## 2 Materials and Methods
### Petrophysical Properties Computed from Micro-Tomography Images
#### 2.3.2 X-ray Tomography Data Processing  
After image acquisition, treatments were made before computing rock parameters. They were carried out on a homemade imaging software, previously developed and used [43]. Filters to smooth the pixels histogram (mean filter) and to correct brightness were applied to improve images quality. Then, a segmentation based on a region growing method was used to isolate the void phase of the image. In mono-mineral media, two or three phases are usually observed: the void phase (which can be divided into the macropores phase and the micro-phase, composed of micropores and micro-grains) and the matrix. The macropores phase is composed of pores larger than the pixel size while the microropores and micro-grains smaller than the pixel size are part of the micro-phase. In this study, as previously shown in Figure 1, Normandie has smaller pores than Lexos, which has smaller pores than Euville. In Figure 2, a 2D slice of each rock type is shown. The three structures are very different: Normandie presents a lot of micropores, Euville displays heterogeneous macro-pores, while Lexos shows some homogeneous macro-pores with micropores around. The software used can only display a two-phase segmentation. One phase is composed of macropores (pores larger than the pixel size) while the other phase is considering to be matrix. This segmentation gives a porosity based on macropores, labeled \(\phi_{macro}\), which is consequently largely underestimated. However, adding micropores (pores smaller than 12 \(\upmu\)m) with macropores will induce the incorporation of micro-grains into the void phase, and the corresponding porosity (\(\phi_{total}\)) will be consequently overestimated. Calculations have been made on images segmented with the macro-porosity. In order to obtain a consistent segmentation, the macro-void porosity has been correlated with the porosity from the centrifugation calculations. The pores proportion with a diameter smaller than 12 \(\upmu\)m allow us to know the proportion of the porosity which is not visible on the images.  
Once segmentation is complete, calculations can be executed. Porosity, from the macro-phase and micro-phase, is calculated from the segmentation step. Permeability is calculated from skeleton determination, which corresponds to the path that links the center of each connected pore. The time spent to path through the sample is measured. From this skeleton, the hydraulic tortuosity \(\tau_{h}\) is calculated [15] and displayed in Equation (9). It has been shown that \(\tau_{h}\) and \(\tau_{e}\) (seen in Equation (6)) are similar and comparable [19; 44].  
\[\tau_{h}=\frac{l_{eh}}{l} \tag{9}\]  
where \(l_{eh}\) is the effective path length and \(l\) is the sample length. From the tortuosity, the formation factor (Equation (4)) and the cementation index (Equation (5)) can be deduced.  
The software can also compute some parameters that cannot be obtained by non-destructive laboratory techniques. The pore volume and the pore surface are computed,which corresponds to the void-rock interface. By dividing the pore surface by the pore volume (\(\hat{\xi}\)), a relative pore size is given. A high ratio states for an intricate pore system [15] and a huge quantity of small pores. Moreover, pore size distribution is displayed by two probabilistic methods. The first one is based on the insertion of increasing chord lengths into the void phase [36]. The other method is on the insertion of spheres which have a diameter of one pixel. The sphere size is increasing until it touches the matrix, and the software counts the number of spheres of a given size that fit into the pores, and outputs a pore size distribution. Finally, statistics calculations on connected components allow knowing the proportion of the percolating phase into the volume sample (\(PV\)), which is equivalent to the proportion of the sample used by a fluid to pass through.