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 $var element = document.getElementById("moose-equation-de65cec8-c30c-46d6-bcaa-f593c30dd5ca");katex.render("k_{\\mathrm{r, w}} = \\left(S_{\\mathrm{eff}}\\right)^{(2 + 3 \\lambda)/\\lambda},", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ and the relative permeability of the non-wetting phase is $var element = document.getElementById("moose-equation-1b41ec9b-2169-4a1c-908e-9d60687d4082");katex.render("k_{\\mathrm{r, nw}} = (1 - S_{\\mathrm{eff}})^2 \\left[1 - \\left(S_{\\mathrm{eff}}\\right)^{(2 + \\lambda)/\\lambda}\\right],", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ where $var element = document.getElementById("moose-equation-1cc9df01-9cd3-4b12-8aa1-101f35d6a73c");katex.render("\\lambda", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is a user-defined exponent. When $var element = document.getElementById("moose-equation-c6662447-218f-45d1-b469-aa2f2a7ac282");katex.render("\\lambda = 2", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ , 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]
  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
[]

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]
  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
[]

Figure 2: Brooks-Corey relative permeability Test case 2

Corey

PorousFlowRelativePermeabilityCorey

The relative permeability of the phase is given by Corey (1954) $var element = document.getElementById("moose-equation-6494b684-2238-4e96-ba6a-5989da23596a");katex.render("k_{\\mathrm{r}} = S_{\\mathrm{eff}}^{n},", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ where $var element = document.getElementById("moose-equation-b39249c3-1c86-4c37-acae-14bcd9dd4ccd");katex.render("n", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is a user-defined quantity. Originally, Corey (1954) used $var element = document.getElementById("moose-equation-7660cc72-0d29-4ce7-a836-d8ae627e03ff");katex.render("n = 4", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ 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]
  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
[]

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]
  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
[]

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]
  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
[]

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]
  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
[]

Figure 6: Corey relative permeability Test case 4

van Genuchten

The relative permeability of the wetting phase is given by Genuchten (1980) $var element = document.getElementById("moose-equation-4b7354d5-2554-469a-9c7b-3ec280958d45");katex.render("k_{\\mathrm{r}} = \\sqrt{S_{\\mathrm{eff}}} \\left(1 - (1 - S_{\\mathrm{eff}}^{1/m})^{m} \\right)^{2}.", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ This has the numerical disadvantage that its derivative as $var element = document.getElementById("moose-equation-11b4579e-fefd-4312-9f84-c4e2c9ae066e");katex.render("S_{\\mathrm{eff}}\\rightarrow 1", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ tends to infinity. This means that simulations where the saturation oscillates around $var element = document.getElementById("moose-equation-cdc7cbea-dc3a-48a8-8f3d-e7fe057c60dd");katex.render("S_{\\mathrm{eff}}=1", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ 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: $var element = document.getElementById("moose-equation-df6ec554-9974-4021-be61-b02e91ee4d9d");katex.render("k_{\\mathrm{r}} = \\begin{cases} \\textrm{van Genuchten} & \\textrm{for } S_{\\mathrm{eff}} < S_{c} \\\\ \\textrm{cubic} & \\textrm{for } S_{\\mathrm{eff}} \\geq S_{c}. \\end{cases}", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ Here the cubic is chosen so that its value and derivative match the van Genuchten expression at $var element = document.getElementById("moose-equation-4584d9e5-407f-46eb-8beb-619d5b77f525");katex.render("S=S_{c}", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ , and so that it is unity at $var element = document.getElementById("moose-equation-2f53ab37-fe81-4861-9d4e-b0a8c8c9f2ad");katex.render("S_{\\mathrm{eff}}=1", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ .

For the non-wetting phase, the van Genuchten expression is $var element = document.getElementById("moose-equation-20c36f6d-3499-43ec-8d88-888faccb8460");katex.render("k_{\\mathrm{r}} = \\sqrt{S_{\\mathrm{eff}}} \\left(1 - (1 - S_{\\mathrm{eff}})^{1/m} \\right)^{2m} \\ .", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$

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]
  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
[]

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]
  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
[]

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