# A Nodal Analysis Based Monitoring of an Electric Submersible Pump Operation in Multiphase Flow
## 5 Results
### Model Assessment  
This section compares standard errors, normality, and homogeneity of the critical GVFs predicted using nodal analysis and the empirical models developed by [PERSON], [PERSON], [PERSON], and [PERSON] (the correlation model). The normalized liquid rate obtained from the nodal analysis was also compared with [PERSON] (2010) (the empirical model). The average deviation error was calculated based on the nodal analysis for each model; thus, 4-13%, 5-13%, 5-14%, 13-17%, and 9-54% increases were observed in the [PERSON] (1986), [PERSON] (1999), [PERSON] (1989), [PERSON] (1998), and [PERSON] (2010) models, respectively (Figure 14). Better result matching was observed at the GVF above 0.8; unexpectedly high standard deviation error in the Zhou model resulted from the low production rate associated with the dead well conditions. Since the Zhou model relies on the pump design capacity, such as maximum flow rate, its prediction can be erroneous, especially in a low rate well. The Zhou model performs better at a high production rate well condition (for example, less than 10% average error was observed when ESP is operating in natural well condition). Other models are consistent, with less than 17% error in all well conditions. The Shapiro-Wilk normality test was performed at a 95% confidence level. Except for the [PERSON] model, other correlations were normally distributed in the abnormal range predicted using the nodal analysis (Tables 4 and 5). We performed a Chi-square test to evaluate the homogeneity of the results. It was found that the models' results are homogenous (Table 6). Figure 11: Critical gas fractions at various PIPs (derived at intake nodes). Case 1: an ESP deployed in a dead well and in (**a**) a single-phase well at a fixed GOR = 100 SCF/STB and a water cut = 0; (**b**) a multiphase well at a fixed GOR = 2218 SCF/STB and a water cut = 92.9%; case 2: ESP deployed in natural flowing well; (**c**) a single-phase well at a fixed GOR = 100 SCF/STB and a water cut = 0; (**d**) a multiphase well at a fixed GOR = 2218 SCF/STB and a water cut = 92.9%. Figure 12: Normalized liquid rate at various PIPs (derived at intake nodes and GVF in abnormal range). Case 1: an ESP deployed in a dead well and in (**a**) a single-phase well at a fixed GOR = 100 SCF/STB and a water cut = 0; (**b**) a multiphase well at a fixed GOR = 2218 SCF/STB and a water cut = 92.9%; case 2: ESP deployed in natural flowing well (**c**) a single-phase well at a fixed GOR = 100 SCF/STB and a water cut = 0; (**d**) a multiphase well at a fixed GOR = 2218 SCF/STB and a water cut = 92.9%. Figure 13: Field examples showing variations in ESP performance. (**a**) An increased GVF causes a sudden gas lock in the ESP. Reprinted with permission from Ref. [34]. Copyright 2019 [PERSON]. (**b**) Automatic ESP shutdown after a gas lock [34]. (**c**) An ESP gas lock. Reprinted with permission from Ref. [39]. Copyright 2018 [PERSON], [PERSON], [PERSON] and [PERSON]. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Correlation**} & \multicolumn{2}{c}{**Case 1a**} & \multicolumn{2}{c}{**Case 1b**} & \multicolumn{2}{c}{**Case 2a**} & \multicolumn{2}{c}{**Case 2b**} \\ \cline{2-9}  & **W** & \(p\)**-Value** & **W** & \(p\)**-Value** & **W** & \(p\)**-Value** & **W** & \(p\)**-Value** \\ \hline \multirow{2}{*}{
\begin{tabular}{c} Nodal analysis \\ [PERSON] (2010) \\ \end{tabular} } & 0.925 & 0.300 & 0.951 & 0.700 & 0.994 & 0.990 & 0.985 & 0.965 \\  & 0.913 & 0.300 & 0.852 & 0.300 & 0.995 & 0.700 & 0.982 & 0.965 \\ \hline \hline \end{tabular}  
\begin{tabular}{c} W: statistic test value, Case 1a: single-phase dead well, Case 1b: two-phase dead well, Case 2a: single-phase natural flowing well, and Case 2b: two-phase natural well.  
\end{table} 
Table 6: Results of the Chi-square homogeneity test (both normalized liquid rate and critical GVF obtained in the abnormal operating range). Figure 14: Average of standard errors of different models.