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Theories of Multiphase Flow in Wells
- 1. “MULTI-PHASE FLOW IN WELLS” CREATED BY TAREK AL-SAATI 1
- 2. 2 Multiphase Flow in Wells Multiphase flow (the simultaneous flow of two or more phases of fluid) will occur in almost all the 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 paper we will describe the upward two‐phase vertical flow in terms of predicting the flow regimes and calculating the pressure drops in gas‐liquid two‐phase flow in wells. • 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. When two or more phases occur in a pipe, they tend to flow at different in‐situ velocities. These in‐ situ velocities depend on the density and viscosity of the phase. Typically, in upward two‐phase flow, the lighter phase will be moving faster than the denser phase. This causes a “slip” or holdup effect. Thus, because of this fact, called the holdup phenomena, 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. One measure of the holdup phenomenon that is commonly used in production log interpretation is the “slip velocity”. Slip velocity is defined as the difference between the average in‐situ velocities of the two phases. It is not an independent property from holdup, but is simply another way to represent the holdup phenomenon. In order to show the relation between holdup and slip velocity, the definition of the superficial velocity must be introduced. The superficial velocity of a phase is defined as the volumetric flow rate of the phase divided by the cross‐sectional area of the pipe (as though that phase alone is flowing through the pipe). In other words, the superficial velocity of a phase would be the average in‐situ velocity of the 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 velocity of a phase is the superficial velocity of that phase divided by its holdup. 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. • 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 quantitative description of the phase distribution. In gas‐liquid, vertical, upward flow, four flow regimes are now generally agreed upon in the two phase literature: bubble, slug, churn, and annular flow. These occur as a progression with increasing gas rate for a given liquid rate. A brief description of these flow regimes is as follows. 1. UBubble flowU: Dispersed bubbles of gas in a continuous liquid phase. 2. USlug flowU: At higher gas rates, the bubble coalesce into larger bubbles, called Taylor bubbles, that eventually fill the entire pipe cross section. Between the large gas bubbles are slugs of liquid that contain smaller bubbles of gas entrained in the liquid. 3. UChurn FlowU: With a further increase in gas rate, the larger gas bubbles become unstable and collapse, resulting in churn flow, a highly turbulent flow pattern with both phases dispersed. Churn flow is characterized by oscillatory, up and down motions of a liquid. 4. UAnnular FlowU: At higher gas rates, gas becomes the continuous phase, with liquid flowing in an annuals coating the surface the surface of the pipe and with liquid droplets entrained in the gas phase.
- 4. 4 In this paper we will consider correlations used to calculate the pressure drop in gas‐liquid two‐phase, vertical, flow in wells. Since the flow properties may change significantly along the pipe (mainly the gas density and velocity) in gas‐liquid flow, we must calculate the pressure gradient for a particular location in the pipe; the overall pressure drop is then obtained with a pressure traverse calculation procedure. One of the most commonly used two‐phase correlations is the modified Hagedorn and Brown method (Brown, 1977), and the Beggs and Brill method (Beggs and Brill, 1973). The first of these was developed for vertical, upward flow and is recommended only for near‐vertical wellbores; the Beggs and Brill Correlation can be applied for any wellbore inclination and flow direction. TThhee MMooddiiffiieedd HHaaggeeddoorrnn aanndd BBrroowwnn MMeetthhoodd The modified Hagedorn and Brown method (mH‐B) is an empirical two‐phase flow