Well Tests to Characterize Idealized Lateral Heterogeneities by Vasi Passinos K 1 ,S 1 K 2 ,S 2
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…