CE 111 – Geotechnical Engineering 1 (Soil Mechanics)
Seepage
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flow lines
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Seepage Force – force applied by water on the soil.
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Quicksand Condition - occurs when the seepage pressure,
which acts in the upward direction, overcomes the downward
direction pressure due to submerged weight of soil, and the
sand grains are forced apart. The result is that the soil has no
capability to support a load.
Laplace Equation
𝒌𝟐 𝒛
)
𝒌𝟏 𝑯𝟐 + 𝒌𝟐 𝑯𝟏
𝑓𝑜𝑟 0 ≤ 𝑧 ≤ 𝐻1
𝒉 = 𝒉𝟏 (𝟏 −
𝒌𝟏
) (𝑯𝟏 + 𝑯𝟐 − 𝒛)]
𝒌𝟏𝑯𝟐 + 𝒌𝟐 𝑯𝟏
𝑓𝑜𝑟 𝐻1 ≤ 𝑧 ≤ 𝐻1 + 𝐻2
𝒉 = 𝒉𝟏 [(
Flow Nets
Flow Net – The combination of equipotential lines and
Flow line – Line along which a water particle will travel
from upstream to the downstream side in the permeable
soil medium.
Equipotential lines – line along which the potential head
at all points is equal
Flow channel – The strip between any two adjacent flow
lines
Equipotential Drop – The drop in the piezometric level
between any two adjacent equipotential lines
Boundary Conditions
1. The upstream and downstream surfaces of the permeable
layer (lines ab and de) are equipotential lines
2. Because ab and de are equipotential lines, all the flow
lines intersect them at right angles
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.
4. The equipotential lines intersect acd and fg at right angles
Graphical Method of Flow Net Construction
The graphical method is the method in which the flow net
is constructed by an intensive trial and error procedure. It is
the simplest and the quickest method of flow net construction.
Since the method includes trial and error proceedings; a lot of
practice is required for achieving accurate results. The
following are the steps for constructing a flow net:
1. First of all, smooth curves representing the flow lines that
meet the specified requirements are first marked.
2. Then, the equipotential lines are drawn such that they cut
or intersect the flow lines at right angles. It must be
ensured that the equipotential lines are drawn such that
the fields form approximate curvilinear squares.
3. Any defect that may be present must be identified and
duly rectified.
4. The flow nets will be finally satisfactory for practical uses
when the fields are curvilinear squares.
Key points that must be considered:
•
The boundary conditions must be duly established.
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It must be ensured that each flow line cut the
equipotential lines at right angles to each other.
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The space enclosed by the adjacent equipotential lines
and the flow lines must be curvilinear squares.
Determination of Quantity of Seepage
𝑯𝑵𝒇
𝒒=𝒌
𝑵𝒅
q – total rate of flow through all the channels
k – hydraulic conductivity
H – head difference between the upstream and downstream
sides
Nf – number of flow channels
Nd – Number of potential drops
Two-Dimensional Flow
Isotropic Soil - Property in all directions are equal, k=kx=kz
Anisotropic Soil - Does not have the same physical properties
when the direction of measurement is changed, so, for the
permeability, kx≠kz. For an anisotropic soil,
𝐪 = √𝐤 𝐱 𝐤 𝐳
𝐇𝐍𝐟
𝐍𝐝
a.
Seepage through Earth Dams
3.
4.
5.
Step 1: Obtain α.
Step 2: Calculate Δ and then 0.3 Δ.
Step 3: Calculate d.
𝑳=
𝒅𝟐
6.
𝑯𝟐
𝒅
−√
−
𝒄𝒐𝒔 𝜶
𝒄𝒐𝒔𝟐 𝜶 𝒔𝒊𝒏𝟐 𝜶
Step 4: With known values of α and d, calculate L from:
Step 5: With known value of L, calculate q from:
𝒒 = 𝒌(𝒕𝒂𝒏 𝜶)(𝑳 𝒔𝒊𝒏 𝜶)
7.
How high (above the ground surface) will the water
rise if piezometers are placed at points a, b, c, and
d?
b. What is the rate of seepage through flow channel II
per unit length (perpendicular to the section
shown)?
Given that β=45°, α=30°, B=3m, H=6m, height of dam=7.6 m,
and k=61x10-6, calculate the seepage rate, q, in m 3/day/m
length.
Refer to the constant-head permeability test arrangement in a
two-layered soil as shown in Figure 1. During the test, it was
seen that when a constant head of h1 = 200 mm was maintained,
the magnitude of h2 was 80 mm. If k1 is 0.004 cm/sec, determine
the value of k2 given H1 = 100 mm and H2 = 150 mm.
Refer to Figure 4. Given:
• H1 = 6 m
•D=3m
• H2 = 1.5 m
• D1 = 6 m
Draw a flow net. Calculate the seepage loss per meter length
of the sheet pile (at a right angle to the cross section shown).
Draw a flow net for the single row of sheet piles driven into a
permeable layer as shown in Figure 4. Given:
• H1 = 3 m
• D = 1.5 m
• H2 = 0.5 m
• D1 = 3.75 m
An earth dam is shown in Figure 3. Determine the seepage rate,
q, in m3/day/m length. Given: α1 = 35°, α2 = 40°, L1 = 5 m, H
= 7 m, H1 = 10 m, and k = 3x10-4 m/sec. Use Schaffernak’s
solution.
Design of Filters
When seepage water flows from a soil with relatively fine
grains into a coarser material, there is danger that the fine soil
particles may wash away into the coarse material. Over a
period of time, this process may clog the void spaces in the
coarser material.
For proper selection of the filter material, two conditions
should be kept in mind: (Terzaghi and Peck)
Condition 1:
𝑫𝟏𝟓(𝑭)
≤ 𝟒 𝒕𝒐 𝟓
𝑫𝟖𝟓(𝑺)
Condition 2:
Figure 1 - Example Problem No. 1
𝑫𝟏𝟓(𝑭)
≥ 𝟒 𝒕𝒐 𝟓
𝑫𝟏𝟓(𝑺)
Where:
D15(F) = diameter through which 15% of filter material will
pass
D15(S) = diameter through which 15% of soil to be protected
will pass
D85(S) = diameter through which 85% of soil to be protected
will pass
Example Problems:
1. Refer to figure 1. Given: H1=305 mm., H2=508 mm., h1=610
mm., h=508 mm., Z=203 mm., k1=0.066 cm/sec, and diameter
of the soil specimen is D=76 mm. Determine the rate of flow of
water through the two-layered soil.
2. A flow net for flow around a single row of sheet piles in a
permeable soil layer is shown. We are given that
kx=kz=k=5x10-3cm/sec.
Figure 2 - Example Problem No. 2
Figure 3 - Example Problem No. 3
Figure 4