^{1}

^{1}

^{2}

^{1}

^{2}

The complex thermal-hydraulic-mechanical (THM) coupling is the key issue to the energy extraction from a geothermal reservoir, where fractures are the main channels for fluid circulation and heat transfer. However, the effects of matrix deformation-induced aperture variation and fracture roughness on heat recovery efficiency are unclear. In this paper, a fully coupling THM model based on a discrete fracture network is proposed to explore these coupling effects. First, the fracture roughness and the fracture aperture variation with effective stress are introduced. Second, the water flow and heat transfer in the matrix and fractures as well as the deformation of the geothermal reservoir are individually formulated for a fractured geothermal reservoir. Third, the model is validated with analytical solution for its thermal-hydraulic (TH) coupling effect and literature data for its hydraulic-mechanical (HM) coupling effect. Finally, the features of heat transfer and fluid flow in the fractured geothermal reservoir are comparatively analyzed through four scenarios. The simulation results indicate that the discrete fracture network severely impacts the pressure distribution and temperature advance. The aperture variation induced by solid deformation can enhance heat transfer efficiency, and the fracture roughness can reduce the heat transfer efficiency.

A general operation method for energy extraction from geothermal reservoirs is to inject cold fluid in and to pump hot fluid out [

Different models have been proposed for the THM coupling in an enhanced geothermal system (EGS). First, TH coupling models were developed for the forced convection in geothermal exploitations. They investigated the heterogeneity of porous media [

A fractured geothermal reservoir can be modeled by either continuum-based models (CM) or discrete fracture models (DFM) [

The heat extraction performance of a geothermal system has been evaluated based on DFN models [

This paper develops a THM coupling model to consider the effects of aperture variation and fracture roughness for the heat energy extraction from a geothermal reservoir. Firstly, the discrete fracture network is generated through the Monte Carlo method. The fracture characteristics such as location, length, and orientation are included. Secondly, the aperture variation is described by a function of fracture aperture with effective stress. Further, a roughness factor is introduced into Darcy’s law to describe the effect of fracture roughness on fluid flow in a fracture. Thirdly, a full THM coupling model is established for the heat extraction process from a fractured geothermal reservoir and is validated by two degraded submodels. Finally, the distribution of pressure and temperature in the fractured reservoir is numerically simulated. Their coupling effects are comparatively analyzed.

A fracture network is constructed by randomly generating fractures through the algorithm of the Monte Carlo method. The trace length, orientation, and location of fractures are the three key parameters to a two-dimensional fracture network. Compared with trace length, fracture aperture is too tiny to be expressed on the graph of the fracture network although it severely impacts the fluid seepage. These three key parameters are determined through the following procedure.

Previous studies have shown that the trace length of fractures in geothermal reservoirs obeys a power-law distribution [

The fractures in the geothermal reservoir can be grouped into different sets according to their orientations [

The midpoint of a fracture is used to indicate the location of a fracture. The coordinates of the fracture center is fixed by generating random numbers based on uniform distribution. The fracture centers are sequentially generated. If one fracture is too close to other fractures (<2 m in the domain of

Two types of fracture networks are generated in Figure

Four fracture networks generated by the MATLAB code.

Geometric parameters used for DFN generation.

DFN | Parameter | Fracture length | Fracture number | Orientation | Center point |
---|---|---|---|---|---|

Distribution | Power-law distribution | Designated | Gaussian distribution | Uniform distribution | |

(a-1) | Set 1 | 40-200 m | 120 | 0°/0^{2} | |

Set 2 | 20-150 m | 150 | 90°/0^{2} | ||

(a-2) | Set 1 | 40-200 m | 60 | 0°/0^{2} | |

Set 2 | 20-150 m | 75 | 90°/0^{2} | ||

(b-1) | Set 1 | 40-200 m | 120 | 20°/0^{2} | |

Set 2 | 20-150 m | 150 | 130°/0^{2} | ||

(b-2) | Set 1 | 40-200 m | 60 | 20°/0^{2} | |

Set 2 | 20-150 m | 75 | 130°/0^{2} |

This fully coupling model has five separate yet interacting governing equations: a thermo-elastic mechanical equation, two flow equations (for the fractures and rock matrix), two heat transfer equations in the fractures and rock matrix. This model is mainly based on the following assumptions: (1) The rock matrix is continuous, isotropic, and homogeneous. (2) Surrounding rocks are impermeable. The fluid loss is not considered. (3) Both the matrix and fractures are saturated with single-phase liquid. (4) The fluid flow in both the matrix and fractures follows Darcy’s law. (5) The local thermal equilibrium is assumed at the interface between the rock matrix and the working fluid.

