# The influence of particle size distribution on parameters characterizing the spatial structure of porous beds

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## Abstract

The article proposes functions linking the standard deviation of a particle distribution in a porous bed consisting of spherical particles with various parameters characterising the spatial structure of the bed. The porosity, the inner surface, the specific surface and the geometrical tortuosity were analysed. In the first stage, a set of virtual beds was created with the use of the Discrete Element Method. The Radius Expansion Method was applied to generate virtual beds with different standard deviations. 150 virtual beds were created (25 standard deviations, 3 repetitions with different settings of the random number generator, 2 values of the radius expansion factor). In the second stage, the spatial structure of all virtual beds was analysed. The geometrical tortuosity was calculated with the use of the so-called Path Tracking Method; other parameters were calculated with the use of analytical formulas. The impact of the standard deviation on the parameters characterising the spatial structure of the granular bed was described by approximation functions, which can be used in order to obtain these parameters based on the particle size distribution for others porous beds.

## Keywords

Granular beds Particle distribution Tortuosity Discrete Element Method Path Tracking Method## 1 Introduction

The prediction of the pressure drop, occurring during fluid flows through porous media, is one of the most important problems in the widely understood science and engineering. For example, this issue plays significant role in geology, civil engineering, agriculture, food industry, chemistry and many other areas. So-called porous beds, i.e. granular beds consisting of spherical or quasi-spherical particles, play an important role here—the article relates to such kind of the physical matter.

*p*—pressure (Pa),

*x*—a coordinate along which the pressure drop occurs (m), \(A(\varPhi )\) and \(B(\varPhi )\)—two generalized parameters, dependent on the \(\varPhi\) set of parameters characterizing the spatial structure of the porous medium, \(\mu\)—dynamic viscosity of the fluid [kg/(ms)], \(\rho\)—density of the fluid (kg/m\(^3\)), \(v_f\)—filtration velocity (m/s),

*T*—temperature (\(^{\circ }\)C).

*d*—the representative particle diameter (which may be defined in many ways [41]) (m), \(\phi\)—the porosity (−), \(\varepsilon\)—the packing coefficient (−),

*e*—the void ratio (−), \(V_s\)—the volume of the solid part of the porous body (m\(^3\)), \(V_p\)—the volume of the pore part of the porous body (m\(^3\)),

*V*—the total volume of the porous body (m\(^3\)), \(\tau\)—the tortuosity (−), \(L_0\)—the depth of the porous bed (m), \(L_p\)—the length of the pore channel (m), \(\tau _f\)—the tortuosity factor (−), \(S_0\)—the specific surface of the solid body (in the Kozeny [22] or Carman meaning [6]) (1/m), \(S_p\)—the inner surface of the porous body (m\(^2\)).

Parameters characterising the spatial structure of a porous bed [41]

No. | Parameter | Definition |
---|---|---|

1 | Porosity | \(\phi =\lim _{\varDelta V \rightarrow V_g} \frac{\varDelta V_p}{\varDelta V}\) |

2 | Packing coefficient | \(\varepsilon =\lim _{\varDelta V \rightarrow V_g} \frac{\varDelta V_S}{\varDelta V}\) |

3 | Void ratio | \(e=\frac{V_p}{V_s}\) |

4 | Tortuosity | \(\tau =\frac{L_p}{L_0}\) |

5 | Tortuosity factor | \(\tau _f=\tau ^2\) |

6 | Inner surface | \(S_p=S_s=\sum \limits _{i=1}^{n_s} {S_i}=\sum \limits _{i=1}^{n_s} {\pi d_i^2}\) |

7 | Specific surface (by Kozeny) | \(S_{0,Kozeny}=\frac{S_p}{V}\) |

8 | Specific surface (by Carman) | \(S_{0,Carman}=\frac{S_p}{V_s}\) |

In the context of the described work it is important that the elements of the \(\varPhi\) set are usually treated as constant values, representative for the whole porous bed. Such approach is used in all formulas cited in the description of the Eq. (1). However, these parameters may depend on other factors. In the investigations of granular beds consisting of spherical or quasi-spherical objects, the particle size distribution may be such factor. This is the issue that we study in this paper.

