4,299 research outputs found
Expanded mixed multiscale finite element methods and their applications for flows in porous media
We develop a family of expanded mixed Multiscale Finite Element Methods
(MsFEMs) and their hybridizations for second-order elliptic equations. This
formulation expands the standard mixed Multiscale Finite Element formulation in
the sense that four unknowns (hybrid formulation) are solved simultaneously:
pressure, gradient of pressure, velocity and Lagrange multipliers. We use
multiscale basis functions for the both velocity and gradient of pressure. In
the expanded mixed MsFEM framework, we consider both cases of separable-scale
and non-separable spatial scales. We specifically analyze the methods in three
categories: periodic separable scales, - convergence separable scales, and
continuum scales. When there is no scale separation, using some global
information can improve accuracy for the expanded mixed MsFEMs. We present
rigorous convergence analysis for expanded mixed MsFEMs. The analysis includes
both conforming and nonconforming expanded mixed MsFEM. Numerical results are
presented for various multiscale models and flows in porous media with shales
to illustrate the efficiency of the expanded mixed MsFEMs.Comment: 33 page
On pore-scale modeling and simulation of reactive transport in 3D geometries
Pore-scale modeling and simulation of reactive flow in porous media has a
range of diverse applications, and poses a number of research challenges. It is
known that the morphology of a porous medium has significant influence on the
local flow rate, which can have a substantial impact on the rate of chemical
reactions. While there are a large number of papers and software tools
dedicated to simulating either fluid flow in 3D computerized tomography (CT)
images or reactive flow using pore-network models, little attention to date has
been focused on the pore-scale simulation of sorptive transport in 3D CT
images, which is the specific focus of this paper. Here we first present an
algorithm for the simulation of such reactive flows directly on images, which
is implemented in a sophisticated software package. We then use this software
to present numerical results in two resolved geometries, illustrating the
importance of pore-scale simulation and the flexibility of our software
package.Comment: 15 pages, 6 figure
A machine learning approach for efficient uncertainty quantification using multiscale methods
Several multiscale methods account for sub-grid scale features using coarse
scale basis functions. For example, in the Multiscale Finite Volume method the
coarse scale basis functions are obtained by solving a set of local problems
over dual-grid cells. We introduce a data-driven approach for the estimation of
these coarse scale basis functions. Specifically, we employ a neural network
predictor fitted using a set of solution samples from which it learns to
generate subsequent basis functions at a lower computational cost than solving
the local problems. The computational advantage of this approach is realized
for uncertainty quantification tasks where a large number of realizations has
to be evaluated. We attribute the ability to learn these basis functions to the
modularity of the local problems and the redundancy of the permeability patches
between samples. The proposed method is evaluated on elliptic problems yielding
very promising results.Comment: Journal of Computational Physics (2017
A multiscale flux basis for mortar mixed discretizations of reduced Darcy-Forchheimer fracture models
In this paper, a multiscale flux basis algorithm is developed to efficiently
solve a flow problem in fractured porous media. Here, we take into account a
mixed-dimensional setting of the discrete fracture matrix model, where the
fracture network is represented as lower-dimensional object. We assume the
linear Darcy model in the rock matrix and the non-linear Forchheimer model in
the fractures. In our formulation, we are able to reformulate the
matrix-fracture problem to only the fracture network problem and, therefore,
significantly reduce the computational cost. The resulting problem is then a
non-linear interface problem that can be solved using a fixed-point or
Newton-Krylov methods, which in each iteration require several solves of Robin
problems in the surrounding rock matrices. To achieve this, the flux exchange
(a linear Robin-to-Neumann co-dimensional mapping) between the porous medium
and the fracture network is done offline by pre-computing a multiscale flux
basis that consists of the flux response from each degree of freedom on the
fracture network. This delivers a conserve for the basis that handles the
solutions in the rock matrices for each degree of freedom in the fractures
pressure space. Then, any Robin sub-domain problems are replaced by linear
combinations of the multiscale flux basis during the interface iteration. The
proposed approach is, thus, agnostic to the physical model in the fracture
network. Numerical experiments demonstrate the computational gains of
pre-computing the flux exchange between the porous medium and the fracture
network against standard non-linear domain decomposition approaches
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