Session K14: Porous Media Flows: General (8:45am - 9:30am CST)

On optimization of pleat packing density in a pleated membrane filter

Pleated membrane filters are widely used to remove undesired impurities from a fluid. A filter membrane is sandwiched between porous support layers, then pleated and packed into an annular cylindrical cartridge with a central hollow duct for outflow. While this arrangement offers a high ratio of surface filtration area to volume, the filter performance is not as efficient as a flat filter with the same surface area. This stems from several possible hypotheses including additional resistance from the packing density of the pleats, the complex flow within the pleated membrane, and possible damage of the membrane during the pleating process. In this work, we present a model to investigate the effects of axial variations of the 3D filter cartridge. We also introduce a more detailed description of the geometry that accounts for the cylinder's curvature. Using asymptotic methods to simplify the flow throughout the cartridge makes it possible to investigate how the number of pleats or pleat packing density affects the performance of pleated membrane filters, where the ultimate goal of this study will be to find an optimal number of pleats to achieve optimum filtration performance.   

Numerical investigation of multi-stability in the unstable flow of a polymer solution through porous media

The flow of viscoelastic polymeric fluids through porous media is common in industrial applications such as oil recovery and groundwater remediation. Polymeric stresses can lead to an elastically induced instability of the flow. Here, we numerically study the flow of a polymeric fluid in a channel consisting of multiple diverging and converging physical constraints, mimicking the pore bodies and throats of an ordered porous medium. The inertial effects are negligible due to small Reynolds numbers and the flow characteristics are determined by the Weissenberg number, a dimensionless ratio of elastic to viscous forces. At small Weissenberg numbers, stable eddies appear on the top and the bottom of each pore. Conversely, at large Weissenberg numbers, strong flow fluctuations due to high polymeric stresses lead to stretching and relaxation of the polymeric chains inside the pores. The stretched polymeric chains inside the pore facilitate eddy formation, whereas relaxed chains lead to eddy free regions. The eddy formed by stretched chains and eddy free region induced by relaxed chains lead to multiple distinct unstable flow structures inside a pore. We quantify the eddy area and correlations between the flow patterns of different pairs of pores, as well as polymeric stress and pressure drop across the tortuous channel to better understand the mechanism behind the observed flow patterns.   

Membrane filtration with multiple species of particles

Membrane filtration is widely used in many applications, ranging from industrial processes to everyday living activities. Fouling is an unavoidable part of filtration and understanding the particle fouling mechanism is critical for improving the filtration performance and avoiding filtration failure, hence this is a topic of much ongoing research. Experimental studies can be very valuable, but are expensive and time-consuming, therefore theoretical studies offer potential as a cheap and predictive way to improve on current filter designs. The majority of theoretical research focuses on filtration of suspensions that consist of chemically homogeneous particles. In this work we propose a model for filtration of a suspension containing an arbitrary number of particle species, each with different affinities for the filter membrane. We present preliminary results showing how the presence of additional species can change filtration outcomes.   

Optimising the flow through a concertinaed filtration membrane

Membrane filtration is a vital industrial process, with applications including air purification and blood filtration. Recently, new techniques in the manufacturing process have been developed which enable the production of specific pore structures. Since this type of membrane can be precisely manufactured, it is important to understand how the flux through such filters can be optimised through design choice. We develop a model for the flow through a concertinaed filtration membrane composed of angled porous membranes and dead-ends, motivated by the direct-flow device designed by Smart Separations Ltd. We determine how the geometric and operating parameters affect the flow through the device, with the aim of determining the optimal setup to maximise the flux for a given pressure drop. We present results for a membrane of fixed angle and physical properties, and find that there can exist multiple membrane positions that maximise the flux. We also present more general results for membranes of arbitrary thickness and permeance. We show that while the maximal flux achievable depends on the membrane thickness and permeance, the optimal membrane configuration is always in one of two canonical setups, as determined by the trade-off in the underlying physics of the problem.   

