Research paperA macro–microscopic coupled consolidation model for saturated porous media with compressible constituents
Introduction
Consolidation analysis of fluid-saturated porous media is one of the major themes in geotechnical engineering. Consolidation theory was originally developed by Terzaghi in the study of a one-dimensional problem (Terzaghi, 1943). The theory exploits an uncoupled form in which it is assumed that the total stress is held constant throughout the consolidation process and the strain is attributable only to the variation of the pore fluid pressure (Osman, 2010, Soares, 2019). Consolidation theory was later generalized by Biot for three-dimensional problems (Biot, 1941). Various formulations of Biot’s model have been employed in consolidation analysis (Zienkiewicz and Shiomi, 1984, Sabetamal et al., 2011, Navas et al., 2018, Monforte et al., 2019). Interest in this research field is motivated by the capability of Biot’s consolidation theory to describe the coupled behavior between the fluid flow and the deformation of the soil skeleton (Ing and Nie, 2002, Griffiths and Huang, 2010). Over the years, Biot’s consolidation model has been widely evaluated using laboratory measurements of saturated porous media (Nur and Byerlee, 1971, Coussy, 2004, Cheng, 2016).
It has been stated that the porosity, i.e., the volume fraction of the fluid constituent, plays an important role in the consolidation process. For instance, the permeability of soil together with the compression parameters, such as the drained bulk modulus, are both porosity dependent (Carman, 1956, Kupkova, 1993, Lu et al., 1999, Chapuis, 2011, Dehghani et al., 2018). This hydromechanically coupled behavior when the solid constituent is assumed incompressible has been widely studied (Gibson et al., 1967, Gibson et al., 1981, Huang et al., 2012, Dumais and Konrad, 2018). However, porosity-related information for saturated porous media with compressible solid constituents is missing in Biot’s consolidation model (Wilmanski, 1998, de Boer, 2006, Lopatnikov and Gillespie, 2010, Serpieri and Rosati, 2011, Serpieri and Travascio, 2017, Liang et al., 2017, Ehlers, 2018).
There are several approaches to address the problem. A coupled relation between the extrinsic and intrinsic strains has been added to the consolidation model (Serpieri and Travascio, 2017). Serpieri and Travascio (2017) stated that the porosity can be further calculated when the extrinsic and intrinsic strains are obtained. A combination of the porosity equilibrium equation proposed by Cheng (2016) and Biot’s model has been exploited in the consolidation problem considering the changing porosity (Liang et al., 2017). Szalwinski et al. (2020) proposed a porosity rate equation in order to track the variance of the porosity in the consolidation process.
Reviewing the aforementioned achievements, the emphasis was focused on combining the consolidation model with an additional porosity evolution equation. Different from these approaches, the purpose of this paper is to introduce a macro–microscopic coupled consolidation model for saturated porous media with compressible constituents. The proposed model employs a form in order to incorporate the coupled behavior between the macroscopic and microscopic compressibility. The study indicates that the evolution of the porosity for saturated porous media with compressible constituents can be obtained by the proposed model. The derivations are based on porous media theory and restricted to the infinitesimal strain theory. The analytical solutions for a one-dimensional consolidation problem are also presented, and two examples are given in order to demonstrate how to use the proposed model.
The remainder of the paper is organized as follows. Section 2 introduces the basic ideas in porous media theory. A simple derivation of the macro–microscopic coupled constitutive relations is presented in Section 3. In Section 4, a macro–microscopic coupled consolidation model is developed and compared with Biot’s consolidation model. The analytical solutions for the one-dimensional consolidation problem based on the macro–microscopic coupled consolidation model are discussed in Section 5. In Section 6, two examples are given to demonstrate how the proposed model works. The limitations of the proposed model are discussed and further developments are also suggested in Section 7.
Section snippets
Mechanical backgrounds
The purpose of this section is to introduce the basic ideas of porous media theory. For detailed descriptions of porous media theory, readers should be referred to de Boer, 1996, de Boer, 2000, de Boer, 2006 and Ehlers, 2002, Ehlers, 2009, Ehlers, 2018.
Simple derivations for the macro–microscopic coupled constitutive model
This section first briefly reviews the decoupled constitutive relations provided by Carroll and Katsube (1983). This decoupled model provides further insight into the theory by Biot (Katsube, 1985). However, as this decoupled model is established at the microscopic scale, it is difficult to directly use in practice. For this reason, we introduce a macro–microscopic coupled constitutive model based on the decoupled model. It is worth mentioning that both the decoupled form and the
Macro–microscopic coupled consolidation model
In this section, the governing equations of a three-dimensional consolidation theory will be derived based on the balance equations, the macro–microscopic coupled constitutive relations, and Darcy’s law. The independent variables herein include the deformation of the solid skeleton , the volumetric strain of the real solid material , and the hydrostatic pressure of the real fluid (pore pressure) . In this work, porous media is assumed to be in an initially undeformed state and in
Analytical solutions for one-dimensional problems
Consider a saturated soil layer with infinite horizontal extent and height undergoing pressure-controlled consolidation. The bottom surface is fixed with no vertical displacement and is assumed to be impermeable. The top surface is assumed to be free drained, and a linearly increasing load is applied. Schematic diagrams of the one-dimensional consolidation model and the loading curve are shown in Fig. 1, Fig. 2.
The governing equations for the one-dimensional consolidation problem can be
Examples for one-dimensional consolidation problem
To provide guidelines to the readers, we will demonstrate how to track the continuous evolution of the porosity in a one-dimensional consolidation problem. Two examples are provided in this section. In example 1, we verify the feasibility and accuracy of the proposed analytical solutions by comparing them with existing results. Example 2 involves a detailed process in order to illustrate how to obtain porosity information based on the macro–microscopic consolidation model. For both examples,
Discussions
The macro–microscopic coupled consolidation model is established based on Carroll and Katsube’s constitutive relations (Carroll and Katsube, 1983). As suggested by Detournay and Cheng (1993), the application of Carroll and Katsube’s constitutive relations should be restricted to the porous media which is assumed to be micro-homogeneous. Thus, the application of the proposed model should be limited to micro-homogeneous porous media. The microhomogeneity assumption has been evaluated for
Conclusions
This paper proposed a macro–microscopic coupled consolidation model for saturated porous media with compressible constituents. The derivation is conducted within the theoretical framework of porous media theory. A study of the analytical solutions for the one-dimensional consolidation problem based on the proposed model is also provided. Moreover, two examples of the one-dimensional consolidation problem are presented in order to offer guidelines for the application of the proposed model.
List of symbols
Elasticity tensor for macro–microscopic coupled constitutive model Total Cauchy stress tensor Fluid partial Cauchy stress tensor Solid partial Cauchy stress tensor Deviatoric Cauchy stress tensor Interaction force for the fluid phase Interaction force for the solid phase Displacement vector of the solid skeleton Acceleration vector of the fluid phase Acceleration vector of the solid phase Velocity vector of the fluid phase Velocity vector of the solid phase Relative velocity
CRediT authorship contribution statement
Jia-Yu Liang: Conceptualization, Methodology, Validation, Writing – original draft, Writing – review & editing. Yue-Ming Li: Supervision, Writing – review & editing.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (11772251). The authors appreciate the constructive comments provided by two reviewers.
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