basics of groundwater hydrology in geotechnical engineering: permeability - part b

Basics of groundwater hydrology in geotechnical engineering

Part B

Prepared by Dr O. Hamzao hamza at hotmail dot com_

Lecture reference: OH GA03 BLecture reference: OH_GA03_B

Permeability – Part B Dr O.Hamza

ContentContent

• Introduction• Quasi one dimensional and radial flow• Quasi-one-dimensional and radial flow • Field determination of coefficient of permeability

S mmar• Summary • Example problems


Introduction

Darcy’s lawDarcy’s law

ikA ikAq =

where

k i ffi i t f bilit ithk is coefficient of permeability with dimensions of velocity (length/time)

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

(Ref. Geotechnical on the Web)

In a saturated porous media, the rate of flow of water q (volume/time) th h ti l ‘A’ i f d t b ti l t h d li

q is flow rate t Time

through cross-sectional area ‘A’ is found to be proportional to hydraulic gradient ‘ i ’


Introduction

Aquifer and Darcy’s lawAquifer and Darcy’s law

Aquifer is a term used to designate a porous geological formation that: contains water at full saturation- contains water at full saturation

- permits water to move through it under ordinary field conditions


Introduction

Aquifer and Darcy’s lawAquifer and Darcy’s law

The horizontal flow rate q is constant. For an aquifer of width B and varying thickness taquifer of width B and varying thickness t, Darcy's Law indicates that

q = A k i= B t k i

or

Hydraulic gradient varies inversely with aquifer

Where flow occurs in a confined aquifer whose thickness varies gently with

thickness

e e o occu s a co ed aqu e ose t c ess a es ge t y tposition the flow can be treated as being essentially one-dimensional.


Quasi one dimensional and radial flowQuasi-one-dimensional and radial flow• Cylindrical flow: confined aquifer • Cylindrical flow: groundwater lowering • Spherical flow


Quasi-one-dimensional and radial flow

Cylindrical flow: confined aquiferCylindrical flow: confined aquifer

Pumping aquiferPumping 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 beconfined aquifer is one of the one-dimensional problem which can be analysed in cylindrical coordinates.




Darcy's Law still applies, with hydraulic gradient dh/dr and area A varying with radius: A = 2 r t

In this case pore pressure or head varies only with radius r

A = 2πr.t

In this case pore pressure or head varies only with radius r.




Integrating between the borehole and at variable di tdistance r:

where ro is the radius of the borehole and h0the constant head in the borehole.



Cylindrical flow: groundwater loweringCylindrical flow: groundwater lowering

Pumping from a borehole can be used for deliberate groundwater lowering in order todeliberate groundwater lowering in order to facilitate excavation.



Cylindrical flow: groundwater loweringCylindrical flow: groundwater lowering

This is an example of quasi-one-dimensional radial flow with flow thickness t=h Then A=2πr h and Groundwater loweringt=h. Then A=2πr.h and Groundwater lowering

Original level of

Integrating between the borehole and at variable distance r:

Original level of water table

Drawdownat variable distance r:

The radius of influence

Drawdown



Spherical flowSpherical flow

Darcy's Law still applies, with hydraulic gradient dh/dr and area A varying with radius: A=4πr²

head varies only with radius r

where r0 is the radius of the piezometer and h0 the constant head in the piezometer

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

with radius r.

piezometer being used for in-situ determination of permeability) is a one-dimensional problem which can be analysed in spherical coordinates.


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



Field measurement of the permeabilityField 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 ymeasurement?




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.




Well-Pumping testIf the pumping causes a drawdown in an unconfined (i.e. open surface) soil stratum then the quasi-one dimensional flow equationthen the quasi one dimensional flow equation is applied.

Integrating between the two test limits and rearranging the equation:

Impermeable

Ob ti l b h l

Impermeable

(Assuming the pumping causes a drawdown in an unconfined (i.e. open surface) soil stratum then) Observational boreholesunconfined (i.e. open surface) soil stratum then)




Well-Pumping testIf the soil stratum is confined and of thickness t d i t t d thand remains saturated then

Confined stratum



Empirical relations for the coefficient of permeabilityEmpirical relations for the coefficient of permeability

Empirical relations

k = function of (other parameters)

p

Permeability of all soils is strongly influenced by the density of packing ofPermeability 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.




P bilit f l il

For fairly uniform sand Hazen (1930)

Permeability of granular soils

proposed the following relation between the coefficient of permeability k (m/s) and the effective particle size D10 (in mm) (theeffective particle size D10 (in mm) (the particle size than which 10% soil is finer):

2C Dk

where C is a constant approximately equal to

210C.Dk =

pp y q0.01 (see the figure beside)

Hazzan equation and data relating coefficient of permeability and effective grain size of granular soilsHazzan equation and data relating coefficient of permeability and effective grain size of granular soils




P bilit f ft l

S i h H d D i h (1982) h t d th t th

Permeability of soft clays

Samarasinghe, Huang and Drnevich (1982) have suggested that the coefficient of permeability of clays can be given by the equation:

n

e1eCk

n

+=

hwhere e is void ratioC and n are constant to be determined experimentally

Consolidation of soft clay may involve a significant decrease in void ratio and therefore of permeability.


