How to use Kuzmin-Turek stabilization in PorousFlow simulations

This page is part of a set of pages devoted to discussions of numerical stabilization in PorousFlow. See:

Kuzmin and Turek (Kuzmin and Turek, 2004) describe a method of stabilising advection while minimising artificial numerical diffusion. In this page "Kuzmin and Turek" is abbreviatved to "KT". Basic features of KT's approach are detailed in a worked example and the results are compared with full upwinding, RDG and no stabilization in the numerical diffusion page.

This page is for users who want to use KT stabilization in PorousFlow simulations.

note

The Kuzmin-Turek stabilization is new and users are strongly encouraged to experiment with it and report their findings to the moose-users gmail group to iron out any problems and so we can collectively gain experience.

Using the Action system

Both the PorousFlowFullySaturated and PorousFlowUnsaturated Actions support KT stabilization (the PorousFlowBasicTHM Action does not support any numerical stabilization as it is typically unneeded). KT stabilization may be included simply by specifying the stabilization and flux_limiter_type parameters. For instance:

[PorousFlowFullySaturated]
  porepressure = porepressure
  coupling_type = Hydro
  gravity = '0 0 0'
  fp = the_simple_fluid
  mass_fraction_vars = tracer_concentration
  stabilization = KT
  flux_limiter_type = superbee

/opt/civet/build_0/moose/modules/porous_flow/examples/tutorial/06_KT.i

# Darcy flow with a tracer
[Mesh]
  [annular]
    type = AnnularMeshGenerator
    nr = 10
    rmin = 1.0
    rmax = 10
    growth_r = 1.4
    nt = 4
    dmin = 0
    dmax = 90
  []
  [./make3D]
    type = MeshExtruderGenerator
    extrusion_vector = '0 0 12'
    num_layers = 3
    bottom_sideset = 'bottom'
    top_sideset = 'top'
    input = annular
  [../]
  [./shift_down]
    type = TransformGenerator
    transform = TRANSLATE
    vector_value = '0 0 -6'
    input = make3D
  [../]
  [./aquifer]
    type = SubdomainBoundingBoxGenerator
    block_id = 1
    bottom_left = '0 0 -2'
    top_right = '10 10 2'
    input = shift_down
  [../]
  [./injection_area]
    type = ParsedGenerateSideset
    combinatorial_geometry = 'x*x+y*y<1.01'
    included_subdomain_ids = 1
    new_sideset_name = 'injection_area'
    input = 'aquifer'
  [../]
  [./rename]
    type = RenameBlockGenerator
    old_block_id = '0 1'
    new_block_name = 'caps aquifer'
    input = 'injection_area'
  [../]
[]

[GlobalParams]
  PorousFlowDictator = dictator
[]

[Variables]
  [./porepressure]
  [../]
  [./tracer_concentration]
  [../]
[]

[ICs]
  [./tracer_concentration]
    type = FunctionIC
    function = '0.5*if(x*x+y*y<1.01,1,0)'
    variable = tracer_concentration
  [../]
[]

[PorousFlowFullySaturated]
  porepressure = porepressure
  coupling_type = Hydro
  gravity = '0 0 0'
  fp = the_simple_fluid
  mass_fraction_vars = tracer_concentration
  stabilization = KT
  flux_limiter_type = superbee
[]

[BCs]
  [./constant_injection_porepressure]
    type = DirichletBC
    variable = porepressure
    value = 1E6
    boundary = injection_area
  [../]
  [./constant_outer_porepressure]
    type = DirichletBC
    variable = porepressure
    value = 0
    boundary = rmax
  [../]
  [./injected_tracer]
    type = DirichletBC
    variable = tracer_concentration
    value = 0.5
    boundary = injection_area
  [../]
[]

[Modules]
  [./FluidProperties]
    [./the_simple_fluid]
      type = SimpleFluidProperties
      bulk_modulus = 2E9
      viscosity = 1.0E-3
      density0 = 1000.0
    [../]
  [../]
[]

[Materials]
  [./porosity]
    type = PorousFlowPorosity
    porosity_zero = 0.1
  [../]
  [./permeability_aquifer]
    type = PorousFlowPermeabilityConst
    block = aquifer
    permeability = '1E-14 0 0   0 1E-14 0   0 0 1E-14'
  [../]
  [./permeability_caps]
    type = PorousFlowPermeabilityConst
    block = caps
    permeability = '1E-15 0 0   0 1E-15 0   0 0 1E-16'
  [../]
[]

[Preconditioning]
  active = basic
  [./basic]
    type = SMP
    full = true
    petsc_options = '-ksp_diagonal_scale -ksp_diagonal_scale_fix'
    petsc_options_iname = '-pc_type -sub_pc_type -sub_pc_factor_shift_type -pc_asm_overlap'
    petsc_options_value = ' asm      lu           NONZERO                   2'
  [../]
  [./preferred_but_might_not_be_installed]
    type = SMP
    full = true
    petsc_options_iname = '-pc_type -pc_factor_mat_solver_package'
    petsc_options_value = ' lu       mumps'
  [../]
[]

[Executioner]
  type = Transient
  solve_type = Newton
  end_time = 1E6
  dt = 1E5
  nl_rel_tol = 1E-14
[]

[Outputs]
  exodus = true
[]

and

[PorousFlowUnsaturated]
  porepressure = porepressure
  coupling_type = Hydro
  gravity = '0 0 0'
  fp = the_simple_fluid
  relative_permeability_exponent = 3
  relative_permeability_type = Corey
  residual_saturation = 0.1
  van_genuchten_alpha = 1E-6
  van_genuchten_m = 0.6
  stabilization = KT
  flux_limiter_type = None

/opt/civet/build_0/moose/modules/porous_flow/examples/tutorial/08_KT.i

# Unsaturated Darcy-Richards flow
[Mesh]
  [annular]
    type = AnnularMeshGenerator
    nr = 10
    rmin = 1.0
    rmax = 10
    growth_r = 1.4
    nt = 4
    dmin = 0
    dmax = 90
  []
  [./make3D]
    input = annular
    type = MeshExtruderGenerator
    extrusion_vector = '0 0 12'
    num_layers = 3
    bottom_sideset = 'bottom'
    top_sideset = 'top'
  [../]
  [./shift_down]
    type = TransformGenerator
    transform = TRANSLATE
    vector_value = '0 0 -6'
    input = make3D
  [../]
  [./aquifer]
    type = SubdomainBoundingBoxGenerator
    block_id = 1
    bottom_left = '0 0 -2'
    top_right = '10 10 2'
    input = shift_down
  [../]
  [./injection_area]
    type = ParsedGenerateSideset
    combinatorial_geometry = 'x*x+y*y<1.01'
    included_subdomain_ids = 1
    new_sideset_name = 'injection_area'
    input = 'aquifer'
  [../]
  [./rename]
    type = RenameBlockGenerator
    old_block_id = '0 1'
    new_block_name = 'caps aquifer'
    input = 'injection_area'
  [../]
[]

