Conserved quantities#

Nearly all ATS process kernels are conservation equations, where there is a formal conserved quantity, that, upon convergence, is conserved to tolerance. These are always an “extensive” quantity.

Water content in ATS is always measured on [mol] and therefore includes a factor of the cell volume. Energy in ATS is always measured in [MJ]. Unlike nearly all other variables, this is not SI, and is done so because this makes for fairly evenly balanced equations between a coupled flow and energy problem.

Richards Equation water content (liquid only)#

Richards water content evaluator: the standard form as a function of liquid saturation.

\[\Theta = n s \phi V\]

Specified with evaluator type: “richards water content”

field-evaluator-type-richards-water-content-spec

DEPENDENCIES:

  • “porosity”

  • “molar density liquid”

  • “saturation liquid”

  • “cell volume”

Liquid+Gas water content#

Water content for liquid + water vapor.

\[\Theta = (n_l s_l + n_g s_g \omega) \phi V\]

Specified with evaluator type: “liquid+gas water content”

field-evaluator-type-liquid-gas-water-content-spec

DEPENDENCIES:

  • “porosity”

  • “molar density liquid”

  • “molar density gas”

  • “saturation liquid”

  • “saturation gas”

  • “mol frac gas”

  • “cell volume”

Liquid+Ice water content#

Water content for liquid + water vapor.

\[\Theta = (n_l s_l + n_i s_i) \phi V\]

Specified with evaluator type: “liquid+ice water content”

field-evaluator-type-liquid-ice-water-content-spec

DEPENDENCIES:

  • “porosity”

  • “molar density liquid”

  • “molar density ice”

  • “saturation liquid”

  • “saturation ice”

  • “cell volume”

Liquid+Ice+Gas water content#

Three phase water content: vapor, liquid, and ice.

\[\Theta = (n_l s_l + n_i s_i + n_g s_g \omega_g ) \phi V\]

Specified with evaluator type: “three phase water content”

field-evaluator-type-three-phase-water-content-spec

DEPENDENCIES:

  • “porosity”

  • “molar density liquid”

  • “saturation liquid”

  • “molar density ice”

  • “saturation ice”

  • “molar density gas”

  • “saturation gas”

  • “molar fraction gas”

  • “cell volume”

Surface water content#

Snow or canopy water content#

Most other water contents can be formed as `Multiplicative`_ evaluators. See below for a few examples:

Multiplicative evalutator for snow-water_content:

<ParameterList name="snow-water_content" type="ParameterList">
  <Parameter name="field evaluator type" type="string" value="multiplicative evaluator" />
  <Parameter name="evaluator dependencies" type="Array(string)" value="{snow-cell_volume, snow-water_equivalent, surface-molar_density_liquid}" />
  <Parameter name="units" type="string" value="mol" />
</ParameterList>

Multiplicative evaluator for canopy-water_content:

<ParameterList name="canopy-water_content" type="ParameterList">
  <Parameter name="field evaluator type" type="string" value="multiplicative evaluator" />
  <Parameter name="evaluator dependencies" type="Array(string)" value="{canopy-cell_volume, canopy-water_equivalent, surface-molar_density_liquid}" />
  <Parameter name="units" type="string" value="mol" />
</ParameterList>

Richards energy#

Energy content evaluator for a standard Richards equation’s water content.

Energy associated with a soil and pore-water system, in [KJ]

\[E = V * ( \phi u_l s_l n_l + (1-\phi_0) u_r \rho_r )\]

Specified with evaluator type: “richards energy”

Note this equation assumes that porosity is compressible, but is based on the uncompressed rock grain density (not bulk density). This means that porosity is the base, uncompressible value when used with the energy in the grain, but the larger, compressible value when used with the energy in the water.

Note that this ignores energy in the gas phase.

field-evaluator-type-richards-energy-spec

DEPENDENCIES:

  • “porosity” The porosity, including any compressibility. [-]

  • “base porosity” The uncompressed porosity (note this may be the same as porosity for incompressible cases) [-]

  • “molar density liquid” [mol m^-3]

  • “saturation liquid” [-]

  • “internal energy liquid” [KJ mol^-1]

  • “density rock” Units may be either [kg m^-3] or [mol m^-3]

  • “internal energy rock” Units may be either [KJ kg^-1] or [KJ mol^-1], but must be consistent with the above density.

  • “cell volume” [m^3]

Liquid+Gas energy#

Energy content evaluator for a standard Richards equation, including energy in the gas phase.

