Basics of groundwater hydrology in geotechnical engineering: Permeability - Part B
1. Basics of groundwater hydrology in geotechnical engineering
Part B
Prepared by Dr O. Hamza
o_hamza at hotmail dot com
Lecture reference: OH GA03 B
OH_GA03_B
Permeability – Part B Dr O.Hamza
2. Content
• Introduction
• Quasi-one-dimensional
Quasi one dimensional and radial flow
• Field determination of coefficient of permeability
• Summary
S mmar
• Example problems
Permeability – Part B Dr O.Hamza
3. Introduction
Darcy’s law
q = Aki
where
k i coefficient of permeability with
is ffi i t f bilit ith
(Ref. Geotechnical on the Web)
dimensions of velocity (length/time)
Q Quantity of water
q is flow rate = = -----------------------
t Time
In a saturated porous media, the rate of flow of water q (volume/time)
through cross-sectional area ‘A’ is found to be proportional to hydraulic
th h ti l i f dt b ti lt h d li
gradient ‘ i ’
Permeability – Part B Dr O.Hamza
4. Introduction
Aquifer and Darcy’s law
Aquifer is a term used to designate a porous geological formation that:
- contains water at full saturation
- permits water to move through it under ordinary field conditions
Permeability – Part B Dr O.Hamza
5. Introduction
Aquifer and Darcy’s law
The horizontal flow rate q is constant. For an
aquifer of width B and varying thickness tt,
Darcy's Law indicates that
q=Aki
=Btki
or
Hydraulic gradient varies inversely with aquifer
thickness
Where flow occu s in a co
e e o occurs confined aqu e whose t c ess varies ge t y with
ed aquifer ose thickness a es gently t
position the flow can be treated as being essentially one-dimensional.
Permeability – Part B Dr O.Hamza
6. Quasi-one-dimensional
Quasi one dimensional and radial flow
• Cylindrical flow: confined aquifer
• Cylindrical flow: groundwater lowering
• Spherical flow
Permeability – Part B Dr O.Hamza
7. Quasi-one-dimensional and radial flow
Cylindrical flow: confined aquifer
Pumping aquifer
Confined aquifer
Steady-state pumping from a well which extends the full thickness of a
confined aquifer is one of the one dimensional problem which can be
one-dimensional
analysed in cylindrical coordinates.
Permeability – Part B Dr O.Hamza
8. Quasi-one-dimensional and radial flow
Cylindrical flow: confined aquifer
Darcy's Law still applies, with hydraulic
gradient dh/dr and area A varying with radius:
A = 2πr.t
2 rt
In this case pore pressure or head varies only with radius r
r.
Permeability – Part B Dr O.Hamza
9. Quasi-one-dimensional and radial flow
Cylindrical flow: confined aquifer
Integrating between the
borehole and at variable
distance r:
di t
where ro is the radius of the borehole and h0
the constant head in the borehole.
Permeability – Part B Dr O.Hamza
10. Quasi-one-dimensional and radial flow
Cylindrical flow: groundwater lowering
Pumping from a borehole can be used for
deliberate groundwater lowering in order to
facilitate excavation.
Permeability – Part B Dr O.Hamza
12. Quasi-one-dimensional and radial flow
Cylindrical flow: groundwater lowering
This is an example of quasi-one-
dimensional radial flow with flow thickness
t=h Then A=2πr h and
t=h. A=2πr.h Groundwater lowering
Original level of
water table
Integrating between the borehole and
at variable distance r: Drawdown
The radius of influence
Permeability – Part B Dr O.Hamza
13. Quasi-one-dimensional and radial flow
Spherical flow
Darcy's Law still applies, with hydraulic gradient dh/dr
and area A varying with radius: A=4πr²
where r0 is the radius of the piezometer
and h0 the constant head in the piezometer head varies only
with radius r
r.
Variation of pore pressure around a point source or side (for example, a
piezometer being used for in situ determination of permeability) is a one
in-situ one-
dimensional problem which can be analysed in spherical coordinates.
