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A coupled system model of partial differential equations is presented in this paper, which concerns the variation of the pressure and temperature, velocity, and density at different times and depths in high temperature-high pressure (HTHP) gas-liquid two-phase flow wells. A new dimensional splitting technique with Eulerian generalized riemann problem (GRP) scheme is applied to solve this set of conservation equations, where Riemann invariants are introduced as the main ingredient to resolve the generalized Riemann problem. The basic data of “X well” (HTHP well), 7100 m deep, located in Southwest China, is used for the case history calculations. Curve graphs of pressures and temperatures along the depth of the well are plotted at different times. The comparison with the results of Lax Friedrichs (LxF) method shows that the calculating results are more fitting to the values of real measurement and the new method is of high accuracy.

The prediction of pressure and temperature of transient gas-liquid flow in a wellbore is important but difficult for well completion test because they are characterized by the dependence of pressure, density, velocity, and other flow parameters on both time and space. As for pressure prediction research, there exist empirical formulas, such as those given by Beggs and Brill [

Concerning both pressure and temperature in HTHP wells, Wu et al. have presented a coupled system model of differential equations in [

We found an algorithm solving model with generalized Riemann problem (GRP) scheme, which is an analytic extension of the Godunov scheme in [

In this paper, we use GRP method for solving this problem and get more accurate prediction of pressure and temperature compared with those obtained from the existing correlations such as LxF method in [

Considering the two-phase flow system shown in Figure

The two-phase flow.

Consider the flow model shown in Figure

Control volume 1.

Under transient conditions, applied to the control volume in Figure

Control volume 2.

Substituting (

As shown in Figure

Control volume 3.

Substituting (

For the transient flow, it leads directly to the energy equation in terms of temperature. As shown in Figure

The radial transfer of heat.

According to the energy balance law, the heat variation flowing on control volume that is equal to the combination heat of inflow and outflow, and the heat transferring to the second dimension, we get the energy balance equation of transient flow:

Finally, we obtain the coupled system model of partial differential equations:

We unify the conservation equations (

We define the equally spaced grid points, the interface points, and the cells as

We assume that the data at time

The second-order Godunov scheme for (

The local wave configuration is usually piecewise smooth and consists of rarefaction waves, shocks, and contact discontinuities. As the general rarefaction waves are considered, the initial data can be regarded as a perturbation of the Riemann initial data

The GRP scheme assumes piecewise linear data for the flow variables, which leads to the generalized Riemann problem for (

The initial structure of the solution is determined by the associated Riemann solution, denoted by

Set the step length. In this paper,

Obtain each point’s inclination:

The in situ liquid volume fraction (holdup) in (

Calculate the following parameters by Liao and Feng in [

For piecewise given initial data

Determine

In this simulation, we study a pipe in X well located in Sichuan Basin, Southwest China. All the needed parameters are given in [^{3}; depth of the well is 7100 m; friction coefficient is 1.2; ground temperature is 160°C; ground thermal conductivity parameter is 2.06; ground temperature gradient is 0.0218°C/m. Parameters of pipes are given in Table

Parameters of pipes.

Diameter | Thickness | Weight | Expansion | Coefficient | Young | Modulus |
---|---|---|---|---|---|---|

88.9 | 9.53 | 18.9 | 0.0000115 | 215 | 0.3 | 1400 |

88.9 | 7.34 | 15.18 | 0.0000115 | 215 | 0.3 | 750 |

88.9 | 6.45 | 13.69 | 0.0000115 | 215 | 0.3 | 4200 |

73 | 7.82 | 12.8 | 0.0000115 | 215 | 0.3 | 600 |

73 | 5.51 | 9.52 | 0.0000115 | 215 | 0.3 | 150 |

Parameters of azimuth, inclination, and vertical depth.

