The optimization of open-loop shallow geothermal systems, which includes both design and operational aspects, is an important research area aimed at improving their efficiency and sustainability and the effective management of groundwater as a shallow geothermal resource. This paper investigates various approaches to address optimization problems arising from these research and implementation questions about GWHP systems. The identified optimization approaches are thoroughly analyzed based on criteria such as computational cost and applicability. Moreover, a novel classification scheme is introduced that categorizes the approaches according to the types of groundwater simulation model and the optimization algorithm used. Simulation models are divided into two types: numerical and simplified (analytical or data-driven) models, while optimization algorithms are divided into gradient-based and derivative-free algorithms. Finally, a comprehensive review of existing approaches in the literature is provided, highlighting their strengths and limitations and offering recommendations for both the use of existing approaches and the development of new, improved ones in this field.

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

Possible thermal interference (interactions) between neighboring GWHP systems.

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.

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

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

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

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

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.

The high-level optimization problem concerning GWHP systems can be formulated as follows:

In this generalized problem, we differentiate between two types of optimization variables: design variables

The objective function

The simulation model

In addition to the simulation model, other inequality

Depending on how certain elements are specified in the generalized problem (Eq. 1), the resulting optimization problems can be classified according to different criteria:

In this study, the term “optimization approach” is considered to encompass not only the specific methodology used to solve a given optimization problem, such as the choice of an algorithm, but also the way in which the problem is formulated, which includes the selection of a groundwater simulation model. The classification of optimization approaches is shown in Fig.

Proposed classification of the optimization approaches.

Finally, it should be mentioned that the four introduced classes do not encompass all conceivable approaches, since combined approaches also exist. For example, the solution obtained from Class II can serve as an initialization for the optimization process in Class I. However, within the context of this study, the division into four distinct classes seems both logical and practical, since there are substantial differences between these classes. The following section reviews previous research studies on the optimization of GWHP systems.

Despite the increasing importance of GWHP optimization as a research area, the number of existing studies on this topic remains limited.

To date, only one research study has been identified that applies the approach of Class I, i.e. the PDECO framework. This study

There is also only one research study that implements the approach belonging to Class II. In

No research studies have been identified that apply the approach of Class IV, which involves the combination of simplified models with DFO algorithms. It should be noted that other studies on GWHP optimization exist, focusing on aspects such as optimizing the components of a heat pump or determining optimal control strategies. However, these studies do not consider underground processes and are therefore outside the scope of this work. Furthermore, there are other research studies (e.g.

The primary factors for comparing optimization approaches, i.e. their respective classes, are the computational cost and applicability criteria. Figure

Qualitative comparison of the optimization approaches.

The horizontal axis represents the complexity (fidelity) of the groundwater simulation model used in these approaches. In the context of this study, the complexity of a simulation model refers to the level of detail in representing physical phenomena in the subsurface, such as the propagation of thermal plumes, that are relevant to the optimization problem under consideration. Assuming that the required input data, such as groundwater parameters, are available in sufficient quality, more complex simulation models are more accurate, i.e. closer to reality. However, it is important to recognize that data on groundwater parameters and conditions are often limited, which limits the use of complex models. Model complexity is essentially limited by the available data, making the use of highly complex models impractical in the absence of the necessary data. Nevertheless, simpler PDE simulation models, such as a 2D model with uniform groundwater conditions, are applicable even with restricted data availability and generally offer higher accuracy than analytical models with identical input data. Since simulation models are an integral component of optimization approaches, their complexity directly affects the applicability of the obtained optimization results. For instance, the results of an approach that uses a complex groundwater simulation model provided with high-quality data can be applied in practice with greater confidence than the results of an approach based on less accurate models.

In the context of computational costs, two key aspects deserve attention: the convergence rate and the computational cost associated with the evaluation of each candidate solution (a unique combination of optimization variables). The former quantifies the number of optimization iterations required to reach the optimal solution, while the latter describes the run-time required for each model simulation used to evaluate the current candidate solution within the optimization iterations. In general, gradient-based algorithms significantly outperform derivative-free algorithms in terms of convergence rate and therefore it is recommended to use gradient-based algorithms when gradient information is readily available and can be obtained at a reasonable cost

In terms of the complexity/fidelity of the simulation model used in optimization approaches, it is evident that the classes employing PDE models (I and III) outperform those employing simplified models (II and IV). The complexity of the simulation model directly influences the validity of the optimization results, thereby affecting the applicability of the corresponding classes. Consequently, it is reasonable to use approaches from different classes for different application scenarios. For instance, the classes with more complex models (I and III) are suitable for detailed planning of large GWHP systems, while the other two classes (II and IV) can be applied for initial assessments of potential negative interactions between neighboring systems or estimations of geothermal potential on a larger scale.

While there are a limited number of research studies (see Sect.

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.

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

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

It is important to note that the classification and comparison of optimization approaches presented in Sect.

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.

The choice of a simulation model used in an optimization approach has a significant impact on its applicability. For example, approaches using PDE models are more suitable for detailed design of large-scale GWHPs, while simplified models offer practical advantages for assessing the geothermal potential of large areas. However, it is important to note that the degree of model complexity is limited by the availability of hydrogeological data.

The existing research on GWHP optimization is limited, with only a few studies addressing this topic.

Existing approaches have certain limitations and do not cover all relevant applications and research questions in GWHP optimization. One of the main limitations is the high computational cost, which limits the number of optimization parameters and the size of the simulation domain that can be effectively considered. In addition, some approaches are limited in applicability due to the use of simplified groundwater simulation models. Moreover, applications such as optimizing the number and placement of wells in large GWHP systems or simultaneous optimization of pumping rates and well placements remain unexplored. Consequently, there is an ongoing need to develop new and improve existing approaches to address these limitations and fill the research gaps.

The efficient optimization approaches developed for GWHP systems have the potential to be extended to other shallow geothermal applications as well as to other optimization problems where the underlying physical phenomena are described by PDEs. At the same time, approaches from other areas can be adapted and used for GWHP optimization in the future.

No data sets were used in this article.

SH: Conceptualization, Writing – original draft, Methodology, Investigation, Visualization. FB: Writing – reviewing and editing. KZ: Writing – reviewing and editing. TH: Writing – reviewing and editing, Supervision.

The contact author has declared that none of the authors has any competing interests.

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This article is part of the special issue “European Geosciences Union General Assembly 2023, EGU Division Energy, Resources & Environment (ERE)”. It is a result of the EGU General Assembly 2023, Vienna, Austria, 23–28 April 2023.

No external funding was received for conducting this study. We would like to thank Jannis Epting for reviewing the paper and providing valuable comments that improved its quality.

This work was supported by the Technical University of Munich (TUM) in the framework of the Open Access Publishing Program.

This paper was edited by Michael Kühn and Giorgia Stasi, and reviewed by Jannis Epting and one anonymous referee.