Ci a ion: De Ca lo, L.; Fa zamian, M.;
Tu u o, A.C.; Capu o, M.C.
Time-Lapse ERT, Momen Analysis,
and Nume ical Modeling o
Es ima ing he Hyd aulic
Conduc i i y o Unsa u a ed Rock.
Wa e 2023,15, 332. h ps://doi.o g/
10.3390/w15020332
Academic Edi o : M. Le en Ka as
Recei ed: 29 No embe 2022
Re ised: 29 Decembe 2022
Accep ed: 6 Janua y 2023
Published: 12 Janua y 2023
Copy igh : © 2023 by he au ho s.
Licensee MDPI, Basel, Swi ze land.
This a icle is an open access a icle
dis ibu ed unde he e ms and
condi ions o he C ea i e Commons
A ibu ion (CC BY) license (h ps://
c ea i ecommons.o g/licenses/by/
4.0/).
wa e
A icle
Time-Lapse ERT, Momen Analysis, and Nume ical Modeling
o Es ima ing he Hyd aulic Conduc i i y o Unsa u a ed Rock
Lo enzo De Ca lo 1,* , Mohammad Fa zamian 2,3 , An onie a Celes e Tu u o 1
and Ma ia Clemen ina Capu o 1
1Wa e Resea ch Ins i u e, Na ional Resea ch Council o I aly, 70132 Ba i, I aly
2Ins i u o Nacional de In es igação Ag á ia e Ve e iná ia, 2780-157 Oei as, Po ugal
3Cen e o Geog aphical S udies (CEG), IGOT, Uni e sidade de Lisboa, 1600-276 Lisbon, Po ugal
*Co espondence: lo enzo.deca lo@cn .i
Abs ac :
In ecen yea s, geophysical echniques ha e been inc easingly used o moni o low and
anspo p ocesses in he Ea h c i ical zone (ECZ). Among hese, elec ical esis i i y omog aphy
(ERT) is a powe ul ool used o p edic hyd ological pa ame e s and s a e a iables ha in luence
he men ioned p ocesses in he adose zone because o he s ong co ela ion be ween elec ical
and hyd ological p ope ies o he il e ing medium. The e ha e been many ield es s conside ing
geophysical p ospec ing in soils, whe e poin scale hyd ological senso s measu emen s a e ypically
collec ed h ough senso s o geophysical da a alida ion; on he con a y, when he unsa u a ed
zone is made o ha d ocks, he ins alla ion o such senso s is no a i ial issue owing o he ex eme
di icul ies o gua an ee con ac be ween senso s and he su ounding medium. In his con ex ,
he geophysical da a combined wi h app op ia e nume ical analysis echniques can e ec i ely
o e come he lack o in o ma ion o he unsa u a ed subsu ace, which is o he wise unp edic able
wi h adi ional me hods. In he p oposed case s udy, hyd ogeophysical da a we e collec ed o
p o ide a quan i a i e es ima ion o he hyd aulic conduc i i y o sands one h ough an in eg a ed
app oach based on he momen analysis echnique and nume ical modeling.
Keywo ds:
unsa u a ed zone; hyd ogeophysical moni o ing; hyd aulic conduc i i y; momen analysis
1. In oduc ion
Hyd aulic conduc i i y is he key pa ame e o he knowledge and unde s anding o
he low and anspo p ocesses ha ake place in he unsa u a ed zone. Many p ocesses
ha in ol e g oundwa e managemen and p o ec ion, i.e., he e alua ion o he aqui e
echa ge a e o he assessmen o aqui e ulne abili y om oxic subs ances, a e s ic ly
ela ed o he hyd aulic conduc i i y o he il e ing medium. A he same ime, he
ag icul u al managemen p ac ices depend on he accu a e knowledge o he hyd aulic
p ope ies o he soil.
As known, he hyd aulic conduc i i y o an unsa u a ed medium is a unc ion o he
wa e con en o he wa e po en ial [
1
]. Few di ec measu emen s o hyd aulic conduc i i y
unc ion a e ca ied ou in a labo a o y. The s eady-s a e me hod was pe o med using
a pe meame e [
2
], while he hyd aulic conduc i i y unc ion was de e mined using he
ansien me hod [3].
In ield condi ions, na u al p ocesses such as ponded in il a ion, copious ain all, o
i iga ion lead o a nea -sa u a ed s a e owing o he ai apped in a signi ican ac ion o
he po e space, in ol ing he ield-sa u a ed hyd aulic conduc i i y, K s [4].
In il ome e es s we e es ed on bo h soils and ocks o p o ide a eliable es ima ion
o such a pa ame e . Di e en ypes o in il ome e s we e used on soil o measu e K
s
[
5
–
8
],
while se e al c i icisms inc ease he di icul ies in pe o ming such es s on ocks [
9
–
11
]
because o he gap e ec be ween he agile p obes and he igid medium ha could a ec
he unce ain y o he measu emen s.
