*3.3. Hypotheses and Test Analysis*

By using some hypotheses, it is supposed that the diffusion process is controlled by the first Fick's law (1856), which, with a 1D geometry, gives:

$$\mathbf{J}\_{\mathbf{x}}(\mathbf{x}, \mathbf{t}) = -\mathbf{D} \frac{\partial \mathbf{c}(\mathbf{x}, \mathbf{t})}{\partial \mathbf{x}} \tag{4}$$

Jx is the molar surface flow in mol·s−1·m<sup>−</sup>2; D is the effective diffusion coefficient in m2·s<sup>−</sup>1; c is the gas concentration in mol·m<sup>−</sup>3.

The goal is then to measure J through the sample to obtain D under known or measured concentrations at the upstream and downstream sample sides. To be applied, Equation (3) needs a knowledge of the concentration 'c(x,t)' through the sample, which evolves with time. Under a constant molar flow Jx, the downstream concentration increases linearly with

time. At this stage, the mass balance equation coupled with Equation (3) leads to a linear concentration profile in the sample. This will be the main hypothesis used in the results.

The room temperature is constant and controlled at 22 ◦C or 295 K. Assuming that helium is a perfect gas (Patm V=nuRT), it is found that nu = 40.8 moles per unit volume (1 m3); thus, the concentration at the upstream side is cu = 40.8 mol/m3.

The molar flux through the sample cross-section will be:

$$
\boldsymbol{\upvarphi}(\mathbf{x}, \mathbf{t}) = \mathbf{J}\_{\mathbf{x}}(\mathbf{x}, \mathbf{t}) \text{ A} \tag{5}
$$

<sup>ϕ</sup> molar flux through surface A in mol·s<sup>−</sup>1; A sample cross-section in m2.

For the calculations, it was supposed that the downstream helium concentration can be neglected in front of the upstream one, i.e., cd << cu. This leads to:

$$\frac{\partial \mathbf{c}(\mathbf{x}, \mathbf{t})}{\partial \mathbf{x}} \approx \frac{-\mathbf{c}\_{\mathbf{u}}}{\mathbf{L}} \tag{6}$$

L is the sample length.

Under a stationary flow, 'p', the rate of helium particles' increase into the downstream reservoir, is directly linked to ϕ:

$$\boldsymbol{\upmu} = \mathbf{p} \, \frac{\mathbf{V\_d}}{\mathbf{V\_m}} \, \tag{7}$$

Vd downstream reservoir volume in m3; Vm molar volume (at Patm) in m3·mol<sup>−</sup>1; 'p' is in s<sup>−</sup>1.

The following is thus obtained:

$$\mathbf{D} = \frac{\mathbf{p} \mathbf{L} \mathbf{V\_d}}{\mathbf{A} \mathbf{c}\_\mathbf{u} \mathbf{V\_m}} \tag{8}$$

with

Vd = 1.09 × <sup>10</sup>−<sup>3</sup> <sup>m</sup>3; Vm = 24.05 × <sup>10</sup>−<sup>3</sup> <sup>m</sup>3·mol<sup>−</sup>1; cu = 40.8 mol·m<sup>−</sup>3.

#### **4. Results**

#### *4.1. Gas-Permeability Results*

4.1.1. Results with Argon

Gas-permeability tests were performed with argon gas on dry material after the porosity measurements. Three injection pressures were used: 0.5, 1 and 1.5 MPa, in order to quantify the potential Klinkenberg effect [4]. This effect, also known as the 'slipping effect', may occur when the mean gas free path 'λ' is close to or lower than the mean pore size. As 'λ' is increasing when the gas pressure is decreasing, this effect is often visible on material with small pores (like concrete) and/or during tests with a weak injection pressure. As a result, if this effect is present, the measured permeability is apparent and higher than the intrinsic one. A very well-known correction was brought by Klinkenberg [4] in order to take this effect into account:

$$\mathbf{K\_{\rm app}} = \mathbf{K\_{\rm int}} \left( 1 + \frac{\beta}{\mathbf{P\_m}} \right) \tag{9}$$

Kapp is the apparent permeability (m2);

Kint is the intrinsic permeability;

β is the Klinkenberg coefficient and Pm is the mean test pressure:

$$\mathbf{P\_m} = \frac{1}{\mathcal{L}} \int\_0^\mathcal{L} \mathbf{P(x)dx} \tag{10}$$

From relation (8), it can be seen that three different injection pressures are sufficient to assess and correct the Klinkenberg effect.

