Estimation of Recovery Efficiency in High-Temperature Aquifer Thermal Energy Storage Considering Buoyancy Flow

Abstract

With their high storage capacity and energy efficiency as well as the compatibilities with renewable energy sources, high-temperature aquifer thermal energy storage (HT-ATES) systems are frequently the target today in the design of temporally and spatially balanced and continuous energy supply systems. The inherent density-driven buoyancy flow is of greater importance with HT-ATES, which may lead to a lower thermal recovery efficiency than conventional low-temperature ATES. In this study, the governing equations for HT-ATES considering buoyancy flow are nondimensionalized, and four key dimensionless parameters regarding thermal recovery efficiency are determined. Then, using numerical simulations, recovery efficiency for a sweep of the key dimensionless parameters for multiple cycles and storage volumes is examined. Ranges of the key dimensionless parameters for the three displacement regimes, that is, a buoyancy-dominated regime, a conduction-dominated regime, and a transition regime, are identified. In the buoyancy-dominated regime, recovery efficiency is mainly correlated to the ratio between the Rayleigh number and the Peclet number. In the conduction-dominated regime, recovery efficiency is mainly correlated to the product of a material-related parameter and the Peclet number. Multivariable regression functions are provided to estimate recovery efficiency using the dimensionless parameters. The recovery efficiency estimated by the regression function shows good agreement with the simulation results. Additionally, well screen designs for optimizing recovery efficiency at various degrees of intensity of buoyancy flow are investigated. The findings of this study can be used for a quick assessment and characterization of the potential HT-ATES systems based on the geological and operational parameters.

Key Points

• Four key dimensionless parameters for the high-temperature aquifer thermal energy storage systems are identified

• The displacement processes are classified into a buoyancy-dominated regime, a conduction-dominated regime, and a transition regime

• Multivariable regression functions are demonstrated for the estimation of thermal recovery efficiency

1 Introduction

With increasing concern about potential consequences of climate change and the depletion of fossil fuels, the enhanced usage of renewable energies (e.g., solar energy, wind energy, surplus heat from incinerators and power plants, etc.) become part of the design of energy supply systems. The intermittent availability of renewable energies and the seasonal fluctuations of energy demands make the requests for energy storage systems. High-temperature aquifer thermal energy storage (HT-ATES) is an attractive energy storage approach with high storage efficiency and capacity (Fleuchaus et al., 2018).

1.1 High Temperature Aquifer Thermal Energy Storage

Though the concepts are similar, there are distinctions between the HT-ATES and the low-temperature aquifer thermal energy storage (LT-ATES). LT-ATES is used for house/building cooling and warming, using shallow aquifers with a storage temperature below 30°C. In contrast, HT-ATES is versatile, using deeper (saline) aquifers, with the storage temperature generally above 60°C. The main advantages of HT-ATES compared to LT-ATES are: (a) HT-ATES is compatible with multiple renewable energy sources, for example, solar, geothermal, biomass, incineration plants, surplus heat from industry, etc (Fleuchaus et al., 2020); (b) at higher storage temperature the stored heat can directly be used as the source without additional heat pumps, leading to higher energy efficiency (Schout et al., 2014).

The principle concept of an HT-ATES system is shown in Figure 1. The main component of HT-ATES is a groundwater well-doublet (a hot well and a cold/compensation well) for water circulation between the storage aquifer and the energy systems. Groundwater is the carrier medium for heat between the surface facilities and the aquifer. During summer (or a period with a surplus of renewable energy), groundwater is abstracted from the cold well, charged with the surplus heat from renewable or non-renewable sources, and reinjected into the hot well for storage. During winter (or a period with high energy demand), the pump direction is reversed for heat production from the hot well, and cold surface water is injected back into the cold well. The produced heat can be used for, for example, district heating, power generation, etcetera.

Figure 1

Sketch of the HT-ATES, with the flow directions for energy storage and production periods, shown in the left and right subfigure, respectively.

Around 3000 LT-ATES projects have successfully been implemented worldwide (Fleuchaus et al., 2018). In contrast, few HT-ATES projects have been developed, and numerous pilot HT-ATES plants are currently under development (McLing et al., 2022; Sheldon et al., 2021). The main challenges that impede the application of HT-ATES include factors such as (a) clogging due to particles and precipitation of minerals in the heat exchangers, wells, and aquifers (Holmslykke & Kjøller, 2023), (b) corrosion of ATES components and damage to the aquifer (Schout et al., 2014), (c) significant heat losses during transport in boreholes and aquifers, etcetera (Drijver, 2011). This study focuses on the aspect of the heat recovery efficiency from the aquifer.

1.2 Factors Affecting Recovery Efficiency

Recovery efficiency, defined as the ratio between the heat injected and recovered, is a critical metric for ATES systems. Numerous factors can affect recovery efficiency of ATES, for example, aquifer heterogeneity (Collignon et al., 2020; Sommer et al., 2013), geological layering (Bridger & Allen, 2014), regional groundwater flow (Gao et al., 2019; Yapparova et al., 2014), well spacing and thermal interferences (Ganguly et al., 2017; Kim et al., 2010), mono-well setup (Zeghici et al., 2015), changes in the salinity of the injected water (Liu et al., 2020; van Lopik et al., 2016), existence of natural fractures (Chen et al., 2024; Obembe et al., 2018), spatial patterns for the distribution of multiple ATES systems (Sommer et al., 2015), etcetera. Sensitivity analysis of the hydraulic and thermal parameters was implemented by Jeon et al. (2015) and Heldt et al. (2024), and the parameters were ranked by the importance to recovery efficiency.

For HT-ATES, heat loss may result from not only the diffusive heat transport, driven by the gradient in temperature or the mechanical dispersion (Tang & Van Der Zee, 2022) but also by thermal stratification (Sheldon et al., 2021). The injected hot water tends to flow above the ambient water due to the lower density, namely buoyancy flow (Heldt et al., 2021). Hellstrom et al. (1979) solved the calculation of the tilting rate of the front analytically for idealized cases considering a sharp and vertical thermal front separating the injected water and aquifer water. They deduced a characteristic tilting time, indicating the time for buoyancy tilting of an initial vertical front to about 60°. The front tilting rate was found to depend on the temperature level, aquifer permeability, and aquifer thickness. When the time of the ATES cycles is shorter than the characteristic tilting time, the effects of buoyancy on recovery efficiency are expected to be small. Field experiments by Auburn University (Buscheck et al., 1983; Molz et al., 1983) showed that HT-ATES with high injection temperatures up to 81°C resulted in a low recovery efficiency of 0.45. The main reason was the buoyancy flow associated with density contrast between the injected and ambient water at the high injection temperature. Their experiments were well predicted by the numerical simulation by Tsang et al. (1981). A small-scale heat injection field test for HT-ATES was conducted by Heldt et al. (2021) with a storage temperature higher than 70°C. The temperature monitored by the sensor network fitted the modeling results well, considering the advective and conductive heat transport and temperature-induced convection. They claimed that the intensity of buoyancy flow is highly sensitive with respect to the vertical hydraulic conductivity. Beernink et al. (2024) studied the relative contribution of conduction, dispersion, and buoyancy flow on the heat loss in HT-ATES by numerical simulation for a wide range of operational and hydrogeological conditions. They found that the buoyancy flow is mainly affected by storage temperature and hydraulic conductivity of the aquifer.

1.3 Estimation of Recovery Efficiency

Numerous research studies attempted to estimate recovery efficiency of ATES by either correlation functions (between the key parameters and the recovery efficiency) or analytical solutions of the heat transport equations. These results could be used to assess and characterize potential HT-ATES systems quickly.

Doughty et al. (1982) presented the dimensionless form of the heat transfer equations for ATES, and they identified several key dimensionless groups that govern the system's thermal behavior, that is, the Peclet number (ratio of advective transport to diffusive transport), the aspect ratio (between the thermal radius and aquifer thickness), and two material related parameters. They provided the relationships between the key dimensionless groups and recovery efficiency in graphical form. Tang and Rijnaarts (2023) derived dimensionless analytical solutions for the computation of thermal recovery of ATES systems with transient pumping rates, multiple cycles, various well configurations, and thermal plume geometries. However, the derivations of the non-dimensionless governing equation by Doughty et al. (1982) and the analytical solution by Tang and Rijnaarts (2023) both neglected the density-driven (natural) convection since they mainly focused on LT-ATES where the effects of natural convection can be assumed to be small.

