Effects of Bubbles on the Hydraulic Conductivity of Porous Materials – Theoretical Results

In a porous material, both the pressure drop across a bubble and its speed are nonlinear functions of the fluid velocity. Nonlinear dynamics of bubbles in turn affect the macroscopic hydraulic conductivity, and thus the fluid velocity. We treat a porous medium as a network of tubes and combine critical path analysis with pore-scale results to predict the effects of bubble dynamics on the macroscopic hydraulic conductivity and bubble density. Critical path analysis uses percolation theory to find the dominant (approximately) one-dimensional flow paths. We find that in steady state, along percolating pathways, bubble density decreases with increasing fluid velocity, and bubble density is thus smallest in the smallest (critical) tubes. We find that the hydraulic conductivity increases monotonically with increasing capillary number up to Ca 10^–2, but may decrease for larger capillary numbers due to the relative decrease of bubble density in the critical pores. We also identify processes that can provide a positive feedback between bubble density and fluid flow along the critical paths. The feedback amplifies statistical fluctuations in the density of bubbles, producing fluctuations in the hydraulic conductivity.     

Influence of Viscous Fingering on Dynamic Saturation–Pressure Curves in Porous Media

We report on results from primary drainage experiments on quasi-two-dimensional porous models. The models are transparent, allowing the displacement process and structure to be monitored in space and time during primary drainage experiments carried out at various speeds. By combining detailed information on the displacement structure with global measurements of pressure, saturation and the capillary number Ca, we obtain a scaling relation relating pressure, saturation, system size and capillary number. This scaling relation allows pressure–saturation curves for a wide range of capillary numbers to be collapsed on the same master curve. We also show that in the case of primary drainage, the dynamic effect in the capillary pressure–saturation relationship observed on partially water saturated soil samples might be explained by the combined effect of capillary pressure along the invasion front of the gaseous phase, and pressure changes caused by viscous effects in the wetting fluid phase.     

An Analytical Model of Porosity–Permeability for Porous and Fractured Media

The classic Kozeny–Carman equation (KC) uses parameters that are empirically based or not readily measureable for predicting the permeability of unfractured porous media. Numerous published KC modifications share this disadvantage, which potentially limits the range of conditions under which the equations are applicable. It is not straightforward to formulate non-empirical general approaches due to the challenges of representing complex pore and fracture networks. Fractal-based expressions are increasingly popular in this regard, but have not yet been applied accurately and without empirical constants to estimating rock permeability. This study introduces a general non-empirical analytical KC-type expression for predicting matrix and fracture permeability during single-phase flow. It uses fractal methods to characterize geometric factors such as pore connectivity, non-uniform grain or crystal size distribution, pore arrangement, and fracture distribution in relation to pore distribution. Advances include (i) modification of the fractal approach used by Yu and coworkers for industrial applications to formulate KC-type expressions that are consistent with pore size observations on rocks. (ii) Consideration of cross-flow between pores that adhere to a fractal size distribution. (iii) Extension of the classic KC equation to fractured media absent empirical constants, a particular contribution of the study. Predictions based on the novel expression correspond well to measured matrix and fracture permeability data from natural sandstone and carbonate rocks, although the currently available dataset for fractures is sparse. The correspondence between model calculation results and matrix data is better than for existing models.     

Effects of small world connection on the dynamics of a delayed ring network

This paper presents a detailed analysis on the dynamics of a ring network with small world connection. On the basis of Lyapunov stability approach, the asymptotic stability of the trivial equilibrium is first investigated and the delay-dependent criteria ensuring global stability are obtained. The existence of Hopf bifurcation and the stability of periodic solutions bifurcating from the trivial equilibrium are then analyzed. Further studies are paid to the effects of small world connection on the stability interval and the stability of periodic solution. In particular, some complex dynamical phenomena due to short-cut strength are observed numerically, such as: period-doubling bifurcation and torus breaking to chaos, the coexistence of multiple periodic solutions, multiple quasi-periodic solutions, and multiple chaotic attractors. The studies show that small world connection may be used as a simple but efficient “switch” to control the dynamics of a system.     

Impact of the Buoyancy–Viscous Force Balance on Two-Phase Flow in Layered Porous Media

Motivated by geological carbon storage and hydrocarbon recovery, the effect of buoyancy and viscous forces on the displacement of one fluid by a second immiscible fluid, along parallel and dipping layers of contrasting permeability, is characterized using five independent dimensionless numbers and a dimensionless storage or recovery efficiency. Application of simple dimensionless models shows that increased longitudinal buoyancy effects increase storage efficiency by reducing the distance between the leading edges of the injected phase in each layer and decreasing the residual displaced phase saturation behind the leading edge of the displacing phase. Increased transverse buoyancy crossflow increases storage efficiency if it competes with permeability layering effects, but reduces storage efficiency otherwise. When both longitudinal and transverse buoyancy effects are varied simultaneously, a purely geometrical dip angle group defines whether changes in storage efficiency are dominated by changes in the longitudinal or transverse buoyancy effects. In the limit of buoyancy-segregated flow, we report an equivalent, unidimensional flow model which allows rapid prediction of storage efficiency. The model presented accounts for both dip and layering, thereby generalizing earlier work which accounted for each of these but not both together. We suggest that the predicted storage efficiency can be used to compare and rank geostatistical realizations, and complements earlier heterogeneity measures which are applicable in the viscous limit.     

