863 research outputs found
An HMM--ELLAM scheme on generic polygonal meshes for miscible incompressible flows in porous media
We design a numerical approximation of a system of partial differential
equations modelling the miscible displacement of a fluid by another in a porous
medium. The advective part of the system is discretised using a characteristic
method, and the diffusive parts by a finite volume method. The scheme is
applicable on generic (possibly non-conforming) meshes as encountered in
applications. The main features of our work are the reconstruction of a Darcy
velocity, from the discrete pressure fluxes, that enjoys a local consistency
property, an analysis of implementation issues faced when tracking, via the
characteristic method, distorted cells, and a new treatment of cells near the
injection well that accounts better for the conservativity of the injected
fluid
Stable Crank-Nicolson Discretisation for Incompressible Miscible Displacement Problems of Low Regularity
In this article we study the numerical approximation of incompressible
miscible displacement problems with a linearised Crank-Nicolson time
discretisation, combined with a mixed finite element and discontinuous Galerkin
method. At the heart of the analysis is the proof of convergence under low
regularity requirements. Numerical experiments demonstrate that the proposed
method exhibits second-order convergence for smooth and robustness for rough
problems.Comment: Enumath 200
Unified convergence analysis of numerical schemes for a miscible displacement problem
This article performs a unified convergence analysis of a variety of
numerical methods for a model of the miscible displacement of one
incompressible fluid by another through a porous medium. The unified analysis
is enabled through the framework of the gradient discretisation method for
diffusion operators on generic grids. We use it to establish a novel
convergence result in of the approximate
concentration using minimal regularity assumptions on the solution to the
continuous problem. The convection term in the concentration equation is
discretised using a centred scheme. We present a variety of numerical tests
from the literature, as well as a novel analytical test case. The performance
of two schemes are compared on these tests; both are poor in the case of
variable viscosity, small diffusion and medium to small time steps. We show
that upstreaming is not a good option to recover stable and accurate solutions,
and we propose a correction to recover stable and accurate schemes for all time
steps and all ranges of diffusion
Central Schemes for Porous Media Flows
We are concerned with central differencing schemes for solving scalar
hyperbolic conservation laws arising in the simulation of multiphase flows in
heterogeneous porous media. We compare the Kurganov-Tadmor, 2000 semi-discrete
central scheme with the Nessyahu-Tadmor, 1990 central scheme. The KT scheme
uses more precise information about the local speeds of propagation together
with integration over nonuniform control volumes, which contain the Riemann
fans. These methods can accurately resolve sharp fronts in the fluid
saturations without introducing spurious oscillations or excessive numerical
diffusion. We first discuss the coupling of these methods with velocity fields
approximated by mixed finite elements. Then, numerical simulations are
presented for two-phase, two-dimensional flow problems in multi-scale
heterogeneous petroleum reservoirs. We find the KT scheme to be considerably
less diffusive, particularly in the presence of high permeability flow
channels, which lead to strong restrictions on the time step selection;
however, the KT scheme may produce incorrect boundary behavior
- …