# Porous flow in a viscous medium

Governing equations

At the pressure and temperature conditions of the creeping mantle, « Fluids » (hydrous melts or aqueous fluids) are expected to migrate by porous flow though an interconnected network of channels formed along the grain edges [1].

In 1984, D. McKenzie [2] derived the governing equations that describe the transport of a liquid through a porous viscously deformable matrix based on the continuum mechanics theory. The liquid characterized by a very low viscosity relatively to that of the matrix is called the fluid phase and the matrix the solid phase.

These equations are :
– Conservation of mass for both phases :
$\qquad \dfrac{\partial (\rho_f \phi)}{\partial t} + \nabla \cdot (\rho_f \phi \boldsymbol{v}_f) = \Gamma$
$\qquad \dfrac{\partial (\rho_s \phi)}{\partial t} + \nabla \cdot (\rho_s (1-\phi) \boldsymbol{v}_s) = -\Gamma$
– Conservation of momentum :
$\phi(\boldsymbol{v}_f - \boldsymbol{v}_s) = -\dfrac{K}{\mu}(\boldsymbol{\nabla} P + \rho_f \boldsymbol{g})$
$\nabla \cdot (2 \eta \dot{\boldsymbol{\epsilon}}) - \boldsymbol{\nabla} P + \nabla (\xi \nabla \cdot \boldsymbol{v_s}) = (\rho_f \phi + \rho_s (1- \phi) \boldsymbol{g})$
where subscripts $f$ and $s$ denote values for, respectively, the fluid and the solid. $\rho$ is density, $\boldsymbol{v}$ is velocity, $\phi$ is porosity or fluid fraction, and $\Gamma$ is rate of mass transfer between the phases. $K$ is permeability, $\mu$ is fluid viscosity, $p$ total pressure, $\dot{\boldsymbol{\epsilon}}$ deformation rate, and $\boldsymbol{g}$ gravity. $\eta$ and $\xi$ are, respectively, the solid shear and bulk viscosities.

Numerical strategy

The McKenzie’s equations are re-written in a form more suitable for a numerical resolution [see 3].

Two systems are defined assuming that densities of both phases are constant :
– an « incompressible » Stokes-like solid system :
$\nabla \cdot (2 \eta \dot{\boldsymbol{\epsilon}}) - \boldsymbol{\nabla}p = \phi \Delta \rho \boldsymbol{g}$
$\nabla \cdot \boldsymbol{v_s} = \dfrac{{\cal P}}{\xi}$
– a « compressible » fluid system.
$\dfrac{\partial \phi}{\partial t} + \boldsymbol{v}_s \cdot \boldsymbol{\nabla} \phi = (1-\phi) \dfrac{{\cal P}}{\xi} + \dfrac{\Gamma}{\rho_s}$
$\dfrac{{\cal P}}{\xi} - \nabla \cdot \left[ \dfrac{K}{\mu} (\boldsymbol{\nabla} {\cal P} + \boldsymbol{\nabla} p + \Delta \rho \boldsymbol{g} ) \right] = \Gamma \dfrac{\Delta \rho}{\rho_f\rho_s}$
where $\Delta \rho$ is density contrast. ${\cal P}$ and $p$ are, respectively, compaction and dynamic pressures.

References
[1] Wark, D. A., et al. « Reassessment of pore shapes in microstructurally equilibrated rocks, with implications for permeability of the upper mantle. » Journal of Geophysical Research: Solid Earth 108.B1 (2003).
[2] McKenzie, D. « The generation and compaction of partially molten rock. » Journal of Petrology 25.3 (1984): 713-765.
[3] Katz, R. F., et al. « Numerical simulation of geodynamic processes with the Portable Extensible Toolkit for Scientific Computation. » Physics of the Earth and Planetary Interiors 163.1 (2007): 52-68.

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