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
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
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%.
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/Ah
S = Ss b
Ss = specific storage
Figures from Hornberger et al. (1998)
21.
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
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
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
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)
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
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 ]
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)]
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/4πs'
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/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
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
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
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
