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Calculation of the wellbore temperature and pressure distribution during supercritical CO2 fracturing flowback process
International Journal of Heat and Mass Transfer 139 (2019) 10–16
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Calculation of the wellbore temperature and pressure distribution during supercritical CO2 fracturing flowback process Haizhu Wang a,⇑, Xiaojiang Li b, Kamy Sepehrnoori c, Yong Zheng a, Wanjuan Yan a a
State Key Laboratory of Petroleum Resources and Prospecting, China University of Petroleum, Beijing 102249, China Sinopec Research Institute of Petroleum Engineering, Beijing 100101, China c Hildebrand Department of Petroleum and Geosystems Engineering, University of Texas at Austin, Austin, TX 78712, USA b
a r t i c l e
i n f o
Article history: Received 7 December 2018 Received in revised form 5 March 2019 Accepted 21 April 2019 Available online 6 May 2019 Keywords: Supercritical CO2 Fracturing Flowback Wellbore temperature Wellbore pressure
a b s t r a c t Supercritical CO2 fracturing is a new type of waterless technology developed in recent years. When the supercritical CO2 flows back after fracturing, the formation water is lifted by the upward movement of CO2. Since the pressure gradually decreases along the wellbore and the temperature also decreases due to the CO2 expansion, it is easy to result in the formation of hydrate under the conditions of high pressure and low temperature. In order to prevent the clogging of the wellbore due to CO2 hydrate, the wellbore temperature and pressure need to be predicted and regulated. Based on the Span-Wagner CO2 gas state equation and the Fenghour gas transport equation, combined with the classical wellbore flow heat transfer model, a supercritical CO2 fracturing flowback wellbore flow model with heat source and sink considered is developed, the dual coupling solution of axial and radial borehole is realized by iterating the pressure and temperature and coupling of tubing-annulus-formation. The results show that the wellbore pressure and temperature both decrease from the bottom to the top of the well, which is similar to the flow in oil and gas production. The parameters such as discharge output, tubing size, formation temperature gradient and pressure gradient have a great influence on the wellbore temperature and pressure. The reduction of the discharge output and the increase of the tubing size can effectively keep the wellbore temperature high and reduce the risk of CO2 hydrate formation. The discharge time has almost no effect on the wellbore pressure, and only slightly affects the wellbore temperature. The results can provide guidance for the study of supercritical CO2 fracturing flowback. Ó 2019 Elsevier Ltd. All rights reserved.
1. Introduction (depending on the journal, you do not need to number the sections; however, do not start with 0 anyway) Supercritical CO2 fluid is a unique kind of fluid. Its density is close to water, viscosity is close to gas, and diffusion coefficient is high [1–3]. The reservoir clay expansion and groundwater pollution can be avoided, and water-based fracturing fluid can also be replaced to save a lot of water resources when the CO2 fluid used in unconventional oil and gas reservoir fracturing. In addition, it can help CO2 permanent storage while efficiently develop oil and gas resources [4–6]. In the supercritical CO2 fracturing process, a large amount of CO2 is injected into the formation through the tubing and the annulus [7]. CO2 is injected into the wellbore through a highpressure piston pump in a liquid state (generally about 10 to 0 °C) on the ground. The low-temperature liquid CO2 goes through the wellbore and is gradually heated by the formation. When the ⇑ Corresponding author. E-mail address: [email protected] (H. Wang). https://doi.org/10.1016/j.ijheatmasstransfer.2019.04.109 0017-9310/Ó 2019 Elsevier Ltd. All rights reserved.
temperature reaches or exceeds 31.1 °C, under the condition of wellbore pressure (generally higher than 7.38 MPa), it becomes supercritical, and then supercritical CO2 fracturing is carried out [8]. During the whole CO2 injection process, there is no water in the wellbore. After entering the formation, due to the high temperature of the formation, even if the CO2 encounters water, the CO2 hydrate is hard to form because the high pressure, low temperature and water environment is unachievable [9–11]. After the fracturing is completed, the fracturing fluid needs to be returned to the surface. During the flowback process, part of the CO2 in the formation mixes with the formation water into the wellbore. Since the pressure in the wellbore is lower than the formation pressure, the CO2 gradually expands and absorbs heat, leading to temperature drop. During the ascending process of the mixture of CO2 and water along the wellbore, the pressure gradually decreases, the volume of CO2 continues to expand, and the temperature continues to decrease. When the temperature is lowered to a certain extent, CO2 hydrate is formed in the wellbore. Thus, the wellbore is blocked and fracturing fluid flowback is affected.
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H. Wang et al. / International Journal of Heat and Mass Transfer 139 (2019) 10–16
At present, there is extensive research on the heat transfer problem of supercritical CO2 drilling and fracturing. However, it is focused on the injection process of drilling and fracturing [12– 17]. There are no related reports on the prediction of wellbore temperature and pressure in the CO2 flowback process. Therefore, based on the Span-Wagner CO2 gas state equation [18] and the Fenghour gas transport equation [19], this article presents a supercritical CO2 fracturing backflow wellbore flow model considering heat source sink, combined with the classical wellbore flow heat transfer model. The pressure and temperature iterations are used to couple the tubing-annulus-formation in the calculation model to achieve the axial and radial double coupling solution. The results of the study are expected to provide basis for the design of flowback parameters in supercritical CO2 fracturing.
