Heat Transport Boundary Conditions - Overview

By default, all model boundaries in FEFLOW are assumed to be impermeable for heat flux, i.e., no energy can flow into the model or out of the model. Exceptions are flow boundary conditions where water enters or leaves the model without the specification of a heat transport boundary condition. Their handling depends on whether the convective or divergence form of the transport equation is used. At boundaries that are not assumed as impermeable, boundary conditions have to be specifically set. Boundary conditions can be placed both at outer model borders and inside the model.

Boundary conditions are defined on a nodal basis. However, some of them (2nd kind/flux and 3rd kind/transfer boundary conditions) have to be applied to more than one node to be fully functional.

The following four types of boundary conditions for mass transport are available in FEFLOW. All of them can be used as time-constant or in combination with a time series. The application of all the boundary conditions can be constrained by additional physical constraints (constraint conditions).

Boundary Condition	Short Description	Examples
Temperature BC	Fixed temperature (1^st kind/Dirichlet boundary condition)	Well-known temperature at inflowing boundary
Heat-flux BC	Fixed heat flux across a model boundary (2^nd kind/Neumann boundary condition)	Temperature of groundwater recharge
Heat-transfer BC	Fixed reference temperature with additional heat transmission coefficient (3^rd kind/Cauchy boundary condition)	Hot tank/building in the aquifer Air temperature
Heat nodal sink/source BC	Fixed extraction/injection of thermal energy at a single node	Borehole heat exchangers in 2D regional models
Borehole heat exchanger	Varying extraction/injection of thermal energy at multiple overlying nodes	Borehole heat exchanger

For defining boundary conditions, inflows are considered as negative, outflows as positive.

FEFLOW only allows one type of heat transport boundary condition to be set at a node. Inputting another type of boundary condition will erase the previously defined condition.

A couple of more specific boundary condition types can be added additionally via the context menu of the Heat transport section in the Boundary Conditions (BC) branch of the Data panel. These are by default hidden.

Symbol	Boundary Condition	Short Description
	Heat-flux BC (Integral)	Integral heat-flux boundary condition for unconfined models with free & movable surface
	Heat-transfer BC (Integral)	Integral heat-transfer boundary condition for unconfined models with free & movable surface

Heat Transport Constraints - Overview

The application of all boundary conditions in FEFLOW can be constrained by physical limits. The combination of boundary conditions and time-constant or time-varying constraint conditions allows the representation of a broad variety of specific boundary properties. Examples in heat transport include heat transport conditions depending on hydraulic head.

Most constraint conditions are complementary constraints, i.e. boundary conditions of a temperature type (temperature BC, heat-transfer BC) can be constrained by a minimum and maximum heat flow, boundary conditions of a heat flux type (heat-flux BC, heat nodal sink/source BC) can be constrained by a minimum and maximum temperature.

For all heat transport boundary conditions, an additional hydraulic-head constraint is available.

For defining constraints, inflows are considered as positive, outflows as negative. Thus a constraint of maximum heat flow = 0 J/d means that no heat inflow is allowed at the corresponding boundary condition.