An Analytical Solution for Characterizing Mine Water Recharge of Water Source Heat Pump in Abandoned Coal Mines

Abstract

Due to tremendous mining operations, large quantities of abandoned mines with considerable underground excavated space have formed in China during the past decades. This provides huge potential for geothermal energy production from mine water in abandoned coal mines to supply clean heating and cooling for buildings using heat pump technologies. In this study, an analytical model describing the injection pressure of mine water recharge for water source heat pumps in abandoned coal mines is developed. The analytical solution in the Laplace domain for the injection pressure is derived and the influences of different parameters on the injection pressure are investigated. This study indicates that a smaller pumping rate results in a smaller injection pressure, while smaller values of the hydraulic conductivity and the thickness of equivalent aquifer induce larger injection pressures. The well distance has insignificantly influenced the injection pressure at the beginning, but a smaller well distance leads to a larger injection pressure at later times. Additionally, the sensitivity analysis, conducted to assess the behavior of injection pressure with concerning changes in each input parameter, shows that the pumping rate and the hydraulic conductivity have a large influence on injection pressure compared with other parameters.

1. Introduction

Over the past several decades, the massive consumption of fossil fuels has caused excessive greenhouse gas emissions around the whole world, leading to severe environmental problems such as global warming and the frequent occurrence of extreme weather [1]. In order to tackle climate change and its negative impacts, it is critical to take firm action to increase the proportion of clean energy while weakening the share of traditional fossil fuels [2,3,4,5]. A long-term goal is that the proportion of renewable energy will increase to 32% by the year of 2030, as set by the European Union [6]. The shares of electricity generated by renewable energy and non-water renewable energy have accounted for 19.35% and 10.68%, respectively, which far surpassed the prediction proportion for 2035 performed by the U.S. Energy Information Administration [7]. For a long time, coal has been China’s main energy source, accounting for over 50% of the primary energy consumption and production structure, but the utilization of enormous quantities of coal has induced many problems, including large amounts of greenhouse gas emissions and various environmental issues. According to the Paris Climate Agreement, national governments are required to increase the share of clean energy and reduce reliance on fossil fuels, particularly coal [8]. Under these new circumstances, it is estimated that the total energy consumption of China will reach a peak with 5.8 billion tons of standard coal and the CO₂ emissions will reach 10.58 billion tons by the year of 2030 [9].

With the proposal of a carbon emissions peak and carbon neutrality in China, the adoption of clean energy from abandoned coal mines has gained tremendous interest in recent years [8,10,11,12,13], especially when extracting geothermal energy from the mine water in abandoned coal mines, so as to provide heating or cooling for buildings by adopting heat pump technologies (as shown in Figure 1) [14,15,16,17]. During coal exploitation, huge amounts of underground excavated voids (goafs, tunnels, shafts, roadways, etc.) are formed. The closure of coal mines results in the termination of the typical drainage work of mine water. The end of the draining of mine water will gradually induce so-called groundwater rebound [18,19]. These underground excavated voids will be filled with large quantities of mine water, which will assimilate and store the heat released from the interior layer of earth and provide great opportunities for geothermal energy extraction [20,21].

In China, around one-third of mines are water-abundant and most coal mines are located below 800–1000 m, which is greatly compatible with the depth of hydrothermal geothermal energy [19]. Additionally, China closed 5300 coal mines from 2016 to 2020 [19] and it has been estimated that the number of closed coal mines will reach 15,000 by the year 2030 [10]. According to incomplete statistics, the closure of coal mines provided approximately 8 × 10⁷ m³ of underground excavated space resources between 2016 and 2020 [22]. Therefore, tremendous underground mine water storage space from abandoned coal mines in China will provide huge development and utilization possibilities for geothermal energy extraction. Most importantly, the exploitation of geothermal energy from abandoned coal mines not only can avoid problems of safety and mine water environmental pollution, but can also reduce the consumption of fossil fuels, which will lead to the achievement of carbon emission reductions [15,19].

