*3.1. Gas Permeability*

Most of the tests performed in the laboratory are 1D-flow-type tests on cylindrical samples (Figure 1). The gas pressure is P1 at the upstream sample side and P0 at the downstream side. Using Darcy's law and steady flow [2], we obtain:

$$P(\mathbf{x}) = \sqrt{\mathbf{P}\_1^2 \left(1 - \frac{\mathbf{x}}{\mathbf{L}}\right) + \mathbf{P}\_0^2 \frac{\mathbf{x}}{\mathbf{L}}} \tag{1}$$

**Figure 1.** Schematic diagram of the steady-flow test method.

Q1 is the volumetric gas flowrate at the upstream sample side [2]:

$$\mathbf{Q}\_1 = \frac{\mathbf{K}\_{\text{app}} \mathbf{A}}{2\mu \mathbf{L}} \frac{\mathbf{P}\_1^2 - \mathbf{P}\_0^2}{\mathbf{P}\_1} \tag{2}$$

Kapp is the apparent gas permeability (apparent due to a potential Klinkenberg effect—see further), A is the sample cross-section, L is the sample length, and μ is the gas viscosity.

The flowrate Q1 has to be measured to find the apparent gas permeability. Different methods can be used for this purpose: direct measurement with flowmeters (for example Brooks or Bronkhorst) or a measurement based on small pressure variation techniques (often used to calibrate the usual mass flowmeters). This second method was specially developed in our laboratory for materials with a very low permeability. Figure 2 presents a scheme of the system designed and used for this purpose.

**Figure 2.** The device used for gas-permeability experiments.

The device is composed of a buffer reservoir R1 and a tube reservoir R2, respectively, connected at the upstream and downstream sample sides. The gas is first injected from a big gas tank at constant pressure P1 (or Pi). The valve C1 is closed as soon as a steady flow is assumed and R1 is now feeding the sample with gas. The first possibility is then to measure the incoming flow rate Q1. It is in fact the mean flowrate Q1 mean during a time Δt for which there is a decrease ΔP1 of pressure P1. Assuming that there is a steady flow during Δt at a mean injection pressure P1 mean = P1 − <sup>Δ</sup>Pi/2, it can be easily shown [3] that:

$$\mathbf{Q}\_1^{\text{mean}} = \frac{\mathbf{V}\_1 \Delta \mathbf{P}\_1}{\mathbf{P}\_1^{\text{mean}} \Delta \mathbf{t}} \tag{3}$$

The apparent permeability Kapp can then be deduced from relation 2 in which P1 = P1 mean and Q1 = Q1 mean. This method is called the quasi-steady flow method at high pressure because it is applied at the upstream sample side. Experiments were also conducted in the laboratory with electronic mass flowmeters when it was possible. They provided results that were compared to those given upstream by the quasi-steady method. The same results were virtually obtained with a difference in permeability of often less than 1%, as long as the ΔP1 decrease did not exceed 5% of P1.

V1 is the volume of the R1 reservoir, which includes the tubing volume between R1 and the sample. This volume is obtained with an accurate calibration.
