# Rate of Penetration Prediction Method for Ultra-Deep Wells Based on LSTM-FNN
## 3 Field Data Processing
### Feature Correlation Analysis  
Through feature selection, on the one hand, the correlation between various parameters and the ROP can be clarified; on the other hand, irrelevant parameters can be removed to reduce the complexity and improve the calculation efficiency of the model. The depth neural network model has a strong nonlinear fitting ability, and the Spearman and Pearson test methods are mostly used to test the linear relationship between multiple parameters and the ROP. For the nonlinear correlation test, this paper introduces the mutual information feature selection method.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Items** & **Parameter** \\ \hline Geological & Gamma Ray (GR), Spontaneous Potential (SP), Acoustic (AC) \\ \hline \multirow{8}{*}{Engineering (OT), Inter Density of Drilling Fluid (OC), Multi Density of Drilling Fluid (OC), Multi Conductivity (IC), Outlet Conductivity (OC), Pump Impulse Speed (PIS)} \\ \cline{1-1}  & \\ \cline{1-1}  & \\ \cline{1-1}  & \\ \cline{1-1}  & \\ \cline{1-1}  & \\ \cline{1-1}  & \\ \hline Trajectory & Deviation Angle, Borehole Curvature \\ \cline{1-1}  & \\ \cline{1-1}  & \\ \cline{1-1}  & \\ \cline{1-1}  & \\ \cline{1-1}  & \\ \cline{1-1}  & \\ \cline{1-1}  & \\ \hline \hline \end{tabular}
\end{table}
Table 1: Drilling parameters.  
Mutual information \(I\) (\(X\): \(Y\)) of continuous random variables \(X\) and \(Y\) is as follows:  
\[I(X;Y)=H(X)+H(Y)-H(X,Y)=\int\ \int\ p(x,y)\log\frac{p(x,y)}{p(x)\,p(y)}dxdy \tag{11}\]  
In Equation (11), \(H(X)\) and \(H(Y)\) are the unconditional entropy of variables \(X\) and \(Y\), respectively, \(H(X,Y)\) is the joint entropy of variables \(X\) and \(y\), and \(p(x,y)\) is the joint probability distribution of \(X\) and \(Y\); \(p(x)\) and \(p(y)\) are marginal probability distributions, respectively.  
Mutual information is easily affected by the amount of data. In order to eliminate the influence of the amount of data on the value of mutual information, it is necessary to normalize the mutual information, as shown in the following formula:  
\[\text{NMI}(X;Y)=2\frac{I(X,Y)}{H(X)+H(Y)} \tag{12}\]  
The value range of NMI is [0, 1]. If the NMI value of \(X\) and \(Y\) is 0, it means that the two variables are independent variables or mutually independent variables; that is, they have no information contained in each other. On the contrary, the closer the mutual information value is to 1, the higher the degree of correlation between the two variables, and the more the same amount of information is contained.  
This paper focuses on the correlation analysis of engineering parameters. As shown in Figure 4, the NMI of the rotary table speed, riser pressure, outlet flow and outlet conductivity are about 0.08, which is relatively small, indicating that the correlation between the above parameters and mechanical ROP is low and should be removed.  
Figure 4: Mutual information correlation coefficient.  
At the same time, the NMI value of the main pump impulse and inlet flow remains large (0.81) and the NMI value of the inlet flow and ROP is greater than that of the main pump impulse and ROP. In order to reduce the characteristic dimension of the ROP prediction model, the total pump impulse speed is removed.