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Much attention is now focused on utilizing ground heat pumps for heating and cooling buildings, as well as water heating, refrigeration and other thermal tasks. Modeling such systems is important for understanding, designing and optimizing their performance and characteristics. Several heat transfer models exist for ground heat exchangers. In this review article, challenges of modelling heat transfer in vertical heat exchangers are described, some analytical and numerical models are reviewed and compared, recent related developments are described and the importance of modelling these systems is discussed from a variety of aspects, such as sustainability of geothermal systems or their potential impacts on the ecosystems nearby.
Measurements show that, below a certain depth in the ground, the temperature fluctuations observed near the surface of the ground diminish, and the temperature remains relatively constant (e.g., at about 6–42 °C in various states in the US) throughout the year [
Below a certain depth, therefore, the ground generally remains warmer than the outside air in winter and cooler in summer. The relatively cool ground may be used as a sink in summer to store the extracted heat from a conditioned space via a ground heat pump (GHP). In winter, the process may be reversed, and the heat pump can extract heat from the relatively warm ground and transport it into the conditioned space. Compared to a conventional air source heat pump (ASHP), which circulates outdoor air to exchange heat, a ground heat pump exchanges heat by circulating a fluid in the ground. The ground has a lower temperature than the outdoor air in the cooling mode and a higher temperature than the outdoor air in the heating mode. Consequently, the temperature lift across a GHP is lower than that of an air source heat pump for both heating and cooling. Thus, the efficiency of the heat pump, which depends directly on the temperature difference between the circulating fluid and the room, is enhanced for a GHP. Therefore, due to concern about greenhouse gas emissions and high energy prices, the placement of heat loops in the ground is an increasingly common practice for heating and cooling residential, commercial, institutional, recreational and industrial structures. Low temperature geothermal energy has the potential to contribute significantly to mitigating both of these problems.
A geothermal heating and cooling system consists of three main components: a ground heat pump, a ground heat exchanger (GHE) and a distribution system, such as air ducts. GHEs are commonly classified as open loop (groundwater heat pump (GWHP)) or closed loop (ground coupled heat pump (GCHP)), with a third category for those not belonging to either. In an open loop system, ground water from a waterbearing layer is pumped from an aquifer through one well, passed through the heat pump where heat is added to or extracted from a heat carrier and then discharged either onto the surface or to another well in the aquifer. Because the system water supply and discharge are not connected, the loop is “open” [
Various models have been reported for heat transfer in BHEs that are mostly used in design of BHEs, including sizing borehole depth and determining borehole numbers and analysis of
The main objective of modelling GHEs is to determine the temperature of the fluid running in the Utube that exchanges heat between the soil and the heat pump. Under certain operating conditions and building loads, the size and number of the GHE needed to deliver or extract heat to/from the ground is determined according to the acceptable range of the temperature variations of the running fluid. Modelling and simulation of the heat exchange function in GHEs can also be used to evaluate the temperature rise in the soil surrounding these systems and the migration of thermal plumes away from them. This knowledge will guide proper site characterization, system design, construction and operation so that these systems are sustainable and impact the environment as well as other neighbouring systems as little as reasonably possible.
Similar to most human activities, studies show the potential of geothermal heat exchangers for causing environmental impacts. While little research has been done regarding the impact of geothermal systems on the local environment, research on the movement of thermal plumes shows the potential for impact. Migration of thermal plumes away from these systems and changes in temperature from either closed or open loop systems or due to changes in ground water flow patterns from openloop systems may cause undesirable temperature rises in nearby temperaturesensitive ecosystems where small temperature differences are important. For example, temperature disturbances in the ground caused by the operation of geothermal systems may result in disruption to sensitive life stages of aquatic organisms. Similar environmental effects are observed for heat loop and waterline projects (rivers and lakes) [
What is unknown at this point is whether the environmental impacts of geothermal systems are acceptable considering the fact that they can reduce fossil fuel consumption and, therefore, lower greenhouse gas emissions and if geothermal systems can be developed in a manner that has reasonably small potential for impacting the environment.