d ðqmÞ ¼ 0 dz
ð1Þ
d dp sw pd ðqm2 Þ ¼ qgsinh dz dz Ap
ð2Þ
where z is the depth of the well, m; m is the flow velocity, m/s; g is the acceleration of gravity, m/s2; h is the inclination angle of the well, °; sw is the shear stress at the borehole wall (oil pipe wall), MPa; d is the inner diameter of the tubing, m; Ap is the crosssectional area inside the tubing, m2. Substituting the mass conservation equation into the momentum equation and substituting the wall friction term, the pressure drop equation of the CO2 fluid flowing upward in the wellbore is obtained:
dp dm qm2 ¼ qgsinh qm f dz dz 2d
2. Physical and mathematical model During the supercritical CO2 fracturing and flowback process, the supercritical CO2 in the reservoir enters the wellbore forced by formation pressure and flows to the wellhead. Furthermore, the temperature and the pressure of the CO2 fluid in the tubing gradually decreases with the decreasing depth and the geothermal gradient, as shown in Fig. 1. With the change of temperature and pressure, the supercritical CO2 becomes liquid CO2, and finally changes to gaseous CO2 in the wellhead. In Fig. 1, the CO2 in the tubing flows from the bottom to the top of the well, and the annulus is also filled with CO2. The following assumptions are made in the calculation process: (1) heat transfer in the wellbore is steadystate, while the heat transfer in the formation is unsteady; (2) only radial heat transfer is considered outside the oil tubing; (3) the radiation heat transfer and phase change are ignored. (4) the tubing, casing and wellbore are concentric; (5) the wellbore is nearly circular. The annular mud cements well with no gas breakthrough. The Span-Wagner equation, based on Helmholtz free energy, is used to calculate the thermos-dynamic properties, such as density and specific heat capacity of CO2. The transport properties, such as viscosity and thermal conductivity of CO2, are calculated by Fenghour and Vessovic models. More details about the described models can be referred to Refs. [18–21]. For one micro-element in one-dimensional steady flow, the mass conservation equation and the momentum equation are
CO2
where f is the Darcy drag coefficient. For open systems, the energy conservation equation for steadystate flow is
d 1 d q qm e þ m2 ¼ ðpmÞ qmgsinh dz 2 dz Ap dz
Formation
Cement Tubing
Casing
Annulus
Formation
ð4Þ
where e is the internal energy per unit mass of CO2 fluid, J/kg. Heat transfer between the wellbore and the formation can be characterized by the heat transfer equation [22]
q ¼ pdUðT t T ei Þdz
ð3:6Þ
where q is the heat transfer between the fluid in the tubing and the formation in the radial direction, J/s; U is the total heat transfer coefficient, W/(m2 K); Tt is the temperature of the CO2 fluid in the tubing, K; Tei is formation temperature, K. Combined with the pressure drop Eq. (3), the heat transfer equation for the upward flow of CO2 fluid in the wellbore [23]
dT t U pd U pd Tt ¼ T ei þ þ wcp wcp dz
dpfr 1 dp dp þg þ cp q dz dz cp qdz
ð5Þ
where w is the mass flow rate, kg/s; T t is the temperature of the CO2 fluid in the tubing, K; T ei is the original formation temperature, K; pfr is the friction loss of the CO2 fluid, MPa; g is the Joule-Thomson coefficient, K/MPa. Considering the well structure of the tubing and casing, the total heat transfer coefficient is expressed as [24]:
Fig. 1. Schematic of supercritical CO2 flowback in the wellbore.
ð3Þ
U¼
1 d=2 dto 1 d=2 dco d=2 dwb df ðtÞ þ ln ln þ ln þ þ þ ht ks ha ks kc 2ke d dci dco
1 ð7Þ
where ht is the forced convective heat transfer coefficient of the CO2 fluid in the tubing, W/(m2 K); ks is the tubing/casing thermal conductivity, W/(m K); dto is the outer diameter of the tubing, m; ha is the annulus fluid Natural convective heat transfer coefficient, W/(m2 K); dci is the inner diameter of the casing, m; kc is the thermal conductivity of the cement ring, W/(m K); dwb is the diameter of the wellbore, m; dco is the outer diameter of the casing, m; f ðtÞ is a dimensionless temperature function for the unsteady heat transfer of the formation [25], dimensionless; ke is the thermal conductivity of the formation. 3. Model coupling solution The physical properties of CO2 are very sensitive to temperature and pressure. When solving for an element of temperature or pressure, the influence of the change of another element on the physical property parameters must be considered at the same time. Therefore, the pressure drop model and heat transfer model must
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be solved by coupling iteration. In addition, in order to accurately solve the key parameter of the heat transfer model, such as the total heat transfer coefficient, in the radial direction, it is also necessary to couple the tubing-annulus-formation to calculate the internal and external temperatures of different pipe walls and solve it together with the temperature and pressure in the wellbore [24,26]. The calculation process is shown in Fig. 2. 4. Wellbore temperature and pressure distribution in CO2 flowback process The calculation results in the process of the supercritical CO2 flowback in the wellbore after fracturing are shown in Table 1. The influence of the coupling between the tubing and the casing on the annulus is not considered in the calculation. The temperature and pressure distribution of the wellbore is shown in Figs. 3 and 4. Similar to the wellbore flow in oil and gas production, the wellbore pressure and temperature gradually decrease from the bottom to the top of the well, the pressure varies linearly with the depth of the well, while the temperature changes nonlinearly. Under the calculation conditions (ground temperature gradient: 3 K/100 m, formation pressure gradient: 1.2 MPa/100 m, flow rate: 500 t/d, discharge time: 10 h), the CO2 fluid in the whole wellbore is in a supercritical state. Thus, there is no risk of CO2 hydrate formation.
