Generally, the total pressure drop (∆Ptotal ) of a fluid depends on the kinetic and potential energy which total pressure drop can be expressed in term of a static pressure drop( ∆P_{static}_{)}, the momentum pressure drop (∆P_{mom}) and the friction pressure drop ( ∆P_{frict}).

The static pressure drop equation is related to the vary in elevation of the vertical depth.

where z is the depth of the production well. The homogeneous density is

The Lockhart and Martinelli correlation (1949) void fraction can be obtained from the steam quality of x as

This method shows the relation between the void fraction, density and viscosity which are

varied in the production well.

The momentum pressure drop is a covariant of the steam quality term.

The method of Lockhart and Martinelli correlation (1949) is the original well-known method that used for two-phase prediction.In order to calculate the friction pressure is derived from the correlation of the Lockhart and Martinelli model.

The function is estimated based on a two-phase multiplier for the liquid phase (l) and vapor phase(g), respectively

for ∆p_{g}_{ }can be applied with (x^{2})

The friction factor can be defined in terms of Reynold number by Blasius equation:

where is Reynold number is

The viscosity is calculated based on the quality average between two-phase flow which is obtained from

The single -phase friction of the liquid f_{l} and the steam f_{g} are corresponded with two-phase multipliers

Liquid | Gas | C |

Turbulent | Turbulent | 20 |

Laminar | Turbulent | 12 |

Turbulent | Laminar | 10 |

Laminar | Laminar | 5 |

Table 1. Values of C

where is X_{tt}_{ }is the Martinelli parameter for both phase in the turbulent regimes defined as

The correlation of Lockhart and Martinelli is applicable to the vapor quality range 0<x<1.

The most problematic term is vapor quality (x) which can be obtained from heat flux correlation.

where the heat flux( q) is

The heat flux is based on the enthalpy of two-phase flow which proposed by Gunger et Winterton (1986)

All these relation depend on temperature, pressure, and void fraction.

Figure 5 shows the pressure in term of the changing of depth.

The graph indicates that the variation of pressure versus the well depth from the land surface to the bottom hole. The trend of the pressure was upward while the increasing of the well depth. To sum up, it was expected to have the decreasing trend of pressure as a function of depth taking the significant effects of the saturation temperature.

Figure 5 The predicted temperature and the saturation temperature

It is obvious that the predicted temperature was willing to archive the saturation point at 250 C with very close to the bottom hole as shown in figure 5. On the other hand, the analysis of vapor quality (x) reported in figure 6 showed that error vapor quality (x) calculation along the saturation pressure. It provided the exceeding vapor quality values over 1. Generally, the vapor quality should not get more than 1.

Figure 6. the saturation temperature contribution in the production well as a function of vapor quality (x)