[PPT] - Upscaling dissolution dissolution Upscaling mechanisms in porous PowerPoint Presentation

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M. Quintard

Upscaling Upscaling “dissolution” “dissolution” mechanisms in porous media mechanisms in porous media

M. Quintard

Financialsupport:IFP,CNRS/INSU/PNRH,SNECMA,DGA

Institut deMécanique desFluides deToulouse

F.Golfier,Y.Aspa,C.Cohen,G.Vignoles,R.Lenormand,B.Bazin,D.Ding,S.

Véran,F.Fichot,J.Belloni,B.Goyeau,D.Gobin,C.Pierre,F.Plouraboué

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Outline Outline Outline

Background Pore-scale model and effective surface Darcy-scale models Stability Large-scale models? Conclusions Background Pore-scale model and effective surface Darcy-scale models Stability Large-scale models? Conclusions

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Introduction Introduction Introduction

Dissolution:

geochemistry, karsts, salt mines, NAPL, petrol. Engng, aerospace industry…

Problems:

– Multiple-scale analysis – History effects – Instabilities

Dissolution:

geochemistry, karsts, salt mines, NAPL, petrol. Engng, aerospace industry…

Problems:

– Multiple-scale analysis – History effects – Instabilities

HaLongBay Igue de Planagrèze

Daccordetal.,1993

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Example 1: acidizing treatment Example 1: Example 1: acidizing acidizing treatment treatment

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Example 2: ablation of composite structures Example 2: ablation of composite Example 2: ablation of composite structures structures

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Upscaling Surface Heterogeneities: The concept of Effective Surface Upscaling Upscaling Surface Surface Heterogeneities Heterogeneities: : The The concept concept of

f Effective Surface

Effective Surface

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Non-ablative case: various approaches Non Non-

ablative case: various approaches

ablative case: various approaches

micro micro meso meso micro meso !"#

Effectivesurface,

effectiveBC(jump conditions) Mesoscale modelling,GTE

Effectivesurface,effectiveBC

Domain decomposition DirectSimulation

micro

c c + ɶ

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Extension of Wood et al. (2000) Extension of Wood et al. (2000) Extension of Wood et al. (2000)

Flow:Blasius

. . inthefluiddomain . ( ) at ( ) attopofB.L. ( ) ( ) c D c D c k c c y h C c x x c x x l

γκ

∇ = ∇ ∇ − ∇ = Σ = = = = = + u n x

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Extension of Wood et al. (2000) Extension of Wood et al. (2000) Extension of Wood et al. (2000)

with . . inthefluiddomain . ( ) at and . . inthefluiddomain . at c c c c D c D c k c c D c D c k c k c k c

γκ γκ Σ Σ

= + ∇ = ∇ ∇ − ∇ = Σ ∇ = ∇ ∇ − ∇ = + − Σ u n x u n ɶ ɶ ɶ ɶ ɶ ɶ ɶ

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Extension of Wood et al. (2000) Extension of Wood et al. (2000) Extension of Wood et al. (2000)

( ) ... with . . inthefluiddomain . at c s c s D s D s k s k k s

γκ Σ

= + ∇ = ∇ ∇ − ∇ = + − Σ x u n ɶ ɶ ɶ

eff

k k k s

Σ Σ

= + ɶ

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Results for circular patches (far from entrance region) Results for circular patches (far from Results for circular patches (far from entrance region) entrance region)

/ Da kl D =

/ 3 / 1

m f

K k k Sc D ν = = = = ɶ

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Results Results Results

two limit cases

– Da <<1, c(x) = C0 keff = 〈k〉 – Da >>1, keff = k* (harmonic mean of the reactivities)

general case

– influence of geometry – slight influence of Re (for low Re… )

two limit cases

– Da <<1, c(x) = C0 keff = 〈k〉 – Da >>1, keff = k* (harmonic mean of the reactivities)

general case

– influence of geometry – slight influence of Re (for low Re… )

