- variableThe name of the variable that this Kernel operates on
C++ Type:NonlinearVariableName

Description:The name of the variable that this Kernel operates on

- PorousFlowDictatorThe UserObject that holds the list of PorousFlow variable names.
C++ Type:UserObjectName

Description:The UserObject that holds the list of PorousFlow variable names.

# PorousFlowFullySaturatedMassTimeDerivative

Fully-saturated version of the single-component, single-phase fluid mass derivative wrt time

Consider a fully-saturated, single-phase, single-component fluid in a THM simulation. The time-derivatve terms from the fluid governing equation are (1) where and all other nomenclature is described here. Using the THM evolution of porosity, along with the assumption that the fluid bulk modulus, , and its volumetric thermal expansion coefficient, , are constant, and the fluid density is given by (2) the time derivative terms may be written as (3) Here is the so-called Biot Modulus: (4) and is an effective volumetric thermal expansion coefficient: (5) Notice that disregarding the premultiplication by , the above time-derivative terms in Eq. (3) would be linear in the variables , displacement, and , if and were constant.

In standard poro-mechanics it is usual to calculate and at the initial stage of simulation, and keep them fixed forever afterwards. Of course this is an approximation since and were derived using the explicit assumption of a porosity that depended on , , and , but it makes finding analytical solutions much easier. Therefore, PorousFlow offers the following Materials and Kernels. Using these Materials and Kernel allows immediate and precise comparison with analytical and numerical solutions of poro-mechanics.

PorousFlowConstantBiotModulus Material, which computes given by Eq. (4) during the initial stage of simulation, and keeps it fixed thereafter. It requires a Porosity Material, but that Material's Property is only used during the initial computation.

PorousFlowConstantThermalExpansionCoefficient Material, which computes given by Eq. (5) during the the initial stage of simulation, and keeps it fixed thereafter. It requires a Porosity Material, but that Material's Property is only used during the initial computation.

The

`PorousFlowFullySaturatedMassTimeDerivative`

Kernel, which computes one of the following contributions, depending upon the`coupling_type`

flag:

for fluid-flow-only problems;

for TH problems;

for HM problems;

for THM problems.

The `PorousFlowFullySaturatedMassTimeDerivative`

Kernel does not employ lumping, which is largely unecessary in this single-phase, single-component situation. This means only `quad-point'' Materials are needed. In fact, when using all the FullySaturated flow Kernels (see governing equations) standard Materials evaluated at the quadpoints are needed, which saves on computation time and input-file length.`

In each case, the initial pre-multiplication by is optional (indicated by the parenthases around ).

When the Kernel is pre-multiplied by , which is the default, it is computing the time derivative of fluid mass. This allows the Kernel to be easily used with the remainder of PorousFlow: the BCs, the Postprocessors, the AuxKernels, and the DiracKernels are all based on mass.

When the pre-multiplication is not performed, this Kernel is computing the time derivative of fluid volume. This has two great advantages:

the time-derivatives are linearised, resulting in excellent nonlinear convergence;

comparing with results from poro-mechanics theory is straightforward.

However, this means additional care must be taken.

The flow Kernel PorousFlowFullySaturatedDarcyBase should also not pre-multiply by density.

The BCs, Postprocessors, AuxKernels, and DiracKernels must be written in such a way to operate in the fluid-volume scenario rather than the default fluid-mass scenario.

The flag

`consistent_with_displaced_mesh`

should be set`false`

in the PorousFlowVolumetricStrain Material.

## Input Parameters

- blockThe list of block ids (SubdomainID) that this object will be applied
C++ Type:std::vector

Options:

Description:The list of block ids (SubdomainID) that this object will be applied

- biot_coefficient1Biot coefficient
Default:1

C++ Type:double

Options:

Description:Biot coefficient

- coupling_typeHydroThe type of simulation. For simulations involving Mechanical deformations, you will need to supply the correct Biot coefficient. For simulations involving Thermal flows, you will need an associated ConstantThermalExpansionCoefficient Material
Default:Hydro

C++ Type:MooseEnum

Options:Hydro ThermoHydro HydroMechanical ThermoHydroMechanical

Description:The type of simulation. For simulations involving Mechanical deformations, you will need to supply the correct Biot coefficient. For simulations involving Thermal flows, you will need an associated ConstantThermalExpansionCoefficient Material

- multiply_by_densityTrueIf true, then this Kernel is the time derivative of the fluid mass. If false, then this Kernel is the derivative of the fluid volume (which is common in poro-mechanics)
Default:True

C++ Type:bool

Options:

Description:If true, then this Kernel is the time derivative of the fluid mass. If false, then this Kernel is the derivative of the fluid volume (which is common in poro-mechanics)

- displacementsThe displacements
C++ Type:std::vector

Options:

Description:The displacements

### Optional Parameters

- enableTrueSet the enabled status of the MooseObject.
Default:True

C++ Type:bool

Options:

Description:Set the enabled status of the MooseObject.

- save_inThe name of auxiliary variables to save this Kernel's residual contributions to. Everything about that variable must match everything about this variable (the type, what blocks it's on, etc.)
C++ Type:std::vector

Options:

Description:The name of auxiliary variables to save this Kernel's residual contributions to. Everything about that variable must match everything about this variable (the type, what blocks it's on, etc.)

- use_displaced_meshFalseWhether or not this object should use the displaced mesh for computation. Note that in the case this is true but no displacements are provided in the Mesh block the undisplaced mesh will still be used.
Default:False

C++ Type:bool

Options:

Description:Whether or not this object should use the displaced mesh for computation. Note that in the case this is true but no displacements are provided in the Mesh block the undisplaced mesh will still be used.

- control_tagsAdds user-defined labels for accessing object parameters via control logic.
C++ Type:std::vector

Options:

Description:Adds user-defined labels for accessing object parameters via control logic.

- seed0The seed for the master random number generator
Default:0

C++ Type:unsigned int

Options:

Description:The seed for the master random number generator

- diag_save_inThe name of auxiliary variables to save this Kernel's diagonal Jacobian contributions to. Everything about that variable must match everything about this variable (the type, what blocks it's on, etc.)
C++ Type:std::vector

Options:

Description:The name of auxiliary variables to save this Kernel's diagonal Jacobian contributions to. Everything about that variable must match everything about this variable (the type, what blocks it's on, etc.)

- implicitTrueDetermines whether this object is calculated using an implicit or explicit form
Default:True

C++ Type:bool

Options:

Description:Determines whether this object is calculated using an implicit or explicit form

### Advanced Parameters

- vector_tagstimeThe tag for the vectors this Kernel should fill
Default:time

C++ Type:MultiMooseEnum

Options:nontime time

Description:The tag for the vectors this Kernel should fill

- extra_vector_tagsThe extra tags for the vectors this Kernel should fill
C++ Type:std::vector

Options:

Description:The extra tags for the vectors this Kernel should fill

- matrix_tagssystem timeThe tag for the matrices this Kernel should fill
Default:system time

C++ Type:MultiMooseEnum

Options:nontime system time

Description:The tag for the matrices this Kernel should fill

- extra_matrix_tagsThe extra tags for the matrices this Kernel should fill
C++ Type:std::vector

Options:

Description:The extra tags for the matrices this Kernel should fill

### Tagging Parameters

## Input Files

- modules/porous_flow/test/tests/jacobian/mass01_fully_saturated.i
- modules/porous_flow/test/tests/pressure_pulse/pressure_pulse_1d_fully_saturated_2.i
- modules/porous_flow/test/tests/poro_elasticity/mandel_fully_saturated.i
- modules/porous_flow/test/tests/poro_elasticity/terzaghi_fully_saturated_volume.i
- modules/porous_flow/test/tests/poro_elasticity/pp_generation_unconfined_fully_saturated.i
- modules/porous_flow/test/tests/poro_elasticity/pp_generation_unconfined_fully_saturated_volume.i
- modules/porous_flow/test/tests/poro_elasticity/mandel_fully_saturated_volume.i

