Two-phase flow plays a predominant role in various industries especially geothermal reservoirs model, wellbore, and pipelines. The flow combines of phases which are gas/liquid, gas/solid, liquid/solid and immiscible fluids. The flow model can provide the fluid properties such as the temperature profile, the pressure gradient, and flow pattern. Due to the flow in the production well was evaluated the flow characteristics of the vertical pipe. The flow pattern in vertical upflow pipe is showed in the figure 3. 

Figure 3. Flow pattern in vertical upflow, the flow assurance site





The first step of the calculation was to obtain the temperature profile because temperature is the first initial value use for determination of exploration geothermal gradient mapping. There are several methods available for precisely  predicting temperature gradient along the production well. 

With a one-dimensional steady state, Carslaw and Jaeger (1959) was proposed to use the thermal conduction equation.

with the boundary conditions (z=0) are

T(0) = T0

The temperature profile can be determined by direct integration of the temperature gradient with the determination of other parameters.


The details of the parameters were used for calculating the temperature shown in Table 1.
Table 1 the initial measured of Bouillante 3 
The predicted temperature gradient of the BO-3 production well is shown in figure 3. The graph indicates the relation wall temperature from the surface down to the bottom hole of the reservoirs. The temperature range at 500 m depth is around 260 C.
Figure 3. The predicted wall temperature of the BO-3 
Figure 4 The comparison temperature profile with depth in each Bouillante

Two-Phase Flow Pressure Drop


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( ∆Pstatic), the momentum pressure drop (∆Pmom) and the friction pressure drop ( ∆Pfrict). 

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 ∆pg can be applied with (x2)

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 fl and the steam fg 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 Xtt 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)