Most elements of the \(\varPhi\) set may be calculated analytically, if only the number of particles in the bed and their sizes are known. The type of particle distribution is not directly considered in such calculations, but of course it is concealed in above-mentioned data. From the whole \(\varPhi\) set, only the tortuosity has to be obtained in another way. In the literature, many formulas destined for calculation of this quantity may be found. In the [39] we checked 21 such relations and in most cases tortuosity is treated as a direct function of the porosity. An additional parameter (a shape factor) was introduced only in two cases. We did not found any paper investigating the relationships between the kind and parameters of the particle distribution and the tortuosity. It was the main motivation to perform research described in this article. Study like that has the potential to propose formulas allowing calculation of porous bed parameters based on variance of size of particles forming the investigated bed, with no need of experiments, that can be time-consuming, cost-intensive or simply impossible. It should be noted that in the literature investigations in which a particle distribution of a porous bed is taken into account may be found (e.g. [4, 7, 30, 49]). However, the Authors focus usually on the general effect, like permeability or filtration coefficient (values of \(A(\varPhi )\) and \(B(\varPhi )\) terms), not on the elements of the \(\varPhi\) set. Papers related to investigations of mono- and poly-disperse granular systems are known too, even related to their inner geometrical structure [28, 29, 33]. However, it is very difficult to compare the results of these investigations with our data on a quantitative level.

The need for the calculation of the tortuosity determines the methods used in the investigations. To reach the article aim, we use the so-called Path Tracking Method (PTM). This method needs the data on the locations and sizes of all particles forming a bed. Such data may be obtained experimentally (e.g. by the use of the computed tomograpfy and image analysis) or numerically (e.g. by the use of the Discrete Element Method, DEM). Due the fact that it is impossible to freely change the standard deviation in a real bed, in our study the second approach is used. In DEM simulations, the particle size distribution may be freely defined. To keep the resemblance to real granular beds, the porosity, the average particle diameter and the basic value of the standard deviation were taken from an earlier experiments [41].

The key question is whether the virtual beds generated in DEM simulations may be used to analyze the geometry of real granular beds. DEM is a very complicated technique and many different factors (the applied contact models, numerical schemes, time approach and many others) may influence the final spatial arrangement of the particles. To answer this question, all possible elements of the \(\varPhi\) set were calculated in two ways: **direct**, based on the discrete form of the cumulative curve of particle distributions; and **indirect**, based on the data obtained from DEM simulations. This approach allows to show differences between these two ways.

The methodology of the presented investigations consists of four main steps: (a) preparation of a set of cumulative curves of particle distributions (with Gaussian distribution); (b) analytical calculations of all possible elements of the \(\varPhi\) set in function of the standard deviation; (c) preparation of a set of virtual beds with the use of the Discrete Element Method (where the type and parameters of the particle distribution is specified); (d) use of our own algorithms destined for analysis the spatial structure of granular beds consisting of spherical particles (where the so-called Path Tracking Method [36, 43] is the most important). In the last stage, all results are compared and the conclusions are drawn.

## 2 Materials and methods

### 2.1 Materials

A set of virtual beds consisting of spherical particles and created with the use of the Discrete Element Method is the object of our investigations. We assume: (a) Gaussian distribution of the particle size; (b) one average particle diameter—equal to 6.072 (mm) (\(d_e\)); (c) one basic value of the standard deviation equal to 0.05 (mm) (\(\sigma _e\)); (d) one bed porosity (the so-called target porosity, \(\phi _e\)) equal 0.413 (−). These values are taken from previous studies [41], in which real beds with such parameters were investigated.

### 2.2 Methods

#### 2.2.1 Distributions of diameters

The main aim of this task is to create a set of cumulative curves of particle size distributions with a constant average diameter (equal to \(d_e\)) and different values of standard deviations (\(\sigma\)).

Classic mathematical formulas for normal distribution are sufficient for generating theoretical, discrete distribution of the particles forming the bed. By the theoretical distribution, we mean the set of bins including numbers of spheres with a fixed diameter. Distribution is calculated in the range of \(\pm \, 4 \sigma _e\) from \(d_e\). In case of normal distribution, it means that \(\approx\) 99.994 \(\%\) of the population falls inside that range (15,787 must be the minimum total population to expect the first case falling outside that range [13]).

*i*-th (where \(i = 1,2,\ldots ,n_{f}\)) subrange has an attributed diameter:

*N*is the total number of spheres in the bed under creation, then the number of spheres \(n_i\) for the

*i*-th bin is calculated as:

#### 2.2.2 Discrete Element Method

The main aim of this task is to obtain independent sets of virtual granular beds with one porosity, one average diameter and different values of the standard deviation. By the term “virtual bed” we mean here the data on position in the space (x, y and z coordinate) and the size (radius or diameter) of every particle forming the bed.

