Violent expansion of a rising Taylor bubble

This paper studies the rise of a Taylor bubble in a long, vertical liquid-filled conduit such as the riser of an oil production or exploration well. It is shown that, as the bubble rises and expands due to the gradually decreasing hydrostatic pressure, a situation may be reached in which the character of the expansion abruptly transitions from moderate to very violent. The phenomenon can be particularly dangerous with very deep wells, in which a “gas kick,” i.e., the intrusion of gaseous hydrocarbons in the riser, can result in a violent and destructive “blowout.” As an example, with its expansion, a 1-m-long bubble rising in a 1000-m-long tube causes the liquid to spill out of the top of the tube with a velocity of the order of cm/s for the better part of an hour, until it rises to the order of 10 m/s in the last several seconds. This process is considered to have been responsible for the DeepWater Horizon accident in the Gulf of Mexico in 2010. The same phenomenon underlies volcanic eruptions of the Strombolian type, which are characterized by the intermittent ejection of material up to heights of tens or even hundreds of meters. In this paper the process is studied with a quasiequilibrium model, a dynamic model, and a drift-flux model, all of which give results in substantial agreement with each other. The quasiequilibrium model leads to a very simple relation which, while approximate, gives a robust criterion for the occurrence of violent bubble expansions. The results of an illustrative calculation with OpenFOAM are also presented.

Figure 1

Sketch of the situation considered in the quasiequilibrium model of Sec. 2 and some nomenclature. Initially the liquid free surface is at the top of the tube (left, $t=0)$ so that the liquid spills out as the bubble expands (right, $t>0)$.

Figure 2

Graphical representation of the stability boundary Eq. (11) $\mathrm{\Gamma }=1;h_0$ and ${\ell }_0$ are the initial depth and length of the bubble; $h_t=p_t/\left(\rho g\right)$, with $p_t$ the pressure at the top of the tube, is the head at the top of the tube. For initial conditions above the curve the critical length ${\ell }_c$ given by Eq. (7) is reached as the bubble rises in the tube, which marks the onset of the uncontrollable expansion of the bubble. This process is illustrated in the figures that follow and, in particular, in Fig. 8.

Figure 3

The hashed region shows the control volume used with Eqs. (16) and (17): the lower boundary is a cross section of the tube tangent to the bubble top; the upper boundary is the liquid free surface, which remains at the tube top in the case sketched on the left, while it rises as the bubble expands in the case on the right. The dashed lines denote the initial positions of the bubble and of the liquid free surface.

Figure 4

Comparison of the present model (left) and the model of Ref. [10] for case (b) of Ref. [11]. The lines are, in ascending order, the positions of the bubble base, of the bubble top and of the free surface of the liquid, all vs. time. The circles are the experimental data from Ref. [11]. In the experiment the Galilei number was 91.3, the Morton number 0.08, and the Eötvös number 179. The vertical dashed line marks the time $t_c$ estimated from Eq. (14) at which the critical bubble length Eq. (13) is reached.

Figure 5

Time dependence of the three terms in the energy balance Eq. (49) with $\mathrm{\Pi }=1,\text{Ga}=50,{\widehat{h}}_0=5,{\widehat{\ell }}_0=16.7$, and $\mathrm{\Gamma }=100$. The top line (continuous, purple) is the rate of work done by the pressure forces (left-hand side of the equation). The other lines are the rate of change of the kinetic energy of the liquid in the tube (long dashes, green), the rate of change of the potential energy (dotted, blue) and the rate of energy dissipation (dash-dots, red). The sum of the three lower lines equal the rate of pressure work within numerical error.

Figure 6

Total kinetic energy ejected from the tube in dependence of the dimensionless initial bubble depth ${\widehat{h}}_0$ (left) and of the ratio ${\widehat{h}}_0/{\widehat{\ell }}_0$ of the initial depth to the initial bubble length, all with $\mathrm{\Pi }=1$ and $\text{Ga}=50$ fixed. In ascending order, the lines correspond to increasing values of the parameter $\mathrm{\Gamma }$, an approximate measure of the initial bubble enthalpy.

Figure 7

Total kinetic energy ejected from the tube in dependence of the parameter $\mathrm{\Gamma }$, an approximate measure of the initial bubble enthalpy, for various values of $\mathrm{\Pi }$ defined in Eq. (47) and of the Galilei number Ga.

Figure 8

A sequence of snapshots equally spaced in time showing the rapid expansion of the bubble as it approaches the free surface as calculated with OpenFOAM. The Galilei number is 51, the Morton number 0.6 and the Eötvös number 160. Note the constant rise velocity of the bubble base.

Figure 9

Comparison of the dynamic model of Sec. 3 (lines) with the OpenFOAM results (black circles) and the experimental data for the same case of Ref. [11] shown in Fig. 4. The lines are, in ascending order, the positions of the bubble base, of the bubble top and of the free surface of the liquid. The vertical dashed line marks the time $t_c$ estimated from Eq. (14) at which the critical bubble length Eq. (13) is reached.

Figure 10

Comparison of the drift flux model as implemented with a fixed frame (lines) and with a moving frame (circles). The tube height is 8 m and its diameter 0.1 m; the Galilei number has the value of $5.8\times {10}^4$. The dashed line has been computed with 1000 cells and the solid line with 8000; the moving frame calculation has been carried out with 1000 cells. The lower and upper sets of lines mark the position of the bubble base and top, respectively. The vertical dashed line marks the time $t_c$ estimated from Eq. (14) at which the critical bubble length Eq. (7) is reached.

Figure 11

Comparison of the drift-flux model (solid lines, red) with the dynamic model of Sec. 3 for a situation in which the liquid spills out of the top of the tube for the same case as in the previous figure. The lower and upper pairs of lines mark the position of the bubble base and top, respectively. The vertical dashed line marks the time $t_c$ estimated from Eq. (14) at which the critical bubble length Eq. (7) is reached.

Figure 12

Comparison of the drift-flux model (solid lines, red) with the dynamic model of Sec. 3 in the final stage of the bubble rise and expansion. In the situation simulated the liquid spills out of the top of the tube. The tube height is 1000 m and its diameter 0.1 m; the Galilei number has the value of $5.8\times {10}^4$. The vertical dashed line marks the time $t_c$ estimated from Eq. (14) at which the critical bubble length Eq. (7) is reached.

Figure 13

Liquid velocity as it spills out of the tube top in the final stages of the bubble rise and expansion for the case of the previous figure. It will be appreciated that the velocity remains well below 0.1 m/s for about 45 min and rises to nearly 10 m/s in the last 3 s or so.