## 4 Consistent Tangent Stiffness
This last test was considered to compare the performances of the implicit time integration with its consistent unsymmetric tangent matrix (as shown in Section 4) to those obtained with its symmetrized version, as it was proposed very recently in [15], where it was suggested to approximate the tangent matrix with its symmetric part, so that the global linear system of equations could be solved more efficiently at each iteration. The symmetric tangent matrix was computed starting from the non-symmetric one as \(\frac{1}{2}\bigg{[}\bigg{(}\frac{\partial\sigma}{\partial\epsilon}\bigg{)}+ \bigg{(}\frac{\partial\sigma}{\partial\epsilon}\bigg{)}^{T}\bigg{]}\), where \(\frac{\partial\sigma}{\partial\epsilon}\) is given in (60). Figure 10 shows the number of global nonlinear iterations for each time step. From the plots, it is clear that, if, on the one hand, the computation of the consistent tangent matrix was more complex, on the other hand, it guaranteed fewer iterations for each time step than its symmetrized counterpart. A further confirmation can be obtained by Figure 11, where the total analysis time is shown. As a reference, the time value for the implicit and symmetrized case was set equal to 100. The consistent unsymmetric tangent matrix guaranteed the lowest computing time, while the explicit case was slightly higher. The case of the implicit integration with an unsymmetric consistent tangent matrix was considered also in conjunction with a quasi-Newton (q-N) scheme provided in Abaqus(tm) as a variant of a BFGS scheme [32], where the tangent matrix was not recomputed at each iteration, but it was substituted by a series of secant matrices [30]. In this case as well, the faster convergence rate provided by the consistent Newton-Raphson scheme led to a globally l

Figure 3.1 Subsidence evaluation for a real case study. Time evolution of the vertical displacement (normalized) at the point of maximum subsidence. \begin{table}
\begin{tabular}{c c c c c c} \hline \(\kappa^{*}\) & \(\lambda^{*}\) & \(\mu^{*}\) & \(M\) & \(\
u\) & OCR \\ \hline
0.00619 & 0.0575 & 0.00106 & 1.33 & 0.33 & 1.339 \\ \hline \end{tabular}
\end{table} 
Table 3: Parameters for the V-N law for the real case study. ## 6 Conclusions

A rigorous implicit backward Euler integration of the viscoplastic model proposed by [11] was presented. Even though several different implementations of this model have been proposed in the literature, including explicit and semi-implicit time integrations, a fully implicit and rigorous backward Euler implementation was still missing, with the notable exception of [15], where, however, the analytical expression of the consistent tangent matrix was not provided. In this work, besides a comprehensive description of the implicit integration formulation, the analytical derivation of the consistent unsymmetric tangent stiffness matrix for the V-N model was described in detail, together with its validation strategy. The derived model was implemented as a user-defined material subroutine in the commercial finite element code Abaqus(tm) Standard. The expected advantages of the implicit formulation in terms of robustness with respect to an explicit formulation were assessed and validated by means of numerical tests carried out both at the material point level and at the reservoir scale. A strategy for the numerical validation of the consistent tangent matrix was described in detail and implemented, confirming the correctness of its analytical derivation. Finally, the performances of an implementation with a consistent tangent matrix were assessed in comparison with those obtained with the same implicit integration coupled to a symmetrized tangent matrix, as in [15], confirming the superiority of the first with respect to the latter in terms of the number of iterations to convergence and of global computing

Figure 11: Real case test. Total computing time. Figure 10: Real case test. Number of nonlinear iterations per time step for symmetric and unsymmetric operators. time.