# Equivalent Circulation Density Analysis of Geothermal Well by Coupling Temperature

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

_{f}< p

_{a}< p

_{frac}. The annular pressures, such as static pressure, circulation pressure, initiating circulation pressure, viscous pressure and inertial pressure, are regulated through modifying drilling fluid systems, the operation procedure, and back pressure, at any point in the annulus for various well conditions [3]. The temperature at the bottom hole is much higher than the wellhead of the geothermal well; this temperature influences the density and properties of the drilling fluid, thus cannot be neglected. The calculated equivalent circulation density (ECD) with surface measured parameters will receive the improper result which raises the risk of lost circulation/blowout. The variation of the drilling fluid parameters with temperature will affect the pressure balance at the bottom hole, so wellbore temperature, drilling fluid density and rheology should be involved when building the wellbore pressure model of high temperature geothermal wells. Fluctuation pressure (surge and swag pressure) generated by a downward or upward pipe movement or running casing poses a great threat for safe drilling and induces lost circulation, kick and borehole collapse; the parameters that can control the fluctuation pressure are another problem to be studied.

## 2. Methods for Temperature and Pressure Distribution in the Wellbore

#### 2.1. Temperature Model of Wellbore

#### 2.1.1. Assumed Condition of Model

#### 2.1.2. Mathematical Equations

_{f}is the formation temperature near the wellbore, °C; ${\rho}_{f}$ is the formation density, kg/m

^{3}; ${c}_{f}$ is the specific heat capacity of formation, J/(kg·°C); ${k}_{f}$ is the thermal conductivity of formation, W/(m·K); r is the radius, m.

_{D,}the Kabir and Hasan Model [22] is used,

_{D}= 1, the temperature at the interface of the wellbore and formation is

_{D}, as

_{D}was used to arrive at the following expression,

_{a}is the drilling fluid temperature in the annulus, °C; T

_{p}is the drilling fluid temperature in the pipe, °C; L is the well depth, m; Q is the drilling fluid flow rate, m

^{3}/s; ${T}_{\epsilon i}$ is the initial formation temperature, °C; U

_{a}is the total convection coefficient between the annulus and formation, W/(m·K); C

_{m}is the specific heat of the drilling fluid, W/(m·K); U

_{p}is the total convection coefficient between the annulus and the pipe, W/(m·K); ${\rho}_{m}$ is the drilling fluid density at the depth of L in the well, kg/m

^{3}. ${r}_{w}$ is the inner radius of the wellbore, m; ${r}_{c}$ is the outer radius of the casing, m; ${r}_{cm}$ is the radius of cement, m; ${r}_{pi}$ is the inner radius of the pipe, m; ${r}_{po}$ is the outer radius of the pipe, m. ${k}_{t}$ is the thermal conductivity of the pipe, W/(m·K); ${k}_{c}$ is the thermal conductivity of the casing, W/(m·K); ${k}_{cm}$ is the thermal conductivity of cement, W/(m·K); ${h}_{a}$ is the convection heat transfer coefficient of the annulus, W/(m

^{2}·K); ${h}_{pi}$ is the convection heat transfer coefficient of the inner pipe, W/(m

^{2}·K); ${h}_{po}$ is the convection heat transfer coefficient of the outer pipe, W/(m

^{2}·K).

#### 2.1.3. Initial and Boundary Conditions

_{in}is the inlet temperature of the drilling fluid, °C; T

_{out}is the outlet temperature of the drilling fluid, °C.

_{1}, β

_{2}are modeling coefficients; G is the geothermal gradient, °C/100 m; ${T}_{s}$ is the surface temperature, °C.

#### 2.2. Pressure Model in the Wellbore during Circulation

#### 2.2.1. Pressure Gradient of Circulating Drilling Fluid in the Annulus

#### 2.2.2. Hydrostatic Pressure Gradient

_{m}is the drilling fluid density at depth z, kg/m

^{3}.

