flow in fractal fractured-porous media: macroscopic model

NM2PorousMedia-2014International Conference on Numerical and Mathematical Modeling of Flow and Transport in Porous Media Dubrovnik, Croatia, 29 September - 3 October 2014

Flow in Fractal Fractured-porous Media:Macroscopic Model with Super-memory,Macroscopic Model with Super memory,

Appearance of Non-linearity and Instability

Mikhail Panfilov

Laboratoire d’Energétique et de Mécanique Théoriqueet Appliquée, CNRS /Université de Lorraine

1

Published in

Panfilov M., and Rasoulzadeh M. A ppearance of the nonlinearity from the nonlocality in diffusion through

l l f d dmultiscale fractured porous media. Computational Geosciences, v. 17: 269 – 286 (2013)DOI 10.1007/s10596‐012‐9338‐7DOI 10.1007/s10596 012 9338 7

l d h f l d h kRasoulzadeh M., Panfilov M., and Kuchuk F. Effect of memory accumulation in three‐scale fractured‐porous media. International Journal of Heat and Mass Transfer, v. 76: 171 – 183 (2014)International Journal of Heat and Mass Transfer, v. 76: 171 183 (2014)

2Laboratoire d’Energétique et de Mécanique Théoriqueet Appliquée, CNRS /Université de Lorraine

Essence of the problemEssence of the problem


2 - two-scale medium2

21Diff. coef ~Diff coef

1

2Diff. coef

Microscopic equation:

/ a /tb x p x p

Macroscopic model:

t

1

1

0

t

t tB P A P B K t Pd

4

0

2 - multi-scale medium

Microscopic equation:

/ a /tb x p x p

Macroscopic model:

??????

5

OBJECTIVES

‐ to develop the effective model for infinite number of scales

t d l l ith f l l ti th ff ti t‐ to develop an algorithm of calculating the effective parameters

6

APPLICATIONS

Coalbed methaneCoalbed methane

Largeur = 1,6 cm

2 * l = 2,2 cm

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NM2PorousMedia-2014Dubrovnik, Croatia, 29 September - 3 October 2014

multiscale self similar2 - multiscale self-similar

media

media


22 - 2-scale

medium: 121a

21

2

~a

9

3212 - 3-scale

medium: 2

21a

21

2

~a

22 ~aa

3a

4

432212 - 4-scale

medium: 3

21a

321

2

~a

22 ~aa

2a

3a

23

4

~aa

4

Local coordinates for scale n

Medium Periodicity cell at scale nPeriodicity cell at scale n

For domain : the « slow » variable is( )iy

12For domain : the « slow » variable is

the « fast » variable is ( 1)iy

y

Transport Equations

( ) ( )p ( ) ( ) ,i i ipb a p xt

( ) ( 1)i ip pa a

1, 1,

,i i i i

a an n

0p

Condition of high heterogeneity:( )

2( 1) 1

ia ( 1)

( )

1

1

i

iab

( 1) ~ 1ib 13

Two types of domains

14


Method of homogenizationMethod of homogenization


RECURRENT TWO-SCALE HOMOGENIZATION

‐ Splitting into the series of recurrent two‐scale homogenizationsSplitting into the series of recurrent two scale homogenizations

‐ Assumption about the general form of the averaged equations for any scale;

‐ Two‐scale homogenization at an arbitrary scale

‐ Closure of the recursion: two‐scale homogenization at the lowest scale

Main assumptions:Main assumptions: (1) boundary layers between two scales may be neglected ;

(2) contact condition may be accepted in the natural form: 1

1;i i

i i iP pA a P p

16, 1 , 1

, 1 , 1

; i i i ii i i i

A a P pn n


Averaged Medium n

Medium n 1Averaged Medium n

4Medium

. . .

