chapter 3-radial flow into wellbore

Chapter 3 Chapter 3

Steady Radial Flow to a Steady Radial Flow to a fully penetrating Wellfully penetrating Well

Contents Contents

3.1 Summary 3.1 Summary

3.2 Steady Radial Flow in Aquifers3.2 Steady Radial Flow in Aquifers

3.3 Steady Radial Flow in a Leaky, 3.3 Steady Radial Flow in a Leaky, Confined Aquifer Confined Aquifer

3.4 Steady Radial Flow in Multiple 3.4 Steady Radial Flow in Multiple Well SystemsWell Systems

3.1 SummaryTypes of Water Wells

A Fully Penetrating Well :

a well that is screened over the entire thickness of the aquifer

A Partially Penetrating Well :

a well whose length of water entry is less than the aquifer it penetrates

Radial Flow : flow towards a well

3.1 SummaryTypes of Water Wells

3.1 SummaryDrawdown

The drawdown at a given point is the distance the water level is lowered .

( ) ( ) ( )tyxHyxHtyxs ,,0,,,, 10 −=

Drawdown curve shows the variation of drawdown with distance from the well .

Cone of depression : a conic shape the drawdown curve describes

3.1 SummaryWell Losses

Well Losses :

A well loss is caused by flow through the well screen and flow inside of the well to the pump intake .

3.1 SummaryBasic Assumptions

Basic assumptions that apply to all situations described in this chapter :

The aquifer is homogeneous and isotropic

All geologic formations are horizontal and

have infinite horizontal extent

The potentiometric surface of the aquifer is horizontal and not changing with time prior to the start of the pumping

Darcy’s law is validWater is released instantaneously from

the aquifer as the hydraulic head is lowered



The pumping well and the observation wells are fully penetrated

The pumping well has an infinitesimal diameter

Ground-water flow is horizontalThe aquifer is bounded on the bottom

by a confining layer

3.2 Steady Radial Flow in Aquifers Confined Aquifer

Besides the basic assumptions, some additional assumptions are :

The well is pumped at a constant rate

Equilibrium has been reached, that is to say, there is no further change in drawdown with time


ww rrwhenhH

RrwhenHH

==

== 0

Radial symmetry, radial flow problem can be expressed in polar coordinates:

0=⎟⎠⎞

⎜⎝⎛

drdHr

drd

(1)


As the flow of water through any circular section of the aquifer toward the well is the area of the circular section times the hydraulic conductivity, k, times the hydraulic gradient. That is :

CdrdHr =

QdrdHKrMQr =••= π2 KM

QCπ2

=

(2)


KMQ

drdHr

π2=

∫∫ =R

r

H

h ww rdr

KMQdHπ2

0

www r

RKMQshH ln

20 π==− (3)


w

w

rRhHKMQ

lg73.2 0 −=

Equations (3) and (4) are called Dupuit Equation.

Here: R is the radius of influence of the pumping well.

Or:

(4)


For pumping well with observation well(s):

M


1

212 ln

2 rr

KMQhhπ

=−

ww r

rKMQhh ln

2π=−

Thiem Equation

For pumping well with observation well(s):

With two observation wells:

With one observation well at a distance of r :

(5)

(6)


w

w

rR

rr

ww hHhhlnln

)( 0 −+=

From equations (3) and (5), we have:

(7)

Equation (7) indicates that under steady flow conditions, the distribution of Gw head around a pumping well without relation to K and Q.

3.2 Steady Radial Flow in Aquifers unconfined Aquifer

Some additional assumptions are : The aquifer is unconfined and underlain

by a horizontal aquicludeThe well is pumped at a constant rateEquilibrium has been reached ; that is ,

there is no further change in drawdown with timeGw flows horizontally (r>1.5H)

3.2 Steady Radial Flow in Aquifers Unconfined Aquifer

ww rrwhenhh

Rrwhenhh

==

== 0

Radial symmetry, radial flow problem can be expressed in polar coordinates:

02

=⎟⎟⎠

⎞⎜⎜⎝

⎛dr

dhrdrd

(8)

wwww r

RKQsshhh ln)2( 0

220 π

=−=−


w

w

rRshKQ

lg

)2(366.1 0 −=

(9)

(10)

Or:

Equations (9) and (10) are called Dupuit equationfor unconfined aquifers


ww r

rKQhh ln22

π=−

1

221

22 ln

rr

KQhhπ

=−

With one observation well

With two observation wells

(11)

(12)


w

w

rR

rr

ww hhhhlnln

)( 220

22 −+=

From equations (9) and (11), we have:

(13)

Equation (13) indicates that under steady flow conditions, the distribution of Gw head around a pumping well without relation with K and Q. When r>h0, it is correct, but when r<h0, hc<hreal

3.2 Steady Radial Flow in AquifersCertain cases

Application of Dupuit Equation under certain geo-conditions:

1. Unconfined aquifer with huge thickness: Dupuit equation can be re-organized as:

wwww r

RKQsshhh ln)2( 0

220 π

=−=−

www r

RhhK

Qhh ln)( 0

0 +=−π


When h0-hw<< h0:

www r

RKhQshh ln

2 00 π

≈=−

00 2hhh w ≈+

？


2. Confined-Unconfined aquifer:

ww r

aKQhM ln22

π=−

When r <a:


aR

KMQMH ln

20 π=−

( )w

w

rRhMMhKQ

lg2366.1

220 −−

=

When r>a:

3.2 Steady Radial Flow in Aquifers Certain cases

3. Recharging well:

Confined well:

w

w

rRHhKMQ

lg73.2 0−

=

wrR

w hhKQlg

)(366.120

2 −=

Unconfined well:

3.2 Steady Radial Flow in AquifersApplications of Dupuit Equation

Determination of hydraulic conductivity:

For confined aquifer:

ww rR

MsQK ln366.0=

1

2

21

ln)(

366.0rr

ssMQK−

=


For unconfined aquifer:

www rR

sshQK ln

)2(732.0

0 −=

1

2

21210

ln))(2(

732.0rr

sssshQK

−−−=


21

1221 lglglnss

rsrsR−−

=

))(2(lg)2(lg)2(lg

21210

12022101

sssshrshsrshsR

−−−−−−

=

Determination of radius of influence:

For confined aquifer:

For unconfined aquifer:


Forecasts or estimates drawdown or flux.

drawdown flux

3.2 Steady Radial Flow in AquifersDiscussions of Dupuit Equation

1.The relationship between the diameter of a well and the flux

Dupuit :

rw 2rw

q=(1+10%)q0

The real situation just like right figure sw

3.2 Steady Radial Flow in AquifersDiscussion of Dupuit Equation

1.For a increase of rw, the higher k, the more increasing of q is reached.

2.For a constant K, more increasing of q can be obtained for a certain increase of rw for big sw,

3.For a same K and a constant sw, rwincreases from different value reaches different results.


Seepage Face

(水跃)

ws hh >

As a result of the seepage face , the actual water table lies above the computed one .


Functions of Seepage Face:

1.Make sure that water nearby the well getting into it

2.Keep enough cross sectional area for Q (entering well)

As a result of the seepage face , the actual water table lies above the computed one .

3.3 Steady Radial Flow in a Leaky, Confined Aquifer

Besides the basic assumptions , the additional assumptions are :

The aquifer is bounded on the top by an aquitard .

The aquitard is overlain by an unconfined aquifer , in which the water table is initially horizontal and doesn’t fall during pumping of the aquifer .

No water is released from the

storage in the aquitard when the

aquifer is pumped .

Flowing of Ground-water in the

aquitard is vertical.



wrrwhenKMQ

drdsr

rwhens

BHH

rH

rrH

=−=

∞→=

=−

+∂∂

+∂∂

,2

0

012

02

2

π


⎟⎠⎞

⎜⎝⎛≈

BrK

KMQs 02π

Hantush-Jacob equation

3.3 Steady Radial Flow in a Leaky, Confined AquiferCurve-Matching Method

⎟⎠⎞

⎜⎝⎛=

BrK

KMQs 02π

Under a logarithmic scale, we have:

BBrr ⋅=

TQ

BrKs

π2lg]lg[lg 0 +⎟

⎠⎞

⎜⎝⎛=

BBrr lglglg +⎟⎠⎞

⎜⎝⎛=

3.3 Steady Radial Flow in a Leaky, Confined AquiferCurve-Matching Method

On logarithmic paper, the relation between s and r is same to that of and .

⎟⎠⎞

⎜⎝⎛

BrK 0

Br

Superimpose one plot on the other , then select a match point , find the corresponding values of the coordinate .

From them , we can obtain the parameters .

3.3 Steady Radial Flow in a Leaky, Confined Aquiferd

Curve-Matching Method

[s],[ r], [ ] and [r/B] :

][

][

BrrB =

)]([][2 0 B

rKs

QTπ

−=

)(0 BrK

Leakage factor

3.3 Steady Radial Flow in a Leaky, Confined AquiferdStraight-Line Metod

In the area nearby the pumping well. As r/B<<1. So: for K0(x)

rB

TQ

s 123.1ln2π

≈

)/123.1ln()(0 xxK ≈So:

When r/B<0.35, difference error <5% and When r/B<0.1, difference error <1%


So:

)89.0lg(230.2123.1ln

2 Br

TQ

rB

TQ

sππ

−==

That is to say, under semi-logarithmic coordinates , the relation between s and r is linear .


Where r0 is the diameter, the straight line intersects the zero-drawdown axis.

089.0 rB =

iQT

π230.2

−=

Where i is the slope of the straight line

3.4 Steady Radial Flow in Multiple Well Systems

Principle of superposition :

If H1,H2,H3,….Hn are solutions of a equation which is a linear equation for hydraulic head H. Then:

∑ ==

n

i iiHCH1

Is also a solution of the equation. Here Ci are constants.

Individual and composite drawdown curves for three wells in a line


Group Well System :

For a group of wells forming a well field , the cones of depression of pumping wells overlap , each well is said to interfereinterfere with the others because of the increased drawdown and pumping lift created .


Principle of superposition : Because the Laplace equation is linear , at any given point in a confined aquifer , the total drawdown is the sum of the individual drawdowns for each well, known as linear linear superpositionsuperposition .

ij

jn

j

jn

jiji r

RT

Qss ln

211∑∑==

==π


For multiple well systems in an unconfined aquifer resting on a horizontal impervious surface :

ij

jn

j

ji r

RK

QhH ln

1

220 ∑

=

=−π


Example

For two wells with a distance of L

In a confined aquifer :

In an unconfined aquifer :

( )LrRKMsQQ

w

w221 ln

2π==

( )( )LrR

hHKQQw

w2

220

21 ln−

==π

chapter 3-radial flow into wellbore

Documents

steady radial flow

radial flow problem

flow of water

steady flow conditions

q r h hw

aquifer dh r dr

observation wellq r

r rw dr r3 q r h