Computational upscaling of inertia effects from porescale to mesoscale

This paper is included in the Proceedings, Part 1, of the International Conference on Computational Science 2009 (ICCS 2009) held in Baton Rouge, LA, USA, May 25-27, 2009.We propose algorithms for computational upscaling of flow from porescale (microscale) to lab scale (mesoscale). In particular, we solve Navier-Stokes equations in complex pore geometries and average their solutions to derive properties of flow relevant at lab scale such as permeability and inertia coefficients. We discuss two variants of tra-ditional discretizations: a simple algorithm which works well in periodic isotropic media and can be used when coarse approximations are needed, and a more complex one which is well suited for nonisotropic geometries. Convergence of solutions and averaging techniques are major concerns but these can be relaxed if only mesoscopic parameters are needed. The project is a proof-of-concept computational laboratory for porous me-dia which delivers data needed for mesoscale simulations by performing microscale computational simulations

Modeling, Analysis and Simulation of Multiscale Preferential Flow - 8/05-8/10 - Final Report

The research agenda of this project are: (1) Modeling of preferential transport from mesoscale to macroscale; (2) Modeling of fast flow in narrow fractures in porous media; (3) Pseudo-parabolic Models of Dynamic Capillary Pressure; (4) Adaptive computational upscaling of flow with inertia from porescale to mesoscale; (5) Adaptive modeling of nonlinear coupled systems; and (6) Adaptive modeling and a-posteriori estimators for coupled systems with heterogeneous data

Pore-to-core simulations of flow with large velocities using continuum models and imaging data

We consider computational modeling of flow with small and large velocities at porescale and at corescale, and we address various challenges in simulation, upscaling, and modeling. While our focus is on voxel-based data sets from real porous media imaging, our methodology is verified first on synthetic geometries, and we analyze various scaling and convergence properties. We show that the choice of a voxel-based grid and REV size can lead up to 10-20% difference in calculated conductivities. On the other hand, the conductivities decrease significantly with flow rates, starting in a regime usually associated with the onset of inertia effects. This is accompanied by deteriorating porescale solver performance, and we continue our experiments up until about 50% reduction in conductivities, i.e., to Reynolds number just under 1. To account for this decrease, we propose a practical power-based fully anisotropic non-Darcy model at corescale for which we calculate the parameters by upscaling.Keywords: Upscaling, Inertia effects, Anisotropy, Forchheimer model, Flow in porous media, 76S05, 76M45, Navier–Stokes equations, Convergence, Porescale simulations, 76M50, Non-Darcy flo

Modeling non-Darcy flows in realistic pore-scale proppant geometries

The ability to evaluate the effective permeability of proppant packs is useful in predicting the efficiency of hydraulic fracture installations. In this paper we propose a computational approach combining microimaging data from X-ray computed microtomography, the simulations of flow at pore-scale, and an upscaling process which identifies the effective model parameters at the core-scale. With this computational approach applied to proppant pack we confirm the reduction in the fracture conductivity and subsequent reduction in the productivity of a hydraulically fractured reservoir due to the high flow rates and to the migration of fine particles resulting in pore throat bridging

Modeling Flow and Transport at Pore Scale with Obstructions

In this thesis we study mathematical and computational models for phenomena of flow and transport in porous media in the presence of changing pore scale geometries. The differential equations for the flow and transport models at Darcy scale involve the coefficients of permeability, porosity, and tortuosity which depend on the pore scale geometry. The models we propose help to understand how the presence of obstruc- tions impacts the Darcy scale models. The particular changes in pore scale geometry we consider are due to the formation of obstructions to the flow, and come from two important applications of interest, biofilm clogging and gas hydrate crystal plugging up the pores. The direct simulations or experiments of these processes at pore scale is generally unfeasible or impractical. We propose two computationally efficient mathematical and computational models to simulate the formation of the obstructions. The first method extends the phase separation model based on the Allen-Cahn equation; in our variant we add volume constraints and additional localization functions. The second method we propose is a Markov Chain Monte Carlo method inspired by the Ising model; here we use heuristics to choose the particular coefficients which guide the formation of obstructions of a particular type. After we generate independent realizations of the obstructed geometries, we solve flow and transport problems at pore scale. Next we use the technique called upscaling which carries the information to larger scale by averaging, and we are able to derive the ensemble of Darcy scale properties for a collection of generated pore scale geometries with obstructions. We show how these techniques can be used in synthetic geometries as well as in geometries obtained from imaging. In addition, we see that the permeability coefficient is not merely a function of porosity, but is rather highly dependent on the type of obstruction growing at the pore scale

