stochastic reservoir: the scalar buckley-leverett...

17/12/2003 - Pau momas group Peppino Terpolilli Total-Pau

Stochastic reservoir: thescalar Buckley-Leverett model

- CFD Stavanger 217/12/2003

OUTLINE

•• Why stochasticity Why stochasticity ??•• MathematicalMathematical issues issues•• Some Some models: models: DeadDead--oiloil,,BuckleyBuckley--LeverettLeverett………………•• New New approach approach for for upscalingupscaling•• ProgramProgram•• ConclusionsConclusions

- CFD Stavanger 317/12/2003

Stochastic model

Hard Data:Hard Data:

•• wells wells : : core core : : geologygeology, scanning, scanning logslogs petrophysic petrophysic geological schemegeological scheme•• scale problemsscale problems

- CFD Stavanger 417/12/2003

Stochastic model

Soft Data:Soft Data:

•• extension:extension: geophysic geophysic, , geologygeology

•• scale problems and uncertaintyscale problems and uncertainty ((geostatisticgeostatistic))

Eponte

Axe du trou

Rmc

Rxo

Rt

Rés

istiv

ité

Rxo

Rt

Rw

RmfSw

Sxo

Rs

Rm

Zone noncontaminée

(formationvierge)

Zone lavée (envahie)

