Injection ala Buckley and Leverett

PorousFlow is compared with a simple single-phase one-dimensional Buckley-Leverett problem (Buckley and Leverett, 1942). The single-phase fluid moves in a region m. A fully-saturated front initially sits at position , while the region is initially unsaturated. With zero suction function , there is no diffusion of the sharp front, and it progresses towards the right (increasing ).

# Buckley-Leverett 1-phase.
# The front starts at (around) x=5, and at t=50 it should
# have moved to x=9.6.  The version below has a nonzero
# suction function, and at t=50, the front sits between
# (about) x=9.6 and x=9.9.  Changing the van-Genuchten
# al parameter to 1E-4 softens the front so it sits between
# (about) x=9.7 and x=10.4, and the simulation runs much faster.
# With al=1E-2 and nx=600, the front sits between x=9.6 and x=9.8,
# but takes about 100 times longer to run.

[Mesh]
  type = GeneratedMesh
  dim = 1
  nx = 150
  xmin = 0
  xmax = 15
[]

[GlobalParams]
  PorousFlowDictator = dictator
  compute_enthalpy = false
  compute_internal_energy = false
[]

[Variables]
  [./pp]
    [./InitialCondition]
      type = FunctionIC
      function = 'max((1000000-x/5*1000000)-20000,-20000)'
    [../]
  [../]
[]

[Kernels]
  [./mass0]
    type = PorousFlowMassTimeDerivative
    fluid_component = 0
    variable = pp
  [../]
  [./flux0]
    type = PorousFlowAdvectiveFlux
    fluid_component = 0
    variable = pp
    gravity = '0 0 0'
  [../]
[]

[BCs]
  [./left]
    type = DirichletBC
    variable = pp
    boundary = left
    value = 980000
  [../]
[]

[AuxVariables]
  [./sat]
    family = MONOMIAL
    order = CONSTANT
  [../]
[]

[AuxKernels]
  [./sat]
    type = MaterialStdVectorAux
    variable = sat
    execute_on = timestep_end
    index = 0
    property = PorousFlow_saturation_qp
  [../]
[]

[UserObjects]
  [./dictator]
    type = PorousFlowDictator
    porous_flow_vars = 'pp'
    number_fluid_phases = 1
    number_fluid_components = 1
  [../]
  [./pc]
    type = PorousFlowCapillaryPressureVG
    m = 0.8
    alpha = 1e-3
  [../]
[]

[Modules]
  [./FluidProperties]
    [./simple_fluid]
      type = SimpleFluidProperties
      bulk_modulus = 2e6
      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
  [../]
  [./permeability]
    type = PorousFlowPermeabilityConst
    permeability = '1E-10 0 0  0 1E-10 0  0 0 1E-10'
  [../]
  [./relperm]
    type = PorousFlowRelativePermeabilityCorey
    n = 2
    phase = 0
  [../]
  [./porosity]
    type = PorousFlowPorosityConst
    porosity = 0.15
  [../]
[]

[Preconditioning]
  active = andy
  [./andy]
    type = SMP
    full = true
    petsc_options_iname = '-ksp_type -pc_type -snes_atol -snes_rtol -snes_max_it'
    petsc_options_value = 'gmres bjacobi 1E-10 1E-10 20'
  [../]
[]

[Functions]
  [./timestepper]
    type = PiecewiseLinear
    x = '0    0.01 0.1 1   1.5 2   20  30  40  50'
    y = '0.01 0.1  0.2 0.3 0.1 0.3 0.3 0.4 0.4 0.5'
  [../]
[]

[Executioner]
  type = Transient
  end_time = 50
  [./TimeStepper]
    type = FunctionDT
    function = timestepper
  [../]
[]

[VectorPostprocessors]
  [./pp]
    type = LineValueSampler
    start_point = '0 0 0'
    end_point = '15 0 0'
    num_points = 150
    sort_by = x
    variable = pp
  [../]
  [./sat]
    type = LineValueSampler
    start_point = '0 0 0'
    end_point = '15 0 0'
    num_points = 150
    sort_by = x
    variable = sat
  [../]
[]

[Outputs]
  file_base = bl01
  [./csv]
    type = CSV
    sync_only = true
    sync_times = '0.01 50'
  [../]
  [./exodus]
    type = Exodus
    execute_on = 'initial final'
  [../]
[]
(modules/porous_flow/test/tests/buckley_leverett/bl01.i)

This is a difficult problem to simulate numerically as maintaining the sharp front is hard. The front's speed is independent of the relative permeability, since the fluid is flowing from a fully-saturated region (where ). This problem is therefore a good test of the full upwinding.

In the simulation listed above, the pressure at the left boundary is kept fixed at MPa, while the right boundary is kept fixed at Pa, so the difference is MPa. The medium's permeability is set to and its porosity is . It is not possible to use a zero suction function in the MOOSE implementation, but using the van Genuchten parameters Pa and approximates it. The fluid viscosity is Pa.s.

The initial condition is (1) which is shown in Figure 1. With the suction function defined above this gives (2)

Figure 1: Initial setup of the Buckley-Leverett problem where porepressure is a piecewise linear function. The region on the left is fully saturated, while the region on the right has saturation 0.000006. During simulation the value of porepressure on the left boundary is held fixed.

Good approximations for the pressure and the front position may be determined from (3) which has solution (4) For the parameters listed above, the front will be at position m at s. This solution is only valid for zero capillary suction. A nonzero suction function will tend to diffuse the sharp front.

With coarse meshes it is impossible to simulate a sharp front, of course, since the front is spread over at least one element.

Figure 2 shows the results from a MOOSE simulation with a uniform mesh of size m. At s the front in this simulation sits at m as desired. However, the simulation takes seconds to complete due to the very low values of saturation obtained for van Genuchten Pa. Other simulations give similar results but run much faster. For instance, the test suite listed above uses the van-Genuchten parameter Pa, but the front diffuses a little into the unsaturated region (the front sits between m and m at s).

Figure 2: The MOOSE solution of the Buckley-Leverett problem at time 50s. Left: saturation. Right: porepressure. The front sits at x 9.6m as expected from the analytical solution.

References

  1. E. Buckley and M. Leverett. Mechanism of fluid displacement in sands. Transactions of the AIME, 146:107–116, 04 1942. doi:10.2118/942107-G.[BibTeX]