6.1 Saturated Poroelastic Half-Space under Harmonic Loading
6.1.3 Verification of the Unbounded Boundary Treatment
The aim of this section is to test and verify the suggested infinite domain modelling pro-cedure in Section 4.4.1 by comparing the numerical results with analytical solutions of the balance equations. Herein, the investigation is restricted to the materially incompress-ible biphasic model with linear elastic behaviour of the solid constituent, where for this particular case an analytical solution is available. Moreover, the discussion addresses the efficiency and the challenges of the proposed treatment, cf. Heider et al. [87].
In addition to the abbreviations in Table 6.1, this section proceeds from further abbre-viations given in Table 6.5 for the sake of a simple and abstract representation of the implemented schemes.
Abbr. Description
FE-IE finite element discretisation of the near field and infinite elements for the far field together with VDB scheme, cf. Figure 6.1, 2-d FE-IE
FE-fix finite element discretisation with finite (fixed) boundaries, cf. Figure 6.1, 2-d FE Table 6.5: Abbreviations related to the infinite domain modelling
The spatial discretisation with the FE-IE method is carried out using mixed-order (QL) interpolations of both the finite and the infinite elements as discussed in Section 4.4.1 . In this regard, the numeration and location of the connecting nodes must coincide in order to insure the continuity across the FE-IE interface. The time discretisation is performed using the implicit monolithic TR-BDF2 scheme with constant time-step size ∆t = 10−3s for all cases of study.
In order to examine the efficiency of the suggested FE-IE treatment, the domain is trun-cated at a certain distance l1 from the top before the propagating pressure waves are damped out. Thus, based on the analytical treatment, it is shown in Figure 6.7 how far the pressure waves can propagate inside the domain for different values of the permeabil-ity kF. Due to the assumed incompressibility of the pore fluid, only one type of p-waves appears, which damps out after a certain distance.
The performance of the proposed infinite domain treatment depends on a number of factors. In addition to the chosen damping relations and parameters (cf. Section 4.4.1), the implementation of the IE for the quasi-static behaviour of the far field and the pore-pressure approximation with the FEM inside the IE directly influence the accuracy of
0 2 4 6 8 10 12 14
-3 -2 -1 0
00 11
kF1 = 10−1 kF2 = 10−2 k3F= 10−3 case (2),l1= 8 m case (1),l1= 3 m
x2
x2
x2
3m 8m
case(1) case(2)
coordinatex2[m]
displ. uS2 [ 10−4 m ]
Figure 6.7: Analytical solution of the solid displacement for different values of the perme-ability kF at t= 0.15 s (left) and the corresponding truncated domains (right)
the results. In this context, the IE extension distance (or the pole of the mapped IE) represented by l2 in Figure 6.1, 2-d FE-IE, affects the efficiency of the quasi-static IE treatment in an attempt to improve the fit of the far-field response to the considered decay patterns, cf., e. g., Marques & Owen [127] or Zienkiewicz et al. [188] for more details. In this connection, a large increase of the IE extension leads to a coarse mesh for the pore-pressure approximation inside the IE, which is undesired especially under low permeability conditions. A comparable situation is found if for the approximation of a steep pore-pressure gradient, a fine FE mesh is required. However, for the shear-wave propagation through the solid skeleton, the influence of the pore-pressure discretisation is of less importance.
In the following discussion, two cases are compared using different values of the perme-ability kF and the IE extension l2. The specific values are given in Table 6.6 .
Case kF [m/s ] l1 [m ] l2 [m] Abbr.
