Many geological materials, such as shale, mudstone, carbonate rock, limestone and rock salt are multi-porosity porous media in which pores of different scales may co-exist in the host matrix. When fractures propagate in these multi-porosity materials, these pores may enlarge and coalesce and therefore change the magnitude and the principal directions of the effective permeability tensors. The pore-fluid inside the cracks and the pores of host matrix may interact and exchange fluid mass, but the difference in hydraulic properties of these pores often means that a single homogenized effective permeability tensor field is insufficient to characterize the evolving hydraulic properties of these materials at smaller time scale. Furthermore, the complexity of the hydro-mechanical coupling process and the induced mechanical and hydraulic anisotropy originated from the micro-fracture and plasticity at grain scale also makes it difficult to propose, implement and validate separated macroscopic constitutive laws for numerical simulations. This article presents a hybrid data-driven method designed to capture the multiscale hydro-mechanical coupling effect of porous media with pores of various different sizes. At each scale, data-driven models generated from supervised machine learning are hybridized with classical constitutive laws in a directed graph that represents the numerical models. By using sub-scale simulations to generate database to train material models, an offline homogenization procedure is used to replace the up-scaling procedure to generate cohesive laws for localized physical discontinuities at both grain and specimen scales. Through a proper homogenization procedure that preserves spatial length scales, the proposed method enables field-scale simulations to gather insights from meso-scale and grain-scale micro-structural attributes. This method is proven to be much more computationally efficient than the classical DEM-FEM or FEM 2 approach while at the same time more robust and flexible than the classical surrogate modeling approach. Due to the usage of bridging-scale technique, the proposed model may provide multiple opportunities to incorporate different types of simulations and experimental data across different length scales for machine learning. Numerical issues will also be discussed.

An adaptively stabilized finite element scheme is proposed for a strongly coupled hydro-mechanical problem in fluid-infiltrating porous solids at finite strain. We first present the derivation of the poromechanics model via mixture theory in large deformation. By exploiting assumed deformation gradient techniques, we develop a numerical procedure capable of simultaneously curing the multiple-locking phenomena related to shear failure, incompressibility imposed by pore fluid, and/or incompressible solid skeleton and produce solutions that satisfy the inf-sup condition. The template-based generic programming and automatic differentiation (AD) techniques used to implement the stabilized model are also highlighted. Finally, numerical examples are given to show the versatility and efficiency of this model. Darcy's velocity [7], by using inf-sup stable finite elements (e.g., Talyor-Hood, Raviart-Thomas finite elements) [9,[11][12][13][14], or by applying stabilization procedures to cure the otherwise unstable finite elements [7,[15][16][17][18]. The displacement-Darcy-velocity coupling scheme is relatively easy to implement. However, computation time will be significantly increased because of the extra degrees of freedom added for the pore fluid velocity. Implementing inf-sup stable displacement-pressure finite element requires only one extra degree of freedom for the pore pressure. However, inf-sup stable finite elements require special meshing, data structure, and preprocessing and postprocessing tools to accommodate the need to have different basis functions for the displacement and pore pressure solutions. This difficulty is accompanied with a significant increase in computation cost, as higher-order mixed finite elements lead to larger systems of equations and require more integration points to perform numerical integration, as pointed out in [14].An equal-order displacement-pore pressure mixed finite element formulation is computationally more efficient and does not require significant modification to data structures (except adding an additional degree of freedom to all nodes), and it is therefore more feasible for both code development and maintenance. The price for using the equal-order mixed finite element method, however, is that the mixed finite element formulation must be stabilized by either adding additional terms or introducing enhanced shape functions to eliminate the spurious mode encountered as a result of a failure to satisfy the inf-sup condition. In recent years, effort has been invested to develop stabilization procedures for poromechanics problem under the small strain assumption, for example, [7,[16][17][18]. However, there are numerous occasions in which porous solids experience significant deformation such that a finite strain formulation becomes essential. Examples include water-saturated soil near critical state [7] and hydrated biological tissue during normal physiological activities [19]. In those situations, a stabilized formulation for large deformation poromechanics inheriting the same ...

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A stabilized thermo-hydro-mechanical (THM) finite element model is introduced to in-6 vestigate the freeze-thaw action of frozen porous media in the finite deformation range. By applying 7 mixture theory, frozen soil is idealized as a composite consisting of three phases, i.e., solid grain, un-8 frozen water and ice crystal. A generalized hardening rule at finite strain is adopted to replicate how 9 the elasto-plastic responses and critical state evolve under the influence of phase transitions and heat 10 transfer. The enhanced particle interlocking and ice strengthening during the freezing processes and 11 the thawing-induced consolidation at the geometrical nonlinear regimes are both replicated in numer- 12ical examples. The numerical issues due to lack of two-fold inf-sup condition and ill-conditioning of 13 the system of equations are addressed. Numerical examples for engineering applications at cold re-14 gion are analyzed via the proposed model to predict the impacts of changing climate on infrastructure 15 at cold regions.16

SummaryAn adaptively stabilized monolithic finite element model is proposed to simulate the fully coupled thermo‐hydro‐mechanical behavior of porous media undergoing large deformation. We first formulate a finite‐deformation thermo‐hydro‐mechanics field theory for non‐isothermal porous media. Projection‐based stabilization procedure is derived to eliminate spurious pore pressure and temperature modes due to the lack of the two‐fold inf‐sup condition of the equal‐order finite element. To avoid volumetric locking due to the incompressibility of solid skeleton, we introduce a modified assumed deformation gradient in the formulation for non‐isothermal porous solids. Finally, numerical examples are given to demonstrate the versatility and efficiency of this thermo‐hydro‐mechanical model. Copyright © 2015 John Wiley & Sons, Ltd.

Under a pure tensile loading, cracks in brittle, isotropic, and homogeneous materials often propagate such that pure mode I kinematics are maintained at the crack tip. However, experiments performed on geo-materials, such as sedimentary rock, shale, mudstone, concrete and gypsum, often lead to the conclusion that the mode I and mode II critical fracture energies/surface energy release rates are distinctive. This distinction has great influence on the formation and propagation of wing cracks and secondary cracks from pre-existing flaws under a combination of shear and tensile or shear and compressive loadings. To capture the mixed-mode fracture propagation, a mixed-mode I/II fracture model that employs multiple critical energy release rates based on Shen and Stephansson, IJRMMS, 1993 is reformulated in a regularized phase field fracture framework. We obtain the mixedmode driving force of the damage phase field by balancing the microforce. Meanwhile, the crack propagation direction and the corresponding kinematics modes are determined via a local fracture dissipation maximization problem. Several numerical examples that demonstrate mode II and mixed-mode crack propagation in brittle materials are presented. Possible extensions of the model capturing degradation related to shear/compressive damage, as commonly observed in sub-surface applications and triaxial compression tests, are also discussed. c

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