I'm trying to calculate the time for the liquid in two connected reservoirs (open to the air) to equilibrate. The tricky part is the connecting region, which consists of a porous medium. I understand that without the porous medium, I can apply Torricelli's law to account for the change of pressure head over time. However, Torricelli's law doesn't apply to flow through porous medium. I also understand that I can apply Darcy approximation for flow through a porous medium. But Darcy law doesn't take into account for the change in pressure term in the equation.
What I have tried:
1) I tried derive the solution by combining Torricelli's law and Darcy law. But the solution turned out to be weird where I get t = Constant* ln(ΔH) where the Constant consists of area of reservoir, cross sectional area of opening, permeability of the porous medium, length, viscosity, density, gravitational force. So this method failed.
2) I tried looking for unsteady flow solution for Darcy equation. Turned out that there are a few ways such as VANS approximation, Virtual Mass approximation and Volume-average energy equation. To be honest, I failed to follow through the whole process of derivation and I also realized that the solution which the paper provided may not be what I have been looking for. The unsteady state flow that they describe is the initial fluctuation of velocity profile but not a change due to decreasing pressure head.
My question is, how can I proceed to calculate the time for the two reservoirs to equilibrate?
Known parameters: - Initial ΔH - Permeability of the porous column (assumed to be a constant throughout) - Area of reservoir - Cross-sectional area of connecting part - Viscosity - Density - Length
Calculation as suggested by Chester:
The problem is with ln(0) and also if my ΔH is actually smaller than 1, then it will result in a negative value.
*Sorry that I uploaded this as a picture as I am not sure how to insert equations and special characters here.