2.2.1. Governing Equation of the Soil

The thermal model of the soil domain around the horizontal GHE is developed based on the Cartesian coordinates as it is easy to handle when applying the soil's internal source term, in which the value varies with the increase in the soil depth. The governing equation of the soil domain around the horizontal GHE is given as:

$$\frac{1}{\alpha\_s} \frac{\partial T\_s}{\partial t} = \frac{\partial^2 T\_s}{\partial \mathbf{x}^2} + \frac{\partial^2 T\_s}{\partial y^2} + \frac{H\_s}{k\_s} \tag{1}$$

The thermal model of the soil domain around the vertical GHE is developed by considering the cylindrical coordinates. The cylindrical coordinates are selected because these represent the shape of the borehole, thus the boundary conditions can be easily defined and prescribed. As a result, an accurate result can be obtained. The governing equation of the soil domain around the vertical GHE is given as

$$\frac{1}{\alpha\_s} \frac{\partial T\_s}{\partial t} = \frac{\partial^2 T\_s}{\partial r^2} + \frac{1}{r} \frac{\partial T\_s}{\partial r} + \frac{\partial^2 T\_s}{\partial z^2} + \frac{H\_s}{k\_s} \tag{2}$$

An internal heat source term was used to take into account the effect of seasonal changes in soil temperature [21,23], expressed by the following equation:

$$H\_s = \rho\_s c\_s \frac{\Delta T\_s}{\Delta t} \tag{3}$$

The direct measurement or analytical approach can be used to determine deviations in soil temperature during the process of seasonal change. The experimental results are preferred as they represent the actual soil temperature. However, it is sometimes hard to obtain the measurement

data, especially for a specific location and a certain depth. Thus, the analytical equation presented by Baggs [24] was used in this work:

$$T(\mathbf{x},t) = (T\_{\mathrm{m}} \pm \Delta T\_{\mathrm{m}}) + 1.07k\_{\mathrm{v}}A\_{\mathrm{s}}e^{\left(-0.00316\mathbf{x}\left(\frac{1}{a}\right)^{0.5}\right)} \cos\left[\frac{2\pi}{365}\left(t - t\_{0} - 0.1834\mathbf{x}\left(\frac{1}{a}\right)^{0.5}\right)\right] \tag{4}$$

2.2.2. Governing Equation of the Temperature Exchange in the Grout for the Vertical GHE

The governing equation of the grout is given by a similar equation to the one above:

$$\frac{1}{\alpha\_{\mathcal{S}}} \frac{\partial T\_{\mathcal{S}}}{\partial t} = \frac{\partial^2 T\_{\mathcal{S}}}{\partial r^2} + \frac{1}{r} \frac{\partial T\_{\mathcal{S}}}{\partial r} + \frac{\partial^2 T\_{\mathcal{S}}}{\partial z^2} \tag{5}$$
