soil mechanics i 3 – water in soils - univerzita karlovalabmz1.natur.cuni.cz/~bhc/s/sm1/sm1_3_ ·...

SM1_3 November 28, 2017 1

Soil Mechanics I

3 – Water in Soils

1. Capillarity, swelling

2. Seepage

3. Measurement of hydraulic conductivity

4. Effective stress in the ground


WATER IN SOIL - affects soil behaviour (e.g., consistency, consistency limits)

Adsorbed water

Mechanisms of water adsorption to clay surfaces: hydrogen bonds, ion hydration, osmosis, dipole attraction...

The effects neglected in the basic SM

'Free' water

Effect of gravity - effective stress, capillarity, seepage

Influence of Water - Basics


WATER IN SOILS

[2]



Shrinking

w > wS – saturated soil (?)

Suppose the Terzaghi principle of effective stresses valid and u < 0:

for hC= 50 m → u = - 500 kPa → σ' = 500 kPa (+ σ)

Swelling

Due to

mineralogy (smectites)

partial saturation

unloading

Disintegration of “cohesive” soil on submerging in water

elimination of capillary forces

elimination of cementation

Capillarity



GT Practice – the role of water

Hydrostatic pressure

basis for computing effective stresses (only with no seepage, i.e. no hydraulic gradients)

Steady state seepage

the pore pressure is generally different from the hydrostatic pressure

the pressure heads does not have to correspond to phreatic water table

Laplace equation – a flow net

Consolidation

dissipation of excess pore pressures



Steady flow – H. DARCY (1856)

q = A k i

q = flow quantity (volume per time unit)

i = hydraulic gradient i = - Δh/Δx

v = k i

k = hydraulic conductivity (coefficient of permeability; coefficient of filtration)

Steady Flow – Darcy's Law

v = seepage velocity

Real velocity vreal

= q / (n A) = v / n


Permeability Κ: a property of the porous medium, independent of the permeating fluid

Κ = k μ / γ [m2]

Κ = permeability

k = hydraulic conductivity, i.e. coefficient of Darcy's law

μ = dynamic viscosity [N×s×m-2]

(μ =kinematic viscosity × ρ)

γ = unit weight of permeating fluid



Re = v d

ef ρ

w / μ

v = seepage velocity

μ = dynamic viscosity [N×s×m-2]

Change from laminar to turbulent flow at critical velocityv

cr = R

e cr μ /(ρ

w d

ef)

Initial gradient



Hydraulic conductivity (≡ filtration coefficient ≡ coefficient of permeability)

Constant Head Permeameter

Q = A v t; Q = volume of water; v = discharge velocity; t = time

v = k i = k h / L

k = Q L / (h A t)

At low permeability (h. conductivity: k < 10-6 ms-1), the device cannot be used – insufficient readability / accuracy

→ the set-up needs to be refined:

closed system for discharge water, or

triaxial apparatus / flexible wall permeameter (+avoiding preferential flow at the rigid wall)

Another alternative – Falling Head Permeameter (accuracy not good though)

Measurement of Hydraulic Conductivity

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Hydraulic conductivity (≡ filtration coefficient ≡ coefficient of permeability)

Falling Head Permeameter

q in

= - a dh/dt

q out

= A k i

q in

= q out

- a dh/dt = A k h / L

- a ∫dh / h = k A / L ∫ dt

- a (ln h2 – ln h

1)= k A (t

2-t

1) / L

a (ln h1 – ln h

2) = k A (t

2-t

1) / L

k = a L ln(h1/h

2) / (A Δt)


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Measurement of Hydraulic Conductivity – Lab Class

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Hydraulic conductivity (≡ filtration coefficient ≡ permeability coefficient)

In situ

For example

[ → k = q / (2π D H) ln(R / r) ]

Indirect determination

For example FOR SANDS: Hazen k [ms-1] = 0,01D10

2 [mm]


