Geofluids

Volume 2017, Article ID 8652560, 16 pages

https://doi.org/10.1155/2017/8652560

## The Dependency of Diffusion Coefficients and Geometric Factor on the Size of the Diffusing Molecule: Observations for Different Clay-Based Materials

^{1}Belgian Nuclear Research Centre (SCK•CEN), Mol, Belgium^{2}KU Leuven, Heverlee, Belgium^{3}RWTH Aachen University, Aachen, Germany

Correspondence should be addressed to Elke Jacops; eb.neckcs@spocaje

Received 7 June 2017; Accepted 29 October 2017; Published 27 December 2017

Academic Editor: Ian Clark

Copyright © 2017 Elke Jacops et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In order to investigate in more detail the relation between the size of diffusing molecules and their diffusion coefficients (and geometric factors), diffusion experiments with gases of different size and tritiated water (HTO) have been performed on different clayey samples (Boom Clay, Eigenbilzen Sands, Opalinus Clay, Callovo-Oxfordian Clay, and bentonite with different dry densities). We observed that, for unreactive gases in clayey materials, the effective diffusion coefficient varies with the size of the diffusing molecule and this variation can be described by an exponential or a power law function. The variation of the geometric factor can also be described by an exponential function. The observed experimental relations can be used to estimate diffusion coefficients; by measuring experimentally in clay the effective diffusion coefficient of two unreactive dissolved gases with a different size, the diffusion coefficients of other dissolved gases (with a size in between the two measured gases) can be estimated by using the fitted exponential relationship.

#### 1. Introduction

Clay-based materials are considered by many countries in their concepts for the safe disposal of high- and intermediate-level radioactive waste, either as the material of choice in the engineered barrier system or because of the choice of argillaceous formations to host the repository. Examples of European countries where argillaceous formations are being explored as potential host formations are Belgium, Switzerland, and France [1–3]. The use of clay-based materials as engineered barriers is studied in, for example, Switzerland [4], France [5], and Sweden [6]. The clays under consideration all have a high sorption capacity for many radionuclides [7–9], a low hydraulic conductivity [10, 11], and interesting self-sealing properties [12, 13].

In Belgium, no formal decision has been taken yet on a host formation, but for R&D purposes, the Belgian Radioactive Waste Management Organization (ONDRAF/NIRAS) considers Boom Clay (BC) as a potential host formation for a geological disposal facility. In France, the National Agency for Radioactive Waste Management (Andra) selected the Callovo-Oxfordian Clay Formation in the east of France as a potential host formation [14] and in Switzerland the National Cooperative for the Disposal of Radioactive Waste (NAGRA) proposed the Jurassic Opalinus Clay (OPA) as a host. As engineered barrier, mainly bentonite is studied. A frequently used type of bentonite is MX80, which is studied as backfill material by Andra [5, 15], NAGRA [4], and the Swedish Nuclear Fuel and Waste Management Co. (SKB) [16, 17].

In the context of nuclear waste disposal, the transport of dissolved gases in compacted clays is an area, which receives a high amount of interest. First, the production of gas within a geological repository is unavoidable. Mainly anaerobic corrosion of metals will lead to the production of hydrogen. If the rate of gas generation is larger than the diffusive flux into the clay, a free gas phase will form, which might have a negative effect on the performance of the barriers. In order to compute a comprehensive and reliable balance between gas generation versus gas dissipation, correct estimates for gas diffusion coefficients of dissolved gases are essential. Moreover, the produced hydrogen may be converted to other gases like CH_{4} due to, for example, microbial activity. Thus, also the diffusion coefficient of methane needs to be established.

Secondly, naturally occurring noble gases such as He and Ar can act as natural tracers whose profiles can be used to constrain transport properties on the scale of the formation [18–20]. As diffusion is considered to be the dominant transport mechanism, it should be possible to represent these natural tracer profiles by diffusion models, but the availability of reliable diffusion coefficients for He and Ar is limited.

Measuring reliable diffusion coefficients of gases is not evident [21] and only a limited amount of data is available in the literature [22]. Recently, diffusion coefficients of dissolved gases (He and Ar) have been measured in Boom Clay, Callovo-Oxfordian Clay, and Opalinus Clay and results have been reported by Jacops et al. [22] and Jacops et al. [23].

In case of hydrogen, Jacops et al. [24] have shown that diffusion experiments with hydrogen often suffer from microbial activity: here methanogenic microbes convert H_{2} into CH_{4}, making the accurate determination of diffusion coefficients impossible [24]. Similar observations have been reported by Vinsot et al. [25].

