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Operator splitting methods for porous media flow

Initial Situation

The multi-phase 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 non-linear 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 two-scale OS methods using an implicit finite volume scheme for the diffusive part and a semi-Lagrangian method for the advective part are implemented and compared. The two-scale 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 two-scale OS method is still of equal quality compared to that of one-scale OS discretization on fine grids. Moreover, the forward semi-Lagrangian method is also studied and applied in the two-scale OS approach in order to avoid and improve the drawback from the backward semi-Lagrangian method.

os-bkw-a-t4.gif mos-bkw-a-t4.gif mos-fwd-a-t4.gif

Saturation profiles for one-scale backward, two-scale 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 non-linear shock fronts for large time steps compared to the standard OS approach. The COS will be applied to the two- and three-phase flow problems in porous media in higher dimension.


Author(s) Title Appeared In Year Type Download
Ingeborg Gjerde, Kundan Kumar, Jan Nordbotten, Barbara Wohlmuth Splitting method for elliptic equations with line sources accepted for publication in M2AN 2019 Preprint Abstract
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Rainer Helmig, Martin Schneider, Ettore Vidotto, Barbara Wohlmuth Streamline method for resolving sharp fronts for complex two-phase flow in porous media Comput. Geosci. 2018 Article in Journal Abstract
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Yufei Cao, Rainer Helmig, Barbara Wohlmuth A two-scale operator-splitting method for two-phase flow in porous media Advances in Water Resources 2011 Article in Journal Abstract
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Yufei Cao, Yufei Cao, Alexander Weiss, Alexander Weiss, Barbara Wohlmuth, Barbara Wohlmuth A two-scale operator splitting for flow problems in porous media Proceedings of ECMOR XI, Bergen, Norway, 8-11 September 2008 Article in Conference Proceedings Abstract