Assessment of the Exploitation Potential of High-Temperature Geothermal Resources in the First Deep Heat Storage of Yangbajing

Abstract

Well spacing and reinjection rate are two critical parameters controlling the efficiency and sustainability of hot dry rock geothermal development. Taking the Yangbajing geothermal field in Tibet as the geological setting, permeability experiments were conducted on fractured rock masses under multiple operating conditions, and a three-dimensional fully coupled thermo-hydro-mechanical numerical model was established to systematically evaluate the effects of different well spacing–reinjection rate combinations on heat extraction performance. The experimental results show that axial stress is the dominant factor governing specimen deformation and seepage characteristics. Permeability decreases with increasing axial stress, exhibiting an initial sharp decline followed by a gradual reduction. The effect of temperature varies with axial stress level. Under low to moderate axial stress, permeability decreases monotonically with increasing temperature, whereas under high axial stress, it first decreases and then increases. The simulation results indicate that the production temperature remains relatively stable during the early stage of exploitation and subsequently declines, with the rate of decline increasing significantly as the reinjection rate increases or the well spacing decreases. In addition, an exponential positive relationship is identified between well spacing and the optimal reinjection rate. When a 10% decline in production temperature is adopted as the shutdown criterion, the optimal reinjection rate increases from 60 m³/h to 150 m³/h as the well spacing increases from 500 m to 800 m. Based on the simulation results, the theoretical installed capacity of the first deep reservoir in the Yangbajing geothermal field is preliminarily estimated to reach 31.8 MW.

1. Introduction

With the sustained growth of the global economy and society, traditional fossil energy sources such as coal and petroleum are being depleted at an accelerating rate [1,2,3]. Against this background, the development of clean and renewable energy to replace conventional fossil fuels has become an urgent priority and a major direction for future energy development [4,5]. Owing to its abundant reserves, cleanliness, and renewability, geothermal energy has emerged as a major research focus in the energy sector in recent years and has attracted increasing attention from the academic community [6,7].

Among the available technological pathways, enhanced geothermal systems (EGSs) are regarded as the mainstream approach for geothermal resource development [8,9]. However, despite nearly five decades of development, the EGS has yet to achieve large-scale commercial deployment because of several persistent technical bottlenecks, including the high pressures required for hydraulic fracturing, limited controllability of fracture propagation, difficulties in establishing interwell connectivity, and high development costs [10,11]. To overcome these challenges, Zijun Feng et al. [12] proposed a fault-controlled hot dry rock development scheme, in which natural fault fracture zones are directly utilized as geothermal reservoirs. This approach avoids the need for hydraulic-fracturing-based reservoir stimulation and instead realizes injection–production circulation simply by drilling wells to intersect the fault zone. Consequently, both engineering complexity and development costs can be substantially reduced, while the high hydraulic conductivity of the fracture zone can further improve heat conversion efficiency and production performance.

As products of tectonic activity, fault fracture zones are commonly filled with fragmented rock masses characterized by irregular particle morphology and a loose structural fabric. In recent years, extensive studies have been conducted, both domestically and internationally, on the seepage behavior of fragmented rock masses. Kogure [13] investigated seepage through crushed rock under different hydraulic gradients, identified the critical hydraulic gradient governing flow regime transition, and established its quantitative relationship with the effective particle size of the rock. Nakagawa et al. [14] developed a finite element model that incorporates the two-way coupling between rock skeleton deformation and pore water pressure, and further established a numerical procedure capable of predicting the evolution of pore water pressure. Li [15] formulated a nonlinear dynamical equation for fluid–solid coupled seepage in fragmented rock masses, clarified the dynamic response relationships among pore water pressure, seepage velocity, and volumetric strain, and revealed the bifurcation characteristics of the coupled system. Through laboratory experiments, Shuncai Li [16] demonstrated that pore structure alteration induced by pore water pressure and particle corner breakage caused by interparticle interaction are the key factors controlling the seepage behavior of fragmented rock masses, and first reported the unusual phenomenon of a negative non-Darcy flow deviation factor. Kan [17] conducted seepage experiments on fragmented rock masses under cyclic loading and unloading conditions, and clarified the influence of cyclic loading on permeability evolution.

Although previous studies have preliminarily revealed the seepage behavior of fragmented rocks under the effects of stress and pore water pressure, seepage in fault-controlled geothermal exploitation is also strongly influenced by the dynamic evolution of the geothermal field. Moreover, most existing studies have not adequately considered the coupled interaction between pore water pressure and rock mass deformation, which limits the applicability of their conclusions. Therefore, under the multi-field coupled conditions associated with fault-controlled geothermal development, the seepage characteristics of fragmented rock masses still require more in-depth and systematic investigation.

In the exploitation of geothermal resources hosted in hot rock formations, the occurrence state of deep high-temperature rocks and engineering-induced disturbances together constitute an integrated system [18]. Owing to the combined effects of complex deep geological conditions and circulating fluids, the rock mass is continuously subjected to the coupling of the thermal, seepage, and stress fields. Therefore, systematic investigation of thermo-hydro-mechanical (THM) coupling is of great importance. Noorishad [19] was the first to establish the governing equations for THM coupling in saturated porous media based on consolidation theory and thermoelasticity, thereby laying the theoretical foundation for subsequent studies. Building on this framework, many researchers have further conducted systematic investigations into coupled processes in porous media, the thermal extraction response of fractured reservoirs, and the dynamic evolution of permeability, progressively advancing the theoretical framework of multi-field coupling in geothermal reservoirs [20,21,22,23,24]. Based on a single-porosity THM model, Cao et al. [25] demonstrated that rock deformation during geothermal exploitation can significantly alter reservoir permeability, thereby inducing thermal breakthrough and reducing heat exchange efficiency.

