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Boise State UniversityScholarWorks
CGISS Publications and Presentations Center for Geophysical Investigation of the ShallowSubsurface (CGISS)
12-16-2011
3-D Transient Hydraulic Tomography inUnconfined Aquifers with Fast Drainage ResponseMichael CardiffBoise State University
Warren BarrashBoise State University
Copyright 2011 by the American Geophysical Union. DOI: 10.1029/2010WR010367
3-D transient hydraulic tomography in unconfined aquiferswith fast drainage responseM. Cardiff1,2 and W. Barrash1
Received 4 January 2011; revised 28 October 2011; accepted 31 October 2011; published 16 December 2011.
[1] We investigate, through numerical experiments, the viability of three-dimensionaltransient hydraulic tomography (3DTHT) for identifying the spatial distribution ofgroundwater flow parameters (primarily, hydraulic conductivity K) in permeable,unconfined aquifers. To invert the large amount of transient data collected from 3DTHTsurveys, we utilize an iterative geostatistical inversion strategy in which outer iterationsprogressively increase the number of data points fitted and inner iterations solve the quasi-linear geostatistical formulas of Kitanidis. In order to base our numerical experimentsaround realistic scenarios, we utilize pumping rates, geometries, and test lengths similar tothose attainable during 3DTHT field campaigns performed at the Boise HydrogeophysicalResearch Site (BHRS). We also utilize hydrologic parameters that are similar to thoseobserved at the BHRS and in other unconsolidated, unconfined fluvial aquifers. In additionto estimating K, we test the ability of 3DTHT to estimate both average storage values(specific storage Ss and specific yield Sy) as well as spatial variability in storage coefficients.The effects of model conceptualization errors during unconfined 3DTHT are investigatedincluding: (1) assuming constant storage coefficients during inversion and (2) assumingstationary geostatistical parameter variability. Overall, our findings indicate that estimationof K is slightly degraded if storage parameters must be jointly estimated, but that this effectis quite small compared with the degradation of estimates due to violation of ‘‘structural’’geostatistical assumptions. Practically, we find for our scenarios that assuming constantstorage values during inversion does not appear to have a significant effect on K estimatesor uncertainty bounds.
Citation: Cardiff, M., and W. Barrash (2011), 3-D transient hydraulic tomography in unconfined aquifers with fast drainage response,
Water Resour. Res., 47, W12518, doi:10.1029/2010WR010367.
1. Introduction[2] Three-dimensional (3-D) hydraulic tomography (HT)
consists of a series of pumping tests in which both pumpingand measurement can take place at discrete, isolated depths.By collecting data at a variety of lateral locations and a vari-ety of isolated depths, 3DHT allows for the estimation of3-D hydraulic parameters (e.g., hydraulic conductivity K,and specific storage Ss). This is in stark contrast to ‘‘tradi-tional’’ pumping tests where water is pumped at an openwellbore and water level changes are measured at surround-ing wells, which do not measure vertical variations in headand are thus only capable of estimating depth-integrated oraveraged parameters (transmissivity T and storativity S),even if they are analyzed in a tomographic fashion; we willrefer to this as 2DHT. Another key differentiator between3DHT and traditional pumping tests is the time and equip-ment costs required for 3-DHT surveys in the field. 3DHT
with 3-D stimulations of the aquifer requires a method forpumping from a discrete interval either with permanent ortemporary installations, for example, packer and port sys-tems that can be moved within an existing wellbore. In addi-tion, either emplaced pressure sensors at depths within theaquifer (e.g., direct push sensors Butler et al. [2002]), or hy-draulic separation of intervals in existing wellbores (e.g.,packer and port systems) must be installed at observationlocations in order to obtain information on depth variationsof head within the aquifer.
[3] Analyzing pressure response data (i.e., head orchanges in head) from pumping tests in a tomographic fash-ion has been the subject of study for over 15 years, and theliterature in this area is extensive. As noted in earlier discus-sions [Cardiff, 2010], approaches to HT ‘‘differ in the typeof aquifer stimulations they perform (largely 2-D when usingfully penetrating wells or 3-D when using packed-off inter-vals), the type of forward model employed during inversion(2-D, 3-D, axisymmetric 1-D/2-D), the types of heterogeneityassumed (layered 1-D, 2-D cross sectional, 2-D map view,3-D), constraints on the heterogeneity (geostatistical, struc-tural, or otherwise), and the types of auxiliary data utilized(geophysical, core sample, etc.).’’ In order to present a per-spective about the state of HT research to date, Table 1 is anattempt to summarize and classify the research in 2DHT and3DHT. In order to limit the size of this table, we consider
1Center for Geophysical Investigation of the Shallow Subsurface(CGISS), Department of Geosciences, Boise State University, Boise,Idaho, USA.
2Department of Geosciences, University of Wisconsin-Madison, Madi-son, Wisconsin, USA.
Copyright 2011 by the American Geophysical Union.0043-1397/11/2010WR010367
W12518 1 of 23
WATER RESOURCES RESEARCH, VOL. 47, W12518, doi:10.1029/2010WR010367, 2011
Tab
le1.
Sum
mar
yof
Prio
r2D
HT
and
3DH
TSt
udie
s
Aut
hors
[Yea
r]T
ype
ofSt
udy
Pum
ping
Stim
ulat
ion
Typ
e
Hea
dR
espo
nse
Mea
sure
men
tT
ype
Tru
ePa
ram
eter
Dis
trib
utio
na,b,
cH
ead
Res
pons
eD
ata
Use
ddPa
ram
eter
sE
stim
ated
e,f
Para
met
erC
onst
rain
ts/
Reg
ular
izer
s/A
ssum
edPr
ior
Info
rmat
iong
Oth
erD
ata
Sour
ces
Use
din
Inve
rsio
n
Got
tlieb
and
Die
tric
h[1
995]
Num
eric
al2-
D,F
ully
Pene
trat
ing
y2-
D,F
ully
Pene
trat
ing
y2-
DK
(x,z
):Fa
cies
-Bas
ed0-
DSs
:C
onst
ant
Tra
nsie
nt2-
DK
(x,z
)K
(x,z
):ze
roth
-ord
erT
ikho
nov
Reg
ular
izat
ion
Yeh
etal
.[19
96]
Num
eric
al2-
D,F
ully
Pene
trat
ing
z2-
D,F
ully
Pene
trat
ing
z2-
DK
(x,y
):A
niso
trop
icG
eost
atis
tical
Stea
dySt
ate
2-D
K(x
,y)
K(x
,y):
Ani
sotr
opic
Geo
stat
istic
alV
asco
etal
.[19
97]
Num
eric
al2-
D,F
ully
Pene
trat
ing
y2-
D,F
ully
Pene
trat
ing
y2-
DK
(x,z
):G
eost
atis
tical
0-D
Ssh
Tra
nsie
nt,S
elec
ted
time
step
s2-
DK
(x,z
)K
(x,z
):fir
st-o
rder
Tik
hono
vR
egul
ariz
atio
nSo
me
anal
yses
ofre
solu
tion
incl
ude
trac
erda
taV
asco
etal
.[19
97]
Fiel
d2-
D,F
ully
Pene
trat
ing
y2-
D,F
ully
Pene
trat
ing
yT
rans
ient
,Sel
ecte
dtim
est
eps
2-D
K(x
,z)
K(x
,z):
first
-ord
erT
ikho
nov
Reg
ular
izat
ion
Snod
gras
san
dK
itani
dis
[199
8]N
umer
ical
2-D
,Ful
lyPe
netr
atin
gz
2-D
,Ful
lyPe
netr
atin
gz
2-D
K(x
,y):
Faci
es-B
ased
Stea
dySt
ate
2-D
K(x
,y)
K(x
,y):
Geo
stat
istic
ali
Zim
mer
man
etal
.[19
98]
Num
eric
al2-
D,F
ully
Pene
trat
ing
z2D
,Ful
lyPe
netr
atin
gz
2-D
K(x
,y):
Com
bine
dG
eost
atis
ticalþ
Faci
es0-
DSs
Tra
nsie
ntV
aria
blej
Var
iabl
ej
Zim
mer
man
etal
.[19
98]
Num
eric
al2-
D,F
ully
Pene
trat
ing
z2-
D,F
ully
Pene
trat
ing
z2-
DK
(x,y
):C
ombi
ned
Geo
stat
istic
alþ
Faci
es0-
DSs
h
Tra
nsie
ntV
aria
blek
Var
iabl
ek
Vas
coet
al.[
2000
]N
umer
ical
2-D
,Ful
lyPe
netr
atin
gz
2-D
,Ful
lyPe
netr
atin
gz
2-D
K(x
,y):
Geo
stat
istic
al0-
DSs
hT
rans
ient
,Pul
setr
avel
time
met
rics
2-D
K(x
,y)
K(x
,y):
zero
th-
and
first
-or
der
Tik
hono
vV
asco
etal
.[20
00]
Fiel
d2-
D,F
ully
Pene
trat
ing
z2-
D,F
ully
Pene
trat
ing
zT
rans
ient
,Pul
setr
avel
time
met
rics
2-D
K(x
,y)
K(x
,y):
zero
th-
and
first
-or
der
Tik
hono
vYe
han
dLi
u[2
000]
Num
eric
al2-
D,F
ully
Pene
trat
ing
z2-
D,F
ully
Pene
trat
ing
z2-
DK
(x,y
):A
niso
trop
icG
eost
atis
tical
Stea
dySt
ate
2-D
K(x
,y)
K(x
,y):
Ani
sotr
opic
Geo
stat
istic
alPo
intm
easu
rem
ents
ofK
(2)
Yeh
and
Liu
[200
0]N
umer
ical
3-D
3-D
3-D
K(x
,y,z
):A
niso
trop
icG
eost
atis
tical
Stea
dySt
ate
3-D
K(x
,y,z
)K
(x,y
,z):
Ani
sotr
opic
Geo
stat
istic
alPo
intm
easu
rem
ents
ofK
(2)
Vas
coan
dK
aras
aki
[200
1]N
umer
ical
2-D
,Ful
lyPe
netr
atin
gy
2-D
,Ful
lyPe
netr
atin
gy
2-D
K(x
,z):
Faci
es-B
ased
Tra
nsie
nt,S
elec
ted
time
step
s2-
DK
(x,z
)K
(x,z
):ze
roth
-ord
erT
ikho
nov
Reg
ular
izat
ion
Vas
coan
dK
aras
aki
[200
1]Fi
eld
3-D
3-D
Tra
nsie
nt,S
elec
ted
time
step
s2-
DK
(x,z
)K
(x,z
):ze
roth
-an
dfir
st-
orde
rT
ikho
nov
Boh
ling
etal
.[20
02]
Num
eric
al3-
Dl
3-D
l2-
DK
(r,z
):A
niso
trop
icG
eost
atis
tical
0-D
Ss:
Con
stan
t
Tra
nsie
nt1-
DK
(z)
0-D
SsK
(z):
No
cons
trai
nts
onz
vari
abili
ty
Boh
ling
etal
.[20
02]
Num
eric
al3-
Dl
3-D
l2-
DK
(r,z
):A
niso
trop
icG
eost
atis
tical
0-D
Ss:
Con
stan
t
Tra
nsie
nt,S
tead
ySh
ape
1-D
K(z
)K
(z):
No
cons
trai
nts
onz
vari
abili
ty
Liu
etal
.[20
02]
Lab
orat
ory
2-D
,Ful
lyPe
netr
atin
gy
2-D
,Ful
lyPe
netr
atin
gy
2-D
K(x
,z):
Faci
es-B
ased
m
2-D
Ss(x
,z):
Faci
es-B
ased
mSt
eady
Stat
e2-
DK
(x,z
)K
(x,z
):A
niso
trop
icG
eost
atis
tical
Poin
tmea
sure
men
tof
K(1
)Li
uet
al.[
2002
]L
abor
ator
y2-
D,F
ully
Pene
trat
ing
y2-
D,F
ully
Pene
trat
ing
y2-
DK
(x,z
):Fa
cies
-Bas
edm
2-D
Ss(x
,z):
Faci
es-B
ased
mSt
eady
Stat
e2-
DK
(x,z
)K
(x,z
):A
niso
trop
icG
eost
atis
tical
Poin
tmea
sure
men
tof
K(1
)Li
uet
al.[
2002
]N
umer
ical
2-D
,Ful
lyPe
netr
atin
gy
2-D
,Ful
lyPe
netr
atin
gy
2-D
K(x
,z):
Faci
es-B
ased
Stea
dySt
ate
2-D
K(x
,z)
K(x
,z):
Ani
sotr
opic
Geo
stat
istic
alPo
intm
easu
rem
ent
ofK
(1)
Liu
etal
.[20
02]
