Transient Pressure and Temperature Analysis of a Deepwater Gas Well during a Blowout Test

: On one hand, a blowout test can clean the bottom of the well, and on the other, it can learn the productivity of the well, which is important work before putting the well into production and also the main basis for production allocation of the well. The accurate prediction of the blowout test process provides a theoretical basis for the design of a reasonable blowout test system and the determination of well cleaning time. During deepwater blowout tests, gas and liquid ﬂows are unsteady in pipes, and ﬂow parameters change over time. At present, accurately predicting changes in ﬂuid temperature, pressure, liquid holdup, and other parameters in a wellbore during an actual blowout process using the commonly used steady-state prediction methods is difﬁcult, and determining whether a test scheme is reasonable is impossible. Therefore, based on the conservation of mass, momentum, and energy during the blowout test process, in this study, formation, wellbore, and nozzle ﬂows were coupled for the ﬁrst time, and a time and space of unsteady pressure drop and a heat transfer differential equation system was established; furthermore, using the Newton–Raphson method, the equations were solved. Finally, the simulation of the transient ﬂow of the blowout test was completed. Considering a measured deepwater gas well A as an example, the blowout test process was simulated, and the variations in the wellbore ﬂow parameters were analyzed. Comparing the simulation result with the test data, we concluded the following. (1) During the blowout process, the wellbore temperature gradually increased; pressure at the bottom of the wellbore decreased; and pressure at the wellhead increased; and (2) the established model agreed well with the actual production data, and the average error of the wellhead pressure and temperature was less than 5%. Considering the high production capacity of deepwater gas wells, the use of large-sized tubing and nozzles to spray is recommended, which can improve the speed of clearing wells and prevent the formation of hydrate.


Introduction
At present, deepwater is one of the focuses of oil and gas resource development and has broad prospects, but deepwater oil and gas exploitation has been handicapped by the complexity of the environment. Due to environmental complexities in deepwater gas wells, fluid flow parameters change with time during a two-phase flow (i.e., from the formation to the wellbore and in the wellbore), thereby making the flow complicated. Therefore, accurately predicting changes in flow parameters such as wellbore fluid temperature, pressure, and liquid holdup during an actual blowout process is difficult, and researching the two-phase transient flow during blowouts is necessary.

Mathematics Model
The wellbore flow is an unsteady two-phase flow in the test. The model is assumed as follows: (1) the gas-liquid flow in the pipe is one-dimensional and unsteady; (2) the gas is compressible, whereas the liquid is incompressible; (3) a high production and homogeneous flow during the blowout test; and (4) the downward direction of the wellbore flow is defined as the positive direction of the z-axis.
According to the principles of the conservations of mass, momentum, and energy, the control equations are ∂ρ m ∂t Considering the high gas production in the test, the transient simulation of the gasliquid flow is simplified to a homogeneous flow. Rendeiro and Kelso (1988) modified the correction of the gas relative density, and the mixture relative density and mixture density are expressed as follows: where the natural gas deviation coefficient, Z m , can be calculated using the Dranchuk-Abu-Kassem relation: Z m = 1 + A 1 + A 2 /T pr + A 3 /T 3 pr + A 4 /T 4 pr + A 5 /T 5 pr ρ mr + A 6 + A 7 /T pr + A 8 /T 2 pr ρ 2 mr −A 9 A 7 /T pr + A 8 /T 2 pr ρ 5 mr + A 10 1 + A 11 ρ 2 mr ρ 2 mr /T 3 pr exp −A 11 ρ 2 mr (6)

Steady-State Heat Transfer
Heat transfer in deepwater wells comprises two sections: seawater and formation sections. The heat transfer in the wellbore is steady, and the heat transfer during the formation is unsteady and can be described via a transfer heat-conduction time function. The wellbore structure and length of the differential cell, dz, are shown in Figure 1.
According to the law of heat conduction from the wellbore to formation and unsteady heat dissipation, the radial heat gradient equation of formation can be established. Moreover, according to the law of convection heat transfer from the wellbore to seawater, the radial heat gradient equation of seawater can also be established. Combining the conservation of energy and enthalpy gradient equations, a general formula to calculate the wellbore temperature gradient of offshore oil and gas wells is formed: For the formation section: If the fluid heat-transfer coefficient in tubing, the heat conductivity of tubing and casing offer negligible resistance to heat flow, U to1 can be approximated by For the seawater section: L r = 2πr to U to2 C Pm w t  According to the law of heat conduction from the wellbore to formation and unsteady heat dissipation, the radial heat gradient equation of formation can be established. Moreover, according to the law of convection heat transfer from the wellbore to seawater, the radial heat gradient equation of seawater can also be established. Combining the conservation of energy and enthalpy gradient equations, a general formula to calculate the wellbore temperature gradient of offshore oil and gas wells is formed:

Transient Heat Transfer
During the test, the output fluid dissipates heat to the wellbore, and the temperature decreases gradually from the bottom to wellhead. Furthermore, the cement and string are continuously heated by the high-temperature fluid so that the temperature difference between the fluid and wellbore constantly decreases, and the fluid temperature continuously changes with time. Therefore, the heat loss of the wellbore fluid during the test blowout is a transient process ( Figure 2).