In summary, substantial progress has been made in the study of seepage in fragmented rocks and THM coupling in porous media. However, most existing models focus on idealized fractures or hydraulically fractured fracture networks, whereas systematic simulations of the thermo-hydro-mechanical evolution of natural fault zones during long-term heat extraction, as well as their influence on heat recovery performance, remain limited. In addition, coordinated optimization analyses of key development parameters, such as well spacing and reinjection rate, are still lacking, making it difficult to provide direct guidance for engineering design. Based on the hydrothermal system model of the Yangbajing geothermal field proposed by Academician Duoji [26], this study takes the development of geothermal resources in the first deep reservoir of the Yangbajing field as its engineering background. By combining laboratory experiments with numerical simulations and considering the influence of a stochastic discrete fracture network on reservoir development, the effects of key parameters, including well spacing and reinjection rate, on geothermal exploitation are systematically investigated, with the aim of providing guidance for the implementation of the development scheme for the first deep reservoir in the Yangbajing geothermal field.

2. Experiments and Results

2.1. Specimens and Test System

As products of tectonic activity, fault fracture zones are typically filled with fragmented rock masses characterized by irregular particle morphology and loose structural fabrics. To accurately simulate the physical–mechanical properties and seepage behavior of fault fracture zones, crushed granite was selected in this study as a simplified analogue material for natural fault fracture zones in the experiments. Raw granite blocks were first subjected to coarse crushing using a PE-60 × 100 jaw crusher (Jiangsu Pengfei Group Limited, Nantong, China) and then to controlled fine crushing with a roll crusher (Qingdao Yosion Intelligent Technology Co., Ltd., Qingdao, China) to produce angular fragments with uneven fracture surfaces. The crushed material was subsequently mechanically sieved using standard sieves to ensure adequate particle-size classification. In seepage experiments on fragmented rock masses, particle-size selection must balance the dimensions of the reactor and the measurement accuracy. Excessively large particles tend to induce a pronounced wall effect, causing the fluid to preferentially flow along the vessel wall, whereas excessively small particles result in overly dense specimen packing and excessively low permeability, which may exceed the measurement range of the sensors. Repeated experimental validation indicated that a particle-size range of 0.5–4 mm can effectively minimize boundary effects and ensure the reliability of the measured data. Accordingly, crushed granite with three characteristic particle-size ranges, namely 0.5–1 mm (fine particles), 1–2 mm (medium particles), and 2–4 mm (coarse particles), was ultimately selected as the test specimens for the seepage experiments in this study, as shown in Figure 1.

This test was conducted using a self-developed servo-controlled multifunctional high-temperature triaxial rock mechanics testing system. The system consists mainly of a triaxial pressure loading system, a fluid injection system, and other auxiliary components. It can simulate the temperature, stress, and seepage field conditions of deep rock formations, and is therefore suitable for systematically investigating the seepage characteristics of fractured rock masses under in situ conditions. The high-temperature triaxial pressure reactor was redesigned and manufactured according to the experimental requirements and was used exclusively to apply axial pressure. No independent lateral confining pressure was applied to the specimen; instead, lateral confinement was provided by the reactor wall. A schematic diagram of the testing system is shown in Figure 2.

2.2. Test Procedures

Given that the seepage characteristics of rocks are governed by complex THM coupling effects, investigating any single factor in isolation cannot adequately reflect seepage behavior under real geological conditions. Therefore, a systematic experimental approach was adopted to conduct cross-combined investigations of the stress field, temperature field, and seepage field. On this basis, the seepage characteristics of fractured rock masses were examined by controlling factors such as stress state, fluid flow, and temperature distribution under different coupled working conditions, thereby providing a reliable experimental basis for the development of a comprehensive seepage model. In addition, considering the potentially significant influence of the heterogeneity of rock particle-size distribution in deep fault fracture zones on seepage pathways, fractured rocks with three particle sizes were specifically selected as the test objects in order to improve the engineering applicability of the conclusions. This design is expected to systematically reveal the seepage behavior under the dual effects of multi-field coupling and material heterogeneity, and to provide reliable support for the establishment of a seepage model that is more consistent with actual geological conditions. The specific experimental scheme is presented in

Table 1. Experimental Plan.

Particle Size (mm)	Mass (g)	Axial Pressure (MPa)	Temperature (°C)	Injection Flow Rate (mL/min)
2–4	35	25, 50, 75, 100, 150, 200	20, 100, 150, 200, 250	10, 15, 20, 25
1–2	35	25, 50, 75, 100, 150, 200	20, 100, 150, 200, 250	10, 15, 20, 25
0.5–1	35	25, 50, 75, 100, 150, 200	20, 100, 150, 200, 250	10, 15, 20, 25

.

The specific operating procedures were as follows:

(1)

The specimen was cleaned to remove surface dust and then dried, after which 35 g of the specimen was weighed.

(2)

The absolute volume V_b of the granite fragments was measured using the water displacement method. The specimen was then loaded into the reactor and repeatedly compacted with a rubber hammer to ensure uniform packing. A small amount of fragments was slightly added or removed to make the specimen surface approximately level.

(3)

The reactor was installed in the testing machine, and the functions of the heating, sensing, and data acquisition systems, as well as the patency of the pipelines, were checked. Hoop pressure was then applied to ensure reactor sealing, after which axial pressure was loaded to the target value required by the experiment at a loading rate of 0.05 MPa/s and maintained at a stable level. Owing to the lateral deformation of the specimen under axial pressure and its thermal expansion at elevated temperatures, lateral confinement was provided by the rigid constraint of the reactor wall.

(4)

The reactor was heated to the specified temperature using a heating jacket and insulated for 2 h. After the temperature and stress became stable, constant-flow water injection was initiated, while displacement and strain data were synchronously collected and the water pressure and flow rate were recorded.

2.3. Test Results

2.3.1. Calculation of Porosity and Permeability

The porosity of intact rock is mainly controlled by primary deposition or secondary dissolution, whereas the porosity of fragmented rock masses is primarily governed by the packing and supporting arrangement of particles, which may be termed packing porosity. To determine the packing porosity of granite fragments, with the internal pores of individual particles neglected, this study adopted a physical measurement method based on the volume-difference principle. First, the absolute volume of the fragments was determined using the water displacement method. The total packed volume of the fragments in the reactor was then calculated from the displacement data obtained during the experiment. On this basis, the packing porosity, which characterizes the void space between fragments, can be calculated using Equation (1):

$$\phi ={\displaystyle \frac{V_t-V_b}{V_t}}\times 100\%$$

(1)

where V_b is the absolute volume of the fragments; V_t is the Packing volume of the fragments in the reactor.

