## 4 Discussion and Conclusions  
In this work, we develop a gPCE-based framework to quantify and reduce the prediction uncertainty in geomechanical problems. The idea is to exploit the low computational cost of a gPCE-based surrogate model to--(i) perform a global sensitivity analysis and capture the material parameters mostly affecting the expected prediction, and (ii) update the variation ranges of both the state variables of interest and the governing parameters via data assimilation. We use an ensemble-based approach, such as the ES and its variation in the MDA form. Due to the high computational cost of the forward model runs required by the ensemble approaches, the proposed gPCE-ES and gPCE-MDA algorithms, if able to effectively approximate the full model behaviour, can allow for the use of large ensembles and successive assimilations at an affordable cost. The proposed approach is tested in a large-scale geomechanical model dealing with the prediction of land subsidence due to a hydrocarbon production program from a deep reservoir. To describe the constitutive behaviour of the deep rocks, we use a strongly non-linear visco-elasto-plastic model based on the VN law [3; 43]. We show that high-accuracy surrogate models can be achieved via gPCE. Through the computation of the Sobol' indices, in the specific application considered herein we demonstrate that the material parameters mainly impacting on the outcome of interest, that is, land subsidence in time, are those controlling the rock deformability in loading conditions and the size of the yield surface, \(\lambda^{*}\) and \(R\). The initial variability ranges of such parameters are progressively reduced through data assimilation considering: (i) standard ES, (ii) gPCE-ES, and (iii) gPCE-MDA. The gPCE-ES provides highly accurate results comparable to those obtained with the native ES approach but at a significantly lower computational cost. It is worth recalling that the high efficiency of gPCE is mostly due to the low dimension of the parameter space. In fact, the gPCE approach suffers from the so-called \"curse of dimensionality\" due to the exponential increase of the cost with the number of independent stochastic variables. The gPCE-ES is used to test the ES convergence rate at an increasing size of the ensemble. By fast polynomial evaluation we can analyse ensembles up to 20,000 realizations, which are not affordable without a surrogate model. In general, the parameter mean approaches the \"true\" solution with increasing MC realizations. Figure 9 shows that ensemble sizes greater than \(\sim\)2000 realizations do not provide any significant improvement on the update of the state variable, that is, the mean-ensemble value of the surface displacements coincides with the observation vector. This also implies a limitation to the best match of the parameter. Figure 8: Mean and standard deviation of the updated ensembles obtained with gPCE-MDA in case of constant and decreasing \(\alpha_{MDA}\) for parameter (**a**) \(\lambda^{*}\) and (**b**) \(R\) considering the “true” parameter set \(\mathbf{p}^{\mathbf{t}}=(0.005,1.5)\). The blue bar refers to the results obtained with the gPCE-ES algorithm. We analyse the gPCE-ES convergence rate by using two parameter sets to define the \"true\" reference observations, with values located in the middle and on the boundary of the expected variability ranges. Different results are obtained, with a poor constraint for \(R\) when the \"true\" value is closer to the upper bound of the variability range. Then, we test again these sets with the gPCE-MDA approach by repeatedly assimilating the same observations a number of times with an inflated measurement error covariance matrix. We use both constant and decreasing values for the inflating factors, without any significant difference between the two approaches. The quality of the results for constraining \(R\) is significantly improved.