Lecture Packet 8. Pump Test Analysis

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1.72, Groundwat er Hydrology Prof. Charles Harvey

Le ct u r e Pa ck e t # 8 : Pu m p Te st Ana lysis

The idea of a pum p t est is t o st ress t he aquifer by pum ping or inj ect ing wat er and t o not e t he drawdown over space and t im e.

Hist ory

• The earliest m odel for int erpret at ion of pum ping t est dat a was developed by Thiem ( 1906)

( Adolf and Gunt her) for

o Const ant pum ping rat e o Equilibr ium condit ions

o Confined and unconfined condit ions

• Theis ( 1935) published t he first analysis of t ransient pum p t est for

o Const ant pum ping rat e o Confined condit ions

• Since t hen, m any m et hods for analysis of t ransient well t est s have been designed for increasingly com plex condit ions, including

o Aquit ard leakage ( st udy of hydrogeology has becom e m ore a st udy of

aquit ards and less of aquifers)

o Aquit ard st orage o Wellbore st orage o Part ial w ell penet rat ion o Anisot ropy

o Slug t est s

o Recirculat ing w ell t est s ( w at er is not rem oved)

I t is im port ant t o not e t he assum pt ions for a given analysis. St e a dy Ra dia l Flow in a Confine d Aqu ife r

Assum e:

• Aquifer is confined ( t op and bot t om )

• Well is pum ped at a const ant rat e

• Equilibr ium is reached ( no drawdown change w it h t im e)

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Consider Darcy’s law t hrough a cylinder, radius r, wit h flow t ow ard w ell.

drawdown is effect ively zero.

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St e a dy Ra dia l Flow in a n Un con fine d Aquife r

Assum e:

• Aquifer is unconfined but underlain by an im perm eable horizont al unit .

• Well is pum ped at a const ant rat e

• Equilibr ium is reached ( no drawdown change w it h t im e)

• Wells are fully screened and

• There is only one pum ping w ell

Q

aquiclude well well

K

r2

r1

h1

h2

Observat ion Pum ping

aquifer

Wat er t able before pum ping

Wat er t able during pum ping

Radial flow in t he unconfined aquifer is given by

K

Q

=

(2

π

rh

)

dh

and rearrange as

hdh =

Q dr

dr

2 π

K r

I nt egrat e from r1, h1 t o r2, h2

h₂ r₂

jdh =

Q

∫

dr

∫

₂

_π

_K

_r

h₁ r₁

h

₂2

−

h

₁2

=

Q

r

2

2 π

K

ln

r

₁ or not ing t hat T = Kb

2

1 2 2 1 2 2

ln

)

(

r

h

Q

K

−

=

π

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2

• I f t he great est difference in head in t he syst em is < 2% , t hen you can use t he confined equat ion for an unconfined syst em .

• For r < 1.5hm ax ( t he full sat urat ed t hickness) t here w ill be errors using t his equat ion because of vert ical flow near t he well.

• I f difference in head is > 2% but < 25% use t he confined equat ion wit h t he follow ing correct ion for draw down

−

h

₁

)

new

h

“ m easured” reduced com pared t o confined aquifer case

(

h

ext ent , const ant t hickness, hom ogeneous, isot ropic.

• Pot ent iom et ric surface is horizont al before pum ping, is not changing w it h t im e before pum ping, all changes due t o one pum ping w ell

• Darcy’s law is valid and groundw at er has const ant propert ies where only 1 space dim ension is needed

aquifer h( r,t ) Depr ession Drawdown

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We want a solut ion t hat gives h( r,t ) aft er w e st art pum ping. To solve equat ion we need init ial condit ions ( I Cs) and boundary condit ions ( BCs)

I C: h( r,0) = h0

⎛

BCs: h(

∞

,t ) = h0 and limr_Æ0

⎜

r

∂

h

⎞

⎟

=

Q

for t > 0

⎝ ∂

r

⎠

2 π

T

w hich is j ust t he applicat ion of Darcy’s law at t he w ell

Th e is Equa t ion – t r a nsie n t r a dia l flow

1935 – C.V. Theis solves t his equat ion ( wit h C.I . Lubin from heat conduct ion)

Q

∞

e

−u

du

exponent ial int egral in m at h t ables but for our case it is “ well funct ion”

h

₀

−

h

(

r

,

t

)

=

∫

4 π

T

_u

u

Å

2

S

r

where

u =

4 Tt

,

h

₀

−

h

(

r

t

)

=

Q

W

(

u

)

, w ell funct ion is W( u)

4 π

T

Det erm ining T and S from Pum ping Test

I nverse m et hod: Use solut ion t o t he PDE t o ident ify t he param et er values by m at ching sim ulat ed and observed heads ( dependent variables) ; e.g., m easure aquifer drawdown response given a know n pum ping rat e and get T and S.

1. I dent ify pum ping w ell and observat ion w ells and t heir condit ions ( e.g., fully screened) .

2. Det erm ine aquifer t ype and m ake a quick est im at e t o predict w hat you t hink will happen during pum ping t est .

3. Theis Met hod: Arrange Theis equat ion as follows:

∆h

= ⎡

_⎢

114 .

6 Q

⎤

_⎥

W

(

u

)

( in USGS unit s) and

⎣

T

⎦

2

⎡

87

1 .

Sr

⎤

1 r

⎡

⎢

T

⎤

t

= ⎢

=

t

⎣

87

1 .

S

⎦⎥

u

Æ

⎣

T

⎦

⎥

u

log

∆

h

=

log

_⎢

⎡

114 .

6 Q

⎤

_⎥

+

log

W

(

u

)

⎣

T

⎦

2

⎤

⎡

87

1 .

Sr

1 log

t

=

log

_⎢

_{⎥ +}

log

⎣

T

_⎦

u

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5. Plot drawdown vs. t im e on log- log paper of sam e scale. ( t his is from dat a at a single observat ion w ell)

6. Superim pose t he field curve on t he t ype curve, keeping t he axes parallel. Adj ust t he curves so t hat m ost of t he dat a fall on t he t ype curve. You t rying t o get t he const ant s ( bracket ed t erm s) t hat m ake t he t ype curve axes t ranslat e int o your axes.

7. Select a m at ch point ( any convenient point w ill do like W( u) = 1.0 and 1/ u = 1.0) , and read off t he values for W( u) and 1/ u. Then read off t he values for drawdown and t .

8. Com put e t he values of T and S from :

Using USGS Unit s Using Self- Consist ent Unit s

(

114 6

.

QW

u

)

QW

u

)

T

=

T

=

(

h

₀

−

h

)

4 π

(

h

₀

−

h

)

Ttu

4 Ttu

S

=

S

=

₂

.

1 87

r

2

r

I f in USGS unit s of draw dow n ( ft ) , Q ( gpm ) , T ( gpd/ ft ) , r ( ft ) , t ( days) , S ( decim al fract ion) .

,

h

₀

−

h

(

r

t

)

=

Q

W

(

u

)

4 π

T

2

.

1 87

r

S

u =

Tt

To predict drawdown – drawdown vs. dist ance or t im e

• Put in r, S, T, t and solve for u

• Find W( u) based on u and t able

• Mult iply Q/ T and fact or

The analyt ic solut ion describes:

• Geom et ric charact erist ics t o t he cone of depression, st eepening t oward t he w ell

• For a given aquifer cone of depression increases in dept h and ext ent w it h t im e

• Drawdown at a t im e and locat ion increases linearly wit h pum ping rat e

• Drawdown at a t im e and locat ion is great er for sm aller T

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M odifie d N on e qu ilibr iu m Solut ion – Ja cob M e t h od

Over one log cycle you get change in drawdown

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3 .

2 Q

T =

( self- consist ent unit s)

4 π

∆

[

h

₀

−

h

]

Pr oce du r e

1. Plot on sem i- log paper – t on log scale and drawdown on arit hm et ic scale. 2. Pick off t w o values of t im e and t he corresponding values of drawdowns ( over