To generate a virtual bed consisting of spherical particles, the so-called Radius Expansion Method (REM) implemented in the non-commercial YADE code was used [51]. This name denotes a specific way of generating a particle cloud in a DEM model. In this method, an initial cloud of small particles (without any contacts) with a given particle size distribution is randomly generated, and then all particles increase in size until the target porosity is reached [51]. The radiuses increases with the formula \(r^{t+dt} = k r^t\), where *k* is the so-called radius expansion factor and *t* is time. The term “cloud” means here a set of objects (spherical in our model) located in the calculation domain. It should be emphasized that the random generation of objects in a cloud is applied very often [3, 32, 46, 51]. The number of contacts between particles increases during calculations and it is fixed at a constant level after some time. The REM may be used to create clouds with particles having one specific diameter or different diameters. In the second case, a discrete cumulative curve of particle fractions must be defined (Sect. 2.2.1).

A characteristic feature of the REM is that the volume of the space in which the initial cloud is generated, must be very precisely defined. To control this volume, an additional parameter (\(l_c\)) has to be calculated in every single case. In other case, the final cloud of particles obtains the correct porosity as well as the features of the particle distribution, but the diameter values will be incorrect.

*i*-th bin (m\(^3\)), \(V_i\)—the volume of a single particle belonging to the

*i*-th bin (m\(^3\)), \(Y_i\)—the fraction of particles in the

*i*-th bin (−), \(d_i\)—the diameter of particle belonging to the

*i*-th bin (m). In our investigations we assume that the total number of spheres (

*N*) is constant for all virtual beds.

The other possibility to create a virtual bed with a defined particle distribution is the use of a triaxial compression. In this approach, a cloud of particles is created (the particle sizes and distribution are correct in that moment) and then the cloud is compressed by the walls to obtain the target porosity. The particle distribution is introduced in the same way as in the previous method. The main drawback of this method is a long simulation time, counted in hours, not minutes, like in the REM method. Due to the fact that a big number of virtual beds had to be generated, we decided to use the REM approach. Since this method is often applied in many different studies [8, 17, 23, 45, 48, 50], new conclusions related to the consequences of its use seems to be desirable.

#### 2.2.3 Path Tracking Method

The main aim of this task is to obtain values of the geometrical tortuosity for every virtual bed prepared with the use of the Discrete Element Method. The geometrical tortuosity is calculated with the use of the Path Tracking Method (PTM) in a variant called Regular Grid Method (RGM) [36, 42]. The other parameters are calculated analytically, but with the use of the same software, the so-called PathFinder code [31].

The Regular Grid Method defines a way of using the PTM algorithm [36].

#### 2.2.4 Approximation functions

*y*is the parameter of the porous bed being under consideration. Models were fitted to data using Levenberg-Marquardt algorithm [27]. Then, Akaike Information Criterion (AIC) was used to compare models [2, 40]. Hence, formulas given below as linking porous bed parameters with variance of particle size distribution are those indicated by AIC as best describing analysed relationship. Moreover, residual analysis (whiteness test of residuals) was performed for each model indicated by AIC in order to check, whether chosen approximation function properly explains changes in analysed value [25].

Approximation functions considered in the statistical analysis

No. | Function | Kind |
---|---|---|

1 | \(y = p_1 + p_2 \sigma\) | Linear |

2 | \(y = p_1 + p_2 \sigma + p_3 \sigma ^2\) | Polynomial |

3 | \(y = p_1 + p_2 \sigma + p_3 \sigma ^2 + p_4 \sigma ^3\) | Polynomial |

4 | \(y = p_1 + p_2 e^{p_3 \sigma }\) | Expotential |

5 | \(y = p_1 + p_2 \sigma ^{p_3}\) | Expotential |

6 | \(y = p_1 + \frac{p_2}{\sigma }\) | Hyperbolic |

## 3 Results and discussion

### 3.1 Discrete form of particle size distribution

*N*) was set to 10,000 (−), what is less than the maximum safe value calculated in Sect. 2.2.1. Examples of such functions are visible in Fig. 2. The \(Y^{sum}\) symbol denotes the cumulative fraction of all particles with a diameter equal or less than \(d_i\).

### 3.2 Direct analytical calculations

*V*, \(l_c\), \(S_p\), \(S_{0,Carman}\), \(S_{0,Kozeny}\) as well as other quantities (see Eq. 2) for every standard deviation may be directly calculated. The mathematical formulas visible in Table 1 and Eqs. (7)–(10) were used. In this approach, firstly \(V_s\) with the use of the Eq. (9) is calculated and next, taking into account the target porosity \(\phi _e\), the volume of the pore part (\(V_p\)) as well as the total volume of the bed (

*V*) are determinated. After obtaining this data, the porosity is calculated. Knowing all diameters, the inner surface is also calculated. At the end, the fit functions were obtained for all parameters listed above. Results of the direct calculations are collected in Table 3. The ”hat” symbol was added to distinguish these functions from indirect results shown in the rest of the paper.