_{m}is the drilling fluid density under the condition of p and T, kg/m

^{3}; ρ

_{m}

_{0}is the drilling fluid density measured at the surface, kg/m

^{3}; p is the experiment pressure, Pa; p

_{0}is the atmosphere pressure, MPa; T is the experiment temperature, °C; T

_{0}is the temperature at the surface, the value is 15 °C; γ

_{p}, γ

_{pp}, γ

_{T}, γ

_{TT}and γ

_{pT}are coefficients which should be determined, for different drilling fluids, from density measurement at elevated pressures and temperatures. As the effect of pressure on density is insignificant relative to the effect of temperature in the geothermal well, the effect factor to density simply considers temperature, and if we expand Equation (11) in a Taylors series, and neglect the pressure terms and all higher order terms, Equation (11) can be conversed as follows,

#### 2.2.3. Friction Pressure Loss

_{Re}should be calculated as follows,

_{y}is the sheer stress at the yield point, Pa; K is the velocity coefficient; d is the inner diameter of the pipe, m; d

_{1}is the outer diameter of the pipe, m; d

_{2}is the inner diameter of the casing, m.

#### 2.2.4. Iterative Method to Solve the Pressure and Temperature Models

#### 2.3. Surge Pressure

#### 2.3.1. Initiating Circulation Pressure

_{g}is the gel strength, Pa.

_{f}/dL gives

#### 2.3.2. Inertial Effects

^{3}.

_{p}is the tripping acceleration, m/s

^{2}.

#### 2.3.3. Viscous Pressure Due to Vertical Pipe Movement

_{p}, is shown in Figure 3. Note that the velocity profile inside the inner pipe caused by a vertical pipe movement is identical to the velocity profile caused by pumping fluid down the inner pipe. If the mean fluid velocity in the pipe is expressed relative to the pipe wall, the viscous pressure in pipes can be expressed as follows,

_{0}and υ

_{0}, can be evaluated as the boundary conditions

## 3. Result and Discussion

_{2}geothermometer. The mean temperature gradient is 14 °C/100 m. The thermodynamics properties of materials in the wellbore and formation are listed in Table 1 and the fresh water from the river was used as the drilling fluid. According to the previous study [31], the height of the ZK212 to the zero point was −11.87 m, which means that the pore pressure was lower than the hydrostatic pressure. The density of the water varies with temperature, shown as follows,

^{3}/h. The Thermodynamics parameters and sizes of each section in the well are shown in Table 3 and Table 4 respectively.

#### 3.1. Temperature and ECD Distribution in the Wellbore

^{3}, 98 mPa·s, 1.731 W/(m·K) and 1256 J/(kg·°C) respectively. The drilling fluid turns into the Bingham model, and the calculated N

_{Re}is 384. As shown in Figure 9a, the temperature of the Bingham model in the wellbore is much higher than the temperature of the primary Newtonian model, so it is important to note that the temperature in the wellbore will shift after adding additives. Figure 9b shows that the ECD of the drilling fluid to which additives are added has a wide range of variation which is from 976 kg/m

^{3}to 1042 kg/m

^{3}. The position of highest ECD with additives is at the wellhead and it is higher relative to that of primary drilling fluid without additives.

#### 3.2. Surge Pressure Analysis

^{2}, and the maximum rate of the pipes is 2 m/s. In this study, the stable method is adopted to analyze the surge method in the well.

## 4. Conclusions

## Acknowledgements

## Author Contributions

## Conflicts of Interest

## References

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**Figure 7.**Temperature (

**a**) and ECD (

**b**) in the annulus with the geothermal gradient (circulating 8 h).

**Figure 8.**Temperature (

**a**) and ECD (

**b**) in the annulus with the outer diameter of the pipe (circulating 8 h).

**Figure 9.**Temperature (

**a**) and ECD (

**b**) in the annulus with the rheological model of the pipe (circulating 8 h).

**Table 1.**The pressure combination in different condition [24].

Conditions Partial Pressure | Hydrostatic Pressure | Frictional Pressure Loss | Initiating Circulation Pressure | Viscous Pressure | Inertial Wave | |
---|---|---|---|---|---|---|

