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. 164:107038, 2023) 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.

Appendices

Appendix A. Numerical solver

A.1 Numerical computation of h(t, 0)

As stated in the “2.2” section, the numerical computation of h(t, 0) is not straightforward. The time being fixed, let’s write $h_1=h(t,\Delta x)$ (known) and $h_0=h(t,0)$ (unknown). A downwind scheme for $\partial h/\partial x$ and the boundary condition $q(0)=1$ together with (1) lead to the following problem on $h_0$:

$$\begin{aligned} {\begin{matrix} 1=&{}\tilde{\delta }\frac{n}{(n+1)(2n+1)}Y_d(h_0)^{1+1/n}\times \\ &{}\left| S-\frac{h_1-h_0}{\Delta x}\right| ^{1/n} \left( (2n+1)h_0-nY_d(h_0)\right) \end{matrix}} \end{aligned}$$

(26)

with

$$\begin{aligned} \tilde{\delta }=sgn\left( S-\frac{h_1-h_0}{\Delta x}\right) , \end{aligned}$$

(27)

and

$$\begin{aligned} Y_d(h_0) = \max \left( h_0-\frac{B}{|S-\frac{h_1-h_0}{\Delta x}|},0\right) . \end{aligned}$$

(28)

This problem is highly non-linear. However, it is constant equal to 0 whenever $Y_d(h_0)=0$. On the other part of the real line, it can be checked that the function of $h_0$ defined by (26) is actually strictly increasing and continuous, implying the uniqueness of the solution of our problem. Because of the power law involved in (26), an external solver is required to numerically solve the problem. In order to accelerate its resolution, we provide to the solver the point after which $Y_d>0$ (so that it does not get stuck in a constant region). Using (28) and $h_0\ge 0$, we derive the formula:

$$\begin{aligned} h_0>\frac{h_1-S\Delta x}{2}+\sqrt{\left( \frac{h_1-S\Delta x}{2}\right) ^2+B\Delta x}. \end{aligned}$$

(29)

Note that in the special case $n=1$, the problem (26) is equivalent to finding the root of a polynomial of degree 7. However, in practice, it is not faster than using the methodology presented above.

A.2 Numerical stability condition

We develop the expression (1) of the model, so as to obtain explicitly an advection-diffusion form. Denoting V(x) the transport coefficient and D(x) the diffusion coefficient, (1) is equivalent to:

$$\begin{aligned} \frac{\partial }{\partial t} h(x,t) + V(x)\frac{\partial }{\partial x} h(x,t) - D(x)\frac{\partial ^2}{\partial x^2} h(x,t) = 0 \end{aligned}$$

(30)

with $\delta =\text{ sgn }(S-h_x)$,

$$\begin{aligned} V(x) = \delta Y^{1/n}|S-h_x|^{1/n}h, \end{aligned}$$

(31)

and

$$\begin{aligned} {\begin{matrix} &{} D(x)= \frac{-1}{(1+n)(1+2n)}Y^{1/n}|S-h_x|^{1/n-1}\times \\ &{} \left( 2nh\frac{B}{|S-h_x|}+(1+n)h^2+ 2n^2\frac{ B^2}{(S-h_x)^2}\right) . \end{matrix}} \end{aligned}$$

(32)

At each time iteration, as mentioned in the main text, the time step is determined as:

$$\begin{aligned} \Delta t = \min \left( \frac{\Delta x}{2\max _x(V(x))}, C_d\frac{(\Delta x)^2}{\max _x(D(x))}\right) \end{aligned}$$

(33)

where (the classical) $C_d = 0.5$ ensures stability for most simulations. One can lower $C_d$, e.g., $C_d = 0.05$ to stabilize the simulation if needed. Note that for the current paper, where the solutions are not computed after the time of wall-touch, $C_d = 0.5$ leads to stable simulations for all studied parameters (B, S).

Appendix B. Noisy synthetic profiles

The supplementary figure, mentioned in the main text, concerning the study of noisy profiles with 10% noise.

Fig. 7

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 10% of noise. The three colors correspond to three different (B, S) used in Fig. 4