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Typical values

Gravel 10-1 to 10-3 ms-1

Sand 10-2 to 10-4 ms-1

Fine Sand 10-5 ms-1

Silt 10-6 ms-1

Sandy Loam 10-6 to 10-8 ms-1

Clay <10-8 ms-1


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Equation for Seepage

isotropy: kx=k

y=k

z

Δh = 0

δ2h / δx2 + δ2h / δy2 + δ2h / δz2 = 0

Seepage

Hydraulic head does not have to correspond with the phreatic surface ← seepage

.....the actual height in the piezometer ← definition of equipotential lines

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A flow net in 2D

saturated soil, GWT at the surface, steady percolation

Seepage

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A flow net in 2D

(long dam / embankment)

Seepage

Hydraulic head does not have to correspond with the phreatic surface ← seepage

.....the actual height in the piezometer ← definition of equipotential lines

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Influence of seeping water on soil grains: Seepage Forces - DRAG

v = k i

Loss of hydraulic head due to drag effect of water

Δp = γ

w Δh

Δ S = Δp × area = γw Δh Δy Δz =

= γw Δh Δy Δz Δx/Δx =

= γw i (Δx Δy Δz) =

= γw i ΔV

Force acting on the soil skeleton: S = γw i V

Force acting on the soil skeleton in the unit volume: p = γ

w i

Seepage

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Influence of seeping water on soil grains: Seepage Forces - DRAG

Bernoulli equation:

γw (z + u/γ

w + v2/(2g) + h

s) = const

v2/(2g) can be neglected (v small)

Loss of energy between two cross section (distance s):

ΔE = γw Δh

s = (z

2 +u

2/γ

w - (z

1 +u

1/γ

w))

ΔE / Δs = γw Δh

s / Δs

= γ

w i

The loss of energy: p = γw i

Seepage

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Flow net

Boundary conditions:(constant) water levels = equipotentials (GA; CF)impermeable boundaries = flow lines (AB; BC; DE; detto axis of symmetry EF)

Upward seepage → possibility of hydraulic failure - „boiling sand“; „piping“

[1])

Seepage

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hcr ... critical height, i.e. height at liquefaction

neglecting friction on sides; cross section area A

thrust = water pressure on A: u A= hcr γ

w A

equilibrium:

hcr γ

w A = A z γ

sat ( γ

sat ≡ γ)

hcr = z γ / γ

w

icr = (h

cr – z) / z = (z γ / γ

w – z) / z = γ / γ

w – 1 = (γ - γ

w)/γ

w

icr = (γ – γ

w) / γ

w

Seepage

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What is the necessary embedment depth t of the sheet pile wall?

icr = (γ – γ

w) / γ

w

icr ≈ 1

i= (H + h) / (h + t + t) < 1 = icr

H < 2 t

t > ½ H

Seepage

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Capillary height hC

Downward Force: W= ρwg V = ρ

wg h

C π d2 / 4

Upward Force: π d T cosα

“surface tension of water” T = 7×10-5kNm-1

Equilibrium:

ρw g h

C π d2 / 4 = π d T cosα

hC

= 4 T cosα / (ρw

g d)

clean water vs glass → α = 0

hC

= 4 T / (ρw

g d) = 4 T / ( γw d)

hC

[m] ≈ 3×10-5 / (d [m]) [m]

For example: d = 1μm → hC = 30m

Capillarity

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Capillary height hC

depends on PORE SIZE

Theoretical values for soils

(α = 0 a capillary tube of constant diameter d)

silt d ≈ 1mm → hC = 30 mm

fine silt d ≈ 1μm → hC = 30 m

clay d ≈ 10nm →hC = 3 km

Realistic values for soils

sand hC = 0,03 – 0,1 m

loamy sand hC = 0,5 – 2 m

loam (silt) hC = 2 – 5 ( – 10) m

clay hC = tens of metres

Capillarity

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In Capillary Fringe – soil is saturated

Principle of effective stress is valid, u < 0

hC= 50 m → u = - 500 kPa → σ' = σ + 500 kPa

Unsaturated Zone

Three Phase Medium – Terzaghi's principle QUESTIONABLE (the expression for pore pressure not clear to date?)

ua- u

w= T

(1/r

m-1/r) ”capillary suction”

r is the radius of meniscus

Bishop:

u = χ uw+ (1 - χ) u

a

σ' = σ – (χ uw+ (1 - χ) u

a)

σ' = σ – ua + χ (u

a - u

w)

χ function of S, way of loading...

