The lateral seepage from a river bank into a river produced by unconfined flow from a porous aquifer with a well-defined water table is portrayed in Fig. 15.6. The aquifer is assumed to be isotropic and homogeneous, and it is underlain by an impermeable stratum. The flow pattern in the combined unsaturated and saturated layers of such an aquifer is not so easily identified as in a completely saturated confined aquifer. The free surface of the water table has an increasingly downward slope to meet the river bank at point A, the top of a seepage surface. The water-table slope reflects a hydraulic gradient and the line of the water table describes a stream line if no recharge occurs from the ground surface. In effect, the water table is the upper boundary of a flow net in the saturated part of the aquifer. The river bed/bank interface is an equipotential line and hence all streamlines meeting it must turn to meet BC at right angles. The seepage surface AB is not an equipotential and the water table flowline is tangential to the river bank at A (Cedergren, 1997).

To be able to calculate the flow to the river, several approximations are generally made. These approximations, developed by Dupuit and Forchheimer, make the major assumptions:

(a) the hydraulic gradient dh/dx is equal to the slope of the water table;

(b) the related specific discharge is constant throughout the depth of flow; and

(c) the line ED may be taken to act as a constant head boundary with water table height hD.

Thus, the streamlines are taken to be horizontal and the equipotential lines vertical.

Making these assumptions for steady-flow conditions, the discharge per unit width (Q) over the depth of saturation (h), with hydraulic conductivity (K), is given at all sections by:

The integral of hydraulic conductivity over the depth of saturation (here Kh) is also known as the transmissivity, T, of the aquifer. In a real aquifer, there would be nonhorizontal flow components near the water table and near ABC. However, making the Dupuit-Forchheimer assumption of a horizontal flow pattern, satisfactory results are obtained for the discharge when the water table slope is small and the variation in the unconfined aquifer depth relative to that depth is small.

An extension of the seepage flow in an unconfined aquifer can include recharge from water percolating downwards through the unsaturated zone. This is a problem encountered by drainage engineers designing channels for leading off surplus water. In Fig. 15.7, drainage from the surface is assumed to give a uniform recharge rate R at the ground surface of an isotropic, homogeneous aquifer of hydraulic conductivity K. It is often required to know the depth of the water table relative to the water level in the river or channel. With steady-state conditions and adopting the Dupuit-Forchheimer assumptions, the Darcian velocity or specific discharge q at distance x, from Darcy's law, is q = -Kdh/dx. Then the flow Q at x per unit width for the aquifer depth h is Q = -Khdh/dx. However, from mass balance at steady state, the total recharge at any point is also given by Q = Rx. If all the flow in the aquifer is assumed to come from the recharge and none across the boundary ED at x = 0, then:

dx dx

Uniform rainfall rate

Water table

Ground si

Uniform rainfall rate

Water table

Ground si

a h2> RIVER

a h2> RIVER

Fig. 15.7 Recharging unconfined flow contributing lateral inflow to a river.

Integrating between h and h2 with x going from 0 to a (see Fig. 15.7) then:

whence:

Thus, h1, the water table depth at distance a from the river can be found, knowing the water level in the channel, the infiltration rate and K (ignoring the seepage surface). Under the steady recharge assumption, baseflow to the river per unit length of channel, Q, can be expressed in terms of recharge as:

Example: Calculation of water table heights in an unconfined aquifer

In Fig. 15.7, it is required to know the height of the water table at 150 m from the river when irrigation water is applied at a constant rate of 0.8 mm day-1 assuming a steady state is attained with the uniform irrigation rate, K = 4.3 m day-1 and a drain level at 12 m above the impermeable stratum. Substituting into (15.16):

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