As early as the 1980s, the innovative concept of extracting geothermal energy from mine water in abandoned mines was proposed in Springhill, Canada [23]. In the past decades, the exploitation of geothermal energy from abandoned coal mines has been widely used [14,24,25,26,27,28]. Madiseh and Abbasy analyzed the feasibility of closed geothermal heat exchangers installed in underground mine stopes and evaluated the performance of heat transport for such systems [29]. Chudy investigated the geothermal energy possibilities of mine water in the Nowa Ruda region of Poland, employing an open-loop heat pump system with a rejection system. Their findings indicate that this system has the capacity to generate 10 GWh of geothermal energy, effectively reducing carbon dioxide emissions by 4.05 tons [30]. Meanwhile, Wang et al. introduced a new case of an actual abandoned coal mine in Shandong, China to investigate the intricate flow and heat transfer dynamics within the interconnected network of roadways [31]. Also, further analysis was conducted on heat recovery potential under varying operational conditions. Furthermore, Jardon et al. performed an investigation into the potential recovery of geothermal energy from mine water’s thermal value by leveraging water-to-water heat pumps. Their findings indicated that mine water reservoirs in Central Asturias possess an annual energy supply capacity approximating 260,000 thermal MWh [32]. Furthermore, in order to gain a comprehensive overview of geothermal recovery from abandoned mines, Ramos et al. illustrated the implementation procedures and established minimum requirements for heat pump installations in abandoned mines within the Upper Harz region based on 18 projects worldwide [15]. Moreover, based on unique non-isothermal and non-iso-solutal hydrodynamics, Bao and Liu formulated a scientific framework for modeling transient heat extraction from mine water in a single mine shaft through an open-loop system [33]. The simulations explain that the impact of heat extraction on thermohaline stratification stability is negligible under normal pumping rates. Al-Habaibeh et al. investigated the feasibility of extracting energy from an abandoned coal mine by employing a single mine shaft ground source heat pump (GSHP) system [17]. The performance analysis demonstrated that this new system is an effective means of providing sustainable energy for the heating of buildings and can maintain a high and stable coefficient of performance (COP).

Additionally, a considerable amount of theoretical models have been developed to address the issues of flow and heat transfer in many studies. Based on the recurrent calculation algorithm, Rodriguez and Diaz presented a semi-empirical analytical model which can be employed to assess the temperature increment of a fluid flowing through mine galleries [34]. The numerical modellings of fluid flow and heat transfer to investigate the hydraulic and thermal behavior in an open vertical shaft associated with a flooded mine were carried out by Hamm and Sabet [35]. Raymond and Therrien developed a three-dimensional numerical model for simulating heat advection, heat conduction, and mechanical heat dispersion in the underground tunnels and mining shafts of a flooded mine [36]. Guo et al. presented a numerical model incorporating key parameters that can be used to evaluate geothermal resources within abandoned coal mines [16]. The calculation results suggested the importance of fracture permeability and fractured zone height in assessing the geothermal energy potential of similar mine reservoirs. Sun et al. investigated the effects of heat transfer on the heat exchange system of the pipes at an abandoned mine by establishing a flow-heat coupling numerical model. The results showed that the angle of the single-row borehole has a remarkable influence on the heat exchange performance of the system [37]. A 3D numerical calculation model of multi-physical filed coupling for an abandoned mine water source heat pump system based on COMSOL 6.2 Multiphysics software was developed by Zhang et al. [38]. Based on the established model, thermal breakthrough for 14 distinct pumping-recharge well layout schemes was numerically simulated. The results showed that the pumping flow rate, well spacing, hydraulics, and aquifer thickness have significant impacts on thermal breakthrough.