The sustainability of geothermal heat pump systems at their design efficiency is now being questioned due to ‘thermal pollution’ from the system itself, adjacent systems or the urban environment. Studies from Manitoba, where the carbonate rock aquifer beneath Winnipeg has been exploited in thermal applications since 1965, indicate that in many cases these systems are not sustainable or not sustainable at the design efficiency [
In heating or cooling dominated climates, an annual energy imbalance is placed on the ground loop due to heating, cooling and hot water production. For example, Manitoba has a heating dominated climate and there are concerns regarding the longterm thermal performance of the ground loop. Longterm thermal performance of such ground loop systems with imbalanced energy input and outputs in the ground may result in large temperature rises in the region that the loop is installed. Thermal imbalances could cause significant issues with a heat pump’s longterm sustainable performance if not properly considered at the design phase [
Thermal disturbances in the soil associated with GHEs are likely to extend beyond property boundaries and affect adjacent properties. Therefore, with increasing interest in installing such systems in the ground and their potential dense population in coming years, procedures and regulations need to be implemented to prevent disputes between neighbours with potentially interacting systems and their possible negative effects on the performance of existing nearby systems. As stated by Ferguson [
Careful management of geothermal developments to ensure fair access to the subsurface for thermal applications is likely needed. This will require a greater understanding of subsurface heat flow and input from the scientific and technical communities. These concerns have not been well addressed in all cases. Research is needed to allow the investigation of system performance and environmental impact in an integrated manner, so that the best way of utilizing geothermal systems in an environmentally sensitive and sustainable manner can be determined.
The heat transfer modelling in GHEs is complicated since their study involves transient effects in a time range of months or even years. Because of the complexities of this problem and its long time scale, the heat transfer in GHEs is usually analyzed in two separated regions (
The transient borehole wall temperature is important for engineering applications and system simulation. It can be determined by modeling the region outside the borehole by various methods, such as the line source theory. Based on the borehole wall temperature, the fluid inlet and outlet temperatures can be evaluated by a heat transfer analysis inside the borehole. In other words, the regions inside and outside borehole are coupled by the temperature of borehole wall. The heat pump model can utilize the fluid inlet and outlet temperatures for the GHE, and, accordingly, the dynamic simulation and optimization design for a GCHP system can be implemented. This is the basic idea behind the development of the tworegion vertical GHE model. Based on how heat transfer from the circulating fluid to the surrounding soil is simulated, these methods can be divided into analytical and numerical. Semianalytical techniques have also been utilized to describe temperature distributions inside the boreholes as well as outside of them. In these methods, usually analytical solutions are combined with the numerical methods or analytical expressions requiring numerical integrations for evaluation of temperature rise and heat flows. Eskilson and Claesson [
Crosssection of a vertical ground heat exchanger (GHE). The fluid is ascending in one pipe and descending in the other.
In the analytical approaches, heat transfer inside the borehole wall,
The thermal analysis in the borehole seeks to define the inlet and outlet temperatures of the circulating fluid according to borehole wall temperature, its heat flow and the thermal resistance inside the borehole. The latter quantity is determined by thermal properties of the grouting material, the arrangement of flow channels and the convective heat transfer in the tubes. If the thermal resistance between the borehole wall and inner fluid is determined, the GHE fluid temperature can be calculated. Neglecting natural convection, moisture flow and freezing, the borehole thermal resistance can be calculated assuming steadystate heat conduction in the region between the circulating fluids and a cylinder around the borehole when the running time is greater than the critical time, that is
In all analytical models for inside of borehole, the axial heat flows in the grout and pipe walls are considered negligible, as the borehole dimensional scale is small compared with the infinite extent of the ground beyond the borehole [
In some models, such as the Equivalent Diameter method [
Thermal resistances in the borehole.
Many of the models for heat transfer analysis inside the borehole are summarized in
Comparison of various methods in the heat analysis inside the borehole.
1D (Equivalent diameter) [ 
1D (Shape factor) [ 
2D
[ 
Quasi 3D
[ 


Utube disposal  N  Y  Y  Y 
Quantitative expressions of the thermal resistance in the crosssection  N  N  Y  Y 
Thermal interference  N  N  N  Y 
Extinction between the entering and exiting pipes  N  N  N  Y 
Axial convection by fluid flow  N  N  N  Y 
Axial conduction in grout  N  N  N  N 
Several simulation models for the heat transfer outside the borehole are available. The main objective of heat transfer analysis outside the borehole is to determine the transient borehole surface temperature, which is the key to the heat transfer analysis inside the borehole. The models vary in the way the problem of heat conduction in the soil is solved and the way the interference between boreholes is treated.
