Explicit construction of effective flux functions for Riemann solutions

0.25 MB·

by  Pablo Castañeda

Explicit construction of effective flux functions
for Riemann solutions
PabloCastan˜eda
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J

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AbstractFora familyofRiemannproblemsforsystemsofconservationlaws,we

construct a flux function that is scalar and is capable of describing the Riemann
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P solutionoftheoriginalsystem.
A
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h
t 1 Introduction
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[ Weareinterestedininjectionproblemsleadingtoflowinporousmedia,whichare

1 modeledbysystemsofconservationlaws;asurveyofthemathematicaltheoryfor
v suchflowmaybefoundin[2,6,21]andreferencestherein.Inthisworkwefocus
4 ontheRiemannproblemsandtheirsolutionsviatheWaveCurveMethod(cf.[1,8])
4
andontheconstructionofeffectivefluxfunctions(EFF)allowingtounderstandthe
3
wholesystemasasinglescalarconservationlaw,see[8,18].
4
0 The setting for such a construction is given for a fixed state in physical space.
. Thus we can develop the construction of a wave group the lifting of which will
1
givean effectiveflux function.Suchfunctionscan betreated asthe flux functions
0
7 of a scalar conservation law in a certain parametrized coordinate. This lifting is
1 thecrucialpartintheconstruction.However,wewillshowthatthisfunctionisnot
: unique.Weonlyhaveuniquenessasaclassoffunctions,therepresentationofwhich
v
i willbetheeffectivefluxfunctionforeachstate,eachstartingeigenvaluefamilyand,
X
thechosencoordinatesystem.
r Analogouseffectivefluxfunctionshavebeenusedsatisfactorilyinmanyworks.
a
There are implicit uses in [1, 2, 3, 19] and, explicit constructions in [8, 10]. An-
other potentialapplicationis modelingspecialflux functionsin experimentaldata
forwhichclassicalknownmodelsareinappropriate(e.g.[13]vs.[20]).
PabloCastan˜eda
DepartmentofMathematics,ITAM
R´ıoHondo1,CiudaddeMe´xico01080,Mexico.
e-mail:[email protected]
1
2 PabloCastan˜eda
2 The 2 2 system ofconservationlaws
×
Letuswriteasasystemtheconservationlawsusedalongthiswork.Wefocusina
systemwithtwoequations,sinceitisonlyimportanttotakemorethanoneequation.
Theextensiontoanynumberofequationswouldbenatural.
Let u (x,t), u (x,t) to be the conserved quantities at distance x along the real
1 2
axis,attimet.Typicallyinonespatialdimensionasetofequationsgoverningthe
system
¶ U ¶ F ¶ u /¶ t+¶ f /¶ x=0
+ =0, or 1 1 (1)
¶ t ¶ x ¶ u2/¶ t+¶ f2/¶ x=0,
forx IR,t 0,representingtheconservationofU =(u ,u ).Theflowfunctions
1 2
∈ ≥
characterizethesystemandaredenotedasthevectorF(U)=(f (u ,u ), f (u ,u )).
1 1 2 2 1 2
WedenoteasD thespaceofstatesU,ingeneralweconsiderD IR2.
⊂
Ofspecialinterestinapplicationsandnumericalcalculationsconsistsintheclas-
sificationofthesolutionstructureofthesystemofPDE(1)withdiscontinuousdata:
UL if x<0,
U(x,t=0)= (2)
(UR if x>0.
TheRiemannproblemconsistsofsystem (1)with initialRiemanndata(2), which
wewilldenoteasRP(UL,UR),withleftandrightvaluesULandUR,respectively.
Strictly hyperbolicsystems of conservationlaws, i.e.,where the eigenvaluesof
the Jacobian matrix of the flux function are real and distinct, provide a relatively
well understood framework for the solution of Riemann problems [11]. Here we
assumethatthesystemisnotnecessarilystrictlyhyperbolic.
A remarkable observation is that in many problems for flow in porous media,
thereisanextraconservedquantityorfluid,thusanextraequationforsystem(1).