Near-Critical Gas–Liquid Flow in Porous Media: Monovariant Model, Analytical Solutions and Convective Mass Exchange Effects

We examine a class of hydrocarbon reservoirs whose thermodynamic state remains close to the critical point during the all period of reservoir exploitation. Such a situation is typical for the so-called gas–condensate systems, in which the liquid phase is formed from gas when pressure decreases. Due to proximity to critical point, the mixture contains many components which are neutral with respect to the phase state. This determines a low thermodynamic degree of freedom of the system. As the results, the mathematical flow model allows a significant reduction in the number of conservation equations, whatever the number of chemical components. In the vicinity of a well, the system may be reduced to one transport equation for saturation. This nonlinear model yields exact analytical solutions when the flow is self-similar. In more general case of flow, we develop partially linearized solutions which are shown to be sufficiently exact. The spectrum of examined cases covers the flow in a medium with a sharp heterogeneity and a sharp variation in the flow rate. A significant relative gas flow past liquid gives rise to a convective mass exchange phenomenon which appears highly different from that observed in static. In the case of a medium discontinuity, the convective mass exchange gives rise to a phenomenon of condensate saturation billow formation. A sharp variation in the flow rate leads to a hysteretic behavior of the saturation field.     

A Benchmark Solution for Infiltration and Adsorption of Polluted Water Into Unsaturated–Saturated Porous Media

We discuss the numerical modeling of the infiltration of contaminated water into unsaturated porous media. A system with contaminant transport, dispersion, and adsorption is considered. The mathematical model for unsaturated flow is based on Richards nonlinear and degenerate equation. Nonlinear adsorption is represented by adsorption isotherms and kinetic rates. An accurate numerical method is constructed in 1D which can be a good candidate for the solution of inverse problems to determine model parameters in the adsorption part of the model. Our numerical solution is based on the method of lines (MOL method) where space discretization leads to the corresponding system of ODEs. We substantially use the numerical modeling of interfaces, separating fully saturated, partially saturated, and dry zones in the underground. Finally, in a series of numerical experiments and in comparisons with HYDRUS (?imunek et al., The HYDRUS-1D software package for simulating the one-dimensional movement of water, heat, and multiple solutes in variably/saturated media, version 2.0, Rep. IGWMC-TPS-70, 202 pp., Int. Groundwater Model. Cent., Colo. Sch of Mines, Golden, Colo), we demonstrate the effectiveness of our method.     

Double-Diffusive Natural Convection Flows with Thermosolutal Symmetry in Porous Media in the Presence of the Soret–Dufour Effects

The double-diffusive natural convection past a vertical plate embedded in a fluid-saturated porous medium is considered in the boundary-layer and Boussinesq approximations. It is assumed that the Soret–Dufour cross-diffusion effects are significant. The heat and mass fluxes on the plate are prescribed as functions of the surface coordinate x. The general similarity reduction of the problem for power-law and exponential variation of the wall fluxes is given. In the case of thermosolutal symmetry, when the similar temperature and concentration fields become coincident, exact analytical as well as numerical solutions are reported and discussed in some detail. For the flows without thermosolutal symmetry, the final similarity equations have been solved numerically, by paying attention to the influence of the Soret and Dufour numbers on the departure from thermosolutal symmetry. The reported results focus on the wall temperatures and concentrations, whose reciprocals are Nusselt and Sherwood numbers, respectively.     

The cross-correlations of stock markets based on DCCA and time-delay DCCA

In this paper, the Detrended Fluctuation Analysis (DFA) and Detrended Cross-Correlation Analysis (DCCA) are used to investigate the stock markets. The DFA method is a widely-used method for the determination and detection of long-range correlations in stock time series. DCCA is a recently developed method to quantify the cross-correlations of two non-stationary time series. We report the results of correlation and cross-correlation behaviors in US and Chinese stock markets by using the DFA and DCCA methods, respectively. The DCCA shows that there exists some crossovers in the cross-correlation fluctuation function versus time scale of stock absolute returns. The cross-correlations in Chinese stock markets are stronger than those between Chinese and US stock markets. After documenting the equal-time cross-correlations using DCCA method, we study the dynamics of cross-correlations of stock series based on a time-delay. The time-dependence of the underlying cross-correlations is monitored using a time window by step of 1 day. An interesting finding is that the cross-correlation exponents and crossovers demonstrate periodical uncertainty changing with the time-delay.     