In this experiment, the steady-state method was employed, with water used as the seepage medium. The calculation formula of Darcy’s law for one-dimensional steady-state liquid seepage is given as follows:

$$K={\displaystyle \frac{Q\mu L}{A\Delta P}}$$

(2)

where K is the permeability of the fractured granite specimen; μ is the dynamic viscosity of the seepage fluid; Q is the total flow rate through the specimen; L is the length of the specimen; A is the cross-sectional area of the specimen; and ΔP is the pressure difference across the two ends of the specimen.

2.3.2. Axial Strain and Porosity of the Specimen

Based on the axial strain and porosity variations in the three particle-size specimens under different conditions, as obtained from the displacement monitoring system, axial pressure can be identified as the dominant factor controlling specimen deformation. Figure 3 presents the evolution of axial strain and porosity with axial pressure for the three particle-size specimens under the representative temperature condition of 150 °C.

Analysis of Figure 3 shows that, under the same axial pressure, the strain of the specimen increases with increasing fragment particle size. For specimens composed of larger fragments, the total number of interparticle contact points is smaller. Under the same external load, each contact point is therefore subjected to higher stress, making particle sliding, crushing, or embedding more likely, and thus resulting in greater displacement. In addition, the initial porosity of coarse-fragment packing is generally higher, providing more space for particle rotation, slippage, and rearrangement under axial pressure, which further contributes to increased strain. For brittle granite, larger fragments are also more susceptible to local crushing or the generation of new microcracks at contact points, and the damage-induced deformation of the particles themselves further increases the overall strain of the specimen.

When no axial pressure is applied, the porosities of the three particle-size specimens calculated using Equation (1) are as follows: 0.406 for particle sizes of 0.5–1 mm, 0.437 for 1–2 mm, and 0.471 for 2–4 mm. After the application of the same axial pressure, however, the packing porosity of the 1–2 mm specimen is higher than that of the 2–4 mm specimen, despite the latter having the highest initial porosity. This is because the initial particle skeleton of the coarse-grained (2–4 mm) specimen is relatively loose. Under the same axial pressure, it undergoes a greater degree of deformation, accompanied by pronounced particle rearrangement and particle crushing. The resulting fine debris fills the voids between the larger particles, leading to a substantial reduction in porosity. In contrast, the medium-grained (1–2 mm) specimen has a relatively denser initial packing structure and better interparticle coordination, which gives it greater structural stability under the same axial pressure and a lower degree of particle crushing, thereby resulting in a relatively higher porosity.

2.3.3. The Influence of Different Factors on the Seepage Characteristics of Fragmented Specimens

According to the experimental results, the permeability variations in granite fragments under different working conditions exhibit a highly consistent trend. Therefore, the permeability data obtained under representative working conditions were selected for presentation. Figure 4 shows the variation curves of the seepage characteristics of the 1–2 mm particle-size specimen at a flow rate of 15 mL/min.

As shown in Figure 4, the permeability of the specimen decreases significantly with increasing axial pressure. This behavior is mainly attributed to changes in the packing structure of the granite fragments induced by the increase in effective stress. As the axial pressure rises, the effective stress acting on the fragment skeleton correspondingly increases, leading to further compression and rearrangement of the particles, compaction of the pore space, and a reduction in porosity. Meanwhile, the tortuosity of the pore channels increases, resulting in greater flow resistance during fluid migration, while the conductivity of the effective seepage pathways progressively weakens, as reflected by the decrease in permeability. When the axial pressure further increases to 200 MPa, the granite fragments enter a high-stress compaction stage. At this stage, the pore structure has already been substantially compressed, the remaining deformable space is extremely limited, and the seepage channels tend to approach a “saturated compression” state. Consequently, the influence of further increases in axial pressure on pore geometry and channel connectivity becomes significantly weaker, and the rate of permeability reduction correspondingly decreases and gradually stabilizes.

As shown in Figure 4, temperature also has a significant effect on the permeability of granite fragments. An increase in temperature causes thermal expansion of the rock skeleton, which leads to the contraction of the original seepage channels between fragments and a reduction in packing porosity, thereby increasing the resistance to fluid migration within the fragmented specimen. Although the temperature increase reduces the dynamic viscosity of the fluid, which is favorable for permeability enhancement, the negative effect of seepage-channel compression induced by thermal expansion is significantly greater than the positive effect associated with the reduction in dynamic viscosity. Consequently, the overall permeability still exhibits a decreasing trend.

However, when the axial pressure increases to 200 MPa, the variation in permeability becomes relatively small. To illustrate this trend more clearly, Figure 5 presents the permeability variations under different flow-rate conditions at an axial pressure of 200 MPa.

As shown in Figure 5, the permeability of granite fragments no longer decreases monotonically with increasing temperature; instead, it exhibits a trend of first decreasing and then increasing. Under this high axial-pressure condition, the constraining effect of pressure on the granite fragments is strengthened, which significantly suppresses the extent of their thermal expansion.

As the temperature increases from room temperature to 150 °C, the relatively weak thermal expansion of the rock skeleton still exerts a compaction effect on the seepage channels, whereas the dynamic viscosity of the fluid does not decrease markedly within this temperature range. As a result, the permeability-enhancing effect associated with the reduction in fluid viscosity cannot offset the reduction caused by seepage-channel compression, and the permeability therefore continues to decrease.

With a further increase in temperature, the dynamic viscosity of the fluid decreases significantly and continuously, and its positive effect on permeability gradually becomes dominant. Once this effect exceeds the compressive effect of rock thermal expansion on the seepage channels, the permeability begins to increase.

2.3.4. Permeability Model

Based on the full-condition permeability data obtained from the experiments, the coupled relationships of fractured granite permeability with axial pressure, temperature, and flow rate were quantitatively characterized, and the experimental data were fitted accordingly. The resulting permeability model is expressed as follows:

$$K=1.52\times {10}^{-11}\cdot P^{-0.78}\cdot T^{0.42}\cdot Q^{0.35}$$

(3)

where K is the permeability of the fractured granite (m²); P is the axial stress (MPa), corresponding to the in situ stress conditions of the deep geothermal reservoir; T is the reservoir temperature (°C); and Q is the seepage flow rate (mL/min), corresponding to the fluid flow conditions in the geothermal injection–production circulation system. The fitting accuracy of the model is verified by calculating the coefficient of determination R². Substituting the data yields R² = 0.92, indicating good agreement between the fitting results and the experimental data. Equation (3) provides a basis for the subsequent numerical simulation of geothermal reservoir permeability.