Transient Heat Transfer
During the test, the output fluid dissipates heat to the wellbore, and the temperature decreases gradually from the bottom to wellhead. Furthermore, the cement and string are continuously heated by the high-temperature fluid so that the temperature difference between the fluid and wellbore constantly decreases, and the fluid temperature continuously changes with time. Therefore, the heat loss of the wellbore fluid during the test blowout is a transient process ( Figure 2). We introduced the Hasan and Kabir heat storage coefficient into Equation (3) and combined it with the wellbore steady-state heat transfer, Equation (7); an explicit equation for calculating the transient temperature of the wellbore fluid is obtained as follows: Partial differential equation: We introduced the Hasan and Kabir heat storage coefficient into Equation (3) and combined it with the wellbore steady-state heat transfer, Equation (7); an explicit equation for calculating the transient temperature of the wellbore fluid is obtained as follows: Partial differential equation: Introduced heat storage coefficient: Explicit equation: For different wells, the heat storage coefficient can be fitted using test data. Combined with the conservations of mass and momentum in the transient process, the wellbore pressure and temperature coupling model is obtained as follows:

Temperature during Production Adjustment
Because production is often unstable, it requires frequent production adjustment. In this case, using an ordinary temperature transient model cannot predict the temperature change accurately, which leads to a large pressure deviation. To improve the prediction accuracy of the real blowout test process, developing the transient superposition correlation of the wellbore temperature is necessary.
Assuming a virtual initial temperature rise time, the temperature is calculated through the steady-state heat transfer model in the period of an increasing production. The temperature change is divided into a superposition of the well shut-in and opening processes in the period of a decreasing production. The corresponding transient superposition is shown in Figure 3.
accuracy of the real blowout test process, developing the transient superposition correlation of the wellbore temperature is necessary.
Assuming a virtual initial temperature rise time, the temperature is calculated through the steady-state heat transfer model in the period of an increasing production. The temperature change is divided into a superposition of the well shut-in and opening processes in the period of a decreasing production. The corresponding transient superposition is shown in Figure 3. (1) Increasing production The temperature T1 can be calculated as follows: (1) Increasing production The temperature T 1 can be calculated as follows: The virtual time ∆t 1 can be calculated as follows: Therefore, the temperature T 2 at time t 2 can be calculated as follows: (2) Decreasing production Temperature changes ∆T 2,close at t 2 time shut-in process can be calculated as follows: Temperature changes ∆T 2,open in the well opening process can be calculated as follows: Finally, the temperature T 2 at time t 2 can be calculated as follows:

Blowout Test Process and Model Solution
(1) Blowout test process When the well is closed, the induced fluid is on the top of the test fluid in the wellbore. However, when the gas well is open, the upper induced fluid relieved pressure; wellhead fluid flowed first; and bottom fluid flowed later, leading to a two-phase variable mass flow in the wellbore, the blowout test process is shown in Figure 4. During the upstream period, the gas-liquid fluid flowed from the formation into the wellbore and continuously released heat to the formation and seawater sections. When blown out for a while, a steady flow stage was developed. Therefore, the transient flow model was coupled by "wellhead nozzle flow + induced fluid pipe flow + well fluid pipe flow + formation seepage." When the well is closed, the induced fluid is on the top of the test fluid in the wellbore. However, when the gas well is open, the upper induced fluid relieved pressure; wellhead fluid flowed first; and bottom fluid flowed later, leading to a two-phase variable mass flow in the wellbore, the blowout test process is shown in Figure 4. During the upstream period, the gas-liquid fluid flowed from the formation into the wellbore and continuously released heat to the formation and seawater sections. When blown out for a while, a steady flow stage was developed. Therefore, the transient flow model was coupled by "wellhead nozzle flow + induced fluid pipe flow + well fluid pipe flow + formation seepage." In summary, based on the above-mentioned phenomena, the two-phase flow transient model comprises the transient conservations of mass, momentum, and energy, and the coupling of the wellbore flow, wellhead throttling, and formation productivity is considered. The influence of the heat transfer difference between the formation and seawater In summary, based on the above-mentioned phenomena, the two-phase flow transient model comprises the transient conservations of mass, momentum, and energy, and the coupling of the wellbore flow, wellhead throttling, and formation productivity is considered. The influence of the heat transfer difference between the formation and seawater is involved in the calculation of the fluid density in the wellbore using a pseudo single-phase model.
Deliverability equation: The flow rate in the blowout was high, and the wellhead nozzle flow was suitable for the multiphase flow formula (i.e., the Sachdeva-Model of subcritical flow).
(1) Input basic parameters: well depth H, tubing diameter dti, formation pressure p r , fluid extraction index I L , average wellbore temperature T ave , external pressure P 0 , nozzle size d chock , test fluid level L h , and time interval dt. A simple flowchart of the model solution process is shown in Figure 6.

