Transcript Applied Hydrogeology V

Прикладная
Гидрогеология
Yoram Eckstein, Ph.D.
Fulbright Professor 2013/2014
Tomsk Polytechnic University
Tomsk, Russian Federation
Spring Semester 2014
Useful links
http://www.onlineconversion.com/
http://www.digitaldutch.com/unitconverter/
http://water.usgs.gov/ogw/basics.html
http://water.usgs.gov/ogw/pubs.html
http://ga.water.usgs.gov/edu/earthgwaquifer.html
http://water.usgs.gov/ogw/techniques.html
http://water.usgs.gov/ogw/CRT/
VI. Groundwater Flow
to Wells
Type of Water Wells
Production Wells
Injection Wells
Remediation Wells
Pumping wells
Injection wells
Ground-water flow to wells
Extract water
Remove contaminated water
Lower water table for constructions
Relieve pressures under dams
Injections – recharges
Control salt-water intrusion
Our purpose of well studies
Compute the decline in the water level, or
drawdown, around a pumping well whose
hydraulic properties are known.
Determine the hydraulic properties of an
aquifer by performing an aquifer test in
which a well is pumped at a constant rate
and either the stabilized drawdown or the
change in drawdown over time is measured.
Wells in Confined and
Unconfined Aquifers
In unconfined aquifers, pumping will
result in drawdown of the water table
In confined aquifers, pumping will
cause drawdown of the potentiometric
surface
All pores in the confined aquifer will
still remain saturated
Wells in Confined and
Unconfined Aquifers
All pores in the confined aquifer remain
fully saturated
Wells in Confined and
Unconfined Aquifers
The pores within the cone of depression
in an unconfined aquifer are dewaterd
Wells in Confined and
Unconfined Aquifers
http://dli.taftcollege.edu/streams
/geography/Animations/ConeDep
ression.html
Wells in Confined and
Unconfined Aquifers
If qr is the radial groundwater flux at a distance r
from the well, it follows from the mass balance
equation that the total radial flow towards the well
should be equal to the pumping rate:
Q = −2πrbqr
Wells in Confined and
Unconfined Aquifers
Using Darcy’s law to express the groundwater flux
Q = −2πrbqr
becomes
Q = 2𝝅rb
=
𝒅𝒉
−𝑲
𝒅𝒓
=
𝒅𝒉
𝟐𝝅𝒓𝑻
𝒅𝒓
b·K = T
Wells in Confined and
Unconfined Aquifers
Q = 2𝝅rb
𝒅𝒓 =
𝒅𝒉
−𝑲
𝒅𝒓
𝒅𝒓
𝟐𝝅𝑻
𝒓
=
and
𝒅𝒉
𝟐𝝅𝒓𝑻 𝒅𝒓
𝒉=
hence:
𝑸
𝟐𝝅𝑻
𝒍𝒏 𝒓 + 𝒄
where c is an
integration
constant and
therefore:
Wells in Confined and
Unconfined Aquifers
𝒉 = 𝒉𝒐 −
𝑸
𝟐𝝅𝑻
Thiem
equation for
steady-state
flow of water
into a well in
confined
aquifer
𝒍𝒏
𝒓𝒐
𝒓
or
𝒔 = 𝒉𝒐 − 𝒉 =
𝑸
𝟐𝝅𝑻
𝒍𝒏
𝒓𝒐
𝒓
Wells in Confined and
Unconfined Aquifers
In the case of an
unconfined
aquifer thickness
(Ho) equals the
elevation of the
water table (ho)
above the aquifer
bottom. However, within the cone of depression
H < Ho
Wells in Confined and
Unconfined Aquifers
The mass balance
equation is:
𝑄 = −2𝜋𝑟𝐻𝑞𝑟
using Darcy’s law
to express the
groundwater flux and assuming that s<<Ho
this becomes:
𝑑ℎ
𝑑𝐻
𝑄 = −2𝜋𝑟𝐻 −𝐾
= 2𝜋𝑟𝐾𝐻
𝑑𝑟
𝑑𝑟
Wells in Confined and
Unconfined Aquifers
𝑸 = 𝟐𝝅𝒓𝑲𝑯
𝑸
𝟐𝝅𝑲
𝒓𝟐
𝒓𝟏
𝒅𝑯
𝒅𝒓
𝟏
𝒅𝒓 =
𝒓
further integration yields:
𝒓𝟐
𝒉𝒅𝒉
𝒓𝟏
Wells in Confined and
Unconfined Aquifers
𝒓𝟐 𝟏
𝑸
𝟐𝝅𝑲 𝒓𝟏 𝒓
𝑸
𝟐𝝅𝑲
𝒅𝒓 =
𝒍𝒏
𝒓𝟐
𝒓𝟐
Dupuit equation
for steady-state flow
of water into a well
in unconfined aquifer
𝒓𝟐
𝒉𝒅𝒉
𝒓𝟏
=
𝟐
𝒉𝟐 −𝒉𝟏
𝟐
or:
𝟐
which is
Steady Flow in an
Unconfined Aquifer
𝒅𝒉
𝒒 = −𝑲𝒉
𝒅𝒙
′
Integrate both sides of the
equation
Dupuit
Equation
′
𝒒 =
𝟏
𝟐
𝑲
𝒉𝟐 𝟐 − 𝒉𝟏 𝟐
𝑳
L is the flow length
Steady flow through an unconfined aquifer
resting on a horizontal impervious surface
Steady Flow in an
Unconfined Aquifer
 Flow lines are assumed to be horizontal and
parallel to impermeable layer
 The hydraulic gradient of flow is equal to the slope
of water. (slope very small)
Wells in Our Considerations
Fully penetrate the aquifer
The flow is radial symmetric
The aquifer is homogeneous and isotropic
Basic Assumptions
The aquifer is bounded on the bottom by a
confining layer.