It should be noted that Equation (3) is an empirical relationship obtained by fitting the seepage experimental data of crushed granite, and it can reasonably describe the coupled effects of axial stress, temperature, and flow rate on permeability under the experimental conditions considered in this study. Given the complexity of the structural composition and occurrence conditions of natural fault fracture zones, this equation is currently mainly applicable to the particle-size range and experimental conditions investigated herein, and its applicability to more complex geological conditions still requires further verification.

3. Numerical Simulation and Results

3.1. Basic Assumptions

To characterize the THM coupling during geothermal exploitation, the following basic assumptions were adopted:

(1)

The dense surrounding rock around the geothermal reservoir is treated as a homogeneous and isotropic porous continuum, and its mechanical behavior is assumed to obey the small-strain theory.

(2)

Fluid flow is assumed to strictly follow Darcy’s law.

(3)

The inertial effects and body forces associated with fluid flow and rock mass deformation are neglected.

(4)

The thermophysical properties of the rock mass, including density, thermal conductivity, and specific heat capacity, are assumed to be independent of temperature.

(5)

The solid skeleton and the fluid are assumed to remain in local thermal equilibrium.

(6)

For heat transfer in the rock mass, only heat conduction and thermal effects are considered, while thermal radiation is neglected.

3.2. Governing Equations

The governing equation of the seepage field is derived based on the fluid mass balance equation and Darcy’s law:

$${\displaystyle \frac{\partial {\rho }_{\mathrm{w}}\phi }{\partial t}}-\nabla \cdot ({\displaystyle \frac{k}{u}}{\rho }_{\mathrm{w}}\nabla p)=Q_{\mathrm{s}}$$

(4)

where Q_s is the fluid source term, ρ_w is the fluid density, ρ_wφ is the fluid mass per unit volume of rock mass, φ denotes porosity which is related to the volumetric strain of the rock mass, μ is the dynamic viscosity of the fluid, k is permeability, p is the fluid pressure, and t is the time variable.

According to the fundamental laws of thermodynamics, the governing equation of the temperature field for fluid flow in the geothermal reservoir can be obtained as:

$${\displaystyle \frac{\partial ({\rho }_{\mathrm{s}}{c}_{\mathrm{s}}(1-\phi )\mathsf{\Delta }T+{\rho }_{\mathrm{w}}{c}_{\mathrm{w}}\phi \mathsf{\Delta }T)}{\partial t}}=T{\alpha }_{\mathrm{T}}(1-\phi )K{\displaystyle \frac{\partial {\epsilon }_{\mathrm{V}}}{\partial t}}+\nabla \cdot ({\rho }_{\mathrm{w}}{c}_{\mathrm{w}}({\displaystyle \frac{k}{u}}\nabla p)T)+\nabla \cdot [({\lambda }_{\mathrm{s}}(1-\phi )+{\lambda }_{\mathrm{w}}\phi )\nabla T]+Q_{\mathrm{T}}$$

(5)

where ρ_s is the density of the rock skeleton, K is the bulk modulus of the rock mass, T is temperature, c_s is the specific heat capacity at constant pressure of the rock skeleton, α_T is the thermal expansion coefficient of the rock mass, λ_s is the thermal conductivity of the rock skeleton, ε_V is the volumetric strain of the rock skeleton, and the heat source intensity of the rock mass is (1 − φ)QT_s. c_w is the specific heat capacity at constant pressure of the fluid, λ_w is the thermal conductivity of the fluid, and the heat source intensity of the fluid is φQT_w.

The left-hand side of the equation represents the rate of change in internal energy of the system. The first term on the right-hand side is the rate of change in thermal strain energy, the second term is the convective term, the third term is the heat conduction term, and the fourth term is the heat source term, where Q_T = (1 − φ)QT_s + φQT_w.

Although the temperature of the first deep geothermal reservoir in Yangbajing is relatively high, the formation as a whole is still dominated by elastic deformation under small-strain conditions, without pronounced plastic yielding or damage evolution. Therefore, an isotropic linear elastic constitutive model was adopted for the stress-field analysis in this study, with inertial effects neglected. Accordingly, the governing equation for hot rock deformation can be expressed as follows:

$$G{u}_{i,kk}+{\displaystyle \frac{G}{1-2v}}{u}_{k,ki}-{\alpha }_{\mathrm{T}}K{T}_{,i}-\alpha p_{,i}+f_{,i}=0$$

(6)

where u_i is the displacement component, f_i is the body force component, G is the shear modulus, ν is Poisson’s ratio, α is the Biot coefficient with α = 1 − K/K_s, K_s represents the bulk modulus of the rock matrix, −α_TK_,i denotes the effect of temperature variation on rock mass deformation, and −αp_,i denotes the effect of fluid seepage on rock mass deformation. The above equation can adequately characterize the coupled interactions among temperature variation, fluid seepage, and elastic deformation of the rock mass at the reservoir scale, thereby satisfying the requirements of the present study for analyzing the mechanical response of a fault-controlled geothermal reservoir during long-term exploitation.

In this study, the governing equation for porosity is given as follows [27].

$$\phi =1-(1-{\phi }_0)\exp [-{\displaystyle \frac{1}{K_{\mathrm{s}}}}(p-p_0)+{\alpha }_{\mathrm{T}}(T-T_0)-({\epsilon }_{\mathrm{V}}-{\epsilon }_{\mathrm{V}0})]$$

(7)

where p₀ is the initial fluid pressure, T₀ is the initial rock temperature, and ε_V0 is the initial volumetric strain of the rock skeleton.

