Simulation of nitrogen adsorption-desorption isotherms. Hysteresis as an effect of pore connectivity

The sorption mechanisms in porous materials have been of long standing interest and debate. Concretely, the involved hysteresis during nitrogen adsorption-desorption processes and its interpretation has represented an important challenge for experimentalists and theoreticians. Moreover, a better understanding of the different observed hysteretic behaviors and the elapsed are as is still remaining. Such a scenario has motivated us to study the pore connectivity and pore size distribution effects upon the hysteresis in nitrogen adsorption-desorption isotherms of a porous solid. Our simulation has been carried out over the entire range of connectivity assuming three-dimensional pores randomly inscribed in a two-dimensional lattice with different occupation probabilities, related with pore connectivity, including the percolation threshold. The adsorption and desorption curves as a function of pressure have been simulated taking into account the monolayer-multilayer formation process and the capillary condensation-evaporation phenomena by using the Broekhoff de Boer and Kelvin equations, respectively. Results show that the employed methodology allows reproducing different types of hysteresis experimentally observed where the hysteretic behavior is strongly dependent upon both connectivity and the broadening of the pore size distribution. Specifically, the hysteresis loop area exhibits a maximum at the percolation threshold and then it decreases monotonically above the threshold.     

Displacement of residual nonwetting fluid from porous media

Percolation theory of transport in random composites is used to explain the correlation between the residual saturation of nonwetting phase in porous media after displacement by a wetting phase and the capillary number, this number being a measure of the ratio of Darcy-law viscous force in the wetting liquid to interfacial tension force in curved menisci between the two phases. Statistical concepts of percolation theory give estimates of the length distribution of blobs created when the nonwetting phase loses continuity because of displacement by the wetting phase. These estimates agree with the few experimental data. Simple blob mobilization theory and experiments establish that the capillary number required to mobilize a blob is inversely proportional to its length in the direction of the Darcy-law pressure gradient; this and the predictions of percolation theory account for the observed capillary number correlation.     

NETWORK SIMULATION OF RELATIVE PERMEABILITY CURVES USING A BOND CORRELATED-SITE PERCOLATION MODEL OF PORE STRUCTURE

This paper deals with the conductivity and relative conductivity properties of irregular 3-D networks of pores that represent the continua of the oil phase and the aqueous phase respectively, during steady slate two phase flow in porous media. The relative conductivity properties presented, correspond to the saturation history defined by the drainage, imbibition and secondary drainage capillary pressure curves respectively. Use has been made of the pore accessibility history of a 20 × 20 × 20 network and a 10 × 10 × 10 nodes core portion of the network is used to write the flow equations. A set of 1001 linear equations is solved using the Preconditioned Conjugate Gradients Method for the conductivities of the wetting phase and the non-wetting phase respectively, as a function of network saturation and saturation history. The effects of pore throat size distribution and pore body size distribution on relative permeability behaviour has been investigated. Furthermore, the effect of conductivity function q(D) proportional to Dⁿ (n = 0, 1, 2, 3, 4) on relative permeability behaviour was investigated, where D stands for pore throat diameter and n is an exponent depending on pore geometry.

The results of this work are very significant in elucidating the following points that are not clearly stated in the literature: 1) using the bypassing as the only trapping mechanism, the primary drainage and secondary drainage relative permeability curves are in agreement with experimental findings; 2) more realistic displacement mechanisms in secondary imbibition are required to have better agreement with experimental findings; 3) the correlated network models after the site type problem of percolation theory are realistic models of pore structure; 4) the conductivity function q(D) proportional to D³ is the most appropriate pore throat conductivity function because of lamelar like pore geometries; and 5) accurate prediction of the effective permeability requires knowledge of the porosity and the detailed pore geometry in the pore network, in addition to pore size distributions used in the network simulation.     

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.     

Capillary Pressure Estimation Using Statistical Pore Size Functions

Capillary pressure curves, which have been employed for a long period of time by researchers interested in pore size distribution, are commonly obtained from experimental measurements. The dynamic capillary pressure that influences the flow is affected by many factors including the pore size characteristics and pore scale dynamics. Hence, it is important to investigate the variation of the estimated pore size distribution with capillary number. In this study, a glass type micromodel is considered as the porous media sample. A parametric probability density function is proposed to express the pore size distribution of the porous model, which is also measured using an image analysis technique. The capillary pressure saturation mathematical model is developed by integrating the pore size distribution function. Model parameters with a physical significance are estimated by fitting the model to the measured capillary pressure data at different capillary numbers. The results of capillary pressure obtained are well matched to the measured values. The results show that the trends of the extracted pore size distribution curves have similar trends, but they are not exactly the same. Therefore, the dynamic capillary pressure data alone are not sufficient for estimation of the pore size distribution. As a related development, the prediction of the capillary pressure curves based on measured pore size distributions is also presented. The proposed probability distribution function has the flexibility of representing a wide variety of pore size distributions.     

