1. Introduction

Recently, unconventional reservoirs have gained popularity in oil and gas production. The criteria for unconventional hydrocarbon reservoirs are identical to their lithological, reservoir, and production characteristics in terms of (Aguilera and Harding, 2008):

• Low permeability of untypical reservoir lithology;
• Over pressure or extensive over pressure;
• Gas saturation and/or light oil condensate;
• Absence of water in the production process.

Tight gas reservoir is one of these resources. Tight gas is the term commonly used to refer to low permeability reservoirs, which produce mainly dry natural gas. Tight gas reservoir is often defined as a gas bearing sandstone or carbonate matrix, which exhibits in situ permeability to gas less than 0.10 millidarcys. Many “ultra tight” gas reservoirs may have in situ permeability down to 0.0010 millidarcys (Naik, 2003).

The large volume and long-term potential, attractive gas prices, and unprecedented interest in the world’s markets, bring the tight gas reservoirs in the forefront of energy in the future (Arevalo-Villagran et al., 2002).

It is intransitive to obtain petrophysical parameters such as permeability to model these reservoirs. The reservoir description of these low permeability tight gas reservoirs is credible, when laboratory permeability data are reliable. Permeability is a principal parameter to control the fluid flow. The permeability measurement at a high confining pressure in a laboratory is a common way to estimate the permeability structure at depths (Tadayoni and Valadkhani). A method for permeability measurement is a steady state method using gas as pore fluid. In this procedure, the pressure gradient is constant and is proportional to the fluid velocity.

where, k is the absolute core permeability and  is the viscosity. For the cores, which have permeabilities less than 0.1 md, a steady state flow is not valid, when the flowing fluid is liquid. Hence gas is used for determining the permeability in these low permeability cores. However, in experience, the value of permeability determined with this method is higher than the real value of permeability. Klinkenberg in the 1940’s presented the explanation of this treatment. He introduced the phenomenon of gas slippage as the reason (Tadayoni and Valadkhani). Klinkenberg found that the permeability of gas is a function of the mean core pressure and the type of gas.

where, k_a, k, b, and p represent the apparent permeability, liquid permeability, slippage factor, and pore pressure respectively. Gas flow in tight gas sands is mostly affected by some phenomena, which make deviations from Darcy’s law. Gas slippage is a non-Darcy effect associated with a non-laminar gas flow in porous media. These effects happen when the size of the medium rock pore throat radius approaches the size of the mean free path of the gas molecules, and thereby causing the velocity of single gas molecules to accelerate or “slip”, when contacting rock surfaces.

Coring and core analysis are very expensive and time consuming and there are few numbers of drilled cores in each field. Hence measuring Klinkenberg permeability in laboratory will be limited and cannot be applied to the whole sections of the field. On the other hand, well log measurements are usually available for all drilled wells and provide continuous information about reservoir rock and fluid properties across the well. Therefore, finding methods which can create proper relations between well log measurements and core-based Klinkenberg permeability and have good generalization capability in predicting unseen data can be a good way to predict Klinkenberg permeability.

One method to predict Klinkenberg effect is using regression methods. Multiple linear regression (MLR) is one of the primary regression methods, which exhibits acceptable results in predicting target variable from several predictive variables. Moreover, single tree and tree boost techniques are powerful tools for regression tasks. Tree boost models often can provide more accurate results compared to single tree models. Over the past few years, this technique has emerged as one of the most powerful methods for predictive data mining.

Artificial intelligence methods like artificial neural network (ANN) and support vector machine (SVM) have shown superior capability in classification and regression tasks. Usage of these methods reduces obstacles associated with costs and the generalization of the developed models for the prediction of permeability (Baziar et al., 2014; Al Anazi et al., 2009; Al Bulushi et al., 2007; Amari and Wu, 1999; Aminian and Ameri 2005; Aminian et al., 2001; Anifowose and Abdulraheem, 2010; Anifowose et al., 2011; Asadisaghandi and Tahmasebi, 2011; Bhatt, 2002; Carrasquilla et al., 2008; Chang et al., 1997; Goda et al., 2007; Huang et al., 1996; Ibrahim and Potter, 2004; Karimpouli et al.,2010; Mollajan and Memarian, 2013; Saffarzadeh and Shadizadeh, 2012; Shokir 2004; Sun et al., 2001; Wiener et al., 1995; Wong et al., 1995; Wong et al., 2000). Furthermore, Anifowose et al., in a comprehensive study, employed adaptive neuro-fuzzy inference system hybrid models to predict reservoir properties including Klinkenberg permeability.

