Transport Phenomena In Porous Media Research Articles

Modification of the fundamental theorem for transport phenomena in porous media

To examine convection in porous media, numerous analyses and numerical simulations have been conducted based on the macroscopic governing equations. In deriving the macroscopic governing equations, the phenomena intrinsic to porous media, such as Darcy’s flow resistance, Forchheimer’s flow resistance, and dispersion, are modeled using the theorem of the local volume average of a gradient (or a divergence). The theorem has been widely accepted as fundamental in the theory of convection in porous media; however, certain questions relating to the correctness of the theorem (the continuous derivative of the volume average and the pressure correction for Darcy’s law) have been raised. In this study, we modify the conventional theorem for the local volume average of a gradient (or a divergence) to solve the aforementioned questions. First, we introduce the concept of a point mass to describe the reference point for the movement of the fluid phase, and we derive the Reynolds transport theorem in the macroscopic field that corresponds to the continuum of porous media. Then, we examine the definition of divergence at a point to obtain the relation between the microscopic description that employs the velocity vector u and quantity B of the fluid particle and the macroscopic description that employs the reference velocity vector u0 and the reference quantity B0 of the point-mass particle, and we derive the modified theorem for the local volume average of a gradient (or a divergence). Furthermore, we derive the governing equations for porous media with the aid of the modified theorem.

Review on penetration and transport phenomena in porous media during slot die coating

ABSTRACTA distinctive field in the coatings industry is coating of porous media, which has broad applications including paper, textiles, electronics, filtration, and energy sectors. Fluid penetration is an important issue during direct coating of a liquid bead on porous media, which is driven by the pressure from an external flow field and the surface tension in the porous media. Generally, during the coating process, some level of penetration is desirable to obtain specific material properties, but inadequate or excessive penetration is detrimental. To help control the level of penetration, understanding relationships between operating parameters and penetration are highly desirable. In this article, the current state of academic research on modeling penetration in porous media during common coating processes, especially the slot die coating process, is reviewed. Specifically, the challenges, basic ideas, advantages, and disadvantages of macroscale, microscale, and pore‐network models on penetration in porous media are discussed. This article concludes with some recommendations for future work. © 2017 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2017, 55, 1669–1680

Numerical simulation of three-dimensional double-diffusive natural convection in porous media by boundary element method

The paper presents numerical results for three-dimensional double-diffusive natural convection in a cubic enclosure fully filled with fluid saturated porous media. Two opposite vertical walls of the enclosure are subjected to different values of temperature and concentration, which causes buoyant and diffusive flow in the porous media domain. Mathematical model is based on the Brinkman-extended Darcy formulation as a governing momentum equation, which is coupled with the energy and species equations. The three-dimensional boundary element method based solver was used to solve the obtained set of partial differential equations. The existing numerical algorithm primarily derived for the pure fluid flow simulations was adopted to simulate transport phenomena in porous media. It is based on the combination of single and subdomain boundary element method, which solves the velocity-vorticity formulation of the governing equations. In the paper the influence of some governing parameters, specially the Rayleigh number, Darcy number and buoyancy coefficient are investigated in order to analyze the heat and mass transfer through porous enclosure. The numerical code is verified by comparison of the results with available previous numerical data found in the literature.

A study on pressure-driven gas transport in porous media: from nanoscale to microscale

Gas flow in porous media can be seen in various engineering devices such as catalytic converters and fuel cells. It is important to understand transport phenomena in porous media for improvement of the performance of such devices. Porous media with pores as small as the mean free path of gas molecules are used in such devices as proton exchange membrane fuel cells. It is difficult to measure molecular transport through such small pores in the experimental approach. In addition, even when using theoretical or numerical approaches, gas flow through nanoscale pores must be treated by the Boltzmann equation rather than the Navier–Stokes equations because it cannot be considered as a continuum. Thus, conventional analyses based on the continuum hypothesis are inadequate and the transport phenomena in porous media with nanoscale pores are not yet clearly understood. In this study, we represented porous media by randomly arranged solid spherical particles and simulated pressure-driven gas flow through the porous media by using the direct simulation Monte Carlo (DSMC) method based on the Boltzmann equation. DSMC simulations were performed for different porosities and different sizes of solid particles of porous media. It was confirmed that Darcy’s law holds even in the case of porous media with micro-/nanoscale pores. Using the obtained results, we constructed expressions to estimate the pressure-driven gas transport in porous media with micro-/nanoscale pores and porosity ranging from 0.3 to 0.5. The flow velocities estimated by using the constructed expressions agreed well with those obtained in the DSMC simulations.

