# Porous Flow Tutorial Page 11. Two-phase THM borehole injection

Hold onto your hats, here we're going to build a two-phase THM simulation from scratch. Our starting point is the borehole-reservoir-caprock geometry developed in Page 00.

• We would like to simulate cold CO injection into an initially water-filled reservoir.

• There are two phases, and two components. The liquid phase is filled with water only, while the "gas" phases is filled with CO only ("gas" is in quotes because it may be supercritical).

• There is capillarity between the water and CO.

• The fluids have certain nontrivial relative permeability curves.

• Heat is advected with the fluids as well as conducted through the fluid-rock system.

• The rock can deform elastically.

• There is no gravity (for simplicity as this effects initial stresses and fluid pressures, which is not difficult to include but the input file becomes even longer).

• Full coupling between the fluids, the temperature and the displacements is used.

• The porosity and permeability are functions of the porepressures, saturations, temperature and displacements

• High precision equations of state are used for the water and CO.

• At the injection area, CO is injected at a constant rate and at a constant temperature. It pushes mechanically against the borehole wall

• At the outer boundary fluid is removed slowly using a PorousFlowSink.

• In this model there are no vertical displacements

## Variables

Let's step through the input file. This simulation uses the water and gas porepressures as the independent fluid pressures. A perfectly valid alternative would be to use water saturation and gas porepressure. A scaling factor is needed for the temperature and displacement variables as discussed in the convergence page. The Variables are given initial conditions appropriate to a deep CO sequestration site:

[Variables]
[./pwater]
initial_condition = 20E6
[../]
[./pgas]
initial_condition = 20.1E6
[../]
[./T]
initial_condition = 330
scaling = 1E-5
[../]
[./disp_x]
scaling = 1E-5
[../]
[./disp_y]
scaling = 1E-5
[../]
[]

(modules/porous_flow/examples/tutorial/11.i)

## Dictator

The PorousFlowDictator is given the variable names as well as the number of fluid phases and components.

  [./dictator]
type = PorousFlowDictator
porous_flow_vars = 'pwater pgas T disp_x disp_y'
number_fluid_phases = 2
number_fluid_components = 2
[../]

(modules/porous_flow/examples/tutorial/11.i)

## GlobalParams

The global parameters consist of various parameters that are required by several PorousFlow objects, and using GlobalParams allows the input file to remain organised and a little more succinct:

[GlobalParams]
displacements = 'disp_x disp_y disp_z'
gravity = '0 0 0'
biot_coefficient = 1.0
PorousFlowDictator = dictator
[]

(modules/porous_flow/examples/tutorial/11.i)

## Kernels

The Kernels block is rather big. Please refer to the governing equations page to see what equations are being modelled here. Virtually all the power of PorousFlow is being used. The only things that are missing are the following.

• Chemical reactions and desorption are not used.

• Fluid diffusion and dispersion is not active because water only exists in the liquid phase and CO in the gas phase.

• There is no radioactive decay.

• There is no plasticity, which also means there is no plastic heating.

• The solid mechanics is assumed quasi-static.

The Kernels block is

[Kernels]
[./mass_water_dot]
type = PorousFlowMassTimeDerivative
fluid_component = 0
use_displaced_mesh = false
variable = pwater
[../]
[./flux_water]
fluid_component = 0
use_displaced_mesh = false
variable = pwater
[../]
[./vol_strain_rate_water]
type = PorousFlowMassVolumetricExpansion
fluid_component = 0
use_displaced_mesh = false
variable = pwater
[../]
[./mass_co2_dot]
type = PorousFlowMassTimeDerivative
fluid_component = 1
use_displaced_mesh = false
variable = pgas
[../]
[./flux_co2]
fluid_component = 1
use_displaced_mesh = false
variable = pgas
[../]
[./vol_strain_rate_co2]
type = PorousFlowMassVolumetricExpansion
fluid_component = 1
use_displaced_mesh = false
variable = pgas
[../]
[./energy_dot]
type = PorousFlowEnergyTimeDerivative
use_displaced_mesh = false
variable = T
[../]
use_displaced_mesh = false
variable = T
[../]
[./conduction]
type = PorousFlowHeatConduction
use_displaced_mesh = false
variable = T
[../]
[./vol_strain_rate_heat]
type = PorousFlowHeatVolumetricExpansion
use_displaced_mesh = false
variable = T
[../]
type = StressDivergenceTensors
temperature = T
variable = disp_x
thermal_eigenstrain_name = thermal_contribution
use_displaced_mesh = false
component = 0
[../]
[./poro_x]
type = PorousFlowEffectiveStressCoupling
variable = disp_x
use_displaced_mesh = false
component = 0
[../]
type = StressDivergenceTensors
temperature = T
variable = disp_y
thermal_eigenstrain_name = thermal_contribution
use_displaced_mesh = false
component = 1
[../]
[./poro_y]
type = PorousFlowEffectiveStressCoupling
variable = disp_y
use_displaced_mesh = false
component = 1
[../]
[]

(modules/porous_flow/examples/tutorial/11.i)

## AuxVariables and AuxKernels

Some AuxVariables are defined that need further explanation.

• The solid mechanics needs 3 displacement variables. In the assumptions above, the vertical displacement is always zero. The best way of defining it is as an AuxVariable without an AuxKernel so that it will always be zero, but then it may be coupled into various MOOSE objects (using the displacements = in the GlobalParams)

• CO is pumped into this model with a constant rate. To achieve this, the CO pressure in the borehole must gradually increase over time. This means the fluid in the borehole will push against the borehole wall with increasing pressure. Hence this pressure must be recorded and coupled into the mechanical BCs. It is recorded in the effective_fluid_pressure AuxVariable.

• The mass fractions of each component in each phase must be defined, even if they are fixed for always. Since they are unchanging they are most conveniently represented by AuxVariables with certain initial conditions and no AuxKernels.

• The other AuxVariables are useful for visualising the results.

[AuxVariables]
[./disp_z]
[../]
[./effective_fluid_pressure]
family = MONOMIAL
order = CONSTANT
[../]
[./mass_frac_phase0_species0]
initial_condition = 1 # all water in phase=0
[../]
[./mass_frac_phase1_species0]
initial_condition = 0 # no water in phase=1
[../]
[./sgas]
family = MONOMIAL
order = CONSTANT
[../]
[./swater]
family = MONOMIAL
order = CONSTANT
[../]
[./stress_rr]
family = MONOMIAL
order = CONSTANT
[../]
[./stress_tt]
family = MONOMIAL
order = CONSTANT
[../]
[./stress_zz]
family = MONOMIAL
order = CONSTANT
[../]
[./porosity]
family = MONOMIAL
order = CONSTANT
[../]
[]

[AuxKernels]
[./effective_fluid_pressure]
type = ParsedAux
args = 'pwater pgas swater sgas'
function = 'pwater * swater + pgas * sgas'
variable = effective_fluid_pressure
[../]
[./swater]
type = PorousFlowPropertyAux
variable = swater
property = saturation
phase = 0
execute_on = timestep_end
[../]
[./sgas]
type = PorousFlowPropertyAux
variable = sgas
property = saturation
phase = 1
execute_on = timestep_end
[../]
[./stress_rr]
type = RankTwoScalarAux
variable = stress_rr
rank_two_tensor = stress
point1 = '0 0 0'
point2 = '0 0 1'
execute_on = timestep_end
[../]
[./stress_tt]
type = RankTwoScalarAux
variable = stress_tt
rank_two_tensor = stress
scalar_type = HoopStress
point1 = '0 0 0'
point2 = '0 0 1'
execute_on = timestep_end
[../]
[./stress_zz]
type = RankTwoAux
variable = stress_zz
rank_two_tensor = stress
index_i = 2
index_j = 2
execute_on = timestep_end
[../]
[./porosity]
type = PorousFlowPropertyAux
variable = porosity
property = porosity
execute_on = timestep_end
[../]
[]

(modules/porous_flow/examples/tutorial/11.i)

## Boundary conditions

The boundary conditions for the displacements are roller on the sides, fixed at the top and bottom (an arbitrary choice made by the creator of this input file) and Pressure boundary conditions on the injection_area:

  [./roller_tmax]
type = PresetBC
variable = disp_x
value = 0
boundary = tmax
[../]
[./roller_tmin]
type = PresetBC
variable = disp_y
value = 0
boundary = tmin
[../]
[./pinned_top_bottom_x]
type = PresetBC
variable = disp_x
value = 0
boundary = 'top bottom'
[../]
[./pinned_top_bottom_y]
type = PresetBC
variable = disp_y
value = 0
boundary = 'top bottom'
[../]
[./cavity_pressure_x]
type = Pressure
boundary = injection_area
variable = disp_x
component = 0
postprocessor = constrained_effective_fluid_pressure_at_wellbore
use_displaced_mesh = false
[../]
[./cavity_pressure_y]
type = Pressure
boundary = injection_area
variable = disp_y
component = 1
postprocessor = constrained_effective_fluid_pressure_at_wellbore
use_displaced_mesh = false
[../]

(modules/porous_flow/examples/tutorial/11.i)