Typicallythe extraquantityandthe phasesin (1)add upto one.In the same way,
thereisanextrafluxfunctionthatdependsonthesephases,seeforexample[6,8,19]
where u , u , and u can represent the saturation of water, gas and oil. Here the
1 2 3
statespaceisthesaturationtriangledefinedbyU satisfying0 u ,u ,u +u 1
1 2 1 2
≤ ≤
andtheconstraintu +u +u =1.Forthismodelanextrafunctionsatisfies f =
1 2 3 3
1 f f thatgivesathirdredundantequation.Thisisanimportantfactsincethe
1 2
− −
parametrizationoftheEFFcanbegiveninanyofthosecoordinates.
2.1 TersereviewofFundamental Waves
Equations (1) have solutions that propagate as nonlinear waves. Because of self-
similarityofthedataandthePDE,thesolutionsofaRiemannproblemdependon
x/t andconsistofcenteredrarefactionwaves,shockwavesandsectorsofconstant
states,seee.g.[14,15,18].Thecharacteristicspeedsarethetwoeigenvaluesofthe
Jacobianmatrix
ExplicitconstructionofeffectivefluxfunctionsforRiemannsolutions 3
¶ (f (U), f (U)) ¶ F(U)
1 2
J(S) := = .
¶ (u ,u ) ¶ U
1 2
Whentheeigenvaluesaredistinctandrealwesaythatthesystemisstrictlyhyper-
bolic.Sometimes,itloses hyperbolicityatparticularstates, as thecases registered
in [4, 17, 19] for umbilic and quasi-umbilic points. For distinct eigenvalues, the
smallerandlargerarecalledtheslow-andthefast-familycharacteristicspeed.Ac-
tually,theseeigenvaluescanbeequalonacurvesoronlargersets,see[22].
System(1)hassmoothsolutionscalled(slow-andfast-family)rarefactionwaves.
TheyarisebysolvinganODE,namely,
dU/dx =r (U), (3)
k
k=s or f (sloworfast), definedbytheeigenvectorsofthe JacobianmatrixJ(U),
with
J(U) x I r (U)=0, (4)
k
{ − }
where U(x ), for x =x/t, is the profile of the forward rarefaction, provided x is
monotone increasing; it is called backward for x monotone decreasing. The k-
integral curve of a stateUo, denoted by R (Uo), consists of all statesU for each
k
ofwhichU(x )solvestheinitialvalueproblem(3)forthegivenkandinitialcondi-
tionUo,eitherbackwardandforward.
This system also admits solutions in the form of moving jump discontinuities.
In orderto respect conservationof u and u , the fluxesin and out of the moving
1 2
discontinuity must balance. In terms of the state Uo =(uo,uo) on the left of the
1 2
discontinuity,thestateU =(u ,u )ontherightofthediscontinuity,andthepropa-
1 2
gationspeeds ,thisbalanceisexpressedasF(U) s U=F(Uo) s Uo,or
− −
f (U) s u = f (Uo) s uo,
f1(U)−s u1 = f1(Uo)−s u1o. (5)
2 − 2 2 − 2
Eqs. (5) are the Rankine-Hugoniot(RH) conditions. The Rankine-Hugoniotlocus
ofa stateUo, denotedbyH(Uo), consistsofallstatesU foreach ofwhichthere
existsavalues =s (Uo,U)suchthattheRHconditions(5)aresatisfied.
Thetwoformerlociinthespaceofstateswillbethebasisfortheconstructionof
thewavegroupandfortheeffectivefluxfunction.Werecallthatinpracticewemust
select a criterion for the “physically admissible” discontinuitiesthat appear in the
solutions.ThiswillbeLiu’scriterionincludingalsotheLax’sadmissibilitycriteria,
namely:
Shock admissibility. A discontinuity with propagation speed s =s (UL,UR) be-
tweenaleftstateUL andarightstateUR isadmissibleifitsatisfiesLiu’sadmissi-
bilitycriterion[16],wheneveritisapplicable.WeuseLax’sadmissibilitycriterion
[14]inordertoclassifythepropagationspeedasfollows:
l (UR)<s <l (UL), s <l (UR), forslow-familyshocks,
s s f (6)
l (UR)<s <l (UL), l (UL)<s , forfast-familyshocks.
f f s
4 PabloCastan˜eda
Moreover,inthepreviousdefinitionsweallowoneoftheinequalitiestobecomean
equality;henceouradmissibilitycriterion.Thenomenclatureslow-andfast-family
originatesfromthe1-Laxand2-Laxshockwaves,[14].
