Limit Models of Pore Space Structure of Porous Materials for Determination of Limit Pore Size Distributions Based on Mercury Intrusion Data
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Abstract
This paper proposes the application of capillary and chain random models of pore space structure for determination of limit pore diameter distributions of porous materials, based on the mercury intrusion curves. Both distributions determine the range in which the pore diameter distribution of the investigated material occurs and defines the degree of inaccuracy of the method based on the mercury intrusion data caused by the indeterminacy of the sample shape and its pore space architecture. We derived equations describing the quasistatic process of mercury intrusion into the porous layer and porous ball with a random chain pore space structure and analysed the influence of the model parameters on the mercury intrusion curves. It was shown that the distribution of link length in the chain model of the pore space, random location of chain capillaries in the sample and the length distribution of the capillaries do not influence significantly the intrusion process. Therefore, a simple model of the mercury intrusion into the layer is proposed in which chain links of the pore space have random diameters and constant length. This model is used as a basic model of the intrusion process into a sample of any shape and size and with homogeneous and isotropic chain pore space architecture. The thickness of the layer then represents the mean length of chain capillaries in the sample. It was also proved that the capillary and chain models of pore space architecture are limit models of the network model identified in this paper with the pore architecture of the investigated sample. This justifies the application of both models for determination of limit cumulative distributions of pore diameters in porous materials based on the mercury intrusion data.
Keywords
Mercury porosimetry Capillary potential curve Chain and capillary models of pore structure Limit pore diameter distributions1 Introduction
The volumeweighted distribution of pore diameters is the fundamental characteristic of the microscopic pore space structure of porous materials (Scheidegger 1957; Dullien 1979). It defines the volume fraction of pore space points with the assigned pore diameter in the whole pore volume of the sample and makes it possible to assess the basic macroscopic parameters of permeable porous materials, i.e. volume porosity, specific internal surface area, pore area distribution and even permeability and tortuosity (León y León 1998). These parameters play an important role in many physical and chemical processes appearing in such materials, e.g. filtration, transport of mass and energy, wave propagation or chemical reactions.
There are three basic methods used for determination of pore diameter distributions in porous materials: the direct geometrical method based on analysis of 3D microscopic images of the pore space, the indirect method applying the NMR techniques and the method of mercury porosimetry. Geometrical determination of pore diameter distribution consists in assignation to each pore space point (voxel) the maximum diameter of the ball inscribed into the pore space, including this point (Hildebrand and Rüegsegger 1997). It is usually calculated using methods of morphological image analysis (e.g. Shih 2009), and its accuracy is mainly contingent upon resolution of the analysed image. In the case of images created by Xray computed microtomography, their resolution can achieve the value of 1 µm. For materials with pores of diameters greater than the image resolution, this method can be used as a “gold standard” (e.g. Arns 2004).
Investigations of porous materials by the NMR techniques consist in measurement of relaxation times of the induced magnetic field of the hydrogen nuclei contained in fluids filling pore space and are based on dependence of this process on quantity, physical properties and the state of fluids in the sample of the material (Coates et al. 1999). This enables direct determination of the presence and quantity of different fluids in the sample, and indirect evaluation of their volume fraction and properties, e.g. viscosity. Determination of pore diameter distributions in porous materials by the NMR loggings has also indirect character. In this case, the linear dependence of the relaxation time on the volumetosurface ratio of pores filled with fluid containing hydrogen nuclei is used (Brownstein and Tarr 1979; Cohen and Mendelson 1982; Mendelson 1990; Sorland et al. 2007). The range of pore sizes determined in this way depends on the capabilities of the NMR logging for measurements of short relaxation times, and the obtained distributions are qualitatively consistent with that obtained from the microscopic image analysis. However, due to variety of factors influencing the relaxation time, proper interpretation of the NMR logging measurements requires detailed knowledge of the properties of both the skeleton and the fluid in the pore space (Kleinberg and Horsfield 1990; Arns 2004).
Mercury porosimetry is the standard method of experimental determination of the pore diameter distribution in permeable porous materials (León y León 1998; Webb and Orr 1997; Winslow 1984; Giesche 2006). This method applies the nonwetting property of mercury for the surface of almost all materials and consists in determination of the socalled capillary potential curve that relates the volume of mercury intruded into a porous sample with progressively increasing mercury pressure. Interpretation of this curve is based on the assumption that at increasing pressure, mercury is intruded against the capillary forces into the pores of decreasing diameters. This is equivalent to the assumption that the pore structure of the investigated material can be modelled as a bundle of capillaries with random distribution of diameters, crossing the whole sample. In that case, the Washburn formula of menisci equilibrium in a cylindrical capillary can be directly applied for interpretation of the experimental data. This formula relates the liquid pressure with the diameter of the capillary. As a result, the relation between the relative volume of mercury intruded into the porous sample and the diameter of the capillary is obtained, which is interpreted as a cumulative curve of volumeweighted pore diameter distribution in a sample of the investigated porous material.
