Gravitational Stability of a Moving Water-Steam Interface in Hydrothermal Systems

The stability of a hydrothermal system is considered. In this system a water layer overlies a superheated-steam layer in a reservoir with relatively low permeability located between two parallel high-permeability strata. The solution of the steady-state bounded problem with a phase transition interface separating the water and steam zones is obtained on the assumption that the convective energy transfer is small as compared with the conductive transfer. An investigation of the normal stability of the phase transition interface shows that stable configurations almost always exist in the geothermal system considered on the permeability range bounded from above by the value k 10^-15 m². Thus, the criterion of predominance of conductive transfer, which is simultaneously the criterion of existence of the fundamental solution, turns out to be the criterion of stability of the phase interface in the geothermal system considered. The fairly high value of the permeability satisfying this criterion makes it possible to explain the existence of stable natural geothermal reservoirs in which a water layer overlies a steam layer.     

Transition to instability of the interface in geothermal systems

High-temperature geothermal reservoir in porous media is under consideration, consisting of two high-permeability layers, which are separated by a low-permeability stratum. The thermodynamic conditions are assumed to imply that the upper and lower high-permeability layers are filled in by water and by vapour, respectively. In these circumstances the low-permeability stratum possesses the phase transition interface, separating domains occupied by water and vapour. The stable stationary regimes of vertical phase flow between water and vapour layers in the low-permeability stratum may exist. Stability of such regimes where the heavier fluid is located over the lighter one is supported by a heat transfer, caused by a temperature gradient in the Earth's interior. We give the classification of the possible types of transition to instability of the vertical flows in such a system under the condition of smallness of the advective heat transfer in comparison with the conductive one. It is found that in the non-degenerate case there exist three different scenarios of the onset of instability of the stationary vertical phase transition flows. Two of them are accompanied by the bifurcations of the destabilizing vertical flow, leading to appearance of horizontally non-homogeneous regimes with non-constant shape of the interface. The bifurcations correspond to the simple resonance and $1:1$-resonance, which typically arise in reversible systems.     

Stability of a water-vapor phase transition interface in geothermal systems

The problem of normal stability of a phase transition interface for vertical flows in a geothermal reservoir with the water layer lying above the vapor layer is solved. The problem is formulated with account for convective energy transfer, which ensures applicability for arbitrary permeability values. Typical examples of variation of the phase transition interface stability parameters in response to changes in permeability are considered for various geothermal reservoirs.     

Interaction of two water-bearing horizons separated by a slightly permeable band

Consider the unsteady axially symmetric filtration in a stratified aquifer consisting of two highly permeable strata separated by a band of low permeability. The upper stratum has a cover of low permeability (cover rock) containing the water table, while an impermeable layer lies under the lower one (see figure). The filtration is everywhere elastic. An analogous problem has been considered [1–5] either subject to the condition that the pressure in one stratum remains unaltered, or with the permeability of the cover rock neglected, or on the assumption that the passage through the band occurs under a high differential. Shestakov [6] considered this problem on the assumption that the cover rock is impermeable and obtained an approximate solution for the case in which the two strata are equal in permeability.Notation r radial coordinate - z vertical coordinate - h head reckoned from the cover - h₀ initial head - k filtration coefficient - m stratum thickness - m₁ thickness of cover - T filtration permeability - ⁰ elastic water-release factor - ₁ release factor of cover - a pressure-permeability factor - ₁ coefficient for transfer via cover - ₃ coefficient for transfer via band.     