All geologic formations are horizontal and have
infinite horizontal extent.
The potentiometric surface of the aquifer is
horizontal prior to the start of the pumping.
Basic Assumptions (cont.)
The potentiometric surface of the aquifer is not
recharging with time prior to the start of the
pumping.
All changes in the position of the potentiometric
surface are due to the effect of the pumping alone.
Basic Assumptions (cont.)
The aquifer is homogeneous and isotropic.
All flow is radial toward the well.
Ground water flow is horizontal.
Darcy’s law is valid.
Basic Assumptions (cont.)
Ground water has a constant density and viscosity.
The pumping well and the observation wells are
fully penetrating.
The pumping well has an infinitesimal diameter
and is 100% efficient.
A completely confined
aquifer
Addition assumptions:
The aquifer is confined top and bottom.
The is no source of recharge to the aquifer.
The aquifer is compressible.
Water is released instantaneously.
Constant pumping rate of the well.
Polar (or
radial
coordinate
system)
Theis’ Solution to
Transient Flow Equation
2
2
𝜕 ℎ 𝜕 ℎ
𝑆 𝜕ℎ
+
=
𝜕𝑥 2 𝜕𝑦 2
𝑇 𝜕𝑡
𝑄1 − 𝑄2 =2𝜋 𝐾𝑏
= 2𝜋𝑇
𝜕ℎ
𝑟2
𝜕𝑟 2
𝜕ℎ
𝑟
𝜕𝑟 2
−
− 2𝜋 𝐾𝑏
𝜕ℎ
𝑟
𝜕𝑟 1
𝜕ℎ
𝑟1
𝜕𝑟 1
= ∆𝑆
=
Theis’ Solution to
Transient Flow Equation
2𝜋𝑇
𝜕ℎ
𝑟
𝜕𝑟
𝜕ℎ
− 𝑟
𝜕𝑟
2
= ∆𝑆 =
1
𝜕 2 ℎ 𝜕ℎ
𝜕ℎ
= 2𝜋𝑇 𝑟 2 +
∆𝑟 = 𝑆 2𝜋𝑟∆𝑟
=
𝜕𝑟
𝜕𝑟
𝜕𝑡
𝜕 2 ℎ 1 𝜕ℎ 𝑆 𝜕ℎ
= 2+
=
𝜕𝑟
𝑟 𝜕𝑟 𝑇 𝜕𝑡
Theis’ Solution to
Transient Flow Equation
Two-dimensional flow with no vertical components:
2
2
𝜕 ℎ 𝜕 ℎ
𝑆 𝜕ℎ
+
=
𝜕𝑥 2 𝜕𝑦 2
𝑇 𝜕𝑡
thus
2
𝜕 ℎ 1 𝜕ℎ
𝑆 𝜕ℎ
+
=
2
𝜕𝑟
𝑟 𝜕𝑟
𝑇 𝜕𝑡
Theis’ Solution to
Transient Flow Equation
Two-dimensional flow with no vertical components:
𝜕 2 ℎ 1 𝜕ℎ
𝑆 𝜕ℎ
+
=
2
𝜕𝑟
𝑟 𝜕𝑟
𝑇 𝜕𝑡
𝑄
𝑠 = ℎ𝑜 − ℎ =
4𝜋𝑇
∞ −𝑢
𝑒
𝑢
𝑄
du =
𝑊(𝑢)
𝑢
4𝜋𝑇
Theis’ Solution to
Transient Flow Equation
𝑄
𝑠 = ℎ𝑜 − ℎ =
4𝜋𝑇
∞ −𝑢
𝑒
𝑢
𝑄
du =
𝑊(𝑢)
𝑢
4𝜋𝑇
h0 = initial hydraulic head (L; m or ft)
h = hydraulic head (L; m or ft)
s = h0 – h = drawdown (L; m or ft)
Q = constant pumping rate (L3/T; m3/d or ft3/d)
W(u) = well function.
