Uncertainty Quantification of geochemical and mechanical compaction in layered sedimentary basins

Abstract

In this work we propose an Uncertainty Quantification methodology for sedimentary basins evolution under mechanical and geochemical compaction processes, which we model as a coupled, time-dependent, nonlinear, monodimensional (depth-only) system of PDEs with uncertain parameters. While in previous works (Formaggia et al. 2013, Porta et al., 2014 we assumed a simplified depositional history with only one material, in this work we consider multi-layered basins, in which each layer is characterized by a different material, and hence by different properties. This setting requires several improvements with respect to our earlier works, both concerning the deterministic solver and the stochastic discretization. On the deterministic side, we replace the previous fixed-point iterative solver with a more efficient Newton solver at each step of the time-discretization. On the stochastic side, the multi-layered structure gives rise to discontinuities in the dependence of the state variables on the uncertain parameters, that need an appropriate treatment for surrogate modeling techniques, such as sparse grids, to be effective. We propose an innovative methodology to this end which relies on a change of coordinate system to align the discontinuities of the target function within the random parameter space. The reference coordinate system is built upon exploiting physical features of the problem at hand. We employ the locations of material interfaces, which display a smooth dependence on the random parameters and are therefore amenable to sparse grid polynomial approximations. We showcase the capabilities of our numerical methodologies through two synthetic test cases. In particular, we show that our methodology reproduces with high accuracy multi-modal probability density functions displayed by target state variables (e.g., porosity).

Density Estimation in Uncertainty Propagation Problems Using a Surrogate Model

“…The density estimation error of Algorithm 4.1 has two components -the error of the spline approximation (line 3) and that of the histogram method (line 7). 7 The accuracy of the histogram method in line 7 depends on the number of bins L and on the 5 For a review of cubic splines that are strictly local, see [4]. 6 Algorithm 4.1 can be performed with any choice of grid points.…”

“…For clarity, we present it only with a uniform grid. 7 In terms of density estimators, this can be explained by the L…”

“…In such cases, the solution of the (otherwise deterministic) model becomes random, and so one is interested in computing its statistics. This problem, sometimes known as forward uncertainty propagation (UQ), arises in various areas such as biochemistry [33,35], fluid dynamics [6,21,31,35], structural engineering [48], hydrology [7], and nonlinear optics [42].…”

Simplex Stochastic Collocation for Piecewise Smooth Functions With Kinks

“…Because each summand consists of p + 1 factors, each summand is of order O(n −(p+1)/d ). Thereby we can estimate the products in (2) and obtain…”

“…Other alternative approaches for handling discontinuities include enriching the polynomial approximation basis, which generally requires some a priori knowledge of the discontinuity, domain decomposition, also known as multi-element approximations in this context, or discontinuity detection algorithms, see e.g. [2,16] for current references. Note that non-smooth functions with kinks can be smoothed by integration [5,6] over one dimension if the location of the kink is known.…”