A key parameter for natural convection processes in porous media is the Rayleigh number (Ra), which indicates the ratio of heat transport rate by natural convection to that by thermal diffusion (Nield & Bejan, 2013). The Rayleigh number was used in correlation functions in several research studies to estimate recovery efficiency from HT-ATES. Based on numerical simulation of HT-ATES with specific combinations of parameters, Gutierrez-Neri et al. (2011) first found a correlation between the Rayleigh number and recovery efficiency. Schout et al. (2014) extended the work of Gutierrez-Neri et al. (2011) by performing a sensitivity analysis on several parameters, for example, aquifer thickness, permeability, storage temperature, etcetera. They found that the proposed correlation cannot fit a more comprehensive set of data obtained from the sensitivity analysis. They claimed that the reason is the missing of the critical parameters of horizontal aquifer permeability and storage volume in the definition of the Rayleigh number. In this case, they proposed a modified Rayleigh number, which can correlate well with the recovery efficiency for all the simulated cases with two equations for different aquifer thicknesses. Sheldon et al. (2021) further extended the work of Schout et al. (2014), and they provided an improved version of the recovery efficiency–Rayleigh number relationship valid for HT-ATES operating at both annual circles and short-term cycles (e.g., daily cycles) and storage temperatures of up to 300°C. Nevertheless, the injection and production processes during HT-ATES involve both free convection due to density contrasts and forced convection due to the pumping at the well, namely mixed convection (Ward et al., 2007). Holzbecher and Yusa (1995) investigated the patterns of groundwater flow induced by geothermal sources by numerical simulation. They clearly showed that the Rayleigh number is not the only parameter characterizing the mixed convection. They claimed that forced convection should be prescribed by the boundary velocity and be related to the Peclet number. Though their study provided clues, the derivation of the key parameters for the mixed convection systems is not clear.

To the authors' knowledge, the nondimensionalization of HT-ATES systems considering buoyancy flow and the prediction of recovery efficiency using the resulting key parameters has not yet been published. This study mainly aims to address this knowledge gap.

1.4 Objectives

This study considers typical HT-ATES setups with a vertical fully penetrating well without thermal interference from other wells. The dimensionless form of the governing equations for mass and heat transport in HT-ATES is presented, and the key parameters affecting thermal recovery efficiency are discussed. This study aims to characterize the displacement processes in HT-ATES based on the key parameters derived from the non-dimensionless equations. Furthermore, numerical simulations are implemented for the sensitivity analysis of the key parameters. The correlation functions between recovery efficiency and the key parameters are investigated, and the accuracy of these correlation functions to estimate recovery efficiency of HT-ATES is evaluated. Besides, well screen designs for optimizing recovery efficiency at various intensities of buoyancy flow are also investigated.

The paper is organized as follows: Section 2 explains the theory; the details of the numerical model are presented in Section 3; in Section 4 results are discussed; and Section 5 lists the main conclusions.

2 Theory

2.1 Conceptual Model

The basic concept of HT-ATES involves the simulation of both a hot well and a cold well (or compensation well), as shown in Figure 1. In this study, the hot and cold wells are assumed to be far enough apart so that there is no thermal interference. The effect of the cold well on the groundwater flow near the hot well is also assumed to be small. In this case, we focus on investigating thermal recovery efficiency at the hot well, and the consideration of the compensation/cold well is not necessary. The conceptual model is shown in Figure 2a. The aquifer (Ω₁) is assumed to have a thickness of H, bounded by the non-permeable caprock and bedrock (Ω₂ and Ω₃). The bedrock is assumed to have the same properties as the caprock (e.g., density, heat capacity, thermal conductivity). The aquifer and the caprock/bedrock are assumed homogeneous and isotropic at the Darcy scale. Hot water is injected or abstracted at the well (Γ₁). The pumping well fully penetrates the aquifer. Due to the radial symmetry of the conceptual model, the domain is assumed to be two-dimensional, with coordinates of r and z.

Figure 2

Sketch of the conceptual model: subfigure (a) shows the domains and the boundaries. The governing equations for each domain (Ω₁, Ω₂, and Ω₃) and the boundary conditions (Γ₁, Γ₂, Γ₃, Γ₄, and Γ₅) are explained in Section 2.2. The dimension of the model used for numerical simulation is explained in Section 3.2; subfigure (b) shows the example of the computational mesh in the sub-region given by the red frame, as explained in Section 3.3.4.

2.2 Governing Equations

Based on the Boussinesq approximation, we assume density changes have a minimal effect in the continuity equations, and it only affects the gravity term in the momentum balance equation. Groundwater flow and heat transport in the aquifer (Ω₁) can be described by the following three governing equations, representing the conservation of mass, momentum, and energy, respectively:

$$$\nabla \cdot \mathbf{q}=0$$$ (1)

$$$\mathbf{q}=-K\frac{1}{{\mu }_{\mathrm{w}}}\nabla \left(p+{\rho }_{\mathrm{w}}\text{gz}\right)$$$ (2)

$$${\rho }_{\mathrm{a}}{C}_{\mathrm{p},\mathrm{a}}\frac{\partial T}{\partial t}+{\rho }_{\mathrm{w}}{C}_{\mathrm{p},\mathrm{w}}\mathbf{q}\cdot \nabla T-{\lambda }_{a}{\nabla }^{2}T=0$$$ (3)

where q is the Darcy flux, K is the permeability, μ_w the viscosity of water, p is the pressure, ρ_w and ρ_a are the density of water and aquifer, g is the gravitational constant, C_p,w and C_p,a are the heat capacity of water and aquifer, T is temperature, and λ_a is the thermal conductivity of the aquifer. The subscript “a” refers to aquifer, including both solid and water. The heat transport in the caprock and the bedrock (Ω₂ and Ω₃) is governed by:

$$${\rho }_{\mathrm{c}}{C}_{\mathrm{p},\mathrm{c}}\frac{\partial T}{\partial t}-{\lambda }_{c}{\nabla }^{2}T=0$$$ (4)

where ρ_c and C_p,c are density and heat capacity of the caprock, and λ_c is the thermal conductivity of the caprock. The following boundary conditions are employed:

$$$\text{thermal}\,\text{insulation}\,\text{on}\,\text{surface}\,{{\Gamma }}_{4}\cup {{\Gamma }}_{5}\nabla T\cdot \boldsymbol{n}=0$$$ (5)

$$$\text{constant}\,\text{temperature}\,\text{on}\,\text{surface}\,{{\Gamma }}_{1}\quad T={T}_{\text{inj}}$$$ (6)

$$$\text{no}\,\text{flow}\,\text{across}\,\text{surface}\,{{\Gamma }}_{2}\cup {{\Gamma }}_{3}\quad \boldsymbol{q}\cdot \boldsymbol{n}=0$$$ (7)

$$$\text{constant}\,\text{pressure}\,\text{on}\,\text{surface}\,{{\Gamma }}_{4}\quad p={p}_{0}-{\rho }_{0}\text{gz}$$$ (8)

$$$\text{inflow}\,\text{across}\,\text{surface}\,{{\Gamma }}_{1}\quad \boldsymbol{q}\cdot \boldsymbol{n}=\frac{{Q}_{\text{inj}}}{2\pi H{r}_{\mathrm{b}}}$$$ (9)

where n is the normal vector to the boundary, T_inj is the temperature of injected hot water, p₀ is the pressure at the surface, ρ₀ is the initial water density, Q_inj is the injection rate, and r_b is the well radius. At the interfaces between aquifer and caprock/bedrock:

$$${T\vert }_{z+}={T\vert }_{z-}\quad {{\Gamma }}_{2}\cup {{\Gamma }}_{3}$$$ (10)

$$${{\lambda }_{\mathrm{c}}\frac{\partial \mathrm{{T}}}{\partial z}\vert }_{z+}={{\lambda }_{\mathrm{a}}\frac{\partial \mathrm{{T}}}{\partial z}\vert }_{z-}\quad {{\Gamma }}_{2}$$$ (11)

$$${{\lambda }_{\mathrm{a}}\frac{\partial \mathrm{{T}}}{\partial z}\vert }_{z+}={{\lambda }_{\mathrm{c}}\frac{\partial \mathrm{{T}}}{\partial z}\vert }_{z-}\quad {{\Gamma }}_{3}$$$ (12)

In this case, temperature and heat flux are ensured to be continuous at the interface between the aquifer and the caprock/bedrock (Doughty et al., 1982).

