hydro chapter_7_groundwater_by louy Al hami

12/26/2012 1

Chapter 7

Civil Engineering Department

Prof. Majed Abu-Zreig

Hydraulics and Hydrology – CE 352

Groundwater Hydraulics

Hydrologic cycle

Occurrence of Ground Water

• Ground water occurs when water recharges a porous subsurface geological formation “called aquifers” through cracks and pores in soil and rock

• it is the water below the water table where all of the pore spaces are filled with water.

• The area above the water table where the pore spaces are only partially filled with water is called the capillary fringe or unsaturated zone.

• Shallow water level is called the water table

Groundwater Basics -

Definitions

Recharge

Natural

• Precipitation

• Melting snow

• Infiltration by streams and lakes

Artificial • Recharge wells

• Water spread over land in pits, furrows, ditches

• Small dams in stream channels to detain and deflect water

http://capp.water.usgs.gov/GIP/gw_gip/recharge.html

Aquifers

Definition: A geological unit which can store and supply significant quantities of water.

Principal aquifers by rock type:

Unconsolidated

Sandstone

Sandstone and Carbonate

Semiconsolidated

Carbonate-rock

Volcanic

Other rocks

Example Layered Aquifer System

Bedient et al., 1999.

Other Aquifer Features

Groundwater occurrence in confined and

unconfined aquifer

Potentiometric Surfaces

Eastern Aquifer

Growndwater

basins

in Jordan

Unconfined Aquifers

• GW occurring in aquifers: water fills partly an

aquifer: upper surface free to rise and decline:

UNCONFINED or water-table aquifer: unsaturated

or vadose zone

• Near surface material not saturated

• Water table: at zero gage pressure: separates saturated

and unsaturated zones: free surface rise of water in a

well

Confined Aquifer

• Artesian condition

• Permeable material overlain by relatively

impermeable material

• Piezometric or potentiometric surface

• Water level in the piezometer is a measure of

water pressure in the aquifer

Groundwater Basics -

Definitions • Aquifer Confining Layer or Aquitard

– A layer of relatively impermeable material which restricts vertical

water movement from an aquifer located above or below.

– Typically clay or unfractured bedrock.

Aquifer Characteristics

• Porosity

– The ratio of pore/void volume

to total volume, i.e. space

available for occupation by air

or water.

– Measured by taking a known

volume of material and adding

water.

– Usually expressed in units of

percent.

– Typical values for gravel are

25% to 45%.

Bedient et al., 1999.,

Typical Values of Porosity

Aquifer Properties • Porosity: maximum amount of water that a rock

can contain when saturated.

• Permeability: Ease with which water will flow through a porous material

• Specific Yield: Portion of the GW: draining under influence of gravity:

• Specific Retention: Portion of the GW: retained as a film on rock surfaces and in very small openings:

• Storativity: Portion of the GW: draining when the piezometric head dropped a unit depth

Figures from Hornberger et al. (1998)

Unconfined aquifer

Specific yield = Sy

Confined aquifer

Storativity = S

b

h h

Storage Terms

S = V / A h

S = Ss b

Ss = specific storage


• Hydraulic Conductivity – Measure of the ease with which water can flow through an

aquifer.

– Higher conductivity means more water flows through an

aquifer at the same hydraulic gradient.

– Measured by well draw down or lab test.

– Expressed in units of mm/day, ft/day or gpd/ft2.

– Typical values for sand/gravel are 2.5 cm/day to 33 m/day

m1 (1 to 100 ft/day).

– Typical values for clay are 0.3 mm/day (0.001 ft/day). That

is why is is an aquifer confining layer.

• Transmissivity (T = Kb) is the rate of flow through a

vertical strip of aquifer (thickness b) of unit width

under a unit hydraulic gradient

•


• Hydraulic gradient – Steepness of the slope of the water table.

– Groundwater flows from higher elevations to lower elevations

(i.e. downgradient).

– Measured by taking the difference in elevation between two

wells and dividing by the distance separating them.

– Expressed in units of ft/ft or ft/mi.

– Typical values for groundwater are .0001 to .01 m/m.


