1. Unit 04 : Advanced Hydrogeology
Hydraulic Testing

2. 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

3. Pumping Well Terminology
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 ho h s Q

4. 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

5. 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

6. Steady Radial Confined Flowho h s Q r b
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
Rearranging h = - Q r
2pKb r
Integrating h = - Q ln(r) + c
2pKb
BC specifies h = ho at r = R
Using BC ho = - Q ln(R) + c
2pKb
Eliminating constant (c) gives
s = ho – h = Q ln(r/R)
This is the Thiem Equation

7. Steady Unconfined Radial Flowho h s Q r
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
Rearranging hh = - Q r
2pK r
Integrating h2 = - Q ln(r) + c pK
BC specifies h = ho at r = R
Using BC ho2 = - Q ln(R) + c pK
Eliminating constant (c) gives
ho2 – h2 = Q ln(r/R)
This is the Thiem Equation

8. 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:
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

9. 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:
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) pb (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

10. 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)
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

11. Unsteady Radial Confined Flowho h s Q r b
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
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
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

12. 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 t 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 t 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

13. 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 t
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
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

14. Theis Plot : 1/u vs W(u)

15. Theis Plot : Log(t) vs Log(s)

16. Theis Plot : Log(t) vs Log(s)
s=0.17m
[1,1]
Type
Curve
t=51s

17. Theis Analysis 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 m3/s; r = 120 m; td = 51 s and sd = 0.17 m
T = (0.032)/(12.56 x 0.17) = m2/s = 1300 m2/d
S = (0.032 x 51)/(3.14 x 120 x 120 x 0.17) = 2.1 x 10-4

18. where g is Euler’s constant (0.5772)
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 pT Tt pT r2S
s = ho - h = Q ln( 2.25Tt ) = 2.3 Q log( 2.25Tt )
4pT r2S pT r2S
The Cooper-Jacob simplification expresses drawdown (s) as a linear function of ln(t) or log(t).

19. Cooper-Jacob Plot : Log(t) vs s

20. Cooper-Jacob Plot : Log(t) vs s
to = 84s
Ds =0.39 m

21. Cooper-Jacob Analysis
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
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 m3/s; r = 120 m; to = 84 s and Ds = 0.39 m
T = (2.3 x 0.032)/(12.56 x 0.39) = m2/s = 1300 m2/d
S = (2.3 x x 84)/(1.78 x 3.14 x 120 x 120 x 0.39) = 1.9 x 10-4

22. 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” behaviour and deviations from the Theis and Cooper-Jacob analytical solutions for radial unsteady flow

23. Recharge Effect : Recharge > Well Yield
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).

24. Recharge Effect : Recharge < Well Yield
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.

25. 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.

26. Recharge Effect : Leakage Rate
High Leakage
Low Leakage
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.

27. Hantush Type Curves Theis Curve
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.

28. Barrier Effect : No Flow Boundary
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.

29. 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.

30. 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

31. tc = 3.75p(dc2 – dp2) / (Q/s) = 15 Va /Q
Casing Storage Q
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 s dp dc

32. s = 2.3 Q log( 2.25Tt ) = 2.3 Q log( 2.25Tt ) – 4.6Q log(r)
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 r2S pT S pT
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 distance-axis.
T = 4.6Q and S = 2.25Tt
4pDs ro2

33. Distance-Drawdown Graph

34. Aquifer Characteristics
For the example: t = 0.35 days and Q = 1100 m3/d
T = 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.

35. 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.

36. Theoretical Pumped Well Drawdown

37. 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.

38. 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

39. 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.

40. Determining ro
Unconfined Well
Confined Well

41. 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)

42. Delayed-Yield Response
Unconfined response is complex with theory developed by Boulton, Dagan, Steltsova, Rushton and Neuman. uA uB h
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.

43. Neuman Type Curves
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.

44. Partial Penetration
Partial penetration effects occur when the intake of the well is less than the full thickness of the aquifer

45. 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).

46. 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 b 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 (1/200)} = 0.67
Other factor are provided by Muskat (1937), Hantush (1964), Huisman (1964), Neuman (1974) but they are harder to use.

47. Partial Penetration Alternative
Multiple screened sections distributed over the entire saturated thickness functions more efficiently for the same open area.

48. 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.

49. 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.

50. Recovery Curve
Drawdown 10 m
Recovery 10 m
Pumping Stopped
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.

51. 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 Tt
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 Tt’
The total drawdown for t > tr is:
s’ = s + sr = Q (W(u) - W(u’))
4pT

52. 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 r2S r2S
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 r2S
For t > tr, the recovery sr is the difference between the observed drawdown s’ and the extrapolated pumping drawdown (s).

53. Time-Recovery Graph to’ = 0.12 hrs Dsr = 4.6 m
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’)

54. 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’) pDsr 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’ = 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.

55. Time-Residual Drawdown Graph
Ds’ = 5.2 m
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.

56. 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.

57. Residual Drawdown for Real Aquifers
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

58. DST
The drill stem test, used widely in 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.
Drill
Stem
Valve
Packer
Perforated
Section
Packer
Gauge

59. 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
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
t / t’ po
p (kPa) Dp

60. Slug Test
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 mm dia. piezometers so the volume of the slug is typically only 1-10 litres
Displacer
Displaced
Head
Initial
Head

61. Slug Analysis 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)
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
Tube or
Casing
2ra L
2rw

62. Hvorslev Analysis K = ra2 ln(L) ln(h) 2L(t - to) rw ho
Plot time against log (h/ho)
Measure basic time lag To when ln(h0/h) = 1
K = ra2 ln (L)
2LTo rw
Time lag To occurs when: h = e-1ho = 0.37ho
If To = 1000 secs for a 50 mm dia. x 1 m length Casagrande piezometer with 38 mm dia access tube K = 2 x 10-6 m/s
Tube or
Casing
2ra
2rw L
0.6
0.5
0.4
0.3
0.2
0.1
Time, t - to h ho To

63. 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)

64. Image Wells
Recharge boundaries at distance (r) are simulated by a recharge image well at an equal distance (r) across the boundary.
Barrier boundaries at distance (r) are simulated by a pumping image well at an equal distance (r) across the boundary. r

65. s = sp  si = Q [W(u)  W(ui)]
General Solution
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 Tt
and rp,ri are the distances from the pumping and image wells respectively. r rp ri
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.

66. 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 pT
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

67. 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 r2 r1 r1

68. 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

69. 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.
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. t1 t2 s

70. 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

71. st = s + sw = Q W(u) + CQ2 = BQ + 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 = =
st pTCQ / [ln (2.25Tt /S) - 2 ln(rw)] 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 =
st B + CQ

72. Step-Drawdown Test
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. s1 s2 s3
Drawdown, s s4 s5
Time, t

73. Step-Drawdown Test Analysis
Step-drawdown tests are analysed by plotting the reciprocal of specific capacity (s/Q) against the pumping rate (Q).
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 = m/m3/d
If C = 4 x 10-5, then CQ2 = 18.2 m
The well efficiency is 33.3/( ) = 65% C
s/Q (m/m3/d) B
Q (L/s)

74. Well Yield
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.

75. 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 steady-conditions prevail
Survey elevations of all well measurement reference points

76. 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

77. 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) 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

78. 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

79. Measuring Drawdown Pumped wells Observation 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

80. 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)