2.3 Nondimensionalization

We derive the dimensionless form of the governing equations to reveal the key parameter groups for the HT-ATES system. The characteristic length is chosen as the thickness of the aquifer. The non-dimensional independent variables are defined as:

$$${r}^{\ast }=\frac{r}{H},{z}^{\ast }=\frac{z}{H},{t}^{\ast }=t\frac{{D}_{\mathrm{a}}}{{H}^{2}}$$$ (13)

where the dimensionless term is given with superscript *, and D_a is the thermal diffusivity of the aquifer. The thermal diffusivities of the aquifer and the caprock are calculated by:

$$${D}_{\mathrm{a}}=\frac{{\lambda }_{\mathrm{a}}}{{\rho }_{\mathrm{a}}{C}_{\mathrm{p},\mathrm{a}}},{D}_{\mathrm{c}}=\frac{{\lambda }_{\mathrm{c}}}{{\rho }_{\mathrm{c}}{C}_{\mathrm{p},\mathrm{c}}}$$$ (14)

The non-dimensional pressure and temperature are given as:

$$${\,p}^{\ast }=\frac{p+{\rho }_{0}gz}{{\Delta }\rho gH}$$$ (15)

and

$$${T}^{\ast }=\frac{T-{T}_{0}}{{\Delta }T}$$$ (16)

where ΔT and Δρ are the difference between the temperature and density of the injected water and the initial temperature and density, and ΔT and Δρ are calculated as:

$$${\Delta }T={T}_{\text{inj}}-{T}_{0},{\Delta }\rho ={\rho }_{\text{inj}}-{\rho }_{0}$$$ (17)

where ρ_inj the density of injected hot water, and T₀ is the initial aquifer temperature. The relation between the fluid temperature and density is described by a thermal expansion coefficient (β), which is defined as:

$$$\rho -{\rho }_{0}=-{\rho }_{0}\beta \left(T-{T}_{0}\right)$$$ (18)

By incorporating Equations 16-18, one obtains:

$$$\frac{\rho -{\rho }_{0}}{{\Delta }\rho }={T}^{\ast }$$$ (19)

Furthermore, the non-dimensional Darcy flux is given as:

$$${\boldsymbol{q}}^{\ast }=\frac{\boldsymbol{q}}{{q}_{\mathrm{c}}}=\frac{{q}_{\mathrm{r}}}{{q}_{\mathrm{c}}}{\boldsymbol{n}}_{\mathrm{r}}+\frac{{q}_{\mathrm{z}}}{{q}_{\mathrm{c}}}{\boldsymbol{n}}_{\mathrm{z}}$$$ (20)

where q_r and q_z are the components of Darcy flux in direction of r and z, n_r and n_z are the unit vectors in positive r and z directions, and q_c is a characteristic Darcy flux calculated as:

$$${q}_{c}=\frac{{Q}_{\text{inj}}}{2\pi {H}^{2}}$$$ (21)

Then, we substitute the above-defined non-dimensional variables into the governing equations. By substituting Equations 13 and 20 into Equation 1, one obtains:

$$$\frac{1}{{r}^{\ast }}\frac{\partial \left({r}^{\ast }{q}_{\mathrm{r}}/{q}_{\mathrm{c}}\right)}{\partial {r}^{\ast }}+\frac{\partial \left({q}_{\mathrm{z}}/{q}_{\mathrm{c}}\right)}{\partial {z}^{\ast }}=0$$$ (22)

By substituting Equations 13-19 and 20 into Equation 2, one obtains:

$$${\boldsymbol{q}}^{\ast }=-\frac{K{\increment}\rho g}{{\mu }_{\mathrm{w}}{q}_{\mathrm{c}}}\left(\frac{\partial {p}^{\ast }}{\partial {r}^{\ast }}{\boldsymbol{n}}_{\mathrm{r}}+\frac{\partial {p}^{\ast }}{\partial {z}^{\ast }}{\boldsymbol{n}}_{\mathrm{z}}+{T}^{\ast }{\boldsymbol{n}}_{\mathrm{z}}\right)$$$ (23)

By substituting Equations 13-15 and 20 into Equation 3, one obtains:

$$$\frac{\partial {T}^{\ast }}{\partial {t}^{\ast }}+\frac{{\rho }_{\mathrm{w}}{C}_{\mathrm{p},\mathrm{w}}}{{\rho }_{\mathrm{a}}{C}_{\mathrm{p},\mathrm{a}}}\frac{H{q}_{\mathrm{c}}}{{D}_{\mathrm{a}}}{\boldsymbol{q}}^{\ast }\left(\frac{\partial {T}^{\ast }}{\partial {r}^{\ast }}{\boldsymbol{n}}_{\mathrm{r}}+\frac{\partial {T}^{\ast }}{\partial {z}^{\ast }}{\boldsymbol{n}}_{\mathrm{z}}\right)-\frac{1}{{r}^{\ast }}\frac{\partial }{\partial {r}^{\ast }}\left({r}^{\ast }\frac{\partial {T}^{\ast }}{\partial {r}^{\ast }}\right)-\frac{{\partial }^{2}{T}^{\ast }}{\partial {{z}^{\ast }}^{2}}=0$$$ (24)

By substituting Equations 13 and 14 into Equation 13-14, one obtains:

$$$\frac{\partial {T}^{\ast }}{\partial {t}^{\ast }}-\frac{{D}_{\mathrm{c}}}{{D}_{\mathrm{a}}}\frac{1}{{r}^{\ast }}\frac{\partial }{\partial {r}^{\ast }}\left({r}^{\ast }\frac{\partial {T}^{\ast }}{\partial {r}^{\ast }}\right)-\frac{{D}_{\mathrm{c}}}{{D}_{\mathrm{a}}}\frac{{\partial }^{2}{T}^{\ast }}{\partial {{z}^{\ast }}^{2}}=0$$$ (25)

Additionally, we introduce two dimensionless numbers (Peclet number and Rayleigh number), a dimensionless parameter (θ) indicating the ratio of the volumetric heat capacity of water to that of the aquifer, and a dimensionless parameter (γ) indicating the ratio of the thermal diffusivity of the caprock to that of the aquifer:

$$$Pe=\frac{H{q}_{\mathrm{c}}}{{D}_{\mathrm{a}}},Ra=\frac{K{\increment}\rho gH}{{\mu }_{\mathrm{w}}{D}_{\mathrm{a}}},\theta =\frac{{\rho }_{\mathrm{w}}{C}_{\mathrm{p},\mathrm{w}}}{{\rho }_{\mathrm{a}}{C}_{\mathrm{p},\mathrm{a}}},\gamma =\frac{{D}_{\mathrm{c}}}{{D}_{\mathrm{a}}}$$$ (26)

Besides, the operators (e.g., gradient, divergence, Laplace operator) may be given with the superscript * when the operators are acting on dimensionless independent variables, for example:

$$${{\nabla }^{\ast }T}^{\ast }=\frac{\partial {T}^{\ast }}{\partial {r}^{\ast }}{\boldsymbol{n}}_{\mathrm{r}}+\frac{\partial {T}^{\ast }}{\partial {z}^{\ast }}{\boldsymbol{n}}_{\mathrm{z}}\,\text{and}\,{{\nabla }^{\ast }}^{2}{T}^{\ast }=\frac{1}{{r}^{\ast }}\frac{\partial }{\partial {r}^{\ast }}\left({r}^{\ast }\frac{\partial {T}^{\ast }}{\partial {r}^{\ast }}\right)+\frac{{\partial }^{2}{T}^{\ast }}{\partial {{z}^{\ast }}^{2}}$$$ (27)

