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The Finite Element Method in the Static and Dynamic Deformation and Consolidation in Porous Media

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Abstract

The authors start with an introduction to the concepts involved in physics giving the equations of flow through porous media and the deformation characteristics of soils and rocks. Succeeding chapters deal with the practical implications of these phenomena and explain the application of theory in both experimental and field work. Details are given of actual incidents, such as the subsidence experienced in Venice and Ravenna. The authors have also formulated a consolidation code, which is detailed at the end of the book, and provide instructions on how to modify the given program.
... in which, is the mass density of the two-phase porous media, which can be defined as: 38 = (1 − n) s + n w (15) where n is the porosity; s and w are the mass densities of the solid and fluid phases. ü is the acceleration; b is the force density applied by an external force field. ...
... In order to solve the governing equations, it is necessary to establish the initial and boundary conditions within the solution domain (Ω) and on the boundary (Γ). The initial conditions for the pore pressure field at time t = 0 are provided as 38 : ...
... Equation (38) can also be reformulated as: 7 ...
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The hybrid finite element-peridynamic (FEM-PD) models have been evidenced for their exceptional ability to address hydro-mechanical coupled problems involving cracks. Nevertheless, the non-local characteristics of the PD equations and the required inversion operations when solving fluid equations result in prohibitively high computational costs. In this paper, a fast explicit solution scheme for FEM-PD models based on matrix operation is introduced, where the graphics processing units (GPUs) are used to accelerate the computation. An in-house software is developed in MATLAB in both CPU and GPU versions. We first solve a problem related to pore pressure distribution in a single crack, demonstrating the accuracy of the proposed method by a comparison of FEM-PD solutions with those of PD-only models and analytical solutions. Subsequently, several examples are solved, including a one-dimensional dynamic consolidation problem and the fluid-driven hydraulic fracture propagation problems in both 2D and 3D cases, to comprehensively validate the effectiveness of the proposed methods in simulating deformation and fracture in saturated porous media under the influence of hydro-mechanical coupling. In the presented numerical results, some typical strong dynamic phenomena, such as stepwise crack advancement, crack branching, and pressure oscillations, are observed. In addition, we compare the wall times of all the cases executed on both the GPU and CPU, highlighting the substantial acceleration performance of the GPU, particularly when tackling problems with a significant computational workload.
... When applying the Galerkin method with the shape functions N u and N p for the fields of displacement and pore pressure, the governing equation of the fluid flow in saturated porous media becomes (Lewis et al., 1998): ...
... and (Lewis et al., 1998): ...
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Numerical simulations demonstrate the occurrence of both sub- and supershear ruptures in dry and saturated porous media during mode I fracture. Initially, the Finite Element Method is applied to investigate a case of sub-shear velocity in a dry beam on an elastic foundation subjected to sinusoidal loading. Subsequently, the Extended Finite Element Method is employed for fluid injection in both an infinite saturated porous medium and a plate. The results reveal a transition from smooth behavior to a stepwise pattern and eventually to forerunning as the injection rate increases. Notably, this progression has been observed for the first time. Finally, a hybrid Finite Element/Peridynamic model is utilized to explore both dry and fully saturated media under mechanical loading. In the case of saturated media, fluid injection is also considered. Both methods document velocities surpassing the shear wave. The noteworthy aspect is that these results, obtained through entirely different numerical methods and constitutive relations, justify confidence in the assertion that, in mode I fracture, speeds exceeding the shear wave and even the dilatation wave speeds can be achieved through forerunning. The loading condition investigated may be relevant in geophysics.
... where is the effective dispersion tensor of the diffusing phase and can be proven that = since 39 : ...
... This improves the accuracy in representing the displacement field compared to linear shape functions. 39,52 Finally, the set of discretized coupled governing equations end up to an ordinary time-dependent differential equation system stated as: ...
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This paper presents a comprehensive computational model for analyzing thermo-hydro-mechanical coupled processes in unsaturated porous media under frost actions. The model employs the finite element method to simulate multi-phase fluid flows, heat transfer, phase change, and deformation behaviors. A new soil freezing characteristic curve model is proposed to consider the suctions from air-water capillary pressure and water-ice cryosuction. A total pore pressure with components from liquid water pressure, air pressure, and ice pressure is used in the effective stress law. Vapor and dry air are considered miscible gases, utilizing the ideal gas law and Dalton's law. The governing equations encompass the linear momentum balance equation, the energy balance equation, and mass conservation equations for water species (ice, liquid, and vapor) and dry air. Weak forms are formulated based on primary variables of displacement, water pressure, air pressure, and temperature. The spatial discretization is achieved through the finite element method, while temporal discretization employs the fully implicit finite difference method, resulting in a system of fully coupled nonlinear equations. To verify the proposed computational model, a numerical implementation is developed and validated against a set of experimental data from the literature. The successful verification demonstrates the robustness of the model. A detailed discussion of the contributions from phase change strain and different sources of pore pressure is also addressed. K E Y W O R D S finite element method, frost actions, pore pressure, thermo-hydro-mechanical coupling, unsaturated soils This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.
