# Merging criteria for defining pores and constrictions in numerical packing of spheres

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## Abstract

The void space of granular materials is generally divided into larger local volumes denoted as pores and throats connecting pores. The smallest section in a throat is usually denoted as constriction. A correct description of pores and constrictions may help to understand the processes related to the transport of fluid or fine particles through granular materials, or to build models of imbibition for unsaturated granular media. In the case of numerical granular materials involving packings of spheres, different methods can be used to compute the pore space properties. However, these methods generally induce an over-segmentation of the pore network and a merging step is usually applied to mitigate such undesirable artifacts even if a precise delineation of a pore is somewhat subjective. This study provides a comparison between different merging criteria for pores in packing of spheres and a discussion about their implication on both the pore size distribution and the constriction size distribution of the material. A correspondence between these merging techniques is eventually proposed as a guide for the user.

## Keywords

Delaunay tessellation Voronoï graph Void space Granular materials## 1 Introduction

A granular medium includes a set of large volumes of voids between solid particles (pores) connected by throats. The narrowest sections in these throats are generally denoted as constrictions. Pores and constrictions constitute a partition of the void space helpful to define respectively its morphology and its topology [28, 45]. Such a partition can also help to build imbibition models for unsaturated materials [16, 23], models for the coefficient of permeability [7, 8] for fluid-calculation, or geometrical filtration models for studying the migration of particles through granular media [21, 32, 34, 35].

There are different techniques for pore space characterization: through experiments [15, 19, 40, 42, 46], using analytical approaches [22, 29, 36] or numerical approaches [17, 26, 28]. To overcome some limitations associated with experimental methods, the Discrete Element Method (DEM) (among others [10]) can be helpful to draw some main tendencies for packings of spheres with a given grading and density. The pore space of such a packing can be extracted by combining the DEM with spatial partitioning techniques: the Delaunay tessellation [2, 28, 41] or its dual structure, the Voronoï diagram [14, 30, 47] among others.

While the Delaunay tetrahedra constitute volumetric entities that cover pores or parts of pores, a Voronoï graph can complement the definition of the pore structures. Due to the duality of Delaunay and Voronoï decompositions, the Voronoï nodes should correspond to the centers of the inscribed void spheres of the Delaunay tetrahedra and their distance to the surrounding solid spheres to the radius of these inscribed void spheres. When applying a Voronoï computation that is based on the Euclidean distance to the solid spheres as described by Lindow et al. [20], the edges between the Voronoï nodes are curved and run along the maximal distance to the surrounding solid spheres. Then, they describe the median path joining pore centers. The centers of constrictions are located where the distance to the surrounding spheres is minimal along the edge. In terms of duality, this is where the edges cut the common facet of the tetrahedra of the connecting Voronoï nodes and, thus, correspond to the constrictions found in the Delaunay tessellation (Fig. 2).

Even if the equivalence of results for the constriction size distribution (CSD) extracted from a Delaunay tessellation and a Voronoï graph has been proven in the past for a packing of spheres [44], the question arises whether an excessive artificial partition of the void space is generated by both mathematical techniques and how to handle it.

Indeed, using a Delaunay tessellation, Al-Raoush et al. [2] found that the inscribed void sphere confined in each tetrahedron is not necessarily entirely included inside that tetrahedron, and two inscribed void spheres attached to these two neighboring tetrahedra may overlap. It signifies that the opening size between two adjacent tetrahedra may be high enough to indicate a strong interconnection between them. As a result, the tetrahedral tessellation would tend to abusively subdivide a complete pore structure into zones.

Because different techniques may lead to different pore structures and, as a consequence, to a different set of pore and constriction sizes, this study aims to better understand the implications of using a given technique for merging pores on the properties of the poral space in packing of spheres. The problem that arises here is that no definite poral structure can be derived for a packing since the boundaries of a so-called pore is vague by nature. Within these limits, this paper tries to draw some advantages and limits of two techniques for merging pores. The influence of the proposed criteria for merging on the pore structures is also addressed and, as a guide for the user, a correspondence between the criteria associated to both techniques for merging pores is given.

## 2 Generation of numerical samples

The open-source code Yade-DEM [39] was used here to generate numerical samples composed of spheres. In this DEM code, the contacts between particles are deformable while the particles are considered as infinitely rigid bodies [10].