correlation based on the original work of Hagedorn and Brown (1965). This method is a correlation for liquid holdup; the modifications of the original method include using the no‐slip holdup when the original empirical correlation predicts a liquid holdup value less than the no‐slip holdup and the use of the Griffith correlation for the bubble flow regime. These correlations are selected based on the flow regime as follows: usedisexistsFlowBubble usedisexistNOTDOESFlowBubble ncorrelatio ncorrelatio Griffith )BmH( 13.02218.0071.1 2 ⇒ ⇒ ⇒ −⇒ ≥⎟ ⎠ ⎞ ⎜ ⎝ ⎛ −= λ λ gB gB B L L L IF IF D um p f AND LB is greater or equal to 0.13. IF the calculated value of LB is less than 0.13, LB is set to 0.13. The input volume fractions, λl and λg, are parameters used in describing two‐phase flow. They are defined as GasofctionFraInputThe LiquidofctionFraInputThe = = = = u u u u m sg g m s λ λ l l The mixture velocity used in H‐B is the sum of the superficial velocities, VelocityMixtureThe VelocitylSuperficiaGas VelocitlSuperficiaLiquid y =+ = = = = = uuu q u q u sgsm g sg s A A l l l TTwwoo––PPhhaassee PPrreessssuurree GGrraaddiieenntt MMooddeellss
- 5. 5 FFllooww RReeggiimmeess ootthheerr tthhaann bbuubbbbllee ffllooww:: TThhee OOrriiggiinnaall HHaaggeeddoorrnn‐‐BBrroowwnn CCoorrrreellaattiioonn The liquid holdup, and hence, the average density, is obtained from a series of charts using four dimensionless numbers. Where: NumberViscosityLiquid NumberDiameterPipe NumberVelocityGas NumberVelocityLiquid 4 3 4 4 115726.0 872.120 938.1 938.1 = = = = = = = = σρ σ ρ ρ σ ρ μ σ l l l l l ll N N uN uN L D sggv sv D D = inside pipe diameter (ft) usl = superficial liquid velocity (ft/s) usg = superficial gas velocity (ft/s) µL = liquid viscosity (cp) ρL = liquid density (lb/ftP 3 P) σ = gas / liquid surface tension (dynes/cm) Various combinations of these parameters are then plotted against each other to determine the liquid holdup. The first curve provides a value for CNL. This CNL value is then used to calculate the dimensionless group, . can then be obtained from a plot )( 1.0575.0 1.0 Davg vl NPN CNPN L ψ l y .vs yl ψ )( 1.0575.0 1.0 Davg vl NPN CNPN L 14.2 380.0 N NN D Lvg.vsψFinally, the third curve is a plot of
- 6. Therefore, the in‐situ liquid volume fraction, which is denoted by ψ, is calculated by: ψ ψ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = l l y y ONCE AGAIN First, CNL is obtained from the following figure 6 Then the following group is calculated From the following figure HHaaggeeddoorrnn aanndd BBrroowwnn CCoorrrreellaattiioonn ffoorr CCNNLL ((11996655)) )( 1.0575.0 1.0 Davg vl NPN CNPN L
- 7. Then the following group is calculated pressure. catmospheri the is a and wanted,isgradient pressure wherelocation theat pressure obsolute the is Here. p p ψ l y get we 14.2 380.0 N NN D Lvg Figure.thefromreadisFinally, )( fornCorrelatioBrownandHagedorn ψψ then is up The hold liquid ψ ψ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = l l y y equation following the from calculated then is The density mixture lll yy ρρρ )1( g +−= as defined , using a on based is The number Reynolds mixture a factor friction fanninggradient pressure frictional )1(Re μμ ρ y g y l ll muD N − = OR, in terms of mass flow rate and using field units, 102.2 )1( 2 Re μμ y g y l ll D m N − − × = & 7
- 9. Where: ( ) GradientPressureEnergyKineticThe GradientPressureFrictionalThe GradientPressureEnergyPotentialThesin 2 2 2 2 KE PE = = = Δ Δ= = = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ z gu dz dp g f dz dp g g dz dp cm D c m c u F ρ ρ ρ θ Gas‐Liquid Flow through Chokes The flow rate from almost all flowing wells is controlled with a wellhead choke, a device that places a restriction in the flow line (Fig. A). A variety of factors may make it desirable to restrict the production rate from a flowing well, including the prevention of coning or sand production, satisfying production rate limits set by regulatory authorities, and meeting limitations of rate or pressure imposed by surface equipment. Figure A 9
