Relative permeability test descriptions

The details of the capillary pressure curve implementations can be found in here.

Brooks-Corey

The Brooks and Corey (1966) relative permeability model is an extension of the previous Corey (1954) formulation where the relative permeability of the wetting phase is given by $var element = document.getElementById("moose-equation-381ed318-18b4-4a34-8ae7-0595abd2821a");katex.render("k_{\\mathrm{r, w}} = \\left(S_{\\mathrm{eff}}\\right)^{(2 + 3 \\lambda)/\\lambda},", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ and the relative permeability of the non-wetting phase is $var element = document.getElementById("moose-equation-9b76929a-8ffb-49e1-8bb8-4a8c307a9c0c");katex.render("k_{\\mathrm{r, nw}} = (1 - S_{\\mathrm{eff}})^2 \\left[1 - \\left(S_{\\mathrm{eff}}\\right)^{(2 + \\lambda)/\\lambda}\\right],", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ where $var element = document.getElementById("moose-equation-dd830561-4ab3-4a5a-898c-6bcdec3837f6");katex.render("\\lambda", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is a user-defined exponent. When $var element = document.getElementById("moose-equation-30558f78-c3ae-4b92-9c34-559ec541d18b");katex.render("\\lambda = 2", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ , this formulation reduces to the original Corey (1954) form.

Test 1

The input file for this test:

# Test Brooks-Corey relative permeability curve by varying saturation over the mesh
# Exponent lambda = 2 for both phases

[Mesh<<<{"href": "../../../../syntax/Mesh/index.html"}>>>]
  type = GeneratedMesh
  dim = 1
  nx = 20
[]

[GlobalParams<<<{"href": "../../../../syntax/GlobalParams/index.html"}>>>]
  PorousFlowDictator = dictator
[]

[Variables<<<{"href": "../../../../syntax/Variables/index.html"}>>>]
  [p0]
    initial_condition<<<{"description": "Specifies a constant initial condition for this variable"}>>> = 1e6
  []
  [s1]
  []
[]

[AuxVariables<<<{"href": "../../../../syntax/AuxVariables/index.html"}>>>]
  [s0aux]
    family<<<{"description": "Specifies the family of FE shape functions to use for this variable"}>>> = MONOMIAL
    order<<<{"description": "Specifies the order of the FE shape function to use for this variable (additional orders not listed are allowed)"}>>> = CONSTANT
  []
  [s1aux]
    family<<<{"description": "Specifies the family of FE shape functions to use for this variable"}>>> = MONOMIAL
    order<<<{"description": "Specifies the order of the FE shape function to use for this variable (additional orders not listed are allowed)"}>>> = CONSTANT
  []
  [kr0aux]
    family<<<{"description": "Specifies the family of FE shape functions to use for this variable"}>>> = MONOMIAL
    order<<<{"description": "Specifies the order of the FE shape function to use for this variable (additional orders not listed are allowed)"}>>> = CONSTANT
  []
  [kr1aux]
    family<<<{"description": "Specifies the family of FE shape functions to use for this variable"}>>> = MONOMIAL
    order<<<{"description": "Specifies the order of the FE shape function to use for this variable (additional orders not listed are allowed)"}>>> = CONSTANT
  []
[]

[AuxKernels<<<{"href": "../../../../syntax/AuxKernels/index.html"}>>>]
  [s0]
    type = PorousFlowPropertyAux<<<{"description": "AuxKernel to provide access to properties evaluated at quadpoints. Note that elemental AuxVariables must be used, so that these properties are integrated over each element.", "href": "../../../../source/auxkernels/PorousFlowPropertyAux.html"}>>>
    property<<<{"description": "The fluid property that this auxillary kernel is to calculate"}>>> = saturation
    phase<<<{"description": "The index of the phase this auxillary kernel acts on"}>>> = 0
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = s0aux
  []
  [s1]
    type = PorousFlowPropertyAux<<<{"description": "AuxKernel to provide access to properties evaluated at quadpoints. Note that elemental AuxVariables must be used, so that these properties are integrated over each element.", "href": "../../../../source/auxkernels/PorousFlowPropertyAux.html"}>>>
    property<<<{"description": "The fluid property that this auxillary kernel is to calculate"}>>> = saturation
    phase<<<{"description": "The index of the phase this auxillary kernel acts on"}>>> = 1
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = s1aux
  []
  [kr0]
    type = PorousFlowPropertyAux<<<{"description": "AuxKernel to provide access to properties evaluated at quadpoints. Note that elemental AuxVariables must be used, so that these properties are integrated over each element.", "href": "../../../../source/auxkernels/PorousFlowPropertyAux.html"}>>>
    property<<<{"description": "The fluid property that this auxillary kernel is to calculate"}>>> = relperm
    phase<<<{"description": "The index of the phase this auxillary kernel acts on"}>>> = 0
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = kr0aux
  []
  [kr1]
    type = PorousFlowPropertyAux<<<{"description": "AuxKernel to provide access to properties evaluated at quadpoints. Note that elemental AuxVariables must be used, so that these properties are integrated over each element.", "href": "../../../../source/auxkernels/PorousFlowPropertyAux.html"}>>>
    property<<<{"description": "The fluid property that this auxillary kernel is to calculate"}>>> = relperm
    phase<<<{"description": "The index of the phase this auxillary kernel acts on"}>>> = 1
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = kr1aux
  []
[]

[Functions<<<{"href": "../../../../syntax/Functions/index.html"}>>>]
  [s1]
    type = ParsedFunction<<<{"description": "Function created by parsing a string", "href": "../../../../source/functions/MooseParsedFunction.html"}>>>
    expression<<<{"description": "The user defined function."}>>> = x
  []
[]

[ICs<<<{"href": "../../../../syntax/ICs/index.html"}>>>]
  [s1]
    type = FunctionIC<<<{"description": "An initial condition that uses a normal function of x, y, z to produce values (and optionally gradients) for a field variable.", "href": "../../../../source/ics/FunctionIC.html"}>>>
    variable<<<{"description": "The variable this initial condition is supposed to provide values for."}>>> = s1
    function<<<{"description": "The initial condition function."}>>> = s1
  []
[]

[Kernels<<<{"href": "../../../../syntax/Kernels/index.html"}>>>]
  [p0]
    type = Diffusion<<<{"description": "The Laplacian operator ($-\\nabla \\cdot \\nabla u$), with the weak form of $(\\nabla \\phi_i, \\nabla u_h)$.", "href": "../../../../source/kernels/Diffusion.html"}>>>
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = p0
  []
  [s1]
    type = Diffusion<<<{"description": "The Laplacian operator ($-\\nabla \\cdot \\nabla u$), with the weak form of $(\\nabla \\phi_i, \\nabla u_h)$.", "href": "../../../../source/kernels/Diffusion.html"}>>>
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = s1
  []
[]

[UserObjects<<<{"href": "../../../../syntax/UserObjects/index.html"}>>>]
  [dictator]
    type = PorousFlowDictator<<<{"description": "Holds information on the PorousFlow variable names", "href": "../../../../source/userobjects/PorousFlowDictator.html"}>>>
    porous_flow_vars<<<{"description": "List of primary variables that are used in the PorousFlow simulation.  Jacobian entries involving derivatives wrt these variables will be computed.  In single-phase models you will just have one (eg 'pressure'), in two-phase models you will have two (eg 'p_water p_gas', or 'p_water s_water'), etc."}>>> = 'p0 s1'
    number_fluid_phases<<<{"description": "The number of fluid phases in the simulation"}>>> = 2
    number_fluid_components<<<{"description": "The number of fluid components in the simulation"}>>> = 2
  []
  [pc]
    type = PorousFlowCapillaryPressureConst<<<{"description": "Constant capillary pressure", "href": "../../../../source/userobjects/PorousFlowCapillaryPressureConst.html"}>>>
    pc<<<{"description": "Constant capillary pressure (Pa). Default is 0"}>>> = 0
  []
[]

[Materials<<<{"href": "../../../../syntax/Materials/index.html"}>>>]
  [temperature]
    type = PorousFlowTemperature<<<{"description": "Material to provide temperature at the quadpoints or nodes and derivatives of it with respect to the PorousFlow variables", "href": "../../../../source/materials/PorousFlowTemperature.html"}>>>
  []
  [ppss]
    type = PorousFlow2PhasePS<<<{"description": "This Material calculates the 2 porepressures and the 2 saturations in a 2-phase situation, and derivatives of these with respect to the PorousFlowVariables.", "href": "../../../../source/materials/PorousFlow2PhasePS.html"}>>>
    phase0_porepressure<<<{"description": "Variable that is the porepressure of phase 0 (the liquid phase)"}>>> = p0
    phase1_saturation<<<{"description": "Variable that is the saturation of phase 1 (the gas phase)"}>>> = s1
    capillary_pressure<<<{"description": "Name of the UserObject defining the capillary pressure"}>>> = pc
  []
  [kr0]
    type = PorousFlowRelativePermeabilityBC<<<{"description": "Brooks-Corey relative permeability", "href": "../../../../source/materials/PorousFlowRelativePermeabilityBC.html"}>>>
    phase<<<{"description": "The phase number"}>>> = 0
    lambda<<<{"description": "The Brooks-Corey exponent of the phase"}>>> = 2
  []
  [kr1]
    type = PorousFlowRelativePermeabilityBC<<<{"description": "Brooks-Corey relative permeability", "href": "../../../../source/materials/PorousFlowRelativePermeabilityBC.html"}>>>
    phase<<<{"description": "The phase number"}>>> = 1
    lambda<<<{"description": "The Brooks-Corey exponent of the phase"}>>> = 2
    nw_phase<<<{"description": "Set true if this is the non-wetting phase"}>>> = true
  []
[]

[VectorPostprocessors<<<{"href": "../../../../syntax/VectorPostprocessors/index.html"}>>>]
  [vpp]
    type = LineValueSampler<<<{"description": "Samples variable(s) along a specified line", "href": "../../../../source/vectorpostprocessors/LineValueSampler.html"}>>>
    warn_discontinuous_face_values<<<{"description": "Whether to return a warning if a discontinuous variable is sampled on a face"}>>> = false
    variable<<<{"description": "The names of the variables that this VectorPostprocessor operates on"}>>> = 's0aux s1aux kr0aux kr1aux'
    start_point<<<{"description": "The beginning of the line"}>>> = '0 0 0'
    end_point<<<{"description": "The ending of the line"}>>> = '1 0 0'
    num_points<<<{"description": "The number of points to sample along the line"}>>> = 20
    sort_by<<<{"description": "What to sort the samples by"}>>> = id
  []
[]

[Executioner<<<{"href": "../../../../syntax/Executioner/index.html"}>>>]
  type = Steady
  solve_type = NEWTON
  nl_abs_tol = 1e-8
[]

[BCs<<<{"href": "../../../../syntax/BCs/index.html"}>>>]
  [sleft]
    type = DirichletBC<<<{"description": "Imposes the essential boundary condition $u=g$, where $g$ is a constant, controllable value.", "href": "../../../../source/bcs/DirichletBC.html"}>>>
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = s1
    value<<<{"description": "Value of the BC"}>>> = 0
    boundary<<<{"description": "The list of boundary IDs from the mesh where this object applies"}>>> = left
  []
  [sright]
    type = DirichletBC<<<{"description": "Imposes the essential boundary condition $u=g$, where $g$ is a constant, controllable value.", "href": "../../../../source/bcs/DirichletBC.html"}>>>
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = s1
    value<<<{"description": "Value of the BC"}>>> = 1
    boundary<<<{"description": "The list of boundary IDs from the mesh where this object applies"}>>> = right
  []
[]

[Outputs<<<{"href": "../../../../syntax/Outputs/index.html"}>>>]
  csv<<<{"description": "Output the scalar variable and postprocessors to a *.csv file using the default CSV output."}>>> = true
  execute_on<<<{"description": "The list of flag(s) indicating when this object should be executed. For a description of each flag, see https://mooseframework.inl.gov/source/interfaces/SetupInterface.html."}>>> = timestep_end
[]

Figure 1: Brooks-Corey relative permeability Test case 1

Test 2

The input file for this test:

# Test Brooks-Corey relative permeability curve by varying saturation over the mesh
# Exponent lambda = 2 for both phases
# Residual saturation of phase 0: s0r = 0.2
# Residual saturation of phase 1: s1r = 0.3

[Mesh<<<{"href": "../../../../syntax/Mesh/index.html"}>>>]
  type = GeneratedMesh
  dim = 1
  nx = 20
[]

[GlobalParams<<<{"href": "../../../../syntax/GlobalParams/index.html"}>>>]
  PorousFlowDictator = dictator
[]

[Variables<<<{"href": "../../../../syntax/Variables/index.html"}>>>]
  [p0]
    initial_condition<<<{"description": "Specifies a constant initial condition for this variable"}>>> = 1e6
  []
  [s1]
  []
[]

[AuxVariables<<<{"href": "../../../../syntax/AuxVariables/index.html"}>>>]
  [s0aux]
    family<<<{"description": "Specifies the family of FE shape functions to use for this variable"}>>> = MONOMIAL
    order<<<{"description": "Specifies the order of the FE shape function to use for this variable (additional orders not listed are allowed)"}>>> = CONSTANT
  []
  [s1aux]
    family<<<{"description": "Specifies the family of FE shape functions to use for this variable"}>>> = MONOMIAL
    order<<<{"description": "Specifies the order of the FE shape function to use for this variable (additional orders not listed are allowed)"}>>> = CONSTANT
  []
  [kr0aux]
    family<<<{"description": "Specifies the family of FE shape functions to use for this variable"}>>> = MONOMIAL
    order<<<{"description": "Specifies the order of the FE shape function to use for this variable (additional orders not listed are allowed)"}>>> = CONSTANT
  []
  [kr1aux]
    family<<<{"description": "Specifies the family of FE shape functions to use for this variable"}>>> = MONOMIAL
    order<<<{"description": "Specifies the order of the FE shape function to use for this variable (additional orders not listed are allowed)"}>>> = CONSTANT
  []
[]

[AuxKernels<<<{"href": "../../../../syntax/AuxKernels/index.html"}>>>]
  [s0]
    type = PorousFlowPropertyAux<<<{"description": "AuxKernel to provide access to properties evaluated at quadpoints. Note that elemental AuxVariables must be used, so that these properties are integrated over each element.", "href": "../../../../source/auxkernels/PorousFlowPropertyAux.html"}>>>
    property<<<{"description": "The fluid property that this auxillary kernel is to calculate"}>>> = saturation
    phase<<<{"description": "The index of the phase this auxillary kernel acts on"}>>> = 0
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = s0aux
  []
  [s1]
    type = PorousFlowPropertyAux<<<{"description": "AuxKernel to provide access to properties evaluated at quadpoints. Note that elemental AuxVariables must be used, so that these properties are integrated over each element.", "href": "../../../../source/auxkernels/PorousFlowPropertyAux.html"}>>>
    property<<<{"description": "The fluid property that this auxillary kernel is to calculate"}>>> = saturation
    phase<<<{"description": "The index of the phase this auxillary kernel acts on"}>>> = 1
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = s1aux
  []
  [kr0]
    type = PorousFlowPropertyAux<<<{"description": "AuxKernel to provide access to properties evaluated at quadpoints. Note that elemental AuxVariables must be used, so that these properties are integrated over each element.", "href": "../../../../source/auxkernels/PorousFlowPropertyAux.html"}>>>
    property<<<{"description": "The fluid property that this auxillary kernel is to calculate"}>>> = relperm
    phase<<<{"description": "The index of the phase this auxillary kernel acts on"}>>> = 0
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = kr0aux
  []
  [kr1]
    type = PorousFlowPropertyAux<<<{"description": "AuxKernel to provide access to properties evaluated at quadpoints. Note that elemental AuxVariables must be used, so that these properties are integrated over each element.", "href": "../../../../source/auxkernels/PorousFlowPropertyAux.html"}>>>
    property<<<{"description": "The fluid property that this auxillary kernel is to calculate"}>>> = relperm
    phase<<<{"description": "The index of the phase this auxillary kernel acts on"}>>> = 1
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = kr1aux
  []
[]

