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Stokes Flow in a Slowly Varying Two-Dimensional Periodic Pore
Abstract
This article presents a series solution to the velocity in a two-dimensional long sinusoidal channel. The approach is based on a stream function formulation of the Stokes problem and a series expansion in terms of the width to the length ratio, which is considered small. Results show how immobile zones may appear and even flow separation and nonturbulent eddies, even in the absence of prima facie dead-end pores. It is shown that the flow tends to concentrate in strips connecting pore throats.
... Hunt and Sahimi 2017;Pyrak-Nolte and Nolte 2016;Xu et al. 2015;Zimmerman and Bodvarsson 1996). Although most problems that arise in these engineering applications are at large scale, a sound knowledge of the controlling mechanism of fluid flow at the pore-scale is essential to provide important insights for solving problems at the larger scale (Berkowitz 2002;Kitanidis and Dykaar 1997). ...
... Wang (1978) derived the perturbation expansion to Stokes flow in a two-dimensional periodic fracture by assigning a pressure difference between the inflow and outflow boundaries using the ratio of mean aperture to fracture length as the perturbation parameter. Kitanidis and Dykaar (1997) derived a similar perturbation solution to Stokes flow in a fracture bounded by two sinusoidal walls. Basha and El-Asmar (2003) conducted and evaluated the perturbation analysis for fracture flow in various fracture wall configurations with the same perturbation parameter. ...
... Basha and El-Asmar 2003;Hasegawa and Fukuoka 1980;Nicholl et al. 1999) and eddy formation at local positions (e.g. Hasegawa and Izuchi 1983;Kitanidis and Dykaar 1997). In this work, perturbation solutions to two-dimensional Navier-Stokes equations were derived for both the PBC and the FBC. ...
Flow in fractures or channels is of interest in many environmental and geotechnical applications. Most previously published perturbation analyses for fracture flow assume that the ratio of the flow in the fracture aperture direction to the flow in the fracture length direction is of the same order as the ratio of mean fracture aperture to fracture length, and hence, the dominant flow is in the fracture length direction. This assumption may impose an overly strict requirement for the flow in the fracture length direction to be dominant, which limits the applicability of the solutions. The present study uses the ratio of aperture variation to length as the perturbation parameter to derive perturbation solutions for flow in two-dimensional fractures under both the pressure boundary condition (PBC) and the flow rate boundary condition (FBC). The solutions are cross-validated with direct numerical solutions of the Navier–Stokes equations and with solutions from published perturbation analyses using the geometry of two-dimensional symmetric wedges and fractures with sinusoidally varying walls. The study shows that compared with the PBC solution, the FBC solution is in a closer agreement with simulation results and provides a better estimate of the fracture transmissivity especially when the inertial effects are more than moderate. The improvement is due mainly to the FBC solution providing a more accurate quantification of the inertial effects. The solutions developed in this study provide improved means of analysing the hydraulic properties of fractures/channels and can be applied to complex flow conditions and fracture geometries.
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... Dykaar [37]. This asymptotic solution provides viscous corrections to the permeability coefficient K in Darcy's law and reveals a non-trivial structure of the flow on local-level (non-inertial flow recirculation within corrugations). ...
... where a 1 (x) = hh − 4(h ) 2 and a 2 (x) = hφ − 6h φ depend on both shape functions h(x) and φ(x). Note, that for the mirror-symmetric channel (φ(x) ≡ 0) this term gives exactly the 2nd order viscous correction from Kitanidis and Dykaar (see Eq. (37) in [37]). ...
... where a 1 (x) = hh − 4(h ) 2 and a 2 (x) = hφ − 6h φ depend on both shape functions h(x) and φ(x). Note, that for the mirror-symmetric channel (φ(x) ≡ 0) this term gives exactly the 2nd order viscous correction from Kitanidis and Dykaar (see Eq. (37) in [37]). ...