Heat and mass transfer characteristics of simulated high moisture flue gases

The heat transfer process occurring in a condensing heat exchanger where noncondensible gases are dominant in volume is different from the condensation heat transfer of the water vapor containing small amount of noncondensible gases. In the process the mass transfer due to the vapor condensation contributes an important part to the total heat transfer. In this paper, the Colburn-Hougen method is introduced to analyze the heat and mass transfer process when the water vapor entrained in a gas stream condenses into water on the tube wall. The major influential factors of the convective-condensation heat transfer coefficient are found as follows: the partial pressure of the vapor p _v , the temperature of the outer tube wall T _w , the mixture temperature T _g , Re and Pr. A new dimensionless number Ch, which is defined as condensation factor, has been proposed by dimensional analysis. In order to determine the relevant constants and investigate the convection-condensation heat and mass transfer characteristics of the condensing heat exchanger of a gas fired condensing boiler, a single row plain tube heat exchanger is designed, and experiments have been conducted with vapor-air mixture used to simulate flue gases. The experimental results show that the convection-condensation heat transfer coefficient is 1.52 times higher than that of the forced convection without condensation. Based on the experimental data, the normalized formula for convention-condensation heat transfer coefficient is obtained. A heat transfer area m² - Ch condensation factor - c _p specific heat at constant pressure, J/(kg·K) - G mass flux Kg/(m²·s) - h heat transfer coefficient W/(m²·K) - J J-factor - Nu Nusselt number - pa pressure - Pr Prandtl number - Q heat transfer rate - q heat flux W/m² - r latent heat, kJ/kg - Re Reynolds number - Sc Schmidt number - T temperature, C or K - heat conductivity m W/(m·K) - density, kg·m³ - g gas - h moistened hot air - i interface - v vapor - w water     

Effect of surface permeability on the structure of a separated turbulent flow and heat transfer behind a backward-facing step

The structure and heat transfer in a turbulent separated flow in a suddenly expanding channel with injection (suction) through a porous wall are numerically simulated with the use of two-dimensional averaged Navier–Stokes equations, energy equations, and v ²–f turbulence model. It is shown that enhancement of the intensity of the transverse mass flux on the wall reduces the separation region length in the case of suction and increases the separation region length in the case of injection up to complete boundary layer displacement. The maximum heat transfer coefficient as a function of permeability is accurately described by the asymptotic theory of a turbulent boundary layer.     

Numerical study of boundary layer transition in flowing film evaporation on horizontal elliptical cylinder

A numerical study of the onset of longitudinal transition between turbulent and laminar regimes during the evaporation of a water film is presented. These water film streams along a horizontal elliptical tube under the simultaneous effects of gravity, pressure gradients, caused by the vapor flow and curvature, and viscous forces. At the interface of water vapor, the shear stress is supposed to be negligible. Outside the boundary layer, the vapor phase velocity is obtained from potential flow. In the analysis Von Karmans turbulence model is used and the inertia and convection terms are retained. Transfers equations are discretised by using the implicit Keller method. The effects of an initial liquid flow rate per unit of length, Froude number, temperature difference between the wall and the liquid–vapor interface and ellipticity on the transition position have been evaluated. The transition criterion has been given in term of the critical film Reynolds number (Re)_C.     

<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">1</Emphasis></Superscript> Stability of Multi-Dimensional Discrete-Velocity Boltzmann Equations

We study the L ¹ stability of multi-dimensional discrete-velocity Boltzmann equations. Under suitable smallness assumption on initial data, we show that bounded mild solutions are L ¹ stable. For a stability estimate, we employ Bonys multi-dimensional analysis for total interactions over characteristic planes.     

Correlation of Water Vapor Permeability with Microstructure Characteristics of Building Materials Using Robust Chemometrics

This paper studies various microstructure parameters of natural and artificial building materials and aims to their correlation to the water vapor permeability. Three categories of building materials were investigated: stones, bricks, and plasters. Mercury intrusion porosimetry was applied in order to obtain the materials microstructure characteristics, a variety of pore size distributions and pore structure measurements, such as total porosity. The water vapor permeability of materials was determined experimentally according to ASTM standard E96-00. A robust principal component regression approach, coupled with multiple outlier detection, was applied in order to correlate water vapor permeability values to pore size distributions. A good quality correlation model was found by utilizing relative specific pore volume and relative specific pore surface distributions, whereas using pore structure measurements, such as total porosity, the correlation results were very poor. From the results, specific ranges of pore size distribution, corresponding to pores radius sizes greater than $10\,\upmu \text{ m }$ 10 μ m and between 1.778 and $0.421\,\upmu \text{ m }$ 0.421 μ m , contribute to the water vapor permeability of the materials.     

