Smith's Elements of Soil Mechanics. Ian Smith
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Equations (2.12) and (2.14) represent respectively the flow into and out of the element in the x direction, so that the net rate of increase of water within the element, i.e. the rate of change of the volume of the element, is (2.12) –(2.14).
Similar expressions may be obtained for flow in the y and z directions. The sum of the rates of change of volume in the three directions gives the rate of change of the total volume:
(2.15)
Under the laminar flow conditions that apply in seepage problems, there is no change in volume and the above expression must equal zero:
This is the general expression for three‐dimensional flow. In many seepage problems, the analysis can be carried out in two dimensions, the y term usually being taken as zero so that the expression becomes:
(2.17)
If the soil is isotropic, kx = kz = k and the expression is:
(2.18)
An isotropic soil is a soil whose material properties are the same in all directions.
It should be noted that these expressions only apply when the fluid flowing through the soil is incompressible. This is more or less the case in seepage problems when submerged soils are under consideration, but in partially saturated soils considerable volume changes may occur and the expressions are no longer valid.
2.9 Potential and stream functions
The Laplacian equation just derived can be expressed in terms of the two conjugate functions ϕ and 𝜓.
If we put
then
hence
(2.19)
Also, if we put
then
hence
(2.20)
𝜙 and 𝜓 are known respectively as potential and stream functions. If 𝜙 is given a particular constant value then an equation of the form h = a constant can be derived (the equation of an equipotential line); if 𝜓 is given a particular constant value then the equation derived is that of a stream or flow line.
Direct integration of these expressions to obtain a solution is possible for straightforward cases. However, in general, such integration cannot be easily carried out and a solution obtained by a graphical method in which a flow net is drawn has been used by engineers for many decades. Nowadays, however, much use is made of computer software to find the solution using numerical techniques, such as the finite difference and finite element methods. Nevertheless, the method for drawing a flow net by hand is given in Section 2.10.3 for readers interested in learning the techniques involved. The finite difference technique is described in Chapter 13 where it is applied to the numerical determination of consolidation.
2.10 Flow nets
The flow of water through a soil can be represented graphically by a flow net; a form of curvilinear net made up of a set of flow lines intersected by a set of equipotential lines.
Flow lines
The paths which water particles follow in the course of seepage are known as flow lines. Water flows from points of high to points of low head and makes smooth curves when changing direction. Hence, we can draw by hand or by computer, a series of smooth curves representing the paths followed by moving water particles.
Equipotential lines
As the water moves along the flow line, it experiences a continuous loss of head. If we can obtain the head causing flow at points along a flow line, then by joining up points of equal potential, we obtain a second set of lines known as equipotential lines.
2.10.1 Flow quantities
Referring back to Section 2.2.4, it is seen that the potential drop between two adjacent equipotentials divided by the distance between them is the hydraulic gradient. It attains a maximum along a path normal to the equipotentials and, in isotropic soil, the flow follows the paths of the steepest gradients so that flow lines cross equipotential lines at right angles.
Figure 2.8 shows a typical flow net representing seepage through a soil beneath a dam. The flow is assumed to be two‐dimensional, a condition that covers a large number of seepage problems encountered in practice.
From Darcy's law q = Aki, so if we consider unit width of soil and if Δq = the unit flow through a flow channel (the space between adjacent flow lines), then:
where b = distance between the two flow lines.
In Fig. 2.8, the element ABCD is bounded by the same flow lines as element A1B1C1D1 and by the same equipotentials as element A2B2C2D2.
For any element in the net, Δq = bki = bkΔh/l, where
Δh = head loss between the two equipotentials
l = distance between the equipotentials (see Fig. 2.9).
Fig. 2.8 Flow net