The basic advantage of mercury porosimetry, determining its common application for characterization of porous materials, is the wide range of measured pore diameters covering six orders of magnitude, from 3 nm to 360 μm. This advantage is also the shortcoming of the method, because it requires application of high pressures that in some cases may deform the measured sample, influencing the obtained results. The obvious shortcoming of mercury porosimetry is the common use of the capillary model of pore space structure of the investigated material for interpretation of mercury intrusion data. This model does not consider the situations that often occur in real porous material, when the large pores are joined with others by narrow necks. This makes it impossible to fill such pores with mercury at the pressure appropriate for their diameter. Therefore, the pore size distribution determined by the capillary model underestimates the volume of large pores, ascribing it to the volume of small pores. In consequence, the obtained distributions may be burdened by significant error. Many authors have also criticized direct application of the Washburn formula for the interpretation of the capillary potential curves.
Many attempts have been made to develop microscopic models of the pore space structure of real porous materials that would include their characteristic spatial connectivity, inhomogeneity and randomness, and would extend the capabilities for modelling processes in which the surface phenomena play an important role. This particularly concerns quasistatic processes of liquid imbibition and drainage. Generally, these models can be divided into analytical and computational models.
The analytical models are mainly those that apply the approach based on percolation theory (Broadbent and Hammersley 1957; Adler 1992; Berkowitz and Balberg 1992; Berkowitz and Ewing 1998) and, in particular, invasion percolation theory (Lenormand and Bories 1980; Wilkinson and Willemsen 1983). In these models, the pore space is represented by the regular network of sites and bonds, and the main emphasis is focused on the analysis of the influence of the geometrical randomness of the medium on the largescale course of processes in the pore space. A different approach is presented by Chizmadzhev et al. in their monograph (Chizmadzhev et al. 1971), in which the pore space is modelled as a random graph, and the description of the capillary transport is based on the methods of statistical analysis. In this case, a very complex, multiparametric description is obtained that limits its practical usability.
The computational models can be divided into three types (Bhattad et al. 2011): direct, granular and network. In the direct models, the pore space is reconstructed directly from a 3D microscopic image of a sample of porous material, obtained, for example, by the microcomputer tomography method. In turn, in the granular models the pore space is generated computationally by regular or random packing of simple geometrical elements (e.g. balls) with constant or random sizes representing the skeleton of porous material. In both cases, modelling of processes in the pore space consists in numerical solution of differential equations using the finite element, finite volume or lattice Boltzmann methods. Due to its geometrical and computational complexity, such modelling is timeconsuming and can be used only for relatively small samples. This complexity increases even more when processes with surface phenomena are modelled.
An alternative method of porescale modelling that substantially increases computational efficiency and reduces the complexity of the local pore geometry, is the approach based on network models of the pore space (Fatt 1956; Blunt 2001; Martins et al. 2009; Bhattad et al. 2011; Raeesi et al. 2013; Xiong et al. 2016). In these models, the pore space of real materials is represented by a regular or random network of interconnected, geometrically simple elements (e.g. spheres, cuboids or capillaries) of random sizes. Then, the pore description of processes takes the simple form of a Poiseuilletype equation. Considering the complexity and the manner of the construction of the pore network, one can distinguish various generations of models (Bhattad et al. 2011). In the firstgeneration network models, the pore space is composed of interconnected cylindrical capillaries, whereas in models of the second and third generations, the pore space is represented by a network of large chambers (e.g. spheres and cuboids) interconnected by narrow bonds of various cross sections. In the case of thirdgeneration models, the pore network is created directly from threedimensional (e.g. microtomographic) pore space images of a sample of porous material using the methods of morphological image analysis. This concerns both the network structure and size distributions of its components. In models of the first and second generations, in turn, the capillary pressure data and porosity are used to tune the numerical network model (Raeesi et al. 2013). This is done using optimization methods by generation of size distributions of the network components that fit the simulated capillary pressure curve and porosity to their experimental values. In this case, the computational complexity of the simulation process is partially transferred to the stage of network generation of the model.