SummarySummary

• 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 gradient and occurs p y gtowards the lower total head.• Flow rate is proportional to the hydraulic gradient and can be affected by the geometry of the poresgeometry of the 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).


Quizzes and example problemsQuizzes and example problems

Work on:Work on:

• Quizzes: quiz 3 to 6 *Example problems: *• Example problems: *

problem 3 and problem 4

* Note. quiz 1 and problem 1 and 2 are covered in Part A ofNote. quiz 1 and problem 1 and 2 are covered in Part A of Permeability lecture


Quiz 3

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 2 h t i li ti it ti th t i ht lt i t f h d• For case 2, what is a realistic situation that might result in a set of 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.

datumdatumdatum

Case 2Case 1


Quiz 3


Quiz 3

S l tiSolution:

h

Flowhw

hz

h Flow

datumdatumdatum

Case 2Case 1

z

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.


Quiz 4


Quiz 4

An inclined permeameter tube is filled with three layer of soil of different permeabilities as shown in the figurepermeabilities as shown in the figure.

(i) Formulate q in terms of the different dimensions and permeabilities for each soil element

(ii) D t i th h d l (Δh) b t h il l t i(ii) Determine the head loss (Δh) between each soil element assuming 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 Dpoints A, B, C, and D (with respect to the datum)

(v) Plot the various(v) Plot the various heads versus horizontal distance.


Quiz 4


Quiz 4

(i) Flow rate q in each soil element is equal:(i) Flow rate q in each soil element is equal:

LΔhAkAkiq ==

32

222

1

111 ...q

LΔhAkq

LΔhAkq

L

===

321

321

21

ΔhΔhΔhΔhqqqq++=

===

321


Quiz 4


Quiz 4

(ii) Flow rate q in each soil element is equal:(ii) Flow rate q in each soil element is equal:

321 qqqq ===

Quiz 4


Quiz 4

(iii) Flow rate q in each soil element is equal:(iii) Flow rate q in each soil element is equal:

321 qqqq ===

Quiz 4


Quiz 4

(iv) Heads(iv) Heads

Quiz 4Working on Quizzes and Example problems

Quiz 4

(v) Plotting(v) Plotting

NOTE: It is coincident th t ll h dthat all heads appears in a straight line.

Quiz 5


Quiz 5

The site consists of an unconfined aquifer and a confined aquifer separated by a 5 thi k fi i l W t i th fi d if i f h d t5-m thick confining layer. Water in the unconfined aquifer is fresh, and water in the confined aquifer is saline. Two nested piezometers have been drilled, one penetrating the unconfined aquifer (P1), and one penetrating the confined aquifer (P )aquifer (P2).

Land surface elevation: 68.1 m Temperature of water in P1 and P2: 16° CDepth to P1: 21.2 m Depth to P2: 38.6 mDepth to water in the well at P1: 4.3 m Depth to water in the well at P2: 4.9 mUnit 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)2 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?


Quiz 5


Quiz 5

4.3t21.2

4.9

68.1 m38.6 m

Datum


Quiz 5


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 mFor the equivalent fresh-water pressure head pressure must be equal:For the equivalent fresh water pressure head, pressure must be equal:

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 mso h f = h + h f = (68 1 – 38 6 ) + 34 4 = 63 9 mso, h2-fresh hz2 + hw2-frish (68.1 38.6 ) + 34.4 63.9 m

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 aquifernot issue the permit. (In addition, if you inject waste into the lower aquifer it will further increase the pressure head and increase the upward gradient.)


Quiz 6


Quiz 6

A il fil i t f th l ith ti h i th t bl b lA soil profile consists of three layers with properties shown 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 1 0x10-61 3 2x10 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/sFor the flow normal to the layers: kz=6.1x10^-8 m/s


Problem 3 Field measurement of the coefficient of permeability

Working on Quizzes and Example problem

Problem 3. Field measurement of the coefficient of permeability

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 observationwater 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 ffi i t f bilit f th d il

Impermeable

coefficient of permeability of the sandy soil


Problem 3 Field measurement of the coefficient of permeability

Working on Quizzes and Example problem

Problem 3. Field measurement of the coefficient of permeability

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/day60 x 24 259.2 m3/day

r1= 12.5m and r2 = 25m h1= 4.25m and h2= 6.5m

Impermeable

ln(r2/r1) = 0.693

Note ‘ln’ is the logarithm to base e, also called the natural logarithm.


Problem 4 E i i l l ti f th ffi i t f bilit


Problem 4. Empirical relations of the coefficient of permeability

For a clay soil the following are given:

Void ratio 1.1 0.9k ( / ) 0 302 10 7 0 12 10 7

For a clay soil, the following are given:

Use the following empirical relation:

k (cm/s) 0.302 x 10-7 0.12 x 10-7

eCkn

=Use the following empirical relation:

proposed by Samarasinghe, Huang and Drnevich (1982) to estimate the coefficient of permeability of the clay at a void ratio of 1 2

e1Ck

+=

coefficient of permeability of the clay at a void ratio of 1.2.

Hint: form two equations with two unknowns C and n by substituting the experimental values given in the table in the equation.


basics of groundwater hydrology in geotechnical engineering: permeability - part b

Documents

b permeability

spherical flow permeability

radial flow cylindrical

confined aquifer permeability

dimensional radial flow

bulk permeability

i coefficient of permeability

rate of flow