CE424 GEOTECHNICAL ENGINEERING - 1: SOIL MECHANICS

Civil Engineering Department

WATER THROUGH SOILS

UST Manila

Outline of this Lecture

Permeability in Soils Bernoulli’s Equation Darcy’s Law Hydraulic Conductivity Hydraulic Conductivity Tests

Flow of water in Soil

Due to the existence of the inter-connected voids, soils are permeable.

The permeable soils will allow water flow from points of high energy to points of low energy.

Permeability is the parameter to characterize the ability of soil to transport water.

See VIDEO

Permeability in Soils

Permeability is the measure of the soil’s ability to permit water to flow through its pores or voids

It is one of the most important soil properties of interest to geotechnical engineers

Importance in Geotechnical Design:

It influences the rate of settlement of a saturated soil under load.

The design of earth dams is very much based upon the permeability of the soils used.

Importance in Geotechnical Design:

The stability of slopes and retaining structures can be greatly affected by the permeability of the soils involved.

Filters made of soils are designed based upon their permeability.

Knowledge of the permeability properties of soil is necessary to:

Estimating the quantity of underground seepage:

Solving problems involving pumping seepage water from construction excavation;

Stability analyses of earth structures and earth retaining walls subjected to seepage forces.

Bernoulli’s equation

From fluid mechanics we know that, according to Bernoulli’s equation…

The total pressure in terms of water head is formed from 3 parts:

Hydraulic Gradient & Darcy’s Law

Bernoulli’s Equation states that the total head, h at a point in water under motion can be given by the sum of the pressure head (p/γw), velocity head (v2/2g) and elevation head (z).

where p = pressure v = velocity g = acceleration due to gravity γw = unit weight of water

Hydraulic Gradient & Darcy’s Law

In case of flow of water through a porous soil medium, the seepage velocity is small such that the velocity head component in the Bernoulli’s equation can be neglected.

The total head can therefore be adequate represented by:

zphw

+=γ

Hydraulic Gradient & Darcy’s Law

Note: Piezometer is a device used to measure pore water pressure

hA

PA/γw

ZA

flow

L

∆h

PB/γw

ZB

HB

Hydraulic Gradient & Darcy’s Law

The relationship between pressure, elevation and total heads for flow of water through soil is presented in the figure below.

hA

PA/γw

ZA

flow

L

∆h

PB/γw

ZB

HB

Hydraulic Gradient & Darcy’s Law

Piezometer tubes are installed at points A and B The levels to which water rises in the tubes are known

as the piezometric levels of points A and B.

hA

PA/γw

ZA

flow

L

∆h

PB/γw

ZB

HB

Hydraulic Gradient & Darcy’s Law

The pressure head at a point is the length of the vertical column of water in the piezometer installed at that point.

hA

PA/γw

ZA

flow

L

∆h

PB/γw

ZB

HB

Hydraulic Gradient & Darcy’s Law

The elevation head of a point is the vertical distance measured form any arbitrary horizontal datum plane to that point.

hA

PA/γw

ZA

flow

L

∆h

PB/γw

ZB

HB

Hydraulic Gradient & Darcy’s Law

The loss of head between two points A and B is given by

hA

PA/γw

ZA

flow

L

∆h

PB/γw

ZB

HB

)()( Bw

BA

w

ABA ZPZPhhh +−+=−=∆

γγ

Hydraulic Gradient & Darcy’s Law

The head loss ∆h can be expressed in a non-dimensional

form as: Where i = hydraulic gradient L = distance between points A and B

(the length of flow over which the loss of head occurred)

Lhi ∆

=

Darcy's Law (1856):

In 1856, Darcy published a simple equation for the discharge

velocity of water through saturated soil. where v = discharge velocity, the quantity of water

flowing in unit time through a unit gross cross-sectional area of soil at right angles to the direction of flow.

k = coefficient of permeability i = hydraulic gradient

kiv =

Darcy's Law (1856):

Darcy's law, which gives a linear relation between the discharge velocity and the hydraulic gradient, is valid in the range of laminar flow of fluid through pore spaces in the case of soils with particle sizes less than 1mm.

Darcy's Law (1856):

In stones, gravels and very coarse sands, turbulent flow conditions may exist; in that case, Darcy's law can be rewritten as:

The experimental value of φ usually ranges between 0.65 and 1.

φkiv =

Darcy's Law (1856):

Darcy’s law states that how fast the groundwater flow in the aquifer depends on two parameters:

how large is the hydraulic gradient of the water

head (i=dH/dx); and

the parameter describing how permeable the aquifer porous medium –the coefficient of permeability (hydraulic conductivity) k.

Darcy's Law (1856):

The minus sign in the equation denotes that the direction of flow is opposite to the positive direction of the gradient of the head.

Darcy’s law

The physical description of groundwater flow in soil is the Darcy’s law.

The fundamental premise for Darcy’s law to work are: the flow is laminar, no turbulent flows; fully saturated; the flow is in steady state, no temporal variation.

Hydraulic conductivity

Hydraulic conductivity is expressed in cm/sec or m/sec, and discharge is in m3.

It needs to be pointed out that the length is expressed in mm or m, so, in that sense, hydraulic conductivity should be expressed in mm/sec rather than cm/sec.

However, geotechnical engineers continue to use cm/sec as the unit for hydraulic conductivity

Coefficient of Permeability, K

The coefficient of permeability, or hydraulic conductivity, k, is a product of Darcy’s Law.

