THM-modelling benchmark initiative on the effects of temperature on the disposal of heat-generating radioactive waste in clay formations

RESEARCH PAPER
THM-modelling benchma k ini ia i e on he e ec s o empe a u e
on he disposal o hea -gene a ing adioac i e was e in clay
o ma ions
E ic Simo
8,1
•Ch is ophe de Lesquen
2
•Rocio Paola Leon-Va gas
1
•Minh-ngoc Vu
2
•Simon Raude
3
•
Ginge El Tabbal
3
•A naud Dizie
4
•Su esh See ha am
4
•As a Na kuniene
5
•F e
´de
´ ic Collin
6
•
Hangbiao Song
6
•An onio Gens
7
•Fei Song
7
•Alexand u-Bogdan Ta omi
10
•Thomas Nagel
8,9
•
Jo
¨ g Buchwald
8,9
Recei ed: 22 Decembe 2023 / Accep ed: 1 Decembe 2024
The Au ho (s) 2025
Abs ac
Unde s anding he he mo-hyd o-mechanical (THM) beha iou o clay o ma ions, which a e being conside ed as po en ial
hos s o he disposal o adioac i e was e in Eu ope, is impo an o he easibili y and he sa e disposal o hea -gene a ing
adioac i e was e in deep geological clayey o ma ions. To ensu e he eliabili y o nume ical models and ools used o
p edic he THM e olu ion o such sys ems o e long pe iods o ime, i is necessa y o e i y and alida e hese ools
h ough benchma king ini ia i es. As pa o he join Eu opean P og amme on Radioac i e Was e Managemen (EURAD),
he In luence o Tempe a u e on Clay-based Ma e ial Beha iou (HITEC) wo k package in i ed se en eams om ac oss
Eu ope o pa icipa e in a benchma k ini ia i e o assess he expe ise and capabili ies o hese eams and hei nume ical
ools o p edic THM e olu ion in di e en clay ma e ials. The esul s o his benchma k showed ha all eams in ol ed
we e able o adequa ely model he THM beha iou o hea -gene a ing eposi o y sys ems in clay o ma ions, and he ools
hey used can be conside ed e i ied o he solu ion o coupled THM equa ions unde a iable bounda y condi ions
ele an o nuclea was e disposal.
Keywo ds Boom clay Callo o–Ox o dian clay Geological disposal High-le el adioac i e was e Opalinus Clay 
Sa e y assessmen The mo-hyd o-mechanical modelling
1 In oduc ion
A gillaceous o ma ions a e conside ed ac oss Eu ope as a
po en ial hos o he geological disposal o adioac i e
was e. This is due o hei sel -sealing p ope ies, hei low
pe meabili y, hei small molecula di usion and hei
e en ion capaci y o adionuclides. The mo-hyd o-me-
chanical p ocesses (THM p ocesses) play an impo an ole
in he e olu ion o a eposi o y in clay o ma ions. The
exca a ion o he eposi o y mine leads o a damaged zone
in he su ounding hos geological o ma ion. This exca-
a ion-damaged zone is cha ac e ized by a highe hyd aulic
conduc i i y ha enhances hyd aulic low wi hin he
eposi o y. Mo eo e , he hea gene a ed by he was e leads
o he mally induced s esses in he ock and po en ially
u he inc eases he exca a ion-damaged zone. The hea
p opaga ion also gi es ise o excess po e p essu es in he
hos clayey o ma ion due o he dila ion o wa e in he
po ous hos medium ha may lead o ensile damage e en
in he a ield o he eposi o y sys em. A deep unde -
s anding o he THM beha iou o po en ial hos clayey
o ma ions is he e o e impo an o he design and sa e y
assessmen o eposi o y sys ems in hese o ma ions.
The assessmen o he long- e m sa e y o eposi o y
sys ems equi es he use o nume ical models and ools
capable o ep oducing he THM beha iou o he clay hos
E ic Simo, Ch is ophe de Lesquen, Rocio Paola Leon-
Va gas, Minh-ngoc Vu, Simon Raude, Ginge El Tabbal,
A naud Dizie , Su esh See ha am, As a Na kuniene, F e
´de
´ ic
Collin, Hangbiao Song, An onio Gens, Fei Song, Alexand u-
Bogdan Ta omi , Thomas Nagel, Jo
¨ g Buchwald ha e
con ibu ed equally o his wo k.
Ex ended au ho in o ma ion a ailable on he las page o he a icle
123
Ac a Geo echnica
h ps://doi.o g/10.1007/s11440-024-02502-w(0123456789().,- olV)(0123456789().,- olV)
ock ha has been obse ed h ough expe imen al in es i-
ga ions and p edic ing he THM e olu ion o he eposi o y
sys ems o e he assessmen pe iod o up o one million
yea s. I ollows ha he eliabili y o such p edic i e ools
should be e i ied and alida ed [14]. Ve i ica ion is he
ini ial s ep owa ds alida ion, which will subsequen ly
in ol e he compa ison wi h in si u da a om ield
expe imen s-endea ou s. By e i ica ion, one means he
ma hema ical co ec ness o a gi en nume ical model. To
e i y such models, one can, o example, compa e he
ou pu o nume ical simula ions agains hei analy ic
coun e pa s o a gi en p oblem.
Gi en he inc eased complexi y o nume ical models
employed in he ealm o THM coupled sa e y assessmen
o deep geological eposi o ies in clay, he e a e li le
analy ic solu ions agains which compu e codes can be
e i ied. Ensu ing he alidi y o simula ion esul s and he
subsequen eliabili y o decision-making become he e o e
inc easingly challenging. This p edicamen is u he
compounded when mul iple nume ical codes a e employed,
adhe ing o he wo-pe son ule du ing nume ical analyses.
Reg e ably, no comp ehensi e guidelines o s anda ds
ha e been published o add ess he quali y o esea ch and
comme cial codes cu en ly u ilized in he sa e y assess-
men o eposi o ies wo ldwide.
Se e al benchma king ini ia i es ha e been ca ied ou
in e na ionally in a ious esea ch p ojec s o add ess e -
i ica ion by code compa ison on speci ic opics ele an o
he sa e y assessmen o eposi o y, see DECOVALEX
[1–3], BENVASIM [13,16], Reac i e T anspo Model
Benchma king [10,12]. In he scope o he p ojec HITEC
as pa o he Eu opean join p og amme on adioac i e
was e (EURAD), se en modelling eams om ac oss
Eu ope we e in i ed o pa icipa e in a benchma k ini ia i e
o assess he expe ise and capabili ies o hose eams and
hei nume ical ools o p edic he THM e olu ion in di -
e en clay hos ocks in he nea ield and he a ield o
he disposal zone. They p esen ealis ic nume ical
benchma ks o ypical sa e y assessmen p oblems aiming
a demons a ing ha he di e en codes commonly
employed in Eu ope yield consis en esul s. Consequen ly,
based on he indings, hese codes can be deemed e i ied
wi hin he amewo k o ‘‘Valida ion & Ve i ica ion’’
(V&V). The esul s o his benchma king ini ia i e a e he
subjec o he p esen pape .
2 P esen a ion o he wo k package HITEC
o EURAD
HITEC is he ac onym o ‘‘In luence o Tempe a u e on
Clay-based Ma e ial Beha iou ’’ ha is he se en h wo k
package (WP) o he EURAD P ojec . The EURAD p ojec
is a Eu opean join p og amme on adioac i e was e
managemen ha helps o de elop a obus and sus ained
science, echnology and knowledge managemen p o-
g amme ha suppo s he imely implemen a ion o
adioac i e was e managemen (RWM) ac i i ies o he EU
membe s a es and se es o os e mu ual unde s anding
and us be ween Join P og amme pa icipan s [8]. The
wo k package HITEC o EURAD aims o de elop and
documen an imp o ed he mo-hyd o-mechanical (THM)
unde s anding o clay-based ma e ials (hos ocks and
bu e s) exposed a high empe a u es ([100 C) o ha ing
expe ienced high- empe a u e ansien s o ex ended
du a ions. The WP’s aison d’e
ˆ e is o e alua e whe he o
no ele a ed empe a u e limi s (o 100 150 C) a e ea-
sible o a a ie y o geological disposal concep s o hea -
gene a ing adioac i e was e. HITEC s udies clay hos ock
o ma ions exposed o empe a u es o up o 120 C, doc-
umen s and es ablishes he possible ex en o ele a ed
empe a u e damage in he nea o a ield and also indi-
ca es he likely consequences o any such damage. The WP
also looks a ben oni e bu e s and de e mines he em-
pe a u e in luence on bu e swelling p essu e, hyd aulic
conduc i i y, e osion o anspo p ope ies o see when
he bu e sa e y unc ions s a o be unaccep ably
impai ed [8].
Fo he disposal o hea -gene a ing adioac i e was e, i
is impo an o unde s and he consequences o he hea
