Chapter 28, Problem 30P

Under a number of simplifying assumptions, the steady state height of the water table in a one-dimensional, unconfined groundwater aquifer (Fig. P28.30) can be modeled with the following second-order ODE,

$Kh\frac{-d^{2}h}{d{x}^2}+\text{N=0}$

FIGURE P28.30

An unconfined or “phreatic” aquifer.

where $x=\text{distance}\left(\text{m}\right),K=\text{hydraulic conductivity }\left(\text{m/d}\right),h=\text{height}$ of the water table $\left(\text{m}\right)$, $\bar{h}=\text{the}$ average height of the water table $\left(\text{m}\right)$, and $\text{N=information rate }\left(\text{m/d}\right)$.

Solve for the height of the water table for $x=0$ to 1000 m where $h\left(0\right)=10\text{m and }h\left(1000\right)=5$m. Use the following parameters for the calculation: $K=1\text{m/d and N=0}\text{.0001 m/d}\text{.}$ Set the average height of the water table as the average of the boundary conditions. Obtain your solution with (a ) the shooting method and (b ) the finite difference method $\left(\Delta x=100\text{m}\right)$.