[GlobalParams]
  PorousFlowDictator = dictator
[]

[Variables]
  [./porepressure]
  [../]
[]

[PorousFlowUnsaturated]
  porepressure = porepressure
  coupling_type = Hydro
  gravity = '0 0 0'
  fp = the_simple_fluid
  relative_permeability_exponent = 3
  relative_permeability_type = Corey
  residual_saturation = 0.1
  van_genuchten_alpha = 1E-6
  van_genuchten_m = 0.6
  stabilization = KT
  flux_limiter_type = None
[]

[BCs]
  [./production]
    type = PorousFlowSink
    variable = porepressure
    fluid_phase = 0
    flux_function = 1E-2
    use_relperm = true
    boundary = injection_area
  [../]
[]

[Modules]
  [./FluidProperties]
    [./the_simple_fluid]
      type = SimpleFluidProperties
      bulk_modulus = 2E9
      viscosity = 1.0E-3
      density0 = 1000.0
    [../]
  [../]
[]

[Materials]
  [./porosity]
    type = PorousFlowPorosity
    porosity_zero = 0.1
  [../]
  [./permeability_aquifer]
    type = PorousFlowPermeabilityConst
    block = aquifer
    permeability = '1E-14 0 0   0 1E-14 0   0 0 1E-14'
  [../]
  [./permeability_caps]
    type = PorousFlowPermeabilityConst
    block = caps
    permeability = '1E-15 0 0   0 1E-15 0   0 0 1E-16'
  [../]
[]

[Preconditioning]
  active = basic
  [./basic]
    type = SMP
    full = true
    petsc_options = '-ksp_diagonal_scale -ksp_diagonal_scale_fix'
    petsc_options_iname = '-pc_type -sub_pc_type -sub_pc_factor_shift_type -pc_asm_overlap'
    petsc_options_value = ' asm      lu           NONZERO                   2'
  [../]
  [./preferred_but_might_not_be_installed]
    type = SMP
    full = true
    petsc_options_iname = '-pc_type -pc_factor_mat_solver_package'
    petsc_options_value = ' lu       mumps'
  [../]
[]

[Executioner]
  type = Transient
  solve_type = Newton
  end_time = 1E5
  dt = 1E5
  nl_rel_tol = 1E-14
[]

[Outputs]
  exodus = true
[]

Using Kernels and UserObjects

It is also fairly straightforward to swap from full-upwinding (or no stabilization) to KT stabilization. Only two changes are needed in the input file.

Kernels

Instead of PorousFlowAdvectiveFlux or PorousFlowHeatAdvection or another type of PorousFlow Kernel, the PorousFlowFluxLimitedTVDAdvection Kernel must be used. For instance, a full-upwind simulation's Kernels are:

[Kernels]
  [./mass_dot]
    type = PorousFlowMassTimeDerivative
    fluid_component = 0
    variable = pp
  [../]
  [./advection]
    type = PorousFlowAdvectiveFlux
    fluid_component = 0
    variable = pp
  [../]
  [./energy_dot]
    type = PorousFlowEnergyTimeDerivative
    variable = temp
  [../]
  [./heat_advection]
    type = PorousFlowHeatAdvection
    variable = temp
  [../]

/opt/civet/build_0/moose/modules/porous_flow/test/tests/heat_advection/heat_advection_1d.i

# 1phase, heat advecting with a moving fluid
[Mesh]
  type = GeneratedMesh
  dim = 1
  nx = 50
  xmin = 0
  xmax = 1
[]

[GlobalParams]
  PorousFlowDictator = dictator
  gravity = '0 0 0'
[]

[Variables]
  [./temp]
    initial_condition = 200
  [../]
  [./pp]
  [../]
[]

[ICs]
  [./pp]
    type = FunctionIC
    variable = pp
    function = '1-x'
  [../]
[]

[BCs]
  [./pp0]
    type = DirichletBC
    variable = pp
    boundary = left
    value = 1
  [../]
  [./pp1]
    type = DirichletBC
    variable = pp
    boundary = right
    value = 0
  [../]
  [./spit_heat]
    type = DirichletBC
    variable = temp
    boundary = left
    value = 300
  [../]
  [./suck_heat]
    type = DirichletBC
    variable = temp
    boundary = right
    value = 200
  [../]
[]

[Kernels]
  [./mass_dot]
    type = PorousFlowMassTimeDerivative
    fluid_component = 0
    variable = pp
  [../]
  [./advection]
    type = PorousFlowAdvectiveFlux
    fluid_component = 0
    variable = pp
  [../]
  [./energy_dot]
    type = PorousFlowEnergyTimeDerivative
    variable = temp
  [../]
  [./heat_advection]
    type = PorousFlowHeatAdvection
    variable = temp
  [../]
[]

[UserObjects]
  [./dictator]
    type = PorousFlowDictator
    porous_flow_vars = 'temp pp'
    number_fluid_phases = 1
    number_fluid_components = 1
  [../]
  [./pc]
    type = PorousFlowCapillaryPressureVG
    m = 0.6
    alpha = 1.3
  [../]
[]

[Modules]
  [./FluidProperties]
    [./simple_fluid]
      type = SimpleFluidProperties
      bulk_modulus = 100
      density0 = 1000
      viscosity = 4.4
      thermal_expansion = 0
      cv = 2
    [../]
  [../]
[]

[Materials]
  [./temperature]
    type = PorousFlowTemperature
    temperature = temp
  [../]
  [./porosity]
    type = PorousFlowPorosityConst
    porosity = 0.2
  [../]
  [./rock_heat]
    type = PorousFlowMatrixInternalEnergy
    specific_heat_capacity = 1.0
    density = 125
  [../]
  [./simple_fluid]
    type = PorousFlowSingleComponentFluid
    fp = simple_fluid
    phase = 0
  [../]
  [./permeability]
    type = PorousFlowPermeabilityConst
    permeability = '1.1 0 0 0 2 0 0 0 3'
  [../]
  [./relperm]
    type = PorousFlowRelativePermeabilityCorey
    n = 2
    phase = 0
  [../]
  [./massfrac]
    type = PorousFlowMassFraction
  [../]
  [./PS]
    type = PorousFlow1PhaseP
    porepressure = pp
    capillary_pressure = pc
  [../]
[]

[Preconditioning]
  [./andy]
    type = SMP
    full = true
    petsc_options_iname = '-ksp_type -pc_type -snes_atol -snes_rtol -snes_max_it'
    petsc_options_value = 'gmres bjacobi 1E-15 1E-10 10000'
  [../]
[]

[Executioner]
  type = Transient
  solve_type = Newton
  dt = 0.01
  end_time = 0.6
[]

[VectorPostprocessors]
  [./T]
    type = LineValueSampler
    start_point = '0 0 0'
    end_point = '1 0 0'
    num_points = 51
    sort_by = x
    variable = temp
  [../]
[]

[Outputs]
  [./csv]
    type = CSV
    sync_times = '0.1 0.6'
    sync_only = true
  [../]
[]

while an identical simulation using KT stabilization has Kernels:

[Kernels]
  [./mass_dot]
    type = PorousFlowMassTimeDerivative
    fluid_component = 0
    variable = pp
  [../]
  [./fluid_advection]
    type = PorousFlowFluxLimitedTVDAdvection
    variable = pp
    advective_flux_calculator = fluid_advective_flux
  [../]
  [./energy_dot]
    type = PorousFlowEnergyTimeDerivative
    variable = temp
  [../]
  [./heat_advection]
    type = PorousFlowFluxLimitedTVDAdvection
    variable = temp
    advective_flux_calculator = heat_advective_flux
  [../]

/opt/civet/build_0/moose/modules/porous_flow/test/tests/heat_advection/heat_advection_1d_KT.i

# 1phase, heat advecting with a moving fluid
# Using the Kuzmin-Turek stabilization scheme
[Mesh]
  type = GeneratedMesh
  dim = 1
  nx = 50
  xmin = 0
  xmax = 1
[]

[GlobalParams]
  PorousFlowDictator = dictator
  gravity = '0 0 0'
[]

[Variables]
  [./temp]
    initial_condition = 200
  [../]
  [./pp]
  [../]
[]

[ICs]
  [./pp]
    type = FunctionIC
    variable = pp
    function = '1-x'
  [../]
[]

[BCs]
  [./pp0]
    type = DirichletBC
    variable = pp
    boundary = left
    value = 1
  [../]
  [./pp1]
    type = DirichletBC
    variable = pp
    boundary = right
    value = 0
  [../]
  [./spit_heat]
    type = DirichletBC
    variable = temp
    boundary = left
    value = 300
  [../]
  [./suck_heat]
    type = DirichletBC
    variable = temp
    boundary = right
    value = 200
  [../]
[]

[Kernels]
  [./mass_dot]
    type = PorousFlowMassTimeDerivative
    fluid_component = 0
    variable = pp
  [../]
  [./fluid_advection]
    type = PorousFlowFluxLimitedTVDAdvection
    variable = pp
    advective_flux_calculator = fluid_advective_flux
  [../]
  [./energy_dot]
    type = PorousFlowEnergyTimeDerivative
    variable = temp
  [../]
  [./heat_advection]
    type = PorousFlowFluxLimitedTVDAdvection
    variable = temp
    advective_flux_calculator = heat_advective_flux
  [../]
[]

[UserObjects]
  [./dictator]
    type = PorousFlowDictator
    porous_flow_vars = 'temp pp'
    number_fluid_phases = 1
    number_fluid_components = 1
  [../]
  [./pc]
    type = PorousFlowCapillaryPressureVG
    m = 0.6
    alpha = 1.3
  [../]
  [./fluid_advective_flux]
    type = PorousFlowAdvectiveFluxCalculatorSaturated
    flux_limiter_type = superbee
  [../]
  [./heat_advective_flux]
    type = PorousFlowAdvectiveFluxCalculatorSaturatedHeat
    flux_limiter_type = superbee
  [../]
[]

[Modules]
  [./FluidProperties]
    [./simple_fluid]
      type = SimpleFluidProperties
      bulk_modulus = 100
      density0 = 1000
      viscosity = 4.4
      thermal_expansion = 0
      cv = 2
    [../]
  [../]
[]

[Materials]
  [./temperature]
    type = PorousFlowTemperature
    temperature = temp
  [../]
  [./porosity]
    type = PorousFlowPorosityConst
    porosity = 0.2
  [../]
  [./rock_heat]
    type = PorousFlowMatrixInternalEnergy
    specific_heat_capacity = 1.0
    density = 125
  [../]
  [./simple_fluid]
    type = PorousFlowSingleComponentFluid
    fp = simple_fluid
    phase = 0
  [../]
  [./permeability]
    type = PorousFlowPermeabilityConst
    permeability = '1.1 0 0 0 2 0 0 0 3'
  [../]
  [./relperm]
    type = PorousFlowRelativePermeabilityCorey
    n = 2
    phase = 0
  [../]
  [./massfrac]
    type = PorousFlowMassFraction
  [../]
  [./PS]
    type = PorousFlow1PhaseP
    porepressure = pp
    capillary_pressure = pc
  [../]
[]

[Preconditioning]
  [./andy]
    type = SMP
    full = true
    petsc_options_iname = '-ksp_type -pc_type -snes_atol -snes_rtol -snes_max_it'
    petsc_options_value = 'gmres bjacobi 1E-15 1E-10 10000'
  [../]
[]

[Executioner]
  type = Transient
  solve_type = Newton
  dt = 0.01
  end_time = 0.6
[]

[VectorPostprocessors]
  [./T]
    type = LineValueSampler
    start_point = '0 0 0'
    end_point = '1 0 0'
    num_points = 51
    sort_by = x
    variable = temp
  [../]
[]

[Outputs]
  file_base = heat_advection_1d_KT
  [./csv]
    type = CSV
    sync_times = '0.1 0.6'
    sync_only = true
  [../]
[]

Notice that the PorousFlowFluxLimitedTVDAdvection Kernels require an advective_flux_calculator input. This is a UserObject, which is the subject of the next section.

UserObjects

The set of PorousFlow advective flux calculator UserObjects compute the Residual and Jacobian entries that are used by the PorousFlowFluxLimitedTVDAdvection Kernels. The reason that these entries are computed by UserObjects instead of Kernels is that the KT scheme requires information about porepressures, etc, that are 2 nodes away from the current node. This information gets assembled and processed to determine upwinding directions and suitable stabilization approaches before the residual and Jacobian are computed. The process is explained in the worked example of KT stabilization page. The upshot is that a UserObject must be employed instead of the usual MOOSE Kernel-only approach.

