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Wellhead backpressure control strategies and outflow response characteristics for gas kick during managed pressure drilling
Journal of Natural Gas Science and Engineering 75 (2020) 103164
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Wellhead backpressure control strategies and outflow response characteristics for gas kick during managed pressure drilling Youqiang Liao a, b, Xiaohui Sun a, Baojiang Sun a, *, Zhiyuan Wang b, **, Jianbo Zhang a, b, Wenqiang Lou a, b a b
Key Laboratory of Unconventional Oil & Gas Development (China University of Petroleum (East China)), Ministry of Education, Qingdao 266580, PR China School of Petroleum Engineering, China University of Petroleum (East China), Qingdao 266555, PR China
A R T I C L E I N F O
A B S T R A C T
Keywords: Gas kick MPD Gas-liquid-solid flow model Wellhead back-pressure control Outflow response characteristics
This study proposes a transient gas-liquid-solid multiphase flow model for gas kick during MPD, considering the effect of dynamic wellhead back-pressure, temperature field, and velocity relation of different phases. Based on this model, the gas kick control strategies and outflow response characteristics during MPD are thoroughly investigated. The simulated results reveal that, with the upward migration and expansion of the invading gas, the hydrostatic pressure and frictional pressure in the annulus changes accordingly, resulting in a non-linear rela tionship between the wellbore back-pressure and the bottom-hole pressure. Moreover, the effects of an underbalance pressure at the bottom hole, the pressure balance relationship between the formation and the bottom hole, the kick detection level, the well depth on the wellhead back-pressure control and the response behaviors of the outlet flow rate are discussed. The results of this investigation can provide engineering guidance for MPD to address the issue of gas kick.
1. Introduction The tread in oil and gas exploration and the development of associ ated technologies has gradually evolved towards drilling in formations with complex formation pressure systems and a narrow safety density window since the 21st century (Abimbola et al., 2015; Fu et al., 2019). As a result, the number of well kick, leakage, and even blowout acci dents, as well as the difficulty of accident handling and control have increased (Sun et al., 2011; Fu et al., 2020). Against this background, managed pressure drilling technology (MPD) has emerged (Stamnes et al., 2009; Kinik et al., 2015; Udegbunam et al., 2015). This technology aims to achieve fine control of the balance between the bottom hole pressure and the formation pressure or flow rate at the inlet and outlet (Das et al., 2008). It has good adaptability to formations with a narrow safe operating window, such as easy leakage and overflow formation (He et al., 2017). In addition to effectively reducing the frequency of well kick and lost circulation accidents, it also avoids conventional shut-in operations even after the occurrence of gas kick to realize early moni toring and treatment of well kick (Sule et al., 2018). To reduce formation damage during MPD, compared with
conventional drilling, drilling fluid with a lower density is usually used for drilling during MPD and the bottom hole pressure is slightly higher or lower than formation pore pressure (Song et al., 2016). However, due to the uncertainty of formation information, it is very likely that gas influx will occur in abnormally high-pressure gas reservoirs (Sun et al., 2018). Once the high-pressure gas invades the wellbore and returns upwards with the drilling fluid, the gas expands continuously with a decrease in the pressure, and the outlet flow rate and pit gain gradually increase. Karimi and Van (2015), Omrani et al. (2016) and Jiang et al. (2019) investigated the early detection of gas kick during MPD. Their results revealed that when the outlet flow rate and pit gain increase to a certain value, the occurrence of overflow can be detected. A greater wellhead back-pressure can then be applied to suppress the expansion of the invading gas and to prevent the further invasion of the gas. The flow in the annulus becomes a complex gas-liquid-solid three-phase flow due to gas influx and the pressure distribution in the entire wellbore changes with the gas migration (Liao et al., 2019b). Therefore, in the process of MPD, the accurate and rapid assessment of the wellhead back pressure in terms of determining whether the bottom hole pressure is balanced with the formation pressure or exceed a certain safety margin, is of great
* Corresponding author. ** Corresponding author. E-mail addresses: [email protected] (B. Sun), [email protected] (Z. Wang). https://doi.org/10.1016/j.jngse.2020.103164 Received 31 August 2019; Received in revised form 2 January 2020; Accepted 10 January 2020 Available online 18 January 2020 1875-5100/© 2020 Elsevier B.V. All rights reserved.
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significance in the initial control of the overflow and the safe discharge of the gas (He et al., 2017). The existing studies on MPD technology are mainly divided into three categories: more intelligent pressure control equipment, more accurate hydrodynamic model and more perfect control method. With respect to the former, Weatherford has developed high-precision outlet flowmeters and fully-automatic pressure control chokes (Nas, 2010, 2012), and Schlumberger and Halliburton have developed a PWD-based constant bottom hole pressure method (CBHP MPD) (Balanza et al., 2015). With respect to the hydrodynamic model, Bacon et al. (2012) have examined the transient response of compressible, multiphase flow in the annulus using mass conservation over a control volume. Consid ering the gas solubility in the drilling fluid, He et al. (2015), Udegbunam et al. (2015) and Zhou et al. (2018) proposed a transient multiphase hydrodynamic model for kick control of MPD. Gravdal et al. (2009) presented an improved kick management approach during MPD via real-time pore-pressure estimation. This methodology facilitates the automatic estimation of the formation pore pressure based on real-time measurements when a gas kick occurs during MPD. Based on the coupled conservation equations of mass, momentum, and energy in association with appropriate closure relationships, Ma et al. (2016) present a novel multi-phase model that can simulate the transient multi-phase flow behavior of a liquid and a gas in a well. Li and Yin et al. (2014, 2017) proposed a transient multiphase model considering the migration of acid gas in vertical and horizontal wellbores. Sun et al. (2011, 2018, 2019) presented an MPD method based on the calculation of the wellbore pressure using a fully coupled multiphase flow model in which the coupling effect between the reservoir and wellbore, the gas solubility in the drilling fluid and hydrate phase transformation were both consid ered. With respect to the control method, Davoudi et al. (2010) and Smith and Patel (2012) presented initial responses to gas kicks during MPD. However, few studies on multiphase flow behavior and the response characteristics of outlet flow in the MPD process have been published to date. It has been only cursorily considered that the trend of the outlet flow increase can be restrained by increasing the wellhead pressure after the occurrence of overflow. Therefore, the gas-liquid-solid three-phase flow behaviors under dynamic wellhead back-pressure, the control rules of wellhead back-pressure, and the response characteristics of the outlet flow rate at different formation conditions need to be further studied. In this report, we first present a gas-liquid-solid three-phase flow model for gas kick during MPD. Subsequently, the response character istics of gas migration and the outlet flow rate under the condition of dynamic wellhead back-pressure during the overflow process are thor oughly studied. In addition, a control method for wellhead backpressure after gas influx during MPD is obtained. Finally, the effects of an under-balance pressure at the bottom hole, the pressure balance relationship between the formation and the bottom hole, the kick detection level, the well depth on the wellhead back-pressure control and the response behaviors of the outlet flow rate are discussed. The results of this investigation can provide engineering guidance for MPD to address gas kick.
Fig. 1. Schematic layout of an MPD system during gas kick.
have not monitored the overflow and drilling is still conducted with the initial wellhead back pressure (Gravdal et al., 2009). (2) MPD after the detection of the influx refers to the process of rapidly increasing wellhead back-pressure and bottom hole pressure after the increase in flow rate is detected at the surface, and then quickly preventing the overflow (Idris et al., 2019). (3) Managed pressure circulating discharge of intrusive gases in volves the circulation of gases out of the wellbore, based on maintaining a constant bottom hole pressure. In this process, the flow in the wellbore is the complex gas-liquid-solid three-phase flow and with the upward migration and expansion of the gas, the outlet flow rate, hydrostatic pressure, volume fraction of each phase and frictional pressure, etc., will change continually, resulting in a dynamic wellhead back-pressure control. Therefore, in the process of MPD, when drilling into high-pressure gas formation, natural gas enters the wellbore, and a complex gasliquid-solid three-phase flow is formed in the wellbore. A reasonable and accurate wellbore multiphase flow model is of immense significance to the study of the real-time wellhead back pressure control and the response rules of the outlet flow rate. To establish the wellbore multiphase flow model during MPD, the following assumptions are made (He et al., 2017; Wang et al., 2017; Zhang et al., 2019): (1) Drilling fluids and cuttings are incompressible and their flow in the wellbore is one-dimensional; (2) Both gas and liquid phases have the same pressure and temperature; (3) Only gas enters the wellbore, excluding oil and water, and gas influx occurs at the bottom of the well; (4) The gas solubility is not considered in this study;
2. Gas–liquid–solid multiphase flow model for gas kick during MPD The MPD can be summarized as three processes from gas influx to the completion of gas kick treatment: high-pressure gas influx, MPD after the detection of the influx and managed pressure circulating discharge of intrusive gases (as shown in Fig. 1). (1) The high-pressure gas influx process involves the influx of gas into the wellbore, while the increase of the outlet flow rate and pit gain are reduced because the intrusive gas hardly expands in the bottom hole. Therefore, in this process, the MPD operators
2.1. Mass–balance equations As shown in Fig. 1, in the process of conventional MPD, the fluidsolid two-phase flow is observed in the wellbore and the cuttings are generally uniformly distributed in the drilling fluid due to their small 2
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size (Ma et al., 2016; Zhou et al., 2018). Therefore, the mixture in the annulus can be considered as a pseudo-homogeneous liquid. However, when drilling into high-pressure gas formation, the gas invades into the wellbore and migrates upward with the drilling fluid. In this instance, the flow in the wellbore becomes a complex gas-liquid-solid three-phase flow, which is quite different from the original liquid-solid flow due to the invasion, migration, and expansion of the gas phase. According to the conservation of mass, the continuity equations for the gas, liquid, and solid phases can be obtained as follows (Liao et al., 2019b): � � ∂ Aa Eg ρ g ∂ Aa Eg ρ g v g (1) ¼ þ qg ∂t ∂z
Mohammadzadeh et al. (2016), Ghasemikafrudi and Hashemabadi, 2016 and Pang et al. (2018) studied the cuttings transport in a wellbore considering the effects of drill string rotation, drilling fluid properties, and well deviation using CFD simulations. These models are so complex that it is difficult to apply them directly to the field of engineering. Therefore, the empirical correlated cuttings transport equation is often used in research and engineering practice. The empirical correlation of the cuttings slip velocity is presented as reported in (Mohammadzadeh et al., 2016):
∂ðAa El ρl Þ ∂ðAa El ρl vl Þ ¼ ∂t ∂z
(2)
∂ðAa Ec ρc Þ ∂ðAa Ec vc ρc Þ ¼ ∂t ∂z
(3)
where Dc is the cuttings diameter, m; CD is drag function taken from the Schiller-Nauman relation as reported in (Mohammadzadeh et al., 2016): ( 1 þ 0:15Re0:687 Rec � 1000 c CD ¼ (9) 0:0183Rec Rec > 1000
ulc ¼
where Aa is the annulus area, m2; Eg , El , Ec are the volume fraction of the gas, drilling mud, and cuttings, respectively; ρg , ρl , ρc are the density of
Rec ¼
2.2. Velocity relation of different phases The drift-flux model was first proposed by Zuber and Findlay (1965) and has been widely used to describe the phenomenon of gas-liquid two-phase flow in wellbores. (5)
In this model, C0 is the distribution parameter, dimensionless; vm is the average velocity of the mixture, m/s; vgr is the slippage speed, m/s. The flow pattern independent drift flux correlation presented by Wang et al. (2016) can be used to calculate gas slip velocity based on a full-scale experiment as follows: 2 � 1þ
vgr ¼ 1:53
�2 þ
Re 1000
1:2 1þ
� gσ ρl
ρ2l
ρg
�
0:2E4g �2
(6)
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�ffiffi gD ρl ρg Eg 1 þ 0:35
ρl
Eg
�0:25
vc j
(10)
μl
Table 1 Basic parameters and flow test data of Muspac 53 well (Ghobadpouri et al., 2017).
