THE EFFECTS OF MEDIUM THICKNESS ON THE ISLAND SIZE DISTRIBUTION IN IMMISCIBLE DISPLACEMENT FLOW IN POROUS MEDIA

The invasion percolation algorithm is used to simulate two-fluid immiscible displacement of a wetting fluid by a non-wetting fluid in various porous media represented by two-dimensional and three-dimensional networks of interconnected capillaries. Trapping of the displaced fluid occurs, thereby creating isolated islands. The effects of the thickness of the porous medium on the island size distribution are studied for capillary displacements for the case in which buoyancy effects are negligible. It was found in a previous study that the number of islands of size s scales approximately as s~" in two-dimensional porous media, where a is a function of the fluid viscosity ratio. The present work reveals that there is a cross-over behavior between the two-dimensional and the three-dimensional problems.     

Evaluation of the capillary pressure curve techniques for determining pore size distribution — a network approach

Porous media are inter-connected networks of void spaces having different shapes and sizes. Attempts at developing analytical expressions describing fluid flow through them them have not been satisfactory to date for the lack of a satisfactory characterization of the structure of void spaces. The classical model, of bundle of tubes, is too simplified a model to be realistic and useful in most real situations. In the present work the flow behaviour in porous media was modelled by a network of inter-connected tubes of different sizes. An attempt was made to evaluate and synthesize various techniques of relating pore-size distribution to the capillary pressure behaviour of a medium. The validity of the “ink-bottle” effect and the techniques of Fatt and Meyer were tested by comparing the pore-size distributions obtained from capillary pressure data with the actual tube-size distribution of the network. It was seen that the existing techniques of obtaining pore-size distribution yield poor results even in the case of highly idealized network representations of porous media. Some modifications for making these techniques more realistic are discussed.The extent of inter-connections between the pores in a network was found to be a very important factor in influencing the shape of the capillary pressure (P_c) curves. This effect has been tested by allowing 2, 6, 10 and 14 inter-connections (on the average) between pores in the network, thus simulating parallel and intersecting tube models.In order to improve the reliability of results, it must be ascertained that there is, indeed, a one-to-one correspondence between pore-size distributions and capillary pressure curves.     

THE INFLUENCE OF POROUS MEDIA ON THE FLOW OF POLYMER MELTS IN CAPILLARIES†

Porous media placed in the entrance of capillaries were found to reduce the pressure drop across the capillaries (?P_c) by a factor of two or three for polystyrene. The reduction in ?P_c was found to be a function of the distance of the porous media from the capillary entrance, the type of porous media, the length of the capillary, and the rheological properties of the polymer melt. No significant reduction in ?P_c was observed for a polymer melt such as polyethyleneterephthalate (PET) which is nearly devoid of memory. The apparent shear rate for the onset of melt fracture was extended by a factor of three when polystyrene passed through the porous media before entering the capillary. No significant difference in die swell values was observed with the use of porous media in the entrance of the capillaries. The mechanism which accounts for these phenomena is believed to be associated with the break up of the entanglement network in the porous medium which temporarily changes the rheological properties of the polymer melt.     

Visualizing multiphase flow and trapped fluid configurations in a model three‐dimensional porous medium

We report an approach to fully visualize the flow of two immiscible fluids through a model three‐dimensional (3‐D) porous medium at pore‐scale resolution. Using confocal microscopy, we directly image the drainage of the medium by the nonwetting oil and subsequent imbibition by the wetting fluid. During imbibition, the wetting fluid pinches off threads of oil in the narrow crevices of the medium, forming disconnected oil ganglia. Some of these ganglia remain trapped within the medium. By resolving the full 3‐D structure of the trapped ganglia, we show that the typical ganglion size, as well as the total amount of residual oil, decreases as the capillary number Ca increases; this behavior reflects the competition between the viscous pressure in the wetting fluid and the capillary pressure required to force oil through the pores of the medium. This work thus shows how pore‐scale fluid dynamics influence the trapped fluid configurations in multiphase flow through 3‐D porous media. © 2013 American Institute of Chemical Engineers AIChE J, 59:1022‐1029, 2013     

Role of capillary and viscous forces in mobilization of residual oil

Experimental results are reported for a series of microemulsion displacement tests carried out in a consolidated core of medium permeability (1.75 μm²). The interfacial tension and viscosity were varied from 0.02 to 32.0 mN/m and 2.0 to 26.0 mN · s/m², respectively. The amount of oil unrecovered is correlated as a function of capillary number, N_c. Results are compared with those obtained by other workers who used porous media of low permeability and the relative importance of capillary and viscous forces at various stages of oil mobilization is discussed.     

PORE EVOLUTION AND SOLVENT TRANSPORT DURING DRYING OF GELLED SOL-GEL COATINGS: PREDICTING ‘SPR1NGBACK’*

ABSTRACT

This paper reports predictions of drying phenomena in deformable porous gel coatings (i.e. a porous solid elastic network filled with air or solvent). Initially, a gelled coating is saturated with solvent, but as it dries, liquid-vapor menisci begin to recede into larger pores and the gel becomes a partially-saturated porous medium. The tensile capillary pressure in the liquid causes a compressive deformation on the solid skeleton and a consequent reduction in thickness and pore-size of the coating. A theory coupling the large deformation of the solid skeleton to capillary pressure in the interstitial liquid is used to predict the course of drying of dip-coated porous gel coatings. The theory predicts a ‘springback’ effect in late stages of drying as the effects of capillary pressure diminish, which matches with experimental observations.     

Dynamic modeling of drainage through three-dimensional porous materials

The interplay of viscous, gravity and capillary forces determines the flow behavior of two or more phases through porous materials. In this study, a rule-based dynamic network model is developed to simulate two-phase flow in three-dimensional porous media. A cubic network analog of porous medium is used with cubic bodies and square cross-section throats. The rules for phase movement and redistribution are devised to honor the imbibition and drainage physics at pore scale. These rules are based on the pressure field within the porous medium that is solved for by applying mass conservation at each node. The pressure field governs the movement and flow rates of the fluids within the porous medium. Film flow has been incorporated in a novel way. A pseudo-percolation model is proposed for low but non-zero capillary number (ratio of viscous to capillary forces). The model is used to study primary drainage with constant inlet flow rate and constant inlet pressure boundary conditions. Non-wetting phase front dynamics, apparent wetting residuals (S_wr), and relative permeability are computed as a function of capillary number (N_ca), viscosity ratio (M), and pore-throat size distribution. The simulation results are compared with experimental results from the literature. Depending upon the flow rate and viscosity ratio, the displacement front shows three distinct flow patterns—stable, viscous fingering and capillary fingering. Capillary desaturation curves (S_wr vs. N_ca) depend on the viscosity ratio. It is shown that at high flow rates (or high N_ca), relative permeability assumes a linear dependence upon saturation. Pseudo-static capillary pressure curve is also estimated (by using an invasion percolation model) and is compared with the dynamic capillary pressure obtained from the model.     

Two-phase flow in porous media

Two-phase flow in porous media depends on many factors, such as displacement vs steady two-phase flow, saturation, wettability conditions, wetting fluid vs non-wetting fluid is displacing, the capillary number, interfacial tension, viscosity ratio, pressure gradient, uniformly wetted vs mixed-wet pore surface, uniform vs distributed pore throats, small vs large pores, well-connected pores vs pores connected by small throats, etc. These parameters determine how the two fluids are distributed in the pores, e.g. whether they flow in seperate channels or side-by-side in the same channels, either with both fluids being continous or only one fluid being continous and the other discontinuous. In displacement, the capillary number and the viscosity ratio determine whether the displacement front is sharp, or if there is either capillary or viscous fingering.     