3Averaged Medium

3

2

Medium Averaged Medium

2Medium

Averaged Medium

1Medium

17


Step 2Step 2

Step 1

18

ASSUMPTION ABOUT THE GENERAL FORM OF THE AVERAGED EQUATIONSAVERAGED EQUATIONS

f h ( )First step of homogenization (1 2):

By the analogy, for any step i-1 i:

19

ASSUMPTION ABOUT THE GENERAL FORM OF THE AVERAGED EQUATIONSAVERAGED EQUATIONS

The last equation may be presented in the closed form w.r.t.

2 ( 1) ( 1)(1 ) * *i i

i i i ii i

P Pb A P b P Lt t t

Κ

Kernels and :Kernels and :

1 11, t

The objective is to determine them for any i

20

Arbitrary step “i”


21

Formulation of the problem at scale ip

HomogenizationHomogenization …22

ASYMPTOTIC TWO-SCALE HOMOGENIZATION AT EACH STEP

( ) ( ) ( 1),i i iy y y ( ) slow variableiy ( 1) ( 1) ( )

slow variablefaste variable: /i i i

yy y y

( ) ( ) ( 1)

1i i iy y y

Asymptotique expansion w.r.t.

Additi l diti f i di it t

( 1)iy

Additional condition of periodicity w.r.t.

Condition of existence of periodic solutions Averaged equations

y

First approximation Cell problems

23

Upscaled equationsp q

24

Comparison with the general form

Result of recurrent homogenization:

General form:

25

Determination of kernels and

Four coupled recurrent equations 26

Physical meaning of functions

27


Infinite number of scalesInfinite number of scales


28

Stabilization w.r.t. i when i

If the limit behaviour exists, then the iterative system of averaged models should prouve a stabilization w.r.t. i

oras

Li i k lLimit kernels

Pressures and will also tend to the couple of limit pressures and

iP 1iP P Ppressures and P P

29

Limit homogenized model

iP 1iP P PPressures and tend to the couple of limit pressures and iP 1iP P P

Effective kernel:

30

Appearance of non-linearity in the effective and local kernelslocal kernels

i f h L l for in terms of the Laplace transform:

is the volume of a block / the volume of a fracture 31

Approximation for the kernels

exacte:exacte:

If in the right‐hand side: , then ( ) ( )K t t ( ) 1K s Then

oror

32

Results of simulation for iterative kernelsResults of simulation for iterative kernels


33

Local kernels

34

Effective kernels

35


Comparison with microscaleComparison with microscale numerical simulationsnumerical simulations


36

Medium and mesh (Comsol Multiphysics)

37

Results of simulation for 3-scale medium

38PressureStreamlines

Testing the numerical code: 2-scale medium

Three-scale medium

Effect of memory accumulation

The difference in pressure between fractures and blocks

‐ For two‐scale medium :

‐ For three‐scale medium :

Delay between Influence ofDelay between fracture 3 and block 2

Influence of medium 1 on the interaction b t 3 d 2between 3 and 2

Memory42

Memory accumulation


Case of thin fracturesCase of thin fractures


43

Two-scale medium: fractional derivatives

2 2 2 1 1/ 2 2t tB P A P B D P

1 1t PD P d

01tD P d

t t

For infinite number of scale, it is impossible to obtain the pmacroscopic model, as the medium is non self‐similar

44

Three-scale medium

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Conclusion

1 Appearance of the effect of memory accumulation (“super‐memory”)1. Appearance of the effect of memory accumulation ( super‐memory )

2. Appearance of the nonlinearity in the structure of the integral kernels

3. Good approximation of the 3‐scale and more media by the limit infinite‐scale modelscale model


46

Instability

Numerically it was observed that the limit model becomes unstable below a threshold value of the permeability ratio andunstable below a threshold value of the permeability ratio and volume fraction of blocks

47

Instability

Numerically it was observed that the limit model becomes unstable below a threshold value of the permeability ratio andunstable below a threshold value of the permeability ratio and volume fraction of blocks

48

NM2PorousMedia-2014International Conference on Numerical and Mathematical Modeling of Flow and Transport in Porous Media Dubrovnik, Croatia, 29 September - 3 October 2014

Thank you for your attention


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flow in fractal fractured-porous media: macroscopic model

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