A Microdilatational Mixture Theory for Saturated Porous Media: From Virtual Power to Aquitard Compaction and Land Subsidence

Ghiasi Tarzi, Amir; Robati, Amir; Kazerooni Sadi, Mohammad Ali

doi:10.22034/ijumes.2026.2109321.1427

A Microdilatational Mixture Theory for Saturated Porous Media: From Virtual Power to Aquitard Compaction and Land Subsidence

Document Type : Original Article

Authors

Amir Ghiasi Tarzi

Amir Robati

Mohammad Ali Kazerooni Sadi

Department of Civil Engineering, Ke.C., Islamic Azad university, Kerman, Iran

10.22034/ijumes.2026.2109321.1427

Abstract

Groundwater withdrawal drives the compaction of fine-grained aquitards and the resulting land subsidence; however, heavily pumped aquifer systems continue to compact after the excess pore pressure has dissipated, a residual, creep-like settlement that classical Terzaghi–Biot consolidation cannot reproduce and that is usually described by empirical rheological laws. This study aims to provide a thermodynamically consistent, grain-scale origin for this behavior. A microdilatational mixture theory is developed for a saturated medium composed of a solid skeleton, whose grains carry a dissipative dilatational degree of freedom, and a non-polar pore fluid. The balance laws and boundary conditions are derived from the principle of virtual power, the constitutive equations are closed by the Coleman–Noll procedure within the Theory of Porous Media, and the linearized one-dimensional problem is solved with an implicit finite-difference scheme verified against Terzaghi’s analytical solution. The grain-scale dilatation relaxes on its own time scale τφ, distinct from the drainage time tc, so that consolidation proceeds in two stages: primary consolidation at the unrelaxed modulus, followed by secondary compression toward the relaxed modulus. For a San Joaquin Valley-type aquitard (h = 20 m, Δh = 30 m), the primary and ultimate settlements are approximately 28 and 72 cm, and under seasonal pumping the internal length scale and relaxation raise the peak phase lag from the classical finite-layer value of 46.6° to approximately 57°. The theory reduces exactly to Biot–Terzaghi poroelasticity and recovers the Taylor–Merchant/Gibson–Lo rheological model in its local limit; its distinctive, testable signatures are the length-scale effects on pore-pressure profiles and on the seasonal phase lag.

Graphical Abstract

A Microdilatational Mixture Theory for Saturated Porous Media: From Virtual Power to Aquitard Compaction and Land Subsidence

Highlights

· A microdilatational mixture theory for saturated porous media is derived from the principle of virtual power and closed thermodynamically by the Coleman–Noll procedure.

· It adds a single dissipative grain-scale dilatational degree of freedom, with its own relaxation time, to classical poroelasticity.

· In one-dimensional consolidation the theory yields two-stage settlement, primary drainage followed by secondary compression that continues after the excess pore pressure has dissipated.

· The theory reduces exactly to Biot–Terzaghi poroelasticity, to elastic materials with voids, and, in its local limit, to the Taylor–Merchant and Gibson–Lo rheological models.

· Its internal length scale reshapes the pore-pressure boundary layer and produces a seasonal phase lag beyond the classical diffusion bound, the distinguishing, testable signatures of aquitard compaction and land subsidence.

Keywords

Aquitard compaction

land subsidence

microdilatation continuum

secondary compression

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. https://doi.org/10.1016/B978-0-12-240803-8.50017-7

Bowen, R. M. (1980). Incompressible porous media models by use of the theory of mixtures. International Journal of Engineering Science, 18(9), 1129–1148. https://doi.org/10.1016/0020-7225(80)90114-7

Chaussard, E., Bürgmann, R., Shirzaei, M., Fielding, E. J., & Baker, B. (2014). Predictability of hydraulic head changes and characterization of aquifer-system and fault properties from InSAR-derived ground deformation. Journal of Geophysical Research: Solid Earth, 119(8), 6572–6590. https://doi.org/10.1002/2014JB011266

Chiriţă, S., & Galeş, C. (2008). A mixture theory for microstretch thermoviscoelastic solids. Journal of Thermal Stresses, 31(11), 1099–1124. https://doi.org/10.1080/01495730802250847

Coleman, B. D., & Noll, W. (1963). The thermodynamics of elastic materials with heat conduction and viscosity. Archive for Rational Mechanics and Analysis, 13(1), 167–178. https://doi.org/10.1007/BF01262690

