Point and line sources and sinks

This page may be read in conjunction with the theory behind Dirac Kernels in PorousFlow.

Geometric tests

The test suite contains many core tests that demonstrate:

  • when a point sink is placed at a node, it withdraws fluid (or heat) from only that node, for instance

# fully-saturated situation with a poly-line sink at one
# of the nodes.  Because there is no fluid flow, the
# other nodes should not experience any change in
# porepressure.
# The poly-line sink has length=2 and weight=0.1, and
# extracts fluid at a constant rate of 1 kg.m^-1.s^-1.
# Therefore, in 1 second it will have extracted a total
# of 0.2 kg.
# The porosity is 0.1, and the elemental volume is 2,
# so the fluid mass at the node in question = 0.2 * density / 4,
# where the 4 is the number of nodes in the element.
# In this simulation density = dens0 * exp(P / bulk), with
# dens0 = 100, and bulk = 20 MPa.
# The initial porepressure P0 = 10 MPa, so the final (after
# 1 second of simulation) is
# P(t=1) = 0.950879 MPa
#
[Mesh]
  type = GeneratedMesh
  dim = 2
  nx = 1
  ny = 1
  xmin = 0
  xmax = 2
  ymin = 0
  ymax = 1
[]

[GlobalParams]
  PorousFlowDictator = dictator
[]

[Variables]
  [./pp]
    initial_condition = 1E7
  [../]
[]

[Kernels]
  [./mass0]
    type = PorousFlowMassTimeDerivative
    fluid_component = 0
    variable = pp
  [../]
[]

[UserObjects]
  [./dictator]
    type = PorousFlowDictator
    porous_flow_vars = 'pp'
    number_fluid_phases = 1
    number_fluid_components = 1
  [../]
  [./pls_total_outflow_mass]
    type = PorousFlowSumQuantity
  [../]
  [./pc]
    type = PorousFlowCapillaryPressureVG
    m = 0.5
    alpha = 1e-7
  [../]
[]

[Modules]
  [./FluidProperties]
    [./simple_fluid]
      type = SimpleFluidProperties
      bulk_modulus = 2e7
      density0 = 100
      thermal_expansion = 0
    [../]
  [../]
[]

[Materials]
  [./temperature]
    type = PorousFlowTemperature
  [../]
  [./ppss]
    type = PorousFlow1PhaseP
    porepressure = pp
    capillary_pressure = pc
  [../]
  [./ppss_qp]
    type = PorousFlow1PhaseP
    at_nodes = false
    porepressure = pp
    capillary_pressure = pc
  [../]
  [./massfrac]
    type = PorousFlowMassFraction
  [../]
  [./simple_fluid]
    type = PorousFlowSingleComponentFluid
    fp = simple_fluid
    phase = 0
  [../]
  [./porosity]
    type = PorousFlowPorosityConst
    porosity = 0.1
  [../]
[]

[DiracKernels]
  [./pls]
    type = PorousFlowPolyLineSink
    fluid_phase = 0
    point_file = pls01_21.bh
    line_length = 2
    SumQuantityUO = pls_total_outflow_mass
    variable = pp
    p_or_t_vals = '0 1E7'
    fluxes = '1 1'
  [../]
[]

[Postprocessors]
  [./pls_report]
    type = PorousFlowPlotQuantity
    uo = pls_total_outflow_mass
  [../]

  [./fluid_mass0]
    type = PorousFlowFluidMass
    execute_on = timestep_begin
  [../]

  [./fluid_mass1]
    type = PorousFlowFluidMass
    execute_on = timestep_end
  [../]

  [./zmass_error]
    type = FunctionValuePostprocessor
    function = mass_bal_fcn
    execute_on = timestep_end
  [../]

  [./p00]
    type = PointValue
    variable = pp
    point = '0 0 0'
    execute_on = timestep_end
  [../]
  [./p01]
    type = PointValue
    variable = pp
    point = '0 1 0'
    execute_on = timestep_end
  [../]
  [./p20]
    type = PointValue
    variable = pp
    point = '2 0 0'
    execute_on = timestep_end
  [../]
  [./p21]
    type = PointValue
    variable = pp
    point = '2 1 0'
    execute_on = timestep_end
  [../]
[]

[Functions]
  [./mass_bal_fcn]
    type = ParsedFunction
    value = abs((a-c+d)/2/(a+c))
    vars = 'a c d'
    vals = 'fluid_mass1 fluid_mass0 pls_report'
  [../]
[]

[Preconditioning]
  [./usual]
    type = SMP
    full = true
    petsc_options = '-snes_converged_reason'
    petsc_options_iname = '-ksp_type -pc_type -snes_atol -snes_rtol -snes_max_it -ksp_max_it'
    petsc_options_value = 'bcgs bjacobi 1E-10 1E-10 10000 30'
  [../]
[]

[Executioner]
  type = Transient
  end_time = 1
  dt = 1
  solve_type = NEWTON
[]

[Outputs]
  file_base = pls01
  exodus = false
  csv = true
  execute_on = timestep_end
[]
(modules/porous_flow/test/tests/dirackernels/pls01.i)
  • when a point sink that is proportional to mobility (or relative permeability, etc) is placed in an element where some nodes have zero mobility (or relative permeability, etc), then fluid (or heat) is not extracted from those nodes, for instance