#### modules/porous_flow/test/tests/jacobian/mass01_fully_saturated.i

```
# FullySaturatedMassTimeDerivative
[Mesh]
type = GeneratedMesh
dim = 3
nx = 1
ny = 1
nz = 1
[]
[GlobalParams]
PorousFlowDictator = dictator
displacements = 'disp_x disp_y disp_z'
[]
[Modules]
[./FluidProperties]
[./the_simple_fluid]
type = SimpleFluidProperties
thermal_expansion = 0.5
bulk_modulus = 1.5
density0 = 1.0
[../]
[../]
[]
[Variables]
[./pp]
[../]
[./T]
[../]
[./disp_x]
[../]
[./disp_y]
[../]
[./disp_z]
[../]
[]
[ICs]
[./disp_x]
type = RandomIC
min = -0.1
max = 0.1
variable = disp_x
[../]
[./disp_y]
type = RandomIC
min = -0.1
max = 0.1
variable = disp_y
[../]
[./disp_z]
type = RandomIC
min = -0.1
max = 0.1
variable = disp_z
[../]
[./pp]
type = RandomIC
variable = pp
min = 0
max = 1
[../]
[./T]
type = RandomIC
variable = T
min = 0
max = 1
[../]
[]
[BCs]
# necessary otherwise volumetric strain rate will be zero
[./disp_x]
type = PresetBC
variable = disp_x
value = 0
boundary = 'left right'
[../]
[./disp_y]
type = PresetBC
variable = disp_y
value = 0
boundary = 'left right'
[../]
[./disp_z]
type = PresetBC
variable = disp_z
value = 0
boundary = 'left right'
[../]
[]
[Kernels]
[./mass0]
type = PorousFlowFullySaturatedMassTimeDerivative
variable = pp
coupling_type = ThermoHydroMechanical
biot_coefficient = 0.9
[../]
[./dummyT]
type = TimeDerivative
variable = T
[../]
[./grad_stress_x]
type = StressDivergenceTensors
variable = disp_x
component = 0
[../]
[./grad_stress_y]
type = StressDivergenceTensors
variable = disp_y
component = 1
[../]
[./grad_stress_z]
type = StressDivergenceTensors
variable = disp_z
component = 2
[../]
[]
[UserObjects]
[./dictator]
type = PorousFlowDictator
porous_flow_vars = 'pp disp_x disp_y disp_z T'
number_fluid_phases = 1
number_fluid_components = 1
[../]
[./simple1]
type = TensorMechanicsPlasticSimpleTester
a = 0
b = 1
strength = 1E20
yield_function_tolerance = 1.0E-9
internal_constraint_tolerance = 1.0E-9
[../]
[]
[Materials]
[./elasticity_tensor]
type = ComputeIsotropicElasticityTensor
bulk_modulus = 2.0
shear_modulus = 3.0
[../]
[./strain]
type = ComputeSmallStrain
[../]
[./stress]
type = ComputeLinearElasticStress
[../]
[./vol_strain]
type = PorousFlowVolumetricStrain
[../]
[./temperature]
type = PorousFlowTemperature
temperature = T
[../]
[./ppss]
type = PorousFlow1PhaseFullySaturated
porepressure = pp
[../]
[./massfrac]
type = PorousFlowMassFraction
[../]
[./simple_fluid]
type = PorousFlowSingleComponentFluid
fp = the_simple_fluid
phase = 0
[../]
[./porosity]
type = PorousFlowPorosityConst # only the initial vaue of this is ever used
porosity = 0.1
[../]
[./biot_modulus]
type = PorousFlowConstantBiotModulus
biot_coefficient = 0.9
fluid_bulk_modulus = 1.5
solid_bulk_compliance = 0.5
[../]
[./thermal_expansion]
type = PorousFlowConstantThermalExpansionCoefficient
biot_coefficient = 0.9
fluid_coefficient = 0.5
drained_coefficient = 0.4
[../]
[]
[Preconditioning]
[./check]
type = SMP
full = true
#petsc_options = '-snes_test_display'
petsc_options_iname = '-ksp_type -pc_type -snes_atol -snes_rtol -snes_max_it -snes_type'
petsc_options_value = 'bcgs bjacobi 1E-15 1E-10 10000 test'
[../]
[]
[Executioner]
type = Transient
solve_type = Newton
dt = 1
end_time = 2
[]
[Outputs]
exodus = false
[]
```

#### modules/porous_flow/test/tests/pressure_pulse/pressure_pulse_1d_fully_saturated_2.i

```
# Pressure pulse in 1D with 1 phase - transient
# using the PorousFlowFullySaturatedDarcyBase Kernel
# and the PorousFlowFullySaturatedMassTimeDerivative Kernel
[Mesh]
type = GeneratedMesh
dim = 1
nx = 20
xmin = 0
xmax = 100
[]
[GlobalParams]
PorousFlowDictator = dictator
[]
[Variables]
[./pp]
initial_condition = 2E6
[../]
[]
[Kernels]
[./mass0]
type = PorousFlowFullySaturatedMassTimeDerivative
variable = pp
[../]
[./flux]
type = PorousFlowFullySaturatedDarcyBase
variable = pp
gravity = '0 0 0'
[../]
[]
[UserObjects]
[./dictator]
type = PorousFlowDictator
porous_flow_vars = 'pp'
number_fluid_phases = 1
number_fluid_components = 1
[../]
[]
[Modules]
[./FluidProperties]
[./simple_fluid]
type = SimpleFluidProperties
bulk_modulus = 2e9
density0 = 1000
thermal_expansion = 0
viscosity = 1e-3
[../]
[../]
[]
[Materials]
[./temperature_qp]
type = PorousFlowTemperature
[../]
[./ppss_qp]
type = PorousFlow1PhaseFullySaturated
porepressure = pp
[../]
[./massfrac]
type = PorousFlowMassFraction
[../]
[./simple_fluid_qp]
type = PorousFlowSingleComponentFluid
fp = simple_fluid
phase = 0
[../]
[./porosity]
type = PorousFlowPorosityConst
porosity = 0.1
[../]
[./biot_modulus]
type = PorousFlowConstantBiotModulus
fluid_bulk_modulus = 2E9
[../]
[./permeability]
type = PorousFlowPermeabilityConst
permeability = '1E-15 0 0 0 1E-15 0 0 0 1E-15'
[../]
[]
[BCs]
[./left]
type = DirichletBC
boundary = left
value = 3E6
variable = pp
[../]
[]
[Preconditioning]
[./andy]
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
[]
[Postprocessors]
[./p005]
type = PointValue
variable = pp
point = '5 0 0'
execute_on = 'initial timestep_end'
[../]
[./p015]
type = PointValue
variable = pp
point = '15 0 0'
execute_on = 'initial timestep_end'
[../]
[./p025]
type = PointValue
variable = pp
point = '25 0 0'
execute_on = 'initial timestep_end'
[../]
[./p035]
type = PointValue
variable = pp
point = '35 0 0'
execute_on = 'initial timestep_end'
[../]
[./p045]
type = PointValue
variable = pp
point = '45 0 0'
execute_on = 'initial timestep_end'
[../]
[./p055]
type = PointValue
variable = pp
point = '55 0 0'
execute_on = 'initial timestep_end'
[../]
[./p065]
type = PointValue
variable = pp
point = '65 0 0'
execute_on = 'initial timestep_end'
[../]
[./p075]
type = PointValue
variable = pp
point = '75 0 0'
execute_on = 'initial timestep_end'
[../]
[./p085]
type = PointValue
variable = pp
point = '85 0 0'
execute_on = 'initial timestep_end'
[../]
[./p095]
type = PointValue
variable = pp
point = '95 0 0'
execute_on = 'initial timestep_end'
[../]
[]
[Outputs]
file_base = pressure_pulse_1d_fully_saturated_2
print_linear_residuals = false
csv = true
[]
```

