CIVE.5370 EXPERIMENTAL SOIL MECHANICSSoil Permeability
BERNOULLI’S EQUATION
Zg
vu hw
2
2
Where:
h = Total Headu = Pressurev = Velocityg = Acceleration due to Gravityw = Unit Weight of Water
CIVE.5370 EXPERIMENTAL SOIL MECHANICSSoil Permeability
Zg
vu hw
2
2
Therefore:
Zu hw
v ≈ 0(i.e. velocity of water in soil is negligible).
v ≈ 0
BERNOULLI’S EQUATION IN SOIL
CIVE.5370 EXPERIMENTAL SOIL MECHANICSSoil Permeability
Figure 5.1. Das FGE (2005).
CHANGE IN HEADFROM POINTS A
& B (H)BA h hh
B
w
BA
w
A ZuZu h
BA h hh
Lh i
BA h hh
h can be expressed in non-dimensional form
Where:i = Hydraulic GradientL = Length of Flow between
Points A & B
CIVE.5370 EXPERIMENTAL SOIL MECHANICSSoil Permeability
Figure 5.2. Das FGE (2005).
VELOCITY (v) VS. HYDRAULIC GRADIENT (i)General relationship shown in Figure 5.2
Three Zones:1. Laminar Flow (I)2. Transition Flow (II)3. Turbulent Flow (III)
For most soils, flow is laminar. Therefore:
v i
CIVE.5370 EXPERIMENTAL SOIL MECHANICSSoil Permeability
DARCY’S LAW (1856)
Where:v = Discharge Velocity (i.e. quantity of water in
unit time through unit cross-sectional areaat right angles to the direction of flow)
k = Hydraulic Conductivity (i.e. coefficient ofpermeability)
i = Hydraulic Gradient* Based on observations of flow of water through clean sands
CIVE.5370 EXPERIMENTAL SOIL MECHANICSSoil Permeability
FACTORS AFFECTING PERMEABILITYPermeability is not a fundamental soil property but depends upon a number of factors:
• Particle size distribution• Particle shape and texture• Mineralogical composite
• Void ratio• Degree of saturation• Soil fabric
• Nature of fluid• Type of Flow• Temperature
Invariable for a given soil
Dependent upon placing and treatment of the soil
Relate to the permeabilityTemp. Correction:
CIVE.5370 EXPERIMENTAL SOIL MECHANICSSoil Permeability
Figure 5.3. Das FGE (2005).
DISCHARGE AND SEEPAGE VELOCITIES
svvAvAq Where:
q = Flow Rate(quantity of water/unit time)
A = Total Cross-sectional AreaAv = Area of Voidsvs = Seepage Velocity
CIVE.5370 EXPERIMENTAL SOIL MECHANICSSoil Permeability
nv
eev
VV
VV
vV
VVvv
LAAAv
AAAvv
vAAAvq
s
v
s
v
v
svs
v
sv
v
svs
svsv
11
)(
)()()(
Figure 5.3. Das FGE (2005).
DISCHARGE AND SEEPAGE VELOCITIES
CIVE.5370 EXPERIMENTAL SOIL MECHANICSSoil Permeability
HYDRAULIC CONDUCTIVITY: LABORATORY TESTINGConstant Head(ASTM D2434)
Falling Head(no ASTM)
Figure 5.4. Das FGE (2005). Figure 5.5. Das FGE (2005).
CIVE.5370 EXPERIMENTAL SOIL MECHANICSSoil Permeability
Constant Head(ASTM D2434)
AhtQLk
tkiAAvtQ )(Where:
Q = Quantity of water collectedover time t
t = Duration of water collection
tLhkAQ
Figure 5.4. Das FGE (2005).
HYDRAULIC CONDUCTIVITY: LABORATORY TESTING
CIVE.5370 EXPERIMENTAL SOIL MECHANICSSoil Permeability
LABORATORY TESTING: CONSTANT HEAD
CIVE.5370 EXPERIMENTAL SOIL MECHANICSSoil Permeability
Falling Head(No ASTM)
2
110303.2
hhLog
AtaLk
dtdhaA
Lhkq
hdh
AkaLdt
Where:A = Cross-sectional area of Soila = Cross-sectional area of Standpipe
Figure 5.5. Das FGE (2005). 2
1loghhe
AkaLt
Integrate from limits 0 to t
Integrate from limits h1 to h2
after rearranging above equation
after integration
or
HYDRAULIC CONDUCTIVITY: LABORATORY TESTING
CIVE.5370 EXPERIMENTAL SOIL MECHANICSSoil Permeability
LABORATORY TESTING: FALLING HEAD
CIVE.5370 EXPERIMENTAL SOIL MECHANICSSoil Permeability
210sec)/( cDcmk
Uniform Sands - Hazen Formula(Hazen, 1930):
Where:c = Constant between 1 to 1.5D10 = Effective Size (in mm)
eeCk
1
3
1
Sands – Kozeny-Carman(Loudon 1952 andPerloff and Baron 1976):
Where:C = Constant (to be determined)e = Void Ratio
85.024.1 kek
Sands – Casagrande(Unpublished):
Where:e = Void Ratiok0.85 = Hydraulic Conductivity @ e = 0.85
e
eCkn
12
Normally Consolidated Clays(Samarasinghe, Huang, and Drnevich, 1982):
Where:C2 = Constant to be determined experimentallyn = Constant to be determined experimentallye = Void Ratio
HYDRAULIC CONDUCTIVITY: EMPIRICAL RELATIONSHIPS
CIVE.5370 EXPERIMENTAL SOIL MECHANICSSoil Permeability
after Casagrande and Fadum (1940) and Terzagi et al. (1996).
SOILPERMEABILITY
ANDDRAINAGE
CIVE.5370 EXPERIMENTAL SOIL MECHANICSSoil Permeability
From FHWA IF-02-034 Evaluation of Soil and Rock Properties.
Good drainage
10-8 10-910-710-610-510-410-310-210-11.0101102 10-8 10-910-710-610-510-410-310-210-11.0101102
Poor drainage Practically impervious
Clean gravel Clean sands, Clean sand and gravel mixtures
Impervious sections of earth dams and dikes
“Impervious” soils which are modif ied by the effect of vegetation and weathering; f issured, weathered clays; f ractured OC clays
Pervious sections of dams and dikes
Very f ine sands, organic and inorganic silts, mixtures of sand, silt, and clay glacial till, stratif ied clay deposits, etc.
“Impervious” soils e.g., homogeneous clays below zone of weathering
Drainage property
Application in earth dams and dikes
Type of soil
Direct determination of coefficient of permeability
Indirect determination of coefficient of permeability
*Due to migration of f ines, channels, and air in voids.
Direct testing of soil in its original position (e.g., well points). If properly conducted, reliable; considerable experience required. (Note: Considerable experience
also required in this range.)
Constant Head Permeameter; little experience required.
Constant head test in triaxial cell; reliable w ith experience and no leaks.
Reliable;Little experiencerequired
Falling Head Permeameter;Range of unstable permeability;* much experience necessary to correct interpretation
Fairly reliable; considerable experience necessary (do in triaxial cell)
Computat ion:From the grain size distribution(e.g., Hazen’s formula). Only applicable to clean, cohesionless sands and gravels
Horizontal Capillarity Test:Very little experience necessary; especially useful for rapid testing of a large number of samples in the f ield w ithout laboratory facilities.
Computat ions:from consolidation tests; expensive laboratory equipment and considerable experience required.
10-8 10-910-710-610-510-410-310-210-11.0101102 10-8 10-910-710-610-510-410-310-210-11.0101102
COEFFICIENT OF PERMEABILITYCM/S (LOG SCALE)
SOIL PERMEABILITY AND DRAINAGE
CIVE.5370 EXPERIMENTAL SOIL MECHANICSSoil Permeability
LAPLACE'S EQUATION OF CONTINUITYSteady-State
Flow around an impervious
Sheet Pile Wall
Consider water flow at Point A:
vx = Discharge Velocity in x Direction
vz = Discharge Velocity in z Direction
Y Direction Out Of PlaneFigure 5.11. Das FGE (2005).
x
z
y
CIVE.5370 EXPERIMENTAL SOIL MECHANICSSoil Permeability
dxdydzzvv
dzdydxxv
v
zz
xx
Consider water flow at Point A(Soil Block at Pt A shown left)
Rate of water flow into soil block in x direction:
vxdzdyRate of water flow into soil block in z direction:
vzdxdy
Figure 5.11. Das FGE (2005).
Rate of water flow out of soil block in x,z directions:
LAPLACE'S EQUATION OF CONTINUITY
CIVE.5370 EXPERIMENTAL SOIL MECHANICSSoil Permeability
0
0
zv
xv
ordxdyvdzdyv
dxdydzzvvdzdydx
xv
v
zx
zx
zz
xx
Consider water flow at Point A(Soil Block at Pt A shown left)
Figure 5.11. Das FGE (2005).
Total Inflow = Total Outflow
LAPLACE'S EQUATION OF CONTINUITY
CIVE.5370 EXPERIMENTAL SOIL MECHANICSSoil Permeability
02
2
2
2
zhk
xhk
zhkikv
xhkikv
zx
zzzz
xxxx
Consider water flow at Point A(Soil Block at Pt A shown left)
Figure 5.11. Das FGE (2005).
Using Darcy’s Law (v=ki)
LAPLACE'S EQUATION OF CONTINUITY
CIVE.5370 EXPERIMENTAL SOIL MECHANICSSoil Permeability
FLOW NETS: DEFINITION OF TERMSFlow Net: Graphical Construction used to calculate groundwater flow through soil. Comprised of Flow Lines and Equipotential Lines.Flow Line: A line along which a water particle moves through a permeable soil medium.Flow Channel: Strip between any two adjacent Flow Lines.Equipotential Lines: A line along which the potential head at all points is equal.
NOTE: Flow Lines and Equipotential Lines must meet at right angles!
CIVE.5370 EXPERIMENTAL SOIL MECHANICSSoil Permeability
Figure 5.12a. Das FGE (2005).
FLOW NETSFLOW AROUND
SHEET PILE WALL
CIVE.5370 EXPERIMENTAL SOIL MECHANICSSoil Permeability
Figure 5.12b. Das FGE (2005).
FLOW NETSFLOW AROUND
SHEET PILE WALL
CIVE.5370 EXPERIMENTAL SOIL MECHANICSSoil Permeability
1.The upstream and downstream surfaces of the permeable layer (i.e. lines ab and de in Figure 12b Das FGE (2005)) 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 (i.e. line fg in Figure 12b Das FGE (2005)) is a flow line, as is the surface of the impervious sheet pile (i.e. line acd in Figure 12b Das FGE (2005)).
4.The equipontential lines intersect acd and fg(Figure 12b Das FGE (2005)) at right angles.
FLOW NETS: BOUNDARY CONDITIONS
CIVE.5370 EXPERIMENTAL SOIL MECHANICSSoil Permeability
Figure 5.13. Das FGE (2005).
FLOW NETSFLOW UNDER ANIMPERMEABLE
DAM
CIVE.5370 EXPERIMENTAL SOIL MECHANICSSoil Permeability
nqqqq ...321
Figure 5.14. Das FGE (2005).
q k h1 h2l1
l1 k h2 h3
l2
l2 k h3 h4
l3
l3 ...
h1 h2 h2 h3 h3 h4 ... HNd
Rate of Seepage ThroughFlow Channel (per unit length):
Using Darcy’s Law (q=vA=kiA)
Potential DropWhere:
H = Head DifferenceNd = Number of Potential Drops
FLOW NETS: DEFINITION OF TERMS
CIVE.5370 EXPERIMENTAL SOIL MECHANICSSoil Permeability
Figure 5.12b. Das FGE (2005).
d
f
NHN
kq
If Number of Flow Channels = Nf, then the total flow for all channels per unit length is:
Therefore, flow through one channel is:
dNHkq
FLOW NETSFLOW AROUND SHEET PILE WALL EXAMPLE
CIVE.5370 EXPERIMENTAL SOIL MECHANICSSoil Permeability
GIVEN:
Flow Net in Figure 5.17.Nf = 3Nd = 6kx=kz=5x10-3 cm/sec
DETERMINE:
a. How high water will rise in piezometers at points a, b, c, and d.
b. Rate of seepage through flow channel II.
c. Total rate of seepage.
Figure 5.17. Das FGE (2005).
FLOW NETSFLOW AROUND SHEET PILE WALL EXAMPLE
CIVE.5370 EXPERIMENTAL SOIL MECHANICSSoil Permeability
mmm 56.06
)67.15(
SOLUTION:
Potential Drop = dN
H
At Pt a:Water in standpipe =(5m – 1x0.56m) = 4.44m
At Pt b:Water in standpipe =(5m – 2x0.56m) = 3.88m
At Pts c and d:Water in standpipe =(5m – 5x0.56m) = 2.20m
Figure 5.17. Das FGE (2005).
FLOW NETSFLOW AROUND SHEET PILE WALL EXAMPLE
CIVE.5370 EXPERIMENTAL SOIL MECHANICSSoil Permeability
FLOW NETSFLOW AROUND SHEET PILE WALL EXAMPLE
SOLUTION:
dNHkq
k = 5x10-3 cm/seck = 5x10-5 m/sec
q = (5x10-5 m/sec)(0.56m)q = 2.8x10-5 m3/sec/m
fd
f qNN
HNkq
q = (2.8x10-5 m3/sec/m) * 3q = 8.4x10-5 m3/sec/m
Figure 5.17. Das FGE (2005).
CIVE.5370 EXPERIMENTAL SOIL MECHANICSSoil Permeability
FLOW NETS: RULES FOR CREATINGFLOW NETS (FROM UTEXAS)
1. Head drops between adjacent equipotential lines must be constant (or, in those rare cases where this is not desirable, clearly stated, just as in topographic contour maps)!
2. Equipotential lines must match known boundary conditions.
3. Flow lines can never cross.
Flow Line
Equi.
CIVE.5370 EXPERIMENTAL SOIL MECHANICSSoil Permeability
FLOW NETS: STEPS FOR DRAWING(LADD, MIT)
1. Draw problem in ink.2. Draw in known equipotential and flow
boundary lines.3. Sketch 2 or 3 flow lines.4. Draw corresponding equipotential
lines (check for squares and ┴ intersections)
5. Keep adjusting (and adjusting, and adjusting…)
Flow Line
Equi.
CIVE.5370 EXPERIMENTAL SOIL MECHANICSSoil Permeability
FLOW NETS: RULES FOR CREATINGFLOW NETS (FROM UTEXAS)
1. Refraction of flow lines must account for differences in hydraulic conductivity.
2. For isotropic media.a) Flow lines must intersect equipotential
lines at right angles.b) The flow line-equipotential polygons
should approach curvilinear squares, as shown in the Figure to the right.
Flow Line
Equi.
CIVE.5370 EXPERIMENTAL SOIL MECHANICSSoil Permeability
FLOW NETS: RULES FOR CREATINGFLOW NETS (FROM UTEXAS)
6. The quantity of flow between any two adjacent flow lines must be equal.
7. The quantity of flow between any two stream lines is always constant.
Flow Line
Equi.
CIVE.5370 EXPERIMENTAL SOIL MECHANICSSoil Permeability
FLOW NETS: DRAWING PROCEDURE(AFTER HARR (1962, P. 23)
1. Draw the boundaries of the flow region to scale so that all equipotential lines and flow lines that are drawn can be terminated on these boundaries.
2. Sketch lightly three or four flow lines, keeping in mind that they are only a few of the infinite number of curves that must provide a smooth transition between the boundary flow lines. As an aid in spacing of these lines, it should be noted that the distance between adjacent flow lines increases in the direction of the larger radius of curvature.
3. Sketch the equipotential lines, bearing in mind that they must intersect all flow lines, including the boundary streamlines, at right angles and that the enclosed figures must be (curvilinear) squares.
CIVE.5370 EXPERIMENTAL SOIL MECHANICSSoil Permeability
FLOW NETS: DRAWING PROCEDURE(FROM HARR (1962, P. 23)
4. Adjust the locations of the flow lines and the equipotential lines to satisfy the requirements of step 3. This is a trail-and-error process with the amount of correction being dependent upon the position of the initial flow lines. The speed with which a successful flow net can be drawn is highly contingent on the experience and judgment of the individual. A beginner will find the suggestions in Casagrande (1940) to be of assistance.
5. As a final check on the accuracy of the flow net, draw the diagonals of the squares. These should also form smooth curves that intersect each other at right angles.
CIVE.5370 EXPERIMENTAL SOIL MECHANICSSoil Permeability
FLOW NETS: RULES(FROM HUMBOLDT UNIVERSITY GEOLOGY 556 COURSE)
1. In a homogeneous isotropic system, flow lines, and equipotentials are always perpendicular and form curvilinear "squares".
2. Equipotentials are always normal to an impermeable boundary.
3. Flow lines are always parallel to an impermeable boundary.
4. Equipotentials are always parallel to a constant head boundary.
5. Flow lines are always normal to a constant head boundary.
CIVE.5370 EXPERIMENTAL SOIL MECHANICSSoil Permeability
FLOW NETS: SUGGESTED PROCEDURE(FROM HUMBOLDT UNIVERSITY GEOLOGY 556 COURSE)
1. First identify boundary conditions (Which boundaries are impermeable? Which are constant head?)
2. Next think: Where is water entering the system? Where can it leave?
3. Always look for any symmetry in the boundary conditions.
4. Decide on the number of flow tubes you want to use.5. Draw a trial flow line and then draw in other flow lines
to define all the flow tubes; some trial and error sketching may be necessary.
CIVE.5370 EXPERIMENTAL SOIL MECHANICSSoil Permeability
FLOW NETS: SUGGESTED PROCEDURE(FROM HUMBOLDT UNIVERSITY GEOLOGY 556 COURSE)
6. Where flow tubes constrict, higher head gradients (more closely spaced equipotentials) are needed to move the same quantity of water through the flow tube.
7. Fit together the curvilinear squares by drawing in the equipotentials. As you do this, you may have to revise the positions of some of the flow lines. Trial-and-error is the order of the day.
CIVE.5370 EXPERIMENTAL SOIL MECHANICSSoil Permeability
FLOW NETS: EXAMPLES
Unconfined groundwater flow nets on a slope
Wrong Wrong
Correct!
CIVE.5370 EXPERIMENTAL SOIL MECHANICSSoil Permeability
FLOW NETS: EXAMPLES
Cross-sectional flow net of a homogeneous and isotropic aquifer (Hubbert, 1940).
CIVE.5370 EXPERIMENTAL SOIL MECHANICSSoil Permeability
FLOW NETS: EXAMPLES
Contour map of the piezometric surface near Savannah, Georgia, 1957, showing closed contours resulting from heavy local groundwater pumping (from Bedient, after USGS Water-Supply Paper 1611).
CIVE.5370 EXPERIMENTAL SOIL MECHANICSSoil Permeability
FLOW NETS: DAM EXAMPLES
CIVE.5370 EXPERIMENTAL SOIL MECHANICSSoil Permeability
FLOW NETS: DAM EXAMPLES
CIVE.5370 EXPERIMENTAL SOIL MECHANICSSoil Permeability
FLOW NETS: ASSIGNMENT #2