Mud cake

Boue de forage

Exa-Plans 04-1997

profil radial de résistivité

Rayon

Rmc

Rm

WIRE.CALI_1IN6 16

PETROLAN.GR_1GAPI0 200

PARAMETERS.GR_MA_1GAPI0 200

PARAMETERS.GR_CL_1GAPI0 200

1925

1950

1975

DEPTHMETRES

PETROLAN.RHOB_11.95 2.95

PETROLAN.RHOB_CL_11.95 2.95

PETROLAN.NP_10.45 -0.15

PETROLAN.NP_CL_10.45 -0.15

PETROLAN.RXO_10.1 1000

RS0.1 1000

PETROLAN.RT_10.1 1000

PARAMETERS.RT_CL_10.1 1000

PETROLAN.TEMP_1DEGF0 200

PARAMETERS.TEMP_W_1DEGF0 200

PARAMETERS.TEMP_MF_1DEGF0 200

PRECALC.FTEMPDEGF0 200

PETROLAN.RW_L_1OHMM0.01 1

PETROLAN.RW_A_1OHMM0.01 1

PARAMETERS.RW_1OHMM0.01 1

PARAMETERS.RW_DEF_1OHMM0.01 1

PETROLAN.RMF_L_10.01 1

PETROLAN.RMF_A_10.01 1

PARAMETERS.RMF_10.01 1

PARAMETERS.RMF_DEF_10.01 1

PETROLAN.VCL_GR_1V/V0 1

PETROLAN.VCL_RT_10 1

PETROLAN.VCL_NP_10 1

PETROLAN.VCL_ND_10 1

PETROLAN.VCL_NDCOR_10 1

PETROLAN.VCL_10 1

PETROLAN.ERRF_METH_1G/C30 20

PETROLAN.NUM_METH_1G/C30 20

PETROLAN.IDM_1G/C30 100

PETROLAN.RHO_DSOL_1G/C32.5 3

PETROLAN.RHO_MA_1G/C32.5 3

PARAMETERS.ROMAMIN_1G/C32.5 3

PARAMETERS.RHOB_MA_1G/C32.5 3

PARAMETERS.ROMAMAX_1G/C32.5 3

PETROLAN.RHO_HC_1G/C30 1

PARAMETERS.RHOHCMIN_1G/C30 1

PARAMETERS.RHOB_HC_1G/C30 1

PARAMETERS.RHOHCMAX_1G/C30 1

1925

1950

1975

DEPTHMETRES

PETROLAN.SWE_1V/V1 0

PETROLAN.SXOE_1V/V1 0

PETROLAN.SXOMAX_1V/V1 0

PETROLAN.SXOMIN_1V/V1 0

PETR

OLA

N.B

AD

HO

LE_F

LAG

PETROLAN.PHIE_1V/V0.5 0

PETROLAN.VOL_XWATER_1V/V0.5 0

PETROLAN.VOL_UWATER_1V/V0.5 0

PETROLAN.PHIE_1V/V1 0

PETROLAN.VCL_1V/V0 1

PETROLAN.VOL_XWATER_1V/V1 0

PETROLAN.VOL_UWATER_1V/V1 0

PETROLAN.M_DCL_1V/V0 1

PETROLAN.PHIT_1V/V1 0

PETROLAN.PERM_1MD0.1 10000

- CFD Stavanger 717/12/2003

••

6.625 km 6.625 km

PP PS3.0tPP

0.5

2D/4C Post-stack time migration

- CFD Stavanger 817/12/2003

- CFD Stavanger 917/12/2003

CASE B

Model size 118 31 23 (42209 active cells) Major parameters: Faults transmissivities - Xytrans

- CFD Stavanger 1017/12/2003

CASE H - new HM with RFT observations

- CFD Stavanger 1117/12/2003

- CFD Stavanger 1217/12/2003

Darcy law

•• NavierNavier-Stokes -Stokes equationsequations::

•• Darcy lawDarcy law::

•• isis the the matrix matrix of of permeabilitypermeability: : porousporous media media characteristiccharacteristic

1v v v p v ft

νρ

∂+ ∇ = − ∇ + ∆ +

∂

pKq ∇−=µ

K

- CFD Stavanger 1317/12/2003

Darcy law

•• Extended Darcy lawExtended Darcy law::

•• relative relative permeability permeability of phase pof phase p

•• the the depthdepth

( )rpp p p

p

Kkq p g Dρ

µ=− ∇ + ∇

rpk

D

- CFD Stavanger 1417/12/2003

Darcy law

•• Continuum Continuum mechanicsmechanics:: at at a REVa REV located at located at : :

saturation: fraction of pore volume saturation: fraction of pore volume relative relative permeabilitypermeability capillary capillary pressure pressure REV REV

( )cp S

( )rk S

x

, ,o w gS

- CFD Stavanger 1517/12/2003

Kr-pc

- CFD Stavanger 1617/12/2003

Kr-pc

- CFD Stavanger 1717/12/2003

Math issues

•• For single-phaseFor single-phase flows Darcy law leads to flows Darcy law leads tolinear equationlinear equation::

•• For For multimulti-phase-phase flow we recover nonlinear flow we recover nonlinearequtionsequtions: : hyperbolichyperbolic, , degenerate parabolicdegenerate parabolicetc…..etc…..

( ( ). )pC div K x p ft

µ ∂Φ − ∇ =

∂

- CFD Stavanger 1817/12/2003

Math issues

•• The The mathematical mathematical model model is is a system of PDEa system of PDEwith appropriate with appropriate initial initial and boundaryand boundaryconditionsconditions

•• the coefficients of the the coefficients of the equations equations are are poorlypoorlyknown known stochastic approach stochastic approach

•• geologygeology + + stochasticstochastic = = geostatisticgeostatistic

( , )K x ω

⇒

- CFD Stavanger 1917/12/2003

Uncertainty

•• SPDE:SPDE:

•• These problems These problems are are difficultdifficult::

experimental experimental design design approachapproach ‘ Grand projet incertitude ’ ‘ Grand projet incertitude ’ Industrial tools Industrial tools

( ( , ). )p div K x p ft

ω∂− ∇ =

∂

- CFD Stavanger 2017/12/2003

Uncertainty

•• We need We need to to compute compute too too much flowmuch flowsimulationsimulation

•• Grids Grids are large: millions of are large: millions of cellscells

•• Hindered Hindered ensemble-ensemble-based predictionbased prediction

- CFD Stavanger 2117/12/2003

Uncertainty

•• WorkWork in in progress progress : :

Zhang Zhang,, Tchelepi Tchelepi

Tom RussellTom Russell, , JarnanJarnan

Cho Cho , , Lindquist Lindquist

J J Glimm Glimm et al…. et al….

