For a given porous medium with known porosity, particle size and other necessary material properties, how can I arrive at an equation which would express the time required for the given porous medium to fully saturate? The fluid properties and flow conditions (flow rate, pressure drop, etc.) are known. Can I express Darcy's equation as a time-dependent equation?
The easiest way is to combine the continuity equation with conservation equation. The derivation of continuity equation for a porous media are widely presented in all statistics mechanics textbook.
The most rigorous way is to write down the Navier Stokes equation in the appropriate coordinate system, then simplify it according to your valid possible assumption, and solve it.
Darcy's law is a special case of Navier Stokes equations, it may not be a wise way to start with.
https://www.sciencedirect.com/science/article/pii/B9780123838483000037
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