Notice the constrained_effective_fluid_pressure_at_wellbore. This is almost the effective_fluid_pressure AuxVariable defined above, evaluated at the injection_area. But there is a problem at the first timestep, because this uses Material properties that are not properly initialised. So a little bit of deception is used. Firstly, the AuxVariable is evaluated at the injection_area and put into a Postprocessor:

[Postprocessors]
[./effective_fluid_pressure_at_wellbore]
type = PointValue
variable = effective_fluid_pressure
point = '1 0 0'
execute_on = timestep_begin
use_displaced_mesh = false
[../]

(modules/porous_flow/examples/tutorial/11.i)

Then a Function is made that returns either the value of this Postprocessor or 20E6 (the initial reservoir pressure)

[Functions]
[./constrain_effective_fluid_pressure]
type = ParsedFunction
vars = effective_fluid_pressure_at_wellbore
vals = effective_fluid_pressure_at_wellbore
value = 'max(effective_fluid_pressure_at_wellbore, 20E6)'
[../]
[]

(modules/porous_flow/examples/tutorial/11.i)

Finally, the value of this Function is placed into the Postprocessor used in the Pressure BC:

  [./constrained_effective_fluid_pressure_at_wellbore]
type = FunctionValuePostprocessor
function = constrain_effective_fluid_pressure
execute_on = timestep_begin
[../]

(modules/porous_flow/examples/tutorial/11.i)

The boundary conditions for temperature is a simple PresetBC on the injection_area:

  [./cold_co2]
type = PresetBC
boundary = injection_area
variable = T
value = 290  # injection temperature
use_displaced_mesh = false
[../]

(modules/porous_flow/examples/tutorial/11.i)

The boundary conditions for the fluids at the injection_area is just a constant injection of CO with rate kg.s.m:

  [./constant_co2_injection]
type = PorousFlowSink
boundary = injection_area
variable = pgas
fluid_phase = 1
flux_function = -1E-4
use_displaced_mesh = false
[../]

(modules/porous_flow/examples/tutorial/11.i)

At the outer boundary, water and CO are removed if their porepressures rise above their initial values. A PorousFlowPiecewiseLinearSink is used, with an imaginary boundary at fixed porepressure sitting at a distance of m outside the model. The procedure of constructing this sink is described fully in the boundaries documentation. The input-file blocks are

  [./outer_water_removal]
type = PorousFlowPiecewiseLinearSink
boundary = rmax
variable = pwater
fluid_phase = 0
pt_vals = '0 1E9'
multipliers = '0 1E8'
PT_shift = 20E6
use_mobility = true
use_relperm = true
use_displaced_mesh = false
[../]
[./outer_co2_removal]
type = PorousFlowPiecewiseLinearSink
boundary = rmax
variable = pgas
fluid_phase = 1
pt_vals = '0 1E9'
multipliers = '0 1E8'
PT_shift = 20.1E6
use_mobility = true
use_relperm = true
use_displaced_mesh = false
[../]

(modules/porous_flow/examples/tutorial/11.i)

## Fluid properties

High-precision equations of state are used for both the water and the CO. Before running the simulation, these are tabulated, and the tabulated versions are used by MOOSE in all computations, which shortens the simulation time:

[Modules]
[./FluidProperties]
[./true_water]
type = Water97FluidProperties
[../]
[./tabulated_water]
type = TabulatedFluidProperties
fp = true_water
temperature_min = 275
pressure_max = 1E8
interpolated_properties = 'density viscosity enthalpy internal_energy'
fluid_property_file = water97_tabulated_11.csv
[../]
[./true_co2]
type = CO2FluidProperties
[../]
[./tabulated_co2]
type = TabulatedFluidProperties
fp = true_co2
temperature_min = 275
pressure_max = 1E8
interpolated_properties = 'density viscosity enthalpy internal_energy'
fluid_property_file = co2_tabulated_11.csv
[../]
[../]

(modules/porous_flow/examples/tutorial/11.i)

## Materials

The capillary pressure relationship is defined by the UserObject:

  [./pc]
type = PorousFlowCapillaryPressureVG
alpha = 1E-6
m = 0.6
[../]

(modules/porous_flow/examples/tutorial/11.i)

As explained on Page 09 and Page 10, there are a set of "fundamental Materials" that compute all porepressures, saturations, temperature and mass fractions as Material properties (as well as their gradients, and the derivatives with respect to the primary Variables, etc). In the case at hand, these fundamental Materials are:

[Materials]
[./temperature]
type = PorousFlowTemperature
temperature = T
[../]
[./saturation_calculator]
type = PorousFlow2PhasePP
phase0_porepressure = pwater
phase1_porepressure = pgas
capillary_pressure = pc
[../]
[./massfrac]
type = PorousFlowMassFraction
mass_fraction_vars = 'mass_frac_phase0_species0 mass_frac_phase1_species0'
[../]
[./water]
type = PorousFlowSingleComponentFluid
fp = tabulated_water
phase = 0
[../]
[./co2]
type = PorousFlowSingleComponentFluid
fp = tabulated_co2
phase = 1
[../]
[./relperm_water]
type = PorousFlowRelativePermeabilityCorey
n = 4
s_res = 0.1
sum_s_res = 0.2
phase = 0
[../]
[./relperm_co2]
type = PorousFlowRelativePermeabilityBC
nw_phase = true
lambda = 2
s_res = 0.1
sum_s_res = 0.2
phase = 1
[../]
[./porosity]
type = PorousFlowPorosity
fluid = true
mechanical = true
thermal = true
porosity_zero = 0.1
reference_temperature = 330
reference_porepressure = 20E6
thermal_expansion_coeff = 15E-6 # volumetric
solid_bulk = 8E9 # unimportant since biot = 1
[../]
[./permeability_aquifer]
type = PorousFlowPermeabilityKozenyCarman
block = aquifer
poroperm_function = kozeny_carman_phi0
phi0 = 0.1
n = 2
m = 2
k0 = 1E-12
[../]
[./permeability_caps]
type = PorousFlowPermeabilityKozenyCarman
block = caps
poroperm_function = kozeny_carman_phi0
phi0 = 0.1
n = 2
m = 2
k0 = 1E-15
k_anisotropy = '1 0 0  0 1 0  0 0 0.1'
[../]
[./rock_thermal_conductivity]
type = PorousFlowThermalConductivityIdeal
dry_thermal_conductivity = '2 0 0  0 2 0  0 0 2'
[../]
[./rock_internal_energy]
type = PorousFlowMatrixInternalEnergy
specific_heat_capacity = 1100
density = 2300
[../]

[./elasticity_tensor]
type = ComputeIsotropicElasticityTensor
youngs_modulus = 5E9
poissons_ratio = 0.0
[../]
[./strain]
type = ComputeSmallStrain
eigenstrain_names = 'thermal_contribution initial_stress'
[../]
[./thermal_contribution]
type = ComputeThermalExpansionEigenstrain
temperature = T
thermal_expansion_coeff = 5E-6 # this is the linear thermal expansion coefficient
eigenstrain_name = thermal_contribution
stress_free_temperature = 330
[../]
[./initial_strain]
type = ComputeEigenstrainFromInitialStress
initial_stress = '20E6 0 0  0 20E6 0  0 0 20E6'
eigenstrain_name = initial_stress
[../]
[./stress]
type = ComputeLinearElasticStress
[../]

[./effective_fluid_pressure]
type = PorousFlowEffectiveFluidPressure
[../]
[./volumetric_strain]
type = PorousFlowVolumetricStrain
[../]
[]

[Postprocessors]
[./effective_fluid_pressure_at_wellbore]
type = PointValue
variable = effective_fluid_pressure
point = '1 0 0'
execute_on = timestep_begin
use_displaced_mesh = false
[../]
[./constrained_effective_fluid_pressure_at_wellbore]
type = FunctionValuePostprocessor
function = constrain_effective_fluid_pressure
execute_on = timestep_begin
[../]
[]

[Functions]
[./constrain_effective_fluid_pressure]
type = ParsedFunction
vars = effective_fluid_pressure_at_wellbore
vals = effective_fluid_pressure_at_wellbore
value = 'max(effective_fluid_pressure_at_wellbore, 20E6)'
[../]
[]

[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 = 1E3
[./TimeStepper]
dt = 1E3
growth_factor = 1.2
optimal_iterations = 10
[../]
nl_abs_tol = 1E-7
[]

[Outputs]
exodus = true
[]

(modules/porous_flow/examples/tutorial/11.i)

The water and CO densities, viscosities, enthalpies, and internal energies (as well as derivatives of these) are computed by

# Two-phase borehole injection problem
[Mesh]
type = AnnularMesh
dim = 2
nr = 10
rmin = 1.0
rmax = 10
growth_r = 1.4
nt = 4
tmin = 0
tmax = 1.57079632679
[]

[MeshModifiers]
[./make3D]
type = MeshExtruder
extrusion_vector = '0 0 12'
num_layers = 3
bottom_sideset = 'bottom'
top_sideset = 'top'
[../]
[./shift_down]
type = Transform
transform = TRANSLATE
vector_value = '0 0 -6'
depends_on = make3D
[../]
[./aquifer]
type = SubdomainBoundingBox
block_id = 1
bottom_left = '0 0 -2'
top_right = '10 10 2'
depends_on = shift_down
[../]
[./injection_area]
combinatorial_geometry = 'x*x+y*y<1.01'
included_subdomain_ids = 1
new_sideset_name = 'injection_area'
depends_on = 'aquifer'
[../]
[./rename]
type = RenameBlock
old_block_id = '0 1'
new_block_name = 'caps aquifer'
depends_on = 'injection_area'
[../]
[]