Two gradings are studied: a narrowly graded material (UG) and a gap-graded material (GG). The former grading is the one studied in previous studies [28, 38], ranging from 3 to 12 mm as shown in Fig. 4a, with a coefficient of uniformity (\(C_u\)) equal to 1.7. The coefficient of uniformity measures the extent of particle diameters and is defined as the ratio of the diameter corresponding to 60\(\%\) finer by weight to the diameter corresponding to 10\(\%\) finer by weight. The latter grading is the one studied by Reboul et al. [29] and is given in Fig. 4b. The minimum and maximum diameters (\(D_0\) and \(D_{100}\)) for this material are respectively equal to 0.7 and 10 mm, and \(C_u\) is equal to 3.6. Since different techniques for the sample creation may lead to different structures for the packing [4, 31], a deposit under gravity of particles, which is a technique that reflects the process used in actual experiments, is preferred.

To create the sample, a loose cloud of spheres with a prescribed particle size distribution is initially generated in a box having a horizontal size equal to that of the final sample but with a larger vertical size (about twice as more as the horizontal size). The base of the box is a square of 40 mm width (approximately 4\(D_{100}\)). The lateral boundaries of the box are associated to periodic conditions in order to avoid wall effects in the final samples [1, 27]. If a sphere overlaps any other existing spheres, another position is attributed to this sphere.

After this stage of particle generation, the packing is subjected to gravity which induces the spheres to fall freely in the box according to Newton’s laws. Interactions between particles may occur as particles collide. The process is ended when all particles reach a quasi-static equilibrium state. The equilibrium is supposed to have been found when the unbalanced forces (mean resultant forces at contact divided by the mean contact force over the sample) goes below 0.05.

In this study, particle density is taken equal to the one of glass beads, and typical values for \(K_n\) and \(K_t\) of such materials are chosen. For the specific contacts between a particle and the bottom wall of the box, the contact friction coefficient (\(\mu \)) is set to 0. Dissipation in the system is introduced by means of a global damping (\(\alpha \)) proportional to the acceleration forces [10].

In the case of UG material, 650 particles have been used while for GG material, packings with 25,000 particles were generated to obtain representative statistical data. For each grading, two samples are generated corresponding to the loosest state (respectively UGL and GGL) and to the densest state (respectively UGD and GGD). The inter-particle friction coefficient is set to 0.3 which is approximately equal to the value obtained by experimental test on spherical glass beads [3]. The resulting maximum porosities for UGL and GGL match the targeted values obtained through experiments by Biarez and Hicher [6], for the same coefficient of uniformity and the same particle aspect ratio of 1 (difference between the largest dimension and the smallest dimension of a particle). These authors used the ASTM standard to determine both the maximum and minimum porosity of actual granular materials having different gradings and particle aspect ratios.

The densest state is also obtained by gravitational deposition as described in [9, 12], but with a contact friction value between particles equal to zero. In fact, setting the friction to zero is favorable for particle rearrangements, which in turn leads to the compaction of the packing [37, 43, 49]. The minimum porosity reached with such process is equal to that obtained previously by Reboul et al. [28] for the same material (UG) though the process of creation of this dense sample was different. In their work, spheres are initially released under gravity to create a loose sample and then, the densest state is obtained by means of shearing cycles with a contact friction value between particles equal to zero.

It must be noted that such typical DEM densification processes lead to density states which are generally looser than that obtained for actual materials using the ASTM process [5]. Tables 1 and 2 summarize the set of parameters used to obtain the numerical samples and their induced final properties.

Since the top and bottom boundaries of the sample are not periodic, any computation of the poral characteristics of the packing is carried out within a volume smaller than the total sample volume. While the vertical lateral limits of this measurement volume correspond to the periodic boundaries. The limits of the top and bottom volume exclude then a zone of thickness equal to \(D_{100}\) of the granular material. Finally, we checked that the final volume used for the statistics of the void space is greater than the representative elementary volume.

