La lecture à portée de main
Découvre YouScribe en t'inscrivant gratuitement
Je m'inscrisDécouvre YouScribe en t'inscrivant gratuitement
Je m'inscrisDescription
Sujets
Informations
Publié par | pefav |
Nombre de lectures | 18 |
Langue | English |
Poids de l'ouvrage | 1 Mo |
Extrait
MultiphaseflowinporousmediausingtheVAG
scheme
RobertEymard,CindyGuichard,Raphae`leHerbinandRolandMasson
Abstract
WepresenttheuseoftheVertexApproximateGradientschemeforthe
simulationofmultiphaseflowinporousmedia.Theporousvolumeisdistributed
tothenaturalgridblocksandtothevertices,henceleadingtoanewfinitevolume
mesh.Thentheunknownsinthecontrolvolumesmaybeeliminated,anda27-
pointschemeresultsontheverticesunknownsforahexahedralstructuredmesh.
Numericalresultsshowtheefficiencyoftheschemeinvarioussituations,including
misciblegasinjection.
W1Introduction
Simulationofmultiphaseflowinporousmediaisacomplextask,whichhasbeen
theobjectofseveralworksoveralongperiodoftime,seethereferencebooks[12]
and[3].Severaltypesofnumericalschemeshavebeenproposedinthepastdecades.
Thosewhichareimplementedinindustrialcodesaremainlybuiltuponcellcentred
approximationsanddiscretefluxes,inaframeworkwhichisalsothatofthemethod
weproposehere.Letusbrieflysketchthisframework.The3Dsimulationdomain
ismeshedbycontrolvolumes
X
2
M
.Letusdenoteby
L
thediffusionmatrix
(whichisapossiblyfullmatrixdependingonthepointofthedomain).
Foreachcontrolvolume
X
2
M
,thesetofneighbors
Y
2
N
X
isthesetof
allcontrolvolumesinvolvedinthemass
R
balancein
X
,whichmeansthatthefol-
lowingapproximationformulaisused:
X
Ñ
−
LÑ
p
d
x
'
å
Y
2
N
X
F
X
;
Y
(
P
)
;
where
P
=(
p
Z
)
Z
2
M
isthefamilyofallpressureunknownsinthecontrolvolumes,and
R.Eymard
Universite´Paris-Est,e-mail:robert.eymard@univ-mlv.fr
C.Guichard
Universite´Paris-EstandIFPEnergiesnouvelles,e-mail:cindy.guichard@ifpenergiesnouvelles.fr
R.Herbin
Universite´Aix-Marseille,e-mail:raphaele.herbin@latp.univ-mrs.fr
R.Masson
IFPEnergiesnouvelles,e-mail:roland.masson@ifpenergiesnouvelles.fr
1
2R.Eymardetal.
wheretheflux
F
X
;
Y
(
P
)
,betweencontrolvolumes
X
and
Y
,isalinearfunctionofthe
componentsof
P
whichensuresthefollowingconservativityproperty:
F
X
;
Y
(
P
)=
F
Y
;
X
(
P
)
:
(1)
Suchalinearfunction,whichisexpectedtovanishonconstantfamilies,maybe
definedby
ZF
X
;
Y
(
P
)=
å
a
X
;
Y
p
Z
;
(2)
Z
2
M
X
;
Y
wherethefamily
(
a
XZ
;
Y
)
Z
2
M
X
;
Y
and
M
X
;
Y
M
aresuchthat
å
Z
2
M
X
;
Y
a
XZ
;
Y
=
0.
Assuming
N
c
constituentsand
N
a
phases,thediscretebalancelawsthenread
NaF
X
(
A
(
n
+
1
)
A
(
n
)
)+
M
(
n
+
1
)
;
a
F
(
n
+
1
)
;
a
=
0
;
8
i
=
1
;:::;
N
c
;
d
t
X
;
iX
;
i
a
å
=
1
å
X
;
Y
;
iX
;
Y
Y
2
N
X
(3)
F
(
X
;
n
+
Y
1
)
;
a
=
F
X
;
Y
(
P
(
n
+
1
)
;
a
)
r
(
X
;
n
+
Y
1
)
;
a
g
−
(
x
Y
x
X
)
;
8
a
=
1
;:::;
N
a
;
where
n
isthetimeindex,
d
t
isthetimestep,
F
X
istheporousvolumeofthecon-
trolvolume
X
2
M
,
A
X
;
i
representstheaccumulationofconstituent
i
inthecontrol
volume
X
perunitporevolume(assumedtotakeintoaccountthedependenceof
theporositywithrespecttothepressure),
M
a
X
;
Y
;
i
istheamountofconstituent
i
trans-
portedbyphase
a
fromthecontrolvolume
X
tothecontrolvolume
Y
(generally
computedbytakingtheupstreamvaluewithrespecttothesignof
F
X
;
Y
),
P
a
isthe
familyofthepressureunknownsofphase
a
,
g
isthegravityacceleration,
r
a
X
;
Y
is
thebulkdensityofphase
a
betweencontrolvolumes
X
and
Y
and
x
X
isthecenterof
controlvolume
X
.Inadditiontotheserelations,thedifferencesbetweenthephase
pressuresareruledbycapillarypressurelaws.Thermodynamicalequilibriumand
standardclosurerelationsareused.
Whenapplyingscheme(3),oneshouldbeverywaryoftheuseofconformal
finiteelementsinthecaseofhighlyheterogenousmedia.Indeed,assumingthatthe
controlvolumesarevertexcenteredwithverticeslocatedattheinterfacesbetween
differentmedia,thentheporousvolumeconcernedbytheflowofverypermeable
mediumincludesthatofnonpermeablemedium.Thismayleadtosurprisingly
wrongresultsonthecomponentsvelocities.Apossibleinterpretationofthesepoor
resultsisthat,whenseenasasetofdiscretebalancelaws,thefiniteelementmethod
providesthesameamountofimpermeableandpermeableporousvolumeforthe
accumulationtermforanodelocatedataheterogenousinterface.
Wepresentinthispapertheuseofanewscheme,calledVertexApproximate
Gradient(VAG)scheme[8,9],whichcanbeimplementedin(3)sothatthecompo-
nentsvelocitiesarecorrectlyapproximated,thankstoaspecialchoiceofthecontrol
volumesandofthediscretefluxes,whichrespecttheform(2).Thepurposeofre-
spectingtheform(3)-(2)istobeabletoeasilyplugitintoanexistingreservoircode,
sayMulti-PointFluxApproximation(MPFA),bysimplyredefiningthecontrolvol-
Zumesandthecoefficients
a
X
;
Y
ofthediscreteflux.
MultiphaseflowinporousmediausingtheVAGscheme3
Althoughpartofthisschemeisvertexcentered,weshowthatthesolutionob-
tainedonaveryheterogeneousmediumwithacoarsemeshremainsaccurate.This
isagreatadvantageofthisscheme,whichisalsoalwayscoercive,symmetric,and
leadstoa27-stencilonhexahedralstructuredmeshes.InadditiontheVAGscheme
isveryefficientonmesheswithtetrahedrasincetheschemecanthenbewritten
withthenodalunknownsonly,thusinducingareductionofthenumberofdegrees
offreedombyafactor5comparedwithcellcenteredfinitevolumeschemessuchas
MPFAschemes[1,2,4,5].
2Presentationofthescheme
TheVAGschemeisdescribedin[8,9],anditsgradientschemepropertiesarerelated
tothosepresentedin[7];thereforewefocushereonthespecifitiesoftheuseofthis
schemeforamultiphaseflowsimulationoftheform(3).Let
M
beageneralmesh
of
W
,definedbyaset
G
ofgridblocksandtheset
V
oftheirvertices;thisisa
meshofcontrolvolumesinthesenseoftheprecedingsection:acontrolvolumeis
eitheragridblock
K
2
G
oravertex
v
2
V
.Inparticular,aporousvolumemustbe
associatedtoeachcontrolvolume,
i.e.
toeachgridblockandtoeachvertex.Finally
aflux
F
X
;
Y
fromthecontrolvolume
X
tothecontrolvolume
Y
mustbespecified.
Anygivengridblock
K
2
G
has,say,
N
K
vertices;letusdenoteby
V
K
V
theset
ofthesevertices.Wewishtodefineafluxbetweenneighbouringcontrolvolumes
X
=
K
and
Y
=
v
2
V
K
,andbetweenneighbouringcontrolvolumes
X
=
v
2
V
K
and
Y
=
K
2
G
v
=
∙
Y
=
K
2
G
suchthat
v
2
V
K
g
;tothispurpose,weintroducea
localdiscretegradient
Ñ
K
;
v
(
P
K
)
2
R
3
(see[8,9]fortheprecisedefinitions),which
onlydependsonthevalues
P
K
=(
P
K
;
v
)
v
2
V
K
=(
p
v
p
K
)
v
2
V
K
.Wethenintroducethe
0matrices
(
A
vK
;
v
)
v
;
v
0
2
V
K
,whicharedefinedbythefollowingrelation
K0jjå
L
K
Ñ
K
;
v
P
K
−
Ñ
K
;
v
Q
K
=
åå
A
vK
;
v
P
K
;
v
0
Q
K
;
v
;
8
P
K
;
Q
K
2
R
V
K
:
N
Kv
2
V
K
v
2
V
K
v
0
2
V
K
Thefluxfromcontrolvolume
X
=
K
tocontrolvolume
Y
=
v
isthengivenby
0v;vF
X
;
Y
(
P
)=
F
K
;
v
(
P
)=
å
A
K
(
p
v
0
p
K
)
;
0vKV2whichisofthesameformas(2);using(1),weget
F
Y
;
X
(
P
)=
F
X
;
Y
(
P
)
.Letusnow
turntothedefinitionofporousvolumesforall
X