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Multiphase flows in porous media are important in many natural and industrial processes. Pore-scale models for multiphase flows have seen rapid development in recent years and are becoming increasingly useful as predictive tools in both academic and industrial applications. However, quantitative comparisons between different pore-scale models, and between these models and experimental data, are lacking. Here, we perform an objective comparison of a variety of state-of-the-art pore-scale models, including lattice Boltzmann, stochastic rotation dynamics, volume-of-fluid, level-set, phase-field, and pore-network models. As the basis for this comparison, we use a dataset from recent microfluidic experiments with precisely controlled pore geometry and wettability conditions, which offers an unprecedented benchmarking opportunity. We compare the results of the 14 participating teams both qualitatively and quantitatively using several standard metrics, such as fractal dimension, finger width, and displacement efficiency.We find that no single method excels across all conditions and that thin films and corner flow present substantial modeling and computational challenges.

Dilute species transport in generalized Newtonian fluids (GNFs) is typically described using explanatory empirical approaches assuming a traditional Fickian form, which is an approach that lacks predictive ability for systems and conditions not specifically investigated. Dilute species transport was investigated for a wide range of Cross and Carreau fluids flowing through a set of monodisperse and polydisperse sphere pack porous media. Both microscale and macroscale simulations were performed to demonstrate that GNF fluid flow can be predicted based upon Newtonian characterization of the media and rheological characterization of the fluid. Dilute species transport was shown to have a Fickian limit with dispersivity dependent on the porous media, fluid properties, and the flow rate in a nonlinear fashion. Dimensionless analysis and symbolic regression was used to deduce an explanatory and predictive function to describe dispersivity in terms of relevant system properties, enabling prediction of dilute species transport for GNFs flowing through porous media that does not require any non‐Newtonian experiments or parameter estimation.

Systems dominated by heterogeneity over a multiplicity of scales, like porous media, still challenge our modeling efforts. The presence of disparate length- and time-scales that control dynamical processes in porous media hinders not only models predictive capabilities, but also their computational efficiency. Macrosopic models, i.e., averaged representations of pore-scale processes, are computationally efficient alternatives to microscale models in the study of transport phenomena in porous media at the system, field or device scale (i.e., at a scale much larger than a characteristic pore size). We present an overview of common upscaling methods used to formally derive macroscale equations from pore-scale (mass, momentum and energy) conservation laws. This review includes the volume averaging method, mixture theory, thermodynamically constrained averaging, homogenization, and renormalization group techniques. We apply these methods to a number of specific problems ranging from food processing to human bronchial system, and from diffusion to multiphase flow, to demonstrate the methods generality and flexibility in handling different applications. The primary intent of such an overview is not to provide a thorough review of all currently available upscaling techniques, nor a complete mathematical treatment of the ones presented, but rather a primer on some of the tools available for upscaling, the basic principles they are based upon, and their specific advantages and drawbacks, so to guide the reader in the choice of the most appropriate method for particular applications and of the most relevant technical literature.

The thermodynamically constrained averaging theory (TCAT) is a comprehensive theory used to formulate hierarchies of multiphase, multiscale models that are closed based upon the second law of thermodynamics. The rate of entropy production is posed in terms of the product of fluxes and forces of dissipative processes. The attractive features of TCAT include consistency across disparate length scales; thermodynamic consistency across scales; the inclusion of interfaces and common curves as well as phases; the development of kinematic equations to provide closure relations for geometric extent measures; and a structured approach to model building. The elements of the TCAT approach are shown; the ways in which each of these attractive features emerge from the TCAT approach are illustrated; and a review of the hierarchies of models that have been formulated is provided. Because the TCAT approach is mathematically involved, we illustrate how this approach can be applied by leveraging existing components of the theory that can be applied to a wide range of applications. This can result in a substantial reduction in formulation effort compared to a complete derivation while yielding identical results. Lastly, we note the previous neglect of the deviation kinetic energy, which is not important in slow porous media flows, formulate the required equations to extend the theory, and comment on applications for which the new components would be especially useful. This work should serve to make TCAT more accessible for applications, thereby enabling higher fidelity models for applications such as turbulent multiphase flows.