Number | Measured | Inclination | Azimuth | Vertical depth |
---|---|---|---|---|

1 | 0 | 0 | 120.33 | 0 |

2 | 303 | 1.97 | 121.2 | 302.87 |

3 | 600 | 1.93 | 120.28 | 599.73 |

4 | 899 | 0.75 | 126.57 | 898.59 |

5 | 1206 | 1.25 | 124.9 | 1205.45 |

6 | 1505 | 1.04 | 124.62 | 1504.32 |

7 | 1800 | 0.49 | 123.75 | 1799.18 |

8 | 2105 | 2.49 | 125.27 | 2104.04 |

9 | 2401 | 1.27 | 123.13 | 2399.91 |

10 | 2669 | 2.44 | 120.12 | 2667.79 |

11 | 3021 | 0.14 | 127.39 | 3019.63 |

12 | 3299 | 1.18 | 122.60 | 3297.50 |

13 | 3605 | 2.05 | 123.25 | 3603.36 |

14 | 3901 | 0.16 | 121.45 | 3899.22 |

15 | 4183 | 2.92 | 121.24 | 4181.09 |

16 | 4492 | 2.73 | 129.22 | 4489.95 |

17 | 4816.07 | 1.98 | 121.61 | 4813.87 |

18 | 5099.07 | 2.74 | 129.93 | 5096.74 |

19 | 5394.07 | 0.13 | 120.46 | 5391.61 |

20 | 5706.07 | 0.63 | 129.59 | 5703.47 |

21 | 5983.07 | 2.09 | 120.14 | 5980.34 |

22 | 6302.07 | 2.69 | 122.91 | 6299.19 |

23 | 6597.07 | 2.45 | 129.41 | 6594.06 |

24 | 6911.12 | 0.15 | 124.88 | 6907.96 |

Through the simulation, we use GRP method to calculate the prediction of pressure and temperature of the oil in the pipe and draw a sensitive analysis for the results. We compare the results of pressure and temperature calculated for the well head at 1200 s by GRP and LxF scheme with the measurement results, which also shows that GRP scheme is more accurate in the real calculation. We obtain series of results contained in tables and figures and analyze these results as follows.

When the bottom pressure is 70 MPa, temperatures are plotted in Figure

Temperature at different depths on 300 s, 900 s, 1200 s, and 3600 s.

Depth | Time | |||
---|---|---|---|---|

300 s | 900 s | 1200 s | 3600 s | |

0 | 81.22 | 115.29 | 124.19 | 132.27 |

300 | 85.45 | 121.38 | 127.55 | 133.34 |

600 | 92.67 | 125.54 | 131.76 | 136.56 |

900 | 95.54 | 129.48 | 134.96 | 138.87 |

1200 | 101.16 | 133.77 | 137.58 | 141.94 |

1500 | 106.49 | 136.66 | 140.77 | 143.15 |

1800 | 111.76 | 140.67 | 143.23 | 145.17 |

2100 | 116.98 | 143.54 | 146.29 | 147.39 |

2400 | 121.86 | 145.66 | 148.49 | 149.14 |

2700 | 126.89 | 148.45 | 150.78 | 151.93 |

3000 | 131.55 | 151.78 | 152.99 | 153.85 |

3300 | 136.85 | 153.74 | 154.86 | 155.91 |

3600 | 140.88 | 154.02 | 154.46 | 157.87 |

3900 | 144.67 | 157.12 | 158.73 | 159.45 |

4200 | 148.25 | 159.34 | 160.87 | 161.88 |

4500 | 152.74 | 161.53 | 161.65 | 162.65 |

4800 | 155.77 | 162.55 | 162.72 | 163.45 |

5100 | 159.75 | 163.42 | 163.49 | 164.56 |

5400 | 162.36 | 164.56 | 164.87 | 165.24 |

5700 | 164.32 | 165.74 | 165.45 | 166.57 |

6000 | 166.36 | 166.56 | 167.67 | 167.97 |

6300 | 167.91 | 168.77 | 168.87 | 169.65 |

6600 | 168.23 | 169.45 | 169.57 | 169.81 |

6900 | 170.24 | 170.56 | 171.78 | 171.52 |

Temperature distribution at different depths.

It is established that, when depth is constant, the pressure shown in Figure

Pressure at different depths on 300 s, 900 s, 1200 s, and 3600 s.

