Anomalous solute migrations in fractured rocks are governed by geometric characteristics and flow regimes. Although existing inverse models can describe this behavior, the underlying physics for quantifying key transport coefficients remains largely unexplored. Here, we investigate the quantitative impacts of geometric heterogeneity and flow regimes on solute transport in rock fractures. We conduct numerical experiments to simulate water flow and conservative solute transport in 3D fractures with varying geometric features and Reynolds numbers. Our results show that the non-Fickian transport is prevalent across the entire flow regime, with Darcy flows attributed to geometric heterogeneity and non-Darcian flows influenced by additional eddy zones. We employ the mobile-immobile (MIM) domain model and continuous time random walk (CTRW) model to inversely model simulated breakthrough curves. Inverse analyses demonstrate that both models effectively characterize anomalous transport behaviors. The fitted transport coefficients of the MIM model exhibit stronger quantitative relationships with aperture and roughness parameters, as well as Reynolds number, compared to the CTRW model. By incorporating parameterized transport coefficients, we propose physics- and statistics-based models to directly predict anomalous transport behaviors under different flow regimes. These prediction models accurately reproduce solute transport processes of all simulated cases with acceptable errors. The feasibility of directly predicting solute transport under varying flow regimes using geometric information is thus validated. Our study not only supports the study of substance migration based on geometric structure features, but also serves as a foundation for investigating geological activities based on substance migration information. The migration of substances within geological formations is governed by their geometric pore structure, while information about substance migration can reveal the geometric characteristics of these formations. In this context, these two aspects form an interdependent feedback system. Investigating quantitative relationships and models between the geometry of geological formations and substance migration is thus crucial for understanding various geological processes related to geometric structure changes and hydrogeochemical signals (i.e., chemical information carried by groundwater or surface water, which reflects various natural and anthropogenic geological processes) evolution. Here, we explore the impact of both geometric structure and hydrodynamic conditions on solute transport in rock fractures. We conduct extensive direct numerical simulations to determine the inherent quantitative relationships between key transport coefficients and geometric parameters under different flow regimes. Based on these relationships, we establish upscaling models for directly predicting solute transport processes using only geometric parameters within both physics- and statistics-based frameworks. This research has significant implications for predicting solute migrations using measurable physical properties and offers potential applications for studying geological activities related to fractured rocks through analyzing released hydrogeochemical signals. Non-Fickian transport behaviors in 3D fractures under different Re were characterized within physics- and statistic-based frameworks Transport coefficients within mobile-immobile and continuous time random walk frameworks were systematically parameterized by geometric and hydrodynamic parameters Directly predicting solute transport via geometric parameters under different Re is feasible in both physics- and statistic-based frameworks