... 13,18,19 In the theory of porous media and continuum mechanics, frozen soils are commonly modeled as a multiphasic mixture composed of soil grains, crystal ices, and liquid water (and gas if present), with ice and liquid water able to transfer between each other when the thermal condition changes. 20 Although the occurrences of ice in porous media are diversified (e.g., pore-filling ice, cementation ice, concretion ice, ice lens, ice wedge, etc.), 10,21,22 simplified frozen soil models for macroscale modeling usually consider the crystal ices within pores to be connected, homogeneous, and bear loads jointly with soil grains and liquid water following the effective stress principle. [23][24][25] The thermodynamic equilibrium of pore ice and water within frozen soils is typically characterized by a Clausius-Clapeyron equation, [26][27][28] whereas the ice variations under changing thermodynamic conditions are frequently described by a physically consistent or phenomenologically defined soil freezing characteristic curve (SFCC). ...
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Climate warming accelerates permafrost thawing, causing warming‐driven disasters like ground collapse and retrogressive thaw slump (RTS). These phenomena, involving intricate multiphysics interactions, phase transitions, nonlinear mechanical responses, and fluid‐like deformations, and pose increasing risks to geo‐infrastructures in cold regions. This study develops a thermo‐hydro‐mechanical (THM) coupled single‐point three‐phase material point method (MPM) to simulate the time‐dependent phase transition and large deformation behavior arising from the thawing or freezing of ice/water in porous media. The mathematical framework is established based on the multiphase mixture theory in which the ice phase is treated as a solid constituent playing the role of skeleton together with soil grains. The additional strength due to ice cementation is characterized via an ice saturation‐dependent Mohr–Coulomb model. The coupled formulations are solved using a fractional‐step‐based semi‐implicit integration algorithm, which can offer both satisfactory numerical stability and computational efficiency when dealing with nearly incompressible fluids and extremely low permeability conditions in frozen porous media. Two hydro‐thermal coupling cases, that is, frozen inclusion thaw and Talik closure/opening, are first benchmarked to show the method can correctly simulate both conduction‐ and convection‐dominated thermal regimes in frozen porous systems. The fully THM responses are further validated by simulating a 1D thaw consolidation and a 2D rock freezing example. Good agreements with experimental results are achieved, and the impact of hydro‐thermal variations on the mechanical responses, including thaw settlement and frost heave, are successfully captured. Finally, the predictive capability of the multiphysics MPM framework in simulating thawing‐triggered large deformation and failure is demonstrated by modeling an RTS and the settlement of a strip footing on thawing ground.
... When applying classical finite elements to discretize equation (1.1), the resulting method is not robust in the limit of incompressible materials, where → ∞. Moreover, when coupling equation (1.1) to a porous or thermal material, instabilities appear when using lowest-order finite elements [4]. The situation is hardly better when considering finite volume or finite element methods, where the simplest stencils (e.g. ...
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We construct a simple and robust finite volume discretization for linearized mechanics, Stokes and poromechanics, based only on co-located, cell-centered variables. The discretization has a minimal stencil, using only the two neighboring cells to a face to calculate numerical stresses and fluxes. We fully justify the method theoretically in terms of stability and convergence, both of which are robust in terms of the material parameters. Numerical experiments support the theoretical results, and shed light on grid families not explicitly treated by the theoretical results.
... Due to the versatility of the FEM, it has been used for solving scientific and engineering problems of many different kinds (Zienkiewicz, 1977;Lewis and Schrefler, 1998). This means that it is possible to use the FEM for solving hydrothermal mineralizing problems involved in the convective flow of pore-fluid within porous rocks (Zhao et al., 1997;Zhao et al., 2002;Zhao et al., 2008b). ...
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The Laochang Pb-Zn deposit can be typically considered as a hydrothermal mineralizing deposit in the Gejiu ore district. Although extensive studies were conducted to understand the mineralizing system associated with the Laochang Pb-Zn deposit through using the traditional geoscience methods, the mineralizing process involved in this deposit has not been justified in a strictly scientific manner to date. In this article, the hydrothermal mineralizing mechanism of the Laochang Pb-Zn deposit is computationally simulated through using the dual length-scale approach associated with the finite element method (FEM). The related computationally simulating outcomes have revealed the following understanding: 1) the pore-fluid convection provides a continuous source of mineralizing fluid and material for the Laochang Pb-Zn deposit; 2) the convective flow of pore-fluid is the primary dynamic mechanism, which controls the temperature, chemical species and pore-fluid velocity distributions in the Laochang Pb-Zn deposit; 3) the localized structure plays a key role in controlling the localized pore-fluid flow pattern, which can further control the location and grade of the orebody in the Laochang Pb-Zn deposit; 4) the dual length-scale approach associated with the FEM is very useful for dealing with the computational simulation of the hydrothermal mineralizing mechanism involved in the Laochang Pb-Zn deposit.
... Gaspar [14] introduces a stabilized method for a Biot model with secondary consolidation by using the finite difference method on staggered grids. Lewis and Schrefler [19] use the finite element method to study the Biot model with secondary consolidation but not overcome the "locking phenomenon". In this paper, following the idea of [11], we reformulate the Biot model with secondary consolidation into a fluid coupled problem. ...
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