A new unsteady-state equation for the design of subsurface drainage systems

In this study, a new unsteady-state equation is proposed for calculating the drain spacing of subsurface drainage systems.

We consider the one-dimensional Boussinesq equation

                                                                                (1)

for  0<x<L and t>0, where L is the drain spacing (m) and t the time. Here Z(x,t) is the transient groundwater table, K is the saturated hydraulic conductivity, and S the specific yield for a homogeneous soil.

The Equation (1) is considered together with the following initial and boundary conditions:

                                                                                  (2)

where D describes the distance of the drains (placed at x=0 and x=L) from the impervious layer. The function f(x) can be constant, polynomial or trigonometric (Figure 1).

Figure 1. The geometry of the drainage problem.

Assuming f(x)=m₀sin⁡(πx/L) we observe that f(0)=f(L)=0 and f'(L/2)=0 so that the boundary conditions in (2) are satisfied and in addition f (L/2)=m₀ resulting in Z(L/2,0)=D+m₀.

By linearizing Equation (1) we obtain a linear partial differential equation of the form ∂Z/∂t-α(∂²Z)/(∂x² )=0 where α=K(D+m₀/2)/S.

We propose to solve it using the Variational Iteration Method which provides the solution in a series form and converges after a few iterations.

Performing two iterations, we get the following equation to estimate the spacing L between the drains given the height m decrease in the middle (L/2), for a specific time interval T                                       

                                                                     (3)

From the two positive solutions of the quadratic Equation (3) for L², the acceptable solution is given by

                                                                                         (4)

which is valid only if  2m-m₀≥0⇒m≥m₀/2, meaning the above formula is applicable when the height m in the middle is bigger or equal than its half initial value m₀.

The comparison of the proposed equation with the widely used Glover-Dumm equation showed relative error differences smaller than 5%.