Num
eric
al2-
D,F
ully
Pene
trat
ing
y2-
D,F
ully
Pene
trat
ing
y2-
DK
(x,z
):Fa
cies
-Bas
edSt
eady
Stat
e2-
DK
(x,z
)K
(x,z
):A
niso
trop
icG
eost
atis
tical
Poin
tmea
sure
men
tof
K(1
)V
asco
and
Fin
ster
le[2
004]
Fiel
d2-
D,F
ully
Pene
trat
ing
y2-
D,F
ully
Pene
trat
ing
yT
rans
ient
,Pul
setr
avel
time
met
rics
2-D
K(x
,z)
K(x
,z):
first
-ord
erT
ikho
nov
Reg
ular
izat
ion
Tra
cer
trav
eltim
em
etri
csLi
etal
.[20
05]
Num
eric
al2-
D,F
ully
Pene
trat
ing
z2-
D,F
ully
Pene
trat
ing
z2-
DK
(x,y
):A
niso
trop
icG
eost
atis
tical
2-D
Ss(x
,y):
Ani
sotr
opic
Geo
stat
istic
al
Tra
nsie
nt,M
omen
tsof
draw
dow
n2-
DK
(x,y
)2-
DSs
(x,y
)K
(x,y
):G
eost
atis
tical
Ss(x
,y):
Geo
stat
istic
W12518 CARDIFF AND BARRASH: 3-D UNCONFINED HYDRAULIC TOMOGRAPHY W12518
2 of 23
Liet
al.[
2005
]N
umer
ical
2-D
,Ful
lyPe
netr
atin
gz
2-D
,Ful
lyPe
netr
atin
gz
2-D
K(x
,y):
Ani
sotr
opic
Geo
stat
istic
al0-
DSs
Tra
nsie
nt,M
omen
tsof
draw
dow
n2-
DK
(x,y
)0-
DSs
K(x
,y):
Geo
stat
istic
al
Liet
al.[
2005
]N
umer
ical
2-D
,Ful
lyPe
netr
atin
gz
2-D
,Ful
lyPe
netr
atin
gz
2-D
K(x
,y):
Ani
sotr
opic
Geo
stat
istic
al0-
DSs
Tra
nsie
nt,M
omen
tsof
draw
dow
n2-
DK
(x,y
)2-
DSs
(x,y
)K
(x,y
):G
eost
atis
tical
Ss(x
,y):
Geo
stat
istic
al
Liet
al.[
2005
]N
umer
ical
2-D
,Ful
lyPe
netr
atin
gz
2-D
,Ful
lyPe
netr
atin
gz
2-D
K(x
,y):
Ani
sotr
opic
Geo
stat
istic
al2-
DSs
(x,y
):A
niso
trop
icG
eost
atis
tical
Tra
nsie
nt,M
omen
tsof
draw
dow
n2-
DK
(x,y
)0-
DSs
K(x
,y):
Geo
stat
istic
al
Liet
al.[
2005
]N
umer
ical
2-D
,Ful
lyPe
netr
atin
gz
2-D
,Ful
lyPe
netr
atin
gz
2-D
K(x
,y):
Ani
sotr
opic
Geo
stat
istic
al2-
DSs
(x,y
):A
niso
trop
icG
eost
atis
tical
Tra
nsie
nt,M
omen
tsof
draw
dow
n2-
DK
(x,y
)2-
DSs
(x,y
)K
(x,y
):G
eost
atis
tical
Ss(x
,y):
Geo
stat
istic
al
Zhu
and
Yeh
[200
5]N
umer
ical
1-D
,Ful
lyPe
netr
atin
gy,
z1-
D,F
ully
Pene
trat
ing
y,z
1-D
K(x
):G
eost
atis
tical
1-D
Ss(x
):G
eost
atis
tical
Tra
nsie
nt,S
elec
ted
time
step
s1-
DK
(x)
1-D
Ss(x
)K
(x):
Geo
stat
istic
alSs
(x):
Geo
stat
istic
alPo
intm
easu
rem
ent
ofK
(1)
Poin
tmea
sure
men
tof
Ss(1
)Zh
uan
dYe
h[2
005]
Num
eric
al3-
D3-
D3-
DK
(x,y
,z):
Ani
sotr
opic
Geo
stat
istic
al3-
DSs
(x,y
,z):
Ani
sotr
opic
Geo
stat
istic
al
Tra
nsie
nt,S
elec
ted
time
step
s3-
DK
(x,y
,z)
3-D
Ss( x
,y,z
)K
(x,y
,z):
Ani
sotr
opic
Geo
stat
istic
alSs
(x,y
,z):
Ani
sotr
opic
Geo
stat
istic
al
Poin
tmea
sure
men
tof
K(1
)Po
intm
easu
rem
ent
ofSs
(1)
Vas
coan
dK
aras
aki
[200
6]N
umer
ical
2-D
,Ful
lyPe
netr
atin
gy
2-D
,Ful
lyPe
netr
atin
gy
2-D
K(x
,z):
Faci
es-B
ased
0-D
SsT
rans
ient
,Pul
setr
avel
time
met
rics
2-D
K(x
,z)
K(x
,y):
zero
th-
and
first
-or
der
Tik
hono
vV
asco
and
Kar
asak
i[2
006]
Fiel
d3-
D3-
DT
rans
ient
,Pul
setr
avel
time
met
rics
2-D
K(x
,z)
K(x
,z):
zero
th-
and
first
-or
der
Tik
hono
vZh
uan
dYe
h[2
006]
Num
eric
al2-
D,F
ully
Pene
trat
ing
z2-
D,F
ully
Pene
trat
ing
z2-
DK
(x,y
):G
eost
atis
tical
2-D
Ss(x
,y):
Geo
stat
istic
alT
rans
ient
,Sel
ecte
dtim
est
eps
2-D
K(x
,z)
2DSs
(x,z
)K
(x,y
,z):
Geo
stat
istic
alSs
(x,y
,z):
Geo
stat
istic
alZh
uan
dYe
h[2
006]
Num
eric
al2-
D,F
ully
Pene
trat
ing
z2-
D,F
ully
Pene
trat
ing
z2-
DK
(x,y
):G
eost
atis
tical
2-D
Ss(x
,y):
Geo
stat
istic
alT
rans
ient
,Mom
ents
ofdr
awdo
wn
2-D
K(x
,z)
2-D
Ss(x
,z)
K(x
,y,z
):G
eost
atis
tical
Ss(x
,y,z
):G
eost
atis
tical
Zhu
and
Yeh
[200
6]N
umer
ical
2-D
,Ful
lyPe
netr
atin
gz
2-D
,Ful
lyPe
netr
atin
gz
2-D
K(x
,y):
Geo
stat
istic
al2-
DSs
(x,y
):G
eost
atis
tical
Tra
nsie
nt,S
elec
ted
time
step
s2-
DK
(x,z
)2-
DSs
(x,z
)K
(x,y
,z):
Geo
stat
istic
alSs
(x,y
,z):
Geo
stat
istic
alZh
uan
dYe
h[2
006]
Num
eric
al2-
D,F
ully
Pene
trat
ing
z2-
D,F
ully
Pene
trat
ing
z2-
DK
(x,y
):G
eost
atis
tical
2-D
Ss(x
,y):
Geo
stat
istic
alT
rans
ient
,Mom
ents
ofdr
awdo
wn
2-D
K(x
,z)
2-D
Ss(x
,z)
K(x
,y,z
):G
eost
atis
tical
Ss(x
,y,z
):G
eost
atis
tical
Boh
ling
etal
.[20
07]
Fiel
d3-
D3-
DT
rans
ient
1-D
K(z
)0-
DSs
K(z
):1,
5,7,
10,o
r14
Equ
al-t
hick
ness
laye
rs.
Boh
ling
etal
.[20
07]
Fiel
d3-
D3-
DT
rans
ient
,Ste
ady
Shap
e1-
DK
(z)
K(z
):1,
5,7,
10,o
r14
Equ
al-t
hick
ness
laye
rs.
Boh
ling
etal
.[20
07]
Fiel
d3-
D3-
DT
rans
ient
,Sel
ecte
dtim
est
eps
1-D
K(z
)0-
DSs
K(z
):5,
7,10
,or
13Ir
regu
lar
thic
knes
sla
yers
Lay
erth
ickn
esse
sin
form
edby
zero
-of
fset
GPR
data
Boh
ling
etal
.[20
07]
Fiel
d3-
D3-
DT
rans
ient
,Ste
ady
Shap
e1-
DK
(z)
K(z
):5,
7,10
,or
13Ir
regu
lar
thic
knes
sla
yers
Lay
erth
ickn
esse
sin
form
edby
zero
-of
fset
GPR
data
Bra
uchl
eret
al.[
2007
]N
umer
ical
3-D
l3-
Dl
1-D
K(z
):Fa
cies
-Bas
ed0-
DSs
Tra
nsie
nt,P
ulse
trav
eltim
em
etri
cs2-
DD
(r,z
)D
(r,z
):C
oars
e,ev
en-d
eter
min
edgr
id
Tab
le1.
(con
tinue
d)
Aut
hors
[Yea
r]T
ype
ofSt
udy
Pum
ping
Stim
ulat
ion
Typ
e
Hea
dR
espo
nse
Mea
sure
men
tT
ype
Tru
ePa
ram
eter
Dis
trib
utio
na,b,
cH
ead
Res
pons
eD
ata
Use
ddPa
ram
eter
sE
stim
ated
e,f
Para
met
erC
onst
rain
ts/
Reg
ular
izer
s/A
ssum
edPr
ior
Info
rmat
iong
Oth
erD
ata
Sour
ces
Use
din
Inve
rsio
n
W12518 CARDIFF AND BARRASH: 3-D UNCONFINED HYDRAULIC TOMOGRAPHY W12518
3 of 23
Bra
uchl
eret
al.[
2007
]N
umer
ical
3-D
l3-
Dl
1-D
K(z
):Fa
cies
-Bas
ed0-
DSs
Tra
nsie
nt,P
ulse
trav
eltim
em
etri
cs2-
DD
(r,z
)D
(r,z
):C
oars
e,ev
en-d
eter
min
edgr
idB
rauc
hler
etal
.[20
07]
Num
eric
al3-
Dl
3-D
l2-
DK
(r,z
):Fa
cies
-Bas
ed0-
DSs
Tra
nsie
nt,P
ulse
trav
eltim
em
etri
cs2-
DD
(r,z
)nD
(r,z
):C
oars
e,ev
en-d
eter
min
edgr
idIl
lman
etal
.[20
07]
Num
eric
al2-
D,F
ully
Pene
trat
ing
y2-
D,F
ully
Pene
trat
ing
y2-
DK
(x,z
):Fa
cies
-Bas
edSt
eady
Stat
e2-
DK
(x,z
)K
(x,z
):A
niso
trop
icG
eost
atis
tical
Poin
tmea
sure
men
tsof
KIl
lman
etal
.[20
07]
Lab
orat
ory
2-D
,Ful
lyPe
netr
atin
gy
2-D
,Ful
lyPe
netr
atin
gy
2-D
K(x
,z):
Faci
es-B
ased
m
2-D
Ss(x
,z):
Faci
es-B
ased
mSt
eady
Stat
e2-
DK
(x,z
)K
(x,z
):A
niso
trop
icG
eost
atis
tical
Liet
al.[
2007
]Fi
eld
2-D
,Ful
lyPe
netr
atin
gz
2-D
,Ful
lyPe
netr
atin
gz
Tra
nsie
nt,M
omen
tsof
draw
dow
n2-
DK
(x,y
)2-
DSs
(x,y
)K
(x,y
):G
eost
atis
tical
o
Ss(x
,y):
Geo
stat
istic
alo
Liet
al.[
2007
]Fi
eld
2-D
,Ful
lyPe
netr
atin
gz
2-D
,Ful
lyPe
netr
atin
gz
Tra
nsie
nt,M
omen
tsof
draw
dow
n2-
DK
(x,y
)0-
DSs
K(x
,y):
Geo
stat
istic
alo
Liu
etal
.[20
07]
Lab
orat
ory
2-D
,Ful
lyPe
netr
atin
gy
2-D
,Ful
lyPe
netr
atin
gy
2-D
K(x
,z):
Faci
es-B
ased
m
2-D
Ss(x
,z):
Faci
es-B
ased
mT
rans
ient
,Sel
ecte
dtim
est
eps
2-D
K(x
,z)
2-D
Ss(x
,z)
K(x
,z):
Ani
sotr
opic
Geo
stat
istic
alSs
(x,z
):A
niso
trop
icG
eost
atis
tical
Smal
l-sc
ale
data
(cor
e,sl
ugte
sts)
and
laye
rth
ickn
esse
sus
edto
estim
ate
vari
ogra
mva
rian
ces,
corr
elat
ion
leng
ths
Stra
face
etal
.[20
07a]
Fiel
d2-
D,F
ully
Pene
trat
ing
z2-
D,F
ully
Pene
trat
ing
zT
rans
ient
,Sel
ecte
dtim
est
eps
2-D
K(x
,y)
0-D
Ssh
K(x
,y):
Geo
stat
istic
alSP
data
join
tlyin
vert
edSt
rafa
ceet
al.[
2007
b]Fi
eld
2-D
,Ful
lyPe
netr
atin
gz
2-D
,Ful
lyPe
netr
atin
gz
Tra
nsie
nt,S
elec
ted
time
step
s2-
DK
(x,y
)2-
DSs
(x,y
)K
(x,y
):G
eost
atis
tical
Ss(x
,y):
Geo
stat
istic
alYe
han
dZh
u[2
007]
Num
eric
al1-
D,F
ully
Pene
trat
ing
y,z
1-D
,Ful
lyPe
netr
atin
gy,
z1-
DK
(x):
Geo
stat
istic
alSt
eady
Stat
e1-
DK
(x)
K(x
):G
eost
atis
tical
Yeh
and
Zhu
[200
7]N
umer
ical
2-D
,Ful
lyPe
netr
atin
gy
2-D
,Ful
lyPe
netr
atin
gy
2-D
K(x
,y):
Geo
stat
istic
alSt
eady
Stat
e2-
DK
(x,z
)K
(x,z
):G
eost
atis
tical
Con
serv
ativ
ean
dpa
rtiti
onin
gtr
acer
data
sequ
entia
llyin
clud
edF
iene
net
al.[
2008
]N
umer
ical
2-D
,Ful
lyPe
netr
atin
gz
2-D
,Ful
lyPe
netr
atin
gz
2-D
K(x
,y):
Geo
stat
istic
alSt
eady
Stat
e2-
DK
(x,z
)K
(x,z
):G
eost
atis
tical
i
Fie
nen
etal
.[20
08]
Num
eric
al2-
D,F
ully
Pene
trat
ing
z2-
D,F
ully
Pene
trat
ing
z2-
DK
(x,y
):Fa
cies
-Bas
edSt
eady
Stat
e2-
DK
(x,z
)K
(x,z
):G
eost
atis
tical
i,p
Fie
nen
etal
.[20
08]
Num
eric
al2-
D,F
ully
Pene
trat
ing
z2-
D,F
ully
Pene
trat
ing
z2-
DK
(x,y
):Fa
cies
-Bas
edSt
eady
Stat
e2-
DK
(x,z
)K
(x,z
):G
eost
atis
tical
i,p
Hao
etal
.[20
08]
Num
eric
al2-
D,F
ully
Pene
trat
ing
y2-
D,F
ully
Pene
trat
ing
y2-
DK
(x,z
):Fa
cies
-Bas
ed2-
DSs
(x,z
):Fa
cies
-Bas
edT
rans
ient
,Sel
ecte
dtim
est
eps
2-D
K(x
,z)
K(x
,z):
Geo
stat
istic
al
Hao
etal
.[20
08]
Num
eric
al2-
D,F
ully
Pene
trat
ing
y2-
D,F
ully
Pene
trat
ing
y2-
DK
(x,z
):Fa
cies
-Bas
ed2-
DSs
(x,z
):Fa
cies
-Bas
edT
rans
ient
,Sel
ecte
dtim
est
eps
2-D
K(x
,z)
2-D
Ss(x
,z)
K(x
,z):
Geo
stat
istic
alSs
(x,z
):G
eost
atis
tical
Hao
etal
.[20
08]
Num
eric
al2-
D,F
ully
Pene
trat
ing
y2-
D,F
ully
Pene
trat
ing
y2-
DK
(x,z
):Fa
cies
-Bas
ed2-
DSs
(x,z
):Fa
cies
-Bas
edT
rans
ient
,Sel
ecte
dtim
est
eps
2-D
K(x
,z)
2-D
Ss(x
,z)
K(x
,z):
Geo
stat
istic
alSs
(x,z
):G
eost
atis
tical
Hao
etal
.[20
08]
Num
eric
al2-
D,F
ully
Pene
trat
ing
y2-
D,F
ully
Pene
trat
ing
y2-
DK
(x,z
):Fa
cies
-Bas
ed2-
DSs
(x,z
):Fa
cies
-Bas
edT
rans
ient
,Sel
ecte
dtim
est
eps
2-D
K(x
,z)
2-D
Ss(x
,z)
K(x
,z):
Geo
stat
istic
alSs
(x,z
):G
eost
atis
tical
Illm
anet
al.[
2008
]N
umer
ical
2-D
,Ful
lyPe
netr
atin
gy
2-D
,Ful
lyPe
netr
atin
gy
2-D
K(x
,z):
Faci
es-B
ased
Stea
dySt
ate
2-D
K(x
,z)
K(x
,z):
Ani
sotr
opic
Geo
stat
istic
alIl
lman
etal
.[20
08]
Num
eric
al2-
D,F
ully
Pene
trat
ing
y2-
D,F
ully
Pene
trat
ing
y2-
DK
(x,z
):Fa
cies
-Bas
edSt
eady
Stat
e2-
DK
(x,z
)K
(x,z
):A
niso
trop
icG
eost
atis
tical
Slug
test
data
used
for
cond
ition
ing
Illm
anet
al.[
2008
]N
umer
ical
2-D
,Ful
lyPe
netr
atin
gy
2-D
,Ful
lyPe
netr
atin
gy
2-D
K(x
,z):
Faci
es-B
ased
Stea
dySt
ate
2-D
K(x
,z)
K(x
,z):
Ani
sotr
opic
Geo
stat
istic
alC
ore
sam
ples
used
for
cond
ition
ing
Tab
le1.