Among the many permeability models proposed by previous researchers, the Kozeny–Carman equation is the most widely used [28]. The equation relating permeability and porosity derived from it is as follows:

$${\displaystyle \frac{k}{k_0}}={({\displaystyle \frac{\phi }{{\phi }_0}})}^3$$

(8)

In this study, tight coupling is achieved through the dynamic feedback of key multi-field parameters. Fluid pressure in the seepage field directly drives rock mass deformation, which in turn influences seepage efficiency through changes in porosity. The temperature field affects rock mass deformation through thermal expansion, while the flowing fluid transports heat by convection; temperature variations also influence fluid properties and seepage processes. Porosity is jointly regulated by fluid pressure and temperature, and its relationship with permeability is described on the basis of the Kozeny–Carman equation, thereby providing feedback to the seepage field. This framework fully characterizes the interactive and co-evolutionary coupling mechanism among multiple physical fields in high-temperature rock masses. A three-dimensional fully coupled THM numerical model was employed in this study to simulate the geothermal exploitation process, and the governing equations were solved using the finite element method (FEM). The governing equations were discretized in space by FEM and solved in time using a backward difference scheme. The numerical simulations were implemented on the COMSOL Multiphysics 5.8 platform, using the “Darcy’s Law”, “Heat Transfer in Porous Media”, and “Solid Mechanics” interfaces to construct the THM coupling model. This approach can reasonably capture the spatial migration of geothermal fluids within the reservoir, as well as the coupled evolution of the temperature and stress fields.

3.3. Model Validation

Bing Bai [29] analytically solved the one-dimensional thermoelastic consolidation model using finite Fourier transform and its inverse transform, and derived analytical expressions for the evolution of temperature, pore water pressure, and displacement inside the soil column, as shown in Equation (9):

$$\begin{array}{l}T\left(z,t\right)=T_{\mathrm{a}}+{\displaystyle \frac{2}{h}}{\displaystyle \sum _{\infty }^{n=0}}\left\{{\displaystyle \frac{1}{\xi }}\left({\varphi }_{22}\exp (-{\gamma }_{11}{\xi }^{2}t)+{\varphi }_{21}\exp (-{\gamma }_{22}{\xi }^{2}t)\right)\sin \left[\xi \left(h-z\right)\right]\right\}\\ p\left(z,t\right)={\displaystyle \frac{2}{h}}{\displaystyle \sum _{\infty }^{n=0}}\left\{{\displaystyle \frac{1}{\xi }}\left({\varphi }_{12}\exp (-{\gamma }_{11}{\xi }^{2}t)+{\varphi }_{11}\exp (-{\gamma }_{22}{\xi }^{2}t)\right)\sin \left[\xi \left(h-z\right)\right]\right\}\\ \upsilon \left(z,t\right)=-{\displaystyle \frac{Fz}{M}}+{\displaystyle \frac{2}{hM}}{\displaystyle \sum _{\infty }^{n=0}}\left\{{\displaystyle \frac{1}{{\xi }^2}}\left({\psi }_1\exp (-{\gamma }_{11}{\xi }^{2}t)+{\psi }_2\exp (-{\gamma }_{22}{\xi }^{2}t)\right)\cos \left[\xi \left(h-z\right)\right]\right\}\end{array}$$

(9)

where T_a is the top boundary temperature, h is the height of the geometric model, z is the position variable, F is the normal top pressure, and M is Young’s modulus. ξ is the characteristic wave number determined by ξ = (2n + 1) π/2h, (n = 0, 1, 2 …). ϕ_2i and ϕ_1i are the coefficients of the temperature field and pore pressure field, respectively, and Ψ_i is the coefficient of the displacement field. The calculation methods for the above parameters can be found in Ref. [29].

This is a typical THM coupling problem. The numerical method is used to solve this problem, and the model is shown in Figure 6.

The soil column has a height of 2 m and a width of 0.2 m, and its physical properties are listed in

Table 2. Input Parameters of the soil column THM coupling Model.

Parameter Name	Value
Fluid density/(kg·m−3)	1000
Fluid compressibility/Pa−1	1.1 × 10−10
Thermal expansion coefficient/K−1	3 × 10−7
Fluid specific heat capacity/J·(kg·K)−1	4200
Elastic modulus/MPa	60
Soil column density/(kg·m−3)	2600
Soil column specific heat capacity/J·(kg·K)−1	2840
Poisson’s ratio	0.4
Soil column thermal conductivity/W·(m·K)−1	0.5
Porosity	0.4

. A vertical load of 0.1 MPa is applied at the top of the soil column. The initial temperature and initial pore water pressure of the soil column are 283.15 K and 0.1 MPa, respectively, while the temperature at the top surface is 333.15 K. The left and right boundaries of the model are impermeable and adiabatic, with displacement fixed in the $x$-direction. The bottom boundary is also impermeable and adiabatic, with displacement fixed in the $y$-direction. The top boundary is specified as a free seepage boundary. Under the action of the top load, heat transfer, fluid flow, and deformation occur within the soil column over time.

A corresponding soil column model was established in numerical simulation software to calculate the temperature, pore pressure, and deformation of the soil column after a certain period. Comparisons between the analytical and numerical solutions at depths of 0.5, 1.0, 1.5, and 1.8 m are presented in Figure 7. The two sets of results show good agreement, thereby verifying the stability of the THM coupling model.

3.4. Geometric Model

This study takes the Yangbajing geothermal field in Xizang as the geological setting and constructs a physical model based on the exploration data from Well ZK-4001. The Yangbajing geothermal field is a typical non-volcanic high-temperature geothermal system located at a plate margin and formed by continent–continent collision. Its geothermal reservoir structure is characterized by vertical stratification: the shallow reservoir consists of Quaternary sediments and weathered bedrock crust, whereas the deep reservoir is dominated by fractured geothermal reservoirs. Duo Ji [26] further divided the deep geothermal reservoir into two zones, of which the first deep reservoir is buried at a depth of 950 m with a thickness of 400 m and a maximum temperature of 259.6 °C. Its permeability exhibits the typical characteristics of a fault-controlled geothermal reservoir, indicating favorable heat extraction potential. Therefore, a fully coupled THM numerical simulation was carried out with the first deep geothermal reservoir as the main study object, as shown in Figure 8.

