# Predictions of Rock Temperature Evolution at the Lahendong Geothermal Field by Coupled Numerical Model with Discrete Fracture Model Scheme

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## Abstract

**:**

## 1. Introduction

^{®}v. 5.5, 2019). The Lahendong geothermal field, located in Tomohon, North Sulawesi, Indonesia (Figure 1), was examined for the present case study. The Lahendong geothermal field has four units power plant with a total production capacity of 80 MWe [5]. Unit 1, the first 20-MWe power plant, has been producing electricity since June 2001; Unit 2, the second 20-MWe power plant, has been producing electricity since 2007; and Units 3 and 4 (2 × 20 MWe) have been producing electricity since 2009. It is important to understand the evolution of the reservoir characteristics in order to evaluate the sustainability of the reservoir. The reservoir temperature is divided into two different reservoir categories. There are very high temperatures in the southern zone (350 °C) and modest temperatures in the northern zone (250 °C). The southern and northern zones show dryness ranging from about 80% to 100% and 30% to 50%, respectively [6].

## 2. Conceptual Model

#### 2.1. Geological

#### 2.2. Geophysical

#### 2.3. Well-Log Analysis

## 3. Methodology

- The rock mass is treated as a 3D fractured porous media consisting of the rock matrix and discrete fractures. The discrete fracture model is applied to the fault zone in this study area.
- The fractured porous geothermal reservoir has a single-phase fluid saturation. Therefore, the water and fluid flow both in the rock matrix block and fractures comply with the Darcy’s Law.
- The model ignores the variation of fracture aperture.
- The diameter of the well is small, so that storage is negligible.
- As shown in the published paper [22], the water density and dynamic viscosity are not constant but a function of pressure and temperature.
- The density, porosity, permeability, specific heat, and thermal conductivity of the fractured porous media are assumed to be constant.

#### 3.1. Discrete Fracture Model

#### 3.2. Governing Equations

^{®}v. 5.5, 2019). The differential equations used in this study are described in [23,24]; they are divided into the rock matrix and discrete fractures as explained below.

#### 3.2.1. Fluid Migration

^{2}], η is the dynamic fluid viscosity [Pa·s], ${\rho}_{w}$ is the fluid density [kg/m

^{3}], g is the gravitation [m/s

^{2}], e is the volumetric strain of the rock matrix [-], Q is the source-sink term of the seepage process [s

^{−1}], d

_{f}is the fracture thickness [m], S

_{f}is the fracture storage coefficient [1/Pa], ∇τ is the gradient operator restricted to the fracture’s tangential plane, κ

_{f}is the fracture permeability [m

^{2}], e

_{f}is the volumetric strain of the fracture [-], and Q

_{f}is the flow exchange between the rock matrix and the fractures [s

^{−1}] where expressed by ${Q}_{f}=-\frac{{\kappa}_{f}}{\eta}\frac{\partial p}{\partial n}$.

#### 3.2.2. Rock Mass Temperature

^{3}], Cs is the specific heat capacity of the rock matrix [J/kg/K], Ts is the rock temperature [°C], λ is the thermal conductivity of the rock matrix [W/m/K], and q is the heat source [W/m

^{3}].

_{w}is the specific heat capacity of the fluid [J/kg/K], T

_{f}is the water temperature in the discrete fractures [°C], μ

_{f}is the fluid flow velocity in the discrete fractures [m/s], λ

_{f}is the thermal conductivity of the fluid [W/m/K], and h is the convection efficiency [W/m

^{2}/K].

_{0}is the mass flow rate [kg/s], l is the well length [m], and T

_{in}is the injection temperature [°C].

#### 3.2.3. Rock Mass Stress Field

_{ij,j}is the Cauchy stress tensor and F

_{i}is the body force per unit volume in the x, y, z coordinate in 3D. The deformation equation of the rock matrix is expressed as

_{B}is the Biot-Willis coefficient [-], and α

_{T}is the thermal expansion coefficient [1/K].