T = transmissivity (L2/T; m2/d or ft2/d)
Theis’ Solution to
Transient Flow Equation
𝑄
𝑠 = ℎ𝑜 − ℎ =
4𝜋𝑇
∞ −𝑢
𝑒
𝑢
𝑄
du =
𝑊(𝑢)
𝑢
4𝜋𝑇
𝑟2𝑆
𝑢=
4𝑇𝑡
S = storativity (dimensionless)
t = time since pumping began (T; d)
r = radial distance from the pumping well (L; m
or ft)
Well Function W(u)
∞
𝑊 𝑢 =
𝑢
𝑒 −𝑢
𝑑𝑢 = −0.5772 + 𝑙𝑛𝑢 +
𝑢
2
3
4
𝑛
𝑢
𝑢
𝑢
𝑢
+𝑢 −
+
−
+⋯
2 ∙ 2! 3 ∙ 3! 4 ∙ 4!
𝑛 ∙ 𝑛!
Purpose of Pumping
Well Tests
Determine the hydraulic properties of an
aquifer by performing an aquifer test in
which a well is pumped at a constant rate
and either the stabilized drawdown or the
change in drawdown over time is measured.
Determine the hydraulic properties of a
water well by performing variable-rate
production test.
Pumping Well Terminology
Static Water Level [SWL] (ho)
is the equilibrium water level
before pumping commences
Pumping Water Level [PWL]
Q
s
ho
h
(h) is the water level during
pumping
Drawdown (s = ho - h) is the
difference between SWL and PWL
Well Yield (Q) is the volume of
water pumped per unit time
Specific Capacity (Q/s) is the
yield per unit drawdown
Cone of Depression
High Kh aquifer
Low Kh aquifer
Kh  Kv
 A zone of low pressure is created centred on the pumping well
 Drawdown is a maximum at the well and reduces radially
 Head gradient decreases away from the well and the pattern
resembles an inverted cone called the cone of depression
 The cone expands over time until the inflows (from various
boundaries) match the well extraction
 The shape of the equilibrium cone is controlled by hydraulic
conductivity
Pumping test
in a completely
confined aquifer
ℎ𝑜 − ℎ =
𝑄
𝑊(𝑢)
4𝜋𝑇
𝑢=
𝑟2𝑆
4𝑇𝑡
Known values:
Q – pumping rate
r - distance from
the pumping well
to the observation
well (can be rw)
rw – pumping well radius
Cooper-Jacob simplification
Cooper and Jacob (1946) pointed out that the
series expansion of the exponential integral or
W(u) is:
𝑊 𝑢 = −γ + 𝑙𝑛𝑢 +
𝑢2
𝑢3
𝑢4
𝑢𝑛
+𝑢 −
+
−
+⋯
2 ∙ 2! 3 ∙ 3! 4 ∙ 4!
𝑛 ∙ 𝑛!
where γ is Euler’s constant = 0.5772
For u<< 1 , say u < 0.05 the series can be
truncated:
W(u)  – ln(eγ) - ln(u) = - ln(eγu) = -ln(1.78u)
Cooper-Jacob simplification
For u<< 1 , say u < 0.05 the series can be
truncated:
W(u)  -ln(1.78u)
Thus:
𝑄
1.78 ∙ 𝑟 2 𝑆
𝑄 𝑙𝑛 4𝑇𝑡
𝑠 = ℎ𝑜 − ℎ =
𝑙𝑛
=
4𝜋𝑇
4𝑇𝑡
4𝜋𝑇 1.78 ∙ 𝑟 2 𝑆
𝑄 𝑙𝑛 2.25𝑇𝑡
2.3 ∙ 𝑄
2.25 ∙ 𝑇𝑡
=
=
𝑙𝑜𝑔
2
4𝜋𝑇
𝑟 𝑆
4𝜋𝑇
𝑟2𝑆
The Cooper-Jacob simplification expresses
drawdown (s) as a linear function of ln(t) or
log(t).
Cooper-Jacob Plot: Log(t) vs.
Drawdown (s)
0.0
0.1
0.2
Drawdown (m)
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.E+01
1.E+02
1.E+03
Time since pump started (s)
1.E+04
1.E+05
Cooper-Jacob Plot: Log(t) vs.
Drawdown (s)
to = 84sec.
0.0
0.1
0.2
Drawdown (m)
0.3
0.4
0.5
0.6
Ds =0.40 m
0.7
0.8
0.9
1.0
1.E+01
1.E+02
1.E+03
Time since pump started (s)
1.E+04
1.E+05
Cooper-Jacob Analysis
1. Fit straight-line to data (excluding early and
late times if necessary):
– at early times the Cooper-Jacob
approximation may not be valid
– at late times boundaries may significantly
influence drawdown
2. Determine intercept on the time axis for s=0
3. Determine drawdown increment (Ds) for one
log-cycle
Cooper-Jacob Analysis (cont.)
2.3𝑄
4. For straight-line fit, 𝑇 =
4𝜋∆𝑠
2.25𝑇𝑡𝑜
2.3𝑄𝑡𝑜
and
S=
=
2
𝑟
1.78𝜋𝑟 2 ∆𝑠
5. For the example, Q = 32 L/s or 0.032 m3/s;
r = 120 m; to = 84 s and Ds = 0.40 m
6.
