Hydraulic model of steady state multiphase flow in wellbore annuli

PETROLEUM EXPLORATION AND DEVELOPMENT Volume 41, Issue 3, June 2014 Online English edition of the Chinese language journal Cite this article as: PETROL. EXPLOR. DEVELOP., 2014, 41(3): 399–407.

RESEARCH PAPER

Hydraulic model of steady state multiphase flow in wellbore annuli YIN Bangtang1,*, LI Xiangfang2, SUN Baojiang1, ZHANG Hongquan3 1. School of Petroleum Engineering, China University of Petroleum (Huadong), Qingdao 266580, China; 2. College of Petroleum Engineering, China University of Petroleum (Beijing), Beijing 102249, China; 3. The University of Tulsa, Tulsa, Oklahoma, 74104, United States of America

Abstract: Based on the classification and flow behaviors of two phase flow in wellbore annuli, the hydraulic models for slug flow and annular flow in annuli for vertical or inclined wells were established, and the flow regime transition criteria were also obtained. Based on the flow behavior research of multiphase flow in wellbore annuli, the liquid film zone was used as the control volume, and the effect of the tubing liquid film, casing liquid film and the droplets in gas core area on the mass and momentum transfers were considered. The mass and momentum conservation equations of slug and annular flows were obtained. Then the evaluation criterion of flow pattern transitions were established, including dispersed flow to slug flow, bubble flow to slug flow and slug flow to annular flow. The model prediction results were compared under the experimental conditions from the previous literatures. The predictions of flow pattern, liquid holdup and pressure gradient were compared between the new model and the pipe flow model modified by using the hydraulic diameter. The results show that the flow pattern, liquid holdup and pressure gradient can be predicted by the new model more accurately, and the prediction of liquid holdup and pressure gradient are better. Key words: wellbore annuli; multiphase flow; hydraulic model; steady state; tubing liquid film; casing liquid film

Introduction There will be gas-liquid two phase flow in wellbore annulus in a variety of scenarios during drilling and production, such as gas kick during drilling, underbalanced drilling, and producing through annulus in flowing wells when the production is high. Methods commonly used in the study on two phase flow in wellbores can be divided into two classes: empirical model [1−3] and mechanical model [4−6]. The annular hydraulic models used in empirical models mostly are based on the pipe flow models modified by using the hydraulic diameter [7−14], with big error. In addition, this method ignores the geometric difference between the annulus and pipe, the influence of flow pattern on flow parameters, regards two phase flow as pseudo-single phase flow, or cannot be widely used restricted by experimental conditions. In comparison, considering the flow behavior and property of liquid, the annulus structure and fluid velocity, the mechanical model method establishes a fluid flow equation for each flow pattern separately. Since the 1970s, many researchers have studied the

flow pattern transition model [15−18], but there are few models for the prediction of liquid holdup and pressure drop gradient. The tubing and casing liquid film, which changes the applied force on the cell body, and have a great impact on the flow regime transition, liquid holdup and pressure drop gradient prediction, must be considered in the research of the two phase flow in wellbore annulus. In this paper, based on the flow behavior of slug flow, the hydraulic models of slug flow and annular flow were established, and the transition criterion between various flow patterns is studied, taking into account the effect of tubing and casing liquid film, inclined angle and droplets in gas core area. Finally the prediction results are compared with the experimental data from previous literature [16].

1 Flow pattern classification of gas-liquid two phase flow in wellbore annulus Similar to the pipe flow, the flow patterns of gas-liquid two phase flow in vertical wellbore annulus can be classified into 5 different patterns (Fig. 1): bubble flow, dispersed bubble

Received date: 24 Nov. 2013; Revised date: 06 Apr. 2014. * Corresponding author. E-mail: [email protected] Foundation item: Supported by the National Science and Technology Major Project (2011ZX05056-001-03; 2011ZX05056-001-04); Joint Ph.D. Program of National Construction of High Level College for Postgraduate Study Abroad Project (2011644002); and Fundamental Research Funds for the Central Universities (14CX02167A). Copyright © 2014, Research Institute of Petroleum Exploration and Development, PetroChina. Published by Elsevier BV. All rights reserved.

YIN Bangtang et al. / Petroleum Exploration and Development, 2014, 41(3): 399–407

Fig. 1 Flow pattern distribution of gas-liquid two phase flow in an annulus [16]

flow, slug flow, churn flow and annular flow, of which slug flow and annular flow are very different from those in pipe flow. There are two liquid films in slug flow in annulus, one tubing film, the other casing film. Furthermore, the Taylor bubbles are not axisymmetric any more and there is a high turbulent region behind the Taylor bubble according to the experimental observation of Caetano E F [16]. Annular flow in wellbore annulus occurs when the gas velocity is very high. The gas phase in gas core flows at very high velocity, which may contain liquid droplets. And there is a thin liquid film around the gas core. Due to the annulus structure, there are also two liquid films, the casing and tubing liquid films. The casing liquid film is normally thicker than the tubing liquid film.