Application and Analysis Discussion
We selected deepwater well A to verify the applicability of the estab The basic parameters corresponding to well A are shown in Table 1.

Application and Analysis Discussion
We selected deepwater well A to verify the applicability of the established model. The basic parameters corresponding to well A are shown in Table 1. Before the blowout, the wellbore was filled with the test and induced fluid. We assumed that the density of the induced fluid was equal to that of the test fluid to simplify the calculation. The gas well productivity is described using an exponential equation; select the coefficient n = 0.75 and C = 1.2 × 10 4 m 3 /d·MPa −2n . The results of the interpolation temperature and initial pressure profile are shown in Figure 7. Before the blowout, the wellbore was filled with the test and induced fluid. We assumed that the density of the induced fluid was equal to that of the test fluid to simplify the calculation. The gas well productivity is described using an exponential equation; select the coefficient n = 0.75 and C = 1.2 × 10 4 m 3 /d·MPa −2n . The results of the interpolation temperature and initial pressure profile are shown in Figure 7.

Transient Flow Prediction
According to the actual test conditions, the wellbore pressure, temperature, gas production, liquid holdup, hydrate formation temperature, and other parameters were simulated during the blowout (Figure 8). After opening the well for approximately 2 h (8640 s), the formation started producing gas; the working fluid in the wellbore began to be displaced by the gas; the gas-liquid two-phase interface kept moving up; and the wellhead pressure started to increase. At approximately 14,400 s, the wellhead liquid holdup changed to 0 and working fluid was completely displaced, which agreed with the actual situation. The average error in the wellhead pressure and temperature calculations and measured parameters was less than 5%. The simulations agreed well with the measured values, which showed that the established model met the requirements of computational accuracy. The simulation results were verified via field temperature measurements. The

Transient Flow Prediction
According to the actual test conditions, the wellbore pressure, temperature, gas production, liquid holdup, hydrate formation temperature, and other parameters were simulated during the blowout (Figure 8). After opening the well for approximately 2 h (8640 s), the formation started producing gas; the working fluid in the wellbore began to be displaced by the gas; the gas-liquid two-phase interface kept moving up; and the wellhead pressure started to increase. At approximately 14,400 s, the wellhead liquid holdup changed to 0 and working fluid was completely displaced, which agreed with the actual situation. The aver-age error in the wellhead pressure and temperature calculations and measured parameters was less than 5%. The simulations agreed well with the measured values, which showed that the established model met the requirements of computational accuracy. The simulation results were verified via field temperature measurements. The simulation results showed that the mudline temperature was slightly lower than the hydrate formation temperature at the end of the cleanup, and the hydrate inhibitor was injected in the field operation. The wellbore pressure, temperature, liquid holdup, and mixture density distribution at different times were predicted; the results are shown in Figure 9. After the well opening the test fluid was replaced by airflow and wellbore pressure gradient decreased, wherea the wellhead pressure increased gradually. The gas production and wellbore temperatur increased gradually. The liquid holdup and mixture density decreased gradually and fi nally formed as an annular flow.   The wellbore pressure, temperature, liquid holdup, and mixture density distribution at different times were predicted; the results are shown in Figure 9. After the well opening, the test fluid was replaced by airflow and wellbore pressure gradient decreased, whereas the wellhead pressure increased gradually. The gas production and wellbore temperature increased gradually. The liquid holdup and mixture density decreased gradually and finally formed as an annular flow.

Sensitivity Analysis of the Key Parameters of the Test
(1) Nozzle size In the blowout design, the nozzle size is an essential technological parame nozzle size significantly influences the clearance time, surface equipment, and flo We selected 3-, 6-, 9-, 12-, and 15-mm nozzles for the sensitivity analysis (Figure larger the nozzle size, the higher the gas production, the faster the clearance spe higher the mudline temperature. When selecting the nozzle size, it should be larg

