## 4 Results
### Numerical Flow Experiments  
Numerical flow experiments were carried out on X-CT image H3, derived from sample F3, using the workflow detailed in Section 3.5 (Figure 4). Figure 11 illustrates a representative set of results for the synthetic pressure variations for flow through sub-volumes of this image, in the x-, y-, and z- directions. Three values of greyscale cutoff were adopted, low, mid-range, and high values, which were 7400, 7700, and 8000 for the image as depicted in Figure 5 (and as archived online), representing the range of uncertainty in the identification of the distinction between pore space and mineral grains. Figure 12 illustrates the effect of changing the greyscale threshold, to delineate pore-space, on flow prediction. Many such sub-volumes were analyzed, resulting in a suite of predictions of porosity and permeability, illustrated in Figure 13.  
Plotting total porosity against permeability (Figure 13) indicates that these parameters are correlated. It is also apparent that permeability is uniformly highest for flow parallel to the z-axis, and generally slightly higher parallel to the y-axis than parallel to the x-axis. Such variations are evident from the predicted pressure variations being highest in  
Figure 12: Simulated pressure variations across a sub-volume of sample F3, calculated for flow parallel to the z axis, taking different grey-scale thresholds as delineating pore-space from filled space. (**A**) Using the mid-range grey-scale threshold. (**B**) Using the high grey-scale threshold, which predicts more pore space enabling flow to occur with much smaller pressure variations, giving a higher permeability. Figure 13: Total porosity and permeability for each analyzed sub-volume, for simulated flow parallel to the x, y and z axes, for the three greyscale cutoffs for pore space. Thus, for example, ‘mid-Z’ means mid range cutoff and flow in the z direction. (**a**) Low cutoff, 7400. (**b**) Mid range cutoff, 7700. (**c**) High cutoff, 8000. (**d**) The combined datasets, along with Kozeny-Carman type curve fits, after Equation (4), using parameters in Table 7, for flow in the z-direction (i.e., vertical), the x and y directions, and in all three directions. The data presented in this Figure are available at doi:10.5525/gla.researchdata.1140. the x-direction and lowest in the z-direction in the images in Figure 11. Figure 12 also indicates that as the greyscale threshold value for segmentation of pore-space increases, the predicted permeability also increases. This is because porosity and pore connectivity increase, creating a greater capacity for flow through the pore network. Figure 14 shows how the specified small increase in the greyscale threshold value increases the connected porosity, resulting in lower flow velocities and fewer bottlenecks, whereas Figure 12 shows how this increase translates into a lower pressure drop across the sub-volume and, hence, higher permeability. The interdependence between porosity \(\phi_{\text{i}}\) and permeability \(\text{K}_{\text{i}}\) in a given sub-volume of a granular medium can be modelled [41; 42] as a Kozeny-Carman style relationship with percolation threshold \(\phi_{c}\) (modified from [PERSON] and [PERSON] [43])  
\[\text{K}_{\text{i}}=\text{K}_{\text{O}}\frac{(\phi_{\text{i}}\quad\phi_{c})^{ 3}}{(1+\phi_{c}\quad\phi_{i})^{2}} \tag{4}\]  
where \(\text{K}_{\text{O}}\) is a reference permeability, used as a fitting parameter. This allows representative values of permeability to be estimated as a function of porosity. The individual estimates of \(\phi_{\text{i}}\) and \(\text{K}_{\text{i}}\) for different combinations of greyscale cutoff thresholds and flow directions are plotted in Figure 13d for a logarithmic permeability scale, and in Figure 15 for a linear scale, the parameters for fitting the data for each of the combinations of flow directions being listed in Table 7.