Thermodynamically Constrained Averaging Theory Approach for Modeling Flow and Transport Phenomena in Porous Medium Systems: 4. Species Transport Fundamentals

Analysis of 'Investigating an extended multiphase flow model that includes specific interfacial area', Computer Methods in Applied Mechanics and Engineering, 418:116594, 2024

Comments are provided on the recent paper by Ebadi et al. [3], which demonstrates that the formulated model that was solved contains misconceptions or errors that render the work unsuitable for describing the evolution of interfacial areas in two-fluid porous medium systems. The need for kinematic equations is described and components of a theoretically consistent approach are summarized.

Modeling flow of Carreau fluids in porous media

Carreau fluids occur routinely in porous medium systems for a range of applications, and the dependence of the viscosity for such fluids on the rate of strain tensor poses challenges to modeling at an averaged macroscale. Traditional approaches for macroscale modeling such flows have relied upon experimental observations of flows for generalized Newtonian fluids (GNFs) and a phenomenological approach referred to herein as the shift factor. A recently developed approach based upon averaging conservation and thermodynamic equations from the microscale for Cross model GNFs is extended to the case of Carreau fluids and shown to predict the flow through both isotropic and anisotropic media accurately without the need for GNF-flow experiments. The model is formulated in terms of rheological properties, a standard Newtonian resistance tensor, and a length-scale tensor, which does require estimation. An approach based upon measures of the morphology and topology of the pore space is developed to approximate this length-scale tensor. Thus, this work provides the missing components needed to predict Carreau GNF macroscale flow with only rheological information for the fluid and analysis of the pore morphology and topology independent of any fluid flow experiments. Accuracy of predictions based upon this approach is quantified, and extension to other GNFs is straightforward.

In Silico Prediction of Food Properties: A Multiscale Perspective

Several open software packages have popularized modeling and simulation strategies at the food product scale. Food processing and key digestion steps can be described in 3D using the principles of continuum mechanics. However, compared to other branches of engineering, the necessary transport, mechanical, chemical, and thermodynamic properties have been insufficiently tabulated and documented. Natural variability, accented by food evolution during processing and deconstruction, requires considering composition and structure-dependent properties. This review presents practical approaches where the premises for modeling and simulation start at a so-called “microscopic” scale where constituents or phase properties are known. The concept of microscopic or ground scale is shown to be very flexible from atoms to cellular structures. Zooming in on spatial details tends to increase the overall cost of simulations and the integration over food regions or time scales. The independence of scales facilitates the reuse of calculations and makes multiscale modeling capable of meeting food manufacturing needs. On one hand, new image-modeling strategies without equations or meshes are emerging. On the other hand, complex notions such as compositional effects, multiphase organization, and non-equilibrium thermodynamics are naturally incorporated in models without linearization or simplifications. Multiscale method’s applicability to hierarchically predict food properties is discussed with comprehensive examples relevant to food science, engineering and packaging. Entropy-driven properties such as transport and sorption are emphasized to illustrate how microscopic details bring new degrees of freedom to explore food-specific concepts such as safety, bioavailability, shelf-life and food formulation. Routes for performing spatial and temporal homogenization with and without chemical details are developed. Creating a community sharing computational codes, force fields, and generic food structures is the next step and should be encouraged. This paper provides a framework for the transfer of results from other fields and the development of methods specific to the food domain.

Generalized Newtonian fluid flow in porous media

Single-fluid-phase porous medium systems are typically modeled at an averaged length scale termed the macroscale, and Darcy's law is typically relied upon as an approximation of the momentum equation under Stokes flow conditions. Standard approaches for modeling macroscale single-fluid-phase flow of generalized Newtonian fluids (GNFs) extend the standard Newtonian model based upon Darcy's law using an effective viscosity and assuming that the intrinsic permeability is invariant with respect to fluid properties. This approach results in a need to perform an experiment for a non-Newtonian fluid, the introduction of effective parameters that are not tied to known microscale physics, and uncertainty regarding the dependencies of the fitted empirical parameter on system properties. We use the thermodynamically constrained averaging theory (TCAT) to examine the formulation and closure of a macroscale model for GNF flow that is consistent with microscale conservation principles and the second law of thermodynamics. A direct connection between microscale and macroscale quantities is used to formulate an expression for interphase momentum transfer for GNF flow in porous medium systems. Darcy's law is shown to approximate momentum transfer from the fluid phase to the solid phase. Momentum transfer is found to depend on the viscosity at the solid surface, which is only invariant for Newtonian flow. TCAT is used to derive a macroscale equation for the hydraulic resistance based on accessible fluid and solid properties. This hydraulic resistance may be used in the same way that hydraulic conductivity is typically used to model flow at the macroscale, and it includes parameters that can be calculated a priori, without the need to carry out microscale simulations, or experiments, for any GNF. The TCAT approach is validated for four model isotropic and anisotropic media and five Cross-model fluids. The traditional shift factor and effective viscosity are related to the newly derived TCAT model, shedding new light on this common empirical approach. The results from this work form a basis for the modeling of GNF flow in porous medium systems under Stokes flow, which is predictive given the rheological properties of the GNF and the resistance observed for Newtonian flow.

Microscale modeling of nondilute flow and transport in porous medium systems

Nondilute transport occurs routinely in porous medium systems. Experimental observations have revealed effects that seemingly depend upon density, viscosity, velocity, and chemical activity. Macroscale models based upon averaged behavior over many pores have been relied upon to describe such systems to date, which require parametrization of important physical phenomena in material coefficients. To advance fundamental understanding of these complex systems, we examine nondilute transport from a fundamental microscale, or pore-scale, continuum modeling perspective. We approximate the solution of a model based upon the variable-density Navier-Stokes equations and a nondilute species transport equation. Known dependencies of the densities, viscosities, chemical activity, and diffusion for a salt solution on chemical composition are included in the model. Microscale model solutions are averaged to the macroscale and compared with extant experimental observations. Investigation of the effects of various physical phenomena on the microscale velocity distribution and the observed macroscale dispersion are considered using dimensional analysis and constrained simulations. Simulation results are used to explain observed experimental results in light of underlying mechanisms. Conditions under which the various physicochemical effects investigated are important are revealed.

Toward a New Generation of Two-Fluid Flow Models Based on the Thermodynamically-Constrained Averaging Theory

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.

Averaging Theory for Description of Environmental Problems: What Have We Learned?

Advances in Water Resources has been a prime archival source for implementation of averaging theories in changing the scale at which processes of importance in environmental modeling are described. Thus in celebration of the 35th year of this journal, it seems appropriate to assess what has been learned about these theories and about their utility in describing systems of interest. We review advances in understanding and use of averaging theories to describe porous medium flow and transport at the macroscale, an averaged scale that models spatial variability, and at the megascale, an integral scale that only considers time variation of system properties. We detail physical insights gained from the development and application of averaging theory for flow through porous medium systems and for the behavior of solids at the macroscale. We show the relationship between standard models that are typically applied and more rigorous models that are derived using modern averaging theory. We discuss how the results derived from averaging theory that are available can be built upon and applied broadly within the community. We highlight opportunities and needs that exist for collaborations among theorists, numerical analysts, and experimentalists to advance the new classes of models that have been derived. Lastly, we comment on averaging developments for rivers, estuaries, and watersheds.

Averaging Theory for Description of Environmental Problems: What Have We Learned?

Advances in Water Resources has been a prime archival source for implementation of averaging theories in changing the scale at which processes of importance in environmental modeling are described. Thus in celebration of the 35th year of this journal, it seems appropriate to assess what has been learned about these theories and about their utility in describing systems of interest. We review advances in understanding and use of averaging theories to describe porous medium flow and transport at the macroscale, an averaged scale that models spatial variability, and at the megascale, an integral scale that only considers time variation of system properties. We detail physical insights gained from the development and application of averaging theory for flow through porous medium systems and for the behavior of solids at the macroscale. We show the relationship between standard models that are typically applied and more rigorous models that are derived using modern averaging theory. We discuss how the results derived from averaging theory that are available can be built upon and applied broadly within the community. We highlight opportunities and needs that exist for collaborations among theorists, numerical analysts, and experimentalists to advance the new classes of models that have been derived. Lastly, we comment on averaging developments for rivers, estuaries, and watersheds.

A multiphase model for three-dimensional tumor growth

Several mathematical formulations have analyzed the time-dependent behaviour of a tumor mass. However, most of these propose simplifications that compromise the physical soundness of the model. Here, multiphase porous media mechanics is extended to model tumor evolution, using governing equations obtained via the Thermodynamically Constrained Averaging Theory (TCAT). A tumor mass is treated as a multiphase medium composed of an extracellular matrix (ECM); tumor cells (TC), which may become necrotic depending on the nutrient concentration and tumor phase pressure; healthy cells (HC); and an interstitial fluid (IF) for the transport of nutrients. The equations are solved by a Finite Element method to predict the growth rate of the tumor mass as a function of the initial tumor-to-healthy cell density ratio, nutrient concentration, mechanical strain, cell adhesion and geometry. Results are shown for three cases of practical biological interest such as multicellular tumor spheroids (MTS) and tumor cords. First, the model is validated by experimental data for time-dependent growth of an MTS in a culture medium. The tumor growth pattern follows a biphasic behaviour: initially, the rapidly growing tumor cells tend to saturate the volume available without any significant increase in overall tumor size; then, a classical Gompertzian pattern is observed for the MTS radius variation with time. A core with necrotic cells appears for tumor sizes larger than 150 μm, surrounded by a shell of viable tumor cells whose thickness stays almost constant with time. A formula to estimate the size of the necrotic core is proposed. In the second case, the MTS is confined within a healthy tissue. The growth rate is reduced, as compared to the first case - mostly due to the relative adhesion of the tumor and healthy cells to the ECM, and the less favourable transport of nutrients. In particular, for tumor cells adhering less avidly to the ECM, the healthy tissue is progressively displaced as the malignant mass grows, whereas tumor cell infiltration is predicted for the opposite condition. Interestingly, the infiltration potential of the tumor mass is mostly driven by the relative cell adhesion to the ECM. In the third case, a tumor cord model is analyzed where the malignant cells grow around microvessels in a 3D geometry. It is shown that tumor cells tend to migrate among adjacent vessels seeking new oxygen and nutrient. This model can predict and optimize the efficacy of anticancer therapeutic strategies. It can be further developed to answer questions on tumor biophysics, related to the effects of ECM stiffness and cell adhesion on tumor cell proliferation.

Thermodynamically Constrained Averaging Theory Approach for Modeling Flow and Transport Phenomena in Porous Medium Systems: 8. Interface and Common Curve Dynamics

This work is the eighth in a series that develops the fundamental aspects of the thermodynamically constrained averaging theory (TCAT) that allows for a systematic increase in the scale at which multiphase transport phenomena is modeled in porous medium systems. In these systems, the explicit locations of interfaces between phases and common curves, where three or more interfaces meet, are not considered at scales above the microscale. Rather, the densities of these quantities arise as areas per volume or length per volume. Modeling of the dynamics of these measures is an important challenge for robust models of flow and transport phenomena in porous medium systems, as the extent of these regions can have important implications for mass, momentum, and energy transport between and among phases, and formulation of a capillary pressure relation with minimal hysteresis. These densities do not exist at the microscale, where the interfaces and common curves correspond to particular locations. Therefore, it is necessary for a well-developed macroscale theory to provide evolution equations that describe the dynamics of interface and common curve densities. Here we point out the challenges and pitfalls in producing such evolution equations, develop a set of such equations based on averaging theorems, and identify the terms that require particular attention in experimental and computational efforts to parameterize the equations. We use the evolution equations developed to specify a closed two-fluid-phase flow model.

Thermodynamically Constrained Averaging Theory Approach for Modeling Flow and Transport Phenomena in Porous Medium Systems: 8. Interface and Common Curve Dynamics

This work is the eighth in a series that develops the fundamental aspects of the thermodynamically constrained averaging theory (TCAT) that allows for a systematic increase in the scale at which multiphase transport phenomena is modeled in porous medium systems. In these systems, the explicit locations of interfaces between phases and common curves, where three or more interfaces meet, are not considered at scales above the microscale. Rather, the densities of these quantities arise as areas per volume or length per volume. Modeling of the dynamics of these measures is an important challenge for robust models of flow and transport phenomena in porous medium systems, as the extent of these regions can have important implications for mass, momentum, and energy transport between and among phases, and formulation of a capillary pressure relation with minimal hysteresis. These densities do not exist at the microscale, where the interfaces and common curves correspond to particular locations. Therefore, it is necessary for a well-developed macroscale theory to provide evolution equations that describe the dynamics of interface and common curve densities. Here we point out the challenges and pitfalls in producing such evolution equations, develop a set of such equations based on averaging theorems, and identify the terms that require particular attention in experimental and computational efforts to parameterize the equations. We use the evolution equations developed to specify a closed two-fluid-phase flow model.