Table 1 Basic calculation parameters. Parameter
Numerical
Well depth/m Tubing inner diameter/mm Tubing outer diameter/mm Casing inner diameter/mm Casing outer diameter/mm Wellbore diameter/mm Surface temperature/K Geothermal gradient/(K m1) Formation specific heat/(J kg1 K1) Formation thermal conductivity/(W m1 K1) Flow rate/(t d1) Bottom hole pressure/MPa discharge time/h Formation density/(kg m3) Oil casing thermal conductivity/(W m1 K1) Cement ring thermal conductivity/(W m1 K1) Completion fluid density/(kg m3) Completion fluid specific heat/(J kg1 K1) Finishing fluid thermal conductivity/(W m1 K1) Completion fluid viscosity/(mPa s)
2000 62 73 124.37 137 215.9 288.15 0.03 837 2.09 500 24 10 2600 44.7 0.52 1000 4186.8 0.6 0.6
5. Parameter sensitivity analysis There are many factors affecting the temperature distribution of the wellbore in the process of supercritical CO2 fluid flowback, including flow rate, tubing diameter, formation pressure and temperature gradient, and discharge time. In order to analyze the influence of these parameters on the temperature distribution of the wellbore, the parameter sensitivity calculation is carried out. 5.1. Flow rate Figs. 5 and 6 show the wellbore pressure profile and temperature profile of the small flow rate, respectively. It can be seen that
Start
Input parameter Fig. 3. Wellbore pressure profile.
Yes
1st iteration No
Assume T 0,Tti0,Tto0,Tci0
Based on T i-1, calc. pi
CO2 property
Based on pi,Ttii-1,Ttoi-1,Tcii-1, calc. T i
Pressure model
Heat transfer model
Based on pi,T i,qi, calc. Ttii,Ttoi,Tcii
Convergence? Yes End Fig. 2. calculation flow chart.
No
under small flow rate conditions (<400 t/d), the flow rate change has little effect on the wellbore pressure, and the temperature has an influence to the wellbore. The higher flow rate becomes, the higher temperature of the wellbore is, but the increasing amplitude becomes smaller. And with the increase of well depth, the difference between wellbore pressure and temperature under different flow rate gradually becomes smaller, and the temperature finally reaches the formation temperature. However, as the flow rate increases, the influence of flow rate on pressure gradually becomes larger (Fig. 7). The wellhead pressure difference for different large flow rates is significantly greater than that for small flow rate. Fig. 8 is a cross-sectional view of the wellbore temperature under large flow rate. The figure shows that the large flow rate has little effect on the wellbore temperature. However, as the flow rate increases, the wellbore temperature decreases, which is the opposite to the case of small flow rate. The main reason is that the heat loss from the bottom to the top
H. Wang et al. / International Journal of Heat and Mass Transfer 139 (2019) 10–16
Fig. 4. Wellbore temperature profile.
Fig. 6. Small-flow rate wellbore temperature profile.
Fig. 5. Small-flow rate wellbore pressure profile.
Fig. 7. Large-flow rate wellbore pressure profile.
of the well is caused by two factors. One is that heat is gradually dissipated into the annulus and the formation during the hightemperature CO2 ascent. The second is that from the bottom to the top of the well, the pressure is gradually reduced, the temperature drops when the CO2 fluid expands to work. Under small flow rate conditions, CO2 has sufficient time for heat exchange between the annulus and the formation. This factor plays a dominant role, so that the temperature profile is arranged in order with increasing temperature. Under the condition of large flow rate, the effect of expansion work on temperature occupying a dominant role, causing the temperature profile at this time to be reversed with increasing temperature.
13
In addition, regardless of the flow rate, as the depth of the well increases, the difference in wellbore temperature gradually becomes smaller until it approaches the formation temperature.
5.2. Tubing size Figs. 9 and 10 are the wellbore pressure profile and temperature profile for different tubing sizes. The figures show that the tubing size has an influence on the wellbore temperature and pressure. There is a significant difference between the temperature and pressure curves of the small-sized tubing and the curves of the other two larger-sized tubing. The pressure and temperature of the
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Fig. 8. Large-flow rate wellbore temperature profile.
Fig. 10. Wellbore temperature profile with different tubing sizes.
wellbore pressure, and there are significant differences in wellbore pressures under different formation pressure gradients. The difference has a great influence on the wellbore temperature, especially the wellbore temperature in the upper part of the well. The greater the formation pressure gradient is, the higher the overall temperature and pressure of the wellbore become, and the more the wellbore temperature curve deviates from the formation temperature curve. 5.4. Formation temperature gradient Figs. 13 and 14 show the wellbore pressure profile and temperature profile for different geothermal gradients. The figures show
Fig. 9. Wellbore pressure profile with different tubing sizes.
small-sized tubing at the same depth of the wellbore are smaller than those of the other two large-sized tubing, mainly because of the larger the tubing diameter the larger the side area, and thereby higher heat transfer rate. It causes a large temperature change. Therefore, the use of small-sized tubing can significantly reduce the pressure and temperature in the wellbore. 5.3. Formation pressure gradient Figs. 11 and 12 show the wellbore pressure profile and temperature profile for different formation pressure gradients. The figures show that the formation pressure gradient will affect the overall
Fig. 11. Wellbore pressure profile at different formation pressure gradients.
H. Wang et al. / International Journal of Heat and Mass Transfer 139 (2019) 10–16
Fig. 12. Wellbore temperature profile at different formation pressure gradients.
