A finite difference analysis of Biot's consolidation model

Gaspar, Francisco J.; Lisbona, F. J.; Вабищевич, П. Н.

doi:10.1016/s0168-9274(02)00190-3

Cited by 81 publications

(58 citation statements)

References 9 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We assume no external bulk forces are acting on the porous media so that the first equation is homogeneous, i.e., h(x, t) = 0. For the full formulation of the problem with boundary conditions and a special choice of the initial conditions, we refer to [9].…”

Section: Continuous Problemmentioning

confidence: 99%

“…Numerical Methods for Partial Differential Equations DOI 10.1002/num So-called consistent initial conditions are used at t = 0 above (for more details see discussion and references in [9] …”

Section: Continuous Problemmentioning

confidence: 99%

“…For smooth coefficients (e.g., single layered porous media), second-order convergence in operator norms is proven in [9]. In this article we give a theoretical analysis of the convergence rate of the difference scheme (13) in the case of discontinuous coefficients.…”

Section: Lemma 21 For Any V ∈ H ωU and Q ∈ H ωP We Havementioning

confidence: 99%

“…However, often solutions generated by finite elements and finite differences on collocated grids exhibit nonphysical oscillations at the early stages of the time stepping (i.e., close to the initial state). To avoid this difficulty certain discretization on staggered grid have been suggested and theoretically analyzed in [9,10] in the case when the coefficients of the poroelasticity system are smooth. The situation is quite complicated when the coefficients have discontinuities along material interfaces, e.g., multilayered porous media, especially when it is essential to capture accurately the solution near the interface.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

On convergence of certain finite volume difference discretizations for 1D poroelasticity interface problems

Ewing

Iliev

Lazarov

et al. 2006

Numerical Methods Partial

View full text Add to dashboard Cite

In the article two finite difference schemes for the 1D poroelasticity equations (Biot model) with discontinuous coefficients are derived, analyzed, and numerically tested. A recent discretization [Gaspar et al., Appl Numer Math 44 (2003), 487-506] of these equations with constant coefficients on a staggered grid is used as a basis. Special attention is given to the interfaces and as a result a scheme with harmonic averaging of the coefficients is derived. Convergence rate of O(h 3/2 ) in a discrete H 1 -norm for both the pressure and the displacement is established in the case of an arbitrary position of the interface. Further, rate of O(h 2 ) is proven for the case when the interface coincides with a grid node. Following an approach applied to secondorder elliptic equations in [Ewing et al., SIAM J Sci Comp 23(4) (2001), 1334-1350 we derive a modified and more accurate discretization that gives second-order convergence of the fluid velocity and the stress of the solid. Finally, numerical experiments of model problems that confirm the theoretical considerations are presented.

show abstract

Section: Continuous Problemmentioning

confidence: 99%

Section: Continuous Problemmentioning

confidence: 99%

Section: Lemma 21 For Any V ∈ H ωU and Q ∈ H ωP We Havementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On convergence of certain finite volume difference discretizations for 1D poroelasticity interface problems

Ewing

Iliev

Lazarov

et al. 2006

Numerical Methods Partial

View full text Add to dashboard Cite

show abstract

“…In addition, stabilised finite difference methods using central differences on staggered grids are used by Gaspar et al (2003Gaspar et al ( , 2006 to solve Biot's model. However, the numerical solution of the two-dimensional poroelasticity equations is usually approached using finite element methods (Bause et al 2017;Lewis and Schrefler 1978;Phillips and Wheeler 2007).…”

Section: Introductionmentioning

confidence: 99%

Monte Carlo Assessment of the Impact of Oscillatory and Pulsating Boundary Conditions on the Flow Through Porous Media

Rahrah

Vermolen

2018

Transp Porous Med

View full text Add to dashboard Cite

Stress and water injection induce deformations and changes in pore pressure in the soil. The interaction between the mechanical deformations and the flow of water induces a change in porosity and permeability, which results in nonlinearity. To investigate this interaction and the impact of mechanical vibrations and pressure pulses on the flow rate through the pores of a porous medium under a pressure gradient, a poroelastic model is proposed. In this paper, a Galerkin finite element method is applied for solving the quasi-static Biot's consolidation problem for poroelasticity, considering nonlinear permeability. Space discretisation using Taylor-Hood elements is considered, and the implicit Euler scheme for time stepping is used. Furthermore, Monte Carlo simulations are performed to quantify the impact of variation in the parameters on the model output. Numerical results show that pressure pulses and soil vibrations in the direction of the flow increase the amount of water that can be injected into a deformable fluid-saturated porous medium.

show abstract

Conservative discretizations and parameter‐robust preconditioners for Biot and multiple‐network flux‐based poroelasticity models

Hong

Kraus

Lymbery

et al. 2019

Numerical Linear Algebra App

View full text Add to dashboard Cite

Summary The parameters in the governing system of partial differential equations of multiple‐network poroelasticity models typically vary over several orders of magnitude, making its stable discretization and efficient solution a challenging task. In this paper, we prove the uniform Ladyzhenskaya–Babuška–Brezzi (LBB) condition and design uniformly stable discretizations and parameter‐robust preconditioners for flux‐based formulations of multiporosity/multipermeability systems. Novel parameter‐matrix‐dependent norms that provide the key for establishing uniform LBB stability of the continuous problem are introduced. As a result, the stability estimates presented here are uniform not only with respect to the Lamé parameter λ but also to all the other model parameters, such as the permeability coefficients Ki; storage coefficients cpi; network transfer coefficients βi j,i,j = 1,…,n; the scale of the networks n; and the time step size τ. Moreover, strongly mass‐conservative discretizations that meet the required conditions for parameter‐robust LBB stability are suggested and corresponding optimal error estimates proved. The transfer of the canonical (norm‐equivalent) operator preconditioners from the continuous to the discrete level lays the foundation for optimal and fully robust iterative solution methods. The theoretical results are confirmed in numerical experiments that are motivated by practical applications.

show abstract

A finite difference analysis of Biot's consolidation model

Cited by 81 publications

References 9 publications

On convergence of certain finite volume difference discretizations for 1D poroelasticity interface problems

On convergence of certain finite volume difference discretizations for 1D poroelasticity interface problems

Monte Carlo Assessment of the Impact of Oscillatory and Pulsating Boundary Conditions on the Flow Through Porous Media

Conservative discretizations and parameter‐robust preconditioners for Biot and multiple‐network flux‐based poroelasticity models

Contact Info

Product

Resources

About