Permeability – Part B Dr O.Hamza
14. Determination of coefficient of permeability
• Laboratory measurement of the coefficient of permeability
L b t t f th ffi i t f bilit
• Field measurement of the permeability
• Empirical relations for the coefficient of permeability
Permeability – Part B Dr O.Hamza
15. Determination of coefficient of permeability
Field measurement of the permeability
Field measurement Laboratory measurement
• Field or in-situ measurement of permeability avoids the difficulties involved
in obtaining and setting up undisturbed samples
• Field or in-situ measurement of permeability provides information about bulk
permeability, rather than merely the permeability of a small and possibly
unrepresentative sample.
Is field measurement of permeability better than the lab
y
measurement?
Permeability – Part B Dr O.Hamza
16. Determination of coefficient of permeability
Field measurement of the permeability
Well-Pumping test
Observational boreholes
In a well-pumping test, a number of observation boreholes at radii r1 and
r2 are monitored to measure the pressure heads.
Permeability – Part B Dr O.Hamza
17. Determination of coefficient of permeability
Field measurement of the permeability
If the pumping causes a drawdown in an
unconfined (i.e. open surface) soil stratum Well-Pumping test
then the quasi one dimensional flow equation
quasi-one
is applied.
Integrating between the two test limits
and rearranging the equation:
Impermeable
(Assuming the pumping causes a drawdown in an
unconfined (i.e. open surface) soil stratum then)
Observational b h l
Ob ti l boreholes
Permeability – Part B Dr O.Hamza
18. Determination of coefficient of permeability
Field measurement of the permeability
If the soil stratum is confined and of thickness t Well-Pumping test
and remains saturated th
d i t t d then
Confined stratum
Permeability – Part B Dr O.Hamza
19. Determination of coefficient of permeability
Empirical relations for the coefficient of permeability
Empirical relations
p
k = function of (other parameters)
Permeability of all soils is strongly influenced by the density of packing of
the soil particles which can be simply described through void ratio e.
Several empirical equations for estimation of the coefficient of permeability
have been proposed in the past.
Permeability – Part B Dr O.Hamza
20. Determination of coefficient of permeability
Empirical relations for the coefficient of permeability
Permeability of granular soils
P bilit f l il
For fairly uniform sand Hazen (1930)
proposed the following relation between the
coefficient of permeability k (m/s) and the
effective particle size D10 (in mm) (the
particle size than which 10% soil is finer):
k = C D10
C.D 2
where C is a constant approximately equal to
pp y q
0.01 (see the figure beside)
Hazzan equation and data relating coefficient of
permeability and effective grain size of granular soils
Permeability – Part B Dr O.Hamza
21. Determination of coefficient of permeability
Empirical relations for the coefficient of permeability
Permeability of soft clays
P bilit f ft l
Samarasinghe, H
S i h Huang and D d Drnevich (1982) h
i h have suggested th t th
t d that the
coefficient of permeability of clays can be given by the equation:
en
k=C
1+ e
where
h
e is void ratio
C and n are constant to be determined experimentally
Consolidation of soft clay may involve a significant decrease in void ratio
and therefore of permeability.
Permeability – Part B Dr O.Hamza
22. Summary
• All soils are permeable materials, water being free to flow through the
interconnected pores between the solid particles.
• Water in saturated soil will flow in response to hydraulic g
p y gradient and occurs
towards the lower total head.
• Flow rate is proportional to the hydraulic gradient and can be affected by the
geometry of the pores.
pores
• The hydraulic gradient may be associated with natural flow or induced by
loading the soil (i.e. due to excavation or construction).
• Coefficient of permeability may be determined from laboratory experiments or
from in situe measurements
• Pore water pressure u at any point of the soil is computed from the definition
of the hydraulic head, u = γw(h-hz) (where h is total head and hz is
elevation head).
