A flow net is a graphical representation of the flow of water within a soil mass. Flow net is helpful in many ways like analysing seepage in an earthen dam, finding uplift pressure etc. In this blog, we will discuss in detail how flow nets are made in real-life.
Flow Net Characteristics
Before getting into the main topic it is useful to know some basic points related to flow nets. As said earlier, a flow net represents the flow of water within a soil mass i.e., seepage, in a graphical manner. For this purpose, a flow net uses flow lines and equipotential lines.
A flow line is a line along which a fluid particle flows within a flow line. This is also called a flow path or streamline. There can be any number of flow lines that a fluid particle can take within a soil mass. An equipotential line is a line along which the head loss is the same. If equipotential lines are drawn at equal intervals, it means that the headloss between any two equipotential lines is the same. A system of flow lines and equipotential lines constitutes the flow net. The flow net for flow under a sheet pile is shown below.
Listed below are some of the important characteristics of the flow net.
No two equipotential lines cross each other
Flow cannot cross from one flow line to another
The flow field is the area between two equipotential lines and two flow lines
In the case of two-dimensional flows, the flow line and equipotential line are orthogonal to each other and the flow fields formed are in the shape of elementary squares
In the case of one-dimensional flows, the flow line and equipotential line are perpendicular to each other and the flow field formed is in the shape of a square
With this, it is understood that the flow net gives a pictorial representation of the path taken by a flow particle and the head variation along the path.
Flow Net Applications
There are three important uses of a flow net as listed below.
1. Discharge
The rate of flow can be determined using a flow net. The flow per unit length is obtained as, q/L = k.H.(nf/nd), where nf is the number of flow lines, nd is the number of equipotential lines, H is the total head causing flow, and k is the coefficient of permeability of the soil.
2. Head at any point
Flow net can be used to determine the head at any point. H/nd gives the head drop between two consecutive equipotential lines.
3. Hydraulic gradient
The hydraulic gradient at any point can be found using the flow net. Gradient for any flow field is given by h/l, where h is the head lost in that field and l is the length of the field.
Methods for Flow Net Construction
There are four methods for constructing a flow net, as given below.
Mathematical or analytical method
Electrical flow analogy
Numerical analysis
Models
Graphical solution by sketching
Let us see each of the methods in detail.
1. Mathematical or Analytical Method
This approach is largely used for academic purposes only. This method involves using equations for boundary conditions and solutions of Laplace equations obtained by mathematical procedures. Boundary equations are known only for relatively simple cases, but due to the mathematical calculations present, this method could make simple cases difficult to solve.
For example, the best-known theoretical solution was given by Kozeny for flow through an earthen dam with a filter drain at the base towards the downstream side. This flow net consists of confocal parabolas.
2. Electrical Flow Analogy
Laplace's equation for fluid flow holds for electrical and heat flows. Therefore, the use of electrical models for solving complex fluid flow problems is common. In an electrical model, voltage represents the total head, current to velocity and conductivity to permeability. Ohm's law is analogous to Darcy's law, therefore measuring voltage can locate equipotential lines. Flow patterns can then be sketched.
3. Numerical Analysis
This method involves solving Laplace's equation for two-dimensional flow by numerical techniques in case the mathematical solution is difficult. Relaxation methods involving successive approximation of the total heads at various points in a mesh or network are used. Computer processing could also be used with this method to get a rapid solution.
4. Models
This method involves the construction of a scaled model for studying a flow problem. Earthen dams are most often analysed by this method. Such models are commonly constructed between two parallel glass sheets. By injecting dyes at various points, the flow lines may be traced. The top flow line could be directly determined by this method. Piezometer tubes can be used for the determination of the heads at various points.
However, models are of limited use because of the time and effort required to construct such a model and also because of the difficulties caused by capillarity. The capillary flow in the zone above the top flow line may be significant in a model, but it is of little significance in the prototype. Therefore, models are mostly used for demonstrating the fundamentals of fluid flow.
5. Graphical Solution
This method involves sketching of flow net by trial and error method. This method also has the advantage of the fact that the solution to a two-dimensional flow problem is relatively insensitive to the quality of the flow net. Even a crudely drawn flow net generally permits an accurate determination of seepage, pore pressure and gradient. Moreover, flow nets for most common situations are available in many geotechnical research papers.
In this blog, we have covered some of the basic points about a flow net and methods to construct a flow net. Flow net is an important, easy-to-cover topic in geotechnical engineering covering which could prove useful in the GATE exam.
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