To the best of our knowledge, previous studies concerning geothermal energy extraction from mine water in abandoned coal mines predominantly focused on the suitability of certain locations, technical feasibility, heat transfer modeling, and design optimization. However, studies on the injection pressure of mine water recharge for water source heat pumps in abandoned coal mines remain quite limited. Therefore, the objective of this research is to propose an analytical model for describing the injection pressure of mine water recharge for geothermal energy extraction systems in an abandoned coal mine. This could have theoretical significance for the implementation of geothermal energy extraction from mine water in these abandoned coal mines.

2. Mathematical Model and Analytical Solution

2.1. Mathematical Model

Figure 2 depicts a schematic of a mathematical model for an open-loop geothermal energy extraction system in an abandoned coal mine. To simplify the analysis of the recharge of mine water for such a system, several hypotheses in this study are made as below: (1) the goaf space can be considered as the porous media (equivalent to an aquifer), by the same assumption as in Guo et al. [16]; (2) the equivalent aquifers are regarded as homogeneous and are of a uniform thickness; (3) mine water flow in the equivalent aquifer obeys Darcy’s equation; (4) the release and storage of water induced by changes in hydraulic heads are instantaneous.

2.2. Derivation of the Analytical Solution

Sen [39] proposed that the continuity equation in an annular between radius r and $r+\Delta r$ in a time increment $\Delta t$ can be represented as follows:

$$\Delta t\left[Q\left(r,t\right)-Q\left(r+\Delta r,t\right)\right]=2S_s\pi dr\Delta r\left[s\left(r,t\right)-s\left(r,t+\Delta t\right)\right]$$

(1)

in which Q is the pumping rate [L³T⁻¹], s refers to the drawdown [L], $S_s$ is the specific storage [L⁻¹], r is the radical coordinate [L], and t denotes time [T].

Equation (1) as $\Delta t\to 0$ and $\Delta r\to 0$ becomes the following:

$${\displaystyle \frac{\partial Q\left(r,t\right)}{\partial r}}=2S_s\pi dr{\displaystyle \frac{\partial s\left(r,t\right)}{\partial t}}$$

(2)

where d is the thickness of an equivalent aquifer [L].

The pumping rate for any concentric cylindrical surface with radius r can be defined as follows [39]:

$$Q=2\pi rdq$$

(3)

in which q refers to the specific discharge [LT⁻¹].

Then, substituting Equation (3) into Equation (2), the final form of the continuity equation can be given as follows:

$${\displaystyle \frac{\partial q\left(r,t\right)}{\partial r}}+{\displaystyle \frac{1}{r}}q\left(r,t\right)=S_S{\displaystyle \frac{\partial s\left(r,t\right)}{\partial t}}$$

(4)

Recalling the property of $q\left(r,t\right)={\displaystyle \frac{\partial s}{\partial r}}$ and substituting it into Equation (4), the governing equation for characterizing groundwater flow toward the production or injection well in the underground water reservoir can be expressed as follows:

$$K\left[{\displaystyle \frac{{\partial }^{2}s\left(r,t\right)}{\partial r^2}}+{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial s\left(r,t\right)}{\partial r}}\right]=S_s{\displaystyle \frac{\partial s\left(r,t\right)}{\partial t}}$$

(5)

in which K is the hydraulic conductivity [LT⁻¹].

The boundary conditions can be described as follows, respectively:

$$s\left(r,0\right)=0$$

(6)

and

$$s\left(\infty ,t\right)=0$$

(7)

For an open-loop geothermal energy extraction system in an abandoned coal mine, the well radius is relatively large in practical engineering implementation and the wellbore storage cannot be ignored. Thus, similarly as in Papadopulos and Cooper [40], the wellbore storage is considered in this study and the well boundary condition is given as follows:

$$\underset{r\to r_w}{\lim }r{\displaystyle \frac{\partial s\left(r,t\right)}{\partial r}}={\displaystyle \frac{r_w^2}{2Kd}}{\displaystyle \frac{\partial s_w\left(r,t\right)}{\partial t}}-{\displaystyle \frac{Q}{2\pi Kd}}$$

(8)

where r_w denotes the well radius [L].

