A 3-BHE Array Coupled With Pipe Network

Project file on GitLab

Problem Description

For large-scale Ground Source Heat Pump (GSHP) systems, it is often coupled with an array of multiple Borehole Heat Exchangers in order to extract more heat from the subsurface. When operated over a long period of time, there can be thermal interference occurring among the BHEs. This can lead to thermal imbalance occurring in the subsurface. Since all BHEs are connected with each other through a pipeline network, the heat extraction rate on each individual BHE depends strongly on the inflow and outflow temperatures. As these temperature values were controlled by the pipeline network, it has an intrinsic feature of balancing thermal extraction rates among different BHEs. In order to quantitatively consider the thermal imbalance in the subsurface and the effect of the pipeline network, the calculation of hydraulic and heat transport in the pipe network must be included in the numerical simulation. This benchmark tests the pipe network feature with a 3D numerical model to show the phenomena of heat extraction rate shifting over a heating season. Since only 3 BHEs were included in this test, the percentage of shifted thermal load is limited. However, when dozens of BHEs are involved, the amount of shifted thermal load can be significant.

Model Setup


The BHE used in this Model contains a single U-shape pipe (1U type). The details about its finite element realisation could be found by Diersch et al. (2011). For the subsurface domain, a 50 \(\times\) 50 \(\times\) 72 \(m\) mesh was constructed with prism and line elements. 3 BHEs are situated from -2 \(m\) to a depth of -52 \(m\) in the subsurface, with an adjacent distance of 6 \(m\) from each other. The BHE #1 and BHE #3 are located at the left and right side, while the BHE #2 is installed in the centre. The initial soil temperature of the domain is set with 12 \(^\circ\)C. The top surface is assumed as Dirichlet boundary condition with a fixed temperature of 12 \(^\circ\)C over the entire simulation. The detailed input parameters can be found in the 3bhes_1U.prj file, they are also listed in the following table.

Parameter Symbol Value Unit
Soil thermal conductivity \(\lambda_{s}\) \(2.4\) \(\mathrm{W m^{-1} K^{-1}}\)
Soil density \(\rho_{s}\) \(1120\) \(\mathrm{Kg m^{-3}}\)
Soil specific heat capacity \((\rho c)_{s}\) \(2.0\times10^{6}\) \(\mathrm{J m^{-3}K^{-1}}\)
Length of the BHE \(L\) \(50\) \(\mathrm{m}\)
Diameter of the BHE \(D\) \(0.13\) \(\mathrm{m}\)
Diameter of the U-pipe in BHE \(d\) \(0.013665\) \(\mathrm{m}\)
Wall thickness of the pipe \(b_0\) \(0.003035\) \(\mathrm{m}\)
Wall thermal conductivity \(\lambda_{0}\) \(0.39\) \(\mathrm{W m^{-1} K^{-1}}\)
Grout thermal conductivity \(\lambda_{g}\) \(0.806\) \(\mathrm{W m^{-1} K^{-1}}\)
Grout heat capacity \((\rho c)_{g}\) \(3.8\times10^{6}\) \(\mathrm{J m^{-3}K^{-1}}\)
Circulating fluid density \(\rho_{f}\) \(992.92\) \(\mathrm{Kg m^{-3}}\)
Circulating fluid thermal conductivity \(\lambda_{f}\) \(0.62863\) \(\mathrm{W m^{-1} K^{-1}}\)
Circulating fluid heat capacity \((\rho c)_{f}\) \(4.16\times10^{6}\) \(\mathrm{J m^{-3}K^{-1}}\)
Circulating fluid flow rate \(\mathbf{u}\) \(0.0002\) \(\mathrm{m^{3} s^{-1}}\)
Length of the BHE U-pipe in network \(l\) \(100\) \(\mathrm{m}\)
Roughness coefficient of the pipe \(k_s\) \(0.00001\) \(\mathrm{m}\)


A closed-loop pipeline network system was constructed in TESPy to be coupled with the OGS model. The TESPy software developed by Francesco Witte is capable of simulating coupled thermal-hydraulic status of the network, which is composed of pre-defined components including pipes, heat exchangers and different types of turbo machinery. Interested readers may refer to the online documentation of TESPy for the details introduction of the software. Figure 1 illustrates the basic configuration of the entire network. After lifted by the pump, the circulating refrigerant inflow will be divided into 3 branches by the splitter and then flow into each BHEs. Because of the pipe network configuration, the inflow temperature on each BHE will be the same. The refrigerant flowing out of the BHEs array will firstly be mixed at the merging point and then being extracted for heat extraction through the heat pump. After that, the refrigerant will be circulated back into the pipe network. For the boundary condition, a constant thermal load of 3750 \(W\) is imposed on the heat pump for the entire simulation period, which means an average specific heat extraction rate on each BHE with 25 \(W/m\). The total simulation time is 6 months.

Figure 1: Pipeline network model in TESPy


The evolution of the soil temperature at 1 m distance away from the 3 BHEs are shown in Figure 2. Compared with the BHE #1 and BHE #3, the soil temperature near the BHE located at the centre (BHE #2) shows a deeper draw-down. It indicates that a thermal imbalance is occurring in the center of the BHE array. This imbalance leads to a lower outflow temperature from the BHE #2, which is shown in Figure 3. Figure 4 depicts the evolution of the heat extraction rate of each BHE over the time. Compared to the decrease of the heat extraction rate on the centre BHE #2, the rates on the other two BHEs located at the out sides was gradually increasing. It indicates that the heat extraction rate is shifting from the centre BHE towards the outer BHEs over the heating season.

Figure 2: Evolution of the soil temperature located at the 1 m distance away from each BHE

Figure 3: Evolution of the inflow and outflow refrigerant temperature of each BHE

Figure 4: Evolution of the heat extraction rate of each BHE


[1] Diersch, H. J., Bauer, D., Heidemann, W., Rühaak, W., & Schätzl, P. (2011). Finite element modeling of borehole heat exchanger systems: Part 1. Fundamentals. Computers & Geosciences, 37(8), 1122-1135.

[2] Francesco Witte, Thermal engineering systems in python, 2019. URL: https://doi.org/10.5281/zenodo.2555866. doi:10.5281/zenodo.2555866.

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