Re 1000
��0:25
Dc El ρl jvl
In order to verify the accuracy of the proposed model, as well as the effect of cuttings transport, three-phase flow in the annulus of an underbalanced drilling well (Muspac 53) is simulated. Table 1 shows the in formation about this well (Ghobadpouri et al., 2017). After the gas-liquid flow enters into annulus and handles cuttings, the flow in wellbore will become three-phase flow. Fig. 2 shows the annulus pressure distribution along the well, which were simulated using the model proposed in this study, gas–liquid–solid three-phase flow model of Ghobadpouri et al. (2017), gas-liquid two- phase flow model and field data, respectively. As shown, three-phase flow analysis gives relatively better results compared to two-phase flow analysis for bottom hole pressure. In this regard, the similar finding was also indicated by Perez-Tellez et al. (2003) and Gho badpouri et al. (2017). Additionally, the average error of the gas-liquid-solid three-phase flow simulation presented in this study is approximately 2.14% less than the gas-liquid two-phase flow simulation. Additionally, we also verify the accuracy of the results in the pre diction of transient BHP under dynamic choke pressure through a fullscale experiment performed in a test well located in Tianjin, China by Wang et al. (2016). The experimental wellbore is composed of a 244.5 mm diameter casing and a 127 mm diameter drill string. The gas in jection line has a tubing with a diameter of 60.3 mm that is connected to the experimental wellbore at a depth of 600 m. Water was chosen as the drilling fluid and air was used to represent the formation gas in this
where Kg is the gas permeability in the formation, μm2 ; pp is the for mation pore pressure, MPa; pwf is the bottom hole pressure, MPa; ρg is the density of the gas at the bottom, which can be calculated using the gas state equation proposed by Peng and Robinson (1976), kg= m3 ; μg is the gas viscosity, Pa⋅s; S is the skin factor; t is the time since the reservoir section was initially influenced by the well pressure; φ is the porosity of the reservoir; c is the compressibility of the reservoir fluid; rw is the well radius, m; and γ is the Euler–Mascheroni constant (γ � 0.577).
C0 ¼
(8)
3. Model verification
w
vg ¼ C0 vm þ vgr
ρm ÞD2c ðg vm ⋅rvm Þ 18μl ρl CD
where the Reynolds number for cuttings phase is defined as:
the gas, drilling mud and cuttings, respectively, kg=m3 ; vg , vl , vc are the velocity of the gas, drilling mud and cuttings, respectively, m= s; t is the simulation time, s; z is the distance from the wellhead, m; qg is the mass flow rate of the invasion gas in the high-pressure gas reservoir and wellbore interaction, kg=ðm ⋅sÞ, which can be determined according to the reservoir characteristics. The following model is used to determine the gas influx rate (Gravdal et al., 2009; Xu et al., 2018). � 4πKg pp pwf ρg � �� qg ¼ � (4) μg 2S þ log eγ φ4Kμ gcrt 2 g
ρl ðρc
(7)
Parameter
Value
Parameter
Value
Surface temperature (� C) Geothermal gradient (� C/m)
28 2.83
2605 55.6
Drilling fluid density (kg/m3)
940
149.2 2597 0.310 24
Solid density (kg/m )
2800
Gas molecular weight Gas flow rate in standard condition (m3/min) Liquid flow rate (m3/min) ROP (m/h)
28.02 15.008
Wellbore depth (m) Drill string inner diameter (mm) Annuls inner diameter (mm) Annuls outer diameter (mm) Borehole diameter (mm) Casing shoe depth (m)
0.503 6.0
Choke pressure (MPa) Viscosity (mPa s)
3
where Eg is the gas volume fraction, dimensionless; Re is the three-phase Reynolds number, dimensionless; σ is the liquid surface tension, N/m; D is the equivalent diameter of the annulus, m. With respect to the velocity relation of the liquid and solid phases, 3
152.5 177.8
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Fig. 3. Comparison between the experimental pressure given in the literature and the simulated results.
Fig. 2. Comparison for the pressure profile calculated by the present gas–liquid–solid three-phase flow model, gas–liquid–solid three-phase flow model of Ghobadpouri et al. (2017), gas-liquid two-phase flow model and field data.
Table 3 Basic parameters used in the simulation.
full-scale experiment. The basic parameters in the full-scale experiment is shown as Table 2. Fig. 3 presents the comparison between the experimental pressure given in the literature and the simulated results. The variation of the bottom hole pressure for the two methods is the same, with a difference of less than 10%.
Parameter
4. Results and discussion To thoroughly investigate the control rules of the wellhead back pressure and the response characteristics of the outlet flow rate during gas influx in the process of MPD, a series of numerical simulations were performed using the multiphase flow model. The basic parameters used in the simulation are summarized in Table 3 (He et al., 2017). The thermos–physical parameters of the heat transfer mediums are sum marized in Table 4 (Liao et al., 2019a). It is noteworthy that the Bing ham fluid is used to describe the rheological properties of drilling fluid, and its rheological parameters are obtained by regression fitting of readings (3,6, 100, 200, 300 and 600 rpm) of Couette type viscometer. More in-depth studies on the calculations of the rheological parameters can be found in the literatures presented by Gul and van Oort (2019) and Gul et al. (2020).
Table 2 Basic parameters in the full-scale experiment. Parameter
Value 3
Wellbore depth (m) Casing shoe depth (m) Drill pipe outer diameter (mm) Drill pipe inner diameter (mm) Casing outer diameter (mm)
600 318 127
Injection rate (m /s) Injection temperature (� C) Temperature of surface (� C)
0.02267 25 20
108.6
Geothermal gradient (� C/m)
0.03
244.5
1000
Casing inner diameter (mm)
220.5
Drilling fluid density (kg/ m3) Viscosity (mPa s)
Value 3
Well depth (m) Casing shoe depth (m)
4000 2500
Injection rate (m /s) Injection temperature (� C)
Open hole size (mm)
215.9
Temperature of surface (� C)
20
Drill pipe outer diameter (mm) Drill pipe inner diameter (mm) Casing outer diameter (mm) Casing inner diameter (mm) Drilling fluid density (kg/m3) Plastic viscosity (Pa s) Yield point (Pa)
127 108.6 244.5 220.5 215.9 0.024 12
Geothermal gradient (� C/m) Gas permeability (mD) Formation supply radius (m) Formation pressure (MPa) Rate of penetration, m/h Initial casing pressure (MPa) Kick detecting level (m3)
0.03 88 50 54 7.3 1 0.636
0.03 25
Item
Density (kg/ m3)
Specific heat capacity (J=ðkg ⋅� CÞ)
Heat conductivity (J=ðs ⋅m ⋅� CÞ)
Drilling fluid Formation rock Drill pipe/ Casing Cement sheath
1150 2650
4000 999
0.72 2.09
7810
880
43.2
2000
500
0.71
pressure should be increased rapidly, so that the further occurrence of gas influx can be suppressed. Then, the bottom hole pressure should be maintained for circulating discharge of intrusive gases until the gas kick treatment is completed. This is known as constant bottom hole pressure method (CBHP MPD) (Balanza et al., 2015). Based on this method, Fig. 4 shows the variation curves of the drillpipe pressure, wellhead back-pressure and bottom-hole pressure during gas kick control. The gas in the wellbore continuously expands in the return upward process, and the hydrostatic pressure and frictional resistance in the annulus change accordingly. Therefore, there is a non-linear relationship between the wellbore back-pressure and the bottom-hole pressure. During drilling into the high-pressure gas reservoir, the gas invades the wellbore, which leads to a decrease in the bottom-hole pressure due to the decrease in the hydrostatic pressure. The invading gas hardly expands because the bottom-hole pressure is relatively high; therefore, the bottom hole pressure tends to decrease in an approximately linear manner. When gas kick is detected on the surface, the wellhead backpressure and bottom-hole pressure rapidly increase to prevent further
4.1.1. Control of bottom hole pressure and wellhead backpressure During MPD, three processes are completed from gas influx to gas kick control: high-pressure gas influx, MPD after the detection of the influx, and managed pressure circulating discharge of intrusive gases. When gas kick is detected, the wellhead backpressure and bottom-hole
Value
Parameter
Table 4 Thermal parameters of the heat transfer mediums.