[UserObjects]
[./dictator]
type = PorousFlowDictator
porous_flow_vars = 'pwater pgas T disp_x disp_y'
number_fluid_phases = 2
number_fluid_components = 2
[../]
[./pc]
type = PorousFlowCapillaryPressureVG
alpha = 1E-6
m = 0.6
[../]
[]

[GlobalParams]
displacements = 'disp_x disp_y disp_z'
gravity = '0 0 0'
biot_coefficient = 1.0
PorousFlowDictator = dictator
[]

[Variables]
[./pwater]
initial_condition = 20E6
[../]
[./pgas]
initial_condition = 20.1E6
[../]
[./T]
initial_condition = 330
scaling = 1E-5
[../]
[./disp_x]
scaling = 1E-5
[../]
[./disp_y]
scaling = 1E-5
[../]
[]

[Kernels]
[./mass_water_dot]
type = PorousFlowMassTimeDerivative
fluid_component = 0
use_displaced_mesh = false
variable = pwater
[../]
[./flux_water]
fluid_component = 0
use_displaced_mesh = false
variable = pwater
[../]
[./vol_strain_rate_water]
type = PorousFlowMassVolumetricExpansion
fluid_component = 0
use_displaced_mesh = false
variable = pwater
[../]
[./mass_co2_dot]
type = PorousFlowMassTimeDerivative
fluid_component = 1
use_displaced_mesh = false
variable = pgas
[../]
[./flux_co2]
fluid_component = 1
use_displaced_mesh = false
variable = pgas
[../]
[./vol_strain_rate_co2]
type = PorousFlowMassVolumetricExpansion
fluid_component = 1
use_displaced_mesh = false
variable = pgas
[../]
[./energy_dot]
type = PorousFlowEnergyTimeDerivative
use_displaced_mesh = false
variable = T
[../]
use_displaced_mesh = false
variable = T
[../]
[./conduction]
type = PorousFlowHeatConduction
use_displaced_mesh = false
variable = T
[../]
[./vol_strain_rate_heat]
type = PorousFlowHeatVolumetricExpansion
use_displaced_mesh = false
variable = T
[../]
type = StressDivergenceTensors
temperature = T
variable = disp_x
thermal_eigenstrain_name = thermal_contribution
use_displaced_mesh = false
component = 0
[../]
[./poro_x]
type = PorousFlowEffectiveStressCoupling
variable = disp_x
use_displaced_mesh = false
component = 0
[../]
type = StressDivergenceTensors
temperature = T
variable = disp_y
thermal_eigenstrain_name = thermal_contribution
use_displaced_mesh = false
component = 1
[../]
[./poro_y]
type = PorousFlowEffectiveStressCoupling
variable = disp_y
use_displaced_mesh = false
component = 1
[../]
[]

[AuxVariables]
[./disp_z]
[../]
[./effective_fluid_pressure]
family = MONOMIAL
order = CONSTANT
[../]
[./mass_frac_phase0_species0]
initial_condition = 1 # all water in phase=0
[../]
[./mass_frac_phase1_species0]
initial_condition = 0 # no water in phase=1
[../]
[./sgas]
family = MONOMIAL
order = CONSTANT
[../]
[./swater]
family = MONOMIAL
order = CONSTANT
[../]
[./stress_rr]
family = MONOMIAL
order = CONSTANT
[../]
[./stress_tt]
family = MONOMIAL
order = CONSTANT
[../]
[./stress_zz]
family = MONOMIAL
order = CONSTANT
[../]
[./porosity]
family = MONOMIAL
order = CONSTANT
[../]
[]

[AuxKernels]
[./effective_fluid_pressure]
type = ParsedAux
args = 'pwater pgas swater sgas'
function = 'pwater * swater + pgas * sgas'
variable = effective_fluid_pressure
[../]
[./swater]
type = PorousFlowPropertyAux
variable = swater
property = saturation
phase = 0
execute_on = timestep_end
[../]
[./sgas]
type = PorousFlowPropertyAux
variable = sgas
property = saturation
phase = 1
execute_on = timestep_end
[../]
[./stress_rr]
type = RankTwoScalarAux
variable = stress_rr
rank_two_tensor = stress
point1 = '0 0 0'
point2 = '0 0 1'
execute_on = timestep_end
[../]
[./stress_tt]
type = RankTwoScalarAux
variable = stress_tt
rank_two_tensor = stress
scalar_type = HoopStress
point1 = '0 0 0'
point2 = '0 0 1'
execute_on = timestep_end
[../]
[./stress_zz]
type = RankTwoAux
variable = stress_zz
rank_two_tensor = stress
index_i = 2
index_j = 2
execute_on = timestep_end
[../]
[./porosity]
type = PorousFlowPropertyAux
variable = porosity
property = porosity
execute_on = timestep_end
[../]
[]

[BCs]
[./roller_tmax]
type = PresetBC
variable = disp_x
value = 0
boundary = tmax
[../]
[./roller_tmin]
type = PresetBC
variable = disp_y
value = 0
boundary = tmin
[../]
[./pinned_top_bottom_x]
type = PresetBC
variable = disp_x
value = 0
boundary = 'top bottom'
[../]
[./pinned_top_bottom_y]
type = PresetBC
variable = disp_y
value = 0
boundary = 'top bottom'
[../]
[./cavity_pressure_x]
type = Pressure
boundary = injection_area
variable = disp_x
component = 0
postprocessor = constrained_effective_fluid_pressure_at_wellbore
use_displaced_mesh = false
[../]
[./cavity_pressure_y]
type = Pressure
boundary = injection_area
variable = disp_y
component = 1
postprocessor = constrained_effective_fluid_pressure_at_wellbore
use_displaced_mesh = false
[../]

[./cold_co2]
type = PresetBC
boundary = injection_area
variable = T
value = 290  # injection temperature
use_displaced_mesh = false
[../]
[./constant_co2_injection]
type = PorousFlowSink
boundary = injection_area
variable = pgas
fluid_phase = 1
flux_function = -1E-4
use_displaced_mesh = false
[../]

[./outer_water_removal]
type = PorousFlowPiecewiseLinearSink
boundary = rmax
variable = pwater
fluid_phase = 0
pt_vals = '0 1E9'
multipliers = '0 1E8'
PT_shift = 20E6
use_mobility = true
use_relperm = true
use_displaced_mesh = false
[../]
[./outer_co2_removal]
type = PorousFlowPiecewiseLinearSink
boundary = rmax
variable = pgas
fluid_phase = 1
pt_vals = '0 1E9'
multipliers = '0 1E8'
PT_shift = 20.1E6
use_mobility = true
use_relperm = true
use_displaced_mesh = false
[../]
[]

[Modules]
[./FluidProperties]
[./true_water]
type = Water97FluidProperties
[../]
[./tabulated_water]
type = TabulatedFluidProperties
fp = true_water
temperature_min = 275
pressure_max = 1E8
interpolated_properties = 'density viscosity enthalpy internal_energy'
fluid_property_file = water97_tabulated_11.csv
[../]
[./true_co2]
type = CO2FluidProperties
[../]
[./tabulated_co2]
type = TabulatedFluidProperties
fp = true_co2
temperature_min = 275
pressure_max = 1E8
interpolated_properties = 'density viscosity enthalpy internal_energy'
fluid_property_file = co2_tabulated_11.csv
[../]
[../]
[]

[Materials]
[./temperature]
type = PorousFlowTemperature
temperature = T
[../]
[./saturation_calculator]
type = PorousFlow2PhasePP
phase0_porepressure = pwater
phase1_porepressure = pgas
capillary_pressure = pc
[../]
[./massfrac]
type = PorousFlowMassFraction
mass_fraction_vars = 'mass_frac_phase0_species0 mass_frac_phase1_species0'
[../]
[./water]
type = PorousFlowSingleComponentFluid
fp = tabulated_water
phase = 0
[../]
[./co2]
type = PorousFlowSingleComponentFluid
fp = tabulated_co2
phase = 1
[../]
[./relperm_water]
type = PorousFlowRelativePermeabilityCorey
n = 4
s_res = 0.1
sum_s_res = 0.2
phase = 0
[../]
[./relperm_co2]
type = PorousFlowRelativePermeabilityBC
nw_phase = true
lambda = 2
s_res = 0.1
sum_s_res = 0.2
phase = 1
[../]
[./porosity]
type = PorousFlowPorosity
fluid = true
mechanical = true
thermal = true
porosity_zero = 0.1
reference_temperature = 330
reference_porepressure = 20E6
thermal_expansion_coeff = 15E-6 # volumetric
solid_bulk = 8E9 # unimportant since biot = 1
[../]
[./permeability_aquifer]
type = PorousFlowPermeabilityKozenyCarman
block = aquifer
poroperm_function = kozeny_carman_phi0
phi0 = 0.1
n = 2
m = 2
k0 = 1E-12
[../]
[./permeability_caps]
type = PorousFlowPermeabilityKozenyCarman
block = caps
poroperm_function = kozeny_carman_phi0
phi0 = 0.1
n = 2
m = 2
k0 = 1E-15
k_anisotropy = '1 0 0  0 1 0  0 0 0.1'
[../]
[./rock_thermal_conductivity]
type = PorousFlowThermalConductivityIdeal
dry_thermal_conductivity = '2 0 0  0 2 0  0 0 2'
[../]
[./rock_internal_energy]
type = PorousFlowMatrixInternalEnergy
specific_heat_capacity = 1100
density = 2300
[../]