Applying the Laplace transformation to Equation (5), one can obtain the following:

$${\displaystyle \frac{d^2\overline{s}\left(r,p\right)}{d{r}^2}}+{\displaystyle \frac{1}{r}}{\displaystyle \frac{d\overline{s}\left(r,p\right)}{dr}}-{\displaystyle \frac{S_{s}p}{K}}\overline{s}\left(r,p\right)=0$$

(9)

in which and $\overline{s}$ and p refer to the transient drawdown in the Laplace domain and the Laplace variable, respectively.

Then, in order to simplify Equation (9) with defining $\alpha ={\displaystyle \frac{S_{s}p}{K}}$, Equation (9) can be reduced to:

$${\displaystyle \frac{d^2\overline{s}\left(r,p\right)}{d{r}^2}}+{\displaystyle \frac{1}{r}}{\displaystyle \frac{d\overline{s}\left(r,p\right)}{dr}}-\alpha \overline{s}\left(r,p\right)=0$$

(10)

Equation (10) is a linear differential Bessel equation with second order, and the general solution of Equation (10) can be given as follows:

$$\overline{s}\left(r,p\right)=C_1{I}_0\left(\sqrt{\alpha }r\right)+C_2{K}_0\left(\sqrt{\alpha }r\right)$$

(11)

where $I_0\left(·\right)$ and $K_0\left(·\right)$ denote the modified Bessel functions of the first and second kind with the order of 0, respectively. The integration constants C₁ and C₂ can be obtained based on boundary conditions Equations (7) and (8).

Similarly, when the Laplace transformation is applied to boundary conditions Equation (7), one can obtain:

$$\overline{s}\left(\infty ,p\right)=0$$

(12)

Substituting Equation (12) to Equation (11), one obtains:

$$C_1=0$$

(13)

Equation (11) can be then rewritten as:

$$\overline{s}\left(r,p\right)=C_2{K}_0\left(\sqrt{\alpha }r\right)$$

(14)

Also, by employing the Laplace transformation to Equation (8), one obtains:

$$r_w{\displaystyle \frac{d\overline{s}\left(r_w,p\right)}{dr}}={\displaystyle \frac{r_w^2}{2Kd}}p{\overline{s}}_w\left(r_w,p\right)-{\displaystyle \frac{Q}{2\pi Kd}}{\displaystyle \frac{1}{p}}$$

(15)

Taking the derivative of Equation (14), one obtains:

$${\displaystyle \frac{d\overline{s}\left(r,p\right)}{dr}}=-C_2\sqrt{\alpha }{K}_1\left(\sqrt{\alpha }r\right)$$

(16)

The integration constant C₂ can then be determined by combining Equations (15) and (16) as follows:

$$C_2={\displaystyle \frac{Q}{\pi p\left[p{r}_w^2{K}_0\left(\sqrt{\alpha }{r}_w\right)+2d{r}_{w}K\sqrt{\alpha }{K}_1\left(\sqrt{\alpha }{r}_w\right)\right]}}$$

(17)

Substituting Equation (17) to Equation (14), one can obtain the analytical solution of transient drawdown for the production well in the Laplace domain:

$${\overline{s}}_1\left(r_1,p\right)={\displaystyle \frac{Q{K}_0\left(\sqrt{\alpha }{r}_1\right)}{\pi p\left[p{r}_w^2{K}_0\left(\sqrt{\alpha }{r}_w\right)+2d{r}_{w}K\sqrt{\alpha }{K}_1\left(\sqrt{\alpha }{r}_w\right)\right]}}$$

(18)

where r₁ denotes the radial distance from a certain location in the aquifer to axis of the production well [L] and ${\overline{s}}_1$ is the transient drawdown of the production well in the Laplace domain.