Table 1: Available types of PorousFlow advective flux UserObjects

Equation	UserObject	Use case
$var element = document.getElementById("moose-equation-9c21a0f9-35fb-430c-b18c-02449d602bcc");katex.render("-\\frac{\\rho}{\\mu}k_{ij}(\\nabla_{j}P - \\rho g_{j})", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$	PorousFlowAdvectiveFluxCalculatorSaturated	1-phase, 1-component, fully-saturated
$var element = document.getElementById("moose-equation-3c9594c8-0197-446a-9d49-5154e3d6c6dc");katex.render("-\\chi^{\\kappa}\\frac{\\rho}{\\mu}k_{ij}(\\nabla_{j}P - \\rho g_{j})", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$	PorousFlowAdvectiveFluxCalculatorSaturatedMultiComponent	1-phase, multi-component, fully-saturated
$var element = document.getElementById("moose-equation-28b62603-922a-49f3-bb09-4414cf1617d6");katex.render("-k_{\\mathrm{r}}\\frac{\\rho}{\\mu}k_{ij}(\\nabla_{j}P - \\rho g_{j})", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$	PorousFlowAdvectiveFluxCalculatorUnsaturated	1-phase, 1-component, unsaturated
$var element = document.getElementById("moose-equation-298c1fb9-931d-43d2-a742-6a95ca257f03");katex.render("-k_{\\mathrm{r,}\\beta}\\frac{\\rho_{\\beta}}{\\mu_{\\beta}}k_{ij}(\\nabla_{j}P_{\\beta} - \\rho_{\\beta} g_{j})", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$	PorousFlowAdvectiveFluxCalculatorUnsaturated	multi-phase, multi-component, immiscible
$var element = document.getElementById("moose-equation-2f1eda77-4450-44cd-aa42-e3dd332b6d65");katex.render("-k_{\\mathrm{r,}\\beta}\\chi_{\\beta}^{\\kappa}\\frac{\\rho_{\\beta}}{\\mu_{\\beta}}k_{ij}(\\nabla_{j}P_{\\beta} - \\rho_{\\beta} g_{j})", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$	PorousFlowAdvectiveFluxCalculatorUnsaturatedMultiComponent	multi-phase, multi-component
$var element = document.getElementById("moose-equation-109a2e3e-fd2b-4b4a-8a57-73bcafc76d7f");katex.render("-h\\frac{\\rho}{\\mu}k_{ij}(\\nabla_{j}P - \\rho g_{j})", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$	PorousFlowAdvectiveFluxCalculatorSaturatedHeat	heat, 1-phase
$var element = document.getElementById("moose-equation-f157ee07-3820-4cd6-9d79-f5e33385f6d5");katex.render("-h_{\\beta}k_{\\mathrm{r,}\\beta}\\frac{\\rho_{\\beta}}{\\mu_{\\beta}}k_{ij}(\\nabla_{j}P_{\\beta} - \\rho_{\\beta} g_{j})", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$	PorousFlowAdvectiveFluxCalculatorUnsaturatedHeat	heat, multi-phase

The notation is detailed in the nomenclature: briefly, a $var element = document.getElementById("moose-equation-67cb2deb-e526-4324-b230-e5501b7c675b");katex.render("\\beta", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ subscript denotes a phase, a $var element = document.getElementById("moose-equation-5d9396fa-46af-4e8b-90df-d1d31cd9cbda");katex.render("\\kappa", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ superscript denotes a fluid component, $var element = document.getElementById("moose-equation-be59a4ba-000f-47ca-9309-7e881ef21c2d");katex.render("\\rho", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ is a fluid density, $var element = document.getElementById("moose-equation-6191745c-2a9a-4ad9-9ea6-90bc3110bb60");katex.render("\\mu", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ a fluid viscosity, $var element = document.getElementById("moose-equation-799dfd47-81be-4d19-aab8-a0b5c2a31729");katex.render("k_{ij}", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ the permeability tensor, $var element = document.getElementById("moose-equation-7360fc34-1a71-4374-a8f9-6dfa448ed894");katex.render("P", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ a fluid porepressure, $var element = document.getElementById("moose-equation-9aa25c13-2db5-48b1-b773-00f69075cc7b");katex.render("g_{j}", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ the acceleration of gravity, $var element = document.getElementById("moose-equation-383278f5-9514-46b9-b43b-18e6ae6be42d");katex.render("\\chi", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ a mass fraction, $var element = document.getElementById("moose-equation-822ef503-1a93-4646-9b73-205a93fc1f92");katex.render("k_{\\mathrm{r}}", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ a relative permeability, and $var element = document.getElementById("moose-equation-617ea811-3a67-44ed-8e69-06f9afff8239");katex.render("h", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ an enthalpy.

Which UserObjects you need depends on your simulation. For example, in the case of a single-phase, single-component, fully-saturated anisothermal simulation we need two UserObjects: one for the advection of the fluid, and one for the advection of the heat.

  [./fluid_advective_flux]
    type = PorousFlowAdvectiveFluxCalculatorSaturated
    flux_limiter_type = superbee
  [../]
  [./heat_advective_flux]
    type = PorousFlowAdvectiveFluxCalculatorSaturatedHeat
    flux_limiter_type = superbee
  [../]

/opt/civet/build_0/moose/modules/porous_flow/test/tests/heat_advection/heat_advection_1d_KT.i

# 1phase, heat advecting with a moving fluid
# Using the Kuzmin-Turek stabilization scheme
[Mesh]
  type = GeneratedMesh
  dim = 1
  nx = 50
  xmin = 0
  xmax = 1
[]

[GlobalParams]
  PorousFlowDictator = dictator
  gravity = '0 0 0'
[]

[Variables]
  [./temp]
    initial_condition = 200
  [../]
  [./pp]
  [../]
[]

[ICs]
  [./pp]
    type = FunctionIC
    variable = pp
    function = '1-x'
  [../]
[]

[BCs]
  [./pp0]
    type = DirichletBC
    variable = pp
    boundary = left
    value = 1
  [../]
  [./pp1]
    type = DirichletBC
    variable = pp
    boundary = right
    value = 0
  [../]
  [./spit_heat]
    type = DirichletBC
    variable = temp
    boundary = left
    value = 300
  [../]
  [./suck_heat]
    type = DirichletBC
    variable = temp
    boundary = right
    value = 200
  [../]
[]

[Kernels]
  [./mass_dot]
    type = PorousFlowMassTimeDerivative
    fluid_component = 0
    variable = pp
  [../]
  [./fluid_advection]
    type = PorousFlowFluxLimitedTVDAdvection
    variable = pp
    advective_flux_calculator = fluid_advective_flux
  [../]
  [./energy_dot]
    type = PorousFlowEnergyTimeDerivative
    variable = temp
  [../]
  [./heat_advection]
    type = PorousFlowFluxLimitedTVDAdvection
    variable = temp
    advective_flux_calculator = heat_advective_flux
  [../]
[]

[UserObjects]
  [./dictator]
    type = PorousFlowDictator
    porous_flow_vars = 'temp pp'
    number_fluid_phases = 1
    number_fluid_components = 1
  [../]
  [./pc]
    type = PorousFlowCapillaryPressureVG
    m = 0.6
    alpha = 1.3
  [../]
  [./fluid_advective_flux]
    type = PorousFlowAdvectiveFluxCalculatorSaturated
    flux_limiter_type = superbee
  [../]
  [./heat_advective_flux]
    type = PorousFlowAdvectiveFluxCalculatorSaturatedHeat
    flux_limiter_type = superbee
  [../]
[]

[Modules]
  [./FluidProperties]
    [./simple_fluid]
      type = SimpleFluidProperties
      bulk_modulus = 100
      density0 = 1000
      viscosity = 4.4
      thermal_expansion = 0
      cv = 2
    [../]
  [../]
[]