#### modules/porous_flow/test/tests/poro_elasticity/mandel_fully_saturated.i

```
# Mandel's problem of consolodation of a drained medium
# Using the FullySaturatedDarcyBase and FullySaturatedMassTimeDerivative kernels
# Note the use of consistent_with_displaced_mesh = false in the calculation of volumetric strain
#
# A sample is in plane strain.
# -a <= x <= a
# -b <= y <= b
# It is squashed with constant force by impermeable, frictionless plattens on its top and bottom surfaces (at y=+/-b)
# Fluid is allowed to leak out from its sides (at x=+/-a)
# The porepressure within the sample is monitored.
#
# As is common in the literature, this is simulated by
# considering the quarter-sample, 0<=x<=a and 0<=y<=b, with
# impermeable, roller BCs at x=0 and y=0 and y=b.
# Porepressure is fixed at zero on x=a.
# Porepressure and displacement are initialised to zero.
# Then the top (y=b) is moved downwards with prescribed velocity,
# so that the total force that is inducing this downwards velocity
# is fixed. The velocity is worked out by solving Mandel's problem
# analytically, and the total force is monitored in the simulation
# to check that it indeed remains constant.
#
# Here are the problem's parameters, and their values:
# Soil width. a = 1
# Soil height. b = 0.1
# Soil's Lame lambda. la = 0.5
# Soil's Lame mu, which is also the Soil's shear modulus. mu = G = 0.75
# Soil bulk modulus. K = la + 2*mu/3 = 1
# Drained Poisson ratio. nu = (3K - 2G)/(6K + 2G) = 0.2
# Soil bulk compliance. 1/K = 1
# Fluid bulk modulus. Kf = 8
# Fluid bulk compliance. 1/Kf = 0.125
# Soil initial porosity. phi0 = 0.1
# Biot coefficient. alpha = 0.6
# Biot modulus. M = 1/(phi0/Kf + (alpha - phi0)(1 - alpha)/K) = 4.705882
# Undrained bulk modulus. Ku = K + alpha^2*M = 2.694118
# Undrained Poisson ratio. nuu = (3Ku - 2G)/(6Ku + 2G) = 0.372627
# Skempton coefficient. B = alpha*M/Ku = 1.048035
# Fluid mobility (soil permeability/fluid viscosity). k = 1.5
# Consolidation coefficient. c = 2*k*B^2*G*(1-nu)*(1+nuu)^2/9/(1-nuu)/(nuu-nu) = 3.821656
# Normal stress on top. F = 1
#
# The solution for porepressure and displacements is given in
# AHD Cheng and E Detournay "A direct boundary element method for plane strain poroelasticity" International Journal of Numerical and Analytical Methods in Geomechanics 12 (1988) 551-572.
# The solution involves complicated infinite series, so I shall not write it here
[Mesh]
type = GeneratedMesh
dim = 3
nx = 10
ny = 1
nz = 1
xmin = 0
xmax = 1
ymin = 0
ymax = 0.1
zmin = 0
zmax = 1
[]
[GlobalParams]
displacements = 'disp_x disp_y disp_z'
PorousFlowDictator = dictator
block = 0
[]
[UserObjects]
[./dictator]
type = PorousFlowDictator
porous_flow_vars = 'porepressure disp_x disp_y disp_z'
number_fluid_phases = 1
number_fluid_components = 1
[../]
[]
[Variables]
[./disp_x]
[../]
[./disp_y]
[../]
[./disp_z]
[../]
[./porepressure]
[../]
[]
[BCs]
[./roller_xmin]
type = PresetBC
variable = disp_x
value = 0
boundary = 'left'
[../]
[./roller_ymin]
type = PresetBC
variable = disp_y
value = 0
boundary = 'bottom'
[../]
[./plane_strain]
type = PresetBC
variable = disp_z
value = 0
boundary = 'back front'
[../]
[./xmax_drained]
type = DirichletBC
variable = porepressure
value = 0
boundary = right
[../]
[./top_velocity]
type = FunctionPresetBC
variable = disp_y
function = top_velocity
boundary = top
[../]
[]
[Functions]
[./top_velocity]
type = PiecewiseLinear
x = '0 0.002 0.006 0.014 0.03 0.046 0.062 0.078 0.094 0.11 0.126 0.142 0.158 0.174 0.19 0.206 0.222 0.238 0.254 0.27 0.286 0.302 0.318 0.334 0.35 0.366 0.382 0.398 0.414 0.43 0.446 0.462 0.478 0.494 0.51 0.526 0.542 0.558 0.574 0.59 0.606 0.622 0.638 0.654 0.67 0.686 0.702'
y = '-0.041824842 -0.042730269 -0.043412712 -0.04428867 -0.045509181 -0.04645965 -0.047268246 -0.047974749 -0.048597109 -0.0491467 -0.049632388 -0.050061697 -0.050441198 -0.050776675 -0.051073238 -0.0513354 -0.051567152 -0.051772022 -0.051953128 -0.052113227 -0.052254754 -0.052379865 -0.052490464 -0.052588233 -0.052674662 -0.052751065 -0.052818606 -0.052878312 -0.052931093 -0.052977751 -0.053018997 -0.053055459 -0.053087691 -0.053116185 -0.053141373 -0.05316364 -0.053183324 -0.053200724 -0.053216106 -0.053229704 -0.053241725 -0.053252351 -0.053261745 -0.053270049 -0.053277389 -0.053283879 -0.053289615'
[../]
[]
[AuxVariables]
[./stress_yy]
order = CONSTANT
family = MONOMIAL
[../]
[./tot_force]
order = CONSTANT
family = MONOMIAL
[../]
[]
[AuxKernels]
[./stress_yy]
type = RankTwoAux
rank_two_tensor = stress
variable = stress_yy
index_i = 1
index_j = 1
[../]
[./tot_force]
type = ParsedAux
args = 'stress_yy porepressure'
execute_on = timestep_end
variable = tot_force
function = '-stress_yy+0.6*porepressure'
[../]
[]
[Kernels]
[./grad_stress_x]
type = StressDivergenceTensors
variable = disp_x
component = 0
[../]
[./grad_stress_y]
type = StressDivergenceTensors
variable = disp_y
component = 1
[../]
[./grad_stress_z]
type = StressDivergenceTensors
variable = disp_z
component = 2
[../]
[./poro_x]
type = PorousFlowEffectiveStressCoupling
biot_coefficient = 0.6
variable = disp_x
component = 0
[../]
[./poro_y]
type = PorousFlowEffectiveStressCoupling
biot_coefficient = 0.6
variable = disp_y
component = 1
[../]
[./poro_z]
type = PorousFlowEffectiveStressCoupling
biot_coefficient = 0.6
component = 2
variable = disp_z
[../]
[./mass0]
type = PorousFlowFullySaturatedMassTimeDerivative
biot_coefficient = 0.6
coupling_type = HydroMechanical
variable = porepressure
[../]
[./flux]
type = PorousFlowFullySaturatedDarcyBase
variable = porepressure
gravity = '0 0 0'
[../]
[]
[Modules]
[./FluidProperties]
[./simple_fluid]
type = SimpleFluidProperties
bulk_modulus = 8
density0 = 1
thermal_expansion = 0
viscosity = 1
[../]
[../]
[]
[Materials]
[./temperature]
type = PorousFlowTemperature
[../]
[./elasticity_tensor]
type = ComputeElasticityTensor
C_ijkl = '0.5 0.75'
# bulk modulus is lambda + 2*mu/3 = 0.5 + 2*0.75/3 = 1
fill_method = symmetric_isotropic
[../]
[./strain]
type = ComputeSmallStrain
[../]
[./stress]
type = ComputeLinearElasticStress
[../]
[./eff_fluid_pressure_qp]
type = PorousFlowEffectiveFluidPressure
[../]
[./vol_strain]
type = PorousFlowVolumetricStrain
consistent_with_displaced_mesh = false
[../]
[./ppss]
type = PorousFlow1PhaseFullySaturated
porepressure = porepressure
[../]
[./massfrac]
type = PorousFlowMassFraction
[../]
[./simple_fluid_qp]
type = PorousFlowSingleComponentFluid
fp = simple_fluid
phase = 0
[../]
[./porosity]
type = PorousFlowPorosityConst # only the initial value of this is ever used
porosity = 0.1
[../]
[./biot_modulus]
type = PorousFlowConstantBiotModulus
biot_coefficient = 0.6
solid_bulk_compliance = 1
fluid_bulk_modulus = 8
[../]
[./permeability]
type = PorousFlowPermeabilityConst
permeability = '1.5 0 0 0 1.5 0 0 0 1.5'
[../]
[]
[Postprocessors]
[./p0]
type = PointValue
outputs = csv
point = '0.0 0 0'
variable = porepressure
[../]
[./p1]
type = PointValue
outputs = csv
point = '0.1 0 0'
variable = porepressure
[../]
[./p2]
type = PointValue
outputs = csv
point = '0.2 0 0'
variable = porepressure
[../]
[./p3]
type = PointValue
outputs = csv
point = '0.3 0 0'
variable = porepressure
[../]
[./p4]
type = PointValue
outputs = csv
point = '0.4 0 0'
variable = porepressure
[../]
[./p5]
type = PointValue
outputs = csv
point = '0.5 0 0'
variable = porepressure
[../]
[./p6]
type = PointValue
outputs = csv
point = '0.6 0 0'
variable = porepressure
[../]
[./p7]
type = PointValue
outputs = csv
point = '0.7 0 0'
variable = porepressure
[../]
[./p8]
type = PointValue
outputs = csv
point = '0.8 0 0'
variable = porepressure
[../]
[./p9]
type = PointValue
outputs = csv
point = '0.9 0 0'
variable = porepressure
[../]
[./p99]
type = PointValue
outputs = csv
point = '1 0 0'
variable = porepressure
[../]
[./xdisp]
type = PointValue
outputs = csv
point = '1 0.1 0'
variable = disp_x
[../]
[./ydisp]
type = PointValue
outputs = csv
point = '1 0.1 0'
variable = disp_y
[../]
[./total_downwards_force]
type = ElementAverageValue
outputs = csv
variable = tot_force
[../]
[./dt]
type = FunctionValuePostprocessor
outputs = console
function = if(0.15*t<0.01,0.15*t,0.01)
[../]
[]
[Preconditioning]
[./andy]
type = SMP
full = true
petsc_options_iname = '-ksp_type -pc_type -sub_pc_type -snes_atol -snes_rtol -snes_max_it'
petsc_options_value = 'gmres asm lu 1E-14 1E-10 10000'
[../]
[]
[Executioner]
type = Transient
solve_type = Newton
start_time = 0
end_time = 0.7
[./TimeStepper]
type = PostprocessorDT
postprocessor = dt
dt = 0.001
[../]
[]
[Outputs]
execute_on = 'timestep_end'
file_base = mandel_fully_saturated
[./csv]
interval = 3
type = CSV
[../]
[]
```

