Geometric multigrid methods for Darcy-Forchheimer flow in fractured porous media

In this paper, we present a monolithic multigrid method for the efficient solution of flow problems in fractured porous media. Specifically, we consider a mixed-dimensional model which couples Darcy flow in the porous matrix with Forchheimer flow within the fractures. A suitable finite volume discretization permits to reduce the coupled problem to a system of nonlinear equations with a saddle point structure. In order to solve this system, we propose a full approximation scheme (FAS) multigrid solver that appropriately deals with the mixed-dimensional nature of the problem by using mixed-dimensional smoothing and inter-grid transfer operators. Remarkably, the nonlinearity is localized in the fractures, and no coupling between the porous matrix and the fracture unknowns is needed in the smoothing procedure. Numerical experiments show that the proposed multigrid method is robust with respect to the fracture permeability, the Forchheimer coefficient and the mesh size.Comment: arXiv admin note: text overlap with arXiv:1811.0126

Non-Darcy flow through porous media

Since its introduction, Darcy's law has been implemented as a mathematical tool that allows simple calculation and prediction of low velocity subsurface flows. However, turbulence, non-isothermal conditions, as well as other factors can create conditions where Darcy's law does not accurately describe the head and velocity distributions within a given porous matrix. Darcy's law has been widely applied to analytical and numerical modeling of fluid flow through porous matrix, regardless of the hydrogeologic setting. This study attempts to quantify the error incurred by these models through simultaneous numerical modeling of the mass continuity equation using Darcy's law as well as Forchheimer's relation. To this end, results from steady-state and transient Darcy-based and Forchheimer-based numerical models are presented in this study.Geological Science

HPTAM, a two-dimensional Heat Pipe Transient Analysis Model, including the startup from a frozen state

A two-dimensional Heat Pipe Transient Analysis Model, 'HPTAM,' was developed to simulate the transient operation of fully-thawed heat pipes and the startup of heat pipes from a frozen state. The model incorporates: (a) sublimation and resolidification of working fluid; (b) melting and freezing of the working fluid in the porous wick; (c) evaporation of thawed working fluid and condensation as a thin liquid film on a frozen substrate; (d) free-molecule, transition, and continuum vapor flow regimes, using the Dusty Gas Model; (e) liquid flow and heat transfer in the porous wick; and (f) thermal and hydrodynamic couplings of phases at their respective interfaces. HPTAM predicts the radius of curvature of the liquid meniscus at the liquid-vapor interface and the radial location of the working fluid level (liquid or solid) in the wick. It also includes the transverse momentum jump condition (capillary relationship of Pascal) at the liquid-vapor interface and geometrically relates the radius of curvature of the liquid meniscus to the volume fraction of vapor in the wick. The present model predicts the capillary limit and partial liquid recess (dryout) in the evaporator wick, and incorporates a liquid pooling submodel, which simulates accumulation of the excess liquid in the vapor core at the condenser end

Efficient prediction of broadband trailing edge noise and application to porous edge treatment

Trailing edge noise generated by turbulent flow traveling past an edge of an airfoil is one of the most essential aeroacoustic sound generation mechanisms. It is of great interest for noise problems in various areas of industrial application. First principle based CAA with short response time are needed in the industrial design process for reliable prediction of spectral differences in turbulent-boundary-layer trailing-edge noise due to design modifications. In this paper, an aeroacoustic method is studied, resting on a hybrid CFD/CAA procedure. In a first step RANS simulation provides a time-averaged solution, including the mean-flow and turbulence statistics such as length-scale, time-scale and turbulence kinetic energy. Based on these, fluctuating sound sources are then stochastically generated by the Fast Random Particle-Mesh Method to simulate in a second CAA step broadband aeroacoustic sound. From experimental findings it is well known that porous trailing edges significantly lower trailing edge noise level over a large range of frequencies reaching up to 8dB reduction. Furthermore, sound reduction depends on the porous material parameters, e.g. geometry, porosity, permeability and pore size. The paper presents first results for an extended hybrid CFD/CAA method including porous materials with prescribed parameters. To incorporate the effect of porosity, an extended formulation of the Acoustic Perturbation Equations with source terms is derived based on a reformulation of the volume averaged Navier-Stokes equations into perturbation form. Proper implementation of the Darcy and Forchheimer terms is verified for sound propagation in homogeneous and anisotropic porous medium. Sound generation is studied for a generic symmetric NACA0012 airfoil without lift to separate secondary effects of lift and camber on sound from those of the basic edge noise treatments.Comment: 37 page

