Igneous Rocks
Facies Change
Reef
Marine Clay
BatholithBatholith Country Country rockrock
Dike
Channel sand
Floodplain deposits
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 – 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 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
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.