# Test that the upwinding works correctly.
#
# A poly-line sink sits at the centre of the element.
# It has length=4 and weight=0.5, and extracts fluid
# at a constant rate of
# (1 * relative_permeability) kg.m^-1.s^-1
# Since it sits at the centre of the element, it extracts
# equally from each node, so the rate of extraction from
# each node is
# (0.5 * relative_permeability) kg.s^-1
# including the length and weight effects.
#
# There is no fluid flow.
#
# The initial conditions are such that all nodes have
# relative_permeability=0, except for one which has
# relative_permeaility = 1.  Therefore, all nodes should
# remain at their initial porepressure, except the one.
#
# The porosity is 0.1, and the elemental volume is 2,
# so the fluid mass at the node in question = 0.2 * density / 4,
# where the 4 is the number of nodes in the element.
# In this simulation density = dens0 * exp(P / bulk), with
# dens0 = 100, and bulk = 20 MPa.
# The initial porepressure P0 = 10 MPa, so the final (after
# 1 second of simulation) is
# P(t=1) = 8.748592 MPa
[Mesh]
  type = GeneratedMesh
  dim = 2
  nx = 1
  ny = 1
  xmin = 0
  xmax = 2
  ymin = 0
  ymax = 1
[]

[GlobalParams]
  PorousFlowDictator = dictator
[]

[Variables]
  [./pp]
  [../]
[]

[ICs]
  [./pp]
    type = FunctionIC
    variable = pp
    #function = if((x<1)&(y<0.5),1E7,-1E7)
    function = if((x<1)&(y>0.5),1E7,-1E7)
    #function = if((x>1)&(y<0.5),1E7,-1E7)
    #function = if((x>1)&(y>0.5),1E7,-1E7)
  [../]
[]

[Kernels]
  [./mass0]
    type = PorousFlowMassTimeDerivative
    fluid_component = 0
    variable = pp
  [../]
[]

[UserObjects]
  [./dictator]
    type = PorousFlowDictator
    porous_flow_vars = 'pp'
    number_fluid_phases = 1
    number_fluid_components = 1
  [../]
  [./pls_total_outflow_mass]
    type = PorousFlowSumQuantity
  [../]
  [./pc]
    type = PorousFlowCapillaryPressureVG
    m = 0.5
    alpha = 1e-7
  [../]
[]

[Modules]
  [./FluidProperties]
    [./simple_fluid]
      type = SimpleFluidProperties
      bulk_modulus = 2e7
      density0 = 100
      thermal_expansion = 0
    [../]
  [../]
[]

[Materials]
  [./temperature]
    type = PorousFlowTemperature
  [../]
  [./ppss]
    type = PorousFlow1PhaseP
    porepressure = pp
    capillary_pressure = pc
  [../]
  [./ppss_qp]
    type = PorousFlow1PhaseP
    at_nodes = false
    porepressure = pp
    capillary_pressure = pc
  [../]
  [./massfrac]
    type = PorousFlowMassFraction
  [../]
  [./simple_fluid]
    type = PorousFlowSingleComponentFluid
    fp = simple_fluid
    phase = 0
  [../]
  [./porosity]
    type = PorousFlowPorosityConst
    porosity = 0.1
  [../]
  [./relperm]
    type = PorousFlowRelativePermeabilityFLAC
    at_nodes = true
    phase = 0
    m = 2
    s_res = 0.99
    sum_s_res = 0.99
  [../]
[]

[DiracKernels]
  [./pls]
    type = PorousFlowPolyLineSink
    fluid_phase = 0
    point_file = pls03.bh
    use_relative_permeability = true
    line_length = 4
    SumQuantityUO = pls_total_outflow_mass
    variable = pp
    p_or_t_vals = '0 1E7'
    fluxes = '1 1'
  [../]
[]

[Postprocessors]
  [./pls_report]
    type = PorousFlowPlotQuantity
    uo = pls_total_outflow_mass
  [../]

  [./fluid_mass0]
    type = PorousFlowFluidMass
    execute_on = timestep_begin
  [../]

  [./fluid_mass1]
    type = PorousFlowFluidMass
    execute_on = timestep_end
  [../]

  [./zmass_error]
    type = FunctionValuePostprocessor
    function = mass_bal_fcn
    execute_on = timestep_end
  [../]

  [./p00]
    type = PointValue
    variable = pp
    point = '0 0 0'
    execute_on = timestep_end
  [../]
  [./p01]
    type = PointValue
    variable = pp
    point = '0 1 0'
    execute_on = timestep_end
  [../]
  [./p20]
    type = PointValue
    variable = pp
    point = '2 0 0'
    execute_on = timestep_end
  [../]
  [./p21]
    type = PointValue
    variable = pp
    point = '2 1 0'
    execute_on = timestep_end
  [../]
[]

[Functions]
  [./mass_bal_fcn]
    type = ParsedFunction
    value = abs((a-c+d)/2/(a+c))
    vars = 'a c d'
    vals = 'fluid_mass1 fluid_mass0 pls_report'
  [../]
[]

[Preconditioning]
  [./usual]
    type = SMP
    full = true
    petsc_options = '-snes_converged_reason'
    petsc_options_iname = '-ksp_type -pc_type -snes_atol -snes_rtol -snes_max_it -ksp_max_it'
    petsc_options_value = 'bcgs bjacobi 1E-10 1E-10 10000 30'
  [../]
[]

[Executioner]
  type = Transient
  end_time = 1
  dt = 1
  solve_type = NEWTON
[]

[Outputs]
  file_base = pls03
  exodus = false
  csv = true
  execute_on = timestep_end
[]
(modules/porous_flow/test/tests/dirackernels/pls03.i)

Basic point sources/sinks

The following input file extracts fluid at the rate (kg.s) (1)

# Test PorousFlowSquarePulsePointSource DiracKernel

[Mesh]
  type = GeneratedMesh
  dim = 2
  bias_x = 1.1
  bias_y = 1.1
  ymax = 1
  xmax = 1
[]

[GlobalParams]
  PorousFlowDictator = dictator
[]