- CFD Stavanger 2217/12/2003

Math issue

•• ChoCho , , Lindquist Lindquist ::

ensemble- ensemble-based prediction usingbased prediction using

BuckleyBuckley--Leverett equationLeverett equation

- CFD Stavanger 2317/12/2003

Dead-oil model

•• Two immiscibleTwo immiscible phases: phases: water and oilwater and oil•• one component in one component in each each phasephase•• incompressibilityincompressibility

pressure of the pressure of the oil oil phasephase capillary capillary pressure pressure waterwater//oiloil

and and are constantare constant

c cwp p=

p

wρ oρ

- CFD Stavanger 2417/12/2003

Dead-oil model

•• DeadDead--oil equationsoil equations::

pressure pressure equationequation

saturation saturation equationequation

w oQ Q Q= +

w wS d i v q Qt

∂Φ + =

∂

[ ( ) ] ( ) ( ( ) )rw ro rw rw roc

w o w w

k k k k kdiv K p div K p div K g D Qoµ µ µ µ µ

+ ∇ = ∇ + + ∇ +

- CFD Stavanger 2517/12/2003

Buckley-Leverett

•• No No capillary capillary pressure, no pressure, no gravitygravity, no source, no sourcetermterm in pressure in pressure equationequation::

•• BuckleyBuckley--LeverettLeverett: : dimdim=1=1

0, 0, 0,c w o Tp g Q q q q const= = = + = =

[ ( ) ] 0 ;

( )

rw ro

w o

rw roT

w o

k k pKx x

k k pK qx

µ µ

µ µ

∂ ∂+ =

∂ ∂

∂+ = −

∂

- CFD Stavanger 2617/12/2003

Buckley-Leverett

•• We obtainWe obtain::

withwith::

( ) 0T wS q F St x

∂ ∂Φ + =

∂ ∂

( )

rw

ww

rw ro

w o

k

F S water fractional flowk kµ

µ µ

= =+

- CFD Stavanger 2717/12/2003

Buckley-Leverett

•• A A particular particular case of:case of:

withwith::

Method Method of of characteristicscharacteristics

( ) 0

( , 0) ( )o

S f St x

S x S x

∂ ∂+ =

∂ ∂=

(.)f the flux function

- CFD Stavanger 2817/12/2003

Buckley-Leverett

•• QuasilinearQuasilinear equationequation::

1

0; ( )

( ) 0

(.) (.)o

S f S fa St S x SS Sa St x

a and S are piecew ise C

∂ ∂ ∂ ∂+ = =

∂ ∂ ∂ ∂∂ ∂

+ =∂ ∂

- CFD Stavanger 2917/12/2003

Buckley-Leverett

•• Characteristic curves Characteristic curves ::

•• quasilinear equation means quasilinear equation means S constant S constant alongalongcharacteristic curvescharacteristic curves

•• slope slope of of characteristic curves characteristic curves constant:constant:curves curves are are straight linesstraight lines

•• the value of S in (x,t) the value of S in (x,t) equal equal the value the value at at t=0t=0

( )dx a Sdt

=

- CFD Stavanger 3017/12/2003

Buckley-Leverett

•• When characteristic curves intersectWhen characteristic curves intersect, , weweobtain shoksobtain shoks

•• weack weack solutions, existence but solutions, existence but unicity unicity ??•• Entropy Entropy conditions to select the conditions to select the physicalphysical

solution:solution: Lax Lax, , OleinikOleinik……..……..

- CFD Stavanger 3117/12/2003

Buckley-Leverett

•• Some Some conclusionsconclusions obtained obtained by by ChoCho,,LindquistLindquist::

for for oiloil--cutcut, , peak peak production production uncertainty shortlyuncertainty shortlyafter mean breackthroughafter mean breackthrough

accuracy accuracy of ensembles of ensembles mean results behavemean results behaveaccording to according to ‘ central ‘ central limit theoremlimit theorem ’ ’history matching improve the history matching improve the relative relative errorerror, but, butnot that muchnot that much

- CFD Stavanger 3217/12/2003

upscaling

•• We present now We present now a a new approch new approch for for upscalingupscaling

first applied first applied for single phase flowfor single phase flow

and a possible and a possible strategy to tackle Buckleystrategy to tackle Buckley Leverett Leverett modelmodel

- CFD Stavanger 3317/12/200310/12/20035 -

Upscaling methodsLocal Problems

ω

1 3≤ ≤i

Γ Γ Γ2 = ∪a c

Γ Γ Γ1 = ∪b d

P x suri i= ∂ωLinear B.C.

ConfinedB.C.

