<p>Understanding the relationship between fracture normal stiffness and fluid flow is crucial for assessing hydraulic properties via seismic methods. To quantify this relationship, we derive a novel analytical expression directly linking normal stiffness to flow rate, based on the two-part Hooke's model (TPHM). TPHM conceptualizes fracture aperture into soft and hard parts, which exhibit distinct mechanical and hydraulic responses. The key novelty of this work lies in the unification of these dual contributions into a single closed-form stiffness-flow relationship, providing a fundamental mechanistic explanation beyond previous empirical approaches. The derived closed-form relationship reveals that the increase in normal stiffness is governed solely by the closure of the soft-part aperture, while fluid flow transitions from being dominated by the soft-part aperture at low stress to the hard-part aperture at high stress. Consequently, the model inherently captures three fundamental regimes observed in experiments: strong, weak, and independent dependence of flow rate on stiffness. The proposed relationship is validated against multiple published experimental datasets spanning different lithologies (e.g., granite, sandstone, gneiss), showing consistent and accurate predictions. A key finding is that a linear approximation in log–log space effectively describes the strong-dependence regime at low stress. This work provides a unified, physics-based analytical framework that comprehensively explains the full spectrum of stiffness-flow behaviors in single fractures, moving beyond purely empirical correlations.</p>