p_{s} | p_{f} | p_{g} | p_{V} | p_{i} | ||

0 | Standard static | + | ||||

1 | pumping | + | + | |||

2 | circulation | + | + | |||

3a | Trip out (acceleration) | + | - | - | ||

3b | Trip out (constant) | + | - | |||

3c | Trip out (decelerate) | + | - | + | ||

4a | Trip in (acceleration) | + | + | + | ||

4b | Trip in (constant) | + | + | |||

4c | Trip in (decelerate) | + | + | - |

**Table 2.**The frictional pressure loss of a different rheological model under laminar in the pipe and annulus [26].

Position | Newtonian Model | Bingham Model | Power Law Model |
---|---|---|---|

Pipes | $\frac{d{p}_{f}}{dL}=31.92\frac{\mu \overline{v}}{{d}^{2}}$ | $\frac{d{p}_{f}}{dL}=31.92\frac{{\mu}_{p}\overline{v}}{{d}^{2}}+52.3\frac{{\tau}_{y}}{d}$ | $\frac{d{p}_{f}}{dL}=0.1571\frac{{(0.0254)}^{1+n}K{\overline{v}}^{n}{(\frac{3+1/n}{0.0416})}^{n}}{{(0.3048)}^{n}{d}^{1+n}}$ |

Annulus | $\frac{d{p}_{f}}{dL}=47.88\frac{\mu \overline{v}}{{({d}_{2}-{d}_{1})}^{2}}$ | $\frac{d{p}_{f}}{dL}=47.88\frac{{\mu}_{p}\overline{v}}{{({d}_{2}-{d}_{1})}^{2}}+58.84\frac{{\mathsf{\tau}}_{y}}{{d}_{2}-{d}_{1}}$ | $\frac{d{p}_{f}}{dL}=0.1571\frac{{(0.0254)}^{n+1}K{\overline{v}}^{n}{(\frac{2+1/n}{0.0208})}^{n}}{{(0.3048)}^{n}{({d}_{2}-{d}_{1})}^{1+n}}$ |

Name | Density (kg/m^{3}) | Thermal Conductivity (W/(m·K)) | Thermal Capacity (J/(kg·°C) |
---|---|---|---|

Sandstone | 2231 | 1.869 | 711.76 |

Basalt | 1579 | 2.008 | 879.23 |

Granite | 2641 | 2.821 | 837.36 |

Cement | 2100 | 1.454 | 879.23 |

Casing | 7848 | 45.174 | 460.55 |

Materials | Inner Diameter (mm) | Outer Diameter(mm) |
---|---|---|

pipe | 139 | 159 |

casing | 245 | 311 |

cement | 311 | 406 |

Depth (m) | 300 | 500 | 800 | 1000 | 1200 |

Surge pressure (MPa) | 1.57 | 1.31 | 0.92 | 0.66 | 0.39 |

Diameter (m) | 0.216 | 0.245 | 0.265 | 0.290 | 0.311 |

Surge pressure (MPa) | 1.97 | 0.69 | 0.42 | 0.26 | 0.19 |

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**MDPI and ACS Style**

Zheng, X.; Duan, C.; Yan, Z.; Ye, H.; Wang, Z.; Xia, B.
Equivalent Circulation Density Analysis of Geothermal Well by Coupling Temperature. *Energies* **2017**, *10*, 268.
https://doi.org/10.3390/en10030268

**AMA Style**

Zheng X, Duan C, Yan Z, Ye H, Wang Z, Xia B.
Equivalent Circulation Density Analysis of Geothermal Well by Coupling Temperature. *Energies*. 2017; 10(3):268.
https://doi.org/10.3390/en10030268

**Chicago/Turabian Style**

Zheng, Xiuhua, Chenyang Duan, Zheng Yan, Hongyu Ye, Zhiqing Wang, and Bairu Xia.
2017. "Equivalent Circulation Density Analysis of Geothermal Well by Coupling Temperature" *Energies* 10, no. 3: 268.
https://doi.org/10.3390/en10030268