## 2 Materials and Methods
### Microindentation Testing
It was designed with a customized spherical tungsten carbide tip measuring 2.5 mm in diameter and 1.25 mm in effective depth to facilitate accurate measurements. Additionally, the apparatus incorporated an electromagnetic load sensor capable of handling loads up to 5000 N. The bidirectional loading apparatus was constructed as a hollow square-shaped chamber designed to enable the precise application of two independent perpendicular loads. These loads were exerted onto the rock specimens using two servo-controlled hydraulic jacks incorporated within the apparatus. Considering the burial depth of the shale reservoir in this study (2300-2500 m), the maximum lateral confining loads exerted on the shale sample were limited to 20 MPa, and the lateral confining loads applied to the shale specimen were assumed to be identical. Therefore, a series of lateral confining pressures (i.e., 0, 2, 5, 10, and 20 MPa) were selected to replicate the field stress conditions experienced by the shale in this study. Subsequently, microindentation (i.e., spherical penetration) testing was performed following the application of the desired lateral confining pressures on the shale sample. In preparation for the microindentation testing, it was essential to establish the maximum applied load (\(F_{\rm max}\)). Previous studies involved nano- to microindentation tests on comparable shales sourced from the Longmaxi Formation, validating that when the indentation depth surpassed a defined critical threshold (i.e., \(h_{\rm max}\geq\) 10 \(\upmu\)m), the derived mechanical characteristics could accurately represent the bulk rock properties of the shale [25; 26]. In this study, the \(F_{\rm max}\) of 100 N was determined, and the selection of such a substantial load was driven by three key reasons: (1) to mitigate the potential adverse effects of surface roughness and micropores on the measurements; (2) to consider the critical threshold of indentation depth; and (3) to enable the simulation of higher closure pressure (\(p\)) of approximately 20 MPa, akin to the reservoir pressure. In order to comprehensively evaluate the mechanical properties of shale under varying lateral confining loads, cyclic indentation tests were conducted on the specimen. Figure 5 shows the representative load-displacement curves of cyclic loading-unloading indentation tests. Upon reaching the predetermined maximum load under a loading rate of 100 N/min, the load was subsequently held constant for a duration of 60 s. Subsequently, monotonic unloading commenced at a consistent rate equivalent to that at \(F_{\rm max}\). Unloading was halted once the  
Figure 2:1 The microindentation with bidirectional loading apparatus: (**a**) testing system; (**b**) schematic diagram. load decreased to 5% of \(F_{\text{max}}\), followed by reloading back to \(F_{\text{max}}\) to complete a total of five loading and unloading cycles. To assess the reproducibility of the conducted tests, cyclic indentation tests were performed at five different locations on the surface of the sample for each lateral confining load condition, and the spacing between adjacent indentation points was maintained at a minimum of 20 times the anticipated maximum indentation depth (Figure 5). This approach yielded 25 distinct mechanical parameters (e.g., \(E\), \(HB\), etc.) for the shale sample under each lateral confining load. The reported results in this study are presented as the average values derived from these 25 measurements. The sample surface, which was aligned parallel to the depositional bedding plane, underwent indentation loading. To prepare the surface for indentation measurements, a step-by-step polishing method was carried out using a polisher with sandpapers ranging from #400 to #4000 in grit size. The final polishing stage involved the use of an ultrafine alumina suspension containing 0.3 \(\upmu\)m aluminum powder. Throughout the entirety of the polishing process, meticulous precautions were taken to prevent shale hydration, ensuring that direct contact with water was avoided.