(con
tinue
d)
Aut
hors
[Yea
r]T
ype
ofSt
udy
Pum
ping
Stim
ulat
ion
Typ
e
Hea
dR
espo
nse
Mea
sure
men
tT
ype
Tru
ePa
ram
eter
Dis
trib
utio
na,b,
cH
ead
Res
pons
eD
ata
Use
ddPa
ram
eter
sE
stim
ated
e,f
Para
met
erC
onst
rain
ts/
Reg
ular
izer
s/A
ssum
edPr
ior
Info
rmat
iong
Oth
erD
ata
Sour
ces
Use
din
Inve
rsio
n
W12518 CARDIFF AND BARRASH: 3-D UNCONFINED HYDRAULIC TOMOGRAPHY W12518
4 of 23
Illm
anet
al.[
2008
]N
umer
ical
2-D
,Ful
lyPe
netr
atin
gy
2-D
,Ful
lyPe
netr
atin
gy
2-D
K(x
,z):
Faci
es-B
ased
Stea
dySt
ate
2-D
K(x
,z)
K(x
,z):
Ani
sotr
opic
Geo
stat
istic
alSi
ngle
-hol
ete
sts
used
for
cond
ition
ing
Illm
anet
al.[
2008
]L
abor
ator
y2-
D,F
ully
Pene
trat
ing
y2-
D,F
ully
Pene
trat
ing
y2-
DK
(x,z
):Fa
cies
-Bas
edm
2-D
Ss(x
,z):
Faci
es-B
ased
mSt
eady
Stat
e2-
DK
(x,z
)K
(x,z
):A
niso
trop
icG
eost
atis
tical
Illm
anet
al.[
2008
]L
abor
ator
y2-
D,F
ully
Pene
trat
ing
y2-
D,F
ully
Pene
trat
ing
y2-
DK
(x,z
):Fa
cies
-Bas
edm
2-D
Ss(x
,z):
Faci
es-B
ased
mSt
eady
Stat
e2-
DK
(x,z
)K
(x,z
):A
niso
trop
icG
eost
atis
tical
Slug
test
data
used
for
cond
ition
ing
Illm
anet
al.[
2008
]L
abor
ator
y2-
D,F
ully
Pene
trat
ing
y2-
D,F
ully
Pene
trat
ing
y2-
DK
(x,z
):Fa
cies
-Bas
edm
2-D
Ss(x
,z):
Faci
es-B
ased
mSt
eady
Stat
e2-
DK
(x,z
)K
(x,z
):A
niso
trop
icG
eost
atis
tical
Cor
esa
mpl
esus
edfo
rco
nditi
onin
gIl
lman
etal
.[20
08]
Lab
orat
ory
2-D
,Ful
lyPe
netr
atin
gy
2-D
,Ful
lyPe
netr
atin
gy
2-D
K(x
,z):
Faci
es-B
ased
m
2-D
Ss(x
,z):
Faci
es-B
ased
mSt
eady
Stat
e2-
DK
(x,z
)K
(x,z
):A
niso
trop
icG
eost
atis
tical
Sing
le-h
ole
test
sus
edfo
rco
nditi
onin
gK
uhlm
anet
al.[
2008
]N
umer
ical
2-D
,Ful
lyPe
netr
atin
gz
2-D
,Ful
lyPe
netr
atin
gz
2-D
K(x
,y):
Geo
stat
istic
al2-
DSs
(x,y
):G
eost
atis
tical
Tra
nsie
nt2-
DK
(x,z
)K
(x,y
):G
eost
atis
tical
Kuh
lman
etal
.[20
08]
Num
eric
al2-
D,F
ully
Pene
trat
ing
z2-
D,F
ully
Pene
trat
ing
z2-
DK
(x,y
):G
eost
atis
tical
2-D
Ss(x
,y):
Geo
stat
istic
alT
rans
ient
2-D
K(x
,z)
K(x
,y):
Geo
stat
istic
al
Kuh
lman
etal
.[20
08]
Num
eric
al2-
D,F
ully
Pene
trat
ing
z2-
D,F
ully
Pene
trat
ing
z2-
DK
(x,y
):G
eost
atis
tical
2-D
Ss(x
,y):
Geo
stat
istic
alT
rans
ient
2-D
K(x
,z)
K(x
,y):
Geo
stat
istic
al
Kuh
lman
etal
.[20
08]
Num
eric
al2-
D,F
ully
Pene
trat
ing
z2-
D,F
ully
Pene
trat
ing
z2-
DK
(x,y
):G
eost
atis
tical
2-D
Ss(x
,y):
Geo
stat
istic
alT
rans
ient
2-D
K(x
,z)
K(x
,y):
Geo
stat
istic
al
Kuh
lman
etal
.[20
08]
Num
eric
al2-
D,F
ully
Pene
trat
ing
z2-
D,F
ully
Pene
trat
ing
z2-
DK
(x,y
):G
eost
atis
tical
2-D
Ss(x
,y):
Geo
stat
istic
alT
rans
ient
2-D
K(x
,z)
K(x
,y):
Geo
stat
istic
al
Liet
al.[
2008
]Fi
eld
2-D
,Ful
lyPe
netr
atin
gz
2-D
,Ful
lyPe
netr
atin
gz
Stea
dySt
ate
3-D
K(x
,y,z
)K
(x,y
,z):
Geo
stat
istic
al
Liet
al.[
2008
]Fi
eld
2-D
,Ful
lyPe
netr
atin
gz
2-D
,Ful
lyPe
netr
atin
gz
Stea
dySt
ate
3-D
K(x
,y,z
)K
(x,y
,z):
Geo
stat
istic
alFl
owm
eter
data
join
tlyin
vert
edto
add
3-D
vari
abili
tyin
form
atio
nV
asco
[200
8]N
umer
ical
2-D
,Ful
lyPe
netr
atin
gz
2-D
,Ful
lyPe
netr
atin
gz
2-D
K(x
,y):
Geo
stat
istic
al0-
DSs
Tra
nsie
nt,P
ulse
trav
eltim
em
etri
cs2-
DK
(x,y
)K
(x,y
):ze
roth
-an
dfir
st-
orde
rT
ikho
nov
Vas
co[2
008]
Fiel
d2-
D,F
ully
Pene
trat
ing
z2-
D,F
ully
Pene
trat
ing
zT
rans
ient
,Pul
setr
avel
time
met
rics
2-D
K(x
,y)
0-D
Ssh
K(x
,y):
zero
th-
and
first
-or
der
Tik
hono
vB
ohlin
g[2
009]
Fiel
d3-
D3-
DT
rans
ient
,Sel
ecte
dtim
est
eps
2-D
K(r
,z)q
2-D
Ss(r
,z)q
K(r
,z):
SVD
r
Boh
ling
[200
9]Fi
eld
3-D
3-D
Tra
nsie
nt,S
tead
ySh
ape
2-D
K(r
,z)q
K(r
,z):
SVD
r
Car
diff
etal
.[20
09]
Num
eric
al2-
D,F
ully
Pene
trat
ing
z2-
D,F
ully
Pene
trat
ing
z2-
DK
(x,z
):G
eost
atis
tical
Stea
dySt
ate
2-D
K(x
,z)
K(x
,z):
Geo
stat
istic
al
Car
diff
etal
.[20
09]
Fiel
d2-
D,F
ully
Pene
trat
ing
z2-
D,F
ully
Pene
trat
ing
zSt
eady
Stat
e2-
DK
(x,z
)K
(x,z
):G
eost
atis
tical
i
Cas
tagn
aan
dB
ellin
[200
9]N
umer
ical
2-D
,Ful
lyPe
netr
atin
gy
2-D
,Ful
lyPe
netr
atin
gy
2-D
K(x
,z):
Ani
sotr
opic
Geo
stat
istic
alT
rans
ient
,Pul
setr
avel
time
met
rics
2-D
D(r
,z)
D(r
,z):
14Pi
lotP
oint
smPo
intm
easu
rem
ents
ofK
(10)
Cas
tagn
aan
dB
ellin
[200
9]N
umer
ical
2-D
,Ful
lyPe
netr
atin
gy
2-D
,Ful
lyPe
netr
atin
gy
2-D
K(x
,z):
Ani
sotr
opic
Geo
stat
istic
alT
rans
ient
,Pul
setr
avel
time
met
rics
2-D
D(r
,z)
D(r
,z):
14Pi
lotP
oint
smPo
intm
easu
rem
ents
ofK
(10)
Cas
tagn
aan
dB
ellin
[200
9]N
umer
ical
2-D
,Ful
lyPe
netr
atin
gy
2-D
,Ful
lyPe
netr
atin
gy
2-D
K(x
,z):
Ani
sotr
opic
Geo
stat
istic
alT
rans
ient
,Pul
setr
avel
time
met
rics
2-D
K(r
,z)
0-D
SsD
(r,z
):14
Pilo
tPoi
ntsm
Poin
tmea
sure
men
tsof
K(1
0)C
asta
gna
and
Bel
lin[2
009]
Num
eric
al3-
D3-
D3-
DK
(x,y
,z):
Ani
sotr
opic
Geo
stat
istic
al0-
DSs
Tra
nsie
nt,P
ulse
trav
eltim
em
etri
cs3-
DK
(x,y
,z)
0-D
Ssh
D(r
,z):
14Pi
lotP
oint
smPo
intm
easu
rem
ents
ofK
(10)
Cas
tagn
aan
dB
ellin
[200
9]N
umer
ical
3-D
3-D
3-D
K(x
,y,z
):A
niso
trop
icG
eost
atis
tical
0-D
Ss
Tra
nsie
nt,P
ulse
trav
eltim
em
etri
cs3-
DK
(x,y
,z)
0-D
Ssh
D(r
,z):
14Pi
lotP
oint
sm
Tab
le1.
