Ci a ion: ˇ
Ríha, J.; Julínek, T.;
Duchan, D. Quan i ica ion o
G oundwa e Haza ds Rela ed o
Flu ial Floods ia G oundwa e Flow
Modelling: A Re iew. Wa e 2023,15,
1145. h ps://doi.o g/10.3390/
w15061145
Academic Edi o : Ryan Bailey
Recei ed: 17 Feb ua y 2023
Re ised: 7 Ma ch 2023
Accep ed: 11 Ma ch 2023
Published: 15 Ma ch 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
Re iew
Quan i ica ion o G oundwa e Haza ds Rela ed o Flu ial
Floods ia G oundwa e Flow Modelling: A Re iew
Ja omí ˇ
Ríha , Tomáš Julínek * and Da id Duchan
Ins i u e o Wa e S uc u es, Facul y o Ci il Enginee ing, B no Uni e si y o Technology,
602 00 B no, Czech Republic
*Co espondence: julinek. @ u b .cz
Abs ac :
Flood- ela ed issues include he impac o g oundwa e on lood p o ec ion measu es and
o he subsu ace s uc u es in a p o ec ed a ea. A he same ime, subsu ace elemen s o lood
p o ec ion s uc u es may signi ican ly in luence he na u al g oundwa e egime and a ec exis ing
s uc u es du ing non- lood pe iods. The pape p o ides an o e iew o haza dous ac o s linked o
g oundwa e and speci ies a iables o he quan i ica ion o ela ed haza ds. App op ia e hyd aulic
g oundwa e low models a e p esen ed and discussed, and hei sui abili y o he modelling o
indi idual ypical p oblems and o haza d quan i ica ion is speci ied. The use o models and
he applica ion o CAD and GIS ools o da a p e- and pos -p ocessing is men ioned in b ie and
demons a ed on examples o ypical p ac ical si ua ions.
Keywo ds:
modelling; g oundwa e ; lood p o ec ion; g oundwa e looding; g oundwa e haza ds
1. In oduc ion
Inc eased wa e s ages du ing lood e en s can signi ican ly a ec he egime o
g oundwa e low in aqui e s adjacen o i e s. The ise o he wa e le el in a wa e cou se
du ing a lood esul s in empo a y in il a ion along he banks o he i e and seepage
p opaga ion in o he aqui e in he opposi e di ec ion o ha o no mal low. The piezome -
ic le el ises and may cause wha has been e med a “g oundwa e lood” [
1
,
2
], which may
endange bo h lood p o ec ion elemen s and subsu ace pa s o s uc u es in p o ec ed
u banized a eas [
3
–
5
]. This e ec may esul in he wa e logging o e ain behind lood
p o ec ion ba ie s [6–8].
Du ing he las decades, se e al ex eme loods ook place in Cen al Eu ope. In Ma ch
and Ap il o 2006, loods occu ed due o long- e m p ecipi a ion combined wi h snow
mel ing. In July 1997, Augus 2002, and June 2013, loods ook place as a esul o pe sis en
ain all in la ge a eas [
9
,
10
]. Du ing hese e en s, g oundwa e looding happened a mos
o he a ec ed locali ies p o ec ed by he lood p o ec ion sys ems. Fo example, in he
illage o T oubky, a signi ican di e ence be ween he wa e le el in he i e and he
e ain behind he FPM caused wa e logging o he en i e illage a ea. On he con a y,
pe manen wa e logging occu ed in P ague-Zb asla due o he cons uc ion o he FPM
wi h deep subsu ace pa s ha blocked he na u al d ainage o he Vl a a Ri e .
In he case o he cons uc ion o lood con ol wo ks, hei subsu ace pa s may also
a ec he na u al g oundwa e egime du ing non- lood pe iods, du ing which g oundwa-
e mo es om highe loca ions owa ds s eams which d ain adjacen aqui e s [
11
]. He e,
main aining ee communica ion be ween he i e and aqui e is essen ial o , in e alia,
main aining base low in s eams [
12
–
14
]. The e o e, seepage ba ie s such as cu -o walls
and g ou cu ains should be designed o be pa ially pene able.
Fac o s such as geological composi ion (including an h opogenic laye s), he his o ical
de elopmen o owns in lood plains, i e egula ion (including lood p o ec ion s uc-
u es), quay walls, he d ainage o g oundwa e in o sewe age, e c., ha e o be aken in o
Wa e 2023,15, 1145. h ps://doi.o g/10.3390/w15061145 h ps://www.mdpi.com/jou nal/wa e
Wa e 2023,15, 1145 2 o 23
accoun when assessing g oundwa e e ec s and ela ed haza ds. P ac ically all o he
men ioned ac o s may be implemen ed in o powe ul g oundwa e low models. Cu en
ma hema ical modelling me hods [
15
] and exis ing so wa e enable use s o assess he
impac o lood p o ec ion measu es and hei subsu ace elemen s on he ounda ions o
lood p o ec ion s uc u es and buildings in p o ec ed a eas unde a na u al egime, as well
as du ing lood e en s. Hyd aulic calcula ions allow he e alua ion and quan i ica ion o
lood haza ds and hei ep esen a ion using powe ul compu e g aphics so wa es.
When e alua ing he isks associa ed wi h g oundwa e e ec s, he ulne abili y o
a ec ed s uc u es mus be aken in o accoun and compa ed wi h co esponding haza ds.
The po en ial damages and losses mus be es ima ed as well. Al hough g oundwa e
looding usually does no esul in losses o human li e o heal h, i may cause signi ican
ma e ial losses. Finally, he p obabili y o inciden mus be inco po a ed in o he isk
analysis. The p ope design o subsu ace pa s o lood p o ec ion ba ie s may signi ican ly
educe g oundwa e induced haza d and isk.
The a e o seepage p opaga ion depends on aqui e pa ame e s such as s o a i i y,
ansmissi i y, po osi y, sa u a ion o he soil, e c., which should be de e mined by in e se
modelling echniques [
16
–
19
]. The “calib a ion p ocess” depends on simula ion condi ions
(s eady-s a e o ansien ) and he aqui e ype (con ined o uncon ined). Nume ous
app oaches ha e been applied, such as pa icle swa m op imiza ion [
20
], gene ic algo i hms
o di e en ial e olu ion [17], nonlinea eg ession [21], o clonal selec ion algo i hms [22].