On the Visualization of Localized Porous Media deformation during an Indentation Process

A novel experimental setup was developed to systematically examine the localized densification of a porous foam under indentation, from which the local Darcy Permeability of the porous foam is obtained. The experimental setup consists of a Physik instrument (PI) positioning system and a square flat-punch tip on an isolated optical table. The polyester foam was compressed using indenters of different sizes. Its deformation was precisely captured by a high-speed camera. The results were processed with Digital Image Correlation (DIC). It shows that the densification of the porous foam, which is related to the collapse of the pores, occurs at the vicinity near the solid loading surface and propagates as the indentation proceeds. The porosity of the porous foam due to the densification effect was determined, from which one obtained the spatial-temporal distribution of the Darcy permeability of the porous media. The study presented herein, combining a novel indentation system and a comprehensive analysis of the recorded footage, precisely captures the detailed non-uniform compression of a thin porous layer under indentation, will have a significant impact on the study of transport through soft matters.~ ~   

Sub-REV Homogenization of Flow in Porous Media with Isolated Embedded Fractures

Classical homogenization relies on representative elementary volumes (REV) large enough that asymptotic macroscopic parameters, e.g. effective permeabilites, can be employed to describe the expected or mean behavior. In this way, Darcy's law, which describes the relationship between macroscopic pressure gradient and volumetric flow rate, was derived. In the presence of large features, however, the required REV size may reach the same order as the geometric reference scale of the problem, and thus effective permeabilities obtained from classical homogenization studies may be unsuited. This is in particular the case for reservoirs with isolated, highly conductive fractures. Here, a new sub-REV continuum model to describe the pre-asymptotic flow behavior in such media is presented. The model relies on a nonlocal multi-media description based on coupled integral-differential equations. The only empirical information required for calibration is the effective permeability of an infinitely large domain, e.g. as obtained from classical homogenization. With a series of numerical studies and comparison with Monte Carlo reference data it is demonstrated that the devised sub-REV model accurately captures mean flow rates and pressure profiles for arbitrary domain sizes.   

Effective stress jump across thin permeable surfaces

A reduced-order model, based on multi-scale homogenization, allows one to simulate the hydrodynamic interactions between a rigid membrane and a surrounding incompressible fluid flow. The model, intuitive, robust and computationally cheap, is able to provide a description of the micro- and macroscopic fluid behavior and consists of a constraint to be satisfied by the fluid velocity $\mathbf{u}$, imposed within the fluid domain, over a virtual smooth surface passing through the center of each membrane pore $$\mathbf{u}=-{\bf M}:{\bf{\Sigma}}^--{\bf N}:{\bf{\Sigma}}^+,$$ where $\mathbf{\Sigma}^{\pm}$ denotes the upward and downward fluid stresses and ${\bf M}$, $\bf N$ the upward and downward Navier tensors, which can be computed once and for all at the pore-scale. The model shows that the membrane produces a jump in fluid stresses whose intensity and direction, evaluated solving problems at the microscale, depend on the external flow and on the pore geometry. To assert the validity of the macroscopic model developed, its solution is compared with the solution of the full-scale problem. Finally, known laws describing flows through porous media or over rough surfaces (like Darcy law or Navier slip condition) can be deduced from this model as particular cases.   

Dead-end sites and their influence on anomalous transport in porous media

Structural heterogeneity plays a crucial role in the complex transport of species in various porous media, such as subsurface soil and aquifers. Such heterogeneity may occur due to non-uniformity in the sizes or the shapes of grains that comprise the medium. Dead-end sites refer to cavity-like spaces in concave grain-boundaries that appear intermittently through a porous medium. Using microfluidic experiment and numerical simulation, we investigate the velocity fields of colloidal suspensions inside a model heterogenous porous structure. We characterize the porous structure via image processing and isolate dead-end sites from the remaining pore spaces. The study reveals complex flow structures inside dead-end sites that contribute to the small-scale velocity. The velocity within the dead-end sites follow a power law distribution that relate to the distribution of trapped volumes inside these sites through the entire medium. Further, we show that the presence of these sites strongly influences the breakthrough curve and the anomalous transport of colloidal beads in this system.   