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Ablative case: transient

DNS

Ablative case: transient Ablative case: transient

DNS

DNS

Ablation leads to non-differentiable surfaces Limits of ALE and phase field methods

adapted VOF method (Aspa, 2006)

Ablation leads to non-differentiable surfaces Limits of ALE and phase field methods

adapted VOF method (Aspa, 2006)

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Example 1: steady-state surface Example 1: steady Example 1: steady-

state surface

state surface

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Example 2: porous composites Example 2: porous composites Example 2: porous composites

Weaved Layered

Da≈ 61082 Da≈ 3

Note:T changesthe“diffusion”coefficient

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Keff ? case Da<<1 K Keff

eff ? case Da<<1

? case Da<<1

Projected Areas: Am,p and

Af,p

Steady-state ablation:

uniform velocity implies

Projected Areas: Am,p and

Af,p

Steady-state ablation:

uniform velocity implies

ɶ

, , , f f m f m f p m p f p

A A A k k k A A A = ⇒ =

eff m

k k k = ≈

cos( ) cos( )

f f m m

k k ξ θ ξ θ = ⇒

t

k

=

≠

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Keff ? K Keff

eff ?

?

Limit Cases: simple

models

– Da<<1, max(k) – Da>>1, k-harmonic mean

Intermediate Da

complex simulations

Limit Cases: simple

models

– Da<<1, max(k) – Da>>1, k-harmonic mean

Intermediate Da

complex simulations

Simplified model

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V

pore8scale to Darcy8scale Darcy8scale to core8scale

η

porousmedium

wormhole

ω

V∞

β

σ

Dissolution: Darcy-scale (core-scale) models? Dissolution: Darcy Dissolution: Darcy-

scale (core

scale (core-

scale)

scale) models? models?

Pore-scale: non-local

effects (space and time)

Pore-scale: non-local

effects (space and time)

DirectSimulation:Bekri etal.

(1995),Mercet (2000),Zhanget Smith(2001),…

-etworkmodels (Fredd,Hoefner

etal.)

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Target 1: Dissolution Instabilities (Wormholing) Target 1: Dissolution Instabilities Target 1: Dissolution Instabilities ( (Wormholing Wormholing) )

water"aCl system (Zarcone etal.) HClcalcite system (Fredd etal.,SPE) FLOWRATE

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Target 2: “Optimum flow rate” Target 2: “Optimum flow rate” Target 2: “Optimum flow rate”

1 10 1 10 100 1000 injection flux rate (cm3/hour) injected volume to breakthrough (V/VP) salt concentration of 150g/l 0.00E+00 2.00E-01 4.00E-01 6.00E-01 8.00E-01 1.00E+00 0.00E+00 1.00E+00 2.00E+00 3.00E+00 4.00E+00 5.00E+00 injected volume (V/VP) dimensionless length (L/Lm) Q= 5 cm3/heure Q = 10 cm3/heure Q = 30 cm3/heure Q= 50 cm3/heure Q= 100 cm3/heure Q= 150 cm3/heure Q= 200 cm3/heure Q= 250 cm3/heure

OPTIMUM FLOWRATE

"aCL Concentrationof150g/l

Optimuminjectionrate: minimuminjectedacidvolumetobreakthrough Qopt =f(lengthcore,CNaCl ...)

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Simplified Pore-scale problem Simplified Pore Simplified Pore-

scale problem

scale problem

( ) ( )

. . . . inthefluiddomain at

eq

S

c c D c t c C Aβσ − + ∂ + ∇ = ∇ ∇ ∂ = v "ote(binarycase): .

