Propagation velocity and time laws of backpressure wave in the wellbore during managed pressure drilling

Available online at www.sciencedirect.com

ScienceDirect Natural Gas Industry B 5 (2018) 253e260 www.elsevier.com/locate/ngib

Research Article

Propagation velocity and time laws of backpressure wave in the wellbore during managed pressure drilling* Yan Tie a, Qu Junbo a,b,*, Sun Xiaofeng a, Chen Ye a & Pan Yi b a

College of Petroleum Engineering, Northeast Petroleum University, Daqing, Heilongjiang 163318, China b Liaoning Shihua University, Fushun, Liaoning 113001, China Received 31 August 2017; accepted 25 November 2017 Available online 19 June 2018

Abstract When gas invasion, especially overflow, occurs at the bottom hole in the process of managed pressure drilling (MPD), it is common to apply backpressure on the wellbore by adjusting the backpressure pump and throttle valve, so as to rebuild bottom hole pressure balance. If it is still thought that the wellhead backpressure is loaded to the bottom hole instantaneously, there will be larger errors between the calculated wellbore parameters and the actual wellbore flow parameters, which will result in well control failure and even well blowout. In this paper, a pressure wave propagation equation suitable for the gaseliquid two-phase flow in the annulus was established based on the global averaged gaseliquid two-phase flow model to investigate the propagation velocity and time of backpressure wave in the wellbore. Then, gaseliquid interaction was introduced to carry out coupling solution on the equation set. It is shown that pressure wave velocity increases with the increase of drilling mud density, but decreases with the increase of void fraction and virtual mass force coefficient. It changes drastically at first, and then slows down. What's more, when the void fraction is greater than 0.1 or the virtual mass force coefficient exceeds 0.2, the momentum between gas phase and liquid phase is fully exchanged, and the pressure wave velocity decreases slowly, approaching a stable value. In Well Penglai 9 in the Sichuan Basin, for example, the average time of single pressure wave propagation is about 50 s, and the total propagation time of 4 rounds is about 200 s, which accounts for more than 67% of the total time of system control response. It is indicated that the propagation velocity and time of the pressure wave in the annulus calculated by this method can greatly improve the accuracy of managed pressure response time of MPD drilling system and the control precision of adaptive throttle valve. © 2018 Sichuan Petroleum Administration. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Keywords: Managed pressure drilling (MPD); Gas invasion; Gaseliquid two-phase flow model; Virtual mass force coefficient; Gaseliquid two-phase flow; Pressure wave propagation equation; Pressure wave velocity; Managed response time

0. Introduction

* Project supported by General Program of National Natural Science Foundation of China “Wellbore Multi-phase Flow Coupling Mechanism Research e Formations Based on Precise Backpressure Control” (No.: 51474073), Research on Wellbore Temperature Field and Pressure Field Distribution Laws of Deep Unbalanced Drilling Based on Thermal-fluid Coupling” (No.: 51374077). * Corresponding author. College of Petroleum Engineering, Northeast Petroleum University, Daqing, Heilongjiang 163318, China. E-mail address: [email protected] (Qu JB). Peer review under responsibility of Sichuan Petroleum Administration.

Managed pressure drilling (MPD) is a drilling technology that can precisely control the bottom hole pressure and can effectively solve lost circulation, well kick and other drilling safety problems triggered by a narrow safety density window in deep complex formations [1e3]. Currently, there are mainly three ways to control the bottom hole pressure by MPD [4e6]. The first way is to adjust the drilling mud density. In this way, it is necessary to prepare new drilling mud again, so longer time will be spent, and the quantity of formation fluids invading the wellbore is larger in this period. The second way is to adjust the capacity of the inlet mud pump, and increase the flow velocity

https://doi.org/10.1016/j.ngib.2017.11.009 2352-8540/© 2018 Sichuan Petroleum Administration. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