Relationships between standard deviation and chosen elements of the \(\varPhi\) set

No. | Parameter | Analytical function |
---|---|---|

1 | \(\hat{V}_s(\sigma )\) | \(0.001166 + 9.3 \times 10^{-5} \sigma ^2\) |

2 | \(\hat{V}_p(\sigma )\) | \(0.00082 + 6.55 \times 10^{-5} \sigma ^2\) |

3 | \(\hat{V}(\sigma )\) | \(0.001987 + 0.00016 \sigma ^2\) |

4 | \(\hat{l}_c(\sigma )\) | \(0.0998 + 8.07 \times 10^{-5} \sigma + 0.0025 \sigma ^2\) |

5 | \(\hat{S}_p(\sigma )\) | \(1.153 - 1.76 \times 10^{-9} \sigma + 0.03 \sigma ^2 - 1.13 \times 10^{-8} \sigma ^3\) |

6 | \(\hat{S}_{0,Carman}(\sigma )\) | \(1002.89 - 12.962 e^{1.58 \sigma }\) |

7 | \(\hat{S}_{0,Kozeny}(\sigma )\) | \(588.7 - 7.61 e^{1.58 \sigma }\) |

### 3.3 Virtual beds

*N*) is equal to 10000; (d) the target porosity is equal to \(\phi _e\); (e) every virtual bed is generated three times with the use of different parameters of the random function (the repetitions are identified by numbers 1, 2 and 3); (f) all test are performed two times with different values of the radius expansion factor. When the radius expansion factor is less, then the time of DEM simulation is longer but the target porosity may be obtained more precisely. These tests are identified as A and B. The radius expansion factor was equal to 0.0001 and 0.001 in tests A and B, respectively. The visco-elastic contact model was applied. The damping coefficient was set to 0.2. Mass forces in each direction were set to zero due to the desire to obtain more homogeneous beds. In every simulation the friction angle (denoted by \(\gamma\)) was firstly set to 0.5 [rad], and next it was decreased during calculations. In Fig. 3, examples of virtual beds from DEM simulations are visible.

It should be emphasized that the investigations of virtual beds generated rendomly with the use of the REM and the cumulative curves are not new in the literature [5, 9, 20, 24]. However, in the available literature we did not found comments concerning the changes in the particle distribution if this approach to modelling is applied.

### 3.4 Indirect analytical calculations

*V*) dependent on the parameter \(l_c\) is created. The parameter \(l_c\) is calculated analytically, in the same way as in the direct approach. The total volume of the particles (\(V_s\)) and the volume of the pore space (\(V_p\)) is calculated in every time step on the basis of the current data to obtain the current porosity. The main loop stops when the current porosity is equal or less than the target porosity (\(\phi _e\)). This explains why the values visible in Fig. 7 are always a bit lower than this target porosity. As it can be seen, the porosity value is much more constant in the A test. Besides, the porosity decreases in A test if the standard deviation increases, otherwise than in the B test, where the porosity changes are more non-uniform. The minimum and maximum relative errors (where the target porosity is used as the reference value) in A test were equal to 0.06% and 0.68%, in B test in turn 0.38% and 2.63%. None of considered models was able to describe changes in porosity for the B test. In case of the A test, changes in porosity were best explained with the use of the second degree polynomial model:

*V*and \(l_c\) depend directly on this volume.

Mean and maximum relative errors between fit functions obtained in direct and indirect calculations

Parameter | Mean relative error (%) | Maximum relative error (%) |
---|---|---|

\(V_s\) | 0.18 | 0.30 |

\(V_p\) | 0.26 | 0.43 |

| 0.17 | 0.22 |

\(S_p\) | 0.31 | 1.23 |

\(S_{0,Carman}\) | 0.41 | 2.17 |

\(S_{0,Kozeny}\) | 0.24 | 1.44 |

\(l_c\) | 0.02 | 0.03 |

### 3.5 Geometrical tortuosity

Note, that if the standard deviation is equal to zero, then the tortuosity equals 1.208. We may compare this value with data shown by Wang [46], who used coupled DEM-LBM (Lattice Boltzmann Method) simulations to calculate the so-called hydraulic tortuosity in granular beds consisting of spherical particles. The hydraulic tortuosity obtained by him was equal to 1.1975 for the porosity equal to 0.4. Such compliance is very satisfying, especially since in the both cases the used methodologies were different.

