As a tool for addressing problems of scale, we consider an evolving approach known as the thermodynamically constrained averaging theory (TCAT), which has broad applicability to hydrology. We consider the case of modeling of two-fluid-phase flow in porous media, and we focus on issues of scale as they relate to various measures of pressure, capillary pressure, and state equations needed to produce solvable models. We apply TCAT to perform physics-based data assimilation to understand how the internal behavior influences the macroscale state of two-fluid porous medium systems. A microfluidic experimental method and a lattice Boltzmann simulation method are used to examine a key deficiency associated with standard approaches. In a hydrologic process such as evaporation, the water content will ultimately be reduced below the irreducible wetting-phase saturation determined from experiments. This is problematic since the derived closure relationships cannot predict the associated capillary pressures for these states. We demonstrate that the irreducible wetting-phase saturation is an artifact of the experimental design, caused by the fact that the boundary pressure difference does not approximate the true capillary pressure. Using averaging methods, we compute the true capillary pressure for fluid configurations at and below the irreducible wetting-phase saturation. Results of our analysis include a state function for the capillary pressure expressed as a function of fluid saturation and interfacial area.

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.

JAMES MCCLURE: Microscale Modeling of Fluid Flow in Porous Medium Systems. (Under the direction of Cass T. Miller.) Proper mathematical description of macroscopic porous medium flows is essential for the study of a wide range of subsurface contamination scenarios. Existing mathematical formulations, however, demonstrate inadequacies that preclude the accurate description of many systems. Multi-scale models developed using thermodynamically constrained averaging theory (TCAT) rigorously define macroscopic variables in terms of more well-understood microscopic counterparts, permitting detailed analysis of macroscopic model forms based on microscale simulation and experiment. Within this framework, the primary objectives of microscale modeling are to elucidate important physical mechanisms and to inform both the form of macroscale closure relations as well as associated parameter values. In order to meet these goals, numerical tools must include: (1) simulations that provide accurate microscopic solutions for physical phenomena in large, complex domains; (2) morphological analysis tools that can be used to upscale simulation results to larger scales as dictated by the associated theoretical framework. Development of a numerical toolbox for microscale porous medium studies is considered in line with these objectives, including both implementation and optimization strategies. High-performance implementations of the lattice Boltzmann method are developed to simulate oneand two-phase flows using several computing platforms. A modified marching cubes algorithm is developed to explicitly construct all entities in a two-phase system, including all interfaces between the fluid and solid phases in addition to the three phase contact curve. These entities serve as a numerical skeleton for upscaling multiphase porous medium simulation results to the macroscale. Based on these tools, development of macroscopic constitutive laws is illustrated for a special case of anisotropic flow in porous media. In this example, microscale simulation is used to demonstrate a limitation of existing macroscopic forms for cases in which the momentum resistance depends on the flow direction in addition to the orientation. A modified macroscopic form is proposed in order to properly account for this phenomenon.

Fundamental conservation equations for mass, momentum and energy of chemical species can be combined with thermodynamic relations to obtain secondary forms, such as conservation equations for phases, an internal energy balance and a mechanical energy balance. In fact, the forms of secondary equations are infinite and depend on the criteria used in determining which species-based equations to employ and how to combine them. If one uses these secondary forms in developing an entropy inequality to be used in formulating closure relations, care must be employed to ensure that the appropriate equations are used, or problematic results can develop for multispecies systems. We show here that the use of the fundamental forms minimizes the chance of an erroneous formulation in terms of secondary forms and also provides guidance as to which secondary forms should be used if one uses them as a starting point.

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. DOI:10.1103/PhysRevE.87.033012