Finite Analytic Numerical Method For Multi-Phase Flow In Heterogeneous Porous Media

With the increasing demand of oil resources and overexploitation of conventional oil and gas, unconventional oil and gas, such as heavy oil, shale gas, gas hydrate and so on, have become an important part of energy structure in the 21st century. Characteristics of unconventional oil and gas include huge reserves, concentrated distribution, and increasingly mature development technology. Conventional methods and techniques can not be used for unconventional oil and gas exploration and development, because unconventional oil and gas are very different from conventional oil and gas in burial state and occurrence state. During unconventional reservoir numerical simulation, the heterogeneity of geological parameters, especially the absolute permeability, is one of the greatest difficulties. Geological modeling data are often too enormous to be handled directly by numerical simulators due to excessive number of grids. To satisfy practical needs, computations are usually performed at a larger scale. Therefore, upscaling of the permeability is always necessary. The traditional numerical algorithm usually undervalues the effective permeability of the porous media severely. In order to get more accurate results, traditional numerical algorithm needs to refine the grids. The refinement ratio for the grid cell needs to be increased dramatically to get an accurate result for strong heterogeneous cases. The object of this paper is to construct a more efficient and accurate numerical method for heterogeneous porous media.For the two dimensional two-phase flow in heterogeneous porous media, the total equation can be obtained if combining the water and the oil continuity equation. Ignoring the capillary force gradient, the total seepage equation is simplified as the quasi Laplace’s equation, which is the same as the case for the single-phase flow. The internodal transmissibility can be calculated by using the finite analytic numerical method (FAM), which is constructed based upon the power-law analytical solutions of the quasi Laplace equation around the singularity point. Compared with the traditional algorithm, the internodal transmissibility derived by FAM is not only associated with the absolute permeability of adjacent grids, but also associated with the absolute permeability of the grids near the singularity. Numerical examples show that the FAM makes the convergences much faster as the refinement parameter increases, and the accuracy is independent of the heterogeneity. However, the traditional algorithm will severely underestimate the speed of front and the water breakthrough time. In order to get an accurate result, the refinement ratio for the grid cell needs to increase dramatically for the traditional numerical method.For the two dimensional three-phase flow in heterogeneous porous media, the internodal transmissibility derived from FAM is applied directly. The example of three-phase flow shows that the grids with high permeability will form a flow channel for mass and energy. This channel accelerates the flow of the steam and heat, and the three-phase area will soon be formed. This effect will make the gas breakthrough soon to the production wells. While using the traditional numerical method, this phenomenon can hardly be detected.A numerical study on "big channel" phenomenon is performed based on the results of the steam drive for the porous media with the log-normally distributed permeability. Numerical results from the FAM show that the injection well and production well can form a fast-track, and this will make the gas breakthrough the production wells soon. However the traditional algorithm still needs much longer time to develop the fast-track even with grid subdivision. Therefore, the traditional algorithm greatly underestimate the effective permeability in porous media would be a reasonable explanation for "big channel" phenomenon.In summary, the object of this paper is to construct the FAM for two-dimensional multi-phase flow in heterogeneous porous media. Numerical examples show that FAM has an advantage on the calculation accuracy and computational efficiency, compared with the traditional algorithm. Meanwhile, some numerical simulation phenomena may be explained by the FAM.