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: 4. Species transport fundamentals

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.     

Thermodynamically constrained averaging theory approach for modeling flow and transport phenomena in porous medium systems: 6. Two-fluid-phase flow

This work is the sixth in a series of papers on the thermodynamically constrained averaging theory (TCAT) approach for modeling flow and transport phenomena in multiscale porous medium systems. Building upon the general TCAT framework and the mathematical foundation presented in previous works, the limiting case of connected two-fluid-phase flow is considered. A constrained entropy inequality is developed based upon a set of primary restrictions. Formal approximations are introduced to deduce a general simplified entropy inequality (SEI). The SEI is used along with secondary restrictions and closure approximations consistent with the SEI to produce a general functional form of a two-phase-flow model. The general model is in turn simplified to yield a hierarchy of models by neglecting common curves and by neglecting both common curves and interfaces. The simplest case considered corresponds to a traditional two-phase-flow model. The more sophisticated models including interfaces and common curves are more physically realistic than traditional models. All models in the hierarchy are posed in terms of precisely defined variables that allow for a rigorous connection with the microscale. The explicit nature of the restrictions and approximations used in developing this hierarchy of models provides a clear means to both understand the limitations of traditional models and to build upon this work to produce more realistic models.     

Thermodynamically constrained averaging theory approach for modeling flow and transport phenomena in porous medium systems: 2. Foundation

This paper is the second in a series that details the thermodynamically constrained averaging theory (TCAT) approach for modeling flow and transport phenomena in porous medium systems. In this work, we provide the mathematical foundation upon which the theory is based. Elements of this foundation include definitions of mathematical properties of the systems of concern, previously available theorems needed to formulate models, and several theorems and corollaries, introduced and proven here. These tools are of use in producing complete, closed-form TCAT models for single- and multiple-fluid-phase porous medium systems. Future work in this series will rely and build upon the foundation laid in this work to detail the development of sets of closed models.     

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.     

Thermodynamics and constitutive theory for multiphase porous-media flow considering internal geometric constraints

This paper provides the thermodynamic approach and constitutive theory for closure of the conservation equations for multiphase flow in porous media. The starting point for the analysis is the balance equations of mass, momentum, and energy for two fluid phases, a solid phase, the interfaces between the phases and the common lines where interfaces meet. These equations have been derived at the macroscale, a scale on the order of tens of pore diameters. Additionally, the entropy inequality for the multiphase system at this scale is utilized. The internal energy at the macroscale is postulated to depend thermodynamically on the extensive properties of the system. This energy is then decomposed to provide energy forms for each of the system components. To obtain constitutive information from the entropy inequality, information about the mechanical behavior of the internal geometric structure of the phase distributions must be known. This information is obtained from averaging theorems, thermodynamic analysis, and from linearization of the entropy inequality at near equilibrium conditions. The final forms of the equations developed show that capillary pressure is a function of interphase area per unit volume as well as saturation. The standard equations used to model multiphase flow are found to be very restricted forms of the general equations, and the assumptions that are needed for these equations to hold are identified.     

Thermodynamically constrained averaging theory approach for modeling flow and transport phenomena in porous medium systems: 1. Motivation and overview

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.     

Derivation of different suspension equations in sediment-laden flow from Shannon entropy

In this study the well-known Rouse equation and Barenblatt equation for suspension concentration distribution in a sediment-laden flow is derived using Shannon entropy. Considering dimensionless suspended sediment concentration as a random variable and using principle of maximum entropy, probability density function of suspension concentration is obtained. A new and general cumulative distribution function for the flow domain is proposed which can describe specific previous forms reported in the literature. The cumulative distribution function is tested with a variety sets of experimental data and also compared with previous models. The test results ensure the superiority of the new cumulative distribution function. Further a modified form of the cumulative distribution function is discussed and used to derive the suspension model of Greimann et al. The model parameters are expressed in terms of the Rouse number to show the effectiveness of this study using entropy based approach. Finally a non-linear equation in the Rouse number has been suggested to compute it from the experimental data.     

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.     

A unifying framework for watershed thermodynamics: balance equations for mass,momentum, energy and entropy,and the second law of thermodynamics

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.     

