Density log has a direct relation with compressional and shear wave velocities (Figure 1). According to Equations (3) and (4), as the formation density increases, and increase too (Schlumberger Log Interpretation, 1989).

where is formation density (bulk density), is matrix density and is fluid density.

Neutron Porosity indicates the formation hydrogen index, which is detected by Neutron tool [4] . Neutron Log indicates the formation Porosity. The more the Porosity of the formation is, the less the Velocity passing through the formation (Equation (4) and Figure 2).

where is formation porosity (that is measured by neutron tool), is fluid velocity and is rock matrix velocity. So in intervals with higher Porosity (higher Neutron Porosity), Shear and Compressional Wave Velocities decrease.

Compressional wave velocity could easily be obtained from sonic log and according to the Equation (7). As shown in Figure 3, there is a (good conformity between and versus the depth. Figure 4 also shows a direct and linear relationship between and.

Gamma log measures the formation radioactivity which obtained from Equation (5):

where is density of radioactive minerals, is minerals volume, is a radioactive factor that depends on radioactivity intensity of minerals and is formation density. Thus, as the formation density increases, the gamma decreases and vice versa. As mentioned before, velocity depends on the formation density (Equations (3) and (4)), and as the formation density increases (gamma reduces), the velocity also increases and vice versa [5] . Thus, as the gamma increases, the velocity decreases (Figure 5 and Figure 6).

Figure 7 and Figure 8 show that shallow resistivity log and deep resistivity log have nonlinear relationship with shear wave velocity. The following equation demonstrates the Archie formula [6] :

where is formation resistivity, is formation water resistivity, is formation water salinity and is formation porosity. As is evident, with increasing of porosity, formation resistivity decreases, but we should consider effect of the fluid filling the pores, i.e. when hydrocarbon fills formation pores, resistivity increases and when the fluid filling the pores is water, depend on water salinity, resistivity decreases. As mentioned, the shear

wave velocity is not influenced by this kind of fluid and doesn’t change with change of the fluid. Thus the shear wave velocity depends mainly on porosity effect and lithology type.

There are different empirical equations (e.g. Han et al. 1986, Castagna et al. 1993, Pickett et al.) to predict

using petrophysical log data [7] . We studied various empirical equations, which are a combination of different petrophysical parameters for predicting, In order to find out factors which have the most effect on. In addition, compare the relationship between shear wave velocity and other parameters and petrophysical logs showed that there is a close relationship between velocity of compressional and shear waves, especially in carbonate rocks (Figure 3 and Figure 4). Thus prediction of using compressional wave velocity is more reliable, especially in carbonate rocks. Given to the very close relationship of in carbonate rocks, among various empirical equations, we used Castagna equation to predict shear wave velocity. In the previous section, the effect of other features was determined through cross-plotting of these parameters versus. Generally, these empirical relationships only present good results in similar formations and their validation in other rocks is doubtful until it become calibrated. Thus providing a physical model that shows proper understanding of the shear wave behavior would be useful [8] .

In this study, compressional wave velocity data has been obtained using sonic log data and through Equation (7). In addition Castagna et al. equations, which is used for limestone and dolomite, is presented below (Equations (8) and (9)):

Figure 9 shows the result of comparison of values, which measured and estimated by Castagna relationship for entire the reservoir. This relationship has an input parameter and correlation coefficient of values, which measured and estimated by Castagna relationship, is about 72%.

To increase the accuracy of the prediction using regression, other parameters such as neutron porosity, density, gamma, deep resistivity log and shallow resistivity log were considered for entering to equation provided in the multiple regression. Figure 1, Figure 2 and Figures 5-8 show the effect of density, neutron, gamma, deep resistivity and shallow resistivity.

Multiple regression analysis uses from correlation between different well logging data and desired parameter. To estimate a parameter in multiple regression, initially this parameter is communicated to several other parameters. The advantages of this technique compared to simple regression are its more accuracy and its capability to summarize more information. It should be noted in using this technique that parameters which are selected for multiple regression should be independent from each other and don’t have a high correlation [9] . Although previous empirical studies present a proper instruction to select dependent variables, however new prediction equations should be presented for each new region or field. Therefore to identify input data, first we should determine the correlation coefficients between different well logging data by shear wave velocity. Then data that have the highest correlation with shear wave velocity are selected as input data. For this purpose, the correlation coefficients between different well logging data and shear wave velocity were calculated. Then data with high correlation coefficient were selected. According to Figures 1-8, parameters having the highest correlation coefficient with shear wave velocity are: data of compressional wave velocity, density, neutron porosity, deep and shallow resistivity and gamma ray.