2 Hydraulic model of slug flow in wellbore annulus In 2003, Zhang H Q et al [6, 19] established a unified hydrodynamic model (Zhang Model) for the prediction of flow pattern transition, pressure gradient, and liquid holdup and slug characteristics in gas-liquid pipe flow based on the dynamics of slug flow. He found any other flow patterns could be transited from slug flow, when the liquid slug section did not exist, the slug flow would transit to annular flow; when the liquid film area didn’t exist, the slug flow would transit to bubble or dispersed bubble flow. In this paper, based on the slug flow dynamics, considering the effect of tubing liquid film, casing liquid film and droplets in gas core area on the mass and momentum transfer, the liquid film zone is used as the control unit and the hydraulic model of slug flow in wellbore annulus is established. 2.1

Fig. 2

Slug flow in wellbore annulus

droplet velocity is equal to the slug translational velocity and the liquid is incompressible. For a stable slug flow, the input mass flow rate should equal to the output mass flow rate, H LS ( vT − vS ) = H LFc ( vT − vLFc ) + H LFd ( vT − vLFd ) +

H Lc ( vT − vc )

(1)

For the gas phase, the input mass flow rate at the bottom boundary is equal to the output mass flow rate at the top boundary of the film zone, and can be written as: (1 − H LS )( vT − vS ) = (1 − H LFc − H LFd − H Lc ) ×

( vT − vc )

(2)

The sum of Eqs.(1) and (2) gives, vS = H LFcvLFc + H LFd vLFd + (1 − H LFc − H LFd ) vc

(3)

If it takes Taylor bubble ΔtTB to pass the cross section, the liquid volume change in the liquid film during ΔtTB is, VLF = vLFc H LFc Ac ΔtTB + vLFd H LFd Ac ΔtTB + vc H Lc Ac ΔtTB =

( vLFc H LFc + vLFd H LFd + vc H Lc ) Ac

lF vTB

(4)

If it takes the liquid slug ΔtLS to pass the same cross section, the liquid volume change in the liquid film during ΔtLS is, l VLS = vS H LS Ac ΔtLS = vS H LS Ac S (5) vTB The liquid volume change in the slug unit during ΔtTB+ΔtLS is,

Mass conservation equation

As shown in Fig. 2, the entire casing and tubing liquid film zone of a slug unit is used as the control unit. There are liquid droplets in the Taylor bubble area. It is assumed the liquid − 400 −

VSU = vSL Ac ( ΔtTB + ΔtLS ) = vSL Ac

According to the mass conservation law: VSU = VLF + VLS

lF + lS vTB

(6)

(7)

YIN Bangtang et al. / Petroleum Exploration and Development, 2014, 41(3): 399–407

From Equations (4)—(7), the mass conservation equation of liquid phase in slug unit is obtained, ( lF + lS ) vSL = lS H LSvS + lF ( H LFcvLFc + H LFd vLFd + H Lcvc )

(8)

Similarly, the mass conservation equation of gas phase in slug unit is obtained, ( lF + lS ) vSG = lS (1 − H LS ) vS + lF (1 − H LFc − H LFd − H Lc ) vc

The liquid entrainment fraction in the gas core pressed as: H Lcvc FE = H LFcvLFc + H LFd vLFd + H Lcvc 2.2

ρc =

τ LFc S LFc H LFc Ac

H LFd Ac

The force analysis of casing liquid film is shown in Fig. 3. According to the momentum conservation law: H LFc Ac p1 − H LFc Ac p2 + ρ LF Ac H LFc ( vLFc − vT )( vLFc − vS ) +

τ LFc SLFc H LFc Ac

− ρ LF g sin θ

(12)

Similarly, for tubing liquid film, p2 − p1 ρ LF ( vLFd − vT )( vLFd − vS ) τ Id S Id = + − lF lF H LFd Ac

τ LFd S LFd H LFd Ac

− ρ LF g sin θ

(13)

For Taylor bubble, p2 − p1 ρc ( vT − vc )( vS − vc ) = − lF lF

τ Id SId + τ Ic S Ic

(1 − H LFd − H LFc ) Ac

− ρ c g sin θ

1

+

+ ( ρ c − ρ LF ) g sin θ +

⎤ 1 ⎥=0 (1 − H LFc − H LFd ) Ac ⎥⎦

⎢⎣ H LFc Ac ρ LF ( vLFd − vT )( vLFd − vS ) − ρc ( vc − vT )( vc − vS ) − lF

τ LFd S LFd

Equation (11) can be transformed into: p2 − p1 ρ LF ( vLFc − vT )( vLFc − vS ) τ Ic S Ic = + − lF lF H LFc Ac

τ Id S Id

(1 − H LFc − H LFd ) Ac

τ Ic S Ic ⎢

(10)

(11)

+

⎡

is ex-

Momentum conservation equation

τ Ic S IclF − τ LFc S LFclF − ρ LF H LFclF Ac g sin θ = 0

1 − H LFc − H LFd

Solving Eq.(12) and (14), Eq.(13) and (14) simultaneously, the momentum conservation equation is obtained, ρ LF ( vLFc − vT )( vLFc − vS ) − ρc ( vc − vT )( vc − vS ) − lF

(9) [6]

