Variograms Across Spatial Scales Using Graph Geodesics and Diffusion Distances in Heterogeneous Permeability Fields

Variograms Across Spatial Scales Using Graph Geodesics and Diffusion Distances in Heterogeneous Permeability Fields | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Method Article Variograms Across Spatial Scales Using Graph Geodesics and Diffusion Distances in Heterogeneous Permeability Fields Juan Jose Segura This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8896694/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 16 You are reading this latest preprint version Abstract Classical variogram analysis bins squared increments by Euclidean separation, implicitly treating straight-line distance as the relevant measure of proximity. In heterogeneous permeability fields and other subsurface systems, physical coupling is often controlled by connectivity and preferential pathways, so Euclidean lags can mix connected and disconnected pairs and mask scale-dependent structure. Here we provide a compact, reproducible workflow for variography across spatial scales using two connectivity-aware graph metrics: (i) a graph-geodesic (natural) distance computed as weighted shortest paths on a local adjacency graph whose edge costs decrease with permeability, and (ii) a diffusion distance derived from the heat kernel of the normalized graph Laplacian, which yields a time-parameterized, scale-dependent notion of proximity. Using an uploadable subset of the SPE10 benchmark distributed by the Open Porous Media (OPM) project, we compute empirical variograms of a grade-like attribute $g=\log_{10}(k_x)$ under Euclidean, geodesic, and diffusion distances at multiple diffusion times, and we further examine scale-separated variability using diffusion-wavelet-like detail bands formed by differences of heat-smoothed signals. The results show that connectivity-aware distances can re-parameterize lag structure and expose distinct continuity regimes across scales, providing a practical bridge between geostatistical characterization and multiscale graph signal processing for porous media applications. variogram semivariogram graph-geodesic (natural) distance graph geodesic diffusion distance heat kernel spectral graph wavelets permeability heterogeneity porous media Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Review Version 1 posted Editorial decision: Revision requested 06 May, 2026 Reviews received at journal 16 Apr, 2026 Reviews received at journal 10 Apr, 2026 Reviews received at journal 10 Apr, 2026 Reviews received at journal 06 Apr, 2026 Reviewers agreed at journal 31 Mar, 2026 Reviewers agreed at journal 31 Mar, 2026 Reviewers agreed at journal 31 Mar, 2026 Reviewers agreed at journal 11 Mar, 2026 Reviewers agreed at journal 10 Mar, 2026 Reviewers agreed at journal 09 Mar, 2026 Reviewers invited by journal 09 Mar, 2026 Editor invited by journal 01 Mar, 2026 Editor assigned by journal 21 Feb, 2026 Submission checks completed at journal 21 Feb, 2026 First submitted to journal 16 Feb, 2026 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. 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