Hybrid Multiscale Methods with Applications to Semiconductors, Porous Media, and Materials Science

In this work we consider two multiscale applications with tremendous computational complexity at the lower scale. First, we examine a model for charge transport in semicon- ductor structures with heterojunction interfaces. Due to the complex physical phenomena at the interface, the model at the design scale is unable to adequately capture the behavior of the structure in the interface region. Simultaneously it is computationally intractable to simulate the full heterostructure on the scale required near the interface. Second, we con- sider the problem of the simulation of fluid flow in a dynamically evolving porous medium. The evolution of the medium strongly couples the porescale flow solutions and the macro scale model, requiring a novel approach to communicate the porescale evolution to the macroscale without resorting to the intractable simulation of the fluid flow problem di- rectly on the porescale geometry. We formulate novel methods for these two applications in the multiscale framework. For the semiconductor problem we present iterative sub- structuring domain decomposition methods that decouple the interface computation from the macroscale model. For the fluid flow problem we develop a reduced order three-scale fluid flow model based on a spatial decomposition of the porescale geometry and the offline approximation of a stochastic process describing macroscale permeability paramaterized by the volume fraction of the evolved geometry

A numerical study of inertial flow features in moderate Reynolds number flow through packed beds of spheres

In this work, flow through synthetic arrangements of contacting spheres is studied as a model problem for porous media and packed bed type flows. Direct numerical simulations are performed for moderate pore Reynolds numbers in the range, 10 ≤ Re ≤ 600, where non-linear porescale flow features are known to contribute significantly to macroscale properties of engineering interest. To first choose and validate appropriate computational models for this problem, the relative performance of two numerical approaches involving body conforming and non-conforming grids for simulating porescale flows is examined. In the first approach, an unstructured solver is used with tetrahedral meshes, which conform to the boundaries of the porespace. In the second approach, a fictitious domain formulation (Apte et al., 2009. J Comput. Phys. 228 (8), 2712-2738) is used, which employs non-body conforming Cartesian grids and enforces the no-slip conditions on the pore boundaries implicitly through a rigidity constraint force. Detailed grid convergence studies of both steady and unsteady flow through prototypical arrangements of spheres indicate that for a fixed level of uncertainty, significantly lower grid densities may be used with the fictitious domain approach, which also does not require complex grid generation techniques. Next, flows through both random and structured arrangements of spheres are simulated at pore Reynolds numbers in the steady inertial ( 10 ≲ Re ≲ 200) and unsteady inertial (Re ≈ 600) regimes, and used to analyze the characteristics of porescale vortical structures. Even at similar Reynolds numbers, the vortical structures observed in structured and random packings are remarkably different. The interior of the structured packings are dominated by multi-lobed vortex rings structures that align with the principal axes of the packing, but perpendicular to the mean flow. The random packing is dominated by helical vortices, elongated parallel to the mean flow direction. The unsteady dynamics observed in random and structured arrangements are also distinct, and are linked to the behavior of the porescale vortices. Finally, to investigate the existence and behavior of transport barriers in packed beds, a numerical tool is developed to compute high resolution finite-time Lyapunov exponent (FTLE) fields on-the-fly during DNS of unsteady flows. Ridges in this field are known to correspond to Lagrangian Coherent Structures (LCS), which are invariant barriers to transport and form the skeleton of time dependent Lagrangian fluid motion. The algorithm and its implementation into a parallel DNS solver are described in detail and used to explore several flows, including unsteady inertial flow in a random sphere packing. The resulting FTLE fields unambiguously define the boundaries of dynamically distinct porescale features such as counter rotating helical vortices and jets, and capture time dependent phenomena including vortex shedding at the pore level