Fo he in e p e a ion o inpu da a and modelling esul s, mapping echniques based
on GIS ools, applica ions, and da abases on wa e esou ces (including g oundwa e issues)
ha e been applied [
23
]. E en i emo e-sensing and GIS echniques ha e expe ienced apid
de elopmen in ecen decades [
24
–
28
], hei applica ions o de ailed analysis in he a eas
behind lood p o ec ion measu es a e e y limi ed. Iden ical ools a e also a ailable o
web-based solu ions o g oundwa e low, and he analysis o g oundwa e sou ces and
pollu ion isk [
23
,
29
–
31
]. Cloud-based decision suppo sys ems ha e been p esen ed by
Jones e al. [
32
] based on ecen and his o ic wa e da a, including in e ac i e mapping
capabili ies. A nume ical model o g oundwa e low and he anspo o ma e comple ely
in eg a ed in a web en i onmen was desc ibed by Glass e al. [33].
Unlike lu ial i e looding and su ace wa e looding, g oundwa e lood isk maps
a e no commonly a ailable o he public. G oundwa e isk mapping amewo ks a e
howe e p esen ed, e.g., by Collins e al. [
34
] o Me chán-Ri e a [
35
]. Haza d maps o
upli due o ising g oundwa e du ing loods ha e been p esen ed by Julínek e al. [36].
Mos o he a ailable s udies deal wi h pa icula g oundwa e - ela ed issues, such as
g oundwa e looding [
1
,
37
] o combined modelling in ol ing lu ial looding in e ac ion
wi h g oundwa e low [
34
,
38
–
42
], haza ds in ka s ic lowlands [
43
], and o he s, while he e
is a lack o mo e comp ehensi e summa ies o g oundwa e e ec s in ela ion o loods and
lood p o ec ion measu es (FPM). This pape p o ides a lis o haza dous ac o s ela ed o
he g oundwa e egime du ing lood e en s, speci ies a iables o haza d quan i ica ion,
and p oposes app op ia e modelling echniques. Indi idual ypical cases a e demons a ed
o locali ies in he Czech Republic based on abou 50 case s udies ca ied ou in he egion.
The au ho s hus con ibu e o he isk analysis o g oundwa e ela ed issues du ing he
lood and no- lood pe iods, p o iding classi ica ion and quan i ica ion me hods o haza d
de e mina ion. The link be ween g oundwa e low modelling and ela ed haza ds is
de e mined as well.
2. Me hods
2.1. Ra ionale
The g oundwa e p oblems ela ed o lu ial loods and lood p o ec ion measu es a e
qui e complex. In Figu e 1, a diag am o g oundwa e low du ing a lu ial ( i e ) lood is
shown o a si ua ion wi hou (A) and wi h (B) lood p o ec ion measu es such as le ees
o loodwalls. I can be seen ha in he case o no FPM wi h wa e inunda ion, he po e
p essu e in he aqui e is compensa ed by he weigh o o e banked lood wa e . A e
Wa e 2023,15, 1145 3 o 23
ins alling FPM, he p essu e in he aqui e ha usually ac s on he bed o he ela i ely
impe ious opsoil laye may exceed i s weigh , which may cause upli ailu e o he opsoil
behind he lood p o ec ion a angemen s [
36
]. Such an impac can esul in a haza d bo h
o p o ec i e elemen s and o he subsu ace pa s o s uc u es (e.g., cella s, subsu ace
ga ages, e c.) in he p o ec ed e i o y. The h ea is posed by he ise o he piezome ic
head in he aqui e , by he wa e p essu e on he unde g ound pa s o ci il s uc u es, and
by he seepage below he FPM, which ia inc eased hyd aulic g adien s, may cause in e nal
e osion o he soil close o he ounda ions o lood p o ec ion s uc u es [
44
]. Seepage in o
he p o ec ed a ea is also un a ou able du ing a lood; i inc eases he amoun o “inne
wa e ” ha mus be pumped back in o he s eam (g oundwa e looding). When dealing
wi h haza ds ela ed o g oundwa e e ec s du ing he loods, in con ex wi h clima e
changes, one mus conside an expec ed highe in ensi y o lu ial loods, i.e., highe peak
discha ges and lood du a ions [45].
Wa e 2023, 15, x FOR PEER REVIEW 3 o 25
le ees o loodwalls. I can be seen ha in he case o no FPM wi h wa e inunda ion, he
po e p essu e in he aqui e is compensa ed by he weigh o o e banked lood wa e .
A e ins alling FPM, he p essu e in he aqui e ha usually ac s on he bed o he ela-
i ely impe ious opsoil laye may exceed i s weigh , which may cause upli ailu e o
he opsoil behind he lood p o ec ion a angemen s [36]. Such an impac can esul in a
haza d bo h o p o ec i e elemen s and o he subsu ace pa s o s uc u es (e.g., cella s,
subsu ace ga ages, e c.) in he p o ec ed e i o y. The h ea is posed by he ise o he
piezome ic head in he aqui e , by he wa e p essu e on he unde g ound pa s o ci il
s uc u es, and by he seepage below he FPM, which ia inc eased hyd aulic g adien s,
may cause in e nal e osion o he soil close o he ounda ions o lood p o ec ion s uc-
u es [44]. Seepage in o he p o ec ed a ea is also un a ou able du ing a lood; i inc eases
he amoun o “inne wa e ” ha mus be pumped back in o he s eam (g oundwa e
looding). When dealing wi h haza ds ela ed o g oundwa e e ec s du ing he loods,
in con ex wi h clima e changes, one mus conside an expec ed highe in ensi y o lu ial
loods, i.e., highe peak discha ges and lood du a ions [45].
On he o he hand, du ing a non- lood pe iod, he subsu ace impe ious elemen s o
FPM may cause damming o g oundwa e in he aqui e behind he FPM, wi h an un a ou -
able inc ease in he piezome ic head pe manen ly a ec ing he subsu ace pa s o ci il s uc-
u es in he a ea, o e en wi h wa e logging o he e ain on he p o ec ed loodplain.
Figu e 1. G oundwa e low du ing a lood. (A) no lood p o ec ion, (B) lood p o ec ion. Blue
a ows ep esen wa e low di ec ion, ed a ows change o wa e able/piezome ic head.
Figu e 1.
G oundwa e low du ing a lood. (
A
) no lood p o ec ion, (
B
) lood p o ec ion. Blue a ows
ep esen wa e low di ec ion, ed a ows change o wa e able/piezome ic head.
On he o he hand, du ing a non- lood pe iod, he subsu ace impe ious elemen s
o FPM may cause damming o g oundwa e in he aqui e behind he FPM, wi h an un-
a ou able inc ease in he piezome ic head pe manen ly a ec ing he subsu ace pa s o ci il
s uc u es in he a ea, o e en wi h wa e logging o he e ain on he p o ec ed loodplain.