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of the drilling mud in the annulus so as to increase the friction pressure loss to balance the bottom hole pressure. This way is not only affected by the certified capacity of the mud pump, but also contributes little to the adjustment of bottom hole pressure. The third way is to adjust the backpressure pump and throttle valve so as to apply backpressure on the wellbore [7,8], which is the most effective, fastest and most common way to rebuild bottom hole pressure balance. In the case of high pressure reservoir, under the condition of determined drilling mud system and drilling equipment, when formation fluid invasion, especially gas invasion, is monitored in the annulus, the throttle valve is adjusted to produce backpressure at the wellhead to act on the whole wellbore, so that the bottom hole pressure can be balanced gradually with the formation pressure, thus slowing down until restraining the ongoing invasion of formation gas into the wellbore. Compared with the former two ways, this way has a significant advantage in controlling the bottom hole overflow. The backpressure produced by adjusting the wellhead valve is propagated from wellhead to bottom hole in the form of pressure wave, which requires a certain period of time, that is, the propagation time of pressure wave in wellbore. According to related literature [9], when gas invasion does not occur, the flow in the annulus is a single liquid-phase flow of pure drilling mud, and the pressure wave is propagated from wellhead to bottom hole quickly at a velocity of nearly 1200 m/s. When gas invasion occurs, the formation gas invades the wellbore, and the annulus changes from the original pure liquid-phase to gaseliquid two-phase flow. Due to the compressibility of gas and the momentum exchange of interface between gas and liquid phases, the velocity of pressure wave decreases sharply, and the wellhead backpressure wave often needs tens of seconds and even longer time to arrive at the bottom hole [10], during which more formation fluids invade the wellbore. If it is simply thought that the wellhead backpressure is loaded to the bottom hole instantaneously, without considering the propagation time of pressure wave in wellbore and lag of the MPD control response time, then it will be unable to accurately calculate the volume of formation fluids that have invaded the wellbore, especially gas content, and there will be larger errors between the calculated wellbore multi-phase flow parameters and the actual wellbore flow parameters, which may result in well control failures and even well blowouts or other serious drilling accidents. Therefore, studying the propagation velocity and time of the wellhead backpressure in the annulus in the form of pressure wave after MPD gas invasion can greatly improve the accuracy of managed pressure response time of MPD system and the control precision of adaptive throttle valve, and provide a theoretical support for the precise control of MPD.

annular flow field area, and the continuity equation and momentum equation of the two phases are written respectively; then the interaction between gas and liquid interfaces (mass exchange and momentum transfer) is used to couple the equation sets. It is assumed that the gas phase and the liquid phase are continuous media within their own local flow field areas, and thus other artificial assumptions are not needed. This model is applicable to the annular void fraction caused by different degrees of gas invasion and various flow patterns, and the solution obtained contains more comprehensive wellbore flow information. So it is a relatively perfect model for studying the gaseliquid two-phase flow in wellbore after gas invasion. In the global averaged two-fluid model, the advantages of the time averaging method and space averaging method in the two-fluid model are synthesized, and the shortcomings of the two methods are overcome. This model has the strictest derivation process and the most complete equation structure among multiple averaging methods, which can well model the transient phenomena and the propagation laws of pressure wave in gaseliquid two-phase flow of wellbore after gas invasion. The authors used the global averaged two-fluid model, and introduced the source item for the momentum exchange between gas phase and liquid phase and the shear force of the outer wall of drill rods and the well wall to couple the equation set, and establish a propagation equation set of pressure wave in the annular gaseliquid two-phase flow of wellbore. It is assumed that the annulus of wellbore is a uniform cross section. Then for the gaseliquid two-phase flow of wellbore at any dip angle a (Fig. 1), there is not enough time for the mass and energy between gas and liquid phases to exchange because the propagation process of pressure wave is a transient process. Thus it is believed that there is no phase change or mass and energy exchange between gas and liquid phases. Therefore, the momentum exchange between the two phases is considered primarily. Because the wellbore annulus radius is very small as compared to the axial length of wellbore, it can be ignored [13,14]. Thus the three-dimensional gaseliquid two-phase flow of wellbore can be regarded as the one-

1. Establishment of backpressure wave propagation model in gaseliquid two-phase flow of wellbore 1.1. Backpressure wave propagation model and equation In the two-fluid model [11,12], both gas phase and liquid phase are regarded as a continuous medium filling the whole

Fig. 1. Schematic diagram of the gaseliquid two-phase flow model of wellbore in the annulus.