(con
tinue
d)
Aut
hors
[Yea
r]T
ype
ofSt
udy
Pum
ping
Stim
ulat
ion
Typ
e
Hea
dR
espo
nse
Mea
sure
men
tT
ype
Tru
ePa
ram
eter
Dis
trib
utio
na,b,
cH
ead
Res
pons
eD
ata
Use
ddPa
ram
eter
sE
stim
ated
e,f
Para
met
erC
onst
rain
ts/
Reg
ular
izer
s/A
ssum
edPr
ior
Info
rmat
iong
Oth
erD
ata
Sour
ces
Use
din
Inve
rsio
n
W12518 CARDIFF AND BARRASH: 3-D UNCONFINED HYDRAULIC TOMOGRAPHY W12518
5 of 23
Illm
anet
al.[
2009
]Fi
eld
3-D
3-D
Tra
nsie
nt,S
elec
ted
time
step
s3-
DK
(x,y
,z)
3-D
Ss(x
,y,z
)K
(x,y
,z):
Geo
stat
istic
alSs
(x,y
,z):
Geo
stat
istic
alN
ieta
l.[2
009]
Num
eric
al2-
D,F
ully
Pene
trat
ing
z2-
D,F
ully
Pene
trat
ing
z2-
DK
(x,y
):G
eost
atis
tical
Stea
dySt
ate
2-D
K(x
,y)
K(x
,y):
Geo
stat
istic
alPo
intm
easu
rem
ents
ofK
(27)
Sun
etal
.[20
09]
Num
eric
al2-
D,F
ully
Pene
trat
ing
z2-
D,F
ully
Pene
trat
ing
z2-
DK
(x,y
):Fa
cies
0-D
SsT
rans
ient
,Sel
ecte
dtim
est
epsh
2-D
K(x
,y)
K(x
,y):
Ens
embl
eof
mul
ti-po
intg
eost
atis
tical
imag
esPo
intm
easu
rem
ents
ofK
(30)
Sun
etal
.[20
09]
Num
eric
al2-
D,F
ully
Pene
trat
ing
z2-
D,F
ully
Pene
trat
ing
z2-
DK
(x,y
):Fa
cies
0-D
SsT
rans
ient
,Sel
ecte
dtim
est
epsh
2-D
K(x
,y)
K(x
,y):
Ens
embl
eof
mul
ti-po
intg
eost
atis
tical
imag
esPo
intm
easu
rem
ents
ofK
(15)
Sun
etal
.[20
09]
Num
eric
al2-
D,F
ully
Pene
trat
ing
z2-
D,F
ully
Pene
trat
ing
z2-
DK
(x,y
):Fa
cies
0-D
SsT
rans
ient
,Sel
ecte
dtim
est
epsh
2-D
K(x
,y)
K(x
,y):
Ens
embl
eof
mul
ti-po
intg
eost
atis
tical
imag
esX
iang
etal
.[20
09]
Num
eric
al2-
D,F
ully
Pene
trat
ing
z2-
D,F
ully
Pene
trat
ing
z2-
DK
(x,y
):G
eost
atis
tical
2-D
Ss(x
,y):
Geo
stat
istic
alT
rans
ient
,Sel
ecte
dtim
est
eps
2-D
K(x
,z)
2-D
Ss(x
,z)
K(x
,y):
Ani
sotr
opic
Geo
stat
istic
alSs
(x,y
):A
niso
trop
icG
eost
atis
tical
Xia
nget
al.[
2009
]L
abor
ator
y2-
D,F
ully
Pene
trat
ing
y2-
D,F
ully
Pene
trat
ing
y2-
DK
(x,z
):Fa
cies
-Bas
ed2-
DSs
(x,z
):Fa
cies
-Bas
edT
rans
ient
,Sel
ecte
dtim
est
eps
2-D
K(x
,z)
2-D
Ss(x
,z)
K(x
,z):
Ani
sotr
opic
Geo
stat
istic
alSs
(x,z
):A
niso
trop
icG
eost
atis
tic
Poin
tmea
sure
men
tsof
Kan
dSs
Yin
and
Illm
an[2
009]
Lab
orat
ory
2-D
,Ful
lyPe
netr
atin
gy
2-D
,Ful
lyPe
netr
atin
gy
2-D
K(x
,z):
Faci
es-B
ased
2-D
Ss(x
,z):
Faci
es-B
ased
Tra
nsie
nt,M
omen
tsof
draw
dow
n2-
DK
(x,z
)2-
DSs
(x,z
)K
(x,z
):A
niso
trop
icG
eost
atis
tical
Ss(x
,z):
Ani
sotr
opic
Geo
stat
istic
alB
ohlin
gan
dB
utle
r[2
010]
Num
eric
al2-
D,F
ully
Pene
trat
ing
z2-
D,F
ully
Pene
trat
ing
z2-
DK
(x,y
):Fa
cies
-Bas
ed2-
DSs
(x,y
):Fa
cies
-Bas
edT
rans
ient
2-D
K(x
,y)
2-D
Ss(x
,y)
K(x
,y):
177
Pilo
tpoi
nts
with
regu
lari
zatio
nSs
(x,y
):17
7Pi
lotp
oint
sw
ithre
gula
riza
tion
Boh
ling
and
But
ler
[201
0]N
umer
ical
3-D
3-D
3-D
K(x
,y,z
):G
eost
atis
tical
Stea
dySt
ate
3-D
K(x
,y,z
)K
(x,y
,z):
Coa
rsen
edgr
idan
dsm
alld
egre
eof
zero
th-o
rder
Tik
hono
vto
war
dpr
ior
mod
elB
rauc
hler
etal
.[20
10]
Fiel
d3-
D3-
DT
rans
ient
,Pul
setr
avel
time
met
rics
2-D
D(r
,z)s
D(r
,z):
Coa
rse,
even
-de
term
ined
grid
Illm
anet
al.[
2010
a]L
abor
ator
y2-
D,F
ully
Pene
trat
ing
y2-
D,F
ully
Pene
trat
ing
y2-
DK
(x,z
):Fa
cies
-Bas
ed2-
DSs
(x,z
):Fa
cies
-Bas
edSt
eady
Stat
e2-
DK
(x,z
)K
(x,z
):G
eost
atis
tical
Illm
anet
al.[
2010
b]L
abor
ator
y2-
D,F
ully
Pene
trat
ing
y2-
D,F
ully
Pene
trat
ing
y2-
DK
(x,z
):Fa
cies
-Bas
ed2-
DSs
(x,z
):Fa
cies
-Bas
edSt
eady
Stat
e2-
DK
(x,z
)K
(x,z
):G
eost
atis
tical
Liu
and
Kita
nidi
s[2
011]
Lab
orat
ory
2-D
,Ful
lyPe
netr
atin
gy
2-D
,Ful
lyPe
netr
atin
gy
2-D
K(x
,z):
Faci
es-B
ased
2-D
Ss(x
,z):
Faci
es-B
ased
Stea
dySt
ate
2-D
K(x
,z)
K(x
,z):
first
-ord
erT
ikho
nov
Bra
uchl
eret
al.[
2011
]Fi
eld
3-D
3-D
Tra
nsie
nt,P
ulse
trav
eltim
em
etri
cs2-
DD
(r,z
)2-
DSs
(r,z
)D
(r,z
):C
oars
e,ev
en-
dete
rmin
edgr
idSs
(r,z
):C
oars
e,ev
en-
dete
rmin
edgr
idB
rauc
hler
etal
.[20
11]
Fiel
d3-
D3-
DT
rans
ient
,Pul
setr
avel
time
met
rics
3-D
D(x
,y,z
)3-
DSs
(x,y
,z)
D(x
,y,z
):C
oars
e,ev
en-
dete
rmin
edgr
idSs
(r,z
):C
oars
e,ev
en-
dete
rmin
edgr
idB
erg
and
Illm
an[2
011a
]L
abor
ator
y2-
D2-
D2-
DK
(x,z
):Fa
cies
-Bas
ed2-
DSs
(x,z
):Fa
cies
-Bas
edT
rans
ient
,Sel
ecte
dtim
est
eps
2-D
K(x
,z)
2-D
Ss(x
,z)
K(x
,z):
Geo
stat
istic
alSs
(x,z
):G
eost
atis
tical
Tab
le1.
(con
tinue
d)
Aut
hors
[Yea
r]T
ype
ofSt
udy
Pum
ping
Stim
ulat
ion
Typ
e
Hea
dR
espo
nse
Mea
sure
men
tT
ype
Tru
ePa
ram
eter
Dis
trib
utio
na,b,
cH
ead
Res
pons
eD
ata
Use
ddPa
ram
eter
sE
stim
ated
e,f
Para
met
erC
onst
rain
ts/
Reg
ular
izer
s/A
ssum
edPr
ior
Info
rmat
iong
Oth
erD
ata
Sour
ces
Use
din
Inve
rsio
n
W12518 CARDIFF AND BARRASH: 3-D UNCONFINED HYDRAULIC TOMOGRAPHY W12518
6 of 23
Ber
gan
dIl
lman
[201
1b]
Fiel
d3-
D3-
DT
rans
ient
,Sel
ecte
dtim
est
eps
3-D
K(x
,y,z
)3-
DSs
(x,y
,z)
K(x
,y,z
):G
eost
atis
tical
Ss(x
,y,z
):G
eost
atis
tical
Hua
nget
al.[
2011
]Fi
eld
2-D
2-D
Stea
dySt
ate
2-D
K(x
,y)
K(x
,y):
Geo
stat
istic
al
a Unl
ess
othe
rwis
esp
ecifi
ed,‘
‘Geo
stat
istic
al’’
prio
rin
form
atio
nm
eans
assu
min
ga
seco
nd-o
rder
stat
iona
ryra
ndom
field
with
allv
ario
gram
para
met
ers
fixed
.bW
ithre
gard
sto
desc
ript
ion
ofth
etr
uehe
tero
gene
ity,‘
‘Fac
ies-
Bas
ed’’
isus
edto
desc
ribe
para
met
erfie
lds
whe
reth
em
ajor
com
pone
ntof
vari
abili
tyis
due
toch
ange
sin
prop
ertie
sat
boun
dari
es(e
.g.,
geol
ogic
laye
rs).
Of
cour
se,s
ome
vari
abili
tym
ayal
sobe
incl
uded
with
infa
cies
[e.g
.,H
aoet
al.,
2008
].c Fo
rfie
ldex
peri
men
ts,t
rue
para
met
erdi
stri
butio
nsca
nnot
gene
rally
bede
fined
and
are
thus
notp
opul
ated
(tho
ugh
man
yau
thor
sha
veco
rrel
ated
HT
data
agai
nste
xist
ing
data
).dFo
rhe
adre
spon
seda
taus
ed,‘
‘Tra
nsie
nt’’
alon
ere
fers
toa
full
orne
arly
full
set
oftr
ansi
enth
ead
mea
sure
men
ts(i
.e.,
adr
awdo
wn
curv
e),‘
‘Sel
ecte
dtim
est
eps’
’re
fers
toa
smal
lsu
bset
ofhe
adm
easu
rem
ents
sele
cted
per
tran
sien
tre
cord
,‘‘M
omen
tsof
draw
dow
n’’
refe
rsto
calc
ulat
edte
mpo
ral
mom
ents
oftr
ansi
entr
ecor
ds,a
nd‘‘P
ulse
trav
eltim
em
etri
cs’’
refe
rsto
phas
e,am
plitu
de,a
nd/o
rar
riva
ltim
em
etri
csfo
rpr
es-
sure
puls
es.
e Inm
any
case
sof
2-D
inve
rsio
nor
2-D
sam
ple
prob
lem
s,th
ehy
drol
ogic
para
met
ers
wer
ere
ferr
edto
asT
rans
mis
sivi
ty(T
)an
dSt
orat
ivity
(S).
We
assu
me
inth
ese
case
sth
atth
eaq
uife
rha
sun
ifor
mth
ickn
ess
ban
dca
nth
usbe
wri
tten
as2-
DK¼
2-D
T/b,
and
2-D
Ss¼
2-D
S/b,
f Dre
fers
tohy
drau
licdi
ffus
ivity
,K/S
s.gIn
this
tabl
e,ze
roth
-ord
erT
ikho
nov
regu
lari
zatio
nre
fers
tore
gula
riza
tion
that
cont
ains
ate
rmpe
naliz
ing
dist
ance
betw
een
para
met
eres
timat
es
and
ast
artin
gm
odel
.firs
t-or
der
and
seco
nd-o
rder
Tik
hono
vre
g-ul
ariz
atio
nre
fers
tore
gula
riza
tion
cont
aini
nga
term
that
pena
lizes
first
-der
ivat
ive
and
seco
nd-d
eriv
ativ
em
easu
res
ofth
epa
ram
eter
field
usin
gap
prop
riat
ero
ughe
ning
mat
rice
s.hT
estp
robl
em4.
Inth
isco
mpa
riso
npa
per,
inve
rsio
nm
etho
dsva
ried
byco
ntri
buto
r.i V
aria
nce
ofva
riog
ram
estim
ated
aspa
rtof
inve
rsio
n.j T
estp
robl
em3.