Wa e 2023,15, 332. h ps://doi.o g/10.3390/w15020332 h ps://www.mdpi.com/jou nal/wa e
Wa e 2023,15, 332 2 o 13
Howe e , such expe imen al app oaches do no ake in o accoun he wide spa ial
a iabili y o he hyd ological p ope ies, assuming small-scale a eas as ep esen a i e o he
in es iga ed medium, as well as dis u bances and e o s ha can a ec he measu emen s.
In some cases, uncon en ional la ge in il ome e appa a us ha e been used o include
as a as possible he he e ogenei ies o he subsu ace [
12
] bu , as a ma e o ac , he
p oblem o hyd aulic cha ac e iza ion o e la ge a ea s ill emains a g ea challenge o he
scien i ic communi y.
In he las decade, non-in asi e geophysical echniques a e inc easingly used o
hyd ological issues because hey ha e p o en o be eliable, as well as ela i ely quick,
inexpensi e, and capable o in es iga ing la ge a eas wi h high da a densi y sampling.
Among geophysical echniques, elec ical esis i i y omog aphy (ERT) allows o
imaging s a ic elec ical s uc u es in he subsu ace h ough esis ance measu emen s
collec ed on he g ound su ace o bo eholes. A wide ange o applica ions conce n he use
o ERT in he en i onmen al [13], hyd ological [14–16], and ag icul u al con ex [17,18].
In ecen yea s, ERT su eys pe o med in ime-lapse mode, i.e., epea ing esis i i y
da a collec ion along he same p o iles o a ea a di e en empo al s eps, ha e been
e ealed as a powe ul ool o moni o ing hyd ological p ocesses.
The basic concep ual analogy be ween hyd ological and elec ical models led o he
implemen a ion o hyd o-geophysical app oaches, ypically uncoupled and/o coupled, in
o de o cons ain he hyd ological model wi h geophysical da a and o p o ide a quan i a-
i e es ima ion o he hyd aulic conduc i i y and s a e a iables [
19
–
26
]. The ERT-de i ed
hyd ological s a e a iables a e used o calib a e and cons ain a hyd ological model, hus
iden i ying he co esponding go e ning pa ame e s (i.e., hyd aulic conduc i i y) wi h a
wide ange o nume ical app oaches, bo h s ochas ic and/o de e minis ic.
P edic ion o low and solu e anspo was de ined in e ms o p obabilis ic p ope ies
o concen a ion [
27
]. In hyd ology, se e al case s udies conside ed he momen analysis
app oach o p o ide a measu e o plume condi ion, as well as o e alua e o ecas ing sce-
na ios in a long- e m moni o ing p og am [
28
,
29
]. In en i onmen al ea u es, geophysical
da ase s we e inc easingly used as inpu pa ame e s o s ochas ic momen analysis [
30
,
31
],
owing o he high capabili y o in eg a e hyd ological da a in a eas no in es iga ed wi h
adi ional measu emen s.
In he p oposed case s udy, a hyd ogeophysical da ase , based on alling head in il-
a ion measu emen s combined wi h a ime-lapse ERT su ey, was p ocessed h ough
a combined s ochas ic and de e minis ic app oach in o de o p o ide an ini ial accu a e
es ima ion o he sa u a ed hyd aulic conduc i i y, K
s
, o ock in unsa u a ed condi ions.
Quan i a i e in o ma ion abou he wa e mass and he mo emen o he cen e mass o
he wa e plume we e ex ac ed om ERT-de i ed wa e con en obse a ions by pe o m-
ing he momen analysis echnique. A he same ime, wa e con en dis ibu ion in he
unsa u a ed zone du ing he in il a ion es has been p edic ed h ough se e al simula ion
uns o he alling head es by cons aining he wa e head and a ying he K
s
alue. The
momen analysis was pe o med o each simula ion and compa ed wi h he ERT-de i ed
calcula ions, in o de o p o ide an accu a e es ima ion o K
s
. The goals o his pape a e as
ollows: (1) o e alua e he abili y o he geophysical ool o gain quan i a i e in o ma ion
o he dynamics occu ing in he unsa u a ed ock a ield scale, o he wise unp edic able
wi h adi ional obse a ions; (2) o e i y he eliabili y o s ochas ic echniques, such as
momen analysis, in he moni o ing o unsa u a ed low; and (3) o educe he unce ain y
o such p edic ions by in eg a ing geophysical da ase s in o s ochas ic and de e minis ic
app oaches.
2. Ma e ials and Me hods
The app oach used in he p esen pape combines a s ochas ic and de e minis ic
p ocessing o a hyd ogeophysical da ase o es ima ing he K
s
pa ame e in he unsa u a ed
zone. On one hand, he ime-lapse ERT da ase s collec ed du ing he in il a ion es we e
used o image he esis i i y a ia ions o he subsu ace ha , in u n, we e con e ed
Wa e 2023,15, 332 3 o 13
in o wa e con en h ough A chie’s pa ame e s calib a ed in he labo a o y. On he o he
hand, he unsa u a ed low o he in il a ion dynamics was simula ed o p edic he wa e
con en dis ibu ion in he ocky subsu ace.