Similarly, the according analytical solution of transient drawdown for the injection well in the Laplace domain can also be given as follows:

$${\overline{s}}_2\left(r_2,p\right)={\displaystyle \frac{-Q{K}_0\left(\sqrt{\alpha }{r}_2\right)}{\pi p\left[p{r}_w^2{K}_0\left(\sqrt{\alpha }{r}_w\right)+2d{r}_{w}K\sqrt{\alpha }{K}_1\left(\sqrt{\alpha }{r}_w\right)\right]}}$$

(19)

where r₂ is the radial distance from the axis of injection well to a certain location in the equivalent aquifer. ${\overline{s}}_2$ refers to the transient drawdown of the injection well in the Laplace domain.

Then, applying the superposition principle and combining Equations (18) and (19), the transient drawdowns induced by the production and injection well in the equivalent aquifer can be given as follows:

$$s\left(r,t\right)=s_1\left(r_1,t\right)+s_2\left(r_2,t\right)$$

(20)

Applying the Laplace transform to Equation (20), one can obtain:

$$\overline{s}\left(r,p\right)={\overline{s}}_1\left(r_1,p\right)+{\overline{s}}_2\left(r_2,p\right)$$

(21)

Substituting Equations (18) and (19) to Equation (21), one obtains:

$$\overline{s}\left(r,p\right)={\displaystyle \frac{Q\left[K_0\left(\sqrt{\alpha }{r}_1\right)-K_0\left(\sqrt{\alpha }{r}_2\right)\right]}{\pi p\left[p{r}_w^2{K}_0\left(\sqrt{\alpha }{r}_w\right)+2d{r}_{w}K\sqrt{\alpha }{K}_1\left(\sqrt{\alpha }{r}_w\right)\right]}}$$

(22)

The transient drawdown, same as in Liu et al. [41], can be expressed with pressure as:

$$s\left(r,t\right)={\displaystyle \frac{f_0-f\left(r,t\right)}{\rho g}}$$

(23)

where f and f₀ are the pressure and initial pressure in the equivalent aquifer [ML⁻¹T⁻²], respectively. $\rho$ refers to the density of water [ML⁻³], and g denotes the gravitational acceleration [LT⁻²].

Similarly, applying the Laplace transformation to Equation (23), one obtains:

$$\overline{s}\left(r,p\right)={\displaystyle \frac{f_0-p\overline{f}\left(r,p\right)}{\rho gp}}$$

(24)

where $\overline{f}$ denotes to the injection pressure in the Laplace domain.

Substituting Equation (22) to Equation (24), the analytical solution of pressure in the Laplace domain can be described as follows:

$$\overline{f}\left(r,p\right)={\displaystyle \frac{f_0}{p}}-{\displaystyle \frac{\rho gQ\left[K_0\left(\sqrt{\alpha }{r}_1\right)-K_0\left(\sqrt{\alpha }{r}_2\right)\right]}{\pi p\left[p{r}_w^2{K}_0\left(\sqrt{\alpha }{r}_w\right)+2d{r}_{w}K\sqrt{\alpha }{K}_1\left(\sqrt{\alpha }{r}_w\right)\right]}}$$

(25)

In order to calculate the pressure in the injection well, one has r₁ = l and r₂ = r_w in Equation (25). Then, Equation (25) becomes:

$$\overline{f}\left(r,p\right)={\displaystyle \frac{f_0}{p}}-{\displaystyle \frac{\rho gQ\left[K_0\left(\sqrt{\alpha }l\right)-K_0\left(\sqrt{\alpha }{r}_w\right)\right]}{\pi p\left[p{r}_w^2{K}_0\left(\sqrt{\alpha }{r}_w\right)+2d{r}_{w}K\sqrt{\alpha }{K}_1\left(\sqrt{\alpha }{r}_w\right)\right]}}$$

(26)

where l represents the distance between the production and injection well [L].

