1 Introduction

The common method to determine the permeability is conducted by using conventional core analysis (CCA) (porosity–permeability relationship) to form a nonlinear relationship between porosity and permeability. This relationship can be used to predict the permeability as a function of porosity. In carbonate reservoirs, this relationship can be utilized to predict permeability in un-cored wells using basic log parameters. This does not take into account other important rock/reservoir properties such as grain size, sorting, tortuosity, and diagenesis. In addition, pore throats play a major role in identifying accurate permeability values; this is a major challenge in carbonate environment as there is no robust pore size identification of the pore system. The pore throat size has strong relation with the drilling fluid invasion which in turn affects the measured resistivity values.

The main complexity in predicting permeability is the wide variety of pore system in terms of geometry (intergranular, intragranular, intercrystalline, vuggy, and fracture) and pore sizes classes (macro-porosity, meso-porosity, and micro-porosity). For confident permeability characterization, coring campaign and physical core description should be planned in order to get the data from each well which is quite time-consuming and need expensive laboratory measurements.

Gunter et al. [1] described a technique that combines basic reservoir properties, i.e., bed thickness, porosity, and permeability information for flow unit’s calculations. They applied modified Lorenz plots (MLP) to characterize the reservoir. Modified Lorenz plots (MLP) is a crossplot between flow capacity on the Y-axis (from permeability) and storage capacity on X-axis (from porosity) to determine the flow units in the reservoir. This method of flow unit determination is quite useful because it only requires routine porosity and permeability data (from logs and/or core), is independent of facies information, and utilizes simple crossplotting techniques.

Morris and Biggs [2] developed an empirical correlation to predict permeability at initial water saturation. They defined the correlation of logs calculated porosity and resistivity-based saturation to estimate the permeability. Permeability can be estimated by a model that included the non-Archie behavior which accounts for resistivity measurements on double porosity DPC (dual porosity conductivity) or triple porosity TPC (triple porosity conductivity) micritic and oolitic carbonates [3]. Sequence-stratigraphic framework would be more systematically organized using rock-fabric classification instead of using the direct relationship of porosity and permeability [4].

Lacentre and Carrica [5] proposed a method to predict the permeability from well log responses and conventional core analysis. First, they classified the reservoir using mathematical tools from integration of available information such as petrophysics, lithofacies, electrofacies, and hydraulic flow units. Then, core permeability values are mapped and calibrated with well log data using neural networks. This shows better results than accepted methods. The disadvantage of this method is that it requires adequate number of data.

Many studies showed that estimation of permeability in carbonate formation is considered to be a challenging task due to changes in both depositional environment and diagenesis effects on porosity–permeability relationship. A statistical tool named classification tree analysis was proposed that classified data and separating permeability predictions from well logs based on three different approaches: electrofacies, lithofacies, and hydraulic flow units approach (HFUs) [6].

Clustering modeling training data values can be collected based on specified associated parameters. One of them is multi-resolution graph-based clustering (MRGC). It solves problems that usually occur when log data is relatively constrained with few clusters. This will merge large number of clusters into a small cluster that was assigned from the geological characterization. It also reduces several drawbacks that come from conventional methods [7].

2 Artificial intelligence techniques

2.1 Artificial neural network

The artificial neural network technique is inspired from biological neurons that are found in human brain [8]. ANN has the capability to approximate any nonlinear complex function between input and output parameters. ANN is the oldest and simplest artificial intelligent (AI) technique [9]. ANN models consist of fundamental processing unit, termed as neurons. The neural network models are structured on three components, learning algorithm, transfer function, and network architecture [10]. The artificial neural network model comprises of at least three layers, input layer, hidden layer, and output layer. Hidden layer can be multiple. Each layer connects with other layers with the help of weights. The network performance is solely based on the adjustment of weights between these layers. Hidden layers assigned with transfer function usually ‘log’ or ‘tan’ sigmoidal-type activation function. Output layer is assigned with ‘pure linear’ activation function. Network performance also depends upon the number of neurons in the hidden layer, fewer neuron cause under-fitting and excessive neurons cause over-fitting, so optimization is required for the designing of neurons [11].

2.2 Adaptive neuro-fuzzy inference system

Adaptive neuro-fuzzy inference system (ANFIS) is the combination of neural network and fuzzy logic. It is the type of neural network that uses Sugeno fuzzy inference system [12]. ANFIS has the capability to extract the benefits of both mentioned AI techniques in single platform [13]. In order to get best out of this technique, one should use any evolutionary algorithm to optimize the parameters of ANFIS.