#### modules/porous_flow/test/tests/poro_elasticity/terzaghi_fully_saturated_volume.i

```
# Terzaghi's problem of consolodation of a drained medium
# The FullySaturated Kernels are used, with multiply_by_density = false
# so that this becomes a linear problem with constant Biot Modulus
# Also, since the FullySaturated Kernels are used, we have to
# use consistent_with_displaced_mesh = false in the calculation of volumetric strain
#
# A saturated soil sample sits in a bath of water.
# It is constrained on its sides, and bottom.
# Its sides and bottom are also impermeable.
# Initially it is unstressed.
# A normal stress, q, is applied to the soil's top.
# The soil then slowly compresses as water is squeezed
# out from the sample from its top (the top BC for
# the porepressure is porepressure = 0).
#
# See, for example. Section 2.2 of the online manuscript
# Arnold Verruijt "Theory and Problems of Poroelasticity" Delft University of Technology 2013
# but note that the "sigma" in that paper is the negative
# of the stress in TensorMechanics
#
# Here are the problem's parameters, and their values:
# Soil height. h = 10
# Soil's Lame lambda. la = 2
# Soil's Lame mu, which is also the Soil's shear modulus. mu = 3
# Soil bulk modulus. K = la + 2*mu/3 = 4
# Soil confined compressibility. m = 1/(K + 4mu/3) = 0.125
# Soil bulk compliance. 1/K = 0.25
# Fluid bulk modulus. Kf = 8
# Fluid bulk compliance. 1/Kf = 0.125
# Fluid mobility (soil permeability/fluid viscosity). k = 1.5
# Soil initial porosity. phi0 = 0.1
# Biot coefficient. alpha = 0.6
# Soil initial storativity, which is the reciprocal of the initial Biot modulus. S = phi0/Kf + (alpha - phi0)(1 - alpha)/K = 0.0625
# Consolidation coefficient. c = k/(S + alpha^2 m) = 13.95348837
# Normal stress on top. q = 1
# Initial porepressure, resulting from instantaneous application of q, assuming corresponding instantaneous increase of porepressure (Note that this is calculated by MOOSE: we only need it for the analytical solution). p0 = alpha*m*q/(S + alpha^2 m) = 0.69767442
# Initial vertical displacement (down is positive), resulting from instantaneous application of q (Note this is calculated by MOOSE: we only need it for the analytical solution). uz0 = q*m*h*S/(S + alpha^2 m)
# Final vertical displacement (down in positive) (Note this is calculated by MOOSE: we only need it for the analytical solution). uzinf = q*m*h
#
# The solution for porepressure is
# P = 4*p0/\pi \sum_{k=1}^{\infty} \frac{(-1)^{k-1}}{2k-1} \cos ((2k-1)\pi z/(2h)) \exp(-(2k-1)^2 \pi^2 ct/(4 h^2))
# This series converges very slowly for ct/h^2 small, so in that domain
# P = p0 erf( (1-(z/h))/(2 \sqrt(ct/h^2)) )
#
# The degree of consolidation is defined as
# U = (uz - uz0)/(uzinf - uz0)
# where uz0 and uzinf are defined above, and
# uz = the vertical displacement of the top (down is positive)
# U = 1 - (8/\pi^2)\sum_{k=1}^{\infty} \frac{1}{(2k-1)^2} \exp(-(2k-1)^2 \pi^2 ct/(4 h^2))
[Mesh]
type = GeneratedMesh
dim = 3
nx = 1
ny = 1
nz = 10
xmin = -1
xmax = 1
ymin = -1
ymax = 1
zmin = 0
zmax = 10
[]
[GlobalParams]
displacements = 'disp_x disp_y disp_z'
PorousFlowDictator = dictator
block = 0
[]
[UserObjects]
[./dictator]
type = PorousFlowDictator
porous_flow_vars = 'porepressure disp_x disp_y disp_z'
number_fluid_phases = 1
number_fluid_components = 1
[../]
[]
[Variables]
[./disp_x]
[../]
[./disp_y]
[../]
[./disp_z]
[../]
[./porepressure]
[../]
[]
[BCs]
[./confinex]
type = PresetBC
variable = disp_x
value = 0
boundary = 'left right'
[../]
[./confiney]
type = PresetBC
variable = disp_y
value = 0
boundary = 'bottom top'
[../]
[./basefixed]
type = PresetBC
variable = disp_z
value = 0
boundary = back
[../]
[./topdrained]
type = DirichletBC
variable = porepressure
value = 0
boundary = front
[../]
[./topload]
type = NeumannBC
variable = disp_z
value = -1
boundary = front
[../]
[]
[Kernels]
[./grad_stress_x]
type = StressDivergenceTensors
variable = disp_x
component = 0
[../]
[./grad_stress_y]
type = StressDivergenceTensors
variable = disp_y
component = 1
[../]
[./grad_stress_z]
type = StressDivergenceTensors
variable = disp_z
component = 2
[../]
[./poro_x]
type = PorousFlowEffectiveStressCoupling
biot_coefficient = 0.6
variable = disp_x
component = 0
[../]
[./poro_y]
type = PorousFlowEffectiveStressCoupling
biot_coefficient = 0.6
variable = disp_y
component = 1
[../]
[./poro_z]
type = PorousFlowEffectiveStressCoupling
biot_coefficient = 0.6
component = 2
variable = disp_z
[../]
[./mass0]
type = PorousFlowFullySaturatedMassTimeDerivative
coupling_type = HydroMechanical
biot_coefficient = 0.6
multiply_by_density = false
variable = porepressure
[../]
[./flux]
type = PorousFlowFullySaturatedDarcyBase
multiply_by_density = false
variable = porepressure
gravity = '0 0 0'
[../]
[]
[Modules]
[./FluidProperties]
[./simple_fluid]
type = SimpleFluidProperties
bulk_modulus = 8
density0 = 1
thermal_expansion = 0
viscosity = 0.96
[../]
[../]
[]
[Materials]
[./temperature]
type = PorousFlowTemperature
[../]
[./elasticity_tensor]
type = ComputeElasticityTensor
C_ijkl = '2 3'
# bulk modulus is lambda + 2*mu/3 = 2 + 2*3/3 = 4
fill_method = symmetric_isotropic
[../]
[./strain]
type = ComputeSmallStrain
[../]
[./stress]
type = ComputeLinearElasticStress
[../]
[./eff_fluid_pressure_qp]
type = PorousFlowEffectiveFluidPressure
[../]
[./vol_strain]
type = PorousFlowVolumetricStrain
consistent_with_displaced_mesh = false
[../]
[./ppss]
type = PorousFlow1PhaseFullySaturated
porepressure = porepressure
[../]
[./massfrac]
type = PorousFlowMassFraction
[../]
[./simple_fluid_qp]
type = PorousFlowSingleComponentFluid
fp = simple_fluid
phase = 0
[../]
[./porosity]
type = PorousFlowPorosityConst # only the initial value of this is used
porosity = 0.1
[../]
[./biot_modulus]
type = PorousFlowConstantBiotModulus
biot_coefficient = 0.6
fluid_bulk_modulus = 8
solid_bulk_compliance = 0.25
[../]
[./permeability]
type = PorousFlowPermeabilityConst
permeability = '1.5 0 0 0 1.5 0 0 0 1.5'
[../]
[]
[Postprocessors]
[./p0]
type = PointValue
outputs = csv
point = '0 0 0'
variable = porepressure
use_displaced_mesh = false
[../]
[./p1]
type = PointValue
outputs = csv
point = '0 0 1'
variable = porepressure
use_displaced_mesh = false
[../]
[./p2]
type = PointValue
outputs = csv
point = '0 0 2'
variable = porepressure
use_displaced_mesh = false
[../]
[./p3]
type = PointValue
outputs = csv
point = '0 0 3'
variable = porepressure
use_displaced_mesh = false
[../]
[./p4]
type = PointValue
outputs = csv
point = '0 0 4'
variable = porepressure
use_displaced_mesh = false
[../]
[./p5]
type = PointValue
outputs = csv
point = '0 0 5'
variable = porepressure
use_displaced_mesh = false
[../]
[./p6]
type = PointValue
outputs = csv
point = '0 0 6'
variable = porepressure
use_displaced_mesh = false
[../]
[./p7]
type = PointValue
outputs = csv
point = '0 0 7'
variable = porepressure
use_displaced_mesh = false
[../]
[./p8]
type = PointValue
outputs = csv
point = '0 0 8'
variable = porepressure
use_displaced_mesh = false
[../]
[./p9]
type = PointValue
outputs = csv
point = '0 0 9'
variable = porepressure
use_displaced_mesh = false
[../]
[./p99]
type = PointValue
outputs = csv
point = '0 0 10'
variable = porepressure
use_displaced_mesh = false
[../]
[./zdisp]
type = PointValue
outputs = csv
point = '0 0 10'
variable = disp_z
use_displaced_mesh = false
[../]
[./dt]
type = FunctionValuePostprocessor
outputs = console
function = if(0.5*t<0.1,0.5*t,0.1)
[../]
[]
[Preconditioning]
[./andy]
type = SMP
full = true
petsc_options_iname = '-ksp_type -pc_type -snes_atol -snes_rtol -snes_max_it'
petsc_options_value = 'bcgs bjacobi 1E-14 1E-10 10000'
[../]
[]
[Executioner]
type = Transient
solve_type = Newton
start_time = 0
end_time = 10
[./TimeStepper]
type = PostprocessorDT
postprocessor = dt
dt = 0.0001
[../]
[]
[Outputs]
execute_on = 'timestep_end'
file_base = terzaghi_fully_saturated_volume
[./csv]
type = CSV
[../]
[]
```

