Mining is continuously developing and will continue to develop globally, and soon also in the universe (likely starting on the Moon, and later on other planets in the Solar System), beginning with borehole mining [
1]. Despite the advancement of technologies such as electricity generation through geothermal energy [
2,
3], traditional mining remains a cornerstone of the global economy. Prospects for increased demand for mineral resources continue to drive new technologies in mining and other economic sectors [
4].
Freezing boreholes (FBs) have long been used in the construction of mining shafts, primarily to freeze water in aquifers [
5]. This is necessary, for example, to penetrate layers without flooding the drilled shaft with groundwater [
6,
7,
8] or when drilling tunnels [
9]. In the case of this study, the analyzed FB was used to freeze groundwater during mineshaft construction. During a mineshaft construction, the drilling mud is pumped in a reversed order, with it going down the borehole in the annular space. Freezing groundwater prevents mud losses, thus allowing the drilling process to proceed. Another example is maintaining permafrost during drilling in Arctic and sub-Arctic areas [
10]. These boreholes often reach significant depths.
From a thermodynamic perspective, freezing boreholes operate similarly to borehole heat exchangers (BHEs) due to the fact that both types of boreholes are drilled in order to force a transfer of energy between the borehole and surrounding rock formation, with FBs being used to freeze surrounding groundwater and BHEs to source geothermal heat [
11,
12,
13]. However, they can have much greater depths than typical drilled BHEs, reaching up to 300 m below ground level [
14,
15] with constructions in the form of single, double, or triple U-tubes. Therefore, they can be classified as deep borehole heat exchangers although their definition is not yet precisely defined.
In BHEs and FBs, heat is “extracted” [
16], most often at low temperatures. In this paper, it is considered low temperature when it prevents its direct use for heating [
17]. It can only be useful as a result of using geothermal heat pumps [
18,
19]. In rock formations, the process of extracting heat from the Earth results in freezing groundwater (in FBs) or may have the same effect (in BHEs). However, it is not always a necessity [
20]. Because of the previous existence of FBs and the fact that they are obsolete upon the construction of the main shaft, they should be considered for adaptation to BHEs in order to heat the surrounding infrastructure. The main goal of this paper is to prove that they can indeed work as BHEs. Currently, there are no FBs adapted to BHEs, further proving the feasibility of this paper.
For deep BHEs, coaxial design is preferred because of the depth [
21]. A more conductive rock mass shows that the coaxial borehole heat exchanger provides better thermal performance [
22]. The main reason for coaxial construction is lower pressure losses of heat carriers than in the corresponding U-tube [
23]. U-tube construction is recommended by authors for BHES of depth up to 300 m. Many research studies show that coaxial construction for deep BHEs is favorable [
24,
25,
26]. As an example, the unit heat extraction rates were calculated at the end of the numerical simulation [
23] as 32.8 W/m (single U-tube), 36 W/m (double U-tube) and 39.1 W/m (coaxial). The necessary condition for a coaxial BHE is no leaks. When leaks exist, the U-pipe with grouting is a better solution. The internal pipe should be thermally insulated adequately to the depth of the borehole [
27,
28]. The distribution of temperature in the annulus and inside the internal pipe is shown in
Figure 1.
Figure 1.
Temperature profiles in coaxial DBHE of 3000 m depth for different values of thermal conductivity coefficients of inner column (1—0.01 Wm
−1K
−1, 2—0.12 Wm
−1K
−1, 3—1.16 Wm
−1K
−1, 4—46.1 Wm
−1K
−1) [
33].
It is impossible to count the number of boreholes drilled in the world. Most of them were constructed as wells to access water. There are oil and gas boreholes, which also have greater depths. Boreholes drilled in the Carpathian flysch over 150 years ago were most often located away from buildings. Currently, as a result of urbanization, many of them may be adapted into BHEs. Freezing boreholes can be used for heating (and/or cooling) purposes for buildings connected with underground mines.
1.2. Mathematical Model
The analysis was based on the use of the BoHEx simulator [
35] to determine the number of boreholes utilized for geothermal (geoenergetic) purposes. BoHEx is an original mathematical model used for BHE forecast preparation, based on [
20,
35] adapted for this work. Boreholes functioning as BHEs can have a negative impact on each other, meaning two boreholes share one supply, resulting in a lower supply for both boreholes [
36,
37].
