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Modeling of flow and transport in environmental systems often involves formulation of conservation equations at spatial scales involving tens to hundreds of pore diameters in porous media or the depth of flow in a channel. Quantities such as density, temperature, internal energy, and velocity may not be uniform over these macroscopic length scales. The external gravitational potential causes gradients in density, pressure, and chemical potential even at equilibrium. Despite these complications, it is important to formulate the thermodynamic analysis of environmental systems at the macroscopic scale. Heretofore, this has been accomplished primarily using the approach of rational thermodynamics whereby the thermodynamic dependence of macroscale internal energy on macroscale variables is hypothesized directly without development of any systematic method for transforming microscale energy dependence from the microscale to the macroscale. However when thermodynamic variables are inhomogeneous at the microscale, the functional dependence of macroscale internal energy on macroscale variables is not a simple extension of the microscale case. In the present work, the relation between the definitions of microscale and macroscale intensive thermodynamic variables is established. Expressions for the material derivatives of macroscale internal energy of phases, interfaces, and common lines are derived from and consistent with their microscopic counterparts by integrating to the macroscale. The forms obtained and the consistency required will be important for use in analyses of systems at scales where microscopic heterogeneities cannot be neglected. 相似文献
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This work is the fourth 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 built upon by formulating macroscale models for conservation of mass, momentum, and energy, and the balance of entropy for a species in a phase volume, interface, and common curve. In addition, classical irreversible thermodynamic relations for species in entities are averaged from the microscale to the macroscale. Finally, we comment on alternative approaches that can be used to connect species and entity conservation equations to a constrained system entropy inequality, which is a key component of the TCAT approach. The formulations detailed in this work can be built upon to develop models for species transport and reactions in a variety of multiphase systems. 相似文献
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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. 相似文献
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《Advances in water resources》1998,22(4):367-398
The basic aim of this paper is to formulate rigorous conservation equations for mass, momentum, energy and entropy for a watershed organized around the channel network. The approach adopted is based on the subdivision of the whole watershed into smaller discrete units, called representative elementary watersheds (REW), and the formulation of conservation equations for these REWs. The REW as a spatial domain is divided into five different subregions: (1) unsaturated zone; (2) saturated zone; (3) concentrated overland flow; (4) saturated overland flow; and (5) channel reach. These subregions all occupy separate volumina. Within the REW, the subregions interact with each other, with the atmosphere on top and with the groundwater or impermeable strata at the bottom, and are characterized by typical flow time scales.The balance equations are derived for water, solid and air phases in the unsaturated zone, water and solid phases in the saturated zone and only the water phase in the two overland flow zones and the channel. In this way REW-scale balance equations, and respective exchange terms for mass, momentum, energy and entropy between neighbouring subregions and phases, are obtained. Averaging of the balance equations over time allows to keep the theory general such that the hydrologic system can be studied over a range of time scales. Finally, the entropy inequality for the entire watershed as an ensemble of subregions is derived as constraint-type relationship for the development of constitutive relationships, which are necessary for the closure of the problem. The exploitation of the second law and the derivation of constitutive equations for specific types of watersheds will be the subject of a subsequent paper. 相似文献
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General conservation equations for multi-phase systems: 3. Constitutive theory for porous media flow
Equations which describe single phase fluid flow and transport through an elastic porous media are obtained by applying constitutive theory to a set of general multiphase mass, momentum, energy, and entropy equations. Linearization of these equations yields a set of equations solvable upon specification of the material coefficients which arise. Further restriction of the flow to small velocities proves that Darcy's law is a special case of the general momentum balance. 相似文献
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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. 相似文献
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Recent advances in multi-phase flow theory have shown that the flow of several phases in a porous medium is highly influenced by the interfaces separating these phases. First modeling studies based on this new theory have been performed on a pore scale, as well as on a volume-averaged macro scale using balance equations and constitutive relations that take the role and presence of interfaces into account. However, neither experimental data nor analytical solutions are available on the macro scale so far, although their knowledge is essential for the verification of the new models. 相似文献
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This work is the third in a series of papers on the thermodynamically constrained averaging theory (TCAT) approach to modeling flow and transport phenomena in multiscale porous medium systems. Building upon the general TCAT framework and the mathematical foundation presented in previous works in this series, we demonstrate the TCAT approach for the case of single-fluid-phase flow. The formulated model is based upon conservation equations for mass, momentum, and energy and a general entropy inequality constraint, which is developed to guide model closure. A specific example of a closed model is derived under limiting assumptions using a linearization approach and these results are compared and contrasted with the traditional single-phase-flow model. Potential extensions to this work are discussed. Specific advancements in this work beyond previous averaging theory approaches to single-phase flow include use of macroscale thermodynamics that is averaged from the microscale, the use of derived equilibrium conditions to guide a flux–force pair approach to simplification, use of a general Lagrange multiplier approach to connect conservation equation constraints to the entropy inequality, and a focus on producing complete, closed models that are solvable. 相似文献
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《Advances in water resources》1987,10(1):32-43
The mechanics of incompressible multiphase particulate suspensions in a liquid is considered. The basic conservation laws are derived via a spatial averaging method. The thermodynamics of the multiphase system is studied and the appropriate constitutive equations are developed. The basic equations of motion of various constituents are derived and discussed. The model includes the effects of pressure differences in various phases, diffusion forces, equilibrated forces and virtual mass forces. A simplified version of the model is used and the problem of gravity sedimentation is treated. It is shown that the model leads to a stable uniform settling rate. 相似文献
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《Advances in water resources》2005,28(2):161-180
We give several examples of weaknesses in classical, empirically derived models of transport phenomena in porous medium systems. We also place recent attempts to develop improved multiscale porous medium models using averaging theory in context and note deficiencies in these approaches. These deficiencies are found to arise in part from the manner in which thermodynamics is introduced into a constrained entropy inequality, which is used to guide the formation of closed models. Because of this, we briefly examine several established thermodynamic approaches and outline a framework to develop macroscale models that retain consistency with microscale physics and thermodynamics. This framework will be detailed and applied in future papers in this series. 相似文献
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《Advances in water resources》1999,23(1):15-39
The balance equations for mass and momentum, averaged over the scale of a watershed entity, need to be supplemented with constitutive equations relating flow velocities, pressure potential differences, as well as mass and force exchanges within and across the boundaries of a watershed. In this paper, the procedure for the derivation of such constitutive relationships is described in detail. This procedure is based on the method pioneered by Coleman and Noll through exploitation of the second law of thermodynamics acting as a constraint-type relationship. The method is illustrated by its application to some common situations occurring in real world watersheds. Thermodynamically admissible and physically consistent constitutive relationships for mass exchange terms among the subregions constituting the watershed (subsurface zones, overland flow regions, channel) are proposed. These constitutive equations are subsequently combined with equations of mass balance for the subregions. In addition, constitutive relationships for forces exchanged amongst the subregions are also derived within the same thermodynamic framework. It is shown that, after linearisation of the latter constitutive relations in terms of the velocity, a watershed-scale Darcy's law governing flow in the unsaturated and saturated zones can be obtained. For the overland flow, a second order constitutive relationship with respect to velocity is proposed for the momentum exchange terms, leading to a watershed-scale Chezy formula. For the channel network REW-scale Saint–Venant equations are derived. Thus, within the framework of this approach new relationships governing exchange terms for mass and momentum are obtained and, moreover, some well-known experimental results are derived in a rigorous manner. 相似文献
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We present an extended law of mass-action (xLMA) method for multiphase equilibrium calculations and apply it in the context of reactive transport modeling. This extended LMA formulation differs from its conventional counterpart in that (i) it is directly derived from the Gibbs energy minimization (GEM) problem (i.e., the fundamental problem that describes the state of equilibrium of a chemical system under constant temperature and pressure); and (ii) it extends the conventional mass-action equations with Lagrange multipliers from the Gibbs energy minimization problem, which can be interpreted as stability indices of the chemical species. Accounting for these multipliers enables the method to determine all stable phases without presuming their types (e.g., aqueous, gaseous) or their presence in the equilibrium state. Therefore, the here proposed xLMA method inherits traits of Gibbs energy minimization algorithms that allow it to naturally detect the phases present in equilibrium, which can be single-component phases (e.g., pure solids or liquids) or non-ideal multi-component phases (e.g., aqueous, melts, gaseous, solid solutions, adsorption, or ion exchange). Moreover, our xLMA method requires no technique that tentatively adds or removes reactions based on phase stability indices (e.g., saturation indices for minerals), since the extended mass-action equations are valid even when their corresponding reactions involve unstable species. We successfully apply the proposed method to a reactive transport modeling problem in which we use PHREEQC and GEMS as alternative backends for the calculation of thermodynamic properties such as equilibrium constants of reactions, standard chemical potentials of species, and activity coefficients. Our tests show that our algorithm is efficient and robust for demanding applications, such as reactive transport modeling, where it converges within 1–3 iterations in most cases. The proposed xLMA method is implemented in Reaktoro, a unified open-source framework for modeling chemically reactive systems. 相似文献