Thermodynamically Constrained Averaging Theory Approach for Modeling Flow and Transport Phenomena in Porous Medium Systems: 7. Single-Phase Megascale Flow Models

This work is the seventh in a series that introduces and employs the thermodynamically constrained averaging theory (TCAT) for modeling flow and transport in multiscale porous medium systems. This paper expands the previous analyses in the series by developing models at a scale where spatial variations within the system are not considered. Thus the time variation of variables averaged over the entire system is modeled in relation to fluxes at the boundary of the system. This implementation of TCAT makes use of conservation equations for mass, momentum, and energy as well as an entropy balance. Additionally, classical irreversible thermodynamics is assumed to hold at the microscale and is averaged to the megascale, or system scale. The fact that the local equilibrium assumption does not apply at the megascale points to the importance of obtaining closure relations that account for the large-scale manifestation of small-scale variations. Example applications built on this foundation are suggested to stimulate future work.

Thermodynamically Constrained Averaging Theory Approach for Modeling Flow and Transport Phenomena in Porous Medium Systems: 5. Single-Fluid-Phase Transport

This work is the fifth in a series of papers on the thermodynamically constrained averaging theory (TCAT) approach for modeling flow and transport phenomena in multiscale porous medium systems. The general TCAT framework and the mathematical foundation presented in previous works are used to develop models that describe species transport and single-fluid-phase flow through a porous medium system in varying physical regimes. Classical irreversible thermodynamics formulations for species in fluids, solids, and interfaces are developed. Two different approaches are presented, one that makes use of a momentum equation for each entity along with constitutive relations for species diffusion and dispersion, and a second approach that makes use of a momentum equation for each species in an entity. The alternative models are developed by relying upon different approaches to constrain an entropy inequality using mass, momentum, and energy conservation equations. The resultant constrained entropy inequality is simplified and used to guide the development of closed models. Specific instances of dilute and non-dilute systems are examined and compared to alternative formulation approaches.

Thermodynamically Constrained Averaging Theory Approach for Modeling Flow and Transport Phenomena in Porous Medium Systems: 5. Single-Fluid-Phase Transport

This work is the fifth in a series of papers on the thermodynamically constrained averaging theory (TCAT) approach for modeling flow and transport phenomena in multiscale porous medium systems. The general TCAT framework and the mathematical foundation presented in previous works are used to develop models that describe species transport and single-fluid-phase flow through a porous medium system in varying physical regimes. Classical irreversible thermodynamics formulations for species in fluids, solids, and interfaces are developed. Two different approaches are presented, one that makes use of a momentum equation for each entity along with constitutive relations for species diffusion and dispersion, and a second approach that makes use of a momentum equation for each species in an entity. The alternative models are developed by relying upon different approaches to constrain an entropy inequality using mass, momentum, and energy conservation equations. The resultant constrained entropy inequality is simplified and used to guide the development of closed models. Specific instances of dilute and non-dilute systems are examined and compared to alternative formulation approaches.

Thermodynamically Constrained Averaging Theory Approach for Modeling Flow and Transport Phenomena in Porous Medium Systems: 5. Single-Fluid-Phase Transport

This work is the fifth in a series of papers on the thermodynamically constrained averaging theory (TCAT) approach for modeling flow and transport phenomena in multiscale porous medium systems. The general TCAT framework and the mathematical foundation presented in previous works are used to develop models that describe species transport and single-fluid-phase flow through a porous medium system in varying physical regimes. Classical irreversible thermodynamics formulations for species in fluids, solids, and interfaces are developed. Two different approaches are presented, one that makes use of a momentum equation for each entity along with constitutive relations for species diffusion and dispersion, and a second approach that makes use of a momentum equation for each species in an entity. The alternative models are developed by relying upon different approaches to constrain an entropy inequality using mass, momentum, and energy conservation equations. The resultant constrained entropy inequality is simplified and used to guide the development of closed models. Specific instances of dilute and non-dilute systems are examined and compared to alternative formulation approaches.