Unit 04 : Advanced Hydrogeology
Hydraulic Testing
Well Hydraulics
• A water well is a hydraulic structure that is
designed and constructed to permit economic
withdrawal of water from an aquifer
• Water well construction includes:
– Selection of appropriate drilling methods
– Selection of appropriate completion materials
– Analysis and interpretation of well and aquifer
performance
Pumping Well Terminology
Q
s
ho
h
• Static Water Level [SWL]
(ho) is the equilibrium water
level before pumping
commences
• Pumping Water Level
[PWL] (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
Aquifer Characteristics
• Pump tests allow estimation of transmission and
storage characteristics of aquifers
• Transmissivity (T = Kb) is the rate of flow through a
vertical strip of aquifer (thickness b) of unit width
under a unit hydraulic gradient
• Storage Coefficient (S = Sy + Ssb) is storage
change per unit volume of aquifer per unit change in
head
• Radius of Influence (R) for a well is the maximum
horizontal extent of the cone of depression when the
well is in equilibrium with inflows
Steady Radial Confined Flow
Q
r
s
h
b
ho
• Assumptions
Isotropic, homogeneous,
infinite aquifer, 2-D radial flow
• Initial Conditions
h(r,0) = ho for all r
• Boundary Conditions
h(R,t) = ho for all t
• Darcy’s Law Q = -2prbKh/r • Using BC ho = - Q ln(R) + c
2pKb
• Rearranging h = - Q r
2pKb r
• Eliminating constant (c) gives
• Integrating h = - Q ln(r) + c
s = ho – h = Q ln(r/R)
2pKb
2pKb
• BC specifies h = ho at r = R
This is the Thiem Equation
Steady Unconfined Radial Flow
Q
r
s
h
ho
• Assumptions
Isotropic, homogeneous,
infinite aquifer, 2-D radial flow
• Initial Conditions
h(r,0) = ho for all r
• Boundary Conditions
h(R,t) = ho for all t
• Darcy’s Law Q = -2prhKh/r • Using BC ho2 = - Q ln(R) + c
pK
• Rearranging hh = - Q r
2pK r
• Eliminating constant (c) gives
• Integrating h2 = - Q ln(r) + c
ho2 – h2 = Q ln(r/R)
2
2pK
pK
• BC specifies h = ho at r = R
This is the Thiem Equation
Thiem Equation Assumptions
• The equation for unconfined flow can be rearranged to give:
s = ho – h = Q ln(r/R)
2pK (ho+ h)/2
• Compare this with the confined equation:
s = ho – h = Q ln(r/R)
2pKb
• It is clear than the only difference is that the aquifer thickness b
is replaced by (ho + h)/2
• The implicit assumption in the derivation (2-D flow) implies that s
is small compared with ho and that the (ho + h)/2 does not
deviate significantly from ho
• This assumption may not be valid in the immediate vicinity of a
pumping well in an unconfined aquifer
Thiem Equation Applications
• The equation for unconfined flow can be rearranged to give:
K = Q ln(r/R)
p (ho2 - h2)
• Similarly with the confined equation:
K = Q ln(r/R)
2pb (ho – h)
• The radius of influence R is hard to estimate but any two wells
at different radial distance can be used in the equations
K = Q ln(r2/r1)
and
K = Q ln(r2/r1)
p (h22 – h12)
2pb (h2 – h1)
• This means that for a well producing at a steady rate (Q) with a
steady drawdown, any pair of observation points at different
radial distances can be used to estimate K
Specific Capacity
•
•
•
•
•
For a confined well producing at a steady rate (Q) the specific capacity
is given by:
Q = 2pKb
sw ln(rw/R)
This means that for a confined well producing at a steady rate (Q) the
specific capacity is constant.