4.1. Results
Parameter
Value
1
4
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Fig. 4. Variation curves of drillpipe pressure, wellhead back-pressure and bottom-hole pressure during gas kick control.
gas influx into the formation by adjusting the opening of the choke. During MPD operation, when the gas influx reaches the surface, the hydrostatic pressure in the wellbore decreases significantly with the upward migration and expansion of the gas. Therefore, to maintain the constant pressure at the bottom-hole, it is necessary to increase the wellhead back-pressure. When the pit gain reaches its maximum, i.e., the total volume of the gas in the wellbore is at its maximum and the hydrostatic pressure is at its minimum, the wellhead back-pressure reaches its maximum value. As shown in Fig. 4, the simulation results show that the peak value of the wellhead back-pressure is 0.59 MPa higher than that of the under-balance pressure (defined as the difference between formation pressure and bottom-hole pressure) due to gas influx. Subsequently, as the gas is discharged, the required wellhead backpressure decreases and then remains stable until the gas is completely discharged. Additionally, same as the bottom hole pressure, an approximately linear decrease in the drillpipe pressure is occurred during gas influx. Then, to maintain the constant pressure at the bottom-hole, the drillpipe pressure is maintained at a constant pressure until the gas kick treatment is completed. This principle is also applied in the gas kick circulation with the concept of driller’s method. To avoid the recurrence of gas influx, field operators adjust the wellhead back pressure to keep the drillpipe pressure constant and thereby ensure the bottom hole pressure unchanged. Therefore, the decrease of the hydrostatic pressure caused by inva sive gas expansion should be considered in the actual process of MPD and a safety margin should be added according to the under-balance pressure at the bottom-hole.
Fig. 5. Response characteristics of the outlet flow rate during gas kick control. Note: kick detecting level is defined as the minimum pit gain for the detection of gas kick, which represents the level of overflow detection technology.
decreases gradually. When the outlet flow rate is equal to the inlet flow rate, the pit gain reaches its peak value. Then, the pit gain decreases gradually with the gas discharge. The flow of the inlet and outlet will then be balanced until the gas discharge is completed. 4.2. Discussion In the process of MPD, the control rules for the wellhead backpressure are affected by the under-balance pressure at the bottomhole, the pressure balance relationship at the bottom-hole, the gas kick detection level, and the well depth. A more detailed understanding of the response characteristics of the bottom-hole pressure and outlet flow rate under these conditions plays a very important role in the control of the wellbore pressure and gas kick.
4.1.2. Outflow response characteristics Fig. 5 shows the response characteristics of the outlet flow rate during gas kick control. As shown in this figure, when drilling in abnormally high-pressure gas reservoirs, high-pressure gas invades the wellbore. Initially, the invading gas hardly expands due to the high pressure at the bottom-hole. Therefore, the pit gain and the outlet flow rate increase in an approximately linear relationship. When the pit gain reaches the level of gas kick detection, the increase of the wellhead back-pressure causes the pit gain and outlet flow rate to decrease suddenly in a short time interval. As the gas continuously mi grates upwards and expands, the pit gain and outlet flow rate will further increase at an increasing rate until the gas reaches the wellhead and the outlet flow rate is a maximum. Subsequently, the flow at the outlet is gas-liquid-solid three-phase flow, and the outlet flow rate
4.2.1. Under-balance pressure at the bottom-hole In the process of MPD, the maximum pressure for which the choke can work effectively is approximately 5 MPa. Therefore, considering the additional back-pressure caused by gas expansion, it is of great engi neering significance to study the initial under-balance pressure at the bottom-hole and the corresponding maximum wellhead back-pressure to determine whether or it can exhaust the gas via the MPD instead of 5
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the shut-in operation of the well. Fig. 6 shows the wellhead back-pressure control curves under the different under-balance pressure at the bottom hole of 1.82, 2.82, 3.82, 4.82 and 3.42 MPa. It is determined that under the same level of gas kick detection, the larger the initial bottom-hole pressure difference, the earlier the overflow is discovered, the higher the back-pressure applied at the same time and the higher the peak value of the wellhead backpressure. The simulation results show that when the under-balance pressure of the bottom-hole exceeds 3.42 MPa, the peak value of the wellhead back-pressure reaches the maximum working pressure of the choke. Therefore, if the differential pressure at the time of the kick is less than 3.42 MPa, the kick can be circulated using the MPD system. Otherwise, the well has to be shut-in and the kick must be circulated out using the conventional well control equipment and methods. Fig. 7 shows the response characteristics of the outlet flow rate during gas kick for the under-balance pressure at the bottom hole of 1.82, 2.82, 3.82, and 4.82 MPa. It is evident from the figure that the greater the initial bottom-hole pressure difference, the more intense is the gas influx; correspondingly, the shorter is the time taken to reach the gas kick detection level. Considering the same level of overflow detec tion, that is, the total gas volume in the wellbore is the same when overflow is determined, the peak value of the outlet flow is basically the same. However, the higher the bottom-hole pressure, the shorter is the gas-liquid mixing section and higher is the gas void fraction. Therefore, the valley value of the outlet flow rate will be lower when the gas rea ches the wellhead.
Fig. 7. Response characteristics of outlet flow rate for under-balance pressures of 1.82, 2.82, 3.82, 4.82, and 3.42 MPa at the bottom hole.
4.2.2. Pressure balance relationship between formation and bottom-hole A reasonable wellhead back-pressure plays an extremely important role in controlling the gas kick, protecting the wellhead choke equip ment and reducing the risk of circulating exhaust. In addition, an excessive wellhead back-pressure can control gas influx, but if it exceeds the rated working pressure of the choke, it will lead to failure of backpressure application, resulting in secondary gas influx and even blowout accident. However, if the wellhead pressure is too small, the gas influx cannot be effectively controlled. Fig. 8 and Fig. 9 show the curves for the wellhead back-pressure control and the outlet flow rate response characteristics for the condi tion of an under-balance pressure of 1 MPa, balance and over-balance pressure of 1 MPa at the bottom-hole, respectively. It can be deter mined that corresponding to the bottom-hole pressure, the maximum wellhead back-pressure is required for the over-balance condition. When the over-pressure value is 1 MPa, it exceeds the pressure–bearing ca pacity of the wellhead choke and it is impossible to conduct managed
Fig. 8. Wellhead back-pressure control curves for the conditions of 1 MPa under-balance, and balance and over-balance of 1 MPa at the bottom-hole.
Fig. 9. Response characteristics of the outlet flow rate for the conditions of 1 MPa under-balance, and balance and overbalance of 1 MPa at the bottom-hole.
Fig. 6. Wellhead back-pressure control curves for under-balance pressures of 1.82, 2.82, 3.82, 4.82, and 3.42 MPa at the bottom-hole. 6
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pressure circulating exhaust. For the condition of under-balance at the bottom-hole, gas continuously invades into the wellbore, the peak value of the outlet flow is a maximum, and gas is always contained in the wellhead return, so the gas influx cannot be effectively controlled. 4.2.3. Kick detecting level Fig. 10 and Fig. 11 show the curves for the wellhead back-pressure control and outlet flow rate response characteristics under different gas kick detection levels, respectively. It is evident from these two fig ures that the higher the gas kick detection level, the smaller the pit gain corresponding to the overflow detection, the smaller the maximum wellhead back-pressure required, the lower the difference of the outlet flow rate (the smaller peak value and the larger valley value), the lower the required back-pressure control device, and the shorter the time required for circulating the exhaust. When the pit gain reaches 1.836 m3, it is difficult to deal with the gas kick via MPD because the maximum wellhead back-pressure exceeds the rated working pressure of the wellhead equipment. In this case, only the shut-in killing operation can be used to control a well kick accident. The earlier the overflow is detected, the less the gas that enters into the wellbore, the lower the gas holdup of the annulus, the lower the wellhead back-pressure required in the cycle exhaust stage, and the shorter the gas-liquid mixture section, therefore, the exhaust speed is greater. In the actual process of MPD, it is determined that improving the kick detecting level not only reduces the maximum back-pressure at the wellhead, but also reduces the cycle exhaust time, which is more conducive to the control of gas kick.
Fig. 11. Response characteristics of the outlet flow rate under different gas kick detection levels.
wellbore back pressure is similar. 5. Conclusions In this study, a gas-liquid-solid three-phase flow model for gas kick during MPD is presented to investigate the gas kick control and outflow responses during managed pressure drilling. The effects of the underbalance pressure at the bottom-hole, the pressure balance relationship between the formation and the bottom-hole, the kick detection level and well depth on the wellhead back-pressure control and the response rules of the outlet flow rate were analyzed by conducting numerical simula tions. The main conclusions of this study are as follows:
4.2.4. Well depth Fig. 12 and Fig. 13 show the curves of the wellhead back-pressure control and outlet flow rate response characteristics for different well depths, respectively. It can be determined from the figures that the deeper the well depth, the longer the time for pressure control circu lating exhaust. The maximum wellhead back-pressure increases with the increase of the well depth, although the increase is very small and almost negligible. This is because when overflow occurs, the volume of gas invading the wellbore is the same, and when gas moves to the wellbore, the reduced hydrostatic pressure is similar, therefore the applied
(1) initially, the invading gas hardly expands due to the high pressure at the bottom-hole. Therefore, the increases of the pit gain and the outlet flow rate, and the decrease of the bottom-hole pressure both exhibit an approximately linear relationship. With the
Fig. 10. Wellhead back-pressure control curves for different gas kick detection levels. 7
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(2) The invading gas in the wellbore continuously expands in the return upward process, the hydrostatic pressure and frictional resistance in the annulus changes accordingly, resulting in a nonlinear relationship between the wellbore back-pressure and the bottom-hole pressure. Moreover, due to the decrease of the hy drostatic pressure in the wellbore, the peak value of the wellbore back-pressure is 0.59 MPa higher than that of the under-balance pressure at the bottom-hole (in this case). (3) When the under-balance pressure of the bottom hole exceeds 3.42 MPa, the peak value of the wellhead back-pressure exceeds the maximum working pressure of the choke, therefore the shutin operation can only be used at this time instead of circulating discharge of intrusive gases via MPD. (4) The higher the gas kick detection level, the smaller the maximum wellhead back-pressure required. When the pit gain reaches 1.836 m3, it is difficult to deal with the gas kick via MPD, because the maximum wellhead back-pressure exceeds the rated working pressure of the wellhead equipment. Only the shut-in killing operation can be used to control a well kick accident in this case. (5) The deeper the well depth, the longer the time for pressure con trol circulating exhaust. The maximum wellhead back-pressure increases with the increase of the well depth, although the in crease is very small and almost negligible.
Fig. 12. Wellhead back-pressure control curves for different well depths.
Declaration of competing interest None. CRediT authorship contribution statement Youqiang Liao: Writing - review & editing. Xiaohui Sun: Method ology. Baojiang Sun: Writing - review & editing. Zhiyuan Wang: Writing - original draft. Jianbo Zhang: Formal analysis. Wenqiang Lou: Formal analysis. Acknowledgments The work was supported by the National Natural Science Foundation of China (No. 51890914). The National Natural Science Founda tion–Outstanding Youth Foundation (51622405), the Shandong Natural Science funds for Distinguished Young Scholar (JQ201716), CNOOC-KJ 135 ZDXM 05 LTD 06 SHENHAI, 2016 and CNOOC-KJ 135 ZDXM 24 LTD ZJ 04.
