The most important objective in our study is to find the transition point in each turndown curve,. Therefore in this section it will be specified how to determine the border between the steady and unsteady flow.

First, all the data from the transient simulation in LedaFlow is exported to Matlab.

We introduce the $\Delta P$, which is defined as the difference between the pressure at the riser base, $\Delta P$ = $P_{max}$ - $P_{min}$

Fig. 1 is a dimensionless number ($\frac{\Delta P}{\rho gh}$) versus the total mass flow rate, each line represent a constant gas mass fraction.

In this project, it is defined a threshold of $\frac{\Delta P}{\rho gh}$ in order to determine the border between stable (no severe slugging) and no stable (severe slugging). If $\frac{\Delta P}{\rho gh}$ >0.3, it could considered severe slugging, otherwise, it is not. In addition, this data should read along with the graph $\frac{T_{QL}}{T_{total}}$ versus the mass flow rate to determine finally if it is severe slugging or not.

Fig.1

From Fig.1, it can be observed that as the total mass flow rate increases the $\frac{\Delta P}{\rho gh}$ decreases. This is due to the fact that the greater the mass flow rate, the higher the superficial velocity of each phase and therefore the more energy the fluid has in order to travel easily along the pipeline.

In addition, it can be observed three points, one from each series of data, are near to the transition line. It is a must to read these values along with the following graph in order to determine if they are severe slugging.

As mentioned before, it is not convenient to determine the severe slugging phenomenon with only one variable. Therefore the dimensionless number ($\frac{T_{QL}}{T_{total}}$) is computed. T_{QL} number represents the mean liquid production period in a complete cycle, and T_{total} exhibits the mean complete period. It could be considered as severe slugging if $\frac{T_{QL}}{T_{total}}$ < 0.5.

It is worth to mention that the hypothesis of the threshold value of 0.5, is that if the liquid production period is greater than the no liquid production period, then is not considered severe slugging.

Fig. 2 shows the dimensionless number ($\frac{T_{QL}}{T_{total}}$) versus the mass flow rate.

Fig. 2

From Fig. 2 it can be observed that one out of four points that were in the no severe slugging zone in the dimensionless differential pressure graph changed into the severe slugging zone. However, in this graph the first point of the series of data x=0,0095 (magenta colour) is severe slugging, while in the first graph it was not. This fact might be because the transition border is not a specific line and is rather a zone. Nevertheless, this point must be considered as severe slugging since one of the consequences of this phenomenon is the absence of production, which jeopardise the integrity of the pieces of equipment and process downstream.

Fig. 3 relates the dimensionless pressure differential presented in Fig.1 and the dimensionless liquid production period presented in Fig. 2, by dividing $\frac{\Delta P} {\rho gh}$ over $\frac{T_{QL}}{T_{total}}$.

Fig. 3

Fig.3 is created to show that even with a great value of dimensionless pressure differential (big amplitude), no severe slugging will be considered, because the non liquid production period is very small (neglected).

This figure exhibits the dimensionless total frequency $\frac{t_{QL}}{T_{total}}$. The numerator $t_{QL}$ means the time needed for the liquid to pass through the riser $t_{QL}=\frac{H}{U_{sl}}$ and the denominator is the aforementioned mean total frequency.

Fig.4 represents the relationship between the dimensionless number $\frac{t_{QL}}{T_{total}}$ an the total mass flow rate.

Fig. 4

Theoretically, if $\frac{t_{QL}}{T_{total}}$ <1, a severe slugging of type 1 will be found.

Please refer to the lowest point in the red line (Q=0.2807 Kg/s and x=0.0019), which has clearly a value lower than 1.

The pressure profile of this point is shown in figure 5, where a type of severe slugging 1 is clearly found.

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Fig.5 : represents Q=0.2807kg/s and x=0.0019

For more information about Severe Slugging of type 1, go to this page

If $\frac{t_{QL}}{T_{total}}$ =1, it corresponds to severe slugging of type 2, which is proved in Fig. 6 with which represents

Please refer to the second point from the left of the red line in Fig. 4 (Q=0.5614 and x=0.0019), which has clearly a value equal to 1.

The pressure profile of this point is shown in figure 6, where a type of severe slugging 2 is clearly found.

Fig. 6 represents Q=0.5614 and x=0.0019

For more information about Severe Slugging of type 2, go to this page

By combining all the figures above and with a global analysis, the transition points for each constant gas mass flow rate line can be defined, therefore a flow map can be traced in Fig.7

Fig. 7

Fig. 7 shows the flow map of the downward inclined pipeline-riser system based on the result of the simulations.

Flow pattern has an important influence on the prediction of the multiphase flow parameters, therefore, such a flow pattern map can be used in design and operation phase in an offshore oil-production system.