15
Fig. 14. Wellbore temperature profile at different geothermal gradients.
that the geothermal gradient has a large impact on wellbore pressure and wellbore temperature. Increasing the geothermal gradient will significantly increase the overall wellbore temperature. This is because the formation temperature can largely decide the temperature of the supercritical CO2 fluid in the wellbore. The higher the geothermal gradient is, the more heat the supercritical CO2 fluid carries during the flowback, and the heat dissipation in the flowback is less than that under the low geothermal gradient. The increase in the temperature of the wellbore causes the density of the CO2 fluid to gradually decrease, and the volume expansion of the fluid increases the wellbore pressure.
Fig. 15. Wellbore pressure versus time curve.
5.5. Discharge time
Fig. 13. Wellbore pressure profile at different geothermal gradients.
Figs. 15 and 16 show the relationship between wellbore pressure and temperature with time. The figures show that the discharge time can influence the wellbore temperature change, and has little effect on the wellbore pressure change. Due to the short discharge time, the wellbore temperature and pressure slightly change. Thus, the temperature and pressure of the wellbore can be neglected with time in a short time.
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Acknowledgements This study was supported by the National Natural Science Foundation of China Grant numbers: 51521063, National Natural Science Foundation of China Grant numbers: 51874318, and National Major Science and Technology Projects of China Grant numbers: 2017ZX05039-003. References
Fig. 16. Wellbore temperature versus time curve.
6. Conclusion (1) Similar to the wellbore flow in oil and gas production, the wellbore pressure and temperature gradually decrease from the bottom to the top of the well in the CO2 flowback process. The pressure varies linearly with the depth of the well, and the temperature changes nonlinearly. (2) Under small flow rate conditions, flow rate changes have little effect on wellbore pressure and have an impact on wellbore temperature. Under large flow rate conditions, flow rate has an effect on pressure and temperature. As the flow rate increases, the wellbore temperature is reduced, because the pressure is gradually reduced from the bottom to the top of the well. Due to the expansion of the CO2 fluid, the fluid absorbs heat, resulting in a decrease in temperature. (3) Larger tubing size can have larger heat transfer area. Thus, the influence on the temperature and pressure of the wellbore gradually becomes larger. The pressure and temperature of the small-sized tubing are lower than the pressure and temperature of the large-sized tubing. In the field application, the use of a small-sized tubing can significantly reduce the pressure and temperature of the wellbore. (4) The greater the formation pressure gradient is, the higher the overall temperature and pressure of the wellbore become, and the more the wellbore temperature curve deviates from the formation temperature curve. The geothermal gradient will directly affect the wellbore temperature and further the pressure, due to changes in CO2 temperature, which will affect CO2 density, viscosity and other parameters. (5) The discharge time has almost no effect on the wellbore pressure, and has a slight influence on the wellbore temperature. As the discharge time increases, the wellbore temperature increases slightly. Conflict of interest The authors declare that there no conflicts of interest.
[1] G. Li, H. Wang, Z. Shen, et al., Application investigations and prospects of supercritical carbon dioxide jet in petroleum engineering, J. China Univ. Petrol. 37 (5) (2013) 76–80 (In Chinese). [2] H. Wang, G. Li, Z. Shen, A feasibility analysis on shale gas exploitation with supercritical carbon dioxide, Energy Source Part A 34 (15) (2012) 1426–1435. [3] H. Wang, Z. Shen, G. Li, et al., Shale gas exploitation with supercritical CO2 technology, Eng. Sci. 10 (4) (2012) 13–17. [4] Z. Wang, B. Sun, X. Sun, et al., Phase state variations for supercritical carbon dioxide drilling, Greenhouse Gases Sci. Technol. 6 (1) (2016) 83–93. [5] H. Ni, W. Song, R. Wang, et al., Coupling model for carbon dioxide wellbore flow and heat transfer in coiled tubing drilling, J. Nat. Gas Sci. Eng. 30 (2016) 414–420. [6] H. Wang, G. Li, Z. Shen, et al., Experiment on rock breaking with supercritical carbon dioxide jet, J. Petrol. Sci. Eng. 127 (2015) 305–310. [7] H. Wang, L. Li, Z. He, et al., The expulsive force of the development of CO2 sequestration-application of SC-CO2 fluid in oil and gas extraction. Frontiers, Energy 2 (2017). [8] G. Li, H. Wang, Z. Shen, et al. Supercritical CO2 jet fracturing with coiled tubing. ZL201110078618.6. 2011.3 (Chinese patent). [9] E. Dendy Sloan, Gas hydrates? Review of physical/chemical properties, Energy Fuels 12 (2) (1998) 191–196. [10] G. Chen, C. Sun, Q. Ma, Gas Hydrate Science and Technology, Chemical Industry Press, Beijing, 2008, pp. 457–459. [11] G.D. Holder, G. Corbin, K.D. Papadopoulos, Thermodynamic and molecular properties of gas hydrates from mixtures containing methane, Argon, and Krypton, Ind. Eng. Chem. Fundam. 