#### modules/porous_flow/test/tests/poro_elasticity/pp_generation_unconfined_fully_saturated.i

```
# A sample is constrained on all sides, except its top
# and its boundaries are
# also impermeable. Fluid is pumped into the sample via a
# volumetric source (ie kg/second per cubic meter), and the
# rise in the top surface, porepressure, and stress are observed.
#
# In the standard poromechanics scenario, the Biot Modulus is held
# fixed and the source has units 1/time. Then the expected result
# is
# strain_zz = disp_z = BiotCoefficient*BiotModulus*s*t/((bulk + 4*shear/3) + BiotCoefficient^2*BiotModulus)
# porepressure = BiotModulus*(s*t - BiotCoefficient*strain_zz)
# stress_xx = (bulk - 2*shear/3)*strain_zz (remember this is effective stress)
# stress_zz = (bulk + 4*shear/3)*strain_zz (remember this is effective stress)
#
# In porous_flow, however, the source has units kg/s/m^3. The ratios remain
# fixed:
# stress_xx/strain_zz = (bulk - 2*shear/3) = 1 (for the parameters used here)
# stress_zz/strain_zz = (bulk + 4*shear/3) = 4 (for the parameters used here)
# porepressure/strain_zz = 13.3333333 (for the parameters used here)
#
# Expect
# disp_z = 0.3*10*s*t/((2 + 4*1.5/3) + 0.3^2*10) = 0.612245*s*t
# porepressure = 10*(s*t - 0.3*0.612245*s*t) = 8.163265*s*t
# stress_xx = (2 - 2*1.5/3)*0.612245*s*t = 0.612245*s*t
# stress_zz = (2 + 4*shear/3)*0.612245*s*t = 2.44898*s*t
# The relationship between the constant poroelastic source
# s (m^3/second/m^3) and the PorousFlow source, S (kg/second/m^3) is
# S = fluid_density * s = s * exp(porepressure/fluid_bulk)
#
# Finally, note that the volumetric strain has
# consistent_with_displaced_mesh = false
# which is needed when using the FullySaturated version of the Kernels
# in order to generate the above results
[Mesh]
type = GeneratedMesh
dim = 3
nx = 1
ny = 1
nz = 1
xmin = -0.5
xmax = 0.5
ymin = -0.5
ymax = 0.5
zmin = -0.5
zmax = 0.5
[]
[GlobalParams]
displacements = 'disp_x disp_y disp_z'
PorousFlowDictator = dictator
block = 0
[]
[UserObjects]
[./dictator]
type = PorousFlowDictator
porous_flow_vars = 'porepressure disp_x disp_y disp_z'
number_fluid_phases = 1
number_fluid_components = 1
[../]
[]
[Variables]
[./disp_x]
[../]
[./disp_y]
[../]
[./disp_z]
[../]
[./porepressure]
[../]
[]
[BCs]
[./confinex]
type = PresetBC
variable = disp_x
value = 0
boundary = 'left right'
[../]
[./confiney]
type = PresetBC
variable = disp_y
value = 0
boundary = 'bottom top'
[../]
[./confinez]
type = PresetBC
variable = disp_z
value = 0
boundary = 'back'
[../]
[]
[Kernels]
[./grad_stress_x]
type = StressDivergenceTensors
variable = disp_x
component = 0
[../]
[./grad_stress_y]
type = StressDivergenceTensors
variable = disp_y
component = 1
[../]
[./grad_stress_z]
type = StressDivergenceTensors
variable = disp_z
component = 2
[../]
[./poro_x]
type = PorousFlowEffectiveStressCoupling
biot_coefficient = 0.3
variable = disp_x
component = 0
[../]
[./poro_y]
type = PorousFlowEffectiveStressCoupling
biot_coefficient = 0.3
variable = disp_y
component = 1
[../]
[./poro_z]
type = PorousFlowEffectiveStressCoupling
biot_coefficient = 0.3
component = 2
variable = disp_z
[../]
[./mass0]
type = PorousFlowFullySaturatedMassTimeDerivative
variable = porepressure
coupling_type = HydroMechanical
biot_coefficient = 0.3
[../]
[./source]
type = BodyForce
function = '0.1*exp(8.163265306*0.1*t/3.3333333333)'
variable = porepressure
[../]
[]
[AuxVariables]
[./stress_xx]
order = CONSTANT
family = MONOMIAL
[../]
[./stress_xy]
order = CONSTANT
family = MONOMIAL
[../]
[./stress_xz]
order = CONSTANT
family = MONOMIAL
[../]
[./stress_yy]
order = CONSTANT
family = MONOMIAL
[../]
[./stress_yz]
order = CONSTANT
family = MONOMIAL
[../]
[./stress_zz]
order = CONSTANT
family = MONOMIAL
[../]
[]
[AuxKernels]
[./stress_xx]
type = RankTwoAux
rank_two_tensor = stress
variable = stress_xx
index_i = 0
index_j = 0
[../]
[./stress_xy]
type = RankTwoAux
rank_two_tensor = stress
variable = stress_xy
index_i = 0
index_j = 1
[../]
[./stress_xz]
type = RankTwoAux
rank_two_tensor = stress
variable = stress_xz
index_i = 0
index_j = 2
[../]
[./stress_yy]
type = RankTwoAux
rank_two_tensor = stress
variable = stress_yy
index_i = 1
index_j = 1
[../]
[./stress_yz]
type = RankTwoAux
rank_two_tensor = stress
variable = stress_yz
index_i = 1
index_j = 2
[../]
[./stress_zz]
type = RankTwoAux
rank_two_tensor = stress
variable = stress_zz
index_i = 2
index_j = 2
[../]
[]
[Modules]
[./FluidProperties]
[./simple_fluid]
type = SimpleFluidProperties
bulk_modulus = 3.3333333333
density0 = 1
thermal_expansion = 0
[../]
[../]
[]
[Materials]
[./temperature_qp]
type = PorousFlowTemperature
[../]
[./elasticity_tensor]
type = ComputeElasticityTensor
C_ijkl = '1 1.5'
# bulk modulus is lambda + 2*mu/3 = 1 + 2*1.5/3 = 2
fill_method = symmetric_isotropic
[../]
[./strain]
type = ComputeSmallStrain
displacements = 'disp_x disp_y disp_z'
[../]
[./stress]
type = ComputeLinearElasticStress
[../]
[./eff_fluid_pressure]
type = PorousFlowEffectiveFluidPressure
[../]
[./vol_strain]
type = PorousFlowVolumetricStrain
consistent_with_displaced_mesh = false
[../]
[./ppss]
type = PorousFlow1PhaseFullySaturated
porepressure = porepressure
[../]
[./simple_fluid_qp]
type = PorousFlowSingleComponentFluid
fp = simple_fluid
phase = 0
[../]
[./porosity]
type = PorousFlowPorosityConst # the "const" is irrelevant here: all that uses Porosity is the BiotModulus, which just uses the initial value of porosity
porosity = 0.1
[../]
[./biot_modulus]
type = PorousFlowConstantBiotModulus
biot_coefficient = 0.3
fluid_bulk_modulus = 3.3333333333
solid_bulk_compliance = 0.5
[../]
[]
[Postprocessors]
[./p0]
type = PointValue
outputs = csv
point = '0 0 0'
variable = porepressure
[../]
[./zdisp]
type = PointValue
outputs = csv
point = '0 0 0.5'
variable = disp_z
[../]
[./stress_xx]
type = PointValue
outputs = csv
point = '0 0 0'
variable = stress_xx
[../]
[./stress_yy]
type = PointValue
outputs = csv
point = '0 0 0'
variable = stress_yy
[../]
[./stress_zz]
type = PointValue
outputs = csv
point = '0 0 0'
variable = stress_zz
[../]
[./stress_xx_over_strain]
type = FunctionValuePostprocessor
function = stress_xx_over_strain_fcn
outputs = csv
[../]
[./stress_zz_over_strain]
type = FunctionValuePostprocessor
function = stress_zz_over_strain_fcn
outputs = csv
[../]
[./p_over_strain]
type = FunctionValuePostprocessor
function = p_over_strain_fcn
outputs = csv
[../]
[]
[Functions]
[./stress_xx_over_strain_fcn]
type = ParsedFunction
value = a/b
vars = 'a b'
vals = 'stress_xx zdisp'
[../]
[./stress_zz_over_strain_fcn]
type = ParsedFunction
value = a/b
vars = 'a b'
vals = 'stress_zz zdisp'
[../]
[./p_over_strain_fcn]
type = ParsedFunction
value = a/b
vars = 'a b'
vals = 'p0 zdisp'
[../]
[]
[Preconditioning]
[./andy]
type = SMP
full = true
petsc_options_iname = '-ksp_type -pc_type -snes_atol -snes_rtol -snes_max_it'
petsc_options_value = 'bcgs bjacobi 1E-14 1E-10 10000'
[../]
[]
[Executioner]
type = Transient
solve_type = Newton
start_time = 0
end_time = 10
dt = 1
[]
[Outputs]
execute_on = 'timestep_end'
file_base = pp_generation_unconfined_fully_saturated
[./csv]
type = CSV
[../]
[]
```