Two confining pressures (hydrostatic pressures) had been required by our partner ONDRAF/NIRAS: 2.25 and 4.5 MPa. Such a change in confining pressure can induce a significant permeability variation depending on whether the material is (micro-)cracked [3,5]. A typical result, obtained from sample OB-111, can be seen in Figure 5, and the whole set of intrinsic permeability results is presented in Table 2.

**Figure 5.** Example of Klinkenberg effect: tests performed at 2.25 or 4.5 confining pressure.


**Table 2.** Gas permeability results obtained with argon.

First of all, these results show a good material homogeneity in terms of its gas permeability. The confinement effect is very weak; this means that the material is not significantly cracked, as it is well known [6] that cracks close with a confining pressure, which in return induces a strong (and non-reversible) reduction in permeability. This is not the case here. The Klinkenberg effect is actually present but can be considered as being quite low.

### 4.1.2. Results with Helium

As mentioned before, the diffusion test will be performed with helium and not with argon. The results obtained will be used to evaluate the respective proportion of gas transfer due, respectively, to diffusion and permeation. It is thus important to compare the 'argon permeability' with the 'helium permeability'. One comparative set of tests was therefore performed on sample OB-422. The Klinkenberg effect is more sensitive with helium, as can be seen in Figure 6. This is consistent with the fact that the helium molecule size is lower than the argon molecule size [7]. 'β' is supposed to vary as 1/r, 'r' being the radius of the molecule. 'r' is three times higher for argon than for helium. This ratio (1/3) is more or less respected by the 'β' coefficients presented in Table 3. On the other hand, it is clear that argon's and helium's intrinsic permeabilities are virtually the same, i.e., both can be used, without significant differences, to compare the respective flow resulting from permeation or from diffusion.

**Figure 6.** Permeability and Klinkenberg effect: argon vs. helium.


#### *4.2. Gas-Diffusion Results*

Helium-diffusion tests were carried out right after the permeability experiments. The downstream helium concentration is given in Figure 7. One can observe a very good homogeneity in these results for the four samples, as was also the case for the gas permeability. The four tests were performed at a 2.25 MPa confining pressure, as it was supposed that this pressure did not play a crucial role for permeability measurements. It is clear that, after around 3500 s, a permanent flow rate can be assumed, as the He concentration increases linearly. This provides evidence that for the dry concrete, the diffusion phenomenon is quite rapid. The slope 'p' (presented in §3.3) can be obtained from these results, allowing for the calculation of the diffusion coefficient 'D' with relation (7). The results are presented in Table 4; they lie within the range of the gas-diffusion coefficient often reported for concrete [8].

**Table 4.** Diffusion coefficient for the four samples.


**Figure 7.** Helium proportion at the downstream side of samples.

### *4.3. Equivalent Permeability and Discussion*

As mentioned before, the pressure due to gas production in a radioactive-waste storage should slowly increase. It is thus interesting to evaluate, in the case of a very low pression gradient, the proportion of gas transfer resulting from permeation and diffusion. This calculation was made within some simplified hypotheses. In particular, it was assumed that the diffusion coefficient does not depend on the gas pressure, which is not the case. The first step is to calculate the equivalent permeability due to diffusion. If an experiment is conducted with the gas pressure (pure helium, for example) at injection pressure P1 (upstream sample side) and the drainage pressure at P0 = Patm (air), it is supposed here that the diffusion coefficient helium-air is almost the same as for helium-nitrogen. If the test is interpreted as a permeability test, the downstream volumetric flowrate Q0 is given by:

$$\mathbf{Q}\_0 = \mathbf{A} \frac{\mathbf{K}}{2\mu\mathbf{L}} \frac{\left(\mathbf{P}\_1^2 - \mathbf{P}\_0^2\right)}{\mathbf{P}\_0} \tag{11}$$

