CIVE.5370 Lecture 3 - Soil Permeability - uml.edufaculty.uml.edu/ehajduk/Teaching/CIVE.5370/... ·...

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


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


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



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


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


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:



DISCHARGE AND SEEPAGE VELOCITIES

svvAvAq Where:

q = Flow Rate(quantity of water/unit time)

A = Total Cross-sectional AreaAv = Area of Voidsvs = Seepage Velocity


nv

eev

VV

VV

vV

VVvv

LAAAv

AAAvv

vAAAvq

s

v

s

v

v

svs

v

sv

v

svs

svsv

11

)(

)()()(


DISCHARGE AND SEEPAGE VELOCITIES


HYDRAULIC CONDUCTIVITY: LABORATORY TESTINGConstant Head(ASTM D2434)

Falling Head(no ASTM)

Figure 5.4. Das FGE (2005). Figure 5.5. Das FGE (2005).


Constant Head(ASTM D2434)

AhtQLk

tkiAAvtQ )(Where:

Q = Quantity of water collectedover time t

t = Duration of water collection

tLhkAQ


HYDRAULIC CONDUCTIVITY: LABORATORY TESTING


LABORATORY TESTING: CONSTANT HEAD


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


LABORATORY TESTING: FALLING HEAD


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


after Casagrande and Fadum (1940) and Terzagi et al. (1996).

SOILPERMEABILITY

ANDDRAINAGE


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


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


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


Rate of water flow out of soil block in x,z directions:

LAPLACE'S EQUATION OF CONTINUITY


0

0

zv

xv

ordxdyvdzdyv

dxdydzzvvdzdydx

xv

v

zx

zx

zz

xx



Total Inflow = Total Outflow



02

2

2

2

zhk

xhk

zhkikv

xhkikv

zx

zzzz

xxxx



Using Darcy’s Law (v=ki)



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!


Figure 5.12a. Das FGE (2005).

FLOW NETSFLOW AROUND

SHEET PILE WALL


Figure 5.12b. Das FGE (2005).

FLOW NETSFLOW AROUND

SHEET PILE WALL


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



FLOW NETSFLOW UNDER ANIMPERMEABLE

DAM


nqqqq ...321


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


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


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.




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





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



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.


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.



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.



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.


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.


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.


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.


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.


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.


FLOW NETS: EXAMPLES

Unconfined groundwater flow nets on a slope

Wrong Wrong

Correct!


FLOW NETS: EXAMPLES

Cross-sectional flow net of a homogeneous and isotropic aquifer (Hubbert, 1940).


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).


FLOW NETS: DAM EXAMPLES


FLOW NETS: DAM EXAMPLES


FLOW NETS: ASSIGNMENT #2

Date post:	21-Jun-2020
Category:	Documents
Upload:	others
View:	11 times
Download:	2 times

CIVE.5370 Lecture 3 - Soil Permeability - uml.edufaculty.uml.edu/ehajduk/Teaching/CIVE.5370/... ·...

Documents