Open-loop shallow geothermal systems, also known as groundwater heat pumps (GWHPs), have emerged as a promising solution for decarbonizing the residential heating and cooling sector . The performance of GWHPs is primarily influenced by groundwater temperature , which remains relatively stable throughout the year and is elevated in urban areas due to the subsurface urban heat island effect . These systems harness the thermal energy of the aquifer by extracting groundwater from one or more extraction wells and returning it to the same aquifer via injection wells after heat exchange in a heat pump . Since the temperature of the re-injected water is different from that of the extracted (lower in the heating and higher in the cooling case), this results in thermal plumes in the aquifer that propagate downstream along the groundwater flow direction. If these plumes reach the extraction wells of neighboring downstream GWHPs, this can result in either negative or positive thermal interference . Figure  provides an overview of potential thermal interferences that can occur between neighboring systems, depicting scenarios where the operation of downstream systems can be either degraded (negative interference) or enhanced (positive interference).

It is also important to recognize that GWHPs have a thermal impact on groundwater, which serves as a vital source of drinking water in many places (e.g. ). To mitigate the aforementioned negative interactions and improve the efficiency and sustainability of thermal groundwater use, resource management strategies need to be implemented . This includes optimizing the design, particularly well placement, and operation of GWHP systems, since the propagation of thermal plumes is affected by injection well locations, system operation (pumping rates), and aquifer characteristics. For example, optimal well placement can minimize negative thermal interference between neighboring GWHPs or even maximize their positive interference . Hence, optimization of multiple neighboring systems plays an important role in urban planning strategies aimed at enhancing sustainability. In addition, optimization of GWHP systems is crucial for managing groundwater resources and maintaining the current state of groundwater, i.e. preventing adverse changes in its physical, chemical, and biological properties. Furthermore, it is essential to ensure adequate spacing between wells within the same GWHP to prevent hydraulic and thermal breakthroughs . Thus, optimization of individual systems is also important to maximize their efficiency and sustainability. Optimization of individual GWHP systems and concurrent optimization of multiple neighboring systems are challenging due to the complexity of the resulting optimization problems and the necessity for novel and efficient optimization approaches to solve them.

This paper presents a comprehensive overview of optimization approaches for the design and operation of GWHP systems. The approaches are critically evaluated and compared based on several criteria, and a novel classification scheme is introduced to effectively categorize these approaches. Furthermore, the current status of the approaches found in the literature is presented and possible future research directions are discussed.

GWHP systems affect the groundwater body both hydraulically and thermally , which can also affect its chemical and biological conditions to a minor degree . The hydraulic head increases around injection wells and decreases around extraction wells, which also changes the hydraulic gradient and groundwater flow patterns. Thermal impacts are present due to the previously described thermal plumes. To analyze these impacts of GWHP systems on groundwater conditions, simulation models are commonly used. For a particular system design and operation, a simulation model can quantify its impacts on groundwater and, based on that, analyze the performance of the system. Therefore, simulation models play a crucial role for the computation of the resulting groundwater temperature field and serve as an integral component within the optimization procedures.

These simulation models generally fall into three categories: numerical, analytical, and data-driven. Numerical models use partial differential equations (PDEs) to describe the underlying physical phenomena, i.e. groundwater flow and heat transport in aquifers. The resulting system of PDEs can be solved with general PDE solving software or computational fluid dynamics (CFD) software, but there are also several software packages that include specialized domains of numerical simulation for shallow geothermal resources, such as: FEFLOW – based on the finite element method (FEM) or PFLOTRAN – based on the finite volume method (FVM). Numerical models can incorporate various complex subsurface conditions, including spatially heterogeneous groundwater parameters (e.g. hydraulic conductivity) and conditions (e.g. velocity, temperature, hydraulic head), complex boundary conditions, coupled physical processes, multiple subsurface layers, etc., while simultaneously simulating thermal and hydraulic effects of GWHP systems on the groundwater body. Therefore, they are closest to reality given sufficient quality of input data, but are generally computationally expensive.

The second category of models uses analytical formulas to approximate numerical solutions and is commonly applied to estimate thermal plumes associated with smaller GWHPs, whose energy consumption is less than 45 000 kWh per year . Due to their analytical nature, these models offer significant computational advantages over numerical models. In three prominent analytical models for estimating GWHP thermal plumes were analyzed and compared. These models include the radial heat transport model (RHM) , the linear advective heat transport model (LAHM) , and the planar advective heat transport model (PAHM) . The authors concluded that although analytical solutions are less accurate compared to numerical models, they still have value for evaluating the thermal impact of GWHPs. Analytical solutions are particularly useful for performing initial assessments of potential negative interference between neighboring GWHPs .

Finally, data-driven models are gaining popularity in this area of research, primarily due to the emergence of machine learning. A common example is the use of neural networks (NNs) to predict thermal plumes . Data-driven models, such as NNs, offer the advantage of fast evaluation, but rely on extensive training data and require additional time for the training process. Acquiring this training data is often challenging due to the limited measurement and monitoring of hydrogeological data. One possible solution is the use of physics-informed neural networks (PINNs) that integrate physical laws driven by PDEs, mitigating the need for extensive training data .

