The riparian zone, the last few metres of soil through which water flows before entering a gaining stream, has been identified as a first order control on key aspects of stream water chemistry dynamics. We propose that the distribution of lateral flow of water across the vertical profile of soil water chemistry in the riparian zone provides a conceptual explanation of how this control functions in catchments where matrix flow predominates. This paper presents a mathematical implementation of this concept as well as the model assumptions. We also present an analytical solution, which provides a physical basis for the commonly used power-law flow-load equation. This approach quantifies the concept of riparian control on stream-water chemistry providing a basis for testing the concept of riparian control. By backward calculation of soil-water-chemistry profiles, and comparing those with observed profiles we demonstrate that the simple juxtaposition of the vertical profiles of water flux and soil water chemistry provides a plausible explanation for observed variations in stream water chemistry of several major stream components such as Total Organic Carbon (TOC), magnesium, calcium and chloride. The "static" implementation of the model structure presented here provides a basis for further development to account for seasonal influences and hydrological hysteresis in the representation of hyporheic, riparian, and hillslope processes.