Finally, by substituting Equation 26 into Equations 22-24, applying the dimensionless operators, one obtains the dimensionless form of the governing equations for domain of aquifer (Ω₁):

$$${\nabla }^{\ast }\cdot {\boldsymbol{q}}^{\ast }=0$$$ (28)

$$${\boldsymbol{q}}^{\ast }=-\frac{Ra}{Pe}\left({{\nabla }^{\ast }p}^{\ast }+{T}^{\ast }{\boldsymbol{n}}_{\mathrm{z}}\right)$$$ (29)

$$$\frac{\partial {T}^{\ast }}{\partial {t}^{\ast }}+\theta Pe{\boldsymbol{q}}^{\ast }{\cdot \nabla }^{\ast }{T}^{\ast }-{{\nabla }^{\ast }}^{2}{T}^{\ast }=0$$$ (30)

and by substituting Equation 26 into Equation 25, one obtains the dimensionless form of the governing equations for the caprock and bedrock domains (Ω₂ and Ω₃):

$$$\frac{\partial {T}^{\ast }}{\partial {t}^{\ast }}-\gamma {{\nabla }^{\ast }}^{2}{T}^{\ast }=0$$$ (31)

Similarly, by substituting Equations 13-20 and 21 into Equations 5-12, one obtains the dimensionless boundary conditions:

$$${\nabla }^{\ast }{T}^{\ast }\cdot \boldsymbol{n}=0\quad {{\Gamma }}_{4}\cup {{\Gamma }}_{5}$$$ (32)

$$${T}^{\ast }=1\quad {{\Gamma }}_{1}$$$ (33)

$$${\boldsymbol{q}}^{\ast }\cdot \boldsymbol{n}=0\quad {{\Gamma }}_{2}\cup {{\Gamma }}_{3}$$$ (34)

$$${p}^{\ast }=\frac{{p}_{0}}{{\Delta }\rho gH}\quad {{\Gamma }}_{4}$$$ (35)

$$${\boldsymbol{q}}^{\ast }\cdot \boldsymbol{n}=\frac{H}{{r}_{\mathrm{b}}}\quad {{\Gamma }}_{1}$$$ (36)

At the interfaces between aquifer and caprock/bedrock:

$$${{T}^{\ast }\vert }_{{z}^{\ast }+}={{T}^{\ast }\vert }_{{z}^{\ast }-}\quad {{\Gamma }}_{2}\cup {{\Gamma }}_{3}$$$ (37)

$$${{\lambda }_{\mathrm{r}}\frac{\partial {\mathrm{{T}}}^{\ast }}{\partial {z}^{\ast }}\vert }_{{z}^{\ast }+}={{\lambda }_{\mathrm{a}}\frac{\partial {\mathrm{{T}}}^{\ast }}{\partial {z}^{\ast }}\vert }_{{z}^{\ast }-}\quad {{\Gamma }}_{2}$$$ (38)

$$${{\lambda }_{\mathrm{a}}\frac{\partial {\mathrm{{T}}}^{\ast }}{\partial {z}^{\ast }}\vert }_{{z}^{\ast }+}={{\lambda }_{\mathrm{r}}\frac{\partial {\mathrm{{T}}}^{\ast }}{\partial {z}^{\ast }}\vert }_{{z}^{\ast }-}\quad {{\Gamma }}_{3}$$$ (39)

Based on Equations 28-31, the miscible displacement and heat transport processes for the HT-ATES system are mainly dependent on three dimensionless groups, that is, the ratio between the Rayleigh number and the Peclet number, the product of the parameter θ and the Peclet number, and the dimensionless parameter γ.

According to the definitions of Ra and Pe (Equation 26), both Ra and Pe are in fact Peclet numbers. The Pe relates to the forced flow, and Ra relates to the buoyant flow. They can be transferred into each other by the ratio of forced velocity versus buoyant velocity. The dimensionless group Ra/Pe represents the ratio between the natural convection (driven by buoyancy) and the forced convection (driven by the pumping at the boundary). Besides, the dimensionless group θPe represents the ratio between the advective transport and diffusive transport of the heat. The material-related dimensionless parameter γ measures the ability of heat lost into the caprock and the bedrock through conduction.

The selection of suitable sites and operational parameters is essential for the optimization of recovery efficiency of HT-ATES. The forced flow is a design parameter based on a decision by the operator. The buoyant flow is a result of the aquifer properties and fluid properties. For a given aquifer, the properties are given, and the only man-made influence is via the forced flow, that is, adjusting the injection rate. On the other hand, if different sites are available for a given project and desired injection rate, the estimation of recovery efficiency may be an elegant (since cheap) screening criterion. In this case, the relationships between thermal recovery efficiency and the dimensionless groups and the possibilities to estimate recovery efficiency using these dimensionless groups will be investigated in detail using the numerical simulation in the next section.

3 Numerical Model

The numerical simulations with the sensitivity analysis on the key dimensionless groups are implemented to investigate the displacement regimes and the corresponding thermal recovery efficiency for the HT-ATES system. The implementation of the model is explained in Section 3.1. The model is validated in Section 3.2. Details of the model setup for the estimation of recovery efficiency (e.g., simulation domain setup, boundary conditions, discretization, material properties, etc.) are elaborated in Section 3.3. An additional optimization method using only the upper part of the screen for production processes is discussed in Section 3.4. The flowchart for the numerical simulation procedure is shown in Figure 3.

Figure 3

Flowchart for the numerical simulation procedure.

3.1 Numerical Implementation

The governing equation system is implemented in the commercial software COMSOL Multiphysics™ version 6.1, with an intuitive GUI that includes pre- and post-processing tools. COMSOL is widely applied in scientific research and engineering applications for solving coupled physical processes, with flexibility similar to academic codes in defining parameters, variables, and governing partial differential equations (Gao et al., 2021). The model is implemented in the COMSOL modules of flow and heat transport in porous media. COMSOL applies the finite element method (FEM) for spatial discretization and the backward Euler method for temporal discretization. The computations were performed on a single CPU with eight cores, operating at 4.3 GHz and 128 GB RAM.

3.2 Model Validation

The model is validated against the experimental study by Saeid et al. (2014). The experimental setup is simulated in a two-dimensional domain, as shown in Figure 4a. The domain consists of a Baskarp sand layer, measuring 0.337 × 0.15 m, and two impermeable clay layers, measuring 0.337 × 0.05 m. The parameters about the properties of the sand and the clay are given in Table 1. The hot water (T = 60°C) is injected from the left boundary by a constant head gradient (∆z_h = 0.5 m) to displace the water initially inside the sand layer with an initial temperature of T = 20°C, and the right boundary is the outlet. The temperature measured continuously at the location x = 0.15 m, z = 0.12 m during the injection process is used for the comparison between the simulation and the experimental results. The example of temperature distribution at t = 2,000 s is shown in Figure 4b. The temperature at the location x = 0.15 m, z = 0.12 is plotted in Figure 4c. The simulation results fit the experimental data well, which proves the ability of the numerical model to predict the displacement and heat transport processes.

Figure 4

Model validation: subfigure (a) shows the simulation setup; subfigure (b) shows an example of temperature distribution at t = 2,000 s; and subfigure (c) plots the temperature with respect to time at the location x = 0.15 m, z = 0.12, where the dashed line shows the experimental result from Saeid et al. (2014), and the simulation result using the model of this study is given in circles.

Table 1. Parameters for the Model Validation and the Numerical Simulation

Parameter name	Unit	Symbol	Object	Value
Model validation
Porosity^a	–	ϕ	Baskarp sand	0.36
			Clay	0.05
Permeability^a	(m^2)	K	Baskarp sand	5.22 × 10^−12
Thermal conductivity^a	(Wm^−1K^−1)	λ	Baskarp sand	2.5
			Clay	2
Density^a	(kg/m^3)	ρ	Baskarp sand	2,650
			Clay	1,750
Heat capacity^a	(Jkg^−1K^−1)	Cp	Baskarp sand	830
			Clay	2,000
Initial temperature^a	(K)	T0		293
Injection temperature^a	(K)	Tinj		333
Site Burgwedel
Porosity^b	–	ϕ	Aquifer	0.16
			Caprock/bedrock	0.01
Permeability^b	(m^2)	K	Aquifer	1 × 10^−13
Thermal conductivity^b	(Wm^−1K^−1)	λr	Rock (solid)	2.8
Density^b	(kg/m^3)	ρr	Rock (solid)	2,650
Heat capacity^b	(Jkg^−1K^−1)	Cp,r	Rock (solid)	790
Initial temperature^c	(K)	T0	Aquifer	343
Injection temperature^d	(K)	Tinj		363
Injection rated^d	(L/s)	Qinj		40
Thermal radius^d	(m)	rth		83

• ^a Parameters for model validation obtained from the experimental study by Saeid et al. (2014).
• ^b Data measured from core samples from the site Burgwedel in the “GeoTES” project.
• ^c Data measured from boreholes in the site Burgwedel in the “GeoTES” project.
• ^d Operational parameters designed in the “GeoTES” project.