[./elasticity_tensor]
type = ComputeIsotropicElasticityTensor
youngs_modulus = 5E9
poissons_ratio = 0.0
[../]
[./strain]
type = ComputeSmallStrain
eigenstrain_names = 'thermal_contribution initial_stress'
[../]
[./thermal_contribution]
type = ComputeThermalExpansionEigenstrain
temperature = T
thermal_expansion_coeff = 5E-6 # this is the linear thermal expansion coefficient
eigenstrain_name = thermal_contribution
stress_free_temperature = 330
[../]
[./initial_strain]
type = ComputeEigenstrainFromInitialStress
initial_stress = '20E6 0 0  0 20E6 0  0 0 20E6'
eigenstrain_name = initial_stress
[../]
[./stress]
type = ComputeLinearElasticStress
[../]

[./effective_fluid_pressure]
type = PorousFlowEffectiveFluidPressure
[../]
[./volumetric_strain]
type = PorousFlowVolumetricStrain
[../]
[]

[Postprocessors]
[./effective_fluid_pressure_at_wellbore]
type = PointValue
variable = effective_fluid_pressure
point = '1 0 0'
execute_on = timestep_begin
use_displaced_mesh = false
[../]
[./constrained_effective_fluid_pressure_at_wellbore]
type = FunctionValuePostprocessor
function = constrain_effective_fluid_pressure
execute_on = timestep_begin
[../]
[]

[Functions]
[./constrain_effective_fluid_pressure]
type = ParsedFunction
vars = effective_fluid_pressure_at_wellbore
vals = effective_fluid_pressure_at_wellbore
value = 'max(effective_fluid_pressure_at_wellbore, 20E6)'
[../]
[]

[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 = 1E3
[./TimeStepper]
dt = 1E3
growth_factor = 1.2
optimal_iterations = 10
[../]
nl_abs_tol = 1E-7
[]

[Outputs]
exodus = true
[]

(modules/porous_flow/examples/tutorial/11.i)

These are then Joined into std::vectors, ready to be fed into the Kernels (and other Materials, AuxKernels, etc) using the PorousFlowJoiners:

# Two-phase borehole injection problem
[Mesh]
type = AnnularMesh
dim = 2
nr = 10
rmin = 1.0
rmax = 10
growth_r = 1.4
nt = 4
tmin = 0
tmax = 1.57079632679
[]

[MeshModifiers]
[./make3D]
type = MeshExtruder
extrusion_vector = '0 0 12'
num_layers = 3
bottom_sideset = 'bottom'
top_sideset = 'top'
[../]
[./shift_down]
type = Transform
transform = TRANSLATE
vector_value = '0 0 -6'
depends_on = make3D
[../]
[./aquifer]
type = SubdomainBoundingBox
block_id = 1
bottom_left = '0 0 -2'
top_right = '10 10 2'
depends_on = shift_down
[../]
[./injection_area]
combinatorial_geometry = 'x*x+y*y<1.01'
included_subdomain_ids = 1
new_sideset_name = 'injection_area'
depends_on = 'aquifer'
[../]
[./rename]
type = RenameBlock
old_block_id = '0 1'
new_block_name = 'caps aquifer'
depends_on = 'injection_area'
[../]
[]

[UserObjects]
[./dictator]
type = PorousFlowDictator
porous_flow_vars = 'pwater pgas T disp_x disp_y'
number_fluid_phases = 2
number_fluid_components = 2
[../]
[./pc]
type = PorousFlowCapillaryPressureVG
alpha = 1E-6
m = 0.6
[../]
[]

[GlobalParams]
displacements = 'disp_x disp_y disp_z'
gravity = '0 0 0'
biot_coefficient = 1.0
PorousFlowDictator = dictator
[]

[Variables]
[./pwater]
initial_condition = 20E6
[../]
[./pgas]
initial_condition = 20.1E6
[../]
[./T]
initial_condition = 330
scaling = 1E-5
[../]
[./disp_x]
scaling = 1E-5
[../]
[./disp_y]
scaling = 1E-5
[../]
[]

[Kernels]
[./mass_water_dot]
type = PorousFlowMassTimeDerivative
fluid_component = 0
use_displaced_mesh = false
variable = pwater
[../]
[./flux_water]
fluid_component = 0
use_displaced_mesh = false
variable = pwater
[../]
[./vol_strain_rate_water]
type = PorousFlowMassVolumetricExpansion
fluid_component = 0
use_displaced_mesh = false
variable = pwater
[../]
[./mass_co2_dot]
type = PorousFlowMassTimeDerivative
fluid_component = 1
use_displaced_mesh = false
variable = pgas
[../]
[./flux_co2]
fluid_component = 1
use_displaced_mesh = false
variable = pgas
[../]
[./vol_strain_rate_co2]
type = PorousFlowMassVolumetricExpansion
fluid_component = 1
use_displaced_mesh = false
variable = pgas
[../]
[./energy_dot]
type = PorousFlowEnergyTimeDerivative
use_displaced_mesh = false
variable = T
[../]
use_displaced_mesh = false
variable = T
[../]
[./conduction]
type = PorousFlowHeatConduction
use_displaced_mesh = false
variable = T
[../]
[./vol_strain_rate_heat]
type = PorousFlowHeatVolumetricExpansion
use_displaced_mesh = false
variable = T
[../]
type = StressDivergenceTensors
temperature = T
variable = disp_x
thermal_eigenstrain_name = thermal_contribution
use_displaced_mesh = false
component = 0
[../]
[./poro_x]
type = PorousFlowEffectiveStressCoupling
variable = disp_x
use_displaced_mesh = false
component = 0
[../]
type = StressDivergenceTensors
temperature = T
variable = disp_y
thermal_eigenstrain_name = thermal_contribution
use_displaced_mesh = false
component = 1
[../]
[./poro_y]
type = PorousFlowEffectiveStressCoupling
variable = disp_y
use_displaced_mesh = false
component = 1
[../]
[]

[AuxVariables]
[./disp_z]
[../]
[./effective_fluid_pressure]
family = MONOMIAL
order = CONSTANT
[../]
[./mass_frac_phase0_species0]
initial_condition = 1 # all water in phase=0
[../]
[./mass_frac_phase1_species0]
initial_condition = 0 # no water in phase=1
[../]
[./sgas]
family = MONOMIAL
order = CONSTANT
[../]
[./swater]
family = MONOMIAL
order = CONSTANT
[../]
[./stress_rr]
family = MONOMIAL
order = CONSTANT
[../]
[./stress_tt]
family = MONOMIAL
order = CONSTANT
[../]
[./stress_zz]
family = MONOMIAL
order = CONSTANT
[../]
[./porosity]
family = MONOMIAL
order = CONSTANT
[../]
[]

[AuxKernels]
[./effective_fluid_pressure]
type = ParsedAux
args = 'pwater pgas swater sgas'
function = 'pwater * swater + pgas * sgas'
variable = effective_fluid_pressure
[../]
[./swater]
type = PorousFlowPropertyAux
variable = swater
property = saturation
phase = 0
execute_on = timestep_end
[../]
[./sgas]
type = PorousFlowPropertyAux
variable = sgas
property = saturation
phase = 1
execute_on = timestep_end
[../]
[./stress_rr]
type = RankTwoScalarAux
variable = stress_rr
rank_two_tensor = stress
point1 = '0 0 0'
point2 = '0 0 1'
execute_on = timestep_end
[../]
[./stress_tt]
type = RankTwoScalarAux
variable = stress_tt
rank_two_tensor = stress
scalar_type = HoopStress
point1 = '0 0 0'
point2 = '0 0 1'
execute_on = timestep_end
[../]
[./stress_zz]
type = RankTwoAux
variable = stress_zz
rank_two_tensor = stress
index_i = 2
index_j = 2
execute_on = timestep_end
[../]
[./porosity]
type = PorousFlowPropertyAux
variable = porosity
property = porosity
execute_on = timestep_end
[../]
[]

[BCs]
[./roller_tmax]
type = PresetBC
variable = disp_x
value = 0
boundary = tmax
[../]
[./roller_tmin]
type = PresetBC
variable = disp_y
value = 0
boundary = tmin
[../]
[./pinned_top_bottom_x]
type = PresetBC
variable = disp_x
value = 0
boundary = 'top bottom'
[../]
[./pinned_top_bottom_y]
type = PresetBC
variable = disp_y
value = 0
boundary = 'top bottom'
[../]
[./cavity_pressure_x]
type = Pressure
boundary = injection_area
variable = disp_x
component = 0
postprocessor = constrained_effective_fluid_pressure_at_wellbore
use_displaced_mesh = false
[../]
[./cavity_pressure_y]
type = Pressure
boundary = injection_area
variable = disp_y
component = 1
postprocessor = constrained_effective_fluid_pressure_at_wellbore
use_displaced_mesh = false
[../]

