## 5 Conclusions  
The time-dependent boundary condition is of great interest to geothermal borehole systems (including S-GBS and W-GBS), and the corresponding problem in multiple boreholes has profound applications in geothermal borehole systems. This paper proposed an analytical framework of a transient heat conduction problem subjected to a time-dependent boundary from a single borehole to multiple boreholes. The following conclusions and future recommendations were suggested.  
1. The closed-form analytical solution developed here is verified with a 2-D FEM numerical model (having a mean absolute deviation of below 0.7% in all borehole scenarios) as well as validated against a field-scale experiment in the literature (having an absolute error of below 6.3%) to ensure its accuracy and reliability.
2. The analytical solution is much more computationally efficient to be computed than numerical models, requiring 4\(\sim\)8 times less computational power and 24\(\sim\)55 times less computational time.
3. The analytical framework is capable of evaluating the ground temperature in GBS with any time-varying boundary, regardless the number of boreholes. Case studies for S-GBS and W-GBS are also conducted.
4. It was recommended that future studies could consider: the optimal design of borehole spacing and the number of boreholes used; and the economic assessment of such system from the energy demand/supply.  
Figure 10: Temperature distribution over the \(x\)-coordinate (i.e., the line \(y=0\)) for 3-by-3 boreholes over four seasons: (**a**) Spring (\(t=3\) (mon)); (**b**) Summer (\(t=6\) (mon)); (**c**) Fall (\(t=9\) (mon)); and (**d**) Winter (\(t=12\) (mon)). The S-GBS is given by the red solid line and the W-GBS is given by the green dashed line.  
**Author Contributions:** Conceptualization, M.X., S.A.G.-M. and A.P.S.; methodology, M.X., S.A.G.-M. and A.P.S.; software, M.X.; validation, M.X.; formal analysis, M.X., S.A.G.-M. and A.P.S.; investigation, M.X.; resources, F.H. and A.P.S.; data curation, M.X.; writing--original draft preparation, M.X.; writing--review and editing, M.A.H., F.H., S.A.G.-M., H.M.A., H.A.S., H.A.M.A., G.S.A.H., K.I.A. and A.P.S.; visualization, M.X. and A.P.S.; supervision, A.P.S.; project administration, M.A.H., F.H. and A.P.S.; funding acquisition, M.A.H., S.A.G.-M. and A.P.S. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This research work was funded by the Deputyship for Research & Innovation, Ministry of Education in Saudi Arabia through the project number (IFPRC-091-135-2020) and King Abdulaziz University, DSR, Jeddah, Saudi Arabia.  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** The data will be made available upon request.  
**Acknowledgments:** The authors extend their appreciation to the Deputyship for Research & Innovation, Ministry of Education in Saudi Arabia for funding this research work through the project number (IFPRC-091-135-2020) and King Abdulaziz University, DSR, Jeddah, Saudi Arabia. [PERSON] would like to extend his gratitude to McGill Engineering Doctoral Award (MEDA) and Fonds de Recherche du Quebec--Nature et Technologies (FRQ-NT) Bourses de maitrise en recherche (B1X) & Bourse de doctorat en recherche (B2X) for supporting his doctoral study. The numerical simulations for this study were conducted at the HPC facility under Calcul-Quebec and Compute-Canada with contribution from CFI-JELF.  
**Conflicts of Interest:** The authors declare no conflict of interest.