hydro

1. Civil Engineering Department
Prof. Majed Abu-Zreig
Hydraulics and Hydrology – CE 352
Chapter 7
Groundwater Hydraulics
12/26/2012 1

2. Hydrologic cycle

3. 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

4. Groundwater Basics -
Definitions

5. Recharge
Natural Artificial
• Precipitation • Recharge wells
• Melting snow • Water spread over
• Infiltration by streams land in pits, furrows,
and lakes ditches
• Small dams in
stream channels to
detain and deflect
water

6. 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

7. Example Layered Aquifer System
Bedient et al., 1999.

8. Other Aquifer Features

9. Groundwater occurrence in confined and
unconfined aquifer

10. Potentiometric Surfaces

11. Eastern Aquifer

12. Growndwater
basins
in Jordan

14. 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

15. 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

16. 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.

17. 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%.

18. Typical Values of Porosity
Bedient et al., 1999.,

19. 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

20. Storage Terms
h
h
b
Unconfined aquifer Confined aquifer
Specific yield = Sy Storativity = S
S=V/Ah
S = Ss b
Ss = specific storage
Figures from Hornberger et al. (1998)

22. Aquifer Characteristics
• 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

23. Aquifer Characteristics
• 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.

24. Aquifer Characteristics
• 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).

25. The Water Table
• Water table: the
surface separating
the vadose zone
from the saturated
zone.
• Measured using
water level in well
Fig. 11.1

26. Ground-Water Flow
• Precipitation
• Infiltration
• Ground-water
recharge
• Ground-water flow
• Ground-water
discharge to
– Springs
– Streams and
– Wells

27. Ground-Water Flow
• Velocity is
proportional to
– Permeability
Fast (e.g., cm per day)
– Slope of the water
table
• Inversely
Proportional to
– porosity
Slow (e.g., mm per day)

28. Natural Water
Table Fluctuations
• Infiltration
– Recharges ground
water
– Raises water table
– Provides water to
springs, streams
and wells
• Reduction of
infiltration causes
water table to drop

29. 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

30. Effects of
Pumping Wells
• Pumping wells
– Accelerates flow
near well
– May reverse
ground-water flow
– Causes water table
drawdown
– Forms a cone of
depression

31. Effects of
Pumping Wells
Gaining
• Pumping wells
Stream
– Accelerate flow
– Reverse flow Water Table
Drawdown
Low well
– Cause water Cone of Dry Spring
table drawdown Depression
Gaining
– Form cones of Stream Low well
Low river
depression
Pumping well

32. Effects of
Dry well
Pumping Wells
• Continued water- Losing
Stream
Dry well
table drawdown
– May dry up
springs and wells
– May reverse flow
of rivers (and
may contaminate Dry well
aquifer) Dry river
– May dry up rivers
and wetlands

33. Ground-Water/
Surface-Water
Interactions
• Gaining streams
– Humid regions
– Wet season
• Loosing streams
– Humid regions, smaller
streams, dry season
– Arid regions
• Dry stream bed

34. Darcy column
h h
Q A  Q  K A
x x
h/L = grad h
Q is proportional
to grad h
q = Q/A
Figure taken from Hornberger et al. (1998)

35. Darcy’s Law
Henry Darcy’s Experiment (Dijon, France 1856)
Darcy investigated ground water flow under controlled conditions
h1 h2 Q  h, Q  1 x , Q  A
A h
Q: Volumetric flow rate [L3/T]
Q A: Cross Sectional Area (Perp. to flow)
K: The proportionality constant is added
to form the following equation:
h  h : Hydraulic Gradient
h Slope = h/x x
h h
h1 ~ dh/dx Q A  Q  K A
h2
h x x
x K units [L/T]
x1 x2 x

36. 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 v
– a: the area available to flow
– D: the distance traveled during t Vw
– v : Average linear velocity
• In a porous medium: a = A∙n
– A: cross sectional area (perpendicular to flow)
– n: porous For media of porosity K h
• Q = A∙n∙v v 
n x
• v = Q/(n∙A)=q/n

37. Darcy’s Law (cont.)
• Other useful forms of Darcy’s Law
Used for calculating
dh
Volumetric Flow Rate Q   K A Volumes of groundwater
dx flowing during period of
time
Volumetric Flux Q dh Used for calculating
A=
q  K Q given A
(a.k.a. Darcy Flux or dx
Specific discharge)
Ave. Linear
Q q K dh Used for calculating
Velocity A.n = n = v   average velocity of
n dx
groundwater transport
(e.g., contaminant
Assumptions: Laminar, saturated flow
transport