Wa e 2023,15, 1145 4 o 23
2.2. Haza dous Si ua ions
A haza d is de ined as a si ua ion wi h he po en ial o cause undesi able e ec s.
He e, his is induced by he ac ion o g oundwa e on FPM elemen s and on he p o ec ed
e i o y. The ollowing si ua ions may occu :
Du ing a lood e en :
•
Flood p o ec ion s uc u es a e subjec ed o o ces ac ing on hei subsu ace pa s.
In addi ion o ea h p essu e, o ces caused by he g oundwa e unde lowing he
ounda ions mus be aken in o accoun . The mos c i ical ho izon al (F
h
) and e ical
(F ) wa e p essu e o ces a e shown in Figu e 2a.
•
An inc ease in he piezome ic head in he con ined aqui e behind he FPM may esul
in upli ac ing on impe ious lu ial opsoil (Figu e 2b). Simila ly, p essu e due o he
inc ease o he g oundwa e le el (piezome ic head) may a ec he subsu ace pa s
o he buildings wi h deep ounda ions (Figu e 2c). Such upli can cause up u es
in he opsoil and buildings, esul ing in localized concen a ed leakage, signi ican
de o ma ion o ounda ion slabs, o e en global s uc u al ins abili y. This may also
ini ia e wa e logging o he objec s (Figu e 2d).
•
In cases when pe meable soils c op ou o he e ain, seepage may occu behind he
FPM. This causes loading o soils by a p essu e g adien , which may esul in he in e nal
e osion o suscep ible soils (Figu e 2d) in he o m o ex e nal su usion o boiling. These
p ocesses in he p og ession phase may o en endange he s abili y o he FPM.
•
In case o long- e m loods, in combina ion wi h he pe meable aqui e , wa e logging
o he a ea behind he FPM may occu due o seepage on o he e ain (Figu e 2d).
Du ing a non- lood pe iod,
he undesi able e ec o impe ious elemen s such as
slu y walls may mani es i sel :
•
Damming o he g oundwa e le el (GWL) in an aqui e behind he FPM (Figu e 3a)
may cause a signi ican pe manen ise o g oundwa e le els, esul ing in wa e log-
ging o subsu ace pa s o buildings in he a ea.
•
I g oundwa e esou ces occu behind he FPM, impe ious subsu ace elemen s may
block he na u al bank in il a ion om a i e and hus de e io a e he wa e sou ce,
espec i e o a dec ease he yield o wells (Figu e 3b). This may also esul in a signi ican
dec ease o GWL in he p o ec ed a ea and cause unaccep able o e loading o wells.
A lis o po en ial haza ds caused by changes in he g oundwa e egime due o
looding and he FPM is summa ized in Table 1. In Table 1, he change o he g oundwa e
egime is sho - e m ( empo a y om hou s o single weeks) du ing he lood acco ding o
lood du a ion and long- e m in cases when subsu ace elemen s o he FPM pe manen ly
a ec he g oundwa e low.
Table 1. Lis o po en ial haza ds caused by a change in he g oundwa e egime.
Pe iod Haza d Po en ial Consequences
Flood
Tempo a y inc eased wa e p essu e on subsu ace
pa s o FPM Loss o s abili y o FPM, looding o p o ec ed a ea
Tempo a y inc eased wa e p essu e on aqui e opsoil
behind FPM
Collapse o opsoil laye , in e nal e osion o subbase soil,
collapse o FPM
Tempo a y ise o wa e able/piezome ic head in
p o ec ed e i o y, seepage behind FPM on e ain,
inc eased hyd aulic g adien s below FPM
Soil ins abili y due o seepage, in e nal e osion, loss o
s abili y o FPM
Tempo a y ise o wa e able in p o ec ed e i o y Tempo a y wa e logging o e ain behind FPM
No
lood
Pe manen ise o g oundwa e able/piezome ic head,
damming due o subsu ace elemen s o FPM
Pe manen wa e logging o e ain and s uc u es behind
FPM, inc eased p essu e on unde g ound pa s o s uc u es
Pe manen educ ion o bank in il a ion due o
impe meable subsu ace pa s o FPM
Reduc ion o wa e ex ac ed om g oundwa e esou ces,
g oundwa e le el d awdown, o e loading o wells
Wa e 2023,15, 1145 5 o 23
Wa e 2023, 15, x FOR PEER REVIEW 5 o 25
Figu e 2. Haza ds du ing a lood: (a) load on he ounda ions o he FPM, (b) upli on he opsoil
laye , (c) loading o he subsu ace pa s o ci il s uc u es behind he FPM, (d) seepage on o e ain,
wa e logging, in e nal e osion o soils due o high hyd aulic g adien s behind he FPM.
Figu e 2.
Haza ds du ing a lood: (
a
) load on he ounda ions o he FPM, (
b
) upli on he opsoil
laye , (
c
) loading o he subsu ace pa s o ci il s uc u es behind he FPM, (
d
) seepage on o e ain,
wa e logging, in e nal e osion o soils due o high hyd aulic g adien s behind he FPM.

Wa e 2023,15, 1145 6 o 23
Wa e 2023, 15, x FOR PEER REVIEW 6 o 25
Figu e 3. Haza ds du ing a non- lood pe iod: (a) damming o he g oundwa e on he p o ec ed
a ea side, (b) e ec on he yield o g oundwa e sou ces o subsu ace pa s o he FPM.
Table 1. Lis o po en ial haza ds caused by a change in he g oundwa e egime.
Pe iod Haza d Po en ial Consequences
Flood
Tempo a y inc eased wa e p essu e on subsu -
ace pa s o FPM Loss o s abili y o FPM, looding o p o ec ed a ea
Tempo a y inc eased wa e p essu e on aqui e
opsoil behind FPM
Collapse o opsoil laye , in e nal e osion o sub-
base soil, collapse o FPM
Tempo a y ise o wa e able/piezome ic head
in p o ec ed e i o y, seepage behind FPM on
e ain, inc eased hyd aulic g adien s below FPM
Soil ins abili y due o seepage, in e nal e osion, loss
o s abili y o FPM
Tempo a y ise o wa e able in p o ec ed e i-
o y Tempo a y wa e logging o e ain behind FPM
No lood
Pe manen ise o g oundwa e able/piezome ic
head, damming due o subsu ace elemen s o
FPM
Pe manen wa e logging o e ain and s uc u es
behind FPM, inc eased p essu e on unde g ound
pa s o s uc u es
Pe manen educ ion o bank in il a ion due o
impe meable subsu ace pa s o FPM
Reduc ion o wa e ex ac ed om g oundwa e e-
sou ces, g oundwa e le el d awdown, o e load-
ing o wells
Figu e 3.