An artificial neural network (ANN) is a hugely parallel distributed processor including neurons capable of doing mathematical computations by using aviation functions via a learning algorithm. The knowledge is encoded in the interconnection weights between input, hidden, and output layers (Haykin, 1994).

Recently, support vector machines (SVM’s) have gained popularity in regression and classification because of their privileged generalization performance (Kecman, 2005; Cristianini and Shawe-Taylor, 2000; Smola and Schölkopf, 2004). The formulation of SVM is based on the structural risk minimization (SRM) inductive principle, where the empirical risk minimization (ERM) inductive principle and the Vapnik-Chervonenkis (VC) confidence interval are simultaneously minimized (Vapnik, 1982; Vapnik, 2000; Vapnik and Chervonenkis, 1974).

In this study, four methods including multiple linear regression, tree boost, general regression neural network (GRNN), and support vector machine have been used to predict Klinkenberg permeability in Mesaverde tight gas reservoir located in Washakie basin, USA and the results obtained from all the methods have been compared.

1.1. Multiple linear regressions

A regression with two or more explanatory variables is called a multiple linear regression (MLR). Rather than modeling the mean response as a straight line, as in a simple regression, modeling is conducted as a function of several explanatory variables, in MLR. Multiple linear regression can be thought of as an extension of simple linear regression, where there are p explanatory variables, or simple linear regression can be thought of as a special case of multiple linear regression, where p=1. The term “linear” is used, because in multiple linear regression, it can be assumed that the output is directly related to a linear combination of the explanatory variables.

1.2. Tree boost

“Boosting” is a technique for improving the accuracy of a predictive function by applying the function repeatedly in a series and combining the output of each function with weighting so that the total error of the prediction is minimized. In many cases, the predictive accuracy of such a series greatly exceeds the accuracy of the base function used alone. The tree boost algorithm is functionally similar to decision tree forests, because it creates a tree ensemble; however, a tree boost model consists of a series of trees, whereas a decision tree forest consists of a collection of trees grown in parallel. Tree boost is also known as “stochastic gradient boosting” and “multiple additive regression trees” (MART).

The tree boost algorithm used herein was developed by Friedman (Friedman, 1999) and is optimized for improving the accuracy of models built on decision trees. Graphically, a tree boost model can be represented as shown in Figure 1:

Figure 1

A schematic of tree boost model.

The first tree is fitted to the data. The residuals (error values) from the first tree are then fed into the second tree, which attempts to reduce the error. This process is repeated through a series of successive trees. The final predicted value is formed by adding the weighted contribution of each tree.

Usually, the individual trees are fairly small (typically 3 levels deep with 8 terminal nodes), but the full tree boost additive series may consist of hundreds of these small trees. Tree boost models often have a degree of accuracy which cannot be obtained using a large, single-tree model. Tree boost models can handle hundreds or thousands of potential predictor variables. Irrelevant predictor variables are identified automatically and do not affect the predictive model. Tree boost uses the Huber M-regression loss function (Huber, 1964), which makes it highly resistant to outliers and misclassified cases. The randomization element in the tree boost algorithm makes it highly resistant to over fitting. Tree boost can be applied to regression models and k-class classification problems.

The primary disadvantage of tree boost is that the model is complex and cannot be visualized like a single tree. It is more of a “black box” like a neural network.