[Variables]
  [./pp]
  [../]
[]

[Kernels]
  [./mass0]
    type = PorousFlowMassTimeDerivative
    fluid_component = 0
    variable = pp
  [../]
[]

[UserObjects]
  [./dictator]
    type = PorousFlowDictator
    porous_flow_vars = pp
    number_fluid_phases = 1
    number_fluid_components = 1
  [../]
  [./pc]
    type = PorousFlowCapillaryPressureVG
    m = 0.5
    alpha = 1e-7
  [../]
[]

[Modules]
  [./FluidProperties]
    [./simple_fluid]
      type = SimpleFluidProperties
      bulk_modulus = 2e9
      density0 = 1000
      thermal_expansion = 0
    [../]
  [../]
[]

[Materials]
  [./temperature]
    type = PorousFlowTemperature
  [../]
  [./ppss]
    type = PorousFlow1PhaseP
    porepressure = pp
    capillary_pressure = pc
  [../]
  [./massfrac]
    type = PorousFlowMassFraction
  [../]
  [./simple_fluid]
    type = PorousFlowSingleComponentFluid
    fp = simple_fluid
    phase = 0
  [../]
  [./porosity]
    type = PorousFlowPorosityConst
    porosity = 0.2
  [../]
[]

[Postprocessors]
  [./total_mass]
    type = PorousFlowFluidMass
    execute_on = 'initial timestep_end'
  [../]
[]

[Preconditioning]
  [./smp]
    type = SMP
    full = true
  [../]
[]

[Executioner]
  type = Transient
  solve_type = Newton
  nl_abs_tol = 1e-14
  dt = 200
  end_time = 2000
[]

[Outputs]
  perf_graph = true
  file_base = squarepulse1
  csv = true
  execute_on = 'initial timestep_end'
  [./con]
    output_linear = true
    type = Console
  [../]
[]

[ICs]
  [./PressureIC]
    variable = pp
    type = ConstantIC
    value = 20e6
  [../]
[]

[DiracKernels]
  [./sink1]
    type = PorousFlowSquarePulsePointSource
    start_time = 100
    end_time = 300
    point = '0.5 0.5 0'
    mass_flux = -0.1
    variable = pp
  [../]
  [./sink]
    type = PorousFlowSquarePulsePointSource
    start_time = 600
    end_time = 1400
    point = '0.5 0.5 0'
    mass_flux = -0.1
    variable = pp
  [../]
  [./source]
    point = '0.5 0.5 0'
    start_time = 1500
    mass_flux = 0.2
    end_time = 2000
    variable = pp
    type = PorousFlowSquarePulsePointSource
  [../]
[]
(modules/porous_flow/test/tests/dirackernels/squarepulse1.i)

MOOSE produces the expected result:

Figure 1: Results of a "square-pulsed" extraction and injection of fluid from a fluid-flow simulation.

Theis tests

In the fully-saturated, isothermal case with no mechanical coupling and constant, large fluid bulk modulus, the fluid flow equations reduce to the form conventionally used in groundwater flow: (2) where

  • is the "head", . Here is the fluid density.

  • is the "specific storage". If the rock compressibility is ignored then , where is the fluid bulk modulus

  • is the hydraulic conductivity: , where is the permeability and is the fluid viscosity

This equation is identical to the physics described by the PorousFlowFullySaturatedMassTimeDerivative and the PorousFlowFullySaturatedDarcyBase Kernels, which may be used explicitly, or through the PorousFlowBasicTHM Action. When using these, the specific storage in the above equation is replaced by (reciprocal of the Biot modulus), and by , and by . Finally, remember that this formulation is volume based, not mass based like the remainder of PorousFlow.

Place a constant volumetric sink of strength (m.m.s) acting along an infinite line in an isotropic 3D medium. The situation is therefore two dimensional. Theis provided the solution for the head (3) which is frequently used in the groundwater literature. "Ei" is the exponential integral function, with values for small (where is the Euler number ), and for large .

This is checked using a number of PorousFlow tests (the extraction specified using mass rate, with volume rate, and from single and 2-phase systems). For instance:

# Theis problem: Flow to single sink using BasicTHM
# SinglePhase
# RZ mesh

[Mesh]
  type = GeneratedMesh
  dim = 1
  nx = 20
  xmax = 100
  bias_x = 1.05
[]

[Problem]
  coord_type = RZ
[]

[GlobalParams]
  PorousFlowDictator = dictator
[]

[Variables]
  [./pp]
    initial_condition = 20E6
  [../]
[]

[PorousFlowBasicTHM]
  dictator_name = dictator
  add_darcy_aux = false
  fp = simple_fluid
  gravity = '0 0 0'
  multiply_by_density = false
  porepressure = pp
[]

[Modules]
  [./FluidProperties]
    [./simple_fluid]
      type = SimpleFluidProperties
      viscosity = 0.001
    [../]
  [../]
[]

[Materials]
  [./porosity]
    type = PorousFlowPorosityConst
    porosity = 0.05
  [../]
  [./biot_mod]
    type = PorousFlowConstantBiotModulus
    fluid_bulk_modulus = 2E9
    biot_coefficient = 1.0
  [../]
  [./permeability]
    type = PorousFlowPermeabilityConst
    permeability = '1E-14 0 0 0 1E-14 0 0 0 1E-14'
  [../]
[]

[DiracKernels]
  [./sink]
    type = PorousFlowSquarePulsePointSource
    point = '0 0 0'
    mass_flux = -0.16E-3 # recall this is a volumetric flux because multiply_by_density = false in the Action, so this corresponds to a mass_flux of 0.16 kg/s/m because density=1000
    variable = pp
  [../]
[]

[VectorPostprocessors]
  [./pp]
    type = LineValueSampler
    num_points = 25
    start_point = '0 0 0'
    end_point = '100 0 0'
    sort_by = x
    variable = pp
  [../]
[]

[Preconditioning]
  [./smp]
    type = SMP
    full = true
  [../]
[]

[Executioner]
  type = Transient
  solve_type = Newton
  dt = 200
  end_time = 1E3
  nl_abs_tol = 1e-10
[]

[Outputs]
  perf_graph = true
  [./csv]
    type = CSV
    execute_on = final
  [../]
[]
(modules/porous_flow/test/tests/dirackernels/theis_rz.i)

MOOSE agrees with the Theis solution:

Figure 2: Results of the fully-saturated, single-phase Theis simulation.

