Lecture 07 permeability and seepage (11-dec-2021)

Permeability and Seepage
Dr. Muhammad Safdar, PhD. (Professional Geotechnical Engineer)
Assistant Professor, Earthquake Engineering Center,
Department of Civil Engineering,
UET Peshawar, KP, Pakistan.
E-mail: drsafdar@uetpeshawar.edu.pk
CE-331: Geotechnical Engineering-I
.
Lecture-7

Lecture Contents
Geotechnical Engineering-1
Course Instructor
Dr. M. Safdar, EEC UET Peshawar.
A. Permeability
– Permeability
– Scope of Permeability in Geotechnical Engineering
– Darcy’s Law
– Hydraulic Conductivity of Soil
– Hydraulic Conductivity Determination in Laboratory
– Problem 1 & 2
B. Seepage
– Seepage
– Laplace Equation for two dimensional flow
– Flow nets (Problem-3)
– Uplift pressure due to seepage (Problem-4)
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Part (A) Permeability
Permeability
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Permeability is the measure of the soil’s ability to permit water to flow through its
pores or voids. It is property of soil.
• Soils are permeable due to the existence of interconnected voids through which water
can flow from points of high energy to points of low energy.
• Particle sizes and the structural arrangement of the these particles influence the rate of
flow in soil.
Permeability
Course Instructor
• A soil is highly pervious when water
can flow through it easily (e.g. sand,
gravel, gravel-sand mixtures).
• In an impervious soil, the permeability
is very low and water cannot easily flow
through it (e.g. silts, clays, silt-clay
mixtures).
• Rocks are impermeable.

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 Stability analyses of foundations and foundation excavations coming
in contact with flowing water and/or ground water.
 Stability analyses of seepage through dams and/or embankments.
 Stability analyses of seepage through earth retaining walls.
 Design of drainage systems.
 Water lowering.
 Estimation of well yields and design of tube wells.
Scope of Permeability in Geotechnical Engineering
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The total Head (h) causing a water flow is given by Bernoulli’s equation
Course Instructor
Permeability
• U is pore water pressure
and v is velocity.
• 𝛾 = ρ*g

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The total Head (h) causing a water flow is given by Bernoulli’s equation
In case of water flowing through soil, velocity head can be neglected because the
seepage velocity is small, hence:
The loss of head between two points, A and B, can be given by
Course Instructor
Permeability

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The head loss, Δh , can be expressed in a non dimensional form
Course Instructor
Permeability
Hydraulic gradient(i) between A and B is the total head loss per unit length
i =
𝒉𝑨−𝒉𝑩
𝑳𝑨𝑩

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Darcy’s law provides an equation for finding flow of fluid through porous
medium,
According to Darcy (1856) “the flow rate of water through soil of cross-
sectional area is directly proportional to imposed gradient (slope)”.
Valid for laminar flow conditions. If constant of proportionality k is introduced
then we obtain;
q = kAi
Where: k = coefficient of permeability of soil (cm/sec)
Darcy Law
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 Darcy velocity is a fictitious velocity since it assumes that flow
occurs across the entire cross-section of the soil sample.
 Flow actually takes place only through interconnected pore
channels.
 Discharge velocity < Seepage velocity
Fig 1
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Darcy Law

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Refer to Fig 1
v = n.vs (n = porosity, n < 100%)
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Darcy Law

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Assumptions
The following assumptions are made in Darcy’s law;
• The soil is saturated.
• The flow through soil is laminar.
• The flow is continuous and steady.
• The total cross sectional area of soil mass is considered.
• In fine grained soils voids are very small and flow is necessarily
laminar.
• In fractured rock, stones, gravels, and very coarse sands, turbulent
flow conditions may exist.
Course Instructor
Darcy Law

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Hydraulic conductivity or Coefficient of permeability (k)
The coefficient of permeability, k is defined as “the rate of flow of water under laminar
flow conditions through a porous medium area of unit cross section under unit
hydraulic gradient.”
The coefficient of permeability (k) is obtained from the relation
Where q = discharge, Q = total volume of water, t = time period, h = head causing
flow, L= length of manometer outlets, A = cross-sectional area of specimen
Factors affecting Hydraulic Conductivity:
The porosity or void ratio of soil.
The grain size, shape and distribution.
The degree of saturation.
The viscosity of soil water which varies with temperature.
Hydraulic conductivity
Course Instructor

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For fairly uniform sand (that is, sand with a small uniformity coefficient), Hazen
(1930) proposed an empirical relationship for hydraulic conductivity in the form:
Valid for:
Loose soil
Clean sand
Cu= D60/D10 <2
0.1 mm < D10 < 0.3 mm
Where
C=constant that varies from 1.0 to 1.5
D10= effective size, in mm
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Coefficient of permeability (hydraulic conductivity) of a layer may vary with direction
of flow (kv ≠ kh). Stratified soil deposits, large variation in kv & kh from layers to
layers. Thus equivalent hydraulic conductivity is calculated for layered soil deposits
series.
Flow in Horizontal Direction
Flow in Vertical Direction
Hydraulic conductivity of Stratified soils
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Course Instructor
Flow in Horizontal Direction Flow in Vertical Direction

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Coefficient of permeability (k) is determined in the laboratory using two
methods.
A. Constant head permeability test for coarse grained soils (sands,
gravels and their mixtures).
B. Falling/Variable head permeability test for fine grained soils (silts,
clays and their mixtures).
Hydraulic Conductivity Determination (Laboratory)
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Constant head permeability test for coarse grained soils (sands, gravels and their
mixtures).
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ASTM D2434