The following definition is inspired by the Welge-Ole˘ınik’s construction for a
singleconservationequation.LetUobeastateinphysicalspace;astateU isaslow
(respectively fast) extension of Uo if U belongs to H(Uo) and the shock speed
s (Uo,U) equals the slow characteristic speed l (U) (resp. fast l (U)), i.e., the
s f
shockischaracteristicatU.
TheBethe-WendroffTheorem(cf.[12])guaranteesthatatanextensionpointone
oftherarefactionscurvesstartingatU istangenttotheH(Uo)atU.
2.2 TheWaveCurve Method
Solutions found by Wave Curve Method consist of rarefaction fans, shock dis-
continuities and constant states, a survey is found in [1]. The classical construc-
tion is guaranteed to succeed only when UL and UR are close. The Buckley-
Leverett(BL)solutionexhibitsinflectionpoints(cf.[5,18]),whereequalitiessuch
as (cid:209) l (U) r (U)=0 for k=s or f occur. The BL shows that rarefactionwaves
k k
·
and shock waves of the same family can be adjacent. For adjacency to occur, the
shockspeedmustcoincidewiththesame-familycharacteristicspeedatanedgeof
therarefactionwave,andoneoftheLax’sinequalitiesin(6)becomesanequality.
Ofcourse,whentraversingthesolutionbyincreasingx/tmonotonically,thecor-
respondingwavespeedmustalsoincrease.Consequently,fast-familywavesfollow
slow-family waves. This structural feature was first identified by Liu, [15], under
technical restrictionsand holds in general;see [21]. Wave sequencesare concate-
natedfollowingcertainrules.
Awavegroupisasequenceofwaves,allassociatedtothesamefamily,whichare
adjacent,meaningthatnotwowavesareseparatedbyaconstantstate.AsintheLax
construction(Fig.1),ifthebeginningorendstateofawavegroupisprescribed,then
thestateontheoppositeendofthewavegroupliesonacurveinstatespacecalled
awavecurve.Inthelanguageofwavegroups,theRiemannsolutionconsistsofthe
followingwave sequence,fromleft to right:the left stateUL, a slow-familywave
group,anintermediatestateUM,a fast-familywavegroup,andtherightstateUR.
Noticethattheslow-familywavecurvefromULconsistsofallstatesUM attainable
throughaslow-familywavegroup;similarly,thebackwardfast-familywavecurve
fromURconsistsofallstatesUMattainablethroughafast-familywavegroup.Wave
curvesandanalgorithmtoconstructtheRiemannsolutionweredescribedin[15].
In constructingwave groupswith two or more waves, in [15] Liu introduceda
shockwaveadmissibilitycriterionthatencompassesthe criterionofOle˘ınik,[18].
ThelatterisageneralizationofWelge’sconstruction,theoneweuseforconstruct-
ing the wave group, hence we call Welge point to values where a shock wave is
characteristicforascalarfluxfunction.Thewavecurvemethodisapplicabletosys-
ExplicitconstructionofeffectivefluxfunctionsforRiemannsolutions 5
u2 u2
UR uR
2
UM uM
2
UL uL
2
u x
1
Fig.1 SolutionofRP(UL,UR),ULdeterminestheslow-familywavecurveandURthebackward
fast-familywavecurve.TheirintersectiondeterminesUM.Attherightfigure,thesolutionprofile
atafixedtime,noticethevalueofbothcoordinatesystematu .
2
temswithanynumberofconservationlaws.Theeffectivefluxfunctionisgivenfor
eachwavegroupasweshowsoon.
3 The effective flux function construction
The main idea is the following:for any given fixed state R in D construct a base
curveG :I D andan Effective Flux Function(EFF) f:I IR. The base curve
G isaparam→etrizationofawavegroup;forthe2 2systems→wehaveatleastfour
×
waystostartingsuchacurve.TheEFFf(ℓ)istheliftingofthebasecurvedefined
bythewavegrouponthephysicalspace.