As presented in Equation (26), the term of the Bessel function $K_0\left(·\right)$ makes it difficult to obtain the time domain–analytical solution of Equation (26). However, several numerical inverse methods, such as Stehfest [42,43]; Crump [44]; and De Hoog et al. [45], can be applied to address this problem. In this study, the Stehfest method, which has been revealed to tackle such inverse problems accurately, is applied to invert the Laplace domain solution numerically. In this study, we have developed a MATLAB program to calculate the injection pressure in the time domain by combining the Stehfest method and Equation (26). It is found that accurate and stable results can be obtained by choosing the number of terms of the infinite series n ≥ 18 after performing multiple numerical tests. Thus, in order to reduce the computational cost, n = 18 is used in the following calculations.

3. Results

3.1. Comparison of This Study with the Existing Analytical Solution

To verify the accuracy of the analytical solution in this study (Equation (26)), the analytical solution for injection pressure is validated by comparing it with the analytical solution proposed by Ma et al. [46]. These two analytical solutions (Equation (26) in this study and Ma et al. [46]) for water recharge of a water source heat pump are employed to calculate injection pressure with parameters as follows: Q = 40 m³/h, K = 1.25 m/h, d = 60 m, l = 27 m, r_w = 0.1 m, and S = 0.007 m⁻¹. The values of the aforementioned parameters are referenced from the work of Ma et al. [46]. As shown in Figure 3, the results of injection pressure calculated using the analytical solution in this study are consistent with the results presented in the previous study (e.g., Ma et al. [46]). This illustrates that the analytical solution of injection pressure proposed in this work is reliable and can be used to analyze the variations of injection pressure under different conditions for geothermal energy extraction systems in an abandoned coal mine.

3.2. Injection Pressure versus Time with the Pumping Rate

Figure 4 presents the transient injection pressure distributions for different pumping rates Q that range from 40 to 70 m³/h. Other parameters in this scenario are given as follows: K = 1.25 m/h, d = 10 m, l = 20 m, r_w = 0.1 m, and S = 0.007 m⁻¹. As depicted in Figure 4, the changes in injection pressure are significant when the pumping rate is reduced from 70 to 40 m³/h. Larger pumping rate values result in greater injection pressure. This can be explained by the fact that, given the equivalent aquifer in goaf and the length of pumping screen, which are constant, operations at higher pumping rates will inherently possess a correspondingly higher pumping rate for every unit length of its well screen. This indicates that it needs more power to transmit water for larger pumping rates. Thus, one can see in Figure 4 that greater injection pressures are obtained for larger pumping rates.

3.3. Injection Pressure versus Time with the Well Distance

Figure 5 illustrates the transient injection pressure distributions for different well distances between the production and injection wells from 15 to 30 m. In addition, other parameters used to generate these curves include Q = 40 m³/h, K = 1.25 m/h, d = 10 m, r_w = 0.1 m, and S = 0.007 m⁻¹. It can be found in Figure 5 that injection pressure curves merge for different well distances at initial stages, but a smaller well distance leads to a reduced injection pressure at later times. This behavior is due to the fact that the well interference between the production and injection well is not obvious at early times, while it gradually occurs with the increase in time. Also, shorter well distances will cause stronger well interference and have better hydraulic connection. This means that shorter well distances are beneficial for recharging water. Consequently, a smaller injection pressure can be found at late times for a shorter well distance.

3.4. Injection Pressure versus Time with the Specific Storage

Furthermore, it is also important to analyze the impact of specific storage on injection pressure, as indicated in Figure 6. The parameters in this scenario are as follows: Q = 40 m³/h, K = 1.25 m/h, d = 10 m; l = 20 m, r_w = 0.1 m, S = 0.001 m⁻¹, 0.002 m⁻¹, 0.005 m⁻¹, and 0.01 m⁻¹. It can be found in Figure 6 that a larger value of specific storage leads to a reduced injection pressure at the beginning, while at later times, all injection pressure curves for different specific storages approach the same value. This is also understandable because an equivalent aquifer with larger specific storage has a greater capacity to release water compared with one with a smaller storage coefficient. Meanwhile, since the process of water release from the equivalent aquifer nears completion at late times (t > 1 h), the curves of injection pressure approach the same asymptotic value for different specific storages, indicating that the effect of specific storage on injection pressure can be neglected.

