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To calculate the hydraulic gradient ( i ) in steady – non Darcy regimes, the validity of the binomial relationship is confirmed using dimensional analysis and Navier-Stokes equations. According to St. Venant equations, the experimental data used in the present study are of the diffusion wave type and the changes in flow depth ( h ) in terms of changes in flow rate ( Q ) or flow velocity ( V ) (flow-depth curve) are in the Loop form. In other words, for a value of  V , there are two different values for  h , and given the relationship between  i  and  h , for each value of  V , there are two different values for  i  that in the ascending area of the input hydrograph, the value of  i  is greater than its descending area. For this reason, in the present study, in order to use the binomial relation in unsteady flow conditions, for each of the above ascending and descending areas, a different value for the coefficients  a  and  b  of the binomial relation has been optimized using the Particle Swarm Optimization (PSO) algorithm. The results show that if instead of using a constant value for the coefficients  a, b , two different values are used for the mentioned coefficients, the improvement of the values of the sum of the absolute value deviations (SAD) and the sum of the squared deviation (SSQ) for Small grain size are 39% and 47%, respectively, for Medium grain size are 50 and 74, respectively, and for large grain size are 62% and 80%, respectively.

Study of hydraulic gradient and velocity changes of unsteady flow through coarse porous media