Transport Phenomena in Porous Media and Fractal Geometry

1College of Science, China Jiliang University, Hangzhou 310018, China 2Hubei Subsurface Multi-Scale Imaging Key Laboratory, Institute of Geophysics and Geomatics, China University of Geosciences, Wuhan 430074, China 3Department of Mining and Materials Engineering, McGill University, Montreal, QC, Canada H3A 2A7 4Department of Chemical and Biomolecular Engineering, National University of Singapore, Singapore 117585 5School of Physics, Huazhong University of Science and Technology, Wuhan 430074, China

Understanding the capillary behavior using the extended reduced similar geometry method

Pore scale modeling is used to understand the transport phenomena in porous media. Capillary behavior is an important property in pore scale research. We extend the reduced similar geometry (RSG) method to tangential polygons and simplify the results to concise equations. Based on the RSG method results, the effects of the shape similarity, the shape factor and the contact angle on the capillary behavior are investigated. A new and more accurate parameter is proposed to predict the threshold radius during primary drainage. This parameter is demonstrated to predict the threshold radius accurately for arbitrary polygon shapes and real rock shapes. It also performs better than the traditional predictor when predicting the capillary behavior for concave shapes.

Fluid Dynamics and Mass Transfer in a Perfusion Bioreactor with a Porous Wall

A numerical study is carried out on the fluid dynamics and mass transfer in a microchannel perfusion bioreactor. The bioreactor channel has a porous wall for the co-culture of two types of cells which are distributed randomly and uniformly. A group of dimensionless parameters is proposed, which can be applied to correlate the numerical data and characterize the mass transfer in the bioreactor. The normalized numerical data, for the concentration at the porous-fluid interface and concentration difference between the interface and the base, show satisfactory correlation when presented as a function of the effective distance parameter. Index Terms—Transport phenomena in porous media, numerical methods in fluid flow, mass transfer.

Editorial: Deformation and transport phenomena in porous media

ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und MechanikVolume 94, Issue 7-8 p. 559-559 EditorialFree Access Editorial: Deformation and transport phenomena in porous media Wolfgang Ehlers, Wolfgang Ehlers Universität Stuttgart, Stuttgart Research Centre for Simulation Technology (SimTech)Search for more papers by this authorRainer Helmig, Rainer Helmig Universität Stuttgart, Stuttgart Research Centre for Simulation Technology (SimTech)Search for more papers by this authorChristian Rohde, Christian Rohde Universität Stuttgart, Stuttgart Research Centre for Simulation Technology (SimTech)Search for more papers by this author Wolfgang Ehlers, Wolfgang Ehlers Universität Stuttgart, Stuttgart Research Centre for Simulation Technology (SimTech)Search for more papers by this authorRainer Helmig, Rainer Helmig Universität Stuttgart, Stuttgart Research Centre for Simulation Technology (SimTech)Search for more papers by this authorChristian Rohde, Christian Rohde Universität Stuttgart, Stuttgart Research Centre for Simulation Technology (SimTech)Search for more papers by this author First published: 01 July 2014 https://doi.org/10.1002/zamm.201400559AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat No abstract is available for this article. Volume94, Issue7-8Special Issue: Deformation and transport phenomena in porous mediaJuly 2014Pages 559-559 RelatedInformation

Mathematical Simulation of Mass Transport in Porous Media: An Innovative Method to Match Geometrical and Transport Parameters for Scale Transition

