Khosla’s Theory and Concept of Flow Nets

Many of the important hydraulic structures, such as weirs and barrage, were designed on the basis of
Bligh’s theory between the periods 1910 to 1925. In 1926 – 27, the upper Chenab canal siphons, designed
on Bligh’s theory, started posing undermining troubles. Investigations started, which ultimately lead to
Khosla’s theory. The main principles of this theory are summarized below:
         (a) The seepage water does not creep along the bottom contour of pucca flood as started by Bligh, but
on the other hand, this water moves along a set of stream-lines. This steady seepage in a vertical
plane for a homogeneous soil can be expressed by Laplacian equation:

Where, φ = Flow potential = Kh; K = the co-efficient of permeability of soil as defined by

Darcy’s law, and h is the residual head at any point within the soil.

The above equation represents two sets of curves intersecting each other orthogonally. The resultant

flow diagram showing both of the curves is called a Flow Net.

Stream Lines: 

                        The streamlines represent the paths along which the water flows through the sub-soil.

Every particle entering the soil at a given point upstream of the work, will trace out its own path and will

represent a streamline. The first streamline follows the bottom contour of the works and is the same as

Bligh’s path of creep. The remaining streamlines follows smooth curves transiting slowly from the outline

of the foundation to a semi-ellipse, as shown below.

Equipotential Lines:-

                                       (1) Treating the downstream bed as datum and assuming no water on the

downstream side, it can be easily started that every streamline possesses a head equal to h1 while entering

the soil; and when it emerges at the down-stream end into the atmosphere, its head is zero. Thus, the head

h1 is entirely lost during the passage of water along the streamlines.

Further, at every intermediate point in its path, there is certain residual head (h) still to be dissipated

in the remaining length to be traversed to the downstream end. This fact is applicable to every streamline,

and hence, there will be points on different streamlines having the same value of residual head h. If such

points are joined together, the curve obtained is called an equipotential line.

Every water particle on line AB is having a residual head h = h1, and on CD is having a residual head h = 0, and hence, AB and CD are equipotential lines.

Since an equipotential line represent the joining of points of equal residual head, hence if piezometers were installed on an equipotential line, the water will rise in all of them up to the same level as shown in figure below.

(b) The seepage water exerts a force at each point in the direction of flow and tangential to the streamlines as shown in figure above. This force (F) has an upward component from the point where the streamlines turns upward. For soil grains to remain stable, the upward component of this force should be counterbalanced by the submerged weight of the soil grain. This force has the maximum disturbing tendency at the exit end, because the direction of this force at the exit point is vertically upward, and hence full force acts as its upward component. For the soil grain to remain stable, the submerged weight of soil grain should be more than this upward disturbing force. The disturbing force at any point is proportional to the gradient of pressure of water at that point (i.e. dp/dt). This gradient of pressure of water at the exit end is called the exit gradient. In order that the soil particles at exit remain stable, the upward pressure at exit should be safe. In other words, the exit gradient should be safe.

Critical Exit Gradient:-

                                        This exit gradient is said to be critical, when the upward disturbing force on the grain is just equal to the submerged weight of the grain at the exit. When a factor of safety equal to 4 to 5 is used, the exit gradient can then be taken as safe. In other words, an exit gradient equal to ¼ to 1/5 of the critical exit gradient is ensured, so as to keep the structure safe against piping.

The submerged weight (Ws) of a unit volume of soil is given as:

                           γw (1 – n) (Ss – 1)

                                                             Where, w = unit weight of water.

                                                              Ss = Specific gravity of soil particles

                                                              n = Porosity of the soil material

      For critical conditions to occur at the exit point

                                          F = Ws

                                                           Where F is the upward disturbing force on the grain

                                                            Force F = pressure gradient at that point = dp/dl =  γw ×dh/dl