XVI International Conference on Computational Methods in Water Resources (CMWRXVI) Ingeniørhuset  

Effective anisotropy tensor for the numerical solution of heterogenous porous media flow problems  Paper   

Author:  Gian Paolo Leonardi <leonardi@dmsa.unipd.it> (Dept. of Mathematics  University of Modena  Italy) 
 Fabio Paronetto <fabio.paronetto@unile.it> (Dept. of Mathematics  University of Lecce  Italy) 
 Mario Putti <putti@dmsa.unipd.it> (Dept. Mathematical Methods and Models for Scientific Applications  University of Padua  Italy) 
Presenter:  Gian Paolo Leonardi <leonardi@dmsa.unipd.it> (Dept. of Mathematics  University of Modena  Italy) 
Date:
 20060618
Track:
Special Sessions
Session:
Multiscale methods for flow in porous media 
DOI:  10.4122/1.1000000438 
DOI:  10.4122/1.1000000439 

The numerical solution of strongly heterogeneous and anisotropic porous media flow
equations often requires the definition of discrete interfacial average fluxes.
Standard methods, such as Integrated Finite Differences (IFD) or Finite Volumes (FV),
start from the definition of an average solution gradient and calculate the interface
fluxes by means of appropriate averages of the conductivity tensor K. For a scalar K,
the harmonic average is correct in the sense that it returns the same flux if
multiplied by the average gradient in the direction normal to the control volume
boundary. On the other hand, in terms of asymptotic convergence, any average
procedure yields the correct rate. However, significant differences may arise when
working on a fixed mesh size.
The question of how to define a correct averaging procedure for the anisotropic case
is however still open, in particular when the principal directions of anisotropy
change in space. We derive a procedure based on variational principles of
conservation of flux and energy. The resulting mean matrix calls into play the
tangential component of the gradient and provides standard arithmetic or harmonic
means for tangential and normal gradient components in simple cases. Moreover, the
resulting tensor is shown to coincide with a matrix arising from homogeneization
theory, even though it has been obtained using a different approach.
The effectiveness of the proposed average is tested on a number of numerical examples
on twodimensional domains for both IFD and FV. 