## 3 Numerical Results
### Model Set-Up and Random Parameters  
The proposed approach is tested in a realistic geomechanical analysis concerning the development of a production program from a synthetic hydrocarbon reservoir. For the sake of realism, the model is built by introducing some of the typical features of the off-shore reservoirs buried in the Northern Adriatic basin, Italy [9; 10; 53].  
The model domain covers an area of 50 km \(\times\) 50 km and extends down to 5 km depth. It is discretized into the 3-D FE mesh shown in Figure 1a consisting of 71,734 nodes and 410,030 tetrahedrons. The rock in the reservoir and the hydraulically connected aquifer has a strongly non-linear visco-elasto-plastic behaviour described by the VN model, while the overburden and the underburden, that is the portions of the domain that are not filled by color in Figure 1a, are linear elastic. Homogeneous essential conditions apply on the outer and the bottom boundaries, with the top of the domain modeled as a traction-free surface. We assume the hydrocarbon production program from the reservoir to last for 3 years, then start again after a 2-year stop and be completed after 7 years, with the monitoring continuing for 3 additional years after the production stop. The average pressure variation occurring within the reservoir in time is shown in Figure 1b. The pressure distribution in space, also within the connected aquifer, is predicted by means of a dynamic reservoir flow model. The reader can refer to Reference [13] for more details.  
Model uncertainty is connected to the four parameters of the material constitutive law mentioned in Section 2.1. Based on physical considerations and a possible prior knowledge, if available, a range of variability is associated to such parameters. In particular, we set:  
\[\begin{cases}1.\ \ \ln\left(\lambda^{*}\right)\sim\mathcal{U}\left(-5.60;-4.28\right) \\ 2.\ \rho_{x}\sim\mathcal{U}\left(5;20\right)\\ 3.\ \ \rho_{\mu}\sim\mathcal{U}\left(40;80\right)\\ 4.\ \ R\sim\mathcal{U}\left(1.2;1.5\right)\end{cases} \tag{28}\]  
The range for \(\lambda^{*}\) has been chosen recalling the physical relationship between \(\lambda^{*}\) and the vertical compressibility \(c_{m}\), which has been derived in References [9; 53] for the Northern Adriatic basin. For the other parameters typical values reported in the literature, for example in Reference [3], have been taken into account. Hence, we initialize the gPCE algorithm by defining a vector of \(n=4\) random parameters \(\mathbf{p}=\left(\lambda^{*},\rho_{\kappa},\rho_{\mu},R\right)\). Note that since we employ uniform distributions for the random variables, the Legendre polynomials are used as gPCE basis functions. All the other  
Figure 1: **(a) View of the three-dimensional finite element grid of the regional geomechanical model. The hydrocarbon reservoir is blue, while the connected aquifer is red. The vertical axis is exaggerated by a factor 3. **(b) Average pore pressure variation in time within the reservoir.**modelling assumptions, including initial and boundary conditions, geometry and pressure distribution, are considered as deterministic.  
As quantities of interest, collected in the vector \(\mathbf{U}\) given as output from the full forward geomechanical model \(\mathcal{S}\), we consider the time series of the vertical displacement at a single-point of the top surface showed in Figure 2a, corresponding to land subsidence in time. Measurements of this quantity are supposed to be collected every three months for ten years, for example, by a GPS station. The true observations are collected in the vector \(\mathbf{d}^{\dagger}\). They are synthetically obtained by running the forward model with what is assumed to be the true material parameter set (continuous lines in Figure 2b). Then, the observation vector \(\mathbf{d}\) is obtained by adding an error \(\mathbf{\epsilon}\) sampled from a zero-mean Gaussian noise with standard deviation \(\sigma_{\epsilon}=0.01\) m (dots and squares in Figure 2b).