## 3 Merging techniques for pores

Mechanical and numerical parameters for DEM simulations

Parameter | Magnitude |
---|---|

Normal stiffness (\(K_n\)) | \(10^4\) kN/m |

Tangential stiffness (\(K_t\)) | \(10^4\) kN/m |

Specific weight of spheres (\(\rho \)) | 2530 kg/m\(^3\) |

Global damping (\(\alpha \)) | 0.7 |

Inter-particle friction (\(\mu \)) | 0.3 (loosest state) |

0 (densest state) |

Characteristics of numerical samples

Material | UG | GG |
---|---|---|

Coefficient of uniformity (\(C_u\)) | 1.7 | 3.6 |

\(D_0\)–\(D_{100}\) (mm) | 3–12 | 0.7–10 |

Number of particles | 650 | 25,000 |

Maximum porosity | 0.38 | 0.34 |

Minimum porosity | 0.34 | 0.25 |

### 3.1 Overlapping inscribed void spheres technique

Once the locations and radii of the solid spheres are known, a modified (weighted) tetrahedral tessellation (Delaunay tessellation) of the space is performed [11].

Such a 3D partition induces a specific structure for the pore space. Indeed, each tetrahedron is herein supposed to represent a local pore associated to four exits. The void volume included in each tetrahedron can be derived together with other characteristics including the inscribed void sphere. Then, a statistical study over the whole sample can be computed for the properties of the local pores. Accordingly, constrictions defined as the largest empty discs on the tetrahedron faces can be obtained and the CSD can be deduced by means of a statistical study over the sample. All these characteristics are obtained using optimization algorithms (for the distance mapping) and more details can be found in [2, 28, 29].

Apart from these cases, the inscribed void spheres of two adjacent tetrahedra may just partly overlap and these cases are distinguished from those where the inscribed void spheres are completely separated. In the case of overlapping, a merging of the corresponding adjacent pores is applied (Fig. 5), giving birth to a level 1 (\(L_1\)) characterization of the void space. First, the tetrahedra derived from \({L_0}^{\prime }\) are sorted by increasing order of their inscribed void sphere, then for each tetrahedron, the overlap criterion is checked for the four adjacent tetrahedra. It should be noted here that after merging two neighboring pores, the process of merging is ended and didn’t go beyond the direct neighbor.

A level 2 (\(L_2\)) is also processed where merging is not only applied to the adjacent local pore but also to the next adjacent local pore if the inscribed void sphere of this latter overlaps that of the former pore (Fig. 5). No further level for merging pores is envisioned since in that case, the void space will tend to be characterized in terms of duct.

Level 1 and level 2 can be envisioned as criteria for merging pores in the context of the overlapping inscribed void sphere technique. This technique and the proposed criteria hold some advantages and limits. First, even if this technique seems relevant in the case of packing of spheres where a partition of Delaunay can be processed, it may not be able to address the case of media with elongated particles which may give rise to more elongated pores than in packing of spheres. In that case, by nature, few overlapping inscribed void spheres are expected to be found. In the case of packing of spheres, the proposed two criteria imply that the persistence of a pore is limited in distance which can be both an advantage and a drawback. It implies that a pore can only be defined at a certain local scale involving a pore wall composed of maximum eight particles in the case of \(L_1\) or of tens of particles in the case of \(L_2\).

### 3.2 Pore separation technique

The pore separation technique relies on the elements of the Voronoï graph that was computed from the spheres geometry based on the Euclidean distance to the solid spheres [20]. As described in Sect. 1, a Voronoï node *P* and its respective distance represent a pore and its size \(d_P\) in the initial decomposition, and a constriction *C* with its respective distance describe the narrowest location between two adjacent pores along a Voronoï edge (Fig. 6a).

The separation technique keeps track of this distance information and builds a hierarchical structure of these elements that follows the topology of the distance function. It evaluates the separation of each pair of pores \(P_i\) and \(P_j\) by their constriction \(C_{ij}\) based on the relative diameter difference \({t_\mathrm{diff}}({P_i},{C_{ij}},P_j) = ({d_P -{d_{C_{ij}}}})/{d_P}\) with \({d_P} = \min ({d_{P_i}},{d_{P_j}})\) and \({i} \ne {j}\). The value \(t_\mathrm{diff}\) will be used to merge neighboring pores according to the degree of their separation, which is specified by a user-defined threshold *t*. This approach was initially developed for materials with irregular particles and does not consider sphere overlaps in order to include pairs within elongated pores.