3.3 Model Setups for the Estimation of Recovery Efficiency

The aquifer is assumed at a depth of 1,300 m, as shown in Figure 2a. The simulation domain measures 400 × 200 m. The caprock and the bedrock are both 80 m thick, and the aquifer layer in between is 40 m thick. The properties of the materials for the simulations are listed in Table 1. The properties of the porous media are obtained from the geological survey of the aquifer in the Burgwedel region, Germany, where an HT-ATES system is planned (“GeoTES” project). The well is assumed to have a radius of 1 m, and thus, the simulation domain ranges between r = 1 and r = 401 m. The radius of the whole simulation domain is large enough so that any boundary effects on the flow and heat transport are avoided. Also, the thickness of the caprock/bedrock is large enough so that the top or bottom boundaries do not limit the diffusive transport of the heat during the simulations.

3.3.1 Boundary Conditions

The continuity of temperature and heat transport at the interfaces (Γ₂ and Γ₃) between the aquifer and the caprock/bedrock is ensured by COMSOL. The other boundary conditions are defined according to Equations 5-9. The initial pressure in the domain is calculated using the hydrostatic pressure p₀−zρ₀g, where p₀ = 101,325 pa. The other parameters have been given in Table 1. The sweep of the parameters will be discussed in Section 3.3.2. It is assumed that there is no gap between the production and injection periods so that the injection and production processes follow each other. During the production period, boundary Γ₁ applies a negative value of Q_inj, indicating the outflow.

3.3.2 Sweep of Parameters

Simulations with a sweep of parameters are implemented to investigate the recovery efficiency at different combinations of the dimensionless groups. With the given definitions in Equation 26, the dimensionless groups depend on numerous parameters. Recovery efficiency is reported to be more sensitive to hydraulic parameters than thermal parameters for HT-ATES (Sheldon et al., 2021). In this study, we mainly sweep aquifer permeabilities, pumping rates, and caprock/bedrock thermal conductivities. The dimensionless parameter Ra/Pe is varied by changing aquifer permeability between 1 × 10⁻¹⁰, 3.16 × 10⁻¹¹, 1 × 10⁻¹¹, 3.16 × 10⁻¹², 1 × 10⁻¹², 3.16 × 10⁻¹³, and 1 × 10⁻¹³ m². This range covers permeabilities of porous media formed by unconsolidated sand/silt and consolidated sedimentary rocks, such as sandstones and limestones. Then, the dimensionless parameter θPe is varied by changing the flow rate between 10, 20, 40, and 80 L/s to obtain different degrees of intensity of the forced convection. Two additional flow rates at 4 and 180 L/s are simulated for the first cycle. In practice, the flow rates should be large enough for the HT-ATES to be economically efficient but smaller than the maximum flow rate that the system can provide. The range of flow rate given here is reasonable according to the specification defined for the site Burgwedel (“GeoTES” project). Note that the flow rate changes will also lead to Ra/Pe and θPe changes. Furthermore, the material-related dimensionless parameter γ is varied by changing the thermal conductivity of the caprock/bedrock between 1.8, 2.8, and 3.8 Wm⁻¹K⁻¹ since the thermal conductivity of the rocks does not vary significantly in natural environments.

Furthermore, the duration and the pumping rates during storage and production processes are equal. In this case, thermal recovery efficiency (R) can be calculated as:

$$$R=\frac{\int \nolimits_{0}^{{t}_{p}}\left({T}_{p}-{T}_{0}\right)}{{t}_{p}{\Delta }T}$$$ (40)

where t_p is the duration for hot water storage or production, and T_p is the temperature of the outflow water during the production. T_p can be obtained as the average temperature at the well (Γ₁) during the production. The time period is linearly proportional to the storage volume at a given pumping rate. In this case, recovery efficiency also depends on the period of operation or the storage volume (Doughty et al., 1982). The storage volume can be expressed in terms of an aspect ratio (AR), which is calculated by the thermal radius divided by the aquifer thickness:

$$$AR=\frac{{r}_{\text{th}}}{H}$$$ (41)

where rth is the thermal radius, calculated as:

$$${r}_{\text{th}}=\sqrt{\frac{{Q}_{\text{inj}}{t}_{\text{inj}}\theta }{H\pi }}$$$ (42)

where t_inj is the time of the injection period. Simulations are run with the AR varied between 0.346, 1.038, and 2.076, corresponding to the total storage volume calculated at injection time of 30, 90, and 180 d, respectively, at the injection rate Q_inj = 40 L/s, as given in Table 1.

Another parameter expected to affect recovery efficiency is the number of cycles (Doughty et al., 1982). The residual heat in the aquifer from one cycle affects the initial condition for the subsequent cycle and the recovery efficiency in the cycles afterward. Thus, the simulations are run for the number of cycles (N) up to 10 to evaluate the effects on the number of cycles.

In total, 1,134 simulations were carried out with different combinations of the four dimensionless parameters (groups): Ra/Pe, θPe, AR, γ, and the number of cycles N. The lists of parameters can be found Table S1 in Supporting Information S1.

3.3.3 Temperature Dependent Material Properties

To more accurately predict the recovery efficiency, the linear relationship between density and temperature of water Equation 18 is replaced by a temperature-dependent function in the numerical simulations. Also, the viscosity, heat capacity, and thermal conductivity of water are given as temperature-dependent functions. The fluid density, viscosity, heat capacity, and thermal conductivity are defined according to Zhou et al. (2022) as:

$$${\rho }_{w},{\mu }_{w},{C}_{p,w}{,\lambda }_{w}={\kappa }_{1}^{j}+{\kappa }_{2}^{j}T+{\kappa }_{3}^{j}{T}^{2}+{\kappa }_{4}^{j}{T}^{3}+{\kappa }_{5}^{j}{T}^{4}+{\kappa }_{6}^{j}{T}^{5}+{{\kappa }_{7}^{j}T}^{6}$$$ (43)

where $${\kappa }_{i}^{j}$$ are the parameters i for the fluid properties j. All parameters used in this study are given Table S2 in Supporting Information S1. Furthermore, the product of aquifer density and heat capacity (ρ_aC_p,a) is expressed as:

$$${\rho }_{a}{C}_{\mathrm{p},\mathrm{a}}=\phi {\rho }_{\mathrm{w}}{C}_{\mathrm{p},\mathrm{w}}+(1-\phi ){\rho }_{\mathrm{s}}{C}_{\mathrm{p},\mathrm{s}}$$$ (44)

where ρ_s and C_p,s are the density and the heat capacity of the rock and ϕ is the porosity. The caprock/bedrock has a low porosity of 0.01, as given in Table 1. Similarly, thermal conductivity (λ) of the bulk volume is expressed as:

$$${\lambda }_{a}=\phi {\lambda }_{\mathrm{w}}+(1-\phi ){\lambda }_{\mathrm{s}}$$$ (45)

where λ_w and λ_s are the thermal conductivity of the fluid and the rock, respectively.

3.3.4 Discretization

The computational grid is generated using COMSOL's mesh builder. The simulation domains of the caprock/bedrock (Ω₂ and Ω₃) are discretized into triangular elements with a lateral length of h = 2 m. The elements in the aquifer domain (Ω₁) are refined to h = 1 m to better resolve the advective heat transport in the aquifer. The advancing front algorithm is applied for the tessellation (Gao et al., 2023). A detail of the mesh is shown in Figure 2b, and the grid mesh for the whole system can be found Figure S3 in Supporting Information S1. The complete mesh consists of 92,564 internal elements and 1,448 boundary elements. The temporal discretization of each simulation depends on the injection rate and total storage volume. The temporal discretization is small enough to ensure the convergence of the simulation results.

