This article presents an overview on poroelastodynamic models and some analytical solutions. A brief summary of Biot’s theory and on other poroelastic dynamic governing equations is given. There is a focus on dynamic formulations and the quasi-static case is not considered at all. Some analytical solutions for special problems, fundamental solutions, and Green’s functions are discussed. The numerical realization with two different methodologies, namely the Finite Element Method and the Boundary Element Method, is reviewed. Preprint No 3/2008 Institute of Applied Mechanics

Poroelastodynamics: Linear Models, Analytical Solutions, and Numerical Methods

5 SUMMARY Proceeding from the governing equations describing a saturated poroelastic material with intrinsically 7 incompressible solid and fluid constituents, we compare the monolithic and splitting solution of the different multi-field formulations feasible in porous media dynamics. Because of the inherent solid-fluid momentum 9 interactions, one is concerned with the class of volumetrically coupled problems involving a potentially strong coupling of the momentum equations and the algebraic incompressibility constraint. Here, the 11 resulting set of differential-algebraic equations (DAE) is solved by the finite element method (FEM) following two different strategies: (1) an implicit monolithic approach, where the equations are first 13 discretized in space using stable mixed finite elements and second in time using stiffly accurate implicit time integrators; (2) a semi-explicit-implicit splitting scheme in the sense of a fractional-step method, 15 where the DAE are first discretized in time, split using intermediate variables, and then discretized in space using linear equal-order approximations for all primary unknowns. Finally, a one- and a two-dimensional 17 wave propagation example serve to reveal the pros and cons in regard to accuracy and stability of both solution strategies. Therefore, several test cases differing in the used multi-field formulation, the 19 monolithic time-stepping method, and the approximation order of the individual unknowns are analyzed for varying degrees of coupling controlled by the permeability parameter. In the end, we provide a 21 reliable recommendation which of the presented strategies and formulations is the most suitable for which particular dynamic porous media problem. Copyright q 2009 John Wiley & Sons, Ltd. 23

Comparison of monolithic and splitting solution schemes for dynamic porous media problems

: This document is the final technical report for the project, Computational Methods in Nonlinear Mechanics. The Report summarizes results obtained on this project for the period June 30, 1980 through September 30, 1982. All objectives of the original statement of work have been accomplished. These have included advances in mathematical modelling, numerical analysis, approximation theory, development of computer codes, and the study of related issues connected with the following nonlinear structural problems: contact problems in elasticity, reduced integration and penalty methods for finite element approximations of constrained problems in elasticity and fluid flow, contact problems with friction, nonlinear friction laws, plasticity and metal forming, existence and approximation theories in contact problem in elastostatics, fracture mechanics numerical methods for problems of finite elastic deformation, nonlinear eigenvalue problems, and bifurcation theory. Detailed summaries of results in some of these areas are given together with lists of all papers, reports, books, dissertations and oral presentations produced during the contract period. Suggestions for further research areas are also given. (Author)

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Computational Methods in Nonlinear Mechanics.

http://www.marcelfrehner.ch/resourcen/Frehner_et_al_PEPI_2008.pdf

Comparison of finite difference and finite element methods for simulating two-dimensional scattering of elastic waves

The reflection and scattering of Stoneley guided waves at the tip of a crack filled with a viscous fluid was studied numerically intwodimensionsusingthefinite-elementmethod.Therocksurrounding the crack is fully elastic and thefluidfilling the crack is elastic in its bulk deformation behavior and viscous in its shear deformation behavior. The crack geometry, especially the crack tip, is resolved in detail by the unstructured finite-element mesh. Atthetipofthecrack,theStoneleyguidedwaveisreflected.The amplitude ratio between reflected and incident Stoneley guided wave is calculated from numerical simulations, which provide valuesrangingbetween43%andcloseto100%dependingonthe type of fluid filling the crack water, oil or hydrocarbon gas, the crack geometry elliptical or rectangular, and the presence of a small gas cap at the cracktip.The interference of incident and reflectedStoneleyguidedwavesleadstoanodezeroamplitudeat the tip of the crack.At other positions along the crack, this interferenceincreasestheamplitude.However,theexponentialdecay awayfromthecrackmakestheStoneleyguidedwavedifficultto detectatarelativelyshortdistanceawayfromthecrack.Thepart oftheStoneleyguidedwavethatisnotreflectedisscatteredatthe crack tip and emitted into the surrounding elastic rock as body waves. For fully saturated cracks, the radiation pattern of these elastic body waves points in every direction from the crack tip. TheemittedelasticbodywavescanallowthedetectionofStoneley guided wave-related resonant signals at distances away from thecrackwheretheamplitudeoftheStoneleyguidedwaveitself istoosmalltobedetected.