19 (3) (1980) 282–286. [12] X. Li, G. Li, W. Yu, et al., Thermal effects of liquid-supercritical CO2 arising from fluid expansion in fracturing, SPE J. (2018), SPE-191357-PA. [13] X. Li, G. Li, K. Sepehrnoori, et al., Estimation and analysis of CO2 friction loss in wellbore during liquid-supercritical CO2 fracturing, SPE Prod. Oper. (2018), SPE-191142-PA. [14] L. Yi, X. Li, Z. Yang, et al., Coupled calculation model for transient temperature and pressure of carbon dioxide injection well, Int. J. Heat Mass Transf. 121 (2018) 680–690. [15] X. Li, L. Yi, Z. Yang, et al., Coupling model for calculation of transient temperature and pressure during coiled tubing drilling with supercritical carbon dioxide, Int. J. Heat Mass Transf. 125 (2018) 400–412. [16] Y. Feng, X. Li, K.E. Gray, Development of a 3D numerical model for quantifying fluid-driven interface debonding of an injector well, Int. J. Greenhouse Gas Control 62 (2017) 76–90. [17] H. Huo, H. Wang, H. Ni, et al., Study of critical annulus up-returning velocity of cuttings carried by supercritical CO2 in deviated well, J. CO2 Util. 20 (2017) 105–112. [18] R. Span, W. Wagner, A new equation of state for carbon dioxide covering the from the triple-point temperature to 1100 K at pressures fluid region up to 800 MPa, J. Phys. Chem. Ref. Data. 25 (6) (1996) 1509–1596. [19] A. Fenghour, W.A. Wakeham, V. Vesovic, The viscosity of carbon dioxide, J. Phys. Chem. Ref. Data 27 (1) (1998) 31–44. [20] R.C. Reid, J.M. Prausnitz, B.E. Poling, The properties of gases and liquids M, 4th ed., McGraw-Hill Press, New York, 1987, pp. 42–55. [21] V. Vesovic, W.A. Wakeham, et al., The transport properties of carbon dioxide, J. Phys. Chem. Ref. Data 3 (1990, (19)) 763–807. [22] A.R. Hasan, C.S. Kabir, Wellbore heat-transfer modeling and applications, J. Petrol. Sci. Eng. 86–87 (2012) 127–136. [23] X. Li, G. Li, G. Wang, et al., A unified model for wellbore flow and heat transfer in pure CO2 injection for geological sequestration, EOR and fracturing operations, Int. J. Greenhouse Gas Control 57 (2017) 102–115. [24] X. Li, G. Li, K. Sepehrnoori, et al., Estimation and analysis of carbon dioxide friction loss in wellbore during liquid/supercritical carbon dioxide fracturing, SPE Prod. Oper. (2018), SPE-191142-PA. [25] C.S. Kabir, A.R. Hasan, G.E. Kouba, et al., Determining circulating fluid temperature in drilling, work over, and well control operations, SPE Drill. Complet 11 (2) (1996) 74–79. [26] J. Guo, J. Zeng, A coupling model for wellbore transient temperature and pressure of fracturing with supercritical carbon dioxide, Acta Petrolei Sinica 36 (2) (2015) 203–209.
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Calculation of the wellbore temperature and pressure distribution during supercritical CO2 fracturing flowback process Haizhu Wang a,⇑, Xiaojiang Li b, Kamy Sepehrnoori c, Yong Zheng a, Wanjuan Yan a a
State Key Laboratory of Petroleum Resources and Prospecting, China University of Petroleum, Beijing 102249, China Sinopec Research Institute of Petroleum Engineering, Beijing 100101, China c Hildebrand Department of Petroleum and Geosystems Engineering, University of Texas at Austin, Austin, TX 78712, USA b
a r t i c l e
i n f o
Article history: Received 7 December 2018 Received in revised form 5 March 2019 Accepted 21 April 2019 Available online 6 May 2019 Keywords: Supercritical CO2 Fracturing Flowback Wellbore temperature Wellbore pressure
a b s t r a c t Supercritical CO2 fracturing is a new type of waterless technology developed in recent years. When the supercritical CO2 flows back after fracturing, the formation water is lifted by the upward movement of CO2. Since the pressure gradually decreases along the wellbore and the temperature also decreases due to the CO2 expansion, it is easy to result in the formation of hydrate under the conditions of high pressure and low temperature. In order to prevent the clogging of the wellbore due to CO2 hydrate, the wellbore temperature and pressure need to be predicted and regulated. Based on the Span-Wagner CO2 gas state equation and the Fenghour gas transport equation, combined with the classical wellbore flow heat transfer model, a supercritical CO2 fracturing flowback wellbore flow model with heat source and sink considered is developed, the dual coupling solution of axial and radial borehole is realized by iterating the pressure and temperature and coupling of tubing-annulus-formation. The results show that the wellbore pressure and temperature both decrease from the bottom to the top of the well, which is similar to the flow in oil and gas production. The parameters such as discharge output, tubing size, formation temperature gradient and pressure gradient have a great influence on the wellbore temperature and pressure. The reduction of the discharge output and the increase of the tubing size can effectively keep the wellbore temperature high and reduce the risk of CO2 hydrate formation. The discharge time has almost no effect on the wellbore pressure, and only slightly affects the wellbore temperature. The results can provide guidance for the study of supercritical CO2 fracturing flowback. Ó 2019 Elsevier Ltd. All rights reserved.