38. True flow paths
Linear flow
paths assumed
in Darcy’s law
Average linear velocity
Specific discharge
q = Q/A v = Q/An= q/n
n = effective porosity
Figure from Hornberger et al. (1998)

39. 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

40. Horizontal and Vertical Head Gradients
Freeze and Cherry, 1979.

41. Flow to Wells

42. Steady Radial Flow to a Well-
Confined
Cone of Depression
Q
s = drawdown
r
h

43. Steady Radial Flow to a Well-
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

44. Steady Radial Flow to a Well-
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.

45. Steady Radial Flow to a Well-
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)

46. Steady Radial Flow to a Well-
Unconfined

47. Steady Radial Flow to a Well-
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 - h12)/ln(r2/ r1)
solving for K,
K = [Q/π(h22 - h12)]ln (r2/ r1)
where heads h1 and h2 are observed at adjacent
wells located distances r1 and r2 from the pumping
well respectively.

48. Steady Flow to a Well in a Confined
Aquifer
Q
Ground surface
Pre-pumping
head
Pumping
Drawdown curve well
dh Observation
Q = Aq = (2prb)K wells
dr Confining Layer
dh Q h0
r = r1 hw
dr 2pT b h2
h1
Confined r2 Q
aquifer
Q r2
h2 = h1 + ln( ) Bedrock
2pT r1 2rw
Theim Equation
In terms of head (we can write it in terms of drawdown also)

49. Steady Flow to a Well in a Confined Aquifer
Example - Theim Equation
Q
• Q = 400 m3/hr Ground surface
• b = 40 m.
• Two observation wells, Pumping
well
1. r1 = 25 m; h1 = 85.3 m
2. r2 = 75 m; h2 = 89.6 m
Confining Layer
• Find: Transmissivity (T) h0 r1 hw
b h
2 h1
Confine r2 Q
d
Q r aquifer
h2 = h1 + ln( 2 )
2pT r1
Bedrock
2rw
Q æ r2 ö 400 m 3 /hr æ 75 m ö
T= lnç ÷ = lnç ÷ = 16.3 m /hr
2
2p ( h2 - h1 ) è r1 ø 2p ( 89.6 m - 85.3m) è 25 m ø

50. Steady Flow to a Well in a Confined Aquifer
Steady Radial Flow in a Confined
Aquifer
• Head
Q ærö
h( r ) = h0 + lnç ÷
2pT è R ø
• Drawdown
s(r) = h0 - h( r )
Q æ Rö
s( r ) = lnç ÷
2pT è r ø
Theim Equation
In terms of drawdown (we can write it in terms of head also)

51. Steady Flow to a Well in a Confined Aquifer
Example - Theim Equation
Q
•
Ground surface
1-m diameter well
• Q = 113 m3/hr Drawdown Pumping
well
• b = 30 m
• h0= 40 m Confining Layer
•
h0
Two observation wells, b h
h1
r1 hw
2
1. r1 = 15 m; h1 = 38.2 m Confine r2 Q
d
2. r2 = 50 m; h2 = 39.5 m aquifer
• Find: Head and
Bedrock
2rw
drawdown in the well
Q æ Rö
s( r ) = lnç ÷
2pT èrø
Q æ r2 ö 113m 3 /hr æ 50 m ö
T= lnç ÷ = lnç ÷ = 16.66 m /hr
2
2p ( s1 - s2 ) è r1 ø 2p (1.8 m - 0.5 m) è 15 m ø
Adapted from Todd and Mays, Groundwater Hydrology

52. Steady Flow to a Well in a Confined Aquifer
Example - Theim Equation
Q
Ground surface
Drawdown
@ well
Q r
h2 = h1 + ln( 2 )
Confining Layer
2pT r1
h0 r1 hw
b h
2 h1
Confine r2 Q
d
aquifer
Bedrock
2rw
Q rw 113m 3 /hr 0.5 m
hw = h2 + ln( ) = 39.5 m + ln( ) = 34.5 m
2pT r2 2p *16.66 m /hr 50 m
2
sw = h0 - hw = 40 m - 34.5 m = 5.5 m
Adapted from Todd and Mays, Groundwater Hydrology

53. Steady Flow to Wells in
Unconfined Aquifers

54. Steady Flow to a Well in an Unconfined
Aquifer
dh Q
Q = Aq = (2prh)K Ground surface
dr Pre-pumping
Water level
dh 2 Pumping
= prK Water Table well
dr Observation
wells
r
( )= Q
d h2 h0
h
r1 hw
dr pK Unconfined
2 h1
Q
r2
aquifer
Q æ Rö
h0 - h 2 =
2
lnç ÷
pK è r ø
Bedrock
2rw
Q ærö Q r
h 2
(r) = h0
2
+ lnç ÷ h2 = h1 + ln( 2 )
pK è R ø 2pT r1
Unconfined aquifer Confined aquifer