Calculates energy, in [KJ], via the equation:

\[E = V * ( \phi (u_l s_l n_l + u_g s_g n_g) + (1-\phi_0) u_r \rho_r )\]

Specified with evaluator type: “liquid+gas energy”

Note this equation assumes that porosity is compressible, but is based on the uncompressed rock grain density (not bulk density). This means that porosity is the base, uncompressible value when used with the energy in the grain, but the larger, compressible value when used with the energy in the water.

field-evaluator-type-liquid-gas-energy-spec

DEPENDENCIES:

  • “porosity” The porosity, including any compressibility. [-]

  • “base porosity” The uncompressed porosity (note this may be the same as porosity for incompressible cases) [-]

  • “molar density liquid” [mol m^-3]

  • “saturation liquid” [-]

  • “internal energy liquid” [KJ mol^-1]

  • “molar density gas” [mol m^-3]

  • “saturation gas” [-]

  • “internal energy gas” [KJ mol^-1]

  • “density rock” Units may be either [kg m^-3] or [mol m^-3]

  • “internal energy rock” Units may be either [KJ kg^-1] or [KJ mol^-1], but must be consistent with the above density.

  • “cell volume” [m^3]

Liquid+Ice energy#

Energy content evaluator for a two-phase system, including energy in an ice phase.

Calculates energy, in [KJ], via the equation:

\[E = V * ( \phi (u_l s_l n_l + u_i s_i n_i) + (1-\phi_0) u_r \rho_r )\]

Specified with evaluator type: “liquid+ice energy”

Note this equation assumes that porosity is compressible, but is based on the uncompressed rock grain density (not bulk density). This means that porosity is the base, uncompressible value when used with the energy in the grain, but the larger, compressible value when used with the energy in the water.

Note that this ignores energy in the gas phase.

field-evaluator-type-liquid-ice-energy-spec

DEPENDENCIES:

  • “porosity” The porosity, including any compressibility. [-]

  • “base porosity” The uncompressed porosity (note this may be the same as porosity for incompressible cases) [-]

  • “molar density liquid” [mol m^-3]

  • “saturation liquid” [-]

  • “internal energy liquid” [KJ mol^-1]

  • “molar density ice” [mol m^-3]

  • “saturation ice” [-]

  • “internal energy ice” [KJ mol^-1]

  • “density rock” Units may be either [kg m^-3] or [mol m^-3]

  • “internal energy rock” Units may be either [KJ kg^-1] or [KJ mol^-1], but must be consistent with the above density.

  • “cell volume” [m^3]

Liquid+Ice+Gas energy#

Energy content evaluator for a three-phase, gas, liquid, ice system including the surrounding soil.

Calculates energy, in [KJ], via the equation:

\[E = V * ( \phi (u_l s_l n_l + u_i s_i n_i + u_g s_g n_g) + (1-\phi_0) u_r \rho_r )\]

Specified with evaluator type: “three phase energy”

Note this equation assumes that porosity is compressible, but is based on the uncompressed rock grain density (not bulk density). This means that porosity is the base, uncompressible value when used with the energy in the grain, but the larger, compressible value when used with the energy in the water.

field-evaluator-type-three-phase-energy-spec

DEPENDENCIES:

  • “porosity” The porosity, including any compressibility. [-]

  • “base porosity” The uncompressed porosity (note this may be the same as porosity for incompressible cases) [-]

  • “molar density liquid” [mol m^-3]

  • “saturation liquid” [-]

  • “internal energy liquid” [KJ mol^-1]

  • “molar density ice” [mol m^-3]

  • “saturation ice” [-]

  • “internal energy ice” [KJ mol^-1]

  • “molar density gas” [mol m^-3]

  • “saturation gas” [-]

  • “internal energy gas” [KJ mol^-1]

  • “density rock” Units may be either [kg m^-3] or [mol m^-3]

  • “internal energy rock” Units may be either [KJ kg^-1] or [KJ mol^-1], but must be consistent with the above density.

  • “cell volume” [m^3]

Surface water+ice energy#

Energy content for a surface water, partially frozen system.

The energy associated with ponded water, in [KJ], given by:

\[E = V * ( \eta h u_l n_l + (1 - \eta) h u_i n_i )\]

Specified with evaluator type: “surface ice energy”

field-evaluator-type-surface-ice-energy-spec

DEPENDENCIES:

  • “ponded depth” Height of water above the land surface [m]

  • “unfrozen fraction” The fraction of unfrozen water ranges from 0 to 1. [-]

  • “molar density liquid” [mol m^-3]

  • “internal energy liquid” [KJ mol^-1]

  • “molar density ice” [mol m^-3]

  • “internal energy ice” [KJ mol^-1]

  • “cell volume” [m^2]