Thermo-hydro-mechanical modeling of impermeable discontinuity in saturated porous media with X-FEM technique
Highlights
► The THM model of impermeable discontinuities is presented in saturated porous media. ► The X-FEM method is applied to the thermo-hydro-mechanical governing equations. ► The displacement, pressure and temperature fields are used by strong discontinuities. ► The displacement is enriched by the Heaviside and crack tip asymptotic functions. ► The pressure and temperature are enriched by the appropriate asymptotic functions.
Introduction
Thermo-hydro-mechanical (THM) modeling in porous media is one of the most important subjects in geotechnical and environmental engineering. There are various mathematical formulations proposed by researchers for thermo-hydro-mechanical model of porous saturated–unsaturated media in the literature. A fully coupled numerical model to simulate the slow transient phenomena involving the heat and mass transfer in deforming porous media was developed by Gawin and Klemm [1] and Gawin et al. [2], in which the heat transfer was taken through the conduction and convection into the model. A model in terms of displacements, temperature, capillary pressure and gas pressure was proposed by Schrefler et al. [3] and Gawin and Schrefler [4] in partially saturated deformable porous medium, where the effects of temperature on capillary pressure was investigated for drying process of partially saturated porous media. A finite element formulation of multiphase fluid flow and heat transfer within a deforming porous medium was presented by Vaziri [5] in terms of displacement, pore pressure and temperature, and its application was demonstrated in one-dimensional thermal saturated soil layer. A fully coupled thermo-hydro-mechanical model was applied by Neaupane and Yamabe [6] to describe the nonlinear behavior of freezing and thawing of rock. A general governing equation was proposed by Rutqvist et al. [7] for coupled THM process in saturated and unsaturated geologic formations. An object-oriented finite element analysis was performed by Wang and Kolditz [8] in thermo-hydro-mechanical problems of porous media. A combined THM-damage model was presented by Gatmiri and Arson [9] on the basis of a suction-based heat, moisture transfer and skeleton deformation equations for unsaturated media. A thermal conductivity model of three-phase mixture of gas, water and solid was developed by Chen et al. [10] and Tong et al. [11], [12] for simulation of thermo-hydro-mechanical processes of geological porous media by combining the effects of solid mineral composition, temperature, liquid saturation degree, porosity and pressure on the effective thermal conductivity of porous media. The THM model was proposed by Dumont et al. [13] for unsaturated soils, in which the effective stress is extended to unsaturated soils by introducing the capillary pressure based on a micro-structural model and taking the effects of desaturation and thermal softening phenomenon into the model.
Modeling of discontinuity with finite element method in fractured/fracturing porous media has been attracted by researchers. One of the earlier research works in THM modeling of two-phase fractured media was illustrated by Noorishad et al. [14], where a numerical approach was given for the saturated fractured porous rocks. Boone and Ingraffea [15] presented a numerical procedure for the simulation of hydraulically-driven fracture propagation in poroelastic materials combining the finite element method with the finite difference method. A cohesive segments method was proposed by Remmers et al. [16] for the simulation of crack growth, where the cohesive segments were inserted into finite elements as discontinuities in the displacement field by exploiting the partition-of-unity property. Schrefler et al. [17] and Secchi et al. [18] modeled the hydraulic cohesive crack growth problem in fully saturated porous media using the finite element method with mesh adaptation. The crack propagation simulation was performed by Radi and Loret [19] for an elastic isotropic fluid-saturated porous media at an intersonic constant speed. Hoteit and Firoozabadi [20] proposed a numerical procedure for the incompressible fluid flow in fractured porous media based on the combination of finite difference, finite volume and finite element methods. Segura and Carol [21] proposed a hydro-mechanical formulation for fully saturated geomaterials with pre-existing discontinuities based on the finite element method with zero-thickness interface elements. The cohesive segment method was employed by Remmers et al. [22] in dynamic analysis of the nucleation, growth and coalescence of multiple cracks in brittle solids. Khoei et al. [23] and Barani et al. [24] presented a dynamic analysis of cohesive fracture propagation in saturated and partially saturated porous media. The importance of cohesive zone model in fluid driven fracture was studied by Sarris and Papanastasiou [25], and it was shown that the crack mouth opening has larger value in the case of elastic-softening cohesive model compared to the rigid softening model. A numerical model based on the fully coupled approach was presented by Carrier and Granet [26] for the hydraulic fracturing of permeable medium, where four limiting propagation regimes were assumed.
