1 Introduction
1.1 Two variable versus three variable formulations
1.2 Previous results: infinitesimal strain
1.3 Previous results: finite strain
1.4 Contributions of the current work
2 Poroelasticity theory
2.1 Kinematics
2.2 Volume fractions
2.3 The model
3 The stabilized finite element method
3.1 Weak formulation
3.2 The fully discrete model
3.3 Solution via quasi-Newton iteration at \(t_n, n=1, \dots , N\).
3.3.1 Approximation of \(DG^n\).
4 Implementation details
4.1 Matrix assembly for the Newton iteration
4.2 Stabilization matrix assembly
4.3 Fluid-flux boundary condition
5 Numerical results
5.1 3D unconfined compression problem
Parameter | Description | Value |
---|---|---|
\(\phi _{0}\)
| Initial fluid volume fraction | 0.9 |
\(k_{0}\)
| Dynamic permeability |
\(10^{-3} \; \text{ m }^{3}\,\text{ s }\,\text{ kg }^{-1}\)
|
\(\nu \)
| Poisson’s ratio | 0.15 |
E
| Young’s modulus | 1000 \(\text{ kg }\,\text{ m }^{-1}\,\text{ s }^{-2}\)
|
\(\varDelta t\)
| Time step used in the simulation |
\(4\,\text{ s }\)
|
T
| Final time of the simulation |
\(1000\,\text{ s }\)
|
\(\varUpsilon \)
| Stabilization parameter |
\( 10^{-3}\)
|
Newton iteration |
\(||\mathfrak {u}_{i}^{n}-\mathfrak {u}^{n}_{i-1}||\)
|
\(||\varvec{R}(\mathfrak {u}_{i}^{n},\mathfrak {u}^{n-1})||\)
|
---|---|---|
1 | 0.81 | 0.023202 |
2 | 2.81699e−04 | 0.011276 |
3 | 6.93986e−08 | 1.34048e−06 |
4 | 4.10726e−10 | 7.64882e−09 |
Newton iteration |
\(||\mathfrak {u}_{i}^{n}-\mathfrak {u}^{n}_{i-1}||\)
|
\(||\varvec{R}(\mathfrak {u}_{i}^{n},\mathfrak {u}^{n-1})||\)
|
---|---|---|
1 | 0.81 | 0.0235609 |
2 | 1.28528e−05 | 1.99541e−04 |
3 | 7.71658e−08 | 1.49304e−06 |
4 | 4.6844e−10 | 8.17681e−09 |
5.2 Terzaghi’s problem
Parameter | Description | Value |
---|---|---|
\(\phi _{0}\)
| Initial fluid volume fraction | 0.9 |
\(k_{0}\)
| Dynamic permeability |
\(10^{-5} \; \text{ m }^{3}\,\text{ s }\,\text{ kg }^{-1}\)
|
\(\nu \)
| Poisson ratio | 0.25 |
E
| Young’s modulus | 100 \(\text{ kg }\,\text{ m }^{-1}\,\text{ s }^{-2}\)
|
\(\varDelta t\)
| Time step used in the simulation |
\(0.01\,\text{ s }\)
|
T
| Final time of the simulation |
\(1\,\text{ s }\)
|
\(\varUpsilon \)
| Stabilization parameter |
\(2\times 10^{-5}\)
|
5.3 Swelling test
Parameter | Description | Value |
---|---|---|
\(\phi _{0}\)
| Initial fluid volume fraction | 0.9 |
\(k_{0}\)
| Dynamic permeability |
\(10^{-5} \; \text{ m }^{3}\,\text{ s }\,\text{ kg }^{-1}\)
|
\(\nu \)
| Poisson ratio | 0.3 |
E
| Young’s modulus | 8000 \(\text{ kg }\,\text{ m }^{-1}\,\text{ s }^{-2}\)
|
\(\varDelta t\)
| Time step used in the simulation |
\(0.02\,\text{ s }\)
|
T
| Final time of the simulation |
\(20\,\text{ s }\)
|
\(\varUpsilon \)
| Stabilization parameter |
\(10^{-4}\)
|