Flow from a source in a radial 1D problem admits a similarity solution, where the fluid pressure (and saturation, mass fraction, etc) is a function of the similarity variable . This may be observed in Eq. (3). Using this fact, two-phase immiscible flow may be tested. The following input file describes injection of a gas phase into a fully liquid-saturated model at a constant rate:

# Two phase Theis problem: Flow from single source
# Constant rate injection 0.5 kg/s
# 1D cylindrical mesh

[Mesh]
  type = GeneratedMesh
  dim = 1
  nx = 100
  xmax = 2000
  bias_x = 1.05
[]

[Problem]
  type = FEProblem
  coord_type = RZ
  rz_coord_axis = Y
[]

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

[Variables]
  [./ppwater]
    initial_condition = 20e6
  [../]
  [./sgas]
    initial_condition = 0
  [../]
[]

[AuxVariables]
  [./massfrac_ph0_sp0]
    initial_condition = 1
  [../]
  [./massfrac_ph1_sp0]
    initial_condition = 0
  [../]
[]

[Kernels]
  [./mass0]
    type = PorousFlowMassTimeDerivative
    fluid_component = 0
    variable = ppwater
  [../]
  [./flux0]
    type = PorousFlowAdvectiveFlux
    fluid_component = 0
    variable = ppwater
  [../]
  [./mass1]
    type = PorousFlowMassTimeDerivative
    fluid_component = 1
    variable = sgas
  [../]
  [./flux1]
    type = PorousFlowAdvectiveFlux
    fluid_component = 1
    variable = sgas
  [../]
[]

[UserObjects]
  [./dictator]
    type = PorousFlowDictator
    porous_flow_vars = 'ppwater sgas'
    number_fluid_phases = 2
    number_fluid_components = 2
  [../]
  [./pc]
    type = PorousFlowCapillaryPressureConst
    pc = 1e5
  [../]
[]

[Modules]
  [./FluidProperties]
    [./simple_fluid0]
      type = SimpleFluidProperties
      bulk_modulus = 2e9
      density0 = 1000
      viscosity = 1e-3
      thermal_expansion = 0
    [../]
    [./simple_fluid1]
      type = SimpleFluidProperties
      bulk_modulus = 2e9
      density0 = 10
      viscosity = 1e-4
      thermal_expansion = 0
    [../]
  [../]
[]

[Materials]
  [./temperature]
    type = PorousFlowTemperature
  [../]
  [./ppss]
    type = PorousFlow2PhasePS
    phase0_porepressure = ppwater
    phase1_saturation = sgas
    capillary_pressure = pc
  [../]
  [./massfrac]
    type = PorousFlowMassFraction
    mass_fraction_vars = 'massfrac_ph0_sp0 massfrac_ph1_sp0'
  [../]
  [./simple_fluid0]
    type = PorousFlowSingleComponentFluid
    fp = simple_fluid0
    phase = 0
    compute_enthalpy = false
    compute_internal_energy = false
  [../]
  [./simple_fluid1]
    type = PorousFlowSingleComponentFluid
    fp = simple_fluid1
    phase = 1
    compute_enthalpy = false
    compute_internal_energy = false
  [../]
  [./porosity]
    type = PorousFlowPorosityConst
    porosity = 0.2
  [../]
  [./permeability]
    type = PorousFlowPermeabilityConst
    permeability = '1e-12 0 0 0 1e-12 0 0 0 1e-12'
  [../]
  [./relperm_water]
    type = PorousFlowRelativePermeabilityCorey
    n = 1
    phase = 0
  [../]
  [./relperm_gas]
    type = PorousFlowRelativePermeabilityCorey
    n = 1
    phase = 1
  [../]
[]

[BCs]
  [./rightwater]
    type = DirichletBC
    boundary = right
    value = 20e6
    variable = ppwater
  [../]
[]

[DiracKernels]
  [./source]
    type = PorousFlowSquarePulsePointSource
    point = '0 0 0'
    mass_flux = 0.5
    variable = sgas
  [../]
[]

[Preconditioning]
  [./smp]
    type = SMP
    full = true
    petsc_options = '-snes_converged_reason -ksp_diagonal_scale -ksp_diagonal_scale_fix -ksp_gmres_modifiedgramschmidt -snes_linesearch_monitor'
    petsc_options_iname = '-ksp_type -pc_type -sub_pc_type -sub_pc_factor_shift_type -pc_asm_overlap -snes_atol -snes_rtol -snes_max_it'
    petsc_options_value = 'gmres      asm      lu           NONZERO                   2               1E-8       1E-10 20'
  [../]
[]

[Executioner]
  type = Transient
  solve_type = Newton
  end_time = 1e4
  [./TimeStepper]
    type = IterationAdaptiveDT
    dt = 10
    growth_factor = 2
  [../]
[]

[VectorPostprocessors]
  [./line]
    type = NodalValueSampler
    sort_by = x
    variable = 'ppwater sgas'
    execute_on = 'timestep_end'
  [../]
[]

[Postprocessors]
  [./ppwater]
    type = PointValue
    point =  '4 0 0'
    variable = ppwater
  [../]
  [./sgas]
    type = PointValue
    point = '4 0 0'
    variable = sgas
  [../]
  [./massgas]
    type = PorousFlowFluidMass
    fluid_component = 1
  [../]
[]

[Outputs]
  file_base = theis3
  print_linear_residuals = false
  perf_graph = true
  [./csv]
    type = CSV
    execute_on = timestep_end
    execute_vector_postprocessors_on = final
  [../]
[]
(modules/porous_flow/test/tests/dirackernels/theis3.i)

Figure 3 shows the comparison of similarity solutions calculated with either fixed radial distance or fixed time. In this case, good agreement is observed between the two results for both liquid pressure and gas saturation.

