# Pore Size Distribution in Granular Material Microstructure
## 1 Introduction  
Pore space modeling is a challenging issue for scientists in various disciplines. Granular structures yield complex pore geometries, not easily accessible experimentally and difficult to characterize using non-idealized computational methods. Nonetheless, insight into the pore space is essential to homogenization techniques and the macroscale constitutive modeling of flow through porous media [1; 2; 3]. Fluid flow takes place within inter-connected pores, which govern seepage, drainage, consolidation, and internal stability [4; 5]. In granular filters, which have their drainage capability severely impacted by clogging, it is fundamental to define the pore space precisely [6; 7; 8]. Currently, a variety of approaches are available for computing the pore size distribution (PSD) mainly categorized in Delaunay tessellation, medial axis, and watershed based methods [9; 10; 11; 12; 13; 14; 15; 16; 17; 18].  
In the Delaunay tessellation method [19], a pore is delimited by a tetrahedron with vertices at the centers of four nearest neighboring particles. The effective pore size is then defined as the diameter of the largest inscribed sphere in this tetrahedron. This formulation encompasses two major limitations: (1) reasonable results are only obtainable for packings of spherical particles; and (2) pores bounded by more than four spheres cannot be captured, resulting in a dense tessellation network that often leads to multiple subdivisions of a single pore. In the medial axis approach [9], the skeleton of a continuum pore space is first identified. Subsequently, individual pores are located through a series of merging steps of spherical voids centered along the medial axis. Despite not being restricted by granular arrangement or particle shape, the method suffers from a similar limitation to Delaunay tessellation, in that voids of complex polyhedral shapes cannot be accurately reproduced by the algorithm. In watershed methods [17], the pore space is first converted to an Euclidean distance field [20]. Voxels are then sequentially ascribed to a catchment basin starting at the local maxima until different expansion fronts collide and form the boundaries that constitutethe segmentation. While capable of capturing a range of pore geometries, the method often results in over-segmentation [21], requiring arbitrarily set parameters to achieve successful results [22].  
In this study, a new adaptive method is proposed to discretize the pore space and compute the PSD consistent with a diverse range of void shapes and unrestrained by the need of user-defined parameters. The approach consists of three steps. First, similarly to watershed methods, a binarized solid-void digital image is converted into a new domain by applying the Euclidean distance transform (EDT) [20]. Subsequently, pore centers are identified utilizing a robust non-maximum suppression routine [23]. Lastly, a segmentation process (minimum Euclidean distance assignment) [24] is applied to discriminate individual pores, irrespective of shape or size, in the originally continuous void space. Additionally, pore shape and orientation is explored, using the Principal Component Analysis (PCA) technique proposed by [PERSON] and [PERSON] [25].  
For verification purposes, spherical packings of known PSD were analyzed employing the method introduced herein. Following a successful validation, the tool was then applied to numerically generated packings at different density states. Random loose packings were generated by inserting spheres one at a time and allowing them to roll until a stable arrangement was reached [26; 27; 28]. Denser packings were obtained following a newly formulated procedure for cylindrical domains (Section 2.1), in which randomly placed overlapping spheres are subjected to a series of micro-rearrangements that yield a denser state [29; 30; 31].  
Finally, the soundness of the PSD characterization technique described in this paper is demonstrated through its application to 3D digital models of Ottawa 20-30 sand specimens [32; 33].