A stabilized assumed deformation gradient finite element formulation for strongly coupled poromechanical simulations at finite strain (original) (raw)

A Fully-Conservative Finite Volume Formulation for Coupled Poro- Elastic Problems

2018

Solid mechanics is a research field that deals with the mechanical behavior of a wide variety of materials undergoing external loads. Among the various types of solids, porous materials, for instance, can be found in applications such as soil and rock mechanics, biomechanics, ceramics, etc. These applications are studied in the field of poromechanics, which is a specific branch of the solid mechanics that considers all types of porous materials. An important characteristic of such materials is that they contain a network of interconnected pore channels saturated with a fluid. In most situations the mechanical behavior of the porous matrix and the fluid flow through the pore channels are two tightly coupled phenomena interfering with each other. When the fluid moves from one region to another in the porous matrix it changes the pressure field inside the pore channels, which is perceived by the porous matrix as a force imbalance. As a consequence, the porous matrix tends to deform in ...

A mixed stabilized finite element formulation for finite deformation of a poroelastic solid saturated with a compressible fluid

Archive of Applied Mechanics, 2020

A stabilized finite element formulation is proposed to the study of the finite deformation of a porous solid saturated with a compressible fluid. Unlike previous finite element schemes, the compressibility of the fluid constituent is entirely considered, and particularly, the a porosity-dependent permeability is utilized in the present formulation. As a special case, the formulation for the saturating fluid being an ideal gas is also derived. The displacement of solid skeleton and pore pressure are treated as primary variables to express the resulting coupled nonlinear system of equations. Equal-order C 0 elements are employed to approximate displacement and pressure fields. The stability problem caused by the equal-order interpolation is overcome by using the method of polynomial pressure projections. Two examples are provided to demonstrate the effect of fluid compressibility, as well as the porosity-dependent permeability, on the responses of the material.

Finite element modelling of saturated porous media at finite strains under dynamic conditions with compressible constituents

International Journal for Numerical Methods in Engineering, 2011

Two finite element formulations are proposed to analyse the dynamic conditions of saturated porous media at large strains with compressible solid and fluid constituents. Unlike similar works published in the literature, the proposed formulations are based on a recently proposed hyperelastic framework in which the compressibility of the solid and fluid constituents is fully taken into account when geometrical non-linear effects are relevant on both micro-and macroscales. The first formulation leads to a three-field finite element method (FEM), which is suitable for analysing high-frequency dynamic problems, whereas the second is a simplification of the first, leading to a two-field FEM, in which some inertial effects of the pore fluid are disregarded, hence the second formulation is suitable for studying low-frequency problems. A fully Lagrangian approach is considered, hence all terms are expressed with reference to the material setting; the balance equations for the pore fluid are also expressed in terms of the chemical potential and the mass flux of the pore fluid in order to take the compressibility of the fluid into account. To improve the numerical response in the case of wave propagation, a discontinuous Galerkin FEM in the time domain is applied to the three-field formulation. The results are compared with analytical and semi-analytical solutions, highlighting the different effects of the discontinuous Galerkin method on the longitudinal waves of the first and second kind.

Numerical simulation of dynamic pore fluid solid interaction in fully saturated non-linear porous media

In this paper, a large deformation formulation for dynamic analysis of the pore fluid-solid interaction in a fully saturated non-linear medium is presented in the framework of the Arbitrary Lagrangian-Eulerian method. This formulation is based on Biot's theory of consolidation extended to include the momentum equations of the solid and fluid phases, large deformations and non-linear material behaviour. By including the displacements of the solid skeleton, u, and the pore fluid pressure, p, a (u-p) formulation is obtained, which is then discretised using finite elements. Time integration of the resulting highly nonlinear equations is accomplished by the generalized-α method, which assures second order accuracy as well as unconditional stability of the solution. Details of the formulation and its practical implementation in a finite element code are discussed. The formulation and its implementation are validated by solving some classical examples in geomechanics.