It is well-known that inertial particles tend to focus on preferential trajectories in periodic flows. The goal of this thesis was to study the joint effect of particle focusing and sedimentation on their transport through a model 2D fracture with a periodic corrugation. First, single-phase flow though the fracture has been considered: the classical results of the inertial lubrication theory are revisited in order to include asymmetric fracture geometries. Cubic corrections to Darcy's law have been found analytically and expressed in terms of two geometric factors, describing channel geometry. For weakly-inertial particles in a horizontal channel it has been shown that, when inertia is strong enough to balance out the gravity forces, particles focus to some attracting trajectory inside the channel. The full trapping diagram is obtained, that predicts the existence of such attracting trajectory regime depending on the Froude number and on geometric factors. Numerical simulations confirm the asymptotic results for particles with small response times. The influence of the lift force on particle migration has also been studied. In a vertical channel the lift is induced by gravity and leads to complex trapping diagrams. In the absence of gravity the lift is caused by inertial lead/lag of particles and can lead to chaotic particle dynamics. Finally, for dust particles in a vortex pair it has been shown that particles can be trapped into one or two equilibrium points in a reference frame rotating with the vortices. A full trapping diagram has been obtained, showing that any pair of vortices can trap particles, independently of their strength ratio and the direction of rotation
... Renard and De Marsily, 1997;Wood et al., 2003b ). To date the SMM for a periodic system is parameterized by mea- Kitanidis and Dykaar (1997) . suring the travel time distribution of particles across two successive periodic elements and measuring correlation effects via a transition matrix ( Le Borgne et al., 2008a ). ...
... 8 . Fig. 1 also depicts the streamlines for the flow, calculated using a semi-analytical solution ( Dykaar and Kitanidis, 1996;Kitanidis and Dykaar, 1997 ). The solution assumes Stokes flow, with Reynolds number much less than one, meaning inertial effects are negligible, which is generally a good approximation for porous media. ...
... The solution assumes Stokes flow, with Reynolds number much less than one, meaning inertial effects are negligible, which is generally a good approximation for porous media. Details on the flow solution are available from multiple sources ( Bolster et al., 2009a;Dykaar and Kitanidis, 1996;Kitanidis and Dykaar, 1997 ) and so are not presented here. ...
The Spatial Markov Model (SMM) is an upscaled model that has been used successfully to predict effective mean transport across a broad range of hydrologic settings. Here we propose a novel variant of the SMM, applicable to spatially periodic systems. This SMM is built using particle trajectories, rather than travel times. By applying the proposed SMM to a simple benchmark problem we demonstrate that it can predict mean effective transport, when compared to data from fully resolved direct numerical simulations. Next we propose a methodology for using this SMM framework to predict measures of mixing and dilution, that do not just depend on mean concentrations, but are strongly impacted by pore-scale concentration fluctuations. We use information from trajectories of particles to downscale and reconstruct pore-scale approximate concentration fields from which mixing and dilution measures are then calculated. The comparison between measurements from fully resolved simulations and predictions with the SMM agree very favorably.
... On peut aussi estimer que les hypothèses de granulométrie uniforme et de grains sphériques en empilement régulier sont encore plus déterminantes et que le passage en 2D n'est pas critique. Hasegawa and Izuchi (1983) puis Kitanidis and Dykaar (1997) se sont intéressés à l'écoulement dans un chenal d'ouverture variable dont les parois ont une forme sinusoidale. En partant de la formulation en fonction de courant de l'équation de Stokes en 2D et en 13 1.3. ...
... avec a et b deux constantes et Q le débit passant au travers de la section de surface A. Kitanidis and Dykaar (1997) ont cherché une solution analytique au problème de l'écoulement dans un chenal d'ouverture périodique. La géométrie utilisée est 2D et identique à celle décrite dans la section 4.1.1. ...
... Ils dépendent des paramètres géométriques, de la vitesse et de la position. L'expression de chaque terme étant particulièrement long, le lecteur peut se référer aux équations (33) à (38) de l'article de Kitanidis and Dykaar (1997). Chaque terme est évalué séparément, afin de pouvoir constater l'influence de chacun d'entre eux dans le résultat final. ...
In heterogeneous media, flow and transport usually take place in preferential flow channels. Thus, it is of prime interest to take the associated physical processes into account. However, we are limited by current technology and cannot compute the transport of solute at the field scale using the equations of hydrodynamic. Thus, in order to describe solute transport at the field scale, upscaling is required. In this thesis, we study at the scale of the channels the impact of two effects usually neglected. (i) The first effect occurs when a sufficiently concentrated solute is diluted so that the density of the fluid increase significantly. As the density gradient of the solute plume locally modifies the flow field, the solute transport is impacted. In a horizontal smooth channel, the pre-asymptotic regime of dispersion is impacted and the plume is delayed. (ii) Second, when flow is fast enough (or fluid viscosity low enough), inertial effects can dramatically alter the flow field. We study their impact in channels with periodically varying apertures. When the inertial effects increase, recirculation zones grow at the location of the maximum aperture. Solute can then enter by diffusion into these zones and be trapped for a significant duration. (iii) Finally, the coupling of the effects (i) and (ii) is addressed. We reproduce numerically each effect at the hydrodynamic scale and characterize their impact as a function of the Reynolds number, the Péclet number and a dimensionless number that quantify the impact of the density gradients.
... Dykaar [37]. This asymptotic solution provides viscous corrections to the permeability coefficient K in Darcy's law and reveals a non-trivial structure of the flow on local-level (non-inertial flow recirculation within corrugations). ...
... where a 1 (x) = hh − 4(h ) 2 and a 2 (x) = hφ − 6h φ depend on both shape functions h(x) and φ(x). Note, that for the mirror-symmetric channel (φ(x) ≡ 0) this term gives exactly the 2nd order viscous correction from Kitanidis and Dykaar (see Eq. (37) in [37]). ...
... where a 1 (x) = hh − 4(h ) 2 and a 2 (x) = hφ − 6h φ depend on both shape functions h(x) and φ(x). Note, that for the mirror-symmetric channel (φ(x) ≡ 0) this term gives exactly the 2nd order viscous correction from Kitanidis and Dykaar (see Eq. (37) in [37]). ...
It is well-known that inertial particles tend to focus on preferential trajectories in periodic flows. The goal of this thesis was to study the joint effect of particle focusing and sedimentation on their transport through a model 2D fracture with a periodic corrugation. First, single-phase flow though the fracture has been considered: the classical results of the inertial lubrication theory are revisited in order to include asymmetric fracture geometries. Cubic corrections to Darcy's law have been found analytically and expressed in terms of two geometric factors, describing channel geometry. For weakly-inertial particles in a horizontal channel it has been shown that, when inertia is strong enough to balance out the gravity forces, particles focus to some attracting trajectory inside the channel. The full trapping diagram is obtained, that predicts the existence of such attracting trajectory regime depending on the Froude number and on geometric factors. Numerical simulations confirm the asymptotic results for particles with small response times. The influence of the lift force on particle migration has also been studied. In a vertical channel the lift is induced by gravity and leads to complex trapping diagrams. In the absence of gravity the lift is caused by inertial lead/lag of particles and can lead to chaotic particle dynamics. Finally, for dust particles in a vortex pair it has been shown that particles can be trapped into one or two equilibrium points in a reference frame rotating with the vortices. A full trapping diagram has been obtained, showing that any pair of vortices can trap particles, independently of their strength ratio and the direction of rotation.
... This includes micro uidics [27,34], nutrient transport in bloodow [16,37], single and multiphase transport in porous media [7,4,25,24,31,14,3] and transport in groundwater systems [10,11,15,23]. ...
... We dene a two dimensional geometry with sinusoidal wall boundary, as described in refs. [23,4]. ...
... On the pore scale, Reynolds number is typically small ( [23,2] [23] derived an analytical solution for the ow velocity using a perturbation expansion in . However, their approach is no longer possible when one cannot neglect inertial eects:no obvious analytical approach then exists to solving the nonlinear governing Navier-Stokes equations. ...
Contaminant transport in heterogeneous aquifers occurs mostly in the
networks of intersecting channels. The sinusoidal channel geometry is
relevant to transport moderately-disordered porous media, as well as (to
some extent) to flow and transport in fractures with varying apertures.
