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Register a new userNumerical simulation of transport in porous media: some problems from micro to macro scale
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Quanji Cai
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This paper deals with simulation of flow and transport in porous media such as transport of groundwater contaminants. We first discuss how macro scale equations are derived and which terms have to be closed by models. The transport of tracers is strongly influenced by pore scale velocity structure and large scale inhomogeneities in the permeability field. The velocity structure on the pore scale is investigated by direct numerical simulations of the 3D velocity field in a random sphere pack. The velocity probability density functions are strongly skewed, including some negative velocities. The large probability for very small velocities might be the reason for non-Fickian dispersion in the initial phase of contaminant transport. We present a method to determine large scale distributions of the permeability field from point-wise velocity measurements. The adjoint-based optimisation algorithm delivers fully satisfying agreement between input and estimated permeability fields. Finally numerical methods for convection dominated tracer transports are investigated from a theoretical point of view. It is shown that high order Finite Element Methods can reduce or even eliminate non-physical oscillations in the solution without introducing additional numerical diffusivity.
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In this paper, we first extend the micro-macro decomposition method for multiscale kinetic equations from the BGK model to general collisional kinetic equations, including the Boltzmann and the Fokker-Planck Landau equations. The main idea is to use a relation between the (numerically stiff) linearized collision operator with the nonlinear quadratic ones, the laters stiffness can be overcome using the BGK penalization method of Filbet and Jin for the Boltzmann, or the linear Fokker-Planck penalization method of Jin and Yan for the Fokker-Planck Landau equations. Such a scheme allows the computation of multiscale collisional kinetic equations efficiently in all regimes, including the fluid regime in which the fluid dynamic behavior can be correctly computed even without resolving the small Knudsen number. A distinguished feature of these schemes is that although they contain implicit terms, they can be implemented explicitly. These schemes preserve the moments (mass, momentum and energy) exactly thanks to the use of the macroscopic system which is naturally in a conservative form. We further utilize this conservation property for more general kinetic systems, using the Vlasov-Amp{e}re and Vlasov-Amp{e}re-Boltzmann systems as examples. The main idea is to evolve both the kinetic equation for the probability density distribution and the moment system, the later naturally induces a scheme that conserves exactly the moments numerically if they are physically conserved.
Huge data advent in high-performance computing (HPC) applications such as fluid flow simulations usually hinders the interactive processing and exploration of simulation results. Such an interactive data exploration not only allows scientiest to play with their data but also to visualise huge (distributed) data sets in both an efficient and easy way. Therefore, we propose an HPC data exploration service based on a sliding window concept, that enables researches to access remote data (available on a supercomputer or cluster) during simulation runtime without exceeding any bandwidth limitations between the HPC back-end and the user front-end.
In this work we consider the transport of a surfactant in a variably saturated porous media. The water flow is modelled by the Richards equations and it is fully coupled with the transport equation for the surfactant. Three linearization techniques are discussed: the Newton method, the modified Picard and the L-scheme. Based on these, monolithic and splitting schemes are proposed and their convergence is analyzed. The performance of these schemes is illustrated on four numerical examples. For these examples, the number of iterations and the condition numbers of the linear systems emerging in each iteration are 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.
Global spectral analysis (GSA) is used as a tool to test the accuracy of numerical methods with the help of canonical problems of convection and convection-diffusion equation which admit exact solutions. Similarly, events in turbulent flows computed by direct numerical simulation (DNS) are often calibrated with theoretical results of homogeneous isotropic turbulence due to Kolmogorov, as given in Turbulence -U. Frisch, Cambridge Univ. Press, UK (1995). However, numerical methods for the simulation of this problem are not calibrated, as by using GSA of convection and/or convection-diffusion equation. This is with the exception in A critical assessment of simulations for transitional and turbulence flows-Sengupta, T.K., In Proc. of IUTAM Symp. on Advances in Computation, Modeling and Control of Transitional and Turbulent Flows, pp 491-532, World Sci. Publ. Co. Pte. Ltd., Singapore (2016), where such a calibration has been advocated with the help of convection equation. For turbulent flows, an extreme event is characterized by the presence of length scales smaller than the Kolmogorov length scale, a heuristic limit for the largest wavenumber present without being converted to heat. With growing computer power, recently many simulations have been reported using a pseudo-spectral method, with spatial discretization performed in Fourier spectral space and a two-stage, Runge-Kutta (RK2) method for time discretization. But no analyses are reported to ensure high accuracy of such simulations. Here, an analysis is reported for few multi-stage Runge-Kutta methods in the Fourier spectral framework for convection and convection-diffusion equations. We identify the major source of error for the RK2-Fourier spectral method using GSA and also show how to avoid this error and specify numerical parameters for achieving highest accuracy possible to capture extreme events in turbulent flows.
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