Until today, few researchers have worked on the temporal evolution of local scour at bridge piers from the physical point of view, this question was not taken into account, until recent works like Miller (2003). Usually functional dimensionless models using the maximum equilibrium values (maximum scour depth or equilibrium time magnitudes) registered in a physical model, Franzetti (1994), Melville (1997), (1999) were studied. The present work introduces you to a solution in which a series of physical hypothesis take into account the processes that occur during the formation and evolution of local scour at bridge piers. The relationship that exists between the active action of the horseshoe vortices formed in the upstream face of the pier and the passive action of the crumbled bed wall of the pit predetermines the temporal evolution of the scour in three clear different stages during time. Both phenomena feeding back one to each other forming a quasi chaotic system that evolves in time until it reaches the final equilibrium, behavior that is shown by the Poincaré graphs Moon (2004). This is the starting point to make the formulation of a system of two non linear equations in total derivatives to evaluate with great accuracy the temporal evolution of the local scour for square bridge piers that can be easily extended to other geometries. A set of 4 experiments of long duration have been carried out in square bridge piers at a reduced physical model at the laboratory of the University of Catalonia and have been compared with the proposed mathematical model. Three of the four experiments arrived without problems to the equilibrium condition. The results demonstrate the existence of a physical limit for the temporal evolution and show clearly the phases that take place during the scour processes. To analyze the scour processes, the concept of velocity of dissipation of energy that needs a free vortex in a fluid to pass from a high energy to a low energy state, before dissipated in viscosity Batchelor (1953) and a simple hydrological concept to recreate continuous crumbling of the wall of the pit, was used and implemented. These circumstances form a set of equations independent of the final stabilization value of the variables that permits to evaluate a quasi steady state of the local scour processes without knowing the final condition. This kind of formulation allows to evaluate local scour during a flood using the concept of quasi steady state with clear water flow as a strong hypothesis. An example presented here shows the simple calculation.