[./cold_co2]
type = PresetBC
boundary = injection_area
variable = T
value = 290  # injection temperature
use_displaced_mesh = false
[../]
[./constant_co2_injection]
type = PorousFlowSink
boundary = injection_area
variable = pgas
fluid_phase = 1
flux_function = -1E-4
use_displaced_mesh = false
[../]

[./outer_water_removal]
type = PorousFlowPiecewiseLinearSink
boundary = rmax
variable = pwater
fluid_phase = 0
pt_vals = '0 1E9'
multipliers = '0 1E8'
PT_shift = 20E6
use_mobility = true
use_relperm = true
use_displaced_mesh = false
[../]
[./outer_co2_removal]
type = PorousFlowPiecewiseLinearSink
boundary = rmax
variable = pgas
fluid_phase = 1
pt_vals = '0 1E9'
multipliers = '0 1E8'
PT_shift = 20.1E6
use_mobility = true
use_relperm = true
use_displaced_mesh = false
[../]
[]

[Modules]
[./FluidProperties]
[./true_water]
type = Water97FluidProperties
[../]
[./tabulated_water]
type = TabulatedFluidProperties
fp = true_water
temperature_min = 275
pressure_max = 1E8
interpolated_properties = 'density viscosity enthalpy internal_energy'
fluid_property_file = water97_tabulated_11.csv
[../]
[./true_co2]
type = CO2FluidProperties
[../]
[./tabulated_co2]
type = TabulatedFluidProperties
fp = true_co2
temperature_min = 275
pressure_max = 1E8
interpolated_properties = 'density viscosity enthalpy internal_energy'
fluid_property_file = co2_tabulated_11.csv
[../]
[../]
[]

[Materials]
[./temperature]
type = PorousFlowTemperature
temperature = T
[../]
[./saturation_calculator]
type = PorousFlow2PhasePP
phase0_porepressure = pwater
phase1_porepressure = pgas
capillary_pressure = pc
[../]
[./massfrac]
type = PorousFlowMassFraction
mass_fraction_vars = 'mass_frac_phase0_species0 mass_frac_phase1_species0'
[../]
[./water]
type = PorousFlowSingleComponentFluid
fp = tabulated_water
phase = 0
[../]
[./co2]
type = PorousFlowSingleComponentFluid
fp = tabulated_co2
phase = 1
[../]
[./relperm_water]
type = PorousFlowRelativePermeabilityCorey
n = 4
s_res = 0.1
sum_s_res = 0.2
phase = 0
[../]
[./relperm_co2]
type = PorousFlowRelativePermeabilityBC
nw_phase = true
lambda = 2
s_res = 0.1
sum_s_res = 0.2
phase = 1
[../]
[./porosity]
type = PorousFlowPorosity
fluid = true
mechanical = true
thermal = true
porosity_zero = 0.1
reference_temperature = 330
reference_porepressure = 20E6
thermal_expansion_coeff = 15E-6 # volumetric
solid_bulk = 8E9 # unimportant since biot = 1
[../]
[./permeability_aquifer]
type = PorousFlowPermeabilityKozenyCarman
block = aquifer
poroperm_function = kozeny_carman_phi0
phi0 = 0.1
n = 2
m = 2
k0 = 1E-12
[../]
[./permeability_caps]
type = PorousFlowPermeabilityKozenyCarman
block = caps
poroperm_function = kozeny_carman_phi0
phi0 = 0.1
n = 2
m = 2
k0 = 1E-15
k_anisotropy = '1 0 0  0 1 0  0 0 0.1'
[../]
[./rock_thermal_conductivity]
type = PorousFlowThermalConductivityIdeal
dry_thermal_conductivity = '2 0 0  0 2 0  0 0 2'
[../]
[./rock_internal_energy]
type = PorousFlowMatrixInternalEnergy
specific_heat_capacity = 1100
density = 2300
[../]

[./elasticity_tensor]
type = ComputeIsotropicElasticityTensor
youngs_modulus = 5E9
poissons_ratio = 0.0
[../]
[./strain]
type = ComputeSmallStrain
eigenstrain_names = 'thermal_contribution initial_stress'
[../]
[./thermal_contribution]
type = ComputeThermalExpansionEigenstrain
temperature = T
thermal_expansion_coeff = 5E-6 # this is the linear thermal expansion coefficient
eigenstrain_name = thermal_contribution
stress_free_temperature = 330
[../]
[./initial_strain]
type = ComputeEigenstrainFromInitialStress
initial_stress = '20E6 0 0  0 20E6 0  0 0 20E6'
eigenstrain_name = initial_stress
[../]
[./stress]
type = ComputeLinearElasticStress
[../]

[./effective_fluid_pressure]
type = PorousFlowEffectiveFluidPressure
[../]
[./volumetric_strain]
type = PorousFlowVolumetricStrain
[../]
[]

[Postprocessors]
[./effective_fluid_pressure_at_wellbore]
type = PointValue
variable = effective_fluid_pressure
point = '1 0 0'
execute_on = timestep_begin
use_displaced_mesh = false
[../]
[./constrained_effective_fluid_pressure_at_wellbore]
type = FunctionValuePostprocessor
function = constrain_effective_fluid_pressure
execute_on = timestep_begin
[../]
[]

[Functions]
[./constrain_effective_fluid_pressure]
type = ParsedFunction
vars = effective_fluid_pressure_at_wellbore
vals = effective_fluid_pressure_at_wellbore
value = 'max(effective_fluid_pressure_at_wellbore, 20E6)'
[../]
[]

[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 = 1E3
[./TimeStepper]
dt = 1E3
growth_factor = 1.2
optimal_iterations = 10
[../]
nl_abs_tol = 1E-7
[]

[Outputs]
exodus = true
[]

(modules/porous_flow/examples/tutorial/11.i)

A Corey type of relative permeability is chosen for the liquid phase, and a Brooks-Corey type of relative permeability is chosen for the gas phase. These are then Joined:

# Two-phase borehole injection problem
[Mesh]
type = AnnularMesh
dim = 2
nr = 10
rmin = 1.0
rmax = 10
growth_r = 1.4
nt = 4
tmin = 0
tmax = 1.57079632679
[]

[MeshModifiers]
[./make3D]
type = MeshExtruder
extrusion_vector = '0 0 12'
num_layers = 3
bottom_sideset = 'bottom'
top_sideset = 'top'
[../]
[./shift_down]
type = Transform
transform = TRANSLATE
vector_value = '0 0 -6'
depends_on = make3D
[../]
[./aquifer]
type = SubdomainBoundingBox
block_id = 1
bottom_left = '0 0 -2'
top_right = '10 10 2'
depends_on = shift_down
[../]
[./injection_area]
combinatorial_geometry = 'x*x+y*y<1.01'
included_subdomain_ids = 1
new_sideset_name = 'injection_area'
depends_on = 'aquifer'
[../]
[./rename]
type = RenameBlock
old_block_id = '0 1'
new_block_name = 'caps aquifer'
depends_on = 'injection_area'
[../]
[]

[UserObjects]
[./dictator]
type = PorousFlowDictator
porous_flow_vars = 'pwater pgas T disp_x disp_y'
number_fluid_phases = 2
number_fluid_components = 2
[../]
[./pc]
type = PorousFlowCapillaryPressureVG
alpha = 1E-6
m = 0.6
[../]
[]

[GlobalParams]
displacements = 'disp_x disp_y disp_z'
gravity = '0 0 0'
biot_coefficient = 1.0
PorousFlowDictator = dictator
[]

[Variables]
[./pwater]
initial_condition = 20E6
[../]
[./pgas]
initial_condition = 20.1E6
[../]
[./T]
initial_condition = 330
scaling = 1E-5
[../]
[./disp_x]
scaling = 1E-5
[../]
[./disp_y]
scaling = 1E-5
[../]
[]

[Kernels]
[./mass_water_dot]
type = PorousFlowMassTimeDerivative
fluid_component = 0
use_displaced_mesh = false
variable = pwater
[../]
[./flux_water]
fluid_component = 0
use_displaced_mesh = false
variable = pwater
[../]
[./vol_strain_rate_water]
type = PorousFlowMassVolumetricExpansion
fluid_component = 0
use_displaced_mesh = false
variable = pwater
[../]
[./mass_co2_dot]
type = PorousFlowMassTimeDerivative
fluid_component = 1
use_displaced_mesh = false
variable = pgas
[../]
[./flux_co2]
fluid_component = 1
use_displaced_mesh = false
variable = pgas
[../]
[./vol_strain_rate_co2]
type = PorousFlowMassVolumetricExpansion
fluid_component = 1
use_displaced_mesh = false
variable = pgas
[../]
[./energy_dot]
type = PorousFlowEnergyTimeDerivative
use_displaced_mesh = false
variable = T
[../]
use_displaced_mesh = false
variable = T
[../]
[./conduction]
type = PorousFlowHeatConduction
use_displaced_mesh = false
variable = T
[../]
[./vol_strain_rate_heat]
type = PorousFlowHeatVolumetricExpansion
use_displaced_mesh = false
variable = T
[../]
type = StressDivergenceTensors
temperature = T
variable = disp_x
thermal_eigenstrain_name = thermal_contribution
use_displaced_mesh = false
component = 0
[../]
[./poro_x]
type = PorousFlowEffectiveStressCoupling
variable = disp_x
use_displaced_mesh = false
component = 0
[../]
type = StressDivergenceTensors
temperature = T
variable = disp_y
thermal_eigenstrain_name = thermal_contribution
use_displaced_mesh = false
component = 1
[../]
[./poro_y]
type = PorousFlowEffectiveStressCoupling
variable = disp_y
use_displaced_mesh = false
component = 1
[../]
[]

