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IRTG-project: Numerical methods for multi-phase flow in heterogeneous porous media

Initial Situation

Single- and multi-phase 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 single-phase flow and nonlinear for multi-phase 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 multi-phase flow. Therefore, it is popular in the solution for multi-phase flow in reservoir simulation.

The classical cell-centered finite volume (CCFV) method is a physically intuitive control-volume formulation using the two-point flux approximation (TPFA), which is generally used to approximate elliptic operators in reservoir simulation. However, TPFA does not work properly for general non-K-orthogonal 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 near-well 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 non-orthogonal 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 L-method is studied by us since it has smaller flux stencils, a larger domain of convergence and a larger domain of monotonicity compared to the O-method.

Current Results

The influence of different Dirichlet boundary discretizations on the convergence rate of the multipoint flux approximation (MPFA) L-method is investigated by the numerical comparisons between the MPFA O- and L-method, 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.

l+o-new.jpg fullo-new.jpg

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 L-method for homogeneous media are given and illustrated which yield more insight into the L-method. The original transmissibility-based criterion for choosing the L triangle is reinterpreted by a geometry-based criterion, as shown in Fig.2, which is to compare the length of l1 and l2.


Fig.2. Geometry-based criterion for choosing the L triangle.

In terms of the geometry-based 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 L-method with boundary modifications is also studied and the optimal order H1 and L2 error estimates are derived and proved.

Moreover, the MPFA L-method is applied for the numerical simulation of two-phase 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 L-method.

reference.gif ccfv.gif mpfa.gif

Fig.4. Saturation contours for reference solution, TPFA and MPFA L-method.


Author(s) Title Appeared In Year Type Download
Christian Deusner, Shubhangi Gupta, Matthias Haeckel, Rainer Helmig, Barbara Wohlmuth Testing a thermo-chemo-hydro-geomechanical model for gas hydrate bearing sediments using triaxial compression lab experiments Geochemistry, Geophysics, Geosystems 2017 Article in Journal Abstract
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Corinna Hager, Rainer Helmig, Andreas Lauser, Barbara Wohlmuth A new approach for phase transitions in miscible multi-phase flow in porous media Advances in Water Resources, 34 (8), (2011), 957-966 2011 Article in Journal Abstract
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Yufei Cao, Rainer Helmig, Barbara Wohlmuth Convergence of the multipoint flux approximation L-method for homogeneous media on uniform grids Numerical Methods Partial Differential Equations 27,2 329--350 2011 Article in Journal Abstract
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Yufei Cao, Yufei Cao, Rainer Helmig, Rainer Helmig, Barbara Wohlmuth, Barbara Wohlmuth Geometrical interpretation of the multi-point flux approximation L-method Internat. J. Numer. Methods Fluids 60(11), 1173--1199 2009 Article in Journal Abstract
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Yufei Cao, Yufei Cao, Rainer Helmig, Rainer Helmig, Barbara Wohlmuth, Barbara Wohlmuth The influence of the boundary discretization on the multipoint flux approximation L-method R. Eymard, J.M.Hérard (eds.): Finite Volumes for Complex Applications V ( Problems & Perspectives), Wiley, 257-263 2008 Article in Conference Proceedings Abstract