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《国际泥沙研究》2020,35(4):395-407
A two-dimensional vertical (2DV), Eulerian two-phase model or complete two-fluid model of the free surface flow was developed to simulate water-sediment flow in a local scour hole. In the model, the complete forms of the vertical, two-dimensional, two-fluid Navier-Stokes equations were discretized using a finite volume scheme. This discretization was done based on a standard staggered grid system using a curvilinear network system in compliance with the bed boundaries and water level. At the beginning of the computational cycle, the equations governing the fluid phase were solved based on the two-step projection method with a pressure-correction technique. In the first step, the intermediate fluid velocities were obtained by solving different phases of the momentum equations of the fluid phase using the time-splitting technique. In the second step, pressure was obtained and fluid velocities were updated. In this step a simple discretization method was applied for decreasing the computational complexity. After obtaining all the fluid phase variables at a new time step, the sediment phase momentum equations were solved using the time-splitting technique and sediment velocities were obtained. Then, at the end of the computational cycle, the sediment phase mass equation was solved and the concentrations of both phases were updated. At last, the capacity of the model for simulating of the longitudinal fluid velocity and sediment concentration in a local scour hole was evaluated. Numerical results were found to be in good agreement with experimental data. 相似文献
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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. 相似文献
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High resolution images acquired from X-ray μ-CT are able to map the internal structure of porous media on which multiphase flow properties can be computed. While the resolution of a few micrometers may be sufficient for capturing the pore space of many sandstones, most carbonates exhibit a large amount of microporosity; pores which are below the image resolution and are not resolved at specific resolution. Neglecting the effect of micropores on fluid flow and transport properties of these rocks can cause erroneous results in particular at partial saturations. Current image-based pore scale models typically only consider macropores for simulating fluid flow. In this paper, we quantify the effect of microporosity on the effective permeability of the wetting phase for heterogeneous model structures with varying amount of micro-to-macro porosity. A multi-scale numerical approach is proposed to couple an average effect of micropores with an explicit representation of macropores. The Brinkman equation is solved using a lattice Boltzmann formulation to facilitate the coupling of Darcy and Stokes equations in micropores and macropores, respectively. The results show good agreement between the fine scale solution and the results of the upscaled models in which microporous regions are homogenised. The paper analyses in particular the choice of the momentum sink parameter at low wetting phase saturations. It is shown that this parameter can be found using either a flux-based calculation of permeability of microporous regions or chosen purely on the basis of the effective permeability of these regions. 相似文献
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We present a numerical method for multiphase chemical equilibrium calculations based on a Gibbs energy minimization approach. The method can accurately and efficiently determine the stable phase assemblage at equilibrium independently of the type of phases and species that constitute the chemical system. We have successfully applied our chemical equilibrium algorithm in reactive transport simulations to demonstrate its effective use in computationally intensive applications. We used FEniCS to solve the governing partial differential equations of mass transport in porous media using finite element methods in unstructured meshes. Our equilibrium calculations were benchmarked with GEMS3K, the numerical kernel of the geochemical package GEMS. This allowed us to compare our results with a well-established Gibbs energy minimization algorithm, as well as their performance on every mesh node, at every time step of the transport simulation. The benchmark shows that our novel chemical equilibrium algorithm is accurate, robust, and efficient for reactive transport applications, and it is an improvement over the Gibbs energy minimization algorithm used in GEMS3K. The proposed chemical equilibrium method has been implemented in Reaktoro, a unified framework for modeling chemically reactive systems, which is now used as an alternative numerical kernel of GEMS. 相似文献
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Large amounts of gas can result from anaerobic corrosion of metals and from chemical and biological degradation of organic substances in underground repositories for radioactive waste. Gas generation may lead to the formation of a gas phase bubble and to the migration of radioactive gaseous species. Transport occurs in, at least, in two forms: (1) gas bubble, migration is controlled by advection, dispersion and diffusion in the gas phase, and (2) within water pockets, the dissolved species migrate mainly by diffusion. We consider a two-dimensional system representing an isolated heterogeneous fractured zone. A dipole gas flow field is generated and gas tracers are injected. The delay in the breakthrough curves is studied. A simple method is used to solve the gas species transport equations in multiphase conditions. This method is based on a formal analogy between the equations of gas transport in a two phase system and the equations of solute tracer transport in water saturated systems. We perform a sensitivity analysis to quantify the relevance of the various transport mechanisms. We find that gas tracer migration is very sensitive to gas tracer solubility, which affects gas tracer transport of both mobile and immobile zones, and shows high sensitivity to diffusion in the gas phase, to heterogeneity and to gas pressure, but the largest sensitivity was observed with respect to injection borehole properties, i.e. borehole volume and water filled fraction. 相似文献