Basics of groundwater hydrology in geotechnical engineering: Permeability - Part A

Part A

Prepared by Dr O. Hamzao_hamza at hotmail dot com

Lecture reference: OH GA03 A

Permeability – Part A Dr O.Hamza

Lecture reference: OH_GA03_A

ContentContent

• Soil as porous media• Hydraulic head and gradient• Hydraulic head and gradient • Darcy's law and permeability

L b t d t i ti f ffi i t f bilit• Laboratory determination of coefficient of permeability• Summary • Quizzes and example problems


Soil as porous media• Nature of soil • Seepage velocity• Seepage velocity


Soil as a porous media

Nature of soilNature of soil

So soil can be considered as a porous mediaas a porous media.

(1) Solid (mineral particles)(2) Gas (air, co2),

Soil is a natural particulate earth material which has three intermixed phases:

(3) Liquid (usually water, but also possibly oil, chemical solutions)

Soil is a natural particulate earth material which has three intermixed phases:




A material is said to beA material is said to be

permeable if it contains

continues voids.

Can any porous material be permeable?Can any porous material be permeable?




Void ratio, e = Vv / Vs Degree of saturation, Sr =Vw / Vv

Water content, w = Mw/Ms Porosity, n = Vv/V



Seepage velocity qSeepage velocity q

AAAA

The volume flow rate q is calculated as theThe volume flow rate q is calculated as the product of flow velocity v and total cross sectional area A :sectional area A :

q = v. A

Assume a column of soil and water flowing through. q

BBBB

q



Seepage velocitySeepage velocityAAAAAA

Flow line

The ratio of volume flow rate q to the averageThe ratio of volume flow rate q to the average area of voids Av on a cross section normal to the macroscopic direction of flow is called the seepage velocity vs

qv =v

s Av =

BBBBBB

At the particulate level the water follows a tortuous path through the poresAt the particulate level the water follows a tortuous path through the pores.



Seepage velocitySeepage velocityAAAAAA

Flow line

In general, the velocity of water through soil has been found to be relatively smallsoil has been found to be relatively small (e.g. less than 0.61 m/min).

BBBBBB


H d li h d d h d li di tHydraulic head and hydraulic gradient• Pressure elevation and total headPressure, elevation and total head • Hydraulic gradient


Hydraulic head and hydraulic gradient

Pressure Elevation and total headPressure, Elevation and total head

Pressure head

u

The height of the water column is the pressure head (hw)

ww γ

uh =

Pore pressure at a given point (e.g. point A in the diagram) can be measured by the height of water in a standpipe located at that point.y g

Pore pressures are often indicated in this way on diagrams.




Elevation head

The elevation head (hz) of a point is its height aboveThe elevation head (hz) of a point is its height above the datum line.

hwTotal headThe height above the datum of the water level in theThe height above the datum of the water level in the standpipe is the total head (h).

h = hz + hw

To identify significant differences in pore pressure at different points, we need to eliminate the effect of the points' position A height datum is required from whicheliminate the effect of the points position. A height datum is required from which locations are measured.




Total head, h =Elevation head, hz +

Pressure head, hw +

Velocity head, hv hw

The total head h includes also velocity head hv; this head in soils is usually neglected However the velocity head must be considered when dealing withneglected. However, the velocity head must be considered when dealing with flow through pipe and open channels.



Hydraulic gradientHydraulic gradient

ΔhThe hydraulic gradient is the rate of change of total head along the direction of flow Δs

Δhi =

Datum

It is the difference in total head that are important. The direction of flow depends on these differences Flow of pore water in soils is driven fromdepends on these differences. Flow of pore water in soils is driven from positions of higher total head towards positions of lower total head.




Datum

Example. In each diagram there are two points, a small distance Δs apart, hz1and hz2 above datum.




The hydraulic gradientΔhi =The hydraulic gradientΔs

i =

Δh=0 thus i =0 No flow

Datum

In the first diagram, the total heads are equal. The difference in pore i ti l d t th diff i ltit d f th t i t d th

Δh=0 thus i =0 No flow

pressure is entirely due to the difference in altitude of the two points and the

pore water has no tendency to flowPermeability – Part A Dr O.Hamza



ΔhThe hydraulic gradient

ΔsΔhi =

Δh=0 thus i =0 No flow Δh ≠ 0 thus i = (h2 - h1) / Δs flow

Datum

In the second diagram, the total heads are different. The hydraulic gradient

Δh 0 thus i 0 No flow ( 2 1)

is i ≠ 0 and the pore water tends to flow.