1.3. General regression neural network

General regression neural network (GRNN), as proposed by Specht (Specht, 1991), falls into the category of probabilistic neural networks. This neural network like other probabilistic neural networks needs only a fraction of the training samples which a back-propagation neural network would need (Specht, 1991). The data available from the measurements of an operating system is generally never enough for a back-propagation neural network (Specht, 1990). Therefore, the use of a GRNN is especially advantageous due to its ability to converge to the underlying function of the data with only few training samples available. The additional knowledge needed to get the fit in a satisfying way is relatively small and can be done without additional input by the user. This makes GRNN a very useful tool to perform predictions and comparisons of system performance in practice. Figure 2 depicts the diagram of a GRNN. There is one neuron in the input layer for each predictor variable; the hidden layer has one neuron for each case in the training data set and the neuron stores the values of the predictor variables for the case along with the target value. In the next layer, there are two neurons; one neuron is the denominator summation unit and the other is the numerator summation unit. The decision layer divides the value accumulated in the numerator summation unit by the value in the denominator summation unit and uses the result as the predicted target value.

Figure 2

Structure of GRNN.

1.4. Support vector machine

Support vector machines (SVM’s) are learning machines implementing the structural risk minimization inductive principle to obtain good generalization on a limited number of learning patterns (Vapnik, 2000; Al-Anazi and Gates, 2010b). Support vector machines are characterized by the usage of kernels, the absence of local minima, the sparseness of the solution, and the capacity control obtained by acting on the margin or on the number of support vectors. SVM uses kernel functions, which enable them to operate in a high dimensional, implicit feature space without ever computing the coordinates of the data in that space by simply computing the inner products between the images of all the pairs of the data in the feature space. A version of a SVM for regression has been proposed (Vapnik et al., 1997), which is called support vector regression (SVR).

In the parlance of SVM literature, a predictor variable is called an attribute and a transformed attribute, which is used to define the hyperplane, is called a feature. The task of choosing the most suitable representation is known as feature selection. A set of features describing one case (i.e. a row of predictor values) is called a vector. Thus the goal of SVM modeling is to find the optimal hyperplane, which separates clusters of vector in such a way that the cases with one category of the target variable are on the one side of the plane and the cases with the other category are on the other size of the plane. The vectors near the hyperplane are the support vectors. SVM method has been used in many research field and exhibited reliable performance.

A common disadvantage of non-parametric techniques such as SVM’s is the lack of transparency of the results. The lack of transparency in these methods is inevitable and is due to the nature of these techniques. There is no way to clear the procedure of SVM in obtaining the best regression function.

Support vector machine has two common moduli, namely support vector classifier (SVC) and support vector regression (SVR). In this study, SVR has been employed to predict Klinkenberg permeability.

2. Geological background

The data set for this study are obtained from Measverde tight gas sandstones located in Unita basin in the U.S.A. Mesaverde Group sandstones represent the principal gas productive sandstone unit in the largest Western U.S. tight gas sandstone basins including Washakie, Uinta, Piceance, northern Greater Green River, Wind River, and Powder River. Figure 3represents the location map of Washakie basin.

Figure 3

Location map of Washakie basin.

The Mesaverde group is divided into the regressive deposits of Iles formation and the overlying massively stacked, lenticular non-marine Williams Fork formation. The Iles formation comprises the lower part of the Mesaverde. It contains three marine sandstone intervals, namely the Corcoran, Cozzette, and Rollins. The Williams Fork formation extends from the top of the Rollins to the top of the Mesaverde. The lower part of the Williams Fork contains coals and is commonly referred to as the Cameo coal interval. Most of the sandstones in the Williams Fork are discontinuous fluvial sands. The stratigraphy of the Mesaverde group is shown in Figure 4.

3. Experimental method

To have a robust and accurate model, log information from two wells, A₁ and A₂ were used. Well A₁ has a total of 120 data points and well A₂ has 180 data points. One of characteristics of this research is the small number of training data points, which can be a proper index to evaluate the generalization capability of the methods in the presence of small training dataset. To evaluate the accuracy of the models, the wells were subdivided as shown in Table 1.

Figure 4

Cross section showing the stratigraphy of the Mesaverde group (Cumella and Scheevel, 2008).

Table 1

Different datasets arranged to evaluate the accuracy of models.

