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Register a new userPostprocessing of Non-Conservative Flux for Compatibility with Transport in Heterogeneous Media
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Authors
Lars H. Ods{ae}ter
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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 $L^2$ norm. The application of a weighted norm appears to yield better results for heterogeneous media than the standard $L^2$ 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.
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This work settles the problem of constructing entropy stable non-oscillatory (ESNO) fluxes by framing it as a least square optimization problem. A flux sign stability condition is introduced and utilized to construct arbitrary order entropy stable flux as a convex combination of entropy conservative and non-oscillatory flux. This simple approach is robust which does not explicitly requires the computation of costly dissipation operator and high order reconstruction of scaled entropy variable for constructing the diffusion term. The numerical diffusion is optimized in the sense that entropy stable flux reduces to the underlying non-oscillatory flux. Different non-oscillatory entropy stable fluxes are constructed and used to compute the numerical solution of various standard scalar and systems test problems. Computational results show that entropy stable schemes are comparable in term of non-oscillatory nature of schemes using only the underlying non-oscillatory fluxes. Moreover, these entropy stable schemes maintains the formal order of accuracy of the lower order flux used in the convex combination.
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.
Flow and multicomponent reactive transport in saturated/unsaturated porous media are modeled by ensembles of computational particles moving on regular lattices according to specific random walk rules. The occupation number of the lattice sites is updated with a global random walk (GRW) procedure which simulates the evolution of the ensemble with computational costs comparable to those for a single random walk simulation in sequential procedures. To cope with the nonlinearity and the degeneracy of the Richards equation the GRW flow solver uses linearization techniques similar to the $L$-scheme developed in finite element/volume approaches. Numerical schemes for reactive transport, coupled with the flow solver via numerical solutions for saturation and water flux, are implemented in splitting procedures. Diffusion-advection steps are solved by GRW algorithms using either biased or unbiased random walk probabilities. Since the number of particles in GRW simulations can be as large as the number of molecules involved in chemical reactions, one avoids the cumbersome problem of rescaling particle densities to approximate concentrations. Reaction steps are therefore formulated in terms of concentrations, as in deterministic approaches. The numerical convergence of the new schemes is demonstrated by comparisons with manufactured analytical solutions. Coupled flow and reactive transport problems of contaminant biodegradation described by the Monod model are further solved and the influence of flow nonlinearity/degeneracy and of the spatial heterogeneity of the medium is investigated numerically.
A high-order quasi-conservative discontinuous Galerkin (DG) method is proposed for the numerical simulation of compressible multi-component flows. A distinct feature of the method is a predictor-corrector strategy to define the grid velocity. A Lagrangian mesh is first computed based on the flow velocity and then used as an initial mesh in a moving mesh method (the moving mesh partial differential equation or MMPDE method ) to improve its quality. The fluid dynamic equations are discretized in the direct arbitrary Lagrangian-Eulerian framework using DG elements and the non-oscillatory kinetic flux while the species equation is discretized using a quasi-conservative DG scheme to avoid numerical oscillations near material interfaces. A selection of one- and two-dimensional examples are presented to verify the convergence order and the constant-pressure-velocity preservation property of the method. They also demonstrate that the incorporation of the Lagrangian meshing with the MMPDE moving mesh method works well to concentrate mesh points in regions of shocks and material interfaces.
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.
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