Pore network model

Notwithstanding the natural heterogeneity of the subsurface, we can usefully consider homogeneous (bulk, effective) descriptions for at least some problems, especially for water flow (but less so for contaminant migration see Sect. 10.1). Therefore, two basic approaches to modeling generally are used to describe and quantify flow and transport continuum-based models and pore-network models. We discuss each of these here. 

Rieckmann and Keil (1997) introduced a model of a 3D network of interconnected cylindrical pores with predefined distribution of pore radii and connectivity and with a volume fraction of pores equal to the porosity. The pore size distribution can be estimated from experimental characteristics obtained, e.g., from nitrogen sorption or mercury porosimetry measurements. Local heterogeneities, e.g., spatial variation in the mean pore size, or the non-uniform distribution of catalytic active centers may be taken into account in pore-network models. In each individual pore of a cylindrical or general shape, the spatially ID reaction-transport model is formulated, and the continuity equations are formulated at the nodes (i.e., connections of cylindrical capillaries) of the pore space. The transport in each individual pore is governed by the Max-well-Stefan multicomponent diffusion and convection model. Any common type of reaction kinetics taking place at the pore wall can be implemented. 

A similar model has been applied to the modeling of porous media with condensation in the pores. Capillary condensation in the pores of the catalyst in hydroprocessing reactors operated close to the dew point leads to a decrease of conversion at the particle center owing to the loss of surface area available for vapor-phase reaction, and to the liquid-filled pores that contribute less to the flux of reactants (Wood et al., 2002b). Significant changes in catalyst performance thus occur when reactions are accompanied by capillary condensation. A pore-network model incorporates reaction-diffusion processes and the pore filling by capillary condensation. The multicomponent bulk and Knudsen diffusion of vapors in each pore is represented by the Maxwell-Stefan model. 

Gane PAC. (2004) Absorption properties of coatings A selected overview of absorption criteria derived from recent pore network modelling. / Dispers Sci Technol 25(4) 389. 

The pore connectivity r of two types of silica (highly porous beads, monolithic silicas) was calculated according to the pore network model proposed by Meyers and Liapis. Nt was proportional to the particle porosity in the case of highly porous beads. The differences in the pore connectivity for both types of silica were reflected in the mass transfer kinetics in liquid phase separation processes by measuring the theoretical plate height-linear velocity dependencies. In a future study, monolithic silicas possessing different macro- as well as mesopores will be investigated and compared with the presented results. 

A great number of studies have been published to deal with relation of transport properties to structural characteristics. Pore network models [12,13,14] are engaged in determination of pore network connectivity that is known to have a crucial influence on the transport properties of a porous material. McGreavy and co-workers [15] developed model based on the equivalent pore network conceptualisation to account for diffusion and reaction processes in catalytic pore structures. Percolation models [16,17] are based on the use of percolation theory to analyse sorption hysteresis also the application of the effective medium approximation (EMA) [18,19,20] is widely used. 

Liu, H., Zhang, L., and Seaton, N. (1993). Analysis of sorption hysteresis in mesoporous solids using a pore network model. J. Colloid Interface Sci., 156, 285-93. 

The shape of the capillary portion of the liquid-vapor interfacial area for sand (Fig. 1-1 lb) resembles simulation results of Reeves and Celia (1996) of interfacial areas in pore networks due to capillarity only. The discussion illustrates potential limitations in using cylindrical pore network models (Reeves Celia, 1996) especially for studies of volatile liquids and surfactants, and other multiphase transport processes where interfacial areas play a crucial role (Kim et al., 1997 Karkare Si. Fort, 1996). Furthermore, the overwhelming role of adsorbed liquid films casts doubts on several proposed constitutive relationships between capillary pressure (saturation) and interfacia] area (Skopp, 1985 Hassanizadeh Gray, 1993) most of which were based on assumed cylindrical capillary geometry in the absence of adsorption. 

In general it is clear that PFG spin-echo data can provide information on the pore structure at both the gradient length scale, /q, and the diffusion length scale, y (/JA). The NMR data can be analysed to obtain simple structure factors, which may then be related to particular pore network models. ... 

It is also possible to perform a two-dimensional PFG spin-echo experiment employing two orthogonal magnetic field gradients. This yields a two-dimensional propagator Ps X, Y, A) that corresponds to the joint probability for molecular displacements X and Y in time A. Results have been obtained on a packed bed of glass spheres and on a sandstone, and have been compared with those predicted by numerical simulation of the flow assuming pore network models. ... 