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Constant head permeability test
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Constant head permeability test
Problem-1:
Let a constant head permeability test was done and following data were obtain;
Q = 100 ml, A = 45 cm2
, h = 25 cm, L = 15 cm, t = 37 hrs
Solution:
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Falling/Variable head permeability test for fine grained soils (silts, clays
and their mixtures).
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ASTM D5084

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Falling/Variable head permeability test
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Falling/Variable head permeability test
Problem-2
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Solution

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Part (B) Seepage
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Flow of water through soils is called seepage. It is a phenomena, while
permeability is property of soil.
• Seepage occurs, when there is difference in water levels on the two sides of the
structure such as seepage into a cofferdam or seepage through earth dam or
embankment.
• Estimate the quantity of seepage and the seepage pressure acting on the
structure.
• The soil permeability is the main parameter required in the calculation of seepage
quantity.
• In many cases the flow of water through soil is not in one direction only, nor is it
uniform over the entire area perpendicular to the flow . In such cases, the ground
water flow is generally calculated by the use of graphs, called as flow nets. The
concept of the flow net is based on Laplace's equation of continuity, which
governs the steady flow condition for a given point in the soil mass.
Seepage
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Seepage
Seepage
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To derive the Laplace differential equation of continuity, let us consider a single row of
sheet piles that have been driven into a permeable soil layer, as shown in Figure 1. The
row of sheet piles is assumed to be impervious. The steady-state flow of water from the
upstream to the downstream side through the permeable layer is a two-dimensional flow.
For flow at a point A, we consider an elemental soil block. The block has dimensions dx, dy,
and dz (length dy is perpendicular to the plane of the paper); it is shown in an enlarged
scale in Figure 2.
Course Instructor
Laplace Equation for two dimensional flow
Figure 1
Figure 2

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The rates of outflow from the block in the horizontal and vertical directions are,
respectively,
• Water is incompressible and no volume change in the soil mass.
• Total rate of inflow should equal the total rate of outflow
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From Darcy Law, the discharge velocities can be expressed as;
Thus from the above equations we can write as;
If the soil is isotropic with respect to the hydraulic conductivity that is, kx = kz
Course Instructor

Assumptions in deriving laplace equation for two dimensional flow:
1.The flow is two-dimensional.
2. The flow is steady and laminar.
3. Water and the soil are incompressible.
4. The soil mass is homogeneous and isotropic.
5. The soil is fully saturated and Darcy’s law is valid.
 Consider two solution techniques for Laplace’s equation. One is an
approximate method called flow net sketching, the other is the finite difference
technique.
 The flow net sketching technique is simple and flexible and conveys a picture of
the flow regime.
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Flow Net
A combination of a number of flow lines and equipotential lines is called a flow net.
The solution to Laplace Equation is represented by a family of flow lines and a family of
equipotential, referred as a flow net.
Flow net is a graphical representation of a flow field.
• A flow line is a line along which a water particle will travel from upstream to the
downstream side in a permeable soil. Also called as Stream line. The flow lines
intersect the equipotential lines at right angles.
• An equipotential line is a line along which the potential head at all points is equal.
It is simply a contour of constant total head
Course Instructor
Flow net
Flow line
Equipotential line

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Flow Net Construction
Flow nets are constructed for the calculation of groundwater flow and the
evaluation of heads in the media. To complete the graphic construction of a flow net,
a flow net must meet the following criteria
(1) The boundary conditions must be satisfied, which are (see figure at next slide)
Condition 1: The upstream and downstream surfaces of the permeable layer (lines ab
and de) are equipotential lines.
Condition 2: Because ab and de are equipotential lines, all the flow lines intersect
them at right angles.
Condition 3: The boundary of the impervious layer that is, line fg—is a flow line and
so is the surface of the impervious sheet pile, line acd..
Condition 4: The equipotential lines intersect acd and fg at right angles.
Course Instructor
Flow net

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Flow net

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(2) Flow lines must intersect equipotential lines at right angles.
(3) The area between flow lines and equipotential lines must be curvilinear
squares.
(4) The quantity of flow through each flow channel is constant.
(5) The head loss between each consecutive equipotential line is constant.
(6) Flow lines cannot intersect each other.
(7) Equipotential lines cannot intersect each other.
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Flow net

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Flow of water under dams
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Flow net

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Seepage Calculation from a Flow Net
 In any flow net, the strip between any two adjacent flow lines is called a flow
channel.
 Because there is no flow across the flow lines, thus
Course Instructor
Flow net

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 The drop in the piezometric level between any two adjacent equipotential lines
is the same. This is called the potential drop.
 If the number of flow channels in a flow net is equal to Nf, the total rate of flow
through all the channels per unit length can be given by
Course Instructor
Flow net

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 If the elements are not square i.e. b1/l1=b2/l2=b3/l3 …….= n ( any value
other than 1,which is in case of squares ).
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Flow net

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Problem-3
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Flow net

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Course Instructor
Flow net
Solution

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Pressure at any point, X, at the base of structure under which seepage occurs is
given by :
Since the total head at X , 𝐻𝑥 = pressure head + elevation head
𝐻𝑥 = 𝑃𝑥/ γ𝑤 + 𝑍𝑥
Re-arranging for 𝑃𝑥,
𝑃𝑥 = γ𝑤 𝐻𝑥 − 𝑍𝑥
Where Hx=head at point x
Zx= Height of point x from datum or impervious layer
Hx= HB- Head loss/drop . (No of drops to Point x)
Δh=ΔH/nd
Uplift pressure due to Seepage
Bernoulli’s equation =
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Problem-4
Find the pressure at Point B and at base of the structure given in the
previous example.
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Solution:
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Now to find the total force acting upward on base of structure. Use
Simpson rule/Trapezodial rule.
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Course Instructor
Thank You
Any Question(s) ?
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Lecture 07 permeability and seepage (11-dec-2021)

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