Fig. 13. Response characteristics of outlet flow rate for different well depths.
continuous upward migration and expansion of the invading gas, the pit gain and outlet flow rate further increase, and the speed increases continuously until the gas reaches the wellhead and the outlet flow rate reaches its peak.
Appendix A. Pressure and temperature field equations (1) Momentum–balance equation In the process of MPD, gas influx, migration, and expansion will not only affect the upward velocity of the drilling fluid and cuttings, resulting in changes in the frictional pressure drop in the annulus, but will also reduce the hydrostatic pressure in the annulus due to the decrease in the average density of the mixture. Therefore, the gas kick plays an important role in the distribution of the wellbore pressure. Based on the conservation of momentum, the momentum conservation equation in the annulus can be written as (Sun et al., 2011) � � � ∂ Aa Eg ρg v2g þ Aa El ρl v2l þ Aa Ec ρc v2c þ Aa p ∂ Aa Eg ρg vg þ Aa El ρl vl þ Aa Ec ρc vc þ (A.1) ∂t ∂z � ¼ Aa Fw Aa Eg ρg þ Aa El ρl g cos θ where p is the pressure in the annulus, MPa; Fw is the frictional force, N; g is the gravitational acceleration, m=s2 ; θ is the deviation angle of the wellbore. (2) Temperature field equations 8
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The calculation of the intrusive gas physical parameters is based on the accurate simulation of the temperature field, which is of immense sig nificance to the study of gas migration and distribution in the wellbore. During drilling, a dynamic heat transfer process occurs among the drill string, annulus, and formation. According to the conservation of energy, the heat transfer equations between the drill string, annulus, and formation are as follows (Liao et al., 2019a).
∂Tp ∂Tp 2πrpi Up Ta þ vp ¼ ∂t ∂z Cf ρf Ap ∂Ta ∂Ta ¼ va ∂t ∂z
A Ta
�
� Tp þ BðTe;0
�
ρe Ce
Tp
∂Te ∂2 Te 1 ∂Te þ ¼ Ke ∂t ∂r2 r ∂r
(A.2) (A.3)
Ta Þ
� (A.4)
where Ta , Tp and Te are the temperatures in the annulus, drill pipe and formation, respectively, � C; vp and va are the drilling fluid flow velocity in the drill pipe and annulus, respectively, m/s; Cf and Ce are the specific heat of the drilling fluid and the formation, respectively, J=ðkg ⋅� CÞ; Ke is the heat conductivity of formation, w=ðm ⋅� CÞ. A and B are intermediate variables for heat transfer which can be expressed as follows. �� � � A ¼ 2π rpi Up ðρCÞgls Aa �� � (A.5) B ¼ 2π rci Ua ðρCÞgls Aa where rpi and rci are the inner diameter of the drill pipe and casing, respectively, m; ðρCÞgls is the effective product of the density and specific heat
capacity of the fluid in the annulus, J=ðm3 ⋅� CÞ; Ua is the overall heat transfer coefficient in the annulus and Up is the overall heat transfer coefficient in the drill pipe. The calculation method is as follows. (Hasan and Kabir, 2012; 2012): � � 1 rpi ln rpo rpi rpi Up 1 ¼ þ þ (A.6) kp hpi rpo hpo 8 1 rci lnðrco =rci Þ rco lnðrw =rco Þ > 1 > þ þ > Ua ¼ < hci kc kcem
ð0 < h � Hcas Þ
> > 1 > : Ua 1 ¼ hhole
(A.7) ðHcas < h � Hwell Þ
where, hci , hhole , hpi and hpo are the convection heat transfer coefficient of the inner wall of the casing, inner wall of the hole, inner wall of the drill pipe and the outer wall of the drill pipe, respectively, w=ðm2 ⋅� CÞ; kp , kc and kcem are the heat conductivity of the drill pipe, the casing and the formation, respectively, w=ðm ⋅� CÞ; Hcas and Hwell are the depth of the casing and well, respectively, m; rpo , rco and rw are the inner diameter of drill pipe, casing and wellbore, respectively, m. Appendix B. Discretization scheme and numerical solution The flow in the annulus becomes a complex gas-liquid-solid three-phase flow due to the gas influx, and the pressure distribution in the entire wellbore changes with the gas migration and due to changes in the wellhead back pressure, frictional pressure drop, and hydrostatic pressure. Therefore, for this complex model, a direct solution is extremely difficult. Wang et al. (2016) proposed a pressure-based method to calculate the gas-liquid two-phase flow in the wellbore, which has proven to exhibit good accuracy and convergence. Based on this method, the numerical solution is shown in Fig. 3. Firstly, combining Eqs. (A.2)–(A.7)in the temperature field equations, the temperature fields in the wellbore can be solved. Then, assuming the distribution of pressure in the wellbore, the three-phase mass conservation equation is solved. The mass conservation equations for discretizing is as follows. Aj Ekgj ρkgj
1 k 1 k 1 Aj Ekgj 1 ρkgj 1 Ajþ1 Ekgjþ1 ρkgjþ1 vkgjþ1 Ajþ1 Ekgjþ1 ρgjþ1 vgjþ1 ¼ þ qkgj Δt Δz
(B.1)
Aj Eklj ρklj
Aj Eklj 1 ρklj 1 Ajþ1 Ekljþ1 ρkljþ1 vkljþ1 Ajþ1 Ekljþ11 ρkljþ11 vkljþ11 ¼ Δt Δz
(B.2)
Aj Ekcj ρkcj
Aj Ekcj 1 ρkcj 1 Ajþ1 Ekcjþ1 ρkcjþ1 vkcjþ1 Ajþ1 Ekcjþ11 ρkcjþ11 vkcjþ11 ¼ Δt Δz
(B.3)
where, qkgj
��k 4πKg pp pwf �j ρkgj � �� ¼ � μg 2S þ log eγ φ4Kμ gcrt 2 g
(B.4)
w
Combining Eqs. (5)–(10) that describe the velocity relation of the different phases, the gas velocity, liquid velocity, cuttings velocity, void fraction, 9
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liquid holdup and solid concentration are given as follows: Ekgj ¼
Kg vkgj þ bg
Eklj ¼ 1 vklj ¼
Kl
Ekgj
(B.5)
bl Eklj
(B.7)
Eklj
vkgj ¼ C0 Kg þ Kl Ekcj ¼
(B.6)
Ekcj
� bg þ vgr
(B.8)
bc Aj Ekcj 1 þ Ajþ1 Ekcjþ1 vkcjþ1 bc Aj þ Aj vkcj
vkcj ¼ vklj
(B.9) (B.10)
vlc
where 8 Ajþ1 Ekgjþ1 ρkgjþ1 vkgjþ1 Ekgj 1 ρkgj 1 Δz > > Kg ¼ þ bg þ Mg > k k k > A A ρ ρ > j gj j ρgj gj > > > > < Ajþ1 Ekljþ1 ρkljþ1 vkljþ1 Eklj 1 ρklj 1 Kl ¼ þ bl > > Aj ρklj ρklj > > > > > > > Δz : bg ¼ bl ¼ bc ¼ Δt
(B.11)
Using trial and error, the gas velocity, liquid velocity, cuttings velocity, void fraction, liquid holdup and solid concentration can be calculated. Then the momentum equation is used to correct the trial and error. Finally, the iteration is repeated until the gas velocity, liquid velocity, cuttings velocity, void fraction, liquid holdup and solid concentration satisfy the given convergence conditions.
Fig. B.1. Flow chart of the calculation process.
Appendix C. Supplementary data Supplementary data to this article can be found online at https://doi.org/10.1016/j.jngse.2020.103164.
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Nomenclature Aa Ap C CD Cf Cg C0 D Dc E Fw g hci hhole hpi hpo Hcas Hwell kc kcem Kg P pp pwf r rce rci rco rpi rpo rw Re Rec S t Ta Te Tp ulc Ua Up v va vgr vp z
ρ ðρCÞgls
cross–sectional area of annulus, m2 cross–sectional area of drill pipe, m2 compressibility of the reservoir fluid, 1/Pa drag coefficient, dimensionless specific heat of drilling fluid, J=ðkg ⋅� CÞ specific heat of gas, J=ðkg ⋅� CÞ distribution parameter, dimensionless wellbore diameter, m cuttings diameter, m volume fraction frictional force, N gravitational acceleration, m=s2 convection heat transfer coefficient of inner wall of casing, w=ðm2 ⋅� CÞ convection heat transfer coefficient of inner wall of hole, w=ðm2 ⋅� CÞ convection heat transfer coefficient of inner wall of drill pipe, w=ðm2 ⋅� CÞ convection heat transfer coefficient of outer wall of drill pipe, w=ðm2 ⋅� CÞ depth of casing, m depth of well, m heat conductivity of cuttings, w=ðm ⋅� CÞ heat conductivity of cement sheath, w=ðm ⋅� CÞ gas permeability in the formation, μm2 pressure, Pa formation pore pressure, MPa bottom hole pressure, MPa distance from the wellbore, m radius of the wellbore, m inner diameter of casing, m outer diameter of casing, m inner diameter of drill pipe, m outer diameter of drill pipe, m inner diameter of wellbore, m three phase Reynolds number, dimensionless Reynolds number of cuttings, dimensionless skin factor, dimensionless drilling time, s annulus temperature, � C formation temperature, � C drill pipe temperature, � C relative velocity of cuttings and drilling fluid, m/s overall heat transfer coefficient in the annulus, w=ðm2 ⋅� CÞ overall heat transfer coefficient in the drill pipe, w=ðm2 ⋅� CÞ velocity, m/s velocity of drilling fluid in annulus, m/s slippage speed, m/s velocity of drilling fluid in drill pipe, m/s distance from the wellhead, m density, kg=m3 the effective product of the density and specific heat capacity of the fluid in annulus, J=ðm3 ⋅� CÞ
φ γ
deviation angle liquid surface tension, N/m viscosity, Pa⋅s porosity Euler–Mascheroni constant (γ � 0.577)
Subscripts F g l c m
drilling mud gas liquid cuttings mixture
θ
σ μ
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References
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Contents lists available at ScienceDirect
Journal of Natural Gas Science and Engineering journal homepage: http://www.elsevier.com/locate/jngse
Wellhead backpressure control strategies and outflow response characteristics for gas kick during managed pressure drilling Youqiang Liao a, b, Xiaohui Sun a, Baojiang Sun a, *, Zhiyuan Wang b, **, Jianbo Zhang a, b, Wenqiang Lou a, b a b
Key Laboratory of Unconventional Oil & Gas Development (China University of Petroleum (East China)), Ministry of Education, Qingdao 266580, PR China School of Petroleum Engineering, China University of Petroleum (East China), Qingdao 266555, PR China
A R T I C L E I N F O
A B S T R A C T
Keywords: Gas kick MPD Gas-liquid-solid flow model Wellhead back-pressure control Outflow response characteristics
This study proposes a transient gas-liquid-solid multiphase flow model for gas kick during MPD, considering the effect of dynamic wellhead back-pressure, temperature field, and velocity relation of different phases. Based on this model, the gas kick control strategies and outflow response characteristics during MPD are thoroughly investigated. The simulated results reveal that, with the upward migration and expansion of the invading gas, the hydrostatic pressure and frictional pressure in the annulus changes accordingly, resulting in a non-linear rela tionship between the wellbore back-pressure and the bottom-hole pressure. Moreover, the effects of an underbalance pressure at the bottom hole, the pressure balance relationship between the formation and the bottom hole, the kick detection level, the well depth on the wellhead back-pressure control and the response behaviors of the outlet flow rate are discussed. The results of this investigation can provide engineering guidance for MPD to address the issue of gas kick.
