Introduction of Flow Net
The general solution of the Laplace equation yields two sets of curves orthogonal to each other. One set of curves is known as the flow lines and the other set as equipotential lines.
The net formed by intersecting the possible flow lines and equipotential lines is known as a flow net. Flow net is a graphical representation of seepage flow through a soil mass.
A flow line represents the path traced by an individual water particle. An equipotential line is a contour or line joining points of equal potential or head. The flow lines and equipotential lines cut each other at right angles i.e., they are mutually orthogonal as shown in figure 7.2.
The space between any two adjacent flow lines is called flow channel. The space enclosed between two adjacent flow lines and two successive equipotential lines in called a field.
Properties/Characteristics of Flow Net
Following properties of flow net are considered before preceding for construction and application of flow nets.
- Flow lines and equipotential lines cut each other at right angles i.e., they are mutually orthogonal.
- Each field is an approximate square and in a well-constructed flow net one should be able to draw a circle in a field touching all the four sides.
- In a homogeneous soil, every transition in the shape of the two types of curves will be smooth, being either elliptical or parabolic in shape.
- The rate of flow through each flow channel is same.
- The same potential drop occurs between two successive equipotential lines.
Construction of a Flow Net
The flow net can be obtained by any one of the following methods;
- Graphical method
- Analytical method
- Electrical flow method
- Capillary flow analogy
- Sand model
The graphical method is most extensively used method among the methods stated above. The graphical method of flow net construction involves sketching by trial and error. The hydraulic boundary conditions are examined and keeping in mind the properties of flow net initial sketching is done and by trial and error the flow net is improved to make it acceptable for practical application.
Figure: Step by step construction of flow net
The following steps in general can be adopted while constructing a flow net by geographical method.
- The boundary between soil and water is an equipotential line.
- Any boundary between the soil and an impermeable material is a flow line.
- Sketching may start by constructing the first flow line as shown in the figure 7.3 (a). The flow line is divided into a number of squares and equipotential lines are projected outwards into second flow line.
- The second flow line is drawn to form the squares of, the second flow line Some of the squares may, however be quite irregular. Such squares are called singular squares.
- The procedure is continued until the last flow line is formed. This line must consist of squares and satisfy the boundary conditions otherwise, the first flow line has to be repositioned and the whole procedure is repeated.
Application of Flow Net
A flow net can be used to determine
- Quantity of seepage
- Seepage pressure at a point
- Hydrostatic pressure at a point
- Exit gradient
Determination of seepage
The space between five adjacent flow lines is called a flow channel. Let us consider a field as shown in figure 7.4.
Let, Δq = Rate of discharge through each flow channel
Δh=Head drop (equipotential drop) in the flow field
Δh = h/Nd
H = Total head causing flow
Nd = Number of equipotential drops in entire flow net
Nf = Number of flow channels for the complete flow net.
From Darcy’s law, discharge through the flow field per unit length,
Figure: Use of flow net
where, Δs and Δn are dimensions of field as shown in the figure.
This is a required expression for determining the total discharge.
The ratio is a characteristic of the flow net and it is known as the shape factor (f).
For the figure 7.4, since, Nf = 5 and Nd = 10
q = k x h x 5/10 = 0.5kh
a) Determination of seepage pressure at a point
Seepage pressure at a point, ps is given by;
Ps = hp γw
Where hp is total head at that point given by;
Where ‘n’ is the number of equipotential drops up to point ‘p’. In the figure above, n = 8 for point ‘p’, therefore, total heat at ‘p’ is;
b) Determination of hydrostatic pressure at a point
The pressure at any point is equal to the total head minus the elevation head. For point ‘p’ in the figure, the hydrostatic pressure head is given by,
(hp)p = hp – [- (he)p]
.’. (hp)p = hp + (he)p
(hp)p = pressure head at ‘p’.
(he)p = elevation head at ‘p’
hp = total head at the point
The hydrostatic pressure,
u = (hp)p x γw
c) Determination of exit gradient (Hydraulic gradient)
The average value of exit gradient for any flow field is given by,
Where, Δs is the length of the flow field and Δh is the loss of head.
The hydraulic gradient is generally maximum at the exit near point ‘B’ where the length As is minimum. As the velocity depends upon the exit gradient, it is also maximum at the exit.
d) Computation of Seepage Force
Let us consider a cubical field of side ‘a’. Let F1 and F2 be the force exerted by the seeping water respectively on the upstream and downstream faces of the elements.
F₁ = Area x Seepage pressure
.’ F₁ = a² x h₁ γw
.’. F₂=a² × h₂ γw
The differential force acting on the element is,
(F₁-F₂) = a² γw (h₁ – h₂)
Since, (h₁ – h₂) is the head drop, Δh, we can write,
F = a² γw Δh = a³ γw (Δh/a) = a³ γwi
where, a³ is the volume of the element.
The force per unit volume of the element is i x γw.
where, i = (Δh/a) is hydraulic gradient.