## 3 Directional Hydraulic Fracturing Experiments in Gelatin Samples
### Results and Discussions
The fractured sample surfaces in Test Case 1-3 (\(w=10\%\)) are oblique (Figure 5e), which means HF propagation is more likely to be directionally controlled by the initial notch in a stiffer gelatin sample under a given differential stress condition (\(\sigma_{\text{d}}=1000\) Pa). Figure 4: Results of Test Case 1-2 (\(w=8\%\) and \(\sigma_{\text{d}}=1000\) Pa): (**a**) HF propagates along notch direction for certain distance before re-orientation; (**b**) HF re-orientation observed from another angle; (**c**) fractured gelatin sample taken out from container. Figure 5: Results of Test Case 1-3 (\(w=10\%\) and \(\sigma_{\rm d}=1000\) Pa): (**a**) HF propagates along notch direction; (**b**) HF trajectory observed from another angle; (**c**) fractured gelatin sample observed from the angle in Figure 5a; (**d**) fractured gelatin sample observed from the angle in Figure 5b; (**e**) fracture surfaces of gelatin sample. In Test Scenario 2, the differential stress magnitude changes from 1000 Pa to 1500 Pa to investigate HF re-orientation in gelatin samples with different concentrations under a higher differential stress condition. Figure 6 shows the HF trajectory in Test Case 2-1 (\(w=6\%\) and \(\sigma_{\text{d}}=1500\) Pa). Compared with that in Test Case 1-1 (\(w=6\%\) and \(\sigma_{\text{d}}=1000\) Pa) (Figure 3c-e), the HF propagation plane looks asymmetric and re-orientates more quickly towards the horizontal direction (Figure 6a-c). Note that the difference between the differential stress magnitudes in Test Case 1-1 and Test Case 2-1 is only 500 Pa. This corroborates the hypothesis that the impact of the differential stress on DHF depends on its relative magnitude (e.g., relative to the gelatin elastic modulus) rather than its absolute magnitude. The existence of the differential stress restricts directional propagation of the HF along the initial notch, and the re-orientation distance is determined by the ratio between the differential stress and the gelatin elastic modulus. In addition, the three-dimensional HF geometry is not definitely as symmetric as that in numerical modelling with the homogeneous assumption [18]. The HF geometry tends to become asymmetric in a relatively high differential stress state. Note that gelatin is also a homogenous material, and hence the asymmetric propagation is not due to its heterogeneity. In Test Case 2-2, the gelatin concentration increases to 8%, and the differential stress is held at 1500 Pa (Figure 7). The HF geometry returns to a more symmetric pattern, and the initial notch again gives certain guidance on the HF propagation direction. This phenomenon becomes more obvious in Test Case 2-3, in which the gelation concentration is 10% (Figure 8). Also, it is found that the sample surfaces in these two test cases resemble each other (Figures 7e and 8e). This indicates that a higher gelatin elastic modulus (\(w=8\%\) and \(w=10\%\)) does prevent asymmetric propagation of the HF, but its promotion of DHF can be limited if the differential stress is sufficiently high (compared with that in Test Scenario 1). Figure 6: Results of Test Case 2-1 (\(w=6\%\) and \(\sigma_{\text{d}}=1500\) Pa): (**a**) HF re-orientates towards its favorite propagation plane; (**b**) HF re-orientation observed from another angle; (**c**) asymmetrical HF propagation; (**d**) fractured gelatin sample taken out from container. Figure 7: Results of Test Case 2-2 (\(w=8\%\) and \(\sigma_{\rm d}=1500\) Pa): (**a**) HF initiates from initial notch (plan view); (**b**) HF initiates from initial notch (section view); (**c**) HF propagates obliquely with limited re-orientation; (**d**) HF trajectory observed from another angle; (**e**) fractured gelatin sample taken out from container. The differential stress in Test Scenario 3 grows to 2000 Pa to provide further support for the above conclusions.