The momen analysis calcula ed on bo h he ERT-de i ed wa e con en da ase and
he simula ed one was used o connec ing he wo di e en app oaches in o de o p o ide
an es ima ion o he K
s
. The concep ual scheme o he p oposed app oach is desc ibed in
he lowcha shown in Figu e 1.
Wa e 2023, 15, x FOR PEER REVIEW 3 o 14
2. Ma e ials and Me hods
The app oach used in he p esen pape combines a s ochas ic and de e minis ic p o-
cessing o a hyd ogeophysical da ase o es ima ing he K
s
pa ame e in he unsa u a ed
zone. On one hand, he ime-lapse ERT da ase s collec ed du ing he in il a ion es we e
used o image he esis i i y a ia ions o he subsu ace ha , in u n, we e con e ed in o
wa e con en h ough A chie’s pa ame e s calib a ed in he labo a o y. On he o he
hand, he unsa u a ed low o he in il a ion dynamics was simula ed o p edic he wa e
con en dis ibu ion in he ocky subsu ace.
The momen analysis calcula ed on bo h he ERT-de i ed wa e con en da ase and
he simula ed one was used o connec ing he wo di e en app oaches in o de o p o-
ide an es ima ion o he K
s
. The concep ual scheme o he p oposed app oach is desc ibed
in he lowcha shown in Figu e 1.
Figu e 1. Wo k low diag am desc ibing he in eg a ed hyd ogeophysical app oach used in he p e-
sen case s udy o es ima ing sa u a ed hyd aulic conduc i i y, K
s
.
2.1. P ope ies o he In es iga ed Rock
The ock conside ed in he case s udy is a Pleis ocenic calca eni e, a sedimen a y ca -
bona ic po ous sands one o ma ine o igin, widesp ead in he Medi e anean unsa u a ed
zone. In e ms o hyd aulic p ope ies, he ock is well cha ac e ized, mainly in labo a o y
ac i i ies, as desc ibed by an ex ensi e li e a u e.
Capu o and Nimmo [32] p o ided accu a e es ima ions o wa e e en ion and hy-
d aulic conduc i i y unc ions o calca eni e samples collec ed in di e en ock qua ies.
Tu u o e al. [33] es ed he Da cy–Buckingham law in he cen i ugal ield on po ous
Figu e 1.
Wo k low diag am desc ibing he in eg a ed hyd ogeophysical app oach used in he
p esen case s udy o es ima ing sa u a ed hyd aulic conduc i i y, Ks.
2.1. P ope ies o he In es iga ed Rock
The ock conside ed in he case s udy is a Pleis ocenic calca eni e, a sedimen a y ca -
bona ic po ous sands one o ma ine o igin, widesp ead in he Medi e anean unsa u a ed
zone. In e ms o hyd aulic p ope ies, he ock is well cha ac e ized, mainly in labo a o y
ac i i ies, as desc ibed by an ex ensi e li e a u e.
Capu o and Nimmo [
32
] p o ided accu a e es ima ions o wa e e en ion and hy-
d aulic conduc i i y unc ions o calca eni e samples collec ed in di e en ock qua ies.
Tu u o e al. [
33
] es ed he Da cy–Buckingham law in he cen i ugal ield on po ous ock
media by demons a ing expe imen ally he di ec p opo ionali y be ween lux densi y
and di e en cen i ugal o ces main aining cons an he wa e con en . Tu u o e al. [
34
]
de i ed he whole e en ion cu es o calca eni e ocks by a me cu y in usion po osime y
es o mos o he me cu y con en ange, illing he gaps exhibi ed by he e en ion
cu es ob ained using he s anda d p ocedu es. Capu o e al. [
35
] es ed se e al labo a o y
me hods o he p edic ion o he WRCs pa ame e s c i ical o he nume ical simula ion o
he unsa u a ed low. And iani e al. [
36
] conduc ed alling head in il a ion es s a bench
Wa e 2023,15, 332 4 o 13
scale on egula ly shaped calca eni ic ocks wi h he aim o de ine he hyd aulic beha io
o calca eni es wi h di e en ex u es in unsa u a ed condi ions.
2.2. The Hyd ogeophysical Tes
The expe imen al es conside ed in he p esen pape conce ns a 4D ime-lapse ERT
moni o ing o in il a ion measu emen s pe o med a alling head on he bo om o a
calca eni e qua y. De ails o he expe imen al se up o he hyd ogeophysical es , as
well as elabo a ions o he in il a ion es and geophysical esul s, we e desc ibed in
de ail in [
21
]. The es consis ed o a double injec ion o abou 7 L o wa e in wo di e en
empo al s eps in o an in il ome e ing o 0.50 m in diame e sealed o he ou c opped ock.
Du ing he es , he dec easing o wa e le el was measu ed wi h a p essu e ansduc o
(PTX DRUCK LTD, S oke P io , B omsg o e, UK) and he deepening o he we ing on
was moni o ed by 4D ime-lapse ERT su eys. Twen y- i e pa ial hyd aulic heads, de ined
as echa ge pe iods in he code, measu ed a speci ic imes poin s du ing he in il ome e
es , we e used as cons ained bounda y condi ions o he simula ions.