Sensitivity Analysis of the Key Parameters of the Test
(1) Nozzle size In the blowout design, the nozzle size is an essential technological parameter. The nozzle size significantly influences the clearance time, surface equipment, and flow path. We selected 3-, 6-, 9-, 12-, and 15-mm nozzles for the sensitivity analysis ( Figure 10). The larger the nozzle size, the higher the gas production, the faster the clearance speed, and higher the mudline temperature. When selecting the nozzle size, it should be larger than 10 mm to improve the cleaning speed and mudline temperature as well as prevent hydration. (2) Influence of tubing sizes Based on the basic data of well A, 62-, 76-, 88.3-, and 100.5-mm inner diameter tubin were selected for the sensitivity analysis ( Figure 11). The larger the tubing, the lower th friction resistance, the higher the gas production, and faster the cleaning speed. Howeve affected by the decrease in the flow rate, the rising speed of the mudline temperatu would be slower. To improve the cleaning speed, choosing tubing with a size of 76 or 88 mm is recommended.  (2) Influence of tubing sizes Based on the basic data of well A, 62-, 76-, 88.3-, and 100.5-mm inner diameter tubing were selected for the sensitivity analysis ( Figure 11). The larger the tubing, the lower the friction resistance, the higher the gas production, and faster the cleaning speed. However, affected by the decrease in the flow rate, the rising speed of the mudline temperature would be slower. To improve the cleaning speed, choosing tubing with a size of 76 or 88.3 mm is recommended. (2) Influence of tubing sizes Based on the basic data of well A, 62-, 76-, 88.3-, and 100.5-mm inner diameter tubing were selected for the sensitivity analysis ( Figure 11). The larger the tubing, the lower the friction resistance, the higher the gas production, and faster the cleaning speed. However, affected by the decrease in the flow rate, the rising speed of the mudline temperature would be slower. To improve the cleaning speed, choosing tubing with a size of 76 or 88.3 mm is recommended.

Conclusions
(1) Coupling formation, wellbore, and surface nozzle flows as well as combining a twophase nozzle flow model, two-phase fluid holdup model, formation productivity equation, and deepwater transient heat transfer model, a deepwater blowout wellbore transient flow model was established, which considered the characteristics of the transient flow and determines the reliable blowout test time and test system. (2) As the blowout test continued, the bottomhole pressure decreased slightly and the wellhead pressure increased gradually. Moreover, the wellbore temperature increased gradually; however, the mudline temperature was low. The mudline temperature may be lower than the hydrate formation temperature in the cleaning process, so corresponding measures should be taken to prevent and control the hydrate formation. (3) The new model can be used to simulate the variation in the wellbore fluid level, pressure, temperature, gas production, liquid holdup, and hydrate formation temperature with time and well depth changing during the blowout period. The simulation result of gas well A showed that the calculated data agreed well with the measured values in the test, with an average error in the wellhead pressure and temperature of less than 5%, and the established model had a high calculation accuracy. (4) The sizes of the nozzles and tubing are the essential parameters in the blowout design. They significantly influence the clearance time, surface equipment, and blowout process. The transient model of the blowout test in this paper could effectively determine the nozzle regulation system during a blowout test as well as determine the reasonable injection time of the inhibitor and tubing string size. Application analysis and discussion showed that under the condition of satisfying the requirements of the test operation, choosing large nozzle and tubing sizes is necessary to accelerate the discharge of the wellbore working fluid, increase the wellbore temperature, and prevent hydrate formation.

Conclusions
(1) Coupling formation, wellbore, and surface nozzle flows as well as combining a twophase nozzle flow model, two-phase fluid holdup model, formation productivity equation, and deepwater transient heat transfer model, a deepwater blowout wellbore transient flow model was established, which considered the characteristics of the transient flow and determines the reliable blowout test time and test system. (2) As the blowout test continued, the bottomhole pressure decreased slightly and the wellhead pressure increased gradually. Moreover, the wellbore temperature increased gradually; however, the mudline temperature was low. The mudline temperature may be lower than the hydrate formation temperature in the cleaning process, so corresponding measures should be taken to prevent and control the hydrate formation. (3) The new model can be used to simulate the variation in the wellbore fluid level, pressure, temperature, gas production, liquid holdup, and hydrate formation temperature with time and well depth changing during the blowout period. The simulation result of gas well A showed that the calculated data agreed well with the measured values in the test, with an average error in the wellhead pressure and temperature of less than 5%, and the established model had a high calculation accuracy. (4) The sizes of the nozzles and tubing are the essential parameters in the blowout design.
They significantly influence the clearance time, surface equipment, and blowout process. The transient model of the blowout test in this paper could effectively determine the nozzle regulation system during a blowout test as well as determine the reasonable injection time of the inhibitor and tubing string size. Application analysis and discussion showed that under the condition of satisfying the requirements of the test operation, choosing large nozzle and tubing sizes is necessary to accelerate the discharge of the wellbore working fluid, increase the wellbore temperature, and prevent hydrate formation.