Traditional models of two-fluid flow through porous media at the macroscale have existed for nearly a century. These phenomenological models are not firmly connected to the microscale; thermodynamic constraints are not enforced; empirical closure relations are well known to be hysteretic; fluid pressures are typically assumed to be in a local equilibrium state with fluid saturations; and important quantities such as interfacial and curvilinear geometric extents, tensions, and curvatures, known to be important from microscale studies, do not explicitly appear in traditional macroscale models. Despite these shortcomings, the traditional model for two-fluid flow in porous media has been extensively studied to develop efficient numerical approximation methods, experimental and surrogate measure parameterization approaches, and convenient pre- and post-processing environments; and they have been applied in a large number of applications from a variety of fields. The thermodynamically constrained averaging theory (TCAT) was developed to overcome the limitations associated with traditional approaches, and we consider here issues associated with the closure of this new generation of models. It has been shown that a hysteretic-free state equation exists based upon integral geometry that relates changes in volume fractions, capillary pressure, interfacial areas, and the Euler characteristic. We show an analysis of how this state equation can be parameterized with a relatively small amount of data. We also formulate a state equation for resistance coefficients that we show to be hysteretic free, unlike traditional relative permeability models. Lastly, we comment on the open issues remaining for this new generation of models.

Arsenic (As), chromium (Cr), and vanadium (V) are naturally occurring, redox-active elements that can become human health hazards when they are released from aquifer substrates into groundwater that may be used as domestic or irrigation source. As such, there is a need to develop incisive conceptual and quantitative models of the geochemistry and transport of potentially hazardous elements to assess risk and facilitate interventions. However, understanding the complexity and heterogeneous subsurface environment requires knowledge of solid-phase minerals, hydrologic movement, aerobic and anaerobic environments, microbial interactions, and complicated chemical kinetics. Here, we examine the relevant geochemical and hydrological information about the release and transport of potentially hazardous geogenic contaminants, specifically As, Cr, and V, as well as the potential challenges in developing a robust understanding of their behavior in the subsurface. We explore the development of geochemical models, illustrate how they can be utilized, and describe the gaps in knowledge that exist in translating subsurface conditions into numerical models, as well as provide an outlook on future research needs and developments.

Sharp-front problems arising from equations that tend toward a hyperbolic limit occur routinely in modeling transport phenomena in porous medium systems. Numerical methods to approximate hyperbolic conservation equations are mature for finite volume methods. However, irregularly shaped domains make finite element methods (FEMs) an attractive approach for many subsurface problems. For conforming FEM approaches, sharp fronts continue to pose a computational challenge. We build on recent work to develop and evaluate an entropy viscosity approach. Recent theoretical advancements have resulted in the formulation of entropy inequality equations for many porous medium problems, which we use to compute a physically-based entropy production rate for both linear and nonlinear species transport problems. The entropy production rate is in turn used to add dissipation to the approximation of the hyperbolic operator using an entropy-viscosity formulation. We demonstrate how existing, problem-specific theoretical results can be leveraged in a generic and efficient numerical method to create problem-specific dissipation functions.

The thermodynamically constrained averaging theory (TCAT) has been used to develop a simplified entropy inequality (SEI) for several major classes of macroscale porous medium models in previous works. These expressions can be used to formulate hierarchies of models of varying sophistication and fidelity. A limitation of the TCAT approach is that the determination of model parameters has not been addressed other than the guidance that an inverse problem must be solved. In this work we show how a previously derived SEI for single-fluid-phase flow and transport in a porous medium system can be reduced for the specific instance of diffusion in a dilute system to guide model closure. We further show how the parameter in this closure relation can be reliably predicted, adapting a Green’s function approach used in the method of volume averaging. Parameters are estimated for a variety of both isotropic and anisotropic media based upon a specified microscale structure. The direct parameter evaluation method is verified by comparing to direct numerical simulation over a unit cell at the microscale. This extension of TCAT constitutes a useful advancement for certain classes of problems amenable to this estimation approach.

The thermodynamically constrained averaging theory (TCAT) is an evolving approach for formulating macroscale models that are consistent with both microscale physics and thermodynamics. This consistency requires some mathematical complexity, which can be an impediment to understanding and efficient application of this model-building approach for the non-specialist. To aid understanding of the TCAT approach, a simplified model formulation approach is developed and used to show a more compact, but less general, formulation compared to the standard TCAT approach. This new simplified model formulation approach is applied to the case of binary species diffusion in a single-fluid-phase porous medium system, clearly showing a TCAT approach that is applicable to many other systems as well. Recent extensions to the TCAT approach that enable a priori parameter estimation, and approaches to leverage available TCAT modeling building results are also discussed.

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.