Transport Settings

General Settings

Mathematical formulation of transport equation

FEFLOW contains two options for the formulation of the transport equation, the so-called Convective form (default), and the Divergence form. The main practical difference is in the meaning of flux boundary conditions, which also has implications for out-flowing boundaries.

White Paper Vol. I, Chapter 6

The choice of transport equation formulation influences the meaning of transport boundary conditions of flux and well type. Typically, in the Convective form these kinds of transport-boundary conditions are not applicable if flow-boundary conditions are defined at the same node.

As an example we will regard a boundary section with an influx of contaminant mass via a second kind (Neumann) boundary condition:

Convective form: The assigned mass flux is only driven by dispersion. FEFLOW will try to realize the assigned mass flux by setting a lower concentration along the boundary section in order to achieve the required concentration gradient. Additional mass will enter the model along the boundary section via advection, so that the total mass-flow budget will usually be higher than the assigned boundary-condition value.

Divergence form: The assigned mass flux represents both dispersive and advective mass flux: Compared to the convective form, a lower (more realistic) concentration at the boundary section and a lower mass influx result. The mass-flow budget will be identical to the assigned boundary condition.

The divergence form of the transport equation might be necessary if mass-flux boundary conditions are needed at boundary sections at which the advective part is dominating. On the other hand, the correct calculation of free outflow requires a smooth velocity field. This is often not available at certain locations of the model, e.g., around pumping wells. This can lead to mass accumulation at these locations.

Be aware that boundaries with no mass-transport boundary condition have a natural boundary condition of zero dispersive flux (zero gradient), only the advective flux is computed at these boundaries. If you want to account for the dispersive flux leaving the model, you have to assign mass transport boundary conditions explicitly!

Fluid viscosity

By default, FEFLOW does not consider viscosity differences caused by temperature and/or concentration differences in the model. If viscosity dependencies are incorporated in the calculation, all Conductivity values refer to a predefined reference temperature and concentration. Internally conductivities are then re-calculated for the actual temperature/concentration in each element at a given time.

The predefined relation between temperature, concentration, and viscosity can be replaced by a user-defined equation. For this purpose, an equation editor is available by clicking on the Edit button.

The Fluid viscosity can be displayed current active view via the section Auxiliary Data of the Data panel.

Sorption

Solute dispersion

Besides the standard linear Fickian Bear-Scheidegger dispersion, FEFLOW supports a nonlinear dispersion relation.

The nonlinear dispersion has shown to lead to more accurate results for density-dependent simulations with high-concentration gradients (brine transport).

Sorption type

Equilibrium sorption processes can be taken into account using either the linear Henry isotherm or the non-linear Freundlich or Langmuir isotherms. Depending on the chosen sorption type, different Sorption Coefficients are available in the Data panel.

Flow field direction

For specific applications such as groundwater age calculation or determination of capture zones a backward transport calculation is applied.

When switching to Reverse flow field, the transport simulation is carried out on a flow field where all velocity vectors have been rotated by 180 degrees.

White Paper Vol. I, Chapter 8

Fluid density

Dependency on temperature

By default the relation between temperature and fluid density is considered as linear. This holds only for very small temperature variations in the model. Switching to Variable FEFLOW applies a non-linear relationship. By default, this is a 6th order polynomial reflecting the temperature range between 0° C and 100° C. The relationship can be edited using an equation editor by clicking on Edit button.

Density ratio

By default, FEFLOW applies a single density-difference ratio to the sum of all concentrations in multi-species simulations (constant solutal density ratio). Alternatively, different density ratios can be used for each single species.

Considered density variations

Applying the Boussinesq approximation (default) the density is only considered in the buoyancy term of the Darcy equation. All other density dependencies in the balance terms are ignored. This simplification is applicable to all problems with a small to moderate density variation. Applying the extended Boussinesq approximation, additional terms are considered as density-dependent.

Extended Boussinesq approximation that includes spatial and temporal variation of species concentration and of temperature, as well a diffusive components for fluid, species, and heat.

Reference Values

Concentration

Unit: [M/L³]
Default value: 0 mg/l

The reference concentrations C₀ and C_s are used as a basis for density- and viscosity-dependent flow. All conductivity values input in FEFLOW are assumed to be valid for a concentration C₀. They are modified during a density- and viscosity-dependent simulation according to the current concentration.

The reference concentrations should also reflect the basis that has been used to calculate the Density Ratio.

By default, FEFLOW assumes C₀ = 0 mg/l and C_s according to the maximum concentration in initial or boundary conditions. The buttons may be used to apply other values easily.

Temperature

Unit: [T]
Default value: 10°C

The reference temperature T₀ is used as a basis for density- and viscosity-dependent flow, referencing certain inflow temperatures, and for calculating energy contents.

It may influence the simulation at various stages:

Conductivity
All conductivity values input in FEFLOW are assumed to be valid for a temperature T0. They are modified during a density- and viscosity-dependent simulation according to the current temperature.
Buoyancy
The difference between the current temperature at a node and the reference temperature determines the buoyancy term in density-dependent simulations. Water at nodes with a temperature higher than the reference temperature will be subject to buoyancy, even if there is no temperature gradient within the model. It is therefore recommended to set the reference temperature to the true background temperature of the modelled system.
Inflow Temperature
The reference temperature is used as inflow temperature for water sources based on the Source/Sink (flow) and In-/Outflow on Top/Bottom parameters.
It is not used, however, to define inflow temperature at Fluid-flux BCs. There the inflow temperature is equal to the temperature at the boundary node (no-gradient condition) if no heat-transport boundary condition is set.
Thermal Expansion
The reference temperature should reflect the basis that has been used to calculate the Expansion Coefficient (if constant expansion is used in density-dependent simulations).
Content Analysis
The thermal energy stored in the system calculated in the Content panel is computed relative to the reference temperature.

Density ρ0

Unit: [M/L³]
Default value: 999.793 kg/m³

In order to calculate the Auxiliary Data parameter Fluid Density, the reference density has to be defined for reference conditions (reference concentration and/or reference temperature depending on the problem class). The reference density does not influence the simulation.

Viscosity

Unit: [M/LT]
Default value: 0.003 kg/ms

In order to calculate the Auxiliary Data parameter Fluid Viscosity, the reference viscosity has to be defined for reference temperature. The reference viscosity does not influence the simulation.