1. Introduction (depending on the journal, you do not need to number the sections; however, do not start with 0 anyway) Supercritical CO2 fluid is a unique kind of fluid. Its density is close to water, viscosity is close to gas, and diffusion coefficient is high [1–3]. The reservoir clay expansion and groundwater pollution can be avoided, and water-based fracturing fluid can also be replaced to save a lot of water resources when the CO2 fluid used in unconventional oil and gas reservoir fracturing. In addition, it can help CO2 permanent storage while efficiently develop oil and gas resources [4–6]. In the supercritical CO2 fracturing process, a large amount of CO2 is injected into the formation through the tubing and the annulus [7]. CO2 is injected into the wellbore through a highpressure piston pump in a liquid state (generally about 10 to 0 °C) on the ground. The low-temperature liquid CO2 goes through the wellbore and is gradually heated by the formation. When the ⇑ Corresponding author. E-mail address: [email protected] (H. Wang). https://doi.org/10.1016/j.ijheatmasstransfer.2019.04.109 0017-9310/Ó 2019 Elsevier Ltd. All rights reserved.
temperature reaches or exceeds 31.1 °C, under the condition of wellbore pressure (generally higher than 7.38 MPa), it becomes supercritical, and then supercritical CO2 fracturing is carried out [8]. During the whole CO2 injection process, there is no water in the wellbore. After entering the formation, due to the high temperature of the formation, even if the CO2 encounters water, the CO2 hydrate is hard to form because the high pressure, low temperature and water environment is unachievable [9–11]. After the fracturing is completed, the fracturing fluid needs to be returned to the surface. During the flowback process, part of the CO2 in the formation mixes with the formation water into the wellbore. Since the pressure in the wellbore is lower than the formation pressure, the CO2 gradually expands and absorbs heat, leading to temperature drop. During the ascending process of the mixture of CO2 and water along the wellbore, the pressure gradually decreases, the volume of CO2 continues to expand, and the temperature continues to decrease. When the temperature is lowered to a certain extent, CO2 hydrate is formed in the wellbore. Thus, the wellbore is blocked and fracturing fluid flowback is affected.
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H. Wang et al. / International Journal of Heat and Mass Transfer 139 (2019) 10–16
At present, there is extensive research on the heat transfer problem of supercritical CO2 drilling and fracturing. However, it is focused on the injection process of drilling and fracturing [12– 17]. There are no related reports on the prediction of wellbore temperature and pressure in the CO2 flowback process. Therefore, based on the Span-Wagner CO2 gas state equation [18] and the Fenghour gas transport equation [19], this article presents a supercritical CO2 fracturing backflow wellbore flow model considering heat source sink, combined with the classical wellbore flow heat transfer model. The pressure and temperature iterations are used to couple the tubing-annulus-formation in the calculation model to achieve the axial and radial double coupling solution. The results of the study are expected to provide basis for the design of flowback parameters in supercritical CO2 fracturing.
d ðqmÞ ¼ 0 dz
ð1Þ
d dp sw pd ðqm2 Þ ¼ qgsinh dz dz Ap
ð2Þ
where z is the depth of the well, m; m is the flow velocity, m/s; g is the acceleration of gravity, m/s2; h is the inclination angle of the well, °; sw is the shear stress at the borehole wall (oil pipe wall), MPa; d is the inner diameter of the tubing, m; Ap is the crosssectional area inside the tubing, m2. Substituting the mass conservation equation into the momentum equation and substituting the wall friction term, the pressure drop equation of the CO2 fluid flowing upward in the wellbore is obtained:
dp dm qm2 ¼ qgsinh qm f dz dz 2d
2. Physical and mathematical model During the supercritical CO2 fracturing and flowback process, the supercritical CO2 in the reservoir enters the wellbore forced by formation pressure and flows to the wellhead. Furthermore, the temperature and the pressure of the CO2 fluid in the tubing gradually decreases with the decreasing depth and the geothermal gradient, as shown in Fig. 1. With the change of temperature and pressure, the supercritical CO2 becomes liquid CO2, and finally changes to gaseous CO2 in the wellhead. In Fig. 1, the CO2 in the tubing flows from the bottom to the top of the well, and the annulus is also filled with CO2. The following assumptions are made in the calculation process: (1) heat transfer in the wellbore is steadystate, while the heat transfer in the formation is unsteady; (2) only radial heat transfer is considered outside the oil tubing; (3) the radiation heat transfer and phase change are ignored. (4) the tubing, casing and wellbore are concentric; (5) the wellbore is nearly circular. The annular mud cements well with no gas breakthrough. The Span-Wagner equation, based on Helmholtz free energy, is used to calculate the thermos-dynamic properties, such as density and specific heat capacity of CO2. The transport properties, such as viscosity and thermal conductivity of CO2, are calculated by Fenghour and Vessovic models. More details about the described models can be referred to Refs. [18–21]. For one micro-element in one-dimensional steady flow, the mass conservation equation and the momentum equation are
CO2
where f is the Darcy drag coefficient. For open systems, the energy conservation equation for steadystate flow is
d 1 d q qm e þ m2 ¼ ðpmÞ qmgsinh dz 2 dz Ap dz
Formation
Cement Tubing
Casing
Annulus
Formation
ð4Þ
where e is the internal energy per unit mass of CO2 fluid, J/kg. Heat transfer between the wellbore and the formation can be characterized by the heat transfer equation [22]
q ¼ pdUðT t T ei Þdz
ð3:6Þ
where q is the heat transfer between the fluid in the tubing and the formation in the radial direction, J/s; U is the total heat transfer coefficient, W/(m2 K); Tt is the temperature of the CO2 fluid in the tubing, K; Tei is formation temperature, K. Combined with the pressure drop Eq. (3), the heat transfer equation for the upward flow of CO2 fluid in the wellbore [23]
dT t U pd U pd Tt ¼ T ei þ þ wcp wcp dz
dpfr 1 dp dp þg þ cp q dz dz cp qdz
ð5Þ
where w is the mass flow rate, kg/s; T t is the temperature of the CO2 fluid in the tubing, K; T ei is the original formation temperature, K; pfr is the friction loss of the CO2 fluid, MPa; g is the Joule-Thomson coefficient, K/MPa. Considering the well structure of the tubing and casing, the total heat transfer coefficient is expressed as [24]:
Fig. 1. Schematic of supercritical CO2 flowback in the wellbore.