Yilong Yuan [30] reported that, according to geophysical and geochemical exploration analyses, the area of the Yangbajing reservoir is estimated to be approximately 4.0 km². Accordingly, the geothermal reservoir was modeled at a depth of 950 m below the surface, with a horizontal extent of 2 km × 2 km and a thickness of 400 m. Owing to the lack of measured statistical data on the subsurface fracture system in the Yangbajing area, a stochastic discrete fracture network (DFN) was introduced to provide a more realistic representation of the geological formation. The injection and production wells were arranged to penetrate the central part of the geothermal reservoir, with an open-hole length of 300 m.

3.5. Mesh Discretization

To reduce the computational cost of the numerical model, the actual geological formation was simplified during geometric model construction. Considering that heat extraction mainly occurs within the geothermal reservoir, the thickness of the overlying surrounding rock was appropriately reduced in the model. In addition, only the open-hole section of the wellbore was retained, while the upper wellbore structure was not considered. The model was discretized using a free tetrahedral mesh, resulting in a total of 890,877 mesh elements, as shown in Figure 9.

The specific meshing strategy was as follows. Global mesh control: a relatively coarse base mesh was adopted for the surrounding rock region, with the maximum element size controlled at approximately 100 m in order to limit the overall computational cost. Local mesh refinement: targeted refinement was applied to key regions where the physical fields vary significantly. Near-well regions: the mesh around the production and reinjection wells was refined to 1–3 m to accurately capture the pressure and temperature gradients associated with fluid convergence and divergence. Near the fracture network: local refinement was applied in the regions containing discrete fractures, with the element size controlled at 5–10 m, so as to better characterize the controlling effect of fractures on fluid flow and heat transfer.

3.6. Boundary Conditions

The surrounding boundaries of the model are assumed to consist of tight, low-permeability cap rocks and are therefore specified as no-flow boundaries. The initial temperature field of the model is derived from the drilling data of Well ZK-4001. Considering that the surrounding rock provides a continuous high-temperature heat supply, the lateral and bottom boundaries of the model are specified as isothermal boundaries, with the initial reservoir temperature set to 250 °C. The lateral boundaries are defined as fixed-displacement boundaries in the x-direction, whereas the bottom boundary is defined as a fixed-displacement boundary in the y-direction. The upper boundary of the model is subjected to the pressure of the overlying strata, which is calculated by P_z = ρ_sgz [31], where g = 9.8 N/kg and z is the burial depth in meters. Both the injection well and the production well are specified as fixed-displacement boundaries.

3.7. Initial Conditions

The initial reservoir conditions and thermophysical parameters required for the model are listed in

Table 3. Numerical Simulation Parameters.

Parameter Name	Value
Rock mass density/(kg·m−3)	2620
Rock mass isobaric specific heat capacity/J·(kg·K)−1	924.6
Initial porosity of the geothermal reservoir	0.1
Thermal expansion coefficient of rock mass/K−1	1 × 10−5
Bulk modulus of rock matrix/GPa	54.5
Bulk modulus of rock mass/GPa	8.33
Thermal conductivity of rock matrix/W·(m·K)−1	3.2
Young’s modulus/GPa	10
Poisson’s ratio	0.3
Initial temperature/°C	250
Initial porosity of surrounding rock	0.02
Initial permeability of surrounding rock/m2	1 × 10−18

. Among these parameters, the density, dynamic viscosity, specific heat capacity, and thermal conductivity of the fluid are all treated as temperature-dependent in order to accurately represent the underlying physical processes, whereas the thermophysical properties of the rock mass are approximated as constants.

According to the drilling report of Well ZK-4001, the average temperature of the geothermal reservoir is approximately 250 °C. The initial hydrostatic pressure is assumed to be calculated by p = ρ_wgz, where ρ_w is the density of the geothermal fluid. Serious chemical scaling may occur when the temperature of the geothermal fluid is below 60 °C, which will significantly affect the heat production efficiency and even damage the power generation system. Therefore, the injection water temperature is set at 60 °C [32].

According to the study by Yilong Yuan [30], the optimal injection flow rate is approximately 60 m³/h at a well spacing of 500 m. Taking this value as the design reference point, five injection flow-rate cases were considered: 50, 75, 100, 125, and 150 m³/h. The model predictions of Shufang Wang [33] indicate that a well spacing of 400 m may result in a rapid temperature decline in the production well. Therefore, four well-spacing cases were selected, namely 500, 600, 700, and 800 m. The production pressure was set equal to the formation hydrostatic pressure.

3.8. Results

3.8.1. Temperature Evolution and System Lifetime

When low-temperature fluid is injected through the reinjection well, a pronounced and localized cooling zone initially develops around the well, with a relatively limited extent of thermal influence. As time progresses, the cold fluid gradually migrates preferentially toward the production well along the highly permeable fracture network, causing the cooling zone to propagate along the fractures and continuously expand, as shown in Figure 10.

Eventually, the migration pathways of the low-temperature fluid exhibit distinct preferential flow channels toward the production well along high-permeability fractures. The propagation of the cooling zone is highly consistent with the distribution of high-permeability fractures, directly reflecting the migration pattern of low-temperature fluid controlled by the fractured medium. This result also demonstrates the fluid–reservoir interaction and the dynamic evolution of the temperature field during geothermal exploitation.

To avoid excessive curve overlap caused by plotting multiple cases in a single figure, the production temperature evolutions for different reinjection rates at a fixed well spacing of 500 m and for different well spacings at a fixed reinjection rate of 150 m³/h are presented separately in Figure 11. The production process of geothermal water can generally be divided into two stages: an initial stable stage followed by a subsequent decline stage. The transient variation in production well temperature can be well explained by the spatial evolution of the reservoir temperature field. During the early stage of exploitation, the injected water can be fully heated by the rock matrix, allowing the production well temperature to remain close to the initial reservoir temperature. Once the cooling front reaches the production well, the injected water no longer has sufficient time to exchange heat fully with the rock matrix, resulting in a gradual decline in production temperature during the later stage.

