Geoloop (v1.0)—An Efficient Semi-Analytical Deep Borehole Heat Exchanger Model

Abstract

The open-source Python package Geoloop introduces a novel, semi-analytical model for predicting the performance of deep (>500 m depth) vertical borehole heat exchangers (BHEs), with a focus on capturing depth-dependent variations in subsurface thermal properties, i.e., geothermal gradient and thermal conductivity. Conventional computationally efficient semi-analytical models based on load-aggregation of g-functions often assume uniform subsurface thermal properties. Geoloop addresses this gap by implementing a vertically stacked approach, allowing for realistic simulation of depth-variability in both the subsurface and borehole material properties. The model is benchmarked in the shallow domain against standard depth-uniform g-function implementations (up to 100 m depth) and for deeper conditions with a numerical finite volume model, demonstrating strong agreement and validating its accuracy and efficiency. Simulations for typical Dutch conditions show that deeper BHEs (up to 2000 m) can achieve significantly higher thermal power supply than shallower systems, and results in terms of resulting inlet/outlet temperatures for given heat extraction rates can strongly deviate (>4 °C) from results obtained by depth-uniform assumptions in thermal properties. Application of the model to the Dutch context reveals a non-linear increase in heat extraction potential with depth, surpassing values assumed in common practice by Dutch industry. The results highlight the importance of considering local geological heterogeneity and depth-dependent properties for accurate deep borehole heat exchanger (BHE) performance assessment and system optimization. Geoloop thus offers a robust, versatile platform for advancing the design and analysis of deep vertical BHE systems.

1. Introduction

Closed-loop ground-source heat pump (GSHP) systems are a popular solution for sustainable production of heat and cold [1]. The system consists of two main components: a subsurface heat exchanger that employs the relatively stable subsurface temperature to extract or inject thermal energy, coupled to a heat pump at the surface that amplifies the thermal capacity from the heat exchanger. The system principle is based on heat conduction in the subsurface, through which heat or cold is absorbed by the heat carrier fluid in the closed-loop circuit in the heat exchanger. Different types of heat exchangers exist, such as spiral coils, horizontal loops and vertical borehole heat exchanger (BHE) systems. For BHE systems, different types of tubes are available, such as a single U-shaped pipe (U-tube), multiple U-tubes in the same borehole or a coaxial tube with a smaller pipe inside a larger pipe. All types include one or multiple inlet(s) and outlet(s) through which the heat carrier fluid is circulated through the subsurface. The heat pump transfers the absorbed heat or cold from the carrier fluid to the desired quality energy, transported as fluid in the distribution system in the building.

Numerous tools are available for BHE performance calculations to aid industry in the design process for newly installed systems (e.g., [2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26]). The models that employ a (semi-)analytical approach for prediction of BHE performance often use g-functions. Eskilson [3] first introduced the concept of g-functions: a dimensionless function that describes how the borehole wall temperature evolves over time in response to constant heat injection or extraction by a BHE or a field of BHEs. g-Functions are based on solutions of the heat conduction equation and therefore do not consider advective, convective or radiative heat flow. They neglect transient effects in the borehole itself, which play a role in hourly heat load fluctuations, but still are effective for predicting the intraday to long-term seasonal temperature response, while considering the effect of axial conduction in the ground and the thermal interaction between BHEs [15]. To calculate the heat flow between the fluid and borehole wall, g-functions are often coupled to a circuit of thermal resistances that represents the borehole interior (e.g., [7,8,16,27]).

Eskilson [3] initially derived the g-functions for different BHE field configurations from numerical finite difference calculations of the Finite Line Source (FLS) model, with the boundary condition of a uniform borehole wall temperature along the borehole length. To overcome the long calculation times required by the numerical approach, Cimmino & Bernier [10] proposed a much faster, semi-analytical solution to the FLS model to calculate the g-functions, respecting the same boundary condition. In addition to Eskilson’s work, this method also accounts for time-variation in the heat load and variation in the borehole length. First, they only included the single U-tube design, but this solution was later extended for a BHE configuration with multiple U-tubes [12] or coaxial tubes [16].

Due to the high computational efficiency, g-functions have proven a de facto standard for fast predictions of BHE performance. However, they assume homogeneous BHE design and subsurface thermal properties and temperature conditions. In reality, the subsurface temperature is marked by the geothermal gradient and increasing with depth and the bulk thermal conductivity varies with lithology and increases with depth due to a decrease in porosity [9,28].

Depth dependency in temperature and thermal conduction also affects the relationship between the system performance and BHE design from the perspective of realistic operational conditions. For example, for deeper systems, adjustments to the BHE design (number of pipes, pipe dimensions, (grouting, insulation) materials, etc.) may be desired to work more effectively and efficiently. In the currently available methods, focus is on configurations with U-tubes or coaxial tubes, leaving a methodological gap for optimization of alternative system designs.

The effect of depth-dependent thermal properties can to some extent be approximated with g-functions by adopting averaged depth-uniform properties for temperatures and conductivity. However, this tends to simplify the depth dependency in subsurface behavior, which can impact predicted BHE system performance. Several authors have recognized this limitation before and proposed alternative models. For example, Abdelaziz et al. [9] developed a multilayer FLS model that uses spatial superposition to combine layers with variable subsurface properties and incorporate vertical heat conduction. Guo et al. [25] proposed a semi-analytical model with an iterative solution for borehole wall temperature and heat transfer rate in each segment, with a focus on surface boundary conditions and the effect of ambient temperature. Dong et al. [26] applied an iterative scheme in an analytical solution to the Infinite Line Source (ILS) model for a coaxial BHE. In the context of BHE design optimization and subsurface parameter estimation from a thermal response test (TRT), Urchueguía et al. [24] adapted the composite two-region line source approach that captures the thermal effect within and outside of the drilling radius at short TRT timescales. In application of their approach, based on the ILS model, for TRT analysis of the required test time and uncertainty in obtained thermal properties of grout, subsurface and BHE are reduced. The semi-analytical approach in these studies, although powerful, is focused on characterization of very particular BHE design or heat transfer phenomenological aspects. On the other hand, numerical finite element or finite volume-based methods and tools allow full flexibility in BHE design, subsurface properties and operational boundary conditions. For example, Schulte et al. [13] published BASIMO, a tool that applies a dual-continuum approach with an analytical, one-dimensional thermal resistance and capacity model for the BHE interior coupled to a numerical finite element model to calculate the subsurface heat transport. Schulte et al. [14] also implemented an extension to the analytical solution for the borehole interior, to account for a partly insulated BHE design with depth-dependent grout properties. With a focus on detailed modeling of subsurface conditions, Seib et al. [23] created a finite element model to estimate the effect of a hydraulically conductive fault zone on several middle-deep BHEs in an energy storage system in Germany. Focusing on BHE design optimization and innovation, Javadi et al. [22] used a numerical model to better understand the heat transfer and phase transition in their experimentally tested innovative grouting materials. The main limitations to these numerical methods are their high computational demands and steep learning curve that often limit their practical applicability.

In this paper, we present a novel model that predicts BHE performance and incorporates g-functions in a semi-analytical solution that focuses on capturing the depth variability in heat flow as induced by depth-dependent thermal properties of the subsurface and the BHE materials (Figure 1). The novel model is implemented in Geoloop, an open-source Python package that includes multiple methods for BHE performance calculations. In addition, Geoloop supports sensitivity analysis through stochastic simulations and provides the ability to optimize BHE design and operational constraints. With the implementation of our model in Geoloop, we fill the gap between computationally expensive numerical models and (semi-)analytical alternatives that neglect subsurface heterogeneity and offer a versatile tool that balances computational efficiency and flexibility in system and simulation configuration.

