# A Nodal Analysis Based Monitoring of an Electric Submersible Pump Operation in Multiphase Flow
## 2 Multiphase Surging Empirical Model Review  
As discussed in Section 1, several models were developed to understand the ESP operating behavior during the multiphase condition, such as dynamic and simplified empirical models used to simulate multiphase pumping behavior. However, those models require a complex solution of mass and momentum of the gas-liquid mixture in the entire ESP, and their applicability needs a detailed pump impeller and diffuser design data. Despite the developed models' complexity and theoretical dominance, these models are still used in the industry [14]. This section presents only available critical models of judging stable and unstable (surging) ESP operation; their empirical formulas are summarized in Table 1. \begin{table}
\begin{tabular}{l l} \hline \hline
**Prediction Model** & **ESP Surging Correlation** \\ \hline [PERSON] et al. (1986) [15] & \(\mathcal{O}_{\text{Turpin}}=\big{(}\frac{2000}{\text{{3}PTP}}\big{)}\frac{ \mathbf{Q}_{\text{s}}}{\mathbf{Q}_{\text{t}}}\), \(\mathbf{Assume}\)\(\mathcal{O}_{\text{Turpin}}=\mathbf{1}\), initiation of abnormal operation \\ [PERSON] et al. (1986) [15] & \(\frac{\mathbf{Q}_{\text{s}}}{\mathbf{Q}_{\text{t}}}=\frac{3\text{PIP}}{2000}\), \(\mathbf{\lambda}_{\text{c}}=\frac{Q_{\text{s}}}{\mathbf{Q}_{\text{s}}+\mathbf{ Q}_{\text{t}}}=\frac{\big{(}\frac{\text{PIP}}{2000}\big{)}}{1+\big{(}\frac{\text{PIP}}{2000} \big{)}}\) \\  & \(\mathbf{Q}_{\text{g}}\), \(\mathbf{Q}_{\text{t}}\), are gas and liquid flow rate at the critical gas volume fraction \([\lambda_{\text{c}}]\) \\ \[PERSON] (1989)-[PERSON] (2001) [16] & \(\text{PIP}=\mathbf{935}\big{(}\frac{\mathbf{Q}_{\text{s}}}{\mathbf{Q}_{\text{ t}}}\big{)}^{\frac{1}{22}}\), \(\mathbf{\frac{Q}{\mathbf{Q}_{\text{s}}}}=\big{(}\frac{\text{PIP}}{935}\big{)}^{ \mathbf{1},\mathbf{7},\mathbf{24}}\), \(\mathbf{\lambda}_{\text{c}}=\frac{\mathbf{Q}_{\text{s}}}{\mathbf{Q}_{\text{s}}+ \mathbf{Q}_{\text{t}}}=\frac{\big{(}\big{(}\frac{\text{PIP}}{2000}\big{)}^{ \mathbf{1},\mathbf{724}}\big{)}}{\mathbf{1}+\big{(}\big{(}\frac{\text{PIP}}{2000} \big{)}^{\mathbf{1},\mathbf{724}}\big{)}}\) \\ \[PERSON] (1998) [17] & \(\mathbf{\lambda}_{\text{c}}=\mathbf{0.0187}\text{PIP}^{\mathbf{0.4342}}\) \\ \[PERSON] [PERSON] (1999) [18] & \(\mathbf{\lambda}_{\text{c}}=\mathbf{0.004}\big{(}\text{PIP}-\mathbf{14.7}\big{)}^{ \mathbf{0.6801}}\) \\ \[PERSON] (2003) [19] & \(\frac{\mathbf{Q}_{\text{s}}}{\mathbf{Q}_{\text{s}}}=\big{(}\mathbf{5.580}\big{(} \frac{\mathbf{\rho}_{\text{s}}}{\mathbf{\rho}_{\text{t}}}\big{)}+\mathbf{0.098} \big{)}\Big{(}\frac{\mathbf{Q}_{\text{t}}}{\mathbf{Q}_{\text{mm}}}\Big{)}^{ \mathbf{1},\mathbf{421}}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Models of surging correlations. [PERSON] [15] formulated the first empirical model to predict ESP performance in gas well conditions. Using [PERSON] and [PERSON]'s experimental data [12], he developed a model to predict the surging (abnormal) initiation. His model uses the Turpin correlation (\(\varnothing_{\text{Turpin}}\)) parameter to judge the ESP operating condition, where the \(\varnothing_{\text{Turpin}}\) is a function of gas volume fraction and pump intake pressure. According to the author, the ESP surging condition can be initiated when (\(\varnothing_{\text{Turpin}}>1\)). Due to its simplicity, Turpin correlation is widely used to relate the ESP performance to the in situ gas and PIP to evaluate the limits of stable operation [5]. In 1989, [PERSON] presented an empirical correlation for judging the pump performance in homogenous conditions. [PERSON] agreed with [PERSON] that the PIP plays a significant role in determining critical GVF. In monitoring the ESP operation, the Dunbar model can examine the ESP performance in gas well conditions [11]. Using the mixture of air and water, [PERSON] tested the two mixed flow pumps (GN4000 and GN7000) and one radial flow pump (GN2100). [PERSON] presented the empirical model to compute the maximum GVF at ESP surging initiation point. The model evaluates the critical GVF as a function of the pump operating PIP [17].