ρG (1 − H LFc − H LFd − H Lc ) + ρ LF H Lc

+

τ Ic S Ic

(1 − H LFc − H LFd ) Ac ⎡

τ Id S Id ⎢

1

⎣⎢ H LFd Ac

2.3

+

(15)

+ ( ρ c − ρ LF ) g sin θ +

⎤ 1 ⎥=0 (1 − H LFc − H LFd ) Ac ⎦⎥

(16)

Auxiliary parameters

(1) Slug translational velocity. Nicklin D J [20] proposed that the slug translational velocity could be the function of mixture velocity, vT = CSvS + vTB (17) In Equation (17), CS is the ratio of maximum velocity and average velocity of stable slug flow, which changes with the flow condition. Zhang H Q et al [6] believed that it was 2.0 when the flow was laminar flow and 1.3 when it was turbulence flow. In the transition area (2 000≤Re≤4 000); CS = 2.0 − 0.7 ( Re − 2 000 ) / 2 000 (18) Santos O L A et al [21] established the Taylor bubble rise velocity model for the drilling fluid,

vTB = C1C2 gd c ( ρ L − ρG ) ρ L (14)

(19)

Where C1 = 0.314 3K + 0.2551

Where

C2 = 0.053 2lg Reb + 0.770 8 2−n 9.782vTB ρ L ⎡ d co − dci ⎤ Reb = ⎢ ⎥ K ⎣⎢ 4 ( 2 + 1 n ) ⎦⎥

n

(2) Interfacial friction factor. The frictional force exists when the Taylor bubble are wetted by the liquid film in the slug flow. The interfacial friction factor model established by Andritsos N et al[22] is used for calculating the interfacial friction factor of casing liquid film and Taylor bubble, the tubing liquid film and Taylor bubble.

Where

Fig. 3 Force analysis of casing liquid film in slug flow in wellbore annulus

− 401 −

0.5 ⎡ ⎛ 2δ ⎞ ⎛ v ⎞⎤ f Ic = f c ⎢1 + 15 ⎜ c ⎟ ⎜ SG − 1⎟ ⎥ ⎢⎣ ⎝ d c ⎠ ⎝ vSGt ⎠ ⎥⎦ 0.5 vSGt = 5( ρG0 ρG )

(20)

0.5 ⎡ ⎛ 2δ dp ⎞ ⎛ vSG ⎞⎤ f Id = f c ⎢1 + 15 ⎜ − 1⎟ ⎥ (21) ⎟ ⎜ ⎜ Ddp ⎟ vSGt ⎢ ⎠ ⎥⎦ ⎝ ⎠ ⎝ ⎣ (3) Wall shear stress. The calculation method of the wall

YIN Bangtang et al. / Petroleum Exploration and Development, 2014, 41(3): 399–407

shear stress used in the momentum equation for the slug flow in wellbore annulus is as follows, 2 τ LFc = f LFc ρ LFvLFc 2

τ LFd = f LFd ρ v

2 LF LFd

(22)

2

(23)

τIc = fIcρc ( vc − vLFc ) vc − vLFc 2

(24)

τId = fIdρc ( vc − vLFd ) vc − vLFd 2 Where

f LFc = CReLFc

−n

(25)

fLFd = CReLFd

ρLF H LFc ( vLFc − vT )( vS − vT ) + ρLF H LFd ( vLFd − vT )( vS − vT ) +

−n

The interfacial Reynolds number of casing liquid film and Taylor bubble, the tubing liquid film and Taylor bubble can be calculated by the modified pipe flow model based on the hydraulic diameter. v ρ d 4 ALFcvLFc ρ LF (26) ReLFc = LFc LF LFc = μLF ( SLFc + SIc ) μLF ReLFd =

vLFd ρ LF d LFd

μ LF vc ρcdcc

Rec =

μc

=

ALFc = H LFc Ac

Where

=

4 ALFd vLFd ρ LF ( SLFd + SId ) μLF

(27)

4 AcTvc ρc

(28)

( SIc + SId ) μc ALFd = H LFd Ac

AcT = (1 − H LFc − H LFd ) Ac (4) Pipe wall and interfacial perimeter. The liquid film will creep up the wall of the pipe at high gas flow rate or high inclination angle. This may significantly increase the interfacial and wetted wall perimeters. The wetted wall fraction can be worked out by use of the Grolman correlation [23], 0.25

0.15

⎛ σw ⎞ ⎟ ⎝σ ⎠

Θ = Θ0 ⎜

+

2 ρG 1 ⎛ ρLvSL dR ⎞ ⎜ ⎟ ρL − ρG cosθ ⎝ σ ⎠

2 ⎡ ⎤ vSG ⎢ ⎥ 2 ⎢⎣ (1 − H LFc − H LFd ) gd R ⎥⎦ S LFc = πDcΘ

( dp dz )LF lF + ( dp dz )LS lS ⎛ dp ⎞ (36) ⎜ ⎟ = d z lF + lS ⎝ ⎠T The pressure gradient of liquid film area and liquid slug area both consist of gravity term, acceleration term and friction term, and the friction term includes casing and tubing liquid film friction, and Taylor bubble friction. ( dp dz )LF = ( ρ LF + ρc ) g sin θ +

