IRTGproject: Operator splitting methods for porous media flow
Project manager:  Prof. Dr. rer. nat. B.I. Wohlmuth,
Prof. Dr. Ing. R. Helmig, Prof. M. Espedal 
Projektbearbeiter:  Ph.D. Y. Cao 
Initial Situation
The multiphase flow system using the fractional flow formulation contains a transport equation which includes both the diffusive and the advective processes. As well known, the numerical approximation of a diffusion
equation and of an advection equation are relatively easy, but
a numerical treatment of an equation with these two coupled types is
quite challenging for whichever schemes, explicit or implicit, we apply.
This motivates us to study and apply the operator splitting (OS) method
for nonlinear problems in porous media.
The main idea of the standard OS approach is to split complex operators in evolution
equations into simpler ones which are successively solved in each time step.
For porous media flow a natural splitting is given by the diffusive and the
advective part of the flux. Then for each part, optimal solvers for the
particular evolution equation can be used and different time scales and grid sizes can be applied which gives us a multiple
of choices to search for the best one.
Current Results
Both one and twoscale OS methods using an implicit
finite volume scheme for the diffusive part and a semiLagrangian method
for the advective part are implemented and compared. The twoscale OS approach is to solve the diffusion part on a coarse grid
while on a fine grid for the advection part. This is motivated by the fact that the diffusion and the
advection act on different scales.
From the numerical results, it turns out that
despite of the poor approximation in the diffusive part, the solution of twoscale OS method is still of equal
quality compared to that of onescale OS discretization on fine grids. Moreover, the forward semiLagrangian method is also studied and applied in the twoscale OS approach in order to avoid and improve the drawback from the backward semiLagrangian method.
Saturation profiles for onescale backward, twoscale backward and forward OS methods.

Work Plan
In future work, the corrected operator splitting (COS) method will be the main topic which can correctly resolve the nonlinear shock fronts for large time steps compared to the standard OS approach. The COS will be applied to the two and threephase flow problems in porous media in higher dimension.
Publications
Author(s) 
Title 
Appeared In 
Year 
Type 
Download 
Yufei Cao, Alexander Weiss, Barbara Wohlmuth 
A twoscale operator splitting for flow problems in porous media 
Proceedings of ECMOR XI, Bergen, Norway, 811 September 
2008 
Article in Conference Proceedings 
Abstract
PDF
Bibtex
