A new homogenization scheme is proposed to model the effects of surface phenomena occurring at the pore fluid/solid interface on the effective transport properties of multiscale and reactive geosystems, e.g., clayey geomaterials. This new scheme is a generalization of the so-called differential scheme (DS) used to infer the effective properties of a composite made up of several materials. It is named the fractional differential scheme and is based on two key elements: (i) the concept of realizability of the DS itself and (ii) a fractional integral formulation of the DS for a porous medium considered as a two-component composite. The formulation of the fractional DS introduces two parameters: a cementation exponent m and a fractional order α. The cementation exponent m is related to the microstructure of the material. The fractional order α accounts for the amplitude of the “surface” transport of ions resulting from the physico-chemical interactions between hydrated cations and swelling clay minerals. Both parameters m and α are inverted from two sets of data: electrical conductivity measurements obtained on clayey rocks and effective diffusion coefficient measurements acquired from a Na-montmorillonite geomaterial. The inversion results demonstrate that the fractional DS model is able to capture the dependence of the cation concentration on the effective transport properties of the clayey materials under study. Our results also show that the fractional order α can be considered an indirect indicator of the amplitude of physico-chemical interactions between hydrated cations and swelling clay minerals occurring at the pore fluid/solid interface.

中文翻译：

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多尺度反应性多孔介质有效输运特性的分数微分方案：粘土地质材料的应用

提出了一种新的均质化方案来模拟发生在孔隙流体/固体界面处的表面现象对多尺度和反应性地质系统（例如粘土地质材料）的有效输运特性的影响。这种新方案是所谓的差分方案 (DS) 的推广，用于推断由多种材料组成的复合材料的有效特性。它被命名为分数微分方案，基于两个关键要素：(i) DS 本身的可实现性概念和 (ii) DS 的分数积分公式，用于被视为双组分复合材料的多孔介质。分数 DS 的公式引入了两个参数：胶结指数m和分数阶α。胶结指数m与材料的微观结构有关。分数阶α解释了由水合阳离子和膨胀粘土矿物之间的物理化学相互作用引起的离子“表面”传输的幅度。参数m和α都是从两组数据反演的：在粘土岩上获得的电导率测量值和从 Na-蒙脱石地质材料获得的有效扩散系数测量值。反演结果表明，分数 DS 模型能够捕捉阳离子浓度对所研究粘土材料有效传输特性的依赖性。我们的结果还表明分数阶α 可以被认为是水合阳离子与发生在孔隙流体/固体界面处的膨胀粘土矿物之间物理化学相互作用幅度的间接指标。