∂∂

∂ωPn

sur

P x sur

ii

i i i

= −

=

⎧⎨⎪

⎩⎪

0 Γ

Γ

P w xw H

i i i

i p

= +∈

⎧⎨⎩

1 ( )ω

K K x P P dxij i j* ( ) .= ∇ ∇∫1

ω ω

⎫

⎬

⎪⎪⎪

⎭

⎪⎪⎪

Γa

Γb

Γc

Γd

Stochastic Homog.

on

on

on

( )⎩⎨⎧

Γ∪Γ=∂+=∇−

21..0)(

ωω

onCBinPxKdiv i

3,1 ≤≤ jiPeriodic B.C.

(homogenization) periodic on ω∂

- CFD Stavanger 3417/12/2003

satisfy

constant symmetric matrix H such that

New ApproachConstraints

11 )(1

−−

⎟⎟⎠

⎞⎜⎜⎝

⎛= ∫ωω

dxxKh ∫=ωω

dxxKa )(1

=),,( ωahM

)(xK

)( xK

is symmetrici)

ii)⎪⎪⎩

⎪⎪⎨

⎧

βα ≤<0

Harmonic average Arithmetic average

•

•

•dIRxK ∈∀≤≤

→→→→→ξξβξξξα 22 .)(

22 .→→→→

≤≤ ξξξξ aHh

- CFD Stavanger 3517/12/200310/12/20037 -

New Approach (energy function)

Min

ω

gi gi

general g if i ,

E K K x P P dx f P dxijD

i j i j( ) ( ) .= ∇ ∇ −∫ ∫12 ω ω

E H H U U dx f U dxijD

i j i j( ) .= ∇ ∇ −∫ ∫12 ω ω

1≤ ≤i j m,

1 ≤ ≤i m

[ ]I H E H E KD ijD

ijD

i j

m

( ) ( ) ( ),

= −=∑

1

2

H M h a∈ ( , , )ω

( )⎩⎨⎧

∂==∇−

ωω

onPinfPxKdiv

i

ii)( ( )⎩⎨⎧

∂==∇−

ωω

onUinfUHdiv

i

ii

- CFD Stavanger 3617/12/200310/12/20038 -

New Approach (velocity function)

Min

ω

gi gi

V K K x P dxiD

i

→

= ∇∫( ) ( )1ω ω

1 ≤ ≤i m

H M h a∈ ( , , )ω

V H H U dxiD

i

→

= ∇∫( ) 1ω ω

J H V H V KD iD

iD

i

m

( ) ( ) ( )= −→ →

=∑

1

2

( )⎩⎨⎧

∂==∇−

ωω

onPinfPxKdiv

i

ii)( ( )⎩⎨⎧

∂==∇−

ωω

onUinfUHdiv

i

ii

general g if i , 1 ≤ ≤i m

- CFD Stavanger 3717/12/2003 10/12/200314 -

Special instances

inf

E K K x P P dxijD

i j( ) ( ) .= ∇ ∇∫12 ω

E H H U U dxH

ijD

i jij( ) .= ∇ ∇ =∫1

2 2ω

ω

1 ≤ ≤i j d,

H M h a∈ ( , , )ω

x i x i

1 ≤ ≤i dg x fi i i= =, 0

2

1,

)(2

)( ∑=

⎥⎦

⎤⎢⎣

⎡−=

d

ji

Dij

ijD KE

HHI

ω

( )⎩⎨⎧

∂==∇−

ωω

onPinfPxKdiv

i

ii)( ( )⎩⎨⎧

∂==∇−

ωω

onUinfUHdiv

i

ii

Convexquadraticfunction

- CFD Stavanger 3817/12/200310/12/200315 -

Special instances

∂∂

ωω

αβ

αβαβ

I HH

HE KD D( ) ( )= −

⎡

⎣⎢

⎤

⎦⎥2

1≤ ≤α β, d

∫ ∇∇=⇔=∂∂

ωβααβ

αβ ωdxPPxKH

HHID .)(10)(

Constraints are verifiedby H and I HD ( ) = 0

We find the classical tensor with linear B.C.

2

1,

)(2

)( ∑=

⎥⎦

⎤⎢⎣

⎡−=

d

ji

Dij

ijD KE

HHI

ω

- CFD Stavanger 3917/12/200310/12/200311 -

Proposition: The function is differentiable and we have the following expression for :

New Approach

Proposition: Cost functions are continuous.