Table 1 epo s he main in o ma ion o he in il a ion es , including ime o he s a
and s op o he wo injec ions, hyd aulic heads, and ERT obse a ion. Fo simplici y, only
en hyd aulic heads a e men ioned, co esponding o he alues measu ed a he s a and
s op o he injec ions, and hose obse ed a six ERT obse a ions.
Table 1. Time poin s o he in il ome e es and ERT da a collec ion.
In il a ion Tes Time Poin (hh:mm) Hyd aulic Head (cm) ERT Obse a ion
11:20 0
S a i s injec ion 11:43 3.1
12:03 2.0 1
12:33 1.0 4
S op in il a ion
measu emen s 12:43 0.8
13:23 8
S a second injec ion 14:01 3.3
14:16 2.5 12
14:36 2.1 13
14:59 1.6 14
S op in il a ion
measu emen s 15:09 1.3
15:59 0.2 15
In his pape , he geophysical ou pu s a e embedded in o a s ochas ic and de e minis ic
scheme, as pa o he in eg a ed da a p ocessing desc ibed in Figu e 1.
The expe imen al se up o he hyd ogeophysical es is shown in Figu e 2.
Fo y-eigh s ainless s eel elec odes we e placed a ound he ing wi h 0.40 m in e -
elec odes spacing o collec abou 1200 dipole–dipole esis i i y da a poin s wi h Syscal
P o Swi ch 48 (I is Ins umen s) esis i i y-me e . The da a collec ion was epea ed 15 imes
o e 4.5 h in o de o de ec changes in esis i i y caused by wa e con en a ia ions.
Each acquisi ion ime ook abou 15 min. ERTLab comme cial so wa e (Mul i-Phase
Technologies, LLC, in collabo a ion wi h GeoS udi As ie s. .l.) was used o he in e sion
p ocess. Acco ding o [
25
], a con enien o mula ion o A chie’s law was used o con e
he in e ed esis i i y in o sa u a ion deg ee.
ρ( i)=ρ( 0)Sw( 0)
Sw( i)n
, (1)
whe e
ρ
(
0
) and
ρ
(
1
) a e he esis i i ies o he ock (
Ω·
m) a imes
0
and
1
, espec i ely,
and S
w
(
0
) and S
w
(
1
) a e he sa u a ion deg ees a imes
0
and
1
, espec i ely. This
app oach allows he simpli ica ion o A chie’s equa ion, as he “n” sa u a ion index is
Wa e 2023,15, 332 5 o 13
he only unknown pa ame e o be calib a ed. A chie’s calib a ion was pe o med in a
labo a o y on calca eni e co e samples. In he labo a o y, s a ing om he sa u a ion
condi ion, he esis i i y–wa e con en cu e was eco ded e e y i e minu es du ing he
d ying p ocess.
Wa e 2023, 15, x FOR PEER REVIEW 5 o 14
p ocess. Acco ding o [25], a con enien o mula ion o A chie’s law was used o con e
he in e ed esis i i y in o sa u a ion deg ee.
ρ( )=ρ( )󰇧S( )
S( )󰇨, (1)
whe e ρ( 0) and ρ( 1) a e he esis i i ies o he ock (Ω∙m) a imes 0 and 1, espec i ely,
and Sw ( 0) and Sw ( 1) a e he sa u a ion deg ees a imes 0 and 1, espec i ely. This ap-
p oach allows he simpli ica ion o A chie’s equa ion, as he “n” sa u a ion index is he
only unknown pa ame e o be calib a ed. A chie’s calib a ion was pe o med in a labo -
a o y on calca eni e co e samples. In he labo a o y, s a ing om he sa u a ion condi ion,
he esis i i y–wa e con en cu e was eco ded e e y i e minu es du ing he d ying
p ocess.
Figu e 2. Expe imen al se up o he in il ome e es : (a) echnical d awing o he expe imen al se
up in a plan iew; (b) pic u e o 3D elec odes’ con igu a ion a angemen .
2.3. Fo wa d Hyd ological Modeling: Richa ds’ Equa ion
VS2DRTI code [37,38] was used o simula ing he alling head in il a ion expe i-
men . The code sol es he wo-dimensional app oxima ion o Richa ds’ equa ion o e
ime using he ini e-di e ence me hod, in bo h ca esian and adial coo dina es, hus
p o iding a quasi- h ee-dimensional model o an iso opic sys em, assuming angula
symme y.
The 3D Richa ds’ equa ion o a iably sa u a ed low in a homogenous igid po ous
medium [2] is exp essed in he ollowing o m:
𝜕𝜃(ℎ)
𝜕(𝑡)=𝜕
𝜕𝑥𝐾(ℎ)𝜕ℎ
𝜕𝑥+𝜕
𝜕𝑦𝐾(ℎ)𝜕ℎ
𝜕𝑦+𝜕
𝜕𝑧𝐾(ℎ)𝜕ℎ
𝜕𝑧+1−𝑆, (2)
whe e z is he ele a ion, is ime, θ is he olume ic wa e con en , h is he wa e po en ial,
θ(h) is he wa e e en ion unc ion, K(h) is he hyd aulic conduc i i y unc ion, and S is a
sink e m ep esen ing oo wa e up ake.