Haza ds du ing a non- lood pe iod: (
a
) damming o he g oundwa e on he p o ec ed a ea
side, (b) e ec on he yield o g oundwa e sou ces o subsu ace pa s o he FPM.
2.3. Haza d Quan i ica ion
Today, he quan i ica ion o indi idual haza d cha ac e is ics is ca ied ou p edom-
inan ly by g oundwa e low models. The a iables ob ained as a esul o hyd aulic
modelling a e:
•piezome ic head h;
•
wa e p essu e p, ho izon al (F
h
) and e ical (F
) wa e p essu e o ces ac ing on he
su ace o subsu ace s uc u es, opsoil laye , and he FPM;
•
p essu e, o hyd aulic g adien (g ad p, g ad h), which ac s on he soil as a olume ic
o ce and may cause i s in e nal ins abili y;
•
wa e logging o he a ea is quan i ied by he a ec ed a ea A, whe e wa e seeps
on o e ain;
•seepage amoun Qwhen dealing wi h pumped wa e om wells.
Gene ally, all haza d quan i ie s a e a unc ion o he loca ion (coo dina es) and ime.
The spa io empo al ields o espec i e a iables (p,h, g ad p, g ad h, e c.) a e de e mined by
nume ical modelling. A mo e de ailed desc ip ion o he models can be ound in Sec ion 3.
2.4. Da a Acquisi ion
To ensu e accu a e modelling, i is c ucial o acqui e ele an da a ha may di e
om hose needed o analysing o he ypes o loods. The geological s uc u e and
hyd ogeological p ope ies o he a ea o in e es a e o u mos impo ance, which may no
hold ue o “su ace” loods. A b ie o e iew o he indi idual da a equi ed and hei
acquisi ion is p esen ed below.
Wa e 2023,15, 1145 7 o 23
Geog aphic and geode ic da a desc ibe e ain le els and p o ide in o ma ion abou
s uc u es wi hin he s udied a ea. The e ain le els a e e icien ly ob ained ia he
combina ion o Digi al Te ain Models (DTM) wi h mo e accu a e geode ic land su eying.
The da a abou s uc u es should include he shape o subsu ace s uc u es, including he
ounda ion dep h (le el), and p e e ably he ype and ma e ial o he s uc u e. O special
in e es a e linea condui s such as sewe s o la ge diame e s, in eg a ed pipe galle ies,
unde g ound ailway lines, e c.
FPM design p ojec s ( loodwalls, lood le ees) should be inco po a ed in o e ain
geome y. Subsu ace elemen s such as slu y o cu -o walls and d ainage sys ems mus
be aken in o accoun .
Geological and hyd ogeological da a gained om geological su eying should p o ide
in o ma ion on local geological condi ions and s uc u es. Hyd ogeological in o ma ion
should include a desc ip ion o he g oundwa e egime ( low di ec ion and amoun ,
piezome ic le el in he aqui e , seepage amoun ), pe meabili y alues, and s o age cha ac-
e is ics. These da a a e aken om he moni o ing o obse a ion bo eholes, om pumping
es s, and possibly om he de e mina ion o soil p ope ies (g ain size dis ibu ion, po os-
i y, uni o mi y, e c.).
Hyd ological and hyd aulic da a should speci y wa e le els in su ace wa e bodies
( i e s, lakes), bo h du ing he non- lood pe iod and du ing loods, in he o m o lood
hyd og aphs. The wa e le els which ep esen bounda y condi ions o seepage low a e
equen ly de e mined by open channel hyd aulics me hods.
Addi ional da a should be ob ained o p o ide in o ma ion abou exis ing wa e wells,
pumping amoun s, o he ag icul u al d ainage sys ems, e c.
Mos o he da a a e usually a ailable om adminis a i e bodies ha p o ide da a
o he public, including geological se ices and a chi es, hyd ological se ices, and i e
au ho i ies. In mos cases, he equi ed da a ha e a spa ial cha ac e and may be linked o
spa ial coo dina es. Fo hei analysis, p e-p ocessing, and ep esen a ion, he use o CAD
and/o GIS ools is ecommended.
Unce ain ies in such inpu da a p oduce unce ain ies in he modelling esul s. These
a e discussed in Sec ion 3.6.
3. G oundwa e Flow Modelling
Nume ical g oundwa e low modelling has been a s anda d discipline in con inuum
mechanics since he ea ly 1970s [
15
,
46
]. As he physical backg ound, assump ions, and
ma hema ical o mula ion a e su icien ly desc ibed and discussed in he ele an li e a u e,
his ex con ains only a b ie desc ip ion and he gene al cha ac e is ics o indi idual
models, wi h a ocus on hei use in p ac ical applica ions.
3.1. Modelling P ocedu e and Types o Models
To assess he haza ds men ioned in Sec ion 2, models o he low in he sa u a ed
zone a e p e e en ially used. The modelling p ocedu e comp ises a se o he ollowing
s anda d s eps:
•
The desc ip ion o a eal sys em whe e he a ea o in e es is iden i ied, and manage-
men p oblems and po en ial haza ds a e o mula ed.
•
The objec i es o modelling ha e o be ca e ully de ined oge he wi h expec ed ou comes
(see Sec ion 2). This in ol es he analysis o bo h lood and non- lood si ua ions.
•
The concep ual model consis s o a se o assump ions ela ed o he geome y, shape,
and bounda ies o he domain, as well as aqui e ma e ials and hei p ope ies (homo-
genei y, iso opy, po osi y, hyd aulic conduc i i y, comp essibili y, e c.). Acco ding o
he expec ed cha ac e o he low, he dimension and ime egime (s eady, ansien )
o he model a e de ined (Table 2).
•
The ma hema ical o mula ion (model) is ep esen ed by a se o go e ning equa ions,
plus ini ial and bounda y condi ions.
Wa e 2023,15, 1145 8 o 23
•
The compu e code app op ia e o he p oblem solu ion has o be selec ed [
47
–
49
].
P e- and pos - p ocessing a e necessa y pa s o he da a analysis, p epa a ion, and
p esen a ion. To his end, enginee ing app oaches a e combined wi h e icien pos -
p ocessing me hods which enable he display o spa ial and empo al da a using CAD
sys ems and hema ic maps wi hin GIS ools.