[Materials]
  [./temperature]
    type = PorousFlowTemperature
    temperature = temp
  [../]
  [./porosity]
    type = PorousFlowPorosityConst
    porosity = 0.2
  [../]
  [./rock_heat]
    type = PorousFlowMatrixInternalEnergy
    specific_heat_capacity = 1.0
    density = 125
  [../]
  [./simple_fluid]
    type = PorousFlowSingleComponentFluid
    fp = simple_fluid
    phase = 0
  [../]
  [./permeability]
    type = PorousFlowPermeabilityConst
    permeability = '1.1 0 0 0 2 0 0 0 3'
  [../]
  [./relperm]
    type = PorousFlowRelativePermeabilityCorey
    n = 2
    phase = 0
  [../]
  [./massfrac]
    type = PorousFlowMassFraction
  [../]
  [./PS]
    type = PorousFlow1PhaseP
    porepressure = pp
    capillary_pressure = pc
  [../]
[]

[Preconditioning]
  [./andy]
    type = SMP
    full = true
    petsc_options_iname = '-ksp_type -pc_type -snes_atol -snes_rtol -snes_max_it'
    petsc_options_value = 'gmres bjacobi 1E-15 1E-10 10000'
  [../]
[]

[Executioner]
  type = Transient
  solve_type = Newton
  dt = 0.01
  end_time = 0.6
[]

[VectorPostprocessors]
  [./T]
    type = LineValueSampler
    start_point = '0 0 0'
    end_point = '1 0 0'
    num_points = 51
    sort_by = x
    variable = temp
  [../]
[]

[Outputs]
  file_base = heat_advection_1d_KT
  [./csv]
    type = CSV
    sync_times = '0.1 0.6'
    sync_only = true
  [../]
[]

Clicking on the links in the above table will provide you with more examples. For instance the following set of Kernels and UserObjects illustrates the case of a single tracer in a fully-saturated single-phase fluid:

[Kernels]
  [./mass0]
    type = PorousFlowMassTimeDerivative
    fluid_component = 0
    variable = tracer
  [../]
  [./flux0]
    type = PorousFlowFluxLimitedTVDAdvection
    variable = tracer
    advective_flux_calculator = advective_flux_calculator_0
  [../]
  [./mass1]
    type = PorousFlowMassTimeDerivative
    fluid_component = 1
    variable = porepressure
  [../]
  [./flux1]
    type = PorousFlowFluxLimitedTVDAdvection
    variable = porepressure
    advective_flux_calculator = advective_flux_calculator_1
  [../]
[]

/opt/civet/build_0/moose/modules/porous_flow/test/tests/flux_limited_TVD_pflow/pffltvd_1D.i

# Using flux-limited TVD advection ala Kuzmin and Turek, mploying PorousFlow Kernels and UserObjects, with superbee flux-limiter
# 1D version
[Mesh]
  type = GeneratedMesh
  dim = 1
  nx = 10
  xmin = 0
  xmax = 1
[]

[GlobalParams]
  PorousFlowDictator = dictator
  gravity = '0 0 0'
[]

[Variables]
  [./porepressure]
  [../]
  [./tracer]
  [../]
[]

[ICs]
  [./porepressure]
    type = FunctionIC
    variable = porepressure
    function = '1 - x'
  [../]
  [./tracer]
    type = FunctionIC
    variable = tracer
    function = 'if(x<0.1,0,if(x>0.3,0,1))'
  [../]
[]

[Kernels]
  [./mass0]
    type = PorousFlowMassTimeDerivative
    fluid_component = 0
    variable = tracer
  [../]
  [./flux0]
    type = PorousFlowFluxLimitedTVDAdvection
    variable = tracer
    advective_flux_calculator = advective_flux_calculator_0
  [../]
  [./mass1]
    type = PorousFlowMassTimeDerivative
    fluid_component = 1
    variable = porepressure
  [../]
  [./flux1]
    type = PorousFlowFluxLimitedTVDAdvection
    variable = porepressure
    advective_flux_calculator = advective_flux_calculator_1
  [../]
[]

[BCs]
  [./constant_injection_porepressure]
    type = DirichletBC
    variable = porepressure
    value = 1
    boundary = left
  [../]
  [./no_tracer_on_left]
    type = DirichletBC
    variable = tracer
    value = 0
    boundary = left
  [../]
  [./remove_component_1]
    type = PorousFlowPiecewiseLinearSink
    variable = porepressure
    boundary = right
    fluid_phase = 0
    pt_vals = '0 1E3'
    multipliers = '0 1E3'
    mass_fraction_component = 1
    use_mobility = true
    flux_function = 1E3
  [../]
  [./remove_component_0]
    type = PorousFlowPiecewiseLinearSink
    variable = tracer
    boundary = right
    fluid_phase = 0
    pt_vals = '0 1E3'
    multipliers = '0 1E3'
    mass_fraction_component = 0
    use_mobility = true
    flux_function = 1E3
  [../]
[]

[Modules]
  [./FluidProperties]
    [./the_simple_fluid]
      type = SimpleFluidProperties
      bulk_modulus = 2E9
      thermal_expansion = 0
      viscosity = 1.0
      density0 = 1000.0
    [../]
  [../]
[]

[UserObjects]
  [./dictator]
    type = PorousFlowDictator
    porous_flow_vars = 'porepressure tracer'
    number_fluid_phases = 1
    number_fluid_components = 2
  [../]
  [./pc]
    type = PorousFlowCapillaryPressureConst
  [../]
  [./advective_flux_calculator_0]
    type = PorousFlowAdvectiveFluxCalculatorSaturatedMultiComponent
    flux_limiter_type = superbee
    fluid_component = 0
  [../]
  [./advective_flux_calculator_1]
    type = PorousFlowAdvectiveFluxCalculatorSaturatedMultiComponent
    flux_limiter_type = superbee
    fluid_component = 1
  [../]
[]

[Materials]
  [./temperature]
    type = PorousFlowTemperature
  [../]
  [./ppss]
    type = PorousFlow1PhaseP
    porepressure = porepressure
    capillary_pressure = pc
  [../]
  [./massfrac]
    type = PorousFlowMassFraction
    mass_fraction_vars = tracer
  [../]
  [./simple_fluid]
    type = PorousFlowSingleComponentFluid
    fp = the_simple_fluid
    phase = 0
  [../]
  [./relperm]
    type = PorousFlowRelativePermeabilityConst
    phase = 0
  [../]
  [./porosity]
    type = PorousFlowPorosity
    porosity_zero = 0.1
  [../]
  [./permeability]
    type = PorousFlowPermeabilityConst
    permeability = '1E-2 0 0   0 1E-2 0   0 0 1E-2'
  [../]
[]

[Preconditioning]
  active = basic
  [./basic]
    type = SMP
    full = true
    petsc_options = '-ksp_diagonal_scale -ksp_diagonal_scale_fix'
    petsc_options_iname = '-pc_type -sub_pc_type -sub_pc_factor_shift_type -pc_asm_overlap'
    petsc_options_value = ' asm      lu           NONZERO                   2'
  [../]
  [./preferred_but_might_not_be_installed]
    type = SMP
    full = true
    petsc_options_iname = '-pc_type -pc_factor_mat_solver_package'
    petsc_options_value = ' lu       mumps'
  [../]
[]