The equation for unconfined flow can be rearranged:
Q = pK (ho + hw)
sw ln(rw/R)
Writing hw = ho - sw gives:
Q = - pK (sw + 2ho)
sw
ln(rw/R)
For an unconfined well producing at a steady rate (Q) the specific
capacity reduces with increasing drawdown. The maximum specific
capacity for an unconfined well is given by:
Q = 2pKho
sw ln(rw/R)
Unsteady Radial Confined Flow
Q
r
s
h
ho
b
• PDE 1  (rh ) = S h
r r r
T t
• Solution is more complex than
steady-state
• Change the dependent
variable by letting u = r2S
4Tt
• Assumptions
Isotropic, homogeneous,
infinite aquifer, 2-D radial flow
• Initial Conditions
h(r,0) = ho for all r
• Boundary Conditions
h(,t) = ho for all t
• The ultimate solution is:
ho- h = Q  exp(-u) du
4pT u u
where the integral is called the
exponential integral written as
the well function W(u)
This is the Theis Equation
Theis PDE to ODE
•
•
•
•
•
•
Let a = S/T (to simplify notation
where a is called the hydraulic
diffusivity)
PDE 1  (rh ) = a h
r r r
t
u = r2S = ar2
4Tt 4t
So u = ar = 2u
r 2t
r
And u = -ar2 = -u
t
4t2
t
Rewriting the PDE in terms of u:
1 d (r u dh ) u = a u dh
r du r du r
t du
•
•
•
•
Rewriting partial derivatives in
terms of u:
1 d (r 2u dh ) 2u = -a u dh
r du r du r
t du
Rearranging and cancelling:
d (u dh ) = -ar2 dh = -u dh
du du
4t du
du
Expanding the LHS derivative:
u d (dh ) + dh = -u dh
du du
du
du
d (dh ) = -(u + 1) dh
du du
u du
Writing dh/du as h’ gives an ODE:
dh’ = -(u + 1) h’ or dh’ = -(1 + 1) du
du
u
h’
u
Theis Integration
•
•
•
•
•
•
•
The resulting ODE is:
dh’ = -(1 + 1) du
h’
u
Integrating: ln(h’) = -u – ln(u) + c
Simplifying: ln(h’u) = c – u
Inverting: h’u = exp(c).exp(-u)
To eliminate exp(c), use Darcy’s
Law:
lim rh = -Q = lim 2h’u
r  0 r 2pKb u 0
Remember
rh = rh’u = h’ar2 = 2h’u
r
r
2t
lim h’u = -Q = exp(c)
u 0 4pKb
Simplifying: h’ = -Q exp(-u)
4pT u
•
•
•
•
Recall h’ = dh/du
dh = -Q exp(-u).du
4pT u
Integrating:
h = -Q  exp(-u) du + C
4pTu u
Finally, using h(,t) = ho to
eliminate C:
ho - h = Q  exp(-u) du
4pTu u
The integral is called the
exponential integral but is often
written as the Theis well
function W(u)
s = ho - h = Q W(u)
4pT
Theis Plot : 1/u vs W(u)
10.0
W(u)
1.0
0.1
0.0
1.E-01
1.E+00
1.E+01
1/u
1.E+02
1.E+03
Theis Plot : Log(t) vs Log(s)
Drawdown (m)
10.0
1.0
0.1
0.0
1.E+01
1.E+02
1.E+03
Time since pump started (s)
1.E+04
1.E+05
Theis Plot : Log(t) vs Log(s)
Drawdown (m)
10.0
s=0.17m
1.0
0.1
0.0
1.E+01
1.E+02
1.E+03
Time since pump started (s)
[1,1]
Type
Curve
t=51s
1.E+04
1.E+05
Theis Analysis
1.
2.
3.
4.
5.
6.
7.
Overlay type-curve on data-curve keeping axes parallel
Select a point on the type-curve (any will do but [1,1] is
simplest)
Read off the corresponding co-ordinates on the data-curve
[td,sd]
For [1,1] on the type curve corresponding to [td,sd], T = Q/4psd
and S = 4Ttd/r2 = Qtd/pr2sd
For the example, Q = 32 L/s or 0.032 m3/s; r = 120 m; td = 51
s and sd = 0.17 m
T = (0.032)/(12.56 x 0.17) = 0.015 m2/s = 1300 m2/d
S = (0.032 x 51)/(3.14 x 120 x 120 x 0.17) = 2.1 x 10-4
Cooper-Jacob
• Cooper and Jacob (1946) pointed out that the series expansion
of the exponential integral or W(u) is:
W(u) = – g - ln(u) + u - u2 + u3 - u4 + ..…
1.1! 2.2! 3.3! 4.4!
where g is Euler’s constant (0.5772)
• For u<< 1 , say u < 0.05 the series can be truncated:
W(u)  – ln(eg) - ln(u) = - ln(egu) = -ln(1.78u)
• Thus: s = ho - h = - Q ln(1.78u) = - Q ln(1.78r2S) = Q ln( 4Tt )
4pT
4pT
4Tt
4pT 1.78r2S
s = ho - h = Q ln( 2.25Tt ) = 2.3 Q log( 2.25Tt )
4pT
r 2S
4pT
r 2S
• The Cooper-Jacob simplification expresses drawdown (s) as a
linear function of ln(t) or log(t).
