CHAPTER 4
WELL HYDRAULICS
}{2
2
2
2
2
2
t
hS
z
h
y
h
x
hT
STEADY ONE-DIRECTIONAL FLOW:
02
2
x
h
Law) sDarcy' (from 1
K
VC
x
h
21CXCh
K
VXh
CXh
0 ; 0at 0let 2
A. Confined Aquifer
This states the h decreases linearly, with flow in X direction
.
B. Unconfined Aquifer
• Sol. of Laplace equation for unconfined aquifer
not possible.
• WT. in 2D flow represents a flow line
• Shape of WT determines the flow distribution,
but at the same time flow distribution governs
WT shape.
To obtain the solution, Dupuit Assumptions --
sin 5 ° 0.0872
0.3%
tan 5 ° 0.0875
sin 10 ° 0.1737
1.6%
tan 10 ° 0.1763
sin 20 ° 0.3420
6.4%
tan 20 ° 0.3640
tan sinor dx
dh
ds
dh
ds
dh
dx
sin = tan
1. Velocity of flow is proportional to the tangent of
hyd. grad.
dx
dhKhq
C2XΚh
2q
0at 0 Xhh
2
20h
KC
)(2
)(2
22
0
2
0
2
hhx
Kq
hhx
Kq
2. Flow is horizontal and uniform in a vertical section.
Flux per unit width at a section
If
• This indicates W.T. of parabolic form.
• Dupuit assumptions become increasingly
poor approximations to actual flow.
• Actual W.T. deviates more and more from
computed W.T. in the flow direction.
• W.T. actually approaches the boundary
tangentially above water surface and forms a
seepage face.
sin = tan
This indicates that W.T. is not of parabolic
form; however, for flat slopes, where
It closely predicts W.T. position except near the
outflow.
STEADY RADIAL FLOW TO A WELL:
A. Confined Aquifer
When well is pumped, water is removed from
aquifer surrounding the well and W.T. or P.S.
lowered depending upon the type of aquifer.
Drawdown - Distance the water
level is lowered.
Cone of Depression - 3D
Area of Influence - 2D
Radius of Influence - 1D
Assumptions for Well Flow Equations
bottomaquifer above head cPiezometri h
1. Const. Discharge
2. Fully Penetrating Well
3. Homogeneous, isotropic, horz. aquifer with
infinite horz. extent
4. Water released immediately from aquifer storage
due to W.T. or P.S. decline
For Infinite Aquifer
Value of h must be measured in steady
state condition only. Not a very
practical method of determining K.
t
wo
wo
w
rr
hhTQ
hh
hh
/ln2
02
1
Equilibrium Equation (Thiem Equation) Valid
within the radius of influence
)(2
)/ln(
12
12
hhb
rrQK
h
ln r
(Original P.S.)
h
B. Unconfined Aquifer
dr
dhKrhQ )2(
)/ln(
)(
2
12
2
1
2
2
2
1
2
1
rr
hhKQ
hdhKr
drQ
h
h
r
r
If h is constant, i.e., steady state cond. -
Av. thickness
If aquifer is infinite h2 h0 (orig. static water level) and
h1 hw
C. Well Flow in Uniform Recharge
Equilibrium cond. or steady state cond. can be
reached in unconfined aquifers due to recharge
from rainfall or irrigation.
• Uniform Recharge Rate
= w cfs/ft2
• Well Flow
• Horizontal flow thru vertical cylinder (r < ro)
• Also, flux q
wrQq 2
dr
dhKrhq )2(
Q = r 0 ² w
r 0 = radius of influence
Integrating and Substituting 2
or
w
Q
in ln term,
and multiplying by K
w
)(2
ln22
0 w
w
o rrK
w
r
r
K
Q
If w known, compute r0 for given Q and , or estimate w if
other parameters known, or estimate if w and other
parameters known.