p oduced on he p ope ies o he na u al and enginee ed
clay ba ie s and on hei long- e m pe o mance. Mos
sa e y cases o disposal concep s ha in ol e clay cu -
en ly conside a empe a u e limi o 90 100 C. Being
able o ole a e highe empe a u es, whils s ill ensu ing an
app op ia e pe o mance, would ha e signi ican ad an-
ages (e.g. sho e cooling imes a su ace in e media e
s o age acili ies, mo e e icien packaging, ewe disposal
con aine s, ewe anspo ope a ions, smalle acili y
oo p in s, e c.). This WP in e oga es he alidi y o he
cu en ly applied he mal limi s and also he impo ance o
he accu acy o he assumed adiological was e p ope ies
and consequen ly ma ks a i s s ep owa ds op imiza ion o
he a chi ec u e o deep geological disposal acili ies [8].
Labo a o y expe imen s ca ied ou in he scope o WP
HITEC a e s udying he e ec s o inc eased empe a u e in
he nea ield, such as ac u ing and sel -sealing in he
exca a ed-damaged zone (sub ask 2.1), while o he dedi-
ca ed expe imen s a e looking a he a - ield e ec s, whe e
he mal loading and he gene a ion o o e p essu es may
esul in he loss o in eg i y o he geological ba ie
(sub ask 2.2). Sub ask 2.3 is ocusing on he modelling o
hese e ec s. Two se s o 2D models we e c ea ed o s udy
he impac o empe a u e on he beha iou o he h ee
clayey hos o ma ions (Boom Clay, Callo o–Ox o dian
(COx) clays one and Opalinus Clay (OPA)) in he nea
Ac a Geo echnica
123
ield and in he a ield. Some o he labo a o y expe i-
men s we e also simula ed o imp o e he unde s anding o
hei THM beha iou . Th ee in si u hea ing es s (PRA-
CLAY in he Boom Clay, FE in he Opalinus Clay and
ALC1605 in he COx) we e inally modelled, aking in o
accoun he esul s om he i s wo s eps. The p esen
wo k desc ibes he esul s o 2D benchma k s udies ha
ha e been ealized o es he nume ical capabili ies
a ailable ac oss Eu ope o model he THM beha iou o
he h ee clay hos ocks [8].
3 Benchma k desc ip ion
The benchma king exe cise selec ed o his wo k consis s
o modelling he nea - ield e ec s o exca a ion and
hea ing a ound he disposal galle y in a eposi o y o
adioac i e was e in a clay o ma ion. The disposal concep
chosen he e is inspi ed by he Swiss and F ench eposi o y
concep s. In he Swiss design, he galle y will be back illed
wi h ben oni e, whe eas he F ench one lea es some oom
o acili a e a possible e ie al o he was e packages. In
addi ion, he highly exo he mic packages a e sepa a ed by
spacing bu e s o educe he he mal load. In o de o
ocus on he hos ocks and o be able o compa e he
models, he bounda y condi ion a ound he galle y is se a
he in e ace be ween he galle y and he hos ock [8].
Th ee subcases a e p oposed [8]:
•Iso opic s ess condi ions wi h iso opic he mo-
elas ici y
•Aniso opic s ess condi ions wi h c oss-aniso opic (i.e.
ans e sely iso opic) he mo-elas ici y
•Aniso opic s ess condi ions wi h elas oplas ic/damage
models ( he choice o he model is le o he modelling
eams)
The p esen wo k will ocus on he i s wo subcases
whe eas he hi d case will be he subjec o u he pub-
lica ions. The goal wi h he i s wo subcases is i s o
compa e he nume ical codes on a ixed exe cise by igh ly
imposing he bounda y condi ions and he mechanical
cons i u i e laws and p ope ies.
The i s subcase e e s o a simple elas ic iso opic case,
which conside s iso opic hyd aulic, mechanical and he -
mal hos ock p ope ies as well as an iso opic a - ield
s ess ield. The ully coupled he mal-hyd o-mechanical
(THM) elas ic beha iou o Callo o–Ox o dian clays one,
Boom Clay and Opalinus Clay is analysed. The iso opic
hos ock p ope ies a e collec ed in Table 1. The second
subcase akes in o accoun he aniso opy o he hos ocks’
p ope ies and he in si u s ess ield. The p incipal di ec-
ions a e o ien ed pa allel and pe pendicula o he bedding
plane ha is assumed, o all h ee hos ocks, coinciden
wi h he ho izon al and e ical di ec ions, espec i ely.
Along he e ical p incipal di ec ion, all hos ocks exhibi
lowe in insic pe meabili y, elas ic modulus and he mal
p ope ies. The aniso opic hos ock p ope ies a e col-
lec ed in Table 2. Solid and wa e phase p ope ies a e he
same as in he iso opic elas ic case.
As al eady men ioned, he model consis s o a c oss
sec ion o a hea ing galle y and hos ock pe pendicula o
he galle y axis. Due o symme y, only a qua e o he
galle y c oss sec ion is modelled. The simula ed egion
measu es 100 m in bo h xand ydi ec ions and a plane
s ain condi ion (z x;y;zg¼0) is adop ed. A ep esen a ion
o he domain is shown in Fig. 1 oge he wi h he speci ied
Table 1 Iso opic he mo-hyd o-mechanical pa ame e s o he Cal-
lo o–Ox o dian clays one, Opalinus and Boom Clay [8]
Pa ame e s Uni s COx OPA Boom Clay
Solid phase
densi y qs
kg m32690 2340 2639
Sa u a ed bulk
densi y q
kg m32386 2030 2000
Po osi y n– 0.18 0.13 0.39
In insic pe m. Km22:30 1020 3:01020 2:83 1019
Young’s modulus
E
MPa 7000 6000 300
Poisson’s a io m– 0.3 0.3 0.125
Bio coe icien b– 0.8 0.6 1
The mal
conduc i i y k
Wm1
K1
1.67 1.85 1.47
Linea he mal
expansion
coe icien as
K11:25 1051:71051:0105
Solid speci ic
hea capaci y cp
Jkg1
K1
790 995 769
Table 2 Aniso opic he mo-hyd o-mechanical pa ame e s o he
Callo o–Ox o dian clays one, Opalinus and Boom Clay [8]
Pa ame e s Uni s COx OPA Boom
Clay
Kk o bedding m23:91020 51020 41019
K? o bedding m21:31020 11020 21019
Ek o bedding MPa 8000 8000 400
E? o bedding MPa 5000 4000 200
mk o
bedding ðmkkÞ
– 0.21 0.35 0.125
m? o bedding
ðm?kÞ
– 0.35 0.25 0.25
G? o bedding MPa 2500 2300 80
kk o bedding Wm1K11.88 2.4 1.65
k? o bedding Wm1K11.25 1.3 1.31
Ac a Geo echnica
123
obse a ion poin s. The co esponding coo dina es o hese
poin s a e summa ized in Table 5.
The ini ial and bounda y condi ions a e di ided in o
h ee phases o ep oduce he e olu ion o he THM
beha iou in he nea ield o he galle y om he exca-
a ion o he galle y up o he hea ing o he ock (compa e
Fig. 2).
Exca a ion phase: The o al s ess a he galle y wall
AE is linea ly dec eased in 24 h om he ini ial s ess o
5% o COx and OPA and o 50% o Boom Clay, while
he o al in si u ho izon al s ess emains cons an along CB
and he o al in si u e ical s ess is p esc ibed on CD.
Di ichle bounda y condi ions a e assigned o he symme-
y bounda ies AB and DE. The po e wa e p essu e a he
galle y wall is linea ly educed in 24 h o 0.1 MPa, while
on bounda ies BC and CD a cons an po e p essu e o
4.7 MPa o COx and OPA and 2.25 MPa o Boom Clay
is p esc ibed. Impe ious bounda ies a e assigned on he
symme y bounda ies AB and DE.
Wai ing phase: The bounda y condi ions emain
unchanged wi h espec o he end o he exca a ion phase
and a e held o six mon hs, allowing wa e o d ain
owa ds he unnel. No he mal lux is applied a his s age.
Hea ing phase: The mechanical bounda y condi ions
a e simila o he p e ious phases. Impe ious hyd aulic
bounda y condi ions a e now assigned o he galle y wall.
In his phase, a hea lux o 200 Wm1is applied o he
galle y wall o e a pe iod o en yea s.
The ini ial and bounda y condi ions a e summa ized in
Tables 3and 4. The bounda y condi ions a ound he
galle y a e se a he in e ace be ween he galle y and he
hos ock.
The wa e p ope ies o be conside ed in his bench-
ma king exe cise a e a densi y o 1000 kg m3, a speci ic
hea capaci y o 4180 Jkg1K1and a comp essibili y a
40 Co 4:5104MPa1. The e olu ion o he olu-
me ic he mal expansion coe icien o wa e is gi en as a
polynomial i o empe a u e unde a mosphe ic p essu e
ollowing he Eq. [11]:
aw104½C1¼4106½C4T30:001½C3T2þ
0:1404½C2T0:3795½C1
wi h Tbeing he empe a u e in C.
The e olu ion o he wa e iscosi y as a unc ion o he
empe a u e unde a mosphe ic condi ions is app oxima ed
using Vogel’s o mula:
l¼exp AþB
CþT