2.3 Support vector machine

Support vector machine (SVM) is the type of supervised learning technique that is mostly used for regression and pattern recognition purposes [14]. Support vector machine construct hyperplanes or the group of hyperplanes by separating the data on the basis of clustering mostly used for classification, regression, or other tasks. SVM models are based on statistical learning theory, and they are very useful to capture any nonlinear function [15].

2.4 Permeability prediction using AI techniques

The backpropagation neural network (BPNN) was used to estimate the permeability based on well logs such as gamma ray, hole size, micro-spherically focused log, flushed zone resistivity, deep resistivity, sonic time, bulk density, neutron porosity, and photoelectric capture cross section [16]. BPNN model can predict the reservoir permeability with a high consistency with the core data. It was recommended to deep understand the reservoir lithofacies sequence in order to compare the BPNN model with the conventional regression method (porosity–permeability correlation) [16].

Well logs were used such as neutron porosity, caliber, shallow and deep resistivity, bulk density and spontaneous potential for porosity estimation and neutron porosity, gamma ray, deep resistivity, bulk density, and spontaneous potential for permeability prediction. The fuzzy logic and neural networks results were compared with the regression technique, conventional method to estimate the reservoir properties. The correlation coefficient for permeability and porosity estimation was 0.999 using fuzzy logic and neural network compared to 0.564 and 0.764, respectively, using multiple regressions [17, 18].

Type-2 fuzzy logic was used to predict the permeability using well logs such as sonic time, micro-spherically focused log, neutron porosity, total porosity, bulk density, and water saturation [19]. The estimated permeability results using type-2 fuzzy logic were compared with type-1 fuzzy logic, SVM, and ANN. The developed model can improve the correlation coefficient by 11.7% compared to the model based on type-1 fuzzy logic. In addition, the developed model yielded up to 36.1% enhancement in the correlation coefficient over the other models and up to 62.9% of root-mean-square error.

SVM was used to predict the permeability of gas reservoir using log data (gamma ray, sonic time, neutron porosity, bulk density, micro-spherical focused resistivity, shallow resistivity, and deep resistivity), and the results were compared with GRNN technique [20–22]. The overall correlation coefficient (R) was 0.97 when using SVM to predict the permeability compared to 0.71 when using GRNN.

Permeability was estimated using AI network based on log data such as gamma ray, sonic, neutron porosity, bulk density, water saturation, and resistivity [23]. It was found that ANN can be used to predict the permeability from log data with a correlation coefficient and mean absolute error comparable to regression analysis.

The neural network model was optimized using artificial ant colony algorithm which was initially inspired by the observation of ant [24]. The optimized NN is favorable by petroleum geologist as it provided more accuracy for permeability determination than the simple NN technique. The mean square error was 7.95 and 12.84 for optimized and simple NN model, respectively, when comparing the estimated permeability with the actual core permeability.

It is clear for the literature that the AI techniques can be used to estimate the permeability. Two main issues with the previous work that we will focus on in this study. First, previous AI studies estimated the permeability using many logs data as inputs for the AI models, so the objective of this study is to determine the minimum number of logs that can be used to estimate the permeability with high accuracy. Previously, several log data used in the prediction process that has no relevance to the permeability such as caliper log and photoelectric log. In this study, we will develop a new parameter which is the mobility index. Mobility index is a strong indication for the oil mobility in the reservoir which is a direct indication for the reservoir permeability. Second, they developed a black box of AI models; our objective is to extract a mathematical equation from the ANN model that can be used to predict the permeability with high accuracy and minimum logs data.

3 Laboratory and log data preparation

Core permeability for different samples was measured using gas permeameter, and the gas permeability was corrected to the liquid absolute permeability using Klinkenberg correction [25] as follows:

$$k_{\text{l}} = k_{\text{g}} \left( {1 + \frac{b}{{p_{\text{m}} }}} \right)$$
(1)

where k l = liquid permeability of the rock; k g = apparent permeability calculated from gas flow tests; b = Klinkenberg’s factor, a constant for a particular gas in a particular porous medium; p m = mean flowing pressure of the gas in the flow system, measured in atmospheres. The permeability was measured by nitrogen at net confining pressure of 500 psi.