In the saturated porous matrix, the mass conservation of fluid is

The seepage velocity

Furthermore, the storage term at the first term of Equation (

The porosity dilation-induced fluid variation is treated as the source term

Thus, the governing equation of water flow in the matrix is

The continuity equation of fluid flow in discrete fractures is

A fracture consists of two smooth parallel plates; thus, the volumetric flow rate

We introduce

The following empirical expression is used in our computation:

In this study, fractures are treated as the internal boundaries of the matrix seepage model. The fluid pressure in fractures is the Dirichlet boundary condition of fluid flow in the matrix, while the pore pressure of the matrix is the motivation of the source/sink term of fracture flow. The source or sink term in the governing equation for fracture flow is

Finally, the governing equation for fracture flow is obtained as

As a kind of porous media, the rock matrix has a higher specific surface area. Fluid can exchange heat with the matrix instantly. Hence, the temperature of pore water is assumed to be identical to that of the rock matrix. Then, the heat transfer equation in the matrix can be written as

The heat flux

The heat transfer equation in discrete fractures is

The constitutive equation of poroelasticity for THM coupling is

The Navier equation for displacement

A geothermal reservoir is composed of a rock matrix and fracture network. The rock matrix is usually of low porosity and low permeability. The slight change of porosity has little effect on water flow and heat transfer; thus, the rock matrix is assumed to have constant porosity. The fracture network is the primary channel for fluid flow. Any change of fracture aperture may severely affect water flow. The fracture aperture is related to effective normal stress as [

The fluid property includes water density

These governing equations with the initial and boundary conditions formulate a complete THM coupling model. Generally, these partial differential equations are too complex to derive analytical solutions. This study uses a finite element method (FEM) to solve this problem. Particularly, the FEM solutions are numerically obtained through the COMSOL Multiphysics, a powerful solver for partial differential equations. The solution procedure has four steps. Firstly, rock fractures are generated with the Monte Carlo algorithm. Then, the geometric model of fractured rock mass is imported into COMSOL Multiphysics. Thirdly, both governing equations and boundary conditions are assigned to the geometric model, and model parameters and initial conditions are set. Finally, the distributions of temperature and pressure are solved and visualized.

A realistic geothermal reservoir usually has a three-dimensional (3D) fracture network. A complex 3D fracture network is extremely difficult in implementation and largely increases computation cost. In this study, a 2D fracture network in Figure

FEM mesh model of the geothermal reservoir with initial and boundary conditions.

For the reservoir seepage, the hydraulic pressure is kept as 80 MPa at the injection well and as 66 MPa at the right boundary. The top and bottom boundaries are impermeable. The initial pressure of this reservoir is 70 MPa. For the heat transfer in the reservoir, the water temperature is 30°C at the injection well. The top and bottom boundaries are thermally insulated. The initial temperature of the reservoir is 180°C. For the mechanical equilibrium model, the top and right boundaries are applied a constant normal compression of 90 MPa, which is equivalent to the overburden stress at 3500 m deep. The other two boundaries have zero normal displacement and free tangential displacement. Table

Parameters of the coupling model.

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

Rock matrix density | 2700 | kg·m^{-3} | |

Specific heat capacity of water | 4200 | J·kg^{-1}·K^{-1} | |

Specific heat capacity of the rock matrix | 1000 | J·kg^{-1}·K^{-1} | |

Thermal conductivity of water | 0.7 | W·m^{-1}·K^{-1} | |

Thermal conductivity of the rock matrix | 3 | W·m^{-1}·K^{-1} | |

Porosity | 0.05 | 1 | |

Matrix permeability | m^{2} | ||

Young’s modulus | 30 | GPa | |

Poisson’s ratio | 0.25 | 1 | |

Biot’s coefficient | 1 | 1 | |

Coefficient of volumetric expansion | 1/K | ||

Storage coefficient of the matrix | 1/Pa | ||

Initial aperture of fractures | 0.18 | mm | |

Standard deviation of the fracture surface height | 0.005 | mm |

This fully coupling THM model is divided into two submodels: a TH coupling model for the heat transfer in a single fracture and an HM coupling model for the variation of fracture aperture. Figure

Geometric model of heat transfer in a single fractured rock.

The distribution of fracture temperature and the observation line at the 100th day are plotted in Figure

Comparison between analytical solution and present simulation.

Temperature at the 100th day

Temperature at fixed points

Figure

Fracture distribution diagram between two horizontal wells.

Under the same parameters, both the simulation results in this study and those from Moradi et al. [

Comparison of fracture apertures at different locations.

The distribution of pore pressure has similar characteristics in the four fracture networks. Thus, only the pressure of the fracture network (a-1) is plotted for discussion. As shown in Figure

Pressure distribution of the reservoir at the 3rd year and 30th year.

The temperature distributions of the four fracture networks at the 30th year are presented in Figure

Temperature distributions of the four fracture networks at the 30th year.

If the fractures form a network which connects the injection well with the production well, the cooling area will advance faster along these fractures than in other zones (the dash circle area of Figure

The temperature and flow velocity at the production well are two important parameters to measure EGS performance. Based on the simulation results of the fracture network (a-1), the water temperature in the matrix and fractures along the production well is plotted in Figure

Outlet temperature and flow velocity at the different times.