1. Introduction The tread in oil and gas exploration and the development of associ ated technologies has gradually evolved towards drilling in formations with complex formation pressure systems and a narrow safety density window since the 21st century (Abimbola et al., 2015; Fu et al., 2019). As a result, the number of well kick, leakage, and even blowout acci dents, as well as the difficulty of accident handling and control have increased (Sun et al., 2011; Fu et al., 2020). Against this background, managed pressure drilling technology (MPD) has emerged (Stamnes et al., 2009; Kinik et al., 2015; Udegbunam et al., 2015). This technology aims to achieve fine control of the balance between the bottom hole pressure and the formation pressure or flow rate at the inlet and outlet (Das et al., 2008). It has good adaptability to formations with a narrow safe operating window, such as easy leakage and overflow formation (He et al., 2017). In addition to effectively reducing the frequency of well kick and lost circulation accidents, it also avoids conventional shut-in operations even after the occurrence of gas kick to realize early moni toring and treatment of well kick (Sule et al., 2018). To reduce formation damage during MPD, compared with
conventional drilling, drilling fluid with a lower density is usually used for drilling during MPD and the bottom hole pressure is slightly higher or lower than formation pore pressure (Song et al., 2016). However, due to the uncertainty of formation information, it is very likely that gas influx will occur in abnormally high-pressure gas reservoirs (Sun et al., 2018). Once the high-pressure gas invades the wellbore and returns upwards with the drilling fluid, the gas expands continuously with a decrease in the pressure, and the outlet flow rate and pit gain gradually increase. Karimi and Van (2015), Omrani et al. (2016) and Jiang et al. (2019) investigated the early detection of gas kick during MPD. Their results revealed that when the outlet flow rate and pit gain increase to a certain value, the occurrence of overflow can be detected. A greater wellhead back-pressure can then be applied to suppress the expansion of the invading gas and to prevent the further invasion of the gas. The flow in the annulus becomes a complex gas-liquid-solid three-phase flow due to gas influx and the pressure distribution in the entire wellbore changes with the gas migration (Liao et al., 2019b). Therefore, in the process of MPD, the accurate and rapid assessment of the wellhead back pressure in terms of determining whether the bottom hole pressure is balanced with the formation pressure or exceed a certain safety margin, is of great
* Corresponding author. ** Corresponding author. E-mail addresses: [email protected] (B. Sun), [email protected] (Z. Wang). https://doi.org/10.1016/j.jngse.2020.103164 Received 31 August 2019; Received in revised form 2 January 2020; Accepted 10 January 2020 Available online 18 January 2020 1875-5100/© 2020 Elsevier B.V. All rights reserved.
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Journal of Natural Gas Science and Engineering 75 (2020) 103164
significance in the initial control of the overflow and the safe discharge of the gas (He et al., 2017). The existing studies on MPD technology are mainly divided into three categories: more intelligent pressure control equipment, more accurate hydrodynamic model and more perfect control method. With respect to the former, Weatherford has developed high-precision outlet flowmeters and fully-automatic pressure control chokes (Nas, 2010, 2012), and Schlumberger and Halliburton have developed a PWD-based constant bottom hole pressure method (CBHP MPD) (Balanza et al., 2015). With respect to the hydrodynamic model, Bacon et al. (2012) have examined the transient response of compressible, multiphase flow in the annulus using mass conservation over a control volume. Consid ering the gas solubility in the drilling fluid, He et al. (2015), Udegbunam et al. (2015) and Zhou et al. (2018) proposed a transient multiphase hydrodynamic model for kick control of MPD. Gravdal et al. (2009) presented an improved kick management approach during MPD via real-time pore-pressure estimation. This methodology facilitates the automatic estimation of the formation pore pressure based on real-time measurements when a gas kick occurs during MPD. Based on the coupled conservation equations of mass, momentum, and energy in association with appropriate closure relationships, Ma et al. (2016) present a novel multi-phase model that can simulate the transient multi-phase flow behavior of a liquid and a gas in a well. Li and Yin et al. (2014, 2017) proposed a transient multiphase model considering the migration of acid gas in vertical and horizontal wellbores. Sun et al. (2011, 2018, 2019) presented an MPD method based on the calculation of the wellbore pressure using a fully coupled multiphase flow model in which the coupling effect between the reservoir and wellbore, the gas solubility in the drilling fluid and hydrate phase transformation were both consid ered. With respect to the control method, Davoudi et al. (2010) and Smith and Patel (2012) presented initial responses to gas kicks during MPD. However, few studies on multiphase flow behavior and the response characteristics of outlet flow in the MPD process have been published to date. It has been only cursorily considered that the trend of the outlet flow increase can be restrained by increasing the wellhead pressure after the occurrence of overflow. Therefore, the gas-liquid-solid three-phase flow behaviors under dynamic wellhead back-pressure, the control rules of wellhead back-pressure, and the response characteristics of the outlet flow rate at different formation conditions need to be further studied. In this report, we first present a gas-liquid-solid three-phase flow model for gas kick during MPD. Subsequently, the response character istics of gas migration and the outlet flow rate under the condition of dynamic wellhead back-pressure during the overflow process are thor oughly studied. In addition, a control method for wellhead backpressure after gas influx during MPD is obtained. Finally, the effects of an under-balance pressure at the bottom hole, the pressure balance relationship between the formation and the bottom hole, the kick detection level, the well depth on the wellhead back-pressure control and the response behaviors of the outlet flow rate are discussed. The results of this investigation can provide engineering guidance for MPD to address gas kick.
Fig. 1. Schematic layout of an MPD system during gas kick.
have not monitored the overflow and drilling is still conducted with the initial wellhead back pressure (Gravdal et al., 2009). (2) MPD after the detection of the influx refers to the process of rapidly increasing wellhead back-pressure and bottom hole pressure after the increase in flow rate is detected at the surface, and then quickly preventing the overflow (Idris et al., 2019). (3) Managed pressure circulating discharge of intrusive gases in volves the circulation of gases out of the wellbore, based on maintaining a constant bottom hole pressure. In this process, the flow in the wellbore is the complex gas-liquid-solid three-phase flow and with the upward migration and expansion of the gas, the outlet flow rate, hydrostatic pressure, volume fraction of each phase and frictional pressure, etc., will change continually, resulting in a dynamic wellhead back-pressure control. Therefore, in the process of MPD, when drilling into high-pressure gas formation, natural gas enters the wellbore, and a complex gasliquid-solid three-phase flow is formed in the wellbore. A reasonable and accurate wellbore multiphase flow model is of immense significance to the study of the real-time wellhead back pressure control and the response rules of the outlet flow rate. To establish the wellbore multiphase flow model during MPD, the following assumptions are made (He et al., 2017; Wang et al., 2017; Zhang et al., 2019): (1) Drilling fluids and cuttings are incompressible and their flow in the wellbore is one-dimensional; (2) Both gas and liquid phases have the same pressure and temperature; (3) Only gas enters the wellbore, excluding oil and water, and gas influx occurs at the bottom of the well; (4) The gas solubility is not considered in this study;
2. Gas–liquid–solid multiphase flow model for gas kick during MPD The MPD can be summarized as three processes from gas influx to the completion of gas kick treatment: high-pressure gas influx, MPD after the detection of the influx and managed pressure circulating discharge of intrusive gases (as shown in Fig. 1). (1) The high-pressure gas influx process involves the influx of gas into the wellbore, while the increase of the outlet flow rate and pit gain are reduced because the intrusive gas hardly expands in the bottom hole. Therefore, in this process, the MPD operators
2.1. Mass–balance equations As shown in Fig. 1, in the process of conventional MPD, the fluidsolid two-phase flow is observed in the wellbore and the cuttings are generally uniformly distributed in the drilling fluid due to their small 2
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size (Ma et al., 2016; Zhou et al., 2018). Therefore, the mixture in the annulus can be considered as a pseudo-homogeneous liquid. However, when drilling into high-pressure gas formation, the gas invades into the wellbore and migrates upward with the drilling fluid. In this instance, the flow in the wellbore becomes a complex gas-liquid-solid three-phase flow, which is quite different from the original liquid-solid flow due to the invasion, migration, and expansion of the gas phase. According to the conservation of mass, the continuity equations for the gas, liquid, and solid phases can be obtained as follows (Liao et al., 2019b): � � ∂ Aa Eg ρ g ∂ Aa Eg ρ g v g (1) ¼ þ qg ∂t ∂z
Mohammadzadeh et al. (2016), Ghasemikafrudi and Hashemabadi, 2016 and Pang et al. (2018) studied the cuttings transport in a wellbore considering the effects of drill string rotation, drilling fluid properties, and well deviation using CFD simulations. These models are so complex that it is difficult to apply them directly to the field of engineering. Therefore, the empirical correlated cuttings transport equation is often used in research and engineering practice. The empirical correlation of the cuttings slip velocity is presented as reported in (Mohammadzadeh et al., 2016):
∂ðAa El ρl Þ ∂ðAa El ρl vl Þ ¼ ∂t ∂z
(2)
∂ðAa Ec ρc Þ ∂ðAa Ec vc ρc Þ ¼ ∂t ∂z
(3)
where Dc is the cuttings diameter, m; CD is drag function taken from the Schiller-Nauman relation as reported in (Mohammadzadeh et al., 2016): ( 1 þ 0:15Re0:687 Rec � 1000 c CD ¼ (9) 0:0183Rec Rec > 1000
ulc ¼
where Aa is the annulus area, m2; Eg , El , Ec are the volume fraction of the gas, drilling mud, and cuttings, respectively; ρg , ρl , ρc are the density of
Rec ¼
2.2. Velocity relation of different phases The drift-flux model was first proposed by Zuber and Findlay (1965) and has been widely used to describe the phenomenon of gas-liquid two-phase flow in wellbores. (5)
In this model, C0 is the distribution parameter, dimensionless; vm is the average velocity of the mixture, m/s; vgr is the slippage speed, m/s. The flow pattern independent drift flux correlation presented by Wang et al. (2016) can be used to calculate gas slip velocity based on a full-scale experiment as follows: 2 � 1þ
vgr ¼ 1:53
�2 þ
Re 1000
1:2 1þ
� gσ ρl
ρ2l
ρg
�
0:2E4g �2
(6)
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�ffiffi gD ρl ρg Eg 1 þ 0:35
ρl
Eg
�0:25
vc j
(10)
μl
Table 1 Basic parameters and flow test data of Muspac 53 well (Ghobadpouri et al., 2017).