wi h Tbeing he empe a u e in K and lin mPas1. The
i ing pa ame e s ha e he ollowing alues:
A¼3:719½,B¼578:919 K and C¼137:546 K [9]
Se en modelling eams om ac oss Eu ope we e
eques ed o build a nea - ield 2D gene ic model o simu-
la e he h ee subcases p esen ed abo e. The eams we e
o ganized in such a way ha each hos ock was s udied by
se e al eams and hus allowing a compa ison o esul s
and inc easing con idence in he modelling wo k. The
eams we e ANDRA, BGE, EDF, EIG EURIDICE, LEI,
Uni e si y o Lie
`ge and UPC Ba celona. Table 6gi es an
o e iew o he di e en eams and hei co esponding
codes. In o al six nume ical codes ha e been used by he
di e en eams o he modelling o he di e en
benchma ks.
4 Implemen a ion o he mo-hyd o-
mechanical (THM) models in a ious
nume ical codes
In he implemen a ion o he mo-hyd o-mechanical (THM)
go e ning equa ions o po ous media, he a ious
nume ical codes in ol ed in his benchma k adop di e en
me hodologies based on hei co e nume ical s a egies and
in ended applica ions. Below, we discuss he key ea u es
o he THM equa ions implemen ed in he codes FLAC3D,
OpenGeoSys, COMSOL, LAGAMINE, Code_As e , and
Code_B igh . We use capi al le e s o he di e gence
(Di ) and g adien (G ad) ope a o s o clea ly dis inguish
Fig. 1 Obse a ion poin s in he analysis domain, see Table 5 o
coo dina es [8]
Ac a Geo echnica
123
be ween he Lag angian o mula ion ( ollowing a mo ing
ma e ial poin ) and he Eule ian o mula ion ( ocused on a
ixed spa ial poin ). This con en ion helps o a oid con-
usion and emphasizes he speci ic ame o e e ence
being used in he di e en codes.
4.1 Ene gy balance
In CODE_ASTER he ene gy balance de i ed by [7]as
p esen ed in he p e ious sec ion is exp essed in he
Lag angian amewo k by:
X
p;c
hmp
cmp
c
|fflfflfflfflfflffl{zfflfflfflfflfflffl}
Ene gy ans e
due o mass
þ
_
Q0
|{z}
Ex e nal hea
sou ce
þX
p;c
Di hmp
cMp
c

|fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}
Di e gence o
en halpy lux
þDi q
|fflffl{zfflffl}
Di e gence o
hea lux
X
p;c
Mp
cFm
|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}
Wo k done by
mass o ces
¼H
|{z}
Ene gy
dissipa ion e m
whe e hmp
c ep esen s he speci ic en halpy o componen c
in phase p;mp
cdeno es he mass o componen cin phase p;
_
Q0is he a e o hea addi ion om an ex e nal sou ce;
Di hmp
cMp
c

indica es he di e gence o he en halpy lux
o componen cin phase p; Di qis he di e gence o he
Fig. 2 Bounda y condi ions a di e en s ages o he analyses [8]
Table 3 Summa y o he bounda y condi ions applied on he galle y wall (compa e Fig. 2)[8]
Phases Mechanical condi ions Hyd aulic
condi ions
The mal
condi ions
T0T0þ24 hðÞ: exca a ion elease up o 5 % o he ini ial in si u s ess o COx and
OPA, and up o 50 % o Boom Clay
P0!Pa m
(0.1 MPa) in
24 h
No low a he
bo ehole wall
T0þ24 hðÞT0þ6 mon hsðÞ:
wai ing
a 5 % o he ini ial in si u s ess P0¼Pa m As abo e
T0þ6 mon hsðÞT0þ10 yea sðÞ:
hea ing
As abo e No low Cons . powe
(200 Wm1)
Table 4 Assumed ini ial condi ions in he h ee hos ocks [8]
Pa ame e s Componen s COX OPA Boom Clay
Iso opic case
To al s ess MPa 012.5 12.5 4.5
Po e p essu e MPa P04.7 4.7 2.25
Tempe a u e CT022 22 16.5
Aniso opic case
To al s ess MPa xx 12.4 2.2 3.825
yy 12.7 4.0 4.5
zz 16.4 6.5 3.825
Po e p essu e MPa P04.7 2.1 2.25
Tempe a u e CT022 22 16.5
Ac a Geo echnica
123

