimated by the highest probability minus the next highest probability. Confidence measurements compare the prediction uncertainty of one method. The effects of log combinations on prediction ability are examined by the confidence measurements and overall prediction accuracy. Different logs combinations may also affect facies classification ability and prediction accuracy differently. This study compared different combination of wire line logs using randomly selected subsamples (20% of total sample) to construct conditional probability distribution function (cpdf). The prediction accuracy increases with more logs included (Fig. 2). The enclosed area (surrounded by overall confidence and distinguishing ability curves) increases as uncertainty increases. Including more logs, which are sensitive to lithology, into training set, prediction accuracy will increase for all models. 7. Model prediction comparison One limitation of statistical method is that they require a large amount of statistical data. The performance of estimation is constrained by number of samples and well logs. To compare the effects of logs combinations, seven combinations of all 3 logs are tested in the beta-Bayesian method, MLR and DA and 350 randomly selected samples are used to establish models (sample size will be discussed later). Many random samples were drawn to construct estimation errors for each method (Table 2). Different models have different prediction abilities; adding more

Fig. 3. Influence of sample size and models on prediction accuracy.

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Table 3 Tukey pairwise comparison of sample size's influence on prediction accuracy Tukey

B B

Table 4 Tukey pair wise comparison of model influence on prediction accuracy

Grouping

Mean

N

Size

A A A A A A

0.834 0.829 0.828 0.827 0.823 0.821 0.799 0.716

18 18 18 18 18 18 18 18

800 700 600 500 400 300 200 100

C

Note: Groups with same letter and shade represent one group. Inside a group there is no significant difference at 95% confidence level.

logs into the model improves prediction accuracy. MLR with 3 logs has the best prediction accuracy (83%) for this test case. The BBM prediction accuracy (80.6%) is as good as MLR, which is proved by ANOVA analysis. Another question to answer is how much sample will be enough? As we expected, increased sample size increases the prediction accuracy. The prediction accuracy increases rapidly with sample size and is significantly higher when sample size increases to 200 (about 25% of total dataset; Fig. 3, Table 3). This result is a useful guidance, which infers that one quarter of sample size is possibly adequate for accurate facies classification in this shallow marine reservoir. This number seems pessimistic and impractical considering the expensive core expense and probable long pay zone. However, in real study, core perforation intervals could have more geological and engineering guidance such as facies sedimentary stacking pattern from sequence stratigraphy study, drill cutting records and formation petrophysical properties from well test analysis. Furthermore, perforation should focus on highly uncertainty intervals. Combined Combining these factors, it's practical to narrow the sample size to a practical quantity. Moreover, a Tukey pairwise comparison (Fig. 4, Table 4) shows that there are no significant differences among these methods.

Tukey

Grouping

Mean

N

Method

A A A

0.824 0.815 0.81

48 48 48

MLR DA BBM

The average prediction accuracies for three methods are within same group A. This means all three methods have similar good prediction ability (with 95% of confidence).

With the beta-Bayesian method, overall confidence and distinguishing ability are quite high, near 100% at cored sections. For intervals with less core information, our overall confidence and interval confidence can be either small or high, which depends on the different classifying model and log response. Otherwise, for those intervals with similar log responses, distinguishing ability can fall to zero — the most likely and best alternative classification are nearly equiprobable. Three statistical accuracies are defined to quantify the performance of BBM. Prediction accuracy, geological accuracy, and overall accuracy are calculated with the following equation, APrediction ¼

eii ei:

ð9Þ

AGeological ¼

eii e:j

ð10Þ

Atotal ¼

eii ei. e.j e..

eii e::

ð11Þ

represent number in i row and i column; total marginal number of each classified facies; total marginal number of each geological prior facies; grand total number.

Fig. 4. Comparison of prediction accuracy for different methods. A random sample of 25% of the total sample data is used as training set for all methods. Overall confidence and distinguishing ability curves are used to evaluate the prediction accuracy. The enclosed area (surrounded by overall confidence and distinguishing ability curves) indicates the uncertainty of prediction. Tukey pairwise test indicates no significant difference among three methods.

H. Tang, C.D. White / Journal of Petroleum Science and Engineering 61 (2008) 88–93 Table Table 55 Error matrix for BBM using 800 samples

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The classification models can generate three-dimensional log-derived facies distributions for geologic modeling and reservoir simulation. Acknowledgement The authors would like to thank Dr. Kurnianwan Bobby for providing support on manuscript proof reading and log plotting. We also thank Craft & Hawkins Department of Petroleum Engineering at Louisiana State University for providing financial support for this study. References

Note: Facies 1 and 8, the main reservoir component, has 97 and 98% geological and prediction accuracy. Overall accuracy is 81%.

About 25% facies samples are used to build training set. 800 facies classification and three logs from well bore are used to validate the facies. It's noteworthy that the prediction accuracy is high, 100 and 94% respectively, for facies 1 and 8 the two most important facies from the reserve calculation perspective (Table 5). Prediction errors of facies 2, 3 and 7 could be because of low number of training data and/or similar log responses. Two methods can be used to minimize the facies prediction errors. First, adjusting facies prior P(F) can decrease the sampling bias. The estimation of prior probability can be obtained from data observations or local geologic knowledge. Establishing reliable conditional probability model using direct or indirect relationships between facies and log data can minimize the prediction error. The beta-Bayesian method which uses beta distributions (Weisstein, 1999) to model the conditional probabilities P(X|F) can be used to minimize the error of bin selection. The term X is the log values such as gamma ray, density and neutron. In addition, beta distribution parameters can be adjusted by a nonlinear regression method (Person, 1997) to maximize prediction accuracy. 8. Conclusion In this study, the beta-Bayesian method uses empirical beta distributions to model the distribution of petrophysical properties conditional to facies, eliminating difficulties in bin selection. Petrophysical property distributions are assumed conditionally independent to simplify the use of Bayes rule. Confidence, discrimination ability, and probability logs compare the prediction performance of the statistical methods as well as illustrating influences of log combinations and sample size. Two-way analysis of variance compares prediction accuracy of the models. For a given dataset, there are no significant differences (with 90% confidence) in predictions accuracy among the three methods. Additional logs improve prediction accuracy from 30 to above 80%. Final prediction accuracy is 82 to 90% for these three algorithms. Including about one quarter of the complete core and facies data in model construction provides possible adequate predictions; models were validated against the data subsets which are not used in model construction.

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Glossary Cy (.): the covariance function D2(.): generalized squared distances from cluster center eii: represent number in i row and i column ei.: total marginal number of each classified facies e.j: total marginal number of each geological prior facies e..: grand total number Σ: covariance matrix Greek symbols εr : the relative errors μ^ : the estimated mean δ^ : the estimated variance β. .: regression coefficients