The hierarchical manner arises from specifying tuples \({T_{ij}} = ({P_i},{C_{ij}},{P_j},{t_\mathrm{diff}}({P_i},{C_{ij}},{P_j}))\) that are processed in a particular order. The approach starts from tuples of direct neighbors in the unmerged graph (Fig. 6b) and evaluates them in increasing order of the difference thresholds. Each step assigns the smaller pore to the larger one. Hereafter, the neighbor tuples that contain the newly merged pore will be updated by replacing this pore by the larger representative one as well as by re-computing \(t_\mathrm{diff}\) accordingly.

For example, if \({d_{P_i}} < {d_{P_j}}\), then all neighbor tuples \({T_{ik}}\) with \({k} \ne {j}\) will be converted to \({T_{jk}} = ({P_j}, {C_{jk}}, {P_k}, {t_\mathrm{diff}}({P_j},{C_{jk}},{P_k}))\) to be neighbors of \(P_j\). \(P_i\) and \(C_{ij}\) are labeled on the graph as belonging to \(P_j\) (Fig. 6c) and will be discarded from further considerations. This step is then repeated until all (newly created) tuples with a difference threshold \({t_\mathrm{diff}} \le {t}\) have been processed. More algorithmic details can be found in [18].

The resulting tuples represent hierarchical neighbors rather than direct neighbors, where each pore represents all hierarchically assigned pores. They not only treat local information on the separation but also allow considering the separation between groups of local pores that are less significantly separated. The constriction and the difference relation \(t_\mathrm{diff}\) of such a tuple represent then the most significant separation criterion between the two hierarchically neighbored pores. This can increase their life time as separated pores compared to the direct neighbor relations and avoids inappropriate merge propagation.

Voronoï approaches may produce additional pore centers in the graph that do not correspond to maxima in the distance function. In such cases, the diameter of constriction separating two adjacent pores is equal to that of the smaller pore (\(t=0\%\)). This is similar to what was found with the weighted Delaunay tessellation (see Sect. 3.1). On the graph, such constrictions are then identical to the smaller pore (two edges in the center of Fig. 6a), which will be merged at the very beginning of the hierarchical merge.

## 4 Pore distributions derived from different merging criteria

The dual complexes of the Delaunay/Voronoï decomposition, as already described in Sect. 1, encode the elements of the pore space of a sphere packing. Herein, the Delaunay cells or tetrahedra are the entities that cover the pores or parts of pores and, thus, the appropriate entity to evaluate the pore volumes from both merging techniques.

### 4.1 Overlapping inscribed void spheres technique

As explained previously in Sect. 3.1, the pore size can be measured in terms of the largest inscribed void sphere associated to each tetrahedron but also by considering the sphere having a volume equal to that of the void within a Delaunay cell (\(L_0\)) or within merged Delaunay cells (\(L_1\), \(L_2\)). This latter method is denoted in the following *equivalent void sphere* approach. Using these two definitions for characterizing the pores, the distribution of pore sizes, for UG and GG materials, at loose and dense states, are plotted in Figs. 7 and 8. It is interesting to note that the pore size distributions can be well described by a Log-Normal law in agreement with previous studies [28, 48]. The correlation is almost perfect for \(L_0\) (not shown herein); nevertheless, the statistical model tends to shift the mode towards larger pore diameters and to attenuate its frequency when \(L_1\) and \(L_2\) criteria are considered.

For UGL sample, accordingly to the work of Reboul et al. [28] on the same grading, the distribution computed on the basis of the inscribed void spheres approach has a mode shifted towards the smaller diameters compared to the other distributions obtained by the equivalent void sphere approach, since this former approach disregards a part of the void space. Moreover, the distribution obtained from \(L_1\) is quite different than that corresponding to \(L_0\), and a slight decrease in the modal value is also observed, thus showing that numerous pores around the mode in \(L_0\) have been merged in \(L_1\). On the contrary, no significant difference in the equivalent pore diameter distribution is observed from \(L_1\) to \(L_2\) (Fig. 7a). It tends to indicate that within the framework of this merging technique, the persistence of a pore is mainly limited to an adjacent tetrahedron for a given Delaunay cell.

For GGL sample, the distribution and the modal value corresponding to the inscribed void sphere are only slightly shifted towards the smaller diameters. This may be a consequence of the wide range of particle sizes involved in this material, which tends to generate more flat tetrahedra with inscribed void spheres which are not entirely confined in these tetrahedra. Consequently, this would create the possibility of greater void spheres than in the case of UG material, the volume of which could match the volume of \(L_1\) or \(L_2\) distribution (Fig. 8a). It must be noted that other well graded materials that were studied (not shown herein) also exhibited this pattern. However, the tetrahedral shape seems to be the main configuration represented within the sample, irrespective of the grading and porosity, since the modal values for \(L_0\), \(L_1\) and \(L_2\) are almost similar. The same finding was observed by Reboul et al. [28] for UGL sample.