The available log data are bulk density (RHOB), neutron porosity (NPHI), flushed zone resistivity (log R MSFL), deep resistivity (log R t ), corrected porosity (PHIT), and water saturation (S W), as shown in Fig. 1. For every data point of the logs, core permeability is available as shown in Fig. 1. Three hundred and fifty-six data points are available, which will be used to correlate the core permeability to the log data. The depth shifting was performed by correlating the porosity measured in the laboratory with that from the log (estimated from bulk density). Because R xo , R mc , and h mc are obtained directly from the micro-resistivity measurements by inversion processing, no mud cake thickness corrections are required. The values of R xo can be used directly from the R MSFL with medium and deep-resistivity measurements (or array-resistivity measurements) to derive R t .

Fig. 1
figure 1

Field log data and actual core permeability for case 1

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The first step in the data analysis is to determine the relative importance of each log data to the core permeability. The correlation coefficient was calculated between the core permeability data to each log data. The correlation coefficient (R 2) changes from −1 to 1, where strong relation between any two parameters represents a value of R 2 either close to −1 or 1. Figure 2 shows that the core permeability is a strong function of neutron porosity, total porosity, bulk density, and flushed zone resistivity. The R 2 for neutron porosity, total porosity, bulk density, and flushed zone resistivity is 0.82, 0.83, −0.80, and −0.74, respectively. The core permeability is a good function of water saturation, the R 2 is −0.52 and a week function of deep resistivity, where the R 2 is −0.19 for the log value of the deep resistivity. Figure 2 shows that taking the logged value of flushed zone resistivity (log R MSFL) gives better R 2 than taking the value (R MSFL), the R 2 for the logged value and the normal value is −0.74 and −0.49, respectively. The same result was obtained when calculating the correlation coefficient using the logged values of deep resistivity (log R T ) as compared with the normal value of deep resistivity (R T ), the R 2 is −0.19 and −0.13 for log R T and R T , respectively, Fig. 2. The core permeability is a weak function of the caliper log as the correlation coefficient was found to be 0.39, Fig. 2. Caliper data available and it showed constant reading overall the reservoir sections because the area is hard carbonate. The hole size from the caliper was consistent and constant in the available wells and because of that we excluded the caliper from the analysis.

Fig. 2
figure 2

Relative importance between core permeability with different log data

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4 Artificial intelligence network models

Permeability prediction model was developed using artificial neural network (ANN) based on the available log data, bulk density, neutron porosity, total porosity, water saturation, flushed zone resistivity (log R MSFL), and deep resistivity (log R T ). Figure 3a shows that the developed model can predict the permeability with a correlation coefficient of 0.96 and the mean square error was 0.34 when compared with the actual permeability (case 1) based on all of the logs bulk density, neutron porosity, total porosity, water saturation, flushed zone resistivity (log R MSFL), and deep resistivity (log R T ).

Fig. 3
figure 3

Logarithmic value of permeability prediction using ANN model based on different log data

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In order to evaluate the effect of removing the water saturation from the input data, case 1.1 was prepared in which the bulk density, neutron porosity, total porosity, flushed zone resistivity (log R MSFL), and deep resistivity (log R T ) were used to develop a model using ANN to predict the permeability. Figure 3b shows that the developed ANN bases on these inputs can predict the permeability with a correlation coefficient of 0.95 and a root-mean-square error of 0.40 when compared with the actual permeability data. The obtained results confirmed that water saturation can be removed from the input data and it will not affect the results of the estimated permeability. The correlation coefficient was decreased from 0.96 to 0.95 which is negligible value, and the root-mean-square error was increased from 0.34 to 0.40. It is recommended to remove the water saturation from the inputs data for the ANN model to avoid the uncertainty and error percent of the calculated value of the water saturation and its effect on the predicted permeability.

The calculation of the total porosity is a hard task as it requires the knowledge of the exact lithological composition and fluid density at each point of the given log data. In order to evaluate the effect of taking the total porosity out of the input data of ANN model, case 1.2 was prepared based on four log data namely bulk density, neutron porosity, flushed zone resistivity (log R MSFL), and deep resistivity (log R T ). Figure 3c shows that the developed ANN model can predict the permeability with a correlation coefficient of 0.95 and a root-mean-square error of 0.40. Based on these results, it is recommended to remove the total porosity log from the input data. The change in the correlation coefficient was negligible as it was decreased from 0.96 to 0.95 and the root-mean-square error was increased from 0.34 to 0.40.

To decrease the number of input data from four to three inputs, a new term was developed, which is mobility index. Mobility index is defined as the inverse of square root of normal value of flushed zone resistivity minus the inverse of square root of normal value of the deep resistivity, Eq. 2.

$${\text{Mobility}}\;{\text{index}} = \frac{1}{{\sqrt {R_{\text{MSFL}} } }} - \frac{1}{{\sqrt {R_{T} } }}$$
(2)

where mobility index (Ω m)−0.5, R MSFL is flushed zone resistivity (Ω m) and R T is deep resistivity (Ω m).

