*2.1. Physical Model*

The physical model is described first including a number of simplifications, which are usually needed for quantitative analysis of complex systems or phenomena. A horizontal GHE, as shown in Figure 1a, represents a multi-U-shaped pipe buried in a trench at a specific depth *h* close to the ground surface. A horizontal GHE has a total length *L*, diameter *d*, wall thickness *t*, and pipe spacing *Ls*. The heat transfer process at the horizontal GHE occurs when the fluid at a temperature *Tfi* enters the GHE and exchanges the heat energy with the pipe's inner surface by convection. The heat then conducts through the pipe wall to the surrounding soil. The ground exchanges the heat at the ground surface with the atmosphere through diverse heat transfer mechanisms including reflection, convection, radiation, and evaporation. A vertical GHE, as shown in Figure 1b, could be modelled as a single U-shaped pipe, which is buried inside a borehole having a diameter *D*<sup>b</sup> and depth *z*. The pipe is described by diameter *d*, length *L*, thickness *t*, and shank spacing *Ls*. The borehole is usually filled with a material, which has a relatively higher thermal conductivity than the surrounding soil, in order

to enhance the GHE's thermal performance. The borehole could be confined by different types of soil/rock layers that have different thermal conductivities. The heat transfer at GHE occurs when the fluid is operating at a temperature of *Tfi* and the mass flow rate m is circulated through a U-shaped pipe, exchanges its heat with the pipe's wall by convection. The heat then flows through the pipe's wall, grout, and surrounding soil by conduction sequentially. The convection occurs when the soil surface exchanges the heat with the atmosphere. As mentioned above, a vertical GHE is appropriate for installation in areas with limited land access. It may benefit from the relatively stable ground temperature at the deeper ground depth, and it leads to an improvement in performance. However, this performance decreases with the increase in the operation time, and it may take a season to recover fully. The poor soil conductivity and relatively deep depth of burring are the main reasons for the slow recovery of the soil temperature.

**Figure 1.** (**a**) Schematic of the horizontal geothermal heat exchanger (GHE). (**b**) Schematic of the vertical GHE.

To take advantage of different arrangements and negate their disadvantages, a combined, structured, horizontal–vertical GHE was proposed, see Figure 2. This figure represents a schematic diagram of the proposed combined horizontal–vertical GHE, coupled with a heat pump. The hybrid horizontal vertical GHE can be operated in five different modes through the adjustment of the valves, namely (1) the horizontal GHE only, (2) the vertical GHE only, (3) the horizontal to the vertical GHE, (4) the vertical to the horizontal GHE, and (5) both the horizontal and the vertical GHEs concurrently, by splitting the fluid flow.

**Figure 2.** Schematic of the hybrid horizontal vertical GHE.
