Finite element methods for unsaturated porous solids and their application to dam engineering problems
Introduction
Constitutive modeling of coupled porous continua is often developed in thermodynamic frameworks based on the volumetric fraction concept. In these mixture-theory approaches, volumetric fractions such as saturation degree and porosity are considered as primary variables, together with solid skeleton displacements. However, to avoid uncertainties in the setting of initial and boundary conditions, fluid pressures are normally preferred to volumetric fractions among the primary variables used for numerical formulation of the porous solid problem.
Consistently with these considerations, in [1] we have presented hyperelastic relations for three-phase porous media assuming solid skeleton displacements and fluid pressures as primary variables. These laws are developed in a Biot-type thermodynamic framework considering a unique free energy for the whole porous continuum [2], [3] and investigating their relations with constitutive models obtained from alternative approaches based on mixture theories.
In this work, we propose finite element methods for the resolution of governing equations presented in [1]. The fluid-mass balance equation is discretized in time by a backward Euler scheme and the resulting non-linear coupled equation system is solved by a standard Newton–Raphson iterative procedure. A linearization consistent with the aforementioned time integration scheme is then performed to obtain fluid-term tangent operators. Several reasons for non-linearity and coupling are considered, including those due to a permeability model accounting not only for partial saturation but also for the influence of strain, as typically observed in rock masses.
To increase the solution accuracy in strongly non-linear problems, we also consider the time integration of a mixed form of fluid-content rate equation, improving the mass–conservation properties of the proposed formulation. A complete characterization of tangent operators is presented for the classical assumption of retention behaviour depending only on capillary pressure.
To assess the accuracy of aforementioned constitutive laws and the performance of finite element methods presented herein, we consider several numerical tests. Firstly, we present the two-dimensional simulation of two well-known one-dimensional problems: the desaturation of a sand column and the pressure-driven infiltration of an almost initially dry porous layer. The obtained numerical results are compared with experimental data presented by Liakopoulos [4] for the former test and with the semi-analytical solution of the infiltration problem proposed by Philip [5].
Furthermore, the presented formulation is applied in two problems of interest for dam engineering, that is, the simulation of reservoir bank response to rapid drawdown and the three-dimensional analysis of reservoir operation effects on a concrete dam, considering the interaction with foundation and abutment rock mass. In this way, the finite element formulation proposed herein is employed to extend the results we presented in [6] under the assumptions of full saturation of the rock mass and plane conditions for strain and flow. In the analysis of these last two problems, we also use the numerical formulation of unilateral boundary conditions on fluid flow proposed in [7] to model interfaces between unsaturated porous solids and atmosphere.
An outline of the rest of the paper is as follows. After recalling the weak equations for linear momentum balance of the porous continuum and for mass conservation of both the fluid phases (Section 2.1), the considered constitutive equations are summarized in Section 2.2, including the employed permeability model (Section 2.2.1). The finite element methods and the numerical examples are presented in Sections 3 Finite element formulation, 4 Representative numerical simulations, respectively. Some concluding remarks are reported in Section 5.
Section snippets
Governing equations
In the following, we briefly recall the equations governing the problem of a porous solid subjected to infinitesimal deformations characterized by displacements of the solid skeleton. The porous space is filled by two immiscible fluid phases: a liquid w and a gas g. We assume as positive the compressive fluid pressures and the tensile normal components of stress tensor .
Finite element formulation
In the following, we present finite element methods for the resolution of governing equations recalled in previous sections. Isoparametric interpolations are employed to approximate displacements and pore pressures at a point of a generic finite element :where and are the shape functions used to interpolate nodal displacements and nodal pore pressures , respectively. Approximations of strain and pore pressure gradients are based on
Representative numerical simulations
The finite element methods presented in previous sections have been implemented in code FEAP [17] over a mixed 3-noded triangle and over a mixed 8-noded brick for the resolution of plane and three-dimensional problems, respectively. Both the elements are based on linear interpolations of displacements and pore pressures as well as on a constant interpolation of volumetric strain and stress [14]. A standard Gauss quadrature rule is employed in evaluating residual and tangent terms, considering
Concluding remarks
Numerical results presented in previous sections show that some available experimental data, analytical solutions and typically observed dam behaviours can be effectively reproduced with the finite element methods proposed in this paper.
It is also verified that with respect to continuum tangents, operators consistent with the backward Euler scheme used for time integration of coupled fluid-content rate equation lead to considerable improvements in the Newton–Raphson convergence rate.
Acknowledgements
This research was supported by MIUR in the context of projects “Diagnostic analyses and safety assessment of existing concrete dams” coordinated by G. Maier (PRIN 2004) and “Structural monitoring, diagnostic inverse analyses and safety assessments of existing concrete dams” coordinated by G. Novati (PRIN 2007).
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Supported by MIUR projects coordinated by G. Maier (PRIN 2004) and G. Novati (PRIN 2007).