×

0.8

a 2 + b2

(29) (30)

S LFd = πDdpΘ

(31)

S Ic =

S LFc ( ACDc − ALFc ) + SCDc ALFc ACDc

(32)

S Id =

S LFd ( ACDd − ALFd ) + SCDd ALFd ACDd

(33)

S LFc

( dp

2 f LFc ρ LFvLFc f ρ v2 f ρ v2 + S LFd LFd LF LFd + Sc c c c 2 Ac 2 Ac 2 Ac

dz )LS = ρS g sin θ + ρ LF H LFc ( vLFc − vT )( vLFc − vS ) +

ρ LF H LFd ( vLFd − vT )( vLFd − vS ) +

2 fS 2 ρS ( vSG + vSL ) dR

Annular flow in wellbore annulus can be characterized as a high velocity gas core surrounded by the casing and tubing liquid film, with some liquid droplets in the gas core area (Fig. 4). 3.1

Mass conservation equation

Since the liquid phase includes three parts, casing liquid film, tubing liquid film and liquid droplets in the gas core, the mass conservation of liquid phase is, vSL = H LFcvLFc + H LFd vLFd + vSL FE (39) Since gas phase only exists in the gas core, the mass conversation equation of gas phase is, vSG = (1 − H LFc − H LFd − H Lc ) vc (40) Assuming the gas and liquid phase distribute evenly in the gas core, then vSL FE H Lc = (41) vSL FE + vSG 3.2

Momentum conservation equations

The first term on the left of Equations (15) and (16) denotes

Compared with the casing and tubing diameters, δC and δT are much smaller, so the Equation (34) can be simplified as, H LFd δ dp WT R= = (35) H LFc δ c 2π − WT

(

)

1 2sin −1 R + 2 R 1 − R 2 − R 2 π 1 − R2 (6) Pressure gradient. The average pressure gradient of the whole slug unit can be calculated by WT =

(38)

3 Hydraulic model of annular flow in wellbore annulus

(5) The ratio of casing liquid film holdup to tubing liquid film holdup. The ratio can be calculated by the liquid holdup model for annular flow in annulus and casing and tubing liquid film thickness model proposed by Caetano E F [16]. H LFd δ dp R (1 + δ dp Ddp ) (34) = H LFc δ c (1 − δ c Dc )

Where

(37)

− 402 −

Fig. 4

Annular flow in the wellbore annulus

YIN Bangtang et al. / Petroleum Exploration and Development, 2014, 41(3): 399–407

to the momentum exchange between the liquid slug area and the film zone, the liquid slug area will not exist and the momentum exchange will be 0 when the slug flow transits to annular flow, i.e. the first term on the left of Equations (15) and (16) will be 0. Then the momentum conservation equations are, τ LFc S LFc τ Id S Id − − ( ρ c − ρ LF ) g sin θ − H LFc Ac (1 − H LFc − H LFd ) Ac ⎡

⎤ 1 1 + ⎥=0 H A 1 H H A − − ⎥ LFc LFd ) c ⎦ ⎣⎢ LFc c (

τ Ic S Ic ⎢ τ LFd S LFd H LFd Ac

−

τ Ic SIc

(1 − H LFc − H LFd ) Ac

(42)

Where

+ Fd

4.2

ReG = ρG vc d R μG

−0.2 G

hFc+ = ρ Gδ cvcc* μG

vcc* = τ Ic ρ G

)

− 200 ρ G / ρ LF ⎤ ⎦

(45)

+ * hFd = ρ Gδ dp vcd μG

16 (1 − R ) R e 1 − R4 1 − R2 + 1 − R2 l n R 2

(46)

For turbulence flow,

1 ⎛ 16 ⎞ f CA ⎜ ⎟ ⎝ FCA ⎠

0.45 exp ⎡⎣( 3 000 − R e ) /106 ⎤⎦

=

0.45 exp ⎣⎡( 3 000 − R e ) /106 ⎦⎤ ⎫ ⎧ ⎛ 16 ⎞ ⎪ ⎪ 4lg ⎨ R e f CA ⎜ ⎬ − 0.4 ⎟ F ⎝ CA ⎠ ⎪ ⎪ ⎩ ⎭

(47)

16 (1 − R ) 1 − R4 1 − R2 + 1 − R2 ln R 2

Where

4

FCA =

Transition from bubble flow to slug flow

⎡ ( ρ − ρ ) gσ ⎤ vSG = 0.25vSL + 0.306 ⎢ L 2G ⎥ ρL ⎣ ⎦

(2) Wall friction factor. Caetano E F[16] derived the wall friction factor equation based on the continuity equation, momentum equation and Fanning equation. For laminar flow,

Transition criterion of flow patterns The flow pattern transition criterion is set up based on the

(49)

The dispersed bubbles don’t collide and merge when the gas velocity in the annulus is slow, but rise in a linear path. When the gas velocity is high, the gas bubbles become larger. When the size of the bubbles is more than a critical size, the bubbles begin to deform and move in Z pattern path, and then they collide and merge, forming bubbles with a spherical cap which are similar to the Taylor bubble in the slug flow. Then the transition to slug flow happens. Taitel et al [26] believed the transition happened when the gas content was 25%; and the bubbles would merge if the distance between bubbles was half of the bubble diameter. But Caetano E F [16] concluded that the transition would happen when the gas content was 20% from his experiments. So the transition criterion is,