Unlike the microscopic modelling, there are a few proposals for the macroscopic description of the quasistatic processes of nonwetting liquid intrusion into porous materials. The direct application of theories describing liquid transport in unsaturated porous materials to solve this problem, such as the Richards equation (Richards 1931), produces results identical to those for the medium with a capillary pore space structure. This results from the assumption commonly taken in papers in this field that the capillary pressure is a constitutive quantity and is a unique function of saturation with liquid. Such a constitutive assumption causes that the character of the distribution of both quantities in a porous body is always the same. In this case, the homogeneity of the capillary pressure will always induce a homogeneous distribution of saturation with liquid. For this reason, even the advanced thermodynamic models of twophase flow in porous materials presented in papers by Hassanizadeh and Gray (1990, 1993) do not describe the inhomogeneity of liquid distribution during the quasistatic intrusion processes.
A nonstandard approach to the problem of macroscopic description of such processes is presented in the paper (Cieszko et al. 2015), where balance equations and constitutive relations have been formulated analysing the process of liquid intrusion in the pressurespace continuum, and menisci motion, as a process of diffusive transport in this continuum. In this model, there are parameters and functions describing pore size distributions in the boundary conditions and in the coefficients characterizing diffusive transport of menisci in the pressurespace continuum. The paper (Cieszko 2016) presents generalization of the macroscopic description of the nonstationary processes of capillary transport of liquid and gas in porous material. It was proved in this paper that such processes take place in the fifthdimensional pressure–timespace continuum, and quasistatic processes are the special case of this model.
The purpose of the present paper is to formulate a description of mercury intrusion into a porous sample with a relatively simple random chain pore space structure and to propose a method of determining limit pore diameter distributions in porous materials based on the chain and capillary models and the mercury intrusion data.
In the chain model, the individual pores are cylindrical tubes of random length and diameter distributions that are joined at random in a series forming the capillaries of a stepwise changing cross section. The importance of the chain model for interpretation of the mercury intrusion data consists in the fact that the chain and capillary models for given pore size distribution and porosity form limit models of the network pore space structure in which cylindrical tubes with random size distribution form a spatial network (Cieszko and Kempiński 2006). The network model reflects the structure of pore space in real porous materials. However, due to the complexity of the description of mercury intrusion into a sample with such a pore structure, formulation of an effective model for interpretation of mercury intrusion data is difficult (Chizmadzhev et al. 1971).
Application of both limit models for interpretation of mercury intrusion data makes it possible to determine limit distributions of pore sizes in the investigated material. Both distributions determine the range in which the pore diameter distribution of the investigated material occurs and defines the degree of inaccuracy of the method caused by the indeterminacy of the sample shape and its pore space architecture. Characterization of pore size distribution in this way is additionally justified by the fact that the accuracy of mercury porosimetry is also limited by at least two other physical assumptions made during interpretation of the mercury intrusion data (León y León 1998): the wetting angle is constant, and the pore space is unchanged during mercury intrusion.
This paper consists of six main sections. Equations describing the quasistatic process of mercury intrusion into porous material with a chain pore structure are derived in the first four main sections. This was done in three stages: first, mercury intrusion into the halfspace of porous material with a chain pore space structure was analysed and the integral Volterra equation was obtained for the probability distribution of mercury occurrence in the halfspace. This distribution was used in a section tree to derive the equation for the probability distribution of mercury occurrence in the porous layer during twoside mercury intrusion. Next, an expression describing the saturation of the porous layer with mercury (capillary potential curve) was derived in Sect. 4, and four special cases of the model were obtained. Section 5 shows that the obtained model of mercury intrusion into the porous layer can be used for the description of mercury intrusion into a ball with homogeneous and isotropic chain pore architecture. In this case, particular chains of pores (links) form chords of the ball with random length distribution. The chord length distribution in the ball was used to derive an explicit expression for the ball saturation with mercury.
The obtained results were applied in Sect. 6 to analyse the influence of the model parameters on the capillary potential curve. It was shown that link length distribution in the chain model, location of chain capillaries in the sample and distribution of the capillary length do not influence significantly the capillary potential curve. This proves that a very simple expression describing mercury intrusion into a porous layer with a chain pore space structure and constant link length can be used as a basic description of the capillary potential curve for a porous sample of any shape and size and with chain pore space architecture. The thickness of the layer then represents the mean length of the chain capillaries in the sample.
Section 7 proved that the capillary and chain models of pore space architecture are limit models of the network model identified in this paper with the pore architecture of real porous material. This justifies the application of both models for determination of limit distributions of pore diameters in porous materials, the procedure of which is described in this section.