Higher order time integration methods for two-phase flow

Time integration methods that adapt in both the order of approximation and time step have been shown to provide efficient solutions to Richards' equation. In this work, we extend the same method of lines approach to solve a set of two-phase flow formulations and address some mass conservation issues from the previous work. We analyze these formulations and the nonlinear systems that result from applying the integration methods, placing particular emphasis on their index, range of applicability, and mass conservation characteristics. We conduct numerical experiments to study the behavior of the numerical models for three test problems. We demonstrate that higher order integration in time is more efficient than standard low-order methods for a variety of practical grids and integration tolerances, that the adaptive scheme successfully varies the step size in response to changing conditions, and that mass balance can be maintained efficiently using variable-order integration and an appropriately chosen numerical model formulation.     

Multi-scale entropy analysis of Mississippi River flow

Multi-scale entropy (MSE) analysis was applied to the long-term (131 years) daily flow rates (Q) of the Mississippi River (MR) to investigate possible change in the complexity of the MR system due to human activities since 1940s. Unlike traditional entropy-based method that calculates entropy at only one single scale, the MSE analysis provided entropies over multiple time scales and thus accounts for multi-scale structures embedded in time series. It is found that the sample entropy (S _E) for Q of the MR and its two components, overland flow (OF) and base flow (BF), generally increase as time scale increases. More importantly, it is found that there have been entropy decreases in Q, OF, and BF over large time scales. In other words, the MR may have been losing its complexity since 1940s. We explain that the possible loss in the complexity of the MR system may be due to the major changes in land use and land cover and soil conservation practices in the MR basin since 1940s.     

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.     

Numerical analysis of one-dimensional unsaturated flow in layered soils

The paper deals with numerical solutions to the Richards equation to simulate one-dimensional flow processes in the unsaturated zone of layered soil profiles. The equation is expressed in the pressure-based form and a finite-difference algorithm is developed for accurately estimating the values of the hydraulic conductivity between two neighboring nodes positioned in different soil layers, often referred to as the interlayer hydraulic conductivity. The algorithm is based upon flux conservation and continuity of pressure potential at the interface between two consecutive layers, and does not add significantly to simulation run time. The validity of the model is established for a number of test problems by comparing numerical results with the analytical solutions developed by Srivastava and Yeh²⁹ which hold for vertical infiltration towards the water table in a two-layer soil profile. The results show a significant reduction in relative mass balance errors when using the proposed model. Some specific insights into its numerical performance are also gained by comparisons with a numerical model in which the more common geometric averaging operator acts on the interlayer conductivities.     

TCAT Analysis of Capillary Pressure in Non-equilibrium, Two-fluid-phase, Porous Medium Systems

Standard models of flow of two immiscible fluids in a porous medium make use of an expression for the dependence of capillary pressure on the saturation of a fluid phase. Data to support the mathematical expression is most often obtained through a sequence of equilibrium experiments. In addition to such expressions being hysteretic, recent experimental and theoretical studies have suggested that the equilibrium functional forms obtained may be inadequate for modeling dynamic systems. This situation has led to efforts to express relaxation of a system to an equilibrium capillary pressure in relation to the rate of change of saturation. Here, based on insights gained from the thermodynamically constrained averaging theory (TCAT) we propose that dynamic processes are related to changes in interfacial area between phases as well as saturation. A more complete formulation of capillary pressure dynamics is presented leading to an equation that is suitable for experimental study.     

Estimates of effective permeability for non-Newtonian fluid flow in randomly heterogeneous porous media

An understanding of the interplay between non-Newtonian effects in porous media flow and field-scale domain heterogeneity is of great importance in several engineering and geological applications. Here we present a simplified approach to the derivation of an effective permeability for flow of a purely viscous power–law fluid with flow behavior index n in a randomly heterogeneous porous domain subject to a uniform pressure gradient. A standard form of the flow law generalizing the Darcy’s law to non-Newtonian fluids is adopted, with the permeability coefficient being the only source of randomness. The natural logarithm of the permeability is considered a spatially homogeneous and correlated Gaussian random field. Under the ergodic hypothesis, an effective permeability is first derived for two limit 1-D flow geometries: flow parallel to permeability variation (serial-type layers), and flow transverse to permeability variation (parallel-type layers). The effective permeability of a 2-D or 3-D isotropic domain is conjectured to be a power average of 1-D results, generalizing results valid for Newtonian fluids under the validity of Darcy’s law; the conjecture is validated comparing our results with previous literature findings. The conjecture is then extended, allowing the exponents of the power averaging to be functions of the flow behavior index. For Newtonian flow, novel expressions for the effective permeability reduce to those derived in the past. The effective permeability is shown to be a function of flow dimensionality, domain heterogeneity, and flow behavior index. The impact of heterogeneity is significant, especially for shear-thinning fluids with a low flow behavior index, which tend to exhibit channeling behavior.     