Re 1000
��0:25
Dc El ρl jvl
In order to verify the accuracy of the proposed model, as well as the effect of cuttings transport, three-phase flow in the annulus of an underbalanced drilling well (Muspac 53) is simulated. Table 1 shows the in formation about this well (Ghobadpouri et al., 2017). After the gas-liquid flow enters into annulus and handles cuttings, the flow in wellbore will become three-phase flow. Fig. 2 shows the annulus pressure distribution along the well, which were simulated using the model proposed in this study, gas–liquid–solid three-phase flow model of Ghobadpouri et al. (2017), gas-liquid two- phase flow model and field data, respectively. As shown, three-phase flow analysis gives relatively better results compared to two-phase flow analysis for bottom hole pressure. In this regard, the similar finding was also indicated by Perez-Tellez et al. (2003) and Gho badpouri et al. (2017). Additionally, the average error of the gas-liquid-solid three-phase flow simulation presented in this study is approximately 2.14% less than the gas-liquid two-phase flow simulation. Additionally, we also verify the accuracy of the results in the pre diction of transient BHP under dynamic choke pressure through a fullscale experiment performed in a test well located in Tianjin, China by Wang et al. (2016). The experimental wellbore is composed of a 244.5 mm diameter casing and a 127 mm diameter drill string. The gas in jection line has a tubing with a diameter of 60.3 mm that is connected to the experimental wellbore at a depth of 600 m. Water was chosen as the drilling fluid and air was used to represent the formation gas in this
where Kg is the gas permeability in the formation, μm2 ; pp is the for mation pore pressure, MPa; pwf is the bottom hole pressure, MPa; ρg is the density of the gas at the bottom, which can be calculated using the gas state equation proposed by Peng and Robinson (1976), kg= m3 ; μg is the gas viscosity, Pa⋅s; S is the skin factor; t is the time since the reservoir section was initially influenced by the well pressure; φ is the porosity of the reservoir; c is the compressibility of the reservoir fluid; rw is the well radius, m; and γ is the Euler–Mascheroni constant (γ � 0.577).
C0 ¼
(8)
3. Model verification
w
vg ¼ C0 vm þ vgr
ρm ÞD2c ðg vm ⋅rvm Þ 18μl ρl CD
where the Reynolds number for cuttings phase is defined as:
the gas, drilling mud and cuttings, respectively, kg=m3 ; vg , vl , vc are the velocity of the gas, drilling mud and cuttings, respectively, m= s; t is the simulation time, s; z is the distance from the wellhead, m; qg is the mass flow rate of the invasion gas in the high-pressure gas reservoir and wellbore interaction, kg=ðm ⋅sÞ, which can be determined according to the reservoir characteristics. The following model is used to determine the gas influx rate (Gravdal et al., 2009; Xu et al., 2018). � 4πKg pp pwf ρg � �� qg ¼ � (4) μg 2S þ log eγ φ4Kμ gcrt 2 g
ρl ðρc
(7)
Parameter
Value
Parameter
Value
Surface temperature (� C) Geothermal gradient (� C/m)
28 2.83
2605 55.6
Drilling fluid density (kg/m3)
940
149.2 2597 0.310 24
Solid density (kg/m )
2800
Gas molecular weight Gas flow rate in standard condition (m3/min) Liquid flow rate (m3/min) ROP (m/h)
28.02 15.008
Wellbore depth (m) Drill string inner diameter (mm) Annuls inner diameter (mm) Annuls outer diameter (mm) Borehole diameter (mm) Casing shoe depth (m)
0.503 6.0
Choke pressure (MPa) Viscosity (mPa s)
3
where Eg is the gas volume fraction, dimensionless; Re is the three-phase Reynolds number, dimensionless; σ is the liquid surface tension, N/m; D is the equivalent diameter of the annulus, m. With respect to the velocity relation of the liquid and solid phases, 3
152.5 177.8
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Fig. 3. Comparison between the experimental pressure given in the literature and the simulated results.
Fig. 2. Comparison for the pressure profile calculated by the present gas–liquid–solid three-phase flow model, gas–liquid–solid three-phase flow model of Ghobadpouri et al. (2017), gas-liquid two-phase flow model and field data.
Table 3 Basic parameters used in the simulation.
full-scale experiment. The basic parameters in the full-scale experiment is shown as Table 2. Fig. 3 presents the comparison between the experimental pressure given in the literature and the simulated results. The variation of the bottom hole pressure for the two methods is the same, with a difference of less than 10%.
Parameter
4. Results and discussion To thoroughly investigate the control rules of the wellhead back pressure and the response characteristics of the outlet flow rate during gas influx in the process of MPD, a series of numerical simulations were performed using the multiphase flow model. The basic parameters used in the simulation are summarized in Table 3 (He et al., 2017). The thermos–physical parameters of the heat transfer mediums are sum marized in Table 4 (Liao et al., 2019a). It is noteworthy that the Bing ham fluid is used to describe the rheological properties of drilling fluid, and its rheological parameters are obtained by regression fitting of readings (3,6, 100, 200, 300 and 600 rpm) of Couette type viscometer. More in-depth studies on the calculations of the rheological parameters can be found in the literatures presented by Gul and van Oort (2019) and Gul et al. (2020).
Table 2 Basic parameters in the full-scale experiment. Parameter
Value 3
Wellbore depth (m) Casing shoe depth (m) Drill pipe outer diameter (mm) Drill pipe inner diameter (mm) Casing outer diameter (mm)
600 318 127
Injection rate (m /s) Injection temperature (� C) Temperature of surface (� C)
0.02267 25 20
108.6
Geothermal gradient (� C/m)
0.03
244.5
1000
Casing inner diameter (mm)
220.5
Drilling fluid density (kg/ m3) Viscosity (mPa s)
Value 3
Well depth (m) Casing shoe depth (m)
4000 2500
Injection rate (m /s) Injection temperature (� C)
Open hole size (mm)
215.9
Temperature of surface (� C)
20
Drill pipe outer diameter (mm) Drill pipe inner diameter (mm) Casing outer diameter (mm) Casing inner diameter (mm) Drilling fluid density (kg/m3) Plastic viscosity (Pa s) Yield point (Pa)
127 108.6 244.5 220.5 215.9 0.024 12
Geothermal gradient (� C/m) Gas permeability (mD) Formation supply radius (m) Formation pressure (MPa) Rate of penetration, m/h Initial casing pressure (MPa) Kick detecting level (m3)
0.03 88 50 54 7.3 1 0.636
0.03 25
Item
Density (kg/ m3)
Specific heat capacity (J=ðkg ⋅� CÞ)
Heat conductivity (J=ðs ⋅m ⋅� CÞ)
Drilling fluid Formation rock Drill pipe/ Casing Cement sheath
1150 2650
4000 999
0.72 2.09
7810
880
43.2
2000
500
0.71
pressure should be increased rapidly, so that the further occurrence of gas influx can be suppressed. Then, the bottom hole pressure should be maintained for circulating discharge of intrusive gases until the gas kick treatment is completed. This is known as constant bottom hole pressure method (CBHP MPD) (Balanza et al., 2015). Based on this method, Fig. 4 shows the variation curves of the drillpipe pressure, wellhead back-pressure and bottom-hole pressure during gas kick control. The gas in the wellbore continuously expands in the return upward process, and the hydrostatic pressure and frictional resistance in the annulus change accordingly. Therefore, there is a non-linear relationship between the wellbore back-pressure and the bottom-hole pressure. During drilling into the high-pressure gas reservoir, the gas invades the wellbore, which leads to a decrease in the bottom-hole pressure due to the decrease in the hydrostatic pressure. The invading gas hardly expands because the bottom-hole pressure is relatively high; therefore, the bottom hole pressure tends to decrease in an approximately linear manner. When gas kick is detected on the surface, the wellhead backpressure and bottom-hole pressure rapidly increase to prevent further
4.1.1. Control of bottom hole pressure and wellhead backpressure During MPD, three processes are completed from gas influx to gas kick control: high-pressure gas influx, MPD after the detection of the influx, and managed pressure circulating discharge of intrusive gases. When gas kick is detected, the wellhead backpressure and bottom-hole
Value
Parameter
Table 4 Thermal parameters of the heat transfer mediums.