Linear displacement of a wetting fluid by an immiscible non-wetting fluid in a porous medium: A predictive algorithm

The linear displacement of a wetting fluid by an immiscible non-wetting fluid in a two-dimensional porous medium composed of a network of sites multi-connected by bonds has been simulated mathematically. The algorithm involves Monte Carlo decision making, random walks and principles of the percolation theory. The algorithm described in the present work successfully predicts the three distinct behaviours of immiscible displacement in porous media. This algorithm is tested against experiments available in the literature for two-dimensional porous media. The agreement between the numerical results and the experiments is very good.     

Theoretical consideration of the effect of porosity on thermal conductivity of porous materials

The thermal conductivity of porous materials is theoretically studied in connection with nanoporous materials used in recent semiconductor devices. The effects of porosity and pore size on the thermal conductivity are discussed. The thermal conductivity of insulating materials is determined by the heat capacity of phonons, the average phonon velocity and the phonon mean free path. We investigate the porosity dependence of these quantities, especially by taking into account phonon scatterings by pores, and present an expression for the thermal conductivity as a function of porosity. Our model consideration predicts that the thermal conductivity of nanoporous materials depends on the ratio of the pore size R_p to the phonon mean free path for zero-porosity, l₀. The thermal conductivity for l₀/R_p > 1 decreases steeply with increasing porosity because of effective phonon scatterings by pores. On the other hand, the thermal conductivity for l₀/R_p < 0.1 decreases moderately with increasing porosity because phonon scatterings by pores are no longer effective. On the basis of the present theoretical consideration, we discuss the principal factor dominating the porosity dependence of thermal conductivity in nanoporous materials. We also discuss how one can design nanoporous materials with lower or higher thermal conductivity.     

Some aspects of wetting/dewetting of a porous medium

Various features of wetting/dewetting of porous media are examined. The phenomenon of capillary hysteresis is illustrated by a vertical capillary tube which consists of an alternating sequence of convergent—divergent conical sections. A study of the kinetics of wetting of this tube by a liquid shows that when the velocity of the liquid/vapour meniscus is plotted against the height of penetration, it oscillates about the Washburn velocity—distance curve and performs Haines jumps. A general macroscopic equation is derived for the rate of wetting/dewetting of a porous medium having randomly distributed, finely divided particles or pores. Use is made of the Forchheimer equation, which is an extension of Darcy's equation to higher Reynolds numbers. Dissipative energy terms due to internal fluid calculaton and to irreversible movements of the meniscus strongly affect the initial rate of imbibition, but as the wetting progresses the Reynolds number decreases and Washburn's equation prevails.The application of percolation theory to wetting/dewetting phenomena in porous media is studied. The use of percolation theory by Kirkpatrick and Stinchcombe to find the electrical conductivity of inhomogeneous solid mixtures is adapted to determining the permeability of a porous medium to fluid flow. It is also shown how the relation between the “precolation probability” and the concentration of “unblocked” channels or pores can be applied in calculating the capillary pressure—desaturation curve in drainage. In particular, percolation theory predicts that a threshold pressure or break-through pressure is required before a non-wetting fluid can displace a wetting fluid in a porous medium. It is often convenient to use tree-like or branching lattice networks as models of a porous medium, because these are amenable to exact solutions in regard to percolation probability and permeability. The percolation properties of porous medium models which consist of lattice networks of cylindrical channels with a distribution of cross-sections and also of randomly packed rotund particles are examined and their relevance to wetting/dewetting phenomena discussed.     

