A multi-level parallel solver for rarefied gas flows in porous media

A high-performance gas kinetic solver using multi-level parallelization is developed to enable pore-scale simulations of rarefied flows in porous media. The Bhatnagar–Gross–Krook model equation is solved by the discrete velocity method with an iterative scheme. The multi-level MPI/OpenMP parallelization is implemented with the aim to efficiently utilize the computational resources to allow direct simulation of rarefied gas flows in porous media based on digital rock images for the first time. The multi-level parallel approach is analyzed in detail confirming its better performance than the commonly-used MPI processing alone for an iterative scheme. With high communication efficiency and appropriate load balancing among CPU processes, parallel efficiency of 94% is achieved for 1536 cores in the 2D simulations, and 81% for 12288 cores in the 3D simulations. While decomposition in the spatial space does not affect the simulation results, one additional benefit of this approach is that the number of subdomains can be kept minimal to avoid deterioration of the convergence rate of the iteration process. This multi-level parallel approach can be readily extended to solve other Boltzmann model equations

Computational study of gas transport in shale at pore-scale and beyond

Unconventional gas resources like shale are poised to enter a golden age thanks to the worldwide increased exploitation potential. Yet, the future of these resources is far from assured since it is still subject to many uncertainties, mainly with regard to gas recoverability. The macroscopic flow properties that allow the prediction of production are directly linked to the microscopic flow where underlying rarefaction effects play a major role. This is due to the pore size which is as small as a few nanometers and comparable to the mean free path. Thus, an in-depth understanding of gas transport in ultra-tight porous media is crucial for the accurate determination of flow properties in shale rocks. This thesis is a fundamental research aiming to aid and benefit shale gas exploration and development. The objective of this work is to provide useful insights for such non-equilibrium porous media flows, where the conventional fluid mechanics theory fails. Even though there are multiple heuristic permeability models in the literature, I find them unsuitable to provide reliable apparent permeability estimates since they often include simplifications of the flow mechanisms and matrix complexity. Notably, I hereby establish/prove the limitations of the accuracy of the Navier-Stokes equations to the first order of Knudsen number. We also thoroughly analyse Klinkenberg's slip factor behaviour for a wide range of gas rarefaction, utilising gas kinetic theory, for both simple and complex porous media. Moreover, using controllably random porous media, I systematically quantify the impact of numerous structural characteristics, i.e. porosity, tortuosity, specific surface area, heterogeneity and degree of anisotropy, on both intrinsic and apparent permeability.;One of the key contributions of this work is a new semi-analytical permeability formulation derived using the produced simulation results. This expression, suitable for both isotropic and anisotropic two-dimensional porous media, accounts for the aforementioned properties as well as for continuum and slip flow. The main advantage of the proposed formulation is the fact that it does not entail any experimental or numerical data as input, unlike other established models. Shale is intrinsically multiscale, thus the direct simulation of transport in all scales is not feasible. Upscaling from the pore-scale is indispensable in order to eventually obtain the essential macroscopic properties in the field-scale. For this reason, I examine well-known analytical and numerical upscaling techniques, verifying the sensitivity and accuracy of the latter ones. Studying microscale sample images, we need to consider the appearance of microfractures. The difference of the characteristic length scales between the nanopores and the microfractures requires a hybrid upscaling method such as the Brinkman approach.;The suitability of this model is extensively validated on fractured porous media of interest, especially on the grounds that the exact form of the effective viscosity is still a matter of discussion. We perform this validation comparing numerous direct simulation results with the corresponding ones from the Brinkman solution. Different values of the effective viscosity are investigated, along with a variable permeability model applied at the vicinity of the fluid-porous interface. Due to lack of an appropriate universal treatment of the transition zone of random porous media, we consider effective viscosity equal to fluid viscosity. The accuracy of the Brinkman approach is further examined using several two and three-dimensional random porous media containing fractures, as well as considering rarefied conditions. Although I find that heterogeneity and anisotropy increase the error of the effective permeability derived from the Brinkman approach, generally, the effective permeability extracted from this coarse-scale model compares favourably to its fine-scale counterpart obtained from the Stokes and Boltzmann model equations for porous media flows. Finally, I conclude that neglecting the rarefaction effects leads to a significant underestimation of the effective permeability of fractured ultra-tight porous media.Unconventional gas resources like shale are poised to enter a golden age thanks to the worldwide increased exploitation potential. Yet, the future of these resources is far from assured since it is still subject to many uncertainties, mainly with regard to gas recoverability. The macroscopic flow properties that allow the prediction of production are directly linked to the microscopic flow where underlying rarefaction effects play a major role. This is due to the pore size which is as small as a few nanometers and comparable to the mean free path. Thus, an in-depth understanding of gas transport in ultra-tight porous media is crucial for the accurate determination of flow properties in shale rocks. This thesis is a fundamental research aiming to aid and benefit shale gas exploration and development. The objective of this work is to provide useful insights for such non-equilibrium porous media flows, where the conventional fluid mechanics theory fails. Even though there are multiple heuristic permeability models in the literature, I find them unsuitable to provide reliable apparent permeability estimates since they often include simplifications of the flow mechanisms and matrix complexity. Notably, I hereby establish/prove the limitations of the accuracy of the Navier-Stokes equations to the first order of Knudsen number. We also thoroughly analyse Klinkenberg's slip factor behaviour for a wide range of gas rarefaction, utilising gas kinetic theory, for both simple and complex porous media. Moreover, using controllably random porous media, I systematically quantify the impact of numerous structural characteristics, i.e. porosity, tortuosity, specific surface area, heterogeneity and degree of anisotropy, on both intrinsic and apparent permeability.;One of the key contributions of this work is a new semi-analytical permeability formulation derived using the produced simulation results. This expression, suitable for both isotropic and anisotropic two-dimensional porous media, accounts for the aforementioned properties as well as for continuum and slip flow. The main advantage of the proposed formulation is the fact that it does not entail any experimental or numerical data as input, unlike other established models. Shale is intrinsically multiscale, thus the direct simulation of transport in all scales is not feasible. Upscaling from the pore-scale is indispensable in order to eventually obtain the essential macroscopic properties in the field-scale. For this reason, I examine well-known analytical and numerical upscaling techniques, verifying the sensitivity and accuracy of the latter ones. Studying microscale sample images, we need to consider the appearance of microfractures. The difference of the characteristic length scales between the nanopores and the microfractures requires a hybrid upscaling method such as the Brinkman approach.;The suitability of this model is extensively validated on fractured porous media of interest, especially on the grounds that the exact form of the effective viscosity is still a matter of discussion. We perform this validation comparing numerous direct simulation results with the corresponding ones from the Brinkman solution. Different values of the effective viscosity are investigated, along with a variable permeability model applied at the vicinity of the fluid-porous interface. Due to lack of an appropriate universal treatment of the transition zone of random porous media, we consider effective viscosity equal to fluid viscosity. The accuracy of the Brinkman approach is further examined using several two and three-dimensional random porous media containing fractures, as well as considering rarefied conditions. Although I find that heterogeneity and anisotropy increase the error of the effective permeability derived from the Brinkman approach, generally, the effective permeability extracted from this coarse-scale model compares favourably to its fine-scale counterpart obtained from the Stokes and Boltzmann model equations for porous media flows. Finally, I conclude that neglecting the rarefaction effects leads to a significant underestimation of the effective permeability of fractured ultra-tight porous media