3.5. Injection Pressure versus Time with the Hydraulic Conductivity

The transient injection pressure distributions for different hydraulic conductivity K are depicted in Figure 7. In this case, other relevant parameters remain unchanged as before: Q = 40 m³/h, d = 10 m, l = 20 m, r_w = 0.1 m, and S = 0.007 m⁻¹. The results shown in Figure 7 indicate that changes in injection pressure are remarkable when the hydraulic conductivity K is increased from 0.01 to 0.1 m/h. One can observe that a larger value of hydraulic conductivity leads to a reduced injection pressure. As a consequence, it can be seen that a weaker injection pressure correlated to larger hydraulic conductivity, as illustrated in Figure 7.

3.6. Injection Pressure versus Time with the Thickness of Equivalent Aquifer in Goaf

The distribution curves of transient injection pressure for different thicknesses of equivalent aquifer d from 4 to 16 m are revealed in Figure 8. The values of base parameters chosen for this scenario are as follows: Q = 40 m³/h; K = 1.25 m/h; l = 20 m; r_w = 0.1 m; and S = 0.007 m⁻¹. One can observe from Figure 8 that the injection pressure for the abandoned coal mine water source heat pump system is perceptibly influenced by the thickness of the equivalent aquifer in goaf. The thickness of the equivalent aquifer with a larger value induces a smaller injection pressure. Additionally, it is noted that the changes in injection pressure with smaller thickness of equivalent aquifer (from 4 to 8 m) are notably larger than those for the larger thickness of equivalent aquifer, which is increased from 8 to 16 m. This occurs because the fully penetrating well with a larger thickness of equivalent in goaf has a correspondingly larger well screen, indicating that the pumping rate for a unit length of the well screen is larger for a smaller thickness of equivalent aquifer while the pumping rate remains constant. Also, differences in injection pressure are smaller for larger thicknesses of equivalent aquifer (e.g., from 8 to 16 m) than for smaller thicknesses of equivalent aquifer, as demonstrated in Figure 8.

3.7. Sensitivity Analysis

Sensitivity analyses are conducted to observe the effect of various parameters, including Q, l, S, and K, on the injection pressure for the geothermal energy extraction system in an abandoned coal mine. In this study, the widespread normalized parameter sensitivity analysis proposed by Kabala [47] is implemented, and the coefficient of normalized sensitivity can take the following form:

$$Y_{m,n}=R_n{\displaystyle \frac{\partial X_m}{\partial R_n}}$$

(27)

where Y_m,n represents the normalized sensitivity due to variations in the $m$th parameter R_n at the nth time and X_m refers to a function of R_n and n. A finite-difference equation is employed to approximate the partial derivative term on the right side of Equation (27) [48]:

$${\displaystyle \frac{\partial X_m}{\partial R_n}}={\displaystyle \frac{X_m\left(R_n+\Delta R_n\right)-X_m\left(R_n\right)}{\Delta R_n}}$$

(28)

where the small increment $\Delta$R_n is normally taken as 0.01R_n, similarly as in Yeh [48], Wen et al. [49], and Tu et al. [50,51,52]. In addition, the following parameters are employed in this scenario to perform the sensitivity analysis: Q = 40 m³/h, K = 1.25 m/h, l = 20 m, d = 10 m, r_w = 0.1 m, and S = 0.007 m⁻¹.