#### modules/porous_flow/test/tests/poro_elasticity/pp_generation_unconfined_fully_saturated_volume.i

```
# A sample is constrained on all sides, except its top
# and its boundaries are
# also impermeable. Fluid is pumped into the sample via a
# volumetric source (ie m^3/second per cubic meter), and the
# rise in the top surface, porepressure, and stress are observed.
#
# In the standard poromechanics scenario, the Biot Modulus is held
# fixed and the source has units 1/s. Then the expected result
# is
# strain_zz = disp_z = BiotCoefficient*BiotModulus*s*t/((bulk + 4*shear/3) + BiotCoefficient^2*BiotModulus)
# porepressure = BiotModulus*(s*t - BiotCoefficient*strain_zz)
# stress_xx = (bulk - 2*shear/3)*strain_zz (remember this is effective stress)
# stress_zz = (bulk + 4*shear/3)*strain_zz (remember this is effective stress)
#
# In standard porous_flow, everything is based on mass, eg the source has
# units kg/s/m^3. This is discussed in the other pp_generation_unconfined
# models. In this test, we use the FullySaturated Kernel and set
# multiply_by_density = false
# meaning the fluid Kernel has units of volume, and the source, s, has units 1/time
#
# The ratios are:
# stress_xx/strain_zz = (bulk - 2*shear/3) = 1 (for the parameters used here)
# stress_zz/strain_zz = (bulk + 4*shear/3) = 4 (for the parameters used here)
# porepressure/strain_zz = 13.3333333 (for the parameters used here)
#
# Expect
# disp_z = 0.3*10*s*t/((2 + 4*1.5/3) + 0.3^2*10) = 0.612245*s*t
# porepressure = 10*(s*t - 0.3*0.612245*s*t) = 8.163265*s*t
# stress_xx = (2 - 2*1.5/3)*0.612245*s*t = 0.612245*s*t
# stress_zz = (2 + 4*shear/3)*0.612245*s*t = 2.44898*s*t
#
# Finally, note that the volumetric strain has
# consistent_with_displaced_mesh = false
# which is needed when using the FullySaturated version of the Kernels
# in order to generate the above results
[Mesh]
type = GeneratedMesh
dim = 3
nx = 1
ny = 1
nz = 1
xmin = -0.5
xmax = 0.5
ymin = -0.5
ymax = 0.5
zmin = -0.5
zmax = 0.5
[]
[GlobalParams]
displacements = 'disp_x disp_y disp_z'
PorousFlowDictator = dictator
block = 0
[]
[UserObjects]
[./dictator]
type = PorousFlowDictator
porous_flow_vars = 'porepressure disp_x disp_y disp_z'
number_fluid_phases = 1
number_fluid_components = 1
[../]
[]
[Variables]
[./disp_x]
[../]
[./disp_y]
[../]
[./disp_z]
[../]
[./porepressure]
[../]
[]
[BCs]
[./confinex]
type = PresetBC
variable = disp_x
value = 0
boundary = 'left right'
[../]
[./confiney]
type = PresetBC
variable = disp_y
value = 0
boundary = 'bottom top'
[../]
[./confinez]
type = PresetBC
variable = disp_z
value = 0
boundary = 'back'
[../]
[]
[Kernels]
[./grad_stress_x]
type = StressDivergenceTensors
variable = disp_x
component = 0
[../]
[./grad_stress_y]
type = StressDivergenceTensors
variable = disp_y
component = 1
[../]
[./grad_stress_z]
type = StressDivergenceTensors
variable = disp_z
component = 2
[../]
[./poro_x]
type = PorousFlowEffectiveStressCoupling
biot_coefficient = 0.3
variable = disp_x
component = 0
[../]
[./poro_y]
type = PorousFlowEffectiveStressCoupling
biot_coefficient = 0.3
variable = disp_y
component = 1
[../]
[./poro_z]
type = PorousFlowEffectiveStressCoupling
biot_coefficient = 0.3
component = 2
variable = disp_z
[../]
[./mass0]
type = PorousFlowFullySaturatedMassTimeDerivative
variable = porepressure
multiply_by_density = false
coupling_type = HydroMechanical
biot_coefficient = 0.3
[../]
[./source]
type = BodyForce
function = 0.1
variable = porepressure
[../]
[]
[AuxVariables]
[./stress_xx]
order = CONSTANT
family = MONOMIAL
[../]
[./stress_xy]
order = CONSTANT
family = MONOMIAL
[../]
[./stress_xz]
order = CONSTANT
family = MONOMIAL
[../]
[./stress_yy]
order = CONSTANT
family = MONOMIAL
[../]
[./stress_yz]
order = CONSTANT
family = MONOMIAL
[../]
[./stress_zz]
order = CONSTANT
family = MONOMIAL
[../]
[]
[AuxKernels]
[./stress_xx]
type = RankTwoAux
rank_two_tensor = stress
variable = stress_xx
index_i = 0
index_j = 0
[../]
[./stress_xy]
type = RankTwoAux
rank_two_tensor = stress
variable = stress_xy
index_i = 0
index_j = 1
[../]
[./stress_xz]
type = RankTwoAux
rank_two_tensor = stress
variable = stress_xz
index_i = 0
index_j = 2
[../]
[./stress_yy]
type = RankTwoAux
rank_two_tensor = stress
variable = stress_yy
index_i = 1
index_j = 1
[../]
[./stress_yz]
type = RankTwoAux
rank_two_tensor = stress
variable = stress_yz
index_i = 1
index_j = 2
[../]
[./stress_zz]
type = RankTwoAux
rank_two_tensor = stress
variable = stress_zz
index_i = 2
index_j = 2
[../]
[]
[Modules]
[./FluidProperties]
[./simple_fluid]
type = SimpleFluidProperties
bulk_modulus = 3.3333333333
density0 = 1
thermal_expansion = 0
[../]
[../]
[]
[Materials]
[./temperature_qp]
type = PorousFlowTemperature
[../]
[./elasticity_tensor]
type = ComputeElasticityTensor
C_ijkl = '1 1.5'
# bulk modulus is lambda + 2*mu/3 = 1 + 2*1.5/3 = 2
fill_method = symmetric_isotropic
[../]
[./strain]
type = ComputeSmallStrain
displacements = 'disp_x disp_y disp_z'
[../]
[./stress]
type = ComputeLinearElasticStress
[../]
[./eff_fluid_pressure]
type = PorousFlowEffectiveFluidPressure
[../]
[./vol_strain]
type = PorousFlowVolumetricStrain
consistent_with_displaced_mesh = false
[../]
[./ppss]
type = PorousFlow1PhaseFullySaturated
porepressure = porepressure
[../]
[./simple_fluid_qp]
type = PorousFlowSingleComponentFluid
fp = simple_fluid
phase = 0
[../]
[./porosity]
type = PorousFlowPorosityConst # the "const" is irrelevant here: all that uses Porosity is the BiotModulus, which just uses the initial value of porosity
porosity = 0.1
[../]
[./biot_modulus]
type = PorousFlowConstantBiotModulus
biot_coefficient = 0.3
fluid_bulk_modulus = 3.3333333333
solid_bulk_compliance = 0.5
[../]
[]
[Postprocessors]
[./p0]
type = PointValue
outputs = csv
point = '0 0 0'
variable = porepressure
[../]
[./zdisp]
type = PointValue
outputs = csv
point = '0 0 0.5'
variable = disp_z
[../]
[./stress_xx]
type = PointValue
outputs = csv
point = '0 0 0'
variable = stress_xx
[../]
[./stress_yy]
type = PointValue
outputs = csv
point = '0 0 0'
variable = stress_yy
[../]
[./stress_zz]
type = PointValue
outputs = csv
point = '0 0 0'
variable = stress_zz
[../]
[./stress_xx_over_strain]
type = FunctionValuePostprocessor
function = stress_xx_over_strain_fcn
outputs = csv
[../]
[./stress_zz_over_strain]
type = FunctionValuePostprocessor
function = stress_zz_over_strain_fcn
outputs = csv
[../]
[./p_over_strain]
type = FunctionValuePostprocessor
function = p_over_strain_fcn
outputs = csv
[../]
[]
[Functions]
[./stress_xx_over_strain_fcn]
type = ParsedFunction
value = a/b
vars = 'a b'
vals = 'stress_xx zdisp'
[../]
[./stress_zz_over_strain_fcn]
type = ParsedFunction
value = a/b
vars = 'a b'
vals = 'stress_zz zdisp'
[../]
[./p_over_strain_fcn]
type = ParsedFunction
value = a/b
vars = 'a b'
vals = 'p0 zdisp'
[../]
[]
[Preconditioning]
[./andy]
type = SMP
full = true
petsc_options_iname = '-ksp_type -pc_type -snes_atol -snes_rtol -snes_max_it'
petsc_options_value = 'bcgs bjacobi 1E-14 1E-10 10000'
[../]
[]
[Executioner]
type = Transient
solve_type = Newton
start_time = 0
end_time = 10
dt = 1
[]
[Outputs]
execute_on = 'timestep_end'
file_base = pp_generation_unconfined_fully_saturated_volume
[./csv]
type = CSV
[../]
[]
```