A Stochastic Particle Tracking Scheme for Embedded Discrete Fracture Models

In particle-based transport models for fractured media, first-order kinetic reactions, for example, can be simulated as stochastic transitions of notional particles between discrete states [1]. For particle-tracking in dual continua models, Liu et al. [2] have quantified the probabilities for particle transfer between the matrix and fractures assuming complete mixing within a grid cell. For coupled surface-subsurface set-ups, de Rooij et al. [3] have formulated a path line specific probability for transfer from a 2-D overland domain to a 3-D subsurface domain. We devise a stochastic particle-tracking scheme suited for the Embedded Discrete Fracture Model (EDFM) in a permeable matrix. Here, fractures are treated as lower dimensional manifolds [4, 5], and interfaces between the matrix and fractures are not resolved by the flow field. We formulate the probability of notional particle transfer between the interacting cells of different continua, say, from a matrix cell to a fracture cell and, also, the distribution of residence times before the transfer. The probability is specific to the associated fluid particle's trajectory in the grid cell. The scheme is mass conservative in matrix and fracture continua. Further, it can be incorporated into random walk models for dispersion.   

Flow past a confined array of rigid hairs

A variety of aquatic organisms use appendages covered with arrays of hairs to capture food, smell, and move the fluid around them. At the hair level, the flow is characterized by a low Reynolds number (Re), whose value controls the transport through the array. The array acts either as a rake forcing the fluid around at low Re or small hair spacing, or as a sieve letting the fluid through at higher Re or large hair spacing. To develop sensors inspired by those biological system, we study the influence of confinement on the flow regimes through the porous structure. We investigate the flow past an array of hairs in a rectangular channel, with a developed Poiseuille flow field. Through numerical simulations, we vary the geometry of the array, the flow rate through the channel, and the confinement. We show that the transition between the rake and sieve regimes depends on the geometry of the system. To interpret our results, we consider the flow around an isolated and confined hair.   

Flame-front kinematics in porous media analyzed via 2D simulations

Combustion in porous media is characterized by strongly corrugated flame regimes and conjugate heat transfer. To quantify the flame regimes in these environments, 2D flame profiles within adiabatic porous media are investigated via pore-scale simulations. Premixed laminar methane-air flames are stabilized within 2D arrays of cylinders. Several cylinder configurations with varying degrees of regularity are simulated in order to evaluate the effect of porous media geometry on the flame stabilization. Flame-front corrugations are statistically analyzed, and theory is developed to explain the stabilization regimes observed computationally. Specifically, this work discusses the effects of flame stretching, normal, and tangential diffusion on local laminar flame speeds. Similarities between flame corrugation through obstacles and standard turbulent premixed flame theory are also highlighted.   

A homogenized model for flow and transport through porous media

The major challenge in flow through porous media is to better understand the link between pore-scale microstructure and macroscale flow and transport. For idealized microstructures, the mathematical framework of homogenization theory can be used for this purpose. Here, we consider a 2D microstructure comprising an array of circular posts, the size and spacing of which can vary arbitrarily in the streamwise direction. We use homogenization theory to develop effective continuum equations for macroscale flow and transport that are characterized by the local porosity, an effective local anisotropic flow permeability, and an effective local anisotropic solute diffusivity. These macroscale properties depend nontrivially on both degrees of microstructural geometric freedom (post size and spacing). We take advantage of this dependence to compare scenarios where the same porosity field is constructed with different combinations of post size and spacing. For example, we consider scenarios where the porosity is spatially uniform but the permeability and diffusivity are not. Our results may be useful in the design of filters, or for studying the impact of deformation on transport in soft porous media.   

Validation of Brinkman Equation for a simple shear driven flow over porous media.

The Brinkman equation has found wide popularity in modeling porous media as it overcomes the strict requirements necessary to model exact porous geometries. However there remains conflicting literature on what the Brinkman viscosity should be as well as the physical domains where the Brinkman equation can be a reasonable model. We investigate the accuracy of the Brinkman equation in porous media subjected to an overlying shear flow. This is accomplished through exact solutions of geometry resolved regular porous structures. Several porous structures are considered with cylinders of various cross section geometries and solid volume fractions. The Brinkman equation is then used to obtain velocity profiles that best fit the exact solutions using the Brinkman viscosity as a free parameter. We find that the Brinkman equation can provide excellent matches for the interior portion of the porous domain, but struggles to consistently model the interfacial region of the porous domain. We provide some guidelines for expected values of the Brinkman viscosity. Additionally we report the error between the exact and Brinkman velocity profile providing a quantitative evaluation of the Brinkman equation's ability to faithfully model the flow through the porous media.   

