Heterogeneous models

Realistic models of porous media often feature spatially-varying material properties, especially porosity and permeability. There are several ways to introduce heterogeneity into the model in PorousFlow. In the following, we demonstrate how to create models with heterogeneous rock properties and use these properties in a simulation.

Regular grids

For regular structured grids that can be easily constructed using common meshing tools (or even the meshing capability built in to MOOSE), heterogeneity can simply be read from an external data file. In this example, we consider the 2D model of permeability heterogeneity presented as Case 1 of the tenth SPE comparative problem.

The permeability data for this model is read from an ASCII file containing coordinates and permeability values (in millidarcys):

# SPE 10 comparative problem - model 1
# Data from https://www.spe.org/web/csp/datasets/set01.htm

AXIS X
3.81 11.43 19.05 26.67 34.29 41.91 49.53 57.15 64.77 72.39 80.01 87.63 95.25 102.87 110.49 118.11 125.73 133.35 140.97 148.59 156.21 163.83 171.45 179.07 186.69 194.31 201.93 209.55 217.17 224.79 232.41 240.03 247.65 255.27 262.89 270.51 278.13 285.75 293.37 300.99 308.61 316.23 323.85 331.47 339.09 346.71 354.33 361.95 369.57 377.19 384.81 392.43 400.05 407.67 415.29 422.91 430.53 438.15 445.77 453.39 461.01 468.63 476.25 483.87 491.49 499.11 506.73 514.35 521.97 529.59 537.21 544.83 552.45 560.07 567.69 575.31 582.93 590.55 598.17 605.79 613.41 621.03 628.65 636.27 643.89 651.51 659.13 666.75 674.37 681.99 689.61 697.23 704.85 712.47 720.09 727.71 735.33 742.95 750.57 758.1

AXIS Y
0.381 1.143 1.905 2.667 3.429 4.191 4.953 5.715 6.477 7.239 8.001 8.763 9.525 10.287 11.049 11.811 12.573 13.335 14.097 14.859

DATA
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5.063000e-01 7.623000e-01 8.111000e-01 6.848000e-01 8.802000e-01
8.280000e-02 6.073000e-01 1.998270e+01 1.891070e+01 8.912400e+00
1.546830e+01 2.589600e+00 4.095700e+00 1.603810e+01 5.034400e+00
2.530200e+00 3.252300e+00 8.370500e+00 1.713300e+01 7.048890e+01
3.464520e+01 3.835920e+01 3.609120e+01 6.506500e+01 3.707900e+01
8.961540e+01 2.999440e+01 6.632730e+01 3.901500e+01 2.654400e+01

A PiecewiseMultilinear function is used to interpolate the permeability to the mesh.

[Functions]
  [perm_md_fcn]
    type = PiecewiseMultilinear
    data_file = spe10_case1.data
  []
[]

Constant Monomial AuxVariables are used to store the permeability read from the data file:

[AuxVariables]
  [temperature]
    initial_condition = 50
  []
  [xnacl]
    initial_condition = 0.1
  []
  [porosity]
    family = MONOMIAL
    order = CONSTANT
    initial_condition = 0.2
  []
  [perm_md]
    family = MONOMIAL
    order = CONSTANT
  []
  [perm]
    family = MONOMIAL
    order = CONSTANT
  []
[]

A FunctionAux AuxKernel is used to populate the AuxVariables.

In this example, the permeability in the data file is in millidarcys. As PorousFlow expects permeability in SI units of m $var element = document.getElementById("moose-equation-372519c0-49c7-446c-a7ae-f7f5a45c531b");katex.render("^2", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ , we multiply each permeability value by $var element = document.getElementById("moose-equation-516a8090-958b-4b64-8f4d-85ffa3792ff8");katex.render("9.869233 \\times 10^{-16}", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ and save these values in a new AuxVariable:

[AuxKernels]
  [perm_md]
    type = FunctionAux
    function = perm_md_fcn
    variable = perm_md
    execute_on = initial
  []
  [perm]
    type = ParsedAux
    variable = perm
    coupled_variables = perm_md
    expression = '9.869233e-16*perm_md'
    execute_on = initial
  []
[]

As this interpolation and multiplication is only required at the beginning of the simulation, we set the execute_on parameter to initial only.

Finally, the heterogeneous permeability (in m $var element = document.getElementById("moose-equation-81dffae8-f857-4044-8c0f-ed3299fde40d");katex.render("^2", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ ) can be used in the calculation using a PorousFlowPermeabilityConstFromVar material.

[Materials]
  [permeability]
    type = PorousFlowPermeabilityConstFromVar
    perm_xx = perm
    perm_yy = perm
    perm_zz = perm
  []
[]

The above steps create the following heterogeneous model that can then be used in a simulation:

Figure 1: Heterogeneous permeability for SPE comparative problem case 1

Although this example is for a two-dimensional mesh, the procedure for producing a three-dimensional mesh is identical.

Reservoir models

Often geological models are created using a modelling package to create realistic interpretations of the geology. In this case, some pre-processing of the geological model is often required before it can be used in PorousFlow.

This pre-processing must be performed using software that can take a geological model and convert it into a form that PorousFlow can read. One example of a pre-processing code that may be used is em2ex. This code converts a reservoir model in either Eclipse format or cell centered csv format to an Exodus mesh that can used in PorousFlow.

The following example shows how to use an Exodus mesh created from a reservoir model in a PorousFlow simulation. For this example, we use publicly available data from the SAIGUP project to construct the heterogeneous reservoir model shown in Figure 2 using em2ex.

Figure 2: Heterogeneous permeability for SAIGUP model

The heterogeneous porosity and permeability can then be read from the grid and used in the calculations using the following steps:

First, the mesh (containing the heterogeneous reservoir properties) is read into PorousFlow:

[Mesh]
  type = FileMesh
  file = field.e
[]

Constant monomial AuxVariables are then created. As the reservoir model again contains permeability in millidarcys, additional AuxVariables are also declared to hold the permeability in SI units (m $var element = document.getElementById("moose-equation-8fb7e635-2072-4a61-b1b1-93ffb6ef3774");katex.render("^2", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ ).