[Functions<<<{"href": "../../../../syntax/Functions/index.html"}>>>]
  [s1]
    type = ParsedFunction<<<{"description": "Function created by parsing a string", "href": "../../../../source/functions/MooseParsedFunction.html"}>>>
    expression<<<{"description": "The user defined function."}>>> = x
  []
[]

[ICs<<<{"href": "../../../../syntax/ICs/index.html"}>>>]
  [s1]
    type = FunctionIC<<<{"description": "An initial condition that uses a normal function of x, y, z to produce values (and optionally gradients) for a field variable.", "href": "../../../../source/ics/FunctionIC.html"}>>>
    variable<<<{"description": "The variable this initial condition is supposed to provide values for."}>>> = s1
    function<<<{"description": "The initial condition function."}>>> = s1
  []
[]

[Kernels<<<{"href": "../../../../syntax/Kernels/index.html"}>>>]
  [p0]
    type = Diffusion<<<{"description": "The Laplacian operator ($-\\nabla \\cdot \\nabla u$), with the weak form of $(\\nabla \\phi_i, \\nabla u_h)$.", "href": "../../../../source/kernels/Diffusion.html"}>>>
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = p0
  []
  [s1]
    type = Diffusion<<<{"description": "The Laplacian operator ($-\\nabla \\cdot \\nabla u$), with the weak form of $(\\nabla \\phi_i, \\nabla u_h)$.", "href": "../../../../source/kernels/Diffusion.html"}>>>
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = s1
  []
[]

[UserObjects<<<{"href": "../../../../syntax/UserObjects/index.html"}>>>]
  [dictator]
    type = PorousFlowDictator<<<{"description": "Holds information on the PorousFlow variable names", "href": "../../../../source/userobjects/PorousFlowDictator.html"}>>>
    porous_flow_vars<<<{"description": "List of primary variables that are used in the PorousFlow simulation.  Jacobian entries involving derivatives wrt these variables will be computed.  In single-phase models you will just have one (eg 'pressure'), in two-phase models you will have two (eg 'p_water p_gas', or 'p_water s_water'), etc."}>>> = 'p0 s1'
    number_fluid_phases<<<{"description": "The number of fluid phases in the simulation"}>>> = 2
    number_fluid_components<<<{"description": "The number of fluid components in the simulation"}>>> = 2
  []
  [pc]
    type = PorousFlowCapillaryPressureConst<<<{"description": "Constant capillary pressure", "href": "../../../../source/userobjects/PorousFlowCapillaryPressureConst.html"}>>>
    pc<<<{"description": "Constant capillary pressure (Pa). Default is 0"}>>> = 0
  []
[]

[Materials<<<{"href": "../../../../syntax/Materials/index.html"}>>>]
  [temperature]
    type = PorousFlowTemperature<<<{"description": "Material to provide temperature at the quadpoints or nodes and derivatives of it with respect to the PorousFlow variables", "href": "../../../../source/materials/PorousFlowTemperature.html"}>>>
  []
  [ppss]
    type = PorousFlow2PhasePS<<<{"description": "This Material calculates the 2 porepressures and the 2 saturations in a 2-phase situation, and derivatives of these with respect to the PorousFlowVariables.", "href": "../../../../source/materials/PorousFlow2PhasePS.html"}>>>
    phase0_porepressure<<<{"description": "Variable that is the porepressure of phase 0 (the liquid phase)"}>>> = p0
    phase1_saturation<<<{"description": "Variable that is the saturation of phase 1 (the gas phase)"}>>> = s1
    capillary_pressure<<<{"description": "Name of the UserObject defining the capillary pressure"}>>> = pc
  []
  [kr0]
    type = PorousFlowRelativePermeabilityBC<<<{"description": "Brooks-Corey relative permeability", "href": "../../../../source/materials/PorousFlowRelativePermeabilityBC.html"}>>>
    phase<<<{"description": "The phase number"}>>> = 0
    lambda<<<{"description": "The Brooks-Corey exponent of the phase"}>>> = 2
    s_res<<<{"description": "The residual saturation of the phase j. Must be between 0 and 1"}>>> = 0.2
    sum_s_res<<<{"description": "Sum of residual saturations over all phases.  Must be between 0 and 1"}>>> = 0.5
  []
  [kr1]
    type = PorousFlowRelativePermeabilityBC<<<{"description": "Brooks-Corey relative permeability", "href": "../../../../source/materials/PorousFlowRelativePermeabilityBC.html"}>>>
    phase<<<{"description": "The phase number"}>>> = 1
    lambda<<<{"description": "The Brooks-Corey exponent of the phase"}>>> = 2
    nw_phase<<<{"description": "Set true if this is the non-wetting phase"}>>> = true
    s_res<<<{"description": "The residual saturation of the phase j. Must be between 0 and 1"}>>> = 0.3
    sum_s_res<<<{"description": "Sum of residual saturations over all phases.  Must be between 0 and 1"}>>> = 0.5
  []
[]

[VectorPostprocessors<<<{"href": "../../../../syntax/VectorPostprocessors/index.html"}>>>]
  [vpp]
    type = LineValueSampler<<<{"description": "Samples variable(s) along a specified line", "href": "../../../../source/vectorpostprocessors/LineValueSampler.html"}>>>
    warn_discontinuous_face_values<<<{"description": "Whether to return a warning if a discontinuous variable is sampled on a face"}>>> = false
    variable<<<{"description": "The names of the variables that this VectorPostprocessor operates on"}>>> = 's0aux s1aux kr0aux kr1aux'
    start_point<<<{"description": "The beginning of the line"}>>> = '0 0 0'
    end_point<<<{"description": "The ending of the line"}>>> = '1 0 0'
    num_points<<<{"description": "The number of points to sample along the line"}>>> = 20
    sort_by<<<{"description": "What to sort the samples by"}>>> = id
  []
[]

[Executioner<<<{"href": "../../../../syntax/Executioner/index.html"}>>>]
  type = Steady
  solve_type = NEWTON
  nl_abs_tol = 1e-8
[]

[BCs<<<{"href": "../../../../syntax/BCs/index.html"}>>>]
  [sleft]
    type = DirichletBC<<<{"description": "Imposes the essential boundary condition $u=g$, where $g$ is a constant, controllable value.", "href": "../../../../source/bcs/DirichletBC.html"}>>>
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = s1
    value<<<{"description": "Value of the BC"}>>> = 0
    boundary<<<{"description": "The list of boundary IDs from the mesh where this object applies"}>>> = left
  []
  [sright]
    type = DirichletBC<<<{"description": "Imposes the essential boundary condition $u=g$, where $g$ is a constant, controllable value.", "href": "../../../../source/bcs/DirichletBC.html"}>>>
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = s1
    value<<<{"description": "Value of the BC"}>>> = 1
    boundary<<<{"description": "The list of boundary IDs from the mesh where this object applies"}>>> = right
  []
[]

[Outputs<<<{"href": "../../../../syntax/Outputs/index.html"}>>>]
  csv<<<{"description": "Output the scalar variable and postprocessors to a *.csv file using the default CSV output."}>>> = true
  execute_on<<<{"description": "The list of flag(s) indicating when this object should be executed. For a description of each flag, see https://mooseframework.inl.gov/source/interfaces/SetupInterface.html."}>>> = timestep_end
[]

Figure 2: Brooks-Corey relative permeability Test case 2

Corey

PorousFlowRelativePermeabilityCorey

The relative permeability of the phase is given by Corey (1954) $var element = document.getElementById("moose-equation-2bb2d925-b283-4004-a049-188c7f05c953");katex.render("k_{\\mathrm{r}} = S_{\\mathrm{eff}}^{n},", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ where $var element = document.getElementById("moose-equation-cccf8b0f-49b2-4cd4-b7d5-e433bd968db7");katex.render("n", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is a user-defined quantity. Originally, Corey (1954) used $var element = document.getElementById("moose-equation-892bd965-2ce6-44c3-9703-a7dd1431eefb");katex.render("n = 4", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ for the wetting phase, but the PorousFlow module allows an arbitrary exponent to be used.

Test 1

The input file for this test:

# Test Corey relative permeability curve by varying saturation over the mesh
# Corey exponent n = 1 for both phases (linear residual saturation)
# No residual saturation in either phase

[Mesh<<<{"href": "../../../../syntax/Mesh/index.html"}>>>]
  type = GeneratedMesh
  dim = 1
  nx = 20
[]

[GlobalParams<<<{"href": "../../../../syntax/GlobalParams/index.html"}>>>]
  PorousFlowDictator = dictator
[]

[Variables<<<{"href": "../../../../syntax/Variables/index.html"}>>>]
  [p0]
    initial_condition<<<{"description": "Specifies a constant initial condition for this variable"}>>> = 1e6
  []
  [s1]
  []
[]

[AuxVariables<<<{"href": "../../../../syntax/AuxVariables/index.html"}>>>]
  [s0aux]
    family<<<{"description": "Specifies the family of FE shape functions to use for this variable"}>>> = MONOMIAL
    order<<<{"description": "Specifies the order of the FE shape function to use for this variable (additional orders not listed are allowed)"}>>> = CONSTANT
  []
  [s1aux]
    family<<<{"description": "Specifies the family of FE shape functions to use for this variable"}>>> = MONOMIAL
    order<<<{"description": "Specifies the order of the FE shape function to use for this variable (additional orders not listed are allowed)"}>>> = CONSTANT
  []
  [kr0aux]
    family<<<{"description": "Specifies the family of FE shape functions to use for this variable"}>>> = MONOMIAL
    order<<<{"description": "Specifies the order of the FE shape function to use for this variable (additional orders not listed are allowed)"}>>> = CONSTANT
  []
  [kr1aux]
    family<<<{"description": "Specifies the family of FE shape functions to use for this variable"}>>> = MONOMIAL
    order<<<{"description": "Specifies the order of the FE shape function to use for this variable (additional orders not listed are allowed)"}>>> = CONSTANT
  []
[]

[AuxKernels<<<{"href": "../../../../syntax/AuxKernels/index.html"}>>>]
  [s0]
    type = PorousFlowPropertyAux<<<{"description": "AuxKernel to provide access to properties evaluated at quadpoints. Note that elemental AuxVariables must be used, so that these properties are integrated over each element.", "href": "../../../../source/auxkernels/PorousFlowPropertyAux.html"}>>>
    property<<<{"description": "The fluid property that this auxillary kernel is to calculate"}>>> = saturation
    phase<<<{"description": "The index of the phase this auxillary kernel acts on"}>>> = 0
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = s0aux
  []
  [s1]
    type = PorousFlowPropertyAux<<<{"description": "AuxKernel to provide access to properties evaluated at quadpoints. Note that elemental AuxVariables must be used, so that these properties are integrated over each element.", "href": "../../../../source/auxkernels/PorousFlowPropertyAux.html"}>>>
    property<<<{"description": "The fluid property that this auxillary kernel is to calculate"}>>> = saturation
    phase<<<{"description": "The index of the phase this auxillary kernel acts on"}>>> = 1
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = s1aux
  []
  [kr0]
    type = PorousFlowPropertyAux<<<{"description": "AuxKernel to provide access to properties evaluated at quadpoints. Note that elemental AuxVariables must be used, so that these properties are integrated over each element.", "href": "../../../../source/auxkernels/PorousFlowPropertyAux.html"}>>>
    property<<<{"description": "The fluid property that this auxillary kernel is to calculate"}>>> = relperm
    phase<<<{"description": "The index of the phase this auxillary kernel acts on"}>>> = 0
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = kr0aux
  []
  [kr1]
    type = PorousFlowPropertyAux<<<{"description": "AuxKernel to provide access to properties evaluated at quadpoints. Note that elemental AuxVariables must be used, so that these properties are integrated over each element.", "href": "../../../../source/auxkernels/PorousFlowPropertyAux.html"}>>>
    property<<<{"description": "The fluid property that this auxillary kernel is to calculate"}>>> = relperm
    phase<<<{"description": "The index of the phase this auxillary kernel acts on"}>>> = 1
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = kr1aux
  []
[]

[Functions<<<{"href": "../../../../syntax/Functions/index.html"}>>>]
  [s1]
    type = ParsedFunction<<<{"description": "Function created by parsing a string", "href": "../../../../source/functions/MooseParsedFunction.html"}>>>
    expression<<<{"description": "The user defined function."}>>> = x
  []
[]

[ICs<<<{"href": "../../../../syntax/ICs/index.html"}>>>]
  [s1]
    type = FunctionIC<<<{"description": "An initial condition that uses a normal function of x, y, z to produce values (and optionally gradients) for a field variable.", "href": "../../../../source/ics/FunctionIC.html"}>>>
    variable<<<{"description": "The variable this initial condition is supposed to provide values for."}>>> = s1
    function<<<{"description": "The initial condition function."}>>> = s1
  []
[]

[Kernels<<<{"href": "../../../../syntax/Kernels/index.html"}>>>]
  [p0]
    type = Diffusion<<<{"description": "The Laplacian operator ($-\\nabla \\cdot \\nabla u$), with the weak form of $(\\nabla \\phi_i, \\nabla u_h)$.", "href": "../../../../source/kernels/Diffusion.html"}>>>
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = p0
  []
  [s1]
    type = Diffusion<<<{"description": "The Laplacian operator ($-\\nabla \\cdot \\nabla u$), with the weak form of $(\\nabla \\phi_i, \\nabla u_h)$.", "href": "../../../../source/kernels/Diffusion.html"}>>>
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = s1
  []
[]

[UserObjects<<<{"href": "../../../../syntax/UserObjects/index.html"}>>>]
  [dictator]
    type = PorousFlowDictator<<<{"description": "Holds information on the PorousFlow variable names", "href": "../../../../source/userobjects/PorousFlowDictator.html"}>>>
    porous_flow_vars<<<{"description": "List of primary variables that are used in the PorousFlow simulation.  Jacobian entries involving derivatives wrt these variables will be computed.  In single-phase models you will just have one (eg 'pressure'), in two-phase models you will have two (eg 'p_water p_gas', or 'p_water s_water'), etc."}>>> = 'p0 s1'
    number_fluid_phases<<<{"description": "The number of fluid phases in the simulation"}>>> = 2
    number_fluid_components<<<{"description": "The number of fluid components in the simulation"}>>> = 2
  []
  [pc]
    type = PorousFlowCapillaryPressureConst<<<{"description": "Constant capillary pressure", "href": "../../../../source/userobjects/PorousFlowCapillaryPressureConst.html"}>>>
    pc<<<{"description": "Constant capillary pressure (Pa). Default is 0"}>>> = 0
  []
[]