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

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

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, pp. 1–15). Springer. https://doi.org/10.1007/978-3-7091-0983-0_1

dell’Isola, F., Rosa, L., & Woźniak, C. (1998). A micro-structured continuum modelling compacting fluid-saturated grounds: The effects of pore-size scale parameter. Acta Mechanica, 127(1–4), 165–182. https://doi.org/10.1007/BF01170371

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. https://journals.ub.ovgu.de/index.php/techmech/article/download/1218/1192/3715

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 (M. Epstein & R. E. Smelser, Trans.). Mathematics and Mechanics of Complex Systems, 8(2), 153–190. https://doi.org/10.2140/memocs.2020.8.153 (Original work published 1973)

Gibson, R. E., & Lo, K. Y. (1961). A theory of consolidation for soils exhibiting secondary compression (Publication No. 41). Norwegian Geotechnical Institute.

Haghshenas Haghighi, M., & Motagh, M. (2024). Uncovering the impacts of depleting aquifers: A remote sensing analysis of land subsidence in Iran. Science Advances, 10(19), Article eadk3039. https://doi.org/10.1126/sciadv.adk3039

Herrera-García, G., Ezquerro, P., Tomás, R., Béjar-Pizarro, M., López-Vinielles, J., Rossi, M., Mateos, R. M., Carreón-Freyre, D., Lambert, J., Teatini, P., Cabral-Cano, E., Erkens, G., Galloway, D., Hung, W.-C., Kakar, N., Sneed, M., Tosi, L., Wang, H., & Ye, S. (2021). Mapping the global threat of land subsidence. Science, 371(6524), 34–36. https://doi.org/10.1126/science.abb8549

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. https://doi.org/10.1007/978-3-662-55771-6_57

Ieşan, D., & Quintanilla, R. (2014). On a theory of thermoelastic materials with a double porosity structure. Journal of Thermal Stresses, 37(9), 1017–1036. https://doi.org/10.1080/01495739.2014.914776

Lees, M., Knight, R., & Smith, R. G. (2022). Development and application of a 1D compaction model to understand 65 years of subsidence in the San Joaquin Valley. Water Resources Research, 58(6), Article e2021WR031390. https://doi.org/10.1029/2021WR031390

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

Murad, M. A., Guerreiro, J. N., & Loula, A. F. D. (2001). Micromechanical computational modeling of secondary consolidation and hereditary creep in soils. Computer Methods in Applied Mechanics and Engineering, 190(15–17), 1985–2016. https://doi.org/10.1016/S0045-7825(00)00218-8

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

Ojha, C., Werth, S., & Shirzaei, M. (2019). Groundwater loss and aquifer system compaction in San Joaquin Valley during 2012–2015 drought. Journal of Geophysical Research: Solid Earth, 124(3), 3127–3143. https://doi.org/10.1029/2018JB016083

Poland, J. F. (Ed.). (1984). Guidebook to studies of land subsidence due to ground-water withdrawal (Studies and Reports in Hydrology No. 40). UNESCO. https://unesdoc.unesco.org/ark:/48223/pf0000065167

Qi, J., Xie, Y., Li, C., Guo, H., & Wang, Y. (2025). Creep behavior of clayey soil and its model prediction in the Cangzhou land subsidence area. Scientific Reports, 15, Article 9130. https://doi.org/10.1038/s41598-025-93928-z

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

Smith, R. G., & Vajedian, S. (2025). How quickly does elastic deformation respond to seasonal changes in hydraulic head? Insights from modeling and field data in the western USA. Hydrogeology Journal, 33(2), 611–625. https://doi.org/10.1007/s10040-025-02891-6

Taylor, D. W., & Merchant, W. (1940). A theory of clay consolidation accounting for secondary compression. Journal of Mathematics and Physics, 19(1–4), 167–185. https://doi.org/10.1002/sapm1940191167

Terzaghi, K. (1943). Theoretical soil mechanics. Wiley. https://doi.org/10.1002/9780470172766

Truesdell, C. (1984). 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.

Wilson, R. K., & Aifantis, E. C. (1982). On the theory of consolidation with double porosity. International Journal of Engineering Science, 20(9), 1009–1035. https://doi.org/10.1016/0020-7225(82)90036-2