_{f}is the displacement of the fractures [mm], σ’ is the effective stress of the fractures [Pa], and k is the fracture stiffness [Pa/m]. Subscripts n and s refer to the normal and tangential directions, respectively. The governing equations for calculating the natural state condition are in a stationary condition. This means that the reservoir variable does not change over time to reach the steady flow and pressure field.

#### 3.3. Validation and Calibration of the Numerical Model

#### 3.3.1. Validation of the TH and THM Coupling Model

_{0}is the initial temperature [°C], T

_{in}is the injection temperature [°C], erfc is the complementary error function, and U is a unit step function.

_{f}) of 1 mm. Figure 7 shows the distribution of temperature contours at different times. It shows that the temperature change in the fracture decreases faster than that in the rock matrix. Figure 8 provides more detailed information on the temperature distribution along the fracture. Figure 8a shows the temperature change in the fracture at different times (t = 10 days, 100 days, and 1000 days), while Figure 8b shows the temperature at three different positions in the fracture (x = 10 m, 50 m, and 100 m), respectively. It can be seen that the numerical results are in excellent agreement with the analytical solutions.

#### 3.3.2. Natural State Calibration

## 4. Computational Model

^{2}with a domain of 5000 m × 7000 m × 2800 m. Three geological layers are set, and the fault zone is applied as the discrete fracture model. The finite element mesh system used in the calculation is illustrated in Figure 14. The mesh consists of 416,850 tetrahedral elements and 65,195 triangular elements. A finer mesh is generated for all elements in the model.

#### 4.1. Initial and Boundary Conditions

#### 4.2. Model Parameters

## 5. Results and Discussions

#### 5.1. Natural State Condition

#### 5.2. Evolution of the Reservoir Temperature

_{2}S/H

_{2}gas geothermometer calculations well. The temperature of the reservoir fluctuated at around 270 °C over the given period. This means that there was no significant change in temperature in this reservoir.

#### 5.3. Specific Gross Electrical Power Prediction

_{f}, kJ/kg), specific enthalpy of saturated vapor (h

_{g}, kJ/kg), specific entropy of saturated liquid (s

_{f}, kJ/kg/K), and specific entropy of saturated vapor (s

_{g}, kJ/kg/K) before and after the expansion process. After the temperature or pressure of the reservoir was predicted, the thermodynamic properties of fluid can be defined by using the table of properties for saturated water [45].

## 6. Conclusions

_{2}S/H

_{2}gas geothermometer studies. This suggests that the cold-water flow from the injection well to the production wells was not the main problem.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 4.**Magnetotellurics (MT) survey data in the field area [7].

**Figure 5.**2D fractured porous medium in discrete fracture model [27].

**Figure 7.**Temperature change over time in the range of: (

**a**) t = 1 day; (

**b**) t = 10 days; (

**c**) t = 100 days; (

**d**) t = 1000 days.

**Figure 8.**Temperature distribution: (

**a**) in discrete fracture at different times; (

**b**) in discrete fracture at different distances from the injection well.

**Figure 11.**Comparison of analytical and numerical solution in the soil column at different positions: (

**a**) displacement; (

**b**) pore pressure.

**Figure 14.**Mesh system in model geometry (tetrahedral elements: 416,850; triangular elements: 65,195): (

**a**) rock matrix; (

**b**) discrete fractures.

**Figure 15.**Geothermal field temperature under natural state condition: (

**a**) 3D geometry; (

**b**) cut plane in y = 141,000 m, (

**c**) cut plane in x = 705,500 m.

**Figure 18.**Measured versus simulated results in natural state condition: (

**a**) pressure; (

**b**) temperature. (Measured data are based on Yani [8]).

**Figure 19.**Comparison of reservoir temperatures among initial condition, wellhead temperature, and prediction by the current model.