[Materials<<<{"href": "../../../../syntax/Materials/index.html"}>>>]
  [temperature]
    type = PorousFlowTemperature<<<{"description": "Material to provide temperature at the quadpoints or nodes and derivatives of it with respect to the PorousFlow variables", "href": "../../../../source/materials/PorousFlowTemperature.html"}>>>
  []
  [ppss]
    type = PorousFlow2PhasePS<<<{"description": "This Material calculates the 2 porepressures and the 2 saturations in a 2-phase situation, and derivatives of these with respect to the PorousFlowVariables.", "href": "../../../../source/materials/PorousFlow2PhasePS.html"}>>>
    phase0_porepressure<<<{"description": "Variable that is the porepressure of phase 0 (the liquid phase)"}>>> = p0
    phase1_saturation<<<{"description": "Variable that is the saturation of phase 1 (the gas phase)"}>>> = s1
    capillary_pressure<<<{"description": "Name of the UserObject defining the capillary pressure"}>>> = pc
  []
  [kr0]
    type = PorousFlowRelativePermeabilityCorey<<<{"description": "This Material calculates relative permeability of the fluid phase, using the simple Corey model ((S-S_res)/(1-sum(S_res)))^n", "href": "../../../../source/materials/PorousFlowRelativePermeabilityCorey.html"}>>>
    phase<<<{"description": "The phase number"}>>> = 0
    n<<<{"description": "The Corey exponent of the phase."}>>> = 1
  []
  [kr1]
    type = PorousFlowRelativePermeabilityCorey<<<{"description": "This Material calculates relative permeability of the fluid phase, using the simple Corey model ((S-S_res)/(1-sum(S_res)))^n", "href": "../../../../source/materials/PorousFlowRelativePermeabilityCorey.html"}>>>
    phase<<<{"description": "The phase number"}>>> = 1
    n<<<{"description": "The Corey exponent of the phase."}>>> = 1
  []
[]

[VectorPostprocessors<<<{"href": "../../../../syntax/VectorPostprocessors/index.html"}>>>]
  [vpp]
    type = LineValueSampler<<<{"description": "Samples variable(s) along a specified line", "href": "../../../../source/vectorpostprocessors/LineValueSampler.html"}>>>
    warn_discontinuous_face_values<<<{"description": "Whether to return a warning if a discontinuous variable is sampled on a face"}>>> = false
    variable<<<{"description": "The names of the variables that this VectorPostprocessor operates on"}>>> = 's0aux s1aux kr0aux kr1aux'
    start_point<<<{"description": "The beginning of the line"}>>> = '0 0 0'
    end_point<<<{"description": "The ending of the line"}>>> = '1 0 0'
    num_points<<<{"description": "The number of points to sample along the line"}>>> = 20
    sort_by<<<{"description": "What to sort the samples by"}>>> = id
  []
[]

[Executioner<<<{"href": "../../../../syntax/Executioner/index.html"}>>>]
  type = Steady
  solve_type = NEWTON
  nl_abs_tol = 1e-8
[]

[BCs<<<{"href": "../../../../syntax/BCs/index.html"}>>>]
  [sleft]
    type = DirichletBC<<<{"description": "Imposes the essential boundary condition $u=g$, where $g$ is a constant, controllable value.", "href": "../../../../source/bcs/DirichletBC.html"}>>>
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = s1
    value<<<{"description": "Value of the BC"}>>> = 0
    boundary<<<{"description": "The list of boundary IDs from the mesh where this object applies"}>>> = left
  []
  [sright]
    type = DirichletBC<<<{"description": "Imposes the essential boundary condition $u=g$, where $g$ is a constant, controllable value.", "href": "../../../../source/bcs/DirichletBC.html"}>>>
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = s1
    value<<<{"description": "Value of the BC"}>>> = 1
    boundary<<<{"description": "The list of boundary IDs from the mesh where this object applies"}>>> = right
  []
[]

[Outputs<<<{"href": "../../../../syntax/Outputs/index.html"}>>>]
  csv<<<{"description": "Output the scalar variable and postprocessors to a *.csv file using the default CSV output."}>>> = true
  execute_on<<<{"description": "The list of flag(s) indicating when this object should be executed. For a description of each flag, see https://mooseframework.inl.gov/source/interfaces/SetupInterface.html."}>>> = timestep_end
[]

Figure 3: Corey relative permeability Test case 1

Test 2

The input file for this test:

# Test Corey relative permeability curve by varying saturation over the mesh
# Corey exponent n = 2 for both phases
# No residual saturation in either phase

[Mesh<<<{"href": "../../../../syntax/Mesh/index.html"}>>>]
  type = GeneratedMesh
  dim = 1
  nx = 20
[]

[GlobalParams<<<{"href": "../../../../syntax/GlobalParams/index.html"}>>>]
  PorousFlowDictator = dictator
[]

[Variables<<<{"href": "../../../../syntax/Variables/index.html"}>>>]
  [p0]
    initial_condition<<<{"description": "Specifies a constant initial condition for this variable"}>>> = 1e6
  []
  [s1]
    family<<<{"description": "Specifies the family of FE shape functions to use for this variable"}>>> = LAGRANGE
    order<<<{"description": "Specifies the order of the FE shape function to use for this variable (additional orders not listed are allowed)"}>>> = FIRST
  []
[]

[AuxVariables<<<{"href": "../../../../syntax/AuxVariables/index.html"}>>>]
  [s0aux]
    family<<<{"description": "Specifies the family of FE shape functions to use for this variable"}>>> = MONOMIAL
    order<<<{"description": "Specifies the order of the FE shape function to use for this variable (additional orders not listed are allowed)"}>>> = CONSTANT
  []
  [s1aux]
    family<<<{"description": "Specifies the family of FE shape functions to use for this variable"}>>> = MONOMIAL
    order<<<{"description": "Specifies the order of the FE shape function to use for this variable (additional orders not listed are allowed)"}>>> = CONSTANT
  []
  [kr0aux]
    family<<<{"description": "Specifies the family of FE shape functions to use for this variable"}>>> = MONOMIAL
    order<<<{"description": "Specifies the order of the FE shape function to use for this variable (additional orders not listed are allowed)"}>>> = CONSTANT
  []
  [kr1aux]
    family<<<{"description": "Specifies the family of FE shape functions to use for this variable"}>>> = MONOMIAL
    order<<<{"description": "Specifies the order of the FE shape function to use for this variable (additional orders not listed are allowed)"}>>> = CONSTANT
  []
[]

[AuxKernels<<<{"href": "../../../../syntax/AuxKernels/index.html"}>>>]
  [s0]
    type = PorousFlowPropertyAux<<<{"description": "AuxKernel to provide access to properties evaluated at quadpoints. Note that elemental AuxVariables must be used, so that these properties are integrated over each element.", "href": "../../../../source/auxkernels/PorousFlowPropertyAux.html"}>>>
    property<<<{"description": "The fluid property that this auxillary kernel is to calculate"}>>> = saturation
    phase<<<{"description": "The index of the phase this auxillary kernel acts on"}>>> = 0
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = s0aux
  []
  [s1]
    type = PorousFlowPropertyAux<<<{"description": "AuxKernel to provide access to properties evaluated at quadpoints. Note that elemental AuxVariables must be used, so that these properties are integrated over each element.", "href": "../../../../source/auxkernels/PorousFlowPropertyAux.html"}>>>
    property<<<{"description": "The fluid property that this auxillary kernel is to calculate"}>>> = saturation
    phase<<<{"description": "The index of the phase this auxillary kernel acts on"}>>> = 1
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = s1aux
  []
  [kr0]
    type = PorousFlowPropertyAux<<<{"description": "AuxKernel to provide access to properties evaluated at quadpoints. Note that elemental AuxVariables must be used, so that these properties are integrated over each element.", "href": "../../../../source/auxkernels/PorousFlowPropertyAux.html"}>>>
    property<<<{"description": "The fluid property that this auxillary kernel is to calculate"}>>> = relperm
    phase<<<{"description": "The index of the phase this auxillary kernel acts on"}>>> = 0
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = kr0aux
  []
  [kr1]
    type = PorousFlowPropertyAux<<<{"description": "AuxKernel to provide access to properties evaluated at quadpoints. Note that elemental AuxVariables must be used, so that these properties are integrated over each element.", "href": "../../../../source/auxkernels/PorousFlowPropertyAux.html"}>>>
    property<<<{"description": "The fluid property that this auxillary kernel is to calculate"}>>> = relperm
    phase<<<{"description": "The index of the phase this auxillary kernel acts on"}>>> = 1
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = kr1aux
  []
[]

[Functions<<<{"href": "../../../../syntax/Functions/index.html"}>>>]
  [s1]
    type = ParsedFunction<<<{"description": "Function created by parsing a string", "href": "../../../../source/functions/MooseParsedFunction.html"}>>>
    expression<<<{"description": "The user defined function."}>>> = x
  []
[]

[ICs<<<{"href": "../../../../syntax/ICs/index.html"}>>>]
  [s1]
    type = FunctionIC<<<{"description": "An initial condition that uses a normal function of x, y, z to produce values (and optionally gradients) for a field variable.", "href": "../../../../source/ics/FunctionIC.html"}>>>
    variable<<<{"description": "The variable this initial condition is supposed to provide values for."}>>> = s1
    function<<<{"description": "The initial condition function."}>>> = s1
  []
[]

[Kernels<<<{"href": "../../../../syntax/Kernels/index.html"}>>>]
  [p0]
    type = Diffusion<<<{"description": "The Laplacian operator ($-\\nabla \\cdot \\nabla u$), with the weak form of $(\\nabla \\phi_i, \\nabla u_h)$.", "href": "../../../../source/kernels/Diffusion.html"}>>>
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = p0
  []
  [s1]
    type = Diffusion<<<{"description": "The Laplacian operator ($-\\nabla \\cdot \\nabla u$), with the weak form of $(\\nabla \\phi_i, \\nabla u_h)$.", "href": "../../../../source/kernels/Diffusion.html"}>>>
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = s1
  []
[]

[UserObjects<<<{"href": "../../../../syntax/UserObjects/index.html"}>>>]
  [dictator]
    type = PorousFlowDictator<<<{"description": "Holds information on the PorousFlow variable names", "href": "../../../../source/userobjects/PorousFlowDictator.html"}>>>
    porous_flow_vars<<<{"description": "List of primary variables that are used in the PorousFlow simulation.  Jacobian entries involving derivatives wrt these variables will be computed.  In single-phase models you will just have one (eg 'pressure'), in two-phase models you will have two (eg 'p_water p_gas', or 'p_water s_water'), etc."}>>> = 'p0 s1'
    number_fluid_phases<<<{"description": "The number of fluid phases in the simulation"}>>> = 2
    number_fluid_components<<<{"description": "The number of fluid components in the simulation"}>>> = 2
  []
  [pc]
    type = PorousFlowCapillaryPressureConst<<<{"description": "Constant capillary pressure", "href": "../../../../source/userobjects/PorousFlowCapillaryPressureConst.html"}>>>
    pc<<<{"description": "Constant capillary pressure (Pa). Default is 0"}>>> = 0
  []
[]

[Materials<<<{"href": "../../../../syntax/Materials/index.html"}>>>]
  [temperature]
    type = PorousFlowTemperature<<<{"description": "Material to provide temperature at the quadpoints or nodes and derivatives of it with respect to the PorousFlow variables", "href": "../../../../source/materials/PorousFlowTemperature.html"}>>>
  []
  [ppss]
    type = PorousFlow2PhasePS<<<{"description": "This Material calculates the 2 porepressures and the 2 saturations in a 2-phase situation, and derivatives of these with respect to the PorousFlowVariables.", "href": "../../../../source/materials/PorousFlow2PhasePS.html"}>>>
    phase0_porepressure<<<{"description": "Variable that is the porepressure of phase 0 (the liquid phase)"}>>> = p0
    phase1_saturation<<<{"description": "Variable that is the saturation of phase 1 (the gas phase)"}>>> = s1
    capillary_pressure<<<{"description": "Name of the UserObject defining the capillary pressure"}>>> = pc
  []
  [kr0]
    type = PorousFlowRelativePermeabilityCorey<<<{"description": "This Material calculates relative permeability of the fluid phase, using the simple Corey model ((S-S_res)/(1-sum(S_res)))^n", "href": "../../../../source/materials/PorousFlowRelativePermeabilityCorey.html"}>>>
    phase<<<{"description": "The phase number"}>>> = 0
    n<<<{"description": "The Corey exponent of the phase."}>>> = 2
  []
  [kr1]
    type = PorousFlowRelativePermeabilityCorey<<<{"description": "This Material calculates relative permeability of the fluid phase, using the simple Corey model ((S-S_res)/(1-sum(S_res)))^n", "href": "../../../../source/materials/PorousFlowRelativePermeabilityCorey.html"}>>>
    phase<<<{"description": "The phase number"}>>> = 1
    n<<<{"description": "The Corey exponent of the phase."}>>> = 2
  []
[]

[VectorPostprocessors<<<{"href": "../../../../syntax/VectorPostprocessors/index.html"}>>>]
  [vpp]
    type = LineValueSampler<<<{"description": "Samples variable(s) along a specified line", "href": "../../../../source/vectorpostprocessors/LineValueSampler.html"}>>>
    warn_discontinuous_face_values<<<{"description": "Whether to return a warning if a discontinuous variable is sampled on a face"}>>> = false
    variable<<<{"description": "The names of the variables that this VectorPostprocessor operates on"}>>> = 's0aux s1aux kr0aux kr1aux'
    start_point<<<{"description": "The beginning of the line"}>>> = '0 0 0'
    end_point<<<{"description": "The ending of the line"}>>> = '1 0 0'
    num_points<<<{"description": "The number of points to sample along the line"}>>> = 20
    sort_by<<<{"description": "What to sort the samples by"}>>> = id
  []
[]

[Executioner<<<{"href": "../../../../syntax/Executioner/index.html"}>>>]
  type = Steady
  solve_type = NEWTON
  nl_abs_tol = 1e-8
[]

[BCs<<<{"href": "../../../../syntax/BCs/index.html"}>>>]
  [sleft]
    type = DirichletBC<<<{"description": "Imposes the essential boundary condition $u=g$, where $g$ is a constant, controllable value.", "href": "../../../../source/bcs/DirichletBC.html"}>>>
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = s1
    value<<<{"description": "Value of the BC"}>>> = 0
    boundary<<<{"description": "The list of boundary IDs from the mesh where this object applies"}>>> = left
  []
  [sright]
    type = DirichletBC<<<{"description": "Imposes the essential boundary condition $u=g$, where $g$ is a constant, controllable value.", "href": "../../../../source/bcs/DirichletBC.html"}>>>
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = s1
    value<<<{"description": "Value of the BC"}>>> = 1
    boundary<<<{"description": "The list of boundary IDs from the mesh where this object applies"}>>> = right
  []
[]