ð3Þ
U¼
1 d=2 dto 1 d=2 dco d=2 dwb df ðtÞ þ ln ln þ ln þ þ þ ht ks ha ks kc 2ke d dci dco
1 ð7Þ
where ht is the forced convective heat transfer coefficient of the CO2 fluid in the tubing, W/(m2 K); ks is the tubing/casing thermal conductivity, W/(m K); dto is the outer diameter of the tubing, m; ha is the annulus fluid Natural convective heat transfer coefficient, W/(m2 K); dci is the inner diameter of the casing, m; kc is the thermal conductivity of the cement ring, W/(m K); dwb is the diameter of the wellbore, m; dco is the outer diameter of the casing, m; f ðtÞ is a dimensionless temperature function for the unsteady heat transfer of the formation [25], dimensionless; ke is the thermal conductivity of the formation. 3. Model coupling solution The physical properties of CO2 are very sensitive to temperature and pressure. When solving for an element of temperature or pressure, the influence of the change of another element on the physical property parameters must be considered at the same time. Therefore, the pressure drop model and heat transfer model must
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be solved by coupling iteration. In addition, in order to accurately solve the key parameter of the heat transfer model, such as the total heat transfer coefficient, in the radial direction, it is also necessary to couple the tubing-annulus-formation to calculate the internal and external temperatures of different pipe walls and solve it together with the temperature and pressure in the wellbore [24,26]. The calculation process is shown in Fig. 2. 4. Wellbore temperature and pressure distribution in CO2 flowback process The calculation results in the process of the supercritical CO2 flowback in the wellbore after fracturing are shown in Table 1. The influence of the coupling between the tubing and the casing on the annulus is not considered in the calculation. The temperature and pressure distribution of the wellbore is shown in Figs. 3 and 4. Similar to the wellbore flow in oil and gas production, the wellbore pressure and temperature gradually decrease from the bottom to the top of the well, the pressure varies linearly with the depth of the well, while the temperature changes nonlinearly. Under the calculation conditions (ground temperature gradient: 3 K/100 m, formation pressure gradient: 1.2 MPa/100 m, flow rate: 500 t/d, discharge time: 10 h), the CO2 fluid in the whole wellbore is in a supercritical state. Thus, there is no risk of CO2 hydrate formation.
Table 1 Basic calculation parameters. Parameter
Numerical
Well depth/m Tubing inner diameter/mm Tubing outer diameter/mm Casing inner diameter/mm Casing outer diameter/mm Wellbore diameter/mm Surface temperature/K Geothermal gradient/(K m1) Formation specific heat/(J kg1 K1) Formation thermal conductivity/(W m1 K1) Flow rate/(t d1) Bottom hole pressure/MPa discharge time/h Formation density/(kg m3) Oil casing thermal conductivity/(W m1 K1) Cement ring thermal conductivity/(W m1 K1) Completion fluid density/(kg m3) Completion fluid specific heat/(J kg1 K1) Finishing fluid thermal conductivity/(W m1 K1) Completion fluid viscosity/(mPa s)
2000 62 73 124.37 137 215.9 288.15 0.03 837 2.09 500 24 10 2600 44.7 0.52 1000 4186.8 0.6 0.6
5. Parameter sensitivity analysis There are many factors affecting the temperature distribution of the wellbore in the process of supercritical CO2 fluid flowback, including flow rate, tubing diameter, formation pressure and temperature gradient, and discharge time. In order to analyze the influence of these parameters on the temperature distribution of the wellbore, the parameter sensitivity calculation is carried out. 5.1. Flow rate Figs. 5 and 6 show the wellbore pressure profile and temperature profile of the small flow rate, respectively. It can be seen that
Start
Input parameter Fig. 3. Wellbore pressure profile.
Yes
1st iteration No
Assume T 0,Tti0,Tto0,Tci0
Based on T i-1, calc. pi
CO2 property
Based on pi,Ttii-1,Ttoi-1,Tcii-1, calc. T i
Pressure model
Heat transfer model
Based on pi,T i,qi, calc. Ttii,Ttoi,Tcii
Convergence? Yes End Fig. 2. calculation flow chart.
No
under small flow rate conditions (<400 t/d), the flow rate change has little effect on the wellbore pressure, and the temperature has an influence to the wellbore. The higher flow rate becomes, the higher temperature of the wellbore is, but the increasing amplitude becomes smaller. And with the increase of well depth, the difference between wellbore pressure and temperature under different flow rate gradually becomes smaller, and the temperature finally reaches the formation temperature. However, as the flow rate increases, the influence of flow rate on pressure gradually becomes larger (Fig. 7). The wellhead pressure difference for different large flow rates is significantly greater than that for small flow rate. Fig. 8 is a cross-sectional view of the wellbore temperature under large flow rate. The figure shows that the large flow rate has little effect on the wellbore temperature. However, as the flow rate increases, the wellbore temperature decreases, which is the opposite to the case of small flow rate. The main reason is that the heat loss from the bottom to the top
H. Wang et al. / International Journal of Heat and Mass Transfer 139 (2019) 10–16
Fig. 4. Wellbore temperature profile.
