1,100 research outputs found
An analysis of polynomial chaos approximations for modeling single-fluid-phase flow in porous medium systems
We examine a variety of polynomial-chaos-motivated approximations to a stochastic form of a steady state groundwater flow model. We consider approaches for truncating the infinite dimensional problem and producing decoupled systems. We discuss conditions under which such decoupling is possible and show that to generalize the known decoupling by numerical cubature, it would be necessary to find new multivariate cubature rules. Finally, we use the acceleration of Monte Carlo to compare the quality of polynomial models obtained for all approaches and find that in general the methods considered are more efficient than Monte Carlo for the relatively small domains considered in this work. A curse of dimensionality in the series expansion of the log-normal stochastic random field used to represent hydraulic conductivity provides a significant impediment to efficient approximations for large domains for all methods considered in this work, other than the Monte Carlo method
Uncertainty Quantification of geochemical and mechanical compaction in layered sedimentary basins
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, non-linear,
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).Comment: 25 pages, 30 figure
Bifurcations and dynamics emergent from lattice and continuum models of bioactive porous media
We study dynamics emergent from a two-dimensional reaction--diffusion process
modelled via a finite lattice dynamical system, as well as an analogous PDE
system, involving spatially nonlocal interactions. These models govern the
evolution of cells in a bioactive porous medium, with evolution of the local
cell density depending on a coupled quasi--static fluid flow problem. We
demonstrate differences emergent from the choice of a discrete lattice or a
continuum for the spatial domain of such a process. We find long--time
oscillations and steady states in cell density in both lattice and continuum
models, but that the continuum model only exhibits solutions with vertical
symmetry, independent of initial data, whereas the finite lattice admits
asymmetric oscillations and steady states arising from symmetry-breaking
bifurcations. We conjecture that it is the structure of the finite lattice
which allows for more complicated asymmetric dynamics. Our analysis suggests
that the origin of both types of oscillations is a nonlocal reaction-diffusion
mechanism mediated by quasi-static fluid flow.Comment: 30 pages, 21 figure
Uncertainty-aware Validation Benchmarks for Coupling Free Flow and Porous-Medium Flow
A correct choice of interface conditions and useful model parameters for
coupled free-flow and porous-medium systems is vital for physically consistent
modeling and accurate numerical simulations of applications. We consider the
Stokes--Darcy problem with different models for the porous-medium compartment
and corresponding coupling strategies: the standard averaged model based on
Darcy's law with classical or generalized interface conditions, as well as the
pore-network model. We study the coupled flow problems' behaviors considering a
benchmark case where a pore-scale resolved model provides the reference
solution and quantify the uncertainties in the models' parameters and the
reference data. To achieve this, we apply a statistical framework that
incorporates a probabilistic modeling technique using a fully Bayesian
approach. A Bayesian perspective on a validation task yields an optimal
bias-variance trade-off against the reference data. It provides an integrative
metric for model validation that incorporates parameter and conceptual
uncertainty. Additionally, a model reduction technique, namely Bayesian Sparse
Polynomial Chaos Expansion, is employed to accelerate the calibration and
validation processes for computationally demanding Stokes--Darcy models with
different coupling strategies. We perform uncertainty-aware validation,
demonstrate each model's predictive capabilities, and make a model comparison
using a Bayesian validation metric
Prediction of the intramembranous tissue formation during perisprosthetic healing with uncertainties. Part 1. Effect of the variability of each biochemical factor
A stochastic model is proposed to predict the intramembranous process in periprosthetic healing in the early post-operative period. The methodology was validated by a canine experimental model. In this first part, the effects of each individual uncertain biochemical factor on the bone-implant healing are examined, including the coefficient of osteoid synthesis, the coefficients of haptotactic and chemotactic migration of osteoblastic population and the radius of the drill hole. A multi-phase reactive model solved by an explicit finite difference scheme is combined with the polynomial chaos expansion to solve the stochastic system. In the second part, combined biochemical factors are considered to study a real configuration of clinical acts
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