(1) 10−2 3 0.15 FE-IE (1)1
3 FE-IE (2)1
(2) 10−1 8 1 FE-IE (1)2
8 FE-IE (2)2
Table 6.6: Two considered cases with varying permeability and geometry of the infinite domain model
In case (1) with lower permeability, it is clear from Figure 6.8, left, that the FE-IE treat-ment results in a more accurate solid displacetreat-ment solution than FE-fix. This situation
6.1 Saturated Poroelastic Half-Space under Harmonic Loading 117
-0.75 -0.5 -0.25 0
0 0.1 0.2 0.3
Analytical FE-fix FE-IE (1)1
FE-IE (2)1
timet [s ]
t=0.15s
displ.uS2[10−4 m]
-1 -0.5 0
0 1
2 3
-0.3 -0.2
0.8 1.2
Analytical FE-fix FE-IE (1)1
FE-IE (2)1
coordinate x2 [m ]
displ.uS2[10−4m] x2=1m
Figure 6.8: Displacement time historyuS2 at x2 = 1 m (left), anduS2 solution overx2 at t= 0.15 s (right) for case (1) and FE = 20 elem/m
becomes even more prevalent with the progress of calculation time. In addition, the FE-IE with shorter l2 yields more accurate approximations than that with longer extension.
0 0.5 1 1.5 2
-3 -2 -1 0 1 2 3
1.7 1.9
-0.1 0.1
Analytical FE-fix
near field far field
FE-IE (1)1
FE-IE (2)1
coordinate x2 [m ] porepressurep[103 N/m2 ]
Figure 6.9: Pore-pressure solution pfor case (1) with FE = 20 elem/m at t = 0.15 s The error can be quantified by calculating, for instance, the relative displacement error4 ERRu at a point with coordinatex2 = 1 m and time t= 0.15 s. In this regard, the FE-fix treatment leads to ERRu ≈ 92 %, FE-IE (2)1 yieldsERRu ≈ 4.0 %, and FE-IE (1)1 gives the best solution with ERRu ≈ 0.03 % . Moreover, it is obvious from Figure 6.8 that ERRu changes according to the point position and time of observation.
Although increasing the IE extension has a positive impact on the solid displacement so-lution in the far field (cf. Marques & Owen [127]), the pore-pressure approximation with the FEM inside the IE plays an important role in the overall accuracy of the coupled problem. This coupling between the solid displacement and the pore-pressure solution
4ERRu:=|(uS2−ua)/ua|with ua being the analytical displacement solution and uS2 the numerical displacement solution.
-1.5 -1 -0.5 0 0.1
0 0.1 0.2 0.3
Analytical FE-fix FE-IE (1)2
FE-IE (2)2
timet[s ]
displ.uS2[10−4 m] t=0.15s
-3.5 -3 -2.5 -2 -1.5 -1 -0.5 0
0 1 2 3 4 5 6 7 8
-1 -0.5
1 2
Analytical FE-fix FE-IE (1)2
FE-IE (2)2
coordinate x2 [m]
x2=1m displ.uS2[10−4 m]
Figure 6.10: Displacement time history uS2 at x2 = 1 m (left), and uS2 solution over x2 at t= 0.15 s (right) for case (2) and FE = 10 elem/m
becomes more evident for lower values of the permeability. In this connection, Figure 6.9 shows that the FE-IE (1)1 case with shorter IE extension l2, and thus, a denser FE dis-cretisation of the pore pressure is better than the FE-IE (2)1 case with longer IE extension.
Consequently, at the bottom of the truncated domain (x2 = 0 m), Figure 6.8, right, shows that the coarse pore-pressure mesh leads to a poor FE-IE (2)1 approximation of the solid displacement in the coupled problem.
A similar discussion is performed for case (2) with higher permeability as given in Ta-ble 6.6 . In this regard, Figures 6.10 and 6.11 show that the FE-IE treatment again leads to better results than the FE-fix treatment for both the solid displacement and the pore-pressure fields.
0 0.5 1 1.5 2
-2 0 2 4 6 8
Analytical FE-fix FE-IE (2)2
FE-IE (1)2
coordinatex2 [m]
near field far field
porepressurep[103 N/m2 ]
Figure 6.11: Pore-pressure solution pfor case (2) with FE = 10 elem/m at t= 0.15 s For a quantitative comparison between the different schemes, the relative displacement errorERRu is evaluated, for instance, at a point withx2 = 1 m andt = 0.15 s. The FE-fix treatment leads to ERRu ≈ 33 %, FE-IE (1)2 results in ERRu ≈ 10.4 %, and FE-IE (2)2 yields the best solution with ERRu ≈ 9.0 %. Moreover, Figures 6.10 and 6.11 show that