Modeling large-deforming fluid-saturated porous media using an Eulerian incremental formulation (original) (raw)
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Modelling large-deforming fluid-saturated porous media using an Eulerian incremental formulation
2016
The paper deals with modelling fluid saturated porous media subject to large deformation. An Eulerian incremental formulation is derived using the problem imposed in the spatial configuration in terms of the equilibrium equation and the mass conservation. Perturbation of the hyperelastic porous medium is described by the Biot model which involves poroelastic coefficients and the permeability governing the Darcy flow. Using the material derivative with respect to a convection velocity field we obtain the rate formulation which allows for linearization of the residuum function. For a given time discretization with backward finite difference approximation of the time derivatives, two incremental problems are obtained which constitute the predictor and corrector steps of the implicit time-integration scheme. Conforming mixed finite element approximation in space is used. Validation of the numerical model implemented in the SfePy code is reported for an isotropic medium with a hyperelast...
IMA Journal of Numerical Analysis, 2013
We consider the equation due to Richards which models the water flow in a partially saturated underground porous medium under the surface. We propose a discretization of this equation by an implicit Euler's scheme in time and finite elements in space. We perform the a posteriori analysis of this discretization, in order to improve its efficiency via time step and mesh adaptivity. Some numerical experiments confirm the interest of this approach. Résumé: Nous considérons l'équation dite de Richards qui modélise l'écoulement d'eau dans un milieu poreux partiellement saturé souterrain, situé juste sous la surface. Nouś ecrivons une discrétisation de cetteéquation par schéma d'Euler implicite en temps et eléments finis en espace. Nous en effectuons l'analyse a posteriori, le butétant d'améliorer son efficacité par adaptation du pas de temps et du maillage. Quelques expériences numériques confirment l'intérêt de cette approche.
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 High-Order Discretization of Nonlinear Poroelasticity
Computational Methods in Applied Mathematics
In this work, we construct and analyze a nonconforming high-order discretization method for the quasi-static single-phase nonlinear poroelasticity problem describing Darcean flow in a deformable porous medium saturated by a slightly compressible fluid. The nonlinear elasticity operator is discretized using a Hybrid High-Order method, while the Darcy operator relies on a Symmetric Weighted Interior Penalty discontinuous Galerkin scheme. The method is valid in two and three space dimensions, delivers an inf-sup stable discretization on general meshes including polyhedral elements and nonmatching interfaces, supports arbitrary approximation orders, and has a reduced cost thanks to the possibility of statically condensing a large subset of the unknowns for linearized versions of the problem. Moreover, the proposed construction can handle both nonzero and vanishing specific storage coefficients.
Mixed finite element formulation for non-isothermal porous media in dynamics
2015
We present a mixed finite element formulation for the spatial discretization in dynamic analysis of non-isothermal variably saturated porous media using different order of approximating functions for solid displacements and fluid pressures/temperature. It is known in fact that there are limitations on the approximating functions N and N for displacements and pressures if the Babuska–Brezzi convergence conditions or their equivalent [5] are to be satisfied. Although this formulation complicates the numerical implementation compared to equal order interpolation, it provides competitive advantages e.g. in speed of computation, accuracy and convergence. A fully coupled mathematical [1] and numerical model for the analysis of the thermo-hydromechanical dynamic behaviour of multiphase geomaterials is reduced to a computationally efficient formulation by neglecting the relative acceleration of the fluid phases and the convective terms [2], [3]. The resulting mathematical model is based on ...
Finite-difference analysis of fully dynamic problems for saturated porous media
Journal of Computational and Applied Mathematics, 2011
Finite-difference methods, using staggered grids in space, are considered for the numerical approximation of fully dynamic poroelasticity problems. First, a family of second-order schemes in time is analyzed. A priori estimates for displacements in discrete energy norms are obtained and the corresponding convergence results are proved. Numerical examples are given to illustrate the convergence properties of these methods. As in the case of an incompressible fluid and small permeability, these schemes suffer from spurious oscillations in time, a first order scheme is proposed and analyzed. For this new scheme a priori estimates and convergence results are also given. Finally, numerical examples in one and two dimensions are presented to show the good monotonicity properties of this method.
Modeling the multiphase flows in deformable porous media
MATEC Web of Conferences
This work proposes the nonlinear model for the flow of mixture of compressible liquids in a porous medium with consideration of finite deformations and thermal effects. Development of this model is based on the method of thermodynamically consistent systems of conservation laws. Numerical analysis of the model is based on the WENO-Runge-Kutta method of the high accuracy. The model is developed to solve the problems arising when studying the different-scale fluid dynamic processes. Evolution of the wave fields in inhomogeneous saturated porous media is considered.
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.
On numerical analyses in the presence of unstable saturated porous materials
International Journal for Numerical Methods in Engineering, 2003
The numerical analysis of the dynamic evolution problem concerning an elastic-plastic saturated porous media in the presence of softening (or non-associativity) is considered in the framework of the Biot formulation extended to take into account plastic phenomena. The ÿnite step boundary value problem, obtained by discretization in time of the continuous initial boundary value problem, is studied and the issue of its ill-posedness is particularly addressed. The conditions for the loss of ellipticity are established for the linearized problem solved at each iteration when using the Newton-Raphson scheme. In particular, the roles of the algorithmic properties on this loss of ellipticity are derived in details. The integration scheme of the balance of mass equation plays a major role and it is shown that the uid ow (Darcy's law) does indeed introduce a length scale but in addition to being dependent on the integration time step, it is found to be insu cient for regularization. To illustrate and corroborate the obtained results, a one-dimensional example (exhibiting all the features of the three-dimensional situation) is considered and the corresponding linearized ÿnite step problem is solved in closed form.
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.