# Laminar flow in low-permeability material. Which is faster, gas or a liquid?

I consider a simple low-permeability material $(k \sim 10^{-18}\;\text{m}^2)$. Now I either apply a gas (e.g. nitrogen) pressure or liquid (water) pressure of - let's say $10\;\text{MPa}$ - on one side. I'm assuming that the gas viscosity is about two orders of magnitude smaller than that of water, but on the other hand the gas bulk modulus is not constant and smaller.

Is there a straightforward way to approximately estimate, whether the gas or the liquid will permeate faster through the host matrix? As simple as the question sounds I could not find a satisfactory answer up to now. While some lab results I know of suggest the gas to be faster, a simple Darcy-flow code showed the water to be faster.

Thank you very much guys!

Due to the low permeability of the material it is safe to assume that we have low velocities, thus a Poiseuille flow in the pores. For a quick estimate we can consider the entire porous medium as a single channel of some radius $r$ and length $L$.

If we consider the liquid we use the incompressible form of the Poiseuille equation: $$\tag{1} Q_l = \frac{\pi r^4 \Delta P}{8 \mu_l L}$$

Where $Q_l$ is the liquid flow rate that we want to compare this to the gas flow rate $Q_g$.

As you already mentioned, gases have a variable bulk modulus, i.e. are compressible. For a compressible fluid the Poiseuille equation can still be used, but with a correction factor. If we assume an ideal gas at isothermal conditions this correction factor only depends on the inlet and outlet pressures so we get: $$\tag{2} Q_g = \frac{\pi r^4 \Delta P}{8 \mu_g L} \frac{P_{in}+P_{out}}{2 P_{out}}$$ where the correction factor essentially accounts for the gradual expansion of the gas when going from the high pressure inlet to the low pressure outlet.

Now to answer your question, we can equate (1) and (2) to obtain a ratio of flow rates depending on the gas and liquid viscosities and the in- and outlet pressures: $$\tag{3} \frac{Q_g}{Q_l}=\frac{\mu_l}{\mu_g}\frac{P_{in}+P_{out}}{2 P_{out}}$$

This shows that the effect of compressibility of the gas is in fact to enhance the flow rate even more then the lower viscosity already does. So you can safely assume that the gas will flow much faster.

• Thank you for your reply. Do you happen to have a reference where that correction factor is derived? And secondly, although the volumetric flow rate of the gas is much higher, this doesn't neccessarily mean that filling a fixed volume V with both substances will also show a faster pressure increase for gas, or does it? (Because that pressure increase would be proportional to the bulk modulus, which could again cancel out the effect of the higher gas flow rate) – wtfermi Feb 6 '14 at 7:08
• @wtfermi see this link for the compressibility correction factor derivation: repository.tamu.edu/bitstream/handle/1969.1/…. view pages 26 through 34 (of 243) – Armadillo Mar 4 '15 at 5:36