- 11. Single‐Phase Liquid Flow The flow through a wellhead choke will rarely consist of single‐phase liquid, since the flowing tubing pressure is almost always below the bubble point. However, when this does occur, the flow rate is related to the pressure drop across the choke by ρ pg ACq cΔ = 2 Where: C = the flow coefficient of the choke A = the cross‐sectional area of the choke The flow coefficient through for flow through nozzles is given in Crane’s Figure (Crane, 1957) as a function of e Reynolds number in the choke and the ratio of the choke diameter to the pipe diameter. th The top equation is derived by assuming that the pressure drop through the choke is equal to the kinetic energy pressure drop divided by the square of a drag coefficient. This equation applies for Sub‐critical flow, which will usually be the case for single‐phase liquid flow. For oilfield units, the Single‐Phase Liquid Flow Equation becomes ( ) ρ p DCq Δ = 2 2 800,22 Where q is in bbl/d, D2 is the choke diameter in inches, Δp is in psi, and ρ is in lbm/ft^3. The choke diameter is often referred to as the “bean size,” because the device in the choke that restricts the flow is called the bean. Bean sizes are usually given in 64P th P of an inch. Single‐Phase Gas Flow When a compressible fluid passes through a restriction, the expansion of the fluid is important factor. For isentropic flow of an ideal gas through a choke, the rate is related to the pressure ratio, p2/p1, by (Szilas, 1975) ( ) ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ −⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ −⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = + p p p p T Rg p T pDq g c sc sc g 1 2 1 2 12 1 1 2 2 197.28 2 4 γ γ γ γ γ γ α π Which can be expressed in oil field units as ( ) ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ −⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ −⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ = + p p p p Tp p Dq gsc g 1 2 1 2 12 1 12 64 1 1 505.3 γ γ γ γ γ γ α 11
- 12. Where qg is in MSCF/d, D46 is the choke diameter (bean size) in 64ths of inches, Δp is in psi, T1 is the temperature upstream of the choke in °R, γ is the heat capacity ratio, Cp/ Cv, α is the flow coefficient of the choke, γg is the gas gravity, psc is the standard pressure, and p1 and p2 are the pressure upstream and downstream of the choke, respectively. The Single‐Phase Gas Flow Equations apply when the pressure ratio is equal to or greater than the critical pressure ratio, given by ( ) ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ + =⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − 1 2 1 1 2 γ γ γ p p c When the pressure ratio is less than the critical pressure ratio, p2/p1 should be set to (p2/p1) c and the single gas flow equation is used, since the flow rate is insensitive to the downstream pressure whenever the flow is critical. For air and other diatomic gases, γ is approximately 1.4, and the critical pressure ratio is 0.53. In petroleum engineering operations, it is commonly assumed that flow through a choke is critical whenever the downstream pressure is less than about half of the upstream pressure. Gas‐Liquid Flow Two phase flow through a choke has not been described well theoretically. To determine the flow rate of two phases through a choke, empirical correlations for critical flow are generally used. Some of these correlations for critical flow are generally used. Some of these correlations are claimed to be valid up to pressure ratios of 0.7 (Gilbert, 1954). One means of estimating the conditions for critical two‐phase flow through a choke is to compare the velocity in the choke with the two‐phase sonic velocity, given by Wallis (1969) for homogeneous mixtures as [ ] ⎪⎭ ⎪ ⎬ ⎫ ⎪⎩ ⎪ ⎨ ⎧ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ++= − vv v lcl l gcg g llggc 22 2 1 ρ λ ρ λ ρλρλ Where vc is the sonic velocity of the two‐phase mixture and vgc and v lc are the sonic velocities of the gas and liquid, respectively. The empirical correlations of Gilbert (1954) and Ros (1960) have the same form, namely, ( ) D GLRqA p C B l 64 1 = 12