Figure 9 shows the number of Delaunay cells per pore in the case of \(L_1\), for UG and GG materials, at loose and dense states. Irrespective of the porosity and of the grading, about 50% of Delaunay cells are not affected by the merging. The tetrahedral shape is then predominant, while more complex entities involving three or four tetrahedra (sharing a common face with a central tetrahedron) are poorly represented in the sample.

### 4.2 Pore separation technique

As outlined previously, the pore separation technique is based on the pore-constriction size relations extracted from a Voronoï graph [20]. It computes the Voronoï nodes by determining their four generator spheres, which also are the spheres that build the corresponding tetrahedra in the Delaunay tessellation.

To measure the pore sizes for the comparison, an approximated method was applied to each tetrahedron and its bounding box. A set of quasi-random points (obtained from a Niederreiter sequence [25]) were spatially distributed within the bounding box. Knowing the volume of the bounding box, the void volume is estimated from the number of void points (points located inside the tetrahedron but outside the spherical particles) compared to the number of total points.

The underlying equivalent pore size distributions are given in Fig. 10 for both UGL and GGL samples, and similar observations can be reported when the inscribed void sphere distribution is compared to that derived from \(L_0\) (unmerged cells). Additionally, Fig. 10 shows the behavior for large thresholds (\(t=30\%\)): the difference between the equivalent pore size distributions become highly significant. The mode is very shifted towards the larger diameters. Such a large threshold merges pore clusters that already had been merged for smaller thresholds and, thus, massively increases the corresponding pore size.

### 4.3 Discussion

Relative error (in %) between the pore size distributions derived from different merging criteria for UG

\(t ~(\%)\) | 1 | 2 | 3 | 4 | 5 | 6 | |
---|---|---|---|---|---|---|---|

UGL | \(L_1\) | 3.9 | 3.0 | 3.1 | 3.8 | 5.6 | 8.2 |

\(L_2\) | 9.0 | 6.5 | 4.6 | 2.5 | 1.7 | 2.3 | |

UGD | \(L_1\) | 3.7 | 2.1 | 2.2 | 2.8 | 4.2 | 5.5 |

\(L_2\) | 7.5 | 5.6 | 3.9 | 1.8 | 1.0 | 1.2 |

Relative error (in %) between the pore size distributions derived from different merging criteria for GG

\(t~(\%)\) | 1 | 2 | 3 | 4 | 5 | 6 | |
---|---|---|---|---|---|---|---|

GGL | \(L_1\) | 2.5 | 5.5 | 8.2 | 11.1 | 13.7 | 16.4 |

\(L_2\) | 5.5 | 3.4 | 3.2 | 5.5 | 7.8 | 10.2 | |

GGD | \(L_1\) | 3.5 | 2.2 | 3.0 | 4.7 | 6.4 | 8.1 |

\(L_2\) | 6.3 | 4.3 | 3.1 | 2.2 | 2.4 | 4.0 |

For UG material, higher values of *t* are required to generate pore distributions similar to those obtained by the overlapping inscribed void spheres approach. This discrepancy is most likely caused by the nature of the pore network of the studied materials. Broadly graded material, notably at loosest state, more frequently produce clusters of dense inscribed spheres with large overlap, whereas inscribed spheres seem to be more distant with smaller overlap (higher separation) in the UG material. Furthermore, within broadly graded material, several solid spheres may build large voids containing large clusters of inscribed spheres. Such cases may exceed the expected maximum size given by the neighbor levels in the overlapping spheres approach and produce multiple pore instances within such a void. Late merges during the separation merge in UG material and underestimated volumes in GG during the \(L_1\) and \(L_2\) merges most likely lead to the drifting of the \(L_1 - t\) and \(L_2 - t\) correspondences between the UG and GG materials.