The values of the AuxVariables for porosity and the components of permeability in millidarcys are set using the initial_from_file_var parameter. These AuxVariables are not modified throughout the simulation, so represent the initial heterogeneity of the model.

[AuxVariables]
  [temperature]
    initial_condition = 50
  []
  [xnacl]
    initial_condition = 0.1
  []
  [porosity]
    family = MONOMIAL
    order = CONSTANT
    initial_from_file_var = poro
  []
  [permx_md]
    family = MONOMIAL
    order = CONSTANT
    initial_from_file_var = permX
  []
  [permy_md]
    family = MONOMIAL
    order = CONSTANT
    initial_from_file_var = permY
  []
  [permz_md]
    family = MONOMIAL
    order = CONSTANT
    initial_from_file_var = permZ
  []
  [permx]
    family = MONOMIAL
    order = CONSTANT
  []
  [permy]
    family = MONOMIAL
    order = CONSTANT
  []
  [permz]
    family = MONOMIAL
    order = CONSTANT
  []
[]

Like the previous example, the permeability can be converted to SI units using a ParsedAux AuxKernel for each component.

[AuxKernels]
  [permx]
    type = ParsedAux
    variable = permx
    coupled_variables = permx_md
    expression = '9.869233e-16*permx_md'
    execute_on = initial
  []
  [permy]
    type = ParsedAux
    variable = permy
    coupled_variables = permy_md
    expression = '9.869233e-16*permy_md'
    execute_on = initial
  []
  [permz]
    type = ParsedAux
    variable = permz
    coupled_variables = permz_md
    expression = '9.869233e-16*permz_md'
    execute_on = initial
  []
[]

The heterogeneous porosity and permeabilities can then be used in the calculations:

[Materials]
  [porosity]
    type = PorousFlowPorosityConst
    porosity = porosity
  []
[]

[Materials]
  [permeability]
    type = PorousFlowPermeabilityConstFromVar
    perm_xx = permx
    perm_yy = permy
    perm_zz = permz
  []
[]

Using this process, complex geological models with heterogeneous reservoir properties can be used in PorousFlow.

(modules/porous_flow/examples/reservoir_model/spe10_case1.data)

# SPE 10 comparative problem - model 1
# Data from https://www.spe.org/web/csp/datasets/set01.htm

AXIS X
3.81 11.43 19.05 26.67 34.29 41.91 49.53 57.15 64.77 72.39 80.01 87.63 95.25 102.87 110.49 118.11 125.73 133.35 140.97 148.59 156.21 163.83 171.45 179.07 186.69 194.31 201.93 209.55 217.17 224.79 232.41 240.03 247.65 255.27 262.89 270.51 278.13 285.75 293.37 300.99 308.61 316.23 323.85 331.47 339.09 346.71 354.33 361.95 369.57 377.19 384.81 392.43 400.05 407.67 415.29 422.91 430.53 438.15 445.77 453.39 461.01 468.63 476.25 483.87 491.49 499.11 506.73 514.35 521.97 529.59 537.21 544.83 552.45 560.07 567.69 575.31 582.93 590.55 598.17 605.79 613.41 621.03 628.65 636.27 643.89 651.51 659.13 666.75 674.37 681.99 689.61 697.23 704.85 712.47 720.09 727.71 735.33 742.95 750.57 758.1

AXIS Y
0.381 1.143 1.905 2.667 3.429 4.191 4.953 5.715 6.477 7.239 8.001 8.763 9.525 10.287 11.049 11.811 12.573 13.335 14.097 14.859

DATA
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1.585423e+02 3.947287e+02 3.195416e+02 2.715688e+02 2.770910e+02
4.954243e+02 9.248610e+01 3.628963e+02 4.592500e+01 5.943979e+02
7.297025e+02 6.479113e+02 7.087760e+02 5.040027e+02 5.266553e+02
5.295231e+02 6.271413e+02 5.711511e+02 5.036870e+02 8.684793e+02
4.446296e+02 5.447110e+01 2.631990e+01 6.731440e+01 2.130350e+01
4.430090e+01 9.313310e+01 5.525880e+01 8.350960e+01 3.950380e+01
6.842540e+02 8.519227e+02 8.213070e+02 9.397925e+02 5.594230e+01
3.227130e+01 9.398020e+01 5.397360e+01 1.849060e+01 5.573600e+00
7.708900e+00 1.743620e+01 1.032460e+01 6.383900e+00 4.392600e+00
3.787800e+00 1.003200e+00 1.339930e+01 8.174700e+00 7.898800e+00
1.255850e+01 1.925690e+01 1.695690e+01 1.890860e+01 1.965270e+01
5.739100e+00 1.155230e+01 1.688330e+01 1.014030e+01 4.336500e+00
8.420010e+01 7.633040e+01 4.791890e+01 3.604190e+01 4.680160e+01
2.842620e+01 9.179250e+01 8.212020e+01 6.558840e+01 7.462320e+01
7.560810e+01 6.276540e+01 8.196060e+01 4.817710e+01 8.041430e+01
5.775670e+01 4.036520e+01 5.914030e+01 7.780650e+01 3.116980e+01
6.329620e+01 2.936220e+01 2.492710e+01 8.637770e+01 6.629440e+01
2.748900e+01 6.216578e+02 8.815621e+02 9.559203e+02 6.497747e+02
9.666068e+02 6.384572e+02 9.368089e+02 9.126683e+02 5.892847e+02
2.095080e+01 8.180230e+01 3.271480e+01 6.768050e+01 2.800300e+01
2.641410e+01 4.986140e+01 6.471860e+01 4.816710e+01 6.950000e+01
4.509720e+01 5.082340e+01 5.559720e+01 7.988200e+01 8.284980e+01
6.196550e+01 2.874410e+01 6.442910e+01 7.934360e+01 7.966210e+01
4.535560e+01 2.305690e+01 5.759970e+01 4.597750e+01 5.747640e+01
1.000000e-03 9.436000e-01 2.400000e-03 1.580000e-02 2.650000e-02
1.810000e-01 9.792310e+01 8.543770e+01 7.301460e+01 3.322180e+01
7.056200e+01 7.775640e+01 7.163840e+01 4.740670e+01 8.998730e+01
2.125570e+01 8.186730e+01 4.644820e+01 7.089510e+01 7.053410e+01
8.788960e+01 8.881220e+01 8.571570e+01 2.667420e+01 6.912790e+01
7.694840e+01 3.774125e+02 2.984794e+02 1.547556e+02 1.731793e+02
2.156930e+02 4.606700e+02 2.430335e+02 3.108780e+02 3.814086e+02
4.008700e+00 3.632487e+02 2.796491e+02 2.910099e+02 3.221187e+02
7.565040e+01 4.020711e+02 3.135750e+01 1.743459e+02 1.265232e+02
1.320701e+02 3.909721e+02 2.170093e+02 4.794367e+02 8.616980e+01
3.912849e+02 3.464990e+01 3.464958e+02 8.804140e+01 4.913810e+01
7.259490e+01 6.486560e+01 7.867930e+01 5.971530e+01 7.413730e+01
3.573680e+01 9.143590e+01 8.390020e+01 9.000980e+01 9.394060e+01
3.360500e+01 7.920140e+01 2.262320e+01 7.647790e+01 8.146470e+01
2.086170e+01 8.087280e+01 7.558370e+01 3.527730e+01 5.685840e+01
5.513820e+01 3.372750e+01 4.393800e+00 1.689500e+00 3.917300e+00
1.017060e+01 1.328660e+01 1.464260e+01 1.303100e+01 1.956750e+01
1.283300e+01 9.705800e+00 4.584200e+00 1.895430e+01 4.505400e+00
3.246500e+00 6.492000e-01 5.539000e-01 3.972000e-01 4.069000e-01
2.994000e-01 7.859000e-01 2.494000e-01 8.147000e-01 8.171000e-01
5.000000e+02 6.969910e+02 9.906108e+02 5.312515e+02 9.491307e+02
9.989154e+02 6.765117e+02 7.689616e+02 7.588500e+00 8.443600e+00
8.662630e+01 3.287030e+01 7.655060e+01 4.386060e+01 3.877180e+01
6.039210e+01 9.337100e+01 7.818872e+02 8.966949e+02 8.056482e+02
5.903506e+02 9.231628e+02 9.261812e+02 8.383217e+02 8.264461e+02
3.485304e+02 2.246142e+02 1.298528e+02 4.356608e+02 3.614165e+02
2.443892e+02 3.140622e+02 1.349121e+02 6.708000e+00 2.363923e+02
3.206070e+02 2.029255e+02 3.948556e+02 3.746730e+02 8.480300e+00
1.163070e+01 1.955170e+01 1.770740e+01 1.547910e+01 8.550000e-01
2.503000e-01 8.446000e-01 3.060000e-02 7.860000e-01 6.188000e-01
4.986000e-01 7.301000e-01 4.234000e-01 9.833000e-01 8.010000e-01
5.230000e-02 2.890000e-01 2.440000e-01 9.900000e-03 5.264000e-01
4.540000e-02 9.840000e-01 3.629000e-01 7.156000e-01 1.815000e-01
1.663000e-01 7.718000e-01 8.992000e-01 7.011000e-01 8.090000e-01
5.063000e-01 7.623000e-01 8.111000e-01 6.848000e-01 8.802000e-01
8.280000e-02 6.073000e-01 1.998270e+01 1.891070e+01 8.912400e+00
1.546830e+01 2.589600e+00 4.095700e+00 1.603810e+01 5.034400e+00
2.530200e+00 3.252300e+00 8.370500e+00 1.713300e+01 7.048890e+01
3.464520e+01 3.835920e+01 3.609120e+01 6.506500e+01 3.707900e+01
8.961540e+01 2.999440e+01 6.632730e+01 3.901500e+01 2.654400e+01

(modules/porous_flow/examples/reservoir_model/regular_grid.i)

# SPE 10 comparative problem - model 1
# Data and description from https://www.spe.org/web/csp/datasets/set01.htm
# Simple input file that just establishes gravity equilibrium in the model
#
# Heterogeneous permeability is included by reading data from an external file
# using the PiecewiseMultilinear function, and saving that data to an elemental
# AuxVariable that is then used in PorousFlowPermeabilityConstFromVar

[Mesh]
  type = GeneratedMesh
  dim = 2
  nx = 100
  ny = 20
  xmax = 762
  ymax = 15.24
[]

[GlobalParams]
  PorousFlowDictator = dictator
  gravity = '0 -9.81 0'
  temperature_unit = Celsius
[]

[Variables]
  [porepressure]
    initial_condition = 20e6
  []
[]

[Functions]
  [perm_md_fcn]
    type = PiecewiseMultilinear
    data_file = spe10_case1.data
  []
[]

[BCs]
  [top]
    type = DirichletBC
    variable = porepressure
    value = 20e6
    boundary = top
  []
[]

[AuxVariables]
  [temperature]
    initial_condition = 50
  []
  [xnacl]
    initial_condition = 0.1
  []
  [porosity]
    family = MONOMIAL
    order = CONSTANT
    initial_condition = 0.2
  []
  [perm_md]
    family = MONOMIAL
    order = CONSTANT
  []
  [perm]
    family = MONOMIAL
    order = CONSTANT
  []
[]

[Kernels]
  [mass0]
    type = PorousFlowMassTimeDerivative
    variable = porepressure
  []
  [flux0]
    type = PorousFlowFullySaturatedDarcyFlow
    variable = porepressure
  []
[]

[AuxKernels]
  [perm_md]
    type = FunctionAux
    function = perm_md_fcn
    variable = perm_md
    execute_on = initial
  []
  [perm]
    type = ParsedAux
    variable = perm
    coupled_variables = perm_md
    expression = '9.869233e-16*perm_md'
    execute_on = initial
  []
[]

[UserObjects]
  [dictator]
    type = PorousFlowDictator
    porous_flow_vars = porepressure
    number_fluid_phases = 1
    number_fluid_components = 1
  []
[]

[FluidProperties]
  [water]
    type = Water97FluidProperties
  []
  [watertab]
    type = TabulatedBicubicFluidProperties
    fp = water
    save_file = false
  []
[]

[Materials]
  [temperature]
    type = PorousFlowTemperature
    temperature = temperature
  []
  [ps]
    type = PorousFlow1PhaseFullySaturated
    porepressure = porepressure
  []
  [massfrac]
    type = PorousFlowMassFraction
  []
  [brine]
    type = PorousFlowBrine
    compute_enthalpy = false
    compute_internal_energy = false
    xnacl = xnacl
    phase = 0
    water_fp = watertab
  []
  [porosity]
    type = PorousFlowPorosityConst
    porosity = porosity
  []
  [permeability]
    type = PorousFlowPermeabilityConstFromVar
    perm_xx = perm
    perm_yy = perm
    perm_zz = perm
  []
[]

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

[Executioner]
  type = Transient
  solve_type = Newton
  end_time = 1e5
  nl_abs_tol = 1e-12
  nl_rel_tol = 1e-06
  steady_state_detection = true
  steady_state_tolerance = 1e-12
  [TimeStepper]
    type = IterationAdaptiveDT
    dt = 1e2
  []
[]

[Outputs]
  execute_on = 'initial timestep_end'
  exodus = true
  perf_graph = true
[]

(modules/porous_flow/examples/reservoir_model/regular_grid.i)

# SPE 10 comparative problem - model 1
# Data and description from https://www.spe.org/web/csp/datasets/set01.htm
# Simple input file that just establishes gravity equilibrium in the model
#
# Heterogeneous permeability is included by reading data from an external file
# using the PiecewiseMultilinear function, and saving that data to an elemental
# AuxVariable that is then used in PorousFlowPermeabilityConstFromVar

[Mesh]
  type = GeneratedMesh
  dim = 2
  nx = 100
  ny = 20
  xmax = 762
  ymax = 15.24
[]

[GlobalParams]
  PorousFlowDictator = dictator
  gravity = '0 -9.81 0'
  temperature_unit = Celsius
[]

[Variables]
  [porepressure]
    initial_condition = 20e6
  []
[]

[Functions]
  [perm_md_fcn]
    type = PiecewiseMultilinear
    data_file = spe10_case1.data
  []
[]

[BCs]
  [top]
    type = DirichletBC
    variable = porepressure
    value = 20e6
    boundary = top
  []
[]

[AuxVariables]
  [temperature]
    initial_condition = 50
  []
  [xnacl]
    initial_condition = 0.1
  []
  [porosity]
    family = MONOMIAL
    order = CONSTANT
    initial_condition = 0.2
  []
  [perm_md]
    family = MONOMIAL
    order = CONSTANT
  []
  [perm]
    family = MONOMIAL
    order = CONSTANT
  []
[]

[Kernels]
  [mass0]
    type = PorousFlowMassTimeDerivative
    variable = porepressure
  []
  [flux0]
    type = PorousFlowFullySaturatedDarcyFlow
    variable = porepressure
  []
[]

[AuxKernels]
  [perm_md]
    type = FunctionAux
    function = perm_md_fcn
    variable = perm_md
    execute_on = initial
  []
  [perm]
    type = ParsedAux
    variable = perm
    coupled_variables = perm_md
    expression = '9.869233e-16*perm_md'
    execute_on = initial
  []
[]

[UserObjects]
  [dictator]
    type = PorousFlowDictator
    porous_flow_vars = porepressure
    number_fluid_phases = 1
    number_fluid_components = 1
  []
[]

[FluidProperties]
  [water]
    type = Water97FluidProperties
  []
  [watertab]
    type = TabulatedBicubicFluidProperties
    fp = water
    save_file = false
  []
[]

[Materials]
  [temperature]
    type = PorousFlowTemperature
    temperature = temperature
  []
  [ps]
    type = PorousFlow1PhaseFullySaturated
    porepressure = porepressure
  []
  [massfrac]
    type = PorousFlowMassFraction
  []
  [brine]
    type = PorousFlowBrine
    compute_enthalpy = false
    compute_internal_energy = false
    xnacl = xnacl
    phase = 0
    water_fp = watertab
  []
  [porosity]
    type = PorousFlowPorosityConst
    porosity = porosity
  []
  [permeability]
    type = PorousFlowPermeabilityConstFromVar
    perm_xx = perm
    perm_yy = perm
    perm_zz = perm
  []
[]

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

[Executioner]
  type = Transient
  solve_type = Newton
  end_time = 1e5
  nl_abs_tol = 1e-12
  nl_rel_tol = 1e-06
  steady_state_detection = true
  steady_state_tolerance = 1e-12
  [TimeStepper]
    type = IterationAdaptiveDT
    dt = 1e2
  []
[]

[Outputs]
  execute_on = 'initial timestep_end'
  exodus = true
  perf_graph = true
[]

(modules/porous_flow/examples/reservoir_model/regular_grid.i)

# SPE 10 comparative problem - model 1
# Data and description from https://www.spe.org/web/csp/datasets/set01.htm
# Simple input file that just establishes gravity equilibrium in the model
#
# Heterogeneous permeability is included by reading data from an external file
# using the PiecewiseMultilinear function, and saving that data to an elemental
# AuxVariable that is then used in PorousFlowPermeabilityConstFromVar

[Mesh]
  type = GeneratedMesh
  dim = 2
  nx = 100
  ny = 20
  xmax = 762
  ymax = 15.24
[]

[GlobalParams]
  PorousFlowDictator = dictator
  gravity = '0 -9.81 0'
  temperature_unit = Celsius
[]

[Variables]
  [porepressure]
    initial_condition = 20e6
  []
[]

[Functions]
  [perm_md_fcn]
    type = PiecewiseMultilinear
    data_file = spe10_case1.data
  []
[]

[BCs]
  [top]
    type = DirichletBC
    variable = porepressure
    value = 20e6
    boundary = top
  []
[]

[AuxVariables]
  [temperature]
    initial_condition = 50
  []
  [xnacl]
    initial_condition = 0.1
  []
  [porosity]
    family = MONOMIAL
    order = CONSTANT
    initial_condition = 0.2
  []
  [perm_md]
    family = MONOMIAL
    order = CONSTANT
  []
  [perm]
    family = MONOMIAL
    order = CONSTANT
  []
[]

[Kernels]
  [mass0]
    type = PorousFlowMassTimeDerivative
    variable = porepressure
  []
  [flux0]
    type = PorousFlowFullySaturatedDarcyFlow
    variable = porepressure
  []
[]

[AuxKernels]
  [perm_md]
    type = FunctionAux
    function = perm_md_fcn
    variable = perm_md
    execute_on = initial
  []
  [perm]
    type = ParsedAux
    variable = perm
    coupled_variables = perm_md
    expression = '9.869233e-16*perm_md'
    execute_on = initial
  []
[]

[UserObjects]
  [dictator]
    type = PorousFlowDictator
    porous_flow_vars = porepressure
    number_fluid_phases = 1
    number_fluid_components = 1
  []
[]

[FluidProperties]
  [water]
    type = Water97FluidProperties
  []
  [watertab]
    type = TabulatedBicubicFluidProperties
    fp = water
    save_file = false
  []
[]

[Materials]
  [temperature]
    type = PorousFlowTemperature
    temperature = temperature
  []
  [ps]
    type = PorousFlow1PhaseFullySaturated
    porepressure = porepressure
  []
  [massfrac]
    type = PorousFlowMassFraction
  []
  [brine]
    type = PorousFlowBrine
    compute_enthalpy = false
    compute_internal_energy = false
    xnacl = xnacl
    phase = 0
    water_fp = watertab
  []
  [porosity]
    type = PorousFlowPorosityConst
    porosity = porosity
  []
  [permeability]
    type = PorousFlowPermeabilityConstFromVar
    perm_xx = perm
    perm_yy = perm
    perm_zz = perm
  []
[]

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

[Executioner]
  type = Transient
  solve_type = Newton
  end_time = 1e5
  nl_abs_tol = 1e-12
  nl_rel_tol = 1e-06
  steady_state_detection = true
  steady_state_tolerance = 1e-12
  [TimeStepper]
    type = IterationAdaptiveDT
    dt = 1e2
  []
[]

[Outputs]
  execute_on = 'initial timestep_end'
  exodus = true
  perf_graph = true
[]

(modules/porous_flow/examples/reservoir_model/regular_grid.i)

# SPE 10 comparative problem - model 1
# Data and description from https://www.spe.org/web/csp/datasets/set01.htm
# Simple input file that just establishes gravity equilibrium in the model
#
# Heterogeneous permeability is included by reading data from an external file
# using the PiecewiseMultilinear function, and saving that data to an elemental
# AuxVariable that is then used in PorousFlowPermeabilityConstFromVar

[Mesh]
  type = GeneratedMesh
  dim = 2
  nx = 100
  ny = 20
  xmax = 762
  ymax = 15.24
[]

[GlobalParams]
  PorousFlowDictator = dictator
  gravity = '0 -9.81 0'
  temperature_unit = Celsius
[]

[Variables]
  [porepressure]
    initial_condition = 20e6
  []
[]

[Functions]
  [perm_md_fcn]
    type = PiecewiseMultilinear
    data_file = spe10_case1.data
  []
[]

[BCs]
  [top]
    type = DirichletBC
    variable = porepressure
    value = 20e6
    boundary = top
  []
[]

[AuxVariables]
  [temperature]
    initial_condition = 50
  []
  [xnacl]
    initial_condition = 0.1
  []
  [porosity]
    family = MONOMIAL
    order = CONSTANT
    initial_condition = 0.2
  []
  [perm_md]
    family = MONOMIAL
    order = CONSTANT
  []
  [perm]
    family = MONOMIAL
    order = CONSTANT
  []
[]

[Kernels]
  [mass0]
    type = PorousFlowMassTimeDerivative
    variable = porepressure
  []
  [flux0]
    type = PorousFlowFullySaturatedDarcyFlow
    variable = porepressure
  []
[]

[AuxKernels]
  [perm_md]
    type = FunctionAux
    function = perm_md_fcn
    variable = perm_md
    execute_on = initial
  []
  [perm]
    type = ParsedAux
    variable = perm
    coupled_variables = perm_md
    expression = '9.869233e-16*perm_md'
    execute_on = initial
  []
[]

[UserObjects]
  [dictator]
    type = PorousFlowDictator
    porous_flow_vars = porepressure
    number_fluid_phases = 1
    number_fluid_components = 1
  []
[]

[FluidProperties]
  [water]
    type = Water97FluidProperties
  []
  [watertab]
    type = TabulatedBicubicFluidProperties
    fp = water
    save_file = false
  []
[]

[Materials]
  [temperature]
    type = PorousFlowTemperature
    temperature = temperature
  []
  [ps]
    type = PorousFlow1PhaseFullySaturated
    porepressure = porepressure
  []
  [massfrac]
    type = PorousFlowMassFraction
  []
  [brine]
    type = PorousFlowBrine
    compute_enthalpy = false
    compute_internal_energy = false
    xnacl = xnacl
    phase = 0
    water_fp = watertab
  []
  [porosity]
    type = PorousFlowPorosityConst
    porosity = porosity
  []
  [permeability]
    type = PorousFlowPermeabilityConstFromVar
    perm_xx = perm
    perm_yy = perm
    perm_zz = perm
  []
[]