Drying of suspensions and solutions on inert particle surface in mechanically spouted bed dryer

To eliminate some disadvantages of the conventional spouted bed dryers the mechanically spouted bed （MSB） system was developed. This dryer type is convenient to use inert particles providing an increased surface area for drying of materials of high-moisture content and heat sensitive materials. On three different drying tasks are demonstrated the experimental optimization of process parameters to obtain products of demanded quality. The main object was at drying of AIO（OH） suspension to preserve the particle size under 2.5μm and to obtain product with a moisture content of about 0.05 kg/kg （d.b.）. For this reason a very thin particle coating and intensive abrasion had to be assured. At drying of tomato concentrates the thermoplasticity makes the process very difficult. To jump over the deliquescent and sticky state developed at the critical temperature-moisture content values a very short drying time （8-10 s） must be provided. The third task was to form powder-like product from bovine serum albumin （BSA） solution having very low solid content （2-4%）. The selected process parameters given in this paper resulted in a mean particle size of less than 20 μm while the soluble oreserved orotein content was higher than 90%.     

The influences of both offset and mass moment of inertia of a tip mass on the dynamics of a centrifugally stiffened visco-elastic beam

The present study is concerned with the out-of-plane vibrations of a rotating, internally damped (Kelvin-Voigt model) Bernoulli-Euler beam carrying a tip mass. The centroid of the tip mass, possessing also a mass moment of inertia is offset from the free end of the beam and is located along its extended axis. This system can be thought of as an extremely simplified model of a helicopter rotor blade or a blade of an auto-cooling fan. The differential eigenvalue problem is solved by using Frobenius method of solution in power series. The characteristic equation is then solved numerically. The simulation results are tabulated for a variety of the nondimensional rotational speeds, tip mass, tip mass offset, mass moment of inertia and internal damping parameters. These are compared with the results of a conventional finite element modeling as well, and excellent agreement is obtained. Some numerical results are given in graphical form. The numerical results obtained, indicate clearly that the tip mass offset and mass moment of inertia are important parameters on the eigencharacteristics of rotating beams so that they have to be included in the modeling process.     

Infiltration of Salt Solute in Homogeneous and Saturated Porous Media – An Analytical Solution Evaluated by Numerical Simulations

In many cases various land disposal activities (e.g. infiltration, injection wells) constitute an important potential source of groundwater contamination. Using a 2D physical model, the behaviour of the infiltration of a salt solute, locally injected in a homogeneous and saturated porous medium, has been analysed. Under various experimental conditions (density effects, injection flow rate) the salt solute penetrates the porous media and leads to a steady-state regime inside the mixing zone. By using experimental observations, the basic equations describing the flow and transport phenomena can be simplified and an analytical solution obtained. Its validity is subject to numerical verification. The numerical model, based on the development of the mass balance equation expressed by its conservative form, uses a combination of the mixed hybrid finite element (MHFE) and discontinuous finite element (DFE) methods. The efficiency of this numerical model was previously verified on standard benchmarks, for example Elder's problem and Henry's problem. In the first step, the qualitative good agreement between the experimental and numerical results enabled us to use the numerical model in order to verify some hypotheses resulting from visual observations. Thus, the numerical results reveal the existence of a steady-state regime inside the mixing zones. Nevertheless, both its vertical and longitudinal extensions are less than those observed in the physical model. In the second step, the numerical results enable to establish the validity domain as well as the accuracy of the proposed analytical solution.     

Further Comments on “Combined Forced and Free Convective Flow in a Vertical Porous Channel: The Effects of Viscous Dissipation and Pressure Work”

This note is concerned with the assertion of Barletta and Nield (2009a) that “a fluid with a thermal expansion coefficient greater than that of a perfect gas (β > β _perfect gas) is of marginal or no interest in the framework of convection in porous media”, and that for a remark of Magyari (Transp. Porous Media, 2009) about the forced convection eigenflow solutions, the circumstance β > β _perfect gas does not represent “a sound physical basis”. Here, it is shown, however, that these assertions are in contradiction with the experimentally measured values of β for important technical fluids as e.g., air, nitrogen, carbon dioxide, and ammonia where, in the temperature range between −20 and +100°C, just the inequality β > β _perfect gas holds.     

