Reply to the Comment on “Tu et al., an Analytical Solution of Groundwater Flow in a Confined Aquifer With a Single‐Well Circulation System, Water Resources Research, First Published: 12 June 2020.”

Abstract

We would like to thank Dr. Kabala for his interest in our work and his Comment on it and appreciate the opportunity to clarify and address his concerns. The idea of employing a single-well circulation groundwater heat-pump system (as shown in Figure 1) to provide heating or cooling of buildings was first presented by Sorensen and Reffstrup (1992). Our work is based on the conceptual model of Sorensen and Reffstrup (1992) and the analytical solution developed for this system by Ni et al. (2011). We have to admit that we were not aware of Kabala's (1993) work, partly because it did not appear in the literature with the geothermal exploitation background. Similarly, as Kabala (1993) and Kahler and Kabala (2018) apparently were not aware of the work of Sorensen and Reffstrup (1992), Ni et al. (2011), and related literature relevant to their works.In our study, we have rigorously derived the analytical solution for the governing equation for a single-well circulation system by employing a combination of the Laplace and Fourier cosine transforms, without using the superposition principle and adopting the analytical solution of a partially penetrating well obtained by Hantush (1961). More importantly, our approach provides an opportunity to be further extended to obtain analytical solutions for the governing equations for a single-well circulation system considering factors such as non-Darcian flow (Tu, Wu, Simunek, Zhu, et al., 2020), well skin (Tu et al., 2021), and wellbore storage (Tu et al., in preparation). This allows us to investigate further the effects of these factors on groundwater flow behavior in a single-well circulation system. It would not be possible to obtain such analytical solutions using the superposition principle (as used in Kabala, 1993) unless related analytical solutions already exist.While we admit that we did not sufficiently review the hydrogeological literature on single-well circulation systems, we disagree with Dr. Kabala's Comment that our work is a re-publication of his work (Kabala, 1993). We have clearly used in our work a different approach than Kabala (1993). A more detailed Reply to the original and revised Comment can be found in the following sections. Reply to Comments About the Derivation of the Analytical SolutionWe are certainly not the first to obtain an analytical solution for a single-well circulation groundwater heatpump system. Sorensen and Reffstrup (1992) were the first to present the concept of the single-well circulation groundwater heat-pump system (see Figure 1). This was a new engineering application for extracting

Research on Water Environment Improvement in Urban Rivers and Lakes Connected Areas——Taking “Shanghai Fish” Lake as an Example

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