Darcy’s law and permeability• Darcy's law • Soil permeability• Temperature and permeabilityp p y• Stratified soil and permeability


Darcy’s law and permeability

Darcy’s lawDarcy’s law

v ~ ior

q ~ i

In a saturated porous media, the flow velocity v or the rate of flow of water q (volume/time) through cross-sectional area A is found to be

ti l t h d li di t i

in 1856, a French hydraulic engineer named Henry Darcy published an

proportional to hydraulic gradient i

equation for flow through a porous medium.




k.iqv == k.iA

v

wherewhere

v is flow velocity

k is coefficient of permeability withk is coefficient of permeability with dimensions of velocity (length/time)

q is flow rate = = -----------------------tQ Quantity of water

Timet Time

Higher hydraulic gradient i faster flow ? agree disagree

Higher coefficient of permeability k faster flow ? agree disagree




Higher hydraulic gradient i faster flow ? agree disagree

Higher coefficient of permeability k faster flow ? agree disagree

The coefficient of permeability of a soil is a measure of the water flow conductivity.



Soil permeabilitySoil permeability

G lGravel

Coefficient of permeabilityk

clay

Coefficient of permeability may vary from as large as 1 m/s for clean gravels to a small value of the order of 10-10m/s for very fine soil such as clay-silt (BS

clay

a small value of the order of 10 m/s for very fine soil such as clay-silt (BS 8004, 1986).



Temperature and permeabilityTemperature and permeability

The values of k at 0°C and 10°C are 56% and 77% respectively of the value measured at 20°C.

An alternative permeability K (dimensions: length²) is sometimes used as a more absolute coefficient depending only on the characteristics ofas a more absolute coefficient depending only on the characteristics of the soil skeleton.

(Ref. Geotechnical on the Web)

The flow of water through confined spaces is controlled by its viscosity η and the viscosity is controlled by temperature.



Stratified soil and permeabilityStratified soil and permeability

Representative graphic lithology log and core

photos of an insitu

In nature, soil is stratified



Consider a stratified soil having horizontal layers of thickness t1, t2, t3, etc. with coefficients of permeability k1, k2 k3, etc3 y 1 2 3

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For a single layer, Darcy's law indicates:

iii

hAkAkiq Δ==

indicates:

iii t

q

How can we apply the concept of hydraulic gradient and coefficientHow can we apply the concept of hydraulic gradient and coefficient of permeability?




V ti l flWhich condition of these two is valid for vertical flow?

Vertical flow

(1) total flow rate q = q1= q2= q3..head drop Δh= Δh1 + Δh2+ Δh3

(2) total flow rate q = q1+ q2+ q3..head drop Δh= Δh1 = Δh2 = Δh3

For vertical flow, the flow rate q through area A of each layer is the same.For vertical flow, the flow rate q through area A of each layer is the same. Hence the head drop across a series of layers can be given.




V ti l flHead drop across a series of layers is

Vertical flow

The average coefficient of permeability is

q = q1= q2= q3..Δh= Δh1 + Δh2+ Δh3

For vertical flow, the flow rate q through area A of each layer is the same.For vertical flow, the flow rate q through area A of each layer is the same. Hence the head drop across a series of layers can be given.




Horizontal flowThe flow rate through a layered block of soil of breadth B

The average coefficient of permeability

q = q1+ q2+ q3..Δh Δh1 Δh2 Δh3

For horizontal flow, the head drop Δh over the same flow path length Δs will

Δh= Δh1 = Δh2 = Δh3

, p p gbe the same for each layer. So i1 = i2 = i3 etc.


Determination of coefficient of permeability

L b t t f th ffi i t f bilit• Laboratory measurement of the coefficient of permeability• Field measurement of the permeability• Empirical relations for the coefficient of permeability



Laboratory measurements of permeabilityLaboratory measurements of permeability

PPermeameter

Δh

1. Constant head Permeability test

Recommended for coarse-grained soilsRecommended for coarse grained soils.

The soil sample is subjected to steady total head drop Δh causing a water flows (with ead d op caus g a ate o s ( tflow rate q)

In this test, an element of the soil with length L is placed into a cylindrical container with a sectional area of A.




PPermeameter

1 C t t h d P bilit t t

Δh

1. Constant head Permeability test

q k.iAqv ==Darcy’s law

ΔsΔhi =Hydraulic gradient

ThusThus,A Δh




PPermeameter

2 F lli h d P bilit t t2. Falling head Permeability test

Recommended for fine-grained soils

Hydraulic gradient Δh/L varies with time.

Total head h in standpipe of area ‘a’ is allowed to fall; heads h1 and h2 are p p ; 1 2measured at times t1 and t2.




