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 Neighboring

T1 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

y

h

x

h

t

h

T

S nnn

n

n

Governing Equation

on htyxh )0,,(Initial Condition

when 0),,( tyxhn yx or

r

hrTQ n

rn

0lim2

Boundary Conditions

Analysis – 2-Domain

• Conditions at the contact

tyLxx

h

T

TtyLx

x

h,,,, 2

1

21

Lxhh at 21

L

1 2

Analysis – 3-Domain

• Conditions at the contact

tywLxx

h

T

TtywLx

x

h m

s

ms ,,,,

wLxhh ms at m ms

L w

Method – Analytical• Transient analytical solution using Method

of Images (Fenske, 1984)

),,(

11

,,,,1

11Lyxf

dt

Er

Sr

Ttyxd

tE

ds

12

2112

1,,,,

ST

ST

tESTtyxs

drrd

21

14

rS

tTtd

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

14

LS

tTtdL

21

14

rS

tTtd

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

2

4

6

8

10

12

14

16

0.1 10 1000 100000td

s d

homogeneous No Flow T1/T2=10T1/T2=100 T1/T2=5 T1/T2=0.1

T1/T2=0.01 T1/T2=0.5 CH

0

1

2

0.1 10 1000 100000td

ds d

/dln

(t d)

2-Domain T Contrast – 0.5L

0

2

4

6

8

10

12

14

0.1 10 1000td

sd

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

ds

d/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

(td)

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

012

34567

8910

0.1 10 1000td

sd

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/d

ln(t

d)

Graphical Evaluation – 2-DomainEstimate Aquifer Properties

0

2

4

6

8

10

12

14

16

0.001 0.01 0.1 1 10 100 1000

tdL

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=0.0179

TTLL=0.55=0.55SL=0.25

TE=1SE=0.0179

TTLL=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 t

yx

t

yx

dLdL

d eyxeyxttd

sd2222 2

22222

2

21

ln

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

0

2

4

6

8

10

12

14

0.01 0.1 1 10 100 1000tdL

s d

Distance to the Contact

tc = 7.3

78.11

1

S

TtL c

Streltsova, 1988

0

2

4

6

8

10

12

14

16

0.1 10 1000 100000td

sd

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

(td)

3-Domain T Contrast - 0.125L

3-Domain T Contrast - 0.5L

0

2

4

6

8

10

12

14

0.1 10 1000td

sd

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

LK

wKT

a

sssd

a

sd K

L

w

KC

wKT sss

w

KC s

Strip Transmissivness & Conductance

010 52.1 98831min

min1

.CB.A

B

Cm

CmA

ssdT

18.1 094.0

1maxmax2

BA

B

mm

Ad

C

0.1

1

10

100

1000

10000

0 0.5 1mmin

Tss

d

0.001

0.01

0.1

1

10

1 1.5 2

mmaxC

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 PorosityOverlying 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

Down

Up

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

0

1

2

3

4

5

6

7

8

9

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 1

t/r2

ds

/dln

(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…