Appendix VII

Appendix VII
Tóth’s Analytical Solution for Regional Groundwater Flow

Tóth (1962, 1963) has presented two analytical solutions for the boundary-value problem representing steady-state flow in a vertical, two-dimensional, saturated, homogeneous, isotropic flow field bounded on top by a water table and on the other three sides by impermeable boundaries (the shaded cell of Figure 6.1, reproduced in an xz coordinate system in Figure A7.1).

Figure A7.1 Region of flow for Tóth’s analytical solution.

He first considered the case where the water-table configuration is a straight line of constant slope. For this case, the region of flow in Figure A7.1 is the region ABCEA. Since it is not possible to obtain an analytical solution in a trapezoidal region, Tóth approximated the actual region of flow by the shaded region ABCDA. He projected the hydraulic-head values that exist along the actual water table AE onto the upper boundary AD of the region of solution. The approximation is satisfactory for small \alpha.

The equation of flow is Laplace’s equation:

\frac{\delta^2h}{\delta x^2} + \frac{\delta^2h}{\delta z^2} = 0 (7.1)

For a region bounded by x = s and z = z_0, the boundary conditions are

\frac{\delta h}{\delta x}(0, z) = \frac{\delta h}{\delta x}(s, z) = 0 on AB and CD

\frac{\delta h}{\delta x}(x, 0) = 0 on BC (7.2)

h(x, z_0) = z_0 + cx on AD

where c = tan \alpha.

The analytical solution, obtained by separation of variables, is

h(x, z) = z_0 + \frac{cs}{2} - \frac{4cs}{\pi^2}\sum_{m=0}^{\infty} \frac{\cos[(2m + 1)\pi x/s] \cos h[(2m + 1)\pi z/s]}{(2m +1)^2 \cos h[(2m +1)\pi z_0/s]} (7.3)

This equation satisfies both the equation of flow (A7.1) and the boundary conditions (A7.2). When plotted and contoured, it leads to the equipotential net shown in Figure A7.1. Flow is from the recharge boundary DF through the field to the discharge boundary AF.

Tóth also considered the case where the water-table configuration is specified as a sine curve superimposed on the line AE (to represent hummocky topography). The final boundary condition then becomes

h(x, z_0) = z_0 + cx + \alpha \sin bx on AD (A7.4)

where c = \tan \alpha, a = a'/\cos \alpha, and b = b'/\cos \alpha, a' being the amplitude of the sine curve and b' being the frequency.

The analytical solution for this case takes the form

h(x, z) = z_0 + \frac{cs}{2} + \frac{\alpha}{sb}(1 - \cos bs)  + 2 \displaystyle\sum_{m=1}^{\infty} \left[\frac{ab(1 - \cos bs \cos m\pi)}{b^2 - \frac{m^2\pi^2}{s^2}} + \frac{cs^2}{m^2\pi^2}(\cos m\pi - 1)\right] \times \frac{\cos (m\pi x/s) \cos h (m\pi z/s)}{s \cdot \cos h (m\pi z_0/s)} (A7.5)

The equipotential net described by this function is similar to that of Figure 6.2(b).