Methodology to Assess the Effects of Rice Cultivation Under Flooded Conditions on van Genuchten’s Model Parameters and Pore Size Distribution

The effects of rice (Oryza sativa L.) cultivation under flooded conditions on soil’s physical-hydraulic properties were studied in this article, using a new methodology based on a combined analysis on soil water retention curve (WRC) and pore size distribution (PSD). WRC analysis was carried out through the changes of van Genuchten’s model parameters, the characteristics of WRC at the inflection point, and the specific water capacity curve. Analysis of PSD was performed on the volume changes of porosity fractions through a detailed pore size classification, while different pore size classifications based on their hydraulic and structural characteristics were also used. The methodology was applied using a small dataset obtained from fine-textured Entisol soils which were subjected to rice cultivation under flooded conditions in Axios River plain (Northern Greece). Measurements of WRC were obtained at four depths of the soil profile from two fields, before and after the growing season of rice. The analysis indicated that the van Genuchten’s model parameters (θ _s, a, and n) and the WRC characteristics at the inflection point (pressure head h _i , pore equivalent diameter D _i , and slope S _i ) significantly changed after the growing season following similar patterns, along the soil profile in both fields. The parameters θ _s, a, D _i , and S _i were decreased, while n and h _i were increased. The h _i and a were the most sensitive parameters, while the values of (h _i and 1/a) in each layer before and after the growing season for each field were linearly correlated and shifted to higher values because of compaction, indicating that it could be applied as a tool to evaluate the degree of soil compaction to similarly textured soils. The peaks of the specific water capacity curves were compressed (lower values of slope S _i ) and shifted to lower water potentials (h _i ) that corresponded to pores of equivalent diameter D _i between 2 and 6 μm. The soils had few structural pores (>9 μm) and low air-filled porosity (>30 μm) before the growing season, which presented accessory reduction after the growing season in both fields. Total porosity was reduced at the expense of structural porosity along the soil profile, while the pore size class of 5–3 μm was identified as the threshold where the smaller pores’ volume started to increase in all layers of both fields. The results indicated that the changes in the WRC and the PSD follow specific trends, which can be used in future studies to model temporal variability of soil’s physical-hydraulic properties.     

Can Moderately Rarefied Gas Transport Through Round and Flat Tight Channels of Fractured Porous Media be Described Accurately?

We study the generation and flow of foam through rough-walled, fractured marble rocks that mimic natural fracture systems in carbonate reservoirs. Flow was isolated to the fracture network because of the very low rock permeability of the marble samples and foam generated in situ during co-injection of surfactant solution and gas. The foam apparent viscosities were calculated at steady pressure gradients for a range of gas fractions, and similar to foam flow in porous media, we identified two flow regimes for foam flow in fractures: a high-quality flow regime only dependent on liquid velocity and a low-quality flow regime determined by the gas and liquid velocities. Variations in local fluid saturation during co-injection were visualized and quantified using positron emission tomography combined with computed tomography.

     

Comments on “Abrupt-Interface Solution for Carbon dioxide Injection into Porous Media” by M. Dentz and D. Tartakovsky

Several numerical simulations were conducted to compare the results with analytical solutions given by “Abrupt-Interface Solution for Carbon Dioxide Injection into Porous Media” by M. Dentz and D. Tartakovsky, “Injection and Storage of CO₂ in Deep Saline Aquifers: Analytical Solution for CO₂ Plume Evolution During Injection” by J. M. Nordbotten et al.     

Comment on “Combined Forced and Free Convective Flow in a Vertical Porous Channel: The Effects of Viscous Dissipation and Pressure Work” by A. Barletta and D. A. Nield,Transport in Porous Media,DOI 10.1007/s11242-008-9320-y, 2009

In a recent article by Barletta and Nield (Transport in Porous Media, DOI , 2009), the title problem for the fully developed parallel flow regime was considered assuming isoflux/isothermal wall conditions. For the limiting cases of the forced and the free convection, analytical solutions were reported; for the general case, numerical solutions were reported. The aim of the present note is (i) to give an analytical solution for the full problem in terms of the Weierstrass elliptic P-function, (ii) to illustrate this general approach by two easily manageable examples, and (iii) to rise a couple of questions of basic physical interest concerning the interplay between the viscous dissipation and the pressure work. In this context, the concept of “eigenflow” introduced by Barletta and Nield is discussed in some detail.     

The effects of variable properties on MHD unsteady natural convection heat and mass transfer over a vertical wavy surface

A mathematical model will be analyzed in order to study the effects of variables viscosity and thermal conductivity on unsteady heat and mass transfer over a vertical wavy surface in the presence of magnetic field numerically by using a simple coordinate transformation to transform the complex wavy surface into a flat plate. The fluid viscosity is assumed to vary as a exponential function of temperature and thermal conductivity is assumed to vary linearly with temperature. An implicit marching Chebyshev collocation scheme is employed for the analysis. Numerical solutions are obtained for different values of variable viscosity, variable thermal conductivity and MHD variation parameter. Numerical results show that, variable viscosity, variable thermal conductivity and MHD variation parameter have significant influences on the velocity, temperature and concentration profiles as well as for the local skin friction, Nusselt number and Sherwood number.