4.1. Results
Parameter
Value
1
4
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Fig. 4. Variation curves of drillpipe pressure, wellhead back-pressure and bottom-hole pressure during gas kick control.
gas influx into the formation by adjusting the opening of the choke. During MPD operation, when the gas influx reaches the surface, the hydrostatic pressure in the wellbore decreases significantly with the upward migration and expansion of the gas. Therefore, to maintain the constant pressure at the bottom-hole, it is necessary to increase the wellhead back-pressure. When the pit gain reaches its maximum, i.e., the total volume of the gas in the wellbore is at its maximum and the hydrostatic pressure is at its minimum, the wellhead back-pressure reaches its maximum value. As shown in Fig. 4, the simulation results show that the peak value of the wellhead back-pressure is 0.59 MPa higher than that of the under-balance pressure (defined as the difference between formation pressure and bottom-hole pressure) due to gas influx. Subsequently, as the gas is discharged, the required wellhead backpressure decreases and then remains stable until the gas is completely discharged. Additionally, same as the bottom hole pressure, an approximately linear decrease in the drillpipe pressure is occurred during gas influx. Then, to maintain the constant pressure at the bottom-hole, the drillpipe pressure is maintained at a constant pressure until the gas kick treatment is completed. This principle is also applied in the gas kick circulation with the concept of driller’s method. To avoid the recurrence of gas influx, field operators adjust the wellhead back pressure to keep the drillpipe pressure constant and thereby ensure the bottom hole pressure unchanged. Therefore, the decrease of the hydrostatic pressure caused by inva sive gas expansion should be considered in the actual process of MPD and a safety margin should be added according to the under-balance pressure at the bottom-hole.
Fig. 5. Response characteristics of the outlet flow rate during gas kick control. Note: kick detecting level is defined as the minimum pit gain for the detection of gas kick, which represents the level of overflow detection technology.
decreases gradually. When the outlet flow rate is equal to the inlet flow rate, the pit gain reaches its peak value. Then, the pit gain decreases gradually with the gas discharge. The flow of the inlet and outlet will then be balanced until the gas discharge is completed. 4.2. Discussion In the process of MPD, the control rules for the wellhead backpressure are affected by the under-balance pressure at the bottomhole, the pressure balance relationship at the bottom-hole, the gas kick detection level, and the well depth. A more detailed understanding of the response characteristics of the bottom-hole pressure and outlet flow rate under these conditions plays a very important role in the control of the wellbore pressure and gas kick.
4.1.2. Outflow response characteristics Fig. 5 shows the response characteristics of the outlet flow rate during gas kick control. As shown in this figure, when drilling in abnormally high-pressure gas reservoirs, high-pressure gas invades the wellbore. Initially, the invading gas hardly expands due to the high pressure at the bottom-hole. Therefore, the pit gain and the outlet flow rate increase in an approximately linear relationship. When the pit gain reaches the level of gas kick detection, the increase of the wellhead back-pressure causes the pit gain and outlet flow rate to decrease suddenly in a short time interval. As the gas continuously mi grates upwards and expands, the pit gain and outlet flow rate will further increase at an increasing rate until the gas reaches the wellhead and the outlet flow rate is a maximum. Subsequently, the flow at the outlet is gas-liquid-solid three-phase flow, and the outlet flow rate
4.2.1. Under-balance pressure at the bottom-hole In the process of MPD, the maximum pressure for which the choke can work effectively is approximately 5 MPa. Therefore, considering the additional back-pressure caused by gas expansion, it is of great engi neering significance to study the initial under-balance pressure at the bottom-hole and the corresponding maximum wellhead back-pressure to determine whether or it can exhaust the gas via the MPD instead of 5
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the shut-in operation of the well. Fig. 6 shows the wellhead back-pressure control curves under the different under-balance pressure at the bottom hole of 1.82, 2.82, 3.82, 4.82 and 3.42 MPa. It is determined that under the same level of gas kick detection, the larger the initial bottom-hole pressure difference, the earlier the overflow is discovered, the higher the back-pressure applied at the same time and the higher the peak value of the wellhead backpressure. The simulation results show that when the under-balance pressure of the bottom-hole exceeds 3.42 MPa, the peak value of the wellhead back-pressure reaches the maximum working pressure of the choke. Therefore, if the differential pressure at the time of the kick is less than 3.42 MPa, the kick can be circulated using the MPD system. Otherwise, the well has to be shut-in and the kick must be circulated out using the conventional well control equipment and methods. Fig. 7 shows the response characteristics of the outlet flow rate during gas kick for the under-balance pressure at the bottom hole of 1.82, 2.82, 3.82, and 4.82 MPa. It is evident from the figure that the greater the initial bottom-hole pressure difference, the more intense is the gas influx; correspondingly, the shorter is the time taken to reach the gas kick detection level. Considering the same level of overflow detec tion, that is, the total gas volume in the wellbore is the same when overflow is determined, the peak value of the outlet flow is basically the same. However, the higher the bottom-hole pressure, the shorter is the gas-liquid mixing section and higher is the gas void fraction. Therefore, the valley value of the outlet flow rate will be lower when the gas rea ches the wellhead.
Fig. 7. Response characteristics of outlet flow rate for under-balance pressures of 1.82, 2.82, 3.82, 4.82, and 3.42 MPa at the bottom hole.
4.2.2. Pressure balance relationship between formation and bottom-hole A reasonable wellhead back-pressure plays an extremely important role in controlling the gas kick, protecting the wellhead choke equip ment and reducing the risk of circulating exhaust. In addition, an excessive wellhead back-pressure can control gas influx, but if it exceeds the rated working pressure of the choke, it will lead to failure of backpressure application, resulting in secondary gas influx and even blowout accident. However, if the wellhead pressure is too small, the gas influx cannot be effectively controlled. Fig. 8 and Fig. 9 show the curves for the wellhead back-pressure control and the outlet flow rate response characteristics for the condi tion of an under-balance pressure of 1 MPa, balance and over-balance pressure of 1 MPa at the bottom-hole, respectively. It can be deter mined that corresponding to the bottom-hole pressure, the maximum wellhead back-pressure is required for the over-balance condition. When the over-pressure value is 1 MPa, it exceeds the pressure–bearing ca pacity of the wellhead choke and it is impossible to conduct managed
Fig. 8. Wellhead back-pressure control curves for the conditions of 1 MPa under-balance, and balance and over-balance of 1 MPa at the bottom-hole.
Fig. 9. Response characteristics of the outlet flow rate for the conditions of 1 MPa under-balance, and balance and overbalance of 1 MPa at the bottom-hole.
Fig. 6. Wellhead back-pressure control curves for under-balance pressures of 1.82, 2.82, 3.82, 4.82, and 3.42 MPa at the bottom-hole. 6
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pressure circulating exhaust. For the condition of under-balance at the bottom-hole, gas continuously invades into the wellbore, the peak value of the outlet flow is a maximum, and gas is always contained in the wellhead return, so the gas influx cannot be effectively controlled. 4.2.3. Kick detecting level Fig. 10 and Fig. 11 show the curves for the wellhead back-pressure control and outlet flow rate response characteristics under different gas kick detection levels, respectively. It is evident from these two fig ures that the higher the gas kick detection level, the smaller the pit gain corresponding to the overflow detection, the smaller the maximum wellhead back-pressure required, the lower the difference of the outlet flow rate (the smaller peak value and the larger valley value), the lower the required back-pressure control device, and the shorter the time required for circulating the exhaust. When the pit gain reaches 1.836 m3, it is difficult to deal with the gas kick via MPD because the maximum wellhead back-pressure exceeds the rated working pressure of the wellhead equipment. In this case, only the shut-in killing operation can be used to control a well kick accident. The earlier the overflow is detected, the less the gas that enters into the wellbore, the lower the gas holdup of the annulus, the lower the wellhead back-pressure required in the cycle exhaust stage, and the shorter the gas-liquid mixture section, therefore, the exhaust speed is greater. In the actual process of MPD, it is determined that improving the kick detecting level not only reduces the maximum back-pressure at the wellhead, but also reduces the cycle exhaust time, which is more conducive to the control of gas kick.
Fig. 11. Response characteristics of the outlet flow rate under different gas kick detection levels.
wellbore back pressure is similar. 5. Conclusions In this study, a gas-liquid-solid three-phase flow model for gas kick during MPD is presented to investigate the gas kick control and outflow responses during managed pressure drilling. The effects of the underbalance pressure at the bottom-hole, the pressure balance relationship between the formation and the bottom-hole, the kick detection level and well depth on the wellhead back-pressure control and the response rules of the outlet flow rate were analyzed by conducting numerical simula tions. The main conclusions of this study are as follows:
4.2.4. Well depth Fig. 12 and Fig. 13 show the curves of the wellhead back-pressure control and outlet flow rate response characteristics for different well depths, respectively. It can be determined from the figures that the deeper the well depth, the longer the time for pressure control circu lating exhaust. The maximum wellhead back-pressure increases with the increase of the well depth, although the increase is very small and almost negligible. This is because when overflow occurs, the volume of gas invading the wellbore is the same, and when gas moves to the wellbore, the reduced hydrostatic pressure is similar, therefore the applied
(1) initially, the invading gas hardly expands due to the high pressure at the bottom-hole. Therefore, the increases of the pit gain and the outlet flow rate, and the decrease of the bottom-hole pressure both exhibit an approximately linear relationship. With the
Fig. 10. Wellhead back-pressure control curves for different gas kick detection levels. 7
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Journal of Natural Gas Science and Engineering 75 (2020) 103164
(2) The invading gas in the wellbore continuously expands in the return upward process, the hydrostatic pressure and frictional resistance in the annulus changes accordingly, resulting in a nonlinear relationship between the wellbore back-pressure and the bottom-hole pressure. Moreover, due to the decrease of the hy drostatic pressure in the wellbore, the peak value of the wellbore back-pressure is 0.59 MPa higher than that of the under-balance pressure at the bottom-hole (in this case). (3) When the under-balance pressure of the bottom hole exceeds 3.42 MPa, the peak value of the wellhead back-pressure exceeds the maximum working pressure of the choke, therefore the shutin operation can only be used at this time instead of circulating discharge of intrusive gases via MPD. (4) The higher the gas kick detection level, the smaller the maximum wellhead back-pressure required. When the pit gain reaches 1.836 m3, it is difficult to deal with the gas kick via MPD, because the maximum wellhead back-pressure exceeds the rated working pressure of the wellhead equipment. Only the shut-in killing operation can be used to control a well kick accident in this case. (5) The deeper the well depth, the longer the time for pressure con trol circulating exhaust. The maximum wellhead back-pressure increases with the increase of the well depth, although the in crease is very small and almost negligible.
Fig. 12. Wellhead back-pressure control curves for different well depths.
Declaration of competing interest None. CRediT authorship contribution statement Youqiang Liao: Writing - review & editing. Xiaohui Sun: Method ology. Baojiang Sun: Writing - review & editing. Zhiyuan Wang: Writing - original draft. Jianbo Zhang: Formal analysis. Wenqiang Lou: Formal analysis. Acknowledgments The work was supported by the National Natural Science Foundation of China (No. 51890914). The National Natural Science Founda tion–Outstanding Youth Foundation (51622405), the Shandong Natural Science funds for Distinguished Young Scholar (JQ201716), CNOOC-KJ 135 ZDXM 05 LTD 06 SHENHAI, 2016 and CNOOC-KJ 135 ZDXM 24 LTD ZJ 04.
