Equation Of State

The Equation of State Tab can be accessed from the FEFLOW Problem Settings. It defines the dependency of fluid density and fluid viscosity on temperature, concentration and/or pressure.

Fluid density

For fluid density the following options are available:

1. No dependency on temperature or concentration

2. Linear dependency on temperature or concentration

3. Nonlinear dependency on:

   1. Temperature

   2. Temperature and pressure

   3. Concentration,  temperature and pressure

The first two options have been always available in FEFLOW. From FEFLOW 10.0, the software is able to calculate a nonlinear dependency of density and viscosity on temperature, pressure and concentration, to account for higher pressure and temperature ranges, as explained below.

1. No dependency

The fluid density is not dependent on any temperature or concentration changes. Reference temperature T₀ and/or concentration C₀ are defined and correspond to the reference density ρ₀.

Equation of State window in FEFLOW.

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.

2. Linear dependency

By default the relation between temperature and fluid density is considered as linear. This holds only for very small temperature variations in the model. This holds only for very small temperature variations in the model. For the linear dependency on temperature an Expansion coefficient β  is defined together with a reference density   ρ₀ at a reference temperature T₀. The Expansion coefficient can be defined globally as a fluid property or spatially variable as a material property. Similarly, the linear dependency on concentration is defined by a Density coefficient α  (in older FEFLOW versions called "Density ratio") together with a maximum and reference concentration C₀. The salinity effects can be species-specific or defined for the bulk concentration.

Equation of State window in FEFLOW: linear dependency on temperature and concentration options.

By default, FEFLOW applies a single Density coefficient to the sum of all concentrations in multi-species simulations (constant solute Density coefficient). Alternatively, different Density coefficients can be used for each single species.

Option to activate species-specific Salinity effects - ONLY available for multispecies models.

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.

3. Nonlinear dependency

3.1 Non-linear dependency on temperature  

The non-linear dependency on temperature uses by default a 6^th order polynomial expression which can be edited by clicking the Edit button and provides a good fit for conditions from 0 to 100°C. For higher temperatures the non-linear dependency on temperatures and pressure should be used. Please refer to the FEFLOW tooltip for this option displayed in the GUI for more details.

Nonlinear dependency on temperature option: FEFLOW tooltip.

3.2 Non-linear dependency on temperature and pressure

This option is meant for deep geothermal applications where temperatures exceed 100°C but water remains in single-phase fluid state due to massive pressure increases. In that case a reference absolute pressure p₀ is required. Please refer to the FEFLOW tooltip for this option displayed in the GUI for more details.

Nonlinear dependency on temperature and pressure option: FEFLOW tooltip.

The user can refer to the phase diagram of water below to know the consequences of pressure and temperature changes in a single-phase fluid state.

Phase diagram of water. Copyrights attributed to Wikimedia Commons (see original here).

3.3 Non-linear dependency on concentration, temperature and pressure

If nonlinear salinity effects need to be accounted as well, the last option should be selected. The benchmark data is well reproduced in a concentration range of 0 < C < 300 g/l, where  NaCl is presumed a proxy for salinity.

Nonlinear dependency on concentration, temperature and pressure option: FEFLOW tooltip.

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.

At large depths, viscosity changes may become significant. In FEFLOW viscosity can be treated as constant or depending on process variables and/or parameters. 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. Please note that the parameter μ₀ is used in the below formula to calculate the fluid viscosity and can be set by the user.  

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

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:

1. 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.

2. 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.

3. 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.

4. 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).

5. Content Analysis
   The thermal energy stored in the system calculated in the Content panel is computed relative to the reference temperature.

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.