To evaluate the relative importance of the new term (mobility index) with the actual core permeability, the correlation coefficient was determined and it was found to be 0.8 which indicated a strong function between the mobility index and the actual core permeability.

ANN model was developed based on three inputs: bulk density, neutron porosity, and mobility index (case 2) to estimate the permeability. Figure 4a shows that the developed ANN model was able to predict the permeability with a correlation coefficient of 0.95 and a root-mean-square error of 0.37. Based on these results, it is recommended to use the mobility index term instead of using the two logs data (flushed zone and deep resistivity) to decrease the size of the developed matrix of ANN and hence reducing the number of weights and biases, which eventually makes calculation much easier to predict the permeability.

Fig. 4
figure 4

Logarithmic value of permeability prediction using ANN, ANFIS, and SVM based on bulk density, neutron porosity, and mobility data

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To evaluate the accuracy of ANN model, different AI techniques were applied on the same data (case 2) such as ANFIS and SVM. Figure 4b shows that ANFIS model can be used to predict the permeability with a correlation coefficient of 0.96 and a root-mean-square error of 0.37 compared with the actual core permeability based on mobility index, bulk density, and neutron porosity. Figure 4c shows that SVM model can predict the permeability with a correlation coefficient of 0.95 and a root-mean-root square error of 0.38 when compared with the actual core data based on mobility index, bulk density, and neutron porosity.

The developed three models gave similar results of the predicted permeability with similar values of the correlation coefficient and root-mean-square error. The ANN model yielded the best profile and match of the predicted data with the actual core data, as shown in Fig. 4a. So it was decided to select the ANN model to convert the AI to a white box by extracting a mathematical equation from the model that can be used to predict the permeability without the need of the software. The developed mathematical equation depends only on the mobility index, bulk density, and neutron porosity as inputs.

4.1 Development of the mathematical equation for permeability determination

To build the mathematical equation of a wide range of the parameters, 1223 actual field data points were used, which contained the mobility index, bulk density, and neutron porosity, as shown in Fig. 5. Seventy percent of the data was used for model training and thirty percent was used for testing. Table 1 lists the range of the input parameters and the actual core permeability that were used to build the mathematical model. The range of the mobility index is −0.66–0.32 Ω m−0.5, the range of neutron porosity is 0.004–0.28 fraction, the range of bulk density is 2.18–2.9 g/cm3, and the range of the core permeability is 0.02–2590 md. The wide range of permeability indicates the heterogeneity of the carbonate reservoir.

Fig. 5
figure 5

Total data used to build the ANN model and extract the mathematical equation

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Table 1 Data range of the developed mathematical model for permeability prediction
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The data go into the neural network model are normalized between −1 and 1 by using two points slope, Eqs. 3 and 4.

$$\frac{{Y - Y_{\hbox{min} } }}{{Y_{\hbox{max} } - Y_{\hbox{min} } }} = \frac{{X - X_{\hbox{min} } }}{{X_{\hbox{max} } - X_{\hbox{min} } }}$$
(3)
$$Y = 2 \times \left( {\frac{{X - X_{\hbox{min} } }}{{X_{\hbox{max} } - X_{\hbox{min} } }}} \right) - 1$$
(4)

where Y is the input parameter in the normalized form, Y max = 1, Y min = −1, X max is the maximum value of input data, X min is the minimum value of input data, X is the input parameter to be normalized. For example, the minimum value of the bulk density (X min) is 2.18 g/cm3, and the maximum value of the bulk density (X max) is 2.9 g/cm3, so for the value of the bulk density equal to 2.6 g/cm3, the normalized value will equal to 0.1666. Another example, the minimum (X min) and maximum (X max) value of the neutron porosity are 0.004 and 0.28 fraction, respectively, so for the neutron porosity value of 0.15 fraction, the normalized neutron porosity value will be 0.058.

Figure 6a shows that the ANN model was able to predict the permeability for the training data point (857 data points) with a root-mean-square error of 0.28 based on the mobility index, bulk density, and neutron porosity. Figure 6b shows that the correlation coefficient of the predicted permeability with the actual core permeability was 0.95.

