495 research outputs found
A priori error estimates for the numerical solution of a coupled geomechanics and reservoir flow model with stress-dependent permeability
In this paper we consider the numerical solution of a coupled geomechanics
and a stress-sensitive porous media reservoir flow model.We combine mixed
finite elements for Darcy flow and Galerkin finite elements for elasticity. This work
focuses on deriving convergence results for the numerical solution of this nonlinear
partial differential system. We establish convergence with respect to the L2-norm
for the pressure and for the average fluid velocity and with respect to the H1-norm
for the deformation. Estimates respect to the L2-norm for mean stress, which is
of special importance since it is used in the computation of permeability for poroelasticity,
can be derived using the estimates in the H1-norm for the deformation.
We start by deriving error estimates in a continuous-in-time setting. A cut-off operator
is introduced in the numerical scheme in order to derive convergence. The
spatial grids for the discrete approximations of the pressure and deformation do
not need be the same. Theoretical convergence error estimates in a discrete-in-time
setting are also derived in the scope of this investigation. A numerical example
supports the convergence results
Postprocessing of Non-Conservative Flux for Compatibility with Transport in Heterogeneous Media
A conservative flux postprocessing algorithm is presented for both
steady-state and dynamic flow models. The postprocessed flux is shown to have
the same convergence order as the original flux. An arbitrary flux
approximation is projected into a conservative subspace by adding a piecewise
constant correction that is minimized in a weighted norm. The application
of a weighted norm appears to yield better results for heterogeneous media than
the standard norm which has been considered in earlier works. We also
study the effect of different flux calculations on the domain boundary. In
particular we consider the continuous Galerkin finite element method for
solving Darcy flow and couple it with a discontinuous Galerkin finite element
method for an advective transport problem.Comment: 34 pages, 17 figures, 11 table
Nonlinear nonlocal multicontinua upscaling framework and its applications
In this paper, we discuss multiscale methods for nonlinear problems. The main
idea of these approaches is to use local constraints and solve problems in
oversampled regions for constructing macroscopic equations. These techniques
are intended for problems without scale separation and high contrast, which
often occur in applications. For linear problems, the local solutions with
constraints are used as basis functions. This technique is called Constraint
Energy Minimizing Generalized Multiscale Finite Element Method (CEM-GMsFEM).
GMsFEM identifies macroscopic quantities based on rigorous analysis. In
corresponding upscaling methods, the multiscale basis functions are selected
such that the degrees of freedom have physical meanings, such as averages of
the solution on each continuum.
This paper extends the linear concepts to nonlinear problems, where the local
problems are nonlinear. The main concept consists of: (1) identifying
macroscopic quantities; (2) constructing appropriate oversampled local problems
with coarse-grid constraints; (3) formulating macroscopic equations. We
consider two types of approaches. In the first approach, the solutions of local
problems are used as basis functions (in a linear fashion) to solve nonlinear
problems. This approach is simple to implement; however, it lacks the nonlinear
interpolation, which we present in our second approach. In this approach, the
local solutions are used as a nonlinear forward map from local averages
(constraints) of the solution in oversampling region. This local fine-grid
solution is further used to formulate the coarse-grid problem. Both approaches
are discussed on several examples and applied to single-phase and two-phase
flow problems, which are challenging because of convection-dominated nature of
the concentration equation
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