Upsc Overview 04

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Upscaling of Geocellular Models for

Flow Simulation

Louis J. Durlofsky

Department of Petroleum Engineering Stanford Uni!ersity

"#e!ron$e%aco E$" San &amon "'



2

'cknowledgments

• Yuguang Chen (Stanford University)

• Mathieu Prevost (now at Total)

• Xian-Huan en (ChevronTe!a"o)

• Yal"hin #fendiev (Te!as $%M)

(p#oto )y Eric Flodin*



3

• &ssues and e!isting te"hni'ues

• $dative lo"al-gloal us"aling

• *elo"ity re"onstru"tion and +ultis"ale solution

• ,eneralied "onve"tion-diffusion transort +odel

• Us"aling and flow-ased grids (./ unstru"tured)

• 0utstanding issues and su++ary

+utline



4

&e,uirements-"#allenges for Upscaling

• $""ura"y % 1oustness 2 1etain geologi"al realis+ in flow si+ulation

2 *alid for different tyes of reservoir heterogeneity

2 $li"ale for varying flow s"enarios (well "onditions)• #ffi"ien"y

&n3e"tor

Produ"er

&n3e"tor

Produ"er



5

E%isting Upscaling $ec#ni,ues

• Single-hase us"aling4 flow (Q 5 ) 2 6o"al and gloal te"hni'ues (k → k7 or T 7)

• Multihase us"aling4 transort (oil "ut)

2 Pseudo relative er+eaility +odel (k r3 → k r37)

• 8Multis"ale9 +odeling

2 Us"aling of flow (ressure e'uation) 2 :ine s"ale solution of transort (saturation e'uation)



6

Local Upscaling to "alculate k

• 6o"al ;Cs assu+ed4 "onstant ressure differen"e• &nsuffi"ient for "aturing large-s"ale "onne"tivity in

highly heterogeneous reservoirs

or

6o"al #!tended 6o"al

Solve ∇⋅(k⋅∇ p)<= over lo"al region

for "oarse s"ale k 7 or T 7

,loal do+ain



' /ew 'pproac#

• Standard lo"al us"aling +ethods unsuitale forhighly heterogeneous reservoirs

• ,loal us"aling +ethods e!ist> ut re'uire gloal

fine s"ale solutions (single-hase) and oti+iation

• ?ew aroa"h uses gloal "oarse s"ale solutions to

deter+ine aroriate oundary "onditions for lo"al

k7 or T 7 "al"ulations

2 #ffi"iently "atures effe"ts of large s"ale flow

2 $voids gloal fine s"ale si+ulation

$dative 6o"al-,loal Us"aling



'dapti!e Local0Glo)al Upscaling ('LG*

ell-driven gloal "oarse flow

• Thresholding4 6o"al "al"ulations only in high-flow

regions ("o+utational effi"ien"y)

y

x

Coarse s"ale roerties

k7 or T 7 and us"aled well inde!

6o"al fine s"ale "al"ulation

&nterolated ressure

gives 6o"al ;Cs

Coarse ressure

6o"al fine s"ale "al"ulation

&nterolated ressure

gives lo"al ;Cs

Coarse ressure



$#res#olding in 'LG

Per+eaility Strea+lines Coarse lo"@s

1egions for

6o"al "al"ulations

• $voids nonhysi"al "oarse s"ale roerties (T 7<q "5∆ p ")

• Coarse s"ale roerties effi"iently adated to a givenflow s"enario

• &dentify high-flow region> A ε (ε ∼ =B)|q c|

|q c|max



Multiscale Modeling

( ) 0=∇⋅⋅∇ c p*

k 0)( =⋅∇+∂

∂S

t

S v

• Solve flow on "oarse s"ale> re"onstru"t fines"ale !> solve transort on fine s"ale

• $"tive resear"h area in reservoir si+ulation4

2 /ual +esh +ethod (:/)4 1a+D % Eillough (FF)>

,uGrillot % *erdiDre (FF)> ,autier et al. (FFF)

2 Multis"ale :#M4 Hou % u (FFI)

2 Multis"ale :*M4 Jenny> 6ee % T"helei (K==.> K==L)



11

&econstruction of Fine Scale 1elocity

( ) 0=∇⋅⋅∇

c

p*

k 0)( =⋅∇+∂

∂

S t

S

v

Upscaling,global

coarse scalefow

Solve local ne scale

∇⋅(k ⋅∇ p)=0

Partition coarse

fux to ne

scale

Reconstructe

ne scalev

(ownscaling)

