Optimization of Geometry and Dimensions of Sheet Piles in Diversion Dams by the Use of Conformal Mapping Theory

Movement of water in porous media is governed by Laplace equation. In this paper, the optimum location and the angle of inclination of cutoffs in diversion dams have been obtained by using conformal mapping theory to minimize the hydraulic gradient and uplift pressure. Finite element method is used to compare the results of the two methods too. The complex functions are used to analyze the seepage flow beneath the hydraulic structure. Schwarz-Christoffel transformation is one of the conformal mapping techniques. The physical plane beneath the hydraulic structure which is called z- plane and complex potential plane () are conformaly mapped on auxiliary half plane (t) by using this transformation. Finally, the function between z and planes will be obtained and variation of uplift pressure in different points below the structure and variation of hydraulic gradients through downstream will be calculated. The results of the calculations show that the optimum location of the cutoff to minimize the uplift pressure is the heel of the apron and the angle of inclination in this location is decreased by increasing s/b where s and b are the lengths of cutoffs and hydraulic structure, respectively. The value of the angle of the inclination is 60° for s⁄b=1/3. The optimum location of the cutoff to minimize the exit gradients through downstream is the toe of the apron, the values of the angle of inclination for x⁄b≤0.6 are 10°, 30° and 60° respectively. The calculations are done for the other locations along the apron such as upstream and middle of the apron and the solutions are compared with the finite element method too. The comparison of the two methods shows a good consistency between these methods.