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Stabilized Low-Order Explicit Finite Element Formulations for the Coupled Hydro-Mechanical Analysis of Saturated Poroelastic Media


We developed new stabilized low-order explicit finite element formulations for analysing the fully coupled hydro-mechanical behaviour of fluid-saturated poroelastic media. For space stabilization, the low-order U-p finite element employs two stabilization schemes. One stabilization scheme is based on the polynomial pressure projection technique in the fluid phase. The other one assumes enhanced strain field for the solid-phase terms. For time stabilization, an unconditionally stable explicit integration formula is proposed for discretization in the time domain to eliminate the time-step-size sensitivity of the temporal discretization finite difference format. The performance of the proposed formulations is demonstrated through four numerical examples. The proposed formulations not only agree strongly with the analytical/reference solutions, but also yield unconditionally stable high-precision results that outperform the standard finite element combined finite difference scheme. The modelling results indicate the proposed schemes possess significant advantages in terms of precision and computational efficiency for large timescales and adaptability to space-domain discretization. The proposed schemes have great potential in engineering applications for large timescale problems.

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b :

Body force vector

E :

Young’s modulus

E :

Parameter interpolation matrix

\( {\bf{f}}^{{u}} \) :

Vector of the nodal loads

\( {\tilde{\bf{f}}}^{{u}} \) :

Vector of the nodal loads by static condensation

\( {\bf{f}}^{{p}} \) :

Vector of the flow sources

g :

Gravitational acceleration

G :

Shear modulus

G :

Internal shape function matrix

h :

Element size

H :

Flow stiffness

\( {\bar{\bf{H}}} \) :

Exponential term matrix

\( {\mathbf{I}} \) :

Unit matrix

\( {\text{K}}_{{dr}} \) :

Drained bulk modulus

\( {\text{K}}_{\text{s}} \) :

Bulk modulus of the solid grain

\( {\text{K}}_{f} \) :

Bulk modulus of the fluid

\( {\bf{k}}_{{f}} \) :

Permeability tensor

\( {\text{k}} \) :

Hydraulic conductivity tensor

K :

Elastic stiffness matrix

\( {\tilde{\bf{K}}} \) :

Elastic stiffness matrix by static condensation

\( {\bar{\bf{K}}} \) :

Coefficient matrix

l :

Second-order unit tensor

L :


\( {\bar{\bf{M}}} \) :

Coefficient matrix

m :


n :


n :

Unit normal vector at the domain boundary

\( {\text{n}}_{\text{ele}} \) :

Number of elements

N :


\( {\bf{N}}^{{u}} \) :

Displacement shape functions

\( {\bf{N}}^{{p}} \) :

Pore pressure shape functions

p :

Fluid pore pressure

\( {\text{Q}} \) :

Fluid mass flux from a sink or source

\( {\text{Q}}^{ *} \) :

Biot’s modulus

Q :

Flow–stress coupling matrix

\( {\tilde{\bf{Q}}} \) :

Flow–stress coupling matrix by static condensation

\( {\mathbf{r}}_{0} \) :

Constant parts of external load

\( {\mathbf{r}}_{1} \) :

Constant parts of external load

L :

Flow capacity matrix

S :

Transformation matrix

\( {\tilde{\bf{S}}} \) :

Transformation matrix by static condensation

\( {\bf{S}}^{\text{stab}} \) :

Stabilization term matrix

T :

Exponential matrix

\( {\bf{T}}_{\text{a}} \) :

Substructure of exponential matrix

u :

Displacement vectors

U :

Solid admissible displacement

\( {\bf{v}} \) :

Darcy’s velocity vector of fluid flow

\( V^{\varOmega } \) :

Area of integral domain

w :

Weighted residual parameter

\( {\tilde{\bf{w}}} \) :

Additional weighted residual parameter

x :

Vector of global unknown variables

\( \alpha \) :

Biot’s coefficient

\( {\varvec{\upalpha}} \) :

Vector that contains the internal strain parameters

\( \gamma_{f} \) :

Unit weight of the fluid

\(\varvec \varepsilon \) :

Strain tensor under the assumption of infinitesimal transformation

\( \tilde{\varvec \varepsilon } \) :

Additional strain field

\( \zeta \) :

Coordinate component in the isoparametric space

η :

Coordinate component in the isoparametric space

\( \nu \) :

Poisson’s ratio

θ :

Integration parameter

\(\varvec \varTheta \) :

Solid skeleton stiffness tensor

ϑ :

Stabilization parameter

\( \rho_{{f}} \) :

Fluid density

\( \boldsymbol{\sigma} \) :

Total stress tensor

\( {\boldsymbol{\sigma}}^{{\prime }} \) :

Effective stress tensor

τ 0 :

Constant multiplier

τ :

Downscaling time-step size

φ :

Weighted residual parameter

\( \varGamma \) :

Domain boundary

\( \varGamma_{\sigma } \) :

Solid traction boundary

\( \varGamma_{{u}} \) :

Displacement boundary

\( \varGamma_{{p}} \) :

Pore pressure boundary

\( \varGamma_{{q}} \) :

Velocity (flux) boundary

\( \varUpsilon \) :

Weak forms of the governing equations

\( \varUpsilon^{\text{stab}} \) :

Stabilization term introduced into the variational form

\( \varPsi \) :

Weak forms of the governing equations

\( \varPhi \) :

Weak forms of the governing equations

\( {\varvec{\Omega}} \) :


\( \prod {\left( \cdot \right)} \) :

Projection operator for calculating the domain volume average

\( \bigcup {\left( \cdot \right)} \) :

Union operation symbol


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This work is supported in part by the National Science Foundation of China (51679028), State Key Laboratory for Geomechanics and Deep Underground Engineering, China University of Mining & Technology (SKLGDUEK1804) and the Fundamental Research Funds for the Central Universities (DUT18JC10).

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Correspondence to Gen Li.

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Li, G., Wang, K. Stabilized Low-Order Explicit Finite Element Formulations for the Coupled Hydro-Mechanical Analysis of Saturated Poroelastic Media. Transp Porous Med 124, 1035–1059 (2018).

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  • Hydro-mechanically coupled processes
  • Poroelasticity
  • Low-order finite element
  • Unconditionally stable