Joule–Thomson Effects on the Flow of Liquid Water

We present a revised form of the energy balance for the coupled thermodynamics of liquid water flowing in porous media and give examples of situations where a commonly used formulation based on transport of enthalpy leads to erroneous results. Assuming negligible contribution from kinetic energy as well as sources and sinks such as energy from radioactive decay, total energy conservation is reduced to a balance between changes in internal energy, enthalpy, conductive heat flux, and gravitational potential energy. The Joule–Thomson coefficient is defined as the change in temperature with respect to an increase in pressure at constant enthalpy. Because liquid water has a negative Joule–Thomson coefficient at low temperatures, at a constant gravitational potential water cools as it compresses and heats as it expands. If one ignores the gravitational energy, transport of enthalpy alone leads to water heating by 2 \(^\circ \) C per kilometer as it is brought up from depth. The corrected energy balance transports methalpy, which is enthalpy plus gravitational potential energy. Although the simpler form leads to small changes in the temperature profile for typical simulations, there are several instances where this effect may prove to be important. The most important impact of the erroneous form is probably in the field of geothermal energy production, where the creation of a few degrees of heat in a simulation could lead to miscalculation of power plant efficiencies.     

The Laminar Boundary Layer over a Permeable Wall

An analysis is given of the laminar boundary layer over a permeable/porous wall. The porous wall is passive in the sense that no suction or blowing velocity is imposed. To describe the flow inside and above the porous wall a continuum approach is employed based on the Volume-Averaging Method (S. Whitaker The method of volume averaging). With help of an order-of-magnitude analysis the boundary-layer equations are derived. The analysis is constrained by: (a) a low wall permeability; (b) a low Reynolds number for the flow inside the porous wall; (c) a sufficiently high Reynolds number for the freestream flow above the porous wall. Two boundary layers lying on top of each other can be distinguished: the Prandtl boundary layer above the porous wall, and the Brinkman boundary layer inside the porous wall. Based on the analytical solution for the Brinkman boundary layer in combination with the momentum transfer model of Ochoa-Tapia and Whitaker (Int. J. Heat Mass Transfer 38 (1995) 2635). for the interface region, a closed set of equations is derived for the Prandtl boundary layer. For the stream function a power series expansion in the perturbation parameter is adopted, where is proportional to ratio of the Brinkman to the Prandtl boundary-layer thickness. A generalization of the Falkner–Skan equation for boundary-layer flow past a wedge is derived, in which wall permeability is incorporated. Numerical solutions of the Falkner–Skan equation for various wedge angles are presented. Up to the first order in wall permeability causes a positive streamwise velocity at the interface and inside the porous wall, but a wall-normal interface velocity is a second-order effect. Furthermore, wall permeability causes a decrease in the wall shear stress when the freestream flow accelerates, but an increase in the wall shear stress when the freestream flow decelerates. From the latter it follows that separation, as indicated by zero wall shear stress, is delayed to a larger positive pressure gradient.     