𝑇=
7. 𝑆 =
2.3∙0.032
2.56∙0.40
= 0.01465
2.3∙0.032∙84
1.78∙𝜋∙1202 ∙0.40
𝑚2
𝑠𝑒𝑐
=
= 1.9 ∙ 10−4
𝑚2
1266
𝑑𝑎𝑦
Recharge Effect : Recharge > Well Yield
15
Drawdown (m)
20
25
30
35
1.E+01
1.E+02
1.E+03
1.E+04
1.E+05
1.E+06
Time since pump started (s)
Recharge causes the slope of the log(time) vs drawdown
curve to flatten as the recharge in the zone of influence of the
well matches the discharge. The gradient and intercept can
still be used to estimate the aquifer characteristics (T,S).
Recharge Effect : Recharge < Well Yield
15
Drawdown (m)
20
25
30
35
1.E+01
1.E+02
1.E+03
1.E+04
1.E+05
1.E+06
Time since pump started (s)
If the recharge is insufficient to match the discharge, the
log(time) vs drawdown curve flattens but does not become
horizontal and drawdown continues to increase at a reduced
rate. T and S can be estimated from the first leg of the curve.
Sources of Recharge
Various sources of recharge may cause deviation
from the ideal Theis behaviour.
Surface water: river, stream or lake boundaries may
provide a source of recharge, halting the expansion of
the cone of depression.
Vertical seepage from an overlying aquifer, through
an intervening aquitard, as a result of vertical
gradients created by pumping, can also provide a
source of recharge.
Where the cone of depression extends over large
areas, leakage from aquitards may provide sufficient
recharge.
Recharge Effect: Leakage Rate
15
Drawdown (m)
20
High Leakage
25
Low Leakage
30
35
1.E+01
1.E+02
1.E+03
1.E+04
1.E+05
1.E+06
Time since pump started (s)
Recharge by vertical leakage from overlying (or
underlying beds) can be quantified using analytical
solutions developed by Jacob (1946). The analysis
assumes a single uniform leaky bed.
Barrier Effect: No Flow Boundary
15
Drawdown (m)
20
25
30
35
1.E+01
1.E+02
1.E+03
1.E+04
1.E+05
Time since pump started (s)
Steepening of the log(time) vs. drawdown curve
indicates an aquifer limited by a barrier boundary of
some kind. Aquifer characteristics (T,S) can be
estimated from the first leg.
Potential Flow Barriers
Various flow barriers may cause deviation from the
ideal Theis behaviour.
Fault truncations against low permeability aquitards.
Lenticular pinchouts and lateral facies changes
associated with reduced permeability.
Groundwater divides associated with scarp slopes.
Spring lines with discharge captured by wells.
Artificial barriers such as grout curtains and slurry
walls.
Casing Storage
It has been known for many decades that early time
data can give erroneous results because of removal of
water stored in the well casing.
When pumping begins, this water is removed and the
amount drawn from the aquifer is consequently
reduced.
The true aquifer response is masked until the casing
storage is exhausted.
Analytical solutions accounting for casing storage
were developed by Papadopulos and Cooper (1967)
and Ramey et al (1973)
Unfortunately, these solutions require prior
knowledge of well efficiencies and aquifer
characteristics
Casing Storage
Q
s
dp
dc
Schafer (1978) suggests that an estimate of the
critical time to exhaust casing storage can be
made more easily:
tc = 3.75p(dc2 – dp2) / (Q/s) = 15 Va /Q
where tc is the critical time (d);
dc is the inside casing diameter (m);
dp is the outside diameter of the rising
main (m);
Q/s is the specific capacity of the well
(m3/d/m)
Va is the volume of water removed from
the annulus between casing and rising main.
Note: It is safest to ignore data from pumped
wells earlier than time tc in wells in low-K HSU’s
Jacob-Cooper
Distance-Drawdown Method
If more than three observation wells are used in an
aquifer test, and drawdowns are measured at the
same time in these wells, the Jacob distancedrawdown method can be used to determine aquifer
transmissivity and storativity.
In this method drawdown is plotted on arithmetic
scale as a function of the distance from the pumping
well on the log scale.
A straight line is then drawn through the data points
and extended to the zero-drawdown axis. The
intercept is the distance at which the pumping well is
not affecting the water level and is designated ro
Distance-Drawdown Graph
𝑻=
𝟐.𝟑∙𝑸
𝟐𝝅∆ 𝒉𝒐 −𝒉
𝑺=
𝟐.𝟐𝟓∙𝑻∙𝒕
𝒓𝒐 𝟐
ro = 126 m
0
Drawdown (m)
1
2
Ds = 3.8 m
3
4
5
1
10
Distance (m)
100
1000
Aquifer Characteristics
For the example: t = 0.35 days and Q = 1100 m3/d
2.3∙𝑄
T=
2𝜋∆ ℎ𝑜 −ℎ
2.25∙𝑇∙𝑡
S=
𝑟𝑜 2
=
=
2.3∙1100
= 106 m2/d
2𝜋∙3.8
2.25∙106∙0.35
-3
=
5.3
·
10
1262
 The estimates of T and S from log(time)-drawdown and
log(distance)-drawdown plots are independent of one another
and so are recommended as a check for consistency in data
derived from pump tests.
 Ideally 4 or 5 observation wells are needed for the distancedrawdown graph and it is recommended that T and S are
computed for several different times.
