#
About Gas Permeability and Diffusion through Concrete^{ †}

^{1}

^{2}

^{3}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Material and Sample Preparation

## 3. Experimental Setup and Experimental Conditions

#### 3.1. Gas Permeability

_{1}at the upstream sample side and P

_{0}at the downstream side. Using Darcy’s law and steady flow [2], we obtain:

_{1}is the volumetric gas flowrate at the upstream sample side [2]:

_{app}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.

_{1}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.

_{1}and a tube reservoir R

_{2}, respectively, connected at the upstream and downstream sample sides. The gas is first injected from a big gas tank at constant pressure P

_{1}(or P

_{i}). The valve C

_{1}is closed as soon as a steady flow is assumed and R

_{1}is now feeding the sample with gas. The first possibility is then to measure the incoming flow rate Q

_{1}. It is in fact the mean flowrate Q

_{1}

^{mean}during a time Δt for which there is a decrease ΔP

_{1}of pressure P

_{1}. Assuming that there is a steady flow during Δt at a mean injection pressure P

_{1}

^{mean}= P

_{1}− ΔP

_{i}/2, it can be easily shown [3] that:

_{app}can then be deduced from relation 2 in which P

_{1}= P

_{1}

^{mean}and Q

_{1}= Q

_{1}

^{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 ΔP

_{1}decrease did not exceed 5% of P

_{1}.

_{1}is the volume of the R

_{1}reservoir, which includes the tubing volume between R

_{1}and the sample. This volume is obtained with an accurate calibration.

#### 3.2. Diffusion Test—Principle of the Method

#### 3.3. Hypotheses and Test Analysis

- J
_{x}is the molar surface flow in mol·s^{−1}·m^{−2}; - D is the effective diffusion coefficient in m
^{2}·s^{−1}; - c is the gas concentration in mol·m
^{−3}.

_{x}, 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.

_{atm}V = n

_{u}RT), it is found that n

_{u}= 40.8 moles per unit volume (1 m

^{3}); thus, the concentration at the upstream side is c

_{u}= 40.8 mol/m

^{3}.

- $\mathsf{\phi}$ molar flux through surface A in mol·s
^{−1}; - A sample cross-section in m
^{2}.

_{d}<< c

_{u}. This leads to:

- ${\mathrm{V}}_{\mathrm{d}}$ downstream reservoir volume in m
^{3}; - ${\mathrm{V}}_{\mathrm{m}}$ molar volume (at P
_{atm}) in m^{3}·mol^{−1}; - ‘p’ is in s
^{−1}.

- V
_{d}= 1.09 × 10^{−3}m^{3}; - V
_{m}= 24.05 × 10^{−3}m^{3}·mol^{−1}; - c
_{u}= 40.8 mol·m^{−3}.

## 4. Results

#### 4.1. Gas-Permeability Results

#### 4.1.1. Results with Argon

- K
_{app}is the apparent permeability (m^{2}); - K
_{int}is the intrinsic permeability; - β is the Klinkenberg coefficient and P
_{m}is the mean test pressure:

#### 4.1.2. Results with Helium

#### 4.2. Gas-Diffusion Results

#### 4.3. Equivalent Permeability and Discussion

_{1}(upstream sample side) and the drainage pressure at P

_{0}= 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 Q

_{0}is given by:

_{k}:

_{D}can then be extracted from relation (15):

_{1}. K

_{D}is roughly in the form Cste/P

_{1}when P

_{1}is increased. This means that the proportion of flow due to diffusion will be lesser and lesser as P

_{1}is increased. This is illustrated in Figure 8, which presents the ratio K

_{D}/K, in which K has been chosen as a mean value of 1.5 × 10

^{−16}m

^{2}. This ratio is equivalent to the proportion of gas flow due to diffusion compared to the one due to permeation.

## 5. Conclusions

^{−16}m

^{2}). This homogeneity was also verified for the effective diffusion coefficients (around 4 × 10