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.     

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

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.     

Dispersion in flow through porous media—II. Two-phase flow

In this paper we extend our previous study (Sahimi et al., 1986, Chem. Engng Sci.41, 2103–2122) of dispersion processes in porous media occupied by two fluid phases. We report results of Monte Carlo investigations of dispersion in two-phase flow through disordered porous media represented by square and simple cubic networks of pores of random radii. The percolation theory of Heiba et al. (1982, SPE 11015, 57th Annual Fall Meeting of the Soc. Petrol. Engrs) is used to determine the statistical distribution of phases in the porespace. One of the phases is assumed to be strongly wetting on the porewall in the presence of the other phase. A pore size distribution is chosen which yields through the percolation theory of Heiba et al. network relative permeabilities that are in agreement with the available experimental data.As in one-phase flow dispersion is diffusive in the cases simulated, i.e. it can be described by the convective-diffusion equation. Longitudinal dispersivity in a given phase rises greatly as the saturation of that phase approaches residual (i.e. its percolation threshold); transverse dispersivity also increases, but more slowly. As residual saturation of a phase is neared, the backbone of the subnetwork occupied by the phase becomes increasingly tortuous, with local mazes spotted along it that are highly effective dispersers. Dispersivities are found to be phase, saturation and saturation history dependent.Some limited Monte Carlo experiments with a residence time representation of the effects of deadend paths within a phase or reversible adsorption on the pore walls demonstrate that the approach developed can be extended to study the influence of such delay mechanisms on the dispersion process.     

Influence of the boundary conditions on capillary flow dynamics and liquid distribution in a porous medium

After depositing a wetting liquid onto a porous medium surface, and under the influence of the capillary pressure, the liquid is imbibed into the porous medium creating a wetted imprint. The flow within the porous medium does not cease once all the liquid is imbibed but continues as a secondary capillary flow, where the liquid flows from large pores into small pores along the liquid interface. The flow is solved using the capillary network model, and the influence of the boundary condition on the liquid distribution within the porous medium is investigated. The pores at the porous medium boundaries can be defined as open or closed pores, where an open pore is checked for the potential threshold condition for flow to take place. In contrast, the closed pore is defined as a static entity, in which the potential condition for flow to take place is never satisfied. By defining the pores at distinct porous medium boundaries as open or closed, one is able to obtain a very different liquid distribution within the porous medium. The liquid saturation profiles along the principal flow direction, ranging from constant to steadily decreasing, to the profile with a local maximum, are found numerically. It is shown that these saturation profiles are also related to the geometrical dimension that is perpendicular to the flow principal direction, and changing the boundary type from open to closed allows the liquid distribution within the porous medium to be controlled. In addition to the liquid distribution, the influence of the boundary conditions on capillary pressure and relative permeability is investigated, where both parameters are not influenced by variation of the boundary condition types. © 2011 American Institute of Chemical Engineers AIChE J, 2012     

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.     

Pore-Scale Simulation of Drying of a Porous Media Saturated with a Sucrose Solution

This work presents results of Monte Carlo simulations of isothermal drying of a nonhygroscopic porous media initially saturated with a sugar solution. The porous media is represented by a two-dimensional network of cubic pores connected by throats with a given radius distribution. The considered network had just one open side (the three other sides were sealed) from which water evaporation occurred. Water evaporation, hydraulic flow, and diffusivity of sucrose in water are considered in the physical model. It was considered that drying occurred under isothermal conditions (low drying rates) and that the capillary forces surpass the viscous forces, as in invasion percolation. It was also considered that water evaporation inside the network of pores and throats causes solution concentration, which remains at the corners, allowing hydraulic connection throughout the whole network. At each simulation step, a single meniscus moves through a particular pore segment with the higher displacing force. As drying progresses, air replaces the solution. Determination of the mechanism prevailing at any given drying stage requires calculation of evaporation. In other words, each step of the simulation involves finding the solution to three systems of equations: the vapor pressure field in the vapor phase, the pressure field in the liquid phase, and the solutes' concentration in the liquid phase. Herein, we report results of drying curves calculated as a function of the sucrose and water saturation and of the distribution of liquid, sucrose, and vapor as drying advances. The results presented in this work showed that network models are a powerful tool for investigating the influence of the main mechanisms controlling drying at its different stages; that is, from liquid saturation condition to very low saturation (end of drying). Despite the applied simplifications, the model can capture the main aspects of drying of liquids and solutions present in porous media.     

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.     