#### modules/porous_flow/test/tests/poro_elasticity/mandel_fully_saturated_volume.i

```
# Mandel's problem of consolodation of a drained medium
# Using the FullySaturatedDarcyBase and FullySaturatedFullySaturatedMassTimeDerivative kernels
# with multiply_by_density = false, so that this problem becomes linear
# Note the use of consistent_with_displaced_mesh = false in the calculation of volumetric strain
#
# A sample is in plane strain.
# -a <= x <= a
# -b <= y <= b
# It is squashed with constant force by impermeable, frictionless plattens on its top and bottom surfaces (at y=+/-b)
# Fluid is allowed to leak out from its sides (at x=+/-a)
# The porepressure within the sample is monitored.
#
# As is common in the literature, this is simulated by
# considering the quarter-sample, 0<=x<=a and 0<=y<=b, with
# impermeable, roller BCs at x=0 and y=0 and y=b.
# Porepressure is fixed at zero on x=a.
# Porepressure and displacement are initialised to zero.
# Then the top (y=b) is moved downwards with prescribed velocity,
# so that the total force that is inducing this downwards velocity
# is fixed. The velocity is worked out by solving Mandel's problem
# analytically, and the total force is monitored in the simulation
# to check that it indeed remains constant.
#
# Here are the problem's parameters, and their values:
# Soil width. a = 1
# Soil height. b = 0.1
# Soil's Lame lambda. la = 0.5
# Soil's Lame mu, which is also the Soil's shear modulus. mu = G = 0.75
# Soil bulk modulus. K = la + 2*mu/3 = 1
# Drained Poisson ratio. nu = (3K - 2G)/(6K + 2G) = 0.2
# Soil bulk compliance. 1/K = 1
# Fluid bulk modulus. Kf = 8
# Fluid bulk compliance. 1/Kf = 0.125
# Soil initial porosity. phi0 = 0.1
# Biot coefficient. alpha = 0.6
# Biot modulus. M = 1/(phi0/Kf + (alpha - phi0)(1 - alpha)/K) = 4.705882
# Undrained bulk modulus. Ku = K + alpha^2*M = 2.694118
# Undrained Poisson ratio. nuu = (3Ku - 2G)/(6Ku + 2G) = 0.372627
# Skempton coefficient. B = alpha*M/Ku = 1.048035
# Fluid mobility (soil permeability/fluid viscosity). k = 1.5
# Consolidation coefficient. c = 2*k*B^2*G*(1-nu)*(1+nuu)^2/9/(1-nuu)/(nuu-nu) = 3.821656
# Normal stress on top. F = 1
#
# The solution for porepressure and displacements is given in
# AHD Cheng and E Detournay "A direct boundary element method for plane strain poroelasticity" International Journal of Numerical and Analytical Methods in Geomechanics 12 (1988) 551-572.
# The solution involves complicated infinite series, so I shall not write it here
[Mesh]
type = GeneratedMesh
dim = 3
nx = 10
ny = 1
nz = 1
xmin = 0
xmax = 1
ymin = 0
ymax = 0.1
zmin = 0
zmax = 1
[]
[GlobalParams]
displacements = 'disp_x disp_y disp_z'
PorousFlowDictator = dictator
block = 0
[]
[UserObjects]
[./dictator]
type = PorousFlowDictator
porous_flow_vars = 'porepressure disp_x disp_y disp_z'
number_fluid_phases = 1
number_fluid_components = 1
[../]
[]
[Variables]
[./disp_x]
[../]
[./disp_y]
[../]
[./disp_z]
[../]
[./porepressure]
[../]
[]
[BCs]
[./roller_xmin]
type = PresetBC
variable = disp_x
value = 0
boundary = 'left'
[../]
[./roller_ymin]
type = PresetBC
variable = disp_y
value = 0
boundary = 'bottom'
[../]
[./plane_strain]
type = PresetBC
variable = disp_z
value = 0
boundary = 'back front'
[../]
[./xmax_drained]
type = DirichletBC
variable = porepressure
value = 0
boundary = right
[../]
[./top_velocity]
type = FunctionPresetBC
variable = disp_y
function = top_velocity
boundary = top
[../]
[]
[Functions]
[./top_velocity]
type = PiecewiseLinear
x = '0 0.002 0.006 0.014 0.03 0.046 0.062 0.078 0.094 0.11 0.126 0.142 0.158 0.174 0.19 0.206 0.222 0.238 0.254 0.27 0.286 0.302 0.318 0.334 0.35 0.366 0.382 0.398 0.414 0.43 0.446 0.462 0.478 0.494 0.51 0.526 0.542 0.558 0.574 0.59 0.606 0.622 0.638 0.654 0.67 0.686 0.702'
y = '-0.041824842 -0.042730269 -0.043412712 -0.04428867 -0.045509181 -0.04645965 -0.047268246 -0.047974749 -0.048597109 -0.0491467 -0.049632388 -0.050061697 -0.050441198 -0.050776675 -0.051073238 -0.0513354 -0.051567152 -0.051772022 -0.051953128 -0.052113227 -0.052254754 -0.052379865 -0.052490464 -0.052588233 -0.052674662 -0.052751065 -0.052818606 -0.052878312 -0.052931093 -0.052977751 -0.053018997 -0.053055459 -0.053087691 -0.053116185 -0.053141373 -0.05316364 -0.053183324 -0.053200724 -0.053216106 -0.053229704 -0.053241725 -0.053252351 -0.053261745 -0.053270049 -0.053277389 -0.053283879 -0.053289615'