Each technique holds its own limits and holds some advantages. The overlapping inscribed void spheres approach provides a pre-defined neighborhood limit (\(L_1\) and \(L_2\)) assuming a general maximum pore size at a meso scale. It may be artificial but seems relevant for packing of spheres. Nevertheless, in the case of granular materials with elongated particles, this technique may not be appropriate since one expects to find elongated pores with scarce overlapping inscribed void spheres occurrences. The separation technique allows tuning the pore structure more easily but the definition of the threshold is difficult and requires experience. One can note that this technique is more robust since it can be used for any granular material with any particle shape if a voxelization representation of the material (solids and voids) is available.

## 5 Constriction size distributions derived from different merging criteria

For convenience purposes, only the results corresponding to the UGL and GGL samples are presented in this section, but similar results were also found for UGD and GGD samples.

### 5.1 Overlapping inscribed void spheres technique

The CSDs and the estimated probability density of constriction sizes for \(L_1\) and \(L_2\) merging are given in Figs. 11 and 12 for both samples.

First, it has been noted that the number of constrictions decreases by approximately 40\(\%\) from \(L_0\) to \(L_1\). In fact, the initial computation (\(L_0\)) involves non negligible sets of tetrahedra with overlapping inscribed void spheres that are merged in \(L_1\). About half of them comes from odd configurations (that were removed in \({L_0}^{\prime }\) merging as described in Sect. 3.1), the other half comes from partly overlapping inscribed void spheres. Moreover, \(L_2\) merging just provides few further merged pores than \(L_1\) which means that such cases are not significantly present in the packing of spheres. Accordingly, a shift towards smaller constriction sizes is reported when comparing \(L_1\) CSD with \(L_0\) CSD (Figs. 11a and 12a).

Indeed, further merging criteria belonging to \(L_1\) were also studied [33]. In addition to the overlap condition, these criteria (herein denoted \(L_1(p\%)\), *p* is the threshold denoted as *t* in the pore separation technique) evaluate the degree of separation between pores (see Sect. 3.2). The evolution of the relative number of constrictions corresponding to UG and GG materials, at loose and dense states, is shown in Fig. 13. It can be noted that the configurations involving a low degree of separation (more precisely \(p\le 10\%\)) are more represented within the samples. Furthermore, the decrease in the number of remaining constrictions is most significant in the case of GG material (Fig. 13b) and at loosest state in general.

For GGL, the distribution of constriction sizes exhibits also two distinct modes. In contrast to UGL, the distribution resulting from \(L_1\) or \(L_2\) remains bimodal (Fig. 12b). In such a case, the smaller mode which is approximately not affected by the merging is probably related to constrictions between fine particles in contact while the large one may include constrictions involving at least one particle of diameter greater than the gap. Therefore, the second mode is not destroyed when merging.

Another reason behind can be the large clusters exceeding the \(L_1\) and \(L_2\) neighborhood levels in the GG material as described in Sect. 4.3. Multiple \(L_1\) or \(L_2\) instances are created within such a void while keeping relatively large constrictions connecting these instances.

### 5.2 Pore separation technique

Figures 14 and 15 show the CSDs and the estimated probability density of constriction sizes for respectively UGL and GGL samples. One can note that the CSDs gradually shift towards the smaller diameters as the threshold value *t* increases, and this is accompanied by a progressive decrease in the number of constrictions. Small thresholds mainly merge larger constrictions between pores within voids, which are bounded by smaller constrictions that connect these void clusters. The hierarchical approach strengthens the separation value for the latter ones so that they persist longer. Large thresholds will also merge these cluster-connecting constrictions successively so that the deduced pore structure is highly affected as shown for \(t=30\%\), which can certainly be considered as a significant separation (Figs. 14a, 15a).

In the case of UGL, the probability density of constriction sizes has two modes for \(L_0\), and then becomes unimodal when merging criteria are applied (Fig. 14b). Besides, the first mode is not affected by the merging steps for *t* lower than 5%.

For GGL, the \(L_0\) probability density (Fig. 15b) is similar to that obtained in Fig. 12b but the bimodal character is not as pronounced. However, the distributions still exhibit a bimodal shape when merging, as described in Sect. 5.1.

### 5.3 Discussion

It has been proven in a previous study involving UGL sample that the initial \(L_0\) CSDs derived from the Delaunay and the Voronoï methods are almost congruent [44]. Herein, the proof is also given for the studied GGL sample (Fig. 16) even if it was expected.

The two different merge approaches are based on completely different techniques: overlapping spheres in a local manner and the pore separation in a hierarchical sense. This may lead to different merge behavior.