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

[Executioner]
  type = Transient
  solve_type = Newton
  end_time = 1e5
  nl_abs_tol = 1e-12
  nl_rel_tol = 1e-06
  steady_state_detection = true
  steady_state_tolerance = 1e-12
  [TimeStepper]
    type = IterationAdaptiveDT
    dt = 1e2
  []
[]

[Outputs]
  execute_on = 'initial timestep_end'
  exodus = true
  perf_graph = true
[]

(modules/porous_flow/examples/reservoir_model/field_model.i)

# Field model generated using geophysical modelling tool

[Mesh]
  type = FileMesh
  file = field.e
[]

[GlobalParams]
  PorousFlowDictator = dictator
  gravity = '0 0 -9.81'
  temperature_unit = Celsius
[]

[Problem]
  # Variable porepressure has an initial condition despite the restart
  allow_initial_conditions_with_restart = true
[]

[Variables]
  [porepressure]
    initial_condition = 20e6
  []
[]

[AuxVariables]
  [temperature]
    initial_condition = 50
  []
  [xnacl]
    initial_condition = 0.1
  []
  [porosity]
    family = MONOMIAL
    order = CONSTANT
    initial_from_file_var = poro
  []
  [permx_md]
    family = MONOMIAL
    order = CONSTANT
    initial_from_file_var = permX
  []
  [permy_md]
    family = MONOMIAL
    order = CONSTANT
    initial_from_file_var = permY
  []
  [permz_md]
    family = MONOMIAL
    order = CONSTANT
    initial_from_file_var = permZ
  []
  [permx]
    family = MONOMIAL
    order = CONSTANT
  []
  [permy]
    family = MONOMIAL
    order = CONSTANT
  []
  [permz]
    family = MONOMIAL
    order = CONSTANT
  []
[]

[AuxKernels]
  [permx]
    type = ParsedAux
    variable = permx
    coupled_variables = permx_md
    expression = '9.869233e-16*permx_md'
    execute_on = initial
  []
  [permy]
    type = ParsedAux
    variable = permy
    coupled_variables = permy_md
    expression = '9.869233e-16*permy_md'
    execute_on = initial
  []
  [permz]
    type = ParsedAux
    variable = permz
    coupled_variables = permz_md
    expression = '9.869233e-16*permz_md'
    execute_on = initial
  []
[]

[Kernels]
  [mass0]
    type = PorousFlowMassTimeDerivative
    variable = porepressure
  []
  [flux0]
    type = PorousFlowFullySaturatedDarcyFlow
    variable = porepressure
  []
[]

[UserObjects]
  [dictator]
    type = PorousFlowDictator
    porous_flow_vars = porepressure
    number_fluid_phases = 1
    number_fluid_components = 1
  []
[]

[FluidProperties]
  [water]
    type = Water97FluidProperties
  []
  [watertab]
    type = TabulatedBicubicFluidProperties
    fp = water
    save_file = false
  []
[]

[Materials]
  [temperature]
    type = PorousFlowTemperature
    temperature = temperature
  []
  [ps]
    type = PorousFlow1PhaseFullySaturated
    porepressure = porepressure
  []
  [massfrac]
    type = PorousFlowMassFraction
  []
  [brine]
    type = PorousFlowBrine
    compute_enthalpy = false
    compute_internal_energy = false
    xnacl = xnacl
    phase = 0
    water_fp = watertab
  []
  [porosity]
    type = PorousFlowPorosityConst
    porosity = porosity
  []
  [permeability]
    type = PorousFlowPermeabilityConstFromVar
    perm_xx = permx
    perm_yy = permy
    perm_zz = permz
  []
[]

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

[Executioner]
  type = Transient
  solve_type = Newton
  dt = 1e2
  end_time = 1e2
[]

[Outputs]
  execute_on = 'initial timestep_end'
  exodus = true
  perf_graph = true
[]

(modules/porous_flow/examples/reservoir_model/field_model.i)

# Field model generated using geophysical modelling tool

[Mesh]
  type = FileMesh
  file = field.e
[]

[GlobalParams]
  PorousFlowDictator = dictator
  gravity = '0 0 -9.81'
  temperature_unit = Celsius
[]

[Problem]
  # Variable porepressure has an initial condition despite the restart
  allow_initial_conditions_with_restart = true
[]

[Variables]
  [porepressure]
    initial_condition = 20e6
  []
[]

[AuxVariables]
  [temperature]
    initial_condition = 50
  []
  [xnacl]
    initial_condition = 0.1
  []
  [porosity]
    family = MONOMIAL
    order = CONSTANT
    initial_from_file_var = poro
  []
  [permx_md]
    family = MONOMIAL
    order = CONSTANT
    initial_from_file_var = permX
  []
  [permy_md]
    family = MONOMIAL
    order = CONSTANT
    initial_from_file_var = permY
  []
  [permz_md]
    family = MONOMIAL
    order = CONSTANT
    initial_from_file_var = permZ
  []
  [permx]
    family = MONOMIAL
    order = CONSTANT
  []
  [permy]
    family = MONOMIAL
    order = CONSTANT
  []
  [permz]
    family = MONOMIAL
    order = CONSTANT
  []
[]

[AuxKernels]
  [permx]
    type = ParsedAux
    variable = permx
    coupled_variables = permx_md
    expression = '9.869233e-16*permx_md'
    execute_on = initial
  []
  [permy]
    type = ParsedAux
    variable = permy
    coupled_variables = permy_md
    expression = '9.869233e-16*permy_md'
    execute_on = initial
  []
  [permz]
    type = ParsedAux
    variable = permz
    coupled_variables = permz_md
    expression = '9.869233e-16*permz_md'
    execute_on = initial
  []
[]

