Well Tests to Characterize Idealized Lateral Heterogeneities

by

Vasi PassinosK1,S1

K2,S2

Faults

Steeply Dipping Beds

Igneous Rocks

Facies Change

Reef

Marine Clay

BatholithBatholith Country Country rockrock

Dike

Channel sand

Floodplain deposits

Confined Aquifer Unconfined Aquifer

Conceptual Models

Local NeighboringT1 S1 T2 S2

L L

2-Domain Model 3-Domain Model

Matrix MatrixStrip

Tm SmTm Sm

L Lw

Ts

Ss=Sm

Analysis

2

2

2

2

yh

xh

th

TS nnn

n

n

Governing Equation

on htyxh )0,,(Initial Condition

when 0),,( tyxhn yx or

rh

rTQ n

rn

0

lim2

Boundary Conditions

Analysis – 2-Domain• Conditions at the contact

tyLxxh

TT

tyLxxh

,,,, 2

1

21

Lxhh at 21

L

1 2

Analysis – 3-Domain• Conditions at the contact

tywLxx

hTT

tywLxxh m

s

ms ,,,,

wLxhh ms at m ms

L w

Method – Analytical• Transient analytical solution using Method

of Images (Fenske, 1984)

),,(1

1,,,,111 Lyxf

dtErSrTtyx

dtEds

12

2112

1,,,,STST

tESTtyxs

drrd 2

1

14rStT

td

Methods – Numerical• Transient numerical model using MODFLOW

• 2-Domain – Tr and Sr were varied• 3-Domain - Tr and w of the strip were varied.

• Grid optimized for small mass balance errors

• The properties of the model were selected so that the drawdown and time from the numerical model were dimensionless

Dimensionless Time• Drawdowns were evaluated at three

dimensionless times to illustrate effects during development of drawdown fields.

• Dimensionless time used for type curves

• Dimensionless time used in drawdown fields

21

14LS

tTtdL

21

14rStT

td

2-Domain Model T Contrast

Tr=10

Tr = 1

Tr=0.1

tdLA tdLB tdLC

- 2 0 2 40

2

4

2-Domain Model S Contrast

Sr = 10

Sr = 1

Sr = 0.1

tdLA tdLB tdLC

- 2 0 2 40

2

4

3-Domain Model T Contrast

Tr = 10

Tr = 1

Tr = 0.1

tdLB tdLC tdLD

- 4 - 2 0 2

2-Domain T Contrast – 0.125L

0

24

68

10

1214

16

0.1 10 1000 100000td

s d

homogeneous No Flow T1/T2=10T1/T2=100 T1/T2=5 T1/T2=0.1T1/T2=0.01 T1/T2=0.5 CH

0

1

2

0.1 10 1000 100000td

dsd/d

ln(t d

)

2-Domain T Contrast – 0.5L

0

2

4

6

8

10

12

14

0.1 10 1000td

s d

homogeneous No Flow T1/T2=10T1/T2=100 T1/T2=5 T1/T2=0.1T1/T2=0.01 T1/T2=0.5 CH

0

0.5

1

1.5

2

0.1 10 1000td

dsd/d

ln(t

d)

2-Domain S Contrast – 0.125L

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0.1 10 1000 100000

td

m =

ds d

/dln

(t d)

0

2

4

6

8

10

12

0.1 10 1000 100000

td

s d

S1/S2=1 S1/S2=10 S1/S2=100 S1/S2=0.1 S1/S2=0.01

2-Domain S Contrast – 0.5L

012345678910

0.1 10 1000td

s d

S1/S2=1 S1/S2=10 S1/S2=100 S1/S2=0.1 S1/S2=0.01

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0.1 10 1000td

m =

ds d

/dln

(td)

Graphical Evaluation – 2-DomainEstimate Aquifer Properties

0

2

4

6

8

10

12

14

16

0.001 0.01 0.1 1 10 100 1000tdL

s d to = 0.029 S = 0.017s = 2.3 T = 1

to = 0.42 S = 0.35s = 4.1 T = 0.55

Graphical Evaluation – 2-DomainEstimate Aquifer Properties

0

2

4

6

8

10

12

0.01 0.1 1 10 100 1000

tdL

s d

to = 2.7 S = 0.136s = 4.1 T = 0.55

TE=1SE=0.0179TTLL=0.55=0.55SL=0.25

TE=1SE=0.0179TTLL=0.55=0.55SL=0.136

TTLL=0.55=0.55SL=0.06

TTLL=0.55=0.55SL=0.27

TTLL=0.55=0.55SL=0.021

TTLL=0.55=0.55SL=0.068

TTLL=0.55=0.55SL=0.029

TTLL=0.55=0.55SL=0.021

L

L L

Critical Region• An early semi-log straight line can be

determined by

• The second derivative was compared to plots with a variety of curves. An early SLSL could be identified by a second derivative of 0.2 or less from 0.3<tdL<2.5.

dLdL tyx

tyx

dLdL

d eyxeyxttd

sd2222 2

22222

2

21ln

Critical Region• Observation points confined to a region that

is within 0.3 to 0.5 of the distance between the pumping well and the linear discontinuity

-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0

0.2

0.4

0.6

0.8

1.0

02468101214

0.01 0.1 1 10 100 1000tdL

s d

Distance to the Contact

tc = 7.3

78.11

1

STt

L cStreltsova, 1988

02468

10121416

0.1 10 1000 100000td

s d

homogeneous No Flow T1/T2=10T1/T2=100 T1/T2=5 T1/T2=0.1T1/T2=0.01 T1/T2=0.5 CH

0

0.5

1

1.5

2

0.1 10 1000 100000td

m =

ds d

/dln

(t d)

3-Domain T Contrast - 0.125L

3-Domain T Contrast - 0.5L

02468

101214

0.1 10 1000td

s d

homogeneous No Flow T1/T2=10T1/T2=100 T1/T2=5 T1/T2=0.1T1/T2=0.01 T1/T2=0.5 CH

0

0.5

1

1.5

2

0.1 10 1000td

m =

ds

d/d

ln(t

d)

Strip Transmissivness & Conductance• Hydraulic properties of the strip depend on

strip conductivity and width• Strip K greater than matrix

• Strip K less than matrix

LKwKT

a

sssd

a

sd K

LwK

C

wKT sss

wK

C s

Strip Transmissivness & Conductance

010 52.1 98831minmin1

.CB.A

B

CmCm

AssdT

18.1 094.01max

max2

BA

B

mmAdC

0.1

1

10

100

1000

10000

0 0.5 1mmin

Tss

d

0.001

0.01

0.1

1

10

1 1.5 2mmax

C d

Graphical Evaluation – 3-DomainEstimate Aquifer Properties

0

2

4

6

8

10

12

14

0.001 0.1 10 1000 100000

tdL

s d

to = 0.09 S = 0.054s = 2.3 T = 1

to = 0.028 S = 0.017s = 2.3 T = 1

Determine Properties of Strip• SLSL analysis on the first line will give

T and S of the area near the well.• Take the derivative of time and

determine the maximum or minimum slope.

• Using equations from curve fitting determine Tssd or Cd of the layer.

• Solve for Tss or C

Non-Uniqueness

s

s

Log (t) Log (t)

Dual Porosity Overlying Leaky Layer without storage

Unconfined Aquifer w/delay yield from storage

Overlying Leaky Layer with storage

Streltsova, 1984

Streltsova, 1988 Streltsova, 1984

Neuman, 1975

Field Example

500 feet

DownUp

Ridge

stream

Nstream

fault

Field Case - Site Map

N

500 feet

BW-109

BW2

L

B-4

Felsic

Mafic

Drawdown from Pumping Well

0

5

10

15

20

25

30

35

40

45

50

10 100 1000 10000 100000

t (min)

s

0

0.5

1

1.5

2

10 100 1000 10000

t (min)

m =

ds/

dln(

t)

Drawdown from Piezometers

0123456789

0.0001 0.01 1

t/r2

s

BW-109 BW-2

0

0.5

1

0.0001 0.001 0.01 0.1

t/r2 (min)

m =

ds/

dln(

t)

• Using Semi-Log Straight-Line Analysis :

• Minimum slope using the derivative curve is 0.5

• Tssd=34=Ksw/KaL

• Tss = 24 ft2/min w = 10 to 20 ft

Determining Hydraulic Properties

L = 280 ft Distance to fault

b = 21.5 ft screened thickness

Tm = 0.05 ft2/minSm = 2x10-4 ???

Ts = 26 to 52 ft2/minTs/Tm = 500 to 1000

0

3

6

9

1 10 100 1000 10000

s

0

0.5

1

0.0001 0.001 0.01 0.1 1t/r2

ds/d

ln(t)

Conclusions 2-Domain Model

Using the Jacob method to analyze well tests:• Piezometers r < 0.25L gives T, S of local

region.

• Piezometers r > 0.25L gives average T of both regions.

• Piezometers r > 0.25L unable to predict S

Conclusions – 2-Domain• Piezometers in neighboring region also give

average T of both regions.

• L can be determined from intersecting SLSLs using a piezometer within the critical region

Conclusions 3-Domain Model• Drawdown for low conductivity vertical layer

controlled by conductance.

C=Ks/w

• Drawdown for high conductivity vertical layer controlled by strip transmissivness.

Tss=Ks*w

• Feasible to determine properties of a vertical layer from drawdown curves.

Conclusions

• Analyzing piezometers individually is a poor approach to characterizing heterogeneities.

• Drawdown curves non-unique. Require geological assessment.

Acknowledgments

• Funding– Geological Society of America– Brown Foundation– National Science Foundation

• Others…