Depth | Time | |||
---|---|---|---|---|

300 s | 900 s | 1200 s | 3600 s | |

0 | 42.55 | 46.62 | 50.24 | 51.34 |

300 | 43.23 | 47.53 | 50.64 | 52.67 |

600 | 44.86 | 48.71 | 50.46 | 53.47 |

900 | 45.87 | 49.13 | 51.41 | 54.69 |

1200 | 46.73 | 50.43 | 52.32 | 54.79 |

1500 | 48.46 | 51.24 | 53.36 | 55.53 |

1800 | 49.43 | 52.43 | 54.47 | 56.12 |

2100 | 50.34 | 53.83 | 55.42 | 57.37 |

2400 | 51.96 | 56.92 | 54.37 | 57.85 |

2700 | 53.53 | 57.22 | 55.45 | 58.97 |

3000 | 54.44 | 58.46 | 56.78 | 59.34 |

3300 | 55.24 | 59.97 | 57.47 | 60.95 |

3600 | 56.76 | 59.94 | 58.95 | 61.22 |

3900 | 57.33 | 60.98 | 59.04 | 62.29 |

4200 | 58.93 | 61.22 | 60.29 | 63.33 |

4500 | 59.34 | 62.45 | 61.24 | 64.48 |

4800 | 60.89 | 63.43 | 62.23 | 64.33 |

5100 | 61.56 | 64.19 | 63.22 | 64.78 |

5400 | 63.35 | 65.24 | 64.18 | 65.34 |

5700 | 64.69 | 65.45 | 65.12 | 66.34 |

6000 | 65.45 | 66.79 | 66.15 | 67.56 |

6300 | 66.99 | 67.49 | 67.11 | 67.47 |

6600 | 67.46 | 68.52 | 68.22 | 68.58 |

6900 | 69.28 | 69.46 | 69.92 | 69.55 |

Pressure distribution at different depths.

As shown in Table

Comparative results of the well head at 1200 s.

Well-head | Temperature | Pressure |
---|---|---|

Measurement results | 180.65 | 76.10 |

Results by GRP method (relative error) | 171.78 (5.12%) | 69.92 (8.81%) |

Results by LxF method (relative error) | 169.30 (6.70%) | 69.36 (9.73%) |

In this paper, considering the variation of pressure, temperature, velocity; and density at different times and depths in gas-liquid two-phase flow, we present a system model of partial differential equations according to mass, momentum, and energy. We establish an algorithm solving model with a new difference method with a direct Eulerian GRP scheme which is proven to be efficient for the numerical implementation in this paper. The basic data of the X well (HTHP well), 7100 m deep in Sichuan Basin, Southwest China, was used for case history calculations, and a sensitivity analysis is completed for the model. The gas-liquid’s pressure and temperature curves along the depth of the well are plotted, and the curves intuitively reflect the flow law and the characteristics of heat transfer in formation. The results can provide the technical reliance for the process of designing well tests in high temperature-high pressure gas-liquid two-phase flow wells and dynamic analysis of production. Furthermore, the works in this paper can raise safety and reliability of deep completion test and will yield notable economic and social benefits and avoid or lessen accidents caused by improper technical design.

A total length of conduit (m^{2})

Joule-Thompson coefficient (K/pa)

Heat capacity (J/kg·K)

A hydraulic diameter (m)

Acceleration constant of gravity (m/s^{2})

Formation conductivity (J/m·K)

Pressure (KPa)

Dimensionless radius

Outer radius of conduit (m)

Temperature (K)

Dimensionless time

Temperature of the stratum (K)

Wellbore temperature (K)

Dimensionless wellbore temperature (K)

Temperature of the second surface (K)

Initial temperature of formation (K)

Overall-heat-transfer coefficient (W/m·K)

Velocity (m/s)

A total length of conduit (m)

The distance coordinate in the direction along the conduit

Heat transfer coefficient for natural convection based on outside tubing surface and the temperature difference between outside tubing and inside casing surface

Heat transfer coefficient for radiation based on the outside tubing surface and the temperature difference between the outside tubing and inside casing surface

Thermal conductivity of the casing material at the average casing temperature

Thermal conductivity of the cement at the average cement temperature and pressure

The friction coefficient, dimensionless

Euler constant 1.781

Density (kg/m^{3})

Inclination angle flow conduit.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This research was supported by the Key Program of NSFC (Grant no. 70831005) and the Key Project of China Petroleum and Chemical Corporation (Grant no. GJ-73-0706).