flowtowells_mar8

7/29/2019 flowtowells_mar8

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Flow to Wells (Ch. 5).


2/28

Flow to Wells

For scientific exploration Knowledge of aquifer characteristics

K, S, T,

For resource exploitation Effective, responsible aquifer pumping

Maximum sustainable yield Extent of cone of depression

Connectedness

Others


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Basic Assumptions in Chapter 5

1. aquifer is bounded on the bottom by a confining layer

2. All geologic formations are horizontal and have infinite horizontal extent

3. The potentiometric surface if the aquifer is horizontal prior to the start ofpumping

4. The potentiometric surface of the aquifer is not changing with time prior tothe start of pumping

5. All changes in the position of the potentiometric surface are due to theeffect of the pumping well alone

6. The aquifer is homogeneous and isotropic

7. All flow is radial toward the well

8. Groundwater flow is horizontal

9. Darcys law is valid

10. Groundwater has a constant density and viscosity11. The pumping well and the observational well are fully penetrating; that is,

they are screened over the entire thickness of the aquifer.

12. The pumping well has an infinitesimal diameter and is 100% efficient


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Fig 5.2

Compare with next slide

Fetter,Applied Hydrology, 2001


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Heath, Ground-Water Hydrology, 1983


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Bouwers Fig. 4.1

Bouwer, Groundwater Hydrology, 1978


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Why does potentiometric surface steepen as it approaches the pumping well? If we

assume steady-state, the flux towards the well at any given radius must be equal. Since

radial flow results in ever decreasing cross-sectional area, the gradient must increase to

maintain the same flux.


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A new governing equation

Predicting drawdown and response potentiometric surfaces involve

predicting the spatial and temporal distribution of h, for which we have

equations.

Recall the 3d governing equation

Radial symmetry, resulting from assumptions 6, 2 and 3 allows reduction of

the confined, three-dimensional radial flow into a 2-dimensional problem:

S

h h h hK K K S

x x y y z z t + + =

2 2

2 2

SSh h h

x y K t

+ =


9/28

New Governing Equation

When discussion aquifer productivity we

often use the term transmissivity (T) which

is defined as:

so

T Kb=

2 2

2 2

h h S h

x y T t

+ =


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New Governing Equation

Conversion to radial coordinates:

2 2r x y= +

2 2

2 2

h h S h

x y T t

+ =

2

2

1h h S h

r r r T t

+ =


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A New Governing Equation

With recharge through confining layer

In the next section we will evaluate aboveequation to determine drawdown and aquifercharacteristic for

Steady-state confined Steady-state unconfined

Transient confined

Transient unconfined

t

h

T

S

t

w

r

h

rr

h

=+

+

12

2


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Steady state (equilibrium) vs

Transient (non-equilibrium)

Confined Aquifer

Original potentiometric

QObservation wells

b

t1

t2

tinf

t3

-A cone of depression grows until equilibrium is reach. While cone is

growing we use solutions for transient case. When equilibrium is reached

we use steady-state solutions.


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Topic Organization

Fetter

2 problems

Determining drawdown

Transient confined

Transient unconfined Determining aquifer properties

(K,S)

Steady-state confined

Steady-state unconfined

Transient confined Transient unconfined

Me 2 conditions

Steady-State

Confined drawdown andaquifer properties

Unconfined drawdown andaquifer properties

Transient

Confined drawdown andaquifer properties

Unconfined drawdown andaquifer properties


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Steady State Flow to Wells

When a well is pumped, drawdown and creation of a cone of

depression develops. Eventually, steady state is achieved and the

cone is stable

Pumping rate and aquifer properties determine the shape of the

cone


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Derivation of the Thiem equation (confined)

Equation 5.44 in your text Steady state flow to a well in a confined

aquifer

What is the extent of the cone of depression?

What is the drawdown at any distance h?

=1

2

12

ln

)(2

r

r

hhT

Q


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Thiem Confined

Confined Aquifer

r1

h1

r2

h2

Original potentiometric

=

1

2

12

ln

)(2

r

r

hhTQ

Q Observation wells

b


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Thiem Confined

See example page 168

In addition, estimate the extent of the cone

of depression


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Derivation of the Thiem

equation (unconfined) Equation 5.49 in your text

Steady state radial flow to a well in an

unconfined aquifer

=

1

2

2

1

2

2

ln

)(

r

r

hhK

Q

=1

2

12

ln

)(2

r

r

hhTQ ave


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The extent of the zone

of influence for an

unconfined aquifer is

small compared to thatof a confined aquifer

Heath, Basic Ground-Water Hydrology, 1983

Thiem UnConfined


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Bouwers 4.2

Bouwer, Groundwater Hydrology, 1978

Heath Basic Ground-Water Hydrology 1983


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Example

A 12 well in an unconfined aquifer has been

pumped at 350 gpm until equilibrium has been

achieved. The well penetrates 108 ft below the

original water table. Two observation wells locat57 and 148 ft from the pumped well show

drawdowns of 12 and 7.4 ft respectively

A. Estimate K

B. Estimate T

C. What is the drawdown at the pumped well?

D. How large is the cone of depression?


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Transient (unsteady) flow to a

well in a confined aquiferTheis (1935) equation assumptions:

Fully confined aquifer (top and bottom)

No recharge sourceThe aquifer is compressibleWater released from storage isinstantaneously discharged with a drop in

headWell is pumped at a constant rate


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Remember your friend storativity (S)?

(Be mindful of the differences between confined and unconfined values)

S=gb( +n)



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Derivation of the Theis solution

ANALYTICAL solution to Eq 5.2 subject tothe following boundary and initial

conditions (Eq. 5.12):

h(r,0) = ho for all r (constant piezometric

surface at t = 0)

h(,t) = ho for all t (no drawdown at infinitedistance from the well)

Constant pumping rate at the well: Q = 2rTdh/dr


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Theis Solution

daa

e

T

Qhh

u

a

o

=4

)(4

uWT

Qhho

=

Tt

Sru

4

2

=

)(...!44!33!22

ln5772.0432

uWuuu

uudaa

e

u

a

=

+

+

+=


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Domenico and Schwartz, Physical and Chemical Hydrology, 1998


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Use of the Theis solution

If Q is constant and S and T are known,can solve for drawdown distribution intime: ho-h(r,t)

Alternatively, if the drawdown as afunction of time is known for one or moreobservation wells, given a constant Q at

the well, then one can solve for T and Susing a graphical procedure using a Theistype curve.

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