**Authors**

**Abstract**

The permeability is a main property of oil reservoirs that shows the ability of rocks in the conduct of fluids such as oil, water and gas through the pore spaces of reservoir. The determination of the permeability is a crucial task in reserve estimation, production and development of oil reservoirs. Due to this, an accurate estimate of such important reservoir rock parameter should not be made from log data alone. Thus a judicious combination of core analysis and log data is required to link the most relevant parameters in order to achieve more global relationships to estimate the permeability of a reservoir rocks.

The conventional methods for permeability determination are based on limited core analysis and well test data sets. These methods are however very expensive and time consuming. Furthermore, one or more wells in an oil field may have no core samples. In practice, all exploration data including log, core, and seismic data often reflect complex nonlinear underlying relationships. Also, these data may contain a high value of uncertainty and noise caused by measurement errors. An additional source of uncertainty arises from the mapping of a sparse data set to the entire reservoir domain. Therefore, there is significant uncertainty on the estimated values of the specific petrophysical parameter such as permeability at points between wells using data obtained at the well locations. So, there is a need to use a method could appropriately measure the petrophysical properties of reservoir using available well logs. The methods are currently used to propose permeability models with high generalization performance include empirical correlations like Kozeny-Carmen theory that relates permeability to porosity and the specific area of a porous rock, multi-linear regression, multilayer perception, and fuzzy neural networks. The main limitation of empirical models is that they are developed for a specific formation and perform poorly when estimating permeability in other oil fields. Although multi-linear regression models perform better on unseen data, they often overestimate low values and underestimate high values of petrophysical parameters of the hydrocarbon reservoirs. Alternatively, artificial neural networks (ANNs) have been increasingly applied as a proper computational tool to estimate the required petrophysical properties by identifying the complex non-linear relationships between permeability, porosity, fluid saturations and well log data.

Due to the inherent ability of the ANNs, this study attempts to evaluate the ability of the general regression neural network (GRNN) and to use it for predicting the horizontal and vertical permeability (Kh and Kv) of the gas reservoirs within the Kangan and Dalan formations in the South Pars gas field. To achieve the goal the well logs and core data of three wells are used and the required computational computer codes have been written in MATLAB multi-purpose software environment. The values of digitized well logs data including sonic (DT), gamma ray (GR), compensated neutron porosity (NPHI), density (ROHB), photoelectric factor (PEF), micro spherical focused resistivity log (MSFL), shallow and deep latero-resistivity logs (LLS and LLD) are taken as input, whereas horizontal or vertical permeability (Kh and Kv) are considered as the output of the networks. In order to find the most relevant input variables (logs data) to estimate the permeability, a series of statistical analysis has been done by SPSS statistical software and the obtained correlation matrix showed a strong positive correlation between the permeability and sonic and neutron logs and a strong negative correlation with the density log. The other logs show a low to moderate correlation with permeability.

Among the 250 number of logs and core permeability datasets of the Kangan and Dalan gas reservoirs, 70 percent was randomly divided for training and 30 percent of them was allocated as testing subsets. At the next stage and in order to increase the network resolution to discriminate the high and low values, the data of all input logs were normalized within the interval of -1 to 1, whereas the output of the networks was the logarithm of core derived permeability (i.e. LKh and LKv).

As the smooth factor (SF) is the most important feature in the structure of GRNN, the training of the designed network was done by different values of this factor within the interval 0.1 to 1 and it was found that 0.27 is its optimum value with considering to the RMS error and correlation coefficient (R) of test dataset. The training of the designed network was then implemented by three, four, six, and nine combinations of input variables and it was found that the nine pattern of the input variables (i.e. X, Y, Z, DT, RHOB, NPHI, GR, PEF, MSFL) is the best relevant parameters based on the least RMSE and the highest correlation coefficient (R) values attained during the training and testing process. Consequently the designed multi-layer neural network contains a structure including one input layer with 9 neurons, one hidden layer of radial basis activation function comprising 174 neurons and an output layer containing only one neuron with linear activation function.

The obtained results of the designed networks are then compared to those provided by the multi variables linear regression (MVLR) method. The GRNN results indicate that the average correlation coefficients between core and predicted permeability are 0.95 and 0.902 in comparison to 0.85 and 0.812 of MVLR approach for train and test datasets respectively. Implementations of these methods on test datasets show that the average error of the GRNN technique is (0.65) considerably lower than that of the MVLR method (0.888) for permeability estimation. Hence, it could be concluded that the GRNN approach is faster and is more precise than the MVLR method in prediction of permeability for complex hydrocarbon reservoirs.