2 Modelling of Mercury Intrusion into Porous HalfSpace
2.1 Basic Assumptions
Links of a diameter given by condition (2.1), following the paper by Chizmadzhev et al. (1971); we will call here critical. They divide the set of all other links of the model into two classes. The first class consists of the supercritical links of diameters larger than the critical one. These links may be filled with mercury at a given pressure. The other class consists of the subcritical links with diameters smaller than the critical one, which are impossible to fill with mercury at a given pressure. Using the proposed nomenclature, we can say that during the intrusion process into a sample of porous material, mercury fills only the first few supercritical links of the pore space until the first subcritical link occurs. Due to the random character of the link size distribution, the depth of menisci placement in the pore space will take random values. Only in pores on the sample surface with subcritical diameters do the menisci occur on this surface. Consequently, the mercury distribution in a sample of porous material with a chain pore space structure depends on the shape and size of the sample. Only in samples with capillary pore architecture is this distribution always homogeneous. In samples with chain pore architecture, each chain capillary is filled with mercury independently of the course of this process in the remaining capillaries. Therefore, the description of mercury intrusion into samples with chain pore architecture is much simpler than such description in samples with network pore architecture.
In order to ensure the random character of link location in the halfspace and layer of porous material, we assume that both media have been cut out from an infinite porous medium with a chain pore space structure by planes perpendicular to the axis of the capillaries. Due to the random location of links in the capillaries, all links on the surfaces of the halfspace and the layer will be cut off. Therefore, their length distribution differs from that for links inside the medium.
2.2 Description of Mercury Distribution
We consider a system in which porous material with initially empty pores occupies the halfspace \( z > 0 \) and is in direct contact with the mercury occupying the halfspace \( z < 0 \). The pore space architecture of the porous halfspace is assumed to be formed of link chains with cutoff boundary links. To derive equations describing the quasistatic process of mercury intrusion into the halfspace of porous material, we will first consider a system with uncut boundary links.
The number \( m_{z} \) will be determined considering the set of all capillaries with the first supercritical link. Only in this case will mercury occur inside them. The capillaries filled with mercury at the depth z form a subset of this set.

capillaries with the first link of length \( u > z \),

capillaries with the first link of length \( u < z \).
2.3 Description of Mercury Saturation Distribution
Using probability distributions \( F_{o} \left( z \right) \) and \( F\left( z \right) \) obtained in the previous subsection, we can derive volumetric measures of mercury distributions in both types of porous halfspaces which are directly related to the form of experimental data obtained by the mercury intrusion method, namely saturation of sample with mercury defined as a volume fraction of the pore space occupied by mercury.
3 Modelling of Mercury Intrusion into Porous Layer
We apply the probability \( F\left( z \right) \) describing mercury distribution in the halfspace of porous material, given by (2.12), to derive the expression for the probability \( G\left( z \right) \) of mercury occurrence in a layer of thickness L during the twoside mercury intrusion. Due to the chain architecture of the pore space in the layer, mercury intrusion into pores can be regarded as a twostep process realized first as the lefthandside intrusion and next as the righthandside intrusion or vice versa. During this process, mercury fills the supercritical capillaries which are composed of the supercritical links only, whereas the subcritical capillaries also containing the subcritical links remain partially empty. This means that the lefthandside and righthandside processes of mercury intrusion into subcritical capillaries are independent.
Equality (3.8) relates mercury saturation distribution in the porous layer with the mercury distribution in the surface layer of the porous halfspace.
4 Description of Mercury Intrusion Curve into Porous Layer
We apply expression (3.8) describing mercury distribution in a porous layer to derive the dependence of the layer saturation with mercury during the intrusion process on the mercury pressure p. This dependence is called the capillary potential of porous material and is closely related to the material pore space architecture and its pore size distribution.
Considering relation (2.12) between probabilities \( F\left( z \right) \) and \( F_{o} \left( z \right) \), and the fact that function \( F_{o} \left( z \right) \) is given by the integral Eq. (2.11), the capillary potential of the layer (4.2), in general, will not be given in explicit form. The mathematical complexity of the description of this curve for porous material with the chain pore space architecture is caused by the random distribution of link length.
In the next part of this section, we present special models of the chain pore space structure in the porous layer for which the capillary potential (4.2) has an explicit analytical form. We will consider four cases: the capillary model; the chain model with fixed link length and link nodes on the layer surfaces which we will call periodic; the chain model with fixed link length and cutout surface links which we will call randomperiodic; and the chain model with random link length distribution given by the function for which the integral Eq. (2.11) can be solved.
4.1 Capillary Model
From (4.4), it results that in the model of porous material with capillary pore space architecture, the pore diameter distribution \( \vartheta \left( D \right) \), given by expression (2.21), completely defines the capillary potential curve. This model is commonly used for interpretation of mercury intrusion data.
4.2 Periodic Chain Model
Expression (4.9) for \( N = 1 \) takes form (4.4) describing capillary potential curve of porous layer with capillary pore space architecture.