[AuxVariables]
[./disp_z]
[../]
[./effective_fluid_pressure]
family = MONOMIAL
order = CONSTANT
[../]
[./mass_frac_phase0_species0]
initial_condition = 1 # all water in phase=0
[../]
[./mass_frac_phase1_species0]
initial_condition = 0 # no water in phase=1
[../]
[./sgas]
family = MONOMIAL
order = CONSTANT
[../]
[./swater]
family = MONOMIAL
order = CONSTANT
[../]
[./stress_rr]
family = MONOMIAL
order = CONSTANT
[../]
[./stress_tt]
family = MONOMIAL
order = CONSTANT
[../]
[./stress_zz]
family = MONOMIAL
order = CONSTANT
[../]
[./porosity]
family = MONOMIAL
order = CONSTANT
[../]
[]

[AuxKernels]
[./effective_fluid_pressure]
type = ParsedAux
args = 'pwater pgas swater sgas'
function = 'pwater * swater + pgas * sgas'
variable = effective_fluid_pressure
[../]
[./swater]
type = PorousFlowPropertyAux
variable = swater
property = saturation
phase = 0
execute_on = timestep_end
[../]
[./sgas]
type = PorousFlowPropertyAux
variable = sgas
property = saturation
phase = 1
execute_on = timestep_end
[../]
[./stress_rr]
type = RankTwoScalarAux
variable = stress_rr
rank_two_tensor = stress
point1 = '0 0 0'
point2 = '0 0 1'
execute_on = timestep_end
[../]
[./stress_tt]
type = RankTwoScalarAux
variable = stress_tt
rank_two_tensor = stress
scalar_type = HoopStress
point1 = '0 0 0'
point2 = '0 0 1'
execute_on = timestep_end
[../]
[./stress_zz]
type = RankTwoAux
variable = stress_zz
rank_two_tensor = stress
index_i = 2
index_j = 2
execute_on = timestep_end
[../]
[./porosity]
type = PorousFlowPropertyAux
variable = porosity
property = porosity
execute_on = timestep_end
[../]
[]

[BCs]
[./roller_tmax]
type = PresetBC
variable = disp_x
value = 0
boundary = tmax
[../]
[./roller_tmin]
type = PresetBC
variable = disp_y
value = 0
boundary = tmin
[../]
[./pinned_top_bottom_x]
type = PresetBC
variable = disp_x
value = 0
boundary = 'top bottom'
[../]
[./pinned_top_bottom_y]
type = PresetBC
variable = disp_y
value = 0
boundary = 'top bottom'
[../]
[./cavity_pressure_x]
type = Pressure
boundary = injection_area
variable = disp_x
component = 0
postprocessor = constrained_effective_fluid_pressure_at_wellbore
use_displaced_mesh = false
[../]
[./cavity_pressure_y]
type = Pressure
boundary = injection_area
variable = disp_y
component = 1
postprocessor = constrained_effective_fluid_pressure_at_wellbore
use_displaced_mesh = false
[../]

[./cold_co2]
type = PresetBC
boundary = injection_area
variable = T
value = 290  # injection temperature
use_displaced_mesh = false
[../]
[./constant_co2_injection]
type = PorousFlowSink
boundary = injection_area
variable = pgas
fluid_phase = 1
flux_function = -1E-4
use_displaced_mesh = false
[../]

[./outer_water_removal]
type = PorousFlowPiecewiseLinearSink
boundary = rmax
variable = pwater
fluid_phase = 0
pt_vals = '0 1E9'
multipliers = '0 1E8'
PT_shift = 20E6
use_mobility = true
use_relperm = true
use_displaced_mesh = false
[../]
[./outer_co2_removal]
type = PorousFlowPiecewiseLinearSink
boundary = rmax
variable = pgas
fluid_phase = 1
pt_vals = '0 1E9'
multipliers = '0 1E8'
PT_shift = 20.1E6
use_mobility = true
use_relperm = true
use_displaced_mesh = false
[../]
[]

[Modules]
[./FluidProperties]
[./true_water]
type = Water97FluidProperties
[../]
[./tabulated_water]
type = TabulatedFluidProperties
fp = true_water
temperature_min = 275
pressure_max = 1E8
interpolated_properties = 'density viscosity enthalpy internal_energy'
fluid_property_file = water97_tabulated_11.csv
[../]
[./true_co2]
type = CO2FluidProperties
[../]
[./tabulated_co2]
type = TabulatedFluidProperties
fp = true_co2
temperature_min = 275
pressure_max = 1E8
interpolated_properties = 'density viscosity enthalpy internal_energy'
fluid_property_file = co2_tabulated_11.csv
[../]
[../]
[]

[Materials]
[./temperature]
type = PorousFlowTemperature
temperature = T
[../]
[./saturation_calculator]
type = PorousFlow2PhasePP
phase0_porepressure = pwater
phase1_porepressure = pgas
capillary_pressure = pc
[../]
[./massfrac]
type = PorousFlowMassFraction
mass_fraction_vars = 'mass_frac_phase0_species0 mass_frac_phase1_species0'
[../]
[./water]
type = PorousFlowSingleComponentFluid
fp = tabulated_water
phase = 0
[../]
[./co2]
type = PorousFlowSingleComponentFluid
fp = tabulated_co2
phase = 1
[../]
[./relperm_water]
type = PorousFlowRelativePermeabilityCorey
n = 4
s_res = 0.1
sum_s_res = 0.2
phase = 0
[../]
[./relperm_co2]
type = PorousFlowRelativePermeabilityBC
nw_phase = true
lambda = 2
s_res = 0.1
sum_s_res = 0.2
phase = 1
[../]
[./porosity]
type = PorousFlowPorosity
fluid = true
mechanical = true
thermal = true
porosity_zero = 0.1
reference_temperature = 330
reference_porepressure = 20E6
thermal_expansion_coeff = 15E-6 # volumetric
solid_bulk = 8E9 # unimportant since biot = 1
[../]
[./permeability_aquifer]
type = PorousFlowPermeabilityKozenyCarman
block = aquifer
poroperm_function = kozeny_carman_phi0
phi0 = 0.1
n = 2
m = 2
k0 = 1E-12
[../]
[./permeability_caps]
type = PorousFlowPermeabilityKozenyCarman
block = caps
poroperm_function = kozeny_carman_phi0
phi0 = 0.1
n = 2
m = 2
k0 = 1E-15
k_anisotropy = '1 0 0  0 1 0  0 0 0.1'
[../]
[./rock_thermal_conductivity]
type = PorousFlowThermalConductivityIdeal
dry_thermal_conductivity = '2 0 0  0 2 0  0 0 2'
[../]
[./rock_internal_energy]
type = PorousFlowMatrixInternalEnergy
specific_heat_capacity = 1100
density = 2300
[../]

[./elasticity_tensor]
type = ComputeIsotropicElasticityTensor
youngs_modulus = 5E9
poissons_ratio = 0.0
[../]
[./strain]
type = ComputeSmallStrain
eigenstrain_names = 'thermal_contribution initial_stress'
[../]
[./thermal_contribution]
type = ComputeThermalExpansionEigenstrain
temperature = T
thermal_expansion_coeff = 5E-6 # this is the linear thermal expansion coefficient
eigenstrain_name = thermal_contribution
stress_free_temperature = 330
[../]
[./initial_strain]
type = ComputeEigenstrainFromInitialStress
initial_stress = '20E6 0 0  0 20E6 0  0 0 20E6'
eigenstrain_name = initial_stress
[../]
[./stress]
type = ComputeLinearElasticStress
[../]

[./effective_fluid_pressure]
type = PorousFlowEffectiveFluidPressure
[../]
[./volumetric_strain]
type = PorousFlowVolumetricStrain
[../]
[]

[Postprocessors]
[./effective_fluid_pressure_at_wellbore]
type = PointValue
variable = effective_fluid_pressure
point = '1 0 0'
execute_on = timestep_begin
use_displaced_mesh = false
[../]
[./constrained_effective_fluid_pressure_at_wellbore]
type = FunctionValuePostprocessor
function = constrain_effective_fluid_pressure
execute_on = timestep_begin
[../]
[]

[Functions]
[./constrain_effective_fluid_pressure]
type = ParsedFunction
vars = effective_fluid_pressure_at_wellbore
vals = effective_fluid_pressure_at_wellbore
value = 'max(effective_fluid_pressure_at_wellbore, 20E6)'
[../]
[]