Coefficient of Permeability, K

The “K” depends on several factors such as Fluid viscosity Pore-size distribution Grain-size distribution Void ratio Roughness of mineral particles Degree of soil saturation

Coefficient of Permeability, K

In clayey soils, the structure is an important influence on the coefficient of permeability.

The value of “k” varies over a wide range for different soils.

Some typical values of permeability coefficients are given in the table below.

COEFFICIENT OF PERMEABILITY,K

Typical Values of Permeability Coefficient

Soil Type Permeability Coefficient, k (mm/s)

Clean Gravel 10 to 1000

Coarse Sand 0.1 to 10

Fine Sand 0.01 to 0.1

Silt 0.0001 to 0.01

Clay Less than 0.0001

Effect of Water Temperature on k

Since the viscosity of water is a function of temperature, a change in water temperature will affect the value of the coefficient of permeability.

Effect of Water Temperature on k

It is a general practice to note the temperature of water, T, during the test and use a conversion factor to express the value of k corresponding to 20° C by using the equation:

where nT= viscosity of water at T°C n20 = viscosity of water at 20°C kT = coefficient of permeability at T°C

TT

Ck

nnk

2020 =

Empirical Relation For K

For fairly uniform sand (i.e., small uniformity coefficient), Hazen (1930) has proposed an empirical relation for the coefficient of permeability k (mm/s) in the form:

where c = constant that varies from 10 to 15 D10= effective size, mm k = coefficient of permeability

210cDk =

Empirical Relation For K

Another simple relation was proposed by A.Casagrande for the coefficient of permeability for fine to medium clean sand.

where k = coefficient of permeability at a void ratio e k0.85 = coefficient of permeability at a void

ratio of 0.85

85.024.1 kek =

Equivalent Permeability In Stratified Soil

Depending on the nature of soil deposit the coefficient of permeability of a given layer of

soil may vary with the direction of flow. kv1

kH1

kv2 kH2

kHn kvn

Equivalent Permeability In Stratified Soil

Furthermore, in a stratified soil deposit where the permeability coefficient for a given directional flow changes from layer to layer, an equivalent permeability determination becomes necessary for simplifying calculations.

Equivalent Permeability In Stratified Soil Given the following figure which shows n layers of soil with

flow in the horizontal direction:

kv1 kH1

kv2 kH2

kHn kvn

)...(12211)( nHnHHeqh HkHkHk

Hk +++=

Equivalent Permeability In Stratified Soil

Given the following figure which shows n layers of soil with flow in the vertical direction:

kv1 kH1

kv2 kH2

kHn kvn

H nv

n

vv

eqv

kH

kH

kH

Hk+++

=...

2

2

1

1)(

Laboratory Determination of K

The two standard types of laboratory test procedures for determining the coefficient of permeability of soil are:

the constant head test the variable (falling) head test

Constant Head Test

A typical arrangement of the constant head permeability test is shown in the figure. The water supply at the inlet is adjusted in such a way that the difference of head between the inlet and outlet remains constant during the test.

After a constant rate of flow is established, water is collected in a graduated flask for a known duration.

porous stone

soil sample

porous stone

water supply

graduated flask

Constant Head Test

The total volume of water collected, Q, may be expressed as: where A = cross-sectional area of soil sample t = duration of test

Since i = h / L, where L is the length of the sample.

Constant head tests are suitable for coarse-grained soils that have high coefficients of permeability.

tkiAAvtQ )(==

AhtQLk =

Constant Head Test

Constant Head Test

Constant Head Test

Constant Head Test

Falling Head Test

A typical arrangement of the falling head permeability test is shown in the figure.

Water from a standpipe flows through the soil. Initial head difference h1 at time t = 0 is recorded and water is

allowed to flow through the soil sample so that the final head difference at time t is h2.

soil sample

porous stone

porous stonestand pipe

Falling Head Test

The rate of flow q of the water through the sample at any time t can be given by:

where a = cross-sectional area of the standpipe A = cross-sectional area of the soil sample

dtdhaA

Lhkq −==

Falling Head Test

The coefficient of permeability can be determined using the following equation:

The falling head test is appropriate for fine-grained soils

with low coefficients of permeability.

)(log)(303.22

110 h

hAt

aLk =

Permeability test in the field The figure shows a case where the top permeable layer,

whose coefficient of permeability has to be determined, is underlain by an impermeable layer.

Permeability test in the field In performing the test, water is pumped out at a

constant rate from a test well that has a perforated casing.

Permeability test in the field Several observation wells are made at various

radial distances around the test well.

Permeability test in the field Continuous observations of the water level in the test well

and the observation wells are made after the start of the pumping until steady state is reached.

Permeability test in the field Steady state is established when the water level in

the test well and observation wells becomes constant.

Permeability test in the field

The coefficient of permeability is determined as:

)(

)(log303.22

22

1

2

110

hhr

rqk

−=

π

Permeability test in the field

The average coefficient of permeability for a confined aquifer can also be determined by a pumping test from a perforated well that penetrates the full depth of the aquifer and observing the water table in a number of observation wells at various radial distances.

Permeability test in the field

Permeability test in the field

Pumping is continued at a uniform rate q until a steady state is reached.

The coefficient of permeability in the direction of flow is given as:

)(727.2

)(log2

22

1

2

110

hhHr

rqk

−=

SEEPAGE AND FLOW NETS

END OF PRESENTATION