Modeling the flow of non-Newtonian fluids in packed beds at the pore scale

Flow and transport in porous media are important in many science and engineering applications such as composite materials, subsurface water contamination, packed-bed reactors, and enhanced oil recovery. The general approach to modeling such processes is at the continuum scale. Semi-empirical expressions, such as Darcy\u27s law, are substituted for velocity in the continuity equation, which is then coupled with a momentum, mass, and energy balance. While a continuum approach is acceptable in some cases, additional modeling is required for certain non-linear flows, such as multi-phase flows, inertial flows, non-Newtonian flows, and reactive flows. Pore-scale modeling is a first-principles approach to modeling flow and transport in porous media. In this work, network models that are physically representative of specific unconsolidated media are created. The networks can be used to model a wide range of flows, but the focus here is on polymers and suspensions that exhibit non-Newtonian behavior. The network models are used to model steady flow as well as displacement by less viscous fluids. The transient displacement is used to investigate important viscous fingering patterns. While simple boundary conditions are typically imposed in network modeling (e.g. a pressure gradient in one dimension), a more general approach has been developed where boundary conditions are also imposed by direct coupling to an adjacent continuum region. Important qualitative and quantitative results are obtained from the network model for non-Newtonian fluids. Preferential flow pathways form in the network due to the inherent heterogeneity and interconnectivity in porous media. Quantitative results of Darcy velocity versus applied pressure gradient show different behavior than semi-empirical models (analogous to Darcy\u27s law) for non-Newtonian fluids. The transient displacement patterns for non-Newtonian fluids are also different than for Newtonian fluids. If the fluid exhibits a yield stress, a steady state is reached in which some of the original non-Newtonian fluid is left trapped in the network. The displacement patterns are affected by the boundary conditions, which can be determined from direct coupling to a continuum region

Lattice and Continuum Modelling of a Bioactive Porous Tissue Scaffold

A contemporary procedure to grow artificial tissue is to seed cells onto a porous biomaterial scaffold and culture it within a perfusion bioreactor to facilitate the transport of nutrients to growing cells. Typical models of cell growth for tissue engineering applications make use of spatially homogeneous or spatially continuous equations to model cell growth, flow of culture medium, nutrient transport, and their interactions. The network structure of the physical porous scaffold is often incorporated through parameters in these models, either phenomenologically or through techniques like mathematical homogenization. We derive a model on a square grid lattice to demonstrate the importance of explicitly modelling the network structure of the porous scaffold, and compare results from this model with those from a modified continuum model from the literature. We capture two-way coupling between cell growth and fluid flow by allowing cells to block pores, and by allowing the shear stress of the fluid to affect cell growth and death. We explore a range of parameters for both models, and demonstrate quantitative and qualitative differences between predictions from each of these approaches, including spatial pattern formation and local oscillations in cell density present only in the lattice model. These differences suggest that for some parameter regimes, corresponding to specific cell types and scaffold geometries, the lattice model gives qualitatively different model predictions than typical continuum models. Our results inform model selection for bioactive porous tissue scaffolds, aiding in the development of successful tissue engineering experiments and eventually clinically successful technologies.Comment: 38 pages, 16 figures. This version includes a much-expanded introduction, and a new section on nonlinear diffusion in addition to polish throughou

Dynamic development of hydrofracture

Many natural examples of complex joint and vein networks in layered sedimentary rocks are hydrofractures that form by a combination of pore fluid overpressure and tectonic stresses. In this paper, a two-dimensional hybrid hydro-mechanical formulation is proposed to model the dynamic development of natural hydrofractures. The numerical scheme combines a discrete element model (DEM) framework that represents a porous solid medium with a supplementary Darcy based pore-pressure diffusion as continuum description for the fluid. This combination yields a porosity controlled coupling between an evolving fracture network and the associated hydraulic field. The model is tested on some basic cases of hydro-driven fracturing commonly found in nature, e.g., fracturing due to local fluid overpressure in rocks subjected to hydrostatic and nonhydrostatic tectonic loadings. In our models we find that seepage forces created by hydraulic pressure gradients together with poroelastic feedback upon discrete fracturing play a significant role in subsurface rock deformation. These forces manipulate the growth and geometry of hydrofractures in addition to tectonic stresses and the mechanical properties of the porous rocks. Our results show characteristic failure patterns that reflect different tectonic and lithological conditions and are qualitatively consistent with existing analogue and numerical studies as well as field observations. The applied scheme is numerically efficient, can be applied at various scales and is computational cost effective with the least involvement of sophisticated mathematical computation of hydrodynamic flow between the solid grains