Investigation of volume diffusion hydrodynamics : application to tight porous media

Various engineering problems imply rarefied gas flows that rely in the transition and free molecular regimes, e.g., micro and nano devices. The recent expansion of shale gas production where rarefied conditions are found in reservoirs exposed another area of application with a major importance. Continuum based methods like standard Navier- Stokes equations break down in the transition regime and free molecular regime. In order to model such flows discrete methods are usually adopted. Boltzmann equation can theoretically be used to simulate rarefied gas flows. However, complexity of its collision integral limits its applications mostly to simple cases (i.e., one dimension problems). The direct simulation Monte Carlo method which mimics the Boltzmann equation is the dominant method for simulating rarefied gas flows. It has been tested in several engineering problems, ranging from nano scale flow to re-entry vehicles with very consistent results in comparison with experimental data and analytical solutions. Its computational cost is, however, enormous for complex cases. Observations from Crookes radiometer inspired extending the continuum methods so that they could capture non-equilibrium phenomena in small scales. In the present thesis two different hydrodynamic model are presented. The first one is based on the Korteweg expression and the second one is called “Bi-velocity”. Firstly, the two models are presented in their mathematical forms. The proposed models are then developed in open-source computational fluid dynamics solvers. The models are tested and benchmarked in different rarefied gas flows problems in the whole range of Knudsen number. We used problems that are found in micro and nano systems and tight porous media. Results from the hydrodynamic models are compared against experimental data where available and the direct simulation Monte Carlo method. The two extended hydrodynamic models show improved results in comparison with standard Navier-Stokes

Pore-scale study of rarefied gas flows using low-variance deviational simulation Monte Carlo method

