A General One-dimensional Tracer Transport Equation via the Dual Approach: Direct Integration of the Three-dimensional River Model

One-dimensional advection–dispersion formulations are widely used to model solute fate in rivers, yet they typically rely on restrictive cross-sectional averaging assumptions. Building on the dual approach, which formalizes dimensional reduction without ad hoc closures, we derive a general one-dimensional tracer transport equation by directly integrating the three-dimensional tracer conservation law for open-channel flow. The resulting formulation preserves the effects of cross-sectional heterogeneity through rigorously obtained correction terms that account for shear dispersion and transverse exchange, and it clarifies the conditions under which these terms vanish. We show analytically that the previously proposed one-dimensional equation obtained by combining dual-approach reductions in the horizontal and vertical directions (the “two-horizontal/two-vertical” construction) is a special case of the present derivation. Under standard homogeneity and rapid-mixing limits, our equation reduces to the classical cross-sectional advection–dispersion model, ensuring backward compatibility with established practice. Validation against observed tracer data from a natural river reach demonstrates that the new formulation reproduces concentration evolution and longitudinal spreading more accurately than both the classical one-dimensional model and the combined two-horizontal/two-vertical dual-approach equation. By strengthening the theoretical foundations of dimension-reduced transport and improving empirical fidelity, the proposed equation offers a robust tool for analyzing pollutant dispersion and solute dynamics in fluvial environments and for guiding monitoring and management in rivers with pronounced lateral and vertical structure. Keywords: classical cross-sectional averaging; dual approach; Huong River; one-dimensional tracer transport equation; three-dimensional tracer transport equation; advection–dispersion; shear dispersion.