3.4 Well Screen Optimization Studies

In addition to estimating recovery efficiency, the dimensionless parameter (Ra/Pe) may also be used as an index for optimizing the well screen length to mitigate the disadvantages of buoyant flow. Buscheck et al. (1983) showed that thermal recovery efficiency can be improved by using only the upper part of the screen for production for a specific ATES site with a strong buoyancy flow. In this study, we further investigate the effect of screen length (upper part) on thermal recovery efficiency at various intensities of buoyancy flow characterized by the dimensionless parameter Ra/Pe.

The setup of the numerical model is the same as that discussed in Section 3.3. During injection, the well screen is assumed to have the full well length (40 m) for effectively using aquifer space. While during production, it is assumed the screen only spans across the upper part of the well, with lengths of 5, 10, 15, 20, 25, 30, and 35 m (starting from the top of the aquifer z = −1,300 m). The screen length (ζ) is defined as the length of the screen divided by the length of the well, and in this case ζ ranging between 0.125 and 0.875. For the full screen ζ = 1. The parameters for simulation are according to the basic case given in Table 1, where the aquifer permeability is swapped between 1 × 10⁻¹⁰ and 1 × 10⁻¹³ m² for the cases at various intensities of buoyancy flow. Note that the injection and production rates are the same (at 40 L/s), and thus, the reduced screen length will lead to a faster flow rate at the well during production.

4 Results and Discussion

The displacement regimes during HT-ATES and the recovery efficiency at various Ra/Pe and θPe are first analyzed in Section 4.1. The correlation between recovery efficiency and the dimensionless groups at different aspect ratios, number of cycles, and γ are investigated in Section 4.2. Subsequently, correlations for estimating recovery efficiency based on the dimensionless parameters is proposed in Section 4.3. Finally, the method with optimal screen length to improve recovery efficiency is discussed in Section 4.4.

4.1 Displacement Regimes

The temperature isolines for the simulations of hot water injection at the first cycle and the aspect ratio of AR = 2.076 are plotted in Figure 5. The profiles are arranged according to the logarithm of the dimensionless groups θPe in columns and Ra/Pe in rows. It is observed that the sharpness of the thermal front depends on θPe. The thermal interface becomes significantly smeared with log(θPe) decreasing from 2.88 to 1.28. This indicates that more heat is transferred from the injected hot water to the cold water (initially in the aquifer) by heat conduction. Furthermore, it is also observed that the front morphology mainly depends on Ra/Pe. When log(Ra/Pe) is smaller than −1, the thermal front is vertical and mostly straight, which indicates that the forced convection is much stronger than the natural convection. The front starts to tilt when log(Ra/Pe) is larger than −1. The tilting angle increases with a larger log(Ra/Pe) and a stronger buoyant force. When log(Ra/Pe) is larger than 1, it shows a curved and elongated front. This is because, with increasing buoyancy, hot water is lifted fast (natural convection) and spreads mainly along the top boundary of the aquifer.

Figure 5

Plots of the temperature isolines in the aquifer for the simulations at the end of the first injection and the aspect ratio of AR = 2.076.

The isoline map of the recovery efficiency obtained from the simulations at the first cycle and the aspect ratio of AR = 2.076 is plotted in Figure 6a. It is observed that in the right bottom region with log(Ra/Pe) < −0.5 and log(θPe) > 2, recovery efficiency is higher than in the other regions, up to R = 0.934. This directly results from the low heat conduction and minor buoyancy effects. In contrast, recovery efficiency is significantly lower in the left top region, where the buoyancy and heat conduction are both intensive. Recovery efficiency is down to R = 0.17. Moreover, it is observed that in the top region with log(Ra/Pe) > 0.5, the recovery efficiency isolines are almost horizontal, parallel to the axis of log(θPe). This implies that in the region with strong buoyancy, differences in the intensity of the heat conduction have minor effects on thermal recovery efficiency. The reason for that will be discussed in Section 4.2.1. Besides, in the bottom region with log(Ra/Pe) < −0.5, it is observed that the isolines are mostly vertical, parallel to the axis of log(Ra/Pe). This is due to the neglectable effects of buoyancy, and recovery efficiency is mainly affected by the intensity of the heat conduction. In this case, the displacement processes can be classified into three regimes: a buoyancy-dominated regime, a heat-conduction-dominated regime, and a transition regime. In the transition regime, both buoyancy and heat conductions affect thermal recovery efficiency, leading to inclined isolines. For a given point in the transition regime, the slope of the inclined isolines indicates the ratio of the impact by changing log(θPe) to that by changing log(Ra/Pe) on recovery efficiency. The three regimes are distinguished with the red dashed lines in Figure 6b, where the simulated cases are given as black dots. The two red dashed lines to distinguish the regimes write:

$$$\log \left(\frac{Ra}{Pe}\right)=-0.6875\,\log (\theta Pe)+1.577$$$ (46)

and

$$$\log \left(\frac{Ra}{Pe}\right)=-0.375\,\log (\theta Pe)-0.0215$$$ (47)

Figure 6

Displacement regimes during HT-ATES: subfigure (a) shows the isoline map of the recovery efficiency obtained from the simulations at the first cycle and the aspect ratio of AR = 2.076; subfigure (b) shows the identified three displacement regimes, distinguished the red dashed lines, and the simulated cases are given as black dots.

Please note that the regime distinguishment lines are built not by a normative methodology but by the features of the (recovery efficiency) isolines. The line of Equation 46 outlines the ranges of Ra/Pe and θPe when isolines are mostly horizontal, and the line of Equation 47 outlines the ranges of Ra/Pe and θPe when isolines are mostly vertical. Furthermore, the isoline maps of the recovery efficiency for aspect ratio of AR = 0.346 and AR = 1.038 are also obtained Figure S1 in Supporting Information S1. The major features of the isoline maps at aspect ratios of AR = 0.346 and AR = 1.038 are similar to that in Figure 6a aspect ratio of AR = 2.076. The regime distinguishment lines Equations 46 and 47 are mostly unaffected by the aspect ratio differences.

4.2 Correlations Between Dimensionless Parameters and Recovery Efficiency

4.2.1 Buoyancy Dominated Regime

An example of the transport process in the buoyancy-dominated regime (log(Ra/Pe) = 0.9, log(θPe) = 2.28) at the first cycle is plotted in Figure 7. During the injection process, a recirculation zone is formed close to the bottom of the well, as shown in Figure 7a with the streamlines. The recirculation zone is formed due to the viscous force between the lifting hot water and the stationary cold water. The contact point of the hot water plume with the bottom boundary of the aquifer no longer moves as long as the recirculation zone is formed, as shown in Figure 7a at 90 d and 180 d. The hot water plume develops only along the top boundary of the aquifer. Moreover, during the abstraction process, the preceding recirculation zone disappears, and the cold water enters the bottom part of the well soon after the beginning of abstraction, as shown in Figure 7b. This is the main reason for the early decrease in the production temperature and low recovery efficiency for the buoyancy-dominated regime. Besides, a new recirculation is formed at the tip of the hot water plume due to the viscous force between the sinking cold water and the stationary hot water plume. This hot water plume becomes sightly thinner when the abstraction time is longer, as shown in Figure 7b at 90 and 180 d. The main directions of heat conduction are shown with black arrows in Figure 7b at 180 d. The heat conduction mainly happens in two directions, that is, upward to the caprock and downward to flowing cold water beneath the plume. As the residual hot water plume cannot be recovered, the heat transfer from the residual plume to the caprock by conduction has minimal effects on the total recovered heat. Only the downward conducted heat may affect recovery efficiency. The heat lost due to thermal stratification is much larger than the heat transferred by the downward heat conduction. Thus, the intensity of the natural convection (Ra/Pe), which determines the morphology of the plume, plays a more critical role in thermal recovery efficiency of this regime.

Figure 7

Plot of the injection (a) and the abstraction (b) processes for an example of simulation belonging to the buoyancy-dominated regime (log(Ra/Pe) = 0.9, log(θPe) = 2.28) at the first cycle. The temperature isolines are given in the left figures, and the velocity distributions are given in the right figures (the streamlines in gray show the flow direction) for each time (or thermal radius).

