OBRA PUBLICA EN BIENES DE DOMINIO PUBLICO

Dividing through by xyz and rearranging:

For horizontal, two-dimensional flow in the x-y plane this equation becomes: (7.9) and for describing the flow completely in the three dimensions, the equation is:

(7.10) with Ss the specific storage as previously defined (Ss=S/b). The final equation governs the

changing or transient flow in three dimensions in an anisotropic aquifer.

For a homogeneous, isotropic material, the hydraulic conductivities (K) are equal and constant and Equation 7.10 reduces to:

(7.11) This simplifies further for steady flow when ∂h/∂t=0 and becomes:

(7.12) which is the Laplace equation. The solution of Equation 7.12 gives the hydraulic head h in terms of x, y and z. The solution of the full equation for transient flow in an anisotropic medium gives h in terms of t as well as x, y and z. Both Equation 7.10 and Equation 7.12 are difficult to solve in all but simple situations. They have tested the ingenuity of many researchers. In engineering practice, groundwater movement problems have often been treated by simplifying the boundary conditions. In addition, assumptions of an isotropic aquifer or steady-flow conditions or both are often made to facilitate the solution and yet provide an acceptable precision.

Where conditions allow a problem is reduced to two dimensions, e.g. water movements in a vertical x-z plane through layered aquifers or in the horizontal x-y plane at some convenient level of z. Such simplification enables satisfactory answers to be obtained to problems of well yields or in the assessments of areal groundwater resources by the construction of two-dimensional flow nets.

7.4 Flow Nets

The relative inaccessibility of groundwater compared with surface water means that only point measurements in the ground reservoir are possible. The hydrologist must rely on applying the groundwater flow equations to assess resources. The mathematical representations of groundwater flow form the basis for models of internal drainage and of

changes in storage. Flow nets comprising lines of equipotential and stream or flowlines can represent groundwater movement in two dimensions. In this section, the application of nets or rigid grids is extended to analogue and digital computer models.

7.4.1 Graphical Solutions

A block of saturated land with several vertical piezometers installed is portrayed diagrammatically in Fig. 7.7(a). Piezometer A from ground level 450 m penetrates to a depth of 150 m above datum and the water level rests at 375 m. Using the nomenclature introduced in Chapter 5 for soil moisture, the total potential or hydraulic head h is equal to the sum of the pressure head and the elevation head z (height of pressure measuring point above datum):

Thus knowing the height of the land surface, the length of the piezometer and 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:

In this example where a common datum and piezometer length are used, the hydraulic head h is simply the level at d m below the surface or 450−75 =375 m.

At piezometer B, where the land surface is also 450 m, d is at 150 m and therefore h=300 m. If the distance ∆x between A and B is 300 m, then the mean hydraulic gradient is:

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 specific discharge υAB from A to B of K i or 0.25 K

m s−1_{, where K is the hydraulic conductivity of the medium. In the ‘field’ of piezometers}

in Fig. 7.7(a), 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 rest levels. On a plan view (Fig. 7.7(b)) 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

Fig. 7.7 (a) A ‘field’ of piezometers.

(b) Plan view (x-y plane).

isotropic aquifer, flowlines may be constructed perpendicular to the equipotential lines in the direction of maximum potential gradient downwards. In the example, it is obvious that the groundwater is draining to the corner of the block between B and F, and hence, three flowlines with direction arrows have been drawn on the plan. The pattern of equipotential lines and flowlines constitutes a flow net, of which Fig. 7.7(b) is a very simple example.

Flow nets drawn under certain rules allow flow rates to be calculated very simply. Fig. 7.8 shows a flow net of equipotential lines and flowlines drawn for a two-dimensional groundwater flow. The equipotentials have equal drops of head, ∆h, between any

adjacent pair. Taking a typical cell in which the distance between the equipotenial lines is ∆x, then the velocity of flow through the cell is V=K ∆h/∆x. For unit thickness of aquifer (perpendicular to the flow net), the flow rate through the cell bounded by flow lines ∆y apart, is:

Since ∆q is constant between two adjacent flowlines (no flow can cross them), all the cells between two such flowlines having the same ∆h must have the same width to length ratio, ∆y/∆x. If the flowlines are drawn so that ∆q is the same between all pairs of adjacent flowlines, then the ratio ∆y/∆x will be the same for all the cells in the flow net. In addition, the spacings can be chosen such that ∆y=∆x and all cells then become curvilinear squares. Following such rules, then ∆q=K ∆h, per unit thickness of aquifer.

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

Q=M ∆q=M K H/N

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).

Fig. 7.8 Flow net for two-dimensional