First, we present the depth-dependent semi-analytical model and its numerical implementation in Geoloop. Secondly, we demonstrate the benchmark of the model against the depth-uniform FLS (g-function) solution and a depth-dependent numerical model. Here, we also highlight how neglecting the geothermal gradient affects performance estimates in depth-uniform models. We show that our depth-dependent approach properly identifies heat losses in the shallow subsurface. Lastly, we show a representative case, where we apply our novel model in Geoloop to assess the (local) potential for deep (>500 m depth) BHEs in The Netherlands. In The Netherlands, currently more than 80,000 systems are installed, with about 10,000 installations added each year.

2. Materials and Methods

2.1. Methodology

We use a semi-analytical axisymmetric model for the depth-dependent BHE performance. In the radial direction, the model is separated into two regions: one region includes the heat transfer process between the pipes inside the borehole and the borehole wall, based on steady-state assumptions for heat transfer; the other region includes the transient heat transfer process between the borehole wall and the surrounding subsurface, based on time aggregation of the BHE heat load involving so-called g-functions for the BHE [10,12,16,27]. The model is based on conductive heat flow, without advective heat transfer through fluid flow (except in the heat carrier fluid inside the pipes). For variable definitions and units, see the Nomenclature.

2.1.1. Vertically Stacked Approach

To implement variable (subsurface) thermal properties over depth, the semi-analytical approach is stacked vertically in different segments. Our approach adopts time aggregation of g-functions as a function of soil thermal conductivity which is performed independently for each segment, whereas the g-function of the BHE itself is treated uniformly for all segments and is based on a multi-segmented calculation of the Finite Line Source (FLS) formulation [10], adopting uniform thermal transmissivity in the model. The segment borehole wall temperature $T_b$ as a function of loading history and actual segment load $q_c$ at time t is given by:

$$T_b=R_s\left(t\right)q_c+T_0 ,$$

(1)

where $T_0$ is the (depth dependent) ambient temperature, and the thermal resistance $R_s$(t) is calculated by the load aggregation scheme of (temporal varying) heat loads $Q\left(t\right)$ [29] involving an appropriate convolution of the g-function corresponding to the BHE, based on a multisegmented FLS model [10]. We choose to apply a default g-function for all segments, based on a constant thermal diffusivity of 10⁻⁶ m²/s. The along depth segmentation in the g-function consists of 8 segments, as defined in Cimmino [30], but these segments do not need to align with the conductivity segments in our model, in contrast to the vertically layered approach from Abdelaziz et al. [9]. However, the load aggregation is segment-dependent through a scaling both to potentially variable heat load and soil thermal conductivity in the segment:

$$R_s\left(t\right)={\displaystyle \frac{\mathrm{a}\mathrm{g}\mathrm{g}\left(\frac{Q\left(t\right)}{2\pi k_s},g\left(BHE\right)\right)}{Q\left(t\right)}},$$

(2)

where agg denotes the load aggregation algorithm from Claesson & Javed [29], $k_s$ is the subsurface bulk thermal conductivity and g is the g-function. Effectively, this results in a realistic approximation of thermal transient effects related to the finite dimensions of the borehole, but it evidently compromises the exact solution for vertical heat transfer effects related to variability in thermal conductivity and horizontal effects related to variation in thermal diffusivity. Our justification for the simplified approach is its ease of implementation and major advantages in computational performance, keeping in mind that sacrificing the variability in vertical conductive heat transfer and horizontal transmissivity is minor compared to the thermal transfer effects on the timescales of interest, i.e., 1 to 10 s of years, and the focus towards stand-alone BHE designs.

2.1.2. Segment Thermal Resistivity

The heat transfer in the BHE interior is calculated by a quasi-steady-state solution of a thermal resistance network based on the multipole method [8] (Figure 2). Our segmented approach supports depth variation in pipe and/or grout materials, and underlying properties. For the resistance network, thermal nodes are added to the network for every tube in the BHE, such that any (close to) radially symmetric design, including (multiple) inlets and outlets, can be simulated. It is assumed that the system mass rate $\dot{m}$ is distributed equally over the number of inlet and outlet pipes, based on the assumption that the (set of) inlet and outlet pipe(s) have constant pipe radii in each set.

The quasi-steady-state thermal solution for the heat transfer process in the borehole interior, which describes the fluid temperatures in the pipes for each segment, is given by:

$${\mathbf{T}}_{\mathit{f}}-T_b\mathbf{1}=\Delta {\mathbf{T}}_{\mathit{f}}={\mathbf{R}}_{\mathit{b}}\mathit{q},$$

(3)

where ${\mathbf{T}}_{\mathit{f}}$ is the vector of pipe fluid temperatures, $T_b$ is the borehole wall temperature, 1 is the vector (1, 1, …, 1) of length corresponding to the total number of upward and downward pipes ($N_p$), $\mathit{q}$ corresponds to the heat injection/extraction rates for the pipes (positive is injection) per meter of borehole/pipe length and ${\mathbf{R}}_{\mathit{b}}$ is a square matrix with the resistance values for the pairing pipes.

When rewriting Equation (3):

$$q_c=a{-bT}_b,$$

(4)

with: $a=\sum {{\mathbf{R}}_{\mathit{b}}}^{-\mathbf{1}}{\mathbf{T}}_{\mathit{f}}$

$b=\sum {{\mathbf{R}}_{\mathit{b}}}^{-\mathbf{1}}\mathbf{1}$

$q_c=\sum q_i$

and when substituting Equation (4) in Equation (1):

$$T_b-T_0=R_s\left(t\right)a {- R_s\left(t\right)bT}_b,$$

(5)

we get:

$${ T}_b={\displaystyle \frac{R_s\left(t\right)a+T_0}{1+R_s\left(t\right)b}}$$

(6)

Please note that for coaxial design, the borehole heat flow is solely determined by the temperature difference between the fluid of the outer pipe ($T_{f_o}$) and the borehole wall, and the associated thermal resistivity (${Rd}_{ob}$). Equation (4) therefore becomes:

$$q_c= {{Rd}_{ob}}^{-1} (T_{f_o}-T_b)$$

(7)

and: $a={{Rd}_{ob}}^{-1} T_{f_o}$

$b=$ ${{Rd}_{ob}}^{-1}$.

2.1.3. Stacked Solution for Fluid Temperatures

The stacked segment solution of fluid temperatures $T_{f_{i,k}}$ is formulated using i-index for the pipe and k-index corresponding to the start of the depth segment ordered $\left[k=0,\dots ,k=nz-1\right]$, such that k = 0 corresponds to the inlet and outlet of the BHE. $k=nz$corresponds to the end of the deepest segment. Furthermore, the i-index sets $inlet$ and $outlet$ corresponding to the inlet (down-flowing) and outlet (up-flowing) pipes. The iterative solution is performed starting from an assumption for $T_{f_{i\in inlet,0 }}=T_{in}$ and $T_{f_{i\in outlet,0 }}=T_{out}$ imposed at the top of the BHE, and working downwards towards the bottom of the borehole $T_{f_{i,nz }}$, taking into account the specified pipe flow rates denoted by ${\dot{m}}_i$, with a positive sign in the inlet and negative sign in the outlet pipes. Consequently, the resulting average fluid temperatures of the inlet and outlet at the bottom of the BHE should match, denoted by $T_{\mathrm{m}\mathrm{i}\mathrm{n}}$ and $T_{\mathrm{p}\mathrm{l}\mathrm{u}\mathrm{s}}$ for down-flowing and up-flowing tubes respectively.