• 1eadily erfor+ed in us"aling fra+ewor@



12

&esults2 Performance of 'LG

$veraged fine

Pressure /istriution

Coarse4 e!tended lo"al

Coarse4 $dative lo"al-gloal

Channelied layer (F) fro+ =th SP# CSP

:low rate for se"ified

ressure

• :ine s"ale4 Q < K=BN

• #!tended T 74 Q < IBI

• $6, us"aling4 Q < K=B=3.3

4.3

53.3

54.3

63.3

64.3

3 5 6 7 89teration

:

Q (Fine scale* ; 63.<=

'LG Error2 8>

E%tended local

Error2 =?>

Us"aling KK= × N= → KK × N



13

&esults2 Multiple "#anneli@ed Layers

#!tended lo"al T 7 $dative lo"al-gloal T 7

=th SP# CSP



14

'not#er "#anneli@ed System

== realiations

K= × K= → KL × KL

'LG T T A /BSUk only



15

&esults2 Multiple &eali@ations

• == realiations "onditioned to seis+i" and well data

• 0il-water flow> M<• &n3e"tor4 in3e"tion rate "onstraint> Produ"er4 otto+ hole

ressure "onstraint

• Us"aling4 == × ==→

= × =

!00 reali"ations

Ti+e (days)

; H P

( P S & $ )

:ine s"ale

mean

#0$ con%& int&



16

&esults2 Multiple &eali@ations

Coarse4 Purely lo"al us"aling Coarse4 $dative lo"al-gloal

Ti+e (days)

; H P ( P

S & $ )

'ean (coarse scale)

#0$ con%& int& (coarsescale)

Ti+e (days)

; H P ( P

S & $ )

'ean (ne scale)

#0$ con%& int& (ne scale)



17

&esults (Fo*2 "#anneli@ed System

0 × 0 → ×

Fractional Flow "ur!e

3

3.6

3.8

3.=

3.<

5

5.6

3 3.6 3.8 3.= 3.< 5P19

Fo

:ine s"ale

$6, T 7

#!tended lo"al T 7 :low rates

• :ine s"ale4 Q < NB.=

• #!tended T 74 Q < BI

• $6, us"aling4 Q < NBKN

0il "ut fro+ re"onstru"tion



18

&esults (Sw*2 "#anneli@ed System

*ine scale Sw (0 ×

0)

Reconstructe Sw %rom

extene local T + ( × )

3.4

5.3

3.3

Reconstructe Sw

%rom

-. T + ( × )

*ine scale

streamlines



19

&esults for 7D Systems (SPE 53*

43 c#anneli@ed layers 7 wells pinC;5 pprod;3

$ypical layers

Us"ale fro+

N=×KK=×= → K×LL×=

using different +ethods



20

&esults for Bell Flow &ates 0 7D

$verage errors

• k7 only4 L.O

• #!tended T * ?SU4 KIO

• $dative lo"al-gloal4 .BO



21

&esults for $ransport (Multiscale* 0 7D

fine s"ale

$6, T 7

lo"al T 7 w5nw

standard k7

Produ"er

:o

P*&

fine s"ale

$6, T 7

standard k7

Produ"er K

lo"al T 7 w5nw:o

P*&

• Quality of transort "al"ulation deends on the

a""ura"y of the us"aling



22

1elocity &econstruction !ersus

Su)grid Modeling

• Multis"ale +ethods "arry fine and "oarse grid

infor+ation over the entire si+ulation

• Sugrid +odeling +ethods "ature effe"ts of fine

grid velo"ity via us"aled transort fun"tions4

- Pseudoiation te"hni'ues

- Modeling of higher +o+ents

- ,eneralied "onve"tion-diffusion +odel



23

•Coarse s"ale ressure and saturation e'uations of sa+e

for+ as fine s"ale e'uations

• Pseudo fun"tions +ay vary in ea"h lo"@ and +ay e

dire"tional (even for single set of k rj in fine s"ale +odel)

