## 4 Results
### Optimal Temperature Sensor Location Investigation
Figure 27 shows that the temperature perturbation region is generally larger when the injection pressure is high. For the high injection pressure (80 psi, Figure 27), the top ten sensor pairs are mainly located within the temperature fluctuation region regardless of the bulk flow rate, while the top ten sensor pair locations are more dispersed under the low injection pressure (30 psi, Figure 28). The higher injection pressure of 80 psi is generally preferable because the location of sensors within the temperature perturbation region ensures a higher signal-to-noise ratio. The time delay calculated within this region was expected to be more stable and accurate. The notable ex \begin{table}
\begin{tabular}{c c c c} \hline \hline  & \multicolumn{3}{c}{**Mean Flow Rate (GPM)**} \\  & & **39.52** & **123.84** & **178.30** \\ \hline Injection pressure (psi) & 80 & 0.07 & 0.12 & 0.22 \\  & 30 & 0.07 & 0.20 & 0.20 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Average distance (m) between two sensors based on the top 10 pairs under different mean flow rates and injection pressures. Figure 27: Top 10 temperature sensor pair locations for (**a**) low, (**b**) medium, and (**c**) high bulk flow rates with an 80 psi injection pressure. ception to this was the low-flow-rate condition (39.52 GPM). The results with a high injection pressure (80 psi, Figure 27a) show a greater spread of optimal sensor pair locations than those with a low injection pressure (30 psi injection). This suggests that there may be a relationship between the bulk flow rate to be estimated and the optimal fluid injection pressure. This is an area for future research. Figures 27 and 28 only include the top ten performing pairs, which may obscure the more persistent relationship between the horizontal distance from the injection pipe to the sensors and the CCF flow estimation performance. One main observation from Table 4 is that the top ten pairs have a larger distance between the two sensors when the main flow rate is high. Given the mean flow rate, a larger distance results in a longer delay time between the two sensors. The delay time should be large enough to be detected and used for inferring the CCF flow estimation. Detailed examinations of the correlation between the two distances are conducted below. Instead of selecting the top ten pairs, the 1000 smallest RMSE sensor pairs (out of over 36k pairs) were selected for each bulk flow rate and injection pressure combination. For each sensor pair, only the flow direction coordinate (regarded as x-direction) was considered. All 1000 pairs of the first sensor and second sensor x-locations were plotted in a 3D histogram, weighted by the inverse of the RMSE. Figure 28: Top 10 temperature sensor pair locations for (**a**) low, (**b**) medium, and (**c**) high bulk flow rates with a 30 psi injection pressure. \[n_{x_{1},x_{2},w}=\frac{1}{RMSE_{x_{1},x_{2}}}n_{x_{1},x_{2}} \tag{5}\]  
where \(x_{1}\) and \(x_{2}\) represent the distances from the injection pipe of the first and second sensors; \(n_{x_{1},x_{2}}\) is the frequency of the \(x_{1}\) and \(x_{2}\) sensor locations among the top 1000 pairs; and \(RMSE_{x_{1},x_{2}}\) is the RMSE found from sensors at locations \(x_{1}\) and \(x_{2}\). Thus, \(n_{x_{1},x_{2},w}\) will be large if \(RMSE_{x_{1},x_{2}}\) is small and/or if the \(x_{1}/x_{2}\) location pair frequently appears in the top 1000 pairs. The same \(x_{1}/x_{2}\) location pair may be selected multiple times because multiple y-axis locations are available. Figure 29 shows the weighted optimal locations for the top 1000 sensor pair locations for low (39.52 GPM, Figure 29a), medium (101.95 GPM, Figure 29b), and high (178.30 GPM, Figure 29c) flow conditions. In all cases, a high-pressure injection (80 psi) was used. A low-pressure injection (30 psi) showed similar trends but generally lower overall \(n_{x_{1},x_{2},w}\) values due to typically higher RMSE values.