# Impact of Pore Geometry and Water Saturation on Gas Effective Diffusion Coefficient in Soil
## 2 Materials and Methods
### The Pore-Scale Model  
Gas diffusion in a pore need to overcome the resistance caused both by molecule-wall collision and molecule-molecule collision [40; 41; 42]. The relative dominance of one type of collision over another depends on the ratio of the mean free path \(\lambda\) to the pore diameter \(\delta\), known as Knudsen number \(Kn=\lambda/\delta\). When the Knudsen number is less than 0.01, the gas flow is continuum [43; 44; 42] and the gas transport can be described by the bulk diffusion coefficient. In contrast, if the pore diameter is much smaller than the mean free path, the gas transport is considered in the Knudsen regime and could be described by the Knudsen diffusion coefficient. In a single pore, the effective diffusion coefficient can be described by the dusty gas model as follows, in terms of diffusion coefficients [43; 44; 45; 46; 47; 48; 49]:  
\[d_{eff}\ =\ \frac{d\cdot d_{kn}}{d+d_{kn}}, \tag{1}\]  
Figure 2: Image illustrations of water and pore distributions in the soil (sample 1) under different water saturations \(S\), The left column images show the solid (brown), water (red) and pore (yellow) distributions; the middle column images show the water distribution (red); the right column images show the connected pores (yellow) and unconnected pores (purple): (**a**) \(S=0.825\); (**b**) \(S=0.527\); (**c**) \(S=0.267\). where \(d_{\mathit{eff}}\) is the effective diffusion coefficient of the pore; \(d\) and \(d_{kn}\) are the bulk and Knudsen diffusion coefficient, respectively. Each pore is associated with a bulk diffusion coefficient \(d\) and a Knudsen diffusion coefficient \(d_{kn}\), with the former being a constant and the latter calculated as follows [41; 42; 43; 44]:  
\[d_{kn}\ =\ \frac{\delta}{3}\sqrt{\frac{8 kT}{\pi m}}, \tag{2}\]  
where \(T\) is the temperature, \(k\) is the Boltzmann number, \(m\) is the molecular mass of the gas and \(\delta\) is the pore diameter. In the pore-scale simulation of gas diffusion in soil, we assumed that the system is isothermal and isobaric. We used a constant gas concentration (\(c_{1}\)) on the top and a lower concentration (\(c_{2}\)) on the bottom of the image to impel the gas flow in the soil, and the other four vertical sides were considered as impermeable boundaries. The gas diffusion through the pore geometry of the image is described by the flowing equation [15; 42]:  
\[\frac{\partial c}{\partial t}\ =\ \
abla\cdot d_{\mathit{eff}}\
abla c, \tag{3}\]  
where \(c\) is the gas concentration at each pore voxel in the soil image and \(t\) is the time. \(d_{\mathit{eff}}\) is the effective diffusion coefficient associated with each pore, which can be calculated from Equation (1). Under a specified temperature and pressure, the bulk diffusion coefficient is constant and equal in all the pores, while the Knudsen diffusion coefficient varies spatially depending on the pore diameter, therefore, the effective diffusion coefficient \(d_{\mathit{eff}}\) is spatially variable. We solved Equation (3) numerically using the finite volume method developed by [PERSON] and [PERSON] [50] under the imposed boundary conditions mentioned above. Once the gas diffusion was considered to have attained a steady state, the diffusive flux in each pore voxel was calculated as follows [14; 42]:  
\[q_{i}\ =\ d_{\mathit{eff}}^{i}\
abla c_{i}, \tag{4}\]  
where \(q_{i}\) is the diffusive flux at the pore voxel centered at \(x_{i}\), \(d_{\mathit{eff}}^{i}\) is the effective diffusion coefficient of the pore voxel calculated from Equation (1) and \(\
abla c_{i}\) is the local gas concentration gradient. We calculated the gas concentration and flux at each voxel and then volumetrically averaged them over the whole image. The average diffusive flux and the concentration gradient are assumed to be described by the following equation [14; 42]:  
\[Q\ =\ D_{\mathit{eff}}\
abla C, \tag{5}\]  
where \(Q\) is the average diffusive flux, \(D_{\mathit{eff}}\) is the effective diffusion coefficient of the whole soil and \(\
abla C\) is the average gas concentration gradient.