• Groundwater Velocity – How fast groundwater is moving.

– Calculated by conductivity multiplied by gradient divided by

porosity.

– Expressed in units of ft/day.

– Typical values for gravel or sand are 0.15 to 16 m/day (1 to 50

ft/day).

• Water table: the

surface separating

the vadose zone

from the saturated

zone.

• Measured using

water level in well

The Water Table

Fig. 11.1

• Precipitation

• Infiltration

• Ground-water recharge

• Ground-water flow

• Ground-water discharge to

– Springs

– Streams and

– Wells

Ground-Water Flow

• Velocity is

proportional to

– Permeability

– Slope of the water

table

• Inversely

Proportional to

– porosity

Ground-Water Flow

Fast (e.g., cm per day)

Slow (e.g., mm per day)

• Infiltration

– Recharges ground

water

– Raises water table

– Provides water to

springs, streams

and wells

• Reduction of

infiltration causes

water table to drop

Natural Water

Table Fluctuations

• Reduction of infiltration causes water table to drop

– Wells go dry

– Springs go dry

– Discharge of rivers drops

• Artificial causes

– Pavement

– Drainage

Natural Water

Table Fluctuations

• Pumping wells

– Accelerates flow

near well

– May reverse

ground-water flow

– Causes water table

drawdown

– Forms a cone of

depression

Effects of

Pumping Wells

• Pumping wells

– Accelerate flow

– Reverse flow

– Cause water

table drawdown

– Form cones of

depression Low river

Gaining

Stream

Gaining

Stream

Pumping well

Low well

Low well

Cone of

Depression

Water Table

Drawdown

Dry Spring

Effects of

Pumping Wells

Dry river

Dry well

Effects of

Pumping Wells

Dry well

Dry well

Losing

Stream • Continued water-

table drawdown

– May dry up

springs and wells

– May reverse flow

of rivers (and

may contaminate

aquifer)

– May dry up rivers

and wetlands

Ground-Water/

Surface-Water

Interactions

• Gaining streams

– Humid regions

– Wet season

• Loosing streams

– Humid regions, smaller

streams, dry season

– Arid regions

• Dry stream bed

Figure taken from Hornberger et al. (1998)

Darcy column

h/L = grad h

q = Q/A

Q is proportional

to grad h

x

hAKQ

x

hAQ

Darcy’s Law Henry Darcy’s Experiment (Dijon, France 1856)

AQxQhQ ,1,

x

hAKQ

x

hAQ

Q

Q: Volumetric flow rate [L3/T]

Darcy investigated ground water flow under controlled conditions

h

h1 h2

h

x

h1

Slope = h/x

~ dh/dx h

x h2

x1 x2

K: The proportionality constant is added to form the following equation:

K units [L/T]

A

: Hydraulic Gradient xhh

A: Cross Sectional Area (Perp. to flow)

Calculating Velocity with Darcy’s

Law • Q= Vw/t

– Q: volumetric flow rate in m3/sec

– Vw: Is the volume of water passing through area “a” during

– t: the period of measurement (or unit time).

• Q= Vw/t = H∙W∙D/t = a∙v

– a: the area available to flow

– D: the distance traveled during t

– v : Average linear velocity

• In a porous medium: a = A∙n

– A: cross sectional area (perpendicular to flow)

– n: porous For media of porosity

• Q = A∙n∙v

• v = Q/(n∙A)=q/n

Vw

v

x

h

n

Kv

Darcy’s Law (cont.)

• Other useful forms of Darcy’s Law

dx

dhKq

Q

A =

Q

A.n = q

n = dx

dh

n

Kv

Volumetric Flux (a.k.a. Darcy Flux or

Specific discharge)

Ave. Linear

Velocity

Used for calculating

Q given A


average velocity of

groundwater transport

(e.g., contaminant

transport Assumptions: Laminar, saturated flow

dx

dhAKQ Volumetric Flow Rate


Volumes of groundwater

flowing during period of

time

Figure from Hornberger et al. (1998)

Linear flow

paths assumed

in Darcy’s law

True flow paths

Average linear velocity

v = Q/An= q/n

n = effective porosity

Specific discharge

q = Q/A

Steady Flow to Wells in Confined Aquifers

• Radial flow towered wells

• Aquifers are homogeneous (properties are uniform)

• Aquifers are isotropic (permeability is independent of flow direction)

• Drawdown is the vertical distance measured from the original to the lowered water table due to pumping

• Cone of depression the axismmetric drawdown curve forms a conic geometry

• Area of influence is the outer limit of the cone of depression

• Radius of Influence (ro) for a well is the maximum horizontal extent of the cone of depression when the well is in equilibrium with inflows

• Steady state is when the cone of depression does not change with time

Horizontal and Vertical Head Gradients

Freeze and Cherry, 1979.

Flow to Wells

Steady Radial Flow to a Well-

Confined

Q

Cone of Depression

s = drawdown

h r


Confined

• In a confined aquifer, the drawdown

curve or cone of depression varies with

distance from a pumping well.

• For horizontal flow, Q at any radius r

equals, from Darcy’s law,

Q = -2πrbK dh/dr

for steady radial flow to

a well where Q,b,K are

const


Confined • Integrating after separation of variables, with

h = hw at r = rw at the well, yields Thiem Eqn

Q = 2πKb[(h-hw)/(ln(r/rw ))]

Note, h increases

indefinitely with

increasing r, yet

the maximum head

is h0.


Confined

• Near the well, transmissivity, T, may be

estimated by observing heads h1 and h2

at two adjacent observation wells

located at r1 and r2, respectively, from

the pumping well

T = Kb = Q ln(r2 / r1)

2π(h2 - h1)


Unconfined


Unconfined

• Using Dupuit’s assumptions and applying Darcy’s law

for radial flow in an unconfined, homogeneous,

isotropic, and horizontal aquifer yields:

Q = -2πKh dh/dr

integrating,

Q = πK[(h22 - h1

2)/ln(r2/ r1)

solving for K,

K = [Q/π(h22 - h1

2)]ln (r2/ r1)

where heads h1 and h2 are observed at adjacent

wells located distances r1 and r2 from the pumping

well respectively.

Steady Flow to a Well in a Confined

Aquifer

2rw

Ground surface

Bedrock

Confined

aquifer

Q

h0

Pre-pumping

head

Confining Layer

b

r1

r2

h2

h1

hw

Observation

wells

Drawdown curve

Q

Pumping

well

Q = Aq = (2prb)Kdh

dr

rdh

dr=

Q

2pT

h2 = h1 +Q

2pTln(

r2r1

)

Theim Equation

In terms of head (we can write it in terms of drawdown also)

Example - Theim Equation

• Q = 400 m3/hr

• b = 40 m.

• Two observation wells,

1. r1 = 25 m; h1 = 85.3 m

2. r2 = 75 m; h2 = 89.6 m

• Find: Transmissivity (T)

T =Q

2p h2 - h1( )ln

r2r1

æ

è ç

ö

ø ÷ =

400 m3/hr

2p 89.6 m - 85.3m( )ln

75 m

25 m

æ

è ç

ö

ø ÷ =16.3 m2 /hr

h2 = h1 +Q

2pTln(

r2r1

)2rw

Ground surface

Bedrock

Confine

d

aquifer

Q

h0

Confining Layer

b

r1

r2

h2 h1

hw

Q

Pumping

well

Steady Flow to a Well in a Confined Aquifer

Steady Radial Flow in a Confined

Aquifer

• Head

• Drawdown

h r( ) = h0 +Q

2pTln

r

R

æ

è ç

ö

ø ÷

s r( ) =Q

2pTln

R

r

æ

è ç

ö

ø ÷

s(r) = h0 - h r( )


Theim Equation

In terms of drawdown (we can write it in terms of head also)


• 1-m diameter well

• Q = 113 m3/hr

• b = 30 m

• h0= 40 m

• Two observation wells, 1. r1 = 15 m; h1 = 38.2 m

2. r2 = 50 m; h2 = 39.5 m

• Find: Head and drawdown in the well

2rw

Ground surface

Bedrock

Confine

d

aquifer

Q

h0

Confining Layer

b

r1

r2

h2 h1

hw

Q

Pumping

well Drawdown

Adapted from Todd and Mays, Groundwater Hydrology

T =Q

2p s1 - s2( )ln

r2r1

æ

è ç

ö

ø ÷ =

113m3/hr

2p 1.8 m - 0.5 m( )ln

50 m

15 m

æ

è ç

ö

ø ÷ =16.66 m2 /hr

s r( ) =Q

2pTln

R

r

æ

è ç

ö

ø ÷



2rw

Ground surface

Bedrock

Confine

d

aquifer

Q

h0

Confining Layer

b

r1

r2

h2 h1

hw

Q

Drawdown

@ well

Adapted from Todd and Mays, Groundwater Hydrology

hw = h2 +Q

2pTln(

rwr2

) = 39.5 m +113m3 /hr

2p *16.66 m2 /hrln(

0.5 m

50 m) = 34.5 m

sw = h0 - hw = 40 m- 34.5 m = 5.5 m

h2 = h1 +Q

2pTln(

r2r1

)


Steady Flow to Wells in

Unconfined Aquifers

Steady Flow to a Well in an Unconfined

Aquifer

Q = Aq = (2prh)Kdh

dr

= prKdh2

dr

2rw

Ground surface

Bedrock

Unconfined

aquifer

Q

h0

Pre-pumping

Water level

r1

r2

h2 h1

hw

Observation

wells

Water Table

Q

Pumping

well

rd h2( )

dr=

Q

pK

h02 - h2 =

Q

pKln

R

r

æ

è ç

ö

ø ÷

h2(r) = h02 +

Q

pKln

r

R

æ

è ç

ö

ø ÷

h2 = h1 +Q

2pTln(

r2r1

)

Confined aquifer Unconfined aquifer

Steady Flow to a Well in an Unconfined

Aquifer

2rw

Ground surface

Bedrock

Unconfined

aquifer

Q

h0

Prepumping

Water level

r1

r2

h2 h1

hw

Observation

wells

Water Table

Q

Pumping

well

2 observation wells: h1 m @ r1 m h2 m @ r2 m

K =Q

p h22 - h1

2( )ln

r2r1

æ

è ç

ö

ø ÷

h2(r) = h02 +

Q

pKln

r

R

æ

è ç

ö

ø ÷

h22 = h1

2 +Q

pKln

r2r1

æ

è ç

ö

ø ÷

• Given:

– Q = 300 m3/hr

– Unconfined aquifer

– 2 observation wells,

• r1 = 50 m, h = 40 m

• r2 = 100 m, h = 43 m

• Find: K

K =Q

p h22 - h1

2( )ln

r2r1

æ

è ç

ö

ø ÷ =

300 m3 /hr / 3600 s /hr

p (43m)2 - (40 m)2[ ]ln

100 m

50 m

æ

è ç

ö

ø ÷ = 7.3x10-5 m /sec

Example – Two Observation Wells in an

Unconfined Aquifer

2rw

Ground surface

Bedrock

Unconfined

aquifer

Q

h0

Prepumping

Water level

r1

r2

h2 h1

hw

Observation

wells

Water Table

Q

Pumping

well

Steady Flow to a Well in an Unconfined Aquifer

Pump Test in Confined

Aquifers

Jacob Method

Cooper-Jacob Method of Solution

Cooper and Jacob noted that for small values of r

and large values of t, the parameter u = r2S/4Tt

becomes very small so that the infinite series can be

approx. by: W(u) = – 0.5772 – ln(u) (neglect higher terms)

Thus s' = (Q/4πT) [– 0.5772 – ln(r2S/4Tt)]

Further rearrangement and conversion to decimal logs yields:

s' = (2.3Q/4πT) log[(2.25Tt)/ (r2S)]


A plot of drawdown s' vs.

log of t forms a straight line

as seen in adjacent figure.

A projection of the line back

to s' = 0, where t = t0 yields

the following relation:

0 = (2.3Q/4πT) log[(2.25Tt0)/ (r2S)]

Semi-log plot



So, since log(1) = 0, rearrangement yields

S = 2.25Tt0 /r2

Replacing s' by s', where s' is the drawdown

difference per unit log cycle of t:

T = 2.3Q/4πs'

The Cooper-Jacob method first solves for T and

then for S and is only applicable for small

values of u < 0.01

Cooper-Jacob Example

For the data given in the Fig.

t0 = 1.6 min and s’ = 0.65 m

Q = 0.2 m3/sec and r = 100 m

Thus:

T = 2.3Q/4πs’ = 5.63 x 10-2 m2/sec

T = 4864 m2/sec

Finally, S = 2.25Tt0 /r2

and S = 1.22 x 10-3

Indicating a confined aquifer

Jacob Approximation

• Drawdown, s

• Well Function, W(u)

• Series

approximation of

W(u)

• Approximation of s

s u( ) =Q

4pTW u( )

W u( ) =e-h

hu

¥

ò dh » -0.5772 - ln(u)+ u -u2

2!+

u =r2S

4Tt

W u( ) » -0.5772 - ln(u) for small u < 0.01

s(r,t) »Q

4pT-0.5772 - ln

r2S

4Tt

æ

è ç

ö

ø ÷

é

ë ê ê

ù

û ú ú

s(r,t) =2.3Q

4pTlog10(

2.25Tt

r2S)

Pump Test Analysis – Jacob Method

Jacob Approximation

s =2.3Q

4pTlog(

2.25Tt

r2S)

0 =2.3Q

4pTlog(

2.25Tt0

r2S)

t0

1=2.25Tt0

r2S

S =2.25Tt0

r2


Jacob Approximation

t0

S =2.25Tt0

r2

t1 t2

s1

s2

s

logt2

t1

æ

è ç

ö

ø ÷ = log

10* t1t1

æ

è ç

ö

ø ÷ =1

1 LOG CYCLE

1 LOG CYCLE


Jacob Approximation

S =2.25Tt0

r2=

2.25(76.26 m2/hr)(8 min*1 hr /60 min)

(1000 m)2

= 2.29x10-5

t0

t1 t2

s1

s2

s

t0 = 8 min

s2 = 5 m s1 = 2.6 m s = 2.4 m


Multiple-Well Systems

• For multiple wells with drawdowns that overlap, the principle of superposition may be used for governing flows:

• drawdowns at any point in the area of influence of several pumping wells is equal to the sum of drawdowns from each well in a confined aquifer


Injection-Pumping Pair of Wells

Pump Inject


• The previously mentioned principles also

apply for well flow near a boundary

• Image wells placed on the other side of the

boundary at a distance xw can be used to

represent the equivalent hydraulic condition

– The use of image wells allows an aquifer of

finite extent to be transformed into an

infinite aquifer so that closed-form solution

methods can be applied


•A flow net for a pumping

well and a recharging

image well

-indicates a line of

constant head

between the two wells

Three-Wells Pumping

A

Total Drawdown at A is sum of drawdowns from each well

Q1

Q3

Q2

r


The steady-state drawdown

s' at any point (x,y) is given

by:

s’ = (Q/4πT)ln

where (±xw,yw) are the

locations of the recharge and

discharge wells. For this

case, yw= 0.

(x + xw)2 + (y - yw)2

(x - xw)2 + (y - yw)2


The steady-state drawdown s' at any point (x,y) is given by

s’ = (Q/4πT)[ ln {(x + xw)2 + y2} – ln {(x – xw)2 + y2} ]

where the positive term is for the pumping well and the

negative term is for the injection well. In terms of head,

h = (Q/4πT)[ ln {(x – xw)2 + y2} – ln {(x + xw)2 + y2 }] + H

Where H is the background head value before pumping.

Note how the signs reverse since s’ = H – h

7.5 Aquifer Boundaries

The same principle

applies for well

flow near a

boundary

– Example:

pumping near a

fixed head stream

well near an impermeable boundary

Education

hydro chapter_7_groundwater_by louy Al hami