hea lux ec o q;Mp
c ep esen s he hyd aulic lux o
componen cin phase p, co esponding o win he Eule ian
con igu a ion; Fmdeno es he o ce pe uni mass ac ing on
he componen s due o ex e nal ields; and H ep esen s he
ene gy dissipa ion e m, accoun ing o hea gene a ion and
in e nal dissipa ion.
In LAGAMINE, he ene gy balance equa ion is o mu-
la ed in a weak o m o an a bi a y i ual empe a u e
ield Twi hin he cu en de o med con igu a ion X ,
whose bounda y is deno ed by C
.:
Z
X
_
S
T
|{z}
En halpy e olu ion
T
T;i
|{z}
Hea low
oT
ox
idX
¼Z
X
Q
T
|{z}
Hea sink e m
TdX Z
C
qT

q
T
|{z}
hea lux
TdC
In Code_B igh , he balance o in e nal ene gy o he
medium is de ined by:
o
o ESqSð1/Þ
|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}
Ene gy o he
solid phase
þElqlSl/
|fflfflfflffl{zfflfflfflffl}
Ene gy o he
liquid phase
þEgqgSg/
|fflfflfflfflffl{zfflfflfflfflffl}
Ene gy o he
gas phase
2
6
6
6
6
6
4
3
7
7
7
7
7
5
þdi ic
iþjEs
jþjEl
jþjEg
j

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
Ene gy luxes :
conduc ion and ad ec ion
¼ Q
|{z}
In e nal o ex e nal
ene gy supply
whe e ic
ideno es he conduc i e ene gy lux wi hin he
po ous medium; Q ep esen s sou ces o ene gy, bo h
in e nal and ex e nal; jE
ico esponds o he ad ec i e
ene gy lux esul ing om mass mo emen ; and Esigni ies
he speci ic in e nal ene gy.
In COMSOL, we ha e:
o
o qcpT
|ffl{zffl}
Ene gy s o age e m
ðhea capaci yÞ
2
6
6
6
6
4
3
7
7
7
7
5þdi kg ad TðÞ
|fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}
Di e gence o hea lux
ðconduc ionÞ
þdi qcpuT

|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}
Di e gence o con ec i e
hea ans e
¼Q
|{z}
Hea sou ce
o sink
in FLAC3D he he mal ene gy balance is exp essed as:
Table 5 Coo dina es o obse a ion poin s in me e
Poin s Coo dina es Poin s Coo dina es
P1 (1.25, 0.0) P8 (0.0, 1.9)
P2 (1.9, 0.0) P9 (0.0, 2.5)
P3 (2.5, 0.0) P10 (0.0, 6.25)
P4 (6.25, 0.0) P11 (0.0, 50.0)
P5 (50.0, 0.0) P12 (0.0, 100.0)
P6 (100.0, 0.0) P13 (0.88, 0.88)
P7 (0.0, 1.25) P14 (1.33, 1.33)
P15 (1.77, 1.77)
Table 6 In ol ed modelling eams and hei espec i e codes
O ganiza ion Coun y Nume ical code Boom
Clay
COx OPA
ANDRA F ance COMSOL,
Code_As e
U
BGE Ge many OpenGeoSys,
FLAC3D
UUU
EDF F ance Code_As e UU
EIG
EURIDICE
Belgium COMSOL U
LEI Li huania COMSOL U
ULiege Belgium Lagamine UU
UPC Spain CODE_BRIGHT UU
Table 7 Co espondence o ene gy balance e ms ac oss so wa e
Concep LAGAMINE Code_As e Code_B igh COMSOL FLAC3D OpenGeoSys
Ene gy s o age _
S
TTDi hmp
cMp
c

HESqSð1/ÞþElqlSl/þEgqgSg/qcpTnTqcpT
Hea lux (conduc i e)
T;iqic
ikg ad TqT
ikg ad T
Di e gence o hea lux 
q
TDi qdi ðic
iþjEs
jþjEl
jþjEg
jÞdi ðkg adTÞDi ðqT
iÞdi ðkg adTÞ
Hea lux (ad ec i e) – Pp;chmp
cmp
cjEs
jþjEl
jþjEg
jDi ðqcpuTÞ– di ðqcpuTÞ
Hea sou ce o sink Q
T
_
Q0 QQqT
Q
Ac a Geo echnica
123
Di qT
i
|fflfflfflfflffl{zfflfflfflfflffl}
Hea lux
di e gence
þqT
|{z}
Volume ic hea
sou ce in ensi y
¼onT
o
|{z}
Ra e o hea s o age
pe uni olume
Two p ocess models in OpenGeoSys we e employed in his
benchma king: a non-iso he mal Richa ds low model
coupled wi h mechanics, and a he mo-hyd aulic (TH)
model using he mo-mechanical s o age coe icien s [5].
The i s model ollows a monoli hic app oach, de i ing
hyd aulic-mechanical couplings om h ee-dimensional
e ec i e s ess and omi ing apou di usion based on
benchma k assump ions. The second TH model inco po-
a es mechanical e ec s wi hin he mass balance o he TH
p ocess ia he mo-mechanical s o age coe icien s,
assuming uniaxial s ain (xx ¼yy ¼0) and cons an e -
ical s ess ( zz), while also conside ing he mal s esses
om cons ained ans e se he mal expansion. The ene gy
balance o he non-iso he mal Richa ds low model is
gi en below:
o
o qcpT

|fflfflfflfflffl{zfflfflfflfflffl}
Ene gy s o age
ðhea capaci yÞ
þdi kg adTðÞ
|fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}
Di e gence o
hea luxðconduc ionÞ
þdi qcpuT

|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}
Con ec i e
hea ans e
¼Q
|{z}
Hea sou ce
o sink
Table 7p o ides a concise compa ison o ene gy balance
componen s ac oss six THM simula ion so wa e: LAGA-
MINE, Code_As e , Code_B igh , COMSOL, FLAC3D,
and OpenGeoSys, each employing unique o mula ions.
Ene gy s o age is ep esen ed di e en ly, wi h LAGA-
MINE and Code_B igh conside ing phase-speci ic ene -
gies, while COMSOL, OpenGeoSys, and FLAC3D use
simple o ms such as qcpT. Code_As e employs a
di e gence e m ela ed o en halpy lux. Hea low (con-
duc ion) is consis en ly exp essed ia Fou ie ’s law ac oss
so wa e, commonly as kg ad T, wi h speci ic o ms like
T;iin LAGAMINE and ic
iin Code_B igh . Hea lux
di e gence desc ibes he dis ibu ion o conduc i e hea ,
o en using di e gence ope a o s; Code_B igh uniquely
in eg a es conduc ion and ad ec ion o accoun o mul i-
phase e ec s. Con ec i e hea ans e , no ed in Code_A-
s e , Code_B igh , COMSOL, and OpenGeoSys, includes
hea anspo ed by luid low, while FLAC3D omi s his
e m, ocusing on solid mechanics. Each so wa e includes
a hea sou ce o sink e m, wi h labels as Q
T(LAGA-
MINE),
_
Q0(Code_As e ), Q(Code_B igh ), Q(COMSOL,
OpenGeoSys), and qT
(FLAC3D), allowing o a ange o
ene gy in e ac ions. This o e iew emphasizes each so -
wa e’s adap abili y, om simpli ied single-phase models o
mul i-phase app oaches o comp ehensi e THM
simula ions.
4.2 Fluid mass balance
In LAGAMINE, The luid mass balance equa ion is
exp essed in a weak o m o any admissible i ual po e
p essu e ield p
w:
Z
X
_
M
wp
w
|fflffl{zfflffl}
Time de i a i e
o luid mass

