From Concept to Equation: Integrating the ‘Soil’ Reservoir into the Production Function of GR Models and Proving Temporal Invariance

Lumped conceptual models, particularly those in the Génie Rural model family, are widely recognized as benchmark tools in hydrology. Although the final equations of their production function (the ‘Soil’ reservoir) are available in the literature and open-source packages, their original conceptualization and complete mathematical derivation have rarely been formally published. This paper aims to address this issue by reconstructing the complete logical chain linking conceptual assumptions to operational equations. The hydrological conceptualization is revisited and grounded in an analogy with the filling, extraction, and emptying of a physical reservoir. We derive the ordinary differential equations that govern storage dynamics, detailing the transition from temporal to cumulative flux formulation and providing complete analytical solutions. A step-by-step derivation involving separation of variables, partial fraction decomposition, and integration yields canonical closed-form solutions involving the hyperbolic tangent, as implemented in Génie Rural models. Time-step integration using cumulative variables provides fundamental temporal invariance. In addition to historical reconstruction, this formalization emphasizes important structural features such as mass conservation, numerical robustness, and analytical differentiability. By making these mathematical foundations accessible, this paper provides the hydrological community with a solid basis for future methodological extensions.