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3-D transient hydraulic tomography in unconfined aquifers with fast drainage response M. Cardiff 1,2 and W. Barrash 1 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-dimensional transient hydraulic tomography (3DTHT) for identifying the spatial distribution of groundwater flow parameters (primarily, hydraulic conductivity K) in permeable, unconfined aquifers. To invert the large amount of transient data collected from 3DTHT surveys, we utilize an iterative geostatistical inversion strategy in which outer iterations progressively increase the number of data points fitted and inner iterations solve the quasi- linear geostatistical formulas of Kitanidis. In order to base our numerical experiments around realistic scenarios, we utilize pumping rates, geometries, and test lengths similar to those attainable during 3DTHT field campaigns performed at the Boise Hydrogeophysical Research Site (BHRS). We also utilize hydrologic parameters that are similar to those observed at the BHRS and in other unconsolidated, unconfined fluvial aquifers. In addition to estimating K, we test the ability of 3DTHT to estimate both average storage values (specific storage S s and specific yield S y ) as well as spatial variability in storage coefficients. The effects of model conceptualization errors during unconfined 3DTHT are investigated including : (1) assuming constant storage coefficients during inversion and (2) assuming stationary geostatistical parameter variability. Overall, our findings indicate that estimation of K is slightly degraded if storage parameters must be jointly estimated, but that this effect is quite small compared with the degradation of estimates due to violation of ‘‘structural’’ geostatistical assumptions. Practically, we find for our scenarios that assuming constant storage values during inversion does not appear to have a significant effect on K estimates or 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 pumping and 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 of 3-D hydraulic parameters (e.g., hydraulic conductivity K, and specific storage S s ). This is in stark contrast to ‘‘tradi- tional’’ pumping tests where water is pumped at an open wellbore and water level changes are measured at surround- ing wells, which do not measure vertical variations in head and are thus only capable of estimating depth-integrated or averaged parameters (transmissivity T and storativity S), even if they are analyzed in a tomographic fashion; we will refer to this as 2DHT. Another key differentiator between 3DHT 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 for pumping from a discrete interval either with permanent or temporary 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 the aquifer (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 observation locations in order to obtain information on depth variations of head within the aquifer. [3] Analyzing pressure response data (i.e., head or changes in head) from pumping tests in a tomographic fash- ion has been the subject of study for over 15 years, and the literature in this area is extensive. As noted in earlier discus- sions [Cardiff, 2010], approaches to HT ‘‘differ in the type of aquifer stimulations they perform (largely 2-D when using fully 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 heterogeneity assumed (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 an attempt to summarize and classify the research in 2DHT and 3DHT. In order to limit the size of this table, we consider 1 Center for Geophysical Investigation of the Shallow Subsurface (CGISS), Department of Geosciences, Boise State University, Boise, Idaho, USA. 2 Department 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
23

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  • 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

    http://dx.doi.org/10.1029/2010WR010367

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    W12518 CARDIFF AND BARRASH: 3-D UNCONFINED HYDRAULIC TOMOGRAPHY W12518

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    )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

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    netr

    atin

    gy,

    z1-

    D,F

    ully

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    trat

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    y,z

    1-D

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    ):G

    eost

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    tical

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    eost

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    tical

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    nsie

    nt,S

    elec

    ted

    time

    step

    s1-

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    (x)

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    Ss(x

    )K

    (x):

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    stat

    istic

    alSs

    (x):

    Geo

    stat

    istic

    alPo

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    easu

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    ent

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    uan

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    Num

    eric

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    (x,y

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    Geo

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    istic

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    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

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    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):

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    th-

    and

    first

    -or

    der

    Tik

    hono

    vV

    asco

    and

    Kar

    asak

    i[2

    006]

    Fiel

    d3-

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    DT

    rans

    ient

    ,Pul

    setr

    avel

    time

    met

    rics

    2-D

    K(x

    ,z)

    K(x

    ,z):

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    th-

    and

    first

    -or

    der

    Tik

    hono

    vZh

    uan

    dYe

    h[2

    006]

    Num

    eric

    al2-

    D,F

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    trat

    ing

    z2-

    D,F

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    trat

    ing

    z2-

    DK

    (x,y

    ):G

    eost

    atis

    tical

    2-D

    Ss(x

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    Geo

    stat

    istic

    alT

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    ient

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    ecte

    dtim

    est

    eps

    2-D

    K(x

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    (x,y

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    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

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    trat

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    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

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    [200

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    ical

    2-D

    ,Ful

    lyPe

    netr

    atin

    gz

    2-D

    ,Ful

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    atin

    gz

    2-D

    K(x

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    Geo

    stat

    istic

    al2-

    DSs

    (x,y

    ):G

    eost

    atis

    tical

    Tra

    nsie

    nt,S

    elec

    ted

    time

    step

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    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

