Periodic Porous Media Research Articles

Effective Behavior Near Clogging in Upscaled Equations for Non-isothermal Reactive Porous Media Flow

For a non-isothermal reactive flow process, effective properties such as permeability and heat conductivity change as the underlying pore structure evolves. We investigate changes of the effective properties for a two-dimensional periodic porous medium as the grain geometry changes. We consider specific grain shapes and study the evolution by solving the cell problems numerically for an upscaled model derived in Bringedal et al. (Transp Porous Media 114(2):371–393, 2016. doi: 10.1007/s11242-015-0530-9 ). In particular, we focus on the limit behavior near clogging. The effective heat conductivities are compared to common porosity-weighted volume averaging approximations, and we find that geometric averages perform better than arithmetic and harmonic for isotropic media, while the optimal choice for anisotropic media depends on the degree and direction of the anisotropy. An approximate analytical expression is found to perform well for the isotropic effective heat conductivity. The permeability is compared to some commonly used approaches focusing on the limiting behavior near clogging, where a fitted power law is found to behave reasonably well. The resulting macroscale equations are tested on a case where the geochemical reactions cause pore clogging and a corresponding change in the flow and transport behavior at Darcy scale. As pores clog the flow paths shift away, while heat conduction increases in regions with lower porosity.

An FFT method for the computation of thermal diffusivity of porous periodic media

In this paper, we provide fast Fourier transform (FFT) iterative schemes to compute the thermal diffusivity of a periodic porous medium. We consider the fluid flow through a porous rigid solid due to a prescribed macroscopic gradient of pressure and a macroscopic gradient of temperature. As already proved in the literature, the asymptotic homogenization procedure is reduced to the resolution of two separated problems for the unit cell: (i) the fluid flow governed by the Stokes equations with an applied gradient of pressure, and (ii) the heat transfer by both convection and conduction due to an applied macroscopic gradient of temperature. We develop new numerical approaches based on FFT for the implementation of the cell problems. In a first approach, a simple iterative method based on the primal variable (gradient of temperature) is provided to solve the heat transfer problem. In order to improve the convergence in the range of high values of the prescribed gradient of pressure, we propose a more sophisticated iterative scheme based on the polarization. In order to evaluate their capacities, these FFT algorithms are applied to some specific microstructures of interest including flows past parallel pores (Poiseuille flows) and periodically or randomly distributed cylinders.

Turbulence modeling for flows in wall bounded porous media: An analysis based on direct numerical simulations

Various models are available for simulating turbulent flows in porous media. Models based on the eddy viscosity assumption are often adopted to close the Reynolds stress term. In order to validate the assumptions behind such turbulence models, we studied the dynamics of macroscopic momentum and turbulence kinetic energy in porous media flows by utilizing Direct Numerical Simulation (DNS). The generic porous matrix is composed of regularly arranged spheres. The resulting periodic porous medium is bounded by two walls. The DNS analyses with a Lattice Boltzmann method were performed for various values of the applied pressure gradient, pore size to channel width ratio, and porosity. The DNS results were averaged over time and volume to obtain macroscopic results. The results show that the macroscopic shear Reynolds stress in all Representative Elementary Volumes (REVs), independent of their location, is negligibly small, although the mean velocity gradient takes nonzero values near the wall. The turbulence kinetic energy production rate is generally balanced by the dissipation rate in each REV. The DNS results support a zero-equation turbulence model that accounts for the fact that turbulent structures are restricted in size by the pore scale. The DNS results also suggest that the Brinkman term, which expresses the diffusion of momentum, has an important effect near the wall where the gradient of the shear stress is large. Therefore, the Brinkman term should be taken into account in the macroscopic momentum equation as a component of the total drag. A preliminary macroscopic model for calculating turbulent porous media flows has been proposed and compared with our DNS results.

