## 3 Directional Hydraulic Fracturing Experiments in Gelatin Samples
### Gelatin Elastic Moduli Measured by Indentation Tests  
As an indirect index of fracture propagation behavior, elastic modulus is a basic mechanical property required for analyzing fracture propagation behavior. Therefore, before DHF experiments, indentation tests are conducted to determine the elastic moduli of the gelatin samples (Figure 2). In the indentation test, each cylindrical gelatin specimen has dimensions of 8 cm (in diameter) \(\times\) 12 cm (in height). A cylindrical steel indenter (with a diameter of 1 cm) is connected to a material testing machine to compress the gelatin specimen at a velocity of 5 mm/min [42] for 30 s [17]. The indenter diameter is smaller than one tenth of the specimen height, which minimizes the boundary effect in the indentation test [21; 43]. For each gelatin concentration, five specimens are prepared, and their elastic moduli are separately measured by indentation tests. The highest and lowest values are excluded, and the average of the remaining three test results is calculated as the elastic modulus of the gelatin sample at the given concentration [44]. The final results are listed in Table 2, which show that the elastic modulus of the gelatin increases almost linearly with its concentration. Table 4 gives [PERSON] and [PERSON]'s [31] measurement results of the elastic moduli and fracture toughness of gelatin at different concentrations. It indicates that the fracture toughness of gelatin also changes linearly with its elastic modulus (see the linear regression in Table 4). This means, in each test scenario in Table 2, HF re-orientation trajectories in gelatin samples with different elastic moduli and strengths can be compared. The effect of the gelatin elastic modulus on HF re-orientation will be studied in the following contents.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Mechanism of HF Re-Orientation** & **Magnitude of Differential Stress (MPa)** & **Reference** \\ \hline Oriented perforation & 7.58 & [38] \\ Stress shadow effect & 3.60 to 4.60 & [33] \\ DHF & 5.00 & [14] \\ \hline \hline \end{tabular}
\end{table}
Table 1: Differential stress magnitudes in previous hydraulic fracturing experiments in rock samples and cement.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Test** & **Test Case** & **Differential Stress (Pa)** & **Gelatin Concentration (\%)** & **Elastic Modulus of Gelatin (Pa)** \\ \hline \multirow{3}{*}{1} & 1-1 & 1000 & 6 & 22,365 \\  & 1-2 & 1000 & 8 & 38,475 \\  & 1-3 & 1000 & 10 & 57,375 \\ \hline \multirow{3}{*}{2} & 2-1 & 1500 & 6 & 22,365 \\  & 2-2 & 1500 & 8 & 38,475 \\  & 2-3 & 1500 & 10 & 57,375 \\ \hline \multirow{3}{*}{3} & 3-1 & 2000 & 6 & 22,365 \\  & 3-2 & 2000 & 8 & 38,475 \\ \cline{1-1}  & 3-3 & 2000 & 10 & 57,375 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Experimental schemes of DHF experiments in gelatin samples.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Magnitude of Confining Stress (Pa)** & **Reference** \\ \hline
1400 to 1700 & [27] \\
1400 to 4800 & [28] \\
1000 to 2000 & This study \\ \hline \hline \end{tabular}
\end{table}
Table 3: Confining stress magnitudes in previous hydraulic fracturing experiments in gelatin samples.