A three-parameter runoff probability model for long-slope croplands in the black soil region of Northeast China

The Mangang black soil region of Northeast China, characterized by long or ultra-long slopes (100–4000 m), is experiencing severe water erosion. Soil erosion has worsened, particularly on long slopes, due to precipitation variability and tillage disturbance. Existing models struggle to predict runoff because they fail to capture nonlinear shifts among multiple runoff-generation mechanisms. To address this, a probabilistic runoff model with three parameters ( β , μ , τ ) was developed, coupling infiltration, re-infiltration, and exfiltration with Hortonian, Betsonian, and Duncanian regimes. By extending the Sheridan framework with an exponential exfiltration function, the model estimates runoff coefficients based on slope length and soil saturation. It was validated using rainfall plots, field observations (1–600 m), and literature data, and benchmarked against the Sheridan model and HYDRUS-2D. The model reveals a “hook-shaped” nonlinear relationship between slope length and runoff coefficient (R2 =0.92–0.99, KGE=0.87–0.98), with a turning point (T p , 100–200 m) marking the shift from decreasing to increasing runoff. A new threshold, τ (incomplete saturation slope length), reflects saturation-driven exfiltration feedback, dividing slopes into: (1) Wetting zone (0-T p ): dominated by Hortonian runoff; (2) Transition zone (T p - τ ) with rapid exfiltration and gully initiation; (3) Saturated zone (> τ ): dominated by Duncanian runoff. Compared with HYDRUS-2D, the model improves runoff distance estimation by 25 % and reduces runoff errors by 37–62 %, supporting adaptive land management. • This study explores runoff distribution on long slopes in Northeast China’s black soil region. • It reveals a nonlinear slope length - runoff relationship: first decreasing, then increasing. • A three-parameter “mechanism competition - threshold response” model is proposed. • The study identifies how slope length controls runoff generation processes. • It explains the slope length - hydrology - erosion risk chain more clearly.