Fig. 6
figure 6

Permeability estimation using ANN model for the training data (857 data points)

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Based on the obtained results from the trained data points, the mathematical equation was extracted from the ANN model with the weights and biases that are required to determine the permeability in normalized form (Z), Eq. 5. The denormalized value of the permeability (K) can be obtained using Eq. 6.

$$Z = \left[ {\sum\limits_{j = 1}^{N} {w_{2i} \tan {\text{sig}}\left( {\sum\limits_{j = 1}^{J} {w_{1i,j} x_{j} + b_{1i} } } \right)} } \right] + b2$$
(5)
$$K = 10^{4.7984z - 4.2114}$$
(6)

where X is the input variables (mobility index, neutron porosity, and bulk density) in normalized form; N is the number of neurons (the number of neurons should be optimized to have good match with less error); J is the number of input variables, which in our case they are three (mobility index, neutron porosity, and bulk density); W 1 is weight of hidden layer; W 2 is weight of the output layer; b 1 is bias of the hidden layer, and b 2 is bias of the output layer. Table 2 lists the input parameters for Eq. 5.

Table 2 Coefficients for normalized permeability (Z) Eq. 5
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For example, the permeability is predicted using mobility index, neutron porosity, and bulk density; the value of W 1 will be taken at j = 1 for mobility index, at j = 2 for neutron porosity, and at j = 3 for bulk density. The x j in the previous equations are as follows; x at j = 1 is the mobility index; x at j = 2 is the neutron porosity; and x at j = 3 is the bulk density. For example, the term \(\sum\nolimits_{j = 1}^{J} {w_{1i,j} x_{j} }\) for Z from Table 2 can be calculated as follows; \(\sum\nolimits_{j = 1}^{J} {w_{1i,j} x_{j} } = w_{1,1} x_{1} + w_{1,2} x_{2} + w_{1,3} x_{3}\), where the values of w 1,1, w 1,2, and w 1,3 are −0.6362, 2.1531, and 5.4992, respectively. This will be repeated for the 20 rows of the matrix, and the corresponding values for each row can be used from the tables. The term x represents the input log parameters, i.e., x 1 = mobility index, x 2 = neutron porosity, and x 3 = bulk density.

4.2 Validation of the developed mathematical model

The developed mathematical model was used for testing 366 data points. Figure 7a shows that the root-mean-square error was 0.28 when comparing the predicted permeability with the actual core permeability. Figure 7b shows that the correlation coefficient for the tested data point was 0.95 when using the developed mathematical equation (Eqs. 5, 6) to predict the permeability based on mobility index, neutron porosity, and bulk density.

Fig. 7
figure 7

Permeability prediction for testing data points (366 data points)

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For further testing, the logs data of well A was used for testing using the developed correlation for permeability prediction. Figure 8 shows the input data for the developed equation which are mobility index (ranges from −0.463 to 0.177 Ω m−0.5), neutron porosity (ranges from 0.017 to 0.237 fraction), and bulk density (ranges from 2.25 to 2.68 g/cm3). The available core permeability measurements (357 data points) range from 0.02 to 1083 md.

Fig. 8
figure 8

Log data and actual core permeability for well A

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Figure 9 shows that the new correlation for permeability prediction can be used to predict the reservoir permeability with a root-mean-square error of 0.25 based on the logs data (mobility index, neutron porosity, and bulk density) and the correlation coefficient between the actual core permeability and the predicted permeability was 0.96.

Fig. 9
figure 9

Permeability prediction using the developed mathematical equation for well A

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Based on the obtained results from the testing, it can be concluded that the developed empirical correlation can be used to predict the permeability of heterogeneous carbonate reservoirs with high accuracy based on three log data (mobility index, neutron porosity, and bulk density).

5 Conclusions

Three models of artificial intelligence technique (ANN, ANFIS, and SVM) were developed to predict the permeability of heterogeneous carbonate reservoir based on log data. More than 1500 actual data measurements were used to build and test the developed mathematical equation for permeability determination. Based on the results obtained, the following conclusions can be drawn:

  1. 1.

    Core permeability is strong function of flushed zone resistivity, neutron porosity, bulk density, and total porosity, and it has a good function with water saturation and weak function with deep resistivity.

  2. 2.

    The developed new term (mobility index) is a strong function with the core permeability.

  3. 3.

    Artificial neural network model (ANN) gave comparable results for permeability prediction with the ANFIS and SVM. ANN is better than the ANFIS and SVM as we can extract the mathematical equation and change to a white box.

  4. 4.

    The developed empirical equation can be used to estimate the permeability of heterogeneous carbonate reservoirs based only on three parameters: mobility index, neutron porosity, and bulk density.

  5. 5.

    The developed equation yielded a root-mean-square error <0.28 and a correlation coefficient of 0.95 for permeability prediction from the log data.