Well Efficiency
 The efficiency of a pumped well can be evaluated using
distance-drawdown graphs.
 The distance-drawdown graph is extended to the outer
radius of the pumped well (including any filter pack) to
estimate the theoretical drawdown for a 100% efficient
well.
 This analysis assumes the well is fully-penetrating and
the entire saturated thickness is screened.
 The theoretical drawdown (estimated) divided by the
actual well drawdown (observed) is a measure of well
efficiency.
 A correction is necessary for unconfined wells to allow
for the reduction in saturated thickness as a result of
drawdown.
Causes of Well Inefficiency
Factors contributing to well inefficiency
(excess head loss) fall into two groups:
Design factors




Insufficient open area of screen
Poor distribution of open area
Insufficient length of screen
Improperly designed filter pack
Construction factors


Inadequate development, residual drilling fluids
Improper placement of screen relative to aquifer interval
Theis-Cooper-Jacob Assumptions
Real aquifers rarely conform to the assumptions made
for Theis-Cooper-Jacob non-equilibrium analysis
 Isotropic, homogeneous, uniform thickness
 Fully penetrating well
 Laminar flow
 Flat potentiometric surface
 Infinite areal extent
 No recharge
The failure of some or all of these assumptions leads to
“non-ideal” behavior and deviations from the Theis and
Cooper-Jacob analytical solutions for radial unsteady
flow
Wells in Confined Aquifers
Completely confined aquifer.
Wells in Confined Aquifers
Confined, leaky with no elastic storage in the
leaky layer.
.
Wells in Confined Aquifers
Confined, leaky with elastic storage in the leaky
layer.
A Leaky, Confined Aquifer
A Leaky, Confined Aquifer no water drains from
the confining layer
(but it can drain
through it)
The aquifer is bounded on the top by an aquitard.
The aquitard is overlain by an unconfined aquifer,
know as the source bed.
The water table in the source bed is initially
horizontal.
The water table in the source bed does not fall
during pumping of the aquifer.
A Leaky, Confined Aquifer
– no water drains from the
confining layer
Ground water flow through the aquitard is
vertical.
The aquifer is compressible, and water drains
instantaneously with a decline in head
The aquitard is incompressible, so that no
water is released from storage in the aquitard
when the aquifer is pumped.
Hantush-Jacob Formula
Confined with no elastic storage
h0 – h =
𝑄
W(u,r/B)
4𝜋𝑇
h0 = initial hydraulic head (L; m or ft)
h = hydraulic head (L; m or ft)
h0 – h = drawdown (L; m or ft)
Q = constant pumping rate (L3/T; m3/d or ft3/d)
W(u,r/B) = leaky artesian well function
T = transmissivity (L2/T; m2/d or ft2/d)
B = (Tb’/K’)1/2
Drawdown Formula
Confined with elastic storage
h0 – h =
𝑄
H(u,)
4𝜋𝑇
h0 = initial hydraulic head (L; m or ft)
h = hydraulic head (L; m or ft)
h0 – h = drawdown (L; m or ft)
Q = constant pumping rate (L3/T; m3/d or ft3/d)
H(u,) = a modified leaky artesian well function
T = transmissivity (L2/T; m2/d or ft2/d)
 = r/4B (S’/S)1/2; B = (Tb’/K’)1/2
Unconfined aquifer – 3 phases
of flow to a pumping well
Early stage –
pressure drops
specific storage as a major contribution
flow is horizontal
time-drawdown follows Theis curve
Unconfined aquifer – 3 phases
of flow to a pumping well
Second stage –
water table declines
specific yield as a major contribution
flow is both horizontal and vertical
time-drawdown is a function of Kv/Kh r, b
Unconfined aquifer – 3 phases
of flow to a pumping well
Later stage –
rate of drawdown decreases
flow is again horizontal
time-drawdown again follows Theis curve
S - the specific yield.
Neuman’s assumptions
Aquifer is unconfined.
Vadose zone has no influence on the
drawdown.
Water initially pumped comes from the
instantaneous release of water from elastic
storage.
Eventually water comes from storage due to
gravity drainage of interconnected pores.
Neuman’s assumptions
(cont.)
The drawdown is negligible compared to
the saturated thickness.
The specific yield is at least 10 times the
elastic storativity.
The aquifer may be – but does not have to
be – anisotropic with the radial hydraulic
conductivity different than the vertical
hydraulic conductivity.
Drawdown Formula
unconfined with elastic storage
h0 – h =
𝑄
4𝜋𝑇
W(uA,uB,)
h0 = initial hydraulic head (L; m or ft)
h = hydraulic head (L; m or ft)
h0 – h = drawdown (L; m or ft)
Q = constant pumping rate (L3/T; m3/d or ft3/d)
W(uA,uB,) = the well function for water-table
aquifer
T = transmissivity (L2/T; m2/d or ft2/d)
uA =r2S/(4Tt); uB =r2Sy/(4Tt); =r2Kv/(r2Kh)
Neuman’s type curves
𝑟 2 𝐾𝑣
Γ= 2
𝑟 𝐾ℎ
𝑟2𝑆
𝑢𝐴 =
4𝑇𝑡
𝑟 2 𝑆𝑦
𝑢𝐵 =
4𝑇𝑡
Aquifer Test Software
http://trials.swstechnology.com/demos/AquiferTest_Pro/AquiferTest_
Pro-Trial.msi
Partial Penetration
Partial penetration effects occur when the intake of
the well is less than the full thickness of the aquifer
Effects of Partial Penetration
 The flow is not strictly horizontal and radial.
