A model formulated in terms of both conservation and kinematic equations for phases and interfaces in two-fluid-phase flow in a porous medium system is summarized. Macroscale kinematic equations are derived as extensions of averaging theorems and do not rely on conservation principles. Models based on both conservation and kinematic equations can describe multiphase flow with varying fidelity. When only phase-based equations are considered, a model similar in form to the traditional model for twofluid-phase flow results. When interface conservation and kinematic equations are also included, a novel formulation results that naturally includes evolution equations that express dynamic changes in fluid saturations, pressures, the capillary pressure, and the fluid-fluid interfacial area density in a two-fluid-system. This dynamic equation set is unique to this work, and the importance of the modeled physics is shown through both microfluidic experiments and high-resolution lattice Boltzmann simulations. The validation work shows that the relaxation of interface distribution and shape toward an equilibrium state is a slow process relative to the time scale typically allowed for a system to approach an apparent equilibrium state based upon observations of fluid saturations and external pressure measurements. Consequently, most pressuresaturation data intended to denote an equilibrium state are likely a sampling from a dynamic system undergoing changes of interfacial curvatures that are not typically monitored. The results confirm the importance of kinematic analysis in combination with conservation equations for faithful modeling of system physics.

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Fluid flow through isotropic and anisotropic porous medium systems is investigated for a range of Reynolds numbers corresponding to both Darcy and non-Darcy regimes. A non-dimensional formulation is developed for a Forchheimer approximation of the momentum balance, and lattice Boltzmann simulations are used to elucidate the effects of porous medium characteristics on macroscale constitutive relation parameters. The geometric orientation tensor of the solid phase is posited as a morphological measure of leading-order importance for the description of anisotropic flows. Simulation results are presented to confirm this hypothesis, and parameter correlations are developed to predict closure relation coefficients as a function of porous medium porosity, specific interfacial area of the solid phase, and the geometric orientation tensor. The developed correlations provide improved estimates of model coefficients compared to available estimates and extend predictive capabilities to fully determine macroscopic momentum parameters for three-dimensional flows in anisotropic porous media.

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We formulate a multiple-relaxation-time (MRT) lattice-Boltzmann method (LBM) to simulate two-fluid-phase flow in porous medium systems. The MRT LBM is applied to simulate the displacement of a wetting fluid by a nonwetting fluid in a system corresponding to a microfluidic cell. Analysis of the simulation shows widely varying time scales for the dynamics of fluid pressures, fluid saturations, and interfacial curvatures that are typical characteristics of such systems. Displacement phenomena include Haines jumps, which are relatively short duration isolated events of rapid fluid displacement driven by capillary instability. An adaptive algorithm is advanced using a level-set method to locate interfaces and estimate their rate of advancement. Because the displacement dynamics are confined to the interfacial regions for a majority of the relaxation time, the computational effort is focused on these regions. The proposed algorithm is shown to reduce computational effort by an order of magnitude, while yielding essentially identical solutions to a conventional fully coupled approach. The challenges posed by Haines jumps are also resolved by the adaptive algorithm. Possible extensions to the advanced method are discussed.