PPermeameter2. Falling head Permeability test

The quantity of flow through the sample in time dt: adQ = -adh

dhadtLhkA −=

a

dtikAdQ=

hdh

AkaLdtor −=

LA 1 h2

By integrating between two test limits and re-arranging

hAk

the equation:




Coefficient of permeability k can be indirectly estimated from Oedometer test

Oedometerindirectly estimated from Oedometer test

k c mk = cv.mv.γw

c coefficient of consolidationcv coefficient of consolidationmv one-dimensional compliance


Quizzes and example problemsQuizzes and example problems

Work on:Work on:

• Quizzes: only quiz 1, 2 and 3

• Example problems: only problem 1 and problem 2only problem 1 and problem 2


Quiz 1 (3 minutes)

Working on Quizzes and Example problems

Quiz 1 (3 minutes)

A ll' ti i l t d t l ti 180 b l lA well's tip is located at an elevation 180 m above sea level. The length of the water column in the well is 27 m. What is the total head at the well, if we use the plane at 20 m above sea level as the datum level?

Choose an answer and explain whya)180 mb)187 m

SolutionThe total head is defined as h = hz (elevation) + hw (pressure head).Here h = 180 20 = 160 m h = 27 m h = 187 mHere, hz = 180 - 20 = 160 m, hw = 27 m, h = 187 m


Quiz 2 (2 minutes)


Quiz 2 (2 minutes)

In the well of Problem 1, what is the water pressure, in Pascal (Pa, or kPa = 1000 Pa)? Recall that γw = 9.81 kPa.

Choose an answer and explain whya) 264.87 kPab) 27 kPa

Pressure headw

w γuh =

pore water pressure u = 27 x 9.81 = 264.87 kPa.


Quiz 3 (10 minutes)


A horizontal cylinder of homogonous sand is tested as shown in the figure. I i i t A B C D d E l t d t th t li f th

Quiz 3 (10 minutes)

Imaginary points A, B, C, D and E are located at the centreline of the cylinder. What is the hydraulic gradient expected along the soil.Determine the pressure, elevation and total head at different points to be

bl t l t th h i t l di table to plot them versus horizontal distance.

5 cm

Datum

5 cm

A B C D E

4 cm 5 5 3


Quiz 3


Quiz 3

S l ti

5 cm

Datum

L=10 cm

Δh= -5 cm

Solution5 cm

A B C D E The hydraulic gradient along the soil is

L 10 cm

4 cm 5 5 3

the soil is iBC = Δh / L = -5/10= -0.5

Si thi di t iSince this gradient is constant along the soil:iBC= iCD= iBD= - 0.5iBC = ΔhBC/LBC=(hC-hB)/ LBC

- 0.5 = (hC – 5)/ 5hC = 2 5hC 2.5

Problem 1 Hydraulic head and hydraulic gradient


Problem 1. Hydraulic head and hydraulic gradient

For the soil shown in the figure calculate the total head and pore water pressure at point P and Q. Then determine the direction of the flow. What is the hydraulic gradient between these two points?

5 m

What is the hydraulic gradient between these two points?

Pressure headead

Pressure head

Recall: h (total head) = h (elevation head) + h (pressure head)Recall: h (total head) = hz (elevation head) + hw (pressure head)pressure head hw = height of water above the point


Problem 1 Hydraulic head and hydraulic gradient


Problem 1. Hydraulic head and hydraulic gradient

SolutionTotal head

h = hz (elevation head) + hw (pressure head)hP= -0.5 + 1.7 = 1.2 m

Pore water pressure: Pressure head

hQ= -2.2 + 2.2 = 0 m

uh =Pressure head

thus, pore water pressure at P: uP = 1.7 x 9.81 ≈ 17 kPa; pore water pressure at Q: uQ = 2.2 x 9.81 ≈ 22 kPa

ww γ

h =

p p Q Q

Hydraulic gradientiPQ = Δh / ΔS = (0 -1.2)/ 5 = - 0.24PQ ( )

Direction of flow:Water moves from P to Q

i.e. from high head to low head


Problem 2. Laboratory measurement of the coefficient of Working on Quizzes and Example problems

ypermeability

The results of a constant head permeability test for a cylindrical fine sand sample having a diameter of 150mm and a length of 300mm are as follows:Constant head difference = 500 mmTime of collection of water = 5 minVolume of water collected = 350 cc

ΔhVolume of water collected 350 cc

Find the coefficient of permeability for the soil in m/s.

Note. cc is cubic centimetre, cm3


Problem 2. Laboratory measurement of the coefficient of Working on Quizzes and Example problems

ypermeability

Solution:

Using the equation of constant head permeability testA Δh

where:- Δh is the constant head difference = 500 mm

A Δh

- The cross-section area of the sample, A=πD2/4 = 17662 mm2

The flow rate i e the quantity of water that flows in a unit of time- The flow rate, i.e. the quantity of water that flows in a unit of time, q = volume of water collected (mm) / time of this collection (sec)

= 350x10-6 / (5x60) =

- Thus k = ………….


Basics of groundwater hydrology in geotechnical engineering: Permeability - Part A

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