Inth
isco
mpa
riso
npa
per,
inve
rsio
nm
etho
dsva
ried
byco
ntri
buto
r.kIn
form
atio
nno
tfou
ndin
artic
le.
l 3-D
sim
ulat
ions
used
ara
dial
lysy
mm
etri
cm
odel
,eff
ectiv
ely
only
allo
win
gpu
mpi
ngef
fect
san
dm
easu
rem
entt
ova
ryin
ran
dz.
mT
rue
valu
esun
know
nin
lab
expe
rim
ents
,but
assu
med
tofo
llow
dist
ribu
tion
ofsa
ndpa
ckin
g.nH
ydra
ulic
diff
usiv
ityes
timat
eddu
ring
inve
rsio
n,fo
llow
edby
zona
tion
and
estim
atio
nof
cons
tant
Kan
dSs
.oG
eost
atis
tical
vari
ance
and
corr
elat
ion
leng
thes
timat
edas
part
ofin
vers
ion.
pT
hres
hold
ing
used
tode
linea
tefa
cies
afte
rge
osta
tistic
alin
vers
ion.
qH
eter
ogen
eity
ina
2-D
(x,z
)pl
ane
was
map
ped
to(r
,z)
plan
esfo
rdi
ffer
entt
ests
.r Im
ages
ofhe
tero
gene
ityw
ere
notp
rodu
ced,
butS
VD
was
sugg
este
d,w
hich
wou
ldpr
oduc
em
inim
um-l
engt
hso
lutio
ns(z
erot
h-or
der
Tik
hono
vfo
rth
elim
itof
0da
tava
rian
ce).
s Four
nonj
oint
2-D
inve
rsio
nsw
ere
perf
orm
ed,c
ompa
red
for
cons
iste
ncy
in3-
D.
Tab
le1.
(con
tinue
d)
Aut
hors
[Yea
r]T
ype
ofSt
udy
Pum
ping
Stim
ulat
ion
Typ
e
Hea
dR
espo
nse
Mea
sure
men
tT
ype
Tru
ePa
ram
eter
Dis
trib
utio
na,b,
cH
ead
Res
pons
eD
ata
Use
ddPa
ram
eter
sE
stim
ated
e,f
Para
met
erC
onst
rain
ts/
Reg
ular
izer
s/A
ssum
edPr
ior
Info
rmat
iong
Oth
erD
ata
Sour
ces
Use
din
Inve
rsio
n
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only peer-reviewed papers presenting numerical, laboratory,or field experiments in which a series of pumping tests areused to stimulate an aquifer response and in which a numberof pressure responses are jointly inverted to produce imagesof aquifer heterogeneity. Basic characterization approachessuch as curve matching for individual pumping or slug inter-ference tests are not listed since they do not jointly fit alldata. Likewise, while the governing equations and stimula-tions are related in ERT, pneumatic, and other tomographicmethods, these are not listed since they are not directly con-trolled by the same physical parameters and have differenterrors, uncertainties, stimulation magnitudes, and practicalimplementation constraints. We believe this table capturesthe range of important research in HT during the past 15years, and may help to illuminate areas for future advancesin HT. While every effort was made to ensure the accuracyand completeness of this table (e.g., by contacting at leastone author from each paper in this table), we apologize inadvance for any errors or omissions in this summary. Somelessons can be gleaned from the summary of research pre-sented in Table 1, and are discussed below.
[4] In terms of inversion methods used, geostatisticallybased approaches based on the works of Yeh et al. [1995,1996] or of Kitanidis and Vomvoris [1983] and Kitanidis[1995] appear to be the most popular by far. This can beattributed to a variety of reasons—including software avail-ability, for example—but one key factor may be the factthat these methods have analytical solutions for linear for-ward problems (i.e., they consist of linear optimizations)and have generally been shown to perform well for gradi-ent-based optimization in nonlinear problems. In addition,the use of these methods in a Bayesian interpretation allowsfor calculation of linearized uncertainty metrics for invertedimages, including posterior variance estimates and condi-tional realizations. While research into more computation-ally complex, novel methods for inversion [e.g., Caers,2003; Fienen et al., 2008; Cardiff and Kitanidis, 2009] willalways be valuable for providing alternative interpretationswhen geostatistical assumptions are violated, and for avoid-ing an inversion ‘‘monoculture,’’ the geostatistical approachto the inverse problem appears for the time being to beamong the most practical, realistic, and flexible.
[5] While all aquifers are doubtlessly 3-D, the summaryof research also suggests that analyzing HT data for 3-Dheterogeneity is a daunting challenge, though the computa-tional requirements are more easily met with each passingyear. Whether numerical or field studies, only a handful ofworks have utilized HT data to image 3-D heterogeneity ofaquifer hydraulic conductivity [Yeh and Liu, 2000; Zhu andYeh, 2005; Li et al., 2008; Castagna and Bellin, 2009; Ill-man et al., 2009; Bohling and Butler, 2010; Brauchleret al., 2011; Berg and Illman, 2011b]. Of these, only theworks of Zhu and Yeh [2005], Illman et al. [2009], and Bergand Illman [2011b] have also sought to image 3-D heteroge-neity in aquifer storage parameters. Likewise, as far as we areaware, there are no laboratory studies in which HT data wasutilized to image 3-D heterogeneity in aquifer parameters.
[6] The summary also shows how HT applications havematured recently, in the sense of moving from syntheticexperiments to actual application. While there are relativelyfew papers that present tomographic analyses of actual fielddata from 2DHT or 3DHT data collection campaigns, there
has been a marked increase in field applications in the past5 years. However, there are still relatively few papers inwhich 3-D aquifer pumping stimulations and 3-D pressureresponses have been used as a data source [Vasco and Kara-saki, 2006; Bohling et al., 2007; Bohling, 2009; Illman et al.,2009; Brauchler et al., 2010; Berg and Illman, 2011b], andwe are aware of only three very recent works in which 3DHTfield data was utilized to image full 3-D heterogeneity in aqui-fer parameters [Illman et al., 2009; Brauchler et al., 2011;Berg and Illman, 2011b].
[7] Finally, even though analysis of unconfined aquifersis an important venture (especially for purposes of contami-nant transport monitoring and remediation), the HT papersto date have focused on analyzing confined scenarios orignored changes in aquifer saturated thickness.
[8] As pointed out by Bohling and Butler [2010], the fieldeffort associated with installing 3DHT equipment and oper-ating 3DHT tests can be very high, especially if a large num-ber of tests are required and if the tests must be operated forlong periods of time (e.g., to approximate steady state).Efforts to employ HT in the field and especially in uncon-fined aquifers have also had to deal with numerous con-straints and ‘‘nuisance’’ effects that are often not consideredin numerical experiments, and which may be difficult toanalyze with existing theoretical methods. These include,among others, the following:
[9] 1. Inability to obtain high pumping rates due to cavi-tation concerns.
[10] 2. Surface pump suction limits.[11] 3. Lowering of the water level in-well below the
pumping interval.[12] 4. Existence of unsteady and difficult to characterize
nonpumping stresses such as river stage changes or evapo-transpiration, which may make short testing campaignsdesirable.
[13] 5. Inability to reach ‘‘steady state’’ in a reasonableamount of time per test.
[14] 6. Changes in saturated thickness in unconfinedaquifers due to pumping, and accompanying drawdowncurve response.
[15] The purpose of this paper is to present and test aniterative, practical protocol for 3-D transient hydraulic to-mography (3DTHT) and to investigate the performance ofthe method under realistic field constraints encountered inunconfined aquifers. Specifically, the methodology devel-oped and analysis of the synthetic HT results presented inthis paper are geared toward application of HT at the BoiseHydrogeophysical Research Site (BHRS), an unconfined,high permeability sand-and-gravel aquifer adjacent to theBoise River that serves as a test bed for hydrologic andgeophysical characterization methods [Barrash and Clemo,2002]. The methodology reflects the fact that the nuisanceeffects listed above (low attainable pumping rates, longtimes to achieve steady state, etc.) have been encounteredduring implementation of 3DTHT at the BHRS, and maybe common during actual implementation of 3DTHT as acharacterization method at similar contaminated sites.While the modeling results in this paper focus on a syn-thetic case with known heterogeneity, they utilize designparameters and aquifer parameters that are similar to theBHRS instrumentation and aquifer, respectively. In thissense, this paper evaluates the promise of 3DTHT for
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application at the BHRS and in similar shallow, unconfinedpermeable aquifers—an important area for characteriza-tion, given the widespread use of shallow, unconfined flu-vial aquifers and the relative ease with which these aquiferscan be contaminated.
[16] Our synthetic experiments in this paper are the firstwe are aware of that investigate 3DTHT in an unconfinedaquifer. In addition, we have made efforts to incorporate re-alistic restrictions that have been encountered in 3DTHTinvestigations at the BHRS in order to provide a realisticassessment of the resolution and uncertainty that can beexpected from 3DHT imaging. The restrictions imposedinclude a lack of ‘‘near-field’’ boundary conditions, re-stricted pumping rates, and short pumping tests (required toallow sufficient numbers of tests to be carried out in a rea-sonable period of time). In particular, for realistic field datacollection, short pumping tests may be most useful in thatthey allow greater spatial coverage (due to the ability toperform more testing configurations under set time con-straints), and they reduce the likelihood that other naturalor anthropogenic stresses will contribute significantly to aq-uifer response during the pumping test period. Our applica-tion is perhaps most similar to the work of Zhu and Yeh[2005], the only other published 3DTHT synthetic experi-ment we are aware of that performed 3-D imaging of bothconductivity and storage aquifer parameters. Under theconditions mentioned above, we seek to answer the follow-ing questions:
[17] 1. To what extent is aquifer heterogeneity imagingsuccessful when using relatively short, low flow rate tests?
[18] 2. If storage parameters (Ss and Sy) are relativelyconstant throughout an aquifer, can they be estimated alongwith 3-D K variations via 3DTHT?
[19] 3. If storage parameters vary throughout an aquifer,can the spatial variability in all three parameters (K; Ss;and Sy) be accurately estimated (in the sense that their esti-mates are unbiased and uncertainty estimates are approxi-mately correct)?
[20] In addition to simply assessing, under realisticrestrictions, the inversion of transient 3DTHT data for esti-mates of aquifer hydraulic conductivity and storage param-eters, we also seek to answer some open questions withregard to conceptual modeling errors when inverting earlytime 3DTHT data for unconfined scenarios. Specifically:
[21] 1. If storage parameters vary throughout an aquifer,does assuming constant but unknown values during inver-sion degrade estimates of K?
[22] 2. What errors are introduced by assuming station-ary geostatistics when discrete geologic facies (e.g., layer-ing) are the most prominent form of K variability?
[23] While the answers to these questions are no doubtproblem dependent to some extent, we believe the samplecases contained in this paper begin to answer these ques-tions. Likewise, in order to allow these questions to be morefully explored, the models utilized in this paper are availableon request from the authors. It should be noted at this pointthat the investigation in this paper focuses on the effects ofconceptual modeling errors but assumes that relativelynoise-free, high quality data can be obtained. In that sense,the results presented in this paper represent ‘‘best case’’answers to the questions posed above, and degradation of ac-curacy with large measurement errors should be expected.
2. Statement of the Problem[24] The questions discussed above are investigated for a
synthetic, heterogeneous unconfined aquifer with relativelyhigh permeability (common for sand-and-gravel systems)and using field-attainable pumping rates and measurementconfigurations. The basic description of the assumed gov-erning equations for groundwater flow, and the size anddiscretization of the numerical model, are described belowin sections 2.1 and 2.2, respectively. This model is then uti-lized to analyze transient 3DTHT performance under anumber of analysis cases, as discussed in section 4.1.
2.1. Governing Equations and Numerical Model[25] We consider groundwater flow under saturated but
unconfined conditions, in which the water table is repre-sented as a free surface and in which the drainage dealt withat the free surface is fast enough to be considered ‘‘instanta-neous’’ for the given testing protocol.
[26] Under these approximations, within the saturatedportion of the aquifer, the governing equations for satu-rated, unconfined groundwater flow with minimal spatialdensity gradients applies :
Ss@h@t¼ @
@xK@h@x
� �þ @
@yK@h@y
� �þ @
@zK@h@z
� �þw for 0< z< �;
(1)
where h is hydraulic head [L], Ss is specific storage ½1=L�, Kis hydraulic conductivity ½L=T �, and w represents any sour-ces or sinks of water in terms of volumetric flow rates perunit volume ½½L3=T �=L3�, and where z ¼ 0 represents thebase of the aquifer and z¼ � represents the location of thewater table. The coefficients Ss and K are considered vari-able in space, and the coefficient w may be variable in bothspace and time. A no-flux boundary condition is assumedat the base of the aquifer, i.e.,
@h@z¼ 0 for z¼ 0: (2)
At the lateral boundaries of the aquifer (i.e., for x and ylocations far from the area being studied) we assume con-stant-head boundaries, i.e.,
h¼ ho at �d ; (3)
where ho is a constant head value [L] and Gd represents theset of constant head (Dirichlet) boundaries. While we havechosen to use constant head boundaries, other boundaryconditions may easily be employed. The elevation of thewater table � is treated as a dependent variable, and islinked to the head distribution in that it is the locationwhere h ¼ z (assuming pressure head is measured as a devi-ation from atmospheric pressure). Likewise, h and � arelinked in that a unit drop in � over a unit area results in aproportional volumetric input of Sy [�] to the saturatedzone. To solve the governing equations, we use the popularMODFLOW [Harbaugh, 2005] numerical model underconditions where numerical cells of the model are allowedto drain and water table movement is thus tracked.