Permeability – Part B Dr O.Hamza
23. Quizzes and example problems
Work on:
• Quizzes: quiz 3 to 6 *
• Example problems: *
problem 3 and problem 4
* Note. quiz 1 and problem 1 and 2 are covered in Part A of
Permeability lecture
Permeability – Part B Dr O.Hamza
24. Working on Quizzes and Example problems
Quiz 3
The sets of nested piezometers shown below penetrate a layered aquifer.
•For one of the piezometers, indicate graphically the elevation head, pressure
head, and total head.
• For both cases, indicate the direction of the vertical flow between the layers.
• F case 2, what is a realistic situation th t might result i a set of h d
For 2 h ti li ti it ti that i ht lt in t f heads
such as this?
Note: The wells are drawn with some separation between them to allow you room to
label the heads. Assume, however, that they are truly nested, i.e., that they penetrate
the surface of the aquifer at the same location.
datum
Case 1 Case 2
Permeability – Part B Dr O.Hamza
25. Working on Quizzes and Example problems
Quiz 3
Solution:
S l ti
hw
h Flow Flow
hz
datum
Case 1 Case 2
The situation in Case 2 might happen if the middle layer is being pumped
OR if the middle layer is a zone of incredibly high conductivity.
Permeability – Part B Dr O.Hamza
26. Working on Quizzes and Example problems
Quiz 4
An inclined permeameter tube is filled with three layer of soil of different
permeabilities as shown in the figure
figure.
(i) Formulate q in terms of the different dimensions and permeabilities for
each soil element
(ii) D t
Determine th h d l
i the head loss (Δh) b t
between each soil element assuming
h il l t i
k1=k2=k3
(iii) Re-determine the head loss (Δh) between each soil element assuming
3k1=k2=2k3
(iv) Express the head at
points A, B, C, and D
A B C
(with respect to the
datum)
(v) Plot the various
heads versus
horizontal distance.
Permeability – Part B Dr O.Hamza
27. Working on Quizzes and Example problems
Quiz 4
(i) Flow rate q in each soil element is equal:
Δh
q = Aki = Ak
L
Δh Δh 2
q1 = Ak1 1 q 2 = Ak 2 q 3 = ...
L1 L2
q = q1 = q 2 = q 3
Δh = Δh1 + Δh 2 + Δh 3
Permeability – Part B Dr O.Hamza
28. Working on Quizzes and Example problems
Quiz 4
(ii) Flow rate q in each soil element is equal:
q = q1 = q 2 = q 3
29. Working on Quizzes and Example problems
Quiz 4
(iii) Flow rate q in each soil element is equal:
q = q1 = q 2 = q 3
31. Working on Quizzes and Example problems
Quiz 4
(v) Plotting
NOTE: It is coincident
that ll heads
th t all h d appears
in a straight line.
32. Working on Quizzes and Example problems
Quiz 5
The site consists of an unconfined aquifer and a confined aquifer separated by a
5-m thick
5 thi k confining layer. Water in the unconfined aquifer i f h and water
fi i l W t i th fi d if is fresh, d t
in the confined aquifer is saline. Two nested piezometers have been drilled,
one penetrating the unconfined aquifer (P1), and one penetrating the confined
aquifer (P2)
).
Land surface elevation: 68.1 m Temperature of water in P1 and P2: 16° C
Depth to P1: 21.2 m Depth to P2: 38.6 m
Depth to water in the well at P1: 4.3 m Depth to water in the well at P2: 4.9 m
Unit weight of fresh water at 16° C: 9.99 kN/m3 Unit weight of water in P2: 10.21 kN/m3
• Sketch a diagram (doesn’t have to be to scale) showing the information
described above.
• What is the total head (h1) for P1?
• Determine the pressure head for P2 (hw2-saline), and the equivalent fresh-water
pressure head for P2 (hw2-frish)
w2 frish
• What is the total fresh-water head (h2-fresh) for P2?
