Abstract: With rapid advancement in exploration and production of shale gas reservoirs over the world, the fast simulation of shale gas flows that is required especially in engineering applications has attracted extensive attentions from the engineering and academic communities. In this study, we apply a popular global model reduction method, proper orthogonal decomposition (POD), to speed up the simulation of shale gas reservoirs. However, different from incompressible fluid flows, the compressibility of shale gas induces additional challenges to construct accurate and efficient POD reduced-order model (ROM). First, the compressibility of shale gas increases the nonlinearity of the flow system, the dimension of the projected Darcy-type pressure equation in low-dimensional space still depends on the dimension of the original system, which complicates the computation and worsens the acceleration of the POD-ROM substantially. Second, due to the POD projection term containing the compressibility of shale gas, the additional computational cost is needed to solve the equation of state (EOS) of shale gas to obtain the compressibility factor. To handle these problems, we adopt another model reduction approach, discrete empirical interpolation method (DEIM), to approximate the nonlinearity in pressure equation by only using few selected representative interpolation points over the domain, thus the nonlinearity of the variables and the computation of EOS can be greatly reduced. Combined the POD-ROM with DEIM, a POD-DEIM-ROM for fast simulation of shale gas flows with different gas states in single-continuum porous media is developed. The performances of the proposed model are validated by comparing to the traditional method without acceleration techniques through different numerical cases, the computational speed and numerical accuracy are analyzed in detail. Results show that the proposed model can achieve great acceleration of the simulation (two orders of magnitude) without sacrificing the numerical accuracy obviously. Especially, the influence of the type of EOS (such as ideal gas EOS, Van der Waals EOS, and Peng–Robinson EOS, etc.), the number of POD modes and interpolation points, as well as the permeability filed distribution on the overall performance of the proposed model are investigated.

Abstract: A direct numerical simulation (DNS) code was developed for solving incompressible homogeneous isotropic turbulence with high Reynolds numbers in a periodic box using the Fourier spectral method. The code was parallelized using the Message Passing Interface and OpenMP with two-directional domain decomposition and optimized on the K computer. High resolution DNSs with up to $12288^3$ grid points were performed on the K computer using the code. Efficiencies of 3.84\%, 3.14\%, and 2.24\% peak performance were obtained in double precision DNSs with $6144^3$, $8192^3$, and $12288^3$ grid points, respectively. In addition, a two-path alias-free procedure is proposed and clarified its effectiveness for some number of parallel processes.

Abstract: This work presents Direct Numerical Simulation of mass transfer from buoyancy-driven bubbles rising in a wall-confined vertical channel, by means of a multiple marker level-set method. The Navier-Stokes equations and mass transfer equations are discretized using a finite-volume method on a collocated unstructured mesh, whereas a multiple marker approach is used to avoid the numerical coalescence of bubbles. This approach is implemented in the framework of a mass conservative level-set method, whereas unstructured flux-limiter schemes are used to discretize the convective term of momentum equation, level-set equations, and mass transfer equation, in order to improve the stability of the solver in bubbly flows with high Reynolds number and high-density ratio. The capabilities of this model are proved in the buoyancy-driven motion of single bubbles and bubble swarms in a vertical channel of circular cross-section.

Abstract: A hybrid particle/mesh Vortex Method, called remeshed vortex method, is proposed in this work to simulate three-dimensional incompressible flows. After a validation study of the present method in the context of Direct Numerical Simulations, an anisotropic artificial viscosity model is proposed in this paper in order to handle bi-level simulations. The bi-level approach implies two different mesh sizes for the discretization of the two coupled variables of the problem, namely the vorticity field and the velocity field: the vorticity field is computed on a fine grid while the velocity field is solved on a coarse grid. This approch is proposed with the objective of performing efficient computations based on hybrid GPU-CPU architectures.