Accurate and stablised time integration strategy for saturated porous media dynamics

Abstract

When applying equal-order monolithic schemes for the solution of incompressible fluid saturated porous media dynamics, the resulting pressure field often exhibit spurious oscillations. This is in part due to violation of the inf-sup restriction. Although the mixed order monolithic scheme such as Taylor–Hood element scheme can circumvent this problem and yet use a mixed order monolithic scheme, a reduction of accuracy of both displacements and pressures may happen when the porous media permeability is small. In this paper, we present a new equal-order monolithic scheme that can accurately handle a broader range of permeabilities. We consider the u-p and u-v-p formulations from the theory of porous media (u:solid displacement, v:liquid velocity, p:liquid pressure) for fully saturated materials. We name the new scheme as the fractional steps correction method. Results show that this scheme succeeds in solving both formulations quite well with a spatial discretisation based on either the finite difference or the finite element method.

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Appendices

Appendix A: Matrix form of update algorithms

The matrix form of the update algorithms analysed in this study is presented here. For the monolithic schemes and the first part of fractional steps correction scheme, the matrix form of the update algorithms can be written as

$$\begin{aligned} {[}\mathbf {K}] \begin{bmatrix} {[}\ddot{\mathbf {d}}^{n+1}] \\ {[}\dot{p}_\ell ^{n+1}] \end{bmatrix}= \begin{bmatrix} {[}K]&[Q] \\ {[}\bar{Q}]&[C] \end{bmatrix} \begin{bmatrix} {[}\ddot{\mathbf {d}}^{n+1}] \\ {[}\dot{p_\ell }^{n+1}] \end{bmatrix} = \begin{bmatrix} {[}r_u] \\ {[}r_p] \end{bmatrix} = r.h.s \end{aligned}$$
(A.1)

where [K], [Q], \([\bar{Q}]\) and [C] are the stiffness matrix, coupling matrices and compressibility matrix given by Eqs. (27)–(35) discretised by FDM or FEM, respectively. \([\ddot{\mathbf {d}}^{n+1}]\) is the displacement vector including \([\mathbf {a}^{n+1}, \mathbf {A}^{n+1}]\) and r.h.s stands for right-hand-side of the spacial discritised governing equations including residual displacement vector \([r_u]\) and pressure vector \([r_p]\). These residual vectors can be evaluated by using the displacements and pore pressure at the current time step \(\square ^{n}\).

Based on Eqs. (29), (30) and (36), we can easily observe that the compressibility matrix for the uvp1 model is \([C]_{uvp1} = n_\ell \bar{C}_\ell [\mathbf {I}]\) and the compressibility matrix for the uvp2 and up models is \([C]_{uvp2,up} = n_\ell \bar{C}_\ell [\mathbf {I}]-[\frac{k^\ell _{sat}}{\gamma ^\ell _{ref}}][\mathbf {H}]\), where \([\mathbf {I}]\) and \([\mathbf {H}]\) are the identity matrix and the spacial discretised matrix for the Laplace operator, respectively.

For incompressible fluid, the fluid compressibility coefficient becomes zero, i.e. \(\bar{C}_\ell =0\), which will make the compressibility matrix becomes zero as well \([C]_{uvp1}=[0]\). As a result, the global stiffness matrix of uvp1 will become

$$\begin{aligned} {[}\mathbf {K}_{uvp1}]= \begin{bmatrix} {[}K]&[Q] \\ {[}\bar{Q}]&[\mathbf {0}] \end{bmatrix}. \end{aligned}$$
(A.2)

This matrix form will lead to the Ladyženskaja-Babuška-Brezzi instability and both the uvp2 and up models will have similar matrices in their global stiffness matrix when the porous media is nearly impermeable.

It is worth noting that the second part of the fractional step correction schemes, Eqs. (34) and (40), introduces the displacement gradient terms into their compressibility matrices which prevent them to become null matrices.

