Development, analysis and numerical methods for multicomponent, multiphase flow in porous media.

摘要

In my scientific research I have concentrated on numerical methods for partial differential equations and their applications to multiphase, multicomponent flows in porous media. A fractured porous medium has throughout its extent a system of interconnected fractures dividing the medium into a series of essentially disjointed blocks of porous rock, called matrix blocks. It has two main length of scales of interest: the microscopic scale of the fracture thickness about (10-4 m) and the macroscopic scale of the average distance between fracture planes, i.e., the size of the matrix blocks(about 0.1-1 m). Since the entire porous medium is about (10 3-104m) across, flow can be mathematically simulated only in some averaged sense. The concept of dual porosity (and dual porosity/permeability) has been utilized to model the flow of fluids on its various scales. In this concept, the fracture system is treated as a porous structure distinct from the usual porous structure of the matrix itself. The fracture system is highly permeable, but can store very little fluid, while the matrix has the opposite characteristics. When developing a dual-porosity model, it is critical to treat the flow transfer terms between the fracture and matrix systems.;In the first part of this thesis we have worked on multiphase, multicomponent flow with mass interchange between phases in porous media. The governing equations of a compositional model for three-phase multicomponent fluid flow in multi-dimensional petroleum reservoirs have been cast in terms of a pressure equation and a set of component mass balance equations in this project. The procedure is based on a pore volume constraint for component partial molar volumes, which is different from earlier techniques utilizing an equation of state for phase fluid volumes or saturations. The present technique simplifies the pressure equation, which is written in terms of various pressures such as phase, weighted fluid, global, and pseudo-global pressures. The different formulations resulting from these pressures have been numerically solved; the numerical computations use a scheme based on the mixed finite element method for the pressure equation and the finite volume method for the component mass balance equations. A qualitative analysis of these formulations have been also carried out. The analysis shows that the differential system of these formulations is of mixed parabolic-hyperbolic type, typical for fluid flow equations in petroleum reservoirs. Numerical experiments based on the benchmark problem of the third comparative solution project organized by the society of petroleum engineers(SPE)have been presented.;In the second part we have derived well flow models for various numerical methods used in the discretization of fluid flows and transport in porous media. Numerical simulation of fluid flow and transport processes in the subsurface must account for the presence of wells. The pressure at a grid block that contains a well is different from the average pressure in that block and different from the flowing bottom hole pressure for the well. Various finite difference well models had been developed to account for the difference. This part has been concerned with a systematical derivation of well models for other numerical methods such as standard finite element, control volume finite element, and mixed finite element methods. Numerical results for a simple well example illustrating local grid refinement effects and the seventh comparative solution project organized by the society of petroleum engineers(SPE) have been given to validate these well models. The well models have particular applications to groundwater hydrology and petroleum reservoirs.;Therefore, my dissertation will include the derivation of flow models, their qualitative analysis, the development of numerical methods, and their analysis. Future research will involve the development of computational codes and their parallel versions, and extensions of the mathematical modeling, numerical methods, scientific computing, computer simulation, and the supporting mathematical analysis from ordinary porous media to fractured porous media.