Cooper-Jacob Plot : Log(t) vs 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 s
to = 84s
0.0
0.1
0.2
Drawdown (m)
0.3
0.4
0.5
0.6
Ds =0.39 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.
3.
4.
5.
6.
7.
Determine intercept on the time axis for s=0
Determine drawdown increment (Ds) for one log-cycle
For straight-line fit, T = 2.3Q/4pDs and S = 2.25Tto/r2 =
2.3Qto/1.78pr2Ds
For the example, Q = 32 L/s or 0.032 m3/s; r = 120 m; to = 84
s and Ds = 0.39 m
T = (2.3 x 0.032)/(12.56 x 0.39) = 0.015 m2/s = 1300 m2/d
S = (2.3 x 0.032 x 84)/(1.78 x 3.14 x 120 x 120 x 0.39)
= 1.9 x 10-4
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 “nonideal” behaviour and deviations from the Theis and CooperJacob analytical solutions for radial unsteady flow
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.
Hantush Type Curves
r/B
0.0
0.05
0.1
0.2
10.0
0.5
Theis Curve
1.0
W(u,r/B)
1.0
2.0
0.1
0.0
1.E-01
1.E+00
1.E+01
1.E+02
1.E+03
1.E+04
1/u
Data are fitted in a manner similar to the Theis curve. The parameter
r/B = r( {K’v / b’} / {Khb} )½ increases with the amount of leakage.
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
Distance-Drawdown
• Simultaneous drawdown data from at least three
observation wells, each at different radial distances,
can be used to plot a log(distance)-drawdown graph.
• The Cooper-Jacob equation, for fixed t, has the form:
s = 2.3 Q log( 2.25Tt ) = 2.3 Q log( 2.25Tt ) – 4.6Q log(r)
4pT
r 2S
4pT
S
4pT
• So the log(distance)-drawdown curve can be used to
estimate aquifer characteristics by measuring Ds for
one log-cycle and the ro intercept on the distanceaxis.
T = 4.6Q and S = 2.25Tt
4pDs
r o2
Distance-Drawdown Graph
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
T = 0.366 x 1100 / 3.8 = 106 m2/d
S = 2.25 x 106 x 0.35 / (126 x 126) = 5.3 x 10-3
• 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
distance-drawdown 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.
Theoretical Pumped Well Drawdown
0
dw = 100 mm
dw = 150 mm
dw = 300 mm
2
6
8
0
10
2
Drawdown (m)
Drawdown (m)
4
12
14
16
4
6
8
10
12
14
18
16
0
20
0.01
10
20
30
40
50
60
Distance (m)
0.1
1 Distance (m) 10
100
1000
Unconfined Well Correction
• The adjusted drawdown for an unconfined well is
given by:
sc = (1 - sa ) sa
2b
where b is the initial saturated thickness;
sa is the measured drawdown; and
sc is the corrected drawdown
• For example, if b = 20 m; sa = 6 m; then the corrected
drawdown sc = 0.85sa = 5.1 m
• If the drawdown is not corrected, the Jacob and Theis
analysis underestimates the true transmissivity under
saturated conditions by a factor of sc/ sa.
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
Radius of Influence
• The radius of influence of a well can be determined
from a distance-drawdown plot.
• For all practical purposes, a useful comparative index
is the intercept of the distance-drawdown graph on
the distance axis.
• Radius of influence can be used as a guide for well
spacing to avoid interference.
• Since radius of influence depends on the balance
between aquifer recharge and well discharge, the
radius may vary from year to year.
• For unconfined wells in productive aquifers, the
radius of influence is typically a few hundred metres.
• For confined wells may have a radius of influence
extending several kilometres.
Determining ro
ro = 126 m
ro = 4.1 km
0
Unconfined Well
1
3
Confined Well
4
0
5
2
6
Drawdown (m)
Drawdown (m)
2
7
8
4
6
8
10
12
14
9
16
0
10
20
30
40
50
60
Distance (m)
10
1
10
100
Distance (m)
1000
10000
Unconfined Aquifers
• Most analytical solutions assume isotropic,
homogeneous, confined aquifers or assume
drawdowns are small for the unconfined case.