 Flow-lines curve upwards and downwards as they
approach the intake and flow-paths are consequently
longer.
 The convergence of flow-lines and the longer flow-paths
result in greater head-loss than predicted by the
analytical equations.
 For a given yield (Q), the drawdown of a partially
penetrating well is more than that for a fully penetrating
well.
 The analysis of the partially penetrating case is difficult
but Kozeny (1933) provides a practical method to
estimate the change in specific capacity (Q/s).
Q/s Reduction Factors
Kozeny (1933) gives the following approximate
reduction factor to correct specific capacity (Q/s)
for partial penetration effects:
𝐿
𝜋∙𝐿
𝑟
𝐹 = 1 + 7 ∙ 𝑐𝑜𝑠
𝑏
𝑏
2𝑏 ∙ 2𝐿
where b is the total aquifer thickness (m);
r is the well radius (m); and
L is screen length (m).
The equation is valid for L/b < 0.5 and L/r > 30
Q/s Reduction Factors Example
For a 300 mm diameter well with an aquifer
thickness of 30 m and a screen length of 15 m,
L/b = 0.5 and 2L/r = 200 the reduction factor is:
F = 0.5 × {1 + 7 × 0.707
1
}
200
= 0.67
Other factors are provided by Muskat (1937),
Hantush (1964), Huisman (1964), Neuman (1974)
but they are harder to use.
Partial Penetration
Alternative
Multiple screened sections distributed over
the entire saturated thickness functions more
efficiently for the same open area.
Screen Design
300 mm
diameter well
with single
screened
interval of
15 m in aquifer of 30 m
thickness;
L/b = 0.5 and 2L/r = 200
F = 0.5 × {1 + 7 x cos(0.5p/2)
(1/200)} = 0.67
300 mm
diameter well
with 5 x 3 m
solid sections
alternating
with 5 x 3m screened
sections, in an aquifer of 30
m thickness; there
effectively are five aquifers.
L/b = 0.5 and 2L/r = 40
F = 0.5 × {1 + 7 ×
cos(0.5p/2) (1/40)} = 0.89
This is clearly a much more
efficient well completion.
Recovery Data
 When pumping is halted, water levels rise towards their
pre-pumping levels.
 The rate of recovery provides a second method for
calculating aquifer characteristics.
 Monitoring recovery heads is an important part of the
well-testing process.
 Observation well data (from multiple wells) is
preferable to that gathered from pumped wells.
 Pumped well recovery records are less useful but can be
used in a more limited way to provide information on
aquifer properties.
Recovery Curve
𝒔𝒓
0
Drawdown (m)
2
4
6
Recovery 10 m
Drawdown 10 m
Pumping
Stopped
8
10
12
-6
0
6
12
18
24
30
36
42
48
54
60
66
72
Time (hrs)
The recovery curve on a linear scale appears as an
inverted image of the drawdown curve. The dotted line
represent the continuation of the drawdown curve.
Superposition
 The drawdown (s) for a well pumping at a constant rate
(Q) for a period (t) is given by:
𝑄
𝑠 = ℎ𝑜 − ℎ =
𝑊(𝑢)
4𝜋𝑇
𝑟2𝑆
where 𝑢 =
4𝑇𝑡
 The effects of well recovery can be calculated by adding
the effects of a pumping well to those of a recharge well
using the superposition theorem.
 The drawdown (sr) for a well recharged at a constant rate
(-Q) for a period (t’ = t - tr) starting at time tr is given by:
𝑠𝑟 =
−𝑄
𝑊(𝑢′ )
4𝜋𝑇
𝑠𝑟 is residual drawdown
where
𝑢′ =
𝑟2𝑆
4𝑇𝑡′
Superposition
Drawdown 10 m
Recovery 10 m
Pumping
Stopped
−𝑄
′
𝑠𝑟 =
𝑊(𝑢 )
where
4𝜋𝑇
 The total drawdown for t > tr is:
′
𝑢 =
𝑟2𝑆
4𝑇𝑡′
𝑄
𝑠 ′ = 𝑠 + 𝑠𝑟 =
𝑊 𝑢 − 𝑊 𝑢′
4𝜋𝑇
Residual Drawdown and Recovery
 The Cooper-Jacob approximation can be applied giving:
𝑄 𝑙𝑛 2.25𝑇𝑡
𝑙𝑛 2.25𝑇𝑡′
𝑠 = 𝑠 + 𝑠𝑟 =
−
2
4𝜋𝑇
𝑟 𝑆
𝑟2𝑆
′
 Simplification gives the residual drawdown equation:
𝑄
𝑡
s’ = s + sr =
𝑙𝑛
4𝜋𝑇
𝑡′
 The equation predicting the recovery is:
−𝑄
2.25𝑇𝑡′
𝑠𝑟 =
𝑙𝑛
4𝜋𝑇
𝑟2𝑆
For t > tr, the recovery sr is the difference between the
observed drawdown s’ and the extrapolated pumping
drawdown (s).
Time-Recovery Graph
Est. recovery, s - s' (m)
0.00
2.00
4.00
to’ = 0.12 hrs
6.00
Dsr = 4.6 m
8.00
10.00
12.00
1.E-02
1.E-01
1.E+00
1.E+01
1.E+02
Time after pumping stopped, t' (hrs)
Aquifer characteristics can be calculated from a log(time)recovery plot but the drawdown (s) curve for the pumping
phase must be extrapolated to estimate recovery (s - s’)
Time-Recovery Analysis
 For a constant rate of pumping (Q), the recovery any time
(t’) after pumping stops:
𝑄
−𝑄
𝑇=
=
4𝜋∆ 𝑠 − 𝑠′
−4𝜋∆𝑠𝑟
 For the example, Dsr = 4.6 m and Q = 1100 m3/d so:
T = 1100 / (12.56 x 4.6) = 19 m2/d
 The storage coefficient can be estimated for an observation
4𝑇𝑡𝑜 ′
well (r = 30 m) using: 𝑆
=
𝑟2
 For the example, to’ = 0.12 and Q = 1100 m3/d so:
S = 4 x 19 x 0.12 / (24 x 30 x 30) = 4.3 x 10-4
 It is necessary to use an observation well for this
calculation because well bore storage effects render any
calculation based on rw potentially subject to huge errors.
Time-Residual Drawdown
Graph
Residual Drawdown, s' (m)
0
2
4
6
Ds’ = 5.2 m
8
10
12
1.E+00
1.E+01
1.E+02
1.E+03
Time ratio, t/t'
Transmissivity can be calculated from a log(time ratio)residual drawdown (s’) graph by determining the gradient.
For such cases, the x-axis is log(t/t’) and thus is a ratio.
Time-Residual Drawdown Analysis
 For a constant rate of pumping (Q), the recovery any time
𝑄
(t’) after pumping stops:
𝑇=
4𝜋∆𝑠′
 For the example, Dsr = 5.2 m and Q = 1100 m3/d so:
T = 1100 / (12.56 x 5.2) = 17 m2/d
 Notice that the graph plots t/t’ so the points on the LHS
represent long recovery times and those on the RHS short
recovery times.
 The storage coefficient cannot be estimated for the
residual drawdown plot because the intercept t / t’  1 as
t’  .
 This more obvious, remembering t’ = t - tr where tr is the
elapsed pumping time before recovery starts.
Residual Drawdown for Real
Aquifers
Residual Drawdown, s' (m)
0.00
2.00
4.00
6.00
8.00
10.00
12.00
1.E+00
1.E+01
1.E+02
Time ratio, t/t'
 Theoretical intercept is 1
 >> 1 indicates a recharge effect
 >1 may indicate greater S for pumping than recovery
?consolidation
 < 1 indicates incomplete recovery of initial head - finite
aquifer volume
 << 1 indicates incomplete recovery of initial head - small
aquifer volume
DST
 The drill stem test, used widely in
Drill
Stem
Valve
Packer
Perforated
Section
Packer
Gauge
petroleum engineering, is a recovery
test.
 Packers are used to isolate the HSU of
interest which has been flowing for
some time.
 Initially the bypass valve is open
allowing free circulation.
 When the bypass valve is closed, the
formation pressure is “shut-in” and
begins to recover towards the static
value.
 The Horner plot is a direct analogue of
the residual drawdown plot.
DST Analysis
 Recall that the final form of the recovery equation is:
2.3𝑄
𝑡
s’ = s + sr =
𝑙𝑜𝑔
4𝜋𝑇
𝑡′
 For a DST, the pressure (rather than head) is measured:
2.3𝑄𝜇
𝑡
𝑝𝑜 − 𝑝=
𝑙𝑛
4𝜋𝐾𝑏
𝑡′
 Remembering that p = 𝜌h, T = Kb and K = k𝜌/m
• The Horner-plot has an intercept
po when t / t’ = 1
• This intercept is taken to be the
static formation pressure.
• K can be estimated from the
p (kPa)
po
Dp
100
10
t / t’
1
gradient of the graph: 𝐾 =
2.3𝑄𝜇
4𝜋𝑏∆𝑝
Slug Test
Displaced
Head
Displacer
 The recovery test in a borehole after
withdrawal or injection (or
displacement) of a known volume of
water is called a slug test.
 The slug test is a rapid field method for
estimation of moderate to low K-values
in a single well.
