A new model for soil–water drainage problems

Abstract

Seepage flow is an agent related to the transport and dispersion of contamination in groundwater. Steady two-dimensional seepage flow is governed by Laplace's equation, for which several solution techniques are available. Because computations are complex from a practical point of view, simplified models encompass the Dupuit-Forchheimer approach assuming a horizontal flow. However this approach is inaccurate in seepage problems involving steep drawdowns. In this research, a new theoretical model for 2D seepage flow is proposed based on Fawer's theory for curved flows Castro-Orgaz (Environ Fluid Mech 10(3):2971-2310, 2010), from which a second-order equation results describing the seepage surface. From this development, a numerical solution for the rectangular dam problem based on the second-order model is presented, whereas a simple first-order equation is found to describe flow to drains under a uniform rainfall. The results of this new model are compared with the full 2D solution of Laplace's equation for typical test cases, resulting in an excellent agreement. Keywords Dam · Drains · Flow net · Seepage · Two-dimensional flow List of symbols h Seepage height (m) K Curvature distribution parameter (-) K H Hydraulic conductivity (m/s) n Curvilinear coordinate along equipotential (m) N Length of equipotential (m) 123 428 Environ Fluid Mech (2011) 11:427-435 q Unit discharge m 2 /s L Length of rectangular dam; also half of drain separation (m) R Radius of streamline curvature (m) r Rainfall (m/s) s Curvilinear coordinate along seepage surface (m) t Travel time (s) V Local velocity (m/s) x Horizontal distance (m) y Vertical distance (-) ν Dimensionless curvilinear coordinate (-)

Second‐order two‐dimensional solution for the drainage of recharge based on Picard's iteration technique: A generalized Dupuit‐Forchheimer equation

“…However, this result must be theoretically discussed and highlighted. Castro-Orgaz [2011a] found that the streamline curvature and inclination effects did not affect the water table position for flow to drains. This analysis confirmed that the classical DF equation is a good approach for some water table studies, even though the model did not incorporate the aquifer recharge intensity, thereby precluding a general conclusion.…”

“…For no recharge, equation (17) simplifies to the Dupuit-Fawer equation obtained by Castro-Orgaz [2011a, 2011b, expressing the Laplacian in curvilinear coordinates. Equation (17) is therefore a generalized result for the water table equation in Cartesian coordinates.…”

“…This fact prompted many researchers to search for alternative, still 1-D, improved models [Serrano, 1995;Knight, 2005;Castro-Orgaz, 2011a, 2011b. Knight [2005] and Castro-Orgaz [2011a, 2011b presented more correct 1-D models expressed as differential equations for the water table position requiring numerical handling.…”

“…The classical DF does not consider the existence of seepage surfaces that reduces the accuracy of the estimation of the water table position under aquifer recharge conditions [Youngs, 1990;Knight, 2005]. This fact prompted many researchers to search for alternative, still 1-D, improved models [Serrano, 1995;Knight, 2005;Castro-Orgaz, 2011a, 2011b. Knight [2005] and Castro-Orgaz [2011a, 2011b] presented more correct 1-D models expressed as differential equations for the water table position requiring numerical handling.…”