The recovery efficiency obtained in the simulations for the buoyancy-dominated regime is plotted with respect to log(Ra/Pe) in Figure 8a. A strong linear correlation is observed between log(Ra/Pe) and recovery efficiency at each aspect ratio. Recovery efficiency is lower with a larger log(Ra/Pe), meaning a stronger buoyancy flow. The examples of the residual hot water plume at log(Ra/Pe) equal to −0.1, 0.4, and 0.9 are compared in Figure 8b. The residual hot water plume is observed to be longer at higher log(Ra/Pe), leading to a larger amount of heat being lost. Furthermore, it is also shown in Figure 8a that at the same log(Ra/Pe), the recovery efficiency of AR = 0.346 is slightly higher than AR = 1.038 and AR = 2.076. This is because the percentage of the residual plume volume to the total injected volume is larger at a larger AR. The radial flow velocity is reduced with increasing distance to the well. Thus, in the region far from the well, the ratio between the natural convection and the forced convection is larger (a higher Ra/Pe), and the effects of buoyancy flow become more significant. Thus, the overall effect of buoyancy on recovery efficiency is larger with a higher AR. On the other hand, at high aspect ratios (e.g., AR = 1.038 and AR = 2.076), the difference in the corresponding recovery efficiency is not apparent.

Figure 8

Recovery efficiency in buoyancy-dominated regime: subfigure (a) shows the recovery efficiency plotted with respect to the dimensionless group log(Ra/Pe) at three different aspect ratios; subfigure (b) demonstrates the temperature isolines at the end of the abstraction for simulations at three different log(Ra/Pe) at aspect ratio AR = 2.076.

4.2.2 Conduction Dominated Regime

The correlation between thermal recovery efficiency and log(θPe) for the conduction-dominated regime is investigated. Recovery efficiency is correlated to log(θPe) with a logarithmic function. Recovery efficiency is plotted with respect to ln[log(θPe)] in Figure 9. At each AR, recovery efficiency increases with higher ln[log(θPe)]. This is a direct result of lower heat loss when the intensity of heat conduction is lower. At the same log(θPe), recovery efficiency is generally larger with smaller AR. With a given pumping rate, the contact time and the contact area between the injected hot water and the cold water in the aquifer are both larger at a higher AR so that the amount of heat lost by conduction is larger. It is also observed that the difference in recovery efficiency induced by different AR is larger at a lower log(θPe) (diffusive transport of heat is relatively stronger).

Figure 9

Recovery efficiency plotted with respect to the dimensionless group ln[log(θPe)] at three different aspect ratios.

4.2.3 Effects of Number of Cycles

The temperature isolines at the end of the injection processes of the fifth circle and the tenth circle for the three different regimes are shown in Figure 10a. It is observed that for all three regimes, after multiple cycles of operation, a thick temperature transition zone is built up in both the aquifer and the caprock/bedrock between the storage space and the vicinity. This zone mitigates heat transfer by conduction since the temperature gradient is reduced between the storage space and the vicinity. For the conduction-dominated regime, both the caprock and the bedrock are heated during multiple cycles. During the late stages of production, the temperature of the caprock and the bedrock (near the aquifer boundaries) slightly decreased due to the heat transfer from the caprock/bedrock to the cold water being recovered. This process further increases recovery efficiency. For the buoyancy-dominated regime, it is observed that the caprock is being heated continuously due to the residual hot water plume, while the bedrock remains cold after multiple cycles. By comparing the temperature isolines at the end of the injections of the fifth circle and the tenth circle for all three regimes, it is observed that the space for hot water storage and the location of the residual plume are almost the same for each regime. In contrast, the temperature transition zones become larger. The changes in the size of the temperature transition zone are reduced with a larger number of cycles. The results indicate that even in the buoyancy-dominated regime, there is no apparent migration of the residual hot water plume during multiple cycles. Furthermore, the recovery efficiency is plotted versus the logarithm of the cycle number in Figure 10b. For each regime, there is a strong correlation between recovery efficiency and cycle number with a logarithmic function.

Figure 10

Effects of the number of cycles: subfigure (a) shows examples of simulations (log(θPe) = 2.28) for the three regimes (in columns) at the end of the abstraction process of the ninth cycle (first row) and at the end of the injection and the abstraction process of the tenth cycle (second and third rows); subfigure (b) plots the recovery efficiency with respect to ln(N) for the three examples for different displacement regimes.

4.2.4 Effects of the Dimensionless Parameter γ

The recovery efficiency with respect to the dimensionless parameter γ is plotted in Figure 11 for examples of simulations at log(θPe) = 2.28. For both the conduction-dominated regime (Figure 11a) and the buoyancy-dominated regime (Figure 11b). It is observed that recovery efficiency is decreased with larger γ, which is a direct result of a more considerable heat loss in the caprock/bedrock with larger γ. It is found that recovery efficiency is linearly related to γ at various combinations of cycle number and aspect ratio for both regimes. The effect of γ on recovery efficiency is slightly more significant for the conduction-dominated regime than that of the buoyancy-dominated regime. Overall, the changes in γ (between 0.77 and 1.6) induce a relatively small change in recovery efficiency (lower than 2%), comparing other factors (e.g., displacement regimes, number of cycles, aspect ratio, etc.).

Figure 11

Plot of thermal recovery efficiency with respect to the dimensionless parameter γ, for examples of simulations at log(θPe) = 2.28 for (a) conduction dominated regime at log(Ra/Pe) = −1.1, and (b) buoyancy dominated regime at log(Ra/Pe) = 0.9. In each subfigure, three data series are shown with different combinations of cycle number (N) and aspect ratio (AR).

4.3 Estimation of Recovery Efficiency Using Dimensionless Parameters

In the following, a multivariable regression analysis is carried out for the four dimensionless parameters and the number of cycles and the recovery efficiency for each regime using the data analysis software IBM SPSS Statistics™. The regression function for the buoyancy-dominated regime writes:

$$${R}_{\text{buoyancy}}=0.777-0.007\gamma +0.07\,\mathrm{ln}(N)-0.376\,\log \left(\frac{Ra}{Pe}\right)-0.005AR$$$ (48)

The regression function for the conduction-dominated regime writes:

$$${R}_{\text{conduction}}=0.778-0.013\gamma +0.022\,\mathrm{ln}(N)+0.197\,\mathrm{ln}[\log (\theta Pe)]-0.02AR$$$ (49)

The regression function for the transition regime writes:

$$${R}_{\text{transition}}=0.747-0.006\gamma +0.032\,\mathrm{ln}\,(N)+0.113\,\mathrm{ln}[\log (\theta Pe)]-0.101\,\log \left(\frac{Ra}{Pe}\right)-0.024AR$$$ (50)

Using the regime distinguishment lines provided by Equations 46 and 47 and the correlation functions Equations 48-50, we estimate the recovery efficiency for all the simulated cases. The error of the recovery efficiency estimated by the correlation functions (R_est) is calculated by:

$$$\text{error}=\left(\vert R-{R}_{\text{est}}\vert \right)100\%$$$ (51)

Figure 12 shows examples of the errors at AR = 2.076, γ = 1.2, N = 5, and N = 10. The errors are slightly larger for the buoyancy-dominated regime, especially for the region at the top left, where the recovery efficiency changes significantly due to thermal stratification. Besides, the errors of the recovery efficiency estimated by the correlation functions for all simulation cases are listed Table S1 in Supporting Information S1. It is shown that the correlation functions well predict the recovery efficiency for all three regimes. The average error is 0.72%, 1.21%, and 1.81% for all the simulated cases in the conduction-dominated, transition, and buoyancy-dominated regimes, respectively.

Figure 12

Plots of the errors of the estimated recovery efficiency at AR = 2.076, γ = 1.2, N = 5 (left) and N = 10 (right).

It is worth noting that the regression functions for estimating thermal recovery efficiency are obtained from numerical simulations with the assumption of a homogeneous and isotropic aquifer with no fractures. Thus, the application of the regression functions is limited to the HT-ATES systems with a similar condition, as discussed in Section 2.1. In the cases of heterogeneous, anisotropic, and fractured aquifer systems, the flow patterns and the resulting recovery efficiency may be affected by these factors significantly. The variations in recovery efficiency and the uncertainties caused by these factors (i.e., heterogeneity, anisotropy, fractures) for predicting recovery efficiency of HT-ATES systems still need to be investigated in future research, and these are not considered in this study.