•
The nume ical model should be subjec o calib a ion and e i ica ion based on da a
om g oundwa e le el obse a ions and pumping amoun measu emen s. The
calib a ed and e i ied model may be used o he simula ion o scena ios ha answe
posed ques ions and achie e de ined objec i es.
Table 2. Summa y o he applica ion o g oundwa e low models.
P oblem Type o Model Solu ion Me hod
p elimina y assessmen o he p opaga ion
o a lood wa e in o an aqui e 1D— ansien analy ical me hods o simpli ied bounda y
and ini ial condi ions, nume ical me hods
complex spa ial assessmen o lood wa e
p opaga ion in o a la ge aqui e , assessmen o
piezome ic head and p essu e in he aqui e ,
assessmen o local s abili y o he opsoil and
s uc u es behind FPM, delimi a ion
o wa e logged a eas
2Dh— ansien nume ical me hods
assessmen o he e ec o subsu ace elemen s o
FPM du ing non- lood pe iods, changes in he yield
o a ec ed wa e sou ces, piezome ic head and
p essu e in an aqui e , assessmen o he s abili y o
s uc u es behind FPM, haza d o wa e logging
2Dh—s eady s a e nume ical me hods
de ailed assessmen o he local condi ions in he
icini y o FPM, s abili y o FPM and o he
s uc u es o he peak lood wa e le el scena io,
assessmen o non- lood scena ios
2D —s eady s a e nume ical me hods
solu ions a places wi h complex geome ical
condi ions and a gene al low di ec ion, such as FPM
ha c oss subsu ace condui s, unnels, e c.
3D—s eady s a e nume ical me hods
Fo he solu ion o indi idual cases, he selec ion o an app op ia e ype o model is
c ucial. This conce ns he simpli ica ion o assump ions ela ed o he spa ial and ime
dimensions o he model. The ollowing model ypes may be dis inguished:
•
one-dimensional (1D) g oundwa e low model o cases whe e pa allel seepage in
a la aqui e wi h small hyd aulic g adien s is expec ed (Dupui assump ion)— his
model may be used o bo h con ined and uncon ined aqui e s;
•
wo-dimensional model in he ho izon al plane (2Dh) applicable o la ge and complex
aqui e s wi h small hyd aulic g adien s (Dupui assump ion)— his model may be
used o bo h con ined and uncon ined aqui e s;
•
wo-dimensional model in he e ical plane (2D ), which can be used o pa allel
low wi h signi ican a ia ion in low di ec ion in he e ical plane, bo h o con ined
low and low wi h a ph ea ic su ace;
• h ee-dimensional model (3D) o low bo h in con ined and uncon ined condi ions.
All o he models men ioned abo e may be concei ed as s eady (s a iona y) o dynamic
( ansien ) acco ding o he na u e and egime o seepage. Gene ally, low in he sa u a ed
zone is he subjec o analysis.
In p ac ical cases, he sequence o models used is usually as ollows:
•p elimina y analysis is ca ied ou using a 1D model;
•complex analysis o lood p opaga ion o he aqui e using a ansien 2Dh model;
•
modelling o he condi ions du ing a non- lood pe iod using a s eady s a e 2Dh model;
Wa e 2023,15, 1145 9 o 23
•de ailed analysis o condi ions a he FPM using a s eady s a e 2D model;
•
i necessa y, he de ailed s eady s a e 3D modelling o singula i ies, whe e no dimen-
sional app oxima ions exis , may be conside ed.
3.2. One-Dimensional Model
A one-dimensional ansien model may be used o p elimina y calcula ions a a
small hyd aulic g adien ( he Dupui heo em) and wi h app oxima ely pa allel low. I can
be used o he analysis o he p opaga ion o a lood wa e in o an aqui e du ing a lood.
The go e ning equa ion holds [50]:
∂
∂xT∂h
∂x−S·
∂h
∂ +q=0 (1)
whe e Tis aqui e ansmissi i y (T=k
·
b o con ined aqui e s, T=k
·
H o uncon ined
aqui e s), kis hyd aulic conduc i i y, bis he hickness o he con ined aqui e , and His he
wa e dep h o he uncon ined aqui e . Sis he s o a i i y, which in he case o he con ined
aqui e is equal o he speci ic s o age mul iplied by aqui e hickness b, while in he case o
he uncon ined aqui e , i co esponds o he speci ic yield [
15
], and qis he linea sou ce
(e.g., in il a ion a he uncon ined aqui e ).
The bounda y condi ion (BC) exp esses he known (p esc ibed) piezome ic head
du ing a lood (Di ichle BC):
h(x, )=h(x, )(2)
Fo he ini ial condi ions, i holds ha :
h(x,0)=h0(x)(3)
The ime-dependen bounda y condi ions (2) a he i e side comes om he wa e
s ages du ing a lood, which a e de i ed om open channel hyd aulics o measu emen s
a gauging s a ions. The se ing o he bounda y condi ion and i s loca ion a he d y
side o he FPM is mo e complica ed. I is ecommended ha i be ixed su icien ly a
om he i e o elimina e i s in luence on he a ea close o he FPM. Ano he op ion
in nume ical modelling is o p o ide he compu a ional ( ini e elemen , ini e di e ence)
elemen s adjacen o he BC wi h smalle hyd aulic conduc i i y o a enua e hei e ec on
he seepage condi ions in he aqui e close o he FPM.
The esul o he calcula ions is he piezome ic head, o ph ea ic su ace, as a unc ion
o he ho izon al coo dina e xand ime . In his model, he p essu es, he hyd aulic, and
p essu e g adien s de i ed om piezome ic heads a e no ele an o u he de ailed
analysis and he assessmen o he consequences o single s uc u es. As men ioned abo e,
his model se es p edominan ly o p o ide p elimina y in o ma ion abou he p og ession
o a lood o an aqui e .
1D modelling, due o he e y small compu a ion ime in ol ed, o en includes he
sensi i i y analysis o inpu pa ame e s such as hyd aulic conduc i i y and s o a i i y,
and is also pe o med wi h he aim o se ing he dis ance o he d y-land bounda y
condi ion. The 1D model p o ides he i s idea abou he g oundwa e egime which is o
be in es iga ed in mo e complex models. I enables he se ing o bounda y condi ions and
gi es in o ma ion abou he mos un a ou able combina ion o inpu pa ame e s. I also
enables he p elimina y calib a ion o model pa ame e s i da a om g oundwa e egime
measu emen s a e a ailable.
