# A Hybrid Neural Network Model for Predicting Bottomhole Pressure in Managed Pressure Drilling
## 3 Dataset and Methodology
### Data Pre-Processing
#### 3.1.2 Feature Selection and Data Normalization  
In the process of drilling, BHP is related to many factors. Through correlation analysis, parameters that have a greater impact on BHP are selected, small or irrelevant variables are removed, input parameter dimensions are reduced, and the influence of noise is enlarged by the distance correlation coefficient due to the simultaneous introduction of two or more parameters with high similarity being avoided, as follows:  
\[\mathrm{dCor}_{\mathrm{n}}^{2}(\mathrm{X,Y})=\left\{\begin{array}{cl}\frac{ \mathrm{dCov}_{\mathrm{n}}^{2}(\mathrm{X,Y})}{\sqrt{\mathrm{dCov}_{\mathrm{n}}^ {2}(\mathrm{X})\mathrm{dCov}_{\mathrm{n}}^{2}(\mathrm{Y})}}&\mathrm{dCov}_{ \mathrm{n}}^{2}(\mathrm{X})\mathrm{dCov}_{\mathrm{n}}^{2}(\mathrm{Y})>0\\ 0&\mathrm{dCov}_{\mathrm{n}}^{2}(\mathrm{X})\mathrm{dCov}_{\mathrm{n}}^{2}( \mathrm{Y})=0\end{array}\right. \tag{1}\]  
where \(\mathrm{dCor}_{\mathrm{n}}^{2}(\mathrm{X,Y})\) is the distance correlation coefficient of X and Y, \(\mathrm{dCov}_{\mathrm{n}}^{2}(\mathrm{X,Y})\) is the distance covariance of X and Y, \(\mathrm{dCov}_{\mathrm{n}}^{2}(\mathrm{X})\) is the standard deviation of the distance of X, and \(\mathrm{dCov}_{\mathrm{n}}^{2}(\mathrm{Y})\) is the standard deviation of the distance of Y.  
Parameters with a correlation greater than 0.7 were screened through calculation, and the following 12 input parameters were finally determined, as shown in Table 2, including 6 temporal parameters: well depth, rotary speed, riser pressure, inlet flow rate, outlet flow rate, total pool volume, and 6 non-timing parameters: drilling fluid density, funnel viscosity, fixed vertical depth, back-pressure pump flow rate, outlet density, and sand content.  
The 100,000 filtered data points were divided into train sets and test sets. The train set of 80% (80,000 data points) was used to train the models, and the remaining dataset of 20% (20,000 data points)was used as a test set to test the prediction capabilities of the developed models.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Evaluation Standard** & **Value** & **Evaluation Standard** & **Value** \\ \hline Count & 60,000 & 25\% (Mpa) & 58.26 \\ Mean (Mpa) & 58.61 & 50\% (Mpa) & 58.70 \\ Std (Mpa) & 0.40 & 75\% (Mpa) & 58.70 \\ Min (Mpa) & 57.37 & Max (Mpa) & 59.58 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Data description of BHP.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Input Parameters** & **Output Parameter** \\ \hline \multirow{4}{*}{Temporal parameters} & Well depth, Rotary speed, \\  & Riser pressure, Inlet flow rate, \\  & Outlet flow rate, Total pool \\  & volume \\ \hline \multirow{4}{*}{Non-temporal parameters} & Fixed vertical depth, Drilling \\  & fluid density, \\ \cline{1-1}  & Funnel viscosity, \\ \cline{1-1}  & Back-pressure pump flow rate, \\ \cline{1-1}  & Outlet density, Sand content \\ \hline \hline \end{tabular}
\end{table}
Table 2: Input and output parameters of the model.  
Data normalization can speed up the gradient descent and the optimal solution for the characteristics between different dimensions, in a certain value of the comparability, and avoid excessive numerical problems in the process of training. Therefore, before model training, we need to normalize the processing of all input parameters and make all parameter mapping at the interval \([0,1]\), as shown in the following equation:  
\[\widetilde{\mathrm{x}}=\frac{\mathrm{x}-\mathrm{x}_{\mathrm{min}}}{\mathrm{x }_{\mathrm{max}}-\mathrm{x}_{\mathrm{min}}} \tag{2}\]  
where \(\widetilde{\mathrm{x}}\) is the normalized value of \(\mathrm{x}\), \(\mathrm{x}_{\mathrm{max}}\) and \(\mathrm{x}_{\mathrm{min}}\) are the maximum and minimum values of the variable \(\mathrm{x}\).