* vcd = τ Id ρ G

f CA =

Tsm = 2 f S ρ SvS2 3 1 3 2 − ⎡ ⎤ ⎛ vSG ⎞ 2 ⎥ ⎛ σ ⎞ 5 ⎛ 2 fS 3 ⎞ 5 ⎢ d c = 0.725 + 4.15 ⎜ vS ⎟ ⎟ ⎥⎜ ⎟ ⎜ ⎢ ⎝ vS ⎠ ⎥ ⎝ ρ L ⎠ ⎝ d R ⎠ ⎢⎣ ⎦

(43)

)

WeG = ρG vc2 d R σ

f sm = 0.046 Re

When vSG≥0.1 m/s, the flow pattern transition model from dispersed bubble flow to slug flow established by Zhang H Q [19] are modified by using the hydraulic diameter. 1 (48) H LS = Tsm 1+ 3.16 ( ρ L − ρ G ) gσ If vSG<0.1 m/s, the Barnea model [25] are used,

The perimeter, hydraulic diameter, cross sectional area, the holdup of casing and tubing liquid film will be calculated similarly to the slug flow model. (1) Interfacial friction factor. The interfacial friction factor of casing liquid film and gas core, tubing liquid film and gas core are calculated based on the Asali model modified by Ambrosini et al [24]. fIc = fsm ⎡1 + 13.8WeG0.2 ReG−0.6 hFc+ − 200 ρG / ρLF ⎤ (44) ⎣ ⎦

( (h

Transition from dispersed bubble flow to slug flow

Where

Calculation methods for auxiliary parameters

f Id = f sm ⎡1 + 13.8WeG0.2 ReG−0.6 ⎣

4.1

− ( ρ c − ρ LF ) g sin θ −

⎡ 1 ⎤ 1 τ Id S Id ⎢ + ⎥=0 ⎢⎣ H LFd Ac (1 − H LFc − H LFd ) Ac ⎥⎦ 3.3

formation mechanism of different flow patterns and previous researches.

4.3

1/ 4

(50)

Transition from slug flow to annular flow

At very high gas velocity, the transition from slug flow to annular flow will happen. Most of liquid phase moves upward along the wall in the form of liquid film, and small part is in the form of droplets in gas core. When the film length becomes infinitely long, the transition from slug flow to annular flow occurs, and the momentum exchange between the liquid slug zone and liquid film zone becomes zero. Based on the Zhang H Q transition model [6], the transition criterion from slug flow to annular flow is,

⎡ HLS ( vT − vS ) + vSL ⎤⎦ ( vSG + vSL FE ) − vTvSL FE HLFc + HLFd = ⎣ (51) vTvSG

5

Model verification

Caetano E F [16] studied the flow pattern, liquid holdup and pressure drop gradient of gas- kerosene two-phase flow under different temperature, pressure and apparent velocity in an annulus 76.2 mm in outer diameter and 42.2 mm in inner diameter. According to his experimental conditions, the flow pattern, liquid holdup and pressure drop gradient are predicted

− 403 −

YIN Bangtang et al. / Petroleum Exploration and Development, 2014, 41(3): 399–407

with the new model and Zhang model modified by using the hydraulic diameter. And the results are compared with the experimental results. Taking the slug flow as an example, the solution procedures are as follows: (1) input the basic parameters, calculate vT, lS and FE, evaluate HLS and give the initial value of lF; (2) solve the mass conservation equation based on the auxiliary parameters, obtain VLFc, VLFd, Vc, HLFc, HLFd and HLc; (3) calculate SLFc, SLFd, SIc, SId, fLFc, fLFd, fIc and fId; (4) calculate the friction and HLS; (5) solve lF based on the momentum conservation equation; (6) if lF meets the requirements of iteration precision, the solution is finished, if not, return (1) and restart. Tables 1 and 2 are the comparison of flow pattern prediction results and experimental results in slug flow and annular flow. It can be seen that the new model can predict the flow pattern transition more accurately, and predict slug flow more accurately than annular flow. Table 3 to 6 are the comparison of liquid holdups and pressure gradients predicted by the model proposed in this paper and modified Zhang model and obtained from experiment in slug flow and annular flow. It can be seen that: the error of liquid holdup and pressure gradient predicted by the model proposed in this paper in slug flow and experimental results is less than 15%, and the error of holdup and pressure gradient Table 1

Comparison of flow pattern predicted by models and

that from experiment in slug flow Prediction Prediction results of modified results of the Zhang model new model

Table 2 Comparison of flow pattern predicted from models and experimental results in annular flow vSG/ Test vSL/ No. (m·s−1) (m·s−1)

Experimental results

Prediction results of the new model

Prediction results of modified Zhang model

1

0.104

7.883

AN

AN

SL

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

0.199 0.102 0.199 0.299 0.010 0.042 0.100 0.199 0.399 0.598 0.997 0.003 0.010 0.056 0.112 0.199