The scale-up and scale-down process is of great importance regarding modeling of mass transport phenomena in porous media. Three numerical simulations of porous media for three different scales were developed and validated in this study, analyzing the mass transport phenomena for each scale. More precisely, the first system is a spheres assemblage of various spatial distributions and sizes (mesoscopic), the second is a porous box (macroscopic), and the third is a sphere-in-cell model (microscopic). It was found that the porous structure—shape, size, and positioning—of the spheres at the mesoscopic scale have significant effects on the adsorption efficiency (up to 10%) of highly convective regimes. It was also concluded that a microscopic description at the pore scale is insufficient to adequately describe mass transport phenomena in porous media due to discontinuities of the structure and high local velocity. By comparing the results from the three scales, we obtained a method of matching all geometrical, flow, and transport parameters when a scale transition occurs. The qualitative description of transport phenomena through the three scales and their identical behavior demonstrates the method's effectiveness.

Conditional Methods in Modeling CO2 Capture from Coal Syngas

Gasification of coal or biomass with in-situ CO2 capture is an emerging technology aiming to address the problem of climate change. Development of a CO2 sorbent with desirable properties and understanding the behavior of such a material in carbonation/calcination reactions is an important part of developing the technology. In this paper, we report experimental results describing the carbonation behavior of three synthetic CaO-based sorbents. We also present a physically-based model of the reactive transport processes in sorbent particles, which have complicated pore structures. This modeling is based on the conditional approach (i.e., conditional moment closure (CMC)), which has proven to be successful in modeling reactive transport phenomena in porous media. The model predictions are in good agreement with the experimental data.

Reconsideration at field scale of the relationship between hydraulic conductivity and porosity: the case of a sandy aquifer in South Italy.

To describe flow or transport phenomena in porous media, relations between aquifer hydraulic conductivity and effective porosity can prove useful, avoiding the need to perform expensive and time consuming measurements. The practical applications generally require the determination of this parameter at field scale, while most of the empirical and semiempirical formulas, based on grain size analysis and allowing determination of the hydraulic conductivity from the porosity, are related to the laboratory scale and thus are not representative of the aquifer volumes to which one refers. Therefore, following the grain size distribution methodology, a new experimental relation between hydraulic conductivity and effective porosity, representative of aquifer volumes at field scale, is given for a confined aquifer. The experimental values used to determine this law were obtained for both parameters using only field measurements methods. The experimental results found, also if in the strict sense valid only for the investigated aquifer, can give useful suggestions for other alluvial aquifers with analogous characteristics of grain-size distribution. Limited to the investigated range, a useful comparison with the best known empirical formulas based on grain size analysis was carried out. The experimental data allowed also investigation of the existence of a scaling behaviour for both parameters considered.

Internal Transport Coefficient Measurements in Random Fiber Matrix Heat Exchangers

Experimental determination of transport coefficients, in particular internal heat transfer coefficients, in heterogeneous and hierarchical heat transfer devices such as compact regenerative heat exchangers has posed a persistent challenge for designers. The goal of this study is to (1) present a new general treatment of the experimental determination of such design data, to (2) provide simple correlations for high porosity random fiber matrices for broad design applications, and to (3) illustrate how such measurements close the formidable integro-differential volume averaging theory (VAT) equations governing transport phenomena in porous media. The combined experimental and computational method employed here for determining the internal heat transfer coefficient in the porous structure is based on the VAT model and combines with simple pressure drop measurements to yield the relevant design data for eight different high porosity random fiber samples. The design data are correlated based on a porous media length scale derived from the VAT model governing equations and the transport coefficient correlations obtained are valid for gas flows over a Reynolds number range between 5 and 70. Finally, the correlations are related to explicit, rigorously derived, lower-scale expressions arising from the VAT model. With the illustration of a new experimental tool, and the production of new simple design correlations for high porosity random fiber matrices for regenerative heat transfer applications, within the context of the hierarchical VAT model, future VAT-based simulation studies of such devices may be pursued. Moreover, the nonlocal modeling provided by VAT paves the way to meaningful optimization studies due to its singular ability to provide rigorous modeling and fast numerical solutions for transport phenomena in regenerative compact heat exchangers.