Figure 3: Results of the 2-phase radial injection simulation.

Further tests of this kind, involving multi-component, multi-phase flow, including mutual dissolution of gas and liquid phases may be found here.

Peaceman borehole fluxes

The test suite that checks that the Peaceman flux (4) is correctly implemented. A vertical borehole is placed through the centre of a single element, and fluid flow to the borehole as a function of porepressure is measured. The tests are

  • A production borehole with , with a fully-saturated medium.

# fully-saturated
# production
[Mesh]
  type = GeneratedMesh
  dim = 3
  nx = 1
  ny = 1
  nz = 1
  xmin = -1
  xmax = 1
  ymin = -1
  ymax = 1
  zmin = -1
  zmax = 1
[]

[GlobalParams]
  PorousFlowDictator = dictator
[]

[Variables]
  [./pp]
    initial_condition = 1E7
  [../]
[]

[Kernels]
  [./mass0]
    type = PorousFlowMassTimeDerivative
    fluid_component = 0
    variable = pp
  [../]
[]

[UserObjects]
  [./borehole_total_outflow_mass]
    type = PorousFlowSumQuantity
  [../]
  [./dictator]
    type = PorousFlowDictator
    porous_flow_vars = 'pp'
    number_fluid_phases = 1
    number_fluid_components = 1
  [../]
  [./pc]
    type = PorousFlowCapillaryPressureVG
    m = 0.5
    alpha = 1e-7
  [../]
[]

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

[Materials]
  [./temperature]
    type = PorousFlowTemperature
  [../]
  [./ppss]
    type = PorousFlow1PhaseP
    porepressure = pp
    capillary_pressure = pc
  [../]
  [./ppss_qp]
    type = PorousFlow1PhaseP
    at_nodes = false
    porepressure = pp
    capillary_pressure = pc
  [../]
  [./massfrac]
    type = PorousFlowMassFraction
  [../]
  [./simple_fluid]
    type = PorousFlowSingleComponentFluid
    fp = simple_fluid
    phase = 0
  [../]
  [./porosity]
    type = PorousFlowPorosityConst
    porosity = 0.1
  [../]
  [./permeability]
    type = PorousFlowPermeabilityConst
    permeability = '1E-12 0 0 0 1E-12 0 0 0 1E-12'
  [../]
  [./relperm]
    type = PorousFlowRelativePermeabilityCorey
    at_nodes = true
    n = 2
    phase = 0
  [../]
[]

[DiracKernels]
  [./bh]
    type = PorousFlowPeacemanBorehole

    # Because the Variable for this Sink is pp, and pp is associated
    # with the fluid-mass conservation equation, this sink is extracting
    # fluid mass (and not heat energy or something else)
    variable = pp

    # The following specfies that the total fluid mass coming out of
    # the porespace via this sink in this timestep should be recorded
    # in the pls_total_outflow_mass UserObject
    SumQuantityUO = borehole_total_outflow_mass

    # The following file defines the polyline geometry
    # which is just two points in this particular example
    point_file = bh02.bh

    # First, we want Peacemans f to be a function of porepressure (and not
    # temperature or something else).  So bottom_p_or_t is actually porepressure
    function_of = pressure
    fluid_phase = 0

    # The bottomhole pressure
    bottom_p_or_t = 0

    # In this example there is no increase of the wellbore pressure
    # due to gravity:
    unit_weight = '0 0 0'

    # PeacemanBoreholes should almost always have use_mobility = true
    use_mobility = true

    # This is a production wellbore (a sink of fluid that removes fluid from porespace)
    character = 1
  [../]
[]

[Postprocessors]
  [./bh_report]
    type = PorousFlowPlotQuantity
    uo = borehole_total_outflow_mass
  [../]
  [./fluid_mass0]
    type = PorousFlowFluidMass
    execute_on = timestep_begin
  [../]

  [./fluid_mass1]
    type = PorousFlowFluidMass
    execute_on = timestep_end
  [../]

  [./zmass_error]
    type = FunctionValuePostprocessor
    function = mass_bal_fcn
    execute_on = timestep_end
  [../]

  [./p0]
    type = PointValue
    variable = pp
    point = '0 0 0'
    execute_on = timestep_end
  [../]
[]

[Functions]
  [./mass_bal_fcn]
    type = ParsedFunction
    value = abs((a-c+d)/2/(a+c))
    vars = 'a c d'
    vals = 'fluid_mass1 fluid_mass0 bh_report'
  [../]
[]

[Preconditioning]
  [./usual]
    type = SMP
    full = true
    petsc_options = '-snes_converged_reason'
    petsc_options_iname = '-ksp_type -pc_type -snes_atol -snes_rtol -snes_max_it -ksp_max_it'
    petsc_options_value = 'bcgs bjacobi 1E-10 1E-10 10000 30'
  [../]
[]

[Executioner]
  type = Transient
  end_time = 0.5
  dt = 1E-2
  solve_type = NEWTON
[]

[Outputs]
  file_base = bh02
  exodus = false
  csv = true
  execute_on = timestep_end
[]
(modules/porous_flow/test/tests/dirackernels/bh02.i)
  • An injection borehole with MPa, with a fully-saturated medium.

