## 3 Sample Preparation and Velocity Measurement  
The method of [PERSON] (2014a,b) was adapted to construct artificial tight sandstones with similar mineral components ([PERSON], 2014, 2015), pore structures, and cementation to natural rocks. In this method, powders of silica sand, feldspar, and kaolinite were mixed in a ball mill for 24 h to ensure homogeneity. Then, these mineral powders were mixed with sodium silicate, and the mixture was poured into a mold layer by layer. To form the penny-shaped \"meso-scale\" fractures, touch paper discs were spread on the surface of the mineral powder mixtures when layering the mixture in the mold. Touch paper is a special kind of paper that has been soaked in saltpeter. At high temperatures, the touch paper decomposes into gas and leaves nearly no remains. Thereafter, compression was applied to compact the mineral grains. In this way, a block was formed, and consolidated sodium silicate gave the block its initial mechanical strength. The block was then sintered in a muffle oven at 900 \({}^{\circ}\)C. In the sintering process, the touch paper discs decomposed into gas, leaving penny-shaped voids to simulate these fractures. Three low-porosity artificial rocks with different fracture densities (0, 0.0312, and 0.0624) were fabricated (Figure 1a). Each rock was made in exactly the same way except for the introduction of fractures. Fracture density is controlled by setting the number of fractures per layer (Table 1). The penny-shaped fractures were distributed parallelly in fractured samples with a 3 mm fracture diameter and a 0.06 mm fracture aperture (Figure 1b). All samples were ground into sexangular prisms to make it possible to conduct ultrasonic velocity measurements at 0\({}^{\circ}\), 45\({}^{\circ}\), and 90\({}^{\circ}\) to the fracture normal (Figure 1c). \begin{table}
\begin{tabular}{c c c c} \hline
**Parameters** & **1\#** & **2\#** & **3\#** \\ \hline Number of fracture layers & 49 & 49 & 49 \\ \hline Number of fractures/layers & 0 & 45 & 90 \\ \hline Fracture density & 0 & 0.0312 & 0.0624 \\ \hline Fracture thickness & / & 0.055 mm & 0.055 mm \\ \hline Fracture diameter & / & 3 mm & 3 mm \\ \hline Bulk density (air saturation) & 2.114 g/cc & 2.120 g/cc & 2.085 g/cc \\ \hline Bulk density (water saturation) & 2.231 g/cc & 2.243 g/cc & 2.221 g/cc \\ \hline Total porosity & 11.7\% & 12.3\% & 13.6\% \\ \hline Fracture occupied porosity & 0\% & 0.36\% & 0.72\% \\ \hline Matrix porosity & 11.7\% & 11.94\% & 12.88\% \\ \hline Permeability (D) & 0.0013 & / & / \\ \hline \end{tabular}
\end{table} 
Table 1: The physical parameters of the three samples. Figure 1: (**a**) Three samples with different fracture densities; (**b**) a section of the fractured sample; (**c**) fracture orientation and measurement angle to the symmetry axis; and (**d**) image of the helium porosimetry equipment (Core Lab company, Houston, TX, USA). Tables 1 and 2 show the main parameters of the three synthetic rocks. The porosity was measured using helium porosimetry (Figure 1d). This equipment allows the measurement of either pore or grain volume of a rock sample [36], which can be used to calculate porosity and grain density [14]. This technique is based on the gas expansion method, where the helium gas is initially contained in a pressure chamber with known pressure and volume (V1 and P1). This chamber is connected to a secondary chamber through a valve that, when opened, allows the helium gas to expand into this secondary chamber, typically a matrix cup, which holds the sample, making the pressure drop down to a new value (P2). The whole process is controlled by a microcomputer, and the grain volume (Vg) of sample (V2), can be calculated. In the fractured samples, the background porosities were calculated by subtracting fracture-induced porosity from the total porosity of the sample. The background porosity difference between the three samples was less than 1.2%; this was caused by some inevitable tiny errors during the manufacturing process.