[Outputs<<<{"href": "../../../../syntax/Outputs/index.html"}>>>]
  csv<<<{"description": "Output the scalar variable and postprocessors to a *.csv file using the default CSV output."}>>> = true
  execute_on<<<{"description": "The list of flag(s) indicating when this object should be executed. For a description of each flag, see https://mooseframework.inl.gov/source/interfaces/SetupInterface.html."}>>> = timestep_end
[]

Figure 4: Corey relative permeability Test case 2

Test 3

The input file for this test:

# Test Corey relative permeability curve by varying saturation over the mesh
# Residual saturation of phase 0: s0r = 0.2
# Residual saturation of phase 1: s1r = 0.3

[Mesh<<<{"href": "../../../../syntax/Mesh/index.html"}>>>]
  type = GeneratedMesh
  dim = 1
  nx = 20
[]

[GlobalParams<<<{"href": "../../../../syntax/GlobalParams/index.html"}>>>]
  PorousFlowDictator = dictator
[]

[Variables<<<{"href": "../../../../syntax/Variables/index.html"}>>>]
  [p0]
    initial_condition<<<{"description": "Specifies a constant initial condition for this variable"}>>> = 1e6
  []
  [s1]
    family<<<{"description": "Specifies the family of FE shape functions to use for this variable"}>>> = LAGRANGE
    order<<<{"description": "Specifies the order of the FE shape function to use for this variable (additional orders not listed are allowed)"}>>> = FIRST
  []
[]

[AuxVariables<<<{"href": "../../../../syntax/AuxVariables/index.html"}>>>]
  [s0aux]
    family<<<{"description": "Specifies the family of FE shape functions to use for this variable"}>>> = MONOMIAL
    order<<<{"description": "Specifies the order of the FE shape function to use for this variable (additional orders not listed are allowed)"}>>> = CONSTANT
  []
  [s1aux]
    family<<<{"description": "Specifies the family of FE shape functions to use for this variable"}>>> = MONOMIAL
    order<<<{"description": "Specifies the order of the FE shape function to use for this variable (additional orders not listed are allowed)"}>>> = CONSTANT
  []
  [kr0aux]
    family<<<{"description": "Specifies the family of FE shape functions to use for this variable"}>>> = MONOMIAL
    order<<<{"description": "Specifies the order of the FE shape function to use for this variable (additional orders not listed are allowed)"}>>> = CONSTANT
  []
  [kr1aux]
    family<<<{"description": "Specifies the family of FE shape functions to use for this variable"}>>> = MONOMIAL
    order<<<{"description": "Specifies the order of the FE shape function to use for this variable (additional orders not listed are allowed)"}>>> = CONSTANT
  []
[]

[AuxKernels<<<{"href": "../../../../syntax/AuxKernels/index.html"}>>>]
  [s0]
    type = PorousFlowPropertyAux<<<{"description": "AuxKernel to provide access to properties evaluated at quadpoints. Note that elemental AuxVariables must be used, so that these properties are integrated over each element.", "href": "../../../../source/auxkernels/PorousFlowPropertyAux.html"}>>>
    property<<<{"description": "The fluid property that this auxillary kernel is to calculate"}>>> = saturation
    phase<<<{"description": "The index of the phase this auxillary kernel acts on"}>>> = 0
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = s0aux
  []
  [s1]
    type = PorousFlowPropertyAux<<<{"description": "AuxKernel to provide access to properties evaluated at quadpoints. Note that elemental AuxVariables must be used, so that these properties are integrated over each element.", "href": "../../../../source/auxkernels/PorousFlowPropertyAux.html"}>>>
    property<<<{"description": "The fluid property that this auxillary kernel is to calculate"}>>> = saturation
    phase<<<{"description": "The index of the phase this auxillary kernel acts on"}>>> = 1
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = s1aux
  []
  [kr0]
    type = PorousFlowPropertyAux<<<{"description": "AuxKernel to provide access to properties evaluated at quadpoints. Note that elemental AuxVariables must be used, so that these properties are integrated over each element.", "href": "../../../../source/auxkernels/PorousFlowPropertyAux.html"}>>>
    property<<<{"description": "The fluid property that this auxillary kernel is to calculate"}>>> = relperm
    phase<<<{"description": "The index of the phase this auxillary kernel acts on"}>>> = 0
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = kr0aux
  []
  [kr1]
    type = PorousFlowPropertyAux<<<{"description": "AuxKernel to provide access to properties evaluated at quadpoints. Note that elemental AuxVariables must be used, so that these properties are integrated over each element.", "href": "../../../../source/auxkernels/PorousFlowPropertyAux.html"}>>>
    property<<<{"description": "The fluid property that this auxillary kernel is to calculate"}>>> = relperm
    phase<<<{"description": "The index of the phase this auxillary kernel acts on"}>>> = 1
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = kr1aux
  []
[]

[Functions<<<{"href": "../../../../syntax/Functions/index.html"}>>>]
  [s1]
    type = ParsedFunction<<<{"description": "Function created by parsing a string", "href": "../../../../source/functions/MooseParsedFunction.html"}>>>
    expression<<<{"description": "The user defined function."}>>> = x
  []
[]

[ICs<<<{"href": "../../../../syntax/ICs/index.html"}>>>]
  [s1]
    type = FunctionIC<<<{"description": "An initial condition that uses a normal function of x, y, z to produce values (and optionally gradients) for a field variable.", "href": "../../../../source/ics/FunctionIC.html"}>>>
    variable<<<{"description": "The variable this initial condition is supposed to provide values for."}>>> = s1
    function<<<{"description": "The initial condition function."}>>> = s1
  []
[]

[Kernels<<<{"href": "../../../../syntax/Kernels/index.html"}>>>]
  [p0]
    type = Diffusion<<<{"description": "The Laplacian operator ($-\\nabla \\cdot \\nabla u$), with the weak form of $(\\nabla \\phi_i, \\nabla u_h)$.", "href": "../../../../source/kernels/Diffusion.html"}>>>
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = p0
  []
  [s1]
    type = Diffusion<<<{"description": "The Laplacian operator ($-\\nabla \\cdot \\nabla u$), with the weak form of $(\\nabla \\phi_i, \\nabla u_h)$.", "href": "../../../../source/kernels/Diffusion.html"}>>>
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = s1
  []
[]

[UserObjects<<<{"href": "../../../../syntax/UserObjects/index.html"}>>>]
  [dictator]
    type = PorousFlowDictator<<<{"description": "Holds information on the PorousFlow variable names", "href": "../../../../source/userobjects/PorousFlowDictator.html"}>>>
    porous_flow_vars<<<{"description": "List of primary variables that are used in the PorousFlow simulation.  Jacobian entries involving derivatives wrt these variables will be computed.  In single-phase models you will just have one (eg 'pressure'), in two-phase models you will have two (eg 'p_water p_gas', or 'p_water s_water'), etc."}>>> = 'p0 s1'
    number_fluid_phases<<<{"description": "The number of fluid phases in the simulation"}>>> = 2
    number_fluid_components<<<{"description": "The number of fluid components in the simulation"}>>> = 2
  []
  [pc]
    type = PorousFlowCapillaryPressureConst<<<{"description": "Constant capillary pressure", "href": "../../../../source/userobjects/PorousFlowCapillaryPressureConst.html"}>>>
    pc<<<{"description": "Constant capillary pressure (Pa). Default is 0"}>>> = 0
  []
[]

[Materials<<<{"href": "../../../../syntax/Materials/index.html"}>>>]
  [temperature]
    type = PorousFlowTemperature<<<{"description": "Material to provide temperature at the quadpoints or nodes and derivatives of it with respect to the PorousFlow variables", "href": "../../../../source/materials/PorousFlowTemperature.html"}>>>
  []
  [ppss]
    type = PorousFlow2PhasePS<<<{"description": "This Material calculates the 2 porepressures and the 2 saturations in a 2-phase situation, and derivatives of these with respect to the PorousFlowVariables.", "href": "../../../../source/materials/PorousFlow2PhasePS.html"}>>>
    phase0_porepressure<<<{"description": "Variable that is the porepressure of phase 0 (the liquid phase)"}>>> = p0
    phase1_saturation<<<{"description": "Variable that is the saturation of phase 1 (the gas phase)"}>>> = s1
    capillary_pressure<<<{"description": "Name of the UserObject defining the capillary pressure"}>>> = pc
  []
  [kr0]
    type = PorousFlowRelativePermeabilityCorey<<<{"description": "This Material calculates relative permeability of the fluid phase, using the simple Corey model ((S-S_res)/(1-sum(S_res)))^n", "href": "../../../../source/materials/PorousFlowRelativePermeabilityCorey.html"}>>>
    phase<<<{"description": "The phase number"}>>> = 0
    n<<<{"description": "The Corey exponent of the phase."}>>> = 2
    s_res<<<{"description": "The residual saturation of the phase j. Must be between 0 and 1"}>>> = 0.2
    sum_s_res<<<{"description": "Sum of residual saturations over all phases.  Must be between 0 and 1"}>>> = 0.5
  []
  [kr1]
    type = PorousFlowRelativePermeabilityCorey<<<{"description": "This Material calculates relative permeability of the fluid phase, using the simple Corey model ((S-S_res)/(1-sum(S_res)))^n", "href": "../../../../source/materials/PorousFlowRelativePermeabilityCorey.html"}>>>
    phase<<<{"description": "The phase number"}>>> = 1
    n<<<{"description": "The Corey exponent of the phase."}>>> = 2
    s_res<<<{"description": "The residual saturation of the phase j. Must be between 0 and 1"}>>> = 0.3
    sum_s_res<<<{"description": "Sum of residual saturations over all phases.  Must be between 0 and 1"}>>> = 0.5
  []
[]

[VectorPostprocessors<<<{"href": "../../../../syntax/VectorPostprocessors/index.html"}>>>]
  [vpp]
    type = LineValueSampler<<<{"description": "Samples variable(s) along a specified line", "href": "../../../../source/vectorpostprocessors/LineValueSampler.html"}>>>
    warn_discontinuous_face_values<<<{"description": "Whether to return a warning if a discontinuous variable is sampled on a face"}>>> = false
    variable<<<{"description": "The names of the variables that this VectorPostprocessor operates on"}>>> = 's0aux s1aux kr0aux kr1aux'
    start_point<<<{"description": "The beginning of the line"}>>> = '0 0 0'
    end_point<<<{"description": "The ending of the line"}>>> = '1 0 0'
    num_points<<<{"description": "The number of points to sample along the line"}>>> = 20
    sort_by<<<{"description": "What to sort the samples by"}>>> = id
  []
[]

[Executioner<<<{"href": "../../../../syntax/Executioner/index.html"}>>>]
  type = Steady
  solve_type = NEWTON
  nl_abs_tol = 1e-8
[]

[BCs<<<{"href": "../../../../syntax/BCs/index.html"}>>>]
  [sleft]
    type = DirichletBC<<<{"description": "Imposes the essential boundary condition $u=g$, where $g$ is a constant, controllable value.", "href": "../../../../source/bcs/DirichletBC.html"}>>>
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = s1
    value<<<{"description": "Value of the BC"}>>> = 0
    boundary<<<{"description": "The list of boundary IDs from the mesh where this object applies"}>>> = left
  []
  [sright]
    type = DirichletBC<<<{"description": "Imposes the essential boundary condition $u=g$, where $g$ is a constant, controllable value.", "href": "../../../../source/bcs/DirichletBC.html"}>>>
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = s1
    value<<<{"description": "Value of the BC"}>>> = 1
    boundary<<<{"description": "The list of boundary IDs from the mesh where this object applies"}>>> = right
  []
[]

[Outputs<<<{"href": "../../../../syntax/Outputs/index.html"}>>>]
  csv<<<{"description": "Output the scalar variable and postprocessors to a *.csv file using the default CSV output."}>>> = true
  execute_on<<<{"description": "The list of flag(s) indicating when this object should be executed. For a description of each flag, see https://mooseframework.inl.gov/source/interfaces/SetupInterface.html."}>>> = timestep_end
[]

Figure 5: Corey relative permeability Test case 3

Test 4

The input file for this test:

# Test Corey relative permeability curve by varying saturation over the mesh
# Residual saturation of phase 0: s0r = 0.2
# Residual saturation of phase 1: s1r = 0.3

[Mesh<<<{"href": "../../../../syntax/Mesh/index.html"}>>>]
  type = GeneratedMesh
  dim = 1
  nx = 20
[]

[GlobalParams<<<{"href": "../../../../syntax/GlobalParams/index.html"}>>>]
  PorousFlowDictator = dictator
[]

[Variables<<<{"href": "../../../../syntax/Variables/index.html"}>>>]
  [p0]
    initial_condition<<<{"description": "Specifies a constant initial condition for this variable"}>>> = 1e6
  []
  [s1]
    family<<<{"description": "Specifies the family of FE shape functions to use for this variable"}>>> = LAGRANGE
    order<<<{"description": "Specifies the order of the FE shape function to use for this variable (additional orders not listed are allowed)"}>>> = FIRST
  []
[]

[AuxVariables<<<{"href": "../../../../syntax/AuxVariables/index.html"}>>>]
  [s0aux]
    family<<<{"description": "Specifies the family of FE shape functions to use for this variable"}>>> = MONOMIAL
    order<<<{"description": "Specifies the order of the FE shape function to use for this variable (additional orders not listed are allowed)"}>>> = CONSTANT
  []
  [s1aux]
    family<<<{"description": "Specifies the family of FE shape functions to use for this variable"}>>> = MONOMIAL
    order<<<{"description": "Specifies the order of the FE shape function to use for this variable (additional orders not listed are allowed)"}>>> = CONSTANT
  []
  [kr0aux]
    family<<<{"description": "Specifies the family of FE shape functions to use for this variable"}>>> = MONOMIAL
    order<<<{"description": "Specifies the order of the FE shape function to use for this variable (additional orders not listed are allowed)"}>>> = CONSTANT
  []
  [kr1aux]
    family<<<{"description": "Specifies the family of FE shape functions to use for this variable"}>>> = MONOMIAL
    order<<<{"description": "Specifies the order of the FE shape function to use for this variable (additional orders not listed are allowed)"}>>> = CONSTANT
  []
[]