Finite-element simulations of Stoneley guided-wave reflection and scattering at the tips of fluid-filled fractures

We present a time-discontinuous Galerkin method (DGT) for the dynamic analysis of fully saturated porous media. The numerical method consists of a finite element discretization in space and time. The discrete basis functions are continuous in space and discontinuous in time. The continuity across the time interval is weakly enforced by a flux function. Two applications and several numerical investigations confirm the quality of the proposed space–time finite element scheme. Copyright © 2006 John Wiley & Sons, Ltd.

A time-discontinuous Galerkin method for the dynamical analysis of porous media

A space-time discontinuous Galerkin finite element method is proposed and applied to a convection-dominant single-phase flow problem in porous media. The numerical scheme is based on a coupled space-time finite element discretization allowing for discontinuous approximations in space and in time. The continuities on the element interfaces are weakly enforced by the flux treatments, so that no extra penalty factor has to be determined. The resulting space-time formulation possesses the advantage of capturing the steep concentration front with sharp gradients efficiently. The stability and reliability of the proposed approach is demonstrated by numerical experiments.

A space-time discontinuous Galerkin method applied to single-phase flow in porous media

We present a novel space–time Galerkin method for solutions of second-order time-dependent problems. By introducing the displacement–velocity relationship implicitly, the governing set of equations is reformulated into a first-order single field problem with the unknowns in the velocity field. The resulting equation is in turn solved by a time-discontinuous Galerkin approach (Int. J. Numer. Anal. Meth. Geomech. 2006; 30:1113–1134), in which the continuity between time intervals is weakly enforced by a special upwind flux treatment. After solving the equation for the unknown velocities, the displacement field quantities are computed a posteriori in a post-processing step. Various numerical examples demonstrate the efficiency and reliability of the proposed method. Convergence studies with respect to the h- and p-refinement and different discretization techniques are given. Copyright © 2007 John Wiley & Sons, Ltd.

A new hybrid velocity integration method applied to elastic wave propagation

In the current work, we investigate the dynamic analysis of a two-phase porous material using the space-time discontinuous Galerkin method. The physical model is based on the Theory of Porous Media (TPM). The finite element approximation consists of continuous approximations in space but discontinuous ones in time. The continuity condition between the adjacent time intervals is weakly enforced by the upwind flux treatment. No artificial penalty function is involved. Moreover, the Embedded Velocity Integration technique is applied to reduce the second-order equation in time into a first order one without introducing an additional constraint.

Dynamic analysis of porous materials: Numerical modeling with a space-time FEM

We investigated a space-time Galerkin method applied to the dynamic analysis of fully-saturated porous material. The governing set of equations are derived from the well-studied Theory of Porous Media (TPM). The numerical scheme consists of a coupled finite element discretization in the space-time domain. Discontinuous approximations of the primary variables in time are employed. A natural flux treatment is applied in time to impose the consistency between the adjacent time intervals. A simple space-time decoupled adaptive strategy based on the “jumps” in time and the ZZ error indicator in space is investigated. Numerical experiments demonstrate the efficiency and reliability of the proposed approach. We investigate a discontinuous space-time Galerkin method for the dynamic modeling of porous media. The physical model is based on the thermodynamically-consistent Theory of Porous Media (TPM). In contrast to the well-known Biot’s theory, which is constructed upon a lot of intuitive assumptions, the TPM is based on the rational axioms of the mixture theory extended by the concept of volume fractions. The main advantage of the TPM is that it can be extended straight forwardly to describe more complex physical situations. The physical model of the dynamic analysis has been developed in [4]. In the current work, we solved the governing set of equations with the high-order accurate time-discontinuous Galerkin (DGT) method [1]. Moreover, due to the employment of the Embedded Velocity Integration (EVI) technique [2], we are able to solve a three-fields formulation with the primary variables in the solid velocity vs, the seepage velocity wf and the pore pressure p. The solid displacement us is achieved in a post-processing step according to consistent integration of the velocities vs. The finite element variational form of the three-fields formulation solved by the DGT method has been presented in [3], which will not be repeated here. Moreover, a simple space-time adaptive strategy for the dynamical modeling within the porous material is developed. In the spatial domain, an extended error indicator of Zienkiewicz-Zhu type was developed, in which post-processed values of the characteristic quantities are considered. In the temporal domain, due to the employment of discontinuous approximations in time, a simple error indicator based on the amount of “jumps” in the primary variables is proposed.

Zhiyun Chen

Papers

A time-discontinuous Galerkin method for the dynamical analysis of porous media

A space-time discontinuous Galerkin method applied to single-phase flow in porous media

A new hybrid velocity integration method applied to elastic wave propagation

Dynamic analysis of porous materials: Numerical modeling with a space-time FEM

Dynamic analysis of porous materials: Numerical simulation with an adaptive space‐time FEM