A coupling of mixed and continuous Galerkin finite element methods for poroelasticity I: the continuous in time case

Computational Geosciences, 2007

In this paper, we formulate a finite element procedure for approximating the coupled fluid and mechanics in Biot's consolidation model of poroelasticity. Here, we approximate the pressure by a mixed finite element method and the displacements by a Galerkin method. Theoretical convergence error estimates are derived in a continuous in-time setting for a strictly positive constrained specific storage coefficient. Of particular interest is the case when the lowest-order Raviart-Thomas approximating space or cell-centered finite differences are used in the mixed formulation, and continuous piecewise linear approximations are used for displacements. This approach appears to be the one most frequently applied to existing reservoir engineering simulators.

Robust iterative schemes for non-linear poromechanics

Computational Geosciences, 2018

We consider a non-linear extension of Biot's model for poromechanics, wherein both the fluid flow and mechanical deformation are allowed to be non-linear. We perform an implicit discretization in time (backward Euler) and propose two iterative schemes for solving the non-linear problems appearing within each time step: a splitting algorithm extending the undrained split and fixed stress methods to non-linear problems, and a monolithic L-scheme. The convergence of both schemes is shown rigorously. Illustrative numerical examples are presented to confirm the applicability of the schemes and validate the theoretical results. Index terms-Biot's model L-schemes MFEM convergence analysis coupled problems poromechanics

A stabilized finite element formulation for monolithic thermo-hydro-mechanical simulations at finite strain

International Journal for Numerical Methods in Engineering, 2015

An 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.

A finite element implementation of finite deformation surface and bulk poroelasticity

arXiv (Cornell University), 2023

We present a theoretical and computational model for the behavior of a porous solid undergoing two interdependent processes, the finite deformation of a solid and species migration through the solid, which are distinct in bulk and on surface. Nonlinear theories allow us to systematically study porous solids in a wide range of applications, such as drug delivery, biomaterial design, fundamental study of biomechanics and mechanobiology, and the design of sensors and actuators. As we aim to understand the physical phenomena at a smaller length scale towards comprehending the fundamental biological processes and the miniaturization of devices, the surface effect becomes more pertinent. Although existing methodologies provide the necessary tools to study coupled bulk effects for deformation and diffusion; however, very little is known about fully coupled bulk and surface poroelasticity at finite strain. Here we develop a thermodynamically consistent formulation for multiphysics processes of surface and bulk poroelasticity, specialized for soft hydrated solids, along with a corresponding finite element implementation. Our multiphysical approach captures the interplay between competing processes of finite deformation and species diffusion through the bulk and surface, and provides invaluable insight when surface effects are important.

A hybrid numerical model for multiphase fluid flow in a deformable porous medium

Applied Mathematical Modelling

In this paper, a fully coupled finite volume-finite element model for a deforming porous medium interacting with the flow of two immiscible pore fluids is presented. The basic equations describing the system are derived based on the averaging theory. Applying the standard Galerkin finite element method to solve this system of partial differential equations does not conserve mass locally. A non-conservative method may cause some accuracy and stability problems. The control volume based finite element technique that satisfies local mass conservation of the flow equations can be an appropriate alternative. Full coupling of control volume based finite element and the standard finite element techniques to solve the multiphase flow and geomechanical equilibrium equations is the main goal of this paper. The accuracy and efficiency of the method are verified by studying several examples for which analytical or numerical solutions are available. The effect of mesh orientation is investigated by simulating a benchmark water-flooding problem. A representative example is also presented to demonstrate the capability of the model to simulate the behavior in heterogeneous porous media.

A stabilized finite element method for finite-strain three-field poroelasticity

Computational Mechanics, 2017

We construct a stabilized finite-element method to compute flow and finite-strain deformations in an incompressible poroelastic medium. We employ a three-field mixed formulation to calculate displacement, fluid flux and pressure directly and introduce a Lagrange multiplier to enforce flux boundary conditions. We use a low order approximation, namely, continuous piecewise-linear approximation for the displacements and fluid flux, and piecewise-constant approximation for the pressure. This results in a simple matrix structure with low bandwidth. The method is stable in both the limiting cases of small and large permeability. Moreover, the discontinuous pressure space enables efficient approximation of steep gradients such as those occurring due to rapidly changing material coefficients or boundary conditions, both of which are commonly seen in physical and biological applications.