[AuxKernels<<<{"href": "../../../../syntax/AuxKernels/index.html"}>>>]
  [s0]
    type = PorousFlowPropertyAux<<<{"description": "AuxKernel to provide access to properties evaluated at quadpoints. Note that elemental AuxVariables must be used, so that these properties are integrated over each element.", "href": "../../../../source/auxkernels/PorousFlowPropertyAux.html"}>>>
    property<<<{"description": "The fluid property that this auxillary kernel is to calculate"}>>> = saturation
    phase<<<{"description": "The index of the phase this auxillary kernel acts on"}>>> = 0
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = s0aux
  []
  [s1]
    type = PorousFlowPropertyAux<<<{"description": "AuxKernel to provide access to properties evaluated at quadpoints. Note that elemental AuxVariables must be used, so that these properties are integrated over each element.", "href": "../../../../source/auxkernels/PorousFlowPropertyAux.html"}>>>
    property<<<{"description": "The fluid property that this auxillary kernel is to calculate"}>>> = saturation
    phase<<<{"description": "The index of the phase this auxillary kernel acts on"}>>> = 1
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = s1aux
  []
  [kr0]
    type = PorousFlowPropertyAux<<<{"description": "AuxKernel to provide access to properties evaluated at quadpoints. Note that elemental AuxVariables must be used, so that these properties are integrated over each element.", "href": "../../../../source/auxkernels/PorousFlowPropertyAux.html"}>>>
    property<<<{"description": "The fluid property that this auxillary kernel is to calculate"}>>> = relperm
    phase<<<{"description": "The index of the phase this auxillary kernel acts on"}>>> = 0
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = kr0aux
  []
  [kr1]
    type = PorousFlowPropertyAux<<<{"description": "AuxKernel to provide access to properties evaluated at quadpoints. Note that elemental AuxVariables must be used, so that these properties are integrated over each element.", "href": "../../../../source/auxkernels/PorousFlowPropertyAux.html"}>>>
    property<<<{"description": "The fluid property that this auxillary kernel is to calculate"}>>> = relperm
    phase<<<{"description": "The index of the phase this auxillary kernel acts on"}>>> = 1
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = kr1aux
  []
[]

[Functions<<<{"href": "../../../../syntax/Functions/index.html"}>>>]
  [s1]
    type = ParsedFunction<<<{"description": "Function created by parsing a string", "href": "../../../../source/functions/MooseParsedFunction.html"}>>>
    expression<<<{"description": "The user defined function."}>>> = x
  []
[]

[ICs<<<{"href": "../../../../syntax/ICs/index.html"}>>>]
  [s1]
    type = FunctionIC<<<{"description": "An initial condition that uses a normal function of x, y, z to produce values (and optionally gradients) for a field variable.", "href": "../../../../source/ics/FunctionIC.html"}>>>
    variable<<<{"description": "The variable this initial condition is supposed to provide values for."}>>> = s1
    function<<<{"description": "The initial condition function."}>>> = s1
  []
[]

[Kernels<<<{"href": "../../../../syntax/Kernels/index.html"}>>>]
  [p0]
    type = Diffusion<<<{"description": "The Laplacian operator ($-\\nabla \\cdot \\nabla u$), with the weak form of $(\\nabla \\phi_i, \\nabla u_h)$.", "href": "../../../../source/kernels/Diffusion.html"}>>>
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = p0
  []
  [s1]
    type = Diffusion<<<{"description": "The Laplacian operator ($-\\nabla \\cdot \\nabla u$), with the weak form of $(\\nabla \\phi_i, \\nabla u_h)$.", "href": "../../../../source/kernels/Diffusion.html"}>>>
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = s1
  []
[]

[UserObjects<<<{"href": "../../../../syntax/UserObjects/index.html"}>>>]
  [dictator]
    type = PorousFlowDictator<<<{"description": "Holds information on the PorousFlow variable names", "href": "../../../../source/userobjects/PorousFlowDictator.html"}>>>
    porous_flow_vars<<<{"description": "List of primary variables that are used in the PorousFlow simulation.  Jacobian entries involving derivatives wrt these variables will be computed.  In single-phase models you will just have one (eg 'pressure'), in two-phase models you will have two (eg 'p_water p_gas', or 'p_water s_water'), etc."}>>> = 'p0 s1'
    number_fluid_phases<<<{"description": "The number of fluid phases in the simulation"}>>> = 2
    number_fluid_components<<<{"description": "The number of fluid components in the simulation"}>>> = 2
  []
  [pc]
    type = PorousFlowCapillaryPressureConst<<<{"description": "Constant capillary pressure", "href": "../../../../source/userobjects/PorousFlowCapillaryPressureConst.html"}>>>
    pc<<<{"description": "Constant capillary pressure (Pa). Default is 0"}>>> = 0
  []
[]

[Materials<<<{"href": "../../../../syntax/Materials/index.html"}>>>]
  [temperature]
    type = PorousFlowTemperature<<<{"description": "Material to provide temperature at the quadpoints or nodes and derivatives of it with respect to the PorousFlow variables", "href": "../../../../source/materials/PorousFlowTemperature.html"}>>>
  []
  [ppss]
    type = PorousFlow2PhasePS<<<{"description": "This Material calculates the 2 porepressures and the 2 saturations in a 2-phase situation, and derivatives of these with respect to the PorousFlowVariables.", "href": "../../../../source/materials/PorousFlow2PhasePS.html"}>>>
    phase0_porepressure<<<{"description": "Variable that is the porepressure of phase 0 (the liquid phase)"}>>> = p0
    phase1_saturation<<<{"description": "Variable that is the saturation of phase 1 (the gas phase)"}>>> = s1
    capillary_pressure<<<{"description": "Name of the UserObject defining the capillary pressure"}>>> = pc
  []
  [kr0]
    type = PorousFlowRelativePermeabilityCorey<<<{"description": "This Material calculates relative permeability of the fluid phase, using the simple Corey model ((S-S_res)/(1-sum(S_res)))^n", "href": "../../../../source/materials/PorousFlowRelativePermeabilityCorey.html"}>>>
    scaling<<<{"description": "Relative permeability is multiplied by this factor"}>>> = 0.1
    phase<<<{"description": "The phase number"}>>> = 0
    n<<<{"description": "The Corey exponent of the phase."}>>> = 2
    s_res<<<{"description": "The residual saturation of the phase j. Must be between 0 and 1"}>>> = 0.2
    sum_s_res<<<{"description": "Sum of residual saturations over all phases.  Must be between 0 and 1"}>>> = 0.5
  []
  [kr1]
    type = PorousFlowRelativePermeabilityCorey<<<{"description": "This Material calculates relative permeability of the fluid phase, using the simple Corey model ((S-S_res)/(1-sum(S_res)))^n", "href": "../../../../source/materials/PorousFlowRelativePermeabilityCorey.html"}>>>
    scaling<<<{"description": "Relative permeability is multiplied by this factor"}>>> = 10.0
    phase<<<{"description": "The phase number"}>>> = 1
    n<<<{"description": "The Corey exponent of the phase."}>>> = 2
    s_res<<<{"description": "The residual saturation of the phase j. Must be between 0 and 1"}>>> = 0.3
    sum_s_res<<<{"description": "Sum of residual saturations over all phases.  Must be between 0 and 1"}>>> = 0.5
  []
[]

[VectorPostprocessors<<<{"href": "../../../../syntax/VectorPostprocessors/index.html"}>>>]
  [vpp]
    type = LineValueSampler<<<{"description": "Samples variable(s) along a specified line", "href": "../../../../source/vectorpostprocessors/LineValueSampler.html"}>>>
    warn_discontinuous_face_values<<<{"description": "Whether to return a warning if a discontinuous variable is sampled on a face"}>>> = false
    variable<<<{"description": "The names of the variables that this VectorPostprocessor operates on"}>>> = 's0aux s1aux kr0aux kr1aux'
    start_point<<<{"description": "The beginning of the line"}>>> = '0 0 0'
    end_point<<<{"description": "The ending of the line"}>>> = '1 0 0'
    num_points<<<{"description": "The number of points to sample along the line"}>>> = 20
    sort_by<<<{"description": "What to sort the samples by"}>>> = id
  []
[]

[Executioner<<<{"href": "../../../../syntax/Executioner/index.html"}>>>]
  type = Steady
  solve_type = NEWTON
  nl_abs_tol = 1e-8
[]

[BCs<<<{"href": "../../../../syntax/BCs/index.html"}>>>]
  [sleft]
    type = DirichletBC<<<{"description": "Imposes the essential boundary condition $u=g$, where $g$ is a constant, controllable value.", "href": "../../../../source/bcs/DirichletBC.html"}>>>
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = s1
    value<<<{"description": "Value of the BC"}>>> = 0
    boundary<<<{"description": "The list of boundary IDs from the mesh where this object applies"}>>> = left
  []
  [sright]
    type = DirichletBC<<<{"description": "Imposes the essential boundary condition $u=g$, where $g$ is a constant, controllable value.", "href": "../../../../source/bcs/DirichletBC.html"}>>>
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = s1
    value<<<{"description": "Value of the BC"}>>> = 1
    boundary<<<{"description": "The list of boundary IDs from the mesh where this object applies"}>>> = right
  []
[]

[Outputs<<<{"href": "../../../../syntax/Outputs/index.html"}>>>]
  csv<<<{"description": "Output the scalar variable and postprocessors to a *.csv file using the default CSV output."}>>> = true
  execute_on<<<{"description": "The list of flag(s) indicating when this object should be executed. For a description of each flag, see https://mooseframework.inl.gov/source/interfaces/SetupInterface.html."}>>> = timestep_end
[]

Figure 6: Corey relative permeability Test case 4

van Genuchten

The relative permeability of the wetting phase is given by Genuchten (1980) $var element = document.getElementById("moose-equation-5d3edc84-a2ef-40eb-afcd-4c783358403b");katex.render("k_{\\mathrm{r}} = \\sqrt{S_{\\mathrm{eff}}} \\left(1 - (1 - S_{\\mathrm{eff}}^{1/m})^{m} \\right)^{2}.", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ This has the numerical disadvantage that its derivative as $var element = document.getElementById("moose-equation-bc2ab2df-2cae-4b52-91c9-a4ac1e4ec6ae");katex.render("S_{\\mathrm{eff}}\\rightarrow 1", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ tends to infinity. This means that simulations where the saturation oscillates around $var element = document.getElementById("moose-equation-5e98382a-4e5a-4c94-af5d-c2e25cd113de");katex.render("S_{\\mathrm{eff}}=1", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ do not converge well.

Therefore, a cut version of the van Genuchten expression is also offered, which is almost definitely indistinguishable experimentally from the original expression: $var element = document.getElementById("moose-equation-31cb1777-ffc9-42da-9c74-0b5c290e56b9");katex.render("k_{\\mathrm{r}} = \\begin{cases} \\textrm{van Genuchten} & \\textrm{for } S_{\\mathrm{eff}} < S_{c} \\\\ \\textrm{cubic} & \\textrm{for } S_{\\mathrm{eff}} \\geq S_{c}. \\end{cases}", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ Here the cubic is chosen so that its value and derivative match the van Genuchten expression at $var element = document.getElementById("moose-equation-536820eb-a0ae-4250-98d3-6233aab510c2");katex.render("S=S_{c}", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ , and so that it is unity at $var element = document.getElementById("moose-equation-cbde32df-d172-429c-8b5f-cb148830cd0c");katex.render("S_{\\mathrm{eff}}=1", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ .

For the non-wetting phase, the van Genuchten expression is $var element = document.getElementById("moose-equation-807ebf36-fc28-4487-9d15-30d931e725e0");katex.render("k_{\\mathrm{r}} = \\sqrt{S_{\\mathrm{eff}}} \\left(1 - (1 - S_{\\mathrm{eff}})^{1/m} \\right)^{2m} \\ .", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$

Test 1

The input file for this test:

# Test van Genuchten relative permeability curve by varying saturation over the mesh
# van Genuchten exponent m = 0.5 for both phases
# No residual saturation in either phase

[Mesh<<<{"href": "../../../../syntax/Mesh/index.html"}>>>]
  type = GeneratedMesh
  dim = 1
  nx = 100
[]

[GlobalParams<<<{"href": "../../../../syntax/GlobalParams/index.html"}>>>]
  PorousFlowDictator = dictator
[]

[Variables<<<{"href": "../../../../syntax/Variables/index.html"}>>>]
  [p0]
    initial_condition<<<{"description": "Specifies a constant initial condition for this variable"}>>> = 1e6
  []
  [s1]
  []
[]

[AuxVariables<<<{"href": "../../../../syntax/AuxVariables/index.html"}>>>]
  [s0aux]
    family<<<{"description": "Specifies the family of FE shape functions to use for this variable"}>>> = MONOMIAL
    order<<<{"description": "Specifies the order of the FE shape function to use for this variable (additional orders not listed are allowed)"}>>> = CONSTANT
  []
  [s1aux]
    family<<<{"description": "Specifies the family of FE shape functions to use for this variable"}>>> = MONOMIAL
    order<<<{"description": "Specifies the order of the FE shape function to use for this variable (additional orders not listed are allowed)"}>>> = CONSTANT
  []
  [kr0aux]
    family<<<{"description": "Specifies the family of FE shape functions to use for this variable"}>>> = MONOMIAL
    order<<<{"description": "Specifies the order of the FE shape function to use for this variable (additional orders not listed are allowed)"}>>> = CONSTANT
  []
  [kr1aux]
    family<<<{"description": "Specifies the family of FE shape functions to use for this variable"}>>> = MONOMIAL
    order<<<{"description": "Specifies the order of the FE shape function to use for this variable (additional orders not listed are allowed)"}>>> = CONSTANT
  []
[]

[AuxKernels<<<{"href": "../../../../syntax/AuxKernels/index.html"}>>>]
  [s0]
    type = PorousFlowPropertyAux<<<{"description": "AuxKernel to provide access to properties evaluated at quadpoints. Note that elemental AuxVariables must be used, so that these properties are integrated over each element.", "href": "../../../../source/auxkernels/PorousFlowPropertyAux.html"}>>>
    property<<<{"description": "The fluid property that this auxillary kernel is to calculate"}>>> = saturation
    phase<<<{"description": "The index of the phase this auxillary kernel acts on"}>>> = 0
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = s0aux
  []
  [s1]
    type = PorousFlowPropertyAux<<<{"description": "AuxKernel to provide access to properties evaluated at quadpoints. Note that elemental AuxVariables must be used, so that these properties are integrated over each element.", "href": "../../../../source/auxkernels/PorousFlowPropertyAux.html"}>>>
    property<<<{"description": "The fluid property that this auxillary kernel is to calculate"}>>> = saturation
    phase<<<{"description": "The index of the phase this auxillary kernel acts on"}>>> = 1
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = s1aux
  []
  [kr0]
    type = PorousFlowPropertyAux<<<{"description": "AuxKernel to provide access to properties evaluated at quadpoints. Note that elemental AuxVariables must be used, so that these properties are integrated over each element.", "href": "../../../../source/auxkernels/PorousFlowPropertyAux.html"}>>>
    property<<<{"description": "The fluid property that this auxillary kernel is to calculate"}>>> = relperm
    phase<<<{"description": "The index of the phase this auxillary kernel acts on"}>>> = 0
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = kr0aux
  []
  [kr1]
    type = PorousFlowPropertyAux<<<{"description": "AuxKernel to provide access to properties evaluated at quadpoints. Note that elemental AuxVariables must be used, so that these properties are integrated over each element.", "href": "../../../../source/auxkernels/PorousFlowPropertyAux.html"}>>>
    property<<<{"description": "The fluid property that this auxillary kernel is to calculate"}>>> = relperm
    phase<<<{"description": "The index of the phase this auxillary kernel acts on"}>>> = 1
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = kr1aux
  []
[]

