engleski

Modelling, Analysis and Numerical Simulations of Immiscible Compressible Two-Phase Fluid Flow in Heterogeneous Porous Media

In petroleum engineering and environmental engineering, many processes can be mathematically modelled as multiphase fluid flow in porous media. Carbon dioxide sequestration and long-term nuclear waste storage are examples of such processes. This thesis studies immiscible compressible two-phase fluid flow in porous media. Such flows can be modelled by a set of nonlinear partial differential equations. Multiple formulations and main variable selections are possible, and the choice of the formulation and the main variables strongly influences the PDEs system mathematical analysis and its solving by means of numerical methods. In this thesis, a new formulation for modelling immiscible compressible two-phase flow in heterogeneous porous media is developed and studied. For each phase, the governing equations describing this type of flow are given by the mass balance law and the Darcy-Muscat law. This original system of nonlinear evolutionary partial differential equations is transformed, using the concept of global pressure, to a system of PDEs which is more suitable for mathematical and numerical studies. The new model is fully equivalent to the starting one. In order to make the problem solving more tractable, another model is discussed as well: a simplified model. In this model, further simplifications are performed where applicable: in the fractional flow formulations, the phase pressures are replaced by the global pressure in the calculation of the mass densities. This model was introduced by other authors [25], and an updated version is used in the scope of this thesis. A numerical code based on the vertex centered finite volume method is developed. A comparison of the new model with the simplified fractional flow formulation is performed in two ways: by means of numerical simulations and by comparing the coefficients. The comparisons reveal that the simplified model can be used safely in applications where the mean field pressure is high, capillary pressure is low and the wetting phase is not highly compressible. In the presentation of the numerical simulations, a special attention is paid to the numerical treatment of highly heterogeneous media (multiple rock types) by the vertex centred finite volume method. The model is validated by numerical simulations on the benchmarks proposed by the French research group MoMaS. The benchmarks are related to the flow of water (incompressible) and gas such as hydrogen (compressible), concerning the gas migration through engineered and geological barriers for the deep repository of radioactive waste. In the thesis, the existence result for the new formulation for two-phase compressible flow under realistic assumptions on the data is established. While earlier existence results on the existence of immiscible compressible systems are known, the major difference of the work presented to the earlier results is that the required hypotheses are significantly weakened, so that only physically relevant assumptions are made. In particular, the results presented cover the cases of a singular capillary pressure function, and the discontinuous porosity and absolute permeability tensors.

Two-phase flow; immiscible; compressible; Global pressure; heterogeneous porous media

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