Ecohydrological modelling of flow duration curve in Mediterranean river basins

Flow duration curve provides an important synthesis of the relevant hydrological processes occurring at the basin scale, and, although it is typically obtained from field observations, different theoretical approaches finalized to its indirect reconstruction have been developed in recent years. In this study a recent ecohydrological model for the probabilistic characterization of base flows is tested through its application to a study catchment located in southern Italy, where long historical series of daily streamflow are available. The model, coupling soil moisture balance with a simplified scheme of the hydrological response of the basin, provides the daily flow duration curve. The original model is here modified in order to account for rainfall reduction due to canopy interception and stress its potential applicability to most of the ephemeral Mediterranean basins, where measurements of air temperature and rainfall often represent the only meteorological data available. The model shows a high sensitivity to two parameters related to the transport and evapotranspiration processes. Two different operational approaches for the identification of such parameters are explored and compared: by the first approach, these parameters are considered as time invariant quantities, while, in the second approach, empirical relationships between such parameters and the underlying climatic forcings are first derived and then adopted in the parameters calibration procedure. The model ability in reproducing the empirical flow duration curves and the model sensitivity to climate forcings, here referred as elasticity of the model, are investigated and it is shown how the adoption of the second approach leads to a general improvement of the model elasticity.     

Depth distribution of interrill overland flow and the formation of rills

A Gumbel distribution for maxima is proposed as a model for the depths of interrill overland flow. The model is tested against three sets of field measurements of interrill overland flow depths obtained on shrubland and grassland hillslopes at Walnut Gulch Experimental Watershed, southern Arizona. The model is found to be a satisfactory fit to 81 of the 90 measured distributions. The shape δ and location λ parameters of all fitted distributions are strongly correlated with discharge. However, whereas a common relationship exists between discharge and δ for all depth distributions, the relationships with λ vary systematically downslope. Using the Gumbel distribution as a model for the distribution of overland flow depths, a probabilistic model for the initiation of rills is developed, drawing upon the previous work of Nearing. As an illustration of this approach, we apply this model to the shrubland and grassland hillslopes at Walnut Gulch. It is concluded that the presence of rills on the shrubland, but not on the grassland, is due to the greater runoff coefficient for the shrubland and/or the greater propensity of the shrubland for soil disturbance compared with the grassland. Finally, a generalized conceptual model for rill initiation is proposed. This model takes account of the depth distribution of overland flow, the probability of flow shear stress in excess of local soil shear strength, the spatial variability in soil shear strength and the diffusive effect of soil detachment by raindrops. Copyright © 2005 John Wiley & Sons, Ltd.     

New analytical solutions for groundwater flow in wedge-shaped aquifers with various topographic boundary conditions

The hydraulic head distribution in a wedge-shaped aquifer depends on the wedge angle and the topographic and hydrogeological boundary conditions. In addition, an equation in terms of the radial distance with trigonometric functions along the boundary may be suitable to describe the water level configuration for a valley flank with a gentle sloping and rolling topography. This paper develops a general mathematical model including the governing equation and a variety of boundary conditions for the groundwater flow within a wedge-shaped aquifer. Based on the model, a new closed-form solution for transient flow in the wedge-shaped aquifer is derived via the finite sine transform and Hankel transform. In addition, a numerical approach, including the roots search scheme, the Gaussian quadrature, and Shanks’ method, is proposed for efficiently evaluating the infinite series and the infinite integral presented in the solution. This solution may be used to describe the head distribution for wedges that image theory is inapplicable, and to explore the effects of the recharge from various topographic boundaries on the groundwater flow system within a wedge-shaped aquifer.