Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userInterface spaces based on physics for multiscale mixed methods applied to flows in fractured-like porous media
69
0
0.0
(
0
)
Added by
Franciane Rocha PhD
Publication date
2021
fields
Informatics Engineering
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
It is well known that domain-decomposition-based multiscale mixed methods rely on interface spaces, defined on the skeleton of the decomposition, to connect the solution among the non-overlapping subdomains. Usual spaces, such as polynomial-based ones, cannot properly represent high-contrast channelized features such as fractures (high permeability) and barriers (low permeability) for flows in heterogeneous porous media. We propose here new interface spaces, which are based on physics, to deal with permeability fields in the simultaneous presence of fractures and barriers, accommodated respectively, by the pressure and flux spaces. Existing multiscale methods based on mixed formulations can take advantage of the proposed interface spaces, however, in order to present and test our results, we use the newly developed Multiscale Robin Coupled Method (MRCM) [Guiraldello, et al., J. Comput. Phys., 355 (2018) pp. 1-21], which generalizes most well-known multiscale mixed methods, and allows for the independent choice of the pressure and flux interface spaces. An adaptive version of the MRCM [Rocha, et al., J. Comput. Phys., 409 (2020), 109316] is considered that automatically selects the physics-based pressure space for fractured structures and the physics-based flux space for regions with barriers, resulting in a procedure with unprecedented accuracy. The features of the proposed approach are investigated through several numerical simulations of single-phase and two-phase flows, in different heterogeneous porous media. The adaptive MRCM combined with the interface spaces based on physics provides promising results for challenging problems with the simultaneous presence of fractures and barriers.
rate research
Read More
In this work, we consider a mathematical model for flow in a unsaturated porous medium containing a fracture. In all subdomains (the fracture and the adjacent matrix blocks) the flow is governed by Richards equation. The submodels are coupled by physical transmission conditions expressing the continuity of the normal fluxes and of the pressures. We start by analyzing the case of a fracture having a fixed width-length ratio, called $varepsilon > 0$. Then we take the limit $varepsilon to 0$ and give a rigorous proof for the convergence towards effective models. This is done in different regimes, depending on how the ratio of porosities and permeabilities in the fracture, respectively matrix scale with respect to $varepsilon$, and leads to a variety of effective models. Numerical simulations confirm the theoretical upscaling results.
In this paper, we systemically review and compare two mixed multiscale finite element methods (MMsFEM) for multiphase transport in highly heterogeneous media. In particular, we will consider the mixed multiscale finite element method using limited global information, simply denoted by MMsFEM, and the mixed generalized multiscale finite element method (MGMsFEM) with residual driven online multiscale basis functions. Both methods are under the framework of mixed multiscale finite element methods, where the pressure equation is solved in the coarse grid with carefully constructed multiscale basis functions for the velocity. The multiscale basis functions in both methods include local and global media information. In terms of MsFEM using limited global information, only one multiscale basis function is utilized in each local neighborhood while multiple basis are used in MGMsFEM. We will test and compare these two methods using the benchmark three-dimensional SPE10 model. A range of coarse grid sizes and different combinations of basis functions (offline and online) will be considered with CPU time reported for each case. In our numerical experiments, we observe good accuracy by the two above methods. Finally, we will discuss and compare the advantages and disadvantages of the two methods in terms of accuracy and computational costs.
In porous media, there are three known regimes of fluid flows, namely, pre-Darcy, Darcy and post-Darcy. Because of their different natures, these are usually treated separately in literature. To study complex flows when all three regimes may be present in different portions of a same domain, we use a single equation of motion to unify them. Several scenarios and models are then considered for slightly compressible fluids. A nonlinear parabolic equation for the pressure is derived, which is degenerate when the pressure gradient is either small or large. We estimate the pressure and its gradient for all time in terms of initial and boundary data. We also obtain their particular bounds for large time which depend on the asymptotic behavior of the boundary data but not on the initial one. Moreover, the continuous dependence of the solutions on initial and boundary data, and the structural stability for the equation are established.
In this work, we propose and develop efficient and accurate numerical methods for solving the Kirchhoff-Love plate model in domains with complex geometries. The algorithms proposed here employ curvilinear finite-difference methods for spatial discretization of the governing PDEs on general composite overlapping grids. The coupling of different components of the composite overlapping grid is through numerical interpolations. However, interpolations introduce perturbation to the finite-difference discretization, which causes numerical instability for time-stepping schemes used to advance the resulted semi-discrete system. To address the instability, we propose to add a fourth-order hyper-dissipation to the spatially discretized system to stabilize its time integration; this additional dissipation term captures the essential upwinding effect of the original upwind scheme. The investigation of strategies for incorporating the upwind dissipation term into several time-stepping schemes (both explicit and implicit) leads to the development of four novel algorithms. For each algorithm, formulas for determining a stable time step and a sufficient dissipation coefficient on curvilinear grids are derived by performing a local Fourier analysis. Quadratic eigenvalue problems for a simplified model plate in 1D domain are considered to reveal the weak instability due to the presence of interpolating equations in the spatial discretization. This model problem is further investigated for the stabilization effects of the proposed algorithms. Carefully designed numerical experiments are carried out to validate the accuracy and stability of the proposed algorithms, followed by two benchmark problems to demonstrate the capability and efficiency of our approach for solving realistic applications. Results that concern the performance of the proposed algorithms are also presented.
Accurate numerical simulations of interaction between fluid and solid play an important role in applications. The task is challenging in practical scenarios as the media are usually highly heterogeneous with very large contrast. To overcome this computational challenge, various multiscale methods are developed. In this paper, we consider a class of linear poroelasticity problems in high contrast heterogeneous porous media, and develop a mixed generalized multiscale finite element method (GMsFEM) to obtain a fast computational method. Our aim is to develop a multiscale method that is robust with respect to the heterogeneities and contrast of the media, and gives a mass conservative fluid velocity field. We will construct decoupled multiscale basis functions for the elastic displacement as well as fluid velocity. Our multiscale basis functions are local. The construction is based on some suitable choices of local snapshot spaces and local spectral decomposition, with the goal of extracting dominant modes of the solutions. For the pressure, we will use piecewise constant approximation. We will present several numerical examples to illustrate the performance of our method. Our results indicate that the proposed method is able to give accurate numerical solutions with a small degree of freedoms.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