In the analysis of GHE heat transfer, some complicating factors, such as groundwater movement [
Unlike the area inside the borehole, heat conduction outside the borehole exhibits transient behavior. As a basic problem, the following assumptions are commonly made:
The ground is homogeneous in its thermal properties and initial temperature.
Moisture migration is negligible.
Thermal contact resistance is negligible between the pipe and grout and between the grout and soil.
The effect of ground surface is negligible for the initial 5–10 years (depending on the borehole depth).
Due to its minor order, heat transfer in the axial direction along the borehole, which accounts for the heat flux across the ground surface and down to the bottom of the borehole, is considered negligible. This assumption is valid for a length of the borehole distant enough from the borehole top and bottom. Additionally, heat transfer in the circumferential direction is negligible in this model assuming a single borehole. Therefore, the heat transfer is usually modeled with a onedimensional analysis assuming that the axial and circumferential heat flows are negligible.
The earliest approaches to calculating the heat transfer in the soil surrounding a GHE is Kelvin’s linesource model,
In both analytical models of Kelvin’s theory and the cylindrical source model, the borehole depth is considered infinite, and the axial heat flow along the borehole depth is assumed negligible. Furthermore, when time tends to infinity, the temperature rise of the Kelvin’s theory for an infinite line source tends to infinity, making the infinite model weak for describing heat transfer mechanism in long time steps. Therefore, they can only be used for short time range of operations of GCHP systems. To take into account axial temperature changes for boreholes with finite lengths and in long durations, Eskilson’s approach to the problem of determining the temperature distribution around a borehole is based on combination of analytical and numerical solution techniques. Eskilson [
In the analytical models presented above, a number of assumptions are employed in order to simplify the complicated governing equations. In time varying heat transfer rates and the influence of surrounding boreholes on both long and short time scales, analytical methods are not as suitable as numerical methods. However, due to their much shorter computation times, they are still used widely in designing GHEs.
System simulation models require the ability to operate at short time scales, often less than one minute. Therefore, the dynamic response of the grout material inside the borehole should be considered. This is possible when the model is solved using numerical techniques. Numerical methods have also been used extensively for evaluating the heat conduction inside the borehole and the soil surrounding it [
One of the disadvantages of numerical approaches is their computation time for longterm system performance. The diameters of the Utubes in the borehole are fairly small, on the order of 10^{−}^{2} m, while the size of the solution domain, which depends on the duration of system operation and its heating/cooling load, is approximately on an order of 10 m, making the domain extremely disproportionate. As a result, a large number of mesh elements is required for simulation of a single borehole and its surrounding soil. To achieve an inaccuracy of 2% or less for the steady state heat transfer analysis of boreholes, a minimum number of approximately 18 elements describing any circular shape of a horizontal cross section is needed [
The temperature gradient in the domain between the borehole wall and the far field changes gradually from large to small. Therefore, to reduce computer memory and computational time, the size of the mesh cells is often chosen based on this gradual change. Furthermore, the symmetry about the GHE can be used to save computation time. In such cases, the symmetric portion of the solution domain is replaced by an adiabatic wall boundary condition on the symmetry line. Applying all these techniques, a threedimensional 15 m × 15 m × 60 m domain may require mesh sizes of the order of 1,000,000 elements to simulate multiple boreholes of 50 m length.