Acco ding o he an Genuch en model [39], he soil hyd aulic p ope ies we e de-
sc ibed as ollows:
𝜃=𝜃+𝜃−𝜃
󰇟1+|𝛼ℎ|󰇠, (3)
whe e θ is he wa e con en (m3∙m−3); θ is he esidual wa e con en (m3∙m−3); θs is he
sa u a ed wa e con en (m3∙m−3); h is wa e po en ial (kPa); α is a scale pa ame e in e sely
p opo ional o mean po e diame e (cm−1); and n and m a e he shape pa ame e s o soil
wa e cha ac e is ic, m = 1 − 1/n, 0 < m < 1. Acco ding o [35], an Genuch en’s pa ame e s
we e se as epo ed in Table 2.
Figu e 2.
Expe imen al se up o he in il ome e es : (
a
) echnical d awing o he expe imen al se
up in a plan iew; (b) pic u e o 3D elec odes’ con igu a ion a angemen .
2.3. Fo wa d Hyd ological Modeling: Richa ds’ Equa ion
VS2DRTI code [
37
,
38
] was used o simula ing he alling head in il a ion expe imen .
The code sol es he wo-dimensional app oxima ion o Richa ds’ equa ion o e ime using
he ini e-di e ence me hod, in bo h ca esian and adial coo dina es, hus p o iding a
quasi- h ee-dimensional model o an iso opic sys em, assuming angula symme y.
The 3D Richa ds’ equa ion o a iably sa u a ed low in a homogenous igid po ous
medium [2] is exp essed in he ollowing o m:
∂θ(h)
∂( )=∂
∂xK(h)∂h
∂x+∂
∂yK(h)∂h
∂y+∂
∂zK(h)∂h
∂z+1−S, (2)
whe e zis he ele a ion, is ime,
θ
is he olume ic wa e con en , his he wa e po en ial,
θ
(h) is he wa e e en ion unc ion, K(h) is he hyd aulic conduc i i y unc ion, and Sis a
sink e m ep esen ing oo wa e up ake.
Acco ding o he an Genuch en model [
39
], he soil hyd aulic p ope ies we e de-
sc ibed as ollows:
θ=θ +θs−θ
1+|αh|nm, (3)
whe e
θ
is he wa e con en (m
3·
m
−3
);
θ
is he esidual wa e con en (m
3·
m
−3
);
θs
is he
sa u a ed wa e con en (m
3·
m
−3
); his wa e po en ial (kPa);
α
is a scale pa ame e in e sely
p opo ional o mean po e diame e (cm
−1
); and nand ma e he shape pa ame e s o soil
wa e cha ac e is ic, m= 1
−
1/n, 0 < m< 1. Acco ding o [
35
], an Genuch en’s pa ame e s
we e se as epo ed in Table 2.
O he pa ame e s a e equi ed o be se in he VS2DRTI code, such as he aniso opy a-
io K
hh
/K
zz
(dimensionless), he speci ic s o age S
s
(m
−1
), he e ec i e po osi y
ϕ
(m
3·
m
−3
),
and he ini ial mois u e θcon en (m3·m−3).
Fo he p esen s udy, K
hh
/K
zz
and S
s
we e se equal o 1 and 1.6
×
10
−4
m
−1
, e-
spec i ely, al hough small a ia ions do no cause signi ican changes in he model ou pu .
Mo eo e , e ec i e po osi y
ϕ
was se equal o 0.45 on he basis o p e ious es s pe o med
on co e samples and he ini ial con en
θ
o 0.22 m
3·
m
−3
, as es ima ed om he ERT-de i ed
alue om A chie’s con e sion.

Wa e 2023,15, 332 6 o 13
Table 2. Pa ame iza ion o he model domain.
Pa ame e Scena io A Scena io B Scena io C Scena io D Scena io E
Sa u a ed Khh (cm·min−1)0.1 0.25 0.35 0.75 1
Khh/Kzz 1
Speci ic s o age, Ss(m−1) 1.6 ×10−4
E ec i e po osi y, ϕ0.45
θ, ini ial mois u e con en (m3·m−3)0.22
θ , esidual mois u e con en (m3·m−3)0.02547
α(cm−1)0.07721
n 1.7541
The e o e, once he an Genuch en’s pa ame e s ha e been se in VS2DRTI o all
simula ion uns, he only unknown a iable is he sa u a ed hyd aulic conduc i i y K
s
.
Se e al simula ions scena ios we e un wi h K
s
anging om 0.1 cm
·
min
−1
o 1 cm
·
min
−1
(Table 2), acco ding o he expec ed alues epo ed in he li e a u e.
