## Graphical solutions

The use of flow nets is most easily demonstrated for the case of a confined aquifer where the totally saturated groundwater body is overlain by an impermeable aquiclude. A confined aquifer with several vertical piezometers installed is portrayed diagrammat-ically in Fig. 15.4a. Piezometer A from ground level 450 m penetrates to a depth of 150 m above datum and the water level rests at 375 m. The total potential or hydraulic head h is then equal to the sum of the pressure head and the elevation head z (height of pressure measuring point above a fixed datum).

Thus, knowing the height of the land surface and the length of the piezometer, from a measure of the water depth d, the value of h can be obtained. For A, the piezometer length is 300 m, d is 75 m and z is 150 m so that:

At piezometer B, where the land surface is also 450 m, d is at 150 m and therefore h = 300 m. If the distance Ax between A and B is 300 m, then the mean hydraulic Equipotentials
 Dq a / 4y Dx Dq r t 1

Fig. 15.4 (a) An aquifer with a number of observation wells. (b) Flow net for two-dimensional groundwater flow in plan.

Fig. 15.4 (a) An aquifer with a number of observation wells. (b) Flow net for two-dimensional groundwater flow in plan.

Ah 375-300

Ax 300

Thus at the elevation head, z = 150 m, there is a difference in potential from A to B and therefore there will be a component of Darcian velocity qAB from A to B of 0.25 Ks m s-1, where Ks is the saturated hydraulic conductivity of the medium. In the

'field' of piezometers in Fig. 15.4a, piezometers C and E from different surface levels have h values of 375 m, at piezometer D, h is 400 m, and at B and F, h values are 300 m. Within the block of land there is a three-dimensional surface defined by the hydraulic heads, and this is known as a piezometric surface. This passes through all the water rest levels in the wells. On a plan view (Fig. 15.4b) the points A, C and E lie on an equipotential line (375 m). Through the points B and F on the two-dimensional plan runs the equipotential line of 300 m. Once the equipotential lines have been determined for an isotropic aquifer, streamlines may be constructed perpendicular to the equipotential lines in the direction of maximum potential gradient. In the example, it is obvious that the groundwater is draining to the corner of the block between B and F, and hence, three streamlines with direction arrows have been drawn on the plan. The pattern of equipotential lines and flowlines constitutes a flow net, of which Fig. 15.4b is a very simple example.

Flow nets drawn under certain rules allow flow rates to be calculated very simply. Fig. 15.4b shows a flow net of equipotential lines and streamlines drawn for a two-dimensional groundwater flow. The equipotentials have equal drops of head, Ah, between any adjacent pair. Taking a typical cell in which the distance between the equipotential lines is Ax, then the Darcian velocity of flow through the cell is qv = KAh/Ax. For unit thickness of aquifer (perpendicular to the flow net), the flow rate through the cell bounded by flow lines Ay apart, is:

v Ax Ax

Since Aq is constant between two adjacent flowlines (no flow can cross them), all the cells between two such flowlines having the same Ah must have the same width to length ratio, Ay/Ax. If the streamlines are drawn so that Aq is the same between all pairs of adjacent flowlines, then the ratio Ay/Ax will be the same for all the cells in the flow net. In addition, the spacings can be chosen such that Ay = Ax and all cells then become part of a curvilinear square grid. Following such rules, then Aq = KAh, per unit thickness of aquifer.

If there are N drops of Ah between equipotential boundaries whose potential difference is H, then Ah = H/N. If there are M 'flowtubes' between impermeable boundaries, then the total flow rate (per unit thickness of aquifer), is:

Summarizing the properties and requirements of flow nets in homogeneous, isotropic media:

(a) equipotential lines and flowlines must all intersect at right angles;

(b) constant-head boundaries are equipotential lines;

(c) equipotential lines meet impermeable boundaries at right angles;

(d) if a square grid is used, it should be applied throughout the flow net (although difficulties will arise near sharp corners and towards remote or infinite boundaries).

The steady-state equations that underlie the description of groundwater flows are linear equations. This has a real advantage in the construction of more complex flow nets because it allows the use of the principle of superposition. This means e.g. that, where a pumped well is affecting a regional groundwater flow, the flow nets for each can be constructed separately and added together.

For example, we can consider the effect of adding a pumped well to the flow net of Fig. 15.4b. The flow net for a pumped well in the absence of other disturbances consists of a radial system of M flow tubes converging towards the well. M can be chosen so that the flow rate between each streamline is equal to the flow between streamlines of the original flow net. As the flow converges towards the well, the distances between the streamlines gets smaller. To preserve the correct proportionality, the spacing between equipotentials must also get smaller. This results in a correct reflection of the cone of depression in the piezometric surface or water table normally found around a pumped well (Fig. 15.5b).

Then, having created the two flow nets with the same flow rates and equipo-tential spacing, they are easily superimposed to give the final joint flow net (e.g. Fig. 15.5d). The results will be a good approximation to the effect of a well on the actual system, providing that the effect of the well does not have a significant effect on the boundaries of the flow domain (e.g. if the cone of depression intersects

h+ah

h+ah 3ah Fig. 15.5 Linear superposition of equipotentials: (a) original regional flow net; (b) flow net fora pumped well showing cone of depression in equipotentials; (c) superposition of equipotentials from (a) and (b) before summation; and (d) resulting flow net showing final equipotentials and flow lines.

Ground surface

Water table E

Ground surface

Water table E Equipotential lines

Streamlines

Fig. 15.6 Schematic of a vertical flow net for steady unconfined flow from an aquifer towards a river.

Equipotential lines

Streamlines

Fig. 15.6 Schematic of a vertical flow net for steady unconfined flow from an aquifer towards a river.

and impermeable boundary). There are techniques for dealing with these additional complications (see, e.g. Cedergren, 1997) but it is now more normal to construct a numerical model of an aquifer to deal with complex situations. Once a model has been constructed, exploring the effects of different boundary conditions, complex patterns of hydraulic conductivity, well positions or different pumping rates are easily implemented. 