## 2 Mathematical Model of Reservoir Heating by E-CSS in Horizontal Wells
### Synergistic Heating Model of Electric Heating and Steam Injection  
(1) Horizontal well steam injection phase  
Figure 1 displays the 2D cross-section map of the horizontal wellbore during steam injection. From Figure 1, in the process of steam entering into the horizontal well, on any two-dimensional slice plane perpendicular to the horizontal section of the wellbore, the steam and the electric heater are the two heat sources on the cross-section, where the heat flow rate of steam into the cross-section reservoir is \(\phi_{\text{s}}\), and the heat flow rate of the electric heater into the reservoir via steam is \(\phi_{\text{e}}\). According to the superposition principle of the heat source, during the steam phase of the horizontal well, the temperature rises at any point (\(x\)); the cross section should be equal to the heat transfer and increase of the two heat sources:  
\[\Delta T_{x}=\Delta T_{xs}+\Delta T_{xe} \tag{2}\]  
Substituting Formula (1) into Formula (2) can obtain the elevated temperature at any point \(x\) on the two-dimensional plane:  
\[T_{x1}=-\left(\frac{\phi_{\text{s1}}}{4\pi k}\right)\text{Ei}\left(-\frac{r^{ 2}}{4\lambda\alpha t_{1}}\right)-\left(\frac{\phi_{\text{e1}}}{4\pi k}\right) \text{Ei}\left(-\frac{r^{2}}{4\lambda\alpha t_{2}}\right) \tag{3}\]  
(2) Horizontal well soak phase  
In the horizontal well soak phase, the steam into the horizontal well is stopped; \(\phi_{\text{s}}\) is 0. Therefore, the temperature changes at any point \(x\) on the two-dimensional cross-section  
Figure 1: 2D cross-section map of horizontal wellbore during steam injection. producing the sum of the temperature reduction caused by the outward diffusion of the heat and the temperature increment of the electric heating:  
\[\Delta T_{x2}=T-T_{1}=\left(\frac{\phi_{\rm d2}}{4\pi k}\right)\text{Ei}\left(- \frac{r^{2}}{4\lambda\alpha t_{3}}\right)-\left(\frac{\phi_{\rm e2}}{4\pi k} \right)\text{Ei}\left(-\frac{r^{2}}{4\lambda\alpha t_{3}}\right) \tag{4}\]  
\(T\) is the reservoir temperature at the end of the steam injection, calculated by Formula (3); \(\phi_{\rm d2}\) is the heat diffusion rate of per unit horizontal length of this point, W/m. \(\phi_{\rm e2}\) is the heat input rate of electric heating for the unit horizontal length of the point during the soaking phase, W/m; \(t_{3}\) is the soak well time, day. According to [PERSON]'s law of heat transfer, the heat transfer at any point \(x\) on the two-dimensional cross-section conforms to the radial heat transfer characteristics, and \(\phi_{\rm d2}\) can be represented by the following formula:  
\[\phi_{\rm d2}=\frac{2\pi\Delta T_{x2}}{\frac{1}{k}ln\frac{r}{r_{\rm w}}} \tag{5}\]  
Further transformation of Formula (6) can obtain the two-dimensional phase of the horizontal well soak phase. \[\phi_{b2}=\left(\frac{2\pi\Delta T_{X2}}{\frac{1}{k}ln\frac{r}{r_{\rm w}}} \frac{1}{4\pi k}\right)\text{Ei}\left(-\frac{r^{2}}{4\lambda\alpha t_{3}} \right)-\left(\frac{\phi_{b2}}{4\pi k}\right)\text{Ei}\left(-\frac{r^{2}}{4 \lambda\alpha t_{3}}\right) \tag{6}\]  
Temperature changes at any point \(x\) on the cross-section can be represented by the following formula:  
\[\Delta T_{x2}=\frac{-\left(\frac{\phi_{b2}}{4\pi k}\right)\text{Ei}\left(- \frac{r^{2}}{4\lambda\alpha t_{3}}\right)}{1-\left(\frac{1}{2 ln\frac{r}{r}} \right)\text{Ei}\left(-\frac{r^{2}}{4\lambda\alpha t_{3}}\right)} \tag{7}\]  
Horizontal well production phase  
In the production phase of a horizontal well, as the mixed fluid of hot oil and water is extracted, because the heat of the reservoir is brought out and the continuous heat diffusion pushes outwards, the reservoir begins to cool down. However, in this process, due to the heat compensation effect of the electric heater, the cooling amplitude of the reservoir slows down, and especially the heat compensation effect in the near-well area increases significantly.