3.1 Conservation of mass
To solve for the flow of fluid and solid, and therefore the
redistribution of melt that accompanies compaction in a compacting
layer, we first consider the conservation of mass in the fluid phase:
\begin{equation}
\left(1\right)\ \frac{\partial\phi}{\partial t}\ +\ \frac{\partial}{\partial z}\left[\text{ϕ\ }v_{f}\right]=0,\ \nonumber \\
\end{equation}where \(\phi\) is the porosity or melt fraction, \(v_{f}\) is the fluid
velocity, \(z\) is depth, and \(t\) is time. Similarly, mass
conservation in the matrix is written as:
\begin{equation}
\left(2\right)\ \frac{\partial\left(1-\phi\right)}{\partial t}\ +\ \frac{\partial}{\partial z}\left[\left(1-\phi\right)v_{m}\right]=0,\ \nonumber \\
\end{equation}where \(v_{m}\) is the matrix velocity and the solid and the fluid phase
is assumed to be incompressible.