It was mentioned that in the literature the tortuosity is usually defined as a non-linear function of the porosity. Here we want to check, whether the trend visible in Fig. 17 is not caused by changes of the porosity shown in Fig. 9. For this purpose the available analytical formulas for calculating the tortuosity in granular beds were used. A review of such formulas is available in [39]. In Fig. 19 it can be seen that such analytical formulas do not respond to small changes of the porosity reported earlier (Fig. 9). It confirms that the trend visible in Fig. 17 is a general feature, dependent on the standard deviation and not on the porosity fluctuations. Moreover, since the porosity decreases slightly in function of the standard deviation, the changes in the tortuosity may by even bigger than it can result from the formula (20).

### 3.6 Sensitivity analysis

*i*-th input parameter on the

*j*-th output parameter, \(\varDelta \varphi _i^{in}\) equals change in the value of the

*i*-th input parameter, \(\varDelta \varphi _{j,i}^{out}\) equals change in the value of the

*j*-th output parameter caused by a change in the value of the

*i*-th input parameter.

*i*-th input parameter (for which the value of the

*j*-th output parameter is estimated), \(\bar{\varphi }_i^{in}\) equals base value of the

*i*-th input parameter (from the base model), \(\varphi _{j,i}^{out}\) equals value of the

*j*-th output parameter determined for the current value of the

*i*-th input parameter, \(\bar{\varphi }_{j,i}^{out}\) equals base value of the

*j*-th output parameter (from the base model).

*i*set contains only one parameter (\(\sigma\)) and

*j*set contains six parameters (\(V_s\), \(V_p\),

*V*, \(S_p\), \(S_{0,Carman}\), \(S_{0,Kozeny}\)). We resign to calculate the sensitivity indicator for the characteristic dimension \(l_c\). As the base values we use the values calculated for the standard deviation equal to \(\sigma _e\). In turn, as the current values the data for \(\sigma = 1.15\) (maximum) and \(\sigma = 0.6\) (approx. half of the maximum) are used. Results of calculations are collected in Table 5. The absolute value of the \(|I|_{j,i}\) shows the level of the sensitivity of the investigated variable. The negative sign means that the increase of the standard deviation causes the decrease of the chosen parameter. Without the sensitivity analysis these informations are not obvious.

Values of the normalized sensitivity indicator

Parameter | \(|I|_{0.6} \times 10^{-2}\) | \(|I|_{1.15} \times 10^{-2}\) |
---|---|---|

\(V_s\) | 0.28 | 0.51 |

\(V_p\) | 0.24 | 0.44 |

| 0.26 | 0.48 |

\(S_p\) | 0.09 | 0.09 |

\(S_{0,Carman}\) | − 0.78 | − 0.39 |

\(S_{0,Kozeny}\) | − 0.18 | − 0.39 |

\(\tau\) | 0.28 | 2.03 |

## 4 Conclusions

The following conclusions can be formulated based on the above-discussed topics: (1) Geometrical parameters characterising granular beds (identified in the paper as the set \(\varPhi\)) may be treated as functions of the standard deviation of the particle distribution. These functions may be obtained on the basis of Discrete Element Method and statististical methods; (2) The obtained functions may be used as replacements of constant values in such formulas like e.g. Kozeny–Carman equation, in which the porosity, the tortuosity factor and the specific surface have to be stated; (3) The functions proposed in the paper are developed on the basis of virtual beds and may be treated only as an estimation. However, we hope that the obtained trends and intensivities of changes are close to what is in fact, at least in relation to granular beds consisting of spherical or quasi-spherical particle; (4) Due the fact, that the DEM is an increasingly popular method of modeling different granular systems, the investigations of features, possibilities and limitations of this approach is very important and in our opinion fully justified; (5) A drawback of the Radius Expansion Method is distortion of an originally given particle distribution during the simulations (not commented in the available literature). However, this method seems to be sufficient to perform investigations related to the spatial structure of a porous bed; (6) The quality of a virtual bed depends on the radius expansion factor. This parameter should be relatively small; (7) The sensitivity analysis performed in the article allows estimating the intensity and the direction of changes of all elements of the \(\varPhi\) set when the standard deviation increases; (8) Geometrical tortuosity in granular beds increases slightly and nonlinearly with an increase in the standard deviation of the particle size distribution. Additionally, this relationship seem to be independent on the quality of the virtual bed.

## Notes

### Compliance with ethical standards

### Funding

This study was funded by Polish Ministry of Science and Higher Education in the frames of the statutory research.

### Conflict of interest

Authors declare that they have no conflict of interest.

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