0at

if 1

eq

D c kc c C A Da

βσ

− ∇ = ⇔ = ≈ >> n

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Local equilibrium dissolution:

produces sharp fronts

LNE: Heuristic model classically used in Chemical Engineering

(discussion in Quintard & Whitaker, 1994, 1999)

Local equilibrium dissolution:

produces sharp fronts

LNE: Heuristic model classically used in Chemical Engineering

(discussion in Quintard & Whitaker, 1994, 1999)

β8phase yβ x rβ V σ8phase

nβσ

∫

+ = =

β

β β β β β V A A A

dV c V c C ) ( 1 y x

x

Darcy-scale: Local Non- Equilibrium Models Darcy Darcy-

scale: Local Non

scale: Local Non-

Equilibrium Models

Equilibrium Models

( )

*

. . . additionalterms C C C C t ε α ∂ + ∇ = ∇ ∇ − ∂ + V D +othermacro8scaleequations

A A eq

C c C

β β β

= =

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Upscaling (framework) Upscaling Upscaling (framework) (framework)

Deviations Coupled pore-scale and averaged

equations

Deviations Coupled pore-scale and averaged

equations

1 β β β β

ε − = + v V v ɶ

A A A

c C c

β β β

= + ɶ

( )

1 1

A

A A A A A A A A

C C c dA D C dA c c t V V

βσ βσ

β β β β β β β βσ β β β β βσ

ε ε

           

    ∂     + ∇⋅ + ⋅ − = ∇⋅ ∇ + −∇⋅   ∂      

∫ ∫

V n v w n v ɶ ɶ ɶ

( )

1 1 1

1

A A

A A A A A A A A A c D c

c C c c D c D c dA D c dA t V V

βσ βσ β β β

β β β β β β β β β β β βσ β βσ β

ε ε ε

− − − ⋅∇ ∇⋅ ∇

  ∂   + ⋅∇ + ⋅∇ − ∇⋅ = ∇⋅ ∇ − ∇⋅ − ⋅ ∇ ∂    

∫ ∫

v

v v v n n

ɶ ≪ ɶ ≪

ɶ ɶ ɶ ɶ ɶ ɶ ɶ ɶ

+…..

(QuintardandWhitaker,1999;Golfier etal.,2001)

+

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Upscaling : problems Upscaling Upscaling : problems : problems

Quasi-steady dissolution: terms like

may be neglected in the problem for the deviations

“Closure”= approximate solution of

the coupled equations

– Quasi-steady solution? – But…historical effect remains through the interface evolution

Quasi-steady dissolution: terms like

may be neglected in the problem for the deviations

“Closure”= approximate solution of

the coupled equations

– Quasi-steady solution? – But…historical effect remains through the interface evolution

( )

A βσ β

⋅ − n v w

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A simple example: the tube problem (Graetz’s problem, Pe>>1) A simple example: the tube A simple example: the tube problem problem (

(Graetz’s Graetz’s problem, problem, Pe Pe>>1) >>1)

Classical Problem Classical Problem Dissolution Dissolution

z z

c=Ceq log Sh ( )

fullydev.region entr.reg.

( )

, , .. V D α x

z z

c=Ceq log Sh ( )

fullydev.region entr.reg.

(

)

, , , .. t V D α x

Non8localinspaceandtime?

…see Golfier etal.(2001),Pierreetal.(2005)

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Representation of the deviations: Darcy-Scale equation: + cell problems→

→ → →“effective” properties

Representation of the deviations: Darcy-Scale equation: + cell problems→

→ → →“effective” properties

...

A A A A A

c c c c s c

β β β β β β β β β β

= − = ⋅∇ − + b ɶ 2D and 3D cases (MQ & SW, 94, 99, FG et al., 2002) 2D and 3D cases ( 2D and 3D cases (MQ & SW, 94, MQ & SW, 94, 99, FG et al., 2002) 99, FG et al., 2002)

( ) ( ) ( )

A A A A A A

c c c c t c c

β β β β β β β β β β β β β β β β β

ε α

∗

∂   〈 〉 + ∇⋅ 〈 〉〈 〉 − ∇⋅ 〈 〉 − ⋅∇〈 〉   ∂ = ∇⋅ ⋅∇〈 〉 − 〈 〉 v d u D

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0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 0.0002 0.0004 0.0006 0.0008 0.001 0.0012

$%&'

1 2 3 4 5 6

C C(theor.) flux/C(1/s) flux/C(1/s)(theor.) alpha

Comparison with numerical experiments Comparison with numerical Comparison with numerical experiments experiments

Pe=185 needtheadditionalconvectivetransportterms!

numericalexperiments
verseveralunitcells
solutionoftheclosure

problemsoveraunitcell

* β β β β

α α   − −   =     v u d v

α

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Effective coefficients as a function of the time

evolution of the interface (example of direct simul.: Bekri et al.)?

