<p>From McKenzie's <a href="http://dx.doi.org/10.1093/petrology/25.3.713" rel="nofollow noreferrer">original 1984 paper</a> and from <a href="http://dx.doi.org/10.1016/0012-821X(87)90057-4" rel="nofollow noreferrer">Spiegelman &amp; McKenzie (1987)</a> paper the equations that govern the conservation of momentum and mass of a low-viscosity fluid melt (magma) in a deformable matrix (rocks) are:</p>

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

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

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

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

<p>$k_\phi \sim \alpha^2{\phi^n\over c} $</p>

<p>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, $c$. </p>

CC BY-SA 3.0
2016-11-25T16:04:08.000Z
2016-11-25T16:04:08.000Z
arkaia
<p>The equations you are looking for are called the "McKenzie equations". You should look at the papers of Marc Spiegelman at Columbia University for more information.</p>

CC BY-SA 3.0
2016-10-11T17:58:44.387Z
Could you expand your answer a bit providing the equations and a simple explanation? Thanks
arkaia
CC BY-SA 3.0
2016-11-25T02:51:42.477Z
I think the papers I am referencing do a much better job at explaining what these equations are and do, than I could do here. These are not simple equations; their derivation takes significantly more space than a StackExchange post provides.
Wolfgang Bangerth
CC BY-SA 3.0
2016-10-10T19:15:26.087Z
2016-10-10T19:15:26.087Z
Wolfgang Bangerth
<p>Yes this is a bit broader question but I want to know which processes play their role in movement of magma. If one wants to model magma movement through rocks, which processes should one not miss at all and which equations discuss these processes?. As a starter I can that we can use Navier Stokes equations for velocity fields, continuity equation for pressure distribution and etc. </p>

<numerical-modelling><convection><magmatism>
Which equations describe complete (or near complete) behaviour of magma moving through rocks?
Ather Cheema
# Which equations describe complete (or near complete) behaviour of magma moving through rocks?

(Asked by: username_0 on 2016-10-10)

Yes this is a bit broader question but I want to know which processes play their role in movement of magma. If one wants to model magma movement through rocks, which processes should one not miss at all and which equations discuss these processes?. As a starter I can that we can use Navier Stokes equations for velocity fields, continuity equation for pressure distribution and etc.

**Tags:** numerical-modelling, convection, magmatism


---

## 2 answers

### Answer by username_1 on 2016-11-25. Score: 1

From McKenzie's [original 1984 paper](http://dx.doi.org/10.1093/petrology/25.3.713) and from [Spiegelman & McKenzie (1987)](http://dx.doi.org/10.1016/0012-821X(87)90057-4) 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, $c$.


---

### Answer by username_2 on 2016-10-10. Score: -1

The equations you are looking for are called the "McKenzie equations". You should look at the papers of Marc Spiegelman at Columbia University for more information.

> **Comments:**
> **username_1**: Could you expand your answer a bit providing the equations and a simple explanation? Thanks  
> **username_2**: I think the papers I am referencing do a much better job at explaining what these equations are and do, than I could do here. These are not simple equations; their derivation takes significantly more space than a StackExchange post provides.  

---