A new method for measuring static pressure fluctuations with application to wind-wave interaction

A new technique for in stream static pressure fluctuations sensing is presented. The higher capability of the method with regard the classical one, particularly over laboratory wind wave, is proved. First measurements have been done in a turbulent boundary layer above the air-water interface during the wave generation stage. The results show that, for pure laboratory wind waves at short fetches, a strong coupling exists between air and water motions and that the energy transfer from wind to the waves seems mainly due to the work done by the wave induced pressure fluctuations.List of symbols C wave phase celerity - C _g wave group celerity - Coh coherency function - C _ps pressure coefficient for a static pressure sensing head - C _pt pressure coefficient for a total pressure sensing head - g gravitational acceleration - n frequency - p instantaneous static pressure - p _m measured instantaneous static pressure - p _t instantaneous total pressure - p _tm measured instantaneous total pressure - p (t) static pressure fluctuation - Q _pn quadspectrum between static pressure fluctuations and water level deflections - S _p static pressure fluctuations spectrum - S _pt total pressure fluctuations spectrum - S _n wave spectrum - u instantaneous air velocity vector - U air velocity outside the boundary layer - X fetch - instantaneous incidence angle of the air velocity - instantaneous water level - phase shift - wave energy amplification ratio - ^p non dimensional energy transfer ratio by pressure work - _M non dimensional energy transfer ratio predicted by Miles theory - _a air density - _w water density A version of this paper was presented at the 10th Symposium on Turbulence, University of Missouri-Rolla, September 22–24, 1986     

Strong Static Stability of Liquid Layers

For a class of disjoining pressure potentials we show that liquid layers in equilibrium with positive second variation for nonvanishing variations (called stable by definition) are in fact strong minimizers of the associated energy functional. In a previous paper, explicit stable layers are obtained for some cases of solid configurations with simple geometries, in particular for a single cylindrical or spherical cavity. From this note it follows that all these “stable” equilibria are strong energy minimizers. No smallness for the thickness of the layer is assumed.     

Heat,Water, and Solution Transfer in Unsaturated Porous Media: I -- Theory Development and Transport Coefficient Evaluation

A detailed theory describing the simultaneous transfer of heat, water, and solute in unsaturated porous mediais developed. The theory includes three fully-coupledpartial differential equations. Heat, water, andsolute move in the presence of temperature, T; matricpressure head, _m ; solution osmotic pressure head _o ; and solute concentration C gradients. Thetheory can be applied to describe the mass and energyin radioactive waste repositories, food processing,underground energy storage sites, buried electriccables positions, waste disposal sites, and inagricultural soil. Several transport coefficients forheat, water, and solute are included in the theory. The coefficients are evaluated for a silty clay loamsoil to clarify their dependence on water content (),T, and C. The thermal vapor diffusivity D _Tv first increased as increased to0.22 m³/m³ then decreased with furtherincreases in . D _Tv was 3 orders of magnitudegreater than either isothermal vapor D _mv orosmotic vapor D _ov , diffusivities at of0.20~m³/m³, T of 50°C, and C of 0.001mol/kg. All of the liquid and vapor water transport coefficients increased with increasing T. D _Tv decreased with increasing C to a greater extent thanD _mv and D _ov . The effective thermalconductivity decreased slightly with increasing C. Thesolute diffusion coefficient D _d was 6 to 7orders of magnitude greater than the thermal soluteand salt sieving diffusion coefficients at of0.20~m³/m³, T of 50°C, and C of 0.001 mol/kg.     

Condensation and isothermal water transfer in cement mortar: Part II - transient condensation of water vapor

The transient wetting of a mortar sample swept by a flow of humid air is experimentally studied at temperatures of 30 and 55°C. The water content profile shape and evolution are found to be very different from those which were observed during imbibition. The boundary condition on the exposed wall of the sample is examined. A convenient evolution of the coefficient of diffusion with water content is explored. This coefficient is interpreted in terms of pure vapor diffusion, even at relatively high water contents. But its values at low water content and its temperature dependence are inconsistent. Additional explanations are then considered with the assumption that the vapor condensation in the medium is not an equilibrium process between vapor and liquid phases. The physical origin of such a nonequilibrium process is discussed. A tentative set of transfer and phase change coefficients is proposed in order to describe the experimental data by means of numerical simulation. Then, some aspects of the imbibition processes are re-examined, taking into account the consequences of a nonequilibrium condensation.Nomenclature volumic rate of phase change - D ₀ coefficient of free diffusion of the water vapor in air - D _hv vapor diffusion coefficient of the medium - E, E equivalent air thickness - h relative humidity of gaseous phase - h _c relative humidity at the capillary condensation threshold - h _a relative humidity of the flowing air - h ₀ relative humidity at the air-material interface - h _E equilibrium relative humidity at a given water content - J global massic flux - M molar mass of water - R gas constant - T temperature - t time - x distance from the interface - ₀ total porosity - volumetric water content - _h condensation coefficient (see Equation (8)) - _L mass density of liquid water - vs mass density of saturated water vapor     