This is linked to the number of moles per second nk:

$$\mathbf{m\_k} = \frac{\mathbf{Q\_0}}{\mathbf{V\_m}} \tag{12}$$

In a diffusion test, this quantity is the same as ϕ mentioned before:

$$\frac{\mathbf{Q}\_0}{\mathbf{V}\_\mathbf{m}} = \mathbf{n}\_\mathbf{k} = \boldsymbol{\varrho} = \mathbf{J} \cdot \mathbf{A} = \mathbf{D} \frac{\mathbf{c\_u}}{\mathbf{L}} \mathbf{A} \tag{13}$$

The helium downstream concentration is still neglected, while the upstream concentration is:

$$\mathbf{c}\_{\mathsf{u}} = \mathsf{n}\_{\mathsf{u}} \frac{\mathsf{P}\_{1}}{\mathsf{P}\_{0}} \tag{14}$$

Then:

$$\frac{\mathbf{Q}\_0}{\mathbf{V}\_\mathbf{m}} = \mathbf{D} \frac{\mathbf{n}\_\mathbf{u}}{\mathbf{L}} \mathbf{A} \frac{\mathbf{P}\_1}{\mathbf{P}\_0} = \mathbf{A} \frac{\mathbf{K}}{2\mu\mathbf{L}} \frac{\left(\mathbf{P}\_1^2 - \mathbf{P}\_0^2\right)}{\mathbf{P}\_0} \frac{1}{\mathbf{V}\_\mathbf{m}}\tag{15}$$

The equivalent permeability KD can then be extracted from relation (15):

$$\mathbf{K\_D = 2D\mu n\_u V\_m \frac{P\_1}{\left(P\_1^2 - P\_o^2\right)}} \tag{16}$$

As can be seen in relation (15), the equivalent permeability depends on the injection pressure P1. KD is roughly in the form Cste/P1 when P1 is increased. This means that the proportion of flow due to diffusion will be lesser and lesser as P1 is increased. This is illustrated in Figure 8, which presents the ratio KD/K, in which K has been chosen as a mean value of 1.5 × <sup>10</sup>−<sup>16</sup> m2. This ratio is equivalent to the proportion of gas flow due to diffusion compared to the one due to permeation.

**Figure 8.** Flow ratio due to diffusion compared to the one due to permeation. P1 is the absolute injection gas pressure.

#### **5. Conclusions**

A special device has been designed in order to measure gas diffusion through concrete (or other porous materials). The main goal of this study was to measure both the gas permeability and gas diffusion of an industrial concrete, which could be employed for tunnels intended for radioactive-waste storage. Gas production due to corrosion and water radiolysis should take place in these structures, and the low rate of production would first induce gas transfer at very low pressures, which are generally not used in gas-permeability experiments. Gas-permeability and diffusion tests were then performed on the same set of samples. They revealed that argon and helium permeability is virtually the same when corrected from a (slight) Klinkenberg effect. On the whole, gas permeability was found to be very homogeneous (order of magnitude of 1.5 × <sup>10</sup>−<sup>16</sup> m2). This homogeneity was also verified for the effective diffusion coefficients (around 4 × <sup>10</sup>−<sup>8</sup> m2/s). These coefficients were used to calculate an equivalent permeability KD, which is dependent on the gas injection pressure. This clearly showed that under a low pressure gradient (or injection pressure), diffusion is largely predominant, whereas its induced flow can be neglected as soon as the injection pressure is larger than a few bars. This implies that gas diffusion must be taken into account at the beginning of gas production. Such a study should find a logical extension in the case of partially saturated concrete, which is likely to be encountered in 'in situ' structures.

**Author Contributions:** T.D., L.P. and F.S. conceived and planned the experiments. T.L. and L.P. carried out the experiments. T.L. and F.S. planned and carried out the analyses. S.L., T.L. and F.S. contributed to the interpretation of the results. F.S. and S.L. took the lead in writing the manuscript. All authors have read and agreed to the published version of the manuscript.

**Funding:** The publication cost of this paper was covered with funds from the Polish National Agency for Academic Exchange (NAWA): "MATBUD'2023-Developing international scientific cooperation in the field of building materials engineering" BPI/WTP/2021/1/00002, MATBUD'2023.

**Acknowledgments:** The authors thank ONDRAF/NIRAS for its technical and scientific support.

**Conflicts of Interest:** The authors declare no conflict of interest.