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dimensional two-phase fluid flow along the axial direction of wellbore. It is assumed that there is no mass and energy exchange but only momentum transfer between gas and liquid phases, and this paper mainly deals with the bubbly flow of wellbore, and the annular void fraction 4
0.30. Then, the continuity equation and the momentum equation of gas and liquid phases can be written as follows. 1.1.1. Continuity equation Gas phase: vð4rG Þ vð4rG nG Þ þ ¼0 vt vz Liquid phase: v½ð1  4ÞrL  v½ð1  4ÞrL nL  þ ¼0 vt vz

ð1Þ

ð2Þ

1.1.2. Momentum conservation equation Gas phase:  v v
vp ð4rG nG Þ þ 4rG n2G þ 4 ¼ MGi vt vz vz Liquid phase:

3 CD rL 4jvr jvr ð6Þ 8 rb nd d þ MLi Þ jointly influence the propagaThe two items ðMLi tion of pressure wave. At the same time, the authors consider the influences of factors such as wall shear friction resistance and dip angles of wellbore. Namely: d ¼ MLi



   vvG vvG vvL vvL þ vG þ vL  vt vz vt vz

3 CD rL 4jvr jvr  4rG sin a  tGW 8 rb

3 CD rL 4jvr jvr  ð1  4ÞrL sin a  tLW ð8Þ 8 rb And vr ¼ vGevL. In the equations, 4 represents the void fraction in the annulus of wellbore; similarly, rG: the gas phase density, g/ cm3; rL: the liquid phase density, g/cm3; vG: the gas phase velocity, m/s; vL: the liquid phase velocity, m/s; a: the dip angle of wellbore,  ; Cvm: the virtual mass force coefficient, dimensionless; tGW and tLW: the wall shear friction resistance of gas phase and liquid phase, respectively, N/m2; vr: the relative velocity of gas phase and liquid phase, m/s; CD: the drag force coefficient, dimensionless; and rb: the average bubble radius in bubbly flow, m. MGi and MLi are functions with gaseliquid two-phase flow parameters of wellbore (4, vG, vL) as variables, and are rewritten in the form of the partial derivative as follows: þ

vvG vvL vvG þ Cvm 4rL vL  Cvm 4rL vz vz vt vvL þ f1 þ Cvm 4rL vt

ð3Þ

And the interface momentum exchange item between gas and liquid phases caused by a drag force generated due to different relative velocities [16], which is:



   vvG vvG vvL vvL þ vG þ vL  vt vz vt vz

MGi ¼ Cvm 4rL vG

 v v vp ½ð1  4ÞrL nL  þ ð4Þ ð1  4ÞrL n2L þ ð1  4Þ ¼ MLi vt vz vz In the equations, MGi and MLi respectively represent the source item for momentum exchange caused by the resultant external forces of gas phase and liquid phase. The momentum exchange between gas and liquid phases is the principal factor that influences the propagation of pressure wave in bubbly flow of the wellbore, including the interface momentum exchange item between gas and liquid phases caused by a virtual mass force generated due to different relative accelerations [15], which is:     vvG vvG vvL vvL nd MLi þ vG þ vL ¼ Cvm 4rL  ð5Þ vt vz vt vz

MGi ¼ Cvm 4rL

 MLi ¼ Cvm 4rL

255

ð7Þ

ð9Þ

vvG vvL vvG  Cvm 4rL vL þ Cvm 4rL vz vz vt vvL þ f2 ð10Þ  Cvm 4rL vt Here f1 and f2 are non-differential source items, and

MLi ¼ Cvm 4rL vG

f1 ¼  f2 ¼

3 CD 4rL jvr jvr  4rG sin a  tGW 8 rb

3 CD 4rL jvr jvr  ð1  4ÞrL sin a  tLW 8 vb

ð11Þ ð12Þ

1.2. Solution of the propagation equation of backpressure wave Both gas and liquid phases belong to a compressible fluid phase. The propagation velocity of pressure wave in a pure gas phase and a pure liquid phase can be obtained respectively by using the relationship between pressure and density [17,18]: c2G ¼

vp vrG

ð13Þ

c2L ¼

vp vrL

ð14Þ

here cG and cL respectively represent the velocity of pressure wave in a pure gas phase and a pure liquid phase, m/s. vrk 1 vp Since vrvtk ¼ c12 vp vt and vz ¼ c2k vz ðk ¼ G; LÞ, Equations k (7)e(10) are substituted into Equations (1)e(4) and the following equations are obtained:

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4 vp v4 4vG vp v4 vvG þ rG þ 2 þ rG vG þ 4rG ¼0 c2G vt vt vz cG vz vz

ð15Þ

ðvG  lÞ ðvL  lÞ 4rL 2

2

4 ð1  4Þ þ ½ð1  4ÞrG þ Cvm 4rG rG v2G rL c2L

þ Cvm ð1  4ÞrL  þ ðvG  lÞ 4ð1  4Þ½Cvm rL  ð1  4ÞrG  2

ð1  4Þ vp v4 ð1  4ÞvL vp v4  rL þ  rL v L c2L vt vt vz vz c2L vvL ¼0 þ ð1  4ÞrL vz

ð16Þ

vvG vvL vp  Cvm 4rL þ4 vx vt vt vvG vvL  Cvm 4rL vL ¼ f1 þ ðrG þ Cvm rL Þ4vG vz vz

ð17Þ

ðrG þ Cvm rL Þ4

vvG vvL vp þ ½ð1  4Þ þ Cvm 4rL þ ð1  4Þ vz vt vt vvG vvL þ ½ð1  4Þ þ Cvm 4rL vL ¼ f2  Cvm 4rL vG vz vz  Cvm 4rL

þ ðvL  lÞ2 42 rL ðCvm  1 þ 4Þ ¼ 0 ð22Þ Both gas-phase and liquid-phase flows in the wellbore are low Mach number flows; after simplification of Equation (22) [19], it can be believed that ðvG  lÞ ¼ l and ðvL  lÞ ¼  l, then the equation below is obtained:  4 ð1  4Þ l4 4rL þ ½ð1  4ÞrG þ Cvm 4rG þ Cvm ð1  4ÞrL  rG c2G rL c2L þ l2 4ð1  4Þ½Cvm rL  ð1  4ÞrG 

ð18Þ

þ l2 42 rL ðCvm  1 þ 4Þ ¼ 0 ð23Þ

As a result, the above equation set (18) can be rewritten as the matrix form below, which is a hyperbolic-type equation set in essence. The equation set can be solved by using the method of characteristic line through the introduction of characteristic matrix. Namely:

The equation below is obtained after Equation (23) is solved: h i 912 ð1  4Þ Cvm þ rrGL ð1  4Þ þ 4ðCvm þ 1  4Þ = i l¼± h : 4 þ ð14Þ ½ð1  4Þr þ C 4r þ C ð1  4Þr ; 2 2 vm vm G G L r c r c 8 <

G G

! ! vU vU ! X þY ¼Z ð19Þ vt vz ! ! Here U ¼ ½p; 4; vG ; vL T and Z ¼ ½0; 0; f1 ; f2 T , and the coefficient matrixes X and Y are respectively as follows. 2 4 1 0 0 2 6 rG c G 6 6 ð1  4Þ 6 1 0 0 X¼6 6 rL c2L 6 6 0 ð4rG þ Cvm 4rL Þ Cvm 4rL 40 0

Cvm 4rL

0

3 7 7 7 7 7 7 7 7 5

L L

ð24Þ Mathematically, the eigenvalue l of Equation (24) is the characteristic slope within the plane (z, t) [20,21], that is, l ¼ dz dt . For gaseliquid two-phase flows in the wellbore, the characteristic variable of pressure wave is propagated along the characteristic line dz dt ¼ v±c at a velocity of v ± c, so the pressure wave velocity (c) can be regarded as approximately equal to the eigenvalue of matrix, that is, lz±c, then the expressions of velocity and propagation time of pressure wave of the wellbore in bubbly flow can be obtained: h i 8 912 < ð1  4Þ Cvm þ rrG ð1  4Þ þ 4ðCvm þ 1  4Þ = L i c¼± h : 4 þ ð14Þ ½ð1  4Þr þ C 4r þ C ð1  4Þr ; vm vm G G L r c2 r c2