A 2D low domain o 7
×
1 m (leng h
×
dep h) was buil wi h a cons an g id spacing in
he e ical di ec ion, equal o 0.02 m, and a iable g id spacing in he ho izon al di ec ion,
anging om 0.062 m o 0.143 m below he in il ome e ing. Thus, he domain was
disc e ized in o 2912 nodal poin s. A no- low bounda y condi ion (Di ichle condi ion) was
imposed along he la e al aces o he domain and op su ace ou side he in il ome e ing,
while a ee d ainage bounda y condi ion was used a he bo om. Simula ion ou pu imes
co espond o he ERT obse a ions, i.e.,
1
= 20 min,
4
= 50 min,
8
= 100 min,
12
= 153
min,
13
= 173 min,
14
= 196 min, and
15
= 256 min a e he s a o he i s injec ion, as
epo ed in Table 1.
2.4. Momen Analysis
The basics o he momen analysis is he ollowing equa ion:
Mijk( ) = yΓ
∆θ(x,y,z, )xiyjzkdxdydz (4)
The ze o h and i s spa ial momen s co espond o i+j+k= 0 and 1, espec i ely.
∆θ
is he wa e con en changes based on he esis i i y changes es ima ion in e ed
om he ime-lapse ERT model, a e emo ing he backg ound wa e con en .
Γ
is he
olume o in e es .
Assuming angula symme y o he moni o ed in il a ion dynamics, a 2D modeling
was analyzed in his case s udy o simpli y he calcula ion o he momen analysis.
The 2D sec ion conside ed o he momen s calcula ion c osses he ing h ough i s
cen e and, o a p ecise compa ison o he di e en da ase s, i co esponds o he same
c oss sec ion whe e he ERT-de i ed wa e con en was es ima ed.
The e o e, he ze o h momen , M
00
, is he changes in wa e mass wi hin he domain e-
spec o he backg ound (Equa ion (5)) and ep esen s he wa e s o age along he e e ence
sec ion, exp essed in m3m−3.
M00( ) = x∆θ(x,z)dxdz. (5)
The i s momen , M
01
no malized by he mass M
00
, de ines he e ical cen e o mass
o he plume a a gi en ime, z, exp essed by Equa ion (6).
z=M01
M00 (6)
Wa e 2023,15, 332 7 o 13
3. Resul s
3.1. ERT-De i ed Wa e Con en Ou pu s
Figu e 3shows he ERT-de i ed wa e con en dis ibu ion du ing he in il a ion es
in e ed a se e al ime poin s.
Wa e 2023, 15, x FOR PEER REVIEW 7 o 14
𝑀(𝑡)=𝛥𝜃(𝑥,𝑧)𝑑𝑥𝑑𝑧. (5)
The i s momen , M01 no malized by he mass M00, de ines he e ical cen e o mass
o he plume a a gi en ime, z, exp essed by Equa ion (6).
𝑧=𝑀
𝑀 (6)
3. Resul s
3.1. ERT-De i ed Wa e Con en Ou pu s
Figu e 3 shows he ERT-de i ed wa e con en dis ibu ion du ing he in il a ion es
in e ed a se e al ime poin s.
The backg ound wa e con en (Figu e 3a) highligh s a na ow wa e con en ange
in he subsu ace, om 0.20 o 0.22 m3∙m−3 es ima ed om A chie’s con e sion, by deno -
ing an almos homogeneous ini ial condi ion o he subsu ace, as expec ed. When he i s
injec ion s a s, wa e in il a es below he ing, deepening o e ime un il he end o he
i s injec ion (Figu e 3b–d). The wa e con en obse ed in he backg ound condi ions can
be a ibu ed o copious p ecipi a ion some days be o e he es , leading o high alues o
wa e con en in he uppe po ion o he subsu ace soon a e he s a ing o he i s
injec ion.
As shown in Figu e 3e–h, he second injec ion inc eases he subsu ace wa e con en
in he deepe laye s o he subsu ace (g een ill u ns yellow and hen ed).
Figu e 3. ERT-de i ed wa e con en a di e en ime poin s: (a) be o e he s a ing o he i s injec-
ion; (b) 20 min; (c) 50 min; (d) 100 min a e he i s injec ion; (e) 25 min; ( ) 45 min; (g) 68 min; (h)
128 min a e he second injec ion.
3.2. Momen Analysis De i ed om he ERT Da ase
The ze o h momen , calcula ed om he ERT-de i ed wa e con en ou pu s in e ed
along he 2D e e ence c oss sec ion, is shown in Figu e 4.
Figu e 3.
ERT-de i ed wa e con en a di e en ime poin s: (
a
) be o e he s a ing o he i s
injec ion; (
b
) 20 min; (
c
) 50 min; (
d
) 100 min a e he i s injec ion; (
e
) 25 min; (
) 45 min; (
g
) 68 min;
(h) 128 min a e he second injec ion.