4.3 RandomPeriodic Chain Model
Determination of the explicit form of the expression describing the capillary potential curve (4.2) requires the solution of integral Eq. (2.11) and relation (2.12) to be used.
Both functions satisfy the condition: \( F\left( {ka} \right) = F_{a} \left( {ka} \right) \), for \( k = 1,2, \ldots \).
For the case when the number of links in the layer is equal to one (\( N = 1 \)), the randomperiodic model also reduces to the capillary one, despite the presence of one link node in each capillary of the layer. This is caused by homogeneous distribution of the probability \( F_{L} \left( z \right) \) of mercury occurrence in the layer. In this case: \( F_{L} \left( z \right) = f_{o} \).
4.4 Random Chain Model
From (4.17), it results that to preserve the positive value of parameters \( \alpha \) and \( \beta \) characteristics \( \bar{u} \) and \( \sigma_{u} \) have to satisfy the condition \( 1 > \sigma_{u} /\bar{u} > 1/\sqrt 2 \). Then, the inequality \( \alpha > \beta \) is also satisfied.
Expressions (4.4), (4.9), (4.14) and (4.21) make it possible to analyse the influence of the link length distribution and their diameters in the chain model of pore space architecture on the capillary potential curve of the porous layer. The differences between these descriptions are defined by the form of the expressions in brackets occurring at the quantity \( f_{2} \left( {D^{*} } \right) \). This quantity defines the saturation of layer surfaces with mercury, regardless of which chain model of pore space architecture is considered. Expressions in brackets, in turn, describe the influence of mercury distribution inside the layer on the capillary potential curve. Detailed analysis of the influence of chain model parameters on this curve is presented in Sect. 6.
5 Description of Mercury Intrusion Curve into Porous Ball
We apply expression (4.2) describing the saturation of the porous layer with mercury during the intrusion process to formulate such a description for mercury intrusion into a porous ball with chain pore space architecture. It is assumed that a spherical sample of porous material has a homogeneous and isotropic architecture of link chains (representing pores) and radius R of the ball is much greater than the mean length \( \bar{u} \) of links (\( R \gg \bar{u} \)). In this case, mercury distribution in the ball is inhomogeneous like in samples with network pore space architecture. However, due to the chain pore architecture, each chain capillary is filled with mercury independently of the course of this process in the rest of the capillaries. Therefore, the description of the intrusion process into particular capillaries of porous ball is the same as the description of mercury intrusion into the porous layer.
Expression (5.7) defines the capillary potential curve of a porous ball with chain pore architecture.
6 Analysis of Influence of Model Parameters on the Mercury Intrusion Curve
To determine the sensitivity of the capillary potential curve of a porous ball with chain pore space architecture on model parameters, we will analyse the influence of the following factors: link (pore) length and diameter distributions; random location and length distribution of chain capillaries in a sample; sample size. From integral Eq. (2.11) and expression (2.12) result that link length and random location of chain capillaries in the layer strongly influence the complexity of the chain model. Therefore, these two factors also have a qualitative importance for the model applicability, whereas the influence of the other factors is only of a quantitative character.
In the model proposed in the paper, the description of the ball capillary potential curve with chain pore architecture, given by expression (5.7), is defined by the function describing such curve for the layer, given by expression (4.2). This causes that influence of parameters characterizing random geometry of pores on the ball potential curve is fully represented by the influence of these parameters on the layer potential curve. Therefore, the analysis of influence of these parameters will be carried out on the layer potential curve.
6.1 Basic Assumptions
Considering that quantities \( \sigma_{D} \) and \( \overline{{1/D^{2} }} \) have to take positive values, from expressions (6.2)_{2} and (6.4) the following conditions result: \( m \ge 2 \) and \( n \ge 0 \).
6.2 Influence of Link Length Distribution and Random Location of Capillaries
This means that the capillary potential curves drawn in Fig. 2 for different mean length \( \overline{u} \) of links have also different standard deviations \( \sigma_{u} \).
In Fig. 2, it results that the mercury intrusion curves of a porous layer with the periodic (PM), randomperiodic (RPM) and random (RM) pore space architecture are very close for each values of ratio \( N = L/\overline{u} = L/a \) and the difference between them increases when this ratio decreases. In the case, when \( N \to 1 \), it can be shown that the mercury intrusion curve for the random model considerably differs from those for the capillary, periodic and randomperiodic models which take the same form described by expression (4.4) for the capillary pore architecture. This means that link length distribution and random location of chain capillaries in the layer (or cutout link length distribution on surface of the layer) do not significantly influence the capillary potential curve when the thickness L of the layer is much greater than the mean link length \( \bar{u} \)\( \left( {N = L/\bar{u} = L/a \gg 1} \right) \). Such condition ensures full statistical representation of link chains (pores) in the layer causing that saturation of the layer with mercury for a given pressure depends mainly on the mean placement of menisci in the layer. This placement is fully determined by the mean link length and probability of supercritical link occurrence in the layer defined by formula (2.7).
