International Journal of Urban Management and Energy Sustainability

International Journal of Urban Management and Energy Sustainability

A Microstretch Mixture Theory for Differential Land Subsidence and Earth-Fissure Initiation from Non-uniform Aquifer Drawdown

Document Type : Case Study

Authors
Amir Ghiasi tarzi
Amir Robati
Mohammad Ali Kazerooni sadi
Department of Civil Engineering, Ke.C., Islamic Azad university, Kerman, Iran
10.22034/ijumes.2026.2110389.1432
Abstract
Groundwater pumping lowers aquifer heads unevenly, and where the resulting land subsidence varies laterally it opens damaging earth fissures. Classical poroelasticity treats the soil skeleton as a structureless Cauchy continuum, predicts a sharp horizontal-strain concentration at the margin of the subsidence bowl, and has no intrinsic length scale to set the width of the fissure zone, whereas existing Cosserat descriptions of the skeleton omit the pore fluid. This study develops a microstretch mixture theory for saturated porous media in which the solid skeleton carries grain-scale rotation and dilatation coupled two-way to a non-polar pore fluid. The balance laws are derived from the principle of virtual power and closed by the Coleman–Noll procedure within the Theory of Porous Media, and the linearized theory is applied to the plane-strain consolidation of an aquitard under laterally varying drawdown. The theory introduces two internal length scales. Its Cosserat length caps the peak horizontal tensile strain and spreads it over a fissure zone of finite width, depending on its ratio to the layer thickness: at grain and shear-band scales the classical Biot prediction is recovered, whereas a structural length that is a sizeable fraction of the layer thickness reduces the peak to about seven-tenths of the classical value. The fissure-driving strain is largest early in consolidation, shortly after the head decline. For uniform drawdown the theory reduces exactly to the microdilatational-mixture and Biot–Terzaghi limits. The framework unifies tension- and rotation-based accounts of earth fissuring and yields testable predictions of fissure width, intensity, and timing.

Graphical Abstract

A Microstretch Mixture Theory for Differential Land Subsidence and Earth-Fissure Initiation from Non-uniform Aquifer Drawdown

Highlights

·         A microstretch mixture theory couples grain rotation and dilatation to pore fluid

·         Balance laws follow from virtual power; closure by Coleman–Noll within the TPM

·         A Cosserat length caps peak fissure strain and widens the fissure zone

·         Fissure-driving strain peaks early in consolidation, after the head decline

·         Uniform drawdown recovers the microdilatation and Biot–Terzaghi limits exactly