ð4rL þ Cvm 4rL Þ

G G

L L

ð20Þ 2 4vG 2 6 rG c G 6 6 ð1  4Þv L 6 6 6 rL c2L Y¼6 6 64 6 6 4 4

vG vL 0 0

4

0

3

7 7 7 7 0 4 7 7 7 7 ð4rG vG þ Cvm 4rL vG Þ Cvm 4rL vL 7 7 7 ð1  4ÞrL vL þ 5 Cvm 4rL vG Cvm 4rL vL ð21Þ

By introducing the matrix eigenvalue l of Equation (19), let U ¼ Y  lX and jUj ¼ 0, therefore the equation below is obtained:

ð25Þ h i 912 ð1  4Þ Cvm þ rrGL ð1  4Þ þ 4ðCvm þ 1  4Þ = i T ¼H h : 4 þ ð14Þ ½ð1  4Þr þ C 4r þ C ð1  4Þr ; vm vm G G L r c2 r c2 8 <

G G

L L

ð26Þ here T represents the propagation time of pressure wave, s; and H represents the well depth, m. 1.3. Verification of the propagation velocity of backpressure wave Fig. 2 shows the comparison between the result of Equation (25) and the pressure wave velocity calculated on the basis of the generally accepted Nguyen model [22] and Wallis model [23]. When the void fraction (4)
0.30, the wellbore flow is a

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257

influencing the propagation velocity of backpressure wave in gaseliquid two-phase flow of wellbore. 2.1. Drilling mud density

Fig. 2. Comparison of the pressure wave velocities (c) calculated according to the model in this paper and the Nguyen and Wallis models.

mist flow or a bubbly flow, and the virtual mass force coefficient (Cvm) applicable to the annular bubbly flow is used as the calculating parameter, that is Cvm ¼ 0:5 1þ24 14 . According to Fig. 2, the calculation result of the model in this paper is between that of the Nguyen model and the Wallis model, and it is very close to them, with very high consistency. In the Wallis model, the gaseliquid two-phase flow is simplified as a homogeneous flow model, and the velocity slip between gas and liquid phases is ignored. It is believed that gas and liquid phases form a uniform mixture, and the momentum between gas and liquid phases is fully exchanged; while the Nguyen model does not consider the momentum discontinuity between gas and liquid interfaces at all. In the model of this paper, the momentum exchange between gas and liquid phases and the incomplete degree of momentum exchange are fully considered, including the interphase momentum exchange caused by virtual mass force and drag force between gas and liquid phases, and the influences of factors such as wall shear friction resistance and dip angle of wellbore are also considered, so its calculation result is smaller than that of the Nguyen model and larger than that of the Wallis model. Since the Wallis model also considers the interphase momentum exchange, the calculation model of this paper is much closer to the result of the Wallis model, thus its accuracy is verified. Even a very small amount of gas mixed in the liquid phase (4
0.035) can cause sharp decrease of the pressure wave velocity which almost decreases on a vertical line trend, but within the range of such low void fraction, the results of the three models are completely consistent. When the void fraction of wellbore increases (4 ¼ 0.035e0.300), the pressure wave velocity decreases gradually, but much more slowly than before.

It can be seen from Fig. 3 that when the wellbore void fraction (4) is 0.10, 0.20 and 0.30 respectively, the propagation velocity of pressure wave (c) tends to increase with the increase of drilling mud density. With the increase of drilling mud density, the pressure wave propagation velocity when 4 ¼ 0.10) increases more significantly than that when 4 is 0.20 and 0.30. Essentially, the drilling mud, even if mixed with very little gas, can also significantly reduce the pressure wave propagation velocity. When 4 reaches 0.20, as the drilling mud density increases, the increasing rate of c basically reaches a steady state. Therefore, when 4 is 0.20 and 0.30 respectively, the two curves are basically parallel and very close to each other, which is exactly in line with the variation trend in Fig. 2 that when 4 ranges from 0.20 to 0.30, the pressure wave velocity basically remains constant as void fraction increases. 2.2. Wellbore void fraction According to Fig. 4, as the void fraction increases, the pressure wave velocity decreases. When the pressure wave enters the gas phase from the liquid phase, it has to overcome the surface stress of the bubble film. When it enters the liquid phase from the bubble, for the bubble is easily deformed, the pressure wave is prevented from entering and exiting the bubble, thereby reducing the pressure wave propagation velocity. When the wellbore void fraction (4) is less than or equal to 0.05, the velocity of pressure wave suddenly drops with the increase of void fraction. Especially when pressure wave is close to the pure liquid phase of drilling mud (4 is 0), its propagation velocity is almost mutated. This also shows that even a very small amount of gas mixed in the liquid phase can cause sharp decrease of the pressure wave velocity. When 4 is

2. Factors influencing pressure wave propagation in wellbore Using field MPD data in Refs. [5] and [6] as basic input data of the model, this paper analyzes the main factors

Fig. 3. Influence of drilling mud density (r) on pressure wave velocity (c).