Criteria of hydrodynamic instability of a phase interface in hydrothermal systems

The loss of stability of a vertical phase flow in a geothermal system in which a liquid layer overlies a vapor layer is considered. The loss of stability criteria are obtained in explicit form. It is found that when the physical parameters of the system are varied the transition to phase interface instability can be realized by means of one of the following mechanisms: the transition occurs spontaneously for any perturbation wavenumber (degenerate case); an unstable wavenumber arises at infinity; the instability threshold is determined by a double zero wavenumber. In the latter case the transition to instability is accompanied by simple resonance bifurcation. As a result of this bifurcation, secondary regimes dependent on the horizontal coordinate branch off from the basic regime describing the horizontally-homogeneous vertical phase flows.Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, 2004, pp. 100–109. Original Russian Text Copyright © 2004 by Ilichev and Tsypkin.     

Predicting Vertical Flow Barriers Using Tracer Diffusion in Partially Saturated,Layered Porous Media

Sudden changes in isotopic tracer concentration in pore waters have been interpreted as indicating barriers to vertical advective flow through porous rocks in the subsurface, e.g. step changes in \(^{87}\hbox {Sr}/^{86}\) Sr ratio are often used in the oil and gas industry as a signature of reservoir compartmentalisation. This study shows that this is not necessarily the case. It can take millions of years for such step changes to equilibrate by diffusion if there is no flow resulting from pressure or density gradients even in high permeability, high porosity rocks, particularly if the water saturation is low. Changes in tracer concentration gradients can be good indicators of changes in porosity (or water saturation) between layers. In contrast changes in sorption without a change in porosity are almost impossible to identify. The time taken for concentration gradients to equilibrate is affected by the layer properties but can be quickly estimated from the harmonic average of the effective diffusion coefficient for each layer and a simple analytical expression for a homogeneous system. This was achieved by performing a sensitivity analysis on different layer properties (porosity contrast, saturation contrast, sorption contrast, thickness ratio) using existing analytical solutions for diffusion in layered systems.     

Estimates of Barometric Pumping of Moisture Through Unsaturated Fractured Rock

We present a theory for the motion of water vapor at depth in a discretely fractured permeable medium induced by atmospheric barometric pressure fluctuations, or barometric pumping. The theory involves multiphase mass and energy transport in a fracture/matrix system, with discrete representation of the fracture system. The barometric pressure fluctuations are approximated as periodic in time, with amplitude corresponding to measured values. To simplify the analysis, a single-horizon approximation is applied in which the time-mean gradient is used to evaluate the vertical advective flux in the fractures. Time-periodic solutions are obtained numerically, enabling the calculation of the net efflux of moisture per cycle. The model is applied to material representative of the Yucca Mountain region of southwestern Nevada. The results indicate that the efflux of moisture carried upward from significant depths by barometric pumping is much less than the near surface efflux that is commonly estimated by assuming that air enters the medium dry and is returned to the atmosphere fully saturated with water vapor. This near surface efflux consists primarily of moisture discharged from the upper layer which is frequently replenished by precipitation. Of greater interest to nuclear waste repository design and estimations of net infiltration in arid regions is the fraction of the total moisture efflux that comes from significant depths. This deep transport is quantified by the fracture/matrix transport model described here. Although the transport by barometric pumping from depth is small compared to the total moisture expelled from the surface layer, it is an order of magnitude greater than the vertical moisture flux carried from depth by diffusion.     