• There are three distinct parts to the time drawdown
curve in an unconfined aquifer:
– early time response follows Theis equation with the confined
“elastic” storage corresponding to storativity (bSs)
– intermediate times respond as a leaky aquifer with vertical
flow in the vicinity of the pumped well with storage release
controlled by the aquifer Kh/Kv ratio
– late time response follows Theis equation with gravity
drainage providing storage corresponding to the specific
yield (Sy)
Delayed-Yield Response
uB
uA
h
• Unconfined response
is complex with theory
developed by
Boulton, Dagan,
Steltsova, Rushton
and Neuman.
• Neuman (1975) defines a well function W(uA,uB,h) where each
parameter corresponds to a different time phase:
• early-time response is controlled by uA = r2S/4Tt
• intermediate-times are controlled by h = r2Kv/Khb2
• late-time response is controlled by uB = r2Sy/4Tt
• The Hantush leakage parameter (r/B) is closely related to h
• (r/B)2 = r2Kv’/Khbb’ where the Kv’ and b’ parameters refer to the
leaky bed. If these leaky-bed parameters become aquifer
parameters (r/B)2 = h.
Neuman Type Curves
h
10.0
0.002
0.01
0.04
0.25
1.0
W(uA,uB,h)
1.0
4.0
0.1
0.0
1.E-01
1.E+00
1.E+01
1.E+02
1.E+03
1.E+04
1.E+05
1.E+06
1/u
• The Neuman type curves are fitted to data in a manner similar to
that for Theis curves. Higher values of h indicate more rapid
gravity drainage.
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 flowpaths 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:
F = L {1 + 7 cos(pL) ( r )}
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
• For a 300 mm dia. 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 x {1 + 7 x 0.707 (1/200)} = 0.67
• Other factor 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 dia.
well with single
screened
interval of 15 m
in aquifer of
thickness 30 m.
L/b = 0.5 and 2L/r = 200
F = 0.5 x {1 + 7 x cos(0.5p/2)
(1/200)} = 0.67
300 mm dia.
well with 5 x 3 m
solid sections
alternating with
5 x 3m
screened
sections, in an
aquifer of thickness 30 m.
There effectively are five
aquifers.
L/b = 0.5 and 2L/r = 40
F = 0.5 x {1 + 7 x 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
0
Drawdown (m)
2
4
Drawdown 10 m
6
Recovery 10 m
Pumping
Stopped
8
10
12
-6
0
6
12
18
24
30
36
42
48
54
60
66
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.
72
Superposition
• The drawdown (s) for a well pumping at a constant rate (Q) for a
period (t) is given by:
s = ho - h = Q W(u) where u = r2S
4pT
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’) where u’ = r2S
4pT
4Tt’
• The total drawdown for t > tr is:
s’ = s + sr = Q (W(u) - W(u’))
4pT
Residual Drawdown and Recovery
• The total drawdown for t > tr is:
s’ = s + sr = Q (W(u) - W(u’))
4pT
• The Cooper-Jacob approximation can be applied giving:
s’ = s + sr = Q (ln(2.25Tt) - ln(2.25Tt’))
4pT
r 2S
r 2S
• Simplification gives the residual drawdown equation:
s’ = s + sr = Q ln(t)
4pT t’
• The equation predicting the recovery is:
sr = - Q ln(2.25Tt’)
4pT
r 2S
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:
T=
Q
= -Q = Q
4pD(s - s’)
- 4pDsr 4pDsr
• 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 well
(r = 30 m) using:
S = 4Tto’
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 (t’)
after pumping stops:
T= Q
4pDs’
• 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 petroleum
engineering, is a recovery test.
Drill
Stem
• Packers are used to isolate the HSU of
interest which has been flowing for some
time.
Valve
• Initially the bypass valve is open allowing free
Packer
circulation.
• When the bypass valve is closed, the
Perforated
formation pressure is “shut-in” and begins to
Section
recover towards the static value.
• The Horner plot is a direct analogue of the
Packer
residual drawdown plot.
