## Unconfined Aquifers

As defined in Section 9.1, an unconfined aquifer is a water-bearing layer whose upper boundary is exposed to the open air (atmospheric pressure), as shown in Figure 9.4.1, known as the phreatic surface. Problems with such a boundary condition are difficult to solve, and the vertical component of flow is often neglected. The Dupuit-Forchheimer assumption to neglect the variation of the piezometric head with depth (d^/dz = 0) means that the head along any vertical line is constant (^ = h). Physically, this assumption is not true, of course, but the slope of the phreatic surface is usually small so that the variation of the head horizontally (3^/3x, d^/dy) is much greater than the vertical value of d^/dz. The basic differential equation for the flow of groundwater in an unconfined aquifer can be derived from Darcy's law and the continuity equation.

### Discharge Potential and Continuity Equation

The discharge vector, as defined in Section 9.3, is the product of the specific discharge q and the thickness of the aquifer H. For an unconfined aquifer, the aquifer thickness h varies, and thus

Since h = ^ and K is a constant, Equation 9.4(1) becomes

the discharge potential for unconfined flow introducing as 1

where Cu is an arbitrary constant. Now Equations 9.4(2) can be rewritten as

These equations are the same as those derived for confined flow, Equation 9.3(26).

The continuity equation for unconfined flow, without regard for inflow or outflow along the upper boundary due to precipitation or evaporation, is the same as that for confined flow as

9x 9y

Basic Differential Equation

The governing equation for unconfined flow is obtained when Equation 9.4(4) is substituted into Equation 9.4(5)

The governing equation for both confined and uncon-fined flows is the same, in terms of the discharge potential, and problems can be solved in the same manner mathematically. The only difference between confined and unconfined flows lies in the expression for \$ as

ii=ii

Phreatic or free surface

ii=ii

Phreatic or free surface

FIG. 9.4.1 Unconfined aquifer.

FIG. 9.4.1 Unconfined aquifer.

K^>2 + Cu for unconfined flow

One-Dimensional Flow

The simplest example of unconfined flow is that of an unconfined aquifer between two long parallel bodies of water, such as rivers or canals, as shown in Figure 9.4.2. In this case, ^ is a function of x only, and the differential Equation 9.4(6) reduces to

FIG. 9.4.2 One-dimensional flow in an unconfined aquifer. d2\$

FIG. 9.4.2 One-dimensional flow in an unconfined aquifer. d2\$

with the general solution

Constants A and B can be found from the boundary conditions x = 0, \$ = \$1 B = \$1

0 0