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An Improved Semi-Analytical Solution for Verification of Numerical Models of Two-Phase Flow in Porous Media

Radek Fuík^a, Jií Mikyka^a, Michal Bene^a,* and Tissa H. Illangasekare^b

^a Department of Mathematics, Faculty of Nuclear Science and Physical Engineering, Czech Technical University in Prague, Trojanova 13, 120 00 Prague, Czech Republic
^b Center for Experimental Study of Subsurface Environmental Processes (CESEP), Division of Environmental Science and Engineering, Colorado School of Mines, 1400 Illinois Street, Golden CO 80401, CO

View larger version (15K): [in this window] [in a new window]   Figure 1. Graphs of the function and F1 for different choices of S0 illustrate why the original iterative process fails after the first iteration. Brooks-Corey model uses = 2.

View larger version (8K): [in this window] [in a new window]   Figure 2. Illustration of successive approximations of F in the L norm (using the decadic logarithmic scale) and the value of A. We used the method A (48), Setup 1 with Si = 0 and S0 = 0.99. The iterative process is terminated by A = 10–15.

View larger version (20K): [in this window] [in a new window]   Figure 3. Experimental convergence of the functions F to FBL, F' to F'BL and F'' to F''BL as S0 1 for Setup 1 with Si = 0, using method A (48). The last figure depicts the evolution of the function G as S0 1. Note that the function G and F'' are related by (60), i.e. they differ only by a factor involving A2.

View larger version (21K): [in this window] [in a new window]   Figure 4. Evolution of A in the modified iterative process, method A; test Setup 1, Si = 0. As S0 approaches 1, the iterative process requires higher number of iterations to reach convergence of A. In the very proximity of S0 = 1, the value of A is far from convergence even after 108 iterations. The situation for the method B is analogous.

View larger version (25K): [in this window] [in a new window]   Figure 5. McWhorter exact solutions (the method A) and Buckley-Leverett analytical solutions for various S0; test Setup 1, Si = 0. As S0 1, the unidirectional displacement solution (R=1) approaches the Buckley-Leverett solution, while the head of the bi-directional displacement solution (R=0) advances negligibly.