Here we investigate the spreading of a finite amount of solute entering
such a channel of periodically-varying aperture. In channels of uniform
apertures (parallel plate), when solute buoyancy is negligible, the
advection and diffusion processes eventually lead to the well-known
asymptotic Taylor-Aris dispersion regime. After this asymptotic regime
has been reached, the solute progresses along the fracture at the
average fluid velocity, according to a one-dimensional longitudinal
advection-diffusion process. The corresponding diffusive term features
an apparent dispersion coefficient instead of the molecular diffusion
coefficient. In many real applications the relevant channels do not have
constant aperture. Prior works have shown that deviation from the
parallel plate geometry can significantly alter the behavior, leading to
relative increases or even decreases in the apparent dispersion. These
studies assume small Reynolds number and thus that flow is governed by
the Stokes equation. While this is very often a reasonable assumption,
the Reynolds number can sometimes be of order unity or larger such that
inertial effects are no longer negligible. Increased inertial effects
lead to the presence of recirculation zones, which represent immobile
regions that can have a significant impact on solute transport and in
particular on the asymptotic dispersion. We address flow and geometry
configurations for which inertial effects affect flow and transport. In
these conditions, flow can no longer be predicted by analytical
solutions. We compute the stationary velocity fields based on
Navier-Stokes equations, using a numerical scheme based on a finite
element analysis. The transport problem is then solved numerically by
Lagrangian particle random walk simulations based on the Langevin
equation. Depending on the geometry parameters (ie the aspect ratio of a
cell and the relative amplitude of the aperture fluctuations) and on the
Reynolds number, the size and the position of the recirculation zones
vary. This phenomenon is not only responsible for longer residence times
of the solute in these zones but also for higher velocities in the
middle of the channel. At short times, this leads to unusual spreading
patterns and oscillations in the velocity and apparent dispersion
evolutions over time. At longer time, solute trapping is clearly visible
in the recirculation zones. We characterize the asymptotic dispersion
coefficient as a function of the geometry parameters, the Péclet
number and of the Reynolds number. In our parameters range, a higher
Reynolds number value systematically leads to an increase in the
asymptotic dispersion coefficient. In tracer tests, higher apparent
dispersion coefficient are often interpreted from the parallel plate
model. This study shows that, at sufficiently high Reynolds number
values, the channel uniform aperture inferred in that manner would be
larger than the real mean channel aperture, the discrepancy arising from
the effect of aperture variations.
... While the exact solution is not known, approximate analytical solutions have been obtained as a series expansion in ϵ [47,48]. To fully specify the problem the following boundary conditions are imposed: no-flux and no-slip at the walls, with periodic boundary conditions for the velocity u, together with a constant pressure difference ∆p between the inlet (x = 0) and outlet (x = L) ...
... We will compare our inferred SP results with the 4th-order approximate solution [47,48]. To aid in the comparison all quantities are non-dimensionalized using as characteristic units the viscosity η, the average channel width ⟨w⟩, and the flow rate Q for the corresponding flat parallel plate flow problem, given by the Poiseuille equation [43] ...
We develop a probabilistic Stokes flow framework, using Physics Informed Gaussian Processes, which can be used to solve both forward/inverse flow problems with missing and/or noisy data. The physics of the problem, specified by the Stokes and continuity equations, is exactly encoded into the inference framework. Crucially, this means that we do not need to explicitly solve the Poisson equation for the pressure field, as a physically meaningful (divergence-free) velocity field will automatically be selected. We test our method on a simple pressure driven flow problem, i.e., flow through a sinusoidal channel, and compare against standard numerical methods (Finite Element and Direct Numerical Simulations). We obtain excellent agreement, even when solving inverse problems given only sub-sampled velocity data on low dimensional sub-spaces (i.e., 1 component of the velocity on $1D$ domains to reconstruct $2D$ flows). The proposed method will be a valuable tool for analyzing experimental data, where noisy/missing data is the norm.
... One of the most distinct problems having a wide range of interest is flow through a wavy channel. The literature in this regard displays that a large number of studies have been attempted subject to various flow conditions, however, related to the clear flow situation [10][11][12][13][14][15][16]. Hence, in this regard, we review the developments in the case of clear flow followed by porous media flow. ...
... The task now is to solve the boundary value problem which is described in terms of governing equations and boundary conditions listed in equations (2.6)-(2.11). A perturbation method can be used to find the approximate solution for problems involving a wavy channel [10,11,31,33]. Correspondingly, we assume that the aspect ratio of the channel is small, i.e. δ 2 1, and this allows us to apply perturbation theory. ...