Flow through porous media: a momentum tracer approach

The flow of a fluid through a spatially periodic porous medium is studied using the momentum tracer approach, that is by monitoring the temporal spread of an instantaneous source of momentum (the “momentum tracer”), following its initial introduction into the interstitial fluid. The fluid flow is described macroscopically by a Brinkman-like effective equation, containing three effective tensorial quantities, namely the permeability dyadic, the generalized viscosity tetradic and a coupling triadic term. These phenomenological coefficients are shown to be expressible in terms of the solution of the characteristic eigenvalue problem associated with the (tensorial) Stokes operator, defined within the interstitial region of a single unit cell of the periodic porous medium. The phenomenological coefficients are shown to be independent of the initial position of the momentum tracer as well as of the choice of weight function used in averaging the microscale momentum density over the unit cell. Finally, we show that the analytical expressions for the phenomenological coefficients identically satisfy the symmetry requirements that can be obtained by applying the reciprocal theorem at the macro-scale level.

Homogenization of the Stokes equation with mixed boundary condition in a porous medium

We homogenize stationary incompressible Stokes flow in a periodic porous medium. The fluid is assumed to satisfy a no-slip condition on the boundary of solid inclusions and a normal stress (traction) condition on the global boundary. Under these assumptions, the homogenized equation becomes the classical Darcy law with a Dirichlet condition for the pressure.

Homogenization of a pore scale model for precipitation and dissolution in porous media

In this article, we employ homogenization techniques to provide a rigorous derivation of the Darcy scale model for precipitation and dissolution in porous media. The starting point is the pore scale model in van Duijn & Pop (2004 ), which is a coupled system of evolution equations, involving a parabolic equation which models ion transport in the fluid phase of a periodic porous medium, coupled to an ordinary differential equations modelling dissolution and precipitation at the grains boundary. The main challenge is in dealing with the dissolution and precipitation rates, which involve a monotone but possibly discontinuous function. In order to pass to the limit in these rate functions at the boundary of the grains, we prove strong two-scale convergence for the concentrations at the microscopic boundary and use refined arguments in order to identify the form of the macroscopic dissolution rate, which is again a discontinuous function. The resulting upscaled model is consistent with the Darcy scale model proposed in Knabner et al . (1995 ).

Unfolding homogenization in doubly periodic media and applications

We define a reiterated unfolding operator for a doubly periodic domain presenting two periodicity scales. Then we show how to apply it to the homogenization of both linear and nonlinear problems. The main novelty is that this method allows the use of test functions with one scale of periodicity only and it considerably simplifies the proofs of the convergence results. We illustrate this new approach on a Poisson problem with Dirichlet boundary conditions and on the flow of a power law fluid in a doubly periodic porous medium.

Усредненная модель диффузии в локально периодической пористой среде с нелинейным поглощением на границе

Усредненная модель диффузии в локально периодической пористой среде с нелинейным поглощением на границе

The method of quasiperiodic fields for diffusion in periodic porous media

In this paper, a new theoretical method for smoothing the diffusion equation in periodic porous media is presented. That strategy is named ‘the method of quasiperiodic fields’ because it is based on the postulate that the concentration is a quasiperiodic field throughout the porous medium. The method proceeds in four steps: (i) a first level averaging of the diffusion equation, (ii) the statement of a quasiperiodic problem, (iii) the statement of factorized quasiperiodic problems, and (iv) the development of a closed form for the averaged diffusion equation. A first application of these four steps on the initial diffusion equation in a periodic porous medium provides an averaged diffusion equation in which the effective diffusion coefficient varies periodically at the small scale. A second application of the four steps provides a smoothed diffusion equation in which the effective diffusion coefficient is constant. The final smoothed diffusion equation is stated in terms of the double average of the concentration field.

Darcy’s Law for Flow in a Periodic Thin Porous Medium Confined Between Two Parallel Plates

We study stationary incompressible fluid flow in a thin periodic porous medium. The medium under consideration is a bounded perforated 3D-domain confined between two parallel plates. The distance b ...