Algorithm 1 includes the iterative solution of $T_{in }$and $T_{out}$, which is consistent with the depth-dependent load aggregation and actual load at the considered time $Q_{target}$and $\dot{m}$ at time t. In the algorithm, $c_f$ is the subsurface heat capacity and $qtol$, $ttol$, and tiny are set to 10⁻³, 10⁻⁴, 10⁻¹⁰ respectively. The model accepts two simulation types: it can calculate the system performance directly from an imposed constant or time-variable total heat load ($Q_{target}$), or it can calculate the system performance while maintaining a fixed inlet temperature ($T_{in}$) through time, which effectively bypasses the outer convergence loop.

Algorithm 1: Iterative solution for depth-dependent fluid temperatures.
1:	$T_{in}=5$
2:	$Q=0$
3:	while $\left|Q-Q_{target}\right|>qtol$ do
4:	$T_{out}=T_{in}+0.1\ast sign \left(Q_{target}\right)$
5:	$T_{\mathrm{m}\mathrm{i}\mathrm{n}}=0$
6:	$T_{\mathrm{p}\mathrm{l}\mathrm{u}\mathrm{s}}=1$
7:	while $\left|T_{\mathrm{p}\mathrm{l}\mathrm{u}\mathrm{s}}-T_{\mathrm{m}\mathrm{i}\mathrm{n}}\right|>ttol$ do
8:	for $k=0$to $k=nz-1$ do
9:	$T_{f_{i,k+1}}$= $T_{f_{i,k}}$
10:	repeat
11:	${\mathbf{T}}_{\mathit{f}}=0.5\left(T_{f_{i,k+1}}+T_{f_{i,k}}\right)$
12:	Determine $T_b\left(k\right)$ and $q_i\left(k\right)$ from Equations (4) and (6) based on ${\mathbf{T}}_{\mathit{f}}$
13:	$T_{f_{i,k+1}}=T_{f_{i,k}}-{\displaystyle \frac{q_i\left(k\right)}{{\dot{ m}}_i{c}_{f_i\left(k\right)}}}$
14:	${\mathbf{T}}_{\mathit{f}\mathbf{2}}=0.5\left(T_{f_{i,k+1}}+T_{f_{i,k}}\right)$
15:	until $\left|{\mathbf{T}}_{\mathit{f}}-{\mathbf{T}}_{\mathbf{f}\mathbf{2}}\right|<tiny$
16:	end for
17:	${\mathrm{T}}_{\mathrm{p}\mathrm{l}\mathrm{u}\mathrm{s}}=\sum _{inlet}{T}_{f_{i,nz}}$
18:	${\mathrm{T}}_{\mathrm{m}\mathrm{i}\mathrm{n}}=\sum _{outlet}{T}_{f_{i,nz}}$
19:	Adjust $T_{out}$by a newton iteration step such that $\left(T_{\mathrm{p}\mathrm{l}\mathrm{u}\mathrm{s}}-T_{\mathrm{m}\mathrm{i}\mathrm{n}}\right)\to 0$
20:	end while
21:	Adjust $T_{in}$ by newton iteration towards $\left(Q-Q_{target}\right)\to 0$
22:	end while

The solution in Algorithm 1 is iterated three times to adjust the temporal distribution of the heat extraction ($sign>0$) or injection ($sign<0$) at each segment along the length of the BHE, and uses these to update the time and depth-varying loads for the load aggregation (and underlying g-function evaluation) for the next iteration. The load aggregation calculates $R_s\left(t\right)$for every depth segment $k$ and every timestep $t$, and serves to calculate the temperature evolution in the borehole wall (Equation (6)). Subsequently, the fluid temperatures are calculated according to the iterative scheme. Based on this calculation, the heat load for each depth segment is adjusted according to the depth-dependent (subsurface) thermal properties and depth-dependent $T_0$. The sum of the heat load for the different depth segments converges with the total imposed heat load. For the calculation of $R_s\left(t\right)$ in Equation (6), in the first iteration the load history is set without depth variation, whereas in the next iteration it is set in accordance with the resulting heat load history of each segment.

In the case of a fixed inlet temperature, the same steps are executed. However, in the first iteration an estimate of the heat load is used by the g-function, based on the length of the borehole. Also, the sum of the heat flow for the different depth segments decreases over time due to a decrease in the temperature difference between the fluid and the borehole wall.

2.1.4. Soil Thermal Conductivities

The most important subsurface thermal properties required in the depth-dependent approach for simulations of BHE performance are subsurface temperature and (bulk) thermal conductivity. Using the method proposed by Limberger et al. [31], bulk thermal conductivity can be calculated from lithological data in a data-driven approach. This method uses thermal and mechanical rock properties from Hantschel & Kauerauf [32] to calculate porosity and the thermal conductivity of the rock matrix. A decrease in porosity with depth is included, resulting from rock/soil compaction, and for a mix of lithologies the matrix thermal conductivities are combined in the geometric mean, according to the relative fractions of occurrence. Bulk thermal conductivity is calculated by the geometric mean of matrix and pore-fluid thermal conductivity, under the assumption of hydraulically saturated conditions.

2.2. Geoloop Implementation and Features

We implemented the proposed depth-dependent semi-analytical modeling approach in an open-source Python 3 package: Geoloop. Geoloop provides an extensive modular functionality to define BHE designs, run (stochastic) simulations, and analyze depth dependent effects of (uncertain) subsurface and design parameters, and provides options for design optimization in the face of parasitic losses with respect to fluid circulation. Geoloop is set up in such a way that it does not require any Python coding, running from input defined in JSON configuration files. Geoloop relies for many of its core calculation routines, including evaluation of g-functions, borehole thermal resistivity network matrix, and fluid properties, on functionality available in the pygfunction Python package [21]. This section describes the main innovative features in Geoloop that support the user in optimization of the BHE design. An extensive user manual is included in the Geoloop package documentation.

2.2.1. Modular Framework

Geoloop is implemented in an object-oriented modular framework that combines different model components, configuration management and a (command-line) user interface to support management and reproducibility of simulations. The configuration of the core simulation consists of different sub-configurations that define the input for (optional) sub-modules in the simulation workflow, such as for processing the heat load time-profile or stochastic sampling (Figure 3). Sub-configuration files are linked in the main simulation configuration for integration of the sub-modules in the modeling workflow. For the heat load, flow rate and lithology sub-modules, they can also be used individually in the Geoloop interface for visualization and data (pre-)processing. With this modular framework, sub-components of the overall simulation configuration are easily modified, exchanged and stored. Model objects associated with the different sub-modules are instantiated from (sub-)sets of configuration parameters, which are validated at object initialization using Pydantic [33], providing explicit feedback on missing or inconsistent user input.

Geoloop has a command-line interface and provides dedicated commands for running individual modules, including a single, deterministic simulation, stochastic analyses (Section 2.2.3), and post-processing and plotting routines (Section 2.2.4). In addition to single commands, Geoloop supports batch simulations in which sequences of different modules and/or model configurations are executed automatically. The batch configuration specifies which modules are executed and in which order, while storing the associated configuration files enables users to define, reuse, and adapt complete simulation workflows. Through this modular and configuration-driven design, Geoloop offers a flexible framework that integrates multiple tools and models for BHE performance simulations.

2.2.2. Support for Semi-Analytical and Numerical Models

The framework of the Geoloop package allows for easy switching between BHE simulation models. Different methods are included, including the depth-dependent semi-analytical approach from Section 2.1, the depth-uniform implementation of pygfunction (Section 3.1.1), and a depth-dependent finite volume numerical model (Section 3.1.2). When switching from a depth-dependent model to the depth-uniform pygfunction implementation, the depth-dependent subsurface properties and BHE design aspects are automatically averaged to conform to the depth-uniform method. Furthermore, when using the depth-uniform pygfunction implementation, depth-profiles of fluid temperatures are derived in pygfunction based on an analytical solution of a system of first order linear differential equations [12]. For the evaluation of the BHE g-function from pygfunction [21], for both the depth-dependent and depth-uniform model we adopt default settings involving 8 segments in the FLS formulation for the BHE, as defined in Cimmino & Bernier [10]. The support for the different BHE simulation models allows users to compare different methods and to select the most appropriate model configuration considering case-specific requirements in, for example, computational efficiency or detail in results.