Pseudo &elati!e Permea)ility Models

( ) , 0),( **=∇⋅⋅∇

cc pS k xλ 0),( *=⋅∇+

∂

∂ cc

S t

S xF

)()(

)()(

),( ,=),(

**

*c*

c****

c*

oirowirw

wirwi

i

c

iio

ro

w

rw

μk + μk

μk S f

S f F μ

k k S

=

=+ xvx µ

λ

7 ⇒ us"aled fun"tion

" ⇒ "oarse s"ale p, S



24

Generali@ed "on!ection0Diffusion

Su)grid Model for $wo0P#ase Flow

• Pseudo relative er+eaility des"rition is "onvenient ut

in"o+lete> additional fun"tionality re'uired in general

• ,eneralied "onve"tion-diffusion +odel introdu"es new

"oarse s"ale ter+s

- :or+ derives fro+ volu+e averaging and

ho+ogeniation ro"edures

- Method is lo"al> avoids need to aro!i+ate

- Shares so+e si+ilarities with earlier sto"hasti"

aroa"hes of 6enor+and % "owor@ers (FF> FFF)

)()( yx ji vv ′′



25

•Coarse s"ale saturation e'uation4

Generali@ed "on!ection0Diffusion Model

( )cccc

S S S t

S ∇⋅⋅∇=⋅∇+

∂

∂),(),( xDxG

),()(),( cccc S S f S xmvxG +=

• Coarse s"ale ressure e'uation4

cccc S S S W S ∇⋅++= ),(),()( 21

*xWxλ λ

(+odified "onve"tion m

and diffusion D ter+s)

(+odified for+ for total

+oility> ∇Sc deenden"e)

8ri+itive9 ter+

,C/ ter+

( ) 0),( **=∇⋅⋅∇

cc pS k xλ



26

• D and B2 "o+uted over urely lo"al do+ain4

"alculation of G"D Functions

p = 1

S = 1

p = 0

)()()( S f S f S S vvD −=∇⋅

• m and W 1 "o+uted using e!tended lo"al do+ain4

(D and B2 a""ount for

lo"al sugrid effe"ts)

S S S f S f S ∇⋅−−= )()()( )( Dvvm

(m and W - sugrid effe"ts due

to longer range intera"tions)

target coarse )lock



27

Solution Procedure

• ,enerate fine +odel (== × ==) of res"ried ara+eters

• :or+ unifor+ "oarse grid (= × =) and "o+ute k7 and

dire"tional ,C/ fun"tions for ea"h "oarse lo"@

• Co+ute dire"tional seudo relative er+eailities via total

+oility (Stone-tye) +ethod for ea"h lo"@

• Solve saturation e'uation using se"ond order T*/ s"he+e>

first order +ethod for si+ulations with seudo k rj

fine grid4 l x≥ l z

L x < Lz



28

+il "uts for M ;5 Simulations

• G"D and pseudo +odels agree "losely with fine scale

(seudoiation te"hni'ue sele"ted on this asis)

l % ; 3.64 l @; 3.35σ

;6 side to side flow

533 % 533

53 % 53 (G"D*

53 % 53 (primiti!e*

53 % 53 (pseudo*

+ i l "

u t

P19



29

&esults for $wo0Point Geostatistics

x =0.05, y = 0.01, logk = 2.0

533%533→

53%53 Side Flow

53

3

4

• /iffusive effe"ts only



30

&esults for $wo0Point Geostatistics ("ontd*

x =0.5, y = 0.05, logk = 2.0

533%533 → 53%53 Side Flow

53

3

4

• Per+eaility with longer "orrelation length



31

Effect of 1arying Glo)al "s (M ;5*

p = 1

S = 1

p = 0

3 ≤ t ≤ 3.< P19

p = 1

S = 1 t 3.< P19

p = 0

l % ; 3.64 l @; 3.35σ

;6

+ i l " u t

P19

533 % 533

53 % 53 (G"D*

53 % 53 (primiti!e*

53 % 53 (pseudo*

l % ; 3.64 l @; 3.35σ

;6



32

"orner to "orner Flow (M ; 4*

533 % 533

53 % 53 (G"D*

53 % 53 (pseudo*

+ i l

" u t

P19

$ o t a l

& a t e

P19

l % ; 3.6 l @; 3.36σ

;5.4

• Pseudo +odel shows "onsiderale error> ,C/ +odel

rovides "o+arale agree+ent as in side to side flow



33

Effect of 1arying Glo)al "s (M ; 4*

533 % 533

53 % 53 (G"D*

53 % 53 (pseudo*

+ i l "

u t

P19

$ o t a l &

a t e

P19

l % ; 3.6 l @; 3.36σ

;5.4

• Pseudo +odel overredi"ts oil re"overy> ,C/ +odel

in "lose agree+ent



34

Effect of 1arying Glo)al "s (M ; 4*

l % ; 3.4

l @; 3.36

σ

;5.4

• ,C/ +odel underredi"ts ea@ in oil "ut> otherwise

tra"@s fine grid solution

533 % 533

53 % 53 (G"D*

53 % 53 (pseudo*

+ i l "

u t

P19

$ o t a l

& a t e

P19



35

),( * cc S S ∇λ

( ) 0 **=∇⋅⋅∇

c pk λ Coarse s"ale flow4

Pseudo fun"tions4

,C/ +odel4

T 7 fro+ $6,> deendent on gloal flow

λ 7> m(S c) and D(S c)