^{−8}m

^{2}/s). These coefficients were used to calculate an equivalent permeability K

_{D}, 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

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Andreola, F.; Leonelli, C.; Romagnoli, M.; Miselli, P. Techniques Used to Determine Porosity. Am. Ceram. Soc. Bull.
**2000**, 79, 49–52. [Google Scholar] - Zhang, D.; Agostini, F.; Jeannin, L.; Skoczylas, F. New Insights Brought by Micro-Tomography to Better Understand Gas Transfer Property Variation and Coupling Effects in Salt Rocks. Rock. Mech. Rock Eng.
**2021**, 54, 6457–6480. [Google Scholar] [CrossRef] - Chen, X.; Caratini, G.; Davy, C.A.; Troadec, D. Skoczylas, Coupled transport and poro-mechanical properties of a heat-treated mortar under confinement. Cem. Concr. Res.
**2013**, 49, 10–20. [Google Scholar] [CrossRef] - Klinkenberg, L.J. The permeability of porous media to liquids and gases. In Drilling and Production Practices; American Petroleum Institute: Washington, DC, USA, 1941; pp. 200–213. [Google Scholar]
- Chen, W.; Han, Y.; Agostini, F.; Skoczylas, F.; Corbeel, D. Permeability of a Macro-Cracked Concrete Effect of Confining Pressure and Modelling. Materials
**2021**, 14, 862. [Google Scholar] [CrossRef] [PubMed] - Ding, Q.; Wang, P.; Cheng, Z. Influence of temperature and confining pressure on the mechanical properties of granite. Powder Technol.
**2021**, 394, 10–19. [Google Scholar] [CrossRef] - Degao, H.; Feng, Y.; Zhiguo, S.; Aiwei, Z.; He, Z.; Bin, L. Experimental study about the gas slip flow in Longmaxi shales from the southern Sichuan Basin. Bull. Geol. Sci. Technol.
**2021**, 40, 36–41. [Google Scholar] [CrossRef] - Sercombe, R.; Vidal, C.; Gallé, F. Adenot, Experimental study of gas diffusion in cement paste. Cem. Concr. Res.
**2007**, 37, 579–588. [Google Scholar] [CrossRef] [Green Version]

**Figure 3.**Schematic principle of the diffusion test; n

_{u}is the number of helium moles, c

_{u}and c

_{d}are helium concentrations at the upstream (u) or downstream (d) sides, respectively. V

_{d}is the downstream reservoir volume.

**Figure 8.**Flow ratio due to diffusion compared to the one due to permeation. P

_{1}is the absolute injection gas pressure.

Water kg/m ^{3} | Cement CEM II/B-V kg/m ^{3} | Fly Ash kg/m ^{3} | Coarse Agg. (5–14 mm) kg/m ^{3} | Fine Agg. (0–4 mm) kg/m ^{3} |
---|---|---|---|---|

135 | 335 | 115 | 1252 | 540 |

Sample | Confining Pressure (MPa) | K_{int} (10^{−16} m^{2}) | Coef. β (bar) |
---|---|---|---|

OB-111 | 2.25 4.5 | 1.66 1.45 | 0.4 0.38 |

OB-121 | 2.25 4.5 | 1.73 1.5 | 0.31 0.34 |

OB-321 | 2.25 4.5 | 1.57 1.35 | 0.28 0.35 |

OB-422 | 2.25 4.5 | 1.37 1.16 | 0.31 0.52 |

Sample | Confining Pressure (MPa) | K_{int} (10^{−16} m^{2}) | β (Bars) |
---|---|---|---|

OB-422 Argon | 2.25 | 1.33 | 0.38 |

4.5 | 1.15 | 0.48 | |

OB-422 Helium | 2.25 | 1.32 | 1.02 |

4.5 | 1.16 | 1.09 |

Référence | p (s^{−1}) | D (m^{2}·s^{−1}) |
---|---|---|

OB-111 | 2.5 × 10^{−6} | 4.2 × 10^{−8} |

OB-212 | 2.4 × 10^{−6} | 3.9 × 10^{−8} |

OB-321 | 2.6 × 10^{−6} | 4.4 × 10^{−8} |

OB-422 | 2.4 × 10^{−6} | 3.9 × 10^{−8} |

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**MDPI and ACS Style**

Lamouchi, T.; Levasseur, S.; Potier, L.; Dubois, T.; Skoczylas, F.
About Gas Permeability and Diffusion through Concrete. *Mater. Proc.* **2023**, *13*, 42.
https://doi.org/10.3390/materproc2023013042

**AMA Style**

Lamouchi T, Levasseur S, Potier L, Dubois T, Skoczylas F.
About Gas Permeability and Diffusion through Concrete. *Materials Proceedings*. 2023; 13(1):42.
https://doi.org/10.3390/materproc2023013042

**Chicago/Turabian Style**

Lamouchi, Takwa, Severine Levasseur, Ludovic Potier, Thierry Dubois, and Frédéric Skoczylas.
2023. "About Gas Permeability and Diffusion through Concrete" *Materials Proceedings* 13, no. 1: 42.
https://doi.org/10.3390/materproc2023013042