Ilya D. Mishev IMA Workshop on Compatible Discretizations May 11-16 14, 2004 Why Mixed Finite Elements are not used in the Petroleum Industry and what.

Ilya D. MishevIMA Workshop on Compatible Discretizations

May 11-16 14, 2004

Why Mixed Finite Elements are not used in the Petroleum Industry and what can we do

about it ?

2

Outline Introduction to Petroleum Industry Compositional model (Black Oil model)

– Black Oil is considered as a particular case of Compositional

General framework Examples

3

Introduction to Petroleum IndustryGeologic model

Cores

LogsReservoir model

interpretation

Seismic map

Analogs

4

Compositional ModelPhases - liquid (l), vapor (v), and aqueous (a) -

components - methane, ethane, propane, etc., -

Conservation of mass

Volume V.,,

,,1

11

jj

N

j

iji

N

j

jjTiTi

C

V

i

V

i

VtiW

PP

xSzW

NiR

vU

nU

, ,

n

j

i

i

i

SR

W

Uoverall concentration, component flow rate, sources and sinks,

saturation of phase j.

ij

i

j

xz

porosity,

molar density of phase j,mole fraction of component i,mole fraction of component i in phase j

C

P

NN phases,

components

5

Compositional Model - cont.

.,,,

,

, ljcj

jjj

rjj

PPavjPPP

DgPk

Kv

Darcy law (generalized)

j

jc

j

j

PP

,

v phase velocity,

phase pressure,

capillary pressure,

viscosity of phase j,

Dg

k

j

rj

K absolute permeability,

relative permeability of phase j,

mass density of phase j,

gravitational acceleration,

depth.

6

Compositional Model - cont.Volume balance laws - total volume balance

pore volume - total volume of fluids -

=

PV TV

“pressure equation”

i

N

avlT

N

NNVVVV

C),,,(

,

1 N

moles of comp i

ii

N

i i

TTP

TP

RN

V

t

P

t

V

t

V

PVPVC

U

N

1

),()(

7

Compositional Model - cont.Volume balance laws - liquid phase volume balance

liquid saturation volume of liquid -

=

PV lV

“saturation of oil

equation” ii

N

i i

lll

PlP

lPl

RN

V

t

P

P

VS

t

P

P

V

t

SV

PVPVSC

U

N

1

),,()(

The equations for the other phase saturations are similar.

8


TP

C

N

jiji

Ciii

VV

NiP

NiRt

N

P

,,1,0)(

,,1

1

1

, ,

NUK

U

Simplify - no capillary pressure, no saturation equations.

, ,

.

,,,1,

,,1

1

1

1

1

PPP

N

iiiP

CiiP

N

jjiji

Ciii

fPN

PANifPN

tNiR

t

N

C

C

fN

UKUK

RUN

U

Linearize (typically first discretize then linearize)

9


fuuK

01 p

0)(

pfpK

01

vvuK p ,01

vvuK p

01

vvuK pa

0)(1

vvuK ap

We have to discretize ., pN i

.

,1

PT f

Pa

PA

N

fN

UK fN

UK 111

AP

A

x

x

Wang, Yotov, Wheeler, et. al introduced

(possible problems for non smooth solution)ii Pu

10

Grids

pinchouts

11

General framework

Given general cell centered grid– build dual grid to approximate the fluxes,

– choose approximation space for the pressure,

– define local approximation of the flux on the dual volume,

– exclude fluxes to get the finite volume method

12

Model problem written as a system

Find such that (primal dual MFEM)

General framework

fuuK

01 p

.

01

Qqqfq

p

u

VvvvuK

,

Pp Uu ),(

.

,01

hK K

K K

i i

h

Qqqfq

p

hhhh

hhF

hhhh

i i

nu

VvvvuK

hhhh p P Uu ),( Find such that

13

ExamplesRectangular grid / full tensor (Ware, Parrott, and Rogers)

Dual grid - rectangles, - piecewise constants, - piecewise constant vectors with continuous normals

hh QP ,

hh VU ,

ij

k l

Basis vectors kljlikij eeee ,,,

klkljljlikikijij vvvv eeeev

kle

,,

,,

kl

l

kljl

l

jl

ik

l

ikij

l

ij

kljl

ikij

vv

vv

nvnv

nvnv

M., “Analysis of a new Mixed Finite Volume

Method”,

Comp. Methods Appl. Math. V. 3, 2003

14

Examples (Ware, Parrott, and Rogers)

V V

h

Tlkljkijihhh

pqmnpqmn

klklklklkljljlklikikklijij

jljlklkljljljljlikikjlijij

ikikklklikjljlikikikikijij

ijijklklijjljlijikikijijij

kljlikijhhE E

Ehhh

f

pppppppp

t

pptutututu

pptutututu

pptutututu

pptutututu

ph

nu

pTpu

eeK

eeeevnvvuK

,,,,,

or,where

,0

,0

,0

,0

,,,,0][

1,

,,,,

,,,,

,,,,

,,,,

1

)()(lim][0t0,t

EEE txptxpp nn

15


neighborsji

hhh

qq

pChpp22

h1,

,2,1,1,0

)(~||q|| where

,|||||||||||||| uuuTheorem:

Numerical example:

).sin()sin(),(

,21,21

,4

,)cos()sin(

)sin()cos(,

0

0),(

,),(),(

3222

5221

2

1

yxyxp

xyxdyyxd

Rd

dyxD

RyxRDyx T

K

16


Errors

8 16 32 64 128 256

Mesh points

Erro

r

-error of the pressure, - error of the pressure, -error of the flux, - error of the flux

2L

2L

1H

divH

17

ExamplesVoronoi/Donald mesh / full tensor

18

Unstructured (Voronoi) grids

. onflux the of component normal the eapproximat to constants piecewise

constants, piecewise

hK Ki i

E

h

h

hhhh

hhhE E

Ehhh

D

P

Pqqfq

ph

U

nu

UvnvvuK

.