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Fig. 4. Influence of wellbore void fraction (4) on pressure wave velocity (c).

greater than 0.10, the pressure wave velocity decreases with the increase of void fraction, and the decrease tends to be gentle. The virtual mass force between gas and liquid is mainly influenced by void fraction (Ishii bubbly flow model [12] Cvm ¼ 0:5 1þ24 14 ). The pressure wave velocity is much smaller, even less than 1/2 when the virtual mass force is greater than or equal to 1.0 (Cvm ¼ 1.00) than that when the virtual mass force between gas and liquid is not considered (Cvm ¼ 0). However, when Cvm is 1.00, 5.00 and 10.00 respectively, the pressure wave velocity is very close to each other, with a very small difference, which also proves the characteristics of pressure wave velocity that when the void fraction is low (4
0.05), it changes significantly and when the void fraction is high (4 > 0.10), it changes very gently. This is in line with the variation trend of curves in Fig. 2. 2.3. Influence of gaseliquid virtual mass force coefficient The virtual mass force can reflect the interaction intensity between gas and liquid phases and can significantly influence the sufficiency of momentum exchange between the gas and liquid phases, thus influencing the pressure wave propagation velocity and time in the gaseliquid two-phase flow. According to Fig. 5, at different void fractions, when the virtual mass force coefficient is very small (Cvm
0.10), the pressure wave velocity in the gaseliquid two-phase flow can be greatly reduced even if the force between the gas and liquid phases is weak. When Cvm ¼ 0e1.00, with the increase of the virtual mass force coefficient, the momentum exchange at the interface of the gas and liquid phases greatly increases, and the pressure wave velocity is very sensitive to the interaction force between the gas and liquid phases and the pressure wave propagation velocity drops sharply, almost on a vertical line trend. This is mainly due to the effect of the virtual mass force, which enhances the momentum exchange between the gas and liquid phases, thus reducing the pressure wave velocity. When the interaction force between the gas and liquid phases reaches

Fig. 5. Influence of gaseliquid virtual mass force coefficient (Cvm) on pressure wave velocity (c).

a certain degree (Cvm > 2.0), the momentum between gas and liquid phases is fully exchanged. As the virtual mass force continues to increase, the pressure wave velocity is no longer so sensitive to the magnitude of the interaction forces between phases, so pressure wave velocity changes very slowly and gradually tends to reach a stable value. 3. Example analysis Taking the on-site recorded data from MPD tests of Well Penglai 9 in the Sichuan Basin as an example, the authors used the model in this paper to calculate the propagation time of pressure wave in wellbore. The bottom hole assembly of Well Penglai 9 is PDC drill bits (Ø215.9 mm)  0.24 m, double female joints (Ø430 mm & Ø410 mm)  0.57 m, two backpressure valves (Ø411 mm & Ø410 mm)  0.84 m, quantifiers (Ø411 mm & Ø410 mm)  0.54 m, LWD (Ø410 mm & Ø410 mm)  9.44 m, drill collars (Ø165.1 mm)  8.92 m, stabilizers (Ø212.7 mm)  0.85 m, drill collars (Ø165.1 mm), slope drill pipes (Ø127.0 mm), Kelly protective subs (Ø411 mm & Ø410 mm) and lower cocks (Ø411 mm & Ø410 mm)  0.44 m. The selected drilling mud density is from 1.13 to 1.41 kg/L. The well section of MPD tests is from 2033.45 to 2560.00 m. Gas overflow is seen in wellhead. This shows the occurrence of gas invasion at the bottom hole. The outlet gas flow is between 500 and 2500 m3/h and the capacity of inlet drilling mud pump is 28 L/s. Fig. 6 is the changing curve of wellhead backpressure wave propagation time in the wellbore gaseliquid two-phase flow under different well depths in the well section from 2023.45 to 2560.00 m made by using the model in this paper to calculate at the average void fractions (4) in the annular cross section of 0.05, 0.10, 0.20 and 0.30 respectively. The propagation time of pressure wave (T ) from wellhead to bottom hole is 35e42 s, 44e52 s, 46e59 s and 50e63 s, respectively. In this case, the average propagation time of 50 s is taken. Due to the attenuation of backpressure applied on the wellhead in the form of