[VectorPostprocessors]
  [./tracer]
    type = LineValueSampler
    start_point = '0 0 0'
    end_point = '1 0 0'
    num_points = 11
    sort_by = x
    variable = tracer
  [../]
[]

[Executioner]
  type = Transient
  solve_type = Newton
  end_time = 6
  dt = 6E-2
  nl_abs_tol = 1E-8
  timestep_tolerance = 1E-3
[]

[Outputs]
  [./out]
    type = CSV
    execute_on = final
  [../]
[]

  [./advective_flux_calculator_0]
    type = PorousFlowAdvectiveFluxCalculatorSaturatedMultiComponent
    flux_limiter_type = superbee
    fluid_component = 0
  [../]
  [./advective_flux_calculator_1]
    type = PorousFlowAdvectiveFluxCalculatorSaturatedMultiComponent
    flux_limiter_type = superbee
    fluid_component = 1
  [../]

/opt/civet/build_0/moose/modules/porous_flow/test/tests/flux_limited_TVD_pflow/pffltvd_1D.i

# Using flux-limited TVD advection ala Kuzmin and Turek, mploying PorousFlow Kernels and UserObjects, with superbee flux-limiter
# 1D version
[Mesh]
  type = GeneratedMesh
  dim = 1
  nx = 10
  xmin = 0
  xmax = 1
[]

[GlobalParams]
  PorousFlowDictator = dictator
  gravity = '0 0 0'
[]

[Variables]
  [./porepressure]
  [../]
  [./tracer]
  [../]
[]

[ICs]
  [./porepressure]
    type = FunctionIC
    variable = porepressure
    function = '1 - x'
  [../]
  [./tracer]
    type = FunctionIC
    variable = tracer
    function = 'if(x<0.1,0,if(x>0.3,0,1))'
  [../]
[]

[Kernels]
  [./mass0]
    type = PorousFlowMassTimeDerivative
    fluid_component = 0
    variable = tracer
  [../]
  [./flux0]
    type = PorousFlowFluxLimitedTVDAdvection
    variable = tracer
    advective_flux_calculator = advective_flux_calculator_0
  [../]
  [./mass1]
    type = PorousFlowMassTimeDerivative
    fluid_component = 1
    variable = porepressure
  [../]
  [./flux1]
    type = PorousFlowFluxLimitedTVDAdvection
    variable = porepressure
    advective_flux_calculator = advective_flux_calculator_1
  [../]
[]

[BCs]
  [./constant_injection_porepressure]
    type = DirichletBC
    variable = porepressure
    value = 1
    boundary = left
  [../]
  [./no_tracer_on_left]
    type = DirichletBC
    variable = tracer
    value = 0
    boundary = left
  [../]
  [./remove_component_1]
    type = PorousFlowPiecewiseLinearSink
    variable = porepressure
    boundary = right
    fluid_phase = 0
    pt_vals = '0 1E3'
    multipliers = '0 1E3'
    mass_fraction_component = 1
    use_mobility = true
    flux_function = 1E3
  [../]
  [./remove_component_0]
    type = PorousFlowPiecewiseLinearSink
    variable = tracer
    boundary = right
    fluid_phase = 0
    pt_vals = '0 1E3'
    multipliers = '0 1E3'
    mass_fraction_component = 0
    use_mobility = true
    flux_function = 1E3
  [../]
[]

[Modules]
  [./FluidProperties]
    [./the_simple_fluid]
      type = SimpleFluidProperties
      bulk_modulus = 2E9
      thermal_expansion = 0
      viscosity = 1.0
      density0 = 1000.0
    [../]
  [../]
[]

[UserObjects]
  [./dictator]
    type = PorousFlowDictator
    porous_flow_vars = 'porepressure tracer'
    number_fluid_phases = 1
    number_fluid_components = 2
  [../]
  [./pc]
    type = PorousFlowCapillaryPressureConst
  [../]
  [./advective_flux_calculator_0]
    type = PorousFlowAdvectiveFluxCalculatorSaturatedMultiComponent
    flux_limiter_type = superbee
    fluid_component = 0
  [../]
  [./advective_flux_calculator_1]
    type = PorousFlowAdvectiveFluxCalculatorSaturatedMultiComponent
    flux_limiter_type = superbee
    fluid_component = 1
  [../]
[]

[Materials]
  [./temperature]
    type = PorousFlowTemperature
  [../]
  [./ppss]
    type = PorousFlow1PhaseP
    porepressure = porepressure
    capillary_pressure = pc
  [../]
  [./massfrac]
    type = PorousFlowMassFraction
    mass_fraction_vars = tracer
  [../]
  [./simple_fluid]
    type = PorousFlowSingleComponentFluid
    fp = the_simple_fluid
    phase = 0
  [../]
  [./relperm]
    type = PorousFlowRelativePermeabilityConst
    phase = 0
  [../]
  [./porosity]
    type = PorousFlowPorosity
    porosity_zero = 0.1
  [../]
  [./permeability]
    type = PorousFlowPermeabilityConst
    permeability = '1E-2 0 0   0 1E-2 0   0 0 1E-2'
  [../]
[]

[Preconditioning]
  active = basic
  [./basic]
    type = SMP
    full = true
    petsc_options = '-ksp_diagonal_scale -ksp_diagonal_scale_fix'
    petsc_options_iname = '-pc_type -sub_pc_type -sub_pc_factor_shift_type -pc_asm_overlap'
    petsc_options_value = ' asm      lu           NONZERO                   2'
  [../]
  [./preferred_but_might_not_be_installed]
    type = SMP
    full = true
    petsc_options_iname = '-pc_type -pc_factor_mat_solver_package'
    petsc_options_value = ' lu       mumps'
  [../]
[]

[VectorPostprocessors]
  [./tracer]
    type = LineValueSampler
    start_point = '0 0 0'
    end_point = '1 0 0'
    num_points = 11
    sort_by = x
    variable = tracer
  [../]
[]

[Executioner]
  type = Transient
  solve_type = Newton
  end_time = 6
  dt = 6E-2
  nl_abs_tol = 1E-8
  timestep_tolerance = 1E-3
[]

[Outputs]
  [./out]
    type = CSV
    execute_on = final
  [../]
[]

note

For multi-phase situations you will need an advective flux calculator for each phase and each component, unless the phases are immiscible (each component exists in one phase only).