Multiscale Modeling of Particle Transport in Petroleum Reservoirs

Modeling subsurface particle transport and retention is important for many processes, including sand production, fines migration, and nanoparticle injection. In this study, a pore-scale particle plugging simulator is concurrently coupled with a streamline reservoir simulator to predict the behavior of particles in the subsurface. The coupled simulators march forward in time together. The automated communication between the two models enables the prediction of spatially and time dependent parameters that control the particle transport process. At each time step, the reservoir simulator provides the inlet velocity and particle concentration of the fluid suspension to the pore-scale model which outputs the permeability, porosity, and retention coefficient. This permits the reservoir simulator to include pore-scale physics at selected locations to determine the number of particles retained and the formation damage. The pore-scale simulator tracks the path of individual particles as they are simultaneously injected into the sample and produces an effluent particle concentration curve that is fit with a continuum-scale advection-dispersion model. The advection-dispersion model is matched to the pore-scale data by adjusting two parameters: the dispersion and retention coefficient. The retention coefficient dictates the number of particles retained across a grid block in the reservoir simulator. Incorporating fundamental pore-scale physics into the streamline reservoir simulator improves its predictive ability by updating the particle retention and formation damage of a grid block at each time step

Pore-to-continuum Multiscale Modeling of Two-phase Flow in Porous Media

Abstract Pore-scale network modeling using 3D X-ray computed tomographic images (digital rock technology) has become integral to both research and commercial simulations in recent years. While this technology provides tremendous insight into pore-scale behavior, computational methods for integrating the results into practical, continuum-scale models remain fairly primitive. The general approach is to run pore-scale models and continuum models sequentially, where macroscopic parameters are simulated using the pore-scale models and then used in the continuum models as if they have been obtained from laboratory experiments. While a sequential coupling approach is appealing in some cases, an inability to run the two models concurrently (exchanging parameters and boundary conditions in real numerical time) will prevent using pore-scale image-based modeling to its full potential. In this work, an algorithm for direct coupling of a dynamic pore-network model for multiphase flow with a traditional continuum-scale simulator is presented. The ability to run the two models concurrently is made possible by a novel dynamic pore-network model that allows simultaneous injection of immiscible fluids under either transient or steady-state conditions. The dynamic network algorithm can simulate both drainage and imbibition. Consequently, the network algorithm can be used to model a complete time-dependent injection process that comprises a steady-state relative permeability test, and also allows for coupling to a continuum model via exchange of information between the two models. Results also include the sensitivity analysis of relative permeability to pore-level physics and simulation algorithms. A concurrent multiscale modeling approach is presented. It allows the pore-scale properties to evolve naturally during the simulated reservoir time step and provide a unique method for reconciling the dramatically different time and length scales across the coupled models. The model is tested for examples associated with oil production and groundwater transport in which relative permeability depends on flowrate, thus demonstrating a situation that cannot be modeled using a traditional approach. This work is significant because it represents a fundamental change in the way we might obtain continuum-scale parameters in a reservoir simulation

Pore-scale Modeling of Viscous Flow and Induced Forces in Dense Sphere Packings

We propose a method for effectively upscaling incompressible viscous flow in large random polydispersed sphere packings: the emphasis of this method is on the determination of the forces applied on the solid particles by the fluid. Pore bodies and their connections are defined locally through a regular Delaunay triangulation of the packings. Viscous flow equations are upscaled at the pore level, and approximated with a finite volume numerical scheme. We compare numerical simulations of the proposed method to detailed finite element (FEM) simulations of the Stokes equations for assemblies of 8 to 200 spheres. A good agreement is found both in terms of forces exerted on the solid particles and effective permeability coefficients

Bifurcations and dynamics emergent from lattice and continuum models of bioactive porous media

We study dynamics emergent from a two-dimensional reaction--diffusion process modelled via a finite lattice dynamical system, as well as an analogous PDE system, involving spatially nonlocal interactions. These models govern the evolution of cells in a bioactive porous medium, with evolution of the local cell density depending on a coupled quasi--static fluid flow problem. We demonstrate differences emergent from the choice of a discrete lattice or a continuum for the spatial domain of such a process. We find long--time oscillations and steady states in cell density in both lattice and continuum models, but that the continuum model only exhibits solutions with vertical symmetry, independent of initial data, whereas the finite lattice admits asymmetric oscillations and steady states arising from symmetry-breaking bifurcations. We conjecture that it is the structure of the finite lattice which allows for more complicated asymmetric dynamics. Our analysis suggests that the origin of both types of oscillations is a nonlocal reaction-diffusion mechanism mediated by quasi-static fluid flow.Comment: 30 pages, 21 figure