From the above analysis, it results that the description of the capillary potential curve of a porous layer with random chain pore architecture can be effectively represented by the simple model of constant link length, given by expression (4.9). Therefore, further analysis of the influence of model parameters on the capillary potential curve will be based on the periodic model.
6.3 Influence of Capillary Length Distribution
Similarly, for \( M \to 1 \), it can be shown that the mercury intrusion curve for the porous ball, given by expression (6.6), considerably differs from that for the porous layer, described by expression (4.9). However, when the number \( M \) of links in the mean length of capillaries increases, the influence of their length distributions on the capillary potential curve of the ball decreases and both curves approach to each other. In this case, expression (6.6) describing the mercury intrusion curve into a porous ball can be effectively replaced by the simple expression (4.9) for the intrusion curve into the layer. Then, the thickness of the layer L, present in expression (4.9), should be interpreted as the mean value of capillary length in the ball.
Generalizing the above considerations, we can state that expression (4.9) can be interpreted as an effective model of mercury intrusion curve into a porous sample of any shape and size when its pore structure is chain, isotropic and homogeneous. In this case, the quantity L in this model represents the mean length of chain capillaries (chords) in the sample. This quantity is the mean characteristics of the shape and size of the sample. For ballshaped samples, the mean length \( \overline{L} \) of chain capillaries is related to the ball radius R by (5.2)_{1}.
6.4 Influence of other Model Parameters
Curves in Fig. 4, due to relation \( N = \overline{L} /a = 4R/3a = 2M/3 \), illustrate simultaneously the influence of both parameters \( R \) (or \( \bar{L} \)) and \( a \) on the capillary potential curve. It is visible that an increase in the ball radius (or decrease in the link length) results in transition of the curve to higher pressure values.
The influence of shape parameters \( m \) and \( n \) of link diameter distribution is shown in Fig. 5 for three various values of these parameters. From this figure, it results that parameters \( m \) and \( n \) also play the role of shape parameters for the capillary potential curves.
Considering that for the constant shape parameters \( m \) and \( n \) of link diameter distribution there is a unique linear relation between maximum pore diameter \( D_{o} \) and the mean link diameter \( \bar{D} \), given by relation (6.2)_{1}, the influence of diameter \( \bar{D} \) on the capillary potential curve reduces to the rescaling of the pressure coordinate in Figs. 2, 3, 4 and 5. This because expression (4.9) describing the capillary potential curve, written as a function of dimensionless pressure \( p/\bar{p} \), does not depend explicitly on the mean diameter \( \bar{D} \).
7 Determination of Limit Pore Diameter Distributions
In this section, we show that the capillary potential curves of porous samples with capillary and chain pore space architecture are limit curves for samples with any network pore architecture. This proves that both models of the pore space are limit models of the network one with respect to the capillary potential curve. We also propose the use of the capillary and chain pore space architectures for determination of limit cumulative distributions of pore diameters in real porous materials, based on the mercury intrusion data. Both distributions define the range in which the cumulative distribution of pore diameter of the investigated material occurs.
7.1 Limit Models of Mercury Intrusion Curve
To provide generality of the analysis, we consider a set of cylindrical tubes (links) of random diameter and length described by a probability distribution. These links are used to form three kinds of porous media with the same porosity and statistically homogeneous and isotropic pore space structures and various pore architectures: capillary, chain and network. They have the same pore size distribution. A sample of the same shape (e.g. a ball) is cut out from each of these media and subjected to the process of mercury intrusion. Due to various pore space architectures, they are characterized by different mercury intrusion curves (capillary potential curves).
7.2 Limit Pore Diameter Distributions
For these considerations, we assume that the network model of pore space architecture composed of cylindrical tubes is a good representation of the pore space of real porous materials. In this case, determination of the pore diameter distribution in a sample of porous material based on the mercury intrusion data could be performed as long as the mathematical model of the capillary potential curve for the sample of porous material with the network pore space architecture was available. However, formulation of such an effective model is a very complicated problem (Chizmadzhev et al. 1971).
In this paper, we apply simple limit models of the mercury intrusion curves to propose the procedure for determination of limit cumulative distributions of pore diameters based on the mercury intrusion curves, defining the range of occurrence of cumulative distribution in the investigated material.