The Effect of Wettability on Three-Phase Relative Permeability

We study three-phase flow in water-wet, oil-wet, and fractionally-wet sandpacks. We use CT scanning to measure directly the oil and water relative permeabilites for three-phase gravity drainage. In an analogue experiment, we measure pressure gradients in the gas phase to determine the gas relative permeability. Thus we find all three relative permeabilities as a function of saturation. We find that the gas relative permeability is approximately half as much in a oil-wet medium than in an water-wet medium at the same gas saturation. The water relative permeability in the water-wet medium and the oil relative permeability in the oil-wet medium are similar. In the water-wet medium the oil relative permeability scales as k _roS _o ⁴ for S _o>S _or, where S _or is the waterflood residual oil saturation. With octane as the oil phase, k _roS _o ² for S _o<S _or, while with decane as the oil phase, k _ro falls sharply for S _o<S _or. The water relative permeability in the oil-wet medium resembles the oil relative permeability in the water-wet medium for a non-spreading oil such as decane. These observations can be explained in terms of wetting, spreading, and the pore scale configurations of fluid.     

Luikov equations applicable to sublimation-drying

Present paper presents a derivation of Luikov equations applicable to sublimation-drying. The physical situation and transfer mechanism are elucidated clearly. The coefficients appearing in Luikov equations are given in a more explicit way. Some formulation mistakes in recent publications are indicated.

---

Anwendung der Luikov-Gleichungen auf die Sublimationstrocknung

Zusammenfassung Die Untersuchung bezieht sich auf eine Ableitung der Luikov-Gleichungen, mittels deren sich der Vorgang der Sublimationstrocknung analysieren läßt. Physikalische Anfangssituation und Austauschmechanismen werden klar herausgestellt und die in den Luikov-Gleichungen auftretenden Koeffizienten in expliziter Weise angegeben. Ferner erfolgt Hinweis auf Formulierungsfehler in jüngeren Veröffentlichungen.

---

Nomenclature C M _v/V _f, concentration of vapor, kg/m³ - c _pv specific heat of vapor at constant pressure, J/kg K - c _pw specific heat of adsorbed water at constant pressure, J/kg K - c _s specific heat of solid skeleton, J/kg K - C _s M _s/V _f, concentration of solid skeleton, kg/m³ - C _w M _w/V _f, concentration of adsorbed water, kg/m³ - f V _w/V _f, volumetric fraction of adsorbed water - j _F mass flux of vapor by diffusion (Fick) transfer, kg/m² s - j _D mass flux of vapor by filtration (Darcy) transfer, kg/m² s - j _v total mass flux of vapor, kg/m² s - k permeability, m² - M _s mass of solid skeleton, kg - M _v mass of vapor in pores, kg - M _w mass of adsorbed water, kg - P pressure, Pa - q heat flux, W/m² - R gas constant, J/kg K - T temperature, K - V _f volume of the framework of porous medium, m³ - V _v volume of vapor in porous medium, m³ - V _w volume of the absorbed water, m³ Greek symbols /(c _p), effective thermal diffusivity, m²/s - _m effective vapor diffusivity in porous medium, m²/s - _p R T /, Luikov pressure diffusivity, m²/s - +f, porosity of the porous medium - effective thermal conductivity of porous body, W/m K - dynamic viscosity of vapor, kg/m s - kinematic viscosity, m²/s - Ck/=k/, Luikov filtration motion coefficient, s - V _v/V _f, volumetric fraction of vapor - density of absorbed water, kg/m³ - (c _p) M _v c _pv+M _s c _s+M _w c _pw /V _f=Cc _pv+C _s c _s+fc _pw, effective product of density and specific heat of humid porous body, J/m³K