• Will you issue a permit to inject hazardous waste into the deep aquifer ? Why
or why not?
Permeability – Part B Dr O.Hamza
33. Working on Quizzes and Example problems
Quiz 5
4.3t
4.9
21.2
38.6 m
68.1 m
Datum
Permeability – Part B Dr O.Hamza
34. Working on Quizzes and Example problems
Quiz 5
Fresh water total head for P1 is 68.1 – 4.3 = 63.8 m
Saline pressure head for P2 is 38.6 – 4.9 = 33.7 m
For the equivalent fresh-water pressure head, pressure must be equal:
fresh water head
uSaline = ufirsh
So γSaline x 33.7 = γfrish x hw2-frish
solve for hw2-frish: = γSaline x 33.7 / γfrish
= 10.21 x 33.7 /9.99 = 34.4 m
so,
so h2-fresh = hz2 + hw2-frish = (68 1 – 38 6 ) + 34.4 = 63 9 m
f f (68.1 38.6 34 4 63.9
Thus flow is in an UPWARD direction from the lower aquifer, and you should
not issue the permit (In addition if you inject waste into the lower aquifer
permit. addition,
it will further increase the pressure head and increase the upward
gradient.)
Permeability – Part B Dr O.Hamza
35. Working on Quizzes and Example problems
Quiz 6
A soil profile consists of th
il fil i t f three l
layers with properties shown i th t bl b l
ith ti h in the table below.
Calculate the equivalent coefficients of permeability parallel and normal
to the stratum.
Layer Thickness (m) kx (parallel, m/s) kz (normal, m/s)
1 3 2x10-6
6 1.0x10 6
1 0x10-6
2 4 5x10-8 2.5x10-8
3 3 3x10-5 1.5x10-5
Answers:
For the flow parallel to the layers: kx= 9.6x10^-6 m/s
For the flow normal to the layers: kz=6.1x10^-8 m/s
Permeability – Part B Dr O.Hamza
36. Working on Quizzes and Example problem
Problem 3 Field measurement of the coefficient of permeability
3.
A stratum of sandy soil overlies a horizontal bed of impermeable material;
the surface of which is also horizontal. In order to determine the in situ
permeability of the soil, a test well was driven to the bottom of the stratum.
Two observation boreholes were made at distances of 12.5m and 25m
respectively from the test well.
Water was pumped from the test well at the rate of 3x10-3 m3/s until the
water level became steady. The heights of the water in the two observation
boreholes were then found to be 4.25m and 6.5m above the impermeable
bed.
Find the value, expressed in m3/day, of the Impermeable
coefficient of permeability of the sandy soil
ffi i t f bilit f th d il
Permeability – Part B Dr O.Hamza
37. Working on Quizzes and Example problem
Problem 3 Field measurement of the coefficient of permeability
3.
Key solution
This is a quasi-one dimensional
flow, from which we found that:
where:
q (rate of flow) = 3x10-3 m3/s = 3x10-3 x 60 x
60 x 24 = 259 2 m3/day
259.2
Impermeable
r1= 12.5m and r2 = 25m
h1= 4.25m and h2= 6.5m
ln(r2/r1) = 0.693
Note ‘ln’ is the logarithm to base e, also called the natural logarithm.
Permeability – Part B Dr O.Hamza
38. Working on Quizzes and Example problems
Problem 4 E i i l relations of th coefficient of permeability
4. Empirical l ti f the ffi i t f bilit
For a clay soil, the following are given:
soil
Void ratio 1.1 0.9
k (cm/s)
( /) 0 302 x 10-7
0.302 7 0 12 x 10-7
0.12 7
en
Use the following empirical relation: k=C
1+ e
proposed by Samarasinghe, Huang and Drnevich (1982) to estimate the
coefficient of permeability of the clay at a void ratio of 1 2
1.2.
Hint: form two equations with two unknowns C and n by substituting the
experimental values given in the table in the equation.
Permeability – Part B Dr O.Hamza