[../]
[]
[AuxVariables]
[./stress_yy]
order = CONSTANT
family = MONOMIAL
[../]
[./tot_force]
order = CONSTANT
family = MONOMIAL
[../]
[]
[AuxKernels]
[./stress_yy]
type = RankTwoAux
rank_two_tensor = stress
variable = stress_yy
index_i = 1
index_j = 1
[../]
[./tot_force]
type = ParsedAux
args = 'stress_yy porepressure'
execute_on = timestep_end
variable = tot_force
function = '-stress_yy+0.6*porepressure'
[../]
[]
[Kernels]
[./grad_stress_x]
type = StressDivergenceTensors
variable = disp_x
component = 0
[../]
[./grad_stress_y]
type = StressDivergenceTensors
variable = disp_y
component = 1
[../]
[./grad_stress_z]
type = StressDivergenceTensors
variable = disp_z
component = 2
[../]
[./poro_x]
type = PorousFlowEffectiveStressCoupling
biot_coefficient = 0.6
variable = disp_x
component = 0
[../]
[./poro_y]
type = PorousFlowEffectiveStressCoupling
biot_coefficient = 0.6
variable = disp_y
component = 1
[../]
[./poro_z]
type = PorousFlowEffectiveStressCoupling
biot_coefficient = 0.6
component = 2
variable = disp_z
[../]
[./mass0]
type = PorousFlowFullySaturatedMassTimeDerivative
biot_coefficient = 0.6
multiply_by_density = false
coupling_type = HydroMechanical
variable = porepressure
[../]
[./flux]
type = PorousFlowFullySaturatedDarcyBase
multiply_by_density = false
variable = porepressure
gravity = '0 0 0'
[../]
[]
[Modules]
[./FluidProperties]
[./simple_fluid]
type = SimpleFluidProperties
bulk_modulus = 8
density0 = 1
thermal_expansion = 0
viscosity = 1
[../]
[../]
[]
[Materials]
[./temperature]
type = PorousFlowTemperature
[../]
[./elasticity_tensor]
type = ComputeElasticityTensor
C_ijkl = '0.5 0.75'
# bulk modulus is lambda + 2*mu/3 = 0.5 + 2*0.75/3 = 1
fill_method = symmetric_isotropic
[../]
[./strain]
type = ComputeSmallStrain
[../]
[./stress]
type = ComputeLinearElasticStress
[../]
[./eff_fluid_pressure_qp]
type = PorousFlowEffectiveFluidPressure
[../]
[./vol_strain]
type = PorousFlowVolumetricStrain
consistent_with_displaced_mesh = false
[../]
[./ppss]
type = PorousFlow1PhaseFullySaturated
porepressure = porepressure
[../]
[./massfrac]
type = PorousFlowMassFraction
[../]
[./simple_fluid_qp]
type = PorousFlowSingleComponentFluid
fp = simple_fluid
phase = 0
[../]
[./porosity]
type = PorousFlowPorosityConst # only the initial value of this is ever used
porosity = 0.1
[../]
[./biot_modulus]
type = PorousFlowConstantBiotModulus
biot_coefficient = 0.6
solid_bulk_compliance = 1
fluid_bulk_modulus = 8
[../]
[./permeability]
type = PorousFlowPermeabilityConst
permeability = '1.5 0 0 0 1.5 0 0 0 1.5'
[../]
[]
[Postprocessors]
[./p0]
type = PointValue
outputs = csv
point = '0.0 0 0'
variable = porepressure
[../]
[./p1]
type = PointValue
outputs = csv
point = '0.1 0 0'
variable = porepressure
[../]
[./p2]
type = PointValue
outputs = csv
point = '0.2 0 0'
variable = porepressure
[../]
[./p3]
type = PointValue
outputs = csv
point = '0.3 0 0'
variable = porepressure
[../]
[./p4]
type = PointValue
outputs = csv
point = '0.4 0 0'
variable = porepressure
[../]
[./p5]
type = PointValue
outputs = csv
point = '0.5 0 0'
variable = porepressure
[../]
[./p6]
type = PointValue
outputs = csv
point = '0.6 0 0'
variable = porepressure
[../]
[./p7]
type = PointValue
outputs = csv
point = '0.7 0 0'
variable = porepressure
[../]
[./p8]
type = PointValue
outputs = csv
point = '0.8 0 0'
variable = porepressure
[../]
[./p9]
type = PointValue
outputs = csv
point = '0.9 0 0'
variable = porepressure
[../]
[./p99]
type = PointValue
outputs = csv
point = '1 0 0'
variable = porepressure
[../]
[./xdisp]
type = PointValue
outputs = csv
point = '1 0.1 0'
variable = disp_x
[../]
[./ydisp]
type = PointValue
outputs = csv
point = '1 0.1 0'
variable = disp_y
[../]
[./total_downwards_force]
type = ElementAverageValue
outputs = csv
variable = tot_force
[../]
[./dt]
type = FunctionValuePostprocessor
outputs = console
function = if(0.15*t<0.01,0.15*t,0.01)
[../]
[]
[Preconditioning]
[./andy]
type = SMP
full = true
petsc_options_iname = '-ksp_type -pc_type -sub_pc_type -snes_atol -snes_rtol -snes_max_it'
petsc_options_value = 'gmres asm lu 1E-14 1E-10 10000'
[../]
[]
[Executioner]
type = Transient
solve_type = Newton
start_time = 0
end_time = 0.7
[./TimeStepper]
type = PostprocessorDT
postprocessor = dt
dt = 0.001
[../]
[]
[Outputs]
execute_on = 'timestep_end'
file_base = mandel_fully_saturated_volume
[./csv]
interval = 3
type = CSV
[../]
[]
```