The obtainable low-temperature heating power was also determined, assuming in one scenario the collaboration of boreholes with a water-to-water heat pump, and in another scenario assuming that the working fluid would be a 25% solution of monopropylene glycol (i.e., ground-source heat pump GSHP).
Modeling the heat exchange process between borehole heat exchangers and the formation is a challenging task. The main difficulty arises from large dimensions (several hundred meters) and the time scale (several to several dozen years), as well as the complexity of phenomena (turbulent flow, groundwater flow through various rocks, phase changes in the rock formation, borehole construction, etc.). Even disregarding these aspects, several factors, whose determination and even estimation are very difficult, affect the ability to accurately model the entire system, such as the velocity of deep groundwater movement or the distribution of physical parameters of the rock formation.
To simplify the model’s construction and solution, the analyzed system presented in
Figure 3 is considered as being composed of two separate subdomains (1—pipe heat exchanger, 2—rock formation), similar to [
20]. The first domain is associated with processes occurring inside the borehole (including the borehole wall). In this subdomain, there is a flow of the heat carrier through the pipes, heat penetration into the walls, heat conduction through the pipe walls, sealing, and the borehole wall. The main task in this area is to determine the fluid temperature along the entire length of the channels depending on the borehole temperature. Often, due to the spatial scale, this area is treated as a linear or cylindrical heat source of a stationary nature. In this work, this area was analyzed as a non-stationary, non-linear heat source, allowing for the consideration of phenomena with non-uniform heat extraction [
35]. This mathematical model provides a suitable solution for the problem at hand, which is why it was chosen. The aforementioned BoHEx simulator [
35] uses this model, therefore providing an automated tool.
The second subdomain is associated with the rock formation and extends from the outer surface of the borehole (this surface connects two subdomains with a common boundary condition) to the outer edges of the modelled space. Heat transport dominates here through conduction, or when groundwater flow is present, it becomes mixed, convective-diffusive. Thermal processes in this subdomain, due to the scale and mass of the system, proceed very slowly. In the rock formation area, heat exchange is described using a three-dimensional, non-stationary heat transport equation, which in Cartesian coordinates takes the form:
where:
ρ,
cp,
k denote physical properties—density, specific heat, and thermal conductivity coefficient in each layer of the rock formation.
Most often, information about the lithology of layers is available from well logging, as well as from literature [
38,
39] regarding the variability of these parameters with depth, i.e., in the vertical direction. The source term
s allows for the consideration of parameters such as natural heat sources, phase change heat, or formation water in individual layers.
Assuming no chemical reactions or phase changes in the considered heat carrier, modelling in the first domain boils down to solving the Fourier–Kirchhoff equation. In Cartesian coordinates, this equation takes the form:
To determine the heat transfer coefficient, the Nusselt number relationship for flow through a tube proposed by Seider and Tate, as well as Hausen [
40] was applied. This relationship takes the form of laminar flow (Re < 2100):
where
D,
k,
c,
G,
μ,
μw represent the inner diameter of the pipe, thermal conductivity coefficient, specific heat, mass flow rate, fluid viscosity, and fluid viscosity at the wall temperature, respectively. The thermal resistance resulting from heat conduction through the wall of the pipe and the filling was determined by the formula:
where
ro,
ri,
k denote, for each successive element, the outer radius and inner radius of the pipe, and its thermal conductivity coefficient (material of the pipe). The volumetric flow rate of the circulating heat transfer fluid in the borehole heat exchanger was analyzed in the turbulent flow region.
To solve the model Equations (1) and (2), the control volume method on a Cartesian grid was applied. For the time derivative approximation, a Two-level method was used. The time step was variable. The subdomains were solved iteratively until the required convergence was achieved. The system of linear equations resulting from the discretization was solved using the Conjugate Gradient Squared (CGS) method.
Boundary conditions used during the analysis were as follows:
(c) For
z = z
max
where
qz denotes the natural earth heat stream at a depth of “
z”.
(d) boundary conditions that are considered at the edge of the boundary:
(e) at the border between the two simulated zones, it is assumed: