well tests to characterize idealized lateral heterogeneities by vasi passinos k 1,s 1 k 2,s 2
DESCRIPTION
Igneous Rocks Facies Change Reef Marine Clay Batholith Country rock Dike Channel sand Floodplain depositsTRANSCRIPT
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Well Tests to Characterize Idealized Lateral Heterogeneities
by
Vasi PassinosK1,S1
K2,S2
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Faults
Steeply Dipping Beds
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Igneous Rocks
Facies Change
Reef
Marine Clay
BatholithBatholith Country Country rockrock
Dike
Channel sand
Floodplain deposits
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Confined Aquifer Unconfined Aquifer
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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
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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
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Analysis – 2-Domain• Conditions at the contact
tyLxxh
TT
tyLxxh
,,,, 2
1
21
Lxhh at 21
L
1 2
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Analysis – 3-Domain• Conditions at the contact
tywLxx
hTT
tywLxxh m
s
ms ,,,,
wLxhh ms at m ms
L w
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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
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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
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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
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2-Domain Model T Contrast
Tr=10
Tr = 1
Tr=0.1
tdLA tdLB tdLC
- 2 0 2 40
2
4
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2-Domain Model S Contrast
Sr = 10
Sr = 1
Sr = 0.1
tdLA tdLB tdLC
- 2 0 2 40
2
4
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3-Domain Model T Contrast
Tr = 10
Tr = 1
Tr = 0.1
tdLB tdLC tdLD
- 4 - 2 0 2
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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
)
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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)
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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
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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)
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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
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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
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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
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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
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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
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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
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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
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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)
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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
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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
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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
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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
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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
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Field Example
500 feet
DownUp
Ridge
stream
Nstream
fault
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Field Case - Site Map
N
500 feet
BW-109
BW2
L
B-4
Felsic
Mafic
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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)
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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)
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• 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)
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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
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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
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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.
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Conclusions
• Analyzing piezometers individually is a poor approach to characterizing heterogeneities.
• Drawdown curves non-unique. Require geological assessment.
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Acknowledgments
• Funding– Geological Society of America– Brown Foundation– National Science Foundation
• Others…