[27] It should be noted that while, undoubtedly, water ta-ble response is dependent on both fast and slow unsaturated
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zone drainage, the instantaneous drainage assumption uti-lized by the standard MODFLOW groundwater flow process(and further discussed by Harbaugh [2005]) is a useful andpractical approximation for many unconfined aquifers whereeither one of the two following conditions are met:
[28] The unconfined aquifer is coarse-grained enoughthat during a head drop/pumping test the full effective po-rosity is near instantaneously drained (relative to the speedof the head drop); or
[29] A percentage of the unconfined aquifer’s effectiveporosity drains quickly relative to the speed of the headdrop/pumping test, and the rest of the effective porositycontributes a negligible flux during the time period of thehead drop/pumping test.
[30] That said, while the standard saturated-flow MOD-FLOW approximations are useful and can result in fastermodel runtimes, such an approach is expected to be inaccu-rate for estimating long-term specific yield when delayeddrainage in the vadose zone is important and unaccountedfor [see, e.g., Nwankwor et al., 1984; Narasimhan and Zhu,1993; Endres et al., 2007; Tartakovsky and Neuman, 2007;Moench, 2008; Mishra and Neuman, 2010, 2011]. In thiscase of important delayed drainage, short-term pumpingtests such as those discussed herein are expected to returnlow estimates of true aquifer specific yield and may bethought of as an ‘‘effective’’ specific yield, representative ofvolume balances only at ‘‘early times’’ [Nwankwor et al.,
1984], i.e., time scales comparable to the 3DTHT pumpingtests. In cases where characterization of true aquifer specificyield is of crucial importance, our approach should not beapplied, and a variably saturated flow model should beused, though this is expected to add significant computa-tional effort (due to increased model nonlinearity) and addsthe additional need of estimating pressure/saturation andsaturation/relative permeability curve parameters, whichmust also be considered as possibly spatially variable, andwhose expected spatial distributions are very poorly under-stood at this time.
[31] In this work we will study the identifiability of thehydraulic conductivity variability (K), in particular, underthis conceptual model. The results obtained thus provideinsights into the use of hydraulic tomography in unconfinedaquifers where one of the two conditions listed above aremet. More broadly though, we expect that the use of such amodel may still provide accurate K estimates for aquiferseven when delayed drainage shows nonnegligible effects(see, e.g., the parameter estimation results of Endres et al.[2007]).
2.2. Synthetic Data Source[32] We consider short, 30 min HT experiments in a het-
erogeneous aquifer 60 m � 60 m in lateral extent and 15 mthick with five fully penetrating wells, as shown in Figure 1.The wells are considered to be packed-off so that pressure
Figure 1. Layout of synthetic field site wells and relative size of modeled domain (60 m � 60 m � 15 m).The slice planes show geostatistically based aquifer K heterogeneity used in analysis cases 1–5.
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measurements and pumping can take place at discrete depthintervals. In these tests, pumping at a rate of 0.3 L s�1 takesplace at the central well (named A1) located at (0 m, 0 m),progressively at elevations of 5, 9, and 12 m. These eleva-tions were chosen in order to produce responses representa-tive of drawdown curves when the pumping interval is farbelow the water table, at moderate depth, and near the watertable, respectively. In the surrounding wells (named B1, B2,B3, and B4 and distanced at 2, 3, 4, and 5 m, respectively)drawdown curves are recorded at four different elevations(4, 7, 10, and 13 m). Pumping is performed at A1 only inorder to provide a ‘‘baseline’’ of imaging results when a sin-gle well is used for pumping. Rearrangement of pumpingand observation instrumentation can require labor-intensivemovement of packer and port systems and reassignment ofinstrumentation to wells. Thus, it is worthwhile to considerthe value of a single-well 3DTHT survey when decidingwhether the extra effort associated with additional testingarrangements should be carried out for a given aquifer, avail-able testing infrastructure, and problem to be addressed. Atthe north/south boundaries (positive y/negative y), constant-head values of 14.6 and 14.7 m were applied, respectively,to simulate regional flow across the synthetic site. Constanthead boundaries along the east/west (positive x/negative x)were assigned linearly interpolated values between the northand south boundaries. As shown in Figure 1, the pumpingand monitored wells are located in the center of the model-ing domain and surrounded by a broad heterogeneous extent.This geometry was chosen in order to more realistically rep-resent field uncertainty, where (1) heterogeneity alwaysexists well outside of the given monitoring area, and where(2) near-field constant head or no flux boundaries cannot of-ten be defined. The model geometry and testing strategy areconsistent across tests performed in this paper, though differ-ent types and amounts of aquifer heterogeneity were exam-ined in the various synthetic tests. While relatively limited inlateral extent compared to real aquifers, the pumping well,operating at a low rate of roughly 5 gpm, and observationwells are both located far from the model boundaries, andanalysis of both drawdown and sensitivity matrices indicatesthat boundary conditions have only minimal effects on thesolution. In addition, the short duration of the pumping testsmeans that the drawdowns do not obtain steady state condi-tions (see Figure 2). As the data in this figure are plotted insemilog format, it is also apparent that the drawdown doesnot appear to have clearly attained ‘‘steady shape’’ condi-tions either [Jacob, 1963; Kruseman and deRidder, 1990;Bohling et al., 2002].
[33] The aquifer contains heterogeneity in hydraulic con-ductivity K and, for some analysis cases, heterogeneity inspecific storage Ss and specific yield Sy as well. These param-eters are discretized on a regular, 1 m � 1 m (laterally) �0.6 m (thickness) grid, resulting in approximately 90,000 pa-rameter grid cells. We use the same parameter discretizationduring inversion, meaning each analysis scenario estimatesat least 90,000 parameters and in some cases as many as270,000 (when heterogeneous K, Ss, and Sy are jointlyconsidered).
[34] As discussed above, the aquifer is simulated usingthe popular and well-tested MODFLOW groundwater flowmodel [Harbaugh, 2005]. In order to improve the accuracyof the simulation, the finite difference grid discretization is
further refined relative to the parameter grid (through smallerDELR and DELC spacings) in the vicinity of the pumpingand observation wells, resulting in a MODFLOW modelwith approximately 2 million numerical grid cells. Withinthe model ‘‘natural’’ steady state head is first attained (assum-ing no stresses), followed by a 30 min transient stress periodin which one of the pumping tests is performed. Using thePCG solver with a tolerance of 0.01 mm required roughly2–4 min of runtime for a single model run on a single CPUcore. Sensitivities of observations to parameter values arecalculated using the adjoint-based ADJ process [Clemo,2007], meaning that the number of model runs required forsensitivity matrix evaluation is approximately proportional tothe number of observations being inverted.
3. Inverse Solution Method[35] Synthetic data from the various analysis cases are
inverted to produce images of estimated aquifer heteroge-neity along with uncertainty estimates (covariance matri-ces). To efficiently invert these data, we utilize an iterativescheme that progressively includes more data, by fittingmore data points on each drawdown curve, as outlined inFigure 3. In what we define as an ‘‘outer’’ iteration, a givenset of data is chosen to be inverted, and then supplied to theinner geostatistical inversion loop. The ‘‘inner’’ iterationsrefer to successive applications of the quasi-linear inversionformulas.
3.1. Outer Iterative Methodology[36] During each outer iteration in our inverse method,
we choose a selection of data points from the full set ofrecorded field drawdown curves to fit using our forwardmodel. In the first outer iteration, 2–3 or fewer drawdownpoints may be chosen from each drawdown curve. Thismay be done either manually or using quantitative metrics(in our case, we have simply selected them manually).These data are inverted (in the inner iteration loop) usingthe quasi-linear geostatistical inverse method of Kitanidis[1995], which inverts all supplied data simultaneously. Af-ter inversion of these select data points, a full drawdowncurve is generated for each observation location by the for-ward model and compared to the field data. If each fulldrawdown curve is not acceptably fit, then another tempo-ral data point is chosen from each field drawdown curve(from a more poorly fit section) and added to the next outeriteration of the inversion (see Figure 3). In each new outeriteration, the best estimate of parameters from the previousouter iteration is utilized as an initial guess in order tospeed up convergence. By using fewer data points in earlyinner iterations, the adjoint state-based sensitivity calcula-tions (whose runtime is proportional to the number ofobservations, as described below) are less computationallyintensive.
[37] In essence, our outer iteration scheme is perhapsmost similar to Li et al. [2007] in that results from prioriterations are only used to provide initial guesses for pa-rameters during the following iterations. However, we donot utilize moment-based measures of drawdown in ourscheme as Li et al. did, instead we simply fit several pointson a drawdown curve to match the transient data. One rea-son we do not use the method of moments is that, because
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of our unconfined model, we cannot solve for moments ofdrawdown (at least in the exact sense) using a steady stateflow model as Li et al. did. We thus lose the computationaladvantage of being able to solve a series of steady stateflow models instead of full transient models. However, wenote that Yin and Illman [2009], who fit moments of draw-down during inversion, found their results for estimatingstorage parameters to be inferior to those of Liu et al.[2007], who fit drawdown at selected points in time. It isthus expected that the results obtained using our methodwill result in more accurate storage parameter estimatesthan would be obtained by fitting moments of drawdown.Also, the approach of fitting points on the drawdown curveis more generally applicable than the moments-basedapproach to HT inversion, since in principle it does notneed to rely on the required assumptions for generatingvalid moment-based equations (e.g., that boundary condi-tions are either constant or no-flux).
3.2. Inner Iteration Inversion System[38] Within each outer iteration of our inversion, we rely
on the quasi-linear geostatistical approach of Kitanidis andVomvoris [1983] and Kitanidis [1995], and extensions forlarge-scale problems based on the work of Nowak et al.[2003], to invert the selected data. The basic formulas uti-lized in this method are summarized below.
[39] Consider that one has measurements available fromtests at a field site, as well as a model for simulating thesetests with adjustable parameters. The goal, as in all inver-sion strategies, is to find a set of parameters that fit the datawhile being ‘‘reasonable.’’ In our case, the set of parame-ters are spatially distributed values of K, as well as possiblySs and Sy. In the geostatistical approach we perform theminimization
mins L ¼ mins12½y� hðsÞ�T R�1½y� hðsÞ�
þ 12ðs� X�ÞT Q�1ðs� X�Þ;
(4)
where y is an ðn� 1Þ vector of measured values (the set ofall selected data points from all drawdown curves) ; hð Þ isa forward model ðRm ! RnÞ that takes as input parametervalues and outputs expected measurements; s is a ðm� 1Þvector of distributed parameter values; X is an ðm� pÞ ma-trix of drift functions where Xi;j represents drift function jevaluated at the location of parameter si ; � is a ðp� 1Þvector of unknown drift coefficients (estimated duringinversion); and R and Q represent the data error covariancematrix and expected parameter covariance matrix, respec-tively. Basically, the first term of the minimization seeks toachieve a parameter set that results in a good fit to the
Figure 2. Sample drawdown curves from analysis case 1 for single well and pumping test (test 1,pumping from A1, interval 5 m above datum).
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Figure 3. Flowchart for inversion method showing outer iteration loop (observation selection/curvematching) and inner iteration loop (linearization/inversion).
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observed data, while the second term seeks to find a param-eter set that is ‘‘reasonable’’ given an expected geostatisticaldistribution. Under the assumption of Gaussian measure-ment errors (distributed with covariance matrix R) andprior information suggesting a pluri-Gaussian parameterdistribution with covariance matrix Q, minimization ofLðsÞ is equivalent to maximizing the posterior probabilityof the parameter set given the data. It should be noted that,while R is generally assumed to be a scaled ðn� nÞ identitymatrix (representing independent, identically distributedmeasurement errors with a given variance), Q is generallya full ðm� mÞ matrix with block-Toeplitz structure, assum-ing the parameter values are being estimated on a regular,equispaced grid. Rather than storing and operating on thefull Q matrix, techniques based on the fast Fourier trans-form (FFT) are used to calculate matrix-vector multiplica-tions with Q, and can also be used to generate unconditionalparameter field realizations with covariance Q [see Nowaket al., 2003]. These techniques require only storage of thefirst row of Q, i.e., storage is OðmÞ, and matrix-vector mul-tiplications require computation time proportional toO½m logðmÞ� due to the FFT-based methods used.
[40] In the quasi-linear approach to optimizing (4), weassume that we have a current guess at the parameter set,denoted sc, and we have calculated the expected observa-tions hðscÞ, for this parameter set. When entering the inneriteration loop, sc is set equal to the best parameter estimatesfrom the previous outer iteration; within the inner iterationloop, sc is continually updated, as described below. Under afirst-order Taylor series expansion, the expected observa-tions at a new parameter set sn can be calculated as
hðsnÞ � hðscÞ þHcðsn � scÞ; (5)
where Hc is the Jacobian of h with respect to s evaluated atsc, i.e., Hcði;jÞ ¼ @hi
@sjjsc
.This means that the value of theobjective function L can be approximated as
LðsnÞ �12½y�hðscÞþHcsc�Hcsn�T R�1½y�hðscÞþHcsc�Hcsn�
þ12ðsn�X�ÞT Q�1ðsn�X�Þ:
(6)
Minimizing L with respect to sn under this approximationis a quadratic optimization that can be solved in one stepby setting the (linear) first derivatives equal to 0. The solu-tion is found by solving the following linear system ofequations:
HcQHTc þR HcX
ðHcXÞT 0p�p
" #�
�
" #¼
y�hðscÞþHcsc
0p�1
" #: (7)
Once � and � are found:
sn ¼X�þQHTc �: (8)
In practice, the actual nonlinearity of hð Þ means that snmay not provide the best improvement in the objective
function L. A line search is implemented in which the func-tion L½scþ �ðsn� scÞ� is evaluated for several values of thescalar parameter �, and the � value which produces theminimum L, denoted �, is chosen. Once an optimum � isfound, the problem is linearized around the new best esti-mate of s, i.e., sc is set equal to scþ �ðsn� scÞ, and the pro-gram returns to the beginning of the inner iteration loop.Convergence is declared when either suitably small changesin sc between inner iterations are obtained, and/or whensmall changes in the objective function L between iterationsare observed. Once the inner iterations have converged, theprogram exits the inner iteration loop and returns to theouter iteration loop, where new observations to be fit areselected as necessary.
