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