Physical model of a dissolving porous media in a fluid flow

We investigate the interaction of fluid flow and dissolution of a porous media with experiments and complementary numerical simulation toward understanding the formation of underground caves and channels. We use caramel as the solid matrix because of its relative transparency and rapid dissolution rates. Solid caramel blocks are molded within 3D microfluidic chambers with prescribed shapes to have a well-defined solid matrix, and fluid sources and sinks. The experimental apparatus is transparent, allowing us to image the solid phase as it dissolves in real time. A non-dissolving phase is added to introduce further inhomogeneity in the system. A numerical simulation of the three-phase system and a dissolution rate proportional to the local flow speed and solid fraction of the dissolving phase is demonstrated to capture the evolution of the system with simplified geometries. The calibrated model is then used to understand the evolution of systems as a function of heterogeneity and fluid injection rates. We further discuss the role of gravity in leading to instabilities that cause dissolution patterns with unexpected shapes.   

Transient pressure analysis of multiple fractured wells in stress-sensitive coal seam gas reservoirs with stimulated reservoir volume.

This study is an extension of previous works, which investigates the bottom-hole pressure performances of a multiple fractured well with finite-conductivity hydraulic fractures in a stress-sensitive coal seam gas reservoir with stimulated reservoir volume. The fluid flow in the matrix simultaneously considers adsorption--desorption and diffusion, whereas fluid flow in the natural fractures and the induced fracture networks obeys Darcy's law. The results obtained in this study show that the main flow regimes for the proposed model are bilinear flow between adjacent radial hydraulic fractures, linear flow between adjacent radial hydraulic fractures, pseudo radial flow in stimulated region, and radial flow in un-stimulated region. The effects of the size of stimulated reservoir volume, permeability contrast between stimulated region and un-stimulated region, and the properties of hydraulic fractures on the well-bottom transient pressure responses are demonstrated. The findings of this study is able to gain a better understanding of the transient performances of multiple fractured vertical wells in unconventional reservoirs.   

Mathematical modeling of the flow and fouling in a pleated membrane filter

Pleated membrane filters are essentially thin sheets of porous media sandwiched between two support layers, all of which are housed inside of a cylindrical cartridge. They are in widespread industrial use, since they offer a superior surface area to volume ratio in comparison to equal-area unpleated membrane filters. However their performance characteristics are inferior to those of flat sheet media. We developed a simplified mathematical model, which accounts for the pleated membrane geometry as well as two mechanisms of fouling: (i) adsorption of small particles within membrane pores and (ii) blocking of entire pores by large particles. Using asymptotic analysis based on the small aspect ratio of the pleat, we simplify our model and compare the obtained results to those of equivalent flat sheet media filters.   

Interactions between a confined poroelastic reservoir and a buoyant gravity current

Many geological processes, of which CO$_{2}$ sequestration is an important example, involve the flow of gravity currents through porous media. The long-term behaviour of such buoyant currents strongly depends on the properties of the containing reservoir as well as the interaction between the current and the pressure field of the surrounding fluid. This allows the injected current to experience the full geometry of the reservoir when only the pressure field has reached the boundaries. The reservoir is also affected by the current: the increased pore pressure following injection leads to deformation of the porous medium that, while generally small, may produce ground-level deformation that can be measured remotely to reveal large-scale details of the reservoir's properties. We model the two-phase flow of a buoyant gravity current injected into a fluid-filled poroelastic medium, utilising the large aspect ratio of the domain to vertically average the flow and reservoir properties. With a simple elastic-layer model for the overburden we are able to incorporate the pressure and surface deformation signals in response to the deformation of the reservoir. We apply this model to the In Salah Project to determine reservoir properties and forecast the long-term behaviour post injection.   