Gaseous flow through ultra-tight porous media, e.g. shale and some high-performance insulation materials, is often rarefied, invalidating an analysis by the continuum flow theory. Such rarefied flows can be accurately described by the kinetic theory of gases which utilizes the Boltzmann equation and its simplified kinetic models. While discrete velocity methods have been successful in directly solving these equations, the immense potential of a particle-based solution of the variance-reduced Boltzmann-BGK (Bhatnagar–Gross–Krook) equation for rarefied flows in porous media has not been exploited yet. Here, a parallel solver based on the low variance deviational simulation Monte Carlo method is developed for 3D flows, which enables pore-scale simulations using digital images of porous media samples. The unique advantage of this particle-based formulation is in providing additional insights regarding the multi-scale nature of the flow and surface/gas interactions via two new parameters, i.e. pore and surface activity, respectively. Together, these two parameters can identify key flow properties of the porous media. The computational efficiency and accuracy of the current method has also been analysed, suggesting that this new solver is a powerful simulation tool to quantify flow properties of ultra-tight porous media

GSIS: An efficient and accurate numerical method to obtain the apparent gas permeability of porous media

The apparent gas permeability (AGP) of a porous medium is an important parameter to predict production of unconventional gas. The Klinkenberg correlation, which states that the ratio of the AGP to the intrinsic permeability is approximately a linear function of reciprocal mean gas pressure, is one of the most popular estimations to quantify AGP. However, due to the difficulty in defining the characteristic flow length in complex porous media where the rarefied gas flow is multiscale, the slope in the Klinkenberg correlation varies significantly for different geometries such that a universal expression seems impossible. In this paper, by solving the gas kinetic equation using the general synthetic iterative scheme (GSIS), we compute the AGP in porous media that are represented by Sierpinski fractals and pore body/throat systems. With the abilities of fast convergence to steady-state solution and asymptotic preserving of Navier-Stokes limit, it is shown that GSIS is a promising tool to simulate low-speed rarefied gas flow through complex multiscale geometries. A new definition of the characteristic flow length is proposed as a function of porosity, tortuosity and intrinsic permeability of porous media, which enables to find a unique slope in the Klinkenberg correlation for all the considered geometries. This research also shows that the lattice Boltzmann method using simple wall scaling for the effective shear viscosity is not able to predict the AGP of porous media

HIGH TEMPERATURE FLOW SOLVER FOR AEROTHERMODYNAMICS PROBLEMS

A weakly ionized hypersonic flow solver for the simulation of reentry flow is firstly developed at the University of Kentucky. This code is the fluid dynamics module of known as Kentucky Aerothermodynamics and Thermal Response System (KATS). The solver uses a second-order finite volume approach to solve the laminar Navier– Stokes equations, species mass conservation and energy balance equations for flow in chemical and thermal non-equilibrium state, and a fully implicit first-order backward Euler method for the time integration. The hypersonic flow solver is then extended to account for very low Mach number flow using the preconditioning and switch of the convective flux scheme to AUSM family. Additionally, a multi-species preconditioner is developed. The following part of this work involves the coupling of a free flow and a porous medium flow. A new set of equation system for both free flows and porous media flows is constructed, which includes a Darcy–Brinkmann equation for momentum, mass conservation, and energy balance equation. The volume-average technique is used to evaluate the physical properties in the governing equations. Instead of imposing interface boundary conditions, this work aims to couple the free/porous problem through flux balance, therefore, flow behaviors at the interface are satisfied implicitly

Efficient parallel solver for high-speed rarefied gas flow using GSIS

Recently, the general synthetic iterative scheme (GSIS) has been proposed to find the steady-state solution of the Boltzmann equation in the whole range of gas rarefaction, where its fast-converging and asymptotic-preserving properties lead to the significant reduction of iteration numbers and spatial cells in the near-continuum flow regime. However, the efficiency and accuracy of GSIS has only been demonstrated in two-dimensional problems with small numbers of spatial cell and discrete velocities. Here, a large-scale parallel computing strategy is designed to extend the GSIS to three-dimensional high-speed flow problems. Since the GSIS involves the calculation of the mesoscopic kinetic equation which is defined in six-dimensional phase-space, and the macroscopic high-temperature Navier-Stokes-Fourier equations in three-dimensional physical space, the proper partition of the spatial and velocity spaces, and the allocation of CPU cores to the mesoscopic and macroscopic solvers, are the keys to improving the overall computational efficiency. These factors are systematically tested to achieve optimal performance, up to 100 billion spatial and velocity grids. For hypersonic flows around the Apollo reentry capsule, the X38-like vehicle, and the space station, our parallel solver can get the converged solution within one hour