1≤ ≤α β, d

( )DDD FJI ,,

Proposition: Minimization problems have solutions.

DI

- CFD Stavanger 4017/12/200310/12/200318 -

Properties: G-convergenceDefinition (Spagnolo):

K KG

ε → 0

∀ ∈ −f H 1( )ωIf Pε P0

where

( )⎩⎨⎧

∂==∇−ω

ω

ε

εε

onPinfPxKdiv

0)( ( )

⎩⎨⎧

∂==∇−ω

ωonP

infPxKdiv0

)(

0

00

)(10 ωH

- CFD Stavanger 4117/12/2003 10/12/200319 -

Properties: G-stability

Theorem: Let such that

),,( ωβαε MK ∈

Then:

For linear, confined or periodic boundary conditions

Assumption (A) is satisfied.

0KKG→ε

)(minarg)(minarg0),,(),,(

* HIHIK DMH

DahMH ωβαω

εε∈∈

=∈>∀

*0

* KKG→ε

(A)

Remark:

- CFD Stavanger 4217/12/2003 10/12/200320 -

Conclusion:

But

Properties: Upscaled permeabilities G-convergence

,Ll

=ε aggregation rate

hom*hom KK =

Homogenization theory:

constant tensor

G-stability:

,homKKG→ε

*hom

* KKG→ε

•

•

•

eff

GKK →*

ε

)(hom effKK =We have:

- CFD Stavanger 4317/12/2003

upscaling

•• a possible a possible strategy to tackle Buckleystrategy to tackle Buckley Leverett Leverett model model

using the velocity fieldusing the velocity field

extend the approach by optimization extend the approach by optimization andand controlcontrol

- CFD Stavanger 4417/12/2003

Conclusion

Stochastic model

Ensemble-based prediction

Buckley-Leverett model

New approach for upscaling

- CFD Stavanger 4517/12/2003

Darcy law

•• Continuum Continuum mechanicsmechanics:: at at a REVa REV located at located at : :

porosityporosity: ratio of : ratio of void void to to bulk bulk volumevolume permeabilitypermeability: : Darcy lawDarcy law REVREV

( )K x( )xΦ

x

- CFD Stavanger 4617/12/2003

Darcy law

•• Darcy lawDarcy law:: empirical law empirical law ( (DarcyDarcy in 1856) in 1856)

•• theoretical derivationtheoretical derivation:: Scheidegger Scheidegger, King, King Hubbert Hubbert,, Matheron Matheron

((heuristicheuristic))

Tartar Tartar ( (homogeneization theoryhomogeneization theory))

Stokes Stokes Darcy lawDarcy lawG⎯⎯→

- CFD Stavanger 4717/12/2003

Darcy law

•• Different scaleDifferent scale::

pore pore levellevel: Stokes : Stokes equationsequations lab lab: : measuresmeasures numerical cell numerical cell: : upscalingupscaling field field: : heterogeneityheterogeneity

Darcy law Darcy law Darcy law Darcy lawG⎯⎯→

- CFD Stavanger 4817/12/2003

Black-oil model

•• HypotesisHypotesis::

three three phases: 2 phases: 2 hydrocarbon hydrocarbon phases phases andand waterwater

hydrocarbon hydrocarbon system: 2 components system: 2 components a non-volatile a non-volatile oiloil a volatile a volatile gas gas soluble in the soluble in the oil oil phasephase

- CFD Stavanger 4917/12/2003

Black-oil model

•• PVT PVT behaviourbehaviour: formation volume : formation volume factorfactor

•• wherewhere:: volume of a volume of a fixed fixed mass mass at reservoirat reservoir conditionsconditions

volume of a volume of a fixed fixed mass mass at at stock tankstock tank conditions conditions

( )( )

( )( )

( )( )

; ;o dg g WRC RC RC

g wo WgSTC STCSTC

V V V VBo B B

V VV

+= = =

RCV

STCV

- CFD Stavanger 5017/12/2003

Black-oil model

•• Mass Mass transfer between oil and gas transfer between oil and gas phases:phases:

: : gasgas component in the component in the oil oil phase phase

: : oil oil component in the component in the oil oil phase phase

functions functions of the of the oil oil phase pressure phase pressure

d gS

o S T C

VR

V⎡ ⎤

= ⎢ ⎥⎣ ⎦

dgV

oV

- CFD Stavanger 5117/12/2003

Black-oil model

•• Thermo functions Thermo functions for for oiloil::

020406080

100120140160180200

0 100 200 300 400

P

Rs(

m3/

m3)

1

1,05

1,1

1,15

1,2

1,25

1,3

1,35

1,4

0 100 200 300 400

P (bars)

Bo

(Sm

3/m

3)

0,4

0,5

0,6

0,7

0,8

0,9

1

1,1

1,2

1,3

1,4

muo

(cP)

Bo

muo

- CFD Stavanger 5217/12/2003

Black-oil model

•• WaterWater::

•• oiloil::

•• gaz:gaz:

( ) 0t

Sw K krwd iv P w w gZB w B w w

∂ φ ρ∂ µ

⎡ ⎤⎛ ⎞ − ∇ + =⎜ ⎟ ⎢ ⎥⎝ ⎠ ⎣ ⎦

( ) 0t

=⎥⎦

⎤⎢⎣

⎡+∇−⎟

⎠⎞

⎜⎝⎛ ogZPo

oBoKkrodiv

BoSo ρ

µφ

∂∂

( )

( )

gt

0

Sg So KkrgRs div Pg g ZBg Bo Bg g

KkroRsdiv Po ogZBo o

∂ φ φ ρ∂ µ

ρµ

⎛ ⎞ ⎡ ⎤+ − ∇ +⎜ ⎟ ⎢ ⎥

⎝ ⎠ ⎣ ⎦⎡ ⎤

− ∇ + =⎢ ⎥⎣ ⎦

- CFD Stavanger 5317/12/2003

Black-oil model

•• saturation:saturation:

•• capillarycapillary pressures: pressures:

•• we obtainwe obtain 3 3 equations withequations with 3 3 unknowns unknowns::

1o w gS S S+ + =

w o cowp p p= −

g o c o gp p p= +

, ,o w g o bp S S if p p=

, ,o w s o bp S R if p p>

- CFD Stavanger 5417/12/2003

Black-oil model:boundary conditions

•• BoundariesBoundaries closed closed: no flux : no flux at at the the extreme cellsextreme cells aquiferaquifer: source : source term term in in corresponding cellscorresponding cells•• wellswells:: Dirichlet condition: Dirichlet condition: bottom bottom pressure pressure imposed imposed Neumann condition: production rate Neumann condition: production rate imposedimposed source source terms terms for for perforated cells perforated cells (PI)(PI)

- CFD Stavanger 5517/12/2003

Black-oil model: initial conditions

•• capillary and gravity equilibriumcapillary and gravity equilibrium

•• pressure pressure imposed imposed in in oil oil zone zone at at a a given depthgiven depth

•• oil oil pressure in all pressure in all cells and then cells and then Pc Pc curvescurves

- CFD Stavanger 5617/12/2003 10/12/200312 -

[ ]∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

αα

α β α βωω

α βω

I HH

E H E K

Ux

Ux

dxUx x

dx

Ux x

dx

DijD

ijD

i j

mi j j i

i j

( ) ( ) ( ) *,

= −+ −

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟=

∑∫∫

∫1

Ψ

Ψ

[ ]∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

αβ

α β β αω

α β β αω

α β β αω

I HH

E H E K

Ux

Ux

Ux

Ux

dx

Ux x

Ux x

dx

Ux x

Ux x

dx

DijD

ijD

i j

m

i j i j

j i j i

i j i j

( ) ( ) ( ) *,

= −

+⎛

⎝⎜

⎞

⎠⎟ +

+⎛

⎝⎜

⎞

⎠⎟ −

+⎛

⎝⎜

⎞

⎠⎟

⎛

⎝

⎜⎜⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟=

∑

∫

∫

∫1

Ψ Ψ

Ψ Ψ ⎟⎟

Si α β≠

where is solution of the local problems

and is solution of:

Ui

ψ i

Si α β=

New Approach

( )mi

oninfHdiv

i

ii ≤≤⎩⎨⎧

∂=Ψ=Ψ∇−

10 ω

ω