**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 :

– Conservation of momentum :

where subscripts and denote values for, respectively, the fluid and the solid. is density, is velocity, is porosity or fluid fraction, and is rate of mass transfer between the phases. is permeability, is fluid viscosity, total pressure, deformation rate, and gravity. and 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 :

– a « compressible » fluid system.

where is density contrast. and 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.