The backg ound wa e con en (Figu e 3a) highligh s a na ow wa e con en ange in
he subsu ace, om 0.20 o 0.22 m
3·
m
−3
es ima ed om A chie’s con e sion, by deno ing
an almos homogeneous ini ial condi ion o he subsu ace, as expec ed. When he i s
injec ion s a s, wa e in il a es below he ing, deepening o e ime un il he end o he
i s injec ion (Figu e 3b–d). The wa e con en obse ed in he backg ound condi ions can
be a ibu ed o copious p ecipi a ion some days be o e he es , leading o high alues
o wa e con en in he uppe po ion o he subsu ace soon a e he s a ing o he i s
injec ion.
As shown in Figu e 3e–h, he second injec ion inc eases he subsu ace wa e con en
in he deepe laye s o he subsu ace (g een ill u ns yellow and hen ed).
3.2. Momen Analysis De i ed om he ERT Da ase
The ze o h momen , calcula ed om he ERT-de i ed wa e con en ou pu s in e ed
along he 2D e e ence c oss sec ion, is shown in Figu e 4.
Soon a e he i s injec ion, he inc ease in wa e con en below he in il ome ic
ing, which leads o an inc ease in wa e s o age, can be obse ed. As expec ed, he wa e
s o age g adually inc eases o e ime owing o he double injec ion.
No signi ican di e ence in he slope o he cu e can be obse ed a e he second
injec ion, p obably because o he small change in wa e con en below he in il ome e
ing and he sho empo al window be ween he wo injec ions.
Wa e 2023,15, 332 8 o 13
Wa e 2023, 15, x FOR PEER REVIEW 8 o 14
Soon a e he i s injec ion, he inc ease in wa e con en below he in il ome ic
ing, which leads o an inc ease in wa e s o age, can be obse ed. As expec ed, he wa e
s o age g adually inc eases o e ime owing o he double injec ion.
No signi ican di e ence in he slope o he cu e can be obse ed a e he second
injec ion, p obably because o he small change in wa e con en below he in il ome e
ing and he sho empo al window be ween he wo injec ions.
Figu e 4. Wa e s o age es ima ed h ough he ze o h momen M
00
.
Figu e 5 shows he p edic ion o he mo emen o he mass cen e calcula ed using
he wa e con en alues in e ed om he ime-lapse ERT ou pu s. No in o ma ion can
be obse ed wi hin he i s 0.20 m, which is abou he spa ial esolu ion o he ERT im-
ages. A e he i s injec ion, he mass cen e o he wa e bulb mo es downwa d, each-
ing a maximum dep h o 0.44 m om g ound su ace a ime poin
8
. A e he second
injec ion, he added in il a ed mass wa e causes a ise in he mass cen e , as clea ly ob-
se ed in he shape o he cu e, eaching a alue o 0.35 m a ime
15
.
Figu e 5. Dep h o he mass cen e eached a e wo injec ions.
Figu e 4. Wa e s o age es ima ed h ough he ze o h momen M00.
Figu e 5shows he p edic ion o he mo emen o he mass cen e calcula ed using
he wa e con en alues in e ed om he ime-lapse ERT ou pu s. No in o ma ion can be
obse ed wi hin he i s 0.20 m, which is abou he spa ial esolu ion o he ERT images.
A e he i s injec ion, he mass cen e o he wa e bulb mo es downwa d, eaching a
maximum dep h o 0.44 m om g ound su ace a ime poin
8
. A e he second injec ion,
he added in il a ed mass wa e causes a ise in he mass cen e , as clea ly obse ed in he
shape o he cu e, eaching a alue o 0.35 m a ime 15.
Wa e 2023, 15, x FOR PEER REVIEW 8 o 14
Soon a e he i s injec ion, he inc ease in wa e con en below he in il ome ic
ing, which leads o an inc ease in wa e s o age, can be obse ed. As expec ed, he wa e
s o age g adually inc eases o e ime owing o he double injec ion.
No signi ican di e ence in he slope o he cu e can be obse ed a e he second
injec ion, p obably because o he small change in wa e con en below he in il ome e
ing and he sho empo al window be ween he wo injec ions.
Figu e 4. Wa e s o age es ima ed h ough he ze o h momen M
00
.
Figu e 5 shows he p edic ion o he mo emen o he mass cen e calcula ed using
he wa e con en alues in e ed om he ime-lapse ERT ou pu s. No in o ma ion can
be obse ed wi hin he i s 0.20 m, which is abou he spa ial esolu ion o he ERT im-
ages. A e he i s injec ion, he mass cen e o he wa e bulb mo es downwa d, each-
ing a maximum dep h o 0.44 m om g ound su ace a ime poin
8
. A e he second
injec ion, he added in il a ed mass wa e causes a ise in he mass cen e , as clea ly ob-
se ed in he shape o he cu e, eaching a alue o 0.35 m a ime
15
.
Figu e 5. Dep h o he mass cen e eached a e wo injec ions.
Figu e 5. Dep h o he mass cen e eached a e wo injec ions.
3.3. Nume ical Simula ions
Figu e 6shows he compa ison be ween he ERT-de i ed wa e con en (Figu e 6a)
and h ee di e en scena ios selec ed among i e simula ion uns. Pa icula ly, scena io A
unde es ima es he K
s
(Figu e 6b), scena io E o e es ima es he K
s
(Figu e 6d), and scena io
C app oxima es he ue dis ibu ion o soil mois u e (Figu e 6c).