For example, in the case of 2 phases with 2 components, each potentially existing in both phases, there are 2 PorousFlow governing equations: one for each component. The equation for fluid component zero contains contributions from both phase 0 and phase 1. The equation for fluid component one contains contributions from both phase 0 and phase 1. So the Kernels will look like

[Kernels]
  [./mass_component0]
    type = PorousFlowMassTimeDerivative
    variable = ppwater
    fluid_component = 0
  [../]
  [./flux_component0_phase0]
    type = PorousFlowFluxLimitedTVDAdvection
    variable = ppwater
    advective_flux_calculator = afc_component0_phase0
  [../]
  [./flux_component0_phase1]
    type = PorousFlowFluxLimitedTVDAdvection
    variable = ppwater
    advective_flux_calculator = afc_component0_phase1
  [../]
  [./mass_component1]
    type = PorousFlowMassTimeDerivative
    variable = sgas
    fluid_component = 1
  [../]
  [./flux_component1_phase0]
    type = PorousFlowFluxLimitedTVDAdvection
    variable = sgas
    advective_flux_calculator = afc_component1_phase0
  [../]
  [./flux_component1_phase1]
    type = PorousFlowFluxLimitedTVDAdvection
    variable = sgas
    advective_flux_calculator = afc_component1_phase1
  [../]
[]

/opt/civet/build_0/moose/modules/porous_flow/test/tests/pressure_pulse/pressure_pulse_1d_2phasePS_KT.i

# Pressure pulse in 1D with 2 phases, 2components - transient
# Using KT stabilization

[Mesh]
  type = GeneratedMesh
  dim = 1
  nx = 10
  xmin = 0
  xmax = 100
[]

[GlobalParams]
  PorousFlowDictator = dictator
  gravity = '0 0 0'
[]

[Variables]
  [./ppwater]
    initial_condition = 2e6
  [../]
  [./sgas]
    initial_condition = 0.3
  [../]
[]

[AuxVariables]
  [./massfrac_ph0_sp0]
    initial_condition = 1
  [../]
  [./massfrac_ph1_sp0]
    initial_condition = 0
  [../]
  [./ppgas]
    family = MONOMIAL
    order = FIRST
  [../]
[]

[Kernels]
  [./mass_component0]
    type = PorousFlowMassTimeDerivative
    variable = ppwater
    fluid_component = 0
  [../]
  [./flux_component0_phase0]
    type = PorousFlowFluxLimitedTVDAdvection
    variable = ppwater
    advective_flux_calculator = afc_component0_phase0
  [../]
  [./flux_component0_phase1]
    type = PorousFlowFluxLimitedTVDAdvection
    variable = ppwater
    advective_flux_calculator = afc_component0_phase1
  [../]
  [./mass_component1]
    type = PorousFlowMassTimeDerivative
    variable = sgas
    fluid_component = 1
  [../]
  [./flux_component1_phase0]
    type = PorousFlowFluxLimitedTVDAdvection
    variable = sgas
    advective_flux_calculator = afc_component1_phase0
  [../]
  [./flux_component1_phase1]
    type = PorousFlowFluxLimitedTVDAdvection
    variable = sgas
    advective_flux_calculator = afc_component1_phase1
  [../]
[]

[AuxKernels]
  [./ppgas]
    type = PorousFlowPropertyAux
    property = pressure
    phase = 1
    variable = ppgas
  [../]
[]

[UserObjects]
  [./dictator]
    type = PorousFlowDictator
    porous_flow_vars = 'ppwater sgas'
    number_fluid_phases = 2
    number_fluid_components = 2
  [../]
  [./pc]
    type = PorousFlowCapillaryPressureConst
    pc = 1e5
  [../]
  [./afc_component0_phase0]
    type = PorousFlowAdvectiveFluxCalculatorUnsaturatedMultiComponent
    fluid_component = 0
    phase = 0
    flux_limiter_type = superbee
  [../]
  [./afc_component0_phase1]
    type = PorousFlowAdvectiveFluxCalculatorUnsaturatedMultiComponent
    fluid_component = 0
    phase = 1
    flux_limiter_type = superbee
  [../]
  [./afc_component1_phase0]
    type = PorousFlowAdvectiveFluxCalculatorUnsaturatedMultiComponent
    fluid_component = 1
    phase = 0
    flux_limiter_type = superbee
  [../]
  [./afc_component1_phase1]
    type = PorousFlowAdvectiveFluxCalculatorUnsaturatedMultiComponent
    fluid_component = 1
    phase = 1
    flux_limiter_type = superbee
  [../]
[]

[Modules]
  [./FluidProperties]
    [./simple_fluid0]
      type = SimpleFluidProperties
      bulk_modulus = 2e9
      density0 = 1000
      thermal_expansion = 0
      viscosity = 1e-3
    [../]
    [./simple_fluid1]
      type = SimpleFluidProperties
      bulk_modulus = 2e7
      density0 = 1
      thermal_expansion = 0
      viscosity = 1e-5
    [../]
  [../]
[]

[Materials]
  [./temperature]
    type = PorousFlowTemperature
  [../]
  [./ppss]
    type = PorousFlow2PhasePS
    phase0_porepressure = ppwater
    phase1_saturation = sgas
    capillary_pressure = pc
  [../]
  [./massfrac]
    type = PorousFlowMassFraction
    mass_fraction_vars = 'massfrac_ph0_sp0 massfrac_ph1_sp0'
  [../]
  [./simple_fluid0]
    type = PorousFlowSingleComponentFluid
    fp = simple_fluid0
    phase = 0
  [../]
  [./simple_fluid1]
    type = PorousFlowSingleComponentFluid
    fp = simple_fluid1
    phase = 1
  [../]
  [./porosity]
    type = PorousFlowPorosityConst
    porosity = 0.1
  [../]
  [./permeability]
    type = PorousFlowPermeabilityConst
    permeability = '1e-15 0 0 0 1e-15 0 0 0 1e-15'
  [../]
  [./relperm_water]
    type = PorousFlowRelativePermeabilityCorey
    n = 1
    phase = 0
  [../]
  [./relperm_gas]
    type = PorousFlowRelativePermeabilityCorey
    n = 1
    phase = 1
  [../]
[]

[BCs]
  [./leftwater]
    type = DirichletBC
    boundary = left
    value = 3e6
    variable = ppwater
  [../]
  [./rightwater]
    type = DirichletBC
    boundary = right
    value = 2e6
    variable = ppwater
  [../]
[]

[Preconditioning]
  [./smp]
    type = SMP
    full = true
    petsc_options_iname = '-ksp_type -pc_type -snes_atol -snes_rtol -snes_max_it'
    petsc_options_value = 'bcgs bjacobi 1E-15 1E-20 10000'
  [../]
[]

[Executioner]
  type = Transient
  solve_type = Newton
  dt = 1e3
  end_time = 1e4
[]

[VectorPostprocessors]
  [./pp]
    type = LineValueSampler
    sort_by = x
    variable = 'ppwater ppgas'
    start_point = '0 0 0'
    end_point = '100 0 0'
    num_points = 11
  [../]
[]

[Outputs]
  file_base = pressure_pulse_1d_2phasePS_KT
  print_linear_residuals = false
  [./csv]
    type = CSV
    execute_on = final
  [../]
[]

and the UserObjects will look like

  [./afc_component0_phase0]
    type = PorousFlowAdvectiveFluxCalculatorUnsaturatedMultiComponent
    fluid_component = 0
    phase = 0
    flux_limiter_type = superbee
  [../]
  [./afc_component0_phase1]
    type = PorousFlowAdvectiveFluxCalculatorUnsaturatedMultiComponent
    fluid_component = 0
    phase = 1
    flux_limiter_type = superbee
  [../]
  [./afc_component1_phase0]
    type = PorousFlowAdvectiveFluxCalculatorUnsaturatedMultiComponent
    fluid_component = 1
    phase = 0
    flux_limiter_type = superbee
  [../]
  [./afc_component1_phase1]
    type = PorousFlowAdvectiveFluxCalculatorUnsaturatedMultiComponent
    fluid_component = 1
    phase = 1
    flux_limiter_type = superbee
  [../]

