Transient Radial Flow Toward a Well - New Mexico Tech ... · Transient Radial Flow Toward a Well t h T S y h ... Our goal in aquifer testing is to find T and S; ... Transient Flow

1

Aquifer Equation, based on assumptions becomes a 1D PDE for h(r,t) :

-transient flow in a homogeneous, isotropic aquifer

-fully penetrating pumping well & infinite,

horizontal, confined aquifer of uniform

thickness, thus

essentially horizontal groundwater flow

-flow symmetry: radially symmetric flow

Boundary conditions

-2nd order in space PDE for h(r,t) , need two BC’s in space,

at two r’s, say r1 and r2; in general we’ll pick r1=rw, r2=!

-One Dirichlet BC at infinity and one Neuman at well radius, rw ,

where we assume we know the pumping rate, Qw

t

h

K

Sh

s

!

!="

2

Diffu

sio

n e

quations

t

h

T

S

r

h

r

h

r !

!=

!

!+

!

!2

21

Transient Radial Flow Toward a Well

t

h

T

S

y

h

x

h

!

!=

!

!+

!

!2

2

2

2

During a pumping test, we typically measure drawdown (as opposed to

head), so let’s set up the equation in terms of drawdown.

Assume initial potentiometric surface is horizontal everywhere.

( ) shhthhs !="!=00

t

s

t

h

r

s

r

h

!

!"=

!

!

!

!"=

!

! ,

t

s

T

S

r

s

r

s

r !

!=

!

!+

!

!2

21


!

"

2

To solve, we need boundary conditions and initial conditions

I.C.: drawdown is zero, or head is uniform initially. s = 0 @ t ! 0

B.C.s : 2nd order equation, we need two BCs

Use r = 0, r = "

rbA !2=

Transient Flow Well Test Analysis

r

srT

r

sKbrQ

!

!=

!

!= 2 2 ""

r

sKA

r

hKAQr

!

!=

!

!"== ,0At

Use continuity at the pumping well, treated as a line sink (r = 0)


r

srT

r

sKbrQ

!

!=

!

!= 2 2 ""

T

Q

r

sr

2!=

"

"

T

Q

r

sr

r !2lim

0

"#$

%&'

(

)

)"

(requires constant Q)

s # 0 as r # " (for all t)

3


Solution:

duu

e

T

Qtrs

u

u

4

),( !"

#

=$

where

tT

Sru

4

2

=

duu

e

u

u

!"

#

In mathematics: “The exponential integral”

)( 4

uWT

Qs

!=

In hydrogeology: “The well function” duu

euW

u

u

)( !"

#

=

Theis Equation


tT

Sru

4

2

=r: distance from pumping well to observation well

t: time since pumping started

!"

#

$$$$=

i

i

ii

uuuW

!ln)( % $ = Euler’s constant = 0.577215

Values for W(u) for various values of u are also tabulated in

numerous references

4


)( 4

uWT

Qs

!=

tT

Sru

4

2

=

Our goal in aquifer testing is to find T and S; solving the drawdown

equation shown above for T is problematic—T appears twice.

One way to solve for T is to compare a plot of s versus t to a

“normal” graphic solution (when plotting data from numerous

observation wells, we can typically normalize s by plotting it

as a function of t/r2 instead of t)


“Type curve method” for solving the Theis Equation

•Plot s versus t (or s versus t/r2) on log-log paper

•On another sheet of the same paper, plot W(u) versus 1/u

Why 1/u? Our plot has t in the numerator, but in the

equation defining u, t appears in the denominator

•Match data (keep graph axes parallel)

•Pick “match point”—easiest to pick a match point with

simple numbers—the match point does not have to

fall on the curve.

•Solve equation

tT

Sru

4

2

=

5


(Schwartz and Zhang, 2003)


Q = 500 m3/d

r = 300 m


6

m 78.0s min; 22 t;1.01

;1)( o ====o

o

o

uuW


d

m 151

m) 0.78( 4

d

m500

)( 4

2

3

o

=!==""

ouWs

QT

( )

( )2

2

m 300

1.0min 1440

1dmin 22

d

m514

22 4

4!"

#$%

&!!"

#$$%

&

=!"

#$%

&==

o

o

oo

r

tuT

r

utTS

= 3.46 x 10-6

But S is hard to measure—typically should only

be reported to one significant digit id est 3 x 10-6


Assumptions:•Homogeneous, isotropic aquifer (we used a simplified form of the

continuity equation instead of including the tensor for K)

•Constant water properties

•Infinite aquifer (used to develop boundary condition)

•Aquifer is confined; confining units are impermeable

•All flow is radial and horizontal

•Water is withdrawn instantaneously with decline in head

(not typically the case for leakage from confining units or

low k lenses)

•Well diameter is infinitesimal

(water can be stored in the well bore, leading to

‘delayed’ drawdown)

•Constant discharge from well

7

Head of NMT Hydrology Program

• C.E. Jacob (1965-1970)• Hantush’s mentor, well hydraulics,

theory of leaky aquifers

• 3 faculty, including Frank Titus and

William Brutsaert

• merged with Geology and Geophysics

to form Geosciences Department

• died of heart attack in 1970; Hantush

returned briefly

Simplified approach to transient analysis: The Jacob Approximation

A simplifying assumption that makes solving the Theis equation easier

C. E. Jacob (1940) “Jacob approximation” “Jacob-Cooper equation”

“Cooper-Jacob method” “Cooper-Jacob straight-line method”

!"

#

$$$$=

i

i

ii

uuuW

!ln)( %

...!44!33!22

ln577216.0)(432

+!

"!

+!

"+""=uuu

uuuW


8


)(!

0.01, u For uWii

u

i

i

<<!

"=# $

%

tT

Sru

4

2

= So we need t to be large for u to be small

( )uT

Qln577.0

4s 0.01, u For !!="

#

( ) !"

#$%

&''(

)**+

,--=

tT

Sr

T

Q

4ln781.1ln

4s

2

.


( ) !"

#$%

&''(

)**+

,--=

tT

Sr

T

Q

4ln781.1ln

4s

2

.

( )u

u1

lnln =! ( ) uu log 3.2ln =

( ) !"

#$%

&'(

)*+

,+=

Sr

tT

T

Q2

4ln5615.0ln

4s

-

( )!"

#$%

&=

Sr

tT

T

Q2

4 5615.0ln

4s

'

9


( )!"

#$%

&=

Sr

tT

T

Q2

4 5615.0ln

4s

'

( ) uu log 3.2ln =

Jacob-Cooper simplification


Suppose we measure drawdown at two times: t1, t2

!"

#$%

&'()

*+,

-'+(

)

*+,

-='

122212ln

.252lnln

.252ln

4 t

Sr

Tt

Sr

T

T

Qss

.

[ ]1212

lnln 4

ttT

Qss !=!

"

1

2

12ln

4

t

t

T

Qss

!="

10


1

2

12ln

4

t

t

T

Qss

!="

Pick t2 = 10 t1; head change per log cycle = %slc = s2 - s1

T

Q

t

t

T

Qss

4

3.210ln

4

1

1

12

!!=="


.252

ln 4

0,At t 2o !

"

#$%

&==

Sr

Tt

T

Qs oo

'

.252

ln 0 2

!"

#$%

&=

Sr

Tto

2

.252ln

0!"

#$%

&

= Sr

Tto

ee

( )

.252ln

0

2

4

!"

#$%

&=

Sr

Tto

T

Q

'

!

"

11


2

.252ln

0!"

#$%

&

= Sr

Tto

ee

Sr

Tto

2

.2521 =



Q = 500 m3/d

Semi-log plot (drawdown versus log time)

12


lcs 4

3.2

!=

"

QT

( )( ) d

md

m2

3

51m .811 4

500 3.2==

!T

2

.252

r

TtS

o=

!

S =2.25 51

m2

d( ) 3.4min 1 d

1440min( )

300 m( )2

= 3 x 10"6


tT

Sru

4

2

=

Remember that the Jacob-Cooper method is predicated on

the value of u being less than about 0.01!

( ) ( )( )( )

02.0min100 51 4

10 x 3m 300

min1440

d 1

d

m

62

2==

!

u

According to your textbook, 0.02 is “close” to 0.01, and it is

the maximum value of u over our log cycle of data, so the

Jacob-Cooper method is acceptable.

13


Instead of looking at drawdown in one well with time, we can also look

at drawdown measured at the same time in two or more wells:

“Distance-drawdown method”

s1 (r1)

s2 (r2)Measured at the same time t


r1

r2


!"

#$%

&=

Sr

tT

T

Q2

.252ln

4s

'2

2ln

.252ln

.252ln

i

i

rS

tT

Sr

tT!"#

$%&

'=""

#

$%%&

'

!"

#$%

&+'(

)*+

,--'

(

)*+

,=- 2

2

2

121ln

.252lnln

.252ln

4 r

S

tTr

S

tT

T

Qss

.

!

s1" s

2=

Q

4 # T"ln r

1

2 + ln r2

2[ ]

1

2

21ln

4

2

r

r

T

Qss

!="

14

1

2

21ln

4

2

r

r

T

Qss

!="


T

r

rQ

ss 2

ln

1

2

21

!

""#

$%%&

'

=(

s

r

rQ

T!

""#

$%%&

'

= 2

ln

1

2

(

Distance-drawdown formula


Distance-drawdown formula

Remember that there are two formulae for applying the

Jacob-Cooper method—time-drawdown and distance-drawdown

2

.252

r

TtS

o=

To calculate S, we use a formula similar to that used in the

time-drawdown formula.

We used to because it told us the time at which drawdown was zero;

for a distance-drawdown formula, we want to know the distance at

which s = 0 (we’ll call it ro)—the rest of the derivation is the same as

for the distance-drawdown method, so we’ll jump right to the

formula:

2

.252

or

TtS =

15



Q = 220 gpm

= 29.41 ft3/min

= 42,350 ft3/d

Drawdowns

measured at

t = 220 min


lcs 4

3.2

!=

"

QT

( )( ) d

ftd

ft2

3

987ft 5.71 4

42350 3.2==

!T

2

.252

or

TtS =

tT

Sru

4

2

=

!

S =2.25 987

ft2

d( ) 220min( )

1900 ft( )2

= 9 x 10"5

!

u =1900 ft( )

2

9 x 10"5( )

4 987ft2

d( ) 220min( ) 1 d1440min( )[ ]

= 0.54

16


s

r

rQ

T!

""#

$%%&

'

= 2

ln

1

2

( lch

r

rQ

T!

""#

$%%&

'

= 2

10ln

1

1

(

Distance-drawdown formula Thiem equation


s

r

rQ

T!

""#

$%%&

'

= 2

ln

1

2

( lch

r

rQ

T!

""#

$%%&

'

= 2

10ln

1

1

(

Distance-drawdown formula Thiem equation

Formulation of the DDF is identical to that of the Thiem equation

This means that when Jacob’s approximation applies, the difference in

drawdown between any two points stabilizes—the potentiometric surface

is being drawn down uniformly with time

17


Why doesn’t this apply in early time? Stability is achieved when the

slope of the potentiometric surface has the correct gradient to supply

Q to the well

We are constantly lowering the potentiometric surface—we must do

this to release water from storage. In early time, water is only released

from storage near the well, but as time gets large, water is supplied

from a larger area.

As the volume of aquifer supplying Q increases, less

change in head is needed to yield the same amount of

water, so the cone of depression increases at a

decreasing rate.

18

How do cones of depression vary from “normal” with changes in

aquifer properties, and how does this affect the drawdown hydrograph

of a monitoring well?

How might drawdown hydrographs vary from “normal” in situations

where our pumping test assumptions are not met?

Variations in drawdown

Cone of depression as a function of time

t = 1 min t = 10 min t = 120 min

(Hall, 1996)

T = 1000 gpd/ft;

S = 10-4;

Q = 300 gpm

19


Variations with pumping rate

Q = 100 gpm Q = 300 gpm Q = 600 gpm

T = 1000 gpd/ft;

S = 10-4;

r = 100 ft

(Hall, 1996)


Variations with storativity

S = 0.000001 S = 0.0001

S = 0.1 S = 10-4;

Q = 300 gpm;

r = 100 ft

S = 0.1

0.01

0.0010.0001

0.00001

(Hall, 1996)

20


Variations with transmissivity

T = 500 gpd/ft T = 1, 000 gpd/ft T = 2,500 gpd/ft

S = 1000 gpd/ft;

Q = 300 gpm;

r = 100 ft

(Hall, 1996)

Sensitivity to T and S summarized

Low T: tight, deep cone; High T: wide, shallow cone

Freeze and Cherry (1979)

Low S:

High S:

Cone shape remains

similar, the cone

is just bigger

Low S, creates greater drawdown to produce a given volume of water; cone of

depression shape stays the same

Low T: hard to transmit water, most efficient production is from close in,

creates steeper gradients of s to produce water from close in.

Documents

Transient Radial Flow Toward a Well - New Mexico Tech ... · Transient Radial Flow Toward a Well t h T S y h ... Our goal in aquifer testing is to find T and S; ... Transient Flow