%0 Journal Article %1 graton1935systematic %A Graton, L. C. %A Fraser, H. J. %D 1935 %I The University of Chicago Press %J The Journal of Geology %K 74e20-granularity 76s05-flows-in-porous-media %N 8, Part 1 %P 785--909 %R 10.1086/624386 %T Systematic Packing of Spheres: With Particular Relation to Porosity and Permeability %U https://www.journals.uchicago.edu/doi/10.1086/624386 %V 43 %X Geometrically systematic arrangements of uniform spheres are searchingly investigated in Part I. The relationships disclosed, important wherever orderly distribution of points or particles is involved, are here especially treated as the underlying principles of porosity and permeability. Besides packings denned respectively as "chaotic," "haphazard," and "chance," six cases of simple, systematic packing are recognized; two are orientation variants, but four are independent arrangements and include the two hitherto described: "loosest" and "tightest." Striking symmetries appear, and close analogies to crystal structure, including twinned and tripled arrangements. The unit void of each case is thoroughly explored and illustrated, since in it lies the key to porosity and permeability. The stability of the several cases and their probabilities of being formed indicate that Case 6, tightest or rhombohedral, is most favored, and this is abundantly confirmed by experiment. The commonest natural packing, however, comprises colonies of Case 6 packing strewn in a surrounding mesh of haphazard, the whole constituting chance packing; the practical consequences of such arrangement are discussed. Translation from one packing to another is treated, and its bearing on dilatation outlined. Part II considers permeability in all its relations to porosity, including the elements and degrees of dependence and independence. Geometry of the intersphere voids receives particular attention as affecting fluid flow through them; and plentiful graphs of the surprisingly complex void sections are shown, as well as of their integrated projections which somewhat influence rectilinear flow. Effect of assemblage orientation on flow is emphasized. Since permeability is of vectorial quality, every systematic assemblage of spheres is anisotropic with respect to permeability; therefore, if a single value is to be used for permeability, it must be the mean value.

@article{graton1935systematic, abstract = {Geometrically systematic arrangements of uniform spheres are searchingly investigated in Part I. The relationships disclosed, important wherever orderly distribution of points or particles is involved, are here especially treated as the underlying principles of porosity and permeability. Besides packings denned respectively as "chaotic," "haphazard," and "chance," six cases of simple, systematic packing are recognized; two are orientation variants, but four are independent arrangements and include the two hitherto described: "loosest" and "tightest." Striking symmetries appear, and close analogies to crystal structure, including twinned and tripled arrangements. The unit void of each case is thoroughly explored and illustrated, since in it lies the key to porosity and permeability. The stability of the several cases and their probabilities of being formed indicate that Case 6, tightest or rhombohedral, is most favored, and this is abundantly confirmed by experiment. The commonest natural packing, however, comprises colonies of Case 6 packing strewn in a surrounding mesh of haphazard, the whole constituting chance packing; the practical consequences of such arrangement are discussed. Translation from one packing to another is treated, and its bearing on dilatation outlined. Part II considers permeability in all its relations to porosity, including the elements and degrees of dependence and independence. Geometry of the intersphere voids receives particular attention as affecting fluid flow through them; and plentiful graphs of the surprisingly complex void sections are shown, as well as of their integrated projections which somewhat influence rectilinear flow. Effect of assemblage orientation on flow is emphasized. Since permeability is of vectorial quality, every systematic assemblage of spheres is anisotropic with respect to permeability; therefore, if a single value is to be used for permeability, it must be the mean value.}, added-at = {2025-04-16T08:15:13.000+0200}, author = {Graton, L. C. and Fraser, H. J.}, biburl = {https://www.bibsonomy.org/bibtex/2d87143a3858ed3e381c60481094bacce/gdmcbain}, comment = {doi: 10.1086/624386}, doi = {10.1086/624386}, interhash = {b361092c240b806e2b6de2bb710a676a}, intrahash = {d87143a3858ed3e381c60481094bacce}, issn = {00221376}, journal = {The Journal of Geology}, keywords = {74e20-granularity 76s05-flows-in-porous-media}, month = nov, number = {8, Part 1}, pages = {785--909}, publisher = {The University of Chicago Press}, timestamp = {2025-04-16T08:15:42.000+0200}, title = {Systematic Packing of Spheres: With Particular Relation to Porosity and Permeability}, url = {https://www.journals.uchicago.edu/doi/10.1086/624386}, volume = 43, year = 1935 }