7.938 7.061 11.164 10.954 22.435 22.238 21.503 21.320 18.991 16.938 13.859 12.978 12.987 12.547 12.334 11.981

AN AN AN AN AN AN AN AN AN AN AN AN AN AN AN AN

AN AN SL AN SL AN SL AN AN AN SL AN AN AN SL AN

SL SL SL SL AN AN AN AN AN AN AN AN AN AN AN AN

18

0.399

11.545

AN

AN

AN

Note: SL—Slug flow; AN—Annular flow Table 3

Comparison of liquid holdup predicted by models and

experimental results in slug flow Error of Error of modified Experi- Prediction Prediction results the new Zhang of modified mental results of the model/% model/% Zhang model results new model Liquid holdup/%

Test vSL/ No. (m·s−1)

vSG/ (m·s−1)

Experimental results

1

1.007

0.497

SL

SL

SL

2

0.003

0.594

SL

SL

SL

1

3

0.012

0.586

SL

SL

SL

2

43.0

38.8

44.5

39.6

Test No.

72.1

69.9

69.6

3.0

3.5

86.3

9.8

100.6

86.3

11.1

93.9

4

0.019

0.592

SL

BB

SL

3

5

0.128

0.590

SL

SL

SL

4

44.5

39.8

85.9

10.5

93.3

46.8

45.4

83.1

2.8

77.8

6

0.399

0.577

SL

SL

SL

5

7

0.698

0.600

SL

SL

SL

6

54.9

55.5

76.2

1.2

39.0

8

0.997

0.646

SL

SL

SL

7

63.1

61.4

60.1

2.6

4.7

67.6

64.6

64.1

4.4

5.2

48.6

45.4

91.0

6.7

87.1

62.0

81.3

9

0.012

0.428

SL

SL

SL

8

10

0.399

0.422

SL

SL

SL

9

11

0.698

0.444

SL

SL

SL

10

12

1.495

6.502

SL

SL

SL

11

67.4

66.4

30.8

27.2

23.0

11.6

25.2

13

0.995

9.320

SL

BB

SL

12

14

1.196

9.319

SL

SL

SL

13

20.3

18.1

16.1

10.6

20.3

22.3

19.2

17.4

13.6

21.7

15

0.399

1.304

SL

SL

SL

14

16

0.798

1.292

SL

SL

SL

15

43.0

38.0

54.2

11.6

26.0

50.5

46.1

44.7

8.7

11.5

17

0.010

0.194

SL

SL

SL

16

18

0.042

0.191

SL

SL

SL

17

59.0

60.1

96.9

1.9

64.3

19

0.042

0.428

SL

SL

SL

18

62.0

62.3

96.3

0.5

55.3

48.9

47.3

90.2

3.2

84.5

20

0.199

0.329

SL

SL

SL

19

21

0.199

0.993

SL

SL

SL

20

57.7

60.4

89.3

4.8

55.0

21

42.3

37.9

68.2

10.2

61.4

Note: SL—Slug flow; BB—Bubble flow

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YIN Bangtang et al. / Petroleum Exploration and Development, 2014, 41(3): 399–407

Table 4

Comparison of pressure gradient predicted by models

and experimental results in slug flow

Table 6

Pressure gradient/(Pa·m−1)

Test No.

Error of Error of Experi- Prediction Prediction results the new modified Zhang of modified mental results of the model/% model/% Zhang model results new model

1

5 852.210

5 805.824

2 958.223 3 025.452 3 049.871 3 567.307 4 511.609 5 112.092 5 490.021 3 480.631 5 006.750 5 556.018 5 584.924 4 913.494

7 003.198 7 004.640 6 971.167 6 742.705 6 241.664 4 986.408 5 422.145 7 347.481 6 606.055 5 465.080 4 577.155 3 664.149

4.5 3.2 2.4 3.4 3.1 3.5 8.7 0.3 3.9 2.7

6 575.2

5 700.720

4 254.738

13.3

35.3

15

3 660.4

3 291.939

4 626.338

10.1

16

4 731.2

4 149.894

3 912.427

17

4 552.7

4 675.057

18

4 584.4

19

2 3 4 5 6 7 8 9 10 11 12 13

3 097.3 3 125.0 3 125.0 3 450.2 4 374.2 5 298.3 6 012.1 3 470.0 4 818.4 5 710.7

14

Comparison of pressure gradient predicted by models

and experimental results in annular flow

126.1 124.1 123.1 95.4 42.7 5.9 9.8 111.7 37.1 4.3

Pressure gradient/(Pa·m−1)

Test No.

