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Homogenization and Mathematical Analysis of Immiscible Compressible Two-Phase Flow in Heterogeneous Porous Media by the Concept of the Global Pressure

Mathematical modeling of multiphase fluid flow through porous media is of great practical importance in problems of petroleum and environmental engineering. The variations of the physical properties of the medium occur at distinct space scales, which strongly affects the flow through the heterogeneous porous media and seriously complicates its analysis and the numerical simulations. In this thesis we study a model of the immiscible compressible two-phase flow in porous media in a new, lately established, fully equivalent formulation which uses the notion of the global pressure. The main difficulties in the analysis of this system of evolutionary PDEs for the global pressure and the phase saturation are the nonlinearity, degeneracy and coupling of the equations. Direct numerical and analytical methods for problems of flow in a heterogeneous porous medium are impossible or inefficient due to its considerable inhomogeneity. Therefore the homogenization is used to replace the equations of the original problem by the simpler macroscopic effective equations that well approximate the average behavior of the system. In this thesis we derive three new results on the existence and homogenization for the new global pressure model of immiscible compressible flow in porous media. In comparison to the previous results for this type of flow, which were obtained for certain approximate models, also our hypotheses on the data are significantly relieved so that a physically justified unbounded capillary pressure and the discontinuous porosity and permeability are allowed. First we prove the existence of weak solutions for the immiscible flow of water and gas. The non homogeneous Dirichlet and Neumann boundary data are covered. The proof uses an appropriate regularization, a time discretization and a modified compactness result. Next, we rigorously justify the homogenization process for the immiscible flow of two compressible fluids in a strongly heterogeneous porous medium of a single-rock type. Microscopically, the periodic heterogeneous porous medium is scaled by a small parameter ε representing the typical size of the periodicity blocks with respect to the domain size, and the medium porosity and the permeability are modeled as ε-periodic functions. We pass to the limit as ε → 0 in the microscopic problem by means of the two-scale convergence technique to obtain a nonlinear homogenized problem with effective coefficients which are computed via a cell problem. Finally, in this thesis we also establish the convergence of the homogenization procedure for a double porosity model of the immiscible water-gas flow in a naturally fractured reservoir. This type of porous media consists of a disconnected periodic system of blocks of usual porous media, matrix, which are separated by a net of thin fractures. The matrix keeps most of the fluid while the fractures are notably more permeable. The width of the fractures is assumed to be of the same order as the block size, and the ratio of the permeability of the matrix and the fractures is of order ε^2. Such scaling preserves the flow from the matrix to the fractures. We derive the effective problem for the fracture flow where an additional source-like term arises which depicts the matrix-fracture flow. Its non-explicit form is caused by the nonlinearity and the coupling in the system. On the other hand, in the limit as ε → 0 to each point of the reservoir corresponds a matrix block. The double porosity model comprises the set of the equations for each matrix block, that capture the flow at the medium-size scale. We use the dilation operator in order to obtain the effective matrix problem and identify the fracture source term.

Homogenization; two-phase flow; immiscible; compressible; heterogeneous porous media; global pressure

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