S. Majid Hassanizadeh

Thermodynamic basis of capillary pressure in porous media

Important features of multiphase flow in porous media that distinguish it from single-phase flow are the presence of interfaces between the fluid phases and of common lines where three phases come in contact. Despite this fact, mathematical descriptions of these flows have been lacking in rigor, consisting primarily of heuristic extensions of Darcys law that include a hysteretic relation between capillary pressure and saturation and a relative permeability coefficient. As a result, the standard capillary pressure concept appears to have physically unrealistic properties. The present paper employs microscopic mass and momentum balance equations for phases and interfaces to develop an understanding of capillary pressure at the microscale. Next, the standard theories and approaches that define capillary pressure at the macroscale are described and their shortcomings are discussed. Finally, an approach is presented whereby capillary pressure is shown to be an intrinsic property of the system under study. In particular, the presence of interfaces and their distribution within a multiphase system are shown to be essential to describing the state of the system. A thermodynamic approach to the definition of capillary pressure provides a theoretically sound alternative to the definition of capillary pressure as a simple hysteretic function of saturation.

Removal of viruses by soil passage: overview of modeling, processes, and parameters.

In this article, the modeling of subsurface virus transport under saturated conditions and the factors that affect adsorption and inactivation are evaluated. Both equilibrium and kinetic adsorption are considered. Equilibrium adsorption is found to be of little significance. Adsorption appears to be mainly kinetically limited. At pH 7 and higher, conditions are generally unfavorable for attachment, but viruses may preferentially attach to a minor surface fraction of soil grains that is positively charged. The relation of pH with surface charge and their effects on sticking efficiencies are evaluated. Dissolved organic matter decreases virus attachment by competition for the same binding sites and thus reduces attachment. Bonded organic matter may provide hydrophobic binding sites for viruses and thus enhance attachment. Dissolved organic matter may disrupt hydrophobic bonds. The enhancing and attenuating effects of organic matter are very difficult to quantify and may be responsible for considerable uncertainty when predicting virus removal. Values of inactivation rate coefficients for attached viruses were calculated using data from some batch studies. Enhanced or reduced inactivation is found to be virus-specific and almost independent of adsorption. Temperature is the most important factor that influences virus inactivation. Probably the inactivation rate coefficients of free and attached viruses change similarly with temperature. Some frequently used bacteriophages are evaluated as model viruses. MS2 and PRD1 meet the requirements for worst-case model viruses, at water temperatures less than about 10°C, at pH 6 to 8, and if the soil does not contain too many hydrophobic sites and not too much multivalent cations. Bacteriophage ϕX 174 may be a relatively conservative model virus, because of its low hy drophobic-ity and stability. Together in a cocktail, these three viruses span a range of properties, like size, surface charge, and hydrophobicity. F-specific RNA bacteriophages (FRNAPHs) may be very useful naturally occurring worst-case viruses. FRNAPHs that are present in surface water or treated wastewater that is used for recharging groundwater, consist of stable and poorly adsorbing viruses. An inventory of parameter values from field studies is made. Attachment appears to be the major process that determines virus removal. Still, only very few data are available on attachment and detachment of viruses under field conditions. Removal of viruses by soil passage, log(C/C 0), appears to decline nonlinearly with distance due to heterogeneities within the soil as well as within the population of transported virus particles. Predictions of virus removal at larger distances are severely overestimated if they are based on removal data from column experiments or from short-distance field studies.

Mechanics and thermodynamics of multiphase flow in porous media including interphase boundaries

Abstract The main purpose of this work is to develop a macroscale thermodynamic theory to describe two-phase flow in porous media. Full thermodynamic properties are assigned to the boundary surfaces separating the phases at the microscale. Macroscopic equations of balance for mass, momentum, and energy for each phase and interface along with the averaged entropy inequality are employed as the starting point. A constitutive theory is developed resulting in balance equations and thermodynamics appropriate for modelling multiphase flow in porous media. Volume fractions of phases and areal fractions of interfaces are explicitly included in the theory. Incorporation of the interface equations into the theory allows for a complete description of the problem. The manipulations provide explicit functional dependence of the capillary pressure. An extended form of Darcys law for multiphase flow is obtained from the macroscopic equations of momentum balance. An additional term which accounts for non-uniform fluid saturation at equilibrium appears in the result.

Dynamic Effect in the Capillary Pressure–Saturation Relationship and its Impacts on Unsaturated Flow

Capillary pressure plays a central role in the description of water flow in unsaturated soils. While capillarity is ubiquitous in unsaturated analyses, the theoretical basis and practical implications of capillarity in soils remain poorly understood. In most traditional treatments of capillary pressure, it is defined as the difference between pressures of phases, in this case air and water, and is assumed to be a function of saturation. Recent theories have indicated that capillary pressure should be given a more general thermodynamic definition, and its functional dependence should be generalized to include dynamic effects. Experimental evidence has slowly accumulated in the past decades to support a more general description of capillary pressure that includes dynamic effects. A review of these experiments shows that the coefficient arising in the theoretical analysis can be estimated from the reported data. The calculated values range from 10 4 to 10 7 kg (m s) −1 . In addition, recently developed pore-scale models that simulate interface dynamics within a network of pores can also be used to estimate the appropriate dynamic coefficients. Analyses of experiments reported in the literature, and of simulations based on pore-scale models, indicate a range of dynamic coefficients that spans about three orders of magnitude. To examine whether these coefficients have any practical effects on larger-scale problems, continuum-scale simulators may be constructed in which the dynamic effects are included. These simulators may then be run to determine the range of coefficients for which discernable effects occur. Results from such simulations indicate that measured values of dynamic coefficients are within one order of magnitude of those values that produce significant effects in field simulations. This indicates that dynamic effects may be important for some field situations, and numerical simulators for unsaturated flow should generally include the additional term(s) associated with dynamic capillary pressure.

Unsaturated Flow Theory Including Interfacial Phenomena

The macroscopic porous medium equations for mass, momentum, and energy transport for air, water, and solid phases and the interfaces between these phases are examined in light of the second law of thermodynamics. Attention is focused on the momentum balance for the water phase. Appropriate forms of the momentum balance are obtained, in general, for the slow flow situation and for the case when the water phase completely wets the solid. This last case suggests that the relative wettability of the water and air phases is an important dependent thermodynamic variable which contributes to the hysteretic nature of the capillary pressure versus saturation curve.

High velocity flow in porous media

Experimental observations have established that the proportionality between pressure head gradient and fluid velocity does not hold for high rates of fluid flow in porous media. Empirical relations such as Forchheimer equation have been proposed to account for nonlinear effects. The purpose of this work is to derive such nonlinear relationships based on fundamental laws of continuum mechanics and to identify the source of nonlinearity in equations.Adopting the continuum approach to the description of thermodynamic processes in porous media, a general equation of motion of fluid at the macroscopic level is proposed. Using a standard order-of-magnitude argument, it is shown that at the onset of nonlinearities (which happens at Reynolds numbers around 10), macroscopic viscous and inertial forces are negligible compared to microscopic viscous forces. Therefore, it is concluded that growth of microscopic viscous forces (drag forces) at high flow velocities give rise to nonlinear effects. Then, employing the constitutive theory, a nonlinear relationship is developed for drag forces and finally a generalized form of Forchheimer equation is derived.

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

Abstract 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.

Toward an improved description of the physics of two-phase flow

Abstract The present work incorporates the effects of interface dynamics into the theoretical description of two-phase flow in a porous medium. This advance offers the potential for improved understanding and modeling of multiphase flow processes. To provide background for this work, the traditional approach to describing two-phase flow in porous media is reviewed. The universally employed empirical extension of Darcys Law for single-phase flow to two-phase flow situations is rejected as arbitrary and subject to severe shortcomings. Burial of dynamic effects into relative permeability and capillary pressure hysteresis is shown to be an unsatisfactory theoretical construct for modeling the actual processes occurring in two-phase flow. Examination of the traditional theory at equilibrium shows that interfacial forces actually present in multiphase systems have been overlooked causing the theory to provide contradictory results. To overcome these problems, a general theory of two-phase flow is proposed that is based on the basic principles of mass, momentum, and energy conservation and the second law of thermodynamics. This theory accounts, in a systematic way, for interfacial forces that are known to have an important effect on the movement of fluid phases in a porous medium. A new equation of momentum balance accounting for the presence of interfaces and their energetics is developed. This equation reduces to Darcys Law for the special case of single-phase flow. The extended theory has the potential to describe phenomena unaccounted for by the traditional theory and thus provides a basis for scientific understanding of the physics of two-phase flow. Application of the theory requires experimental study to ascertain the values and precise functional dependence of the constitutive coefficients that arise.

Modeling removal of bacteriophages MS2 and PRD1 by dune recharge at Castricum, Netherlands

Removal of model viruses by dune recharge was studied at a field site in the dune area of Castricum, Netherlands. Recharge water was dosed with bacteriophages MS2 and PRD1 for 11 days at a constant concentration in a 10- by 15-m compartment that was isolated in a recharge basin. Breakthrough was monitored for 120 days at six wells with their screens along a flow line. Concentrations of both phages were reduced about 3 log10 within the first 2.4 m and another 5 log10 in a linear fashion within the following 27 m. A model accounting for one-site kinetic attachment as well as first-order inactivation was employed to simulate the bacteriophage breakthrough curves. The major removal process was found to be attachment of the bacteriophages. Detachment was very slow. After passage of the pulse of dosed bacteriophages, there was a long tail whose slope corresponds to the inactivation rate coefficient of 0.07–0.09 day−1 for attached bacteriophages. The end of the rising and the start of the declining limbs of the breakthrough curves could not be simulated completely, probably because of an as yet unknown process.

A unifying framework for watershed thermodynamics: Constitutive relationships

Abstract 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 Darcys 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.