## 3 Analysis and Discussion
### Numerical Simulation Results  
Post-simulation analyses yielded fracture-specific volumetric flow rates through cross-sectional integration: 3.15 \(\times\) 10\({}^{-6}\) m\({}^{3}\)/h for the 0.50 mm aperture model, 1.58 \(\times\) 10\({}^{-6}\) m\({}^{3}\)/h for the 0.20 mm model, and 1.26 \(\times\) 10\({}^{-6}\) m\({}^{3}\)/h for the 0.05 mm model. The temporal evolution of bottom-surface Ca\({}^{2+}\) concentrations is detailed in Figure 12. The 0.50 mm aperture model achieved concentration stabilization (9.98 mol/m\({}^{3}\)) at 27 h,  
Figure 11: Chemical dissolution rates with and without hydrodynamic conditions. consistent with dynamic dissolution experimental results. Similarly, the b = 0.20 mm and b = 0.05 mm models stabilized at 10.48 mol/m\({}^{3}\) (60 h) and 10.6 mol/m\({}^{3}\) (68 h), respectively, aligning with experimental observations. These results confirm the inverse correlation between fracture aperture and stabilization duration, where narrower apertures prolong equilibration periods while amplifying concentration variation magnitudes, mirroring patterns observed in experiments. The spatial-temporal evolution of HF concentrations is presented in Figure 13, illustrating concentration distributions at three critical stages: initial conditions, leachate breakthrough at the reactor outlet, and steady-state equilibrium. Notably, specimens with varying fracture apertures exhibited distinct breakthrough times at the outlet, aligning temporally with observations from dynamic dissolution experiments. The HF concentration gradients demonstrate a consistent spatial attenuation from the inflow to outflow boundaries, with lower concentrations detected in the rock matrix beneath fractures. Following leachate breakthrough at the reactor outlet, preferential flow through fracture channels can be observed. Further analysis reveals a positive correlation between fracture aperture and HF concentrations at the outlet, consistent with experimental measurements. Additionally, widening fracture apertures amplified concentration differentials between fracture zones and adjacent rock matrices, reflecting enhanced mass transport efficiency in larger flow pathways. The presence of preferential flow pathways fundamentally alters reactant transport dynamics in rock specimens. These findings quantitatively validate the critical role of fracture aperture in governing solute transport mechanisms during acid-rock interactions. Numerical simulations predict the sulfuric acid (H\({}_{2}\)SO\({}_{4}\)) concentration evolution during reactive transport (Figure 14), revealing spatial attenuation patterns analogous to HF but differing significantly in concentration magnitudes and reaction kinetics. The H\({}_{2}\)SO\({}_{4}\) concentration decreases progressively from inflow to outflow boundaries with notably lower concentrations observed in sub-fracture matrix regions. Similarly, preferential flow  
Figure 12: Dynamic variation of Ca\({}^{2+}\) concentration at numerical model base. through fracture channels can be observed when the leachate breaks through to the reactor outlet. Based on the experimental framework, where only the first dissociation step of phosphoric acid was considered for reaction participation, it is hypothesized that sulfuric acid concentration dynamics mirror phosphate's attenuation patterns under identical hydrodynamic and geochemical constraints. Figure 14: H\({}_{2}\)SO\({}_{4}\) simulation results of rock specimens modeled with different fracture apertures. The spatial-temporal evolution of H\({}_{2}\)SO\({}_{4}\) concentrations is presented, illustrating concentration distributions at three critical stages: initial conditions, leachate breakthrough at the reactor outlet, and steady-state equilibrium. Figure 13: HF simulation results of rock specimens modeled with different fracture apertures.