3.3. Posterior Uncertainty Metrics[41] Under the Bayesian viewpoint with respect to for-
mula (4), the parameter values found at the end of iterationrepresent the maximum a posteriori probability (MAP) esti-mates, i.e., the most likely set of parameter values giventhe data and prior information. Under first-order probabilis-tic theory, then, a posteriori estimates of uncertainty in theparameters can also be derived by, basically, taking theinverse of the second derivative (Hessian) of (4) evaluatedat the MAP parameter values [e.g., Kitanidis, 1996]. A con-venient form for the posterior covariance matrix of the pa-rameter estimates, denoted P, is
P ¼ Q�HQ
XT
" #THQHT þ R HX
ðHXÞT 0
" #�1HQ
XT
" #; (9)
where H represents the sensitivity matrix evaluated at theMAP parameter values s. In the case of a linear inverseproblem, the posterior covariance matrix obtained via thesecomputations represents the optimal unbiased estimate ofthe covariance. However, it should be noted that the uncer-tainty estimate provided by this linearized method in the gen-eral case of nonlinear problems is only an approximation ofthe posterior covariance. Under Cramer-Rao theory, this line-arized covariance estimate is generally expected to at leastprovide a lower bound on the actual, nonlinear covariance.
[42] In general P is a large, full ðm� mÞ matrix that isimpractical to compute, store, or operate on. For example,in problems such as those investigated in this paper withOð105Þ parameters, double-precision storage of P wouldrequire a minimum of 40 GB, which cannot be operated onin RAM in most modern PCs. In our approach, at the endof the outer iterations, we calculate approximate (linear-ized) posterior variances for the parameter estimates byonly calculating diagonal entries of P above. While less in-formative than the full posterior covariance matrix, varian-ces can be used to gauge relative uncertainty in obtainedparameter estimates, and to generate approximate condi-tional realizations in cases where MAP estimates of param-eters alone are insufficient (e.g., for follow-on contaminanttransport simulation). For example, we suggest the follow-ing scheme as a method for generating approximate condi-tional realizations:
[43] 1. Generate an unconditional realization of Q usingFFT-based methods [Nowak et al., 2003]. Call this realizationsu. Store the square roots of the diagonal elements of Q (i.e.,the prior standard deviations) in a vector �prior.
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[44] 2. Calculate the posterior standard deviations of s,the parameter estimates, by evaluating the diagonal ele-ments of P and taking the square root of each element. Callthis vector �post.
[45] 3. Calculate the approximate conditional realizationas s þ ð�post=�priorÞ � su, where all products and quotientsare element wise (i.e., Hadamard products).
[46] Following the steps above will generate a condi-tional realization that has a mean of s, correlation structurebased on the variogram used in Q, and variance �post, i.e.,this scheme assumes that while the addition of data has astrong effect on updating the variances of individual pa-rameters, the prior and posterior autocorrelation are not sig-nificantly different. In practice, this method can be used toquickly generate candidate parameter fields that are thenfiltered using acceptance-rejection criteria. The acceptedconditional realizations then represent parameter fields that(1) are consistent with the observed 3DTHT data and (2)have realistic geostatistical variability that can be used tomore accurately simulate, e.g., plausible contaminant trans-port scenarios.
4. Application to Synthetic Data[47] We implement the inversion approach described
above for a number of synthetic numerical test cases whichdiffer both in terms of the distribution of storage parame-ters and in terms of the assumptions utilized during inver-sion. In all cases except case 5, the K distribution utilizedwas the same and is as shown in Figure 1.
4.1. Analysis Cases[48] The basic model setup described above was used to
examine several different scenarios and their impact on theperformance of 3DTHT for estimation of spatially variablehydraulic conductivity. The analysis cases described belowwere all carried out on a modern desktop PC (Corei7 proces-sor, 12 GB of RAM) using MODFLOW as the forwardmodel and using a set of MATLAB functions for performinginversion and visualization. Parallelization was implementedin an ‘‘embarassingly parallel’’ fashion, by assigning bothforward runs [hðscÞ evaluations] and sensitivity calculation(Hc computations) for individual pumping tests to individualCPU cores. Individual model runs required 2–4 min of CPUtime per transient simulation. Sensitivity calculation timingsvaried with the number of observations inverted during eachiteration, but routinely required between 1 and 6 h. In allcases, fewer than 10 sensitivity calculations were requiredfor full convergence (inner and outer iterations). Conver-gence was declared within inner iterations when either theobjective function decrease (change in L) was less than 1%or the maximum relative change in individual sc values periteration was less than 3%. Convergence for the outer itera-tions was declared when the average misfit of all data pointsfrom all drawdown curves was 1 mm or less. As a basis forcomparison, the average of the maximum (final) drawdownmagnitudes across all measurement locations and all pump-ing tests was approximately 1.5 cm, meaning we are assum-ing noise with magnitude of >7% of the signal. For allcases, we chose one early time, one intermediate-time, andone late-time data point from each drawdown curve to be fit,
and found that inclusion of additional data points did not sig-nificantly improve data fit.
[49] To assess the accuracy of the computed parameterestimates and their associated uncertainty metrics for eachcase, we utilized the following measures. First, the rootmean square error (RMSE) is used to assess the overall ac-curacy of the best estimates :
�p ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1m
Xm
i¼1
�ptrueðiÞ � pestðiÞ
�2r
; (10)
where ptrueðiÞ and pestðiÞ are the true and estimated parametervalues [i.e., log10ðKÞ, log10ðSsÞ, or log10ðSyÞ] for grid celli. As an approximate measure of the accuracy of the poste-rior uncertainty estimates, we similarly calculate the nor-malized root mean square error (NRMSE) as
�p ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1m
Xm
i¼1
�ptrueðiÞ � pestðiÞ
�2
Pði;iÞ
vuuut :(11)
In the ideal case where posterior variances are accurate, theNRMSE computed should be close to 1. Since our pumpingtests and measurements are focused in a small area relativeto the overall model, we calculate these metrics for theentire model but also for the ‘‘central area’’ of our model,which we define as a 10 m � 10 m � 15 m prism centeredat the x; y coordinates ð0; 0Þ. Similarly, since data fromthese pumping tests are only dependent on Sy values towardthe top of our model, we calculate RMSE and NRMSE sta-tistics for Sy for the top-most layer of our model as well asfor the overall model.
4.2. Analysis Case 1[50] In analysis case 1 we assume very strong prior infor-
mation in the form of perfect knowledge of the storagecoefficients. Likewise, the storage coefficients are spatiallyuniform within the true model. Case 1 represents a baselineanalysis case in that estimating K is aided by perfect knowl-edge of storage coefficients, and in which the lack of stor-age coefficient variability will eliminate the problem ofK/storage coefficient ‘‘aliasing’’ [see, e.g., Li et al., 2005].In case 1, the only parameters to be estimated are the hetero-geneous values of K throughout the model. Ss and Sy valuesare constant throughout the domain in the true parameterfield, and their values are assumed known during inversion.The true log10ðKÞ field for case 1 is shown in Figure 1, andthe geostatistical parameters of the true field are given inTable 2. Note that the relatively low Sy used in all casesis consistent with drainage response at short times, see,e.g., Neuman [1975] and Moench [1994]. The geostatisticalparameters utilized during inversion are the same as thoseused to generate the heterogeneous parameter field (seeTable 3). However, note that the variance of the realizationgenerated and used as the true parameter field ð0:41Þ, dif-fers from the model value ð0:5Þ which was used as input tothe generation process and was also assumed in the inver-sion process.
[51] The progressive improvement of data fit duringouter iterations is shown in Figure 4. A comparison of theimaging results for case 1 during outer iterations is shown
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in Figure 5, which displays both the ‘‘best estimate’’ of pa-rameter values throughout the domain as well as the esti-mated posterior standard deviation of this estimate, andFigure 6, which shows three slice views comparing the esti-mates obtained (from two different outer iterations) to thetrue parameter field within the central area. The generaltrend visible is that : (1) Slight image improvement is seenas more observations are included in the inversion from theintermediate and early time portion of the drawdowncurve; and (2) detailed heterogeneity in the best estimatesand relatively low uncertainties are obtained near thepumping and observation locations, while far from the ob-servation locations the best estimates trend toward a uni-form, homogeneous value.
[52] The data fit metrics for analysis case 1 are shown inTable 4. Overall we see that RMSE of log10ðKÞ estimatesis much lower in the central area when compared to the fullmodel domain. The NRMSE, which should generally beclose to 1, shows that variances for log10ðKÞ appear to begenerally accurate, but are somewhat underestimated withinthe central area (leading to a NRMSE greater than 1). Webelieve this effect is due to the fact that our the model sen-sitivity is more nonlinear due to higher head gradients inthis area, meaning that the linearized uncertainty boundsestimated are more inaccurate.
4.3. Example of Approximate Conditional RealizationStrategy
[53] While the best estimates obtained by our inversionmethodology outside of the central region could be classi-fied as not ‘‘geologically realistic’’ (see, e.g., Figure 5),they reflect our lack of knowledge about heterogeneousstructures in this area. As noted by Bohling and Butler[2010], there are generally a large number of parameterfields that can be found that are consistent with data col-lected from HT surveys, especially in cases where signifi-cant heterogeneity exists outside the region of the aquiferbeing interrogated. The best estimates obtained by the geo-statistical method, however, represent average featuresacross all possible data-consistent realizations, or at leastthe best approximation of these average features using line-arized theory. The lack of detail in these areas, and accom-panying large uncertainty estimates, can provide guides forfuture data collection during iterative characterization.
[54] In Figure 7 we present a conditional realization gen-erated for case 1 using our proposed approximate strategy,which was selected (filtered) from a group of approximateconditional realizations by the simple technique of ‘‘fittingwithin tolerance’’ of 1 mm, i.e., the average residualsbetween observed data and simulated data from the condi-tional realization was required to be <1 mm. As a basis forcomparison to this level of misfit, average residualsobtained when using a fitted homogeneous K model were6.7 mm.
4.4. Analysis Case 2[55] The same ‘‘true’’ model featuring spatially variable
K but uniform Ss and Sy values was used in case 2, with thekey difference being that the homogeneous values of Ss andSy are estimated in case 2. For this sample case, we foundthat both Ss and Sy parameters were estimated fairly accu-rately, the RMSEs presented in Table 4 correspond to a 4%and 40% error in Ss and Sy, respectively. With respect toestimation of K, the inclusion of estimation of unknown ho-mogeneous values of Ss and Sy does not appear to have a
Table 3. Geostatistical Structural Parameters Assumed DuringInversion of All Analysis Cases
Analysis Case
1, 2, 3 4, 5
log10(K) Assumed Variance (log10(m s�1))2 0.5 0.5Assumed x Correlation Length (m) 15 15Assumed y Correlation Length (m) 12 12Assumed z Correlation Length (m) 3 3
log10(Ss) Assumed Variance (log10(1 m�1))2 N/A 0.08Assumed x Correlation Length (m) N/A 15Assumed y Correlation Length (m) N/A 12Assumed z Correlation Length (m) N/A 3
log10(Sy) Assumed Variance (log10(1 m�1))2 N/A 0.02Assumed x Correlation Length (m) N/A 15Assumed y Correlation Length (m) N/A 12Assumed z Correlation Length (m) N/A 3
Table 2. Summary Statistics for True Parameter Fields Used inAll Analysis Casesa
Analysis Case
1, 2 3, 4 5
log10(K) Generation Mean (log10(m s�1)) �4 �4 N/AGeneration Variance (log10(m s�1))2 0.5 0.5 N/AGeneration x Correlation Length (m) 15 15 N/AGeneration y Correlation Length (m) 12 12 N/AGeneration z Correlation Length (m) 3 3 N/AParameter Field Mean (log10(m s�1)) �3.7 �3.7 �3.6Parameter Field Variance (log10(m s�1)) 0.41 0.41 1.2Minimum Value (log10(m s�1)) �6.22 �6.22 �4.5Maximum Value (log10(m s�1)) �1.28 �1.28 �2
log10(Ss) Generation Mean (log10(1 m�1)) �4.7 �5 N/AGeneration Variance (log10(1 m�1))2 N/A 0.08 N/Ax Correlation Length (m) N/A 15 N/Ay Correlation Length (m) N/A 12 N/Az Correlation Length (m) N/A 3 N/AParameter Field Mean (log10(1 m�1)) �4.7 �4.9 �5.1Parameter Field Variance (log10(1 m�1)) N/A 0.066 0.6Minimum Value (log10(1 m�1)) N/A �5.88 �6Maximum Value (log10(1 m�1)) N/A �3.91 �4
log10(Sy) Generation Mean (�) �1.6 �1.5 N/AGeneration Variance (�) N/A 0.02 N/Ax Correlation Length (m) N/A 15 N/Ay Correlation Length (m) N/A 12 N/Az Correlation Length (m) N/A 3 N/AParameter Field Mean (�) �1.6 �1.4 �1.6Parameter Field Variance (�) N/A 0.016 0.15Minimum Value (�) N/A �1.94 �2Maximum Value (�) N/A �0.96 �1
aFor cases where true parameter field was generated from geostatisticalmodels, generation parameters are given.