Fig. 13. Response characteristics of outlet flow rate for different well depths.
continuous upward migration and expansion of the invading gas, the pit gain and outlet flow rate further increase, and the speed increases continuously until the gas reaches the wellhead and the outlet flow rate reaches its peak.
Appendix A. Pressure and temperature field equations (1) Momentum–balance equation In the process of MPD, gas influx, migration, and expansion will not only affect the upward velocity of the drilling fluid and cuttings, resulting in changes in the frictional pressure drop in the annulus, but will also reduce the hydrostatic pressure in the annulus due to the decrease in the average density of the mixture. Therefore, the gas kick plays an important role in the distribution of the wellbore pressure. Based on the conservation of momentum, the momentum conservation equation in the annulus can be written as (Sun et al., 2011) � � � ∂ Aa Eg ρg v2g þ Aa El ρl v2l þ Aa Ec ρc v2c þ Aa p ∂ Aa Eg ρg vg þ Aa El ρl vl þ Aa Ec ρc vc þ (A.1) ∂t ∂z � ¼ Aa Fw Aa Eg ρg þ Aa El ρl g cos θ where p is the pressure in the annulus, MPa; Fw is the frictional force, N; g is the gravitational acceleration, m=s2 ; θ is the deviation angle of the wellbore. (2) Temperature field equations 8
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The calculation of the intrusive gas physical parameters is based on the accurate simulation of the temperature field, which is of immense sig nificance to the study of gas migration and distribution in the wellbore. During drilling, a dynamic heat transfer process occurs among the drill string, annulus, and formation. According to the conservation of energy, the heat transfer equations between the drill string, annulus, and formation are as follows (Liao et al., 2019a).
∂Tp ∂Tp 2πrpi Up Ta þ vp ¼ ∂t ∂z Cf ρf Ap ∂Ta ∂Ta ¼ va ∂t ∂z
A Ta
�
� Tp þ BðTe;0
�
ρe Ce
Tp
∂Te ∂2 Te 1 ∂Te þ ¼ Ke ∂t ∂r2 r ∂r
(A.2) (A.3)
Ta Þ
� (A.4)
where Ta , Tp and Te are the temperatures in the annulus, drill pipe and formation, respectively, � C; vp and va are the drilling fluid flow velocity in the drill pipe and annulus, respectively, m/s; Cf and Ce are the specific heat of the drilling fluid and the formation, respectively, J=ðkg ⋅� CÞ; Ke is the heat conductivity of formation, w=ðm ⋅� CÞ. A and B are intermediate variables for heat transfer which can be expressed as follows. �� � � A ¼ 2π rpi Up ðρCÞgls Aa �� � (A.5) B ¼ 2π rci Ua ðρCÞgls Aa where rpi and rci are the inner diameter of the drill pipe and casing, respectively, m; ðρCÞgls is the effective product of the density and specific heat
capacity of the fluid in the annulus, J=ðm3 ⋅� CÞ; Ua is the overall heat transfer coefficient in the annulus and Up is the overall heat transfer coefficient in the drill pipe. The calculation method is as follows. (Hasan and Kabir, 2012; 2012): � � 1 rpi ln rpo rpi rpi Up 1 ¼ þ þ (A.6) kp hpi rpo hpo 8 1 rci lnðrco =rci Þ rco lnðrw =rco Þ > 1 > þ þ > Ua ¼ < hci kc kcem
ð0 < h � Hcas Þ
> > 1 > : Ua 1 ¼ hhole
(A.7) ðHcas < h � Hwell Þ
where, hci , hhole , hpi and hpo are the convection heat transfer coefficient of the inner wall of the casing, inner wall of the hole, inner wall of the drill pipe and the outer wall of the drill pipe, respectively, w=ðm2 ⋅� CÞ; kp , kc and kcem are the heat conductivity of the drill pipe, the casing and the formation, respectively, w=ðm ⋅� CÞ; Hcas and Hwell are the depth of the casing and well, respectively, m; rpo , rco and rw are the inner diameter of drill pipe, casing and wellbore, respectively, m. Appendix B. Discretization scheme and numerical solution The flow in the annulus becomes a complex gas-liquid-solid three-phase flow due to the gas influx, and the pressure distribution in the entire wellbore changes with the gas migration and due to changes in the wellhead back pressure, frictional pressure drop, and hydrostatic pressure. Therefore, for this complex model, a direct solution is extremely difficult. Wang et al. (2016) proposed a pressure-based method to calculate the gas-liquid two-phase flow in the wellbore, which has proven to exhibit good accuracy and convergence. Based on this method, the numerical solution is shown in Fig. 3. Firstly, combining Eqs. (A.2)–(A.7)in the temperature field equations, the temperature fields in the wellbore can be solved. Then, assuming the distribution of pressure in the wellbore, the three-phase mass conservation equation is solved. The mass conservation equations for discretizing is as follows. Aj Ekgj ρkgj
1 k 1 k 1 Aj Ekgj 1 ρkgj 1 Ajþ1 Ekgjþ1 ρkgjþ1 vkgjþ1 Ajþ1 Ekgjþ1 ρgjþ1 vgjþ1 ¼ þ qkgj Δt Δz
(B.1)
Aj Eklj ρklj
Aj Eklj 1 ρklj 1 Ajþ1 Ekljþ1 ρkljþ1 vkljþ1 Ajþ1 Ekljþ11 ρkljþ11 vkljþ11 ¼ Δt Δz
(B.2)
Aj Ekcj ρkcj
Aj Ekcj 1 ρkcj 1 Ajþ1 Ekcjþ1 ρkcjþ1 vkcjþ1 Ajþ1 Ekcjþ11 ρkcjþ11 vkcjþ11 ¼ Δt Δz
(B.3)
where, qkgj
��k 4πKg pp pwf �j ρkgj � �� ¼ � μg 2S þ log eγ φ4Kμ gcrt 2 g
(B.4)
w
Combining Eqs. (5)–(10) that describe the velocity relation of the different phases, the gas velocity, liquid velocity, cuttings velocity, void fraction, 9
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Journal of Natural Gas Science and Engineering 75 (2020) 103164
liquid holdup and solid concentration are given as follows: Ekgj ¼
Kg vkgj þ bg
Eklj ¼ 1 vklj ¼
Kl
Ekgj
(B.5)
bl Eklj
(B.7)
Eklj
vkgj ¼ C0 Kg þ Kl Ekcj ¼
(B.6)
Ekcj
� bg þ vgr
(B.8)
bc Aj Ekcj 1 þ Ajþ1 Ekcjþ1 vkcjþ1 bc Aj þ Aj vkcj
vkcj ¼ vklj
(B.9) (B.10)
vlc
where 8 Ajþ1 Ekgjþ1 ρkgjþ1 vkgjþ1 Ekgj 1 ρkgj 1 Δz > > Kg ¼ þ bg þ Mg > k k k > A A ρ ρ > j gj j ρgj gj > > > > < Ajþ1 Ekljþ1 ρkljþ1 vkljþ1 Eklj 1 ρklj 1 Kl ¼ þ bl > > Aj ρklj ρklj > > > > > > > Δz : bg ¼ bl ¼ bc ¼ Δt
(B.11)
Using trial and error, the gas velocity, liquid velocity, cuttings velocity, void fraction, liquid holdup and solid concentration can be calculated. Then the momentum equation is used to correct the trial and error. Finally, the iteration is repeated until the gas velocity, liquid velocity, cuttings velocity, void fraction, liquid holdup and solid concentration satisfy the given convergence conditions.
Fig. B.1. Flow chart of the calculation process.
Appendix C. Supplementary data Supplementary data to this article can be found online at https://doi.org/10.1016/j.jngse.2020.103164.
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Journal of Natural Gas Science and Engineering 75 (2020) 103164
Nomenclature Aa Ap C CD Cf Cg C0 D Dc E Fw g hci hhole hpi hpo Hcas Hwell kc kcem Kg P pp pwf r rce rci rco rpi rpo rw Re Rec S t Ta Te Tp ulc Ua Up v va vgr vp z
ρ ðρCÞgls
cross–sectional area of annulus, m2 cross–sectional area of drill pipe, m2 compressibility of the reservoir fluid, 1/Pa drag coefficient, dimensionless specific heat of drilling fluid, J=ðkg ⋅� CÞ specific heat of gas, J=ðkg ⋅� CÞ distribution parameter, dimensionless wellbore diameter, m cuttings diameter, m volume fraction frictional force, N gravitational acceleration, m=s2 convection heat transfer coefficient of inner wall of casing, w=ðm2 ⋅� CÞ convection heat transfer coefficient of inner wall of hole, w=ðm2 ⋅� CÞ convection heat transfer coefficient of inner wall of drill pipe, w=ðm2 ⋅� CÞ convection heat transfer coefficient of outer wall of drill pipe, w=ðm2 ⋅� CÞ depth of casing, m depth of well, m heat conductivity of cuttings, w=ðm ⋅� CÞ heat conductivity of cement sheath, w=ðm ⋅� CÞ gas permeability in the formation, μm2 pressure, Pa formation pore pressure, MPa bottom hole pressure, MPa distance from the wellbore, m radius of the wellbore, m inner diameter of casing, m outer diameter of casing, m inner diameter of drill pipe, m outer diameter of drill pipe, m inner diameter of wellbore, m three phase Reynolds number, dimensionless Reynolds number of cuttings, dimensionless skin factor, dimensionless drilling time, s annulus temperature, � C formation temperature, � C drill pipe temperature, � C relative velocity of cuttings and drilling fluid, m/s overall heat transfer coefficient in the annulus, w=ðm2 ⋅� CÞ overall heat transfer coefficient in the drill pipe, w=ðm2 ⋅� CÞ velocity, m/s velocity of drilling fluid in annulus, m/s slippage speed, m/s velocity of drilling fluid in drill pipe, m/s distance from the wellhead, m density, kg=m3 the effective product of the density and specific heat capacity of the fluid in annulus, J=ðm3 ⋅� CÞ
φ γ
deviation angle liquid surface tension, N/m viscosity, Pa⋅s porosity Euler–Mascheroni constant (γ � 0.577)
Subscripts F g l c m
drilling mud gas liquid cuttings mixture
θ
σ μ
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References
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