NMR methods offer a noninvasive method of characterizing porous media. A variety of different techniques may be used to obtain useful information on the pore space. For instance, pore sizes may be measured using the freezing point depression technique for mesoporous solids or by relaxation time measurements for macroporous solids. Other pore space information comes from PFG techniques, while direct imaging of the pore space is possible for large pores. The information from studying the pore space can then be incorporated into appropriate pore network models. 

Romero-Zeron, L., Kantzas, A. Evolution of foamed gel confined in pore network models Source Journal of Canadian Petroleum Technology, v 45, n 11, p 51-62, November 2006. 

Pore network models are an example of a discrete model. The earlier pore network models consisted of parallel pores [18] and randomly oriented cross-linked pores [19]. Bethe lattice [20], and regular networks [21] have also been used to represent catalyst structures. Pore network models have been used to analyze the complicated interactions between diffusion and reaction that may occur in catalyst particles, for example Sharatt and Mann [21] used their cubic network... 

The catalyst effectiveness factor rji was calculated from the pore network model of Wood and Gladden [15] under the conditions on which capillary condensation was expected. The pore network model was solved over a range of temperatures from 553 to 580 K and for several pressures in the interval 20-40 bar to create a database of effectiveness factors for input to the macroscopic reactor model. The hydrodesulfurization of 1 mole % diethyl sulfide in an inert dodecane carrier was considered, with a molar gas oil ratio of 4. The catalyst was taken to have a connectivity of 6 and a normal distribution of pore sizes with a mean of 136 A and standard deviation of 28 A. By using the results of the pore network simulation as input to the macroscopic fixed bed reactor model, capillary condensation at the scale of the catalyst pellets was accounted for. 

FIGURE 23.24 Effect of bed entrance pressure upon the conversion of diethyl sulphide in a fixed bed reactor. Lines represent fixed bed reactor simulations with catalyst effectiveness factor calculated from the random pore network model, including the influence of capillary condensation. 

Dadvar, M., Sohrabi, M., and Sahimi, M., Pore network model of deactivation of immobilized glucose isomerase in packed-bed reactors — I two-dimensional simulations at the particle level, Chem. Eng. Set, 56, 2803-2819, 2001. 

In addition, the need has been identified for more realistic pore network models, which can handle the typically observed broad ranges of pore size (fixrm A to pm), and apparent localised spatial variations in pore stmeture and pore size distribution. 

Over a period of time, particularly the last twenty years, researchers have attempted to improve and create models oqrable of describing the influence of porous media in catalytic reaction processes, and they have been aided by the development of computing power and computer modelling techniques. Hence a continual progression has been made from the simple parallel bundle models, which have been the basis of most textbook treatments [1], to stochastic pore network models [2-3] and chamber and throat pore models [4], and more recently fiactal-based models, first introduced by Mann and Wasilewski [5], and subsequently expanded upon by other workers [6-8]. 

The apparent discrepancies observed witl the pore and volume distributions obtained from the random network can easily be resolved by examining the behaviour of an artificially shielded random pore network model. In this case, the edge of the network has an artificial layer of smaller pores across its entire sirface, thus shielding the internal random network structure. In order to set the size of the shielding pores, image analysis has been carried out at the physical edges of axial and radial... 

On initial inspection the results obtained from serial sectioning of LMPA intruded samples appear at odds with the principle theory behind intrusion and retraction as predicted by the Washburn equation. But further inspection shows it is not the Washburn equation, but mercury porosimetry that is at fault. Pore network models have often been used to characterise the behaviour of pore structure in relation to mercury porosimetry. But the model is only as good as the assumptions and the data that it is based iqron. Without artificially shielding the network, the model caimot propa ly detomine the correct psd and cannot derive a more spatially accurate structure that could be used for diffusion and reaction modelling. In order to characterise the pore structure more accurately, we need to introduce some of the elements usually revealed by LMPA intrusion tests. 

The two major pore models that have been used extensively over the years for practical purposes are the parallel-pore model proposed by Wheeler in 1955 [5, 9] and the random-pore model proposed by Wakao and Smith in 1962 [34]. Among the more recent advanced models are the parallel cross-linked pore model [35] and pore-network models [36, 37]. 

Gostick, J.T., loannidis, M.A., Fowler, M.W., and Pritzker, M.D. (2007) Pore network modeling of fibrous gas diffusion layers for polymer electrolyte membrane fuel cells. J. Power Sources, 173, 2TJ 29Q. 

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