w;i
op
w
ox
i
|fflfflffl{zfflfflffl}
Mass low
g adien
0
B
B
B
B
B
B
B
@
1
C
C
C
C
C
C
C
A
dX
¼Z
X
Q
wp
w
|fflffl{zfflffl}
Fluid sink e m
dX Z
C
qw

q
wp
w
|ffl{zffl}
Bounda y
mass lux
dC
He e,
w;i ep esen s he mass low a e, Q
wdeno es a e m
accoun ing o luid sinks, and C
qwis he bounda y segmen
whe e he incoming luid mass pe uni a ea, gi en by 
q
w,is
speci ied. Addi ionally,
_
M
wdeno es he ime a e o change
o luid mass.
The luid mass balance in FLAC3D is de ined by:
Di qi
|fflfflfflffl{zfflfflfflffl}
Di e gence o
luid lux
þq
|{z}
Volume ic luid
sou ce in ensi y
¼o
o
|{z}
Time de i a i e
o luid con en
In COMSOL, he wa e mass (mw) balance equa ion
includes addi ional wa e sou ce e ms o ake in o accoun
he di e en coupling p ocesses (HM, TH and TM) and is
exp essed as ollows:
omw
o
|{z}
Time de i a i e o wa e mass
þdi qwqw
ðÞ
|fflfflfflfflfflffl{zfflfflfflfflfflffl}
Di e gence o mass lux
due o wa e low
¼QH
|{z}
Wa e sou ce
In Code_B igh , The o al mass balance o wa e in he
liquid phase is gi en by:
Ac a Geo echnica
123
o
o xw
lqlSl/þxw
gqgSg/
hi
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
Ra e o change o wa e mass
in liquid and gas phases
þjl;w
i;iþjg;w
i;i

|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}
Di e gence o wa e mass
lux in liquid and gas phases
¼ w
|{z}
Ex e nal wa e supply
He e, xw
land xw
g ep esen he mass ac ion o he com-
ponen (wa e ) ela i e o he o al mass o he liquid and
gas phases, espec i ely. Sland Sga e he sa u a ion
deg ees o he liquid and gas phases, indica ing he ac ion
o po e olume each phase occupies. qland qgdeno e he
densi ies o he liquid and gas phases, espec i ely. jl;w
i
ep esen s he mass lux o wa e wi hin he liquid phase,
while jg;w
i ep esen s he mass lux o wa e wi hin he gas
phase. Las ly, windica es an ex e nal wa e sou ce.
The liquid wa e mass balance equa ion o hyd aulics in
Code_As e is de ined by:
omw
o
|{z}
Ra e o change o
liquid wa e mass con en
þoMw;i
oxi
|fflffl{zfflffl}
Di e gence o
liquid wa e mass low
¼q
|{z}
Volume ic
sou ce e m
In OpenGeoSys, he luid mass balance equa ion is gi en
by:
omw
o
|{z}
Ra e o change o
wa e mass con en
þdi jw
|fflffl{zfflffl}
Di e gence o
wa e mass lux
¼q
|{z}
Volume ic sou ce
o sink e m
whe e:mw: ep esen s he mass o wa e pe uni olume in
he po ous medium. jw: wa e mass lux ec o , de ined by
Da cy’s law. q : olume ic sou ce o sink e m, ep e-
sen ing wa e sou ces o sinks pe uni olume.
Table 8summa izes he co espondence o luid mass
balance e ms ac oss di e en THM codes. In LAGA-
MINE, he weak o m o he luid mass balance includes
explici e ms o he ime de i a i e o luid mass, mass
low g adien , luid sink e m, and bounda y mass lux. In
Code_As e and OpenGeoSys, he o mula ions emphasize
he a e o change o wa e mass con en and di e gence o
wa e mass low, aligning wi h e ms in FLAC3D and
COMSOL whe e luid s o age and anspo mechanisms
a e cen al. Code_B igh in oduces speci ic e ms o he
liquid and gas phase luxes and he ex e nal wa e supply,
necessa y o mul iphase beha iou . These dis inc ions
e lec di e ences in handling bounda y condi ions and
explici o implici sou ce and sink e ms ac oss he codes.
4.3 Mechanical balance equa ion o momen um
conse a ion
The mechanical balance equa ion, cha ac e izing he con-
se a ion o momen um in po ous media, conside s bo h
he e ec i e s esses wi hin he solid ma ix and he body
o ces ac ing upon i . This o mula ion is gene ally alid
ac oss mul iple THM modelling codes and is exp essed as
ollows:
ij;jþbi¼0
whe e ij is he componen o s ess, bi ep esen s body
o ce.
Table 8 Co espondence o luid mass balance e ms ac oss THM codes
Concep LAGAMINE Code_As e Code_B igh COMSOL FLAC3D OpenGeoSys
Time de i a i e o mass _
M
womw=o o
o xw
lqlSl/þxw
gqgSg/
hi
omw=o o =o omw=o
Mass low / lux
w;i=ox
ioMw;i=oxijl;w
i;iþjg;w
i;idi ðqwuÞDi qidi jw
Volume ic sou ce / sink e m Q
wq wQHq q
Bounda y mass lux 
q
w–– –––
Ac a Geo echnica
123
4.4 Challenges in compa ing implemen a ions
Compa ing he implemen a ions o THM p ocesses in
a ious nume ical codes is challenging due o di e ences
in amewo ks (Eule ian s. Lag angian), nume ical
schemes, and handling o mul i-physics couplings. Each
code has unique me hodologies in o mula ing balance
equa ions, applying weak o s ong o ms, and managing
coupling e ec s, especially in mul iphase low, he mal
conduc ion, and mechanical de o ma ion. This a iabili y
s ems om he dis inc i e use cases each so wa e a ge s,
whe he in geo echnical applica ions, en i onmen al engi-
nee ing, o mul i-phase luid anspo in po ous media.
The e o e, benchma king ini ia i es a e essen ial o e i-
ying and compa ing he pe o mance and accu acy o he
THM models ac oss di e en codes.
5 Cons i u i e equa ions o he skele on
In elas ici y heo y, he Hooke’s s ess-s ain ela ion is
o mula ed as ollows: eij ¼Cijkl kl
whe e eand a e espec i ely he s ain and s ess
enso . Cijkl a e he coo dina es o he ou h-o de com-
pliance enso . (C) is he co esponding compliance ma ix
in Kel in no a ion which we will speci y below o he
ma e ials used in his benchma k..
In he ollowing, we a e p esen ing he compliance
ma ices o he ma e ials used in his benchma k.
The Iso opic Elas ic Compliance Ma ix
Fo linea iso opic elas ic ma e ials, he compliance
ma ix CC depends on wo independen ma e ial p ope -
ies, such as he Young’s modulus Eand he Poisson’s a io
m. In he mos gene al iso opic o m, he compliance
ma ix is exp essed as:
C¼
1
Em
Em
E000
m
E
1
Em
E000
m
Em
E
1
E000
000
1
2G00
0000
1
2G0
00000
1
2G
2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
In his exp ession, he shea modulus G is gi en by:
G¼E
2ð1þmÞ
T ans e se Iso opic Elas ic Compliance Ma ix
Fo c oss-aniso opic o ans e se iso opic elas ici y,
i e independen ma e ial pa ame e s a e equi ed o
desc ibe he elas ic beha io :
Ejj;E?;mjjjj;mjj?;and Gjj? He e, he subsc ip s
jjand ?
e e o di ec ions pa allel and pe pendicula o he iso-
opic planes, espec i ely. In his case, he elas ic com-
pliance ma ix is de ined as [17]:
C¼
1
Ekm?k
E?mkk
Ek
000
mk?
Ek
1
E?mk?
Ek
000
mkk
Ekm?k
E?
1
Ek
000
0001
2Gk?
00
0000
1
2Gkk
0
00000
1
2G?k
2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
The symme y o he compliance enso imposes he ol-
lowing ela ionship be ween he Poisson’s a ios and
Young’s moduli:
mk?
Ek¼m?k
E?The shea modulus in he iso opic planes
ðGjjjjÞcan be exp essed as:
The symme y o he s ess and s ain enso s implies
ha he o he shea moduli a e equal:
Gk? ¼Gkk
6 Analy ical benchma k
The i s s ep owa ds model e i ica ion is o compa e he
accu acy o he THM models agains well-de ined analy -
ical solu ions. This sec ion p esen s a benchma k analysis
o he analy ical solu ion o he coupled he mo-hyd o-
mechanical (THM) consolida ion p oblem, o iginally
de i ed by [4] and la e co ec ed o he e ec i e s ess
e m by [6]. The aim o his benchma k is o alida e he
heo e ical o mula ion o he in ol ed THM models
h ough coo dina ed e o s by se e al eams. The bench-
ma k analysis will ocus on h ee p ima y a iables: em-
pe a u e, po e p essu e, and displacemen o he solid
skele on.
P oblem o e iew
The p oblem desc ibes a hea sou ce embedded in a
ully luid-sa u a ed po ous medium. Due o sphe ical
Ac a Geo echnica
123
expansi i y o OPA clay, along wi h a lowe Young’s
modulus, which allows o po e p essu e dissipa ion and
educes he he mal expansi i y con as be ween wa e and
he clay ma ix. This o e p essu e causes s ess elie ,
leading o ensile s ess a he galle y wall wi h a maximum
alue o 4.5 MPa a he end o he simula ion. A P2,
loca ed 0.65 m behind he wall in he ho izon al di ec ion,
a comp essi e s ess o mo e han 4 MPa can be obse ed
a he end o he simula ion.
7.4 Boom Clay: elas ic aniso opic case
The esul s o he benchma k simula ion o Boom Clay,
aking in o accoun aniso opic ma e ial p ope ies, a e
p esen ed in Fig. 9. EUR, ULG and BGE wi h wo p ocess
modell pa icipa ed in his benchma k. In excep ion o
BGE esul s compu ed wi h he hyd o he mal model, he
esul s coming om THM p ocess models a e consis en
o all a iables. Po e p essu e e olu ion coming om he
hyd o he mal model in OpenGeoSys wi h inco po a ion o
he mo-mechanical e ec s o e es ima e he po e
o e p essu e in he hea ing phase when compa ed o he
esul s ob ained wi h THM o mula ions. Fu he in es i-
ga ions a e necessa y o adjus his model. O e all, he
e olu ion o po e p essu e is simila o he iso opic case,
wi h all po e p essu e e olu ion cu es con e ging owa ds
a alue o 2.8 MPa a he end o he simula ion. Rega ding
he mechanical a iables, good consis ency has been
ob ained o he displacemen and s ess esul s. Mul iple
i e a ions we e equi ed o achie e a good consis ency in
his benchma k case. Disc epancies a ose in he ea ly
s ages om ambigui ies in he benchma k speci ica ions
ega ding he aniso opic Poisson a ios. I ’s impo an o
no e ha nuij 6¼ nuji o ans e sely iso opic ma e ials like
a gillaceous ma e ials. To p ope ly calcula e aniso opic
componen s o he Poisson a ios, he o ien a ion o bo h
he nume ical model and iso opic plane mus be aken in o
accoun . E ec i e communica ion be ween eams was key
o esol ing his issue.
As a esul o he mal p essu iza ion, ensile s esses
occu a he galle y wall in he ho izon al di ec ion. The
e ec i e ensile s esses each a alue o nea ly 1 MPa a
he end o he simula ion, which may cause ensile damage
Fig. 9 Benchma k esul s o he elas ic aniso opic case in Boom Clay [8]
Ac a Geo echnica
123

in he ock i he ensile s eng h o he ock is exceeded.
Howe e , his issue is limi ed o he bounda y because
comp essi e s esses a e obse ed a P2.
7.5 Callo o-Ox o dian clays one: elas ic
aniso opic case
The esul s o he subcase 2 o COx clay, assuming
aniso opic ma e ial p ope ies, a e shown in Fig. 10.
Se en eams pa icipa ed in his benchma k (AND, BGE,
EDF, EUR, LEI, ULG and UPC), and o e all, good
ag eemen has been achie ed by all eams. The empe a-
u e, displacemen and de ia o ic s ess e olu ion cu es
a e coinciden a all poin s.
Du ing he hea ing phase, i e eams p edic ed he same
po e p essu e e olu ion end. Resul s o BGE and EDF
de ia e om his end. The eason o his disc epancy has
no ye been ound, bu di e ences in he aniso opic
Poisson a ios used by hese wo eams a e suspec ed. The
e olu ion o he e ec i e e ical and ho izon al s esses
shows small disc epancies a each obse a ion poin s ha
can be ela ed o he di e ence in mesh size and elemen
o mula ion. This will be u he elabo a ed in he discus-
sion sec ion.
The aniso opy o he COx clay appea s o ampli y he
he mally induced po e p essu e, as a maximum alue o
abou 11 MPa was obse ed in he aniso opic case, com-
pa ed o abou 10 MPa in he iso opic case. The e ec i e
ho izon al s esses each app oxima ely 7.5 MPa a he
galle y wall, dec easing o less han 1 MPa a P2, which
is loca ed 0.65 m behind he galle y wall. A P4, which is
loca ed 0.5 m behind he galle y wall, he e ec o he mal
p essu iza ion is lowes .
Fig. 10 Benchma k esul s o he elas ic aniso opic case in COx clays one [8]
Ac a Geo echnica
123
7.6 Opalinus clays one: elas ic aniso opic case
The esul s o he inal benchma k o Opalinus Clay,
aking in o accoun aniso opic ma e ial p ope ies, a e
p esen ed in Fig. 11. The same eams ha wo ked on he
iso opic case o Opalinus Clay also pa icipa ed in his
benchma k. The o e all esul s om all eams a e in good
ag eemen , wi h consis en p edic ions o empe a u e
e olu ion. UPC p edic ed he highes po e p essu e e olu-
ion in his benchma k. EDF and BGE ob ained almos
iden ical po e p essu e e olu ion esul s. In con as o
COx, he esul s o he aniso opic case in Opalinus Clay
show ha less o e p essu e is gene a ed compa ed o he
iso opic case. A maximum o e p essu e alue o abou
6 MPa was ob ained in his benchma k, while he iso opic
case had a maximum alue o 8.5 MPa. This may be
explained by he highe pe meabili y and highe Poisson
a io in he ho izon al di ec ion o he aniso opic Opalinus
Clay.
Fo Opalinus Clay unde aniso opic condi ions, he
displacemen esul s show ha he su ounding egion
mo es away om he galle y wall due o he mal
expansion in he hea ing phase. This can be obse ed a P2,
P3 and P4, wi h P4, loca ed 6.5 m deep in he ock, mo ing
3 mm away om he galle y wall a he end o he simu-
la ion. The e ec i e s ess e olu ion esul ing om he
po e o e p essu e shows ensile s esses a he bounda y o
he galle y wall a P1, which anish a P2. This e ec has
been obse ed in all benchma ks and has been discussed
al eady in p e ious sec ions. The compa a i e assessmen
o all s ess esul s is in good ag eemen o all eams
excep o obse a ion poin P1 whe e UPC p edic less
e ec i e e ical and de ia o ic s esses du ing he wai ing
phase.
8 Discussion
O e all, he esul s o he benchma k s udies conduc ed by
he a ious eams in ol ed showed a high le el o con-
sis ency in he he mo-hyd o-mechanical esponse o he
clay-based ma e ials being s udied. These esul s we e
mo e consis en in he iso opic cases, whea eas some
disc epancies we e obse ed in he aniso opic bench-
ma ks. In he aniso opic cases, he assump ions aken o
Fig. 11 Benchma k esul s o he elas ic aniso opic case in OPA clays one [8]
Ac a Geo echnica
123
compu ing he aniso opic componen s o he Poisson a ios
in espec o he coo dina e sys ems used in he di e en
codes ha e a huge e ec on he ou come o he benchma k.
The e a e esponsible o he disc epancies in he po e
p essu e e olu ion. This can be a oided i he Poisson’s
a ios a e consis en ly de ined in all codes. O he di e -
ences in he esul s ob ained by he di e en eams may be
due o a a ie y o ac o s, including di e ences in he
go e ning equa ions and di e en solu ion schemes and
codes used, and di e ences in he nume ical meshes
employed. A mo e ma hema ical in es iga ion is needed o
quan i y he e ec o hese ac o s. This is edious o pe -
o m as much assump ions in his ega d a e no published
and some nume ical codes used in his s udy a e no open
sou ce.
To be e unde s and he e ec o he mesh size and
elemen o mula ion on he ob ained esul s, a mesh sen-
si i i y s udy has been pe o med. Eigh di e en meshes
we e de eloped in his s udy ha can be di ided in o wo
g oups. The i s g oup consis s o linea meshes whe eas
he second g oup is made o quad a ic coun e pa s de i ed
om he linea meshes. In each g oup, h ee meshes we e
made o quad ila e al elemen s. The second mesh is
de i ed om he i s one wi h 3040 elemen s by di iding
each elemen by ou o ob ain 12160 elemen s. The hi d
mesh is gene a ed in he same manne by di iding each
elemen o he second mesh by ou o ob ain 48460 ele-
men s. The ou h mesh has been gene a ed using iangula
elemen s. All hese meshes we e employed o ca y ou he
iso opic benchma k o Boom Clay using OpenGeoSys by
BGE. The esul s o his s udy a e shown in Fig. 12. Fo all
a iables, a mesh dependency on he esul s is obse ed
especially a he obse a ion poin P2 whe e he g adien s
a e highes . The disc epancy in he po e p essu e esul s is
obse ed du ing he exca a ion (un il 1d) and wai ing
phase (un il 6 mon hs) whe eas in he hea ing phase esul s
Fig. 12 Compa ison o he THM esponse in OPA a he same dis ances om he galle y wall pa allel (P2, P3) and pe pendicula o bedding (P7,
P8) based on UPC esul s
Ac a Geo echnica
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om all meshes a e iden ical. In he exca a ion phase,
some oscilla ions a e obse ed o all quad ila e al meshes
wi h linea shape unc ions. This is also he case o he
quad a ic mesh wi h 3040 quad ila e al cells. I is impo -
an o no ice ha esul s wi h iangula meshes do no
show his a e ac . Thus, iangula elemen o mula ion
seems o be mo e obus o THM simula ion a leas in
OpenGeoSys. This needs o be es ed o o he codes. The
s ess e olu ion cu es shows also some sca e ing o
esul s a obse a ion poin P1 o all linea meshes. This is
p obably he esul o he ex apola ion p oblem due o he
high g adien s nea he galle y wall ha has been no iced
by ULG o he aniso opic cases o Boom Clay and COx
in combina ion wi h he low app oxima ion o de o he
de i ed quan i y s ess. ULG esul s o hese aniso opic
cases a P1 p esen ed in his wo k ha e been ex apola ed
om he in eg a ion poin o he galle y wall o a oid his
issue. The ac ha he meshes o he second g oup do no
show his p oblem u he suppo s his hypo hesis. The
addi ional in eg a ion poin s coming wi h he quad a ic
elemen s and he highe app oxima ion o de o he s esses
help o inc ease he quali y o he ex apola ion o he
esul s o he bounda y wall.
Based on his s udy, one can conclude ha he dis-
c epancy obse ed in he benchma k esul s may be pa ly
explained by he di e en meshes and elemen o mula-
ions employed by he eams. This is especially he case o
esul s sca e ing obse ed nea he galle y a P2 and o a