[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 = 1E3
[./TimeStepper]
dt = 1E3
growth_factor = 1.2
optimal_iterations = 10
[../]
nl_abs_tol = 1E-7
[]

[Outputs]
exodus = true
[]

(modules/porous_flow/examples/tutorial/11.i)

Porosity is chosen to depend on porepressures, saturations (actually just the effective porepressure), temperature and mechanical strain using:

# Two-phase borehole injection problem
[Mesh]
type = AnnularMesh
dim = 2
nr = 10
rmin = 1.0
rmax = 10
growth_r = 1.4
nt = 4
tmin = 0
tmax = 1.57079632679
[]

[MeshModifiers]
[./make3D]
type = MeshExtruder
extrusion_vector = '0 0 12'
num_layers = 3
bottom_sideset = 'bottom'
top_sideset = 'top'
[../]
[./shift_down]
type = Transform
transform = TRANSLATE
vector_value = '0 0 -6'
depends_on = make3D
[../]
[./aquifer]
type = SubdomainBoundingBox
block_id = 1
bottom_left = '0 0 -2'
top_right = '10 10 2'
depends_on = shift_down
[../]
[./injection_area]
combinatorial_geometry = 'x*x+y*y<1.01'
included_subdomain_ids = 1
new_sideset_name = 'injection_area'
depends_on = 'aquifer'
[../]
[./rename]
type = RenameBlock
old_block_id = '0 1'
new_block_name = 'caps aquifer'
depends_on = 'injection_area'
[../]
[]

[UserObjects]
[./dictator]
type = PorousFlowDictator
porous_flow_vars = 'pwater pgas T disp_x disp_y'
number_fluid_phases = 2
number_fluid_components = 2
[../]
[./pc]
type = PorousFlowCapillaryPressureVG
alpha = 1E-6
m = 0.6
[../]
[]

[GlobalParams]
displacements = 'disp_x disp_y disp_z'
gravity = '0 0 0'
biot_coefficient = 1.0
PorousFlowDictator = dictator
[]

[Variables]
[./pwater]
initial_condition = 20E6
[../]
[./pgas]
initial_condition = 20.1E6
[../]
[./T]
initial_condition = 330
scaling = 1E-5
[../]
[./disp_x]
scaling = 1E-5
[../]
[./disp_y]
scaling = 1E-5
[../]
[]

[Kernels]
[./mass_water_dot]
type = PorousFlowMassTimeDerivative
fluid_component = 0
use_displaced_mesh = false
variable = pwater
[../]
[./flux_water]
fluid_component = 0
use_displaced_mesh = false
variable = pwater
[../]
[./vol_strain_rate_water]
type = PorousFlowMassVolumetricExpansion
fluid_component = 0
use_displaced_mesh = false
variable = pwater
[../]
[./mass_co2_dot]
type = PorousFlowMassTimeDerivative
fluid_component = 1
use_displaced_mesh = false
variable = pgas
[../]
[./flux_co2]
fluid_component = 1
use_displaced_mesh = false
variable = pgas
[../]
[./vol_strain_rate_co2]
type = PorousFlowMassVolumetricExpansion
fluid_component = 1
use_displaced_mesh = false
variable = pgas
[../]
[./energy_dot]
type = PorousFlowEnergyTimeDerivative
use_displaced_mesh = false
variable = T
[../]
use_displaced_mesh = false
variable = T
[../]
[./conduction]
type = PorousFlowHeatConduction
use_displaced_mesh = false
variable = T
[../]
[./vol_strain_rate_heat]
type = PorousFlowHeatVolumetricExpansion
use_displaced_mesh = false
variable = T
[../]
type = StressDivergenceTensors
temperature = T
variable = disp_x
thermal_eigenstrain_name = thermal_contribution
use_displaced_mesh = false
component = 0
[../]
[./poro_x]
type = PorousFlowEffectiveStressCoupling
variable = disp_x
use_displaced_mesh = false
component = 0
[../]
type = StressDivergenceTensors
temperature = T
variable = disp_y
thermal_eigenstrain_name = thermal_contribution
use_displaced_mesh = false
component = 1
[../]
[./poro_y]
type = PorousFlowEffectiveStressCoupling
variable = disp_y
use_displaced_mesh = false
component = 1
[../]
[]

[AuxVariables]
[./disp_z]
[../]
[./effective_fluid_pressure]
family = MONOMIAL
order = CONSTANT
[../]
[./mass_frac_phase0_species0]
initial_condition = 1 # all water in phase=0
[../]
[./mass_frac_phase1_species0]
initial_condition = 0 # no water in phase=1
[../]
[./sgas]
family = MONOMIAL
order = CONSTANT
[../]
[./swater]
family = MONOMIAL
order = CONSTANT
[../]
[./stress_rr]
family = MONOMIAL
order = CONSTANT
[../]
[./stress_tt]
family = MONOMIAL
order = CONSTANT
[../]
[./stress_zz]
family = MONOMIAL
order = CONSTANT
[../]
[./porosity]
family = MONOMIAL
order = CONSTANT
[../]
[]

[AuxKernels]
[./effective_fluid_pressure]
type = ParsedAux
args = 'pwater pgas swater sgas'
function = 'pwater * swater + pgas * sgas'
variable = effective_fluid_pressure
[../]
[./swater]
type = PorousFlowPropertyAux
variable = swater
property = saturation
phase = 0
execute_on = timestep_end
[../]
[./sgas]
type = PorousFlowPropertyAux
variable = sgas
property = saturation
phase = 1
execute_on = timestep_end
[../]
[./stress_rr]
type = RankTwoScalarAux
variable = stress_rr
rank_two_tensor = stress
point1 = '0 0 0'
point2 = '0 0 1'
execute_on = timestep_end
[../]
[./stress_tt]
type = RankTwoScalarAux
variable = stress_tt
rank_two_tensor = stress
scalar_type = HoopStress
point1 = '0 0 0'
point2 = '0 0 1'
execute_on = timestep_end
[../]
[./stress_zz]
type = RankTwoAux
variable = stress_zz
rank_two_tensor = stress
index_i = 2
index_j = 2
execute_on = timestep_end
[../]
[./porosity]
type = PorousFlowPropertyAux
variable = porosity
property = porosity
execute_on = timestep_end
[../]
[]

[BCs]
[./roller_tmax]
type = PresetBC
variable = disp_x
value = 0
boundary = tmax
[../]
[./roller_tmin]
type = PresetBC
variable = disp_y
value = 0
boundary = tmin
[../]
[./pinned_top_bottom_x]
type = PresetBC
variable = disp_x
value = 0
boundary = 'top bottom'
[../]
[./pinned_top_bottom_y]
type = PresetBC
variable = disp_y
value = 0
boundary = 'top bottom'
[../]
[./cavity_pressure_x]
type = Pressure
boundary = injection_area
variable = disp_x
component = 0
postprocessor = constrained_effective_fluid_pressure_at_wellbore
use_displaced_mesh = false
[../]
[./cavity_pressure_y]
type = Pressure
boundary = injection_area
variable = disp_y
component = 1
postprocessor = constrained_effective_fluid_pressure_at_wellbore
use_displaced_mesh = false
[../]

[./cold_co2]
type = PresetBC
boundary = injection_area
variable = T
value = 290  # injection temperature
use_displaced_mesh = false
[../]
[./constant_co2_injection]
type = PorousFlowSink
boundary = injection_area
variable = pgas
fluid_phase = 1
flux_function = -1E-4
use_displaced_mesh = false
[../]

[./outer_water_removal]
type = PorousFlowPiecewiseLinearSink
boundary = rmax
variable = pwater
fluid_phase = 0
pt_vals = '0 1E9'
multipliers = '0 1E8'
PT_shift = 20E6
use_mobility = true
use_relperm = true
use_displaced_mesh = false
[../]
[./outer_co2_removal]
type = PorousFlowPiecewiseLinearSink
boundary = rmax
variable = pgas
fluid_phase = 1
pt_vals = '0 1E9'
multipliers = '0 1E8'
PT_shift = 20.1E6
use_mobility = true
use_relperm = true
use_displaced_mesh = false
[../]
[]

[Modules]
[./FluidProperties]
[./true_water]
type = Water97FluidProperties
[../]
[./tabulated_water]
type = TabulatedFluidProperties
fp = true_water
temperature_min = 275
pressure_max = 1E8
interpolated_properties = 'density viscosity enthalpy internal_energy'
fluid_property_file = water97_tabulated_11.csv
[../]
[./true_co2]
type = CO2FluidProperties
[../]
[./tabulated_co2]
type = TabulatedFluidProperties
fp = true_co2
temperature_min = 275
pressure_max = 1E8
interpolated_properties = 'density viscosity enthalpy internal_energy'
fluid_property_file = co2_tabulated_11.csv
[../]
[../]
[]