,0][1

EIK )(xk - scalar coefficient

,2,1pChpp h

(M. “Finite Volume Methods on Voronoi Meshes”,Numer. Meth. PDE,

Herbin, et. al.)

What about hu approximation

19

Unstructured (Voronoi) grids

22

),(,22/1

),(,

edgeE E

Edivhdivhh pCh nuuuu

E

.,22

,0,2,1,0

edgeE D

Ehh

E

pChpp nuuuu

For grids with extra regularity

Hypothesis: The approximation of

could be improved with post-processing.hu

20

Examples (Voronoi/Donald mesh / full tensor)

Dual grid - triangles, - linear piecewise continuous functions, - piecewise constants on Voronoi volumes,

- piecewise constants on triangles with continuous normals

hP

hQ

hh VU ,

jkjkikikijij vvv eeev

,,, jl

l

jkik

l

ikij

l

ij

jlikij

vvv nvnvnv

jlikij

jlikij

jlikij

l

jkjk

l

ikjk

l

ijjk

l

jkik

l

ikik

l

ijik

l

jkij

l

ikij

l

ijij

,1,0,0

,0,1,0

,0,0,1

nenene

nenene

nenene

M. “A New Flexible Mixed Finite Volume Method”, submitted

21

Discrete problem:

Find such that

Examples (Voronoi/Donald mesh / full tensor)

)||||(||||||||||

:Theorem

.,0

,,For

.

,0

,2,1,1,0

1

hK K

1

i i

pChpp

Pqqfq

p

hh

hhhh

klikijh

hhhh

hhhhhh

uuu

MpTuMpTu

eeev

nu

UvvvuK

hhhh Pp Uu ),(

22

Error estimates

,,),(

,),(),(

hhhhhh

hhhhhhhhh

Qqqfpc

gpba

u

Vvvvvu

Find such that hhhh Pp Uu ),(

.,0),(,,||||||||

),(sup)

,,0),(,,||||||||

),(sup)

,||||||||

),(sup)

(.,.)(.,.),(.,.),

01

01

0

hhhhhhhhQhh

hhh

Qq

hhhhhhhhPhh

hhh

Pp

hh

hhh

hhh

aq

quciii

Qqqcp

pbii

inf-supa

i

cba

hhhhhh

hhhhhh

hhhhhh

UuvuVvVv

uUuUv

v

vu

vu

VUu

VVv

VUVvUu

inf

inf

conditions ) ( LBBinf

and continuous

.)dim(dim( hhhh Q )dim(V )Pdim()U

....||||inf||inf|||||| hhhhhhh Phqhhh qpCpphPVVPV |||| vuuu v

23

Extra

24

Black Oil Model

ReservoirConditions

Phases Components

Gas Oil

Water

++

++

+ +

Reservoir Conditions

Standard Conditions

25

Black Oil Model

Phases - liquid, vapor, aqueous

components - oil, gas, water

l v a

o x

g x x

w x

C/P

,,

,

,,

aaw

vvvglllgg

llo

xxvU

vvUvU

26

Black Oil Model



l v a

o x

w x

C/P

,

,

aaw

llo

vU

vU

If only 2 phases exist

total velocity, global pressure

(Chavent, Jaffre)

ft

GP alg

v

vvvKv ),(

No mass transfer

between the phase

27

Black Oil Model



l v a

o x x

g x x x

w x

C/P

,,

,

,,,

,,

aaw

aaagvvvglllgg

vvvollloo

xxxxx

vUvvvU

vvU

28

Examples

pressure

pressurepressure to be eliminatedvelocity

hP

Quadrilateral mesh / full tensor (M. Edwards et. al.)Dual grid - cell-centers connected with the middles of the edges/faces

- piecewise linears (nonconforming space)- piecewise constants on the cell

hP

hQ

hV

hU-piecewise constants with

continuous normals (4 dof),- piecewise constants (8 dof)

29

Examples (quads)

Q

V0

qqfq

p

u

vvKvu

.Q

,V0

hK K

K

4

1 K

i i

h

hhhh

hhF j

hhhh

qqfq

pi

ji

nu

vvKvu

30

Example (quads)

.,,,,

.,i.e,,,,,,,,for

matrices,88,48,0

systemlinear thesolve

,,,,,,,For

.

,V0

,1

,1

,,,,,,,,

K

4

1 Kh

Tkljihhh

kljlikijhT

kljlikijh

h

hh

ljllklkikkkljjljijiikiijh

klkljljlikikijijh

hhF j

hhhh

pppp

ppppuuuu

uuuu

pi

ji

pTpu

pu

MNp

pMNu

ffffffffv

eeeeu

vvKvu

Ilya D. Mishev IMA Workshop on Compatible Discretizations May 11-16 14, 2004 Why Mixed Finite Elements are not used in the Petroleum Industry and what.

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