2.2.3. Sensitivity Analysis, Constraints and Optimization

In addition to deterministic calculations, stochastic simulations provide the opportunity to determine the interrelationship between different system design and operational aspects. These simulations sample a set of BHE configurations, with systematic variation in certain input parameters within a provided value range. As such, the effect of uncertainty in (subsurface) properties can also be considered. Results can be used to gain in-depth understanding of heat flow characteristics in the system and quickly provide insight into the potential effect of adjustments in the BHE design and operational strategy on system performance.

Similarly, the systematic sampling method for parameter values within provided value bounds can be deployed for optimization of the BHE system design, to obtain the maximum power yield with respect to a minimum system efficiency. The system efficiency is described by the Coefficient of Performance (COP) of the fluid circulation pump:

$$COP={\displaystyle \frac{Q_b}{qloop}},$$

(8)

where $Q_b$ represents the extracted/injected subsurface heat and $qloop$ represents the power consumed by the fluid circulation pump. The latter is calculated based on frictional losses during fluid circulation, including the thermo-syphon effect.

In part of our presented analyses, we apply single-parameter optimization by varying the flow rate within the COP constraint, with maximum heat extraction as a target.

2.2.4. Visualization Features

Geoloop features a broad functionality for visualization of simulation results in different types of figures. Depending on the deployed simulation method and the parameter dimensions, simulation input and output parameters are plotted in depth plots, time plots, scatter plots, tornado plots and borehole cross-sections. In addition, Geoloop supports plotting of multiple simulations together for convenient comparison of simulated cases.

3. Results

3.1. Model Validation

We validated the depth-dependent semi-analytical model, and its implementation in Geoloop, in two benchmark cases, comparing inlet, outlet and borehole wall temperature. In the first case we compare temperature results for the depth-dependent implementation (cf. Section 2.1), with temperature results from the standard implementation in pygfunction [21], based on depth-uniform time aggregation of the g-functions, from here on referred to as ‘the standard g-function model or ‘standard g-functions’. With a focus on validating the stacked implementation of the depth-dependent approach and demonstrating its fundamental difference with the standard g-function model under geologically heterogenous conditions, the performance of vertical single U-tube BHE systems was calculated with 100 and 800 m borehole length.

In the second case we made a comparison of temperature output of the depth-dependent semi-analytical model with a depth-dependent numerical model that deploys an axi-symmetrical finite volume method. This numerical model is similar to the Borehole-to-Ground (B2G) model, originally proposed and benchmarked by De Rosa et al. [34] and Ruiz-Calvo et al. [11], for calculating short-term performance of a U-tube. Later it was extended by Cazorla-Marín [18] and Cazorla-Marín et al. [17] for a coaxial tube. We altered the model set-up for practical compatibility with the semi-analytical model, i.e., the thermal nodes in the network are positioned in the middle of the depth segments, and we include over 20 nodes in radial direction starting from the borehole wall, up to 20 m radius and marked by cell sizes exponentially increasing with radial distance from the borehole. In the last cell in the radial direction, a zero heat flow boundary condition applies. For calculating the heat transfer process within the BHE interior, thermal nodes have been added for the pipes, adopting the thermal resistance network from the semi-analytical approach (Figure 2). The heat flow for the pipes at each depth-segment is given by:

$$\mathit{q}= {{\mathbf{R}}_{\mathit{b}}}^{-1}{∆\mathbf{T}}_{\mathit{f}}, {{∆\mathbf{T}}_{\mathit{f}}=\mathbf{T}}_{\mathit{f}}-T_b\mathbf{1},$$

(9)

and the heat flow at the borehole wall is the sum of $\mathit{q}$ for each pipe. With a focus on validating the depth-dependent implementation of subsurface thermal properties, we calculated the performance of a vertical triple U-tube down to 500 m depth, considering a synthetic subsurface bulk thermal conductivity profile and a linear geothermal gradient. In addition, we investigated the effect of axial conduction in a simulation of the same BHE system for a layered subsurface thermal conductivity model.

3.1.1. Benchmark Against Standard (Depth-Uniform) g-Functions

The stacked implementation of g-functions in the depth-dependent semi-analytical model in Geoloop was benchmarked against the standard (depth-uniform) g-function approach in three sets of simulations of a vertical single U-tube (Figure 4). For these simulations, and all other simulations that are discussed in this chapter, we used a thermal conductivity of the backfill material ($k_g$) of 0.844 W/mK, a subsurface thermal diffusivity ($\alpha$) of 10⁻⁶ m²/s, a pipe roughness ($ϵ$) of 10⁻⁶ m and water as a heat carrier fluid. Additional simulation parameters for the benchmark simulations of the depth-dependent semi-analytical model in Geoloop against the standard (depth-uniform) g-function approach are defined in

Table 1. Simulation parameters for the benchmark of the depth-dependent semi-analytical model in Geoloop with the standard (implementation in pygfunction based on depth-uniform time aggregation of the) g-function model.

Parameter	Description	Unit	Simulation Set a	Simulation Set b	Simulation Set c
H	Borehole length	[m]	100	800	800
D	Buried depth	[m]	0	0	0
nInlets	Number of inlet pipes	-	1	1	1
$r_b$	Borehole radius	[m]	0.07	0.07	0.07
pos	Pipe position inside borehole	[x, y]	see Figure 4	see Figure 4	see Figure 4
$r_{out}$	Outer pipe radius	[m]	0.02	0.02	0.02
$r_{in}$	Inner pipe radius	[m]	0.0164	0.0164	0.0164
$k_p$	Pipe thermal conductivity	[W/mK]	0.41	0.41	0.41
$T_g$	Surface temperature	[°C]	11	11	11
$k_s$	Subsurface bulk thermal conductivity	[W/mK]	2.4	2.4	2.4
$Q$	Subsurface heat load	[W]	1000	3000	3000
$m_{flow}$	Mass flow rate	[kg/s]	0.3	0.5	1.5
nyear	Simulated period	[years]	10	10	10
nled	Simulated timestep	[hours]	500	500	500
nsegments	Number of depth segments (Geoloop)	-	10	10	10

.

As the standard (depth-uniform) g-function approach does not allow input of depth-dependent (subsurface) properties, the benchmark simulations include depth-uniform values for the subsurface temperature and thermal conductivity. The fluid and borehole wall temperatures were calculated for a constant heat load.

Figure 5 shows the results from benchmark simulation set a (see Table 1). The output curves from the two models are in perfect agreement, which illustrates the validation of the depth-dependent semi-analytical modeling approach against standard (depth-uniform) g-functions.

Comparison of Fluid Temperatures

Figure 6 shows the calculated fluid and borehole wall temperatures from the three sets of benchmark simulations in Table 1 after one year of operation. Figure 6b shows how the depth-dependent model approach in Geoloop impacts the heat flow along the borehole length, in comparison to the standard (depth-uniform) g-function model. The increase in inlet fluid temperature with depth is smaller in the depth-dependent model compared to the standard (depth-uniform) g-function approach, with a maximum discrepancy of <0.1 °C around 300 m depth. The maximum temperature difference is similar in the outlet pipe, but the depth-dependent model calculates a higher fluid temperature with respect to the standard g-function model. In addition, the borehole wall temperature at the top and bottom of the system is respectively 0.15 °C lower and 0.08 °C higher in the depth-dependent model. This is explained by the different heat transfer mechanisms in the two models: the heat transfer in the standard g-function model is uniform over depth, based on the difference between the depth-average fluid and borehole wall temperatures, whereas the heat transfer in our semi-analytical depth-dependent model scales with the difference between the mean fluid temperature and borehole wall temperature, which is largest in the shallower part of the subsurface compared to deeper parts. This effect is relatively strong in deeper systems, as is illustrated in the comparison between Figure 6a,b; in Figure 6a the fluid temperatures in the two models perfectly align and the discrepancy in the borehole wall temperature is negligible. However, Figure 6c shows that at a higher flow rate, the curves of fluid temperatures calculated by the two methods can also be in perfect agreement with a system length of 800 m.