# fully-saturated
# injection
[Mesh]
  type = GeneratedMesh
  dim = 3
  nx = 1
  ny = 1
  nz = 1
  xmin = -1
  xmax = 1
  ymin = -1
  ymax = 1
  zmin = -1
  zmax = 1
[]

[GlobalParams]
  PorousFlowDictator = dictator
[]

[Variables]
  [./pp]
    initial_condition = 0
  [../]
[]

[Kernels]
  [./mass0]
    type = PorousFlowMassTimeDerivative
    fluid_component = 0
    variable = pp
  [../]
[]

[UserObjects]
  [./dictator]
    type = PorousFlowDictator
    porous_flow_vars = 'pp'
    number_fluid_phases = 1
    number_fluid_components = 1
  [../]
  [./borehole_total_outflow_mass]
    type = PorousFlowSumQuantity
  [../]
  [./pc]
    type = PorousFlowCapillaryPressureVG
    m = 0.5
    alpha = 1e-7
  [../]
[]

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

[Materials]
  [./temperature]
    type = PorousFlowTemperature
  [../]
  [./ppss]
    type = PorousFlow1PhaseP
    porepressure = pp
    capillary_pressure = pc
  [../]
  [./ppss_qp]
    type = PorousFlow1PhaseP
    at_nodes = false
    porepressure = pp
    capillary_pressure = pc
  [../]
  [./massfrac]
    type = PorousFlowMassFraction
  [../]
  [./simple_fluid]
    type = PorousFlowSingleComponentFluid
    fp = simple_fluid
    phase = 0
  [../]
  [./porosity]
    type = PorousFlowPorosityConst
    porosity = 0.1
  [../]
  [./permeability]
    type = PorousFlowPermeabilityConst
    permeability = '1E-12 0 0 0 1E-12 0 0 0 1E-12'
  [../]
  [./relperm]
    type = PorousFlowRelativePermeabilityCorey
    at_nodes = true
    n = 2
    phase = 0
  [../]
[]

[DiracKernels]
  [./bh]
    type = PorousFlowPeacemanBorehole
    variable = pp
    SumQuantityUO = borehole_total_outflow_mass
    point_file = bh03.bh
    function_of = pressure
    fluid_phase = 0
    bottom_p_or_t = 1E7
    unit_weight = '0 0 0'
    use_mobility = true
    character = -1
  [../]
[]

[Postprocessors]
  [./bh_report]
    type = PorousFlowPlotQuantity
    uo = borehole_total_outflow_mass
  [../]

  [./fluid_mass0]
    type = PorousFlowFluidMass
    execute_on = timestep_begin
  [../]

  [./fluid_mass1]
    type = PorousFlowFluidMass
    execute_on = timestep_end
  [../]

  [./zmass_error]
    type = FunctionValuePostprocessor
    function = mass_bal_fcn
    execute_on = timestep_end
  [../]

  [./p0]
    type = PointValue
    variable = pp
    point = '0 0 0'
    execute_on = timestep_end
  [../]
[]

[Functions]
  [./mass_bal_fcn]
    type = ParsedFunction
    value = abs((a-c+d)/2/(a+c))
    vars = 'a c d'
    vals = 'fluid_mass1 fluid_mass0 bh_report'
  [../]
[]

[Preconditioning]
  [./usual]
    type = SMP
    full = true
    petsc_options = '-snes_converged_reason'
    petsc_options_iname = '-ksp_type -pc_type -snes_atol -snes_rtol -snes_max_it -ksp_max_it'
    petsc_options_value = 'bcgs bjacobi 1E-10 1E-10 10000 30'
  [../]
[]

[Executioner]
  type = Transient
  end_time = 0.5
  dt = 1E-2
  solve_type = NEWTON
[]

[Outputs]
  file_base = bh03
  exodus = false
  csv = true
  execute_on = timestep_end
[]
(modules/porous_flow/test/tests/dirackernels/bh03.i)
  • A production borehole with MPa with a fully-saturated medium, with use_mobility=true

# fully-saturated
# production
[Mesh]
  type = GeneratedMesh
  dim = 3
  nx = 1
  ny = 1
  nz = 1
  xmin = -1
  xmax = 1
  ymin = -1
  ymax = 1
  zmin = -1
  zmax = 1
[]

[GlobalParams]
  PorousFlowDictator = dictator
[]

[Functions]
  [./dts]
    type = PiecewiseLinear
    y = '1E-2 1E-1 1 1E1 1E2 1E3'
    x = '0 1E-1 1 1E1 1E2 1E3'
  [../]
[]

[Variables]
  [./pp]
    initial_condition = 0
  [../]
[]

[Kernels]
  [./mass0]
    type = PorousFlowMassTimeDerivative
    fluid_component = 0
    variable = pp
  [../]
[]

[UserObjects]
  [./dictator]
    type = PorousFlowDictator
    porous_flow_vars = 'pp'
    number_fluid_phases = 1
    number_fluid_components = 1
  [../]
  [./borehole_total_outflow_mass]
    type = PorousFlowSumQuantity
  [../]
  [./pc]
    type = PorousFlowCapillaryPressureVG
    m = 0.8
    alpha = 1e-5
  [../]
[]