Homogenization of incompressible generalized Stokes flows through a porous medium

We study the homogenization for families of steady and also unsteady incompressible generalized Stokes systems in a periodic porous medium. We assume that the stress tensor possesses an Orlicz growth and the size of solid parts of the porous medium is comparable to the size of the period. Homogenized systems are established using the two-scale convergence method adopted to Orlicz space setting. We prove the existence and uniqueness of weak solutions of the homogenized systems.

Experimental investigation of 3D woven Cu lattices for heat exchanger applications

Stochastic metallic foams and periodic porous media have been used extensively in heat transfer applications. A relatively new cellular material, 3D woven Cu lattices, show potential for increased thermal performance, due to their high specific surface areas, high thermal conductivity and regular micro-pore distributions. This work investigates the performance of these lattices in both a “standard” and a topology “optimized” architecture using three flow patterns (axial, focused bifurcated and full bifurcated) and two working coolants (water and air). We characterize and compare three performance metrics: pressure drop, average surface temperature and temperature uniformity for the various lattices, flow patterns, and coolants. The optimized weave shows lower pressure drops but higher average surface temperatures and higher temperature variations compared to the standard weave for all flow patterns and both coolants. The bifurcated flow patterns demonstrate lower pressure drops and lower temperature variations but higher average surface temperatures compared to the axial flow pattern for the two weaves and coolants. We also compare the fluidic and thermal performance of the weaves to other common heat dissipation media using the axial flow pattern and both coolants by plotting friction factors, Nusselt numbers and thermal efficiencies as a function of Reynolds numbers that range from 3 to 125. The standard and optimized weaves exhibit relatively high values in flow resistance and heat transfer, and similar values in thermal efficiency compared to other heat exchangers when using water or air. In addition, the weaves provide excellent temperature uniformity in the bifurcated flow patterns, suggesting they are great candidates for applications requiring both high heat removal and uniform temperature distributions such as the cooling of high power laser diodes.

Modeling of Effective Stagnant Thermal Conductivity of Porous Media

Based on one-dimensional analysis of heat conduction, a general formula for the effective stagnant thermal conductivity of spatially periodic porous media is derived without assuming local thermal equilibrium. Furthermore, we discuss the contribution of the contact area between particles to the effective stagnant thermal conductivity in detail, and the modification of the formula is proposed to predict the actual effective stagnant thermal conductivity for the porous media. The present results are in good agreement with experimental results of Nozad et al. (1985, “Heat Conduction in Multi-Phase Systems I: Theory and Experiments for Two-Phase Systems,” Chem. Eng. Sci., 40(5), pp. 843–855) for a packed-sphere bed.

Upscaling of Non-isothermal Reactive Porous Media Flow with Changing Porosity

Abstract Motivated by rock–fluid interactions occurring in a geothermal reservoir, we present a two-dimensional pore scale model of a periodic porous medium consisting of void space and grains, with fluid flow through the void space. The ions in the fluid are allowed to precipitate onto the grains, while minerals in the grains are allowed to dissolve into the fluid, and we take into account the possible change in pore geometry that these two processes cause, resulting in a problem with a free boundary at the pore scale. We include temperature dependence and possible effects of the temperature both in fluid properties and in the mineral precipitation and dissolution reactions. For the pore scale model equations, we perform a formal homogenization procedure to obtain upscaled equations. A pore scale model consisting of circular grains is presented as a special case of the porous medium.

Upscaling nonlinear adsorption in periodic porous media – homogenization approach

We consider the homogenization of a model of reactive flows through periodic porous media involving a single solute which can be absorbed and desorbed on the pore boundaries. This is a system of two convection–diffusion equations, one in the bulk and one on the pore boundaries, coupled by an exchange reaction term. The novelty of our work is to consider a nonlinear reaction term, a so-called Langmuir isotherm, in an asymptotic regime of strong convection. We therefore generalize previous works on a similar linear model. Under a technical assumption of equal drift velocities in the bulk and on the pore boundaries, we obtain a nonlinear monotone diffusion equation as the homogenized model. Our main technical tool is the method of two-scale convergence with drift.