Due to these limitations, several available models are limited to a twodimensional (2D) description of the domain [
To evaluate the longterm temperature response in the soil surrounding multiple borehole systems, a numerical finite volume method in a twodimensional meshed domain is used previously [
One limitation in the previous studies [
One limitation in most of the previous studies is the assumption of uniform heat input along the borehole length to the ground, either when the borehole is assumed as a cylinder or a line source of heat. In order to determine the borehole heat delivery/removal profile along the borehole, the heat transfer model outside the borehole should be coupled to the one inside the borehole. In a recent study, KoohiFayegh and Rosen [
Another limitation in the previous studies is the assumption of steady borehole wall temperature during system operation. When calculating the heat input to the ground, it becomes clear that it varies with the borehole wall temperature. Although the soil temperature at the borehole wall rises as the system operates, it is often assumed that the soil temperature at the borehole wall is constant throughout the operation period. This assumption ignores the drop in heat injection strength when the borehole wall temperature increases and, therefore, underestimates the inlet temperature of the circulating fluid that is required to meet the heat injection needs of the system. Yang
In order to account for higher heat flow rates or thermal interaction between multiple boreholes, the model should be modified to include the transient value of borehole wall temperature. Thus, the nonuniform heat flow rate along the borehole wall becomes transient as well. A model is needed that is able to not only estimate how heat flows in the region surrounding GHEs, but also how a temperature rise in the soil surrounding a borehole caused by the system itself or a neighboring geothermal system can interfere with its heat delivery strength.
Performing energy and moisture balances at the ground surface involves very complex processes, taking into account solar radiation, cloud cover, surface albedo, ambient air temperature and relative humidity, rainfall, snow cover, wind speed and evapotranspiration. Such details provide a proper account of the renewable energy resource. However, due to the complexity of adding all the above heat fluxes in a numerical model, some studies assume the ground surface temperature variation at the ground surface to take the form of a sine wave or Fourier series [
Neglecting the existence of moisture in the soil, the heat flux is described via the conduction heat flow. The coupled heat and moisture flow in a soil system is described with a thermal energy balance coupled with a mass balance. This adds to the complication of the problem since the complete model contains a set of transient simultaneous partial differential equations with many soil parameters that are not readily available. Research shows that the effects of moisture migration are not significant to the operation of a vertical GHE; it is expected that these effects are more pronounced with a horizontal GHE. This is because natural variations of temperature and moisture near the ground surface and operation of the HGHE may create a potentially greater moisture movement. During the cooling season, migration of soil moisture away from the GHE may lead to a drastic drop in soil thermal conductivity and consequently a significantly reduced heat transfer, which has a devastating effect on GHE performance. Therefore, although moisture migration effects can be neglected in early stages of design or conceptual development, not considering them in longterm operation of GCHP systems makes it impossible to assess the performance and potential failure of these systems [
A further complication in the design of groundcoupled heat pump systems is the presence of groundwater. Due to the difficulties encountered both in modeling and computing the convective heat transfer and in learning about the actual groundwater flow in engineering practice, each of the methods presented in the previous sections is based on Fourier’s law of heat conduction and neglect the effects of groundwater flow in carrying away heat. Where groundwater is present, flow will occur in response to hydraulic gradients, and the physical process affecting heat transfer in the ground is inherently a coupled one of heat diffusion (conduction) and heat advection by moving groundwater [
An assessment of the available analytical models demonstrates that they are not capable of estimating the heat delivery/removal strength when the soil surrounding them experiences a temperature rise. In the current study, it is shown that the effect of the temperature rise in the soil surrounding boreholes is not negligible. The distance between two boreholes or two systems of boreholes, the heat flux from the borehole wall and the time of system operation all affect directly the amount of thermal interaction between the systems. However, the effect of these parameters on system operation and heat delivery/removal rate can only be studied in models that account for the change in the borehole wall temperature.
As mentioned in the previous section, in order to account for the sustainability of the system and heat pump efficiency when thermal interaction among boreholes occur, it is important to develop and utilize models that account for the drop in heat delivery strength when the borehole wall temperature increases during the operation time or by another nearby operating system. As a result, the inlet temperature of the circulating fluid needs to be adjusted to a higher/lower one to maintain the heat delivery/removal needs of the system. This analysis is important since ground heat exchangers are coupled to a heat pump that can only work within a certain temperature lift and inlet and outlet circulating temperature ranges. If a system is able to deliver a certain amount of heat to the ground, the increase in the inlet circulating temperature due to temperature rise in the soil caused by a nearby system reflects how thermal interaction affects the sustainability of the system. Furthermore, simulation of heat exchange processes within the system and surrounding environment through local scale assessment, simulation of migration of thermal plumes into the hydrogeological environment through intermediate and regional scale assessment will help gain an estimation of ecological impacts.
The support provided by the Ontario Ministry of Environment through its Best in Science program is gratefully acknowledged.
The authors declare no conflict of interest.