Figure 9 presents temporal distribution curves of the normalized sensitivity for selected parameters. The sensitivity analysis reveals that two parameters, the hydraulic conductivity K and the pumping rate Q, have notable impacts on the injection pressure, but the injection pressure is most sensitive to the hydraulic conductivity, as shown in Figure 9. It is interesting to note that the hydraulic conductivity K shows a negative impact on injection pressure, and this effect approaches a steady state at late times (t > 1 h). Also, the overall behavior of the normalized sensitivity for the pumping rate Q is obviously similar to that of the hydraulic conductivity K, except that the pumping rate Q has a positive impact on injection pressure. Additionally, as illustrated in Figure 9, the influences of the well distance l and the specific storage S on the injection pressure are minimal and can be neglected.

4. Application of the Proposed Model

The application of the proposed analytical model in this study provides a theoretical base for further analyzing the injection pressure of an open-loop geothermal energy extraction system in an abandoned coal mine, considering the effects of non-Darian, well skin, and different aquifer systems. Additionally, the presented analytical solution would be of significance in further estimating the hydraulic parameters of such a system in an abandoned coal mine.

In practical engineering applications, many factors such as well configuration, hydrogeological conditions, thermal geological conditions, costs, and energy consumption need to be considered when designing an open-loop geothermal energy extraction system in an abandoned coal mine. As revealed in Figure 9, the pumping rate in real engineering applications needs to be adjusted according to the efficiency of the geothermal energy extraction system. As indicated in Figure 4, larger pumping rates result in greater injection pressure, which means that, although larger pumping rates could make the system extract more energy, the energy consumption and costs for injection will increase. Therefore, factors such as cost, energy consumption, and efficiency should be comprehensively considered while designing a proper pumping rate for a geothermal extraction system in an abandoned coal mine. Similarly, the well distance is also an important factor to optimize the efficiency of such a geothermal energy extraction system in an abandoned coal mine. It can be seen in Figure 5 that the injection pressure for a smaller well distance is relatively small compared with that for a larger well distance. This suggests that the well interference for a smaller well distance is more significant, which can be very beneficial for water injection, but it will cause thermal breakthrough more quickly and decrease the efficiency of this system. Given that hydraulic conductivity can remarkably affect the injection pressure, it is necessary to investigate hydrological conditions of an abandoned coal mine, especially related to hydraulic conductivity, by employing pumping tests, geophysical surveys, etc.

5. Conclusions

In this study, an analytical model for characterizing mine water recharge for geothermal energy extraction system in an abandoned coal mine has been proposed. The semi-analytical solution for injection pressure is developed leveraging a Laplace transform. With the presented model, the effects of different parameters on injection pressure are analyzed. The specific conclusions of this study can be drawn are as below:

(1)

The injection pressure varies significantly with pumping rate, hydraulic conductivity, and the thickness of equivalent aquifer. A larger pumping rate results in a greater injection pressure, while a larger value of the hydraulic conductivity and the thickness of equivalent aquifer led to a reduced injection pressure.

(2)

The well distance has little impact on the early-stage injection pressure, although it is significant for the late-stage injection pressure. In contrast, larger values of specific storage result in lessened injection pressure at the beginning, but the effects of specific storage on injection pressure can be neglected at a late stage.

(3)

The limitations of the proposed analytical model in this study could also be clarified. The analytical model for injection pressure in an open-loop geothermal energy extraction system in an abandoned coal mine is a relatively idealized model, which do not consider the well skin, inhomogeneity of equivalent aquifer, non-Darcian flow, and different aquifer systems. Additionally, it is noteworthy that although the proposed analytical solution in this study is verified with the existing analytical solution, it would be better in the future to use actual data fromthe field or laboratory for further investigation. In future research, it is necessary to establish a new analytical model considering the effects of non-Darcian flow in such a system. This is more in line with the field reality and will help in improving our understanding of the complex water flow regime in an open-loop geothermal energy extraction system in an abandoned coal mine.