Theheartofourworkresidesintwofacts:(1)whenthestateG (ℓ)isatashock
curve,theRHcondition(5)issatisfied,thustheshockspeedcanbegivenbyanyof
thoserelationsoralinearcombinationofthem,(2)whenthestateG (ℓ)isataninte-
gralcurve,thewavespeedisgivenasl G (ℓ) withk=sor f,thecorresponding
k
familyofsuchararefaction.
(cid:0) (cid:1)
Nowletusexplaintheconstructionofthebasecurveandhowitsliftingisfound.
Aspointedout,wesubdividetheseconstructionsintotwocases.Thisisdoneonly
foraneasierexpositionbecauseitalwaysworksinthesamemanner.
3.1 ConstructionofthebasecurveGGG (((ℓℓℓ)))
InSec.2.2itisexplainedhowtoconstructslow-andfast-familywavegroupswith
integral curves and Hugoniot loci on the state space. For a hyperbolic point, i.e.
6 PabloCastan˜eda
where both characteristic speeds are real and distinct, there are two linearly inde-
pendenteigenvectorsthatdescribethetangentstotheslowandfastwavegroups.
Typically, for each family the respective eigenvalue increases in the direction
ofaneigenvectoranddecreasesintheoppositedirection.Therefore,foraforward
wavegroupconstruction,we haveararefactionwaveinonedirectionandashock
waveintheoppositeone.Therearecaseswherebothdirectionshaveshockorrar-
efactionsasinitialwaves,suchpointsareintheinflectionmanifoldoftherespective
family.
The base curve G :I D will be a parametrization of one wave group. The
→
choice of the interval of interest I and its parametrization is essential in the con-
structionoftheEFFf(ℓ),itslifting,theconstructionofwhichwillbegivensoon.
Firstofall,weneedsomeguidingnotationfortheanalysisthatfollows.AsG (ℓ)
belongstoDwetakethecoordinatesasG (ℓ)=(g (ℓ),g (ℓ)).ThusforU G (ℓ)we
1 2
haveu =g (ℓ)fori=1,2.Actually,ifoneofthecoordinatesg org ism∈onotonic,
i i 1 2
sayg ,thenitispossibletodoareparametrizationℓ=g (ℓ).Thuswecanassume
1 1
thatG (ℓ)=(ℓ,g (ℓ))holdsatleastlocally.(In[8]theparametrizationisgivenwith
2
ℓ as the oil saturation, the third implicit coordinate,and it is easy to see that it is
always possibleto take ℓ as a linear combinationof the coordinatesg andg ; we
1 2
assumethatg (ℓ)=ℓholdsforaneasierexposition.)
1
In the following sections the lifting construction of the EFF f(ℓ) is described.
Nonetheless, it is important to remark that such a lifting has as motivation to be
a function that behaves as a single scalar flux function. Therefore, the Ole˘ınik E-
criterion is applicable (cf. [18]), moreover as such a construction is based on the
envelopeofthefluxfunction,the“mirroreffect”givenin[7]follows.
3.2 EFF construction:the firstwaveis ashock wave
Assume that the base curveG (ℓ) starts with a k-Lax shock wave at the reference
stateR=(uR,uR).AspointedoutinSec.2.2andinFig.1,theforwardwavegroup
1 2
isrelevantfork=sandthebackwardwavegroupfork= f.Intheforwardconstruc-
tionwetakethepartofH(R)thatsatisfies(6)forthechosenkandsuchthatLiu’s
criterionholdsforallpointsbetweenRandU,respectivelyasleftandrightstates,
see[15,16].(Forbackwardconstruction,U andRaretheleftandrightstates.)Thus
for eachU =(u ,u ) in H(R) there exists g (ℓ) such that (ℓ,g (ℓ))=U holds;
1 2 2 2
seeSec.3.1.NoticethatthesetofadmissibleshocksmaystopataBethe-Wendroff
pointU wheres (R,U )=l (U )occurs.(Actuallyk =kmayhold.)
∗ ∗ k ∗ ′
WesettheintervalI=[uR,′u ]assumingthatuR<u .6 (Conversely,I=[u ,uR]
1 ∗1 1 ∗1 ∗1 1
foru <uR.)NowwehavethebasecurveG :I D satisfyingG (ℓ)=(ℓ,g (ℓ))
H(R∗1)the1positionofwhichdeterminesthe cor→respondingshockspeed(5)2. Thu∈s,
weconstructtheliftingas
f : I IR
ℓ −→ f(ℓ):= f (R)+s R,G (ℓ) [ℓ uR], (7)
7−→ 1 − 1
(cid:0) (cid:1)
ExplicitconstructionofeffectivefluxfunctionsforRiemannsolutions 7
thus,noticethatthescalarshockspeedsatisfies
f(ℓ) f (R)
s (uR,ℓ) = − 1 = s R,G (ℓ) , forall ℓ I, (8)
1 ℓ uR ∈
− 1
(cid:0) (cid:1)
sotheRankine-Hugoniotcondition(5)holdsasdesired.
Remark1.Whenℓ isavalueofaWelgepointoftheEFF(alwayswithleftstateuR),
∗ 1
thenU =G (ℓ ) is a Bethe-Wendroffpointof H(R), thus the eigenvectorpoints
∗ ∗
paralleltotheHugoniotlocusandthewavecurvemaybefollowedbyararefaction
curve.(AsstatedbytheBethe-Wendrofftheorem.)
Lemma1.The EFF f(ℓ) in (7) is the first flux function over the base curve, i.e.,
f(ℓ)= f G (ℓ) .
1
Proof. Th(cid:0)eRH(cid:1)Sof(8)isequivalentto[f G (ℓ) f (R)]/[ℓ uR],equatingwith
themiddletermin(8)provesthatf(ℓ)= f1 G (ℓ) −hol1dsin(7).− 1 (cid:3)
1
(cid:0) (cid:1)
TheBethe-WendroffpointatthestateU (cid:0)canb(cid:1)etakenasanewstartingreference
∗
pointforararefactionwave.Justmakesurethatthestartingfamilyk nowmustbe
takenask;theymaynotbethesame.
′
3.3 EFF construction:the firstwaveis ararefactionfan
Assume that the base curveG (ℓ) starts with a k-rarefactioncurveat the reference
state R=(uR,uR). In the forward constructionwe take the R (R) part which has
1 2 k
increasingeigenvaluex =l (U)forU R (R).(Inthebackwardconstructionwe
k k
takethedecreasingeigenvaluedirection.∈)ThusforeachU=(u ,u )inR (R)there
1 2 k
existsg (ℓ)suchthat(ℓ,g (ℓ))=U holds;seeSec.3.1.
2 2
Recall that the rarefaction curve may stop at an inflection point U (where
∗
(cid:209) l (U ) r =0occurs),thenwesettheintervalI=[uR,u ]assumingthatuR<u .
(Coknve∗rse·lyk, I =[u ,uR] for u <uR.) Thus, the bas1e cu∗1rve G :I D s1atisfie∗1s
G (ℓ) = (ℓ,g (ℓ)) ∗1R1(R), the∗1 posi1tion of which determines the c→orresponding
2 k
∈
eigenvalue.Thus,weconstructtheliftingas
f : I IR
ℓ −→ f(ℓ):= f (R)+ ℓ l G (t) dt, (9)
7−→ 1 uR1 k
R (cid:0) (cid:1)
andnoticethatf(ℓ)=l (G (ℓ))holdsasdesired.
′ k
Lemma2.The EFF f(ℓ) in (9) is the first flux function over the base curve, i.e.,
f(ℓ)= f G (ℓ) .
1
Proof. D(cid:0)irect d(cid:1)ifferentiationin (9) shows that f(ℓ)=l (G (ℓ)) holds, notice also
′ k
that(cid:209) G (ℓ)=(1,g (ℓ))isparalleltor (G (ℓ)).Thus,theidentityJ(G (ℓ))(cid:209) G (ℓ)=
2′ k
l (G (ℓ))(cid:209) G (ℓ)holds,see(4),thefirstcoordinateindicatesthat
k
8 PabloCastan˜eda
d ¶ f ¶ f
dℓf1(G (ℓ)) = ¶ u1, ¶ u1 ·(1,g2′(ℓ)) = l k(G (ℓ))
(cid:18) 1 2(cid:19)
is satisfied. Then, as f(ℓ) and f (G (ℓ)) solve the same IVP with initial condition
1
f (R),thefluxesarethesame. (cid:3)
1
TheinflectionatthestateU canbetakenasa newstartingreferencepointfor
∗
a shock wave, just make sure that in (7) the reference state for determining the
HugoniotlocusandtheshockspeedistheoriginalreferencestateRandnotU .