This section provides a comprehensive analysis of two key aspects related to the optimization of GWHP systems. First, in Sect. 3.1, the underlying optimization problems are discussed. Second, in Sect. 3.2, a detailed overview of the approaches for solving these optimization problems is provided. In the following section, we present a generalized problem related to the optimization of GWHP systems, which prepares the way for further analysis in subsequent sections.

While there are a limited number of research studies (see Sect. ) that address the optimization of GWHP systems, the research area remains insufficiently explored, which provides an opportunity to pose new research questions and develop novel optimization approaches. The existing approaches in this field have certain limitations and do not cover all relevant applications. For example, the approach of Class III presented in is limited by the number of optimization variables it can efficiently handle, since the computational cost increases exponentially as the number of variables increases. Similarly, the only study using the approach of Class I does not cover all relevant aspects of GWHP optimization, such as optimizing the number of wells in a large GWHP system, optimizing pumping rates, or simultaneously optimizing pumping rates and well locations.

Class I (PDECO) seems to be the most promising among the four classes because it uses PDE simulation models and has lower computational costs compared to Class III. Here, the complexity level of the PDE model can be selected based on data availability, as discussed in Sect. . However, this class presents significant challenges due to its mathematical complexity and multidisciplinary nature. To overcome the challenges associated with developing new approaches within Class I and facilitate their further advancement, collaboration within multidisciplinary teams will be required in the future.

The limitations of the classes that use simplified simulation models (Class II and IV) are directly related to the limitations of the simulation models employed. Consequently, improving the simplified simulation models directly enhances the approaches within these classes. The main goal is to maintain the simulation models as fast and simple to evaluate while enhancing their closeness to reality. By further improving the accuracy of these simplified models, their scope can be extended to new applications, such as detailed design of large GWHP systems comprising multiple extraction and injection wells. Moreover, the simplified models are well-suited for integration into energy system optimization models (ESOMs), where GWHP systems play an important role . This is because the coupling of a numerical groundwater simulation with an ESOM is impractical and computationally demanding . By using simplified models, the computational cost can be significantly reduced while still capturing the essential aspects of GWHP systems in the broader context of energy system optimization. This further enables the development of automated urban planning tools, which will increase sustainability.

Another important consideration in GWHP optimization is the inherent uncertainty associated with subsurface parameters and conditions. The complex nature of aquifers and the limited availability of measurement and monitoring data contribute to the presence of uncertainties . Incorporating these uncertainties into optimization approaches leads to stochastic programming problems , which constitute a separate field of optimization. The inherent stochastic nature of these problems significantly increases the complexity and computational cost compared to deterministic problems. Stochastic problems are often solved with modified deterministic optimization approaches or by using a deterministic equivalent of the stochastic problem . By minimizing the computational cost of deterministic optimization approaches, researchers can better address the challenges of stochastic problems and develop efficient approaches to such problems. Therefore, it is crucial for the deterministic optimization approaches discussed in this study to reduce their computational cost. Several strategies can be employed to reduce the computational cost of the existing approaches. For instance, the Class III approach may improve its efficiency by using model-based algorithms instead of a genetic algorithm. Similarly, the Class I approach can improve its efficiency by using suitable (re)meshing techniques or by fine-tuning optimization algorithm parameters.

It is important to note that the classification and comparison of optimization approaches presented in Sect.  is not limited to the optimization of GWHP systems, but can be applied to any optimization problem where the underlying physical phenomena are described by PDEs. Moreover, the approaches developed for GWHP systems (see Sect. ) and their future advancements or new approaches in this area can be extended to other applications. First, they can be extended to applications that share the same underlying physics, such as optimization of aquifer thermal energy storage (ATES) systems, calibration of numerical hydro-thermal groundwater simulation models, and optimization of observation well placement. Second, these approaches can be extended to other areas of shallow geothermal energy, including optimization of vertical and horizontal closed-loop shallow geothermal systems and optimization of borehole thermal energy storage (BTES) systems. Lastly, these approaches can be further extended to areas involving different physical phenomena, such as the optimization of wind farms or tidal power plants. Nevertheless, it is important to note that advances in these other areas, particularly in the area of shallow geothermal energy, can reciprocally contribute to the improvement of optimization approaches for GWHP systems. Namely, optimization approaches and principles used in other areas have the potential to be adapted and applied to GWHP systems.

This paper presents a comprehensive analysis and overview of approaches for optimizing the design and operation of GWHP systems. First, the optimization problems arising from this research and practice question were investigated, using a generalized problem as a basis. Then, optimization approaches were identified and compared, and a novel classification of the approaches is proposed. The identified approaches were divided into four distinct classes based on the type of groundwater simulation model used (PDE-based or simplified models) and the optimization algorithm applied (gradient-based or derivative-free). Finally, the paper includes a thorough review of the existing approaches in the literature, highlighting their limitations and outlining opportunities for future improvements.

Based on the analysis performed, several conclusions can be drawn:

Optimization approaches that rely on gradient-based optimization algorithms are preferable, as they consistently outperform derivative-free algorithms.