An example conce ning he ime e olu ion o he piezome ic head in an aqui e adjacen
o he Vl a a Ri e in P ague du ing a lood in 2002 can be seen in Figu es 4and 5. I can
be seen ha he peak piezome ic head ollows he peak o he wa e le el in he Vl a a
Ri e (bounda y condi ion), wi h a ce ain deg ee o a enua ion and ime shi . The a e o
p opaga ion depends on he hyd aulic conduc i i y o he aqui e and on he s o a i i y. As
a ule, in he case o an uncon ined aqui e , he seepage p opaga ion is conside ably slowe
Wa e 2023,15, 1145 16 o 23
Wa e 2023, 15, x FOR PEER REVIEW 18 o 25
Figu e 9. Con ou s o he piezome ic head in a c oss sec ion o FPM.
Figu e 10. Hyd aulic g adien s in a c oss sec ion o FPM (same as Figu e 9). A ows ep esen
g oundwa e low di ec ion.
3.5. 3D Model
3D models a e used pa icula ly o sol ing complica ed de ails whe e he app oxi-
ma ions men ioned abo e a e no accep able. Fo he assessmen o lood p o ec ion
measu es and o he a ec ed s uc u es, i is usually su icien o apply a s eady-s a e so-
lu ion co esponding o he mos un a ou able condi ions selec ed o indi idual scena -
ios. The go e ning equa ion o aniso opic and nonhomogeneous ma e ials holds:
𝜕
𝜕𝑥𝑘𝜕ℎ
𝜕𝑥+𝜕
𝜕𝑦𝑘𝜕ℎ
𝜕𝑦+𝜕
𝜕𝑧𝑘𝜕ℎ
𝜕𝑧=0
(13)
Bounda y condi ions a e analogical o he 2D model. Di ichle BC:
Figu e 9. Con ou s o he piezome ic head in a c oss sec ion o FPM.
Wa e 2023, 15, x FOR PEER REVIEW 18 o 25
Figu e 9. Con ou s o he piezome ic head in a c oss sec ion o FPM.
Figu e 10. Hyd aulic g adien s in a c oss sec ion o FPM (same as Figu e 9). A ows ep esen
g oundwa e low di ec ion.
3.5. 3D Model
3D models a e used pa icula ly o sol ing complica ed de ails whe e he app oxi-
ma ions men ioned abo e a e no accep able. Fo he assessmen o lood p o ec ion
measu es and o he a ec ed s uc u es, i is usually su icien o apply a s eady-s a e so-
lu ion co esponding o he mos un a ou able condi ions selec ed o indi idual scena -
ios. The go e ning equa ion o aniso opic and nonhomogeneous ma e ials holds:
𝜕
𝜕𝑥𝑘𝜕ℎ
𝜕𝑥+𝜕
𝜕𝑦𝑘𝜕ℎ
𝜕𝑦+𝜕
𝜕𝑧𝑘𝜕ℎ
𝜕𝑧=0
(13)
Bounda y condi ions a e analogical o he 2D model. Di ichle BC:
Figu e 10.
Hyd aulic g adien s in a c oss sec ion o FPM (same as Figu e 9). A ows ep esen
g oundwa e low di ec ion.
3.5. 3D Model
3D models a e used pa icula ly o sol ing complica ed de ails whe e he app oxima-
ions men ioned abo e a e no accep able. Fo he assessmen o lood p o ec ion measu es and
o he a ec ed s uc u es, i is usually su icien o apply a s eady-s a e solu ion co esponding
o he mos un a ou able condi ions selec ed o indi idual scena ios. The go e ning equa ion
o aniso opic and nonhomogeneous ma e ials holds:
∂
∂xkx
∂h
∂x+∂
∂yky
∂h
∂y+∂
∂zkz
∂h
∂z=0 (13)

Wa e 2023,15, 1145 17 o 23
Bounda y condi ions a e analogical o he 2D model. Di ichle BC:
h(x,y,z)=h(x,y,z)(14)
A he bounda y wi h p esc ibed lux, he Neumann condi ion holds:
kx
∂h
∂xnx+ky
∂h
∂yny+kz
∂h
∂znz=qp(15)
On he ph ea ic su ace wi h a s eady s a e low, wo bounda y condi ions may be
applied. The i s one may be exp essed as ollows:
h(x,y,z)=zp(x,y)(16)
while he second one co esponds o Equa ion (15) wi h qp= 0.
A he seepage ace, BC has he o m:
h(x,y,z)=zs(x,y)(17)
Mos compu e codes sol e ph ea ic low seepage using he concep o an unsa u a ed
zone, whe e he sa u a ed domain co esponds o he po e p essu es p
≥
p
a
, whe e p
a
is
a mosphe ic p essu e, o p
≥
0 i he a mosphe ic p essu e is used as he e e ence p essu e.
As men ioned abo e, 3D models a e used in cases whe e dimensional simpli ica ions
a e no easible. The e is a la ge a ie y o p ac ical si ua ions whe e he low di ec ion
is qui e gene al and simple models (1D, 2Dh, 2D ) a e no applicable. Such low may
occu in complex geological condi ions combined wi h spa ially a iable ci il s uc u es
in e e ing wi h he low domain [
53
–
55
]. Due o he la ge numbe o compu a ional nodes
in a 3D model, he low domain size is usually minimized while bounda y condi ions a e
de e mined by a p elimina y simpli ied analysis using 2D models. Du ing modelling, i is
easible o di ide he low domain in o homogeneous blocks (mac oelemen s; see Figu e 11),
and o apply he compu a ional mesh (e.g., ini e elemen s) la e on.
Wa e 2023, 15, x FOR PEER REVIEW 20 o 25
Figu e 11. Diag am o laye ed 3D mac oelemen s. Di e en colou s ep esen soil laye s in e aces.
(a) (b)
Figu e 12. Compa ison o piezome ic heads on he base o a su ace laye : (a)—only impe ious
opsoil, (b)—an h opogenic laye on he su ace. Blue ellipse indica es de ail o a le ee wi h he
descending unnel amp shown in Figu e 13.
Figu e 11. Diag am o laye ed 3D mac oelemen s. Di e en colou s ep esen soil laye s in e aces.