Error of Error of Experi- Prediction Prediction results the new modified Zhang of modified mental results of the model/% model/% Zhang model results new model

1

1 304.837

955.950

2

1 508.633

1 176.887

3

1 358.125

924.511

4

1 396.433

1 369.421

5

1 910.228

1 602.693

6

1 189.7

933.771

892.143

21.5

25.0

7

1 455.4

1 482.992

887.917

1.9

39.0

8

1 467.3

1 245.487

870.604

15.1

40.7

9

1 836.1

1 854.375

923.885

1.0

49.7

10

3 331.2

3 716.967

973.935

11.6

70.8

11

4 164.1

3 886.147

1 037.703

6.7

75.1

12

6 277.8

7 105.177

1 308.038

13.2

79.2

13

317.3

359.305

288.365

13.2

9.1

26.4

14

491.8

441.881

292.014

10.2

40.6

12.3

17.3

15

872.5

962.808

405.289

10.4

53.5

7 830.892

2.7

72.0

16

1 205.6

1 469.349

472.750

21.9

60.8

4 878.778

7 776.815

6.4

69.6

17

1 578.4

1 398.479

569.049

11.4

63.9

3 747.6

3 665.271

7 341.401

2.2

95.9

18

2 696.7

2 608.965

746.140

3.3

72.3

20

4 493.2

4 810.484

7 242.767

7.1

61.2

21

3 244.0

3 073.424

5 642.097

5.3

73.9

Table 5

Comparison of liquid holdup predicted by models and

experimental results in annular flow Error of Liquid holdup/% Error of modified Test Experi- Prediction Prediction results the new Zhang No. mental results of the of modified model/% model/% Zhang model new model results 1

11.7

14.2

10.0

22.0

14.2

2

13.9

14.8

10.9

6.7

21.7

3

12.4

15.4

10.2

23.9

18.0

4

10.7

11.2

10.1

5.3

4.8

5

12.6

12.0

10.8

4.4

14.0

6 7

3.0 3.0

2.3 3.5

0.8 1.2

24.7 15.0

74.3 60.7

8

3.0

3.6

1.7

21.3

44.3

9

5.3

5.7

2.2

7.5

57.7

10

10.0

8.1

3.7

19.3

63.3

11

12.7

10.3

5.2

18.6

59.1

12 13 14 15 16

17.9

13.8 4.2 3.3 4.8 9.2

8.9 1.3 2.1 3.1 3.7

22.7

50.4

9.3 7.7 25.4

29.3 40.4 49.4

17

10.5

4.6

18

12.1

6.4

3.0 5.2 7.4

predicted by the model in this paper in annular flow is less than 25%. In contrast, the average error of those predicted by modified Zhang model is about 50%, and the error is smaller when the superficial liquid velocity is large and the superficial gas velocity is small, and larger when the superficial gas velocity is large and the superficial liquid velocity is small. Therefore, the liquid holdup and pressure gradient can be predicted more accurately by the new hydraulic model of slug flow and annular flow, than by the modified Zhang model.

6

Conclusions

The flow behavior and flow pattern of gas-liquid two phase flow in wellbore annuli are analyzed, and compared with the two phase pipe flow. Based on the slug flow dynamics, taking the liquid film zone as the control unit, and considering the effect of the tubing liquid film, casing liquid film and the droplets in gas core area on the mass and momentum transfer, the mechanistic models for slug flow and annular flow in annulus for vertical or inclined wells are established. Then the mass and momentum conservation equations of slug and annular flow are obtained, and the auxiliary parameters are calculated. The evaluation criterion of flow pattern transition is established, considering the flow behavior of different flow pattern in wellbore annulus and the effect of annulus structure. The predicted flow pattern, liquid hold up and pressure gradient by the models are compared with that by the pipe flow model modified by hydraulic diameter. Both the predic− 405 −

YIN Bangtang et al. / Petroleum Exploration and Development, 2014, 41(3): 399–407

tion results are compared with the experimental data from the previous literatures. The results show that the flow pattern, liquid holdup and the pressure gradient can be predicted more accurately by the new model. The error between liquid holdup and pressure gradient predicted by the slug flow model proposed in this paper and experimental results is less than 15%, and the error between liquid holdup and pressure predicted by the annular flow model proposed in this paper and experimental results is less than 25%. In contrast, those predicted by modified pipe flow model, have an average error of around 50%.

Re—Reynolds number; dc—Diameter of gas bubble, m; C1—Influence factor considering the annulus structure; K—Consistency index of drilling fluid, Pa⋅sn; C2—Influence factor considering the non-newtonian property of drilling fluid; Reb—Reynolds number of gas bubble; n—Flow behavior index of drilling fluid; dco—Outer diameter of wellbore annulus, m; dci—Inner diameter of wellbore annulus, m; fc—Interfacial friction factor of Taylor bubble; fIc—Interfacial friction factor of casing liquid film and Taylor bub-

Nomenclature

ble; fId—Interfacial friction factor of tubing liquid film and Taylor bubble;

HLFc—Liquid holdup of casing liquid film, %; vLFc—Casing liquid film velocity, m/s;

δc, δdp—Thickness of casing and tubing liquid film, m;

vT—Slug translational velocity, m/s;

ρG0—Gas density at atmosphere pressure, kg/m3;

HLFd—Liquid holdup of tubing liquid film, %;

fLFc—Interfacial friction factor of casing pipe wall and casing liq-

vLFd—Tubing liquid film velocity, m/s;

uid film; fLFd—Interfacial friction factor of tubing pipe wall and tubing liq-

HLc—Liquid holdup of Taylor bubble, %; vc—Taylor bubble velocity, m/s;

uid film; ReLFc—Reynolds number of interface between casing liquid film

HLS—Liquid holdup of liquid slug, %; vS—Liquid slug velocity, m/s;

and Taylor bubble; C—Local variable;