These methods have errors and inaccuracies in the final response because of using assumptions which needed to derive the answer. These assumptions include homogeneity in depositional environment and existence of simple linear relationship between shear wave velocity and petrophysical parameters, so these six parameters (compressional wave velocity, density, neutron porosity, shallow and deep resistivity and gamma ray) can be used as input to multiple regression.

A multivariate model of the data solves for unknown coefficients of a multivariate equation such as Equation (10):

Weight of input variables to predict is determined by contribution degree of (which characterized by multiple regression). After all variables were included in the model, the following equation with correlation coefficient of 90% was obtained.

The above equation shows that the coefficients of independent variables is high and they are important variables of the regression but coefficients of and is low and they are the weakest variables of the model. In this stage, Low-value variables (low coefficients) were removed from the model and the new model was fitted again and the following equation was obtained:

When all desired parameters were used, was about 90% but when we removed these three factors, the model capability was improved and was increased about 2% and arrived to 92%. Figure 10 shows a good correlation between obtained from Equation 13 and the measured (is about 92) and Figure 11 shows the calculated from multiple regression and obtained from DSI tool versus the depth. The multiple linear regression of the presented variables shows a strong correlation among values predicted from well logging data.

Both regression method and empirical equations method are acceptable for estimating shear wave velocity using logging data for carbonate reservoir, but regression method usually has smaller error compared to empirical equations method. On the other hand, empirical equations haven’t generalizability for different lithologies. Moreover, in these models, shear wave velocity is a function of a few parameters such as compressional wave velocity and porosity. Considering these limitations, a powerful method, that can overcome these shortcomings, is necessary to estimate shear wave velocity. Therefore, it was proposed to implement smart techniques such as neural networks which have been associated with significant successes.

Artificial neural networks are a set of a relatively large number of processing elements (artificial neuron) that are designed in a specific and regular manner and the signals exchange between neurons on the communicational links [10] . Artificial neurons are similar to biological neurons of the human brain. They are parallel processing systems that are used to detect very complex patterns among data and have learning, training and remembering ability and capability of generalizing the results [11] . In this study, initially the available data were processed and inappropriate data were omitted, since they have a negative effect on network training and testing. Then, the data were normalized in (0, 1) interval. The best case for neuron networks is when all inputs and outputs are between 0 and 1. Hence, we normalized the input data (well logging data) and output data (DSI data) in (0, 1) interval to perform the network training in the best possible way. Among all processed data (data from wells No. 1, 2 and 3), data from well No. 1 and well No. 2 were selected as the training set and test and validation set, respectively. Then, a three-layer feed forward neural network (Figure 12) was used to build the model. The network components include neurons and layers. Neurons are organized in layers and each layer is responsible for a specific task. Input layer receives information from the environment and transfers it to the middle layer. Middle layer (hidden) analyses the information entered from the environment into the neural network. Output layer receives the result of analyzed information from middle layer and converts them to a meaningful form and returns them into the environment.

We used try and error method to determine the number of neurons in the middle layer. Accordingly, the best selected network with the highest correlation coefficient and the lowest error has 8 neurons in its hidden layer. Moreover, transfer function from input layer to middle layer is nonlinear tangent sigmoid function (Tansig) and transfer function from the middle layer to the output layer is Purlin linear function.

We used different training algorithms and transfer functions to design a desirable network. Experimentally, it was proved that feed forward networks with error back-propagation algorithm to determine the geomechanical parameters give the best result. Therefore, in order to estimate shear wave velocity, we used error back-propa- gation algorithm (which is a feed forward supervised learning algorithm) with Train LM training function. To avoid excessive network training, a particular set named “validation set” was used. Network validation is performed simultaneously with network training in each course and when validation data error begins to rise, the training is stopped. Mean square error (MSE) curve in terms of the number of training course (Epochs) for training data shows that the network has arrived to the best learning and the lowest error after 198 rounds (Fig- ure 13).

In this study, different input sets with various middle neurons were used to predict. Inputs with the smallest error and the highest correlation coefficient were selected as desirable inputs, and finally the best result was belong to a network with five input parameters and 8 neurons in the middle layer and the values of correlation coefficients for training and testing stages was about and, respectively (Figure 14 & Figure 15). Also Figure 16 shows the calculated from multiple regression and obtained from DSI tool versus the depth. The results demonstrate the high ability of neural networks in predicting geomechanical parameters.

According to the above results, the efficiency of artificial neural network in estimating shear wave velocity is very high. Thus, the measured by DSI tool and calculated from neural network are very close and therefore the results of this network could be generalized to the well No. 3, which hasn’t shear wave velocity.