In modeling of discontinuity based on the standard FEM, the discontinuity is restricted to the inter-element boundaries suffering from the mesh dependency. In such case, the successive remeshing must be carried out to overcome the sensitivity to FEM mesh that makes the computation expensive and cumbersome process. The difficulties confronted in the standard FEM are handled by locally enriching the conventional finite element approximation with an additional function through the concept of the partition of unity, which was introduced in the pioneering work of Melenk and Babuska [27]. The idea was exploited to set up the frame of the extended finite element method by Belytschko and Black [28] and Moës et al. [29]. Indeed, the extended finite element approximation relies on the partition of unity property of finite element shape functions for the incorporation of local enrichments into the classical finite element basis. By appropriately selecting the enrichment function and enriching specific nodal points through the addition of extra degrees-of-freedom relevant to the chosen enrichment function to these nodes, the enriched approximation would be capable of directly capturing the local property in the solution [30], [31], [32]. The X-FEM was originally applied in mesh-independent crack propagation problems, including: the crack growth with frictional contact [33], cohesive crack propagation [34], [35], [36], stationary and growing cracks [37] and three-dimensional crack propagation [38]. An overview of the technique was addressed by Bordas et al. [39] in the framework of an object-oriented-enriched finite element programming. The technique was then employed in elasto-plasticity problems, including: the crack propagation in plastic fracture mechanics [40], the plasticity of frictional contact on arbitrary interfaces [41], [42], the plasticity of large deformations [43], [44], [45] and the strain localization in higher-order Cosserat theory [46]. The X-FEM technique was extended to couple problems by its application in multi-phase porous media. The technique was proposed by de Borst et al. [47] and Rethore et al. [48] for the fluid flow in fractured porous media. The technique was employed in modeling of arbitrary interfaces by Khoei and Haghighat [49] in the nonlinear dynamic analysis of deformable porous media. The method was proposed by Ren et al. [50] in modeling of hydraulic fracturing in concrete by imposing a constant pressure value along the crack faces. The technique was also employed by Lecamipon [51] in hydraulic fracture problems using the special crack-tip functions in the presence of internal pressure inside the crack. The X-FEM was recently employed by Mohamadnejad and Khoei [52], [53] in hydro-mechanical modeling of deformable, progressively fracturing porous media interacting with the flow of two immiscible, compressible wetting and non-wetting pore fluids.
In the present study, the X-FEM technique is presented in thermo-hydro-mechanical modeling of impermeable discontinuities in saturated porous media. The governing equations of thermo-hydro-mechanical porous media is discretized by the X-FEM for the spatial discretization, and followed by a generalized Newmark scheme for time domain discretization. The impermeable discontinuity is modeled by the Heaviside and appropriate asymptotic tip enrichment functions for displacement, pressure, and temperature fields. The outline of the paper is as follows; the governing equations of thermo-hydro-mechanical porous media are presented in Section 2. Section 3 demonstrates the discontinuity behavior of displacement, pressure and temperature in THM medium. The weak form of governing equations are presented in Section 4 together with the spatial and time domain discretization of THM equations on the basis of the X-FEM and generalized Newmark approaches. In Section 5, several numerical examples are analyzed to demonstrate the efficiency of proposed model in saturated porous media. Finally, Section 6 is devoted to conclusion remarks.
Section snippets
Thermo-hydro-mechanical governing equations of saturated porous media
In order to derive the governing equations of thermo-hydro-mechanical porous media, the Biot theory is employed in saturated medium, which is coupled with the heat transfer analysis. The effective stress is an essential concept in the deformation of porous media, which can be defined by σ′ = σ + αpI, where σ′ and σ are the effective stress and total stress, respectively, I is the identity tensor and p is the pore pressure. In this relation, α is the Biot coefficient related to the material
Discontinuities in THM medium
The singularity in a discontinuous porous media can be caused due to the thermal and pressure loading in the vicinity of singular points. Since the governing equation of fluid flow in porous media are similar to the heat transfer equation, the treatment of thermal field near the singular points is assumed to be similar to the fluid phase. By neglecting the effect of transient terms in the heat transfer equation at the vicinity of singular points, Eq. (7) can be transformed to ∇2T = 0 in the
Extended-FEM formulation of THM governing equations
In order to derive the weak form of governing equations (1), (4), (7), the trial functions u(x, t), p(x, t), T(x, t) and the test functions δu(x, t), δp(x, t) and δT(x, t) are required to be smooth enough in order to satisfy all essential boundary conditions and define the derivatives of equations. Furthermore, the test functions δu(x, t), δp(x, t) and δT(x, t) must be vanished on the prescribed strong boundary conditions. To obtain the weak form of governing equations, the test functions δu(x, t), δp(x, t
Numerical simulation results
In order to illustrate the accuracy and versatility of the extended finite element method in thermo-hydro-mechanical modeling of deformable porous media, several numerical examples are presented. The first two examples are chosen to illustrate the robustness and accuracy of computational algorithm for two benchmark problems. The first example illustrates the accuracy of X-FEM model in the heat transfer analysis of a plate with an inclined crack. The second example deals with the
Conclusion
In the present paper, the extended finite element method was developed for numerical modeling of impermeable discontinuity in saturated porous media. In order to derive the thermo-hydro-mechanical governing equations, the momentum equilibrium equation, mass balance equation and the energy conservation relation were applied. The spatial discretization of governing equations of thermo-hydro-mechanical porous media was performed by the X-FEM technique, and followed by the generalized Newmark
Acknowledgment
The authors are grateful for the research support of the Iran National Science Foundation (INSF).
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