<p>In this work, a shallow water flow model incorporating the effects of large hydrostatic pressure is developed and analyzed. This allows us to introduce a theory of the motion of water that becomes dense. When a fluid flow becomes dense due to intrusive mechanisms (assumed to be continuous and increasing) of particles, the hydrostatic pressure increases and thus changes and this is well observed in several real situations. For example, rivers with several branches or junctions where fluids with different densities can flow. In such a situation, the flow of river water can change its concentration regime and alter its pressure (more generally its behavior). In addition to the mechanical equilibrium of the flow consequently, the changing of pressure is strongly associated with the existence of an additional hydrostatic pressure characteristic. Such physical observation has been done by Einstein and El-Samni (Rev Mod Phys 21(3):520–524, 1949) but without any theoretical demonstration and explanation. The existing shallow flow models are insufficient for describing these changing hydrostatic pressures. They loss to maintain mechanical equilibrium of the flow when it changes concentration regime. In addition, they are also inappropriate to representing a weakly stratified flow that can become strongly stratified over time. This study introduces a novel shallow flow theory that is able not only to describe the dynamics of flows that undergo a process of pressure changing over time but also to describe a strongly stratified flow (liquids with large hydrostatic effect) for the first time. This is achieved by incorporating an extra hydrostatic pressure term due to change regime concentration. The proposed model rectifies the shortcomings of several existing models. At the same time it provides a better theoretical framework of the works of Einstein and El-Samni (1949) and it improves the works of Boussinesq in 1877. In light of novel physical considerations and assumptions, we have developed a mathematical model that describes the dynamics of a strong stratified shallow water flow. This new physical concept introduces innovation to fluid mechanics and gas dynamics while improving the Boussinesq assumption (for weakly diluted flow), which was established in 1877 and is widely used in the literature in several situations even when the flow becomes strongly diluted. A brief mathematical analysis of the proposed flow models is presented and several analogies with other well-known models are proved. The presented work deals at this first stage only the hyperconcentration processes of water with free surface.</p>