Design of subsurface drainage network with minimum overall cost using Lagrange multiplier optimization

Subsurface drainage is effective for not only controlling groundwater levels in agricultural lands but also alleviating soil salinity localized at the root zone, especially for arid and semi-arid agro-ecosystems. In this research, an optimization model that minimizes the overall costs of subsurface drainage systems subjected to design constraints was formulated. The Lagrange multiplier method for optimization was utilized to derive a general differential equation for obtaining the most economical design of the subsurface drainage system. Design charts for the minimum cost of the subsurface drainage network ($ ha⁻¹) for lateral diameters (dL) of 2.5, 5, 7.5, 10 and 12.5 cm; hydraulic conductivities (K) of 0.5, 1, 2, 3 and 4 m day⁻¹; different lateral lengths (L_L) of 100, 150, 200, 250, and 300 m; a constant collector length (L_C) of 1000 m (assuming laterals join the collector from both sides); drainage recharge (q) values of 1, 2, 3, 4 and 5 mm day⁻¹; and excavation depths of 1.5 and 2.0 m were developed. The minimum computed subsurface drainage network (SSDN) costs observed at a lateral length (L_L) of 300 m were 205, 246, 288, 331 and 374 ($ ha⁻¹) for lateral diameters of 2.5, 5, 7.5, 10 and 12.5 cm, respectively, corresponding to lateral spacing of 175, 176, 177, 178 and 179 m. The suggested charts for the optimum design of a subsurface drainage system have lower overall costs than using conventional design approaches for pipelines and tube wells. The results will support solutions for designers to implement subsurface drainage systems with less and more affordable expenses in the old lands of the Nile delta of Egypt and other regions with similar agrosystems. Quality control during construction is very necessary for guaranteeing the effective and sustainable performance of subsurface drainage systems for many years.