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

[Materials]
  [./temperature]
    type = PorousFlowTemperature
  [../]
  [./ppss]
    type = PorousFlow1PhaseP
    porepressure = pp
    capillary_pressure = pc
  [../]
  [./ppss_qp]
    type = PorousFlow1PhaseP
    at_nodes = false
    porepressure = pp
    capillary_pressure = pc
  [../]
  [./massfrac]
    type = PorousFlowMassFraction
  [../]
  [./simple_fluid]
    type = PorousFlowSingleComponentFluid
    fp = simple_fluid
    phase = 0
  [../]
  [./porosity]
    type = PorousFlowPorosityConst
    porosity = 0.1
  [../]
  [./permeability]
    type = PorousFlowPermeabilityConst
    permeability = '1E-12 0 0 0 1E-12 0 0 0 1E-12'
  [../]
  [./relperm]
    type = PorousFlowRelativePermeabilityFLAC
    at_nodes = true
    m = 2
    phase = 0
  [../]
[]

[DiracKernels]
  [./bh]
    type = PorousFlowPeacemanBorehole
    variable = pp
    SumQuantityUO = borehole_total_outflow_mass
    point_file = bh02.bh
    fluid_phase = 0
    bottom_p_or_t = -1E6
    unit_weight = '0 0 0'
    use_mobility = true
    character = 1
  [../]
[]

[Postprocessors]
  [./bh_report]
    type = PorousFlowPlotQuantity
    uo = borehole_total_outflow_mass
  [../]

  [./fluid_mass0]
    type = PorousFlowFluidMass
    execute_on = timestep_begin
  [../]

  [./fluid_mass1]
    type = PorousFlowFluidMass
    execute_on = timestep_end
  [../]

  [./zmass_error]
    type = FunctionValuePostprocessor
    function = mass_bal_fcn
    execute_on = timestep_end
  [../]

  [./p0]
    type = PointValue
    variable = pp
    point = '0 0 0'
    execute_on = timestep_end
  [../]
[]

[Functions]
  [./mass_bal_fcn]
    type = ParsedFunction
    value = abs((a-c+d)/2/(a+c))
    vars = 'a c d'
    vals = 'fluid_mass1 fluid_mass0 bh_report'
  [../]
[]

[Preconditioning]
  [./usual]
    type = SMP
    full = true
    petsc_options = '-snes_converged_reason'
    petsc_options_iname = '-ksp_type -pc_type -snes_atol -snes_rtol -snes_max_it -ksp_max_it'
    petsc_options_value = 'bcgs bjacobi 1E-10 1E-10 10000 30'
  [../]
[]

[Executioner]
  type = Transient
  end_time = 1E3
  solve_type = NEWTON
  [./TimeStepper]
    type = FunctionDT
    function = dts
  [../]
[]

[Outputs]
  file_base = bh04
  exodus = false
  csv = true
  execute_on = timestep_end
[]
(modules/porous_flow/test/tests/dirackernels/bh04.i)
  • An injection borehole with with an unsaturated medium, with use_mobility=true

# unsaturated
# injection
[Mesh]
  type = GeneratedMesh
  dim = 3
  nx = 1
  ny = 1
  nz = 1
  xmin = -1
  xmax = 1
  ymin = -1
  ymax = 1
  zmin = -1
  zmax = 1
[]

[GlobalParams]
  PorousFlowDictator = dictator
[]

[Functions]
  [./dts]
    type = PiecewiseLinear
    y = '500 500 1E1'
    x = '4000 5000 6500'
  [../]
[]

[Variables]
  [./pp]
    initial_condition = -2E5
  [../]
[]

[Kernels]
  [./mass0]
    type = PorousFlowMassTimeDerivative
    fluid_component = 0
    variable = pp
  [../]
[]

[UserObjects]
  [./dictator]
    type = PorousFlowDictator
    porous_flow_vars = 'pp'
    number_fluid_phases = 1
    number_fluid_components = 1
  [../]
  [./borehole_total_outflow_mass]
    type = PorousFlowSumQuantity
  [../]
  [./pc]
    type = PorousFlowCapillaryPressureVG
    m = 0.8
    alpha = 1e-5
  [../]
[]

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

[Materials]
  [./temperature]
    type = PorousFlowTemperature
  [../]
  [./ppss]
    type = PorousFlow1PhaseP
    porepressure = pp
    capillary_pressure = pc
  [../]
  [./ppss_qp]
    type = PorousFlow1PhaseP
    at_nodes = false
    porepressure = pp
    capillary_pressure = pc
  [../]
  [./massfrac]
    type = PorousFlowMassFraction
  [../]
  [./simple_fluid]
    type = PorousFlowSingleComponentFluid
    fp = simple_fluid
    phase = 0
  [../]
  [./porosity]
    type = PorousFlowPorosityConst
    porosity = 0.1
  [../]
  [./permeability]
    type = PorousFlowPermeabilityConst
    permeability = '1E-12 0 0 0 1E-12 0 0 0 1E-12'
  [../]
  [./relperm]
    type = PorousFlowRelativePermeabilityFLAC
    at_nodes = true
    m = 2
    phase = 0
  [../]
[]

[DiracKernels]
  [./bh]
    type = PorousFlowPeacemanBorehole
    variable = pp
    SumQuantityUO = borehole_total_outflow_mass
    point_file = bh03.bh
    fluid_phase = 0
    bottom_p_or_t = 0
    unit_weight = '0 0 0'
    use_mobility = true
    character = -1
  [../]
[]

[Postprocessors]
  [./bh_report]
    type = PorousFlowPlotQuantity
    uo = borehole_total_outflow_mass
  [../]

  [./fluid_mass0]
    type = PorousFlowFluidMass
    execute_on = timestep_begin
  [../]

  [./fluid_mass1]
    type = PorousFlowFluidMass
    execute_on = timestep_end
  [../]