Using NMR displacement measurements to probe CO2 entrapment in porous media

Carbon dioxide sequestration in aquifers is seen as a potential climate change mitigation technique. One physical mechanism by which this could occur is capillary trapping of discrete pore‐scale CO₂ bubbles (referred to as ganglia) in the pore space. Nuclear magnetic resonance (NMR) techniques were used to quantify the spatial distribution and pore environment of such CO₂ entrapment in a model porous medium (random glass bead packing). 3D images revealed a relatively macroscopically homogeneous CO₂ entrapment, even though the image resolution is insufficient to resolve individual CO₂ ganglia. Quantification of the pore environment of the CO₂ ganglia was achieved using NMR displacement propagators (displacement probability distributions), acquired both before and after CO₂ entrapment. Lattice Boltzmann (LB) simulations were used to facilitate interpretation of the propagator statistics by considering various pore environments in which CO₂ could become trapped. Comparison with the experimental data suggests that CO₂ is preferentially entrapped in comparatively larger pores. © 2010 American Institute of Chemical Engineers AIChE J, 2011     

Dynamic effects in capillary pressure relationships for two‐phase flow in porous media: Experiments and numerical analyses

Well defined experiments and numerical analyses are conducted to determine the importance of dynamic effect in capillary pressure relationships for two‐phase flow in porous media. Dynamic and quasi‐static capillary pressure‐saturation (P^c‐S_w) and, ?S_w/?t‐t curves are determined. These are then used to determine the dynamic effects, indicated by a dynamic coefficient (τ) in the porous domains which establishes the speed at which flow equilibrium (?S_w/?t = 0) is reached. τ is found to be a nonlinear function of saturation which also depends on the medium permeability. Locally determined τ seems to increase as the distance of the measurement point from the fluid inlet into the domain increases. However, the functional dependence τ‐S_w follows similar trends at different locations within the domain. We argue that saturation weighted average of local τ‐S_w curves can be defined as an effective τ‐S_w curve for the whole domain which follows an exponential trend too. © 2012 The Authors. AIChE Journal, published by Wiley on behalf of the AIChE. This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited. AIChE J, 58: 3891–3903, 2012     

Combined Effect of Magnetic Field and Rotation on Thermosolutal Instability of a Compressible Fluid in Porous Medium

The combined effect of magnetic field and rotation on thermosolutal instability of a compressible fluid in porous medium is considered. The system is found to be stable for (C_p/g)β < 1 where C_p, β, and g stand for specific heat at constant pressure, uniform adverse temperature gradient, and acceleration due to gravity, respectively. The stable solute gradient, magnetic field, and rotation introduce oscillatory modes in the system for (C_p/g)β > 1, which were nonexistent in their absence. For stationary convection, the stable solute gradient and rotation have a stabilizing effect on the system for (C_p/g)β > 1. In the presence of rotation, the magnetic field has a stabilizing (or destabilizing) effect, and the medium permeability has a destabilizing (or stabilizing) effect under certain condition, whereas in the absence of rotation, the magnetic field and rotation have stabilizing and destabilizing effects for (C_p/g)β > 1, respectively, on the system. The sufficient conditions for the existence of overs-lability are obtained.     

The dependence of shear and elongational viscosity on the molecular weight of poly(vinylidene fluoride)

The dependence of shear and elongational viscosity on the molecular weight of poly(vinylidene fluoride) has been studied using a capillary rheometer. The elongational viscosity was evaluated based on Cogswell's method with two types of capillaries: capillary length (L)/capillary diameter (D) = 10 mm/1 mm and L/D = 0 mm/1 mm. We used the ratio P₀/P_L that indicates the contribution of elongational flow to the total flow involving both the shear and elongational flows. P_L and P₀ are the pressure losses in the capillary and the converging flows, respectively. P₀/P_L increased with molecular weight and shear rate. This corresponds to decreasing the number of entanglements of molecular chain under a large displacement, especially high shear. Thus, we suggest using P₀/P_L as the parameter of the entanglement interaction on the molecular chain under a large displacement. © 1999 John Wiley & Sons, Inc. J Appl Polym Sci 71: 2381–2384, 1999     

Radiofrequency and microwave (10 kHz–8 GHz) electrical properties of polypyrrole and polypyrrole–poly(methyl methacrylate) composites

