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
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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
3Laboratoire d’Energétique et de Mécanique Théoriqueet Appliquée, CNRS /Université de Lorraine
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:
??????
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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
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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
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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
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NM2PorousMedia-2014Dubrovnik, Croatia, 29 September - 3 October 2014
Method of homogenizationMethod of homogenization
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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
RECURRENT TWO-SCALE HOMOGENIZATION
Averaged Medium n
Medium n 1Averaged Medium n
4Medium
. . .
3Averaged Medium
3
2
Medium Averaged Medium
2Medium
Averaged Medium
1Medium
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RECURRENT TWO-SCALE HOMOGENIZATION
Step 2Step 2
Step 1
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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:
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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
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Arbitrary step “i”
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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
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Upscaled equationsp q
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Comparison with the general form
Result of recurrent homogenization:
General form:
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Determination of kernels and
Four coupled recurrent equations 26
Physical meaning of functions
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NM2PorousMedia-2014Dubrovnik, Croatia, 29 September - 3 October 2014
Infinite number of scalesInfinite number of scales
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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
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Limit homogenized model
iP 1iP P PPressures and tend to the couple of limit pressures and iP 1iP P P
Effective kernel:
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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
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Results of simulation for iterative kernelsResults of simulation for iterative kernels
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Local kernels
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Effective kernels
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NM2PorousMedia-2014Dubrovnik, Croatia, 29 September - 3 October 2014
Comparison with microscaleComparison with microscale numerical simulationsnumerical simulations
36Laboratoire d’Energétique et de Mécanique Théoriqueet Appliquée, CNRS /Université de Lorraine
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Medium and mesh (Comsol Multiphysics)
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Results of simulation for 3-scale medium
38PressureStreamlines
Testing the numerical code: 2-scale medium
Three-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
NM2PorousMedia-2014Dubrovnik, Croatia, 29 September - 3 October 2014
Case of thin fracturesCase of thin fractures
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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
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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
Laboratoire d’Energétique et de Mécanique Théoriqueet Appliquée, CNRS /Université de Lorraine
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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
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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
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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
Laboratoire d’Energétique et de Mécanique Théoriqueet Appliquée, CNRS /Université de Lorraine
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