Measuring Transport Coefficients in Heterogeneous and Hierarchical Heat Transfer Devices

Experimental determination of transport coefficients, in particular internal heat transfer coefficients, in heterogeneous and hierarchical heat transfer devices such as compact heat exchangers and high surface density heat sinks has posed a persistent challenge for designers. This study presents a unique treatment of the experimental determination of such design data. A new combined experimental and computational method for determining the internal heat transfer coefficient within a heterogeneous and hierarchical heat transfer medium is explored and results are obtained for the case of cross flow of air over staggered cylinders to provide validation of the method. Along with appropriate pressure drop measurements, these measurements allow for thermal-fluid modeling of a heat exchanger by closing the volume averaging theory (VAT)-based equations governing transport phenomena in porous media, which have been rigorously derived from the lower-scale Navier–Stokes and thermal energy equations. To experimentally obtain the internal heat transfer coefficient the solid phase is subjected to a step change in heat generation rate via induction heating, while the fluid flows through under steady flow conditions. The transient fluid phase temperature response is measured. The heat transfer coefficient is then determined by comparing the results of a numerical simulation based on the VAT model with the experimental results. The friction factor is determined through pressure drop measurements, as is usually done. With the lower-scale heat transfer coefficient and friction factor measured, the VAT-based equations governing the transport phenomena in the heat transfer device are closed and readily solved. Several configurations of staggered cylinders in cross flow were selected for this study. Results for the heat transfer coefficient and friction factor are compared to widely accepted correlations and agreement is observed, lending validation to this experimental method and analysis procedure. It is expected that a more convenient and accurate tool for experimental closure of the VAT-based equations modeling transport in heterogeneous and hierarchical media, which comes down to measuring the transport coefficients, will allow for easier modeling and subsequent optimization of high performance compact heat exchangers and heat sinks for which design data does not already exist.

Laplace transform – Finite element method for non Fickian flows

In this paper we consider numerical methods for integro-differential problems based on time discretization via Laplace transformation. We focus our attention in models arising in the context of non Fickian solute transport phenomena in porous media. The mathematical models which describe the evolution of the solute concentrations are characterized by Volterra equations. We present and analyze an hybrid method which combines the Laplace transformation with respect to the time variable with the finite element discretization in the spatial variables. Numerical results illustrate the performance of the method.

Normal Mode Analysis of the High Speed Channel Flow in a Bidisperse Porous Medium

The linear Darcy–Brinkman model of the high speed flow in a bidisperse porous medium proposed by Nield and Kuznetsov (Transport Phenomena in Porous Media, 2005) is revisited in this paper. For the steady unidirectional flow in a parallel plane channel the exact analytical solutions for the fluid velocities are worked out by the normal-mode reduction of the governing equations. The limiting cases of the weak and strong momentum transfer between the flows in the fracture and porous phases are discussed in detail. A comparison to the nonlinear Forchheimer extension of the model proposed recently by Nield and Kuznetsov (Transport Porous Media, 2013) shows that, in the considered parameter range, the nonlinear effect of the Forchheimer drag is negligibly small. Even the simplest zero-momentum transfer solution yields an acceptable approximation.

Laboratory monitoring of <italic>P</italic> waves in partially saturated sand

Energy dissipation is observed on seismic data when a wave propagates through a porous medium, involving different frequency regimes depending on the nature of rock and fluid types. We focus here on the role of partial fluid saturation in unconsolidated porous media, looking in particular at P-wave phase velocity and attenuation. The study consists in running an experiment in a sand-filled tank partially saturated with water. Seismic propagation in the tank is generated in the kHz range by hitting a steel ball on a granite plate. Seismic data are recorded by buried accelerometers and injecting or extracting water controls the partial saturation. Several imbibition/drainage cycles were performed between the water and gas residual saturations. A Continuous Wavelet Transform applied on seismic records allowed us to extract the direct P-wave at each receiver.We observe an hysteresis in phase velocities and inverse quality factors between imbibition and drainage. Phase velocities and inverse quality factors are then jointly inverted to get a final poro-visco-elastic model of the partially saturated sand that satisfactorily reproduces the data. The model formulation consists in generalizing the Biot theory to effective properties of the fluid and medium (permeability and bulk modulus) in order to properly explain the phase velocity variation as a function of the saturation. The strong level of attenuation measured experimentally is further explained by an anelastic effect due to grain to grain sliding, adding to Biot's losses. This study shows that fluid distribution at microscopic scale has strong influence on the attenuation of direct P-waves at macroscopic scale and confirms that seismic prospection may be a powerful tool for the characterization of transport phenomena in porous media.