The conventional methods for permeability determination are based on limited core analysis and well test data sets. These methods are however very expensive and time consuming. Furthermore, one or more wells in an oil field may have no core samples. In practice, all exploration data including log, core, and seismic data often reflect complex nonlinear underlying relationships. Also, these data may contain a high value of uncertainty and noise caused by measurement errors. An additional source of uncertainty arises from the mapping of a sparse data set to the entire reservoir domain. Therefore, there is significant uncertainty on the estimated values of the specific petrophysical parameter such as permeability at points between wells using data obtained at the well locations. So, there is a need to use a method could appropriately measure the petrophysical properties of reservoir using available well logs. The methods are currently used to propose permeability models with high generalization performance include empirical correlations like Kozeny-Carmen theory that relates permeability to porosity and the specific area of a porous rock, multi-linear regression, multilayer perception, and fuzzy neural networks. The main limitation of empirical models is that they are developed for a specific formation and perform poorly when estimating permeability in other oil fields. Although multi-linear regression models perform better on unseen data, they often overestimate low values and underestimate high values of petrophysical parameters of the hydrocarbon reservoirs. Alternatively, artificial neural networks (ANNs) have been increasingly applied as a proper computational tool to estimate the required petrophysical properties by identifying the complex non-linear relationships between permeability, porosity, fluid saturations and well log data.

Due to the inherent ability of the ANNs, this study attempts to evaluate the ability of the general regression neural network (GRNN) and to use it for predicting the horizontal and vertical permeability (Kh and Kv) of the gas reservoirs within the Kangan and Dalan formations in the South Pars gas field. To achieve the goal the well logs and core data of three wells are used and the required computational computer codes have been written in MATLAB multi-purpose software environment. The values of digitized well logs data including sonic (DT), gamma ray (GR), compensated neutron porosity (NPHI), density (ROHB), photoelectric factor (PEF), micro spherical focused resistivity log (MSFL), shallow and deep latero-resistivity logs (LLS and LLD) are taken as input, whereas horizontal or vertical permeability (Kh and Kv) are considered as the output of the networks. In order to find the most relevant input variables (logs data) to estimate the permeability, a series of statistical analysis has been done by SPSS statistical software and the obtained correlation matrix showed a strong positive correlation between the permeability and sonic and neutron logs and a strong negative correlation with the density log. The other logs show a low to moderate correlation with permeability.

Among the 250 number of logs and core permeability datasets of the Kangan and Dalan gas reservoirs, 70 percent was randomly divided for training and 30 percent of them was allocated as testing subsets. At the next stage and in order to increase the network resolution to discriminate the high and low values, the data of all input logs were normalized within the interval of -1 to 1, whereas the output of the networks was the logarithm of core derived permeability (i.e. LKh and LKv).

As the smooth factor (SF) is the most important feature in the structure of GRNN, the training of the designed network was done by different values of this factor within the interval 0.1 to 1 and it was found that 0.27 is its optimum value with considering to the RMS error and correlation coefficient (R) of test dataset. The training of the designed network was then implemented by three, four, six, and nine combinations of input variables and it was found that the nine pattern of the input variables (i.e. X, Y, Z, DT, RHOB, NPHI, GR, PEF, MSFL) is the best relevant parameters based on the least RMSE and the highest correlation coefficient (R) values attained during the training and testing process. Consequently the designed multi-layer neural network contains a structure including one input layer with 9 neurons, one hidden layer of radial basis activation function comprising 174 neurons and an output layer containing only one neuron with linear activation function.

The obtained results of the designed networks are then compared to those provided by the multi variables linear regression (MVLR) method. The GRNN results indicate that the average correlation coefficients between core and predicted permeability are 0.95 and 0.902 in comparison to 0.85 and 0.812 of MVLR approach for train and test datasets respectively. Implementations of these methods on test datasets show that the average error of the GRNN technique is (0.65) considerably lower than that of the MVLR method (0.888) for permeability estimation. Hence, it could be concluded that the GRNN approach is faster and is more precise than the MVLR method in prediction of permeability for complex hydrocarbon reservoirs.

**Keywords**