Effective coefficients as a function of the time

evolution of the interface (example of direct simul.: Bekri et al.)?

– ε ε ε ε (t) – α α α α(t, Pe, …)

α

α α α(ε ε ε ε,Pe) – Dβ

β β β(t, Pe, …)

Dβ

β β β(ε

ε ε ε,Pe) – K(t)

K(ε

ε ε ε) Problem with evolving geometry : Effective

Coefficients (α α α α, Dβ

β β β, K)

Problem with evolving geometry : Problem with evolving geometry : Effective

Effective Coefficients Coefficients ( (α α α α α α α α, D , Dβ

β β β β β β β, K)

, K)

Correlations obtained using:numerical simulation, closure pbs,experiments,…

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Dimensionless equations Dimensionless numbers Dimensionless equations Dimensionless numbers

( ) ( )

β β β β β β β β β β

ε ε ε ε

A ac A A A A A

C

Da

t C Da C Pe C t C ' 1 ' ' ' 1 ' . ' ' ' − = ∂ ∂ − ⋅∇ ⋅ ∇ = ∇ + ∂ ∂ D' V

Damköhler Péclet Acidcapacitynumber

v l Da α = D l Pe v =

( )

σ β β

ρ ε β ε − = 1 C

ac

Transport and Dissolution: numerical model Transport and Dissolution: Transport and Dissolution: numerical model numerical model

2 1

L L F =

2 1

L

D

K = +DarcyBrinkman

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3D example 3D example 3D example

I.C.:randompermeabilityfield

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Stability Analysis Stability Stability Analysis Analysis

2D Reference Simulations

10 88 10 87 10 86 10 85 10 84 10 83 10 82 10 81 1 10 20 30 40 50 60 70 80 90 100 10 88 10 87 10 86 10 85 10 84 10 83 10 82 10 81 1

Injectionvelocity(m/s)

10 20 30 40 50 60 70 80 90 100

Compact Dissolution Stable Front Perturbed front Conical wormhole Uniform Dissolution uniforme Stable front! Ramified Wormholes Dominant Wormholes

Vbt/Vp

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Stability of the linearised problem (Cohen, 2006) Stability Stability of

f the

the linearised linearised problem problem (Cohen, 2006) (Cohen, 2006)

Compact Front

(LE)

Compact Front

(LE)

LNE LNE

Autonomous Case 2Cases:autonomous and non8autonomous

ζ ε0 1

concentration porosity

ζ ε0 1

concentration pressure

x ε0 1

concentration porosit

x ε0 1

concentration porosity pressure

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Case LNE Case LNE Case LNE

( ) ( ) ( )

1/2

² ² 0

autonome

f t dz t f dz

ε ε

ε

Ω Ω

    =      

∫ ∫

⌢

Amplitude Coefficient

Front thickness growth

stabilizing effect

Non8autonomous Case

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Peclet Damköhler Diagram

1.00E+00 1.00E+01 1.00E+02 1.00E+03 1.00E+04 1.00E+05 1.00E+06 1.00E+07 1.00E+08 1.00E+00 1.00E+01 1.00E+02 1.00E+03 1.00E+04 1.00E+05 1.00E+06 1.00E+07 1.00E+08 1.00E+09

