IRTGproject: Numerical methods for multiphase flow in heterogeneous porous media
Project manager: 
Prof. Dr. rer. nat. B.I. Wohlmuth,
Prof. Dr. Ing. R. Helmig

Projektbearbeiter: 
Ph.D. Y. Cao 
Initial Situation
Single and multiphase flow in porous media play an important role in many natural and industrial fields, such as the oil industry
where the flow of oil, water and gas in reservoir is studied. The flow system using the fractional flow formulation contains a pressure equation with elliptic behavior, which is linear for singlephase flow and nonlinear for multiphase flow.
The finite volume method is a numerical discretization technique which can locally inherit
physical conservation laws of original problems. The property of discrete local mass conservation
is desirable to approximate the elliptic operators in the pressure equation for single and multiphase flow. Therefore,
it is popular in the solution for multiphase flow in reservoir simulation.
The classical cellcentered finite volume (CCFV) method is a physically intuitive controlvolume
formulation using the twopoint flux approximation (TPFA), which is generally used to approximate elliptic operators
in reservoir simulation. However, TPFA does not work properly for general non
Korthogonal grids due to the error in its solution which cannot be reduced by refining the grids.
In reservoir simulation, the grids with a high aspect ratio are quite often used, and
the grids with a more complex geometry are preferred at faults or in nearwell regions. To overcome this problem,
the multipoint flux approximation (MPFA) methods were widely studied
in the last decade. It can give a correct discretization of flow equations not
only for general nonorthogonal grids but also for general orientation of the
principal directions of the permeability tensor.
There are many variants of the MPFA method, in which a new MPFA method
called the Lmethod is studied by us since it has smaller flux stencils,
a larger domain of convergence and a larger domain of monotonicity compared to the Omethod.
Current Results
The influence of different Dirichlet boundary discretizations on the convergence rate of
the multipoint flux approximation (MPFA) Lmethod is investigated by the numerical comparisons between the MPFA O and Lmethod, which shows how important it is for this new method to handle Dirichlet boundary conditions in a suitable way. A new Dirichlet boundary strategy is proposed which in some sense can well recover the superconvergence rate of the normal velocity, see Fig.1.
Fig.1. Dirichlet boundary discretizations: "MPFA L: L+O" (left), "MPFA L: full O" (new, right).

A systematic concept and geometrical interpretations of the MPFA Lmethod for homogeneous media are given and illustrated which yield more insight into the Lmethod. The original transmissibilitybased criterion for choosing the L triangle is reinterpreted by a geometrybased criterion, as shown in Fig.2, which is to compare the length of l
_{1} and l
_{2}.
Fig.2. Geometrybased criterion for choosing the L triangle.

In terms of the geometrybased criterion, two illustrations describe the choice range of two L triangles from two points of view.
Fig.3. Two choice regions for determining the L triangle. 
The convergence of the MPFA Lmethod with boundary modifications is also studied and the optimal order H
^{1} and L
^{2} error estimates are derived and proved.
Moreover, the MPFA Lmethod is applied for the numerical simulation of twophase flow in porous media on different quadrilateral grids and numerical results for the pressure and saturation are compared with the results of the TPFA method. The simulation result is shown in Fig.4, which give the numerical comparison among the saturaions from reference solution, TPFA method and MPFA Lmethod.
Fig.4. Saturation contours for reference solution, TPFA and MPFA Lmethod. 
Publications
Author(s) 
Title 
Appeared In 
Year 
Type 
Download 
Yufei Cao, Rainer Helmig, Barbara Wohlmuth 
Geometrical interpretation of the multipoint flux approximation Lmethod 
Internat. J. Numer. Methods Fluids 60(11), 11731199 
2009 
Article in Journal 
Abstract
PDF ^{}
Bibtex

Yufei Cao, Rainer Helmig, Barbara Wohlmuth 
The influence of the boundary discretization on the multipoint flux approximation Lmethod 
R. Eymard, J.M.Hérard (eds.): Finite Volumes for Complex Applications V ( Problems & Perspectives), Wiley, 257263 
2008 
Article in Conference Proceedings 
Abstract
PDF
Bibtex