Viscous flow through a symmetric wavy channel filled with anisotropic porous material is investigated analytically. Flow inside the porous bed is assumed to be governed by the anisotropic Brinkman equation. It is assumed that the ratio of the channel width to the wavelength is small (i.e. δ ² ≪1). The problem is solved up to O ( δ ² ) assuming that δ ² λ ² ≪1, where λ is the anisotropic ratio. The key purpose of this paper is to study the effect of anisotropic permeability on flow near the crests of the wavy channel which causes flow reversal. We present a detailed analysis of the flow reversal at the crests. The ratio of the permeabilities (anisotropic ratio) is responsible for the flow separation near the crests of the wall where viscous forces are effective. For a flow configuration (say, low amplitude parameter) in which there is no separation if the porous media is isotropic, introducing anisotropy causes flow separation. On the other hand, interestingly, flow separation occurs even in the case of isotropic porous medium if the amplitude parameter a is large.
... We note that it is already well known that vortices can form in grooves with smooth edges (Hemmat and Borhan, 1995;Kitanidis and Dykaar, 1997) and, particularly, in grooves with sharp edges (Cao and Kitanidis, 1998;Cieslicki and Lasowska, 1999). ...
... However, a special role that these vortices may play in mass transfer has not yet been appreciated. In fact, Kitanidis and Dykaar (1997) ...
Solute transport through a groove is affected by its vortices. Our laboratory and numerical experiments of dye transport through a single axisymmetric groove reveal evidence of enhanced spreading and mixing by the vortex, i.e., a new kind of dispersion called here the vortex dispersion. The uptake and release of contaminants by vortices in porous media is affected by the flow Reynolds number. The larger the flow Reynolds number, the larger is the vortex dispersion, and the larger is the mass-transfer rate between the mobile zone and the vortex. The long known dependence of the mass-transfer rate between the mobile and "immobile" zones in porous media on flow velocity can be explained by the presence of vortices in the "immobile" zone and their uptake and release of contaminants.
... A sine wave geometry was used to introduce different levels of roughness in the channels. This simple geometry has been widely used to provide an idealized representation of surface roughness to investigate its impacts on fluid flow and chemical transport (Kitanidis and Dykaar, 1997;Bolster et al., 2009;Sund et al., 2015;Deng et al., 2018). Section Methodology details the modeling framework, the mathematical principles, and the simulation setups. ...
In various natural and engineered systems, mineral–fluid interactions take place in the presence of multiple fluid phases. While there is evidence that the interplay between multiphase flow processes and reactions controls the evolution of these systems, investigation of the dynamics that shape this interplay at the pore scale has received little attention. Specifically, continuum scale models rarely consider the effect of multiphase flow parameters on mineral reaction rates or apply simple corrections as a function of the reactive surface area or saturation of the aqueous phase, without developing a mechanistic understanding of the pore-scale dynamics. In this study, we developed a framework that couples the two-phase flow simulator of OpenFOAM (open field operation and manipulation) with the geochemical reaction capability of CrunchTope to examine pore-scale dynamics of two phase flow and their impacts on mineral reaction rates. For our investigations, flat 2D channels and single sine wave channels were used to represent smooth and rough geometries. Calcite dissolution in these channels was quantified with single phase flow and two phase flow at a range of velocities. We observed that the bulk calcite dissolution rates were not only affected by the loss of reactive surface area as it becomes occupied by the non-reactive non-aqueous phase, but also largely influenced by the changes in local velocity profiles, e.g., recirculation zones, due to the presence of the non-aqueous phase. The extent of the changes in reaction rates in the two-phase systems compared to the corresponding single phase system is dependent on the flow rate (i.e., capillary number) and channel geometry, and follows a non-monotonic relationship with respect to aqueous saturation. The pore-scale simulation results highlight the importance of interfacial dynamics in controlling mineral reactions and can be used to better constrain reaction rate descriptions in multiphase continuum scale models. These results also emphasize the need for experimental studies that underpin the development of mechanistic models for multiphase flow in reactive systems.
... Following Wang et al. [4] , we set the cutoff α for secondary roughness to be α cutoff = 0.5 (i.e., 0 <α≤0.5) considering that the bulk flow travels along the fracture length direction [29] . Since flow is predominantly in the central area, 10% of the fracture area near both walls accounts for only ~5% of total flow discharge for a parabolic velocity profile [ 4 , 28 ]. ...