Matrix-oriented implementation for the numerical solution of the partial differential equations governing flows and transport in porous media

In this paper we introduce a new technique for the numerical solution of the various partial differential equations governing flow and transport phenomena in porous media. This method is proposed to be used in high level programming languages like MATLAB, Python, etc., which show to be more efficient for certain mathematical operations than for others. The proposed technique utilizes those operations in which these programming languages are efficient the most and keeps away as much as possible from those inefficient, time-consuming operations. In particular, this technique is based on the minimization of using multiple indices looping operations by reshaping the unknown variables into one-dimensional column vectors and performing the numerical operations using shifting matrices. The cell-centered information as well as the face-centered information are shifted to the adjacent face-center and cell-center, respectively. This enables the difference equations to be done for all the cells at once using matrix operations rather than within loops. Furthermore, for results post-processing, the face-center information can further be mapped to the physical grid nodes for contour plotting and stream lines constructions. In this work we apply this technique to flow and transport phenomena in porous media.

Evaluating transport in irregular pore networks

A general approach for investigating transport phenomena in porous media is presented. This approach has the capacity to represent various kinds of irregularity in porous media without the need for excessive detail or computational effort. The overall method combines a generalized effective medium approximation (EMA) with a macroscopic continuum model in order to derive a transport equation with explicit analytical expressions for the transport coefficients. The proposed form of the EMA is an anisotropic and heterogeneous extension of Kirkpatrick's EMA [Rev. Mod. Phys. 45, 574 (1973)] which allows the overall model to account for microscopic alterations in connectivity (with the locations of the pores and the orientation and length of the throat) as well as macroscopic variations in transport properties. A comparison to numerical results for randomly generated networks with different properties is given, indicating the potential for this methodology to handle cases that would pose significant difficulties to many other analytical models.

An Equation-Type Approach for the Numerical Solution of the Partial Differential Equations Governing Transport Phenomena in Porous Media

A new technique for the numerical solution of the partial differential equations governing transport phenomena in porous media is introduced. In this technique, the governing equations as depicted from the physics of the problem are used without extra manipulations. In other words, there is no need to reduce the number of governing equations by some sort of mathematical manipulations. This technique enables the separation of the physics part of the problem and the solver part, which makes coding more robust and could be used in several other applications with little or no modifications (e.g., multi-phase flow in porous media). In this method, one abandons the need to construct the coefficient matrix for the pressure equation. Alternatively, the coefficients are automatically generated within the solver routine. We show examples of using this technique to solving several flow problems in porous media.

A coupling concept for two‐phase compositional porous‐medium and single‐phase compositional free flow

Domains composed of a porous part and an adjacent free‐flow region are of special interest in many fields of application. So far, the coupling of free flow with porous‐media flow has been considered only for single‐phase systems. Here we extend this classical concept to two‐component nonisothermal flow with two phases inside the porous medium and one phase in the free‐flow region. The mathematical modeling of flow and transport phenomena in porous media is often based on Darcy's law, whereas in free‐flow regions the (Navier‐) ‐Stokes equations are used. In this paper, we give a detailed description of the employed subdomain models. The main contribution is the developed coupling concept, which is able to deal with compositional (miscible) flow and a two‐phase system in the porous medium. It is based on the continuity of fluxes and the assumption of thermodynamic equilibrium, and uses the Beavers‐Joseph‐Saffman condition. The phenomenological explanations leading to a simple, solvable model, which accounts for the physics at the interface, are laid out in detail. Our model can account for evaporation and condensation processes at the interface and is used to model evaporation from soil influenced by a wind field in a first numerical example.