The Effect of a Geothermal Gradient

The fundamental difference in handling of the heat flow along the borehole length between the depth-dependent semi-analytical model in Geoloop and the standard implementation of g-functions (discussed in the previous section) is demonstrated by incorporating a realistic geothermal gradient for a deep borehole. The effect of the geothermal gradient on the performance of a vertical single U-tube down to 800 m is explored in the comparison of a depth-dependent simulation in Geoloop with a simulation in the standard g-function model.

Table 2. Simulation parameters for the comparison of the depth-dependent semi-analytical model in Geoloop, including a geothermal gradient, with the standard (implementation in pygfunction based on depth-uniform time aggregation of the) g-function model.

Parameter	Description	Unit	Depth-Dependent Semi-Analytical Simulation	Standard g-Function Simulation
H	Borehole length	[m]	800	800
D	Buried depth	[m]	0	0
nInlets	Number of inlet pipes	-	1	1
$r_b$	Borehole radius	[m]	0.07	0.07
pos	Pipe position inside borehole	[x, y]	see Figure 4	see Figure 4
$r_{out}$	Outer pipe radius	[m]	0.02	0.02
$r_{in}$	Inner pipe radius	[m]	0.0164	0.0164
$k_p$	Pipe thermal conductivity	[W/mK]	0.41	0.41
$T_g$	Surface temperature	[°C]	10	18
$T_{grad}$	Geothermal gradient	[°C/m]	0.02	0
$k_s$	Subsurface bulk thermal conductivity	[W/mK]	2.4	2.4
$Q$	Subsurface heat load	[W]	3000	3000
$m_{flow}$	Mass flow rate	[kg/s]	0.5	0.5
nyear	Simulated period	[years]	10	10
nled	Simulated timestep	[hours]	500	500
nsegments	Number of depth segments (Geoloop)	-	10	1

defines the simulation configurations. The depth-dependent semi-analytical model uses a geothermal gradient of 0.02 °C/km with a surface temperature of 10 °C. In the standard g-function model, the depth-uniform subsurface temperature is fixed at 18 °C in agreement with the depth-averaged temperature.

Figure 7 shows the calculated fluid and borehole wall temperatures from both simulations in Table 2 after one year of operation. The results depict the effect of the geothermal gradient on the thermal interaction between the heat carrier fluid and the formation along the length of the borehole. As the subsurface temperature increases with depth in the depth-dependent semi-analytical Geoloop simulation, the temperature difference between the injected fluid and the formation also increases. This promotes subsurface heat extraction along the deeper parts of the system. In fact, in the upper 150 m of the simulation heat is injected into the subsurface as the temperature of the injected fluid is higher than in the surrounding formation. The same principle applies in the outlet pipe, where part of the gained heat is lost during upward transport in the upper 450 m. In contrast, with the depth-uniform subsurface temperature in the standard g-function model, the temperature difference between the injected fluid and the formation decreases with depth, which limits the heat extraction in the deeper part of the system and neglects the heat loss at shallow depths. The standard g-function model therefore overestimates the inlet and outlet temperatures at the surface by about 0.3 °C. The impact of heat loss in the shallow subsurface on system performance can only be captured with a depth-dependent model.

3.1.2. Benchmark Against Finite Volume Approach

The depth-dependent implementation of subsurface thermal properties in the semi-analytical model in Geoloop was benchmarked against the depth-dependent numerical model in a simulation of a vertical triple U-tube (Figure 8) with the simulation parameters as defined in

Table 3. Simulation parameters for the benchmark of the depth-dependent semi-analytical model in Geoloop with the numerical model. * Parameters only used in the numerical model.

Parameter	Description	Unit	Value
H	Borehole length	[m]	500
D	Buried depth	[m]	5
nInlets	Number of inlet pipes	-	3
$r_b$	Borehole radius	[m]	0.085
pos	Pipe position in borehole	[x, y]	see Figure 8
$r_{out}$	Outer pipe radius	[m]	0.025
$r_{in}$	Inner pipe radius	[m]	0.0205
$k_p$	Pipe thermal conductivity	[W/mK]	0.41
$T_g$	Surface temperature	[°C]	10
$T_{grad}$	Geothermal gradient	[°C/m]	0.02
$k_s$	Subsurface bulk thermal conductivity	[W/mK]	Synthetic profile of 70% sand and 30% clay
$T_{in}$	Inlet temperature	[°C]	5
$m_{flow}$	Mass flow rate	[kg/s]	3
nyear	Simulated period	[years]	0.1
nled	Simulated timestep	[hours]	1
nsegments	Number of depth segments	-	15
nr *	Number of cells in the radial direction	-	20
$r_{sim}$ *	Simulated radial distance from borehole wall	[m]	20

. The simulations use a synthetic depth profile for the bulk subsurface thermal conductivity, representative of a mixture of 70% sand and 30% clay.

Figure 9 shows the calculated fluid and borehole wall temperatures after 30 days of system operation. There is a slight offset in the results, which is due to the zero heat flow boundary condition at the bottom of the numerical finite volume model that prevents heat extraction from the subsurface below the BHE. As a result, the outlet temperature in the numerical model is 0.02 °C lower compared to the results from the semi-analytical simulation. This also results in a smaller energy yield of 14,275 W in the numerical simulation, compared to 14,430 W in the results from the semi-analytical model in Geoloop. Over longer simulated times, the subsurface thermal front propagates horizontally outward and the system shifts from locally controlled thermal diffusion effects to being dominated by steady-state conductive subsurface heat transport. As such, the thermal effect of the bottom boundary condition in the finite volume method decreases and discrepancies between the two models remain negligible. Therefore, the strong resemblance of the calculated curves of fluid and borehole wall temperatures from the benchmark simulations illustrates the validation of the proposed depth-dependent semi-analytical approach against the numerical finite volume method.

The Effect of Axial Heat Flow

The uniform application of the g-function for all segments in the semi-analytical depth-dependent model captures the depth-dependent variation in horizontal heat transfer related to variable conductivity through the load aggregation in Equation (2) and incorporates vertical heat transfer. However, it does not include effects of vertical heat transfer related to variability in thermal conductivity. It is argued that these effects are minor. We investigated the impact of this simplification by comparing simulation results with the numerical model that incorporates vertical heat transfer between layers. In the simulation of the vertical triple U-tube (Table 3), we replaced the synthetic thermal conductivity profile for a mixture of sand and clay by a three-layer subsurface model with a thermal conductivity ($k_s$) of 1, 4 and 2 W/mK down to 100, 300 and 505 m respectively.

Figure 10 shows the calculated fluid and borehole wall temperatures after 30 days of system operation. The variation in borehole wall temperature along the borehole length reflects the different subsurface thermal conductivity layers. Just above the thermal conductivity layer boundary at 300 m, the borehole wall temperature in the numerical model is slightly higher with respect to the depth-dependent semi-analytical model. Below this layer boundary, the reverse applies to the calculated temperatures. This trend results from the vertical heat flow in the numerical model. In addition, at the bottom of the numerical model, the zero-flow boundary condition causes a similar effect to the results presented in Figure 9. The fluid temperatures calculated by the two models perfectly align. The combined effect of the bottom boundary condition and axial heat flow in the numerical model yields a maximum discrepancy in borehole wall temperature of only 0.06 °C between the two models. Therefore, we consider the impact of the simplified application of a single g-function in the semi-analytical depth-dependent model negligible for the purpose of our analysis timescales.