Wall roughness is often found to have evident influences on flow-related processes in rock fractures. In this work, we first define the primary wall roughness to have dominating effects on the overall flow, whereas the secondary roughness only generates additional flow resistance and have insignificant impacts on the bulk flow. Accordingly, an effective procedure is presented to quantitatively identify primary and secondary roughness, and the validity of the procedure is tested with flow simulations in a two-dimensional rough fracture. The results show that the reconstructed fracture with only primary roughness can still well resemble the general flow behavior in the original fracture with a steady transmissivity increment of 5% for the Reynolds number from 0.01 to 100. We further examine the effects of both primary and secondary roughness on processes associated with heat transfer and solute transport in fracture flow. It is found that the heat transfer and solute transport behaviors are closely related to the flow behavior under the effect of wall roughness. As the flow velocity increases to cause strong flow inertia, larger eddies can occur to affect the heat transfer and solute transport processes. Although the presence of secondary roughness shows limited effects on the overall transport behaviors, secondary roughness is found to give rise to more irregularly-shaped eddies, which result in different local behaviors. The findings in this study can provide new insights into the roughness effect on fracture flow and associated processes. In addition, the roughness quantification presents a straight-forward evaluation of potential deviations when neglecting small-scale secondary roughness, as is often needed when analyzing larger-scale problems.
... In addition, solving numerically the NSE for 3-D rock fractures with fine representation of void geometries is computationally demanding which has restricted its application to mostly small-scale problems (Brush & Thomson, 2003;Wang et al., 2018;Zou et al., 2017). In this context, a further extension to the CL assumption was made by deriving approximate analytical solutions to the NSE using the perturbation method for 2-D fractures with sinusoidally, linearly, and quadratically varying apertures (Basha & El-Asmar, 2003;Hasegawa & Izuchi, 1983;Kitanidis & Dykaar, 1997). Previous studies have shown that the results from the perturbation solution (PS) are consistent with numerical simulation results (Sisavath et al., 2003;Wang et al., 2019). ...
This study presents a nonlinear Reynolds equation (NRE) for single-phase flow through rock fractures. The fracture void geometry is formed by connected wedge-shaped cells at pore scale, based on the measured aperture field. An approximate analytical solution to two-dimensional Navier-Stokes equations is derived using the perturbation method to account for flow nonlinearity for wedge-shaped geometries. The derived perturbation solution shows that the main contributors to the determination of general flow behaviors in local wedges are the degree of aperture variation relative to mean aperture, the ratio of aperture variation to wedge length, the Reynolds number, and the degree of wedge asymmetry. The transmissivity of the entire fracture is then solved with a field of local cell transmissivity that varies along both longitude and latitude directions on the fracture plane. The performance of the proposed NRE is tested against flow experiments and flow simulations by solving numerically the three-dimensional Navier-Stokes equations for three cases of rock fractures with different void geometries. Results from the proposed model are in close agreement with those obtained from simulations and experiments. As the Reynolds number increases, the pressure difference obtained from the NRE demonstrates the same nonlinear behavior as that obtained from the simulations. Overall, the mean discrepancy between the proposed model and flow simulations is 5.7% for Reynolds number ranging from 0.1 to 20, indicating that the proposed NRE can capture the flow nonlinearity in rock fractures.
... Note that for fracture B, θ was not zero even if the applied dh/dl was tiny (Fig. 3b) when BCs' effect was trivial. This kind of eddy is referred to as the Moffatt or Stokes eddies (Kitanidis and Dykaar, 1997;Moffatt, 1964), which can form around the sharp corners in the viscous flow regime and is entirely dependent on the geometry (Cardenas et al., 2009;Chaudhary et al., 2011). ...
Fluid flow in rock fractures is usually theoretically and numerically investigated on the premise of no-slip boundary condition (BC). However, fluid slippage at rock surface is naturally present in some circumstances that might lead to a violation of no-slip BC. How and to what extent slip BC affects non-Darcian fracture flow remains poorly constrained. This study systematically investigated the slippery flow behaviors in rock fractures under sequentially-increasing pressure gradients. Two competing mechanisms impacting the apparent permeability (k) due to fluid slippage were revealed: k was not only enhanced by fluid slippage at fracture walls but also was reduced by the accelerated eddy growth due to enhanced velocity field. The increment in k caused by slip velocity initially dominated over the decrement caused by eddy growth during this competing process, but this dominant situation could reverse given a sufficiently large pressure gradient with fully-developed eddies. Moreover, the slippery fluid flow behavior was tested to be very sensitive to the slip length based on our sensitivity analysis. Therefore, a reasonable estimation of the slip length would be crucial to accurately determine the effects of fluid slippage. This study deepens our understanding of fluid flow in rock fractures when slip BC is present, and might provide theoretical guidance for organic pollution remediation, oil recovery, and geological carbon sequestration in fractured reservoirs.