Forced Imbibition in Stratified Porous Media

Imbibition plays a central role in diverse energy, environmental, and industrial processes, including oil and gas recovery from unconventional reservoir rocks. In many cases, the medium has multiple parallel strata of different permeabilities; however, how this stratification impacts imbibition is poorly understood. We address this gap in knowledge by directly visualizing forced imbibition in three-dimensional (3D) porous media with two parallel strata. We find that imbibition is spatially heterogeneous: for small capillary number Ca, the wetting fluid preferentially invades the fine stratum, while for Ca above a threshold value, the fluid instead preferentially invades the coarse stratum. This threshold value depends on the medium geometry, the fluid properties, and the presence of residual wetting films in the pore space. These findings are well described by a linear stability analysis that incorporates crossflow between the strata. Thus, our work provides quantitative guidelines for predicting and controlling flow in stratified porous media.   

Viscous pressure drop modulates the morphology of a network fractures activated by hydraulic stimulation

Convective transport in low permeability rocks can be enhanced by injection of a fluid to activate pre-existing weak planes (fractures) above a critical fluid pressure given by Mohr's criterion. Using a discrete fracture network (DFN) simulation and complementary averaged equation solutions for a highly heterogeneous rock, we show that the morphology and average transport properties of a cluster of activated fractures depend on the ratio, $F_{N}$, between the standard deviation of the critical pressures and the viscous pressure drop across a fracture. When $F_{N}$ \textless \textless 1, the cluster is well connected, and a linear diffusion equation can be used to describe the cluster's growth. When $F_{N\thinspace }$\textgreater \textgreater $R/l$ where $R$ is the cluster radius and $l$ is the fracture length, a fractal network is formed by an invasion percolation process. In the intermediate regime, 1\textless \textless $F_{N}$\textless \textless $R$/$l$, percolation theory relates the porosity and permeability of the network to the local pressure and an averaged fluid transport equation with pressure-dependent properties describes the cluster growth on length scales much larger than $l F_{N}$. The theory is also applicable to the displacement of a wetting fluid by a more viscous non-wetting fluid in a permeable rock with the capillary number replacing $F_{N}$ in the two-phase flow application.   

Capillary-Rise Dynamics in Porous Materials

In this study we explore the role of partial saturation and accompanying variations in permeability and capillary pressure in capillary rise dynamics into porous material. Experiments show a deviation from the classical Washburn model dynamics after early times and our aim in this work is to investigate this deviation. We use multiphase mixture theory for modeling to capture in a single framework the complex dynamics and are interested in both rigid and deformable porous materials. We hope to compare the results of our model to experimental data.   

Three-dimensional conjugate numerical model of heat and mass transfer of porous food material during convective drying process

Convective drying is the most popular and widely used drying method of food preservation especially seasonal foods. Dried food materials can be stored for longer time and therefore is helpful to feed the world population as well as to save the world capital. A 3-D conjugate numerical model is developed in COMSOL Multiphysics commercial software by assuming food as a hygroscopic, multi-phase, multi-component porous material having interconnected pores of different sizes. The model is developed to determine the heat and moisture distribution in the porous food material by considering surface and internal evaporation, and variable thermophysical properties of food material. The cylindrical shaped Elephant Foot Yam (EFY) is considered as a food sample in this work. It is found that in the starting of drying process, the average temperature of the EFY sample rises quickly due to sensible heating, after that it becomes constant for a while due to moisture evaporation and again it rises steadily till the equilibrium is reached. The entire drying curve is characterized by three falling rate periods along with initial drying rate and very small constant rate period. These two periods are much smaller than the three falling rate periods.   

Evaporation versus imbibition in a porous medium

Predicting and controlling the liquid dynamics in a porous medium is of large importance in numerous technological and industrial situations. We derive a general analytical solution for the dynamics of a liquid front in a porous medium, considering the combined effects of capillary imbibition, gravity and evaporation. We highlight that the dynamics of the liquid front in the porous medium is controlled by two dimensionless numbers: a gravity-capillary number $G$ and an evaporation-capillary number $E$. We analyze comprehensively the dynamics of the liquid front as functions of $G$ and $E$, and show that the liquid front can exhibit seven kinds of dynamics classified in three types of behaviors. For each limiting case, a simplified expression of the general solution is also derived. Finally, estimations of $G$ and $E$ are computed to evidence the most common regimes and corresponding liquid front dynamics encountered in usual applied conditions. This is realized by investigating the influence of the liquid and porous medium properties, as well as of the atmospheric conditions, on the values of the dimensionless numbers.