UET Peshawar CE-402: Khosla's Theory of Seepage Assignment

This assignment solution from the Department of Civil Engineering at the University of Engineering and Technology Peshawar (CE-402: Irrigation Engineering and Water Management) delves into Khosla's Theory of Seepage. It begins by contrasting Khosla's approach with Bligh's theory, highlighting the importance of streamlines and the Laplace equation in understanding water flow through soil. The assignment explains the concepts of flow nets, equipotential lines, and the forces exerted by seeping water, particularly the exit gradient and its impact on soil stability. It details the critical exit gradient and safe exit gradients for various soil types. Furthermore, the solution covers the method of independent variables for determining pressures and exit gradients in hydraulic structures, breaking down complex profiles into simpler ones. It includes corrections for mutual interference of piles, floor thickness, and floor slope. The document provides Khosla charts and correction factors, making it a comprehensive guide to applying Khosla's theory in civil engineering.

Department of Civil Engineering
University of Engineering and Technology Peshawar
CE-402: Irrigation Engineering and
Water Management
Lecturer: Alamgir Khalil
8th Semester (4th Year)
Civil Engineering
Spring 2022
Lecture 10
Theories of Seepage – Khosla’s Theory
1

Department of Civil Engineering
University of Engineering and Technology Peshawar
Theories of Seepage (cont.)
2
Khosla's Theory
➢ Many of the important hydraulic structures, such as weirs and barrages, were
designed on the basis of Bligh's theory between the period 1910 to 1925. In 1926-27,
the upper Chenab canal syphons, 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 :
1) The seeping water does not creep along the bottom contour of impervious floor as
stated by Bligh, but on the other hand, this water moves along a set of streamlines.
This steady seepage in a vertical plane for a homogeneous soil can be expressed by
Laplacian equation.
𝜕2ϕ
𝜕𝑥2 + 𝜕2ϕ
𝜕𝑧2 = 0
where ϕ = Flow potential = Kh where K is the
coefficient of permeability of soil as defined
by Darcy's law, and h is the residual head
at any point within the soil.
Khosla
(1892-1984)

Department of Civil Engineering
University of Engineering and Technology Peshawar
Khosla's Theory (cont.)
3
➢ The Laplace equation represents two sets of curves intersecting each other
orthogonally. One set of lines is called Streamlines, and the other set is called
Equipotential lines. The resultant flow diagram showing both the sets of curves is
called a Flow Net.

Department of Civil Engineering
University of Engineering and Technology Peshawar
Khosla's Theory (cont.)
4
(2) The seepage water exerts a force at
each point in the direction of flow and
tangential to the streamlines
✓ This force (F) has an upward
component from the point where the
streamline 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.
✓ For the soil grain to remain stable, the submerged weight of soil grain should be more
than this upward disturbing force.
✓ 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.

Department of Civil Engineering
University of Engineering and Technology Peshawar
Khosla's Theory (cont.)
5
The submerged weight (𝑊𝑠) of a unit
volume of soil is given as :
For critical conditions to occur at the exit point,
Force 𝐹= pressure gradient at that point =
where 𝛾𝑤 = unit weight of water.
𝑆𝑠 = sp. gravity of soil particles
n = porosity of the soil material.
The upward disturbing force (𝐹) on the grain:
𝑊𝑠 = 𝛾𝑤 1 − 𝑛 (𝑆𝑠 − 1)
𝑑𝑝
𝑑𝑙 = 𝛾𝑤
𝑑ℎ
𝑑𝑙
𝑊𝑠 = 𝐹
𝑑ℎ
𝑑𝑙 = 1 − 𝑛 (𝑆𝑠 − 1) where dh/dl represents the rate of loss
of head or the gradient at the exit end.

Department of Civil Engineering
University of Engineering and Technology Peshawar
Khosla's Theory (cont.)
6
➢ The disturbing force at any point is proportional to the gradient of pressure of
water at that point (i.e. dp/dl). 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: The 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 or 5 is used, the exit gradient can then
be taken as safe. In other words, an exit gradient equal to 1/4 to 1/5 of the critical
exit gradient is ensured, so as to keep the structure safe against piping.

Department of Civil Engineering
University of Engineering and Technology Peshawar
Khosla's Theory (cont.)
7
Values of Khosla's Safe Exit Gradient for different types of Soils
Type of soil Khosla's Safe Exit Gradient
Shingle 1/4 to 1/5 (0.25 to 0.20)
Coarse sand 1/5 to 1/6 (0.20 to 0.17)
Fine sand 1/6 to 1/7 (0.17 to 0.14)
3) Undermining of the floor starts from the downstream end of the d/s impervious
floor, and if not checked, it travels upstream towards the weir wall. The
undermining starts only when the exit gradient is unsafe for the subsoil on which
the weir is founded. It is, therefore, absolutely necessary to have a reasonably deep
vertical cut-off at the downstream end of the d/s impervious floor to prevent
undermining.

Department of Civil Engineering
University of Engineering and Technology Peshawar
Khosla's Theory (cont.)
8
✓ The outer faces of the end sheet piles are much more effective than the inner
ones and the horizontal length of the floor.
✓ The intermediated piles if smaller length than the outer piles are ineffective
except for local redistribution of pressure.
✓ Undermining of floor started from tail end when the hydraulic gradient at the
exit is greater than the critical gradient for a particular soil.
✓ It is absolutely essential to have a reasonably deep vertical cut-off at the
downstream end to prevent piping.
➢ According to Khosla's theory, it was found that the actual uplift pressures were
quite different from those computed by Bligh's theory. This led to the following
provisional conclusions:

Department of Civil Engineering
University of Engineering and Technology Peshawar
Khosla's Theory (cont.)
9
Khosla's method of independent variables for determination of
pressures and exit gradient for seepage below a weir or a barrage.
➢ In order to know as to how the seepage below the foundation of a hydraulic
structure is taking place, it is necessary to plot the flow net. In other words, we
must solve the Laplacian equations. This can be accomplished either by
mathematical solution of the Laplacian equations, or by Electrical analogy method,
or by graphical sketching by adjusting the streamlines and equipotential lines w.r.t.
the boundary conditions.
➢ These are complicated methods and are time consuming. Therefore, for designing
hydraulic structures such as weirs or barrages on pervious foundations, Khosla and
his associates have evolved a simple, quick and an accurate approach, called
Method of Independent Variables.

Department of Civil Engineering
University of Engineering and Technology Peshawar
Khosla’s method of Independent Variables
10
➢ In order to know as how the seepage flow below the foundation of a hydraulic
structure is taking place, Khosla and his associates have evolved a simple, quick and
an accurate approach, called method of independent variables.
➢ In this method, a complex profile like that a weir/barrage is broken into a number of
simple profiles, each of which can be solved mathematically and presented in the
form of curves. These curves help in determining the percentage of pressures at the
various key points.

Department of Civil Engineering
University of Engineering and Technology Peshawar
Khosla’s method of Independent Variables (cont.)
11
➢ The following specific cases of general form were considered in Khosla's Theory
✓ Straight horizontal floor of negligible
thickness with pile at either end, upstream
or at downstream end.
✓ Straight horizontal floor of negligible
thickness with pile at some intermediate
point.
✓ Straight horizontal floor, depressed below
the bed, but with no cut off.

Department of Civil Engineering
University of Engineering and Technology Peshawar
Khosla’s method of Independent Variables (cont.)
12
Khosla's simple profiles for a weir of
complex profile.