In o de o p o ide a quan i a i e es ima ion o K
s
, he momen analysis o each
simula ion scena io was calcula ed o all i e scena ios.
Wa e 2023,15, 332 9 o 13
The compa ison be ween he dep h o he mass cen e es ima ed om he geophysical
and nume ical app oaches shows ha he ERT-de i ed K
s
alue i s well he simula ed one
in he ange 0.25 < Ks< 0.35 cm·min−1(Figu e 7).
Wa e 2023, 15, x FOR PEER REVIEW 9 o 14
3.3. Nume ical Simula ions
Figu e 6 shows he compa ison be ween he ERT-de i ed wa e con en (Figu e 6a)
and h ee di e en scena ios selec ed among i e simula ion uns. Pa icula ly, scena io
A unde es ima es he Ks (Figu e 6b), scena io E o e es ima es he Ks (Figu e 6d), and sce-
na io C app oxima es he ue dis ibu ion o soil mois u e (Figu e 6c).
In o de o p o ide a quan i a i e es ima ion o Ks, he momen analysis o each sim-
ula ion scena io was calcula ed o all i e scena ios.
Figu e 6. Compa ison o he ERT-de i ed wa e con en (a) and he simula ed one o h ee di e en
scena ios: (b) scena io A; (c) scena io C; and (d) scena io E.
The compa ison be ween he dep h o he mass cen e es ima ed om he geophysi-
cal and nume ical app oaches shows ha he ERT-de i ed Ks alue i s well he simula ed
one in he ange 0.25 < Ks < 0.35 cm∙min−1 (Figu e 7).
Figu e 6.
Compa ison o he ERT-de i ed wa e con en (
a
) and he simula ed one o h ee di e en
scena ios: (b) scena io A; (c) scena io C; and (d) scena io E.
Wa e 2023, 15, x FOR PEER REVIEW 10 o 14
Figu e 7. Compa ison be ween he dep h o he mass cen e es ima ed om ERT and simula ions.
4. Discussion and Conclusions
This pape p esen ed an in eg a ed app oach based on hyd ogeophysical da a em-
bedded in o a s ochas ic and de e minis ic model scheme in o de o p o ide quan i a i e
hyd aulic in o ma ion on an unsa u a ed po ous ocky subsu ace. The unde s anding o
he low and anspo p ocesses in he unsa u a ed zone is an exci ing scien i ic challenge
and he p edic ion o hyd aulic p ope ies om geophysical da a is an open ques ion s ill
o be ully sol ed. The added alue poin ed ou in his manusc ip is based on he capa-
bili y o he in eg a ed app oach o in e unsa u a ed hyd ological p ope ies in a complex
domain, when da a a e lacking, o whe e da a collec ion is ex emely di icul o a ec ed
by dis u bances and e o s. In his pape , he achie emen o he objec i es is ob ained
h ough a mul i-s ep elabo a ion p ocess.
The i s s ep is based on he geophysical da a p ocessing, which e ealed he g ea
po en ial o ERT in he cha ac e iza ion and moni o ing o he unsa u a ed low p ocesses
in he ocky subsu ace, as shown in Figu e 3. The ERT-de i ed wa e con en c oss sec-
ions showed a as in il a ion soon a e he i s injec ion (Figu e 3b–d), ollowed by a
slow inc ease in wa e con en a e he second injec ion (Figu e 3e–h), owing o he high
sa u a ion deg ee o he ocky subsu ace.
Geophysical measu emen s alone can lead o misin e p e a ion, a i ac s, o mul iple
models, being based on he in insic non-uniqueness o he geophysical solu ion. The e-
o e, in o de o p oduce a obus hyd ogeophysical model, in he second s ep, he geo-
physical ou comes we e ea ed wi h a s ochas ic app oach, based on he momen analysis
echnique, wi h he aim o ack he mo emen o he injec ed wa e mass.
In pa icula , he dep h o he mass cen e was moni o ed du ing he in il a ion dy-
namics (Figu e 5). The geophysical ool is capable o de ec ing a ia ions in he dep h o
he mass cen e : soon a e he i s injec ion, i inc eases up o abou 0.40 m below g ound
su ace (bgs), hen i mo es sligh ly upwa d a e he second injec ion, eaching a dep h
o 0.35 m bgs a he end o he es .
Changing o he mass cen e dep h o e ime poin ed ou a de ailed esponse o he
unsa u a ed domain, which canno be ob ained using o he adi ional me hods. This is a
0
10
20
30
40
50
60
0 50 100 150 200 250 300 350 400
Dep h om g ound su ace (cm)
Time since s a o injec ion (min)
Ks=0.35 cm/min ERT de i ed Ks=0.1 cm/min
Ks=0.75 cm/min Ks=0.25 cm/min Ks=1 cm/min
Figu e 7. Compa ison be ween he dep h o he mass cen e es ima ed om ERT and simula ions.