Embed Size (px)
Citation preview
Today we will learn about…
• Flow in leaky confined aquifer
• Flow in an unconfined aquifer
• Determining aquifer parameters for non-
equilibrium conditions
• Aquifers in non-ideal conditions
2
UNSTEADY STATE WELL HYDRAULICS:
LEAKY CONFINED AQUIFER
3
Flow in a leaky confined aquifer
4
Q
Potentiometric
surface
bh
Fully penetrating well in an aquifer overlain by a
semipermeable confining layer
0h
,K S
Original potentiometric
surface
'b ',S'K
''b ''K
Leaky confining layer
rHydraulic head, h [m]
Storativity, S [-]
Transmissivity, T [m2/day]
Time, t [days]
Radial distance from the pumping well, r [m]
Recharge to aquifer, q [m/day]
2
2
1h h S h q
r r r T t T
+ = −
( )
0
20
2
''
'1
'
h hq K
b
K h hh h S h
r r r T t Tb
−=
− → + = −
Vertical hydraulic conductivity of the leaky layer
K’ [m/day]
Thickness of the leaky layer, b [m]
Drawdown, h0-h [m]
Case 1. Hantush’s (1956) solution for the case
when no water drains from the confining layer
5
Q
Potentiometric
surface
bh
Fully penetrating well in an aquifer overlain by a
semipermeable confining layer
0h
,K S
Original potentiometric
surface
'b ',S'K
''b ''K
Leaky confining layer
r
Assumptions:
1. The aquifer is confined on the top by an
aquitard
2. The aquitard is overlain by an unconfined
aquifer, known as the source bed
3. The water table in the source bed is
initially horizontal
4. The water table in the source bed does
not fall during pumping of the aquifer
5. Groundwater flow in the aquitard is
vertical
6. The aquitard is incompressible, so that no
water is released from storage in the
aquitard when the aquifer is pumped
7. The aquifer is compressible and water
drains instantaneously with a decline in
head
( )2
' ', , '' '' 100
10 '
S bt or b K bK
bK
( )1/2
0.036 ' '/ ', ,
0.04 / ' 's s
t b S K or
r b KS K S
( ) ( )
( )
22
1/2
30 / 1 10 / ,
/ '/ K' 0.1
w w
w
t r S T r b and
r Tb
−
Any well diameter:
Hantush’s (1956) solution for the case when
no water drains from the confining layer
6
Q
Potentiometric
surface
bh
Fully penetrating well in an aquifer overlain by a
semipermeable confining layer
0h
,K S
Original potentiometric
surface
'b ',S'K
''b ''K
Leaky confining layer
r
The rate at which water is being drawn from elastic storage in the confined
aquifer, qs [m3/day] at a specific time, t [days] since pumping began is
determined from:
( )20
2
'1
'
K h hh h S h
r r r T t Tb
− + = −
( ) ( )0
1/22
, /4
',
4 '
Qh h W u r B
T
r S Tbu B
Tt K
− =
= =
Leaky artesian well function, W(u,r/B)
( )2exp /sq Q Tt SB= −
L sq Q q= −
For a total discharge of Q at time t, the water coming from leakage across
the aquitard is:
Hantush’s (1956) solution for the case when
no water drains from the confining layer
7
Q
Potentiometric
surface
bh
Fully penetrating well in an aquifer overlain by a
semipermeable confining layer
0h
,K S
Original potentiometric
surface
'b ',S'K
''b ''K
Leaky confining layer
r
( ) ( )0
1/22
, /4
',
4 '
Qh h W u r B
T
r S Tbu B
Tt K
− =
= =
If the well is pumped for a long time, all
the water may come from leakage across
the aquitard and none from the elastic
storage (Hantush & Jacob, 1954), this
occurs when: 8 '
'
b St
K
Drawdown in this case: ( ) ( )0 0 /4
Qh h K r B
T− =
Where K0 is a zero-order modified Bessel function of the second kind (see
from table)
Case 2. Hantush’s (1960) solution for the case when some
water comes from elastic storage in the aquitard
8
QPotentiometric
surface
bh
Fully penetrating well in an aquifer overlain by a
semipermeable confining layer
0h
,K S
Original potentiometric
surface
'b ',S'K
''b ''K
Leaky confining layer
r
2 solutions:
1. Early time condition:
Solution:
'b'
10 '
St
K
( ) ( )0 ,4
Qh h H u
T
− =
( )1/2
1/2
2
''/ ,
4 '
4
r TbS S B
B K
r Su
Tt
= =
=
Rate of flow from the storage in the main
aquifer is given by:
Where, the erfc is the complementary
error function = 1-erf(x)
( ) ( )( )( )2
exp
'/ ' '/
sq Q t erfc t
K b S S
=
=
Case 2. Hantush’s (1960) solution for the case when some
water comes from elastic storage in the aquitard
9
QPotentiometric
surface
bh
Fully penetrating well in an aquifer overlain by a
semipermeable confining layer
0h
,K S
Original potentiometric
surface
'b ',S'K
''b ''K
Leaky confining layer
r
2 solutions:
2. Equilibrium state:
Solution:
( )
( ) ( )1/2
8 '/ 3 ''
'/ ' ''/ ''
S S St
K b K b
+ + +
( ) ( )0 0 / ; / 0.014
w
Qh h K r B r B
T− =
In first case, all water comes from the elastic
storage in the aquifer and the aquitard. At
equilibrium, all water comes from drainage from
the overlying source bed.
Case Solution
No leakage
Leakage (no
storage from
aquifer)
Leakage,
storage from
aquitard
Leakage,
storage from
aquitard,
equilibrium
Various solutions for
confined aquifers
10
Hantush, M.S., 1960. Modification of the theory of leaky
aquifers. Journal of Geophysical Research, 65(11), pp.3713-3725.
( ) ( )0 ,4
Qh h H u
T
− =
( ) ( )0 0 /4
Qh h K r B
T− =
( ) ( )0 , /4
Qh h W u r B
T− =
( ) ( )4
o
Qh h W u
T− =
UNSTEADY STATE WELL HYDRAULICS:
UNCONFINED AQUIFER
11
Flow in an unconfined aquifer (Neuman & Witherspoon, 1969)
12
Assumptions in Neuman’s solution to this equation:
1. The aquifer is unconfined
2. Th vadose zone has no influence on the drawdown
3. Water initially pumped comes from instantaneous release of water from elastic storage
4. Eventually water comes from storage due to gravity drainage of interconnected pores
5. The drawdown in negligible compared with the saturated aquifer thickness
6. The specific yield is at least 10 times the elastic storativity
7. The aquifer may be anisotropic with differing radial and vertical hydraulic conductivities
2 2
2 2
rr v s
Kh h h hK K S
r r r z t
+ + =
Saturated thickness of the aquifer, h [m]
Elevation above the base of the aquifer, z [m]
Radial hydraulic conductivity, Kr [m/day]
Vertical hydraulic conductivity, Kz [m/day]
Specific storage, Ss [1/m]
Time, t [days]
Radial distance from pumping well, r [m]
0h h−Q
Original
potentiometric surface
Potentiometric
surface at time ,t
0h
h
r
Fully penetrating well pumping from a confined
aquifer
Flow in an unconfined aquifer: Neuman’s solution
13
Drawdown, h0-h [m]
Pumping rate, Q [m3/day]
Transmissivity, T [m2/day]
Radial distance from pumping well, r [m]
Storativity, S [-]
Specific yield, Sy [-]
0h h−Q
Original
potentiometric surface
Potentiometric
surface at time ,t
0h
h
r
Fully penetrating well pumping from a confined
aquifer
( ) ( )0
2
2
2
2
, ,4
,4
,4
A B
A
y
B
v
h
Qh h W u u
T
r Su for early drawdown
Tt
r Su for later drawdown
Tt
r K
b K
− =
=
=
=
Time, t [days]
Horizontal hydraulic conductivity, Kh [m/day]
Vertical hydraulic conductivity, Kz [m/day]
Initial saturated thickness of the aquifer [m]
DETERMINING AQUIFER PARAMETERS:
NON-EQUILIBRIUM CONDITIONS
14
Non-equilibrium/ transient flow conditions
→ cone of depression continues to grow with time
Recall: A pumping test is an experiment when water is pumped from a well
(stress) at a specified rate and change in water level (response) is measured at
one or more locations surrounding the wells
15
1. To determine the hydraulic characteristics of the aquifer: hydraulic
conductivity, transmissivity, and storativity
2. Identify aquifer boundaries
3. Evaluate performance of the well
4. Water quality of the ground water
Assumptions:
1. The pumping well is screened only in the aquifer being tested
2. All observation wells are screened only in the aquifer being tested
3. The pumping well and observation wells are screened throughout the entire
thickness of the aquifer
Theis graphical solution
16
( )2
,4 4
o
Q r Sh h W u u
T Tt− = =Theis’s nonequilibrium equation:
Rearranging:( )
( ) 2
4,
4 o
Q TutT W u S
h h r= =
−
Steps:
1. Plot W(u) vs. (1/u) on log log scale – call this Theis reverse type curve
2. Plot drawdown (s=ho-h) vs. t on log log scale – call this observation curve
3. Overlay the observation curve over the Theis reverse type curve
4. Keeping the axes on both plots parallel, move the observation curve over the Theis
curve until the data points on the observation curves overlap with the Theis curves.
5. When the overlap occurs, identify any point on the Theis curve, preferably pick the
point where W(u) = 1 and 1/u = 1. This point is called the ‘match’ point.
6. For this point, read the x and y axis values on the observation curves.
7. You have W(u), u, (s=ho-h), and t, these can be substituted in the above equations to find
T and S.
17
Theis reverse non equilibrium type curve
(for fully confined aquifer)
Typical data from a well test
Matching field data to Theis curve to obtain aquifer
parameters
18
Cooper-Jacob straight line time-drawdown method
19
( )
( ) 2
0.5772 ln4
2.3 2.25log
4
o
o
QT u
h h
Q TtT
h h r S
= − −−
→ =
−
After long time:
2
0.054
r Su
Tt=
Ignore higher order terms in well
function:
Steps:
1. Plot drawdown vs. time on semi-log scale (time on log, drawdown on natural scale)
2. Extrapolate the line joining observation points to intersect the drawdown axis at s = 0
3. Read the value of time for this, call this to
4. The slope of the line gives the drawdown per log cycle of time: Δh-ho
5. Estimate transmissivity and storativity as:
( )
2
2.3
4
2.25
o
o
QT
h h
TtS
r
=
−
=
Recovery test:
20
( ) ( )2 2
' ' , & '4 4 4 '
Q r S r Ss W u W u u u
T Tt Tt= − = = Theis’s residual drawdown
estimate:
Applying the approximation:
2.3' log
4 '
Q ts
T t=
Steps:
1. Plot residual drawdown, s’ vs.
logarithm of t/t’
2. Estimate the slope of the
line, or the residual
drawdown, Δs, per log cycle
of t/t’
3. Estimate transmissivity as:
2.3
4 '
QT
s=
What if the pumping well does not go through the full
aquifer width?
21
• 3D flow, vertical flow components
• Both vertical and horizontal hydraulic
conductivity values important
• Effects can be neglected for confined
aquifers if:
1. Observation wells are fully
penetrating
2. Observation wells are located more
than the following distance from the
pumped well: A partially penetrating well in a confined aquifer.
3D flow due to vertical flow components.
1.5 /h vb K K
• Effects of partially penetrating well in unconfined aquifers can be minimized if observation
wells fully penetrates the saturated thickness of the aquifer. Then:
1. Time drawdown curve for observation wells with following distances and time criteria
can be used to approximate the late time Theis curve
2. Time drawdown curve for observation wells with following distances and time criteria
can be used to approximate the early time Theis curve
2/ , /obs h v obs yr b K K t S r T
20.03 / , /obs h v obsr b K K t S r T
AQUIFERS IN NON-IDEAL
CONDITIONS
22
Effect of well interference: wells placed close to each
other
23
• For confined aquifers, Laplace equation holds, and is linear. Therefore:
total drawdown = sum of individual drawdowns from each well
• For unconfined aquifers, the above method will under-estimate the actual drawdown
due to the non-linearity of the general GW equations in the unconfined case
Composite pumping cone for three wells, pumping at different rates, tapping
the same aquifer.
Effect of hydrogeologic boundaries: rivers/lakes
24
Recharge boundaries can be simulated by a recharging image well located opposite to
the river, at the same distance from the river as the original well
A well bounded on one side by a stream,
a recharge boundary.
Effect of hydrogeologic boundaries: barriers
25
Barriers are simulated by locating a discharging well at an equivalent distance away from
the boundary on the opposite side.
A well bounded on one side by an
impermeable boundary, or a barrier.
Effect of hydrogeologic
boundaries: comparison
26
Recharge boundary retards the rate of
drawdown, therefore, the slope of drawdown vs.
time reduces in its presence.
Presence of barriers increases drawdown rates
above the ideal conditions (infinite aquifer), so
slopes are greater than the ideal case.
The observation of drawdown vs. time for long
time periods can be used to understand
whether recharge or barrier boundaries are
present in the aquifer system
Aquifer test design:
1. Where should I drill the pumping well?
– Use information on geology, geophysical surveys, aerial photos, including presence of nearby wells
– Engineering and economic factors may play a greater role in determining the location
– If possible, start by drilling a test/production well, after which the permanent well is installed.
2. Make a borewell log for the well being drilled: a borewell log lists all the geologic formations found at various depths during drilling
3. If possible, also note down the water levels in the well as the well is drilled
4. Use borewell log to determine potential aquifer zones, select one for testing
5. Install pump in the well and a device to measure flow rates (water meter within the pipe line for small rates, orifice weir on discharge pipe for larger rates)
6. Make adequate arrangements to take water away from the test site
7. Maintain constant discharge during pumping, allow variations only within 10% (well development helps here)
8. Account for any other source of water level changes (tidal, recharge sources, etc.)
27
