Which equations describe complete (or near complete) behaviour of magma moving through rocks?
From McKenzie's original 1984 paper and from Spiegelman &amp; McKenzie (1987) paper the equations that govern the conservation of momentum and mass of a low-viscosity fluid melt (magma) in a deformable matrix (rocks) are:

${\partial\over\partial t} (\rho_f\phi) + \nabla \cdot (\rho_f\phi{\bf v})=\Gamma$

${\partial\over\partial t} [\rho_s(1-\phi)] + \nabla \cdot [\rho_s(1-\phi){\bf V]}=-\Gamma$

${\bf v}-{\bf V}=-{k_\phi\over\rho \mu}\nabla \mathscr{P} $

$\nabla \mathscr{P}=\eta\nabla^2{\bf V}+(\zeta+{\eta\over3})\nabla(\nabla\cdot{\bf V})+(1-\phi)\Delta\rho g\hat k $

$k_\phi \sim \alpha^2{\phi^n\over c} $

where $\rho_f$ is the density of the magma, $\phi$ is the porosity or the volume fraction occupied by the magma, ${\bf v}$ is the magma velocity, $\Gamma$ is the "melting function", which gives the rate of mass transfer from matrix to magma, $\rho_s$ is the density of the solid matrix, ${\bf V}$ is the matrix velocity, $k_\phi$, is the permeability, $\mathscr{P}$ is the piezometric pressure (the pressure in excess of hydrostatic pressure, $\mathscr{P}=P-\rho_fgz$), $\Delta\rho=\rho_s-\rho_f$ and $\hat k$ is the unit vector in the $z$ direction with $z$ increasing downwards. The permeability, $k_\phi$, can be expressed as a non-linear function of the grain size $\alpha$, porosity, and a dimensionless coefficient, $c$.