Similarly, interpretation of the experimental curve NM in Fig. 6 by the chain model, given by relation (4.9), is equivalent to the change of parameters of the pore diameter distribution (for the fixed value of the ratio \( L/a \)). This results in transition of the intrusion curve PM for the chain model in the direction of decreasing pressures up to cover with the curve NM. This means transition of the curve NM in Fig. 7 in the direction of increasing diameters. It takes the form of the curve PM in this figure. The values of parameters \( L \) and \( a \) present in the chain model can be estimated based on the volume of the investigated sample and the mean diameter \( \bar{D} \), respectively. Both curves: CM and PM in Fig. 7 define the range of occurrence of the cumulative distribution of pore diameters in the investigated material. This range defines the degree of inaccuracy of the method based on the mercury intrusion data caused by the indeterminacy of the sample shape and its pore space architecture.
8 Final Remarks
In this paper, we considered stochastic models of porous materials with a chain pore space structure composed of cylindrical tubes (links) with random length and diameter distributions and joined at random in series forming the capillaries of a stepwise changing cross section. The integral Volterra equation describing the quasistatic process of mercury intrusion into the halfspace of such material was derived, and the obtained results were used for description of the mercury intrusion curve into a layer and a ball of porous material. Solution of the equation made it possible to obtain analytical expressions for capillary potential curves of porous samples with different chain pore space architectures (capillary, periodic, randomperiodic and random). We have shown that link length distribution, the random location of chain capillaries in the sample and distribution of capillary length do not significantly influence the capillary potential curve. This justifies the use of a simple expression for the capillary potential curve of porous layer with the periodic chain pore architecture as an effective model of mercury intrusion curve into porous sample of any shape and sizes. The thickness of the layer in this model then represents the mean length of chain capillaries in the sample.
It was shown, moreover, that the capillary and chain pore space architectures of porous materials are limit models for network pore architecture with respect to the capillary potential curve. This proves that both models of the pore space are limit models of the network model. We propose application of both limit models for determination of limit cumulative distributions of pore diameters in porous materials investigated by the mercury intrusion method. Both distributions define the range in which the pore diameter distribution of the investigated material occurs. This range defines the degree of inaccuracy of the method based on the mercury intrusion data caused by the indeterminacy of the sample shape and its pore space architecture.
Characterization of pore size distribution in porous materials in this manner is additionally justified by the fact that the accuracy of the mercury porosimetry is also limited by at least two physical assumptions commonly taken during interpretation of the mercury intrusion data: the wetting angle is constant, and the pore space is unchanged during the mercury intrusion.
Considering that interpretation of mercury intrusion data in standard mercury porosimetry is based on the capillary model of pore space architecture and determines the lower limit of pore diameter distribution, the procedure proposed in this paper supplements the mercury porosimetry by also allowing determination of the upper limit distribution estimating the accuracy of this method.
References
 Adler, P.M.: Porous Media. Geometry and Transports. ButterworthHeinemann, Boston (1992)Google Scholar
 Arns, C.H.: A comparison of pore size distributions derived by NMR and XrayCT techniques. Physica A 339, 159–165 (2004)CrossRefGoogle Scholar
 Berkowitz, B., Balberg, I.: Percolation approach to the problem of hydraulic conductivity in porous media. Transp. Porous Med. 9(3), 275–286 (1992)CrossRefGoogle Scholar
 Berkowitz, B., Ewing, R.P.: Percolation theory and network modeling applications in soil physics. Surv. Geophys. 19(1), 23–72 (1998)CrossRefGoogle Scholar
 Bhattad, P., Willson, C.S., Thompson, K.E.: Effect of network structure on characterization and flow modeling using Xray microtomography images of granular and fibrous porous media. Transp. Porous Med. 90, 363–391 (2011)CrossRefGoogle Scholar