Nonlinear Brinkman-type law as a critical case in the polymer fluid filtration

Well-known results on the filtration laws for the Newtonian fluids, obtained by the homogenization technique, served as a motivation to derive similar, analogous counterparts for the specific type of a non-Newtonian fluid. We studied a stationary filtration of the polymer fluid (Ostwald-de Waele model) through the periodic porous medium. Dimensions in the periodic domain are controlled by the two small parameters, one representing the period and the other controlling the ratio of a solid region and a fluid region of the porous medium. We found the specific critical size of the obstacles representing the impermeable part for which homogenization of the nonlinear Stokes flow results in the nonlinear Brinkman-type law by means of -convergence.

Interfacial dynamics of two immiscible fluids in spatially periodic porous media: the role of substrate wettability.

We delineate the contact line dynamics of two immiscible fluids in a medium having spatially periodic porous structures. The flow is driven by an external applied pressure gradient. We bring out the combined consequences of the solid fraction distribution and the substrate wettability on the resulting dynamics of the contact line, by employing phase-field formalism. We capture the sequence of spatiotemporal events leading to formation of liquid bridges by trapping a small amount of displaced phase fluid between two consecutive porous blocks, as dictated by the combinations of substrate wettability and solid fraction. We also demonstrate the existence of a regime of complete interfacial recovery, depending on the parametric space of the governing parameters under concern. Our results essentially demonstrate the intricate mechanisms by virtue of which the wettabilities of the substrates alter the dynamical evolutions of interfaces and the subsequent shapes and sizes of the adsorbed dispersed phases, bearing far-ranging consequences in several practical applications ranging from oil recovery to groundwater flow.

Stabilization of free surface flows in periodic porous media

The flow of an incompressible Newtonian fluid in a periodic porous medium is studied. The equations governing this motion form a free boundary problem involving a potential flow problem and the evolution of a free surface. The flow is driven by gravitational forces and it is stabilized by an external source. The existence of a unique classical solution for large initial data is proved in the sense that there is an unbounded set of initial conditions for which a unique classical solution exists.

Macrotransport analysis of effective mobility of biomolecules in periodic nano-filter polar arrays

Transport of anisotropic biomolecules and/or charged Brownian particles in periodic porous media is of great importance in the fields of biomedicine, water treatment, and environmental engineering etc. In this paper, we present the modeling of transport of biomolecules in periodic polar arrays based on a numerical analysis of effective mobility. Anisotropic biomolecules are transformed to point-sized Brownian particles through introduction of configurational entropy, and the effective charge and effective transport parameters are calculated using macrotransport theory. As an example, the mobility of short dsDNA fragments in a nano-polar array is calculated. It is demonstrated that when the sizes of the gaps between the nano-poles are similar to or smaller than the size of biomolecules, the configurational entropy has a significant effect on the effective velocity. Difference in configurational entropy in the confined space dominates the partitioning of the molecules. In addition, as the effect of entropic barrier decreases with the strength of external electric field, relatively low voltage is preferred in order to achieve better selectivity.

Homogenization of a Conductive, Convective, and Radiative Heat Transfer Problem in a Heterogeneous Domain

We are interested in the homogenization of heat transfer in periodic porous media where the fluid part is made of long thin parallel cylinders, the diameter of which is of the same order as the period. The heat is transported by conduction in the solid part of the domain and by conduction, convection, and radiative transfer in the fluid part (the cylinders). A nonlocal boundary condition models the radiative heat transfer on the cylinder walls. To obtain the homogenized problem we first use a formal two-scale asymptotic expansion method. The resulting effective model is a convection-diffusion equation posed in a homogeneous domain with homogenized coefficients evaluated by solving so-called cell problems where radiative transfer is taken into account. In a second step we rigorously justify the homogenization process by using the notion of two-scale convergence. One feature of this work is that it combines homogenization with a three dimensional (3D) to two dimensional (2D) asymptotic analysis since the ra...