Porous flow¶

Overview¶

The porous_flow module collects the constitutive ingredients for (unsaturated) flow through a porous medium: capillary-pressure correlations, porosity-permeability laws, the effective-saturation transformation, and an advective stress that couples porous flow back to a finite-deformation solid.

The catalog is intentionally a set of small leaves. Most leaves take one or two Scalar inputs and return a Scalar, so they snap together with ComposedModel into the full pressure -> saturation -> permeability chain. The one exception is AdvectiveStress, which consumes the deformation gradient and PK1 stress (R2) to couple porous flow to a finite-deformation solid.

Math¶

Let \(\varphi\) denote the volume fraction of the flowing fluid in the medium and \(\varphi_{\mathrm{max}}\) the maximum fraction it can occupy at full saturation; the residual (immobile) saturation is \(S_r\). The effective saturation is the normalized fraction

\[ S_e = \frac{\varphi/\varphi_{\mathrm{max}} - S_r}{1 - S_r}, \]

so that \(S_e \in [0, 1]\) runs between irreducible and full saturation. Capillary-pressure correlations are written against \(S_e\) rather than \(\varphi\) directly. The catalog ships the two standard correlations:

\[ P_c^{\mathrm{BC}}(S_e) = P_t \, S_e^{-1/p}, \qquad P_c^{\mathrm{vG}}(S_e) = a \left( S_e^{-1/m} - 1 \right)^{1-m}, \]

— the Brooks-Corey law with threshold pressure \(P_t\) and shape exponent \(p\), and the van Genuchten law with shape parameters \((a, m)\). Both diverge as \(S_e \to 0\); both leaves take an optional log-linear extension below a transition saturation \(S_p\) that keeps \(P_c\) finite for the numerics.

Permeability \(K\) is a function of porosity \(\varphi\) (not saturation), and three closures are provided. With reference values \((\varphi_0, K_0)\) they are

(10)¶\[\begin{align} K^{\mathrm{pow}}(\varphi) &= K_0 \left( \frac{\varphi}{\varphi_0} \right)^p, \\ K^{\mathrm{exp}}(\varphi) &= K_0 \exp\!\left[ -a\,(\varphi - \varphi_0) \right], \\ K^{\mathrm{KC}}(\varphi) &= K_0 \, \frac{(\varphi / (1 - \varphi))^n}{(\varphi_0 / (1 - \varphi_0))^m}. \end{align}\]

The first is a plain power law with exponent \(p\), the second an exponential law with scale \(a\), and the third a Kozeny-Carman form parametrized by shape exponents \((n, m)\).

Finally, when the porous flow is coupled to a finite-deformation solid, the advective stress associated with swelling- or phase-change-induced volume change is the scalar

\[ p_s = -\frac{c}{3\, J_s^{5/3} J_t^{2/3}} \, P_{ij} F_{ij}, \]

where \(F\) is the deformation gradient, \(P\) the first Piola-Kirchhoff stress, \(c\) a volume-change coefficient, and \(J_s, J_t\) optional swelling and thermal Jacobians (default to \(1\) when omitted).

Example¶

The example below is a ModelUnitTest for the Brooks-Corey capillary-pressure law with its log-linear extension turned on. It takes the effective saturation \(S\) as the only input and produces the capillary pressure \(P_c\):

# Translated from tests/unit/models/porous_flow/BrooksCoreyCapillaryPressure.i.
[Drivers]
  [unit]
    type = ModelUnitTest
    model = 'model'
    input_Scalar_names = 'S'
    input_Scalar_values = 'S'
    output_Scalar_names = 'Pc'
    output_Scalar_values = 'Pc'
  []
[]

[Tensors]
  [S]
    type = Python
    expr = "Scalar(torch.tensor([0.05, 0.2, 0.65, 0.9], dtype=torch.float64))"
  []
  [Pc]
    type = Python
    expr = "Scalar(torch.tensor([126.0403524130621, 22.92220613, 4.255929933, 2.673595365], dtype=torch.float64))"
  []
[]

[Models]
  [model]
    type = BrooksCoreyCapillaryPressure
    threshold_pressure = 2.3
    exponent = 0.7
    effective_saturation = 'S'
    capillary_pressure = 'Pc'
    log_extension = true
    transition_saturation = 0.1
  []
[]

Explanation¶

The [Models] block declares a single BrooksCoreyCapillaryPressure instance. Two parameters close the constitutive law — threshold_pressure (\(P_t = 2.3\)) and exponent (\(p = 0.7\)) — and the two I/O lines wire the model’s canonical port names to the tensors used by the driver: effective_saturation = 'S' maps the input port to the [S] tensor declared above, and capillary_pressure = 'Pc' names the output that the driver compares against the reference values in [Pc]. Switching log_extension = true together with transition_saturation = 0.1 activates the log-linear extension below \(S_e = 0.1\), which is why the smallest sample point (\(S = 0.05\)) lands on a finite \(P_c \approx 126\) rather than the formal divergence of \(P_t S_e^{-1/p}\).

In a real composition, the upstream of this model is EffectiveSaturation, which converts a raw fluid volume fraction \(\varphi\) to \(S_e\), and the downstream consumer is whatever solver assembles the two-phase flow residual. The porosity-permeability leaves — PowerLawPermeability, ExponentialLawPermeability, and KozenyCarmanPermeability — slot in independently against \(\varphi\), since the three permeability leaves here depend only on porosity \(\varphi\), not on the fluid state. The drop-in alternative for the capillary branch is VanGenuchtenCapillaryPressure, which has the same I/O signature with a different functional shape. For finite-deformation coupling, AdvectiveStress consumes \((F, P)\) plus the optional \(J_s, J_t\) Jacobians and emits a single scalar that the host stress divergence picks up. The full catalog and per-type option list lives in the Syntax catalog under the Models section.

Porous flow¶

Overview¶

Math¶

Example¶

Explanation¶

See also¶