Each training pattern consists of log data, including a gamma ray log (GR), a bulk density log (RHOB), a neutron porosity log (NPHI), a photoelectric factor log (PEF), a resistivity log (RT), and a sonic travel-time log (DT) as the input vectors, and core-based Klinkenberg permeability as a scalar output. DTREG software was used to evaluate the effects of various logs in predicting Klinkenberg permeability and the mentioned logs were qualified as the most effective well logs. It was notable that Rmsfl and Rlld logs were recognized to have minor effects on the prediction of Klinkenberg permeability, and the six mentioned logs were finally qualified as the input data. The relationship between Klinkenberg permeability and each log value has been demonstrated in scatter plots in Figures 5-10.

The accuracy of data was verified before employing them in the models. All the log data were obtained by using modern techniques and accurate instruments. Depth shifting between core and logs was correctly done and log readings at specified depths were exactly matched with the core permeability data.

Figure 5

Scatter plot of Klinkenberg permeability versus DT log.

Figure 6

Scatter plot of Klinkenberg permeability versus GR log.

Figure 7

Scatter plot of Klinkenberg permeability versus NPHI log.

Figure 8

Scatter plot of Klinkenberg permeability versus PEF log.

Figure 9

Scatter plot of Klinkenberg permeability versus RHOB log.

Figure 10

Scatter plot of Klinkenberg permeability versus RT log.

As it can be understood from the above figures, generally Klinkenberg permeability increases as DT log value rises. GR, RT, and NPHI logs relatively represent an inverse relationship with Klinkenberg permeability. PEF log has an irregular relation with Klinkenberg permeability and RHOB log is directly proportional to Klinkenberg permeability.

Regression models (MLR, tree boost, GRNN, and SVM) were constructed. Multiple linear regression was performed by using Microsoft Excel data analysis tool package.

SQBlib open source code was used to build tree boost models (Becker et al., 2013). 600 trees were generated in tree boost series. Each tree in the tree boost series had 10 levels of splits. The tree boost algorithm uses Huber’s M-regression loss function to evaluate error measurements for the regression models (Huber, 1964). This loss function is a hybrid of ordinary least-squares (OLS) and least absolute deviation (LAD). For residuals less than a cutoff point, the squared error values are used; for residuals greater than the cutoff point, the absolute values are used. Huber’s cutoff point was chosen 0.1. A 10-fold cross-validation resampling technique was used to strike the right trade-off between over-fitting and under-fitting.

STATISTICA software was used to construct GRNN models (StatSoft, 1998); GRNN models constructed herein had 4 layers (one input, one hidden, one class, and one decision layer). An optimization algorithm was used to automatically determine the number of neurons in the hidden layer. This algorithm tries building multiple networks with different numbers of neurons in hidden layers and evaluates how well they fit by using cross validation. 12 neurons were selected for the hidden layer. The primary work of training a GRNN network is selecting the optimal sigma values to control the spread of the radial basis function (RBF). The sigma values control the radius of the influence of each point in the model.The conjugate gradient algorithm was used herein to compute the optimal sigma values. In addition, a 10-fold cross validation method was used for the validation.

The online SVR software (Parrella) was utilized to procreate the SVM model. For SVM, the sigmoid kernel function was used, because it has already proved its better performance in comparison with other kernel functions (Al-Anazi and Gates, 2010a; Al-Anazi and Gates, 2010b; Saffarzadeh and Shadizadeh, 2012; Baziar et al., 2014). A 10-fold cross validation method was also used for the validation. The accuracy of an SVM model depends on a true setting of the parameters C, ε, and the kernel parameters. The problem of optimal parameter selection is more complicated by the principle that the complexity of an SVM model depends on all the three parameters. While designing an SVM, the user is confronted with the choice of which kernel to use, and for a given kernel, how to adjust the parameter(s). Two methods were used for finding the optimal parameter values, namely a grid search and a pattern search. A grid search tries values of each parameter across the specified search range using geometric steps. A pattern search (also known as a “compass search” or a “line search”) starts at the center of the search range and makes trial steps in each direction for each parameter. If the fit of the model improves, the search center moves to the new point and the process is repeated; if no improvement is found, the step size is reduced and the search is tried again. The pattern search stops, when the search step size is reduced to a specified tolerance. When using both grid search and pattern search, the grid search is performed first. Once the grid search finishes, a pattern search is performed over a narrow search range surrounding the best point found by the grid search. Hopefully, the grid search will find a region near the global optimum point and the pattern search will then find the global optimum by starting in the right region.

