flow in fractal fractured-porous media: macroscopic model
TRANSCRIPT
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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
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
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Essence of the problemEssence of the problem
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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
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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
8Laboratoire d’Energétique et de Mécanique Théoriqueet Appliquée, CNRS /Université de Lorraine
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22 - 2-scale
medium: 121a
21
2
~a
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3212 - 3-scale
medium: 2
21a
21
2
~a
22 ~aa
3a
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432212 - 4-scale
medium: 3
21a
321
2
~a
22 ~aa
2a
3a
23
4
~aa
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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
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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
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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
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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”
2121Laboratoire d’Energétique et de Mécanique Théoriqueet Appliquée, CNRS /Université de Lorraine
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Formulation of the problem at scale ip
HomogenizationHomogenization …22
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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
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Physical meaning of functions
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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:
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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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Comparison with microscaleComparison with microscale numerical simulationsnumerical simulations
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Medium and mesh (Comsol Multiphysics)
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Results of simulation for 3-scale medium
38PressureStreamlines
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Testing the numerical code: 2-scale medium
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Three-scale medium
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Three-scale medium
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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
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NM2PorousMedia-2014Dubrovnik, Croatia, 29 September - 3 October 2014
Case of thin fracturesCase of thin fractures
43Laboratoire d’Energétique et de Mécanique Théoriqueet Appliquée, CNRS /Université de Lorraine
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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
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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
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