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    trat

    ing

    z2-

    D,F

    ully

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    trat

    ing

    z2-

    DK

    (x,y

    ):G

    eost

    atis

    tical

    2-D

    Ss(x

    ,y):

    Geo

    stat

    istic

    alT

    rans

    ient

    ,Mom

    ents

    ofdr

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    wn

    2-D

    K(x

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    Ss(x

    ,z)

    K(x

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    ):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,

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    10,o

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    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

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    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-

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    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

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    fset

    GPR

    data

    Bra

    uchl

    eret

    al.[

    2007

    ]N

    umer

    ical

    3-D

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    ):Fa

    cies

    -Bas

    ed0-

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    nt,P

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    etri

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    W12518 CARDIFF AND BARRASH: 3-D UNCONFINED HYDRAULIC TOMOGRAPHY W12518

    3 of 23

  • Bra

    uchl

    eret

    al.[

    2007

    ]N

    umer

    ical

    3-D

    l3-

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    1-D

    K(z

    ):Fa

    cies

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    ed0-

    DSs

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    nsie

    nt,P

    ulse

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    eltim

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    etri

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    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

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    nsie

    nt,P

    ulse

    trav

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    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

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    trat

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    y2-

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    (x,z

    ):Fa

    cies

    -Bas

    edSt

    eady

    Stat

    e2-

    DK

    (x,z

    )K

    (x,z

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    niso

    trop

    icG

    eost

    atis

    tical

    Poin

    tmea

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    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

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    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

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    2-D

    Ss(x

    ,z)

    K(x

    ,z):

    Ani

    sotr

    opic

    Geo

    stat

    istic

    alSs

    (x,z

    ):A

    niso

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    icG

    eost

    atis

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    ale

    data

    (cor

    e,sl

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    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

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    Ssh

    K(x

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    Geo

    stat

    istic

    alSP

    data

    join

    tlyin

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    rafa

    ceet

    al.[

    2007

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    eld

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    netr

    atin

    gz

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    ,Ful

    lyPe

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    atin

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    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

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    y,z

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    ,Ful

    lyPe

    netr

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    gy,

    z1-

    DK

    (x):

    Geo

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    alSt

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    eost

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    2-D

    ,Ful

    lyPe

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    ,Ful

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    K(x

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    Geo

    stat

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    e2-

    DK

    (x,z

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    eost

    atis

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    data

    sequ

    entia

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    iene

    net

    al.[

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    ]N

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    2-D

    ,Ful

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    ,Ful

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    gz

    2-D

    K(x

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    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

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    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

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    trat

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    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

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    (x,z

    ):Fa

    cies

    -Bas

    edT

    rans

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    ,Sel

    ecte

    dtim

    est

    eps

    2-D

    K(x

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    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

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    Res

    pons

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    ata

    Use

    ddPa

    ram

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    sE

    stim

    ated

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    Para

    met

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    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

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    (x,y

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    0-D

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    2-D

    K(x

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    K(x

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    Ens

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    mul

    ti-po

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    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

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    DK

    (x,y

    ):Fa

    cies

    0-D

    SsT

    rans

    ient

    ,Sel

    ecte

    dtim

    est

    epsh

    2-D

    K(x

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    K(x

    ,y):

    Ens

    embl

    eof

    mul

    ti-po

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    eost

    atis

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    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

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    ces

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    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 F

    orfie

    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 In

    man

    yca

    ses

    of2-

    Din

    vers

    ion

    or2-

    Dsa

    mpl

    epr

    oble

    ms,

    the

    hydr

    olog

    icpa

    ram

    eter

    sw

    ere

    refe

    rred

    toas

    Tra

    nsm

    issi

    vity

    (T)

    and

    Stor

    ativ

    ity(S

    ).W

    eas

    sum

    ein

    thes

    eca

    ses

    that

    the

    aqui

    fer

    has

    unif

    orm

    thic

    knes

    sb

    and

    can

    thus

    bew

    ritte

    nas

    2-D

    K¼

    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 I

    mag

    esof

    hete

    roge

    neity

    wer

    eno

    tpro

    duce

    d,bu

    tSV

    Dw

    assu

    gges

    ted,

    whi

    chw

    ould

    prod

    uce

    min

    imum

    -len

    gth

    solu

    tions

    (zer

    oth-

    orde

    rT

    ikho

    nov

    for

    the

    limit

    of0

    data

    vari

    ance

    ).s F

    our

    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

    W12518 CARDIFF AND BARRASH: 3-D UNCONFINED HYDRAULIC TOMOGRAPHY W12518

    7 of 23

  • 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 @@x

    K@h@x

    � �þ @@y

    K@h@y

    � �þ @@z

    K@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.

    W12518 CARDIFF AND BARRASH: 3-D UNCONFINED HYDRAULIC TOMOGRAPHY W12518

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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 drawdo


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