Gauge
DST Analysis
• Recall that the final form of the recovery equation is:
ho - h = s’ = 2.3Q log(t)
4pT
t’
• For a DST, the pressure (rather than head) is measured
po - p = 2.3Qm log(t)
4pkb
t’
• Remembering that p = gh,T = Kb and K = kg/m
p (kPa)
po
100
Dp
10
t / t’
1
• 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
gradient of the graph:
• k = 2.3Qm
4pbDp
Slug Test
Displacer
Displaced
Head
Initial
Head
• 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
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 dia.
piezometers so the volume of the slug is typically
only 1-10 litres
Slug Analysis
Tube or
Casing
2ra
•
•
•
•
•
•
•
•
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 csa = pra2
F
is a shape factor = 2pL / ln(L/rw)
Analysis methods include:
– Hvorslev (1951)
– Cooper et al (1967)
L
2rw
The Cooper analysis considers storage but
the Hvorslev analysis is more widely used.
• K = A ln (h) = ra2 ln(L) ln(h)
F(t - to) ho 2L(t - to) rw ho
Hvorslev Analysis
• K=
•
Tube or
Casing
2ra
•
•
•
•
L
2rw
ra2 ln(L) ln(h)
1.0
2L(t - to) rw ho
0.9
0.8
Plot time against log
0.7
(h/ho)
0.6
0.5
Measure basic time lag
To when ln(h0/h) = 1
0.4
h
K = ra2 ln (L)
ho
0.3
2LTo rw
Time lag To occurs
when: h = e-1ho = 0.37ho
0.2
If To = 1000 secs for a
50 mm dia. x 1 m length
Casagrande piezometer
0.1
with 38 mm dia access
tube K = 2 x 10-6 m/s
To
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:
s = sp  si = Q [W(u)  W(ui)]
4pT
where up = rp2S and ui = ri2S
4Tt
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 to ensure that u < 0.05 for all r so the Theis well function is
used: s = Q [W(u)  W(ri2u)] = Q [W(u)  W(au)]
4pT
rp 2
4pT
• For the barrier boundary case:
s = Q [W(u) + W(au)]
4pT
where a = (ri/rp)2 and 0<a<1
• For the recharge boundary
case:
s = Q [W(u) - W(au)]
4pT
where 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
s
t1
s
t2
• 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.
• Assuming that the “dog leg” is created
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
So 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.
Pumping Wells
• The drawdown observed in a pumping well has two
component parts:
– aquifer loss
• drawdown due to laminar flow in the aquifer
– well loss
• drawdown due to turbulent flow in the
immediate vicinity of the well through the
screen and/or gravel pack
• Well loss is usually assumed to be proportional to the
square of the pumping rate:
sw = CQ2
Well Efficiency
• The total drawdown at a pumping well is given by:
st = s + sw = Q W(u) + CQ2 = BQ + CQ2
4pT
• The ratio of the aquifer loss and total drawdown (s/st) is known
as the well efficiency.
s=
W(u)
=
B .
st W(u) + 4pTCQ B + CQ
• Mogg (1968) defines well efficiency at a fixed time (t = 24 hrs).
Thus, writing W(u) as the Cooper-Jacob approximation gives:
s=
1
=
1
.
st 1 + 4pTCQ / [ln (2.25Tt /S) - 2 ln(rw)]
1 + CQ/B(rw)
• Written in this form it is clear that well efficiency reduces with
pumping rate (Q) and increases with well radius (rw), where B is
inversely related to well radius.
• The specific capacity is given by: Q =
1 .
st B + CQ
Step-Drawdown Test
s1
s
Drawdown, s
2
s3
s4
s5
Time, t
• Step-drawdown tests are tests at
different pumping rates (Q)
designed to determine well
efficiency.
• Normally pumping at each
successively greater rate Q1 < Q2 <
Q3 < Q4 < Q5 takes place for 1-2
hours (Dt) and for 5 to 8 steps. The
entire test usually takes place in one
day.
• Equal pumping times (Dt) simplifies
the analysis.
• At the end of each step, the
pumping rate (Q) and drawdown (s)
is recorded.
Step-Drawdown Test Analysis
s/Q (m/m3/d)
• Step-drawdown tests are
analysed by plotting the
reciprocal of specific
capacity (s/Q) against the
pumping rate (Q).
C
B
Q (L/s)
• The intercept of the graph at
Q=0 is B = W(u)/4pT and the
slope is the well loss
coefficient, C.
• B can also be obtained
independently from a Theis
or Cooper-Jacob analysis of
a pump test.
