Study And Application Of Steady-state Single-phase Flow In Porous Media With Permeability In Tensor Form

Steady-state single-phase flow in porous media is fundamental and important in numerical simulation of seepage as well as reservoir simulation. It forms the basis of many other more complex problems as transient flow and multi-phase flow. At present, numerous numerical methods have been developed to solve the steady-state single-phase flow in porous media. These traditional methods play an important role in solving homogeneous and slightly heterogeneous problems. However, as the heterogeneity is so frequent in practical problems, researcher and engineers have to deal with the problems with strong heterogeneity. However, the traditional methods are unsatisfactory in handling strong heterogeneous media due to the poor accuracy and efficiency. Therefore, it is of great importance to construct an accurate and efficient numerical scheme specifically for strong heterogeneous problems.This dissertation extends the finite analytic method to study the steady-state single-phase flow in porous media with permeability in tensor form through mathematical analysis and numerical implementation. The work is summarized as follows:1. The analytical solution of pressure equation in power-law form for porous media with permeability in tensor form is proposed. It is interesting to find out that the index of the power-law solution is only determined by the four neighboring permeabilities. In other words, the power-law index is the characteristic parameter of the neighborhood, which is not affected by the boundary conditions. Further, the localized characteristic of the power-law solution indicates that the decisive influenced area for the problem mainly focus on the neighborhood of the divergence point of the pressure gradient. Therefore, the analytical solution is expected to describe the fingering flow observed in experiment.2. A finite analytical method scheme is constructed for porous media with permeability in tensor form via the power-law solution. Numerical examples demonstrate the high accuracy and convergence speed of the proposed scheme. It also shows that the proposed scheme has good stability because its convergence speed is not affected by the properties. The relative errors of the results have the same convergence speed in different permeability distributions and boundary conditions.3. This dissertation has found that the fluid flows become delocalized for some permeability distributions in the process of constructing the finite analytical scheme, In this situation, the local characteristic of the pressure is not only determined by the neighborhood of divergence point. In addition, any discontinuous point of permeability may cause divergence of the pressure gradient.4. The finite analytic method is applied to solve upscaling problems. First, this dissertation improve the traditional upscaling numerical methods using the finite analytical scheme. Secondly, a variable decomposition is introduced into the auxiliary variables of large scale average theory. With this decomposition, the initial-boundary problems related to the auxiliary variables change to steady-state flow problems. Thus, the finite analytic scheme can be easily applied to solve these problems. Finally, the numerical examples show that the proposed scheme has the characteristics of high precision and fast convergence. It can significantly improve the computational efficiency, that is, generally only 2 or 3 refinements can give fairly accurate equivalent permeability.In summary, the present work is a novel work in the numerical simulation of the reservoir. A novel numerical scheme has been constructed by this finite analytical method. The proposed scheme has a very high accuracy and convergence rate, which can be applied in practical problems.