Wa e 2023,15, 1145 18 o 23
The ollowing example conce ns he complex seepage low below he le ee, which
in e e es wi h subsu ace elemen s o he unnel c ossing he Vl a a i e (P ague) and
g adually ises ou o he e ain in he p o ec ed a ea behind he le ee. In Figu e 11, a laye ed
scheme o mac oelemen s is depic ed. These mac oelemen s co espond o geological and
cons uc ion laye s and a e u he meshed by ini e elemen s wi h a size o 0.5 m, which
ensu es he accu acy o he esul s is su icien . Figu e 12 shows a compa ison o piezome ic
heads o a ious geological condi ions, i.e., an aqui e o e laid by ela i ely impe ious
opsoil (a) and su ace laye s consis ing o an h opogenic back ill (b). In Figu e 13, a de ail o
a le ee wi h a descending unnel is depic ed o scena io (b).
Wa e 2023, 15, x FOR PEER REVIEW 20 o 25
Figu e 11. Diag am o laye ed 3D mac oelemen s. Di e en colou s ep esen soil laye s in e aces.
(a) (b)
Figu e 12. Compa ison o piezome ic heads on he base o a su ace laye : (a)—only impe ious
opsoil, (b)—an h opogenic laye on he su ace. Blue ellipse indica es de ail o a le ee wi h he
descending unnel amp shown in Figu e 13.
Figu e 12.
Compa ison o piezome ic heads on he base o a su ace laye : (
a
)—only impe ious
opsoil, (
b
)—an h opogenic laye on he su ace. Blue ellipse indica es de ail o a le ee wi h he
descending unnel amp shown in Figu e 13.
Wa e 2023, 15, x FOR PEER REVIEW 21 o 25
Figu e 13. De ail o a le ee wi h he descending unnel amp; scena io B acco ding o Figu e 12.
3.6. Discussion o Unce ain ies
In case o g oundwa e low modelling, unce ain ies ela e pa icula ly o:
• The geological composi ion o he a ea, such as he hickness o indi idual laye s (aq-
ui e , opsoil, e c.), which is usually de i ed om a limi ed numbe o bo eholes o
pi s;
• limi ed unde s anding abou o e all hyd ogeological and hyd ological condi ions,
i.e., ime-dependen g oundwa e low egime, he di ec ion and amoun o g ound-
wa e low, in lows and in il a ion o an aqui e — he unce ain ies a e go e ned by
he ex en o moni o ing ne wo k and equency o eadings;
• he knowledge abou geological and hyd ogeological p ope ies o opsoil and aqui-
e soils, namely g anulome y, po osi y, hyd aulic conduc i i y, and s o a i i y,
which a e de i ed om labo a o y and ield es ing, bu in many cases only use em-
pi ical o mulae supplemen ed by single hyd aulic es s (pumping es s);
• he a e o in e ac ion be ween he i e and aqui e , which may be in luenced by
local clogging;
• bounda y condi ions, bo h a he i e side and behind he FPM, a e de i ed om
lood hyd og aphs which a e no ou inely s a is ically assessed in e ms o hei
shape and lood olume;
• in il a ion a es du ing he simula ed e en .
While he geome ical cha ac e is ics o he geological laye s may di e in me es
( ens o %), pe meabili y and s o age cha ac e is ics may be subjec o deg ees o unce -
ain y ha a e o en e y la ge (se e al o de s). To educe he e ec o unce ain ies, he
ollowing echniques may be ecommended.
A sensi i i y analysis is ecommended o assessing how unce ain ies in inpu a i-
ables in luence unce ain ies in ou pu a iables. Some imes he impac o pa ame e s
such as hyd aulic conduc i i y, ansmissi i y, s o a i i y, e c., may be quali a i ely es i-
ma ed om he go e ning equa ions.
The eliabili y o a model may be signi ican ly inc eased by ca e ul calib a ion and
e i ica ion. Du ing he calib a ion, a unique se o model pa ame e s is ound ha p o-
ides a good desc ip ion o he sys em’s beha iou , i.e., ag eemen wi h he piezome ic
head, yields, and o he a iables measu ed in he ield.
A gene al enginee ing app oach is o se “sensi i e” cha ac e is ics and pa ame e s
so as o ensu e conse a i e (“sa e”) esul s a e ob ained. Fo example, he a e o he ise
in he piezome ic head in he aqui e is g ea e wi h highe hyd aulic conduc i i y ( ans-
missi i y) and smalle s o a i i y.
Figu e 13. De ail o a le ee wi h he descending unnel amp; scena io B acco ding o Figu e 12.
Wa e 2023,15, 1145 19 o 23
I is ob ious ha he in e p e a ion o 3D esul s is di icul and should be p ocessed
ia laye s (Figu e 12) and selec ed sec ions (Figu e 13) o he 3D domain. He e, he p e- and
pos -p ocessing abili y o he compu ed code (e.g., [49]) is c ucial.
3.6. Discussion o Unce ain ies
In case o g oundwa e low modelling, unce ain ies ela e pa icula ly o:
•
The geological composi ion o he a ea, such as he hickness o indi idual laye s (aqui e ,
opsoil, e c.), which is usually de i ed om a limi ed numbe o bo eholes o pi s;
•
limi ed unde s anding abou o e all hyd ogeological and hyd ological condi ions, i.e.,
ime-dependen g oundwa e low egime, he di ec ion and amoun o g oundwa e
low, in lows and in il a ion o an aqui e — he unce ain ies a e go e ned by he
ex en o moni o ing ne wo k and equency o eadings;
•
he knowledge abou geological and hyd ogeological p ope ies o opsoil and aqui e
soils, namely g anulome y, po osi y, hyd aulic conduc i i y, and s o a i i y, which
a e de i ed om labo a o y and ield es ing, bu in many cases only use empi ical
o mulae supplemen ed by single hyd aulic es s (pumping es s);
•
he a e o in e ac ion be ween he i e and aqui e , which may be in luenced by
local clogging;
•
bounda y condi ions, bo h a he i e side and behind he FPM, a e de i ed om lood
hyd og aphs which a e no ou inely s a is ically assessed in e ms o hei shape and
lood olume;
•in il a ion a es du ing he simula ed e en .
While he geome ical cha ac e is ics o he geological laye s may di e in me es ( ens
o %), pe meabili y and s o age cha ac e is ics may be subjec o deg ees o unce ain y
ha a e o en e y la ge (se e al o de s). To educe he e ec o unce ain ies, he ollowing
echniques may be ecommended.
A sensi i i y analysis is ecommended o assessing how unce ain ies in inpu a i-
ables in luence unce ain ies in ou pu a iables. Some imes he impac o pa ame e s such
as hyd aulic conduc i i y, ansmissi i y, s o a i i y, e c., may be quali a i ely es ima ed
om he go e ning equa ions.