Multi-scale modeling of inertial flows through propped fractures

Non-Darcy flows are expected to be ubiquitous in near wellbore regions, completions, and in hydraulic fractures of high productivity gas wells. Further, the prevailing dynamic effective stress in the near wellbore region is expected to be an influencing factor for the completion conductivity and non-Darcy flow behavior in it. In other words, the properties (fracture permeability and β-factor) can vary with the time and location in the reservoir (especially in regions close to the wellbore). Using constant values based on empirical correlations for reservoirs/completions properties can lead to erroneous cumulative productivity predictions. With the recent advances in the imaging technology, it is now possible to reconstruct pore geometries of the proppant packs under different stress conditions. With further advances in powerful computing platforms, it is possible to handle large amount of computations such as Lattice Boltzmann (LB) simulations faster and more efficiently. Calculated properties of the proppant pack at different confining stresses show reasonable agreement with the reported values for both permeability and β-factor. These predicted stress-dependent permeability and β-factors corresponding to the effective stress fields around the hydraulic fractured completions is included in a 2D gas reservoir simulator to calculate the productivity index. In image-based flow simulations, spatial resolution of the digital images used for modeling is critical not only because it dictates the scale of features that can be resolved, but also because for most techniques there is at least some relationship between voxel size in the image data and numerical resolution applied to the computational simulations. In this work we investigate this relationship using a computer-generated consolidated porous medium, which was digitized at voxel resolutions in the range 2-10 microns. These images are then used to compute permeability and tortuosity using lattice Boltzmann (LB) and compared against finite elements methods (FEM)simulation results. Results show how changes in computed permeability are affected by image resolution (which dictates how well the pore geometry is approximated) versus grid or mesh resolution (which changes numerical accuracy). For LB, the image and grid resolution are usually taken to be the same; we show at least one case where effects of grid and image resolution appear to counteract one another, giving the mistaken appearance of resolution-independent results. For FEM, meshing can provide certain attributes (such as better conformance to surfaces), but it also adds an extra step for error or approximation to be introduced in the workflow

Relating porous media structure to the Darcy-Forchheimer model

Flow in porous media is an important aspect of many systems, such as fluid separation, heat exchange, underground fluid transport, filtration, and purification. Computational modeling is used in all of these systems to increase the understanding of the system and enable researchers to make optimal decisions regarding the processes within the system. Current tools for modeling flow in porous media require calibration of each system individually, which reduces the quantity and efficiency of the information that simulations can provide. The most common method for modeling flow in porous media, is the Darcy-Forchheimer model. Although this model is accurate and robust, it relies on two coefficients which can only be determined through physical experiments on each individual porous media. These coefficients can be expressed as a product of the fluid properties and the properties of porous media structure; however the variables representing the structure of the porous media are still unable to be determined without physical experiments. For many years determining the relationship between porous media structure and the Darcy-Forchheimer model has been considered impractical, because the scale of porous media made it difficult if not impossible to measure the geometric properties of the material. Additionally, naturally occurring porous media have random structures; thus even if it were feasible to measure the porous media, it would have been difficult to determine the characteristics that most affect flow. Now researchers can both measure and manufacture porous media for specific purposes; however the models have not been updated to allow researchers to take advantage of this technology. Although researchers have the ability to control the exact structure of porous media, the models still lack the ability to help researchers create optimal designs for their systems. This research focuses on understanding the fundamental dynamics of flow in porous media, to enable complex systems to be modeled and developed more easily. Here computational upscaling is used to develop a revised Darcy-Forchheimer equation which includes a relation to the parameters of the porous media. The revised model was developed by simulating several homogeneous structured porous media. The porous media were studied by simulating a periodic unit cell of each porous media to understand the geometric effects. A primary porous media, made of stacked screens was used for the initial analysis. This porous media could be described in as little as two parameters, allowing multiple analyses to be completed without consideration of previous knowledge regarding how flow should behave in porous media. This analysis supported the long held assumption that the Darcy-Forchheimer equation can be divided into a viscous loss term and an inertial loss term. After this primary analysis several less ideal porous media were modeled and analyzed similar to the primary case. A more general relationship that can be used for a wide variety of homogeneous porous media was developed