Theories of Seepage – Khosla’s Theory

This document discusses Khosla’s theory of seepage, which is used in the design of hydraulic structures like weirs and barrages. It explains the principles of the theory, including the movement of seeping water along streamlines and the concept of flow potential. The document also covers the concept of exit gradient and its importance in preventing undermining and piping. Additionally, it explores Khosla’s method of independent variables for determining pressures and exit gradient. Overall, this document provides a comprehensive understanding of Khosla’s theory and its application in water management.

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

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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)

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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.

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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.

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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.

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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.

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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.

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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:

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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.

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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.

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Khosla’s method of Independent Variables (cont.)
12
Khosla's simple profiles for a weir of
complex profile.

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University of Engineering and Technology Peshawar
Khosla’s method of Independent Variables (cont.)
13
Khosla's simple profiles for a weir of
complex profile.

14

15
Khosla Chart for d/s
pile, u/s pile and
depressed floor

16
Khosla Chart for Intermediate Pile

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Khosla’s method of Independent Variables (cont.)
17
➢ The uplift pressure obtained from the superposition of the individual forms are to
be corrected because the individual pressures have been obtained based on the
following assumptions:
1) The floor is of negligible thickness.
2) There is only one pile line.
3) The floor is horizontal.
➢ Because in an actual profile, the above assumptions are not satisfied the following
corrections are needed:
a) correction for the mutual interference of piles ;
b) correction for thickness of floor ;
c) correction for the slope of the floor.

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University of Engineering and Technology Peshawar
Khosla’s method of Independent Variables (cont.)
18
(a) Correction for the Mutual Interference of Piles.
➢ The correction C to be applied as percentage of head due to this effect, is given by
𝐶 = 19 𝐷
𝑏′
𝑑 + 𝐷
𝑏
Where
𝑏′ = The distance between two pile lines.
𝐷= The depth of the pile line, the influence of which has to be
determined on the neighboring pile of depth d.
D is to be measured below the level at which interference is desired.
𝑑= The depth of the pile on which the effect is considered.
𝑏= Total floor length.

Department of Civil Engineering
University of Engineering and Technology Peshawar
Khosla’s method of Independent Variables (cont.)
19
Suppose in the above Fig., we are considering the influence of the pile No. (2) on pile No.
(1) for correcting the pressure at C 1. Since the point C 1 in the rear, this correction shall be
positive. While the correction to be applied to E 2 due to pile No. (1) shall be negative,
since the point E 2 is in the forward direction of flow. Similarly, the correction at C 2 due to
pile No. (3) is positive, and the correction at E3 due to pile No. (2) is negative.
This correction is positive for the points in the rear or back water and subtractive for the
points forward in the direction of flow. This equation does not apply to the effect of an
outer pile on an intermediate pile, if the intermediate pile is equal to or smaller than the
outer pile and is at a distance less than twice the length of the outer pile.

Department of Civil Engineering
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Khosla’s method of Independent Variables (cont.)
20
(b) Correction for the Thickness of Floor
➢ In the standard form profiles, the floor is assumed to have negligible thickness.
Hence, the percentage pressures calculated by Khosla’s equations or graphs shall
pertain to the top levels of the floor. While the actual junction points E and C are at
the bottom of the floor. Hence, the pressures at the actual points are calculated by
assuming a straight-line pressure variation.
Since the corrected pressure at E 1 should be less than the calculated pressure at E 1’,
the correction to be applied for the joint E 1 shall be negative. Similarly, the pressure
calculated C1’ is less than the corrected pressure at C 1, and hence, the correction to be
applied at point C1 is positive.

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Corrections for the thickness of floor
21
𝐶1 = ϕ𝐷 − ϕ𝐶
𝑑1
× 𝑡1
ϕ𝐶1 = ϕ𝐶 + ϕ𝐷 − ϕ𝐶
𝑑1
× 𝑡1
𝐶1 = ϕ𝐷 − ϕ𝐶
𝑑2
× 𝑡2
𝐸1 = ϕ𝐸 − ϕ𝐷
𝑑2
× 𝑡2
𝐸1 = ϕ𝐸 − ϕ𝐷
𝑑3
× 𝑡3
Corrected pressure at 𝐶1 :

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Khosla’s method of Independent Variables (cont.)
22
(c) Correction for the slope of the floor
➢ A correction is applied for a sloping floor, and is taken as positive for the downward
slopes, and negative for the upward slopes following the direction of flow. Values of
correction of standard slopes are tabulated below
**The correction factor given is to be
multiplied by the horizontal length of
the slope and divided by the distance
between the two pile lines between
which the sloping floor is located.
This correction is applicable only to
the key points of the pile line fixed at
the start or the end of the slope.
Slope (H:V) Correction factor
(% of pressure)
1:1 11.2
2:1 6.5
3:1 4.5
4:1 3.3
5:1 2.8
6:1 2.5
7:1 2.3
8:1 2.0

Department of Civil Engineering
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Khosla’s method of Independent Variables (cont.)
23
➢ It has been determined that for a standard form consisting of a floor length (b)
with a vertical cutoff of depth (d), the exit gradient at its downstream end is given
by
𝐺𝐸 = 𝐻
𝑑 ∙ 1
𝜋 𝜆
𝜆 =1 + 1 + 𝛼2
2 𝛼 =𝑏
𝑑
if d = 0; 𝐺𝐸 is infinite. Hence, it becomes essential that a vertical cutoff at the
downstream end must be provided.
Exit Gradient (𝐺𝐸)
Where
𝑑= The depth of the d/s pile
𝑏 = Total floor length.
𝐻= Maximum seepage head

Department of Civil Engineering
University of Engineering and Technology Peshawar
Khosla’s method of Independent Variables (cont.)
24

Department of Civil Engineering
University of Engineering and Technology Peshawar
Khosla’s method of Independent Variables (cont.)
25
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)

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Khosla’s Theory – Example (GARG)
26
Determine the percentage pressures at various key points in Figure. Also determine
the exit gradient and plot the hydraulic gradient line for pond levels on u/s and no
flow on d/s.