Upscaling of two-phase porous-media flows with evolving fluid-fluid interfaces at the pore scale

Abstract

This thesis concerns the upscaling of two-phase/unsaturated flow and transport problems in porous media. Whereas the mathematical models at the pore scale are in general well understood, but due to their complexity, they are not suited for large-scale numerical simulations. Hence, in practice, Darcy-scale models are better suited. The aim of this thesis is to derive multiscale mathematical models that bridge the gap between poreand Darcy-scale models. Asymptotic expansions together with transversal averaging and periodic homogenization are used to derive Darcy-scale models starting from pore-scale models. These resulting models are expressed in terms of effective Darcy-scale quantities like saturation, concentration, pressure and Darcy velocity. The derived Darcy-scale models are important since they describe the averaged behaviour of pore-scale models, but still reproduce the main flow features observed at the pore-scale.

First, we consider a pore-scale model for two-phase/unsaturated flow, defined in a two-dimensional thin strip. We use a sharp-interface approach to model the evolution of the free boundary. We consider three cases: two-phase flow with solute-dependent surface tension (the Marangoni effect), two-phase flow with constant surface tension, and unsaturated flow with constant surface tension. By assuming that the ratio between the width and length of the strip approaches 0, we use formal asymptotic expansion methods and derive the limit of transversally averaged models over the thin strip. Depending on the dimensionless parameters, different upscaled models in various pore-scale regimes are obtained. The resulting models involve Darcy-type laws for the flow, with a concentration dependent surface tension effect (Marangoni effect) and a capillary pressure-saturation dependency involving a second-order derivative of the saturation. These formal results are validated by numerical experiments. We compare the solution of the upscaled models valid in the same capillary regime with the transversally averaged pore-scale quantities, and show that the Marangoni effect influences the overall flow.

Since the porous geometry largely influences the averaged/upscaled quantities and their behavior, we also consider a more general domain, namely a periodically perforated domain. We use a phase-field model for two-phase flow and surfactant transport at the pore scale. Using periodic homogenization theory, we derive a two-scale phase-field model describing the averaged behaviour of the system at the Darcy scale. The resulting twoscale model includes extended Darcy-type laws for the effective velocities, accounting for the concentration-dependent surface tension. The effective quantities are found through the corresponding local (cell) problems at the pore scale. For this two-scale phase-field model, we formulate a numerical scheme and present numerical results highlighting the influence of the solute-dependent surface tension.