Peclet Number Damköhler Number

Nac=4.66E-2 LMEDIUM=10 cm

UNIFORM REGIME WORMHOLING REGIME COMPACT REGIME CONICAL WORMHOLES RAMIFIED WORMHOLES

Dissolution Diagram (Golfier et al., 2001) Dissolution Diagram Dissolution Diagram (

(Golfier Golfier et al., 2001) et al., 2001)

OPTIMUM FLOW RATE

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“Optimum flow rate” “ “Optimum flow rate” Optimum flow rate”

Need “mainly” to calibrate α

α α α

Need “mainly” to calibrate α

α α α

α α A =

whereα0 isthecorrelation

btainedfromtheclosure

problem+simpleunitcell

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Compact dissolution wormholes Conical Dominant wormholes Ramified wormholes

10 20 30 40 50 60 70 80 10-6 10-5 10-4 10-3

injection velocity(m/s) wormholedensity

x = 0.025 m x = 0.035 m x = 0.04 m x = 0.05 m x = 0.075 m x = 0.1 m x = 0.125 m x = 0.15 m x = 0.175 m x = 0.2 m

10 20 30 40 50 60 70 80 10-6 10-5 10-4 10-3

injection velocity (m/s) wormhole density

x = 0.025 m x = 0.035 m x = 0.04 m x = 0.05 m x = 0.075 m x = 0.1 m x = 0.125 m x = 0.15 m x = 0.175 m x = 0.2 m

Confinement Effect: « wormhole

competition », Cohen et al. (2007)

Confinement Effect: Confinement Effect: «

« wormhole wormhole competition competition », Cohen et al. (2007) », Cohen et al. (2007)

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Dominant wormhole growth rate increases with domain height Optimum injection velocity increases with height decreasing

1 10 10 10 10 10 10 10 1 10 10-7 10-6 10-5 10-4 10-3 10-2 Injection velocity V0 (m/s) Vbt/Vp height = 40 cm height = 20 cm height = 10 cm height = 5 cm slope 1/3 1 10 10 10 10 10 10 10 1 10 10-7 10-6 10-5 10-4 10-3 10-2 Injection velocity V0 (m/s) Vbt/Vp height = 40 cm height = 20 cm height = 10 cm height = 5 cm slope 1/3 height = 40 cm height = 20 cm height = 10 cm height = 5 cm slope 1/3 0.05 0.1 0.15 0.2 0.25 10000 20000 30000 40000 50000 time (s) Length of the dominant wormhole (m)

height = 40 cm height = 10 cm height = 5 cm

0.05 0.1 0.15 0.2 0.25 10000 20000 30000 40000 50000 time (s) Length of the dominant wormhole (m)

height = 40 cm height = 10 cm height = 5 cm

Confinement (cont.) Confinement (cont.) Confinement (cont.)

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Effect of geometry: ex. radial Effect Effect of

f geometry

geometry: ex. radial : ex. radial

V0 =4.96·1087 m/s Pe =1.57·1085 Da=6.37·1082 V0 =1.65·1086 m/s Pe =5.23·1085 Da= 1.91·1082 V0 =1.65·1084 m/s Pe =5.23·1084 Da=1.91·1083 V0 =1.65·1082 m/s Pe =5.23·1081 Da= 1.91·1086

see Cohen,2007

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Effect of geometry on optimum flowrate Effect of geometry on optimum Effect of geometry on optimum flowrate flowrate

3D Linear 2D Linear 5 10 15 20 25 30 35 Vbt/Vp 1·10-

7

1·10-

6

1·10-

5

1·10-

4

1·10-

3

1·10-

2

1·10-

1

injection velocity (m/ s) 3D Linear 2D Linear 5 10 15 20 25 30 35 Vbt/Vp 1·10-

7

1·10-

6

1·10-

5

1·10-

4

1·10-

3

1·10-

2

1·10-

1

injection velocity (m/ s)

5 10 15 20 25 30 35 1·10-7 1·10-6 1·10-5 1·10-4 1·10-3 1·10-2 1·10-1 injection velocity (m/s) Vbt/Vp 3D Radial 2D Radial

