1

Flow of Water Through Soils

Bernoulli’s Equation

zg

uhw

++=2

2νγ

where:

h = total head

u = pressure

γw = unit weight of water (9.81kN/m3 or 62.4lbs/ft3)

g = acceleration due to gravity

Z = vertical distance of a given point above or below a datum plane

headElevationheadVelocityheadessureh Pr ++=

2

Flow of Water in Soils

epw

hhZuh +=+=γ

For most soils the velocity of water flow is very small therefore the velocity head term can be neglected

headElevationheadessureh Pr +=

3

FLOW

• Water flow occurs only when there is a change in total head between two points.

• When water flows through soil it will lose head due to friction

Loss of head between Points A and B

⎟⎟⎠

⎞⎜⎜⎝

⎛+−⎟⎟

⎠

⎞⎜⎜⎝

⎛+=−=Δ B

W

BA

W

ABA ZZhhh

γμ

γμ

4

Hydraulic Gradient (i)

• Head loss per unit length is:

Lhi Δ

=where:

i = hydraulic gradient

Δh = head loss between points of interest

L = distance between points of interest

5

In Most Soils

• Flow though most soils can be considered to be laminar

• Therefore a linear relationship between velocity and hydraulic gradient

v α i Flow conditions may be turbulent in fractured rock, stones, gravel, and very coarse sands

Darcy’s law

• 1856 Darcy published a simple equation for the discharge velocity through saturated soils

v = kiv = discharge velocity (L3/L2/T)k = hydraulic conductivity (L/T)

6

Discharge Velocity

• Quantity of water flowing in unit time through a unit gross cross sectional area of soil at right angles to the direction of flow

• Does not account for flow through soil voids

Seepage Velocity

7

Discharge and Seepage Velocity

Seepage Velocity

svvAvAq ==Where:

q= quantity of water flowing per unit time

v= discharge velocity

A= area of soil specimen perpendicular to flow

Av= area of voids in soil section

vs= seepage velocity

8

Seepage Velocity

1

1

−

⎟⎠⎞

⎜⎝⎛

+==

eev

nvvs

where:

n= porosity

e= void ratioSee derivation DAS page 159

nVV

AA vv ==

Seepage Velocity

Note:• Seepage velocity is significantly greater

than the Darcy’s discharge velocity• Seepage velocity is an average velocity

through the pore voids – Higher velocities will occur in small pore

throats– Low velocities in large pore throats

See derivation DAS page 159

9

Hydraulic Conductivity

ηγ

ηρ wKgKk ==

where:

K = absolute permeability (L2)

ρ = density of fluid

g = gravitational constant

η = viscosity of fluid

γw = Unit weight of fluid

Hydraulic ConductivityDepends on:• Type of fluid (water vs molasses)

– Viscosity, density• Pore size distribution• Grain size distribution• Void ratio• Particle shape• Degree of saturation• Size of double layer (clay type)

10

Hydraulic Conductivity

Laboratory Measurement of Hydraulic Conductivity

Cohesionless soils (sand and gravel)• Constant head test• Falling head test

Cohesive soils (silt and clay)• Triaxial cell

11

Constant Head Test

Constant Head TestQ = vAt = kiAt = k(h/L)At

where: Q = quantity of flow (L3)A = cross section area of column (L2)t = duration of water collection (T)

Solve for k:

AhtQLk =

12

Falling Head Test

Note: hydraulic gradient decreases with time

t1

t2

Constant head

No head loss

No head loss

L

Falling Head Test

• Record level h1 at t1=0• Head is allowed to flow such that the final head

difference is at time t = t2 is h2

13

Falling Head Test

• Record level h1 at t1=0• Head is allowed to flow such that the final head

difference is at time t = t2 is h2

column soil intostandpipe qdtdhaq =−=

where:

a = cross sectional area of stand pipe

dh/dt = change in head in change in time= velocity of water falling

Minus sign is used to indicate falling head (decreasing head)

Falling Head Test

ALhkq =soil

where:

k = soil hydraulic conductivity

h = total head loss

L = length of soil column resulting in head loss

A = soil column cross sectional area

Continuity:

qin = qsoil

14

Falling Head TestA

Lhk

dtdha =−

dtLAk

hdha =−

tdLAk

hdha

t

t

h

h ∫∫ =− 2

1

2

1

tLAk

hha =−

2

1ln

h1

h2

t1 t2

2

1log303.2hh

AtaLk =

K Relationships for Granular Soils

Hazen Equationk(cm/s) = cD2

10where:c= a constant that varies from 1.0 to 1.5D10 = effective particle size for 10 percent

passing (mm)

Equation works OK for clean loose sand.

15

K Relationships for Granular Soils

Recommend measuring K in the lab or in the field if it is critical

K field measurement methods:•Bail test•Well pump test•Permeameter•Borehole packer test

16

NEED K

Test the soil…• Granular – constant or falling head• Silt and Clay – triaxial cell• Field pump test

Stratified SoilsHorizontal Flow

vAq =Assume unit length in page

A=1*H

17

Stratified Soils

( )HnHHHeqH

neq

nHnnHH

nn

kkkkH

k

iiiii

iKviKviKv

HvHvHvHvHvq

.....1

......

.....;;ikvconstant K if

1....111 1

321)(

321)(

222111(eq)(eq)H

H

332211

+++=

===

====

+++==

Stratified SoilsVertical Flow

n

n

hhhhhqqqqq

.......

321

4321

+++=====

⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛=

Vn

n

VVV

eqV

kH

kH

kH

kH

HK...

3

3

2

2

1

1)(