## 3 Sample Preparation  
The method of [PERSON] was adapted to build artificial porous sandstones with similar mineral components [17; 18], 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 solution. The mixture was then paved into a mould layer by layer. In order to make the \"meso-scale\" fractures, touch paper discs were spread on the surface of the mineral powder mixtures when layering the mixture in the mould. Touch paper is a special kind of paper which has been soaked in saltpetre. At high temperatures, touch paper decomposes into gas and leaves nearly no remains. In order to pave the mixture in the mould in layers, a special tool was made (Figure 1). The tool consisted of a frame, a glass plate, and a threaded rod. By rotating the threaded rod, the glass plate can be moved vertically. The procedure of paving the mixture is shown in Figure 2. Afterwards, a predetermined number of discs were carefully placed on top of each mixture layer. Then, the next mixture layer was paved and the same number of discs were placed on it (except for the top layer). After this, a uniaxial compressing pressure was applied to the mould for 10 min in order to ensure that the mineral grains were compact. The compressing direction was perpendicular to the fractures and layers. During this process, the pressure was adjusted to control the porosity of the samples. During the next 10 h, the sodium silicate in the mixture consolidated. In this way, a block was formed and the consolidated sodium silicate gave the block initial mechanical strength. Next, the block was demoulded and sintered in a muffle oven at 900 \({}^{\circ}\)C. During sintering, the touch paper discs decomposed, leaving penny-shaped voids as fractures. The final step was to polish the sample surface for the convenience of obtaining ultrasonic velocity measurements. The size of all the cuboid-shaped samples was 69.9 \(\pm\) 0.1 mm in the two directions parallel to the fractures and 49.9 \(\pm\) 0.1 mm in the direction perpendicular to the fractures. In order to produce artificial rocks with the same fracture densities but different porosities, we made the following modifications to the method of [PERSON] (2014) [17,18]: (1) Different amounts of mineral material mixture were layered in the mould (70 mm \(\times\) 70 mm) in 37 layers when building the different porosity samples. (2) The mixture of different samples was compressed into the same thickness (50 \(\pm\) 0.3 mm) by adjusting the applied pressure to form blocks of the same volume. Therefore, the blocks that were made using more mixture would experience higher loading pressure and would have lower porosity. (3) A total of 126 touch paper discs were spread on the surface of each layer (except the top layer) in all the blocks. Fracture density is unitless and is calculated using the following formula:  
\[e=\frac{Na^{3}}{V} \tag{8}\]  
where \(N\) is the number of fractures, \(a\) is the radius of the fractures, and \(V\) is the total volume of the sample. Thus, all the blocks had the same volume (70 mm \(\times\) 70 mm \(\times\) 50 mm)  
Figure 1: (**a**) A tool used to strickle mineral mixture in the mould; (**b**) a structural representation. and contained the same number of fractures (36 layers \(\times\) 126 fractures/layer), making the fracture density practically identical for the different porosities. For each fractured sample, a blank sample (containing no penny-shaped fractures) was made as a reference sample to provide background properties. The blank samples were created using the same method as the fractured samples, excluding the introduction of fractures. Four groups of artificial rocks with different porosities were constructed (Figure 3a). Each group consisted of two samples (one fractured sample containing penny shaped fractures with a fracture density of 0.06 and a corresponding unfratured sample).