When performing scoping calculations on the diffusive mobility and possible build-up of dissolved gases in a geological repository or in a geological formation, reliable gas diffusion coefficients obtained from laboratory experiments are often not available. Hence, these diffusion coefficients are often estimated from measured values of other species like HTO. In case of hydrogen, the diffusion coefficient for helium is often used as a surrogate. In this approach, it is implicitly assumed that the geometric factor of the formation (a factor which describes the effect of the porous network on diffusion and for which the value obtained for HTO is used) is equal for all other gases and species considered (e.g., [26]).

Different approaches for estimating the geometric factor exist: for example, (i) from diffusion experiments (mostly with HTO [26, 27]), (ii) by models (e.g., [28–31]), or (iii) from diffusion simulations on reconstructed clay structures [32, 33]. Archie’s law [28] is a well-known empirical relation relating diffusivity to porosity (but neglecting constrictivity). Variations on Archie’s law are proposed, for example, by Weissberg [34] and Boudreau [29]. Saripalli et al. [30] described a method to calculate tortuosity and constrictivity from the specific surface determined from, for example, N_{2} adsorption measurements. Chou et al. [31] discussed models to calculate tortuosity for variously saturated soil samples, based on their water content. Both Robinet et al. [32] and Keller et al. [33] calculated geometric factors from diffusion simulations on reconstructed mesostructures of, respectively, the Callovo-Oxfordian and the Opalinus Clay. Their developed approach allows the determination of the geometric factor as a function of mineralogy. All these methods calculate the geometric factor as an intrinsic parameter of a specific material, without taking into account the possible effect of the size of the diffusing molecule.

In clay, a relation between the size of the diffusing molecule (expressed by the kinetic diameter) and its diffusion coefficient has been observed [22, 23]. The effect of the size of diffusing molecules on its diffusion coefficient in Callovo-Oxfordian Clay has also been discussed by Dagnelie et al. [35]. For anions, a relation was shown between the anion size (expressed as with the molecular mass) and the ratio of aqueous diffusion coefficients . For cations, a similar relation exists but size is expressed as 1/hydrated radius.

For mortars with a different sand content, the measured geometric factors for Li^{+}, Cl^{−}, and HTO are relatively similar, leading to the conclusion that the formation factor can be used to determine the order of magnitude of the effective diffusion coefficient of other diffusion species [36]. Analyses of these data indicate a variability of the geometric factor with the size of the diffusing molecule, but this is less pronounced due to the similar size of Li^{+}, Cl^{−}, and HTO. The dependence of the geometric factor on the size of the diffusion molecule is demonstrated in diffusion experiments with He, O_{2}, and Xe in mortars (unpublished data).

The main objective of this paper is to investigate in more detail the relation between the size of the diffusing gas molecules and their diffusion coefficients and hence geometric factors in different clayey materials. We also investigate whether this relation can be used to estimate diffusion coefficients of gases, based on their size.

This objective is achieved by performing diffusion experiments with gases of different size, on different clayey samples (Boom Clay, Eigenbilzen Sands, Opalinus Clay, Callovo-Oxfordian Clay, and bentonite (Volclay KWK with different dry densities)).

#### 2. Materials and Methods

##### 2.1. Clay Samples

The Boom Clay is a marine sediment that was deposited in the early Oligocene (Rupelian), 29 to 32 million years ago in the North Sea Basin, at water depths between 50 and 100 m [37]. Shortly after deposition, the accumulated sediment became reducing which is reflected in the common occurrence of framboidal pyrite. The Boom Clay comprises clay minerals (up to 60%, dominated by illite, mixed layered illite-smectite, kaolinite, and traces of chlorite), as well as quartz, K-feldspar, Na-plagioclase, pyrite, and carbonates [38]. It consists of different lithological subunits; more specifically a rhythmic alteration of silty and more clay-rich layers has been observed, as well as the presence of organic- and carbonate-rich layers. Based on these lithological variations, the Boom Clay has been subdivided in four members: the Boeretang Member, the Putte Member, the Terhagen Member, and the Belsele-Waas Member [37]. The sandy unit lying above the Boom Clay is named the Eigenbilzen Sands. The latter consists of dark green, glauconite-rich, clayey, fine-grained to medium-grained sands, with bioturbations [39]. The amount of fine sand increases significantly but the alternation of silty and clayey intervals as observed in the Boom Clay remains [37]. Samples from this formation are in this work described as “clayey sand.”

One of the topics under investigation is the effect of variations in the clay and silt/sand content on the diffusion parameters [23]. Therefore, samples have been selected based on their location in the lithostratigraphic column. The clayey samples originate from the Putte or Terhagen Member, whereas the clayey sand samples originate from the Eigenbilzen Sands. Whether the samples are a representative selection or not is investigated by mineralogical analyses, by grain size analyses, and by measuring the hydraulic conductivity. More information on the origin of the used samples and their orientation with respect to bedding plane can be found in Table 1.