4.4 Optimization of Well Screen Length

With the given operational parameters and aquifer properties, a measure to improve recovery efficiency and mitigate the effects of buoyancy flow is to use only the upper part of the screen for production (Buscheck et al., 1983). The design of the length of the screen applying the dimensionless parameter (Ra/Pe) is demonstrated in this section. The example of simulation with a length of the screen ζ = 0.125 (upper one-eighth of the well) for a buoyancy-dominated case is shown in Figure 13. It is observed that the flow velocity near the well shows significant differences for the production process with the screen at the upper one-eighth of the well, compared to the full screen, as shown in Figure 7. The streamlines (indicating the flow direction) gather to the top of the aquifer for the region near the well Figure 13 (bottom), and the flow velocity is higher (in red) for the upper part of the aquifer than the lower part. The streamlines are not much affected for the regions far away from the well (r > 20 m). In this case, the residual water plume is less for the region near the well Figure 13 (at t = 180 d). The residual water plume remains in the region far from the well. This indicates that changing the well screen length mainly improves the recovery efficiency near the well, and thus, this method is more effective when the thermal radius is small.

Figure 13

Plot of the temperature isolines (top) and flow velocity distribution (bottom) for the production process of a buoyancy-dominated case at log(Ra/Pe) = 0.9 and log(θPe) = 2.28, at 10, 50, and 180d.

Furthermore, the recovery efficiency for seven different cases at various screen lengths ranging between ζ = 0.125 and ζ = 1 are plotted in Figures 14a, 14b, and 14c for the cases belonging to the buoyancy-dominated regime, the transition regime, and the conduction dominated regime, respectively. The recovery efficiency increased with a smaller screen length for the buoyancy-dominated regime (Figure 14a). The improvement of recovery efficiency is more significant when the buoyancy flow is stronger (higher Ra/Pe). The maximum recovery efficiency is observed at the smallest screen length. The recovery efficiency is increased by 8% at screen length ζ = 0.125 for the case with the strongest buoyancy flow at log(Ra/Pe) = 0.9. For the transition regime (Figure 14b), it is observed that the recovery efficiency also increases with lower screen length at log(Ra/Pe) = −0.6. In contrast, the recovery efficiency changes nonmonotonically with respect to screen length at log(Ra/Pe) = −0.1. The maximum recovery efficiency (at log(Ra/Pe) = −0.1) is observed at ζ = 0.25. For the conduction-dominated regime (Figure 14c), it is observed that the recovery efficiency changes nonmonotonically with respect to screen length. When log(Ra/Pe) < −1, the lower screen length can mostly lead to a lower recovery efficiency. This is because when the buoyancy flow (natural convection) is small compared to the forced convection, the front is almost vertical, and the production from only the upper screen will disturb this stable front near the well, leading to a lower recovery efficiency.

Figure 14

Plot of thermal recovery efficiency with respect to the screen length for (a) the buoyancy-dominated cases, (b) the transition cases, and (c) the conduction-dominated cases. Subfigure (d) plots the optimal screen length versus the dimensionless parameter log(Ra/Pe) in circles, and the dashed line shows the correlation function Equation 52.

Finally, the optimal screen length (with the maximum recovery efficiency) is plotted versus the dimensionless group log(Ra/Pe) in Figure 14d. It is observed that the optimal screen length is smaller when log(Ra/Pe) is larger. The correlation between optimal screen length and log(Ra/Pe) can be approximated with a parabolic function:

$$${\zeta }_{\text{opt}}=0.1{\left[\log \left(\frac{Ra}{Pe}\right)-1\right]}^{2}+0.08$$$ (52)

where ζ_opt is the optimal screen length for the production processes.

5 Summary and Conclusions

The governing equation for mass and heat transport during HT-ATES was nondimensionalized. Numerical simulations were carried out to study the displacement regimes and possible correlations between thermal recovery efficiency and the dimensionless parameters. In total, 1,134 simulations were accomplished with a sweep of parameters on aquifer permeability, pumping rate, caprock/bedrock thermal conductivity, storage volume, and number of cycles to obtain results at different combinations of the dimensionless parameters. The main findings of this study are summarized below:

• Based on the derived dimensionless form of the governing equations for mass and heat transport during HT-ATES (considering the buoyancy flow), we found the three key dimensionless groups, that is, Ra/Pe, θPe, and γ. Ra/Pe indicates the ratio between natural convection and forced convection. θPe indicates the ratio between the advective transport and diffusive transport of the heat. γ indicates the ratio between the thermal diffusivity of the caprock/bedrock and that of the aquifer.

• Depending on the thermal front morphology and the displacement characteristics, the displacement processes were distinguished into three regimes: the buoyancy-dominated regime, the conduction-dominated regime, and the transition regime. The regime distinguishment lines were provided with Equations 46 and 47.

• For the buoyancy-dominated regime, differences in the intensity of heat conduction have minimal effects on recovery efficiency, and recovery efficiency was found to be linearly correlated to log(Ra/Pe). For the conduction-dominated regime, the effect of buoyancy is neglectable, and recovery efficiency was found to correlate to log(θPe) with a logarithmic function. For the transition regime, both buoyancy and heat conduction significantly affect recovery efficiency.

• Recovery efficiency increases with decreasing aspect ratio, more cycles, and decreasing γ. It was observed that changes in the location of the stored hot water at the end of injection are small (for all three displacement regimes) for multiple cycles. In contrast, the temperature transition zone between the stored hot and cold water became larger with more cycles. Recovery efficiency correlated to the number of cycles with a logarithmic function.

• Correlation functions (Equations 48-50) for the three displacement regimes were provided to estimate recovery efficiency based on the four dimensionless parameters (i.e., Ra/Pe, θPe, γ, AR) and the number of cycles N. The correlation functions were found to estimate recovery efficiency at a satisfied accuracy.

• The feasibility of the method, that is, using only the upper part of the screen for production processes to improve recovery efficiency, is highly dependent on the intensity of the buoyancy flow and the length of the screen. The optimal screen length (for maximum recovery efficiency) is correlated to log(Ra/Pe) with a parabolic function (Equation 52).

This study showed the possibility of predicting the displacement regimes and estimating recovery efficiency of HT-ATES when the geological and operational parameters are known. These correlation functions and regimes distinguishment lines may play essential roles in designing HT-ATES systems, for example, selecting an ideal aquifer, optimizing the storage temperature, working period, pumping rates, etcetera. This study also showed the correlation function to estimate the optimal well screen length at various intensities of buoyancy flow, and this may be helpful for the improvement of recovery efficiency of HT-ATES systems. Future work is required to assess HT-ATES performance, considering the effects of heterogeneity, anisotropy, and fractures. Besides, based on the results, one may consider raising the forced flow rate to obtain a higher recovery efficiency. But it will in the meantime raises the cost in terms of the viscous forces and required pumping energy. Thus, future work is also required to implement the techno-economic assessments of various measures to mitigate the effects of buoyancy flow (e.g., raising the forced flow rate) for HT-ATES systems.

Nomenclature Symbols

AR

Aspect ratio

C_p

Heat capacity [Jkg⁻¹K⁻¹]

D

Thermal diffusivity [m²/s]

g

Gravity constant [m/s²]

H

Aquifer thickness [m]

h

Mesh size [m]

K

Permeability [m²]

N

Number of cycles

n

Unit vector

Pe

Peclet number

p

Pressure [pa]

Q

Flow rate [L/s]

q

Darcy flux [m/s]

Ra

Rayleigh number

R

Thermal recovery efficiency

r_b

Well radius [m]

rth

Thermal radius [m]

r

Coordinate [m]

T

Temperature [K]

t

Time [s]

z

Coordinate [m]

ζ

Screen length

Ω

Domain

Γ

Boundary

λ

Thermal conductivity [Wm⁻¹K⁻¹]

θ

Dimensionless parameter

β

Thermal expansion coefficient [K⁻¹]

γ

Dimensionless parameter

ϕ

Porosity

κ

Fluid property parameter

μ

Viscosity [pa·s]

ρ

Density [kg/m³].

Subscript

a

Aquifer (solid and water)

c

Caprock/bedrock

inj

Injection

opt

Optimal

p

Production

r

Component in r

s

Solid/rock

w

Water

z

Component in z

0

Initial condition.

Acknowledgments

We acknowledge funding by the Federal Ministry of Education and Research (Bundesministerium für Bildung und Forschung, BMBF), project “GeoTES”, funding reference 06G0917A.

Open Research

Data Availability Statement

The data on which this article is based are available in Gao (2024).