A metamodel for confined yield stress flows and parameters' estimation

Abstract

With the growing demand of mineral consumption, the management of the mining waste is crucial. Cemented paste backfill (CPB) is one of the techniques developed by the mining industry to fill the voids generated by the excavation of underground spaces. The CPB process is the subject of various studies aimed at optimizing its implementation in the field. In this article, we focus on the modelling of the backfill phase where it has been shown in [Vigneaux et al., Cem. Concr. Res. 164 (2023) 107038] that a viscoplastic lubrication model can be used to describe CPB experiments. The aim here is to propose an accelerated method for performing the parameters' estimation of the properties of the paste (typically its rheological properties), with an inverse problem procedure based on observed height profiles of the paste. The inversion procedure is based on a metamodel built from an initial partial differential equation model, thanks to a Polynomial Chaos Expansion coupled with a Principal Component Analysis.

Figures (7)

• Figure 1: 1D model, non-dimensionalized variables. The blue curve is the height $h$ of the material at two successive times (the flow is from left to right). Note that the gravity vector is inclined with an angle $\phi$ since the $x$ axis is assumed to be inclined downslope from the horizontal.
• Figure 2: Study of the convergence ($\Delta x$ is refined, and therefore also $\Delta t$ by \ref{['eq:stabcond']}) of the PDE numerical solver for $B=100$, $S=120$, $n=0.8$. All solutions have been computed at their respective wall-touch time. The reference solution ($h^{ref}$) has been computed using $n_x=9601$ points. The $L^2$ error ($\|h^{ref}(x)-h^{\Delta x,\Delta t}(x)\|_2$) decreases at least like $y=x^{0.6}$.
• Figure 3: Comparison of the results from the PDE direct solver (solid line) and from the metamodel (circle markers), for three sets of $(B,S)$ parameters (colors).
• Figure 4: Comparison between a real height profile (solid line), its noisy version (circle) and the profile determined by the Nelder-Mead algorithm (dotted line). The noisy profiles are created using $2$% of noise. The three colors correspond to three different $(B,S)$ couples randomly chosen. Note that at this level of zoom the line and the dotted line are superimposed, so only one is visible.
• Figure 5: Comparison between a real height profile (solid line), its noisy version (circle) and the profile determined by the Nelder-Mead algorithm (dotted line). The noisy profiles are created using $5$% of noise. The three colors correspond to three different $(B,S)$ used in the figure \ref{['fig:inv_noise_2']}. Note that at this level of zoom the line and the dotted line are superimposed, so only one is visible.