[Functions<<<{"href": "../../../../syntax/Functions/index.html"}>>>]
  [s1]
    type = ParsedFunction<<<{"description": "Function created by parsing a string", "href": "../../../../source/functions/MooseParsedFunction.html"}>>>
    expression<<<{"description": "The user defined function."}>>> = x
  []
[]

[ICs<<<{"href": "../../../../syntax/ICs/index.html"}>>>]
  [s1]
    type = FunctionIC<<<{"description": "An initial condition that uses a normal function of x, y, z to produce values (and optionally gradients) for a field variable.", "href": "../../../../source/ics/FunctionIC.html"}>>>
    variable<<<{"description": "The variable this initial condition is supposed to provide values for."}>>> = s1
    function<<<{"description": "The initial condition function."}>>> = s1
  []
[]

[Kernels<<<{"href": "../../../../syntax/Kernels/index.html"}>>>]
  [p0]
    type = Diffusion<<<{"description": "The Laplacian operator ($-\\nabla \\cdot \\nabla u$), with the weak form of $(\\nabla \\phi_i, \\nabla u_h)$.", "href": "../../../../source/kernels/Diffusion.html"}>>>
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = p0
  []
  [s1]
    type = Diffusion<<<{"description": "The Laplacian operator ($-\\nabla \\cdot \\nabla u$), with the weak form of $(\\nabla \\phi_i, \\nabla u_h)$.", "href": "../../../../source/kernels/Diffusion.html"}>>>
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = s1
  []
[]

[UserObjects<<<{"href": "../../../../syntax/UserObjects/index.html"}>>>]
  [dictator]
    type = PorousFlowDictator<<<{"description": "Holds information on the PorousFlow variable names", "href": "../../../../source/userobjects/PorousFlowDictator.html"}>>>
    porous_flow_vars<<<{"description": "List of primary variables that are used in the PorousFlow simulation.  Jacobian entries involving derivatives wrt these variables will be computed.  In single-phase models you will just have one (eg 'pressure'), in two-phase models you will have two (eg 'p_water p_gas', or 'p_water s_water'), etc."}>>> = 'p0 s1'
    number_fluid_phases<<<{"description": "The number of fluid phases in the simulation"}>>> = 2
    number_fluid_components<<<{"description": "The number of fluid components in the simulation"}>>> = 2
  []
  [pc]
    type = PorousFlowCapillaryPressureConst<<<{"description": "Constant capillary pressure", "href": "../../../../source/userobjects/PorousFlowCapillaryPressureConst.html"}>>>
    pc<<<{"description": "Constant capillary pressure (Pa). Default is 0"}>>> = 0
  []
[]

[Materials<<<{"href": "../../../../syntax/Materials/index.html"}>>>]
  [temperature]
    type = PorousFlowTemperature<<<{"description": "Material to provide temperature at the quadpoints or nodes and derivatives of it with respect to the PorousFlow variables", "href": "../../../../source/materials/PorousFlowTemperature.html"}>>>
  []
  [ppss]
    type = PorousFlow2PhasePS<<<{"description": "This Material calculates the 2 porepressures and the 2 saturations in a 2-phase situation, and derivatives of these with respect to the PorousFlowVariables.", "href": "../../../../source/materials/PorousFlow2PhasePS.html"}>>>
    phase0_porepressure<<<{"description": "Variable that is the porepressure of phase 0 (the liquid phase)"}>>> = p0
    phase1_saturation<<<{"description": "Variable that is the saturation of phase 1 (the gas phase)"}>>> = s1
    capillary_pressure<<<{"description": "Name of the UserObject defining the capillary pressure"}>>> = pc
  []
  [kr0]
    type = PorousFlowRelativePermeabilityVG<<<{"description": "This Material calculates relative permeability of a phase using the van Genuchten model", "href": "../../../../source/materials/PorousFlowRelativePermeabilityVG.html"}>>>
    phase<<<{"description": "The phase number"}>>> = 0
    m<<<{"description": "The van Genuchten exponent of the phase"}>>> = 0.5
  []
  [kr1]
    type = PorousFlowRelativePermeabilityVG<<<{"description": "This Material calculates relative permeability of a phase using the van Genuchten model", "href": "../../../../source/materials/PorousFlowRelativePermeabilityVG.html"}>>>
    phase<<<{"description": "The phase number"}>>> = 1
    m<<<{"description": "The van Genuchten exponent of the phase"}>>> = 0.5
    wetting<<<{"description": "If true, use the van Genuchten form appropriate for a wetting (liquid) phase.  If false, use the non-wetting (gas) expression."}>>> = false
  []
[]

[VectorPostprocessors<<<{"href": "../../../../syntax/VectorPostprocessors/index.html"}>>>]
  [vpp]
    type = LineValueSampler<<<{"description": "Samples variable(s) along a specified line", "href": "../../../../source/vectorpostprocessors/LineValueSampler.html"}>>>
    warn_discontinuous_face_values<<<{"description": "Whether to return a warning if a discontinuous variable is sampled on a face"}>>> = false
    variable<<<{"description": "The names of the variables that this VectorPostprocessor operates on"}>>> = 's0aux s1aux kr0aux kr1aux'
    start_point<<<{"description": "The beginning of the line"}>>> = '0 0 0'
    end_point<<<{"description": "The ending of the line"}>>> = '1 0 0'
    num_points<<<{"description": "The number of points to sample along the line"}>>> = 20
    sort_by<<<{"description": "What to sort the samples by"}>>> = id
  []
[]

[Executioner<<<{"href": "../../../../syntax/Executioner/index.html"}>>>]
  type = Steady
  solve_type = NEWTON
  nl_abs_tol = 1e-7
[]

[BCs<<<{"href": "../../../../syntax/BCs/index.html"}>>>]
  [sleft]
    type = DirichletBC<<<{"description": "Imposes the essential boundary condition $u=g$, where $g$ is a constant, controllable value.", "href": "../../../../source/bcs/DirichletBC.html"}>>>
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = s1
    value<<<{"description": "Value of the BC"}>>> = 0
    boundary<<<{"description": "The list of boundary IDs from the mesh where this object applies"}>>> = left
  []
  [sright]
    type = DirichletBC<<<{"description": "Imposes the essential boundary condition $u=g$, where $g$ is a constant, controllable value.", "href": "../../../../source/bcs/DirichletBC.html"}>>>
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = s1
    value<<<{"description": "Value of the BC"}>>> = 1
    boundary<<<{"description": "The list of boundary IDs from the mesh where this object applies"}>>> = right
  []
[]

[Outputs<<<{"href": "../../../../syntax/Outputs/index.html"}>>>]
  csv<<<{"description": "Output the scalar variable and postprocessors to a *.csv file using the default CSV output."}>>> = true
  execute_on<<<{"description": "The list of flag(s) indicating when this object should be executed. For a description of each flag, see https://mooseframework.inl.gov/source/interfaces/SetupInterface.html."}>>> = timestep_end
[]

Figure 7: van Genuchten relative permeability Test case 1

Test 2

The input file for this test:

# Test van Genuchten relative permeability curve by varying saturation over the mesh
# van Genuchten exponent m = 0.4 for both phases
# Phase 0 residual saturation s0r = 0.1
# Phase 1 residual saturation s1r = 0.2

[Mesh<<<{"href": "../../../../syntax/Mesh/index.html"}>>>]
  type = GeneratedMesh
  dim = 1
  nx = 100
[]

[GlobalParams<<<{"href": "../../../../syntax/GlobalParams/index.html"}>>>]
  PorousFlowDictator = dictator
[]

[Variables<<<{"href": "../../../../syntax/Variables/index.html"}>>>]
  [p0]
    initial_condition<<<{"description": "Specifies a constant initial condition for this variable"}>>> = 1e6
  []
  [s1]
  []
[]

[AuxVariables<<<{"href": "../../../../syntax/AuxVariables/index.html"}>>>]
  [s0aux]
    family<<<{"description": "Specifies the family of FE shape functions to use for this variable"}>>> = MONOMIAL
    order<<<{"description": "Specifies the order of the FE shape function to use for this variable (additional orders not listed are allowed)"}>>> = CONSTANT
  []
  [s1aux]
    family<<<{"description": "Specifies the family of FE shape functions to use for this variable"}>>> = MONOMIAL
    order<<<{"description": "Specifies the order of the FE shape function to use for this variable (additional orders not listed are allowed)"}>>> = CONSTANT
  []
  [kr0aux]
    family<<<{"description": "Specifies the family of FE shape functions to use for this variable"}>>> = MONOMIAL
    order<<<{"description": "Specifies the order of the FE shape function to use for this variable (additional orders not listed are allowed)"}>>> = CONSTANT
  []
  [kr1aux]
    family<<<{"description": "Specifies the family of FE shape functions to use for this variable"}>>> = MONOMIAL
    order<<<{"description": "Specifies the order of the FE shape function to use for this variable (additional orders not listed are allowed)"}>>> = CONSTANT
  []
[]

[AuxKernels<<<{"href": "../../../../syntax/AuxKernels/index.html"}>>>]
  [s0]
    type = PorousFlowPropertyAux<<<{"description": "AuxKernel to provide access to properties evaluated at quadpoints. Note that elemental AuxVariables must be used, so that these properties are integrated over each element.", "href": "../../../../source/auxkernels/PorousFlowPropertyAux.html"}>>>
    property<<<{"description": "The fluid property that this auxillary kernel is to calculate"}>>> = saturation
    phase<<<{"description": "The index of the phase this auxillary kernel acts on"}>>> = 0
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = s0aux
  []
  [s1]
    type = PorousFlowPropertyAux<<<{"description": "AuxKernel to provide access to properties evaluated at quadpoints. Note that elemental AuxVariables must be used, so that these properties are integrated over each element.", "href": "../../../../source/auxkernels/PorousFlowPropertyAux.html"}>>>
    property<<<{"description": "The fluid property that this auxillary kernel is to calculate"}>>> = saturation
    phase<<<{"description": "The index of the phase this auxillary kernel acts on"}>>> = 1
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = s1aux
  []
  [kr0]
    type = PorousFlowPropertyAux<<<{"description": "AuxKernel to provide access to properties evaluated at quadpoints. Note that elemental AuxVariables must be used, so that these properties are integrated over each element.", "href": "../../../../source/auxkernels/PorousFlowPropertyAux.html"}>>>
    property<<<{"description": "The fluid property that this auxillary kernel is to calculate"}>>> = relperm
    phase<<<{"description": "The index of the phase this auxillary kernel acts on"}>>> = 0
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = kr0aux
  []
  [kr1]
    type = PorousFlowPropertyAux<<<{"description": "AuxKernel to provide access to properties evaluated at quadpoints. Note that elemental AuxVariables must be used, so that these properties are integrated over each element.", "href": "../../../../source/auxkernels/PorousFlowPropertyAux.html"}>>>
    property<<<{"description": "The fluid property that this auxillary kernel is to calculate"}>>> = relperm
    phase<<<{"description": "The index of the phase this auxillary kernel acts on"}>>> = 1
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = kr1aux
  []
[]

[Functions<<<{"href": "../../../../syntax/Functions/index.html"}>>>]
  [s1]
    type = ParsedFunction<<<{"description": "Function created by parsing a string", "href": "../../../../source/functions/MooseParsedFunction.html"}>>>
    expression<<<{"description": "The user defined function."}>>> = x
  []
[]

[ICs<<<{"href": "../../../../syntax/ICs/index.html"}>>>]
  [s1]
    type = FunctionIC<<<{"description": "An initial condition that uses a normal function of x, y, z to produce values (and optionally gradients) for a field variable.", "href": "../../../../source/ics/FunctionIC.html"}>>>
    variable<<<{"description": "The variable this initial condition is supposed to provide values for."}>>> = s1
    function<<<{"description": "The initial condition function."}>>> = s1
  []
[]

[Kernels<<<{"href": "../../../../syntax/Kernels/index.html"}>>>]
  [p0]
    type = Diffusion<<<{"description": "The Laplacian operator ($-\\nabla \\cdot \\nabla u$), with the weak form of $(\\nabla \\phi_i, \\nabla u_h)$.", "href": "../../../../source/kernels/Diffusion.html"}>>>
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = p0
  []
  [s1]
    type = Diffusion<<<{"description": "The Laplacian operator ($-\\nabla \\cdot \\nabla u$), with the weak form of $(\\nabla \\phi_i, \\nabla u_h)$.", "href": "../../../../source/kernels/Diffusion.html"}>>>
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = s1
  []
[]

[UserObjects<<<{"href": "../../../../syntax/UserObjects/index.html"}>>>]
  [dictator]
    type = PorousFlowDictator<<<{"description": "Holds information on the PorousFlow variable names", "href": "../../../../source/userobjects/PorousFlowDictator.html"}>>>
    porous_flow_vars<<<{"description": "List of primary variables that are used in the PorousFlow simulation.  Jacobian entries involving derivatives wrt these variables will be computed.  In single-phase models you will just have one (eg 'pressure'), in two-phase models you will have two (eg 'p_water p_gas', or 'p_water s_water'), etc."}>>> = 'p0 s1'
    number_fluid_phases<<<{"description": "The number of fluid phases in the simulation"}>>> = 2
    number_fluid_components<<<{"description": "The number of fluid components in the simulation"}>>> = 2
  []
  [pc]
    type = PorousFlowCapillaryPressureConst<<<{"description": "Constant capillary pressure", "href": "../../../../source/userobjects/PorousFlowCapillaryPressureConst.html"}>>>
    pc<<<{"description": "Constant capillary pressure (Pa). Default is 0"}>>> = 0
  []
[]

[Materials<<<{"href": "../../../../syntax/Materials/index.html"}>>>]
  [temperature]
    type = PorousFlowTemperature<<<{"description": "Material to provide temperature at the quadpoints or nodes and derivatives of it with respect to the PorousFlow variables", "href": "../../../../source/materials/PorousFlowTemperature.html"}>>>
  []
  [ppss]
    type = PorousFlow2PhasePS<<<{"description": "This Material calculates the 2 porepressures and the 2 saturations in a 2-phase situation, and derivatives of these with respect to the PorousFlowVariables.", "href": "../../../../source/materials/PorousFlow2PhasePS.html"}>>>
    phase0_porepressure<<<{"description": "Variable that is the porepressure of phase 0 (the liquid phase)"}>>> = p0
    phase1_saturation<<<{"description": "Variable that is the saturation of phase 1 (the gas phase)"}>>> = s1
    capillary_pressure<<<{"description": "Name of the UserObject defining the capillary pressure"}>>> = pc
  []
  [kr0]
    type = PorousFlowRelativePermeabilityVG<<<{"description": "This Material calculates relative permeability of a phase using the van Genuchten model", "href": "../../../../source/materials/PorousFlowRelativePermeabilityVG.html"}>>>
    phase<<<{"description": "The phase number"}>>> = 0
    m<<<{"description": "The van Genuchten exponent of the phase"}>>> = 0.4
    s_res<<<{"description": "The residual saturation of the phase j. Must be between 0 and 1"}>>> = 0.1
    sum_s_res<<<{"description": "Sum of residual saturations over all phases.  Must be between 0 and 1"}>>> = 0.3
  []
  [kr1]
    type = PorousFlowRelativePermeabilityVG<<<{"description": "This Material calculates relative permeability of a phase using the van Genuchten model", "href": "../../../../source/materials/PorousFlowRelativePermeabilityVG.html"}>>>
    phase<<<{"description": "The phase number"}>>> = 1
    m<<<{"description": "The van Genuchten exponent of the phase"}>>> = 0.4
    s_res<<<{"description": "The residual saturation of the phase j. Must be between 0 and 1"}>>> = 0.2
    sum_s_res<<<{"description": "Sum of residual saturations over all phases.  Must be between 0 and 1"}>>> = 0.3
    wetting<<<{"description": "If true, use the van Genuchten form appropriate for a wetting (liquid) phase.  If false, use the non-wetting (gas) expression."}>>> = false
  []
[]

