Front tracking for two-phase flow in reservoir simulation by adaptive mesh

Mazen Saad
in collaboration with H. Zhang

An algorithm for the  numerical approximation of two-phase  flow in porous media by adaptive mesh is presented.
A convergent and conservative finite volume scheme for an elliptic equation is proposed and  finite differences schemes for a hyperbolic equation on grids with local refinement are constructed and studied.
Hence, an IMPES (IMplicit in Pressure Explicit in Saturation) method is applied in an adaptive composite grid to track the front of a moving solution. An object oriented programming techniques is used.  The computational results for different examples illustrate the efficiency of the proposed algorithm.

(For more information here).


Test 1. Poisson equation

The first example demonstrates the efficiency of the finite volume scheme on a composite grid. To test this scheme we solve numerically a classical equation  of Poisson on a composite mesh :

poisson

ellipticellp



Poisson equation, composite mesh of 5 levels and solution (Number of cells = 1264).
    The figure  depicts the composite mesh of 5 levels, the coarse grid is 10 x 10 grids.
Note that the refined composite grid localizes and closes to the solution.


Test 2. Waterflood simulation

We present one  example treating model in waterflooding simulation.
A one quarter five-spot, two phase, incompressible water injection problem is simulated.
Water is injected at a constant pressure in the top northwest corner, and a constant pressure is specified in the lower southeast corner. All other boundaries are no--flow. The initial water saturation is the critical saturation (i.e. it is considered to be zero).
 

coin


Water is injected at a constant pressure in the top northwest corner, and a constant pressure is specified in the lower southeast corner. All other boundaries are no--flow. The initial water saturation is the critical saturation (i.e. it is considered to be zero).





    The coarse mesh is (10x 10) grids, and four levels of refinement are used. The figures  show the water saturation contours and the composite grid at time 34 and 93 days. Note that the location of the refined area  tracks the shock front quite closely during the time.  One can remark also that the number of cells in the composite grids is far less  then the number of cells in the finest grid (80x80 meshes).


coinellp

Water saturation contours and composite grid for
the Corner example, time = 34.0 days, number of cells = 105.


coinellp

Water saturation contours and composite grid for
the Corner example, time = 93.0 days, number of cells = 1735.