 The procedure is:
 initial head is noted
 the slug is removed, added or displaced
Initial
Head
instantaneously (displacement is best in
this respect)
 head recovery is monitored (usually with
a submerged pressure logging device)
 typical head changes are 2-3 m in 25-50
mm diameter piezometers so the volume
of the slug is typically only 1-10 liters
Slug
Analysis
Tube or
Casing
 ra is access tube internal radius
 rw is perforated section external radius
L
is length of perforated section
 ho is initial head, t = to
 h(t) is head after recovery time t
A
is the tube or casing open area = pra2
F
is a shape factor = 2pL/ln(L/rw)
 Analysis methods include:
 Hvorslev (1951)
2ra
 Cooper et al (1967)
 The Cooper analysis considers storage but the
Hvorslev analysis is more widely used.
L
2rw
𝟐
𝑨
𝒉
𝝅𝒓𝒂
𝑲=
𝒍𝒏
=
𝟐𝝅𝑳
𝑭 𝒕 − 𝒕𝒐
𝒉𝒐
𝑳 𝒕 − 𝒕𝒐
𝒍𝒏
𝒓𝒘
𝑳
𝒉
𝒉
𝒓𝒘
𝒍𝒏
=
𝒍𝒏
𝒉𝒐 𝟐𝑳 𝒕 − 𝒕𝒐
𝒉𝒐
𝒓𝒂 𝟐 𝒍𝒏
Hvorslev
Analysis
𝒓𝒂 𝟐
𝑳
𝒉
𝑲=
𝒍𝒏
𝒍𝒏
𝟐𝑳 𝒕 − 𝒕𝒐
𝒓𝒘
𝒉𝒐
 Plot time against log
Tube or
Casing
(h/ho)
 Measure basic time lag
To when ln(h0/h) = 1
𝑲 =
2ra
L
2rw
𝟐
𝒓𝒂
𝑳
𝒍𝒏
𝟐𝑳𝑻𝒐
𝒓𝒘
 Time lag To occurs
when: h = e-1ho =
0.37ho
 If To = 1000 sec. for a
50 mm diameter × 1 m
length Casagrande
piezometer with 38
mm diameter access
tube K = 2 x 10-6 m/s
1.0
0.9
0.8
0.7
0.6
0.5
h 0.4
ho
To
0.3
0.2
0.1
Time, t - to
Bounded Aquifers
Superposition was used to calculate well recovery
by adding the effects of a pumping and recharge
well starting at different times.
Superposition can also be used to simulate the
effects of aquifer boundaries by adding wells at
different positions.
For boundaries, the wells that create the same
effect as a boundary are called image wells.
This relatively simple application of
superposition for analysis of aquifer boundaries
was for described by Ferris (1959)
Image Wells
 Recharge boundaries at
distance (r) are simulated
by a recharge image well at
an equal distance (r) across
the boundary.
r
r
 Barrier boundaries at
distance (r) are simulated by
a pumping image well at an
equal distance (r) across the
boundary.
r
r
General Solution
ri
r
rp
r
The general solution for adding
image wells to a real pumping well
can be written:
𝑄
𝑠 = 𝑠𝑝 − 𝑠𝑖 =
𝑊 𝑢 𝑝 ± 𝑊 𝑢𝑖
4𝜋𝑇
where: 𝑢𝑝 =
𝑟𝑝 2 𝑆
4𝑇𝑡
and 𝑢𝑖 =
𝑟𝑖 2 𝑆
4𝑇𝑡
and rp,ri are the distances from the
pumping and image wells
respectively.
 For a barrier boundary, for all points on the boundary rp = ri
and the drawdown is doubled.
 For a recharge boundary, for all points on the boundary rp = ri
and the drawdown is zero.
Specific Solutions
Using the Cooper-Jacob approximation is only
possible for large values of r to ensure that u < 0.05
for all r so the Theis’ well function can be used:
𝑸
𝒓𝒊 𝟐 𝒖
𝑸
𝒔=
𝑾 𝒖 ±𝑾
=
𝑾 𝒖 ± 𝑾 𝜶𝒖
𝟐
𝟒𝝅𝑻
𝒓𝒑
𝟒𝝅𝑻
For the barrier boundary
case: a = (ri/rp)2
and 0<a<1
For the recharge boundary
case: a = (ri/rp)2
and 0<a<1
Multiple Boundaries
A recharge boundary and a
barrier boundary at right
angles can be generated by
two pairs of pumping and
recharge wells.
Two barrier boundaries
at right angles can be
generated by
superposition of an array
of four pumping wells.
r2
r1
r2
r1
Parallel Boundaries
A parallel recharge boundary and a barrier boundary
(or any pattern with parallel boundaries) requires an
infinite array of image wells.
r1
r2
Boundary
Location
 For an observation well at distance r1,
measure off the same drawdown (s),
before and after the “dog leg” on a
log(time) vs. drawdown plot.
 Find the times t1 and t2.
s
 Assuming that the “dog leg” is created
t1
s
t2
by an image well at distance r2 , if the
drawdowns are identical then
W(u1) = W(u2) so u1 = u2.
Thus:
r12S/4Tt1 = r22S/4Tt2
r12t2 = r22t1 and r2 = r1(t2 / t1)½
 The distance r2=the radial distance from
the observation point to the boundary.
 Repeating for additional observation
wells may help locate the boundary.