[Kernels]
  [mass0]
    type = PorousFlowMassTimeDerivative
    variable = porepressure
  []
  [flux0]
    type = PorousFlowFullySaturatedDarcyFlow
    variable = porepressure
  []
[]

[UserObjects]
  [dictator]
    type = PorousFlowDictator
    porous_flow_vars = porepressure
    number_fluid_phases = 1
    number_fluid_components = 1
  []
[]

[FluidProperties]
  [water]
    type = Water97FluidProperties
  []
  [watertab]
    type = TabulatedBicubicFluidProperties
    fp = water
    save_file = false
  []
[]

[Materials]
  [temperature]
    type = PorousFlowTemperature
    temperature = temperature
  []
  [ps]
    type = PorousFlow1PhaseFullySaturated
    porepressure = porepressure
  []
  [massfrac]
    type = PorousFlowMassFraction
  []
  [brine]
    type = PorousFlowBrine
    compute_enthalpy = false
    compute_internal_energy = false
    xnacl = xnacl
    phase = 0
    water_fp = watertab
  []
  [porosity]
    type = PorousFlowPorosityConst
    porosity = porosity
  []
  [permeability]
    type = PorousFlowPermeabilityConstFromVar
    perm_xx = permx
    perm_yy = permy
    perm_zz = permz
  []
[]

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

[Executioner]
  type = Transient
  solve_type = Newton
  dt = 1e2
  end_time = 1e2
[]

[Outputs]
  execute_on = 'initial timestep_end'
  exodus = true
  perf_graph = true
[]

(modules/porous_flow/examples/reservoir_model/field_model.i)

# Field model generated using geophysical modelling tool

[Mesh]
  type = FileMesh
  file = field.e
[]

[GlobalParams]
  PorousFlowDictator = dictator
  gravity = '0 0 -9.81'
  temperature_unit = Celsius
[]

[Problem]
  # Variable porepressure has an initial condition despite the restart
  allow_initial_conditions_with_restart = true
[]

[Variables]
  [porepressure]
    initial_condition = 20e6
  []
[]

[AuxVariables]
  [temperature]
    initial_condition = 50
  []
  [xnacl]
    initial_condition = 0.1
  []
  [porosity]
    family = MONOMIAL
    order = CONSTANT
    initial_from_file_var = poro
  []
  [permx_md]
    family = MONOMIAL
    order = CONSTANT
    initial_from_file_var = permX
  []
  [permy_md]
    family = MONOMIAL
    order = CONSTANT
    initial_from_file_var = permY
  []
  [permz_md]
    family = MONOMIAL
    order = CONSTANT
    initial_from_file_var = permZ
  []
  [permx]
    family = MONOMIAL
    order = CONSTANT
  []
  [permy]
    family = MONOMIAL
    order = CONSTANT
  []
  [permz]
    family = MONOMIAL
    order = CONSTANT
  []
[]

[AuxKernels]
  [permx]
    type = ParsedAux
    variable = permx
    coupled_variables = permx_md
    expression = '9.869233e-16*permx_md'
    execute_on = initial
  []
  [permy]
    type = ParsedAux
    variable = permy
    coupled_variables = permy_md
    expression = '9.869233e-16*permy_md'
    execute_on = initial
  []
  [permz]
    type = ParsedAux
    variable = permz
    coupled_variables = permz_md
    expression = '9.869233e-16*permz_md'
    execute_on = initial
  []
[]

[Kernels]
  [mass0]
    type = PorousFlowMassTimeDerivative
    variable = porepressure
  []
  [flux0]
    type = PorousFlowFullySaturatedDarcyFlow
    variable = porepressure
  []
[]

[UserObjects]
  [dictator]
    type = PorousFlowDictator
    porous_flow_vars = porepressure
    number_fluid_phases = 1
    number_fluid_components = 1
  []
[]

[FluidProperties]
  [water]
    type = Water97FluidProperties
  []
  [watertab]
    type = TabulatedBicubicFluidProperties
    fp = water
    save_file = false
  []
[]

[Materials]
  [temperature]
    type = PorousFlowTemperature
    temperature = temperature
  []
  [ps]
    type = PorousFlow1PhaseFullySaturated
    porepressure = porepressure
  []
  [massfrac]
    type = PorousFlowMassFraction
  []
  [brine]
    type = PorousFlowBrine
    compute_enthalpy = false
    compute_internal_energy = false
    xnacl = xnacl
    phase = 0
    water_fp = watertab
  []
  [porosity]
    type = PorousFlowPorosityConst
    porosity = porosity
  []
  [permeability]
    type = PorousFlowPermeabilityConstFromVar
    perm_xx = permx
    perm_yy = permy
    perm_zz = permz
  []
[]

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

[Executioner]
  type = Transient
  solve_type = Newton
  dt = 1e2
  end_time = 1e2
[]

[Outputs]
  execute_on = 'initial timestep_end'
  exodus = true
  perf_graph = true
[]

(modules/porous_flow/examples/reservoir_model/field_model.i)

# Field model generated using geophysical modelling tool

[Mesh]
  type = FileMesh
  file = field.e
[]

[GlobalParams]
  PorousFlowDictator = dictator
  gravity = '0 0 -9.81'
  temperature_unit = Celsius
[]

[Problem]
  # Variable porepressure has an initial condition despite the restart
  allow_initial_conditions_with_restart = true
[]

[Variables]
  [porepressure]
    initial_condition = 20e6
  []
[]