The complex permittivity ? = ?′ ?j ?″ of polypyrrole (PPY) samples and polypyrrole–poly(methyl methacrylate) (PPY‐PMMA) composites is calculated from measurements in the radiofrequency and microwave range (F = 10³–8 × 10⁹ Hz) at room temperature. A relaxation phenomenon is observed in PPY‐PMMA composites with polypyrrole concentration p = 6–12% by weight. The frequency F_max corresponding to the maximum of ?″ appears in the radiofrequency domain and increases with the PPY concentration from 10⁴ to 2 × 10⁶ Hz. This relaxation is caused by space charge moving into the conductive clusters of PPY. At low frequency F ? F_max, the real part of the permittivity ?′ becomes very high. For F ? F_max, PPY‐PMMA composites have a percolative behaviour, the percolation threshold p_c being 3.85%. For F ? F_max, in the microwave domain, an ac component of the conductivity σ_ac appears. σ_ac varies as a power function of the frequency, σ_ac ∝ ω ^x; with x < 1, x independent of p. © 2001 Society of Chemical Industry     

Gravity-driven unsteady-state slug fall in capillaries – modeling and experimental verification

Infiltration of liquid droplets into dry porous media often occurs in industrial and natural settings, which has been widely modeled as liquid slug flow in capillaries. This work focuses on gravity-driven slug motion in vertically oriented capillary tubes. To model the propagation and evolution of the slug, a mathematical model was set on the basis of Newton’s second law and the law of conservation of mass. The model includes terms like slug’s inertia, deposited film, dynamic contact angle, slug’s advancing and receding menisci hysteresis, and it particularly highlights the direct effect of the trailing film on the change of slug mass. In order to verify this model, experiments on water slug with different lengths of initial slugs were conducted in two Pyrex glass capillaries that are partially wettable. It was found that both the length and the velocity of the slug vary during the slug motion in every case. Then the experimental results were simulated with the established model by carefully presetting two fitting parameters, α_a and α_h, that are related to the dynamic contact angle at the advancing meniscus and the thickness of the trailing film, respectively. The good agreement between the experimental and theoretical results demonstrates that the present model is capable of describing the unsteady-state dynamics of slugs fall in porous media.     

SINGLE FLUID FLOW IN POROUS MEDIA

A comprehensive study on single fluid flow in porous media is carried out. The volume averaging technique is applied to derive the governing flow equations. Additional terms appear in the averaged governed equations related to porosity ε, tortuosity τ, shear factor F and hydraulic dispersivity D _h. These four parameters are uniquely contained in the volume averaged Navier-Stokes equation and not all of them are independent. The tortuosity can be related to porosity through the Brudgemann equation, for example, for unconsolidated porous media.

The shear factor models are reviewed and some new results are obtained concerning high porosity cases and for turbulent flows. It is known that there are four regions of flow in porous media: pre-Darcy's flow, Darcy's flow, Forchheimer flow and turbulent flow. The transitions between these regions arc smooth. The first region, the pre-Darcy's flow region represents the surface-interactive flows and hence is strongly dependent on the porous media and the flowing fluid. The other flow regions are governed by the flow strength of inertia. For Darcy's flow, the pressure gradient is found to be proportional to the flow rate. The Forchheimer flow, however, is identified by a strong inertia! effects and the pressure gradient is a parabolic function of flow rate. Turbulent flow is unstable and unsteady flow characterized by chaotic flow patterns. The pressure drop is slightly lower than that predicted using the laminar flow equation.

The hydraulic dispersivity is a property of the porous media. It may be considered as the connectivity of the pores in a porous medium. It characterizes the dispersion of mementum, heat and mass transfer. In this paper, only the dispersion of momentum is studied.

Single fluid flow through cylindrical beds of fibrous mats and spherical particles has been used to show how to solve the single fluid flow problems in porous media utilizing the knowledge developed in this communication. Both the pressure drop and axial flow velocity profiles are computed using the developed shear factor and hydraulic dispersion models. Both the predicted velocity profile and pressure drop compare fairly well with the published experimental data.