Note:
)()(2
ln 22
0
22
0 ww
w
o hhrrK
w
r
r
K
Q
),(
0
0
r
wQfr
independent of h and K
D. Well in a Uniform Flow
• P - Stagnation Point
Uniform Flow
stream lines
• Used in Well Head Protection Plan (WHPA)
•Circular area of influence for radial flow becomes
distorted. Wenzel -
D/S & U/Sheads hyd. &
well;fromr distance aat D/S & U/Sgrads. hyd. &
; discharge
du
du
hh
ii
Q
Radial Flow
Q Ki rh z Kri i
z
h h
z
Q Kr i i h h
KQ
r h h i i
u d u d
u d u d
u d u d
( )( )
*( )
( )( )
( )( )
2
2
thicknessaquifer ;2)( bbhhdu
Q
yibK
x
y ) 2(tan
thickness sat. 0h• For unconfined aquifer,
• For confined aquifer,
• Boundary of the flow area -
• Origin at well
b - aquifer thickness
Q – discharge
i - natural piez. slope
K - Perm
• Boundary asymptotically approaches as
Kbi
Qy
Q
Kbiy
x
yx
L2
2
0)tan(
0 ,
Kbi
Qx
2
• Boundary of contributing area extends to stagnation
point P, where
• Boundary equation, Y and X applicable to unconfined
aquifer, replace b by h0 - sat. aquifer thickness, if
drawdown is small compared to aquifer thickness.
E. Flow to Parallel Streams (Drainage Flow or Base Flow)
)(
)(
2
)(
2
)(
][
2222
2222
2222
xaK
whh
hhxaK
w
hhK
xaw
hdhKdxxW
dx
dhKhwx
a
a
a
h
h
a
x
a
• Recharge rate continuously occurring over the area
h
x
Flux to stream
q Khah
x
x a
x a
Unsteady Radial Flow to a Well
influence of area hSQ
KbTt
h
T
S
r
h
rr
h
12
2
• Extensive Confined Aquifer
• Polar coordinate system
}{2
2
2
2
t
hS
y
h
x
hT
o t r h h
o t h h
o
o
as
at
T
Q
r
h r
o lim
2 ÷
>
...]41.431.321.2
ln5772.0[4
)(4
4
4
available) (Tablesfunction well :
432
2
uuuuu
T
Q
uWT
Q
Tt
Srudu
u
e
T
Q
W(u)wherehhs
u
u
o
• Boundary Conditions
r
Application 1. To find the aquifer parameters or formation
constants S & T
2. To determine drawdown for specified Q, S, T, & t
Assumptions
1. Extensive confined aquifer
2. Homogeneous and isotropic aquifer
3. Well penetrates the entire aquifer
4. Well diameter is small
5. Water is removed instantaneously
from storage with decline in head
A. Theis Method
Tt
sru
T
uQWhhs o
4
:where
4
)(
2
gpmd
mdQ
ftgpd
dmdT
s
,or /ft disch;
,or /ft Trans.,
mor ft drawdown,
33
22
Converting to field units
gpd/ft. Trans.,
Function Welless,dimensionl )(
gpm discharge,
ft. drawdown, )(
)(6.114
)(
0
T
uW
Q
shh
uQWT
hhs o
4
14406.114
4
48.787.1
87.12
Tt
Sru
days time,
coeff., dimensionless storage
ft. well, from distance
t
S
r
)(6.114
uWT
Qs
)( log
6.114loglog)1( uW
T
Qs
uS
T
t
r
87.1
2
uS
T
t
r log
87.1loglog)2(
2
test.aor constant are 87.1
and 6.114
S
T
T
Q
Match the two curves. Locate a match point and obtain
all coordinates. Solve for S & T.
S, r2/t
W(u), u
(1) Insert s, W(u), and Q in Eq. (1) ---- T
(2) Substitute r2/t, u, T in Eq. (2) --- S
For metric system:
2
224
,4
as 4
)(4
)(4
r
TtuS
Tt
Sruu
S
T
t
r
uWs
QTuW
T
Qs
For small r and large t, u is small so that series terms
become negligible after the first two terms.
B. Jacob-Cooper Method
Tt
Sru
4
2
u
sr
T
T
Q
Tt
sr
T
Q
rT
sr
T
Qu
T
Qs
eloglnlog3.2ln
781.1
4ln
44781.1ln
4
4ln781.1ln
4 ln5772.0
4
10
2
2
2
125.2
2
Sr
Tto
2
025.2
r
TtS
or
Thus, a plot of s vs. t forms a st. line.
Plot drawdown, s, from an OBS. well against time, t0
Slope of the line gives S & T values.
All parameters constant except t
Sr
Tt
T
Q
btas
Sr
Tt
T
Qs
2
0
2
25.2log
4
3.20
log
25.2log
4
3.2
1
2
12 log4
3.2
t
t
T
Qsss
1log,101
22 t
tt
t
s
QT
T
Qs
4
32
4
32
: From for one log cycle in Eq 1 system Metric S
Field Units
To avoid large errors, u < 0.01 in this method.
2
03.0
264
r
TtS
s
QT
3.048.7
25.2
2644A
1440x2.3
gpmQ
gpd/ftT
tcoefficien storage
drawdown zeroat time
timeof cycle logper diffdrawdown
0
S
t
s
Distance - Drawdown Method
2
112 log
528
log
r
r
T
Qsss
r
bas
Theis: s vs r2/t ; t - constant; r- variable
Jacob Method:
Need 3 or more observation wells
or
]13.0
[log528
3.0log
2642
rS
Tt
T
Q
Sr
Tt
T
Qs
s
t.f n, wo dk a er b or e z t a e cn a t s i d
3.0
528
2
or
r
TtS
s
QT
log
r
S
Time - drawdown and distance - drawdown methods
provide S & T values, which should be closely agreeable.
r0
Time - drawdown measurements during pumping and
time-recovery measurements during recovery provide two
sets of data from an aquifer test.
• Values of T & S serve to check calculations during
pumping.
• If an obs. well available, take water level recovery data
to obtain T & S.
• Where no obs. available, water level recovery data
from pumped well used to calculate T only.
Recovery Method
Recovery Method
A. Residual Drawdown Method
• Recovery measured in pumped well
• For small r and large t' , integrals approximated by
first two terms in series.
stopped pumping since time t
began pumping since time t
Find T in pumped well
Drawdown Recovery
)()(44
44
22
uWuWT
Qdu
u
edu
u
e
T
Qhhs
tT
sr
u
Tt
sr
u
o
s´
'4'
4
2
2
t
sru
t
sru
over 1 log cycle, log , log 10 = 1
't
t
tt
s
Q
t
t
hh
QT
o
log4
32
log)(4
32
s
QT
4
32
tt of cycle logper recover levelwater
s
Metric System :
s
QT
264or
Field Units:
S can’t be determined from this method
• Recovery measured in observation well
• Plot , recovery, with
• Use Jacob method
B. Time - Recovery Method
ss t
Field Units - : Metric System –
:
:
ss
QT
264
ss
QT
4
32
2
30
r
tTS o
2
252
r
tTS o
Find S & T in observation well
log
calculated
recovery
(s-s´)
(s-s´)
t
t 0
O O
O O
10 100
Time - Recovery Method and Time-drawdown Method
give close values of S and T.
Leaky Aquifers
Drawdown,
S
confined
leaky
log
t
• Due to recharge the top of the curve is flat.
Hantush & Jacob Method for Leaky Aquifers
Q
GS
WT
Aquitard
Leaky
Aquifer
b
h
s
PS
K
K
b
K K
h o
Determine S, T, K´
Assumptions:
1. Leakage is vertical
2. Leakage Drawdown
3. Water level in upper supply aquifer is constant
4. W.T. & P.S. are initially same
BruWT
Qshho / ,
6.114
aquiferleaky afor function well/ , BruW
Tt
Sru
287.1
'/'/ bKT
r
B
r
,
Field Units
K´ = vertical hydraulic gradient of aquitard
b´ = thickness of aquitard
Field Units Metric System
:
:
)/,(6.114
BruWT
Qs )/,(
4BruW
T
Qs
uT
Srt
187.1 2
uT
Srt
12
log
log S
t
•Superimpose the s - t curve on well function curve.
•Select a match point and get BrBruWts /&/, ,u
1 , ,
2
2
2
/'
87.1
)/,( 6.114
r
BrbTK
r
TutS
BruWs
QT
Unconfined Aquifers
oh
hh
2
2
0
• Exact solution difficult because:
- T varies w/ r and t with decline of W.T.
- vertical flow component significant, especially near
well casing.
• If s is small compared to , Theis or Jacob solutions can
be used for unconfined aquifers.
• Jacob suggested that more accurate values of S & T obtained by
subtracting from each drawdown, s.
Bolton Equation
oh
rr
where Ck - correction factor
- well function (table available for )
ohs 5.0
),(12
rtVCkh
Qs
k
o
),( rtV
- varies - 0.30 to 0.16
ho – max. sat. thickness at ro
• For larger drawdowns,
rt ,
oSh
Ktt
kC
varies ,05.0
,505.0
k
k
Ct
oCt
aquifers unconfined in
interest much ofnot
& pumping early
to refers 5t
table or graph available
)( ,75 rfCtk
where (Table or curve available)
minor significance, no eqn.
w
o
o
oiwr
hm
kh
Qhh
t
ln2
,505.0
)(tfm
05.0t
,5.1o
hr >At effect of vertical seepage are negligible
5.122 ln
,75
w
oiwsr
Kt
K
Qhh
t
Unconfined Aquifer
GS
WT S
Observation
Well Q
Delayed drainage
When a well is pumped, water continuously withdraws
from storage within the aquifer as cone of depression
progresses radially outward from the well.
• Since no recharge source is there, no steady-state flow,
and head will continue to decline as long as aquifer is
infinite.
• However, rate of decline of head decreases as cone of
depression spreads.
S,
Storage
Coeff.
T, time of pumping
• Water is released from storage by gravity drainage of
pores in the portion of the aquifer drained by pumping and
by the compaction of aquifer and the expansion of water.
• Gravity drainage of water from sediments is not
immediate; S varies and increases at a diminishing rate
with time.
• First, water is released instantaneously from storage by
compaction of aquifer and expansion of water.
• After a short time, cone of depression grows at a slow
rate as water is released from storage by gravity
drainage reaches the cone.
• Finally, rate of cone expansion increases and cone
continues to expand as gravity drainage keeps pace
with declining water levels.
S – t Curve for Delayed Drainage
log
log
t
S
Well Flow near Aquifer Boundaries
Image
Finite Aq. Infinite Aq.
•Impermeable or negative boundary
•Permeable or positive boundary
Solution of boundary problem in well flow is simplified by
applying the method of images.
Image - an imaginary well introduced to create a hyd flow
system which will be equivalent to the effects of a known
flow system.
a. Well near a stream - Permeable Boundary
• This system is converted to a discharging real well
and a recharge imaginary well in an extensive aquifer.
b. Well near an Impermeable Boundary
c. Aquifer Bounded by Two Impermeable
Boundaries
• I1 and I2 provide required flow, but I3 required balance
drawdowns along the extensions of the boundaries.
d. Impermeable Boundary to a stream
Determination of a Boundary
Discharge.
Real Well
r 3
r 2 r 1
r 4
Discharge.
Image Well a
Obs. Well 1
Obs. Well 2
-Need 2 or more observation wells
Obs. well 3
Tt
Sr
uWuWT
Q
hhhhhh oTo
2
21
201
87.1u
6.114
)()()(
Assume the wells are pumped individually. At a given
time interval
)ln5772.0(ln5772.0
)()(
21
21
21
uu
uwuw
hhhh oo
or u1 = u2
3
2
3
2
2
2
1
2
1
t
r
t
r
t
r
1
212
t
trr
Where t1 - time since pumping began for a given value of
(ho - h) to occur, before the boundary becomes effective.
t2 -time since pumping began, after the boundary
becomes effective, when the divergence of the
drawdown curve caused by the influence of image
well = to particular value of drawdown at t1.
u r2/t
Rate-of-Rise Techniques
Special Techniques:
• Determine local K around a well, without pumping the
well.
Rate-of-Rise Techniques
• Slug Test
• Auger-Hole Method
• Piezometer Method
• Water is suddenly removed by a bucket, bailer, or
cylinder, causing sudden lowering of water levels
around the well.
• Rise of water level with time is measured and K is
obtained.
• Remove enough water to lower water in the well 10 to
50 cm.
Advantages
1. Pumping not needed.
2. Observation wells not required.
3. Tests completed in short time.
4. Provides good preliminary estimate of K.
5. Test useful where continuous Q is difficult, where obs. wells not available, and where interference from other wells.
Disadvantages
1. K measured on small area of aquifer.
2. S generally not evaluated.
Step - Type Pumping Test
Rorabaugh (1953) AGU Tran.
Sternberg (1967) J. Groundwater
211
1o
21
Q Q
...)()()(
tttt
hhhhhh ooo
Q 0
Q 1
Q 2
Q 3
t 1
t 2
t 3
t
Time, t
Discharge, Q
Discharge increased in steps of time. Theis Eq.
Partially Penetrating Wells
Q p
L e
D
2 r w
• A well having length of water entry less than the aquifer is
known as partially penetrating well.
• Flow pattern to such wells differs from radial flow around
fully penetrating wells.
Q p
L e
Q p
L e
• If , then
and if , then
where Q - well discharge
- drawdown at the well
- refers to P.P.W.
QQp hh
p>
hhp
• Average length of flow line in a P.P.W. > that in F.P.W. so
that a greater resistance to flow is encountered. Consider two
wells – P.P.W. and F.P.W.
QQp
h
p
Sp - a dimensionless term,
pwpS
Sr
Tt
T
QS 2
25.2ln
42
wr
D
D
Lef ,
pwo
wop
Srr
rr
Q
Q
/ln
/ln
ratio for P.P.W. > penetration ration
Q
Qb
Drawdown of P.S. at the well
r0= radius of influence
(1)
D
Le
• For screen at top or bottom, use equation 1 and figure
to compute .
• For screen at center, use for obtaining .
• Example:
'20
'50
influence) of (radius 2000 ell;diameter w "122
Le
D
ftrr ow
Q
Qb
2
LepS
10
500 20. = 10 (at center)
45.01029.8
29.8
62.0529.8
29.8
5)4000ln(
)4000ln(
540.050
20 ;100
2/1
50
Q
Q
SD
Le
r
D
p
p
w
Well Losses
ewiwSSS
Drawdown at a well = Aquifer drawdown and drawdown
caused by flow thru well screen and
flow inside the well to pump intake.
Total head loss = formation loss + well loss
Laminar flow Turbulent flow
Since flow in aquifer is laminar,
flow in well screen is turbulent,
where constants
Qsw
but may be > 2 (2 - 4) ,2n
n
iw
n
wfiw
CQBQs
QCQCs
),(, , CBCC Wf
n
w Qs
)/ln(2
1
ln2
wof
n
w
w
oiw
rrT
BC
QCr
r
bK
Qs
• For steady flow in a confined aquifer
• For low Q, well losses may be neglected,
• For high Q, well losses may represent a sizable fraction
of total drawdown.
• For screen size compatible with surrounding porous
media and which is not clogged, well loss caused by
water entering is small than the portion resulting from
axial movement within the well.
iw Drawdown Total
Discharge
) ( capacity specific 1
1
Q f
s
Q
Q C C s
Q n
w f iw
-
Specific Capacity
2;2 nCQBQsiw
• , const. varies depending upon geology
- 0(2000).
Unsteady flow for a confined aquifer
iws
QT Const
n
w
iw CQSr
Tt
T
Qs
2
25.2log
4
3.2
),( tQf
Empirical formulas developed in field
3 cfs
Q = 1 cfs
t, days
iws
Q
Sp. Cap.
•Hence, the concept that
Q ~ siw implying a constant S.C.
Can introduce sizable errors.
1
2
25.2log
4
3.2
1
n
w
iw CQSr
Tt
T
s
Q
Multiple Well System
To determine drawdowns (or interference) in a well field.
nnT
nT
QQQDDDD
DDDD
,.., to duepoint theat drawdown,..
Where
...
2121
21
Determine drawdowns at various points from known Q’s
and add them together.
At a point, Total drawdown.
At a distance of 2D from a well, the effect of partial
penetration is negligible on the flow pattern and
drawdown.
D = Average aquifer thickness