Figure 4. Improvement of drawdown curve fits with successive inclusion of more inverted data points (outer iterationsteps). Main image shows example drawdown curve fits for a single well and test (well B3, all elevations, test 1 pumpingat A1, 5 m above datum). Inset images show crossplot of all observed versus simulated data at given iteration.
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Figure 4
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major effect on K estimate accuracy, though a smallincrease in K RMSE is seen in the central area.
4.5. Analysis Case 3[56] In case 3 we investigate an aquifer with heterogene-
ous storage coefficients as well as heterogeneous K values.In our particular case, we assume for the true field Ss and Sy
values that are correlated with K variability, but with differ-ent means and variances than the K field. As discussed byLi et al. [2005], there is little information available in theliterature about the geostatistical distribution of storage pa-rameters, and their correlation with hydraulic conductivity,for real aquifers. For our sample case, we assume that K,Ss, and Sy are exactly correlated but have different meansand variances. This is analogous to assuming a porosity-controlled system where less consolidated, higher porosityunits will generally have higher K, higher Ss, and higher Sy
values. It should be noted though that this behavior shouldnot always be expected in all conditions. For example,Straface et al. [2007b] found during his analysis that trans-missivity and storativity parameters did not appear to becorrelated at the Montalto Uffugo Scalo field site in Italy.Similarly, Brauchler et al. [2010] found anticorrelationbetween K and Ss at the Stegemühle test site.
[57] The K field in case 3 is the same as the K field incases 1 and 2. During inversion for case 3, we assume that
only K is spatially variable and estimate homogeneous val-ues for both Ss and Sy. K field estimates for this analysiscase are slightly worse than those from case 2, in terms ofRMSE. Likewise, larger NRMSE values for K in case 3 ascompared to case 2 seems to indicate a small degree of ali-asing [see, e.g., Li et al., 2005] due to incorrectly assumingconstant storage values. The storage coefficients estimatedduring inversion appear to be close to the geometric meanof the true, heterogeneous parameter field. For Ss the geo-metric mean of the true parameter field is 1:3� 10�5 m�1,and the estimated homogeneous value was 8:1� 10�6 m�1.For Sy, the geometric mean of the true parameter field is0.037 and the estimated homogeneous value was 0.045.
4.6. Analysis Case 4[58] In case 4 we seek to jointly estimate the spatial vari-
ability in all three parameters (K, Ss, and Sy). We assumeduring inversion that K, Ss, and Sy variability is uncorre-lated (even though, in the true field, the parameters are cor-related) in order to test the ability of inversion toindependently identify these parameters’ variability giventhe data. The true parameter fields used in case 4 areexactly the same as those used in case 3.
[59] In comparison to case 3, we see only very slightchanges in the RMSE of the K estimates, suggesting thatassuming homogeneous Ss and Sy was not detrimental for
Figure 5. Changes in inverted image and uncertainty estimates progressive inclusion of more datathrough outer iterations (case 1). Images represent (a) outer iteration 1 and (b) outer iteration 3. Left-hand side images represent best estimates, and right-hand side images represent posterior standard devia-tion estimates.
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the example considered. In terms of storage estimates,RMSE for Ss and Sy are somewhat smaller than the RMSEfor case 3, suggesting that 3DTHT does have some abilityto estimate heterogeneity in these parameters. However,
the uncertainty estimates obtained by the geostatisticalmethods for both Ss and Sy appear to be highly inaccurate,as shown by NRMSE values for these parameters that arefar from the expected value of unity (Table 4).
Figure 6. Comparison of best estimates obtained at (a) outer iteration 1 and (b) outer iteration 3 against(c) true parameter field. Images show parameter estimates in central 10 m � 10 m � 15 m area along theslices (from left to right, respectively) x ¼ 0 m, y ¼ 0 m, and z ¼ 7:5 m. Filled dots represent pumpinglocations and open dots represent observations.
Table 4. Fit Statistics Calculated for All Analysis Cases
Analysis Case
1 2 3 4 5
log10(K) (log10(m s�1)) RMSE 0.572 0.571 0.571 0.571 0.935NRMSE 0.959 0.941 0.942 0.930 1.525RMSE (central area) 0.360 0.362 0.365 0.366 0.606NRMSE (central area) 1.048 1.018 1.021 1.003 1.668
log10(Ss) (log10(1 m�1)) RMSE N/A 0.016 0.256 0.255 0.776NRMSE N/A 0.214 2.824 1.804 5.488RMSE (central area) N/A 0.016 0.261 0.240 0.699NRMSE (central area) N/A 0.214 2.876 1.697 4.941
log10(Sy) (�) RMSE N/A 0.140 0.128 0.125 0.389NRMSE N/A 0.994 1.195 0.443 1.377RMSE (top layer) N/A 0.140 0.107 0.104 0.038NRMSE (top layer) N/A 0.994 0.999 0.368 0.134
Full Drawdown Curve Head Values (m) RMSE 2.30E-04 2.70E-04 1.00E-03 7.90E-04 1.00E-03
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4.7. Analysis Case 5[60] In case 5 we investigate the effect of the model con-
ceptualization error associated with assuming an incorrect(stationary) geostatistical model. The true log10ðKÞ fieldfor case 5 is different from all other cases, and is shown inFigure 8. The discrete geologic layers in case 5 each have adifferent, homogeneous value for all three parameters K,Ss, and Sy, as given in Table 5. Overall statistics for this pa-rameter field are given in the last column of Table 2.
[61] During inversion, we incorrectly assume the samegeostatistical model as used in case 4. Most significantly,the variance of log10ðKÞ compared with the true statisticsfor the facies-based parameter field is underestimated by afactor of 2.4, and the variances of log10ðSsÞ and log10ðSyÞare both underestimated by a factor of 7.5. (Note, however,that the minimum and maximum values for all parametersare similar to the values from case 4, see Table 2). Thesebiases reflect the fact that, in field practice, prior estimatesof parameter field variances obtained via simple methods—e.g., kriging of pseudo-locally obtained ‘‘homogeneous’’values—may generally underestimate parameter variability(see, e.g., comparisons by Li et al., [2007]). In addition,these biases reflect the fact that stationary geostatisticalmodels may be assumed when true parameter variabilitycontains nonstationary features.
[62] Graphically, estimates of K within the central areaappear fairly accurate (see Figure 8), even though an incor-rect geostatistical model has been assumed, but large-scaletrends outside of the central well area are poorly represented.Comparison between analysis cases 3 and 5 allows us to
investigate the relative errors associated with incorrectlyassuming constant storage coefficients (case 3) versus cor-rectly assuming spatial variability in storage coefficients butincorrectly estimating the geostatistical model (case 5). Wefind that the errors in estimation of K associated with geo-statistical model errors are much greater, for our samplecases, than errors associated with assuming constant storagecoefficients. Large increases in K RMSE are seen in case 5relative to case 4, and likewise the NRMSE for K, Ss, and Sy
reflect the fact that posterior variances are quite poorly esti-mated in this sample case.
5. Discussion and Conclusions[63] We have presented and implemented a geostatisti-
cally based inversion strategy for analyzing data from3DTHT in unconfined aquifers that is able to invert short-term pumping tests where even steady-shape conditions maynot be satisfied. Likewise, our inversion strategy does notrequire constant pumping rates and could be used in scenar-ios where nonpumping stresses also affect the aquiferthroughout the duration of the test. To the best of our knowl-edge, this work represents the first implementation of3DTHT data inversion for unconfined aquifers. The generalmethodology is flexible for implementation in confined sce-narios as well, though.
[64] The suggested strategy was tested on a number ofsample cases of varying complexity. For all cases, thepumping rates and geometry utilized were based on experi-ence implementing 3DTHT at a well-studied, unconsolidated
Figure 7. Conditional realization of log10ðKÞ generated using suggested approximate method. Averagedata misfit for this realization is 0.6 mm.
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unconfined aquifer, the Boise Hydrogeophysical ResearchSite. Likewise, the heterogeneity models utilized are pre-sumed to be realistic for the BHRS and other similarunconfined fluvial aquifers. It should be noted that in theexamined example, the aquifer being studied has relativelyhigh hydraulic conductivity and (if it were confined) highhydraulic diffusivity. A relatively low pumping rate wasutilized which was consistent with obtainable pumpingrates during field experimentation. Furthermore, we havemade an effort to make our study realistic by assuming abroad aerial extent of heterogeneous aquifer surrounding ourwell field. All of these factors cause increased difficulty inimaging hydrologic parameters with HT, but may be com-mon in actual application to unconfined, fluvial aquifers.
[65] Several cases were examined in our syntheticexperiments. In the case where K is heterogeneous but stor-age coefficients are reasonably homogeneous within theaquifer being investigated, we find that both Ss and Sy pa-rameters can be estimated with reasonable accuracy (case2). If Ss and Sy are variable, treating them as homogeneousduring inversion appears to result in estimates that are closeto the geometric mean of the true storage parameter fields(case 3). If Ss and Sy are variable and treated as variableduring inversion (case 4), we find that slight improvementin the parameter estimates can be obtained for the samplecase investigated, but that posterior uncertainty estimatesfor storage coefficients appear to be quite inaccurate, evenwhen correct geostatistics are assumed. Finally, we findthat the assumed geostatistical model used during inversion
appears to have a much greater influence on the accuracyof K estimates and their associated uncertainty measuresthan does treating Ss and Sy as constant values (case 5).While not examined here, a possible area for futureresearch is determining the extent to which unconfined con-ditions will affect inversion results if a confined model isassumed during analysis.
[66] In practice we believe that estimation of K heterogene-ity in permeable unconfined aquifers is minimally impactedby assumptions of constant storage coefficients, when consid-ered relative to other sources of error such as inappropriateuse of a stationary geostatistical model, mis-estimation ofparameter field variances, and mis-estimation of parameterfield correlation lengths. Likewise, given that so little infor-mation exists in the literature about the range of variabilityof storage coefficients (particularly Ss) in natural aqui-fers—and about their correlation or lack thereof with Kvariability—treating Ss and Sy as constant but unknown pa-rameters during inversion seems to be a practical and com-putationally efficient approach for estimating K fields inunconfined aquifers. This agrees with the results of Li et al.[2005], who found for 2-D confined aquifer analysis that‘‘using wrong structural assumptions about the spatial vari-ability of storativity showed significant, but not dramaticdeviations in the estimated log-transmissivity fields.’’
[67] HT provides one method for estimating K variabili-ty that jointly fits all data and acknowledges the variablesensitivity of hydraulic measurements to spatially distrib-uted hydraulic properties. As such, it does not make the‘‘effective’’ homogeneous parameter assumptions that aremade by traditional pumping test and slug test analyses, themeaning of which may be difficult to interpret [see, e.g.,Beckie and Harvey, 2002; Wu et al., 2005]. That said, hy-draulic tomography requires a high degree of computa-tional resources in addition to the installation of wells ormultilevel samplers for measuring depth-dependent pres-sure changes.
[68] In cases where heterogeneity is predominantly 1-D(i.e., layered and only depth dependent), similar maps ofheterogeneity may be obtained by using simpler methods,
Figure 8. Comparison between true and inverted log10ðKÞ distribution for case 5, in which the true pa-rameter field consists of distinct layers. Overall accurate images are obtained in vicinity of wells, withincreasing error at far distances.
Table 5. Parameter Values in True Model for Layered Aquifer,Analysis Case 5
log10(K)(log10(m s�1))
log10(Ss)(log10(1 m�1))
log10(Sy)(�)
Layer 1 (top) �4.00 �6.00 �2.00Layer 2 �2.00 �4.00 �1.00Layer 3 �4.50 �5.50 �1.75Layer 4 (bottom) �2.50 �4.80 �1.25
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such as analyzing core samples, multilevel slug tests, ordirect push instrument responses. However, if aquifer het-erogeneity has significant 2-D or 3-D variability, usingthese techniques may lead to incorrect inferences aboutconnectivity between borehole locations [Liu et al., 2007].In addition, these techniques face other practical difficul-ties, e.g., direct push equipment can be limited by aquifermaterial type and penetration refusal depths, and analysisof core samples is dependent on core recovery. At the otherend of the spectrum, tracer-based studies offer even moreinformation than HT about 3-D connectivity when thetracer tests are analyzed tomographically [e.g., Cirpka andKitanidis, 2001; Pollock and Cirpka, 2008, 2010], but mayrequire time, instrumentation, and computational resourcesthat are well beyond what is required by HT. Studies suchas the one presented here can help site managers to assessthe costs and value of hydraulic tomography relative tothese other characterization methods.
[69] Acknowledgments. We gratefully acknowledge support for thisresearch which was provided by NSF under grants EAR-0710949 andDMS-0934680, and by the US Army RDECOM ARL Army ResearchOffice under grant W911NF-09-1-0534. In addition, the authors would liketo thank the three anonymous reviewers and the Associate Editor (WalterIllman) for their insightful comments, which helped improve the quality ofthis work.
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W. Barrash, Center for Geophysical Investigation of the Shallow Sub-surface (CGISS), Department of Geosciences, Boise State University, 1910University Drive, MG-206, MS 1536, Boise, ID 83725, USA. ([email protected])
M. Cardiff, Center for Geophysical Investigation of the Shallow Subsur-face (CGISS), Department of Geosciences, Boise State University, 1910 Uni-versity Drive, MG-206, MS 1536, Boise, ID 83725, USA. ([email protected])
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