Multiphase flow in porous media using the VAG scheme

Description

Multiphase flow in porous media using the VAG scheme Robert Eymard, Cindy Guichard, Raphaele Herbin and Roland Masson Abstract We present the use of the Vertex Approximate Gradient scheme for the simulation of multiphase flow in porous media. The porous volume is distributed to the natural grid blocks and to the vertices, hence leading to a new finite volume mesh. Then the unknowns in the control volumes may be eliminated, and a 27- point scheme results on the vertices unknowns for a hexahedral structured mesh. Numerical results show the efficiency of the scheme in various situations, including miscible gas injection. 1 Introduction Simulation of multiphase flow in porous media is a complex task, which has been the object of several works over a long period of time, see the reference books [12] and [3]. Several types of numerical schemes have been proposed in the past decades. Those which are implemented in industrial codes are mainly built upon cell centred approximations and discrete fluxes, in a framework which is also that of the method we propose here. Let us briefly sketch this framework. The 3D simulation domain ? is meshed by control volumes X ?M . Let us denote by ? the diffusion matrix (which is a possibly full matrix depending on the point of the domain). For each control volume X ? M , the set of neighbors Y ? NX is the set of all control volumes involved in the mass balance in X , which means that the fol- lowing approximation formula is used: ? ∫ X ? ·??pdx ' ∑Y?NX FX ,Y (

• gradient scheme properties


• flux between


• lower permeability


• volume


• multi-point flux


• vag scheme


• all control


• results obtained


• results

Sujets

• Université
• Masson
• Herbin
• Guichard
• Volume

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:[e-mail]
C.Guichard
Universite´Paris-EstandIFPEnergiesnouvelles,e-mail:[e-mail]
R.Herbin
Universite´Aix-Marseille,e-mail:[e-mail]
R.Masson
IFPEnergiesnouvelles,e-mail:[e-mail]

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