Considérations techniques et économiques sur les projets de digues munies d'un tapis filtrant

Abstract

The article discusses the very common hydraulics problem of seepage through a homogeneous isotropic dam standing on an impervious sub-stratum and with a pervious blanket on its downstream side (Fig. 1), which enables both dam volume and seepage pressure to be recluced [2]. The characteristic non-dimensional parameters involved are the following : l = L/H = Dam width / Upstream head of water α = Upstream facing angle q = Q / KH Non-dimensional discharge. With the potential function φ or the stream function ψ, the steady flow equation is Δφ = 0 or Δψ = 0. This elliptical-type problem is made more complicated by the fact that the boundary conditions are of the composite type and that the position of the free surface region boundary is not known. After a few approximations, Numerov gives the solution in the form of formulae (2) and (3). For α = 90°, formula (2) turns expression (4). Numerical calculation of these functions was accomplished by Shankin [5]. Kozeny considers flow towards a slit and obtains formula (6), which is a better approximation than Dupuit's formula. Casagrancle [6] has put forward a relationship applicable for small angles α. Guevel [7] uses the hodograph method to find discharge and free surface expressions for the case where the latter features a point of inflexion just on the upstream face of the dam. Numerical solution of the problem in the ω = φ + iψ plane with the conventions of Figure 3 requires a successive approximation procedure, as the boundary conclition x = l is unknown to begin with. The solution is first found for the function y, and then for the function x after calculation of the value of x on the boundary (AB). By replacing the derivatives by standard interpolation formulae, one obtains a linear system, which one then solves by the Young-Frankel iteration method. Table 1 compares the direct numerical calculation results with those given by formula (4). Thanks to the remarkable flexibility of numerical calculation, it has been possible to accomplish a systematic study for angles varying from 20° to 90°. The results are listed in Table 2 and shown in Figures 4, ancl 13 to 18. By considering instead of the real dam a hypothetical one of the same volume with a vertical facing, the authors found an approximation for the dam width from the numerical calculation data (see formulae (8) to (14)). for their economic dam volume study, they assumed the downstream free surface to be parabolic in shape (Kozeny's assumption), for which the optimum downstream slope is given by formula (17) and the gain in volume over a design with a vertical downstream facing is ΔΩ = 1/8q. For a symmetrical dam, the minimum volume condition defines the relationship between discharge and angle of inclination, as shown in Figure 9. Figure 10 makes evaluation of the gain in volume casier. An example (Fig. 11) shows how the free surface can be constructed by interpolation. The various formulae and graphs are applied to a practical case. It is concluded that application of a systematic numerical study to a necessarily limited number of cases has enabled the results to be generalised by interpolation and simple quick synthetic formulae to be obtained.