Characterization of Pore Size Distribution by Infrared Scattering in Highly Dense ZnS

A model based upon Mie theory was developed to calculate the amount of spectral infrared scattering caused by the presence of a porous second phase. Various in-line transmission curves were calculated and used to characterize the scattering effects of pore size and concentration. The in-line transmission from 2.5 to 10 fjim of ZnS samples hot-pressed at 137.8,172.3, and 206.7 MPa was measured using Fourier transform infrared spectrophotometry and compared with calculated transmission curves. Good agreement with measured results was obtained only when a size distribution effect was included. From the analysis, a bimodal distribution of pores was found to give the best agreement which duplicates the measured transmission.     

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.     

The Freezing Point of Water in Porous Glass

Lowering of the freezing point of water in porous glass was studied by a DTA technique. The samples contained a spectrum of average pore sizes. The freezing-point lowerings measured from melting and freezing curves correlated, as predicted by simple capillary theory, with the average pore size of the sample. This result is discussed in terms of capillary-condensation theory and theories of frost damage in porous materials.     

Slow immiscible displacement flow in disordered porous media

The immiscible displacement of a wetting fluid by a non-wetting fluid in a disordered porous medium is studied in the capillary region, i.e. when capillary forces are dominant, by using the invasion percolation model to describe the displacement mechanism. The porous medium is represented by a two-dimensional network of interconnected capillaries whose radii follow a uniform size distribution. Disorder is assigned to the medium by considering the probabilities of occurrence of inaccessible pores, p_s, and non-conductive capillaries, p_b. It is found that the dynamic behaviour of the displacing fluid and the fraction of invaded pores depend on the degree of disorder of the medium. The results can provide an interpretation of the effects of the dead-end pore volume on the oil recovery and the displacement behaviour.     

Static and Dynamic Aspects of Liquid Capillary Flow in Thermally Bonded Polyester Nonwoven Fabrics

The weight gain method is employed to study the vertical capillary flow of wetting liquids in polyester nonwoven fabrics with different basis weights. The quantity of liquid absorbed by capillarity in the nonwoven is recorded as a function of time, until saturation. The liquid retention capacity of the nonwovens has been studied from their “saturation level”, i.e. the fraction of pore volume effectively filled with liquid. It is found that this saturation level varies greatly with the type of nonwoven, and generally decreases with nonwoven thickness. Moreover, the expected 100% value is rarely attained even when the sample height is smaller than the Jurin equilibrium height. These observations are attributed to the more heterogeneous pore sizes in very thin nonwovens, where the interconnection of large and small pores inhibits the continued capillary rise of liquid front. The other part of the study concerns the kinetics of liquid capillary flow which has been analyzed by taking into account the contribution of the meniscus in filling the pores. By subtracting this contribution from the mass of liquid absorbed, the new absorption mass is found to vary linearly with the square root of time, in agreement with the Washburn theory. For the thinnest nonwovens, the very small and unrealistic values of Washburn radii deduced from the experimental results do not correspond to the real physical pore sizes, but reflect slow capillary kinetics. This phenomenon is, however, less important when the thickness of the sample increases.     

Modeling at Pore-Scale Isothermal Drying of Porous Materials: Liquid and Vapor Diffusivity

Numerical simulations of isothermal drying of non-hygroscopic liquid-wet rigid porous media are performed. Two- and three-dimensional pore networks represent pore spaces. Two types of mechanisms are considered: evaporation and hydraulic flow. The drying is considered to be a modified form of invasion percolation. Liquid in pore corners allows for a hydraulic connection throughout the network at all times. As drying progresses, liquid is replaced by vapor by two fundamental mechanisms: evaporation and pressure gradient-driven liquid flow. Using a Monte Carlo simulation, evaporation and drainage times are computed. The controlling mechanism is indicated by the shorter calculated time. Initially, the drying is governed by liquid flow, then by a combination of liquid flow and evaporation and finally by local evaporation. Reported here are the distributions of liquid and vapor with drying time, capillary pressure curves, liquid film saturation curves, and liquid diffusivity and vapor diffusivity as a function of liquid saturation.     

Modeling at Pore-Scale Isothermal Drying of Porous Materials: Liquid and Vapor Diffusivity

Numerical simulations of isothermal drying of non-hygroscopic liquid-wet rigid porous media are performed. Two- and three-dimensional pore networks represent pore spaces. Two types of mechanisms are considered: evaporation and hydraulic flow. The drying is considered to be a modified form of invasion percolation. Liquid in pore corners allows for a hydraulic connection throughout the network at all times. As drying progresses, liquid is replaced by vapor by two fundamental mechanisms: evaporation and pressure gradient–driven liquid flow. Using a Monte Carlo simulation, evaporation and drainage times are computed. The controlling mechanism is indicated by the shorter calculated time. Initially, the drying is governed by liquid flow, then by a combination of liquid flow and evaporation and finally by local evaporation. Reported here are the distributions of liquid and vapor with drying time, capillary pressure curves, liquid film saturation curves, and liquid diffusivity and vapor diffusivity as a function of liquid saturation.