The eliabili y o a model may be signi ican ly inc eased by ca e ul calib a ion and
e i ica ion. Du ing he calib a ion, a unique se o model pa ame e s is ound ha p o ides
a good desc ip ion o he sys em’s beha iou , i.e., ag eemen wi h he piezome ic head,
yields, and o he a iables measu ed in he ield.
A gene al enginee ing app oach is o se “sensi i e” cha ac e is ics and pa ame e s
so as o ensu e conse a i e (“sa e”) esul s a e ob ained. Fo example, he a e o he
ise in he piezome ic head in he aqui e is g ea e wi h highe hyd aulic conduc i i y
( ansmissi i y) and smalle s o a i i y.
Be e in o ma ion on he e ec o unce ain ies can be p o ided by he applica ion
o in e al algeb a o by s a is ical modelling p ocedu es. Howe e , in g oundwa e low
modelling, he s ochas ic modelling app oach has no ye become a ool used ou inely by
modelle s on a egula basis [
56
]. This is mos ly due o he insu icien and poo geological
and hyd ogeological da a ha is a ailable. In he case o s ochas ic s udies, only single
pa ame e s a e conside ed o be unce ain [57].
The simpli ica ions o model dimensions and dynamics a e ela ed o he na u e o
he p oblem, he aims o modelling, and he shape and a angemen o he low domain.
Special a en ion should be paid o he analysis, g aphical p esen a ion, and in e p e a ion
o haza ds using con empo a y GIS and CAD sys ems. These a e summa ized in Table 3.
Wa e 2023,15, 1145 20 o 23
Table 3. Lis o po en ial haza ds caused by a change in g oundwa e egime.
Pe iod Haza d In e p e a ion Figu e
Flood
Inc eased wa e p essu e on
subsu ace pa s o FPM and
s uc u es behind FPM,
de ailed analysis
Map o p essu e head, sa e y ac o ,
c oss sec ions wi h piezome ic
con ou s, p essu e diag ams
Figu es 2a,c, 5,8and 9
Inc eased wa e p essu e on aqui e
opsoil behind FPM
Flood wa e p opaga ion diag ams,
maps o piezome ic head, upli
p essu es, sa e y ac o
Figu es 2b, 4,5,6,7a, 8,
12 and 13
Rise o piezome ic head in p o ec ed
a ea, seepage on e ain
Flood wa e p opaga ion diag ams,
c oss sec ion wi h piezome ic
con ou s and hyd aulic g adien s
Figu es 4,5,9and 10
Tempo a y ise o wa e able in
p o ec ed e i o y
Flood wa e p opaga ion diag ams,
map o maximum piezome ic head
di e ences, map o wa e logged a ea
Figu es 4,5and 6a
No lood
Rise o g oundwa e
able/piezome ic head, damming
due o subsu ace pa s o FPM
Map o piezome ic head di e ences
and e ain, di e ences be o e and
a e cons uc ion o FPM
Figu es 6b and 7
Pe manen educ ion o bank
in il a ion due o
impe meable/semipe meable
subsu ace pa s o FPM
Map o di e ences in
ph ea ic/piezome ic su ace, c oss
sec ion h ough wells, d op in yield
Figu es 3,6b, and 7b,c
4. Conclusions
The pape summa izes he p oblems and echniques ela ed o he modelling o
g oundwa e low impac s and haza ds ela ed o lu ial loods.
Two ypical scena ios a e discussed, namely a lood si ua ion and a non- lood pe iod. In
all cases, he s uc u al sa e y o he FPM and a ec ed ci il s uc u es mus be gua an eed.
Expe ience shows ha when inhabi an s a e a ec ed, non- lood pe iods a e mo e sensi i e
han ela i ely sho pe iods o looding. This namely conce ns cases when wa e logging
and simila ha m o inhabi an s and ci il s uc u es may occu . In hese cases, he modelling
esul s should be “sa e ” and echnical p oposals mo e obus . Mo eo e , in u ban a eas,
echnical measu es o en in e e e wi h exis ing in as uc u e placed alongside i e s, such as
oads, subsu ace wa e and sewe mains (Figu e 9), elec ic linings, op ical ib e cables, e c.
The p esen ed s udy aims o ill he gap in a igo ous classi ica ion o g oundwa e
haza ds due o lu ial loods and he a angemen o he FPM. The no el y o he s udy lies
in a o malised analysis o he echnical aspec s o g oundwa e haza ds ela ed o loods.
I can be seen ha lood-p o ec ion p oblems ela ed o g oundwa e a e based on
adi ional g oundwa e low modelling echniques. Howe e , special si e-speci ic ap-
p oaches a e necessa y in he case o haza d iden i ica ion and quan i ica ion, he selec ion
o an app op ia e g oundwa e low model, and he p esen a ion and in e p e a ion o
modelling esul s.
Based on he esul s ob ained, echnical measu es o he a enua ion o haza ds can be
p oposed, such as he app op ia e ype and a angemen o slu y walls, FPM ounda ions,
d ainage sys ems, e c.
The pape summa izes expe ience ob ained o e 30 yea s o g oundwa e low mod-
elling ela ed o he FPM design and assessmen in he e i o y o he Czech Republic,
Slo akia, and Aus ia.
Au ho Con ibu ions:
J.ˇ
R.: Concep ualiza ion, Me hodology, Supe ision, W i ing—o iginal d a ,
alida ion; T.J.: Concep ualiza ion, Fo mal analysis, In es iga ion, Mapping, W i ing—o iginal d a
and edi ing; D.D.: In es iga ion, Valida ion, Mapping, W i ing— e iew and edi ing. All au ho s ha e
ead and ag eed o he published e sion o he manusc ip .
Wa e 2023,15, 1145 21 o 23
Funding:
This esea ch was unded by he Technology Agency o he Czech Republic, p ojec
numbe TH4030087 and In e nal g an agency o he B no Uni e si y o Technology, p ojec numbe
FAST-S-23-8233.
Da a A ailabili y S a emen : No applicable.
Con lic s o In e es :
The au ho s decla e no con lic o in e es . The unde s had no ole in he design
o he s udy; in he collec ion, analyses, o in e p e a ion o da a; in he w i ing o he manusc ip ; o
in he decision o publish he esul s.
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Disclaime /Publishe ’s No e:
The s a emen s, opinions and da a con ained in all publica ions a e solely hose o he indi idual
au ho (s) and con ibu o (s) and no o MDPI and/o he edi o (s). MDPI and/o he edi o (s) disclaim esponsibili y o any inju y o
people o p ope y esul ing om any ideas, me hods, ins uc ions o p oduc s e e ed o in he con en .