Observation of the temporal evolution of reservoir production temperature under different injection flow rates reveals a significant positive correlation between injection flow rate and the rate of production temperature decline. As the injection flow rate increases from 50 m³/h to 150 m³/h, the slope of the temperature decline curve becomes progressively steeper. At year 60, the production temperature at 150 m³/h is approximately 38.08 °C lower than that at 50 m³/h. This is because a higher injection flow rate increases the migration velocity of the cold fluid within the geothermal reservoir and accelerates heat transfer from the reservoir rock to the fluid. At the same time, a higher flow rate expands the migration range of the cold fluid within the reservoir, thereby enlarging the cooling zone. Although a high injection flow rate can enhance heat extraction efficiency in the short term, it also accelerates reservoir cooling and shortens the economic lifetime of the system. Therefore, in practical development, the injection flow rate should be properly controlled to achieve better economic performance.

Analysis of the production well temperature evolution under different well spacings reveals a significant negative correlation between well spacing and the rate of production temperature decline. As the well spacing increases from 500 m to 800 m, the decline in production temperature becomes progressively slower. At year 60, the production temperature at a well spacing of 800 m is approximately 30.8 °C higher than that at 500 m. This behavior can be explained as follows: under a fixed injection flow rate, a larger well spacing results in a longer fluid migration path within the reservoir and a longer heat exchange duration between the fluid and the reservoir medium, allowing the high-temperature fluid to maintain a higher temperature upon reaching the production well. In addition, the propagation of the cooling front is constrained by well spacing. Under larger well-spacing conditions, the cooling front is less likely to propagate rapidly toward the production well region, thereby resulting in a slower decline in production temperature.

The system lifetime varies considerably under different operating conditions. To ensure the long-term viability of the production system, the decline in production temperature over the entire service period should not exceed 10%. Therefore, a 10% decline in production temperature was adopted in this study as the shutdown criterion [34]. The wellbore lifetime was set to 60 years, meaning that the maximum heat extraction period considered in the simulation was 60 years [6]. Accordingly, at a fixed well spacing of 500 m, the system lifetime is 60.0, 46.3, 33.2, 26.3, and 21.6 years for reinjection rates of 50, 75, 100, 125, and 150 m³/h, respectively. Similarly, at a fixed reinjection rate of 150 m³/h, the system lifetime is 21.6, 29.8, 41.7, and 57.6 years for well spacings of 500, 600, 700, and 800 m, respectively.

3.8.2. Heat Production, Power Generation and System Pumping Consumption

The heat production efficiency of the geothermal system is expressed by Equation (8):

$$W_{\mathrm{h}}=c_{\mathrm{w}}m(T_{\mathrm{pro}}-T_{\mathrm{inj}})$$

(10)

where c_w is the heat capacity of the fluid, m is the mass flow rate of the production well, T_pro is the production temperature, and T_inj is the injection temperature.

According to the first and second laws of thermodynamics, the efficiency f defines the thermal energy that can be converted into maximum mechanical work [35].

$$f=1-(T_{\mathrm{rej}}/T_{\mathrm{pro}})$$

(11)

T_rej is the heat rejection temperature. The annual average air temperature of the Yangbajing Geothermal Field is approximately 9 °C [36], so T_rej = 282.15 K is adopted in this study.

Generally, the utilization efficiency of converting maximum mechanical work into electrical energy is about 0.45. Therefore, the power generation W_e can be defined by Equation (10) [35].

$$W_{\mathrm{e}}=0.45\times f\times W_{\mathrm{h}}$$

(12)

The total energy consumption during system operation includes the energy consumed by the pumps in the injection well and production well, calculated as:

$$\begin{array}{l}{W}_{\mathrm{p}1}=Q_{\mathrm{inj}}(P_{\mathrm{inj}}-\rho g{h}_1)/\rho {\eta }_{\mathrm{p}}\\ W_{\mathrm{p}2}=Q_{\mathrm{pro}}(\rho g{h}_2-P_{\mathrm{pro}})/\rho {\eta }_{\mathrm{p}}\end{array}$$

(13)

where W_p1 is the energy consumption of the injection pump, W_p2 is the energy consumption of the production pump, Q_inj and P_inj are the injection flow rate and pressure, respectively, and Q_pro and P_pro are the production flow rate and pressure, respectively. ρgh₁ is the formation hydrostatic pressure near the open-hole section of the injection well, and ρgh₂ is the formation hydrostatic pressure near the open-hole section of the production well. The pump efficiency is 80%. Since the production pressure in the model equals the formation hydrostatic pressure, W_p2 = 0, and the total pump energy consumption W_p = W_p1.

3.8.3. Collaborative Optimization of Well Spacing and Injection Flow Rate

To further elucidate the coupling mechanism between well spacing and injection flow rate, this study establishes a coordinated optimization model for these two parameters by comprehensively evaluating temperature-field evolution and thermal breakthrough characteristics under different spatial configurations, thereby providing a theoretical basis for the design of engineering parameter combinations.

Excluding pumping power consumption, the system lifetime and the cumulative net power generation over the full life cycle of the heat extraction system under different well spacings and injection flow rates are presented in Figure 12.

The effect of injection flow rate on cumulative net power generation over the system life cycle exhibits a nonlinear trend of an initial increase followed by a subsequent decline. As shown in Figure 12a, when the well spacing is 500 m and the injection flow rate is 75 m³/h, the cumulative net power generation reaches its peak value of 4.70 × 10¹⁵ J. Compared with 50 m³/h, a moderate increase in injection flow rate can improve the effective thermal energy extraction rate per unit time by enhancing the circulation efficiency of geothermal fluid within the reservoir. However, as the injection flow rate continues to increase, excessive injection accelerates the decline in production temperature and sharply shortens the system lifetime, causing the cumulative net power generation to gradually decrease with further increases in injection flow rate. By contrast, at a relatively low injection flow rate of 50 m³/h, the production temperature decline does not reach 10% by the end of the designed wellbore service life of 60 years, indicating that the thermal energy of the geothermal reservoir is not fully exploited under this condition and that geothermal resources are therefore not efficiently utilized. To further determine the optimal injection parameters, an additional case with an injection flow rate of 60 m³/h was specifically designed and investigated. The results show that, under this condition, the production temperature declines to approximately 10% exactly at the end of the wellbore service life. Meanwhile, the cumulative net power generation over the system life cycle is calculated to be 4.94 × 10¹⁵ J. It can therefore be concluded that, without causing underutilization of geothermal resources, a relatively low injection flow rate yields better economic performance within the wellbore service life.