The separation technique evaluates the importance of the distance maxima and minima along the Voronoï graph: it fuses local pores that are insignificantly separated to their next larger pore, then evaluates the remaining more important pores among each other building up a hierarchy. That way, it detects significant constrictions and stops merging there for a given hierarchy level, which depends on a user-defined threshold.

Relative error (in %) between the CSD derived from different merging criteria for UG

\(t~(\%)\) | 1 | 2 | 3 | 4 | 5 | 6 | |
---|---|---|---|---|---|---|---|

UGL | \(L_1\) | 1.4 | 1.2 | 1.7 | 2.3 | 3.0 | 3.7 |

\(L_2\) | 4.2 | 3.1 | 2.4 | 1.8 | 1.4 | 1.5 | |

UGD | \(L_1\) | 2.2 | 2.0 | 2.3 | 3.2 | 4.0 | 4.8 |

\(L_2\) | 4.8 | 3.5 | 2.8 | 2.0 | 1.7 | 1.9 |

Relative error (in %) between the CSD derived from different merging criteria for GG

\(t~(\%)\) | 1 | 2 | 3 | 4 | 5 | 6 | |
---|---|---|---|---|---|---|---|

\(t~(\%)\) | 1 | 2 | 3 | 4 | 5 | 6 | |

GGL | \(L_1\) | 2.1 | 2.1 | 2.6 | 3.0 | 3.6 | 4.0 |

\(L_2\) | 4.6 | 3.0 | 1.9 | 1.5 | 1.8 | 2.1 | |

GGD | \(L_1\) | 2.2 | 0.9 | 1.0 | 1.7 | 2.5 | 3.4 |

\(L_2\) | 4.6 | 2.9 | 1.7 | 0.9 | 0.8 | 1.5 |

## 6 Conclusion

In this paper, different void characteristics in packings of spheres are derived from a partition of the space. These characteristics are the distribution of the diameter of the void sphere having a volume equal to that of the pore, which characterizes the morphology of the void space, and the constriction size distribution which characterizes its topology. Since the usual Delaunay or Voronoï partitions may lead to an artificial over-segmentation of the pore space, two different techniques for merging local pores were studied and compared. These techniques, which lie on the computation of the inscribed void sphere associated to a local pore, are the overlapping void spheres technique and the pore separation technique.

In the overlapping void spheres approach, two criteria or levels are defined: a level 1 (\(L_1\)) where two direct neighboring local pores are merged if their respective inscribed void spheres are overlapping, and a second level (\(L_2\)), where the next neighboring local pores can also be merged with the two first ones to create a single pore in case of further overlapping of void inscribed spheres. The pore separation technique hierarchically evaluates the degree of separation indicated by the distance function of the void space for a given threshold *t*. To compare both approaches, different thresholds *t* were tested to show the impact of the merging criterion on the distribution of pore and constriction sizes.

Two materials were studied, one which can be considered as uniformly graded and another one which is widely graded but gap-graded, both in a loose or in a dense state. From a direct computation of pores and constrictions, \(L_1\) merging induces the removal of about 40% of constrictions irrespective of the grading and of the density. \(L_2\) level merging brings fewer new merging of local pores. In the pore separation technique, the removal of constrictions is important for small values of the threshold (\(t\le 1\%\)) irrespective of grading and density, and the rate of removal of further constrictions tends to decrease as *t* increases.

In the case of the uniformly graded material, merging tends to remove the larger constrictions and lets appear a clear single mode for the distributions of constriction sizes. In the case of the studied gap-graded material, two close modes are obtained after merging which is just typical of the studied grading. The same trend is observed with the pore separation technique while *t* is equal or smaller than 5%. For larger threshold values, the constriction size corresponding to the mode tends to shift to the smaller diameters.

A correspondence is found between the two merging techniques irrespective of the considered void characteristics, pores or constrictions, the grading and the density. \(L_1\) merging corresponds to a threshold of about 2% and \(L_2\) to a threshold of about 5%. It can serve as a guide for a user for the definition of pores at a meso scale even if a definite pore structure cannot be obtained due to subjective nature of these bodies.

## Notes

### Acknowledgements

Part of this work belongs to a project funded by *Compagnie Nationale du Rhône* (CNR). F. Seblany and E. Vincens acknowledge CNR for its interest and its financial support.

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflict of interest.

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