Fractal-Fractional Creep Constitutive Model for Shale under Triaxial Stress States

Fractal-Fractional Creep Constitutive Model for Shale under Triaxial Stress States | 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 Research Article Fractal-Fractional Creep Constitutive Model for Shale under Triaxial Stress States Cong Lu, Zhile Han, Shouxin Wang, Chi Chen, Shiqian Xu, Qijun Zeng, and 2 more This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8958049/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract The long-term rheological behavior of shale critically influences wellbore stability and hydraulic fracture conductivity in unconventional reservoirs. Traditional integer-order constitutive models, such as the Nishihara model, frequently fail to accurately capture the complex nonlinear creep characteristics of shale, particularly the rapid transient decay during primary creep and the accelerated deformation during tertiary creep. To address these limitations, this study proposes a creep constitutive model based on the Fractal-Fractional Atangana-Baleanu-Caputo (FF-ABC) derivative. The viscoelastic deformation is described by a Fractal-Fractional Maxwell element, which integrates the memory effects of fractional calculus with the multi-scale structural characteristics of fractal geometry; the viscoplastic deformation is characterized by a fractal power-law damage model activated upon reaching a critical time threshold. Analytical solutions for triaxial stress states are derived using the fractal-fractional integral operator and the elastic-viscoelastic correspondence principle. The triaxial formulation explicitly incorporates the deviatoric stress tensor, enabling prediction of creep strain under variable confining pressures. Model validity is verified against experimental data from triaxial compression tests on shale samples. Results demonstrate that the proposed FF-ABC model achieves substantially higher fitting accuracy ($R^2 > 0.98$) compared to the classical Nishihara model and standard fractional derivative models. Sensitivity analysis indicates that fractal order $\beta_1$ governs the initial transient response, fractional order $\alpha_1$ controls long-term viscoelastic flow, and damage parameter $\beta_2$ determines tertiary creep acceleration. This study establishes a robust, physically meaningful theoretical framework for predicting time-dependent shale deformation in deep subsurface engineering applications. Shale creep Fractal-fractional derivative Atangana-Baleanu operator Constitutive modeling Triaxial stress state Viscoplastic damage Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Posted Version 1 posted 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. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-8958049","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":598836288,"identity":"ce552b87-a3c7-4a0c-8ff4-6af6d999e1be","order_by":0,"name":"Cong Lu","email":"","orcid":"","institution":"Southwest Petroleum University","correspondingAuthor":false,"prefix":"","firstName":"Cong","middleName":"","lastName":"Lu","suffix":""},{"id":598836290,"identity":"7a906e58-f7b3-4fd6-aa79-737bf3174c19","order_by":1,"name":"Zhile 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Traditional integer-order constitutive models, such as the Nishihara model, frequently fail to accurately capture the complex nonlinear creep characteristics of shale, particularly the rapid transient decay during primary creep and the accelerated deformation during tertiary creep. To address these limitations, this study proposes a creep constitutive model based on the Fractal-Fractional Atangana-Baleanu-Caputo (FF-ABC) derivative. The viscoelastic deformation is described by a Fractal-Fractional Maxwell element, which integrates the memory effects of fractional calculus with the multi-scale structural characteristics of fractal geometry; the viscoplastic deformation is characterized by a fractal power-law damage model activated upon reaching a critical time threshold. Analytical solutions for triaxial stress states are derived using the fractal-fractional integral operator and the elastic-viscoelastic correspondence principle. The triaxial formulation explicitly incorporates the deviatoric stress tensor, enabling prediction of creep strain under variable confining pressures. Model validity is verified against experimental data from triaxial compression tests on shale samples. Results demonstrate that the proposed FF-ABC model achieves substantially higher fitting accuracy ($R^2 \u003e 0.98$) compared to the classical Nishihara model and standard fractional derivative models. Sensitivity analysis indicates that fractal order $\\beta_1$ governs the initial transient response, fractional order $\\alpha_1$ controls long-term viscoelastic flow, and damage parameter $\\beta_2$ determines tertiary creep acceleration. 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