Yan T. et al. / Natural Gas Industry B 5 (2018) 253e260

Fig. 6. Pressure wave propagation time (T ) corresponding to different well depths (H ).

pressure wave during propagation in the wellbore gaseliquid two-phase flow and the backpressure losses (often very large) caused by other uncertainties, MPD drilling usually needs at least 4 rounds of backpressure valve adjustment to finally balance the formation pressure at the bottom hole. The total time for the entire managed response process, including detecting well hole overflow, regulating wellhead to generate backpressure and rebalancing bottom hole pressure, is about 300 s, in which the propagation time of pressure wave in the annulus reaches about 200 s, accounting for 67% of the total managed response time. This is in line with the on-spot recorded curve data of PCDS-I MPD system in Well Penglai 9. It can be seen that improving the calculation precision of the propagation velocity and time of pressure wave in wellbore can greatly improve the calculation precision of the total MPD managed response time, so we can more accurately calculate the formation fluid volume that has invaded the annulus of the wellbore, which is very conducive to accurately calculating wellbore annulus flow patterns and flow parameters, thus greatly enhancing the precise control of MPD. 4. Conclusions 1) Based on the global averaged gaseliquid two-phase flow model, considering factors such as gaseliquid interaction, wall shear friction and wellbore dip angle, the propagation equation set of backpressure wave in annulus bubbly flow was established and the virtual mass force coefficient, which represents the degree of momentum exchange between gas and liquid, was introduced to couple and solve the equation set, so the formula for calculating the propagation velocity and time of pressure wave in wellbore was obtained. The results are very close to those in previous studies, thus the correctness of the model and calculation formula in this paper was verified. 2) The influences of major influencing factors such as drilling mud density, wellbore void fraction and gaseliquid virtual mass force coefficient on pressure

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wave velocity were discussed. Under the wellbore bubbly flow pattern, pressure wave velocity increases with the increase of drilling mud density, but decreases with the increase of void fraction and virtual mass force coefficient. It changes drastically at first, and then slows down. Even a very small amount of gas invading the wellbore or very weak interaction between gas and fluid phases can cause a sharp decrease of the pressure wave velocity. When the void fraction is greater than or equal to 0.10 or the virtual mass force coefficient is greater than or equal to 2.0, the momentum between the gas and liquid phases is fully exchanged, and the pressure wave velocity decreases slowly, approaching a stable value. 3) With the field data of MPD test of Well Penglai 9 in the Sichuan Basin as an example, the MPD managed response time based on precise wellhead backpressure control was analyzed. The results show that the average time of single pressure wave propagation to the bottom hole generated by the wellhead valve regulation is about 50 s, and the total propagation time of 4 rounds to finally balance the bottom hole pressure is about 200 s, accounting for more than 67% of the total time of system managed response. It is indicated that the pressure wave propagation time in wellbore is a very important part of the total managed control time. Studying the propagation velocity and time of wellhead backpressure wave in the annulus during MPD gas invasion can greatly improve the accuracy of managed pressure response time of MPD drilling system and the precision of adaptive throttle valve, providing a theoretical support for the precise control of MPD.