/opt/civet/build_0/moose/modules/porous_flow/test/tests/pressure_pulse/pressure_pulse_1d_2phasePS_KT.i

# Pressure pulse in 1D with 2 phases, 2components - transient
# Using KT stabilization

[Mesh]
  type = GeneratedMesh
  dim = 1
  nx = 10
  xmin = 0
  xmax = 100
[]

[GlobalParams]
  PorousFlowDictator = dictator
  gravity = '0 0 0'
[]

[Variables]
  [./ppwater]
    initial_condition = 2e6
  [../]
  [./sgas]
    initial_condition = 0.3
  [../]
[]

[AuxVariables]
  [./massfrac_ph0_sp0]
    initial_condition = 1
  [../]
  [./massfrac_ph1_sp0]
    initial_condition = 0
  [../]
  [./ppgas]
    family = MONOMIAL
    order = FIRST
  [../]
[]

[Kernels]
  [./mass_component0]
    type = PorousFlowMassTimeDerivative
    variable = ppwater
    fluid_component = 0
  [../]
  [./flux_component0_phase0]
    type = PorousFlowFluxLimitedTVDAdvection
    variable = ppwater
    advective_flux_calculator = afc_component0_phase0
  [../]
  [./flux_component0_phase1]
    type = PorousFlowFluxLimitedTVDAdvection
    variable = ppwater
    advective_flux_calculator = afc_component0_phase1
  [../]
  [./mass_component1]
    type = PorousFlowMassTimeDerivative
    variable = sgas
    fluid_component = 1
  [../]
  [./flux_component1_phase0]
    type = PorousFlowFluxLimitedTVDAdvection
    variable = sgas
    advective_flux_calculator = afc_component1_phase0
  [../]
  [./flux_component1_phase1]
    type = PorousFlowFluxLimitedTVDAdvection
    variable = sgas
    advective_flux_calculator = afc_component1_phase1
  [../]
[]

[AuxKernels]
  [./ppgas]
    type = PorousFlowPropertyAux
    property = pressure
    phase = 1
    variable = ppgas
  [../]
[]

[UserObjects]
  [./dictator]
    type = PorousFlowDictator
    porous_flow_vars = 'ppwater sgas'
    number_fluid_phases = 2
    number_fluid_components = 2
  [../]
  [./pc]
    type = PorousFlowCapillaryPressureConst
    pc = 1e5
  [../]
  [./afc_component0_phase0]
    type = PorousFlowAdvectiveFluxCalculatorUnsaturatedMultiComponent
    fluid_component = 0
    phase = 0
    flux_limiter_type = superbee
  [../]
  [./afc_component0_phase1]
    type = PorousFlowAdvectiveFluxCalculatorUnsaturatedMultiComponent
    fluid_component = 0
    phase = 1
    flux_limiter_type = superbee
  [../]
  [./afc_component1_phase0]
    type = PorousFlowAdvectiveFluxCalculatorUnsaturatedMultiComponent
    fluid_component = 1
    phase = 0
    flux_limiter_type = superbee
  [../]
  [./afc_component1_phase1]
    type = PorousFlowAdvectiveFluxCalculatorUnsaturatedMultiComponent
    fluid_component = 1
    phase = 1
    flux_limiter_type = superbee
  [../]
[]

[Modules]
  [./FluidProperties]
    [./simple_fluid0]
      type = SimpleFluidProperties
      bulk_modulus = 2e9
      density0 = 1000
      thermal_expansion = 0
      viscosity = 1e-3
    [../]
    [./simple_fluid1]
      type = SimpleFluidProperties
      bulk_modulus = 2e7
      density0 = 1
      thermal_expansion = 0
      viscosity = 1e-5
    [../]
  [../]
[]

[Materials]
  [./temperature]
    type = PorousFlowTemperature
  [../]
  [./ppss]
    type = PorousFlow2PhasePS
    phase0_porepressure = ppwater
    phase1_saturation = sgas
    capillary_pressure = pc
  [../]
  [./massfrac]
    type = PorousFlowMassFraction
    mass_fraction_vars = 'massfrac_ph0_sp0 massfrac_ph1_sp0'
  [../]
  [./simple_fluid0]
    type = PorousFlowSingleComponentFluid
    fp = simple_fluid0
    phase = 0
  [../]
  [./simple_fluid1]
    type = PorousFlowSingleComponentFluid
    fp = simple_fluid1
    phase = 1
  [../]
  [./porosity]
    type = PorousFlowPorosityConst
    porosity = 0.1
  [../]
  [./permeability]
    type = PorousFlowPermeabilityConst
    permeability = '1e-15 0 0 0 1e-15 0 0 0 1e-15'
  [../]
  [./relperm_water]
    type = PorousFlowRelativePermeabilityCorey
    n = 1
    phase = 0
  [../]
  [./relperm_gas]
    type = PorousFlowRelativePermeabilityCorey
    n = 1
    phase = 1
  [../]
[]

[BCs]
  [./leftwater]
    type = DirichletBC
    boundary = left
    value = 3e6
    variable = ppwater
  [../]
  [./rightwater]
    type = DirichletBC
    boundary = right
    value = 2e6
    variable = ppwater
  [../]
[]

[Preconditioning]
  [./smp]
    type = SMP
    full = true
    petsc_options_iname = '-ksp_type -pc_type -snes_atol -snes_rtol -snes_max_it'
    petsc_options_value = 'bcgs bjacobi 1E-15 1E-20 10000'
  [../]
[]

[Executioner]
  type = Transient
  solve_type = Newton
  dt = 1e3
  end_time = 1e4
[]

[VectorPostprocessors]
  [./pp]
    type = LineValueSampler
    sort_by = x
    variable = 'ppwater ppgas'
    start_point = '0 0 0'
    end_point = '100 0 0'
    num_points = 11
  [../]
[]

[Outputs]
  file_base = pressure_pulse_1d_2phasePS_KT
  print_linear_residuals = false
  [./csv]
    type = CSV
    execute_on = final
  [../]
[]

D. Kuzmin and S. Turek. High-resolution FEM-TVD shcemes based on a fully multidimensional flux limiter. Journal of Computational Physics, 198:131–158, 2004.

@article{KuzminTurek2004,
    author = "Kuzmin, D. and Turek, S.",
    title = "{High-resolution FEM-TVD shcemes based on a fully multidimensional flux limiter}",
    journal = "Journal of Computational Physics",
    volume = "198",
    pages = "131--158",
    year = "2004"
}

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How to use Kuzmin-Turek stabilization in PorousFlow simulations

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