## 3 Study on the ECD Model of the Irregular Eccentric Wellbore
### Temperature Field
#### 3.1.2 Governing Equation  
(1) Inside the casing:  
The heat exchange of the fluid inside the casing includes the thermal transfer of fluid flow during the pumping process (axial), the forced convection heat transfer between the fluid and the casing's inner wall due to fluid-solid coupling (radial), and the energy (viscous dissipation energy) generated by the fluid friction pressure drop. \[Q_{c}-\rho_{1}qC_{1}\frac{\partial T_{c}}{\partial Z}+\pi r_{ci}h_{ci}(T_{w}-T_ {c})=\rho_{1}C_{1}\pi r_{ci}^{2}\frac{\partial T_{c}}{\partial t} \tag{4}\]  
where the following variables are used:  
\(Q_{c}\) is energy generated by friction pressure drop of drilling fluid per unit length of casing, W/m;  
\(\rho_{1}\) is the density of the working fluid inside the casing, kg/m\({}^{3}\);  
\(q\) is the discharge rate of the drilling fluid, m\({}^{3}\)/s;  
\(C_{1}\) is the specific heat capacity of the drilling fluid, W/(m\(\cdot\)K);  
\(T_{c}\) is the temperature of the drilling fluid inside the casing, \({}^{\circ}\)C;  
\(T_{w}\) is the temperature of the casing column, \({}^{\circ}\)C;  
\(r_{ci}\) is the inner diameter of the casing, m;  
\(h_{ci}\) is the convective heat transfer coefficient between the fluid inside the casing and the casing, W/(m\(\cdot\)K). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Fluid** & **Parameters** & **The Fitted Equation** & **R\({}^{2}\)** \\ \hline Drilling fluid & \(k\) & \(k=0.623-0.006\times(T-T_{0})-0.004\times(P-P_{0})\) & \\  & \(n\) & \(n=0.769\) & 0.982 \\  & \(\tau\) & tau = \(1.784+0.015\times(T-T_{0})+0.017\times(P-P_{0})\) & \\ \hline Spacer & \(k\) & \(k=0.69-0.001\times(T-T_{0})\) & 0.98 \\  & \(n\) & \(n=0.573\) & 0.998 \\  & \(\tau\) & tau = \(18.514-0.013\times(T-T_{0})\) & \\ \hline Cement slurry & \(k\) & \(k=0.992-0.003\times(T-T_{0})\) & \\  & \(n\) & \(n=0.739\) & 0.999 \\  & \(\tau\) & tau = \(1.698+0.005\times(T-T_{0})\) & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Fitting flow equation of the drilling fluid, spacer, and cement slurry. (2) Casing string:  
\[\rho_{2}C_{2}\frac{\partial T_{w}}{\partial t}=\lambda_{2}\frac{\partial^{2}T_{w} }{\partial_{2}^{2}}+\frac{2r_{co}h_{co}}{r_{co}^{2}-r_{cl}^{2}}(T_{a}-T_{w})+ \frac{2r_{ci}h_{ci}}{r_{co}^{2}-r_{cl}^{2}}(T_{c}-T_{w}) \tag{5}\]  
where the following variables are used:  
\(\rho_{2}\) is the density of the casing string, kg/m\({}^{3}\);  
\(C_{2}\) is the specific heat capacity of the casing string, J/(kg\(\cdot\)K);  
\(\lambda_{2}\) is the thermal conductivity of the casing, W/(m\(\cdot\)K);  
\(r_{ci}\) is the inner diameter of the casing, m;  
\(r_{co}\) is the outer diameter of the casing, m;  
\(h_{ci}\) is the convective heat transfer coefficient between the fluid inside the casing and the casing itself, W/(m\({}^{2}\cdot\)K);  
\(h_{co}\) is the convective heat transfer coefficient between the annular fluid and the casing, W/(m\({}^{2}\cdot\)K);  
\(T_{a}\) is the temperature of the annular fluid, \({}^{\circ}\)C.  
(3) Annular fluid:  
The heat exchange of the annular fluid includes the thermal transfer of fluid flow during the pumping process, the forced convection heat transfer formed by fluid-solid coupling between the fluid and casing outer wall and the formation in the radial direction, and the energy (viscous dissipation energy) produced by the fluid friction pressure drop. The heat conduction in the axial direction and the natural convection heat transfer are neglected.