• Consisten"y etween T 7 and λ 7 i+ortant for highly

heterogeneous syste+s

"om)ine G"D wit# 'LG T Upscaling

)( * cS λ



36

'LG A Su)grid Model for $ransport (G"D*

t 3.= P19 t ≥ 3.= P19

• Stanford * +odel (layer )

• Us"aling4 ==×.= → =×.

• Transort solved on "oarse s"ale

flow rate oil cut



37

flow si+ulation flow si+ulation

us"aling

gridding

diagnosti"

GocadGocadinterfa"e

"oarse +odel

infoB +as

fine +odel

Unstructured Modeling 0 Borkflow



38

/umerical Discreti@ation $ec#ni,ue

• C*:# +ethod4

2 6o"ally "onservativeR flu! on a fa"e e!ressed as linear

"o+ination of ressures

2 Multile oint and two oint flu! aro!i+ations

• /ifferent us"aling te"hni'ues for MP:$ and TP:$

i j

k

qij < a pi p j " pk BBB or qij < T ij ( pi - p j )

Primal anddual grids

( *



39

7D $ransmissi)ility Upscaling ($PF'*

/ual "ells Pri+al grid

"onne"tion p<

p<=fitted

e!tended

regions

"ell j "ell i

T ij * = -qij A

p j A pi A-



40

Grid Generation2 Parameters

• Se"ify flow-diagnosti"

• ,rid ase"t ratio

• ,rid resolution "onstraint4

2 &nfor+ation +a (flow rate> t ))

2 P a and P b > sa and sb

2 N (nu+er of nodes)

min max

1

property

cumulative frequency

a b

Pa

Pb

min max

Sa

Sb

property

resolution constraint

a b



41

velocity

grid density

Upscaledk*

Unstructured Gridding and Upscaling

(fro+ Prevost> K==.)



42

Flow0ased Upscaling2 Layered System

• 6ayered syste+4 K== ! == ! = "ells

• Us"ale er+eaility and trans+issiility

• 1un k 7-MP:$ and T 7-TP:$ for M <

• Co+ute errors in Q ∆ p and L1 nor+ of ! "

p<= p< =B

=BK

Fl d U li & lt



43

Flow0ased Upscaling2 &esults

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

PVI

Reference (fine)TPFAMPFA

! ! < K nodes N ! N ! . < LN nodes

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Fw

error in !" error in Q5∆ p

TP:$ IBNO -BKO

MP:$ IBFO -KBKO

error in !" error in Q5∆ p

TP:$ CNBO -HBFO

MP:$ KCB.O -.CBIO

PVI

Fw

(fro+ MB Prevost> K==.)



44

Layered &eser!oir2 Flow &ate 'daptation

• ,rid density fro+

flow ratelog * grid sie

sb

sa

= =BK =BL =BN =B =

=BK

=BL

=BN

=B

P*&

: w

referen"eunifor+ "oarse (?<K!!<KL)

flow-rate adated (?<.FL)

Q"<=BK

Q"<=BFF

PVI

Fw

• :low results

(fro+ Prevost> K==.)

(Qf < B=)



45

Summary

• Us"aling is re'uired to generate realisti" "oarse

s"ale +odels for reservoir si+ulation

• /es"ried and alied a new adative lo"al-gloal

+ethod for "o+uting T *

• &llustrated use of $6, us"aling in "on3un"tion with

+ultis"ale +odeling

• /es"ried ,C/ +ethod for us"aling of transort• Presented aroa"hes for flow-ased gridding and

us"aling for ./ unstru"tured syste+s



Future Directions

• Hyridiation of various us"aling te"hni'ues(eBgB> flow-ased gridding $6, us"aling)

• :urther develo+ent for ./ unstru"tured syste+s

• 6in@age of single-hase gridding and us"aling

aroa"hes with two-hase us"aling +ethods

• /yna+i" udating of grid and "oarse roerties

• #rror +odeling

Documents

Upsc Overview 04