[AuxVariables]
  [temperature]
    initial_condition = 50
  []
  [xnacl]
    initial_condition = 0.1
  []
  [porosity]
    family = MONOMIAL
    order = CONSTANT
    initial_from_file_var = poro
  []
  [permx_md]
    family = MONOMIAL
    order = CONSTANT
    initial_from_file_var = permX
  []
  [permy_md]
    family = MONOMIAL
    order = CONSTANT
    initial_from_file_var = permY
  []
  [permz_md]
    family = MONOMIAL
    order = CONSTANT
    initial_from_file_var = permZ
  []
  [permx]
    family = MONOMIAL
    order = CONSTANT
  []
  [permy]
    family = MONOMIAL
    order = CONSTANT
  []
  [permz]
    family = MONOMIAL
    order = CONSTANT
  []
[]

[AuxKernels]
  [permx]
    type = ParsedAux
    variable = permx
    coupled_variables = permx_md
    expression = '9.869233e-16*permx_md'
    execute_on = initial
  []
  [permy]
    type = ParsedAux
    variable = permy
    coupled_variables = permy_md
    expression = '9.869233e-16*permy_md'
    execute_on = initial
  []
  [permz]
    type = ParsedAux
    variable = permz
    coupled_variables = permz_md
    expression = '9.869233e-16*permz_md'
    execute_on = initial
  []
[]

[Kernels]
  [mass0]
    type = PorousFlowMassTimeDerivative
    variable = porepressure
  []
  [flux0]
    type = PorousFlowFullySaturatedDarcyFlow
    variable = porepressure
  []
[]

[UserObjects]
  [dictator]
    type = PorousFlowDictator
    porous_flow_vars = porepressure
    number_fluid_phases = 1
    number_fluid_components = 1
  []
[]

[FluidProperties]
  [water]
    type = Water97FluidProperties
  []
  [watertab]
    type = TabulatedBicubicFluidProperties
    fp = water
    save_file = false
  []
[]

[Materials]
  [temperature]
    type = PorousFlowTemperature
    temperature = temperature
  []
  [ps]
    type = PorousFlow1PhaseFullySaturated
    porepressure = porepressure
  []
  [massfrac]
    type = PorousFlowMassFraction
  []
  [brine]
    type = PorousFlowBrine
    compute_enthalpy = false
    compute_internal_energy = false
    xnacl = xnacl
    phase = 0
    water_fp = watertab
  []
  [porosity]
    type = PorousFlowPorosityConst
    porosity = porosity
  []
  [permeability]
    type = PorousFlowPermeabilityConstFromVar
    perm_xx = permx
    perm_yy = permy
    perm_zz = permz
  []
[]

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

[Executioner]
  type = Transient
  solve_type = Newton
  dt = 1e2
  end_time = 1e2
[]

[Outputs]
  execute_on = 'initial timestep_end'
  exodus = true
  perf_graph = true
[]

(modules/porous_flow/examples/reservoir_model/field_model.i)

# Field model generated using geophysical modelling tool

[Mesh]
  type = FileMesh
  file = field.e
[]

[GlobalParams]
  PorousFlowDictator = dictator
  gravity = '0 0 -9.81'
  temperature_unit = Celsius
[]

[Problem]
  # Variable porepressure has an initial condition despite the restart
  allow_initial_conditions_with_restart = true
[]

[Variables]
  [porepressure]
    initial_condition = 20e6
  []
[]

[AuxVariables]
  [temperature]
    initial_condition = 50
  []
  [xnacl]
    initial_condition = 0.1
  []
  [porosity]
    family = MONOMIAL
    order = CONSTANT
    initial_from_file_var = poro
  []
  [permx_md]
    family = MONOMIAL
    order = CONSTANT
    initial_from_file_var = permX
  []
  [permy_md]
    family = MONOMIAL
    order = CONSTANT
    initial_from_file_var = permY
  []
  [permz_md]
    family = MONOMIAL
    order = CONSTANT
    initial_from_file_var = permZ
  []
  [permx]
    family = MONOMIAL
    order = CONSTANT
  []
  [permy]
    family = MONOMIAL
    order = CONSTANT
  []
  [permz]
    family = MONOMIAL
    order = CONSTANT
  []
[]

[AuxKernels]
  [permx]
    type = ParsedAux
    variable = permx
    coupled_variables = permx_md
    expression = '9.869233e-16*permx_md'
    execute_on = initial
  []
  [permy]
    type = ParsedAux
    variable = permy
    coupled_variables = permy_md
    expression = '9.869233e-16*permy_md'
    execute_on = initial
  []
  [permz]
    type = ParsedAux
    variable = permz
    coupled_variables = permz_md
    expression = '9.869233e-16*permz_md'
    execute_on = initial
  []
[]

[Kernels]
  [mass0]
    type = PorousFlowMassTimeDerivative
    variable = porepressure
  []
  [flux0]
    type = PorousFlowFullySaturatedDarcyFlow
    variable = porepressure
  []
[]

[UserObjects]
  [dictator]
    type = PorousFlowDictator
    porous_flow_vars = porepressure
    number_fluid_phases = 1
    number_fluid_components = 1
  []
[]

[FluidProperties]
  [water]
    type = Water97FluidProperties
  []
  [watertab]
    type = TabulatedBicubicFluidProperties
    fp = water
    save_file = false
  []
[]

[Materials]
  [temperature]
    type = PorousFlowTemperature
    temperature = temperature
  []
  [ps]
    type = PorousFlow1PhaseFullySaturated
    porepressure = porepressure
  []
  [massfrac]
    type = PorousFlowMassFraction
  []
  [brine]
    type = PorousFlowBrine
    compute_enthalpy = false
    compute_internal_energy = false
    xnacl = xnacl
    phase = 0
    water_fp = watertab
  []
  [porosity]
    type = PorousFlowPorosityConst
    porosity = porosity
  []
  [permeability]
    type = PorousFlowPermeabilityConstFromVar
    perm_xx = permx
    perm_yy = permy
    perm_zz = permz
  []
[]

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

[Executioner]
  type = Transient
  solve_type = Newton
  dt = 1e2
  end_time = 1e2
[]

[Outputs]
  execute_on = 'initial timestep_end'
  exodus = true
  perf_graph = true
[]

Regular grids
Reservoir models