... A sound understanding of the conditions and pore-scale processes that 38 physically control the rate of exchange between stagnant and fast-flowing regions is needed to better 39 understand solute spreading and mixing and subsequently the evolution of conservative and reactive 40 transport processes (e.g. Alhashmi et al., 2015;Lichtner and Kang, 2007;Kitanidis and Dykaar, 1997;41 Wirner et al., 2014;Cortis and Berkowitz, 2004;Briggs et al., 2018;Baveye et al., 2018). 42 Solute transport has been widely studied by performing direct numerical simulations at pore scale (see 43 e.g. ...
We derive a novel double-continuum transport model based on pore-scale characteristics. Our approach relies on building a simplified unit cell made up of immobile and mobile continua. We employ a numerically resolved pore-scale velocity distribution to characterize the volume of each continuum and to define the velocity profile in the mobile continuum. Using the simplified unit cell, we derive a closed form model, which includes two effective parameters that need to be estimated: a characteristic length scale and a parameter, R D , given by the ratio of characteristic times that lumps the effect of stagnant regions and escape process. To calibrate and validate our model, we rely on a set of pore-scale numerical simulation performed on a 2D disordered segregated periodic porous medium, taking into account different initial solute distributions. Using a Global Sensitivity Analysis, we explore the impact of the two effective parameters on solute concentration profiles and thereby define a Sensitivity Analysis driven criterion for model calibration. The latter is compared to a classical calibration approach. Our results show that, depending on the initial condition, the mass exchange process between mobile and immobile continua impact on solute profile shape significantly. Our transport model is capable of interpreting both symmetric and highly skewed solute concentration profiles. Effectiveness of the calibration of the two parameters largely depends on the calibration dataset and the selected objective function whose definition can be supported by the implementation of sensitivity analysis. By relying on a sensitivity analysis driven calibration, we are able to provide an accurate and robust interpretation of the concentration profile evolution across different given initial conditions by relying on a unique set of effective parameter values.
... A sound understanding of the conditions and pore-scale processes that physically control the rate of exchange between stagnant and fast-flowing regions is needed to better understand solute spreading and mixing and subsequently the evolution of conservative and reactive transport processes (e.g. Alhashmi et al., 2015;Lichtner and Kang, 2007;Kitanidis and Dykaar, 1997;Wirner et al., 2014;Cortis and Berkowitz, 2004;Briggs et al., 2018;Baveye et al., 2018). ...
We present and derive a novel double-continuum transport model based on pore-scale characteristics. Our approach relies on building a simplified unit cell made up of immobile and mobile continua. We employ a numerically resolved pore-scale velocity distribution to characterize the volume of each continuum and to define the velocity profile in the mobile continuum. Using the simplified unit cell, we derive a closed form model, which includes two effective parameters that need to be estimated: a characteristic length scale and a ratio of waiting times RD that lumps the effect of stagnant regions and escape process. To calibrate and validate our model, we rely on a set of pore-scale numerical simulation performed on a 2D disordered segregated periodic porous medium considering different initial solute distributions. Using a Global Sensitivity Analysis, we explore the impact of the two effective parameters on solute concentration profiles and thereby define a sensitivity analysis driven criterion for model calibration. The latter is compared to a classical calibration approach. Our results show that, depending on the initial condition, the mass exchange process between mobile and immobile continua impact on solute profile shape significantly. By introducing parameter RD we obtain a flexible transport model capable of interpreting both symmetric and highly skewed solute concentration profiles. We show that the effectiveness of the calibration of the two parameters closely depends on the content of information of calibration dataset and the selected objective function whose definition can be supported by of the implementation of model sensitivity analysis. By relying on a sensitivity analysis driven calibration, we are able to provide a good interpretation of the concentration profile evolution independent of the given initial condition relying on a unique set of