[VectorPostprocessors<<<{"href": "../../../../syntax/VectorPostprocessors/index.html"}>>>]
  [vpp]
    type = LineValueSampler<<<{"description": "Samples variable(s) along a specified line", "href": "../../../../source/vectorpostprocessors/LineValueSampler.html"}>>>
    warn_discontinuous_face_values<<<{"description": "Whether to return a warning if a discontinuous variable is sampled on a face"}>>> = false
    variable<<<{"description": "The names of the variables that this VectorPostprocessor operates on"}>>> = 's0aux s1aux kr0aux kr1aux'
    start_point<<<{"description": "The beginning of the line"}>>> = '0 0 0'
    end_point<<<{"description": "The ending of the line"}>>> = '1 0 0'
    num_points<<<{"description": "The number of points to sample along the line"}>>> = 20
    sort_by<<<{"description": "What to sort the samples by"}>>> = id
  []
[]

[Executioner<<<{"href": "../../../../syntax/Executioner/index.html"}>>>]
  type = Steady
  solve_type = NEWTON
  nl_abs_tol = 1e-7
[]

[BCs<<<{"href": "../../../../syntax/BCs/index.html"}>>>]
  [sleft]
    type = DirichletBC<<<{"description": "Imposes the essential boundary condition $u=g$, where $g$ is a constant, controllable value.", "href": "../../../../source/bcs/DirichletBC.html"}>>>
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = s1
    value<<<{"description": "Value of the BC"}>>> = 0
    boundary<<<{"description": "The list of boundary IDs from the mesh where this object applies"}>>> = left
  []
  [sright]
    type = DirichletBC<<<{"description": "Imposes the essential boundary condition $u=g$, where $g$ is a constant, controllable value.", "href": "../../../../source/bcs/DirichletBC.html"}>>>
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = s1
    value<<<{"description": "Value of the BC"}>>> = 1
    boundary<<<{"description": "The list of boundary IDs from the mesh where this object applies"}>>> = right
  []
[]

[Outputs<<<{"href": "../../../../syntax/Outputs/index.html"}>>>]
  csv<<<{"description": "Output the scalar variable and postprocessors to a *.csv file using the default CSV output."}>>> = true
  execute_on<<<{"description": "The list of flag(s) indicating when this object should be executed. For a description of each flag, see https://mooseframework.inl.gov/source/interfaces/SetupInterface.html."}>>> = timestep_end
[]

Figure 8: van Genuchten relative permeability Test case 2

References

R. H. Brooks and A. T. Corey. Properties of porous media affecting fluid flow. J. Irrig. Drain. Div., 92:61–88, 1966.

@article{brookscorey1966,
    author = "Brooks, R. H. and Corey, A. T.",
    title = "Properties of porous media affecting fluid flow",
    journal = "J. Irrig. Drain. Div.",
    volume = "92",
    pages = "61--88",
    year = "1966"
}

A. T. Corey. The interrelation between gas and oil relative permeabilities. Prod. Month., 19:38–41, 1954.

@article{corey1954,
    author = "Corey, A. T.",
    title = "The interrelation between gas and oil relative permeabilities",
    journal = "Prod. Month.",
    volume = "19",
    pages = "38--41",
    year = "1954"
}

M. Th. van Genuchten. A closed for equation for predicting the hydraulic conductivity of unsaturated soils. Soil Sci. Soc., 44:892–898, 1980.

@article{vangenuchten1980,
    author = "van Genuchten, M. Th.",
    title = "A closed for equation for predicting the hydraulic conductivity of unsaturated soils",
    journal = "Soil Sci. Soc.",
    volume = "44",
    pages = "892--898",
    year = "1980"
}

(modules/porous_flow/test/tests/relperm/brooks_corey1.i)

# Test Brooks-Corey relative permeability curve by varying saturation over the mesh
# Exponent lambda = 2 for both phases

[Mesh]
  type = GeneratedMesh
  dim = 1
  nx = 20
[]

[GlobalParams]
  PorousFlowDictator = dictator
[]

[Variables]
  [p0]
    initial_condition = 1e6
  []
  [s1]
  []
[]

[AuxVariables]
  [s0aux]
    family = MONOMIAL
    order = CONSTANT
  []
  [s1aux]
    family = MONOMIAL
    order = CONSTANT
  []
  [kr0aux]
    family = MONOMIAL
    order = CONSTANT
  []
  [kr1aux]
    family = MONOMIAL
    order = CONSTANT
  []
[]

[AuxKernels]
  [s0]
    type = PorousFlowPropertyAux
    property = saturation
    phase = 0
    variable = s0aux
  []
  [s1]
    type = PorousFlowPropertyAux
    property = saturation
    phase = 1
    variable = s1aux
  []
  [kr0]
    type = PorousFlowPropertyAux
    property = relperm
    phase = 0
    variable = kr0aux
  []
  [kr1]
    type = PorousFlowPropertyAux
    property = relperm
    phase = 1
    variable = kr1aux
  []
[]

[Functions]
  [s1]
    type = ParsedFunction
    expression = x
  []
[]

[ICs]
  [s1]
    type = FunctionIC
    variable = s1
    function = s1
  []
[]

[Kernels]
  [p0]
    type = Diffusion
    variable = p0
  []
  [s1]
    type = Diffusion
    variable = s1
  []
[]

[UserObjects]
  [dictator]
    type = PorousFlowDictator
    porous_flow_vars = 'p0 s1'
    number_fluid_phases = 2
    number_fluid_components = 2
  []
  [pc]
    type = PorousFlowCapillaryPressureConst
    pc = 0
  []
[]

[Materials]
  [temperature]
    type = PorousFlowTemperature
  []
  [ppss]
    type = PorousFlow2PhasePS
    phase0_porepressure = p0
    phase1_saturation = s1
    capillary_pressure = pc
  []
  [kr0]
    type = PorousFlowRelativePermeabilityBC
    phase = 0
    lambda = 2
  []
  [kr1]
    type = PorousFlowRelativePermeabilityBC
    phase = 1
    lambda = 2
    nw_phase = true
  []
[]

[VectorPostprocessors]
  [vpp]
    type = LineValueSampler
    warn_discontinuous_face_values = false
    variable = 's0aux s1aux kr0aux kr1aux'
    start_point = '0 0 0'
    end_point = '1 0 0'
    num_points = 20
    sort_by = id
  []
[]

[Executioner]
  type = Steady
  solve_type = NEWTON
  nl_abs_tol = 1e-8
[]

[BCs]
  [sleft]
    type = DirichletBC
    variable = s1
    value = 0
    boundary = left
  []
  [sright]
    type = DirichletBC
    variable = s1
    value = 1
    boundary = right
  []
[]

[Outputs]
  csv = true
  execute_on = timestep_end
[]

(modules/porous_flow/test/tests/relperm/brooks_corey2.i)

# Test Brooks-Corey relative permeability curve by varying saturation over the mesh
# Exponent lambda = 2 for both phases
# Residual saturation of phase 0: s0r = 0.2
# Residual saturation of phase 1: s1r = 0.3

[Mesh]
  type = GeneratedMesh
  dim = 1
  nx = 20
[]

[GlobalParams]
  PorousFlowDictator = dictator
[]

[Variables]
  [p0]
    initial_condition = 1e6
  []
  [s1]
  []
[]

[AuxVariables]
  [s0aux]
    family = MONOMIAL
    order = CONSTANT
  []
  [s1aux]
    family = MONOMIAL
    order = CONSTANT
  []
  [kr0aux]
    family = MONOMIAL
    order = CONSTANT
  []
  [kr1aux]
    family = MONOMIAL
    order = CONSTANT
  []
[]

[AuxKernels]
  [s0]
    type = PorousFlowPropertyAux
    property = saturation
    phase = 0
    variable = s0aux
  []
  [s1]
    type = PorousFlowPropertyAux
    property = saturation
    phase = 1
    variable = s1aux
  []
  [kr0]
    type = PorousFlowPropertyAux
    property = relperm
    phase = 0
    variable = kr0aux
  []
  [kr1]
    type = PorousFlowPropertyAux
    property = relperm
    phase = 1
    variable = kr1aux
  []
[]

[Functions]
  [s1]
    type = ParsedFunction
    expression = x
  []
[]

[ICs]
  [s1]
    type = FunctionIC
    variable = s1
    function = s1
  []
[]

[Kernels]
  [p0]
    type = Diffusion
    variable = p0
  []
  [s1]
    type = Diffusion
    variable = s1
  []
[]

[UserObjects]
  [dictator]
    type = PorousFlowDictator
    porous_flow_vars = 'p0 s1'
    number_fluid_phases = 2
    number_fluid_components = 2
  []
  [pc]
    type = PorousFlowCapillaryPressureConst
    pc = 0
  []
[]

[Materials]
  [temperature]
    type = PorousFlowTemperature
  []
  [ppss]
    type = PorousFlow2PhasePS
    phase0_porepressure = p0
    phase1_saturation = s1
    capillary_pressure = pc
  []
  [kr0]
    type = PorousFlowRelativePermeabilityBC
    phase = 0
    lambda = 2
    s_res = 0.2
    sum_s_res = 0.5
  []
  [kr1]
    type = PorousFlowRelativePermeabilityBC
    phase = 1
    lambda = 2
    nw_phase = true
    s_res = 0.3
    sum_s_res = 0.5
  []
[]

[VectorPostprocessors]
  [vpp]
    type = LineValueSampler
    warn_discontinuous_face_values = false
    variable = 's0aux s1aux kr0aux kr1aux'
    start_point = '0 0 0'
    end_point = '1 0 0'
    num_points = 20
    sort_by = id
  []
[]

[Executioner]
  type = Steady
  solve_type = NEWTON
  nl_abs_tol = 1e-8
[]

[BCs]
  [sleft]
    type = DirichletBC
    variable = s1
    value = 0
    boundary = left
  []
  [sright]
    type = DirichletBC
    variable = s1
    value = 1
    boundary = right
  []
[]

[Outputs]
  csv = true
  execute_on = timestep_end
[]

(modules/porous_flow/test/tests/relperm/corey1.i)

# Test Corey relative permeability curve by varying saturation over the mesh
# Corey exponent n = 1 for both phases (linear residual saturation)
# No residual saturation in either phase

[Mesh]
  type = GeneratedMesh
  dim = 1
  nx = 20
[]

[GlobalParams]
  PorousFlowDictator = dictator
[]

[Variables]
  [p0]
    initial_condition = 1e6
  []
  [s1]
  []
[]

[AuxVariables]
  [s0aux]
    family = MONOMIAL
    order = CONSTANT
  []
  [s1aux]
    family = MONOMIAL
    order = CONSTANT
  []
  [kr0aux]
    family = MONOMIAL
    order = CONSTANT
  []
  [kr1aux]
    family = MONOMIAL
    order = CONSTANT
  []
[]

[AuxKernels]
  [s0]
    type = PorousFlowPropertyAux
    property = saturation
    phase = 0
    variable = s0aux
  []
  [s1]
    type = PorousFlowPropertyAux
    property = saturation
    phase = 1
    variable = s1aux
  []
  [kr0]
    type = PorousFlowPropertyAux
    property = relperm
    phase = 0
    variable = kr0aux
  []
  [kr1]
    type = PorousFlowPropertyAux
    property = relperm
    phase = 1
    variable = kr1aux
  []
[]

[Functions]
  [s1]
    type = ParsedFunction
    expression = x
  []
[]

[ICs]
  [s1]
    type = FunctionIC
    variable = s1
    function = s1
  []
[]

[Kernels]
  [p0]
    type = Diffusion
    variable = p0
  []
  [s1]
    type = Diffusion
    variable = s1
  []
[]

[UserObjects]
  [dictator]
    type = PorousFlowDictator
    porous_flow_vars = 'p0 s1'
    number_fluid_phases = 2
    number_fluid_components = 2
  []
  [pc]
    type = PorousFlowCapillaryPressureConst
    pc = 0
  []
[]

[Materials]
  [temperature]
    type = PorousFlowTemperature
  []
  [ppss]
    type = PorousFlow2PhasePS
    phase0_porepressure = p0
    phase1_saturation = s1
    capillary_pressure = pc
  []
  [kr0]
    type = PorousFlowRelativePermeabilityCorey
    phase = 0
    n = 1
  []
  [kr1]
    type = PorousFlowRelativePermeabilityCorey
    phase = 1
    n = 1
  []
[]

[VectorPostprocessors]
  [vpp]
    type = LineValueSampler
    warn_discontinuous_face_values = false
    variable = 's0aux s1aux kr0aux kr1aux'
    start_point = '0 0 0'
    end_point = '1 0 0'
    num_points = 20
    sort_by = id
  []
[]

[Executioner]
  type = Steady
  solve_type = NEWTON
  nl_abs_tol = 1e-8
[]

[BCs]
  [sleft]
    type = DirichletBC
    variable = s1
    value = 0
    boundary = left
  []
  [sright]
    type = DirichletBC
    variable = s1
    value = 1
    boundary = right
  []
[]

[Outputs]
  csv = true
  execute_on = timestep_end
[]

(modules/porous_flow/test/tests/relperm/corey2.i)

# Test Corey relative permeability curve by varying saturation over the mesh
# Corey exponent n = 2 for both phases
# No residual saturation in either phase

[Mesh]
  type = GeneratedMesh
  dim = 1
  nx = 20
[]

[GlobalParams]
  PorousFlowDictator = dictator
[]

[Variables]
  [p0]
    initial_condition = 1e6
  []
  [s1]
    family = LAGRANGE
    order = FIRST
  []
[]

[AuxVariables]
  [s0aux]
    family = MONOMIAL
    order = CONSTANT
  []
  [s1aux]
    family = MONOMIAL
    order = CONSTANT
  []
  [kr0aux]
    family = MONOMIAL
    order = CONSTANT
  []
  [kr1aux]
    family = MONOMIAL
    order = CONSTANT
  []
[]