References [1] Wang Haige, Ge Yunhua & Shi Lin. Technologies in deep and ultra-deep well drilling: present status, challenges and future trend in the 13th FiveYear Plan period (2016e2020). Nat Gas Ind 2017;37(4):1e8. [2] Zhou Yingcao, Cui Meng & Zha Yongjin. Discussion and prospect of managed pressure drilling technology. Pet Drill Tech 2008;36(4):1e4. [3] Yan Ling, Wu Huisheng & Yan Yan. Application of precise MPD in kick & loss wells. Nat Gas Ind 2015;35(2):59e63. [4] Wang Guo, Fan Honghai, Liu Gang, Gong Peibin, Li Yong & Shi Chunlin. Application of managed pressure drilling technique. Pet Drill Tech 2009;37(l):34e8. [5] Zhou Yingcao, Yang Xiongwen, Fang Shiliang, Liu Wei & Ji Rongyi. Development and field test of PCDS-I precise managed pressure drilling system. Pet Drill Tech 2011;39(4):7e12. [6] Zhou Yingcao, Yang Xiongwen, Fang Shiliang, Zhao Qing, Xiao Runde & Tang Gui. Field test and analysis on effect of auto-controlled pressure drilling system in the well Penglai 9. Oil Drill Prod Technol 2011;33(6):19e22. [7] Xu Xiaofeng, Chen Yufeng, Jin Pengbo, Liu Jinjing, Xu Ji & Cheng Hua. Theory and practice of controlling wellhead back pressure in UBD. Nat Gas Technol 2010;4(4):21e3. [8] Zhang Xingquan, Zhou Yingcao, Liu Wei & Jiang Zhibo. Wellhead backpressure control in under-balanced and managed pressure drilling. Nat Gas Ind 2013;33(10):75e9. [9] Yu Jinhai, Sun Ning & Liu Jian. Analysis on pressure controlling response time of delicate controlled pressure drilling. Oil Drill Prod Technol 2011;33(2):21e4.

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[10] Kong Xiangwei, Lin Yuanhua, Qiu Yijie & Yuan Jiming. Research on gaseliquid two phase pressure response time caused by adjusting throttle valve in drilling operation. Drill Prod Technol 2014;37(5): 39e41. [11] Ishii M & Mishima K. Two-fluid model and hydrodynamic constitutive relations. Nucl Eng Des 1984;82(2/3):107e26. [12] Ishii M & Hibiki T. Thermo-fluid dynamics of two-phase flow. 2nd ed. New York: Springer Publishers; 2011. [13] Liu Xiushan. Multiphase simulation technique of drilling fluid pulse transmission along well bore. Acta Pet Sin 2006;27(4):115e8. [14] Meng Yingfeng, Xu Chaoyang, Wei Na, Li Gao, Li Hongtao & Duan Mubai. Numerical simulation and experiment of the annular pressure variation caused by gas kick/injection in wells. J Nat Gas Sci Eng 2015;22:646e55. [15] Park JW, Drew DA, Lahey RT & Clausse Jr RA. Void wave dispersion in bubbly flows. Nucl Eng Des 1990;121(1):1e10. [16] Park JW, Drew DA & Lahey RT. The analysis of void wave propagation in adiabatic mono-dispersed bubbly two-phase flows using an ensemble averaged two-fluid model. Int J Multiphas Flow 1998;24(7):1205e44.

[17] Xu Jinliang & Chen Tingkuan. A study on the characteristics of pulverized-coal air suspension flow in a horizontal-turning-up bend. J Xi'an Jiaotong Univ 1994;28(5):73e80. [18] Tao Bei, Chen Dehua, Che Chengxuan & Wang Xiuming. Study on the sound velocity in a gaseliquid flow. J Appl Acoust 2015;34(4):373e6. [19] Liu Lei, Wang Yaoshe & Zhou Fangde. Vibration hybrid control of plate with controllable constrained damping layer. Chin J Appl Mech 1999;16(3):22e7. [20] Bai Bofeng, Huang Fei, Guo Liejin & Wang Xianyuan. Effect of interphase forces on the propagation of pressure wave in aireliquid bubbly flow. Nucl Power Eng 2003;24(S2):70e4. [21] Tan Lixin, Xu Weilin, Yang Yongquan, Wei Wenli & Li Jianzhong. Research on the closure of control equations of two-phase flow and mathematical model of airewater two-phase flow. J Shaanxi Water Power 2000;16(4):1e4. [22] Nguyen DL, Winter ERF & Greirer M. Sonic velocity in two-phase system. Int J Multiphas Flow 1981;7(3):311e20. [23] Wallis GB. One-dimension two-phase flow. New York: McGraw-Hill Publishers; 1969.