SEMINARIO DI MATEMATICA APPLICATA

[Giacomo Aletti <giacomo.aletti@unimi.it>]

Apologies for cross-posting,
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Giovedì 3 luglio 2003, ore 15.30
Sala di Rappresentanza, Dipartimento di Matematica (Via C. Saldini n.50,
Milano)



SEMINARIO DI MATEMATICA APPLICATA
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ON THE DOUBLE POROSITY MODEL OF SINGLE PHASE FLOW IN RANDOM MEDIA



Andro Mikelic
LaPCS, UFR Mathématiques
Université Claude Bernard Lyon 1
FRANCE



We consider the linearized equations of slightly compressible single fluid
flow through a highly heterogeneous random porous medium, consisting of two
types of material. Due to the high heterogeneity of the two materials the
ratio of their permeability coefficients is of order epsilon square, where
epsilon is the characteristic scale of heterogeneities. Supposing that the
matrix blocks set of the porous medium consists of random stationary
inclusions, and assuming positive definitness of the effective permeability
tensor associated to the corresponding Neumann problem for the random
fractures system, we obtain the homogenized problem for a random version of
the double porosity model used in geohydrology. It includes as a particular
case the periodic setting, already studied by homogenization theory methods
(see, for example, articles by Arbogast, Douglas and Hornung or by Bourgeat,
Luckhaus and Mikelic). The homogenized problem is obtained by using the
stochastic two scale convergence in the mean, and by means of convergence
results specially adapted to our a priori estimates and to the random
geometry, which do not require extension of solutions to the matrix part.

A careful reduction of our two-scale homogenized system leads to two
auxiliary stochastic problems. The first one corresponds to the flow in
random fractures with Neumann condition at the interface. The second one is
a stochastic parabolic equation defined in the matrix blocks. Finally, we
provide several examples of the random double porosity model in such random
structures as disperse and generalized disperse media, perforated blocks
structure and Voronoi tesselation models.




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La home page del seminario e' http://users.mat.unimi.it/~pavarino/sma/
Il seminario e' organizzato dal Dipartimento di Matematica dell'Universita'
di Milano in collaborazione con MIRIAM (MIlan Research group in Industrial
and Applied Mathematics) e il Dottorato di Ricerca MACRO (MAtematica
Computazionale e Ricerca Operativa) e MSSCI (Matematica, Statistica, Scienze
Computazionali e Informatica).