[AuxKernels]
  [s0]
    type = PorousFlowPropertyAux
    property = saturation
    phase = 0
    variable = s0aux
  []
  [s1]
    type = PorousFlowPropertyAux
    property = saturation
    phase = 1
    variable = s1aux
  []
  [kr0]
    type = PorousFlowPropertyAux
    property = relperm
    phase = 0
    variable = kr0aux
  []
  [kr1]
    type = PorousFlowPropertyAux
    property = relperm
    phase = 1
    variable = kr1aux
  []
[]

[Functions]
  [s1]
    type = ParsedFunction
    expression = x
  []
[]

[ICs]
  [s1]
    type = FunctionIC
    variable = s1
    function = s1
  []
[]

[Kernels]
  [p0]
    type = Diffusion
    variable = p0
  []
  [s1]
    type = Diffusion
    variable = s1
  []
[]

[UserObjects]
  [dictator]
    type = PorousFlowDictator
    porous_flow_vars = 'p0 s1'
    number_fluid_phases = 2
    number_fluid_components = 2
  []
  [pc]
    type = PorousFlowCapillaryPressureConst
    pc = 0
  []
[]

[Materials]
  [temperature]
    type = PorousFlowTemperature
  []
  [ppss]
    type = PorousFlow2PhasePS
    phase0_porepressure = p0
    phase1_saturation = s1
    capillary_pressure = pc
  []
  [kr0]
    type = PorousFlowRelativePermeabilityCorey
    phase = 0
    n = 2
  []
  [kr1]
    type = PorousFlowRelativePermeabilityCorey
    phase = 1
    n = 2
  []
[]

[VectorPostprocessors]
  [vpp]
    type = LineValueSampler
    warn_discontinuous_face_values = false
    variable = 's0aux s1aux kr0aux kr1aux'
    start_point = '0 0 0'
    end_point = '1 0 0'
    num_points = 20
    sort_by = id
  []
[]

[Executioner]
  type = Steady
  solve_type = NEWTON
  nl_abs_tol = 1e-8
[]

[BCs]
  [sleft]
    type = DirichletBC
    variable = s1
    value = 0
    boundary = left
  []
  [sright]
    type = DirichletBC
    variable = s1
    value = 1
    boundary = right
  []
[]

[Outputs]
  csv = true
  execute_on = timestep_end
[]

(modules/porous_flow/test/tests/relperm/corey3.i)

# Test Corey relative permeability curve by varying saturation over the mesh
# Residual saturation of phase 0: s0r = 0.2
# Residual saturation of phase 1: s1r = 0.3

[Mesh]
  type = GeneratedMesh
  dim = 1
  nx = 20
[]

[GlobalParams]
  PorousFlowDictator = dictator
[]

[Variables]
  [p0]
    initial_condition = 1e6
  []
  [s1]
    family = LAGRANGE
    order = FIRST
  []
[]

[AuxVariables]
  [s0aux]
    family = MONOMIAL
    order = CONSTANT
  []
  [s1aux]
    family = MONOMIAL
    order = CONSTANT
  []
  [kr0aux]
    family = MONOMIAL
    order = CONSTANT
  []
  [kr1aux]
    family = MONOMIAL
    order = CONSTANT
  []
[]

[AuxKernels]
  [s0]
    type = PorousFlowPropertyAux
    property = saturation
    phase = 0
    variable = s0aux
  []
  [s1]
    type = PorousFlowPropertyAux
    property = saturation
    phase = 1
    variable = s1aux
  []
  [kr0]
    type = PorousFlowPropertyAux
    property = relperm
    phase = 0
    variable = kr0aux
  []
  [kr1]
    type = PorousFlowPropertyAux
    property = relperm
    phase = 1
    variable = kr1aux
  []
[]

[Functions]
  [s1]
    type = ParsedFunction
    expression = x
  []
[]

[ICs]
  [s1]
    type = FunctionIC
    variable = s1
    function = s1
  []
[]

[Kernels]
  [p0]
    type = Diffusion
    variable = p0
  []
  [s1]
    type = Diffusion
    variable = s1
  []
[]

[UserObjects]
  [dictator]
    type = PorousFlowDictator
    porous_flow_vars = 'p0 s1'
    number_fluid_phases = 2
    number_fluid_components = 2
  []
  [pc]
    type = PorousFlowCapillaryPressureConst
    pc = 0
  []
[]

[Materials]
  [temperature]
    type = PorousFlowTemperature
  []
  [ppss]
    type = PorousFlow2PhasePS
    phase0_porepressure = p0
    phase1_saturation = s1
    capillary_pressure = pc
  []
  [kr0]
    type = PorousFlowRelativePermeabilityCorey
    phase = 0
    n = 2
    s_res = 0.2
    sum_s_res = 0.5
  []
  [kr1]
    type = PorousFlowRelativePermeabilityCorey
    phase = 1
    n = 2
    s_res = 0.3
    sum_s_res = 0.5
  []
[]

[VectorPostprocessors]
  [vpp]
    type = LineValueSampler
    warn_discontinuous_face_values = false
    variable = 's0aux s1aux kr0aux kr1aux'
    start_point = '0 0 0'
    end_point = '1 0 0'
    num_points = 20
    sort_by = id
  []
[]

[Executioner]
  type = Steady
  solve_type = NEWTON
  nl_abs_tol = 1e-8
[]

[BCs]
  [sleft]
    type = DirichletBC
    variable = s1
    value = 0
    boundary = left
  []
  [sright]
    type = DirichletBC
    variable = s1
    value = 1
    boundary = right
  []
[]

[Outputs]
  csv = true
  execute_on = timestep_end
[]

(modules/porous_flow/test/tests/relperm/corey4.i)

# Test Corey relative permeability curve by varying saturation over the mesh
# Residual saturation of phase 0: s0r = 0.2
# Residual saturation of phase 1: s1r = 0.3

[Mesh]
  type = GeneratedMesh
  dim = 1
  nx = 20
[]

[GlobalParams]
  PorousFlowDictator = dictator
[]

[Variables]
  [p0]
    initial_condition = 1e6
  []
  [s1]
    family = LAGRANGE
    order = FIRST
  []
[]

[AuxVariables]
  [s0aux]
    family = MONOMIAL
    order = CONSTANT
  []
  [s1aux]
    family = MONOMIAL
    order = CONSTANT
  []
  [kr0aux]
    family = MONOMIAL
    order = CONSTANT
  []
  [kr1aux]
    family = MONOMIAL
    order = CONSTANT
  []
[]

[AuxKernels]
  [s0]
    type = PorousFlowPropertyAux
    property = saturation
    phase = 0
    variable = s0aux
  []
  [s1]
    type = PorousFlowPropertyAux
    property = saturation
    phase = 1
    variable = s1aux
  []
  [kr0]
    type = PorousFlowPropertyAux
    property = relperm
    phase = 0
    variable = kr0aux
  []
  [kr1]
    type = PorousFlowPropertyAux
    property = relperm
    phase = 1
    variable = kr1aux
  []
[]

[Functions]
  [s1]
    type = ParsedFunction
    expression = x
  []
[]

[ICs]
  [s1]
    type = FunctionIC
    variable = s1
    function = s1
  []
[]

[Kernels]
  [p0]
    type = Diffusion
    variable = p0
  []
  [s1]
    type = Diffusion
    variable = s1
  []
[]

[UserObjects]
  [dictator]
    type = PorousFlowDictator
    porous_flow_vars = 'p0 s1'
    number_fluid_phases = 2
    number_fluid_components = 2
  []
  [pc]
    type = PorousFlowCapillaryPressureConst
    pc = 0
  []
[]

[Materials]
  [temperature]
    type = PorousFlowTemperature
  []
  [ppss]
    type = PorousFlow2PhasePS
    phase0_porepressure = p0
    phase1_saturation = s1
    capillary_pressure = pc
  []
  [kr0]
    type = PorousFlowRelativePermeabilityCorey
    scaling = 0.1
    phase = 0
    n = 2
    s_res = 0.2
    sum_s_res = 0.5
  []
  [kr1]
    type = PorousFlowRelativePermeabilityCorey
    scaling = 10.0
    phase = 1
    n = 2
    s_res = 0.3
    sum_s_res = 0.5
  []
[]

[VectorPostprocessors]
  [vpp]
    type = LineValueSampler
    warn_discontinuous_face_values = false
    variable = 's0aux s1aux kr0aux kr1aux'
    start_point = '0 0 0'
    end_point = '1 0 0'
    num_points = 20
    sort_by = id
  []
[]

[Executioner]
  type = Steady
  solve_type = NEWTON
  nl_abs_tol = 1e-8
[]

[BCs]
  [sleft]
    type = DirichletBC
    variable = s1
    value = 0
    boundary = left
  []
  [sright]
    type = DirichletBC
    variable = s1
    value = 1
    boundary = right
  []
[]

[Outputs]
  csv = true
  execute_on = timestep_end
[]

(modules/porous_flow/test/tests/relperm/vangenuchten1.i)

# Test van Genuchten relative permeability curve by varying saturation over the mesh
# van Genuchten exponent m = 0.5 for both phases
# No residual saturation in either phase

[Mesh]
  type = GeneratedMesh
  dim = 1
  nx = 100
[]

[GlobalParams]
  PorousFlowDictator = dictator
[]

[Variables]
  [p0]
    initial_condition = 1e6
  []
  [s1]
  []
[]

[AuxVariables]
  [s0aux]
    family = MONOMIAL
    order = CONSTANT
  []
  [s1aux]
    family = MONOMIAL
    order = CONSTANT
  []
  [kr0aux]
    family = MONOMIAL
    order = CONSTANT
  []
  [kr1aux]
    family = MONOMIAL
    order = CONSTANT
  []
[]

[AuxKernels]
  [s0]
    type = PorousFlowPropertyAux
    property = saturation
    phase = 0
    variable = s0aux
  []
  [s1]
    type = PorousFlowPropertyAux
    property = saturation
    phase = 1
    variable = s1aux
  []
  [kr0]
    type = PorousFlowPropertyAux
    property = relperm
    phase = 0
    variable = kr0aux
  []
  [kr1]
    type = PorousFlowPropertyAux
    property = relperm
    phase = 1
    variable = kr1aux
  []
[]

[Functions]
  [s1]
    type = ParsedFunction
    expression = x
  []
[]

[ICs]
  [s1]
    type = FunctionIC
    variable = s1
    function = s1
  []
[]

[Kernels]
  [p0]
    type = Diffusion
    variable = p0
  []
  [s1]
    type = Diffusion
    variable = s1
  []
[]

[UserObjects]
  [dictator]
    type = PorousFlowDictator
    porous_flow_vars = 'p0 s1'
    number_fluid_phases = 2
    number_fluid_components = 2
  []
  [pc]
    type = PorousFlowCapillaryPressureConst
    pc = 0
  []
[]

[Materials]
  [temperature]
    type = PorousFlowTemperature
  []
  [ppss]
    type = PorousFlow2PhasePS
    phase0_porepressure = p0
    phase1_saturation = s1
    capillary_pressure = pc
  []
  [kr0]
    type = PorousFlowRelativePermeabilityVG
    phase = 0
    m = 0.5
  []
  [kr1]
    type = PorousFlowRelativePermeabilityVG
    phase = 1
    m = 0.5
    wetting = false
  []
[]

[VectorPostprocessors]
  [vpp]
    type = LineValueSampler
    warn_discontinuous_face_values = false
    variable = 's0aux s1aux kr0aux kr1aux'
    start_point = '0 0 0'
    end_point = '1 0 0'
    num_points = 20
    sort_by = id
  []
[]

[Executioner]
  type = Steady
  solve_type = NEWTON
  nl_abs_tol = 1e-7
[]

[BCs]
  [sleft]
    type = DirichletBC
    variable = s1
    value = 0
    boundary = left
  []
  [sright]
    type = DirichletBC
    variable = s1
    value = 1
    boundary = right
  []
[]

[Outputs]
  csv = true
  execute_on = timestep_end
[]

(modules/porous_flow/test/tests/relperm/vangenuchten2.i)

# Test van Genuchten relative permeability curve by varying saturation over the mesh
# van Genuchten exponent m = 0.4 for both phases
# Phase 0 residual saturation s0r = 0.1
# Phase 1 residual saturation s1r = 0.2

[Mesh]
  type = GeneratedMesh
  dim = 1
  nx = 100
[]

[GlobalParams]
  PorousFlowDictator = dictator
[]

[Variables]
  [p0]
    initial_condition = 1e6
  []
  [s1]
  []
[]

[AuxVariables]
  [s0aux]
    family = MONOMIAL
    order = CONSTANT
  []
  [s1aux]
    family = MONOMIAL
    order = CONSTANT
  []
  [kr0aux]
    family = MONOMIAL
    order = CONSTANT
  []
  [kr1aux]
    family = MONOMIAL
    order = CONSTANT
  []
[]

[AuxKernels]
  [s0]
    type = PorousFlowPropertyAux
    property = saturation
    phase = 0
    variable = s0aux
  []
  [s1]
    type = PorousFlowPropertyAux
    property = saturation
    phase = 1
    variable = s1aux
  []
  [kr0]
    type = PorousFlowPropertyAux
    property = relperm
    phase = 0
    variable = kr0aux
  []
  [kr1]
    type = PorousFlowPropertyAux
    property = relperm
    phase = 1
    variable = kr1aux
  []
[]

[Functions]
  [s1]
    type = ParsedFunction
    expression = x
  []
[]

[ICs]
  [s1]
    type = FunctionIC
    variable = s1
    function = s1
  []
[]

[Kernels]
  [p0]
    type = Diffusion
    variable = p0
  []
  [s1]
    type = Diffusion
    variable = s1
  []
[]

[UserObjects]
  [dictator]
    type = PorousFlowDictator
    porous_flow_vars = 'p0 s1'
    number_fluid_phases = 2
    number_fluid_components = 2
  []
  [pc]
    type = PorousFlowCapillaryPressureConst
    pc = 0
  []
[]

[Materials]
  [temperature]
    type = PorousFlowTemperature
  []
  [ppss]
    type = PorousFlow2PhasePS
    phase0_porepressure = p0
    phase1_saturation = s1
    capillary_pressure = pc
  []
  [kr0]
    type = PorousFlowRelativePermeabilityVG
    phase = 0
    m = 0.4
    s_res = 0.1
    sum_s_res = 0.3
  []
  [kr1]
    type = PorousFlowRelativePermeabilityVG
    phase = 1
    m = 0.4
    s_res = 0.2
    sum_s_res = 0.3
    wetting = false
  []
[]

[VectorPostprocessors]
  [vpp]
    type = LineValueSampler
    warn_discontinuous_face_values = false
    variable = 's0aux s1aux kr0aux kr1aux'
    start_point = '0 0 0'
    end_point = '1 0 0'
    num_points = 20
    sort_by = id
  []
[]

[Executioner]
  type = Steady
  solve_type = NEWTON
  nl_abs_tol = 1e-7
[]

[BCs]
  [sleft]
    type = DirichletBC
    variable = s1
    value = 0
    boundary = left
  []
  [sright]
    type = DirichletBC
    variable = s1
    value = 1
    boundary = right
  []
[]

[Outputs]
  csv = true
  execute_on = timestep_end
[]

Brooks - Corey
Corey
van Genuchten
References