55. Steady Flow to a Well in an Unconfined
Aquifer
Q ærö
2
(r) = h0
2
+
Q
h lnç ÷
pK è R ø
Ground surface
Prepumping
Water level
Pumping
Water Table well
2 observation wells: Observation
wells
h1 m @ r1 m
h2 m @ r2 m
h0 r1 hw
h
2 h1
Q æ r2 ö
Unconfined r2 Q
aquifer
h2 = h1 +
2 2
lnç ÷
pK è r1 ø Bedrock
2rw
æ r2 ö
Q
K= lnç ÷
(
p h2 - h1 è r1 ø
2 2
)

56. Steady Flow to a Well in an Unconfined Aquifer
Example – Two Observation Wells in an
Unconfined Aquifer
Q
• Given:
Ground surface
Prepumping
Water level
– Q = 300 m3/hr Pumping
Water Table well
– Unconfined aquifer Observation
wells
– 2 observation wells,
• r1 = 50 m, h = 40 m h0 r1 hw
• r2 = 100 m, h = 43 m h
2 h1
Unconfined r2 Q
aquifer
• Find: K Bedrock
2rw
Q æ r2 ö 300 m 3 /hr / 3600 s /hr æ100 m ö
K= lnç ÷ = lnç ÷ = 7.3x10 -5 m /sec
( ) [
p h2 - h1 è r1 ø p (43m)2 - (40 m)2
2 2 è 50 m ø ]

57. Pump Test in Confined
Aquifers
Jacob Method

58. 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

59. Cooper-Jacob Method of Solution
A plot of drawdown s' vs. Semi-log plot
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)]

60. Cooper-Jacob Method of Solution

61. Cooper-Jacob Method of Solution
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/4πs'
The Cooper-Jacob method first solves for T and
then for S and is only applicable for small
values of u < 0.01

62. 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/4πs’ = 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

63. Pump Test Analysis – Jacob Method
Jacob Approximation
Q r 2S
• Drawdown, s s ( u) = W ( u) u=
4 pT 4Tt
¥ e -h u2
• Well Function, W(u) W ( u) = ò dh » -0.5772 - ln(u) + u - +
u h 2!
• Series W (u) » -0.5772 - ln(u) for small u < 0.01
approximation of
W(u) Q é æ r 2 S öù
s(r,t) » ê-0.5772 - lnç ÷ú
4 pT ê
ë è 4Tt øú û
• Approximation of s
2.3Q 2.25Tt
s(r,t) = log10 ( 2 )
4 pT r S

64. Pump Test Analysis – Jacob Method
Jacob Approximation
2.3Q 2.25Tt
s= log( 2 )
4 pT r S
2.3Q 2.25Tt
0= log( 2 0 )
4 pT r S
2.25Tt 0
1=
r 2S
2.25Tt 0
S=
r2
t0

65. Pump Test Analysis – Jacob Method
Jacob Approximation
1 LOG CYCLE
æ t2 ö æ10 *t1 ö
logç ÷ = logç ÷ =1
è t1 ø è t1 ø
s2
s
s1
1 LOG CYCLE
t1 t2
2.25Tt 0
S=
r2
t0

66. Pump Test Analysis – Jacob Method
Jacob Approximation
t0 = 8 min
s2 = 5 m
s1 = 2.6 m s2
s = 2.4 m s
s1
t1 t2
t0
2.25Tt 0 2.25(76.26 m 2 /hr)(8 min*1 hr /60 min)
S= 2
=
r (1000 m)2
= 2.29x10 -5

67. 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

68. Multiple-Well Systems

69. Injection-Pumping Pair of Wells
Pump Inject

70. Multiple-Well Systems
• 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

71. Multiple-Well Systems
•A flow net for a pumping
well and a recharging
image well
-indicates a line of
constant head
between the two wells

72. Three-Wells Pumping
Total Drawdown at A is sum of drawdowns from each well
Q2
Q1 A r
Q3

73. Multiple-Well Systems
The steady-state drawdown
s' at any point (x,y) is given
by:
(x + xw) + (y - yw)2
2
s’ = (Q/4πT)ln
(x - xw)2 + (y - yw)2
where (±xw,yw) are the
locations of the recharge and
discharge wells. For this
case, yw= 0.

74. Multiple-Well Systems
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

75. 7.5 Aquifer Boundaries
The same principle
applies for well
flow near a
boundary
– Example:
pumping near a
fixed head stream

76. well near an impermeable boundary