Fig. 6. Small-flow rate wellbore temperature profile.
Fig. 5. Small-flow rate wellbore pressure profile.
Fig. 7. Large-flow rate wellbore pressure profile.
of the well is caused by two factors. One is that heat is gradually dissipated into the annulus and the formation during the hightemperature CO2 ascent. The second is that from the bottom to the top of the well, the pressure is gradually reduced, the temperature drops when the CO2 fluid expands to work. Under small flow rate conditions, CO2 has sufficient time for heat exchange between the annulus and the formation. This factor plays a dominant role, so that the temperature profile is arranged in order with increasing temperature. Under the condition of large flow rate, the effect of expansion work on temperature occupying a dominant role, causing the temperature profile at this time to be reversed with increasing temperature.
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In addition, regardless of the flow rate, as the depth of the well increases, the difference in wellbore temperature gradually becomes smaller until it approaches the formation temperature.
5.2. Tubing size Figs. 9 and 10 are the wellbore pressure profile and temperature profile for different tubing sizes. The figures show that the tubing size has an influence on the wellbore temperature and pressure. There is a significant difference between the temperature and pressure curves of the small-sized tubing and the curves of the other two larger-sized tubing. The pressure and temperature of the
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Fig. 8. Large-flow rate wellbore temperature profile.
Fig. 10. Wellbore temperature profile with different tubing sizes.
wellbore pressure, and there are significant differences in wellbore pressures under different formation pressure gradients. The difference has a great influence on the wellbore temperature, especially the wellbore temperature in the upper part of the well. The greater the formation pressure gradient is, the higher the overall temperature and pressure of the wellbore become, and the more the wellbore temperature curve deviates from the formation temperature curve. 5.4. Formation temperature gradient Figs. 13 and 14 show the wellbore pressure profile and temperature profile for different geothermal gradients. The figures show
Fig. 9. Wellbore pressure profile with different tubing sizes.
small-sized tubing at the same depth of the wellbore are smaller than those of the other two large-sized tubing, mainly because of the larger the tubing diameter the larger the side area, and thereby higher heat transfer rate. It causes a large temperature change. Therefore, the use of small-sized tubing can significantly reduce the pressure and temperature in the wellbore. 5.3. Formation pressure gradient Figs. 11 and 12 show the wellbore pressure profile and temperature profile for different formation pressure gradients. The figures show that the formation pressure gradient will affect the overall
Fig. 11. Wellbore pressure profile at different formation pressure gradients.
H. Wang et al. / International Journal of Heat and Mass Transfer 139 (2019) 10–16
Fig. 12. Wellbore temperature profile at different formation pressure gradients.
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Fig. 14. Wellbore temperature profile at different geothermal gradients.
that the geothermal gradient has a large impact on wellbore pressure and wellbore temperature. Increasing the geothermal gradient will significantly increase the overall wellbore temperature. This is because the formation temperature can largely decide the temperature of the supercritical CO2 fluid in the wellbore. The higher the geothermal gradient is, the more heat the supercritical CO2 fluid carries during the flowback, and the heat dissipation in the flowback is less than that under the low geothermal gradient. The increase in the temperature of the wellbore causes the density of the CO2 fluid to gradually decrease, and the volume expansion of the fluid increases the wellbore pressure.
Fig. 15. Wellbore pressure versus time curve.
5.5. Discharge time
Fig. 13. Wellbore pressure profile at different geothermal gradients.
Figs. 15 and 16 show the relationship between wellbore pressure and temperature with time. The figures show that the discharge time can influence the wellbore temperature change, and has little effect on the wellbore pressure change. Due to the short discharge time, the wellbore temperature and pressure slightly change. Thus, the temperature and pressure of the wellbore can be neglected with time in a short time.
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Acknowledgements This study was supported by the National Natural Science Foundation of China Grant numbers: 51521063, National Natural Science Foundation of China Grant numbers: 51874318, and National Major Science and Technology Projects of China Grant numbers: 2017ZX05039-003. References
Fig. 16. Wellbore temperature versus time curve.
6. Conclusion (1) Similar to the wellbore flow in oil and gas production, the wellbore pressure and temperature gradually decrease from the bottom to the top of the well in the CO2 flowback process. The pressure varies linearly with the depth of the well, and the temperature changes nonlinearly. (2) Under small flow rate conditions, flow rate changes have little effect on wellbore pressure and have an impact on wellbore temperature. Under large flow rate conditions, flow rate has an effect on pressure and temperature. As the flow rate increases, the wellbore temperature is reduced, because the pressure is gradually reduced from the bottom to the top of the well. Due to the expansion of the CO2 fluid, the fluid absorbs heat, resulting in a decrease in temperature. (3) Larger tubing size can have larger heat transfer area. Thus, the influence on the temperature and pressure of the wellbore gradually becomes larger. The pressure and temperature of the small-sized tubing are lower than the pressure and temperature of the large-sized tubing. In the field application, the use of a small-sized tubing can significantly reduce the pressure and temperature of the wellbore. (4) The greater the formation pressure gradient is, the higher the overall temperature and pressure of the wellbore become, and the more the wellbore temperature curve deviates from the formation temperature curve. The geothermal gradient will directly affect the wellbore temperature and further the pressure, due to changes in CO2 temperature, which will affect CO2 density, viscosity and other parameters. (5) The discharge time has almost no effect on the wellbore pressure, and has a slight influence on the wellbore temperature. As the discharge time increases, the wellbore temperature increases slightly. Conflict of interest The authors declare that there no conflicts of interest.
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