Temperature

Flow velocity

The temperature at the production well is basically kept at about 180°C in the first 15 years of water injection. During this period, the EGS can maintain a high-efficiency heat production. After 30 years of cool water injection, the outlet temperature could remain above 140°C. Then, the cooling area gradually expands to the production well. The temperature drop along the production well is not uniform. It is faster in the zones with dense fractures where the faster flow of working fluid can more quickly transfer the thermal heat of the reservoir.

Under THM coupling, the aperture of fractures is in the range of 60

The TH coupling is the core for heat extraction. In the EGS operation, solid deformation and some other factors affect the TH coupling, thus modifying the heat extraction efficiency. In this section, four scenarios in the fully coupling THM model are investigated to comparatively analyze the different coupling effects on the outlet temperature and flow velocity. Table

Different coupling factors in four scenarios.

Coupling factor | Formula term | Scenario 1 (TH) | Scenario 2 THM) | Scenario 3 (TH+roughness) | Scenario 4 (THM+roughness) |
---|---|---|---|---|---|

HM | Initial stress equilibrium | √ | √ | √ | √ |

TH | √ | √ | √ | √ | |

HM for the matrix | ✗ | √ | ✗ | √ | |

HM for the fracture | ✗ | √ | ✗ | √ | |

TM | ✗ | √ | ✗ | √ | |

Fracture roughness | ✗ | ✗ | √ | √ |

The initial stress equilibrium refers to the initial state of fluid-solid static coupling before cool water injection. Hence, the first step is to calculate the initial stress equilibrium. Then, the four models with different couplings are implemented to investigate their different coupling effects on heat transfer.

Based on the simulation results, the temperature distributions of the four scenarios at 30 years are presented in Figure

Temperature distributions of the four scenarios at the 30th year.

The outlet temperature and flow velocity are plotted in Figure

Outlet temperature and flow velocity under different scenarios.

Temperature 25th year

Flow velocity 25th year

The average outlet temperature is calculated by

The summation term refers to the fractures while the integration term refers to the rock matrix. The average outlet temperature is a comprehensive indicator for the evaluation of EGS performance. The outlet temperatures at 25 years are plotted in Figure

Average outlet temperatures of the four scenarios.

This study developed a fully coupling thermal-hydro-mechanical model based on a discrete fracture network. The effects of deformation-induced aperture variation and fracture roughness on heat transfer efficiency in geothermal reservoirs were numerically investigated. Both TH and HM coupling effects in this coupling model were validated by either analytical solution or literature data. The different coupling effects on the distributions of pressure and temperature and heat recovery efficiency were comparatively analyzed through four scenarios. Based on these investigations, the following conclusions can be drawn.

The pressure distribution and temperature advance in the geothermal reservoir are severely impacted by the discrete fracture network. The lower pressure gradient and faster temperature advance usually occur in the zone with denser interconnected fractures

The deformation of the matrix induced by cooler water increases the aperture of the fracture and its permeability. This will enhance the heat transfer efficiency. However, the fracture roughness reduces the fracture flow velocity, thus reducing the heat transfer efficiency

The simple TH model can predict the similar outlet temperature to the fully coupling THM model in our calculation case. This may be due to the fact that the combining coupling effects of deformation and roughness are ignorable in that case. On this sense, a TH model is a simple and practical tool for heat extraction prediction

The conceptual geometric model is shown in Figure

Based on the cubic law of fracture flow, the flow velocity in the fracture is

Since all the parameters in Equation (

As the fracture rock geometry model is symmetrical on the midline and

Further, the heat transfer equation in rock can be rewritten as

The initial conditions are

The boundary conditions are

The governing equations of Equations (

The following dimensionless variables are defined as

Thus, the governing equations can be expressed through these dimensionless variables as

The dimensionless initial conditions are

The dimensionless boundary conditions are

The time-domain variables are transformed into the Laplace domain as

Combining the dimensionless initial conditions of Equations (

The boundary conditions in the Laplace domain can be expressed as

It is clear that Equation (

Substituting Equation (

Substituting Equation (

Solving Equations (

With Equations (

Substituting Equation (

The general solution of Equation (

Substituting Equation (

Hence, the analytical solutions in the Laplace domain are obtained as

Thus, the analytical solutions of temperature distribution are derived as Equations (

The Stehfest method is a numerical approximation for inverse transform to derive the original function in time domain. If the function

Generally,

Data is available on request.

None of the authors have any conflicts of interest.

The authors are grateful to the financial support from the National Natural Science Foundation of China (Grant No. 51674246), the Postgraduate Research & Practice Innovation Program of Jiangsu Province (Grant No. SJKY19_1859), and the Postgraduate Research & Practice Innovation Program of China University of Mining and Technology (Grant No. KYCX19_2170).

_{2}-circulated geothermal reservoir