Multiphase flow—the simultaneous flow of two or more phases of fluid—will occur in almost all oil production wells, in many gas production wells, and in some types of injection wells. In an oil well, whenever the pressure drops below the bubble point, gas will evolve, and from that point to the surface, gas–liquid flow will occur. Thus, even in a well producing from an undersaturated reservoir, unless the surface pressure is above the bubble point, two-phase flow will occur in the wellbore and/or tubing.
Many oil wells also produce significant amounts of water, resulting in oil–water flow or oil–water–gas three-phase flow.
Two-phase flow behavior depends strongly on the distribution of the phases in the pipe, which in turn depends on the direction of flow relative to the gravitational field. In this chapter, upward vertical and inclined flow are described; horizontal and near-horizontal flow are treated in Chapter 8.
7.4.1. Holdup Behavior
In two-phase flow, the amount of the pipe occupied by a phase is often different from its proportion of the total volumetric flow rate. As an example of a typical two-phase flow situation, consider the upward flow of two phases, α and β, where α is less dense than β, as shown in Figure 7-8. Typically, in upward two-phase flow, the lighter phase (α) will be moving faster than the denser phase (β). Because of this fact, called the holdup phenomenon, the in-situ volume fraction of the denser phase will be greater than the input volume fraction of the denser phase—that is, the denser phase is “held up” in the pipe relative to the lighter phase.
Figure 7-8. Schematic of two-phase flow.
This relationship is quantified by defining a parameter called holdup, y, as
where Vβ = volume of denser phase in pipe segment and V = volume of pipe segment. The holdup, yβ, can also be defined in terms of a local holdup, yβl, as
The local holdup, yβl, is a time-averaged quantity—that is, yβl is the fraction of the time a given location in the pipe is occupied by phase β.
The holdup of the lighter phase, yα, is defined identically to yβ as
or, because the pipe is completely occupied by the two phases,
In gas–liquid flow, the holdup of the gas phase, yα, is sometimes called the void fraction.
Another parameter used in describing two-phase flow is the input fraction of each phase, λ, defined as
and
where qα and qβ are the volumetric flow rates of the two phases. The input volume fractions, λα and λβ, are also referred to as the “no-slip holdups.”
Another measure of the holdup phenomenon that is commonly used in production log interpretation is the “slip velocity,” us. Slip velocity is defined as the difference between the average velocities of the
two phases. Thus,
where and are the average in-situ velocities of the two phases. Slip velocity is not an independent property from holdup, but is simply another way to represent the holdup phenomenon. In order to show the relationship between holdup and slip velocity, we introduce the definition of superficial velocity, usα or usβ, defined as
and
The superficial velocity of a phase would be the average velocity of the phase if that phase filled the entire pipe—that is, if it were single-phase flow. In two-phase flow, the superficial velocity is not a real velocity that physically occurs, but simply a convenient parameter.
The average in-situ velocities and are related to the superficial velocities and the holdup by
and
Substituting these expressions into the equation defining slip velocity (7-74) yields
Correlations for holdup are generally used in two-phase pressure gradient calculations; the slip velocity is usually used to represent holdup behavior in production log interpretation.
Example 7-6. Relationship between Holdup and Slip Velocity
If the slip velocity for a gas–liquid flow is 60 ft/min and the superficial velocity of each phase is also 60 ft/min, what is the holdup of each phase?
Solution
From Equation (7-79), since superficial velocity of a phase is q / A,
Solving for yl, a quadratic equation is obtained:
For us = usg = usl = 60 ft/min, the solution is yl = 0.62. The holdup of the gas phase is then yg = 1 – yl = 0.38. The holdup of the liquid is greater than the input fraction (0.5), as is typical in upward gas–liquid flow.
7.4.2. Two-Phase Flow Regimes
The manner in which the two phases are distributed in the pipe significantly affects other aspects of two-phase flow, such as slippage between phases and the pressure gradient. The “flow regime” or flow pattern is a qualitative description of the phase distribution. In gas–liquid, vertical, upward flow, four flow regimes are now generally agreed upon in the two-phase flow literature: bubble, slug, churn, and annular flow. These occur as a progression with increasing gas rate for a given liquid rate. Figure 7-9 (Govier and Aziz, 1977) shows these flow patterns and the approximate regions in which they occur as functions of superficial velocities for air–water flow in a small-diameter pipe. A brief description of these flow regimes is as follows.