Very approximate assumption: χ = Sr

Capillarity

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Capillary water in sand – unconfined compression of wet sand

Assumed: “Bishop's Effective Stress”

σ' = σ – (χ uw+(1-χ) u

a)

Pore Pressure u = χ uw+ (1 - χ) u

a

for χ = S

u = S uw+ (1 - S) u

a

If the air phase continuous (at w = 0.1 should be) then

→ pore pressure u = S uw

→ capillary suction s = - uw

Procedure:

M.C. for total stress; Failure envelope; Determination of capillary cohesion; M.C. for effective stress. From its shift the pore pressure and suction in the sand castle (assuming the Bishop's stress and χ = S)

Capillarity – Lab Class

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Sand without capillary water (dry or saturated)

Angle of repose:

continuous slope failure – ideal plasticity “critical state”

→ τmax

= σ' tg φcr'

Equilibrium:

T = W sin α = τmax

× 1 = W cos α × tg φcr

→ tg α = tg φcr

→ α = φcr

Capillarity – Lab Class

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σv

= ∑ (hi γ

i)

u = hwγ

w = (z - z

w) γ

w

σv' = σ

v – u = ∑(h

i γ

i) – h

wγ

w

detto for increments

Effective vertical stress in the ground

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Δσv' = Δσ

v – Δu

Δσv

> 0

Δu = 0 (drained event)

Δσv' = Δσ

v> 0

→ increase in effective stress

→ deformation

→ settlement under loaded area


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Δσv' = Δσ

v – Δu

Lowering of GWT – increase of effective stress:

before:1u = hw

γw 1σ

v= h

γ

sat

after: 2u=(hw-Δh

w) γ

w 1σ

v= h

γ

sat

(soil remains saturated - capillarity)

Δu = 2u – 1u = - Δhw

γw

Δσv

= 0

Δσv' = - Δu > 0

→ increase of effective stress

→ deformation

→ settlement on lowering GWT


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← increase in effective stress


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σ = 18 × 1 = 18 kPau = 10 × 1 = 10 kPaσ' = 18 – 10 = 8 kPa

σ = 18 × 1 = 18 kPau = 10 × 0,5 = 5 kPaσ' = 18 – 5 = 13 kPa

σ = 18 × 1 + 10 × 100 = 1018 kPau = 10 × 101 = 1010 kPaσ' = 1018 – 1010 = 8 kPa


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http://labmz1.natur.cuni.cz/~bhc/s/sm1/

Atkinson, J.H. (2007) The mechanics of soils and foundations. 2nd ed. Taylor & Francis.

Further reading:

Wood, D.M. (1990) Soil behaviour and critical state soil mechanics. Cambridge Univ.Press.

Mitchell, J.K. and Soga, K (2005) Fundamentals of soil behaviour. J Wiley.

Atkinson, J.H: and Bransby, P.L. (1978) The mechanics of soils. McGraw-Hill, ISBN 0-07-084077-2.

Bolton, M. (1979) A guide to soil mechanics. Macmillan Press, ISBN 0-33318931-0.

Craig, R.F. (2004) Soil mechanics. Spon Press.

Holtz, R.D. and Kovacs, E.D. (1981) An introduction to geotechnical engineering, Prentice-Hall, ISBN 0-13-484394-0

Feda, J. (1982) Mechanics of particulate materials, Academia-Elsevier.)

Literature for SM1

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[1] Atkinson, J.H. (2007) The mechanics of soils and foundations. 2nd ed. Taylor & Francis.

[2] Santamarina, J (2003) in Mitchel, J.K. and Soga, K (2005) Fundamentals of soil behaviour

References – used figures etc.

soil mechanics i 3 – water in soils - univerzita karlovalabmz1.natur.cuni.cz/~bhc/s/sm1/sm1_3_ ·...

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