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Extension of LNE models to complex phase diagrams, and multicomponent systems Extension of LNE models to complex phase Extension of LNE models to complex phase diagrams, and diagrams, and multicomponent multicomponent systems systems

Equilibrium conditions at Aβσ

βσ βσ βσ

Equilibrium conditions at Aβσ

βσ βσ βσ

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

i j j i j j

c c c c

β σ β β β σ σ σ

µ µ         + = +                 ɶ ɶ . . . . . . . . (.., ,...) . . (.., ,...) . . . . . . . . . .

i i i j i j j j c c

c c c c c c

β σ β σ

β σ β σ β β β σ σ σ β σ

µ µ µ µ                       ∂   ∂     + = +             ∂ ∂                                   ɶ ɶ

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Expression for the mass exchange terms Expression for the mass exchange Expression for the mass exchange terms terms

Case 1: diagonal β

β β β and σ σ σ σ

General case? Case 1: diagonal β

β β β and σ σ σ σ

General case?

. . . .

c c

J c J c

σ β σ β

β β β σ σ σ

µ µ                   + = +                         ɶ ɶ exchangetermforspecies

i i i c c i

J i J

σ β σ β

µ µ ÷ −

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Example: ZrO2 / Zr (Belloni, 2008) Example: Example: ZrO2 / Zr ( (Belloni Belloni, 2008) , 2008)

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Example: ZrO2 / Zr (Belloni, 2008) Example: Example: ZrO2 / Zr ( (Belloni Belloni, 2008) , 2008)

Zr mass fraction(2373K,574s)

( ) ( )

( )

* *

. . . ( )

l l l l l l l l l l l s s l l l l l l l ml l l

C C t C t C h C C ε ρ ε ρ ε ρ ε ρ ρ ∂ +∇ ∂ ∂ + = ∂ ∇ ∇ + − v D

+similar equation fors

+Averaging of the mass balanceBCat Als: ( )

* * * *

1 ( ( ) ( ))

s s s s ms s s l s l l ml l l

h C C t C C h C C ε ρ ρ ρ ∂ = − ∂ − + −

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Flow in Heterogeneous Systems Flow in Heterogeneous Systems Flow in Heterogeneous Systems

0.2 0.4 0.6 0.8 1 1 2 3 4 5 6 7 8 9 10

Volume de pore injecté Longueur adimensionnelle (L /L)

w

0.2 0.4 0.6 0.8 1

1 2 3 4 5 6 7 8 Volume de pore injecté

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M. Quintard

fluid zone/porous zone

1D effective medium

1- equation model?

– 1 single equation – easy to implement – loss of information

2- equation or “double-porosity” model?

– 1 equation for each zone – Darcy-scale problem is similar to the pore-scale problem in the case of equilibrium dissolution

pb. with non-locality and history effects

fluid zone/porous zone

1D effective medium

1- equation model?

– 1 single equation – easy to implement – loss of information

2- equation or “double-porosity” model?

– 1 equation for each zone – Darcy-scale problem is similar to the pore-scale problem in the case of equilibrium dissolution

pb. with non-locality and history effects

Core-scale description Core Core-

scale description

scale description

ω

η

l: domain

integration

L

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Core-Scale Volume Fractions: Definitions Core Core-

Scale Volume Fractions:

Scale Volume Fractions: Definitions Definitions

Wormhole volume fraction: ϕ

ϕ ϕ ϕω

ω ω ω

Core-scale porosity Wormhole volume fraction: ϕ

ϕ ϕ ϕω

ω ω ω

Core-scale porosity

*

1

V

dV V ε ε

∞

=

∫

*

(1 )

ω ω

ε ϕ ϕ ε = + − ifLocalEquilibriumdissolution:

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Ex.: 2-equation model (Golfier et al., 2004, 2006) Ex.: 2 Ex.: 2-

equation model (

equation model (Golfier Golfier et al., et al., 2004, 2006) 2004, 2006)

Flow : Transport and Dissolution : Flow : Transport and Dissolution :

( )

* *

1 . 0in 8region

A A A A A A

C C C C t Pe C t C

ϖ β ϖ ϖ ϖ ϖ β β β β ϖ ϖ ϖ β σ η β

ϕ α ϕ β α ρ η ∂ + ∇ = ∇⋅ ⋅∇ − ∂ ∂ = ∂ = V D**

( )

ϖ β ϖ β ϖ β ϖ β

µ V K V .