3.2. Application for Deep Borehole Heat Exchangers in The Netherlands

In The Netherlands, vertical BHEs are typically installed down to a maximum depth of 100–200 m, with some deeper systems down to 300 m. The required depth of the system is determined by the drilling operator, under the assumption of an average yield for a vertical BHE in the Dutch subsurface. However, especially in The Netherlands, where the (shallow) subsurface consists of porous sediments and sedimentary rock [35], the depth-dependent behavior of the subsurface temperature and thermal conductivity is expected to positively impact BHE performance with increasing depth.

In The Netherlands in the shallow (<500 m) subsurface an average temperature gradient of 0.02 °C/m is observed, underlain with a deep geothermal gradient of 0.03 °C/m, based on an analysis of a large dataset of subsurface temperature measurements in The Netherlands [36]. However, compared to subsurface temperature measurements, direct thermal conductivity measurements from the Dutch subsurface are relatively scarce. In contrast, information about other subsurface properties, such as lithology, is readily available at the Geological Survey of The Netherlands [37].

In the application of the semi-analytical depth-dependent model we aim to show that the potential energy yield per meter depth for a vertical BHE is not constant with depth, in contrast to what is currently assumed in industry. Firstly, we asses the potential for deeper BHE systems (100–800 m) in The Netherlands in a set of stochastic and deterministic BHE performance calculations that use the Dutch shallow geothermal gradient and implement synthetic depth profiles of bulk thermal conductivity to show the generic impact of depth variation in subsurface thermal properties on BHE performance. Secondly, we include a representative case of a deep (2000 m) coaxial system in the city of Roermond, in which we compare the semi-analytical depth-dependent approach in Geoloop to the depth-uniform equivalent in the standard (depth-uniform) g-function model. The results illustrate the implications of using location-specific geological information on the relation of the BHE energy potential with design choices and increasing maximum system depth.

3.2.1. Impact of Depth-Dependent Subsurface Properties on System Performance

The generic impact of depth variation in subsurface thermal properties on BHE performance was determined in two stochastic simulations with the simulation parameters as defined in

Table 4. Simulation parameters for stochastic simulations with a synthetic bulk subsurface thermal conductivity profile.

Parameter	Description	Unit	Value
optimize_keys	Parameter(s) to optimize for	-	Flow rate
optimize_keys_bounds	Boundary values for optimization parameters	[kg/s]	0.1–10
$CO{P}_{crit}$	Minimum COP of the fluid circulation pump	-	15
D	Buried depth	[m]	5
H	BHE length	[m]	100–800
nInlets	Number of inlet pipes	-	2
$r_b$	Borehole radius	[m]	0.085
pos	Pipe position in borehole	[x, y]	(0.054, 0), (−0.054, 0), (0, 0.054), (0, −0.054)
$r_{out}$	Outer pipe radius	[m]	0.025
$r_{in}$	Inner pipe radius	[m]	0.0205
$ϵ$	Pipe roughness	[m]	10−6
$k_p$	Pipe thermal conductivity	[W/mK]	0.41
fluid	Heat carrier fluid	-	Water
$k_g$	Thermal conductivity of the backfill	[W/mK]	0.844
$T_g$	Surface temperature	[°C]	10
$T_{grad}$	Geothermal gradient	[°C/m]	0.02
$\alpha$	Subsurface thermal diffusivity	[m2/s]	10−6
$k_s$	Subsurface bulk thermal conductivity	[W/mK]	Synthetic profile of sand or clay
$T_{in}$	Inlet temperature	[°C]	5
nyear	Simulated period	[years]	1
nsegments	Depth segments	-	20
n_samples	Stochastic samples	-	100

. The yearly average power yield was calculated based on a fixed inlet temperature and the optimization algorithm was deployed to optimize flow rate and obtain a system configuration that generates maximum power while respecting a minimum COP for the fluid circulation pump ($CO{P}_{crit}$). The BHE length was sampled in a uniform distribution in the range specified in Table 4.

Each stochastic simulation includes a distinct synthetic subsurface thermal conductivity profile for a subsurface composed exclusively of sand or clay. To account for uncertainty in the subsurface thermal properties, a depth-uniform error of 10% was imposed on the porosity values used for calculating the bulk thermal conductivity. As such, a set of depth profiles with a range of thermal conductivity values was obtained and stochastically sampled in the simulation process. Figure 11 shows the relation between the calculated yearly average power yield at the end of the simulated year and the stochastically sampled BHE length. The scattered nature of the datasets in the figure follows from the uncertainty in the subsurface thermal conductivity. The results clearly show a non-linear increase in yearly average power supply with increasing BHE length that is more significant for the case with sand. This is an effect of the increase in subsurface bulk thermal conductivity and temperature with depth.

In the simulations, a continuous load is imposed on the system as a result of the fixed inlet temperature. In reality, constant operation of a BHE in heating mode would lead to significant system degradation. In addition, the results do not consider the extra power that would be supplied by the heat pump at the building side. The stochastically calculated yearly average power yield should therefore not be considered in absolute terms, but focus lies on the relation between system parameters. Also, optimization of the flow rate based on a minimum COP of the fluid circulation pump limits the maximum potential of the simulated system. However, even when respecting this boundary condition, the non-linear increase in system potential with depth is observed. This finding supports the potential for deeper BHEs and forms the base for future work, in application of the depth-dependent semi-analytical model in Geoloop for BHE design optimization.

The implications for a realistic power demand profile on potential peak power supply of the generic depth variation in subsurface properties were determined in two deterministic simulations with the simulation parameters as defined in

Table 5. Simulation parameters for deterministic simulations with a synthetic bulk subsurface thermal conductivity profile.

Parameter	Description	Unit	Double U-Tube 300 m	Double U-Tube 800 m
D	Buried depth	[m]	5	5
H	BHE length	[m]	300	800
nInlets	Number of inlet pipes	-	2	2
$r_b$	Borehole radius	[m]	0.085	0.085
pos	Pipe position in borehole	[x, y]	(0.044, 0), (−0.044, 0), (0, 0.044), (0, −0.044)	see Table 3
$r_{out}$	Outer pipe radius	[m]	0.025	0.03
$r_{in}$	Inner pipe radius	[m]	0.0205	0.0245
$ϵ$	Pipe roughness	[m]	10−6	10−6
$k_p$	Pipe thermal conductivity	[W/mK]	0.41	0.41
fluid	Heat carrier fluid	-	Water	Water
$k_g$	Thermal conductivity of the backfill	[W/mK]	0.844	0.844
$T_g$	Surface temperature	[°C]	10	10
$T_{grad}$	Geothermal gradient	[°C/m]	0.02	0.02
$\alpha$	Subsurface thermal diffusivity	[m2/s]	10−6	10−6
$k_s$	Subsurface bulk thermal conductivity	[W/mK]	Synthetic profile of sand	Synthetic profile of sand
$Q$	Heat load	[W]	see Figure 12a	see Figure 12a
lp_scale	Heat load scaling factor	-	2.1	9.5
$m_{flow}$	Maximum mass flow rate	[kg/s]	3.7	6
lp_minscaleflow	Minimum flow rate scaling factor	-	0.1	0.1
nyear	Simulated period	[years]	1	1
nled	Simulated timestep	[hours]	1	1
nsegments	Depth segments	-	20	20

. Both simulations use a synthetic subsurface thermal conductivity profile for a subsurface composed of sand. The fluid temperature was calculated based on an hourly variable heat demand. A heat load profile for the year 2020 was used, representative of a single household (one dwelling) in the center of Amsterdam calculated with Ninja Renewables default settings [38,39]. The city of Amsterdam is located in the west of The Netherlands, with a temperate maritime climate (i.e., mild summers and cool winters) which is reflected in the heat load profile. The threshold for heat demand was set to a temperature of 16 °C and an absence of cooling demand was assumed. In this heat load profile for one household, the peak power demand during the winter is approximately 4.3 kW and the yearly average power demand is about 1.3 kW. In total, 11.6 MWh of power is consumed every year.

