Use of Misbach's Non-Darcy Equation in the Design of Ditch Drainage systems

Abstract

The linear relationship between flow velocity and hydraulic gradient given by Darcy's law is restricted to a limited range of Reynolds number and flow velocity. Due to the absence of a single equation satisfying the flow phenomenon in the pre-linear, linear and postlinear zones of flow through porous media, the Misbach's empirical equation [I = aV^m, m>I] for post-linear zone is used for the development of open surface drain spacing model in the present study. A homogeneous differential equation was formulated employing steady state equilibrium concept of flow to ditch drains separated by a distance (S). The drains are excavated closer to the depth of impermeable layer, The analytical solution of the developed differential equation was obtained by substituting the boundary conditions in non-dimensional form to get the equations of the phreatic surface and ditch drain

spacing. The equations were analyzed with different combinations of assumed recharge rate, hydraulic conductivity (the coefficient "a" is inverse of hydraulic conductivity) and the exponent "m" of Misbach 's Equation. It was observed that the spacing obtained using Misbach's non-Darcy equation is always more than that obtained with Hooghoudt's drain spacing formulae, Therefore, use of Darcy's equation for design of ditch drains in sandy and sandy loam soils under post linear flow regime may lead to higher construction costs of the drainage system. The phreatic curve using Misbach's non-Darcy equation lies above the phreatic curve obtained using Hooghoudt's equation. The position of the phreatic curve plotted using non-Darcian equation indicates higher drain spacing and efficient disposal of gravitational water.