Double Porosity Models for Liquid Filtration in Incompressible Poroelastic Media

[0903.0797] Double Porosity Models for Liquid Filtration in Incompressible Poroelastic Media Skip to main content Search Submit Donate Log in Mathematics > Analysis of PDEs arXiv:0903.0797 (math) This paper has been withdrawn by Anvarbek Meirmanov [Submitted on 4 Mar 2009 ( v1 ), last revised 28 Aug 2026 (this version, v2)] Title: Double Porosity Models for Liquid Filtration in Incompressible Poroelastic Media Authors: Anvarbek Meirmanov No PDF available, click to view other formats Abstract: Double porosity models for the liquid filtration in a naturally fractured reservoir is derived from the homogenization theory. The governing equations on the microscopic level consist of the stationary Stokes system for an incompressible viscous fluid, occupying a crack-pore space (liquid domain), and stationary Lame equations for an incompressible elastic solid skeleton, coupled with corresponding boundary conditions on the common boundary "solid skeleton-liquid domain". We suppose that the liquid domain is a union of two independent systems of cracks (fissures) and pores, and that the dimensionless size $\delta$ of pores depends on the dimensionless size $\varepsilon$ of cracks: $\delta=\varepsilon^{r}$ with $r>1$. The rigorous justification is fulfilled for homogenization procedure as the dimensionless size of the cracks tends to zero, while the solid body is geometrically periodic. As the result we derive the well-known Biot -- Terzaghi system of liquid filtration in poroelastic media, which consists of the usual Darcy law for the liquid in cracks coupled with anisotropic Lame's equation for the common displacements in the solid skeleton and in the liquid in pores and a continuity equation for the velocity of a mixture. The proofs are based on the method of reiterated homogenization, suggested by G. Allaire and M. Briane. As a consequence of the main result we derive the double porosity model for the filtration of the incompressible liquid in an absolutely rigid body. Comments: Errors in differential equations Subjects: Analysis of PDEs (math.AP) MSC classes: 35Q30 Cite as: arXiv:0903.0797 [math.AP] (or arXiv:0903.0797v2 [math.AP] for this version) https://doi.org/10.48550/arXiv.0903.0797 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Anvarbek Meirmanov [ view email ] [v1] Wed, 4 Mar 2009 16:14:59 UTC (20 KB) [v2] Fri, 28 Aug 2026 10:55:51 UTC (1 KB) (withdrawn) Full-text links: Access Paper: Withdrawn Current browse context: math.AP < prev | next > new | recent | 2009-03 Change to browse by: math References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab)

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