smalle ex en a P3 as one can be obse ed o ins ance in
he esul s o he aniso opic benchma k cases, see o
example Fig. 10. Figu e 12 shows also ha he esul s o
he po e p essu e e olu ion a e consis en o all he meshes
in he hea ing phase. Thus he disc epancy obse ed in he
po e p essu e esul s canno be explained by i s
dependency on he spa ial disc e iza ion. The eplica ion o
his s udy based on an aniso opic benchma k case may be
necessa y o con i m hese conclusions. Fu he analyses
may be needed o unde s and he o he unde lying mech-
anisms leading o he disc epancies. Howe e , i can be
concluded a his s age ha all o he eams in ol ed in his
s udy we e able o accu a ely model he THM e olu ion o
hea -gene a ing eposi o y sys ems in clay o ma ions and
ha he ools and echniques hey used can be conside ed
e i ied in his con ex based on hei abili y o p oduce
simila esul s.
The aniso opic e ec s obse ed in he benchma k
s udies we e mo e p onounced in he empe a u e e olu ion
and mechanical esponse o he clay-based ma e ials bu
we e no e iden in he po e p essu e e olu ion. Ins ead, he
po e p essu e appea ed o homogenize du ing he hea ing
phase, eaching simila asymp o ic alues a he a ious
obse a ion poin s. This sugges s ha he aniso opic
e ec s may ha e a g ea e in luence on he empe a u e and
mechanical beha iou o he ma e ials, bu no on he po e
p essu e, see compa e P2 s. P8 and P3 s. P9 in Fig. 13.
Fo he iso opic case, he esul s a he poin s si ua ed a
he same dis ance om he galle y wall a e iden ical as
shown in Fig. 13.
Acco ding o he benchma k s udies, an inc ease in
p essu e was obse ed du ing he hea ing phase in all cases.
This phenomenon, known as he mal p essu iza ion, occu s
due o he di e ence in he he mal expansion o wa e and
he clay ma ix. The wa e is cons ained by he less
expansi e clay ma ix, leading o an accumula ion o
p essu e. The esul ing ensile s esses ha may occu in
he clay o ma ion as a esul o he mal p essu iza ion
we e obse ed o be local and o anish quickly wi h dep h
in he ock in he s udied benchma ks. Howe e , i is
Fig. 13 Compa ison o he THM esponse in OPA a he same dis ances om he galle y wall pa allel (P2, P3) and pe pendicula o bedding (P7,
P8) based on UPC esul s
Ac a Geo echnica
123
impo an o no e ha hese esul s will no be he same in a
ypical eposi o y con igu a ion wi h back illed galle ies
and mul iple canis e s disposed o adjacen o one ano he .
Despi e his, he esul s sugges ha hese ensile s esses
may emain local e en in his benchma k case and may
disappea o e ime as he he mal powe o he adioac i e
was e decays and he po e p essu e dec eases. Depending
on he ensile s eng h o he clay, ensile s esses may
cause local damage o he clay.
The mo-hyd o-mechanical (THM) simula ions can be
ime-consuming and compu a ionally demanding, pa icu-
la ly when used o assess he sa e y o eposi o y sys ems
ha equi e he conside a ion o la ge geological o ma-
ions in he nume ical analysis. In hese cases, he use o
nume ically e icien me hods may be essen ial o handle
he compu a ional demands o such simula ions. The TH
model wi h he mo-mechanical s o age coe icien s p o-
posed by [5] and implemen ed in he OpenGeoSys has been
shown o be able o adequa ely ep oduce he po e o e -
p essu e o he iso opic benchma k in Boom Clay. The
model sligh ly o e p edic ed he he mal induced po e
p essu e esponse in he aniso opic case. Fu he wo k is
necessa y o imp o e he model p edic ions in such si ua-
ion. Ne e heless, his model has been ound o esul in a
signi ican speed-up, up o wo o de s o magni ude,
compa ed o adi ional THM simula ions. The ac ual
speed-up depends on he size o he model and he mos
signi ican bene i s a e ypically seen in models wi h a la ge
numbe o deg ees o eedom, whe e complexi y educ ion
is pa icula ly impo an . Howe e , o a de ailed e alua-
ion o he s ess and displacemen e olu ion, ully coupled
THM simula ions emain necessa y [5].
9 Conclusions
The EURAD wo k package HITEC seeks o enhance he
unde s anding o he mo-hyd o-mechanical p ocesses in
clay-based ma e ials subjec ed o ele a ed empe a u es.
To his end, a benchma k ini ia i e was conduc ed o assess
he cu en s a e o modelling THM phenomena in clay
ma e ials. The esul s o his ini ia i e, which in ol ed
eams om ac oss Eu ope, showed ha hese eams and
hei nume ical ools a e capable o accu a ely p edic ing
he THM beha iou o clay-based ma e ials such as Boom
Clay, Opalinus Clay, and Callo o-Ox o dian clay. In gen-
e al, he esul s o he iso opic benchma ks we e highly
consis en , while some disc epancies we e obse ed in he
aniso opic benchma ks, po en ially due o di e ences in
he go e ning equa ions and codes used, as well as mesh
size, elemen o mula ions and in e pola ion algo i hms. In
conclusion, he esul s o his benchma k demons a e he
expe ise and capabili ies o he pa icipa ing eams in
modelling THM phenomena o he sa e y assessmen o
eposi o y sys ems in clay o ma ions. The in ol ed codes
can be seen as e i ied wi hin he amewo k o ‘‘Valida-
ion & Ve i ica ion’’ (V&V). Fu he s udies ha ake in o
accoun e ec s o plas ici y and pe meabili y changes due
o damage a e necessa y o be e ep oduce benchma ks a
eposi o y condi ions. This will be he subjec o u he
publica ion o he in ol ed eams.
Acknowledgemen s The p ojec leading o his wo k has ecei ed
unding om he Eu opean Union’s Ho izon 2020 esea ch and
inno a ion p og amme unde g an ag eemen No. 847593. This pape
desc ibes objec i e echnical esul s and analysis. The au ho s alone
a e esponsible o he con en s o his s udy.
Au ho con ibu ion Fi s au ho w o e he main manusc ip . O he
au ho s con ibu ed by p o iding pa ela ed o hei nume ical
analyses.
Funding Open Access unding enabled and o ganized by P ojek
DEAL.
Da a a ailabili y No da ase s we e gene a ed o analysed du ing he
cu en s udy.
Decla a ions
Con lic o in e es The au ho s decla e no con lic o in e es .
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Au ho s and A filia ions
E ic Simo
8,1
•Ch is ophe de Lesquen
2
•Rocio Paola Leon-Va gas
1
•Minh-ngoc Vu
2
•Simon Raude
3
•
Ginge El Tabbal
3
•A naud Dizie
4
•Su esh See ha am
4
•As a Na kuniene
5
•F e
´de
´ ic Collin
6
•
Hangbiao Song
6
•An onio Gens
7
•Fei Song
7
•Alexand u-Bogdan Ta omi
10
•Thomas Nagel
8,9
•
Jo
¨ g Buchwald
8,9
&E ic Simo
[email p o ec ed]
1
BGE TECHNOLOGY GmbH, Eschens asse 55,
31224 Peine, Ge many
2
ANDRA, 1/7 ue Jean Monne , 92290 Cha
ˆ enay-Malab y,
F ance
3
EDF R&D, 7 B d Gaspa d Monge, 91120 Palaiseau, F ance
4
Belgian Nuclea Resea ch Cen e (SCK CEN), Boe en ang
200, B-2400 Mol, Belgium
5
Li huanian Ene gy Ins i u e, B eslaujos s ., 44403 Kaunas,
Li huania
6
U ban and En i onmen al Enginee ing Resea ch Uni ,
Uni e si e
´de Lie
`ge, Alle
´edelaDe
´cou e e 9, 4000 Lie
`ge,
Belgium
7
Depa men o Ci il and En i onmen al Enginee ing,
Uni e si a Poli ecnica de Ca alunya, Jo di Gi ona 1-3,
08034 Ba celona, Spain
8
Geo echnical Ins i u e, TU Be gakademie F eibe g, Gus a -
Zeune -S aße 1, 09599 F eibe g, Ge many
9
Helmhol z Cen e o En i onmen al Resea ch-UFZ,
Pe mose s aße 15, 04318 Leipzig, Ge many
10
BGE mbH, Eschens asse 55, 31224 Peine, Ge many
Ac a Geo echnica
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