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.