As shown in Figure 12b, the simulation results exhibit the same trend, namely a significant negative correlation between injection flow rate and system lifetime. To determine the optimal injection parameters, an additional case with an injection flow rate of 80 m³/h was introduced. Under this condition, the system lifetime reaches 59.2 years, which is generally consistent with the wellbore service life, while the cumulative power generation increases to 6.25 × 10¹⁵ J. Therefore, considering both system lifetime and cumulative net power generation, an injection flow rate of 80 m³/h was identified as the optimal operating parameter at a well spacing of 600 m. Similarly, as shown in Figure 12c,d, the optimal operating condition was determined to be 110 m³/h at a well spacing of 700 m. When the well spacing is further increased to 800 m, the optimal operating condition is 150 m³/h.

This optimization scheme significantly improves the heat extraction efficiency of the system over its full life cycle while ensuring engineering safety through precise regulation of injection parameters. The results indicate that, within the wellbore service life and under the condition of avoiding geothermal resource underutilization, reducing the fluid migration velocity within the reservoir and appropriately lowering the injection flow rate can enhance heat recovery and yield better economic performance.

3.8.4. Fitting Equation of Well Spacing and Injection Flow Rate

By systematically analyzing the power generation performance at different reinjection rates for each well spacing, the reinjection rate that maximizes cumulative power generation over the system lifetime was identified as the optimal reinjection rate. Under the shutdown criterion of a 10% decline in production temperature, the optimal reinjection rates corresponding to well spacings of 500, 600, 700, and 800 m were determined to be 60, 80, 110, and 150 m³/h, respectively. The characteristic curve describing the nonlinear increasing relationship between reinjection rate and well spacing is shown in Figure 13:

The results indicate that the reinjection rate is constrained by both lower and upper bounds. The lower bound is governed by development efficiency: an insufficient reinjection rate leads to low heat extraction efficiency and a reservoir development period that exceeds the wellbore service life, thereby resulting in resource underutilization. The upper bound is controlled by system sustainability: an excessively high reinjection rate accelerates thermal breakthrough, shortens system lifetime, increases pumping consumption, and significantly reduces economic performance. It should be noted that this equation is an empirical fitting relationship established on the basis of the simulation results of this study. It can adequately characterize the variation pattern between well spacing and the optimal reinjection rate within the current range of operating conditions, thereby providing a useful reference for parameter optimization under similar conditions.

3.8.5. Prospects for the Development of the Yangbajing Geothermal Field

Under the shutdown criterion of a 10% decline in production temperature, the spatial extent of the reservoir temperature-field disturbance corresponding to the optimal reinjection scheme for each well spacing is shown in Figure 14. Because a larger well spacing permits a higher optimal reinjection rate, thereby enhancing geothermal fluid displacement and convective heat transfer, the cold front propagates over a wider range. Consequently, the extent of temperature disturbance increases significantly with increasing well spacing.

To evaluate the development potential of the first deep geothermal reservoir in Yangbajing, a preliminary estimate of the number of heat-extraction units that can be deployed within the entire reservoir was made based on the optimized performance of a single heat-extraction unit and the extent of thermal disturbance. The layout corresponding to the maximum installed capacity is shown in Figure 15. When the well spacing is 800 m and the reinjection rate is 150 m³/h, the installed capacity of a single heat-extraction unit can reach 7.95 MW. On this basis, the theoretical installed capacity of the first deep reservoir in the Yangbajing geothermal field is preliminarily estimated to be approximately 31.8 MW, indicating considerable development potential.

4. Conclusions

Based on the geological characteristics of the first deep geothermal reservoir in Yangbajing, the main conclusions obtained from seepage experiments on fractured rock masses and three-dimensional fully coupled thermo-hydro-mechanical numerical simulations are as follows:

(1)

The seepage behavior of fractured granite is jointly controlled by axial stress, temperature, and flow rate, with axial stress being the dominant factor. Permeability decreases rapidly at first and then more gradually with increasing axial stress. The effect of temperature depends on the stress level: under low to moderate axial stress, permeability decreases with increasing temperature, whereas under high axial stress, it begins to increase once the temperature exceeds 150 °C.

(2)

The production temperature exhibits a two-stage evolution during exploitation, remaining relatively stable in the early stage and declining in the later stage. Increasing the reinjection rate accelerates thermal breakthrough and shortens system lifetime, whereas increasing the well spacing extends the heat exchange path, slows the temperature decline, and prolongs system lifetime.

(3)

When a 10% decline in production temperature is adopted as the shutdown criterion, the optimal reinjection rates corresponding to well spacings of 500, 600, 700, and 800 m are 60, 80, 110, and 150 m³/h, respectively. The regression curve is fitted as Q = 9.81e^(L/298.91) + 7.503, R² = 0.99964. This indicates that the optimal reinjection rate increases with increasing well spacing.

(4)

Based on the optimized results of a single heat-extraction unit and the extent of thermal disturbance, the theoretical installed capacity of the first deep geothermal reservoir in Yangbajing is preliminarily estimated to be approximately 31.8 MW, indicating substantial development potential.

5. Limitations and Future Work

5.1. Limitations

Owing to limitations in the experimental conditions and research scale, this study has not yet systematically investigated fragmented rock over a wider range of particle-size gradations, broader temperature–pressure conditions, and longer-term seepage scenarios. In the numerical simulations, certain simplifications were adopted with respect to the mechanical response of the rock mass, the thermophysical properties of the rock, and the fracture network structure. In addition, full-scale long-term coupled simulations of the multi-well system for the entire geothermal field have not yet been performed, and systematic validation against field monitoring data remains to be completed.

5.2. Future Work

Future research may broaden the experimental parameter range and, in combination with microstructural characterization techniques, further elucidate the seepage evolution mechanisms of fragmented rock masses. In the numerical simulations, more refined constitutive relationships and temperature-dependent rock thermophysical properties may be incorporated, while drilling, geophysical, and fracture statistical data may be used to constrain fracture network parameters, thereby enabling long-term coupled simulations and sensitivity analyses of multi-well systems. In addition, field monitoring data should be used for model calibration and validation, and detailed techno-economic evaluations should be conducted to further enhance the reliability and engineering applicability of the results.