Two-dimensional Flow equation

What is Flow Nets ?
The graphical form and representation of solutions to Laplace equation for two-dimensional flow of seepage water also can be represented as flow nets diagram. Generally two orthogonal sets of curvature form a flow net:

  • Equipotential lines are described as connecting points of equal total head h
  • Flow lines are representing the direction of seepage down towards the hydraulic gradient

The sequential two no of flow lines can never be meet and simultaneously, two equipotential lines also can never meet. The intermediate space between two adjacent or parallel flow lines is defined as a flow channel, and inside the seepage zone which is formed on the side of flownet between any two no adjacent flow lines and two adjacent equipotential lines which is referred to as a field zone. 

Calculation and equation of flow in a channel: 
When a standpipe type piezometers were inserted inside the ground with their tips on a single equipotential line, then the water level would rise up to the same level/height in each standpipe. The pore water pressures would be different with each other because of their different internal elevations.There can be no flow exist along an equipotential line as there is no existance of hydraulic gradient.

Now consider a field of length of L within a flow channel medium. There is also a fall of total head Dh. Now the average hydraulic gradient is

Flow net equation:

As the necessary flow lines are b apart and considering unit length which is perpendicuar to field zone , the flow rate is as follows,

There is also an advantage in case of sketching flow nets in the formation of curvilinear ‘squares’ so that, a circle which can be insrcibed within the each four-sided bounded by two no of equipotential lines and two no flow lines.

In a square representation , b = L , and the concerned flow rate is obtained as follows,

 Dq = k.Dh 

Then the flow rate of water through such type of flow channel is the soil permeability k  and it multiplied by at the uniform interval Dh between the adjacent equipotential lines.

Calculation of total flow
In case of a complete equation, the flow net can be easily drawn with the overall head drop of value h , which divided into Nd so that,the equation can be expressed as Dh = h / Nd
If Nf is the defined as no. of flow channels, then the total flow rate is expressed as,