To compare methods, each one was used to create a predicted value of the Klinkenberg permeability and the difference between each predicted and real value was assessed by the correlation coefficient (r), the average absolute error (AAE), and the standard error of the mean (SEM) as defined in Table 2.

Table 2

Error formulas for comparing methods.

4. Results and discussion

At first, about 200 data points from two wells were selected to primarily evaluate the capability of SVM in predicting Klinkenberg permeability. In this regard, logistic regression as a traditional approach was selected to be compared with SVM. 150 data points were selected as the training data and 50 data points were considered for the testing procedure. Table 3tabulates the error indexes measured in predicting Klinkenberg permeability by SVM and logistic regression. The scatter plots of the predictions made by each method are demonstrated in Figures 11 and 12.

Table 3

Comparison of the error measured between SVM and logistic regression.

Figure 11

Scatter plot of predicting Klinkenberg permeability by SVM.

Figure 12

Scatter plot of predicting Klinkenberg permeability by logistic regression.

The correlation coefficient of the predictions from the MLR, tree boost, GRNN, and SVM are depicted in Figure 13.

Figure 13

Comparison between the correlation coefficient of different methods in six datasets.

It can be understood from Figure 13that all the methods used have similar results and exhibit reliable performance. SVM overly performs better in predicting Klinkenberg rather than MLR, tree boost, and GRNN. It is also notable that GRNN has acceptable performance and predict Klinkenberg permeability better than MLR and tree boost techniques. Figures 14-17 displays the scatter plots of the predicted and core-based Klinkenberg permeability in dataset number 6.

Figure 14

Scatter plot of core-based and Klinkenberg permeability predicted by GRNN.

Figure 15

Scatter plot of core-based and Klinkenberg permeability predicted by MLR.

Figure 16

Scatter plot of core-based and Klinkenberg permeability predicted by SVM.

Figure 17

Scatter plot of core-based and Klinkenberg permeability predicted by tree boost.

Correlation coefficient is not a reliable index to evaluate the accuracy of predictions. In this regard, some other measures of error were employed. To compare the models, the average percentage difference in error between each technique and the actual data was computed through the analysis of AAE and SEM error measurements. Table 4shows the average errors (AAE and SEM) of MLR, tree boost, GRNN, and SVM models. As it can be seen, SVM and GRNN have similar performance in predicting Klinkenberg permeability and exhibit better results compared with the two other techniques. The results show that the SVM method has obtained the lowest average error values compared to MLR, tree boost, and GRNN techniques, except for the AAE error measured for GRNN in dataset 5 and SEM error measured for GRNN in datasets 1 and 6 (see Table 4). Furthermore, it is notable that simpler methods like MLR often present reliable results as advanced methods such as SVM and neural networks. The analysis of these error statistics reveals that SVM and GRNN are proper approaches to the prediction of Klinkenberg permeability.

Table 4

Measures of AAE and SEM obtained from the prediction of Klinkenberg permeability by MLR, tree boost, SVM, and GRNN.

5. Conclusions

In this study, support vector machine, general regression neural network, tree boost, and multiple linear regression methods were used to predict Klinkenberg permeability in Mesaverde tight gas sandstones located in Washakie basin, the USA. Moreover, the performance of these methods in regression tasks and predicting Klinkenberg permeability was compared. The following main conclusions can be drawn in this study:

• Multiple linear regression, tree boost, support vector machine, and general regression neural network are accurate methods to predict Klinkenberg permeability in the tight gas reservoirs;
• Support vector machine outperforms general regression neural network, tree boost, and multiple linear regression methods in predicting the Klinkenberg permeability of the tight gas reservoir;
• General regression neural network model exhibits better efficiency than tree boost and multiple linear regression methods in the prediction of Klinkenberg permeability;
• Support vector machine presents a great generalization capability in the presence of a small number of training data points;
• In comparison to the advanced methods such as SVM and neural networks, simple regression methods like MLR can also lead to reliable results.

Nomenclature