**Figure 20.**Comparison of reservoir temperature evolution prediction in LHD-11 among initial condition, wellhead temperature, enthalpy monitoring: (

**a**) liquid geothermometer data; (

**b**) gas geothermometer data.

Parameter | Unit | Value |
---|---|---|

Initial temperature | °C | 300 |

Injection temperature | °C | 40 |

Thermal conductivity | W/m/K | 3.5 |

Fluid density | kg/m^{3} | 1000 |

Specific heat capacity of the fluid | J/kg/K | 4200 |

Rock density | kg/m^{3} | 2500 |

Specific heat capacity of the rock | J/kg/K | 1000 |

Fracture thickness | m | 5 $\times $ 10^{−4} |

Fluid flow velocity in the discrete fractures | m/s | 0.01 |

Parameter | Unit | Value |
---|---|---|

Fluid density | kg/m^{3} | 1000 |

Soil density | kg/m^{3} | 2600 |

Porosity | - | 0.4 |

Hydraulic conductivity | m/s | 1 $\times $ 10^{−9} |

Thermal conductivity | W/m/K | 0.5 |

Specific heat capacity of the fluid | J/kg/K | 4200 |

Specific heat capacity of the soil | J/kg/K | 800 |

Elastic modulus of the soil | MPa | 60 |

Poisson’s ratio | - | 0.4 |

Thermal expansion coefficient | - | 3 $\times $ 10^{−7} |

Biot-Willis coefficient | - | 1.0 |

Coefficient of compressibility | Pa^{−1} | 1.1 $\times $ 10^{−10} |

Type | Location | |||||
---|---|---|---|---|---|---|

Bottom | Lake | Northern | Southern | Western | Eastern | |

Temperature, °C | 91-0.0853z | 213-0.0801z | 213-0.0801z | 59-0.0199z | ||

Hydraulic head ^{1,2}, m | 767 | 772 | 809 | 837 | 506 | |

Heat flux ^{2}, mW/m^{2} | 100 |

Parameter | Unit | Value | |||
---|---|---|---|---|---|

Post-Tondano Formation | Tondano Formation | Pre-Tondano Formation | Fracture | ||

Density ^{1} | kg/m^{3} | 2630 | 2320 | 2490 | 1800 |

Porosity ^{1} | 0.05 | 0.12 | 0.11 | 0.2 | |

Permeability (x, y, z) ^{2} | m^{2} | 2.1 $\times $ 10^{−15}, 2.1 $\times $ 10^{−15}, 2.1 $\times $ 10^{−13} | 2.3 $\times $ 10^{−13}, 2.3 $\times $ 10^{−13}, 2.3 $\times $ 10^{−11} | 1 $\times $ 10^{−17}, 2 $\times $ 10^{−15}, 5 $\times $ 10^{−13} | 7 $\times $ 10^{−15}, 2 $\times $ 10^{−15}, 2 $\times $ 10^{−15} |

Specific heat ^{2} | J/kg/K | 1000 | 1000 | 1000 | 1000 |

Heat conductivity ^{3} | W/m/K | 2.2 | 2.5 | 2.1 | 1.8 |

Elastic modulus | GPa | 45 | 50 | 65 | |

Poisson’s ratio ^{4} | 0.4 | 0.35 | 0.3 |

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**MDPI and ACS Style**

Qarinur, M.; Ogata, S.; Kinoshita, N.; Yasuhara, H.
Predictions of Rock Temperature Evolution at the Lahendong Geothermal Field by Coupled Numerical Model with Discrete Fracture Model Scheme. *Energies* **2020**, *13*, 3282.
https://doi.org/10.3390/en13123282

**AMA Style**

Qarinur M, Ogata S, Kinoshita N, Yasuhara H.
Predictions of Rock Temperature Evolution at the Lahendong Geothermal Field by Coupled Numerical Model with Discrete Fracture Model Scheme. *Energies*. 2020; 13(12):3282.
https://doi.org/10.3390/en13123282

**Chicago/Turabian Style**

Qarinur, Muhammad, Sho Ogata, Naoki Kinoshita, and Hideaki Yasuhara.
2020. "Predictions of Rock Temperature Evolution at the Lahendong Geothermal Field by Coupled Numerical Model with Discrete Fracture Model Scheme" *Energies* 13, no. 12: 3282.
https://doi.org/10.3390/en13123282