## 3 Results
### Second-Order Effects  
A deeper insight into the dependencies between the different design variables can be gained by evaluating the second-order effects. The second-order effects measure the combined influence of two input quantities which are not explainable by one quantity  
Figure 6: An illustration of how the bar width could effect tortuosity. The bar width \(d\) is changed; bar height \(h\) and spacing \(s\) remain unchanged. The blue arrows show a possible streamline of a fluid particle. The length of the streamline positively correlates with the apparent tortuosity. (**a**) Fluid streamline for a large bar width and thus rather long streamline. (**b**) Fluid streamline for a small bar width, and thus, rather short streamline. alone. Figure 7 shows the computed second-order effects for each output quantity in matrix form. Bright colors indicate a large effect, while dark colors indicate a small effect. The leftmost matrix shows the interaction effects for the flow resistivity \(\Xi\). By inspecting the colorbar, it can be seen that the effects are generally smaller than the main effects (compare Figure 5a), as the largest value is approx. 0.20 for the \(d\)-s (bar width-bar spacing) interaction. This interaction effect seems intuitive, as both quantities are found to have an effect on the flow resistivity on their own; see Figure 5a. Moreover, they exhibit an interaction effect that is similarly large as the effect of the bar spacing alone. This can be explained as follows: an increase in the bar width increases the flow resistivity, while an increase in the bar spacing decreases the flow resistivity when both parameters are investigated on their own. Both effects together can either compensate or amplify each other; thus, the interaction effect is important, too. The remaining interaction effects are rather low compared to the \(s\)-\(d\)-interaction, and thus are not discussed any further here. The second left plot shows the interaction effects on the porosity. In general, it can be verified that the magnitude is lower than the one for the flow resistivity, with a maximum of approx. 0.10. Here, the bar width \(d\) and bar height \(h\) together have the largest effect. This result is counter-intuitive for the following reason. As mentioned before, the porosity of the generic specimen can be roughly estimated by \(\Phi=\frac{s}{s+d}\); thus, the bar height is not expected to have a major influence on the porosity. However, the mentioned formula does not account for the effect that an increased bar height reduces the total number of bar planes in the specimen. This might influence the apparent tortuosity for the entire specimen as well. Therefore, this observed interaction effect does not seem implausible. The matrix shown in the middle of Figure 7 displays the second-order effects for the tortuosity. Generally, it can be seen that the magnitude of the effects for this model parameter are the lowest of all investigated parameters. The most important interaction effect is found for the parameters bar width \(d\) and plane angle \(\varphi\). This could be explained as follows: for the first-order sensitivity index for the bar width \(d\), a large impact on the tortuosity could be already verified and explained, cf. Figure 6, for a plane angle \(\varphi\) of \(90^{\circ}\). For other plane angles smaller than \(90^{\circ}\), the bars perpendicular to the drawing plane in Figure 6 become slanted. Thus, the path segment for the fluid particle with horizontal movement in Figure 6 is even longer; thus, the tortuosity should be smallest for a plane angle of \(90^{\circ}\). This results in a compensating effect when the bar width is increased but the plane angle decreases. Thus, the \(d\)-\(\varphi\) second-order effect accounts for this interaction. With the \(d\)-\(\varphi\) interaction having the largest effect, other interaction effects can be verified in this plot as well, e.g., \(s-h,s-\varphi\) with an impact of approx. 0.037 and \(s-d\) width and approximate value of 0.035.