[Materials]
[./temperature]
type = PorousFlowTemperature
temperature = T
[../]
[./saturation_calculator]
type = PorousFlow2PhasePP
phase0_porepressure = pwater
phase1_porepressure = pgas
capillary_pressure = pc
[../]
[./massfrac]
type = PorousFlowMassFraction
mass_fraction_vars = 'mass_frac_phase0_species0 mass_frac_phase1_species0'
[../]
[./water]
type = PorousFlowSingleComponentFluid
fp = tabulated_water
phase = 0
[../]
[./co2]
type = PorousFlowSingleComponentFluid
fp = tabulated_co2
phase = 1
[../]
[./relperm_water]
type = PorousFlowRelativePermeabilityCorey
n = 4
s_res = 0.1
sum_s_res = 0.2
phase = 0
[../]
[./relperm_co2]
type = PorousFlowRelativePermeabilityBC
nw_phase = true
lambda = 2
s_res = 0.1
sum_s_res = 0.2
phase = 1
[../]
[./porosity]
type = PorousFlowPorosity
fluid = true
mechanical = true
thermal = true
porosity_zero = 0.1
reference_temperature = 330
reference_porepressure = 20E6
thermal_expansion_coeff = 15E-6 # volumetric
solid_bulk = 8E9 # unimportant since biot = 1
[../]

(modules/porous_flow/examples/tutorial/11.i)

Permeability is chosen to follow the Kozeny-Carman relationship:

  [./permeability_aquifer]
type = PorousFlowPermeabilityKozenyCarman
block = aquifer
poroperm_function = kozeny_carman_phi0
phi0 = 0.1
n = 2
m = 2
k0 = 1E-12
[../]
[./permeability_caps]
type = PorousFlowPermeabilityKozenyCarman
block = caps
poroperm_function = kozeny_carman_phi0
phi0 = 0.1
n = 2
m = 2
k0 = 1E-15
k_anisotropy = '1 0 0  0 1 0  0 0 0.1'
[../]

(modules/porous_flow/examples/tutorial/11.i)

The rock thermal conductivity is chosen to be independent of water saturation and to be isotropic (and independent of rock type — reservoir or cap-rock):

  [./rock_thermal_conductivity]
type = PorousFlowThermalConductivityIdeal
dry_thermal_conductivity = '2 0 0  0 2 0  0 0 2'
[../]

(modules/porous_flow/examples/tutorial/11.i)

while the rock internal energy is also constant:

  [./rock_internal_energy]
type = PorousFlowMatrixInternalEnergy
specific_heat_capacity = 1100
density = 2300
[../]

(modules/porous_flow/examples/tutorial/11.i)

The elasticity tensor of the rock (both reservoir and cap-rock) is assumed isotropic with a Young's modulus of 5GPa:

  [./elasticity_tensor]
type = ComputeIsotropicElasticityTensor
youngs_modulus = 5E9
poissons_ratio = 0.0
[../]

(modules/porous_flow/examples/tutorial/11.i)

The strain calculator must take into consideration both the thermal strain (see governing equations) as well as initial effective stress. The initial total stress is assumed to be zero (for simplicity, not because it is physically very likely, but a nonzero value doesn't change the results much), so the initial effective stress is just the initial porepressure

  [./strain]
type = ComputeSmallStrain
eigenstrain_names = 'thermal_contribution initial_stress'
[../]
[./thermal_contribution]
type = ComputeThermalExpansionEigenstrain
temperature = T
thermal_expansion_coeff = 5E-6 # this is the linear thermal expansion coefficient
eigenstrain_name = thermal_contribution
stress_free_temperature = 330
[../]
[./initial_strain]
type = ComputeEigenstrainFromInitialStress
initial_stress = '20E6 0 0  0 20E6 0  0 0 20E6'
eigenstrain_name = initial_stress
[../]

(modules/porous_flow/examples/tutorial/11.i)

The thermal_contribution eigenstrain name has to be provided to the StressDivergenceTensors Kernels, by the way (see above). Stress is just linear elastic:

  [./stress]
type = ComputeLinearElasticStress
[../]

[./effective_fluid_pressure]
type = PorousFlowEffectiveFluidPressure
[../]
[./volumetric_strain]
type = PorousFlowVolumetricStrain
[../]
[]

[Postprocessors]
[./effective_fluid_pressure_at_wellbore]
type = PointValue
variable = effective_fluid_pressure
point = '1 0 0'
execute_on = timestep_begin
use_displaced_mesh = false
[../]
[./constrained_effective_fluid_pressure_at_wellbore]
type = FunctionValuePostprocessor
function = constrain_effective_fluid_pressure
execute_on = timestep_begin
[../]
[]

[Functions]
[./constrain_effective_fluid_pressure]
type = ParsedFunction
vars = effective_fluid_pressure_at_wellbore
vals = effective_fluid_pressure_at_wellbore
value = 'max(effective_fluid_pressure_at_wellbore, 20E6)'
[../]
[]

[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 = 1E3
[./TimeStepper]
dt = 1E3
growth_factor = 1.2
optimal_iterations = 10
[../]
nl_abs_tol = 1E-7
[]

[Outputs]
exodus = true
[]

(modules/porous_flow/examples/tutorial/11.i)

Finally, the effective fluid pressure must be computed because it is needed by the Porosity and the solid-fluid coupling, and the volumetric strain feeds into the Porosity:

# Two-phase borehole injection problem
[Mesh]
type = AnnularMesh
dim = 2
nr = 10
rmin = 1.0
rmax = 10
growth_r = 1.4
nt = 4
tmin = 0
tmax = 1.57079632679

(modules/porous_flow/examples/tutorial/11.i)

## Executioner

For this model, an IterationAdaptiveDT Timestepper is used. This is because the dynamics at early times, particularly the thermal shock induced by sudden application of cool CO at the injection area, means small timesteps are needed, but after some time larger timesteps can be used.

[Executioner]
type = Transient
solve_type = Newton
end_time = 1E3
[./TimeStepper]
dt = 1E3
growth_factor = 1.2
optimal_iterations = 10
[../]
nl_abs_tol = 1E-7
[]

(modules/porous_flow/examples/tutorial/11.i)

## The end

Holey Dooley, you made it to the end, well done!

An animation of the results is shown in Figure 1. The porepressure front moves relatively quickly, followed by the saturation front, and then the temperature front. The solid mechanical deformations are governed mostly by the temperature. By the way, this type of dynamic 2-phase injection problem using PorousFlow has been benchmarked against analytical results with excellent agreement (and will hopefully be written into a PorousFlow Example — we are awaiting permission by funding bodies).

Figure 1: CO saturation, CO porepressure, temperature and hoop stress in the 2-phase CO injection simulation.

## Postscript: the same again in 2D

As mentioned on Page 00, this problem is really an axially-symmetric problem, so may be better modelled by MOOSE using its "RZ" coordinate system. The changes to the input file are minimal. Apart from the mesh generation (discussed in Page 00) the changes are itemized below.

There only need by a disp_r Variable in place of disp_x and disp_y:

  [./disp_r]
scaling = 1E-5
[../]

(modules/porous_flow/examples/tutorial/11_2D.i)
[GlobalParams]
displacements = 'disp_r disp_z'
gravity = '0 0 0'
biot_coefficient = 1.0
PorousFlowDictator = dictator

(modules/porous_flow/examples/tutorial/11_2D.i)
  [./dictator]
type = PorousFlowDictator
porous_flow_vars = 'pwater pgas T disp_r'
number_fluid_phases = 2
number_fluid_components = 2
[../]

(modules/porous_flow/examples/tutorial/11_2D.i)

There are mechanical Kernels only for disp_r, and the StressDivergenceTensors Kernel is modified:

  [./grad_stress_r]
type = StressDivergenceRZTensors
temperature = T
variable = disp_r
thermal_eigenstrain_name = thermal_contribution
use_displaced_mesh = false
component = 0
[../]

(modules/porous_flow/examples/tutorial/11_2D.i)

The stress AuxKernels are modified. In TensorMechanics with RZ coordinates, the 00 component is , the 11 component is and the 22 component is . Therefore, these AuxKernels read

  [./stress_rr_aux]
type = RankTwoAux
variable = stress_rr
rank_two_tensor = stress
index_i = 0
index_j = 0
[../]
[./stress_tt]
type = RankTwoAux
rank_two_tensor = stress
variable = stress_tt
index_i = 2
index_j = 2
[../]
[./stress_zz]
type = RankTwoAux
rank_two_tensor = stress
variable = stress_zz
index_i = 1
index_j = 1
[../]

(modules/porous_flow/examples/tutorial/11_2D.i)

The boundary conditions for the mechanics become simpler:

  [./pinned_top_bottom_r]
type = PresetBC
variable = disp_r
value = 0
boundary = 'top bottom'
[../]
[./cavity_pressure_r]
type = Pressure
boundary = injection_area
variable = disp_r
component = 0
postprocessor = constrained_effective_fluid_pressure_at_wellbore
use_displaced_mesh = false
[../]

(modules/porous_flow/examples/tutorial/11_2D.i)

Finally, the strain calculator needs to be of RZ type:

  [./strain]
type = ComputeAxisymmetricRZSmallStrain
eigenstrain_names = 'thermal_contribution initial_stress'
[../]

(modules/porous_flow/examples/tutorial/11_2D.i)

The reader may check that the 3D and 2D models produce the same answer, although of course the 2D version is much faster due to it having only 176 degrees of freedom compared with the 3D's 1100.