• For Q = 2700 m3/d and s =
33.3 m the B = 0.012 m/m3/d
• If C = 4 x 10-5, then CQ2 =
18.2 m
• The well efficiency is
33.3/(33.3+18.2) = 65%
Well Yield
Well yield
US gpm
< 100
< 170
< 350
< 700
< 1000
< 1800
< 3000
< 3800
< 6000
L/s
<6.4
<11
<22
<44
<64
<110
<190
<240
<380
Nom. pump dia. Opt. casing dia. Min. casing dia.
m3/d
550
950
1900
3800
5500
9800
16000
21000
33000
in
4
5
6
8
10
12
14
16
20
mm
100
130
150
200
250
300
360
410
510
in
6
8
10
12
14
16
20
24
30
mm
150
200
250
300
360
410
510
610
760
in
5
6
8
10
12
14
16
20
24
mm
130
150
200
250
300
360
410
510
610
• The chart is used to select casing sizes for a particular yield.
The main constraint is pumping equipment.
• For example, if the well is designed to deliver 4,000 m3/d, the
optimum casing dia. is 360 mm (2 nom. sizes > pump dia.) and
the minimum 300 mm.
• The drilled well diameter would have to be 410 to 510 mm to
provide at least a 50 mm grout/cement annulus.
Pump Test Planning
• Pump tests will not produce satisfactory estimates of either
aquifer properties or well performance unless the data collection
system is carefully and QA/QC is addressed in the design.
• Several preliminary estimates are needed to design a
successful test:
–
–
–
–
–
–
–
Estimate the maximum drawdown at the pumped well
Estimate the maximum pumping rate
Evaluate the best method to measure the pumped volumes
Plan discharge of pumped volumes distant from the well
Estimate drawdowns at observation wells
Simulate the test before it is conducted
Measure all initial heads several times to ensure that steadyconditions prevail
– Survey elevations of all well measurement reference points
Number of Observation Wells
• Number depends on test objectives and
available resources for test program.
– Single well can give aquifer characteristics (T and
S). Reliability of estimates increases with
additional observation points.
– Three wells at different distances are needed for
time-distance analysis
– No maximum number because anisotropy,
homogeneity, and boundaries can be deduced
from response
Pump Test Measurements
• The accuracy of drawdown data and the results of subsequent
analysis depends on:
– maintaining a constant pumping rate
– measuring drawdown at several (>2) observation wells at different
radial distances
– taking drawdowns at appropriate time intervals at least every min
(1-15 mins); (every 5 mins) 15-60 mins; (every 30 mins) 1-5 hrs;
(every 60 mins) 5-12 hrs; (every 8 hrs) >12 hrs
– measuring barometric pressure, stream levels, tidal oscillations as
necessary over the test period
– measuring both pumping and recovery data
– continuing tests for no less than 24 hours for a confined aquifers
and 72 hours for unconfined aquifers in constant rate tests
– collecting data over a 24 hour period for 5 or 6 pumping rates for
step-drawdown tests
Measuring Pumping Rates
• Control of pumping is normally required as head and
pump rpm changes. Frequent flow rate
measurements are needed to maintain constant rate.
• Lower rates
– periodic measurements of time to fill a container of known
volume
– “v” notch weir - measure head (sensitive at low flows)
• Higher rates
–
–
–
–
–
impellor driven water meter - measure velocity (insensitive)
circular orifice weir - measure head v=(2gh)½
rectangular notch weir - measure head
free-flow Parshall flume (drop in floor) - measure head
cutthroat flume (flat floor) – measure head
Measuring Drawdown
• Pumped wells
– heads are hard to measure due to turbulence and pulsing.
– data cannot reliably estimate storage.
• Observation wells
– smallest possible diameter involves least time lag
– screens usually 1-2 m; longer is better but not critical should
be at same depth as centre of production section
– if too close (< 3 to 5 x aquifer thickness) can be strongly
influenced by anisotropy (stratification)
– if too far away (>200 m unconfined) Dh(t) increases with time
so a longer test is required – boundary and other effects can
swamp aquifer response
Drawdown Instrumentation
• Dipmeters
– let cable hang to remove kinks
– rely on light or buzzer, have spare batteries
• Steel tapes
– read wetted part for water level (chalking helps)
– hard to use where high-frequency readings are needed
• Pressure gauges
– measure head above reference point
– need drawdown estimates to set gauge depth
• Pressure tranducers/data loggers
– hang in well and record at predetermined interval
– can be rented (cheaply) for tests
– remote sites (no personnel) and closest wells (frequency)