The load profile and maximum flow rate were scaled in a manual iterative process, such that a minimum inlet temperature of 4 °C was obtained during peak load in the first half year of operation, for a yearly average COP of the fluid circulation pump ($CO{P}_{circ}$) between 14 and 15. The scaling factor for the load profile represents the amount of dwellings that could be supplied by the system, based on the single dwelling reference profile from Ninja Renewables. The flow rate that is passed to the simulation defines a maximum value, which is scaled for every timestep at a factor similar to how the heat load for that timestep relates to the maximum heat load in the profile. Since the flow rates are in the laminar flow regime, the hydraulic resistance scales proportionally to flow rate, such that COP is therefore constant. To ensure numerical stability, a minimum scaling factor for the flow rate is also defined.

Figure 12 shows the results of the scaled heat load profile and the calculated inlet and outlet temperatures. In both simulations, the single dwelling heat load profile was scaled such that the inlet temperature reaches a threshold of 4 °C during peak supply in the last month of the simulated year. For the 300 m long system, the heat load profile was scaled by a factor of 2.1 (2.1 dwelling), yielding a maximum peak power supply of 9.0 kW. For the 800 m long system, the heat load profile was scaled by a factor of 9.5 (9.5 dwelling), yielding a maximum peak power supply of 40.7 kW. These values for the peak power supply compare directly to the value range assumed in common practice by Dutch industry of 25 to 40 W generated power per meter of BHE depth. For the system down to 305 m, the calculated peak power supply of 9.0 kW corresponds to 30 W/m generated by the BHE, which lies within the assumed range. For the system down to 805 m, the calculated peak power supply of about 40.7 kW corresponds to about 51 W/m generated by the BHE, which exceeds the upper boundary of the assumed range.

Note that in the model, the heat load profile that is representative of the heat demand in the building is directly applied to the BHE system, without correction for the heat generated by the electricity consumption of the heat pump (HP). The calculated power yield could thus be considered as 1/$CO{P}_{HP}$ lower than what would actually be delivered to the building. For a heat pump with a $CO{P}_{HP}$ of 6 and a fluid circulation pump with a $CO{P}_{circ}$ of 15, the COP of the total system would be 4.5. In addition, the absence of a seasonal cooling load in the used load profile is a conservative assumption, since the possible regenerative effect of subsurface heat injection in the cooling mode is not included. Also, the threshold of a minimum inlet temperature of 4 °C is a conservative boundary condition, since heat pumps commonly accommodate lower inlet temperatures, especially for heat carrier fluids that include additives such as glycol. This would imply a higher potential peak power supply and an even bigger potential for deep BHEs.

3.2.2. Implications for a Realistic Subsurface Model on a Deep Coaxial BHE

The impact of using local geological information on the relation of BHE potential with increasing system depth was determined for the city of Roermond in the SE of The Netherlands. The local geology reflects the sedimentary infill of the striking NW-SE Roer Valley Graben, with predominantly Upper Oligocene to Quaternary sediments [35].

We performed three sets (a, b and c) of deterministic simulations (

Table 6. Simulation parameters for three sets (a, b, c) of deterministic simulations of a vertical coaxial system at the location of Roermond. Values correspond to the simulations in Figure 14.

Parameter	Description	Unit	Depth-Dependent Simulation a, b, c	Standard g-Functions Simulation a, b, c
D	Buried depth	[m]	5	5
H	BHE length	[m]	2000	2000
nInlets	Number of inlet pipes	-	1	1
$r_b$	Borehole radius	[m]	0.156	0.156
pos	Pipe position in borehole	[x, y]	0, 0	0, 0
$r_{out}$	Outer pipe radius	[m]	0.15 (outer pipe), 0.08 (inner pipe)	0.15 (outer pipe), 0.08 (inner pipe)
$r_{in}$	Inner pipe radius	[m]	0.135 (outer pipe), 0.072 (inner pipe)	0.135 (outer pipe), 0.072 (inner pipe)
$ϵ$	Pipe roughness	[m]	10−6	10−6
$k_p$	Pipe thermal conductivity	[W/mK]	30	30
$ins{u}_{dr}$	Fraction of pipe radius with insulation	-	0 (a)/0 (b)/0.5 (c)	0 (a)/0 (b)/0.5 (c)
$ins{u}_k$	Insulation thermal conductivity	[W/mK]	- (a)/- (b)/0.026 (c)	- (a)/- (b)/0.026 (c)
$ins{u}_z$	Maximum depth of pipe insulation	[m]	- (a)/- (b)/2000 (c)	- (a)/- (b)/2000 (c)
fluid	Heat carrier fluid	-	Water	Water
$m_{flow}$	Mass flow rate	[kg/s]	20	20
$k_g$	Thermal conductivity of the backfill	[W/mK]	2	2
$T_g$	Surface temperature	[°C]	10(a)/40(b)/10(c)	40
$T_{grad}$	Geothermal gradient	[°C/m]	0.03(a)/0(b)/0.03(c)	0
$\alpha$	Subsurface thermal diffusivity	[m2/s]	10−6	10−6
$k_s$	Subsurface bulk thermal conductivity	[W/mK]	see Figure 13	2.3 (average of Figure 13)
$Q$	Heat load	[W]	150,000	150,000
nyear	Simulated period	[years]	0.085	0.085
nsegments	Depth segments	-	50	1

). Each set includes one simulation with the depth-dependent semi-analytical model in Geoloop and one simulation that uses the standard implementation of g-functions. All simulations calculate the system performance based on a constant heat extraction of 150 kW. The subsurface lithology and derived depth profile of bulk thermal conductivity down to 2 km depth at this location are shown in Figure 13. The lithology profile down to 1000 m is based on borehole data, extended down to 2000 m based on a synthetic mixture of sandstone and claystone.

In the first simulation set (a in Table 6), the depth-dependent model uses the depth-dependent subsurface thermal properties at the location of Roermond and the standard implementation of g-functions uses the depth-averaged subsurface thermal properties according to the maximum system depth. The simulated system includes metal pipes. The second set of simulations (b in Table 6) uses the same input parameters, but the depth-dependent model incorporates a uniform subsurface temperature of 40 °C instead of a geothermal gradient. The third set of simulations (c in Table 6) also uses the same input parameters as the first, but with an insulated inner pipe where half the pipe thickness has a thermal conductivity of 0.026 W/mK.

Figure 14 shows the fluid and borehole wall temperatures after one month of system operation. In Figure 14a, the depth-dependent simulation shows how the geothermal gradient and depth profile in subsurface thermal conductivity promote heat extraction at larger depths, with a larger temperature difference between the fluid in the inlet pipe and the borehole wall. However, the same principle promotes heat loss in the shallow part of the system, where the fluid in the outlet pipe is higher compared to the formation. In the standard g-function model, heat loss in the outlet pipe is underestimated in the shallow part of the system, where the fluid temperature stays below the depth-average approximation of the formation temperature. Therefore, the standard g-function model overestimates the inlet and outlet temperature by about 4 °C after the first month. Figure 14b shows a similar effect, but the combination of a depth-uniform subsurface temperature and depth-variable thermal conductivity profile shows the isolated contribution of the latter to the depth-dependent system behavior. Here, the results for the standard g-function model are the same as in Figure 14a, but without geothermal gradient, the temperature difference between the inlet pipe and the borehole wall is increased in the shallow part and reduced in the deeper part of the depth-dependent simulation. This effect is most pronounced along the bottom half of the system, enabled by a higher subsurface thermal conductivity. However, the calculated inlet and outlet temperature at the top in the depth-dependent simulation are almost the same as in simulation set a, which illustrates that the system is more sensitive to depth-variation in temperature than to depth-variation in thermal conductivity.

