# Time for Two Reservoirs that are Connected with a Porous Medium to Equilibrate

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.

• Darcy's Law does take into account the pressure difference across the porous plug, which is $\rho g \Delta H$. Have you tried using that alone? ie Forget Torricelli's Law. If you are still having difficulty, please show your calculation. Aug 4, 2016 at 18:18
• After giving it a second thought, I think you may be right that I can directly do an integration with Darcy's equation. This is also suggested by Chester. I am going to try that out. Thanks!
– SHL
Aug 5, 2016 at 1:35

The pressure difference across the porous plug is given by Darcy's law:

$$P_L-P_R=\frac{\mu}{k}vL\tag{1}$$where L is the length of the plug, k is the permeabiity of the plug, v is the superficial velocity through the plug, and $\mu$ is the fluid viscosity. The volumetric flow rate through the plug is given by $vA$, where A is the area of the plug. So, $$A_c\frac{d(\Delta H)}{dt}=-2vA\tag{2}$$where $A_c$ is the cross sectional area of each of the two tanks. If we neglect kinetic energy effects, then $$P_L-P_R=\rho g\Delta H\tag{3}$$If you combine these equations, you obtain a differential equation for the time variation of $\Delta H$ involving only the areas, the permeability, the viscosity, the density, g, and the length of the plug.

• Turned out what you have proposed is very similar to what I have done in my approach 1. I have a few doubts here though. First, in equation (2), where does the "2" come from? Second, I still get t= Constant* ln(ΔH) as my solution which doesn;t make sense to me though. I will show my calculation in the question later. Thanks for your help!
– SHL
Aug 5, 2016 at 1:52
• Sorry about your doubts. I have practical experience in flow through porous media. How can I help alleviate your doubts? Aug 5, 2016 at 1:54
• @SHL : The factor of 2 arises because a decrease in height by 1m on one side causes an increase in height of 1m on the other side, hence the difference in height decreases by 2m. Aug 5, 2016 at 2:10
• @Chester Miller : I have uploaded the calculation steps and I couldn't get a reasonable solution. Is there anything that I have done wrong? Thanks.
– SHL
Aug 5, 2016 at 2:41
• @ Sammy Gerbil: Thanks for explaining that to me, that makes a lot of sense!
– SHL
Aug 5, 2016 at 2:43