∗
3.4 ThecompleteEFF construction
InprevioussectionswedepictedtheconstructionofabasecurveG (ℓ),seeSec.3.1,
andtwowaysfortheliftingoff(ℓ)basedonshockcurves(Sec.3.2)oronrarefaction
curves(Sec.3.3).InSec.2.2wepointedoutthatawavecurveisacompositionof
the former waves; it changes types at inflection or Bethe-Wendroff points. In this
section we show the construction of a complete EFF which is actually a smooth
function.
From shock to rarefaction curve. Once we start a base curve with shock waves,
thisisacurvealongH(R)andwouldchangetoanintegralcurvewithinthesame
wave only at a Bethe-Wendroff pointU . Notice that at such a point the equality
∗
f(u )=l (U )holdsforcertaink,thereforethecontinuityforthederivativesofthe
∗1 k ∗
liftingbetweenbothexpressions(7)and(9)holds.
The continuity of the EFF itself holds because of the adding of f (R) in both
1
liftings(7)and(9):fromLemma1wehavethatf(ℓ)= f G (ℓ) holdsinparticular
1
atU ,andsincefrom(9)wehavethatf(u )= f (U )holds,alsotheEFFcontinuity.
∗ ∗1 1 ∗ (cid:0) (cid:1)
From rarefactionto shock curve. The continuityat this transitiondo notseems so
natural;theactualvalueoff(ℓ)in (9)is notknowapriorifora givenℓ. However
asthetransitionmustoccuratapointU belongingtobothR (R)andH(R),thus
∗ k
the values l (U ) and s (R,U ) agree and from the latter and (7), we notice that
k ∗ ∗
f(u )= f (U )holds;hencethesmoothnessatsuchtransitions.
∗1 1 ∗
For a complete construction of an EFF, we notice that transitions from shock
wavestorarefactionwavesandvice-versamayoccurmanytimes.Thisisperfectly
controlledbyourwayofdoingtheliftings;recallthatfromashocktoararefaction,
the family k may have changed to k and that from a rarefaction to a shock, the
′
referencepointRistheoriginalone.Wehaveprovenourmainresult:
Theorem1.LetRtobeafixedstateinD.ConstructawavecurvethroughR.Select
acoordinateℓ,letussayℓ=u ,toparametrizethecurveasG :I D satisfying
1
G (uR)=R.Therefore,anEFFistherespectivefluxfunctionalongG→(ℓ)as
i
f : I IR
ℓ −→ f G (ℓ) .
1
7−→
(cid:0) (cid:1)
ExplicitconstructionofeffectivefluxfunctionsforRiemannsolutions 9
Proof. SeeLemmas1and2. (cid:3)
JustrecallthatsuchanEFFisconstructedbasedonR,thusevenforR G (ℓ),
′
∈
therespectiveEFFmaybedistinct.Inthenextsectionweshowsomeexamples.
4 Some examples andapplicability
Inthissectionweexplaintwoexamples.Thefirstonebasedinthesatisfactoryuse
ofEFFsin[8,10],constructedalongtheso-calledseparatrixasacrucialwavegroup
fortheRiemannsolution.Thesecondoneshowshowchoosingtheparametrization
coordinateisimportantforunderstandinganEFF.