Keywords
Cosserat length scale
earth fissures
microstretch continuum
principle of virtual power
theory of Porous media
Atkin, R. J., & Craine, R. E. (1976). Continuum theories of mixtures: Basic theory and historical development. The Quarterly Journal of Mechanics and Applied Mathematics, 29(2), 209–244. https://doi.org/10.1093/qjmam/29.2.209
Bedford, A., & Drumheller, D. S. (1983). Theories of immiscible and structured mixtures. International Journal of Engineering Science, 21(8), 863–960. https://doi.org/10.1016/0020-7225(83)90071-X
Biot, M. A. (1941). General theory of three-dimensional consolidation. Journal of Applied Physics, 12(2), 155–164. https://doi.org/10.1063/1.1712886
Bowen, R. M. (1976). Theory of mixtures. In A. C. Eringen (Ed.), Continuum physics (Vol. 3, pp. 1–127). Academic Press.
Bowen, R. M. (1980). Incompressible porous media models by use of the theory of mixtures. International Journal of Engineering Science, 18(9), 1129–1148.
Bowen, R. M. (1982). Compressible porous media models by use of the theory of mixtures. International Journal of Engineering Science, 20(6), 697–735.
Budhu, M. (2011). Earth fissure formation from the mechanics of groundwater pumping. International Journal of Geomechanics, 11(1), 1–11. https://doi.org/10.1061/(ASCE)GM.1943-5622.0000060
Budhu, M., & Adiyaman, I. B. (2010). Mechanics of land subsidence due to groundwater pumping. International Journal for Numerical and Analytical Methods in Geomechanics, 34(14), 1459–1478. https://doi.org/10.1002/nag.863
Chiriţă, S., & Galeş, C. (2008). A mixture theory for microstretch thermoviscoelastic solids. Journal of Thermal Stresses, 31(11), 1099–1124.
Coleman, B. D., & Noll, W. (1974). The thermodynamics of elastic materials with heat conduction and viscosity. In W. Noll, The foundations of mechanics and thermodynamics: Selected papers. Springer. https://doi.org/10.1007/978-3-642-65817-4 (Reprinted from Archive for Rational Mechanics and Analysis, 13, 167–178, 1963)
Cowin, S. C., & Nunziato, J. W. (1983). Linear elastic materials with voids. Journal of Elasticity, 13(2), 125–147. https://doi.org/10.1007/BF00041230
dell’Isola, F., & Placidi, L. (2011). Variational principles are a powerful tool also for formulating field theories. In F. dell’Isola & S. Gavrilyuk (Eds.), Variational models and methods in solid and fluid mechanics (CISM International Centre for Mechanical Sciences, Vol. 535). Springer. https://doi.org/10.1007/978-3-7091-0983-0_1
Del Piero, G. (2009). On the method of virtual power in continuum mechanics. Journal of Mechanics of Materials and Structures, 4(2), 281–292. https://doi.org/10.2140/jomms.2009.4.281
Diebels, S. (2001). Evolution of the volume fractions in compressible porous media. In W. Ehlers (Ed.), IUTAM Symposium on Theoretical and Numerical Methods in Continuum Mechanics of Porous Materials (Solid Mechanics and Its Applications, Vol. 87). Springer. https://doi.org/10.1007/0-306-46953-7_3
Diebels, S. (2002). Micropolar mixture models on the basis of the Theory of Porous Media. In W. Ehlers & J. Bluhm (Eds.), Porous media: Theory, experiments and numerical applications (pp. 121–145). Springer. https://doi.org/10.1007/978-3-662-04999-0_3
Diebels, S., & Ehlers, W. (1996). On fundamental concepts of multiphase micropolar materials. Technische Mechanik, 16(1), 77–88.
Ehlers, W. (2002). Foundations of multiphasic and porous materials. In W. Ehlers & J. Bluhm (Eds.), Porous media: Theory, experiments and numerical applications (pp. 3–86). Springer. https://doi.org/10.1007/978-3-662-04999-0_1
Eringen, A. C. (1999). Microcontinuum field theories: I. Foundations and solids. Springer. https://doi.org/10.1007/978-1-4612-0555-5
Forest, S., & Sievert, R. (2006). Nonlinear microstrain theories. International Journal of Solids and Structures, 43(24), 7224–7245. https://doi.org/10.1016/j.ijsolstr.2006.05.012
Galloway, D. L., & Burbey, T. J. (2011). Review: Regional land subsidence accompanying groundwater extraction. Hydrogeology Journal, 19(8), 1459–1486. https://doi.org/10.1007/s10040-011-0775-5
Germain, P. (1973). The method of virtual power in continuum mechanics. Part 2: Microstructure. SIAM Journal on Applied Mathematics, 25(3), 556–575. https://doi.org/10.1137/0125053
Germain, P. (2020). The method of virtual power in the mechanics of continuous media, I: Second-gradient theory. Mathematics and Mechanics of Complex Systems, 8(2), 153–190. https://doi.org/10.2140/memocs.2020.8.153
Green, A. E., & Rivlin, R. S. (1964). On Cauchy’s equations of motion. Zeitschrift für angewandte Mathematik und Physik, 15, 290–292. https://doi.org/10.1007/BF01607019
Hassanizadeh, M., & Gray, W. G. (1979a). General conservation equations for multi-phase systems: 1. Averaging procedure. Advances in Water Resources, 2, 131–144. https://doi.org/10.1016/0309-1708(79)90025-3
Hassanizadeh, M., & Gray, W. G. (1979b). General conservation equations for multi-phase systems: 2. Mass, momenta, energy, and entropy equations. Advances in Water Resources, 2, 191–203.
Holzer, T. L., & Pampeyan, E. H. (1981). Earth fissures and localized differential subsidence. Water Resources Research, 17(1), 223–227. https://doi.org/10.1029/WR017i001p00223
Hütter, G. (2020). Coleman–Noll procedure for classical and generalized continuum theories. In H. Altenbach & A. Öchsner (Eds.), Encyclopedia of continuum mechanics (pp. 316–323). Springer.
Mariano, P. M. (2016). Trends and challenges in the mechanics of complex materials: A view. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 374, 20150341. https://doi.org/10.1098/rsta.2015.0341
Maugin, G. A. (2017). Non-classical continuum mechanics: A dictionary (Advanced Structured Materials, Vol. 51). Springer. https://doi.org/10.1007/978-981-10-2434-4
Mindlin, R. D. (1964). Micro-structure in linear elasticity. Archive for Rational Mechanics and Analysis, 16(1), 51–78. https://doi.org/10.1007/BF00248490
Mühlhaus, H.-B., & Vardoulakis, I. (1987). The thickness of shear bands in granular materials. Géotechnique, 37(3), 271–283. https://doi.org/10.1680/geot.1987.37.3.271
Noll, W. (1974). La mécanique classique, basée sur un axiome d’objectivité. In The foundations of mechanics and thermodynamics: Selected papers. Springer. https://doi.org/10.1007/978-3-642-65817-4
Nunziato, J. W., & Cowin, S. C. (1979). A nonlinear theory of elastic materials with voids. Archive for Rational Mechanics and Analysis, 72(2), 175–201. https://doi.org/10.1007/BF00249363
Poland, J. F. (Ed.). (1984). Guidebook to studies of land subsidence due to ground-water withdrawal (Studies and Reports in Hydrology, Vol. 40). UNESCO.
Quiligotti, S., Maugin, G. A., & dell’Isola, F. (2003). An Eshelbian approach to the nonlinear mechanics of constrained solid–fluid mixtures. Acta Mechanica, 160(1–2), 45–60. https://doi.org/10.1007/s00707-002-0968-z
Terzaghi, K. (1943). Theoretical soil mechanics. John Wiley & Sons. https://doi.org/10.1002/9780470172766
Truesdell, C. (1984). Thermodynamics of diffusion. In Rational thermodynamics (2nd ed.). Springer. https://doi.org/10.1007/978-1-4612-5206-1
Twiss, R. J., & Eringen, A. C. (1971). Theory of mixtures for micromorphic materials—I: Balance laws. International Journal of Engineering Science, 9(10), 1019–1044.
Wang, G.-Y., Zhu, J.-Q., You, G. G., Zhang, D., Yu, J., Gou, F.-G., & Wu, J.-Q. (2025). Earth fissures during groundwater depletion and recovery: A case study at Shitangwan, Wuxi, Jiangsu, China. Hydrology, 12(3), 62. https://doi.org/10.3390/hydrology12030062
Ye, S., Franceschini, A., Zhang, Y., Janna, C., Gong, X., Yu, J., & Teatini, P. (2018). A novel approach to model earth fissure caused by extensive aquifer exploitation and its application to the Wuxi case, China. Water Resources Research, 54(3), 2249–2269. https://doi.org/10.1002/2017WR021872

  • Receive Date 11 February 2026
  • Revise Date 10 April 2026
  • Accept Date 25 May 2026