1 ⋅

− = ∇ = ∇

−

P

needDarcyscalelocalequilibrium!

pb.withregionalvelocities?

( )

η β η β η β η β

µ

A A

V K V .

1 ⋅

− = ∇ = ∇

−

P

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Obtained Correlations: permeability Obtained Correlations: permeability

Wormholing regime Ramifiedregime

K=f(ϕϖ ) K=f(ε*β )

ε*

β

12

3 2 * ϖ

ϕ L = K

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New Model (Cohen et al., 2006) New Model (Cohen et New Model (Cohen et al al., 2006) ., 2006)

VH VM Vtotal =VM +VH VH VM Vtotal =VM +VH

VH containsdominantwormholes VM containsfacedissolutionandshortwormholes

( ) ( )

'

.

H H H M M H H H H H H H

g P P C f V t C φ ψ φ ε φ − = ′ − ′ ′ − ′ ′ ∇ + ∂ ′ ∂

− H H H

P V ′ ∇ ′ − = ′ K

' H ac H

g

t

= ∂ ∂ε

( ) ( )

. = ′ − ′ ′ − ′ ∇

H M H H

P P V ψ φ

( ) ( )

'

.

M M H M M H M M M M M M

g P P C f V t C φ ψ φ ε φ − = ′ − ′ ′ + ′ ′ ∇ + ∂ ′ ∂

− M M M

P V ′ ∇ ′ − = ′ K

' M ac M

g

t

= ∂ ∂ε

( ) ( )

. = ′ − ′ ′ + ′ ∇

H M M M

P P V ψ φ

Media H Media M

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Validation: example from dominant wormhole regime Validation: Validation: example example from from dominant dominant wormhole wormhole regime regime

Section-scale simulation Core-scale simulation

Section-scale Core-scale pressure (Pa) time (s)

X (m) X (m) porosity porosity

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Keq(x), ε(x),C(x),P(x) Acid injection 1st step : Section-scale – dissolution modelling 2nd step : treatment simulation – skin calculation 3rd step : introduction of skin in simulator reservoir – treatment optimisation

Reservoir Scale: goals Reservoir Reservoir Scale Scale: goals : goals

2 ln

e w

kh P Q r B S r π µ ∆ = +

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Example (Cohen, 2006) Example Example (Cohen, 2006) (Cohen, 2006)

global skin evolution

3D radial simulation

19,53 mD ϕ ϕ ϕ ϕ = 0,1857 15,53 mD ϕ ϕ ϕ ϕ = 0,1691 154 mD ϕ ϕ ϕ ϕ = 0,345 13,38 mD ϕ ϕ ϕ ϕ = 0,1544 17,53 mD ϕ ϕ ϕ ϕ = 0,1666

90 m

Acidizing Simulation

formation damage porosity permeability volume of changed porosity local skin evolution 3D radial grid

10 20 30 40 50 60 70 80 90

3
1

1 3 5

skin z (m) t = 0 s t = 26548 s

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Conclusions Conclusions Conclusions

Effective Surface:

– If not limit cases, or if no steady-state:

DNS?

– Coupling with instabilities?

Darcy-scale models:

– LNE model has potential for representing instabilities with a minimum of parameters – Coupling with strong heterogeneities?

Reservoir-scale models?

Effective Surface:

– If not limit cases, or if no steady-state:

DNS?

– Coupling with instabilities?

Darcy-scale models: