Modern building construction requires efficient climate control, preferably using sustainable energy resources. Such resources can be solar collector systems for warm-water support or ambient air or geothermal-based heat pump systems that can be used for cooling in the summer and heating in the winter.

The geothermal applications mostly consist of pipe installations with water or brine as fluid to provide the heat exchange between the subsurface and a heat pump. There are two main types of these so-called closed-loop systems: vertical borehole heat exchangers (BHEs), which are installed in boreholes (see part 1 of this series); and horizontal heat collectors, which are deposited on large areas and in very shallow depths of only one or two meters.

While the vertical BHEs are often simplified as infinite thermal line sources or sinks, the horizontal systems are more complicated from a modeling point of view. This is because they have to be arranged in patterns to cover large surface areas, and the use of this simplification does not match anymore. The total length of the pipes can easily reach 100 meters or more. The flow regime is preferably turbulent because of the resulting lower thermal resistivity between the fluid and the subsurface. You can imagine that a CFD simulation approach of systems with these dimensions is a challenge (despite the computational performance of modern HPC clusters). Fortunately, there is a solution for this kind of task.

Let’s have a look at how the Pipe Flow Module can be used in the growing sector of geothermal heat collector systems.

Thankfully, we do not have to care about the computational difficulties that may arise when simulating long pipe systems. The Pipe Flow Module provides all necessary capabilities to calculate pressure drops, velocities, and foremost — the heat transport between the pipe fluid and the subsurface, using prescribed 3D curved-edge correlation functions. All we need to do is to define the inlet conditions (temperature, velocity) and the pipe, fluid, and subsurface properties. Therefore, we can completely focus on questions like, “What do you think is the best horizontal pipe arrangement underneath your garden?”

Let’s find out.

The heat transfer example, provided in a recent blog post, shows how a prescribed heat extraction rate can be realized in the form of a numerically-calculated inlet temperature boundary condition. By doing so, any desired heat extraction rate can be given and the corresponding inlet temperature can be computed. Assuming that the fluid properties are temperature-independent, a simplified version of the inlet temperature can be written as:

(1)

T_{in}=T_{out}-\frac{P(t)}{\rho_{f} \cdot C_{P,f} \cdot \dot{V}}.

Thus, it is a function of the outlet temperature (), density (), heat capacity (), and volumetric flow rate () of the working fluid in the pipes. The temperature difference between the inlet and outlet temperature is controlled via the heat extracted by the heat pump, , which is a function of time because heat pumps (for house heating purposes) usually do not run all day. A typical heating rate of a one-family house heat pump is 8 kW, where 2 kW of electric power is needed to run it and 6 kW are to be extracted from the subsurface. To reach a daily heat demand of, say, 48 kWh during the winter season in Germany or North America, the heat pump needs to run six hours per day.

*6 kW of heat extraction for six hours per day, in a three-day cycle.*

The number of possibilities of how the heat collector pipes can be arranged in the subsurface is endless. Here, we will have a look at three randomly chosen patterns. Let’s call them snake, snail, and meander designs. The heat collector is embedded in the subsurface with typical thermal properties that you could find in the uppermost soil layer in your garden. The subsurface temperature distribution corresponds to that of the temperature in Germany during the month of January.

*Three different layer designs for a geothermal heat collector system: 1) snake, 2) snail, and 3) meander design.*

The COMSOL Multiphysics feature, *geometry subsequence*, allows us to create all of the three different pipe geometries within the same model. It also allows us to perform a parametric study that solves for each particular sub-geometry. We can compare the different designs by having a look at the resulting outlet temperature as a function of time.

*Comparison of the outflow temperatures of the three collector designs.*

The three collectors each show a different thermal behavior. This is due to the varying distances between the pipes, their different lengths, and thereby the surface available for heat transfer. Even though the freezing point of the water-glycol mixture inside the pipes is at about -13°C, the outflow temperature is — for environmental reasons — legally limited to stay above -5°C in Germany.

The snake design tends to go below this point only after a few days and would, thus, not be the preferred design here. However, a geothermal engineer could now proceed to optimize a design (modify the pipe length, pipe diameter, or the covered area to reach a maximum outlet temperature, for example.) One could, of course, also easily perform long time runs to study the impact of seasonal temperature variations or even use the pipes for cooling purposes in the summer.

Once the optimal design of the geothermal heat collector in your garden is found, why not go on and model the modern underfloor heating system in the house? The procedure is straight-forward. There is only a change of sign because heat is injected here.

*Combined model of a garden heat collector together with an underfloor house heating system. The collector extracts heat from the subsurface at 6 kW, while the house heating system injects heat into the floor of the house at 30°C.*

- Read part 1 of the Geothermal Energy series: Modeling Geothermal Processes with COMSOL Software
- Read part 2 of the Geothermal Energy series: Coupling Heat Transfer with Subsurface Porous Media Flow
- Read about designing heat exchangers
- Explore the Pipe Flow Module

One of the greatest challenges in geothermal energy production is minimizing the prospecting risk. How can you be sure that the desired production site is appropriate for, let’s say, 30 years of heat extraction? Usually, only very little information is available about the local subsurface properties and it is typically afflicted with large uncertainties.

Over the last decades, numerical models became an important tool to estimate risks by performing parametric studies within reasonable ranges of uncertainty. Today, I will give a brief introduction to the mathematical description of the coupled subsurface flow and heat transport problem that needs to be solved in many geothermal applications. I will also show you how to use COMSOL software as an appropriate tool for studying and forecasting the performance of (hydro-) geothermal systems.

The heat transport in the subsurface is described by the heat transport equation:

(1)

(\rho C_p)_{eq} \frac{\partial T}{\partial t} + \rho C_p {\bf u } \cdot \nabla T = \nabla \cdot (k_{eq} \nabla T ) + Q + Q_{geo}

Heat is balanced by conduction and convection processes and can be generated or lost through defining this in the source term, . A special feature of the *Heat Transfer in Porous Media* interface is the implemented *Geothermal Heating* feature, represented as a domain condition: .

There is also another feature that makes the life of a geothermal energy modeler a little easier. It’s possible to implement an averaged representation of the thermal parameters, composed from the rock matrix and the groundwater using the matrix volume fraction, , as a weighting factor. You may choose between volume and power law averaging for several immobile solids and fluids.

In the case of volume averaging, the volumetric heat capacity in the heat transport equation becomes:

(2)

(\rho C_p )_{eq} = \sum_{i} ( \theta_{pi}\rho_{pi}C_{p,pi})+(1-\sum_{i}\theta_{pi})\rho C_p

and the thermal conductivity becomes:

(3)

k_{eq}=\sum_{i} \theta_{pi} k_{pi} + ( 1-\sum_{i} \theta_{pi} ) \rho C_p

Solving the heat transport properly requires incorporating the flow field. Generally, there can be various situations in the subsurface requiring different approaches to describe the flow mathematically. If the focus is on the micro scale and you want to resolve the flow in the pore space, you need to solve the creeping flow or Stokes flow equations. In partially saturated zones, you would solve Richards’ equation, as it is often done in studies concerning environmental pollution (see our past Simulating Pesticide Runoff, the Effects of Aldicarb blog post, for instance).

However, the fully-saturated and mainly pressure-driven flows in deep geothermal strata are sufficiently described by Darcy’s law:

(4)

{\mathbf u} = -\frac{\kappa}{\mu} \nabla p

where the velocity field, , depends on the permeability, , the fluid’s dynamic viscosity, , and is driven by a pressure gradient, . Darcy’s law is then combined with the continuity equation:

(5)

\frac{\partial}{\partial t} (\rho \epsilon_p) + \nabla \cdot ( \rho {\bf u} ) = Q_m

If your scenario concerns long geothermal time scales, the time dependence due to storage effects in the flow is negligible. Therefore, the first term on the left-hand side of the equation above vanishes because the density, , and the porosity, , can be assumed to be constant. Usually, the temperature dependencies of the hydraulic properties are negligible. Thus, the (stationary) flow equations are independent of the (time-dependent) heat transfer equations. In some cases, especially if the number of degrees of freedom is large, it can make sense to utilize the independence by splitting the problem into one stationary and one time-dependent study step.

Fracture flow may locally dominate the flow regime in geothermal systems, such as in karst aquifer systems. The Subsurface Flow Module offers the *Fracture Flow* interface for a 2D representation of the Darcy flow field in fractures and cracks.

Hydrothermal heat extraction systems usually consist of one or more injection and production wells. Those are in many cases realized as separate boreholes, but the modern approach is to create one (or more) multilateral wells. There are even tactics that consist of single boreholes with separate injection and production zones.

Note that artificial pressure changes due to water injection and extraction can influence the structure of the porous medium and produce hydraulic fracturing. To take these effects into account, you can perform poroelastic analyses, but we will not consider these here.

It is easy to set up a COMSOL Multiphysics model that features long time predictions for a hydro-geothermal application.

The model region contains three geologic layers with different thermal and hydraulic properties in a box with a volume V≈500 [m³]. The box represents a section of a geothermal production site that is ranged by a large fault zone. The layer elevations are interpolation functions from an external data set. The concerned aquifer is fully saturated and confined on top and bottom by aquitards (impermeable beds). The temperature distribution is generally a factor of uncertainty, but a good guess is to assume a geothermal gradient of 0.03 [°C/m], leading to an initial temperature distribution T_{0}(z)=10 [°C] – z·0.03 [°C/m].

*Hydrothermal doublet system in a layered subsurface domain, ranged by a fault zone. The edge is about 500 meters long. The left drilling is the injection well, the production well is on the right. The lateral distance between the wells is about 120 meters.*

COMSOL Multiphysics creates a mesh that is perfectly fine for this approach, except for one detail — the mesh on the wells is refined to resolve the expected high gradients in that area.

Now, let’s crank the heat up! Geothermal groundwater is pumped (produced) through the production well on the right at a rate of 50 [l/s]. The well is implemented as a cylinder that was cut out of the geometry to allow inlet and outlet boundary conditions for the flow. The extracted water is, after using it for heat or power generation, re-injected by the left well at the same rate, but with a lower temperature (in this case 5 [°C]).

The resulting flow field and temperature distribution after 30 years of heat production are displayed below:

*Result after 30 years of heat production: Hydraulic connection between the production and injection zones and temperature distribution along the flow paths. Note that only the injection and production zones of the boreholes are considered. The rest of the boreholes are not implemented, in order to reduce the meshing effort.*

The model is a suitable tool for estimating the development of a geothermal site under varied conditions. For example, how is the production temperature affected by the lateral distance of the wells? Is it worthwhile to reach a large spread or is a moderate distance sufficient?

This can be studied by performing a parametric study by varying the well distance:

*Flow paths and temperature distribution between the wells for different lateral distances. The graph shows the production temperature after reaching stationary conditions as a function of the lateral distance.*

With this model, different borehole systems can easily be realized just by changing the positions of the injection/production cylinders. For example, here are the results of a single-borehole system:

*Results of a single-borehole approach after 30 years of heat production. The vertical distance between the injection (up) and production (down) zones is 130 meters.*

So far, we have only looked at aquifers without ambient groundwater movement. What happens if there is a hydraulic gradient that leads to groundwater flow?

The following figure shows the same situation as the figure above, except that now there is a hydraulic head gradient of H=0.01 [m/m], leading to a superposed flow field:

*Single borehole after 30 years of heat production and overlapping groundwater flow due to a horizontal pressure gradient.*

- Download the Geothermal Doublet tutorial
- Explore the Subsurface Flow Module
- Part I of the Geothermal Energy series: Modeling Geothermal Processes with COMSOL Software
- Related papers and posters presented at the COMSOL Conference:
- Hydrodynamic and Thermal Modeling in a Deep Geothermal Aquifer, Faulted Sedimentary Basin, France
- Simulation of Deep Geothermal Heat Production
- Full Coupling of Flow, Thermal and Mechanical Effects in COMSOL Multiphysics® for Simulation of Enhanced Geothermal Reservoirs
- Multiphysics Between Deep Geothermal Water Cycle, Surface Heat Exchanger Cycle and Geothermal Power Plant Cycle
- Modelling Reservoir Stimulation in Enhanced Geothermal Systems

During my time as a PhD student, I had to deal with different kinds of geothermal energy extraction techniques. The focus of my research was on shallow geothermal borehole heat exchangers, but within my team, I also came in contact with open borehole and deep geothermal approaches. To my surprise, I found that almost all of the processes involved in the various geothermal applications could be modeled with the Subsurface Flow Module.

Renewable energies are a growing industry and the geothermal energy branch is a hot topic of active research. Over the past few decades, different techniques were established to extract geothermal heat from shallow to deep subsurface levels. The closed-loop *borehole heat exchanger* (BHE) is a standard approach for lower- and mid-depth applications.

In a BHE, fluid circulates through pipes that are located inside of boreholes, resulting in indirect thermal contact between the fluid and the subsurface. This method was restricted by the closed nature of the heat exchanger with the effective heat exchange area limited by the equipment involved. This could could be significantly increased by extracting groundwater from aquifers, gaining thermal energy through the re-injection of hot fluid a distance from the injection point through hydrothermal doublets.

Enhanced geothermal systems (EGS) were developed to exploit geothermal resources in dry and impermeable rocks or *hard dry rocks* (HDR), by hydraulic fracturing, or fracking. This procedure involves pumping high-pressure water into the desired layer to establish new cracks and widen existing cracks and fractures. Injected water can then travel through the fractures, heat up, and be extracted at a second borehole.

*Geothermal heat extraction methods.*

Heat transport in the subsurface proceeds by convection, dispersion, and conduction. You need to know the thermal properties of the geological layers in order to run simulations. However, we can usually only obtain vague estimations based on geological maps and core samples. Convective heat transport, either naturally driven by buoyancy forces or artificially established through the wells, can play a role in — and even dominate — the processes.

Depending on the local geology, the subsurface flow regime may be situated in fully or partially saturated porous media, or it can proceed in fractures. Although the different techniques for geothermal heat production differ fundamentally, the Subsurface Flow Module provides the required features to simulate the subsurface thermal development. You can easily couple heat transport with the velocity field.

In some situations, the coupling must be applied bi-directionally. If the temperature gradients are high, the temperature dependencies of involved parameters (hydraulic conductivity, for example) are no longer negligible and must be taken into account. Also, poroelastic processes may play a role in some cases, especially when it comes to fracking.

Let’s have a look at an example to demonstrate some of the features that are necessary to simulate geothermal processes. The following model solves for the heat transport around a shallow geothermal installation embedded in a geological domain. The domain is separated into various parts, representing different geological layers with their particular properties. The thermal influence of seasonal temperature changes at the surface is taken into account.

*A three-by-three borehole heat exchanger (BHE) array 135-meter deep, located in layered bedrock. Each BHE extracts heat at 20 W/mK throughout the year. Between 60-70 meters is an aquifer where groundwater flow occurs, causing horizontal convective heat transport. Shown on the right are the borehole wall temperatures of the three BHEs in the middle of the array. Due to the thermal interaction between the heat sinks, the temperature of middle BHE (green line) is lower than the other two. In the region of the aquifer, the BHE further downstream than the other two (red line) is affected by the heat exchange occurring upstream of it, resulting in a lower temperature.*

Simulation is necessary for the prediction of long-term impacts on the BHEs in order to identify whether the pipe will avoid freezing. The most straightforward way to quickly model the BHEs is to neglect the glow and heat transfer within the borehole and apply an appropriate heat flux boundary condition at the walls. In doing so, the borehole becomes a local heat sink and heat will travel towards it. If more than one BHE is installed at a local site, the heat exchangers might start to interact after some time. In particular, if groundwater flow in an aquifer is present, the boreholes will connect thermally. This thermal interaction can cause significant efficiency losses of the whole geothermal system. On the other hand, groundwater flow also increases the thermal recovery rate. Reliable predictions can only be made if the geological data set is accurate enough.

*The heat transfer through the BHEs over a period of time. The top surface temperature changes seasonally between 0-20°C, representing a city in central Europe or the east coast in the US. Notice how the heat pattern elongates in the direction of flow in the region of the aquifer.*

Today, we have introduced how you can use multiphysics simulation in geothermal energy applications. Next up in this blog series we will present a more advanced application, involving coupling heat transfer in heat pipe arrays with subsurface porous media flow. Stay tuned!

- Explore the Subsurface Flow Module
- Download the Geothermal Heating from a Pond-Loop model
- Related papers presented at the COMSOL Conference:
- Oscillatory Thermal Response Test (OTRT) — An Advanced Method for Gaining Thermal Properties of the Subsurface
- Simulation of Deep Geothermal Heat Production
- Coupling Heat Transfer in Heat Pipe Arrays with Subsurface Porous Media Flow for Long Time Predictions of Solar Rechargeable Geothermal Systems
- Finite-Element Evaluation of Thermal Response Tests Performed on U-Tube Borehole Heat Exchangers
- Numerical Simulation of the Thermal Response Test Within COMSOL Multiphysics® Environment
- Numerically Generated g-functions for Ground Coupled Heat Pump Applications
- Use of COMSOL as a Tool in the Design of an Inclined Multiple Borehole Heat Exchanger
- Modeling Horizontal Ground Heat Exchangers in Geothermal Heat Pump Systems
- Modeling of Usage of Air Injection Well in a Geothermal System