For this deep coaxial system, insulation of the inner pipe could strongly reduce heat loss in the shallow subsurface as shown in Figure 14c. In this case, almost all heat gained in the outer, inlet pipe of the system is transported to the surface in the inner, outlet pipe. Compared to the depth-dependent model, the standard g-function model strongly underestimates heat extraction at the bottom half of the system and therefore the inlet and outlet temperature by about 2 °C. Also, the calculated borehole wall temperature is lower, which would impact predictions of the system lifetime. In addition, comparison of the fluid temperature-depth profiles between the first (a) and last (c) set of simulations shows the effect of thermal short-circuiting through the metal inner pipe and the potential gain in mitigating this effect. All figures show that a depth-dependent BHE model is essential to capture the balance between deep heat extraction and shallow heat losses, especially for deep systems and for BHE designs where the fluid temperature approaches the temperature at the borehole wall.

4. Discussion

In the numerical implementation of the stacked semi-analytical model in Geoloop, the number of iterations in the scheme used to solve for depth-dependent fluid temperatures is fixed at three. Model testing confirmed that this is sufficient to ensure convergence of the heat flow in the different model segments to the total imposed heat load. Beyond three iterations, no significant improvement in numerical accuracy was observed, while computational effort increased. With three iterations, the runtime for the benchmark simulation defined in Section 3.1.1 scales in runtime approximately proportionally to the number of segments adopted compared to the default runtimes of pygfunction. For 10 up to 50 segments this results in runtimes of well below a minute on an average laptop for a relatively detailed temporal resolution of output, as shown in Section 3.2. The presented method performs significantly faster than numerical alternatives, which underlines its applicability in stochastic simulations and sensitivity analyses, where large numbers of model evaluations are required.

For the analytical derivation of g-functions based on the Finite Line Source (FLS) model in pygfunction, three types of boundary conditions can be applied: uniform heat extraction rate, uniform borehole wall temperature, or hybrid formulations [10]. In our stacked implementation of the g-functions, only the boundary condition of a uniform borehole wall temperature is applicable.

As shown in Section 3.1.1 and Section 3.2, the BHE performance results from depth-uniform and depth-dependent methods may appear very similar. However, the standard implementation of pygfunction for depth-uniform load aggregation and temperature result needs to be treated with care for subsurface parameters which vary with increasing depth. In particular, when over different parts of the BHE system length heat is injected or extracted (as generally can be the case for deep boreholes up to 2000 m depth studied in this paper), the depth-uniform approach will overestimate system performance. On the other hand, a depth-uniform model can underestimate performance for insulated designs. In addition, the depth-uniform approach cannot take into account depth variation in BHE thermal properties.

In Geoloop, the number of model segments in the depth-dimension is a user-defined variable and is proposed to be limited to around 50. Although many more segments would computationally and technically be feasible, the added value would be limited as the thermal properties of the subsurface and BHE materials are properly averaged over each model segment and results would be hardly affected.

Geoloop currently provides a robust basis for calculating depth-dependent BHE performance, yet several simplifications limit its applicability in complex subsurface or system-scale analyses. With a focus on capturing depth variability in deep (>500 m) BHE performance (i.e., at the scale of 1 to 100 s of meters, and timescales considered), we demonstrated (in Section 3.1.2) that a single g-function in our semi-analytical depth-dependent model is numerically efficient and robust. However, for subsurface conditions with lithological variation at different times and spatial scales (i.e., cm), the effect of variability in vertical heat transfer can increase. Under such conditions, axial heat flow effects could be included by implementation of the computationally more expensive multilayer FLS model from Abdelaziz et al. [9]. Also, incorporation of groundwater flow is relevant for accurately representing heterogeneous thermal and hydrogeological subsurface conditions in sedimentary geological settings. By implementing the analytical Moving Finite Line Source (MFLS) model [40] in our depth-dependent semi-analytical approach, the depth-variable impact of groundwater flow on the subsurface heat extraction rate along the borehole length can be further investigated. In addition, thermodynamic coupling of the BHE system with a full heat pump model would improve the models’ applicability to integrated energy system design and system optimization. The current optimization strategy is applicable to optimization of the BHE configuration, but coupled optimization of design and control of the BHE and heat pump system would require a more sophisticated optimization algorithm integrating performance of the heat pump and deep BHE. These enhancements would improve model accuracy and robustness, particularly for deep BHE applications. For a discussion on the simplifications and limitations of the quasi-steady-state solution of a thermal resistance network based on the multipole method for the borehole interior and general applicability of g-functions derived from the FLS model, we refer to the corresponding literature [8,10,41].

5. Conclusions

In this paper we presented a semi-analytical model for fast, depth-dependent BHE performance calculations. The method uses an iterative solution to solve for depth-dependent fluid temperatures in a vertically stacked approach for load aggregation and a single g-function for the BHE. The depth-dependent approach has high computational speed and is benchmarked against the standard implementation of depth-uniform load aggregation and borehole wall and fluid temperature calculation available in pygfunction and against a numerical finite volume method. All three methods are included in Geoloop, an open-source Python package with a modular framework that facilitates broad functionality for modeling BHE systems and visualization of the results. We demonstrated capabilities for sensitivity analysis of the BHE system through stochastic simulations and optimization of the system design and operational constraints on flow rate in accordance with a target BHE subsurface system COP. We applied the presented model in a representative case for the assessment of the potential of deep BHE systems in The Netherlands. The results demonstrate a clear non-linear increase in potential extraction of heat at larger depths, exceeding industry-standard values. The presented benchmark and example simulations illustrate the versatility of the Geoloop package and prove that the novel semi-analytical depth-dependent approach fills the gap between expensive numerical models and (semi-)analytical alternatives that neglect subsurface heterogeneity.

Deterministic simulations using a synthetic heat load profile for one year of operation and representative of a single household in Amsterdam showed that for a 300 m BHE system, the peak thermal power supply was about 9 kW (30 W/m), within the value range assumed in common practice by Dutch industry. For an 800 m system, the peak thermal power supply reached approximately 40.7 kW (51 W/m), exceeding the upper boundary of the value range assumed in common practice by Dutch industry. These simulations assumed a minimum inlet temperature of 4 °C during peak load and are in accordance with a COP of ca. 15, taking into account power consumption for fluid circulation. The results did not account for heat generated by heat pump electricity consumption or seasonal cooling loads, making the results conservative.

In the city of Roermond, simulations of a deep coaxial BHE have been based on a constant heat extraction of 150 kW for 1 month of operation and incorporating a realistic depth profile of soil composition and saturated bulk thermal conductivity. Three sets of simulations compared the effect of a variable depth-dependent thermal conductivity profile with a depth-averaged equivalent. Results demonstrate that a depth-dependent model is required to accurately capture the impact of depth-dependent subsurface properties on system behavior, as fluid temperatures can be underestimated or overestimated by several degrees in the depth-uniform approximation. This highlights the importance of considering local geological heterogeneity for accurate BHE performance assessment.

The results consistently show that BHEs for depth ranges of 300 to 2000 m in clastic sedimentary environments and a moderate geothermal gradient of 0.02 to 0.03 °C/m, as observed in The Netherlands, are marked by positive perspectives for enhanced heat recovery with increasing depth, provided that BHE designs are adapted to application at greater depths and to reduce heat losses in the shallow domain.