  [./zmass_error]
    type = FunctionValuePostprocessor
    function = mass_bal_fcn
    execute_on = timestep_end
  [../]

  [./p0]
    type = PointValue
    variable = pp
    point = '0 0 0'
    execute_on = timestep_end
  [../]
[]

[Functions]
  [./mass_bal_fcn]
    type = ParsedFunction
    value = abs((a-c+d)/2/(a+c))
    vars = 'a c d'
    vals = 'fluid_mass1 fluid_mass0 bh_report'
  [../]
[]

[Preconditioning]
  [./usual]
    type = SMP
    full = true
    petsc_options = '-snes_converged_reason'
    petsc_options_iname = '-ksp_type -pc_type -snes_atol -snes_rtol -snes_max_it -ksp_max_it'
    petsc_options_value = 'bcgs bjacobi 1E-10 1E-10 10000 30'
  [../]
[]

[Executioner]
  type = Transient
  end_time = 6500
  solve_type = NEWTON
  [./TimeStepper]
    type = FunctionDT
    function = dts
  [../]
[]

[Outputs]
  file_base = bh05
  exodus = false
  csv = true
  execute_on = timestep_end
[]
(modules/porous_flow/test/tests/dirackernels/bh05.i)

Further commentary, and the results demonstrating that MOOSE is correct may be found in the page discussing the theory behind Dirac Kernels in PorousFlow.

Comparison with a steady-state 2D analytic solution

The test

# Comparison with analytical solution for cylindrically-symmetric situation
[Mesh]
  type = FileMesh
  file = bh07_input.e
[]

[GlobalParams]
  PorousFlowDictator = dictator
[]

[Functions]
  [./dts]
    type = PiecewiseLinear
    y = '1000 10000'
    x = '100 1000'
  [../]
[]

[Variables]
  [./pp]
    initial_condition = 1E7
  [../]
[]

[Kernels]
  [./mass0]
    type = PorousFlowMassTimeDerivative
    fluid_component = 0
    variable = pp
  [../]
  [./fflux]
    type = PorousFlowAdvectiveFlux
    fluid_component = 0
    variable = pp
    gravity = '0 0 0'
  [../]
[]

[BCs]
  [./fix_outer]
    type = DirichletBC
    boundary = perimeter
    variable = pp
    value = 1E7
  [../]
[]

[UserObjects]
  [./dictator]
    type = PorousFlowDictator
    porous_flow_vars = 'pp'
    number_fluid_phases = 1
    number_fluid_components = 1
  [../]
  [./borehole_total_outflow_mass]
    type = PorousFlowSumQuantity
  [../]
  [./pc]
    type = PorousFlowCapillaryPressureVG
    m = 0.8
    alpha = 1e-5
  [../]
[]

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

[Materials]
  [./temperature]
    type = PorousFlowTemperature
  [../]
  [./ppss]
    type = PorousFlow1PhaseP
    porepressure = pp
    capillary_pressure = pc
  [../]
  [./massfrac]
    type = PorousFlowMassFraction
  [../]
  [./simple_fluid]
    type = PorousFlowSingleComponentFluid
    fp = simple_fluid
    phase = 0
  [../]
  [./porosity]
    type = PorousFlowPorosityConst
    porosity = 0.1
  [../]
  [./permeability]
    type = PorousFlowPermeabilityConst
    permeability = '1E-11 0 0 0 1E-11 0 0 0 1E-11'
  [../]
  [./relperm]
    type = PorousFlowRelativePermeabilityFLAC
    m = 2
    phase = 0
  [../]
[]

[DiracKernels]
  [./bh]
    type = PorousFlowPeacemanBorehole
    variable = pp
    SumQuantityUO = borehole_total_outflow_mass
    point_file = bh07.bh
    fluid_phase = 0
    bottom_p_or_t = 0
    unit_weight = '0 0 0'
    use_mobility = true
    re_constant = 0.1594  # use Chen and Zhang version
    character = 2 # double the strength because bh07.bh only fills half the mesh
  [../]
[]

[Postprocessors]
  [./bh_report]
    type = PorousFlowPlotQuantity
    uo = borehole_total_outflow_mass
    execute_on = 'initial timestep_end'
  [../]
  [./fluid_mass]
    type = PorousFlowFluidMass
    execute_on = 'initial timestep_end'
  [../]
[]

[VectorPostprocessors]
  [./pp]
    type = LineValueSampler
    variable = pp
    start_point = '0 0 0'
    end_point = '300 0 0'
    sort_by = x
    num_points = 300
    execute_on = timestep_end
  [../]
[]

[Preconditioning]
  [./usual]
    type = SMP
    full = true
    petsc_options = '-snes_converged_reason'
    petsc_options_iname = '-ksp_type -pc_type -snes_atol -snes_rtol -snes_max_it -ksp_max_it'
    petsc_options_value = 'bcgs bjacobi 1E-10 1E-10 10000 30'
  [../]
[]

[Executioner]
  type = Transient
  end_time = 1E3
  solve_type = NEWTON
  [./TimeStepper]
    # get only marginally better results for smaller time steps
    type = FunctionDT
    function = dts
  [../]
[]

[Outputs]
  file_base = bh07
  [./along_line]
    type = CSV
    execute_on = final
  [../]
  [./exodus]
    type = Exodus
    execute_on = 'initial final'
  [../]
[]
(modules/porous_flow/test/tests/dirackernels/bh07.i)

checks whether the Peaceman borehole extracts fluid at the correct rate and the corresponding reduction in porepressure agrees with an analytic solution of Laplace's equation. Details and results demonstrating the correctness of PorousFlow can be found in the page discussing the theory behind Dirac Kernels in PorousFlow.