Fig. 21
figure21

Maximum relative error of surface displacement for different grid/mesh sizes. a With \(k^\ell _{sat}=0.1\) m/s. b With \(k^\ell _{sat}=10\) m/s

Appendix B: Error analysis of mixed order scheme

This section presents the spatial error analysis of the mixed order scheme using the matrix approach. The error analysis of time derivatives is excluded in this section. By considering the discretisation or interpolation error introduced by the FEM, the matrix form for the updated algorithm (Eq. A.1) could be rewritten as follows

$$\begin{aligned} \begin{bmatrix} {[}K]&[Q] \\ {[}\bar{Q}]&[C] \end{bmatrix} \begin{bmatrix} {[}\ddot{\mathbf {d}}^{n+1}] \\ {[}\dot{p_\ell }^{n+1}] \end{bmatrix} + \begin{bmatrix} {[}E_{Ku}] + [E_{Qp}] \\ {[}E_{\bar{Q}u}] + [E_{Cp}] \end{bmatrix} = \begin{bmatrix} {[}r_u] \\ {[}r_p] \end{bmatrix} \end{aligned}$$
(B.1)

where \([E_{Ku}]\) and \([E_{\bar{Q}u}]\) are the discretisation (or interpolation) error vector for displacement and \(E_{Qp}\) and \(E_{Cp}\) are the discretisation (or interpolation) error vector for pore pressure. For mixed order up scheme, it is expected that \([E_{Ku}]\) and \([E_{\bar{Q}u}]\) have third order convergence rate as opposed to \(E_{Qp}\) and \(E_{Cp}\) that have second order convergence rate. After some simple manipulations, Eq. B.1 can be rewritten as

$$\begin{aligned}&\begin{bmatrix} {[}K] - [Q][C]^{-1}[\bar{Q}]&[\mathbf {0}] \\ {[}\mathbf {0}]&[C-[\bar{Q}][K]^{-1}[Q]] \end{bmatrix} \begin{bmatrix} {[}\ddot{\mathbf {d}}^{n+1}] \\ {[}\dot{p_\ell }^{n+1}] \end{bmatrix}\nonumber \\&\quad =\begin{bmatrix} {[}r_u] - [Q][C]^{-1}[r_p]\\ {[}r_p] - [\bar{Q}][K]^{-1}[r_u] \end{bmatrix}\nonumber \\&\qquad - \begin{bmatrix} {[}E_{Ku}] + [E_{Qp}] - [Q][C]^{-1}([E_{\bar{Q}u}] + [E_{Cp}])\\ {[}E_{\bar{Q}u}] + [E_{Cp}] - [\bar{Q}][K]^{-1}([E_{Ku}] + [E_{Qp}]) \end{bmatrix}. \end{aligned}$$
(B.2)

From Eq. B.2 we can see that the pressure error term exists in the solution of displacement. This means that, although quadratic interpolation function has been used, the solution of displacement may still have only second order convergence rate. Figure 21 shows that the convergence rate of the mixed order element schemes depends on which error term is dominated.

From Fig. 21a we can observe that the mixed order up-m-QL-FEM scheme generates larger error than the linear order up-m-LL-FEM scheme when using the same level of mesh and considering relatively low permeability. The observed results are due to the exceeds pore pressure generated by the applied loads that is not able to dissipate fast enough to prevent a high pressure gradient underneath the loading zone. As a result, pressure error \(E_p\) may be larger than the displacement error \(E_u\) which only has second order convergence rate. Additionally, as pointed out by Markert et al. [23], in this case, the mixed order scheme may have larger error due to the poor interpolated pressure at the mid nodes.

From Fig. 21b we can observe that, when the porous media has higher permeability, so that the pressure can dissipate faster without creating a large pressure gradient underneath the loading area. In this case, the up-m-QL-FEM scheme has nearly third order convergence rate in a coarse mesh. This behaviour happens because the error of the up-m-QL-FEM scheme is dominated by the displacement error \(E_u\). Since \(E_u\) has a faster convergence rate, as the mesh size decreases, the pressure error \(E_p\) starts to take over. As a result, the up-m-QL-FEM scheme has around second order convergence rate when using fine meshes.

Appendix C: Efficiency analysis

The efficiency of all schemes is analysed in two parts: (1) by measuring the computation time as in the numerical simulations in Sect. 4; and (2) by comparing the number of components in the global (stiffness) matrix obtained with each method since this number is directly related to the computation time. The second analysis is presented in this appendix via Table 3.

Table 3 Efficiency analysis considering global matrix bandwidth

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Zhang, Y., Pedroso, D.M., Li, L. et al. Accurate and stablised time integration strategy for saturated porous media dynamics. Acta Geotech. 15, 1859–1879 (2020). https://doi.org/10.1007/s11440-019-00879-7

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Keywords

  • Accuracy
  • Fractional schemes
  • Monolithic schemes
  • Stability
  • Theory of porous media