ΔtTB, ΔtLS—Time taken by the Taylor bubble and liquid slug to

dLFc—Interface diameter of casing liquid film and Taylor bubble,

pass the cross section, s; ρLF—Density of liquid film, kg/m3;

m;

Ac—Cross sectional area of wellbore annulus, m2;

μLF—Viscosity of casing liquid film, mPa·s;

lU—Length of slug unit, m;

ReLFd—Reynolds number of interface between tubing liquid film

lF—Length of liquid film area, m;

and Taylor bubble; dLFd—Interface diameter of tubing liquid film and Taylor bubble,

lS—Length of liquid slug area, m; vTB—Tayolor bubble rise velocity, m/s;

m;

VLF—Liquid volume change in the liquid film during ΔtTB, m3;

Rec—Reynolds number of interface between Taylor bubble;

3

VLS—Liquid volume change in the liquid film during ΔtLS, m ;

dcc—Diameter of Taylor bubble, m;

VSU—Liquid volume change in the liquid film during ΔtTB+ΔtLS,

μc—Viscosity of Taylor bubble, mPa·s;

m3;

ALFc—Cross sectional area of casing liquid film, m2;

vSL—Superficial liquid velocity, m/s;

ALFd—Cross sectional area of tubing liquid film, m2;

vSG—Superficial gas velocity, m/s;

AcT—Cross sectional area of Taylor bubble, m2;

FE—Liquid entrainment fraction in the gas core, %;

Θ—Wetted wall fraction; Θ0—Minimum wetted wall fraction corresponding to a flat inter-

p1, p2—Inlet and outlet pressure of slug unit, Pa; τIc—Shear stress of casing liquid film and Taylor bubble, Pa;

face; σ—Surface tension of liquid film, N;

SIc—Perimeter of casing liquid film and Taylor bubble, m; τLFc—Shear stress of casing wall and casing liquid film, Pa;

σw—Surface tension of water, N;

SLFc—Perimeter of casing wall, m;

dR—Equivalent diameter of annulus, m;

τId—Shear stress of tubing liquid film and Taylor bubble, Pa;

a, b—Coefficient;

SId—Perimeter of tubing liquid film and Taylor bubble, m;

Dc—Casing diameter, m;

τLFd—Shear stress of tubing wall and casing liquid film, Pa;

Ddp—Tubing diameter, m;

SLFd—Perimeter of tubing wall, m;

ACDc—Cross sectional area embraced by the wetted wall of casing

g—Acceleration of gravity, 9.8 m/s2;

liquid film and its chord, m2;

θ—Inclination angle, (°);

ACDd—Cross sectional area embraced by the wetted wall of tubing

ρc—Density of Taylor bubble, kg/m3;

liquid film and its chord, m2;

ρG—Gas density in Taylor bubble area, kg/m3;

SCDc—Chord length corresponding to the wetted wall fraction of

ρL—Liquid density in Taylor bubble area, kg/m3;

casing liquid film, m; SCDd—Chord length corresponding to the wetted wall fraction of

CS—Ratio of maximum velocity and average velocity in stable slug flow;

tubing liquid film, m;

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YIN Bangtang et al. / Petroleum Exploration and Development, 2014, 41(3): 399–407

33(12): 2120–2125.

R—Ratio of annulus inner diameter to outer diameter;

[11] Gao Y H, Sun B J, Xiang C S, et al. Gas hydrate problems

(dp/dz)T—Pressure gradient of whole slug unit, Pa/m;

during deep water gas well test. Shenzhen: SUTTC, 2012.

(dp/dz)LF—Pressure gradient of liquid film area, Pa/m;

[12] Wang Zhiyuan. Study on annular multiphase flow pattern

(dp/dz)LS—Pressure gradient of liquid slug area, Pa/m;

transition mechanism considering gas hydrate phase transition.

WeG—Weber number;

Dongying: China University of Petroleum (Huadong), 2009.

ReG—Reynolds number of gas phase; μG—Viscosity of gas phase, mPa·s;

[13] Yang Jin, Tang Haixiong, Liu Zhengli, et al. Prediction model

fsm—Friction factor of smooth pipe;

of casing annulus pressure for deepwater well drilling and

+ Fc + Fd

completion operation. Petroleum Exploration and Develop-

h —Dimensionless thickness of casing liquid film;

ment, 2013, 40(5): 616–619.

h —Dimensionless thickness of tubing liquid film;

[14] Sun B J, Gong P B, Wang Z Y. Simulation of gas kick with

fCA—Friction factor of annulus wall;

high H2S content in deep well. Journal of Hydrodynamics,

FCA—Geometric parameters of friction in annulus; Tsm—Sum of momentum exchange between the pipe wall and the

2013, 25(2): 264–273. [15] Sadatomi M, Sato Y, Saruwatari S. Two-phase flow in vertical

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ρS—Density of liquid slug, kg/m3.

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