Steady-state water table height estimations with an improved pseudo-two-dimensional Dupuit-Forchheimer type model
Highlights
► We present a new approximate 1D model which, in the simplest way, retains the 2D features. ► The model is applied to the rectangular dam problem and the drainage of flows with recharge. ► The existence of the seepage surface is investigated and theoretical limitations discussed in depth. ► The present model is a higher order 1D Dupuit-Forchheimer model. ► The model is further compared to the full 2D solution, resulting in good agreement.
Introduction
Modelling steady seepage flow in unconfined aquifers is still a challenging task, given the difficulty of the precise free surface location. The water table heights are unknown in advance, and are part of the problem solution. To analyse the problem one has to use the incompressible stream function ψ whose derivatives determine the horizontal u and vertical v velocity components in the Cartesian coordinates (Fig. 1) x and y as
Mass conservation for incompressible water flow leads to (Bear, 1972)
The Darcy law for homogeneous, isotropic porous media yieldswhere K is the hydraulic conductivity, ϕ = (p/γ + y) the potential function or hydraulic head, p/γ the pressure head, p the pressure and γ the specific weight of water. An alternative expression for the conservation of mass is, from Eqs. (4), (5) (Bear, 1972)
The characterisation of the flow net requires the integration of either Eq. (3) or Eq. (6) with the corresponding boundary conditions associated to the physical problem (Knight, 2005, Rushton and Youngs, 2010). Eq. (6) requires, therefore, a left and right boundary condition, a seepage condition and a non-linear free surface condition. Its integration yields a full two-dimensional (2D) numerical solution which approximates the exact nonlinear problem, but demands numerical methods, which in many cases are far from simple. An alternative to the numerical solution of Laplace’s equation is the adoption of the complex potential. Polubarinova-Kochina (1962) developed exact analytical solutions for the 2D unconfined flow in porous media. However, her equations were complex and seldom used in practice (Hornung and Krueger, 1985), and did not include rainfall recharge to the aquifer. Most of the classical theory does not give room for hydrologic functions as the recharge. Therefore, the treatment of 2D seepage flow should allow for recharge to its use in practice.
Although 2D models are the most powerful tool, either their demanding numerical efforts or the implied difficulties of the complex analytical solutions triggered the consideration of approximate one-dimensional (1D) models for unconfined seepage flow. For example, the new decomposition and variation iteration approaches permit simple analytical, higher-dimensional solutions (Serrano, 1995, Serrano, 2003).
The Dupuit-Forchheimer approximation has been widely accepted as the simplest approach to solve the problem. In this 1D model, however, only one boundary condition can be imposed yet computation is straightforward. The Dupuit-Forchheimer theory assumes that the horizontal velocity u is uniform in depth, and, thus, streamlines are everywhere horizontal and the equipotential lines are vertical. This fact therefore reduces the validity of the approach to regions of the water table where both the free surface slope and curvature are small.
Youngs (1966) indicated that the Dupuit-Forchheimer theory is exact for the discharge in the rectangular dam problem, but it ignores the existence of seepage surfaces where the water exits from the soil (Youngs, 1990). Youngs and Rushton, 2009a, Youngs and Rushton, 2009b found that the Dupuit-Forchheimer theory yields good estimates of steady-state water table heights due to accretion. Rushton and Youngs (2010) analyzed the limitations of the Dupuit-Forchheimer theory related to the seepage surface for recharge to symmetrically-located downstream boundaries. They found that the inclusion of the seepage surface height as a boundary condition in the Dupuit-Forchheimer model is often accurate enough. However, this simple approach cannot predict the seepage surface. Knight (2005) assumed a parabolic distribution of the potential ϕ with depth, which improved the Dupuit-Forchhemier model for unconfined seepage flow. The model gave better results than the classical Dupuit-Forchheimer theory in both the rectangular dam problem and the groundwater flow to drains, although the seepage surfaces were not accurately described in several cases. Castro-Orgaz (2011) followed Knight’s (2005) path and proposed an alternative approach, resulting in good agreement for 2D problems in some cases.
The purpose of the present work is to present an approximate 1D model which, in the simplest way, retains the 2D features of the flow net, e.g. the inclination and curvature of the streamlines. The model is applied to both the rectangular dam problem and the drainage of flows with recharge. The existence of the seepage surface is then investigated and limitations of the Dupuit-Forchheimer theory are discussed. The present model is seen to be a higher order 1D Dupuit-Forchheimer type model, which is also compared to the full 2D solution, as well as to some other approximate.
Section snippets
Governing equations
Consider steady 2D flow in porous media (Fig. 1a). Eq. (3) can be transformed from Cartesian (x, y) coordinates to natural (s, n) coordinates (Fig. 1b) from the relationshipswhere κ is the streamline curvature and θ the streamline inclination relative to the horizontal plane. After differentiation it results
For natural coordinates ∂ψ/∂s = 0, from which Eq. (9) simplifies towhich is Laplace’s equation along an
Rectangular dam problem
Consider a rectangular dam of length L, upstream flow depth he and downstream flow depth hw (Fig. 2a). Youngs, 1966, Youngs, 1990 and Kashef (1965) demonstrated that the Dupuit equation for discharge is exactly given by
The exact solution of Polubarinova-Kochina (1962) adapted by Hornung and Krueger (1985) is plotted in Fig. 2b for the test case L/he = 1 and hw/he = 0. The discharge is given by Eq. (41), and the Dupuit-Forchheimer solution after integration of Eq. (36) by (
Drainage of recharge to symmetrically-located downstream boundaries
Consider the drainage problem of recharge q, which may be the rainfall, reaching the water table with symmetrically-located downstream boundaries (Fig. 3a) (Rushton and Youngs, 2010). Their results solving the 2D Laplace equation for q/K = 0.4 and ho/L = 0.2 are plotted in Fig. 3b, where ho is the water depth at the downstream boundary and L the half of the length between boundaries. Eq. (35) may be simplified for this case towith Q = qx. An initial value of mM = 65 was adopted to
Conclusions
An approximate one-dimensional model retaining the two-dimensional characteristics of the curvilinear flow net for unconfined flow in aquifers is presented. The model is able to describe the curvature effects in regions where the water table slope and curvature are large, and where classical 1D models based on the Dupuit-Forchheimer approach fail. The developments results in an improved, pseudo-two-dimensional extension of the one-dimensional Dupuit-Forchheimer approach. It was compared with 2D
References (17)
Improving the Dupuit-Forchheimer groundwater free surface approximation
Adv. Water Resour.
(2005)- et al.
Drainage of recharge to symmetrically located downstream boundaries with special reference to seepage faces
J. Hydrol.
(2010) Exact analysis of certain problems of ground-water flow with free surface conditions
J. Hydrol.
(1966)An examination of computed steady-state water table heights in unconfined aquifers: Dupuit-Forchheimer estimates and exact analytical results
J. Hydrol.
(1990)- et al.
Steady-state ditch-drainage of two-layered soil regions overlying an inverted V-shaped impermeable bed with examples of the drainage of ballast beneath railway tracks
J. Hydrol.
(2009) The Dynamics of Fluids in Porous Media
(1972)A new model for soil–water drainage problems
Environ. Fluid Mech.
(2011)- et al.
Numerical Methods for Engineers
(2002)
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