4.1 SimplifiedquadraticCorey model
InEnhancedOilRecovery(EOR)theproposedconservationlawsaregivenbyfrac-
tional flux functionsdue to physicalfeatures as rock and fluid permeabilities(see
e.g.[6,19]).Herewetakeaschematicmodelwithflowfunctionsfor(1)givenby:
Au2 Bu2
f (U) = 1 , f (U) = 2 , (10)
1 Au2+Bu2+Cu2 2 Au2+Bu2+Cu2
1 2 3 1 2 3
whereconstantsA,BandCdependonseveralphysicalquantities.Itwaspointedout
thatu andu arerelatedtowaterandgassaturations,theoilsaturationisrelatedto
1 2
u =1 u u .Theflowfunctions(10)arerelatedtothefluxfunctionforwater
3 1 2
− −
and gas, which came from the so-called quadratic permeability Corey model; an
extra implicit flow function for oil f (U)=Cu2/(Au2+Bu2+Cu2) is sometimes
3 3 1 2 3
useful.Thedomainisthesaturationtrianglegivenbytheconstraints0 u ,u ,u
1 2 3
≤
andu +u +u =1.
1 2 3
4.2 Example1:criticalsolutionalong theseparatrix
AclassicalprobleminEORistheWateralternatingGas(WAG)injection,whichhas
adirectrelationtotheRiemannproblemRP(UL,O)wheretheleftRiemanndatum
UL representsa mixtureofwater andgas,and the leftRiemanndatumO=(0,0)
representsavirginreservoirstatecontainingsolelyoil.
In the saturation triangle there are three base wave curvesreachingO (cf. [2]).
They are lines which can be parametrized by ℓ I as the third (implicit) coordi-
nate u for the interval I =[0,1]. Let G (ℓ)=(∈1 ℓ,0), G (ℓ)=(0,1 ℓ) and,
3 1 2
G (ℓ)=((1 ℓ)B/D,(1 ℓ)A/D)bethebasecurve−sforsuchparametriza−tionwith
3
− −
thedenominatorD=A+B,seeFig.2.Thus,simplecomputationsleadtotheEFFs
10 PabloCastan˜eda
u
2
G
Fig.2 Theshadowedregion
representsthesaturationtri-
angle.ThevertexO,W andG
arerelatedtopureoil,water
and gas, respectively. The
basecurvesG arethelines B
i J
connectingOtoW,GandB
respectivelyfori=1,2and3.
S
ThedashedcurvesareH(R), ∗
bothbranchesofahyperbola.
Thethincontinuous curve u
1
isaslow-familyrarefaction
O R W
connectingJtoS .
∗
f(ℓ)= f (G (ℓ))fori=1,2,3givenexplicitlyby
i 3 i
Cℓ2 Cℓ2 Cℓ2
f (ℓ)= , f (ℓ)= , f (ℓ)= .
1 A(1 ℓ)2+Cℓ2 2 B(1 ℓ)2+Cℓ2 3 AB(1 ℓ)2/D+Cℓ2
− − −
The EFFs abovesatisfy the BL solution,see [5], with S-shapedflux functions.
TheirWelgepointscanbecalculatedbythevalues
ℓ =1 C/(A+C), ℓ =1 C/(B+C), ℓ =1 CD/(AB+CD),
∗1 − ∗2 − ∗3 −
p p p
with relative positionssatisfying ℓ <ℓ ,ℓ . These inequalities guaranteethat the
∗3 ∗1 ∗2
optimalinjectionmixtureforoilproductionoccurswithintheseparatrix,see[8,10].
ForWelge points, theirvaluesℓ also indicatethe locationsofBethe-Wendroff
∗i
pointsU :=G (ℓ ). It is possible to verify that s (U ,O)=l (U ) holds, so the
i∗ ∗i 3∗ s 3∗
rest of the base curveG is a slow rarefaction. The analog Bethe-Wendroff points
3
U ,U ,showthatalongG ,G therarefactionsareofthefastfamily.
1∗ 2∗ 1 2
4.3 Example2:choosing a parametrizationcoordinate
The complete solution for the WAG injection needs a slow wave group.It can be
found in forward direction from any point representingmixture of water and gas.
However, intermediate states lie over one of the base curves of Sec. 4.2, here we
showtheconstructionofEFFoverbackwardslowcurvesforstatesR=(m,0).
The RH relation (5) lead to the RH locus H(R) for all states U in the sat-
uration triangle satisfying f (U) f (R) = [f (U)/u ](u m), since the shock
1 1 2 2 1
speedis f (U)/u ,see(5.b).Asim−plemanipulationshows−thatH(R)istheedge
2 2