 Blunt, M.J.: Flow in porous media: porenetwork models and multiphase flow. Curr. Opin. Colloid Interface Sci 6, 197–207 (2001)CrossRefGoogle Scholar
 Broadbent, S.R., Hammersley, J.M.: Percolation processes I: crystals and mazes. Proc. Camb. Philos. Soc. 53, 629–641 (1957). https://doi.org/10.1017/S0305004100032680 CrossRefGoogle Scholar
 Brownstein, K.R., Tarr, C.E.: Importance of classical diffusion in NMR studies of water in biological cells. Phys. Rev. A 19, 2446 (1979)CrossRefGoogle Scholar
 Cieszko, M.: Macroscopic description of capillary transport of liquid and gas in unsaturated porous materials. Meccanica 51(10), 2331–2352 (2016)CrossRefGoogle Scholar
 Cieszko, M., Kempiński, M.: Determination of limit pore size distributions of porous materials from mercury intrusion curves. Eng. Trans. 54(2), 143–158 (2006)Google Scholar
 Cieszko, M., Chaplya, Y., Kempiński, M.: Continuum description of quasistatic intrusion of nonwetting liquid into a porous body. Contin. Mech. Therm. 27(1–2), 133–144 (2015)CrossRefGoogle Scholar
 Chizmadzhev, YuA, Markin, V.S., Tarasevich, M.R., Chirkov, YuG: Macrokinetics of Processes in Porous Media. Nauka, Moscow (1971). (in Russian) Google Scholar
 Coates, G.R., Xiao, L., Prammer, M.G.: NMR Logging. Principles and Applications. Halliburton Energy Services, Houston (1999)Google Scholar
 Cohen, M.H., Mendelson, K.S.: Nuclear magnetic relaxation and the internal geometry of sedimentary rocks. J. Appl. Phys. 53, 1127 (1982)CrossRefGoogle Scholar
 Dullien, F.A.L.: Porous Media: Fluid Transport and Pore Structure. Academic Press, New York (1979)Google Scholar
 Fatt, I.: The network model of porous media: I. Capillary pressure characteristics. Pet. Trans. AIME 207, 144–159 (1956)Google Scholar
 Giesche, H.: Mercury porosimetry: a general (practical) overview. Part. Part. Syst. Charact. 23(1), 9–19 (2006)CrossRefGoogle Scholar
 Hassanizadeh, S.M., Gray, W.G.: Mechanics and thermodynamics of multiphase flow in porous media including interphase boundaries. Adv. Water Resour. 13, 169–186 (1990)CrossRefGoogle Scholar
 Hassanizadeh, S.M., Gray, W.G.: Thermodynamic basis of capillary pressure in porous media. Water Resour. Res. 29(10), 3389–3405 (1993)CrossRefGoogle Scholar
 Hildebrand, T., Rüegsegger, P.: A new method for the modelindependent assessment of thickness in threedimensional images. J. Microsc. 185(1), 67–75 (1997)CrossRefGoogle Scholar
 Kellerer, A.M.: Chordlength distributions and related quantities for spheroids. Radiat. Res. 98, 425–437 (1984)CrossRefGoogle Scholar
 Kleinberg, R.L., Horsfield, M.A.: Transverse relaxation processes in porous sedimentary rock. J. Magn. Reson. 88(1), 9–19 (1990)Google Scholar
 Lenormand, R., Bories, S.: Description d’un mécanisme de connexion de liaison destine à l’étude du drainage avec piégeage en milieu poreux. C. R. Acad. Sci. Paris 291B, 279–283 (1980)Google Scholar
 León y León, C.A.: New perspectives in mercury porosimetry. Adv. Colloid Interfaces 76–77, 341–372 (1998)CrossRefGoogle Scholar
 Martins, A.A., Laranjeira, P.E., Braga, C.H., Mata, T.M.: Modeling of transport phenomena in porous media using network models. Prog. Porous Media Res. 5, 165–261 (2009)Google Scholar
 Mendelson, K.S.: Percolation model of nuclear magnetic relaxation in porous media. Phys. Rev. B 41, 562 (1990)CrossRefGoogle Scholar
 Raeesi, B., Morrow, N., Mason, G. (2013) Pore network modeling of experimental capillary pressure hysteresis relationships. SCA2013015 (2013)Google Scholar
 Richards, L.A.: Capillary conduction of liquids through porous mediums. J. Appl. Phys. 1, 318 (1931)Google Scholar
 Scheidegger, A.E.: The Physics of Flow Through Porous Media. University Press, Toronto (1957)Google Scholar
 Shih, F.Y.: Image Processing and Mathematical Morphology. Fundamentals and Applications. Taylor and Francis Group, Boca Raton (2009)Google Scholar
 Sorland, G.H., Djurhuus, K., Wideroe, H.C., Lien, J.R., Skauge, A.: Absolute pore size distribution from NMR. Diffus. Fundam. 5, 4.1–4.15 (2007)Google Scholar
 Webb, P.A., Orr, C.: Analytical Methods in Fine Particle Technology. Micromeritics Instrument Corporation, Norcross (1997)Google Scholar
 Winslow, D.N.: Advances in experimental techniques for mercury intrusion porosimetry. Surf. Colloid Sci. 13, 259–282 (1984)CrossRefGoogle Scholar
 Wilkinson, D., Willemsen, J.F.: Invasion percolation: a new form of percolation theory. J. Phys. A: Math. Gen. 16(14), 3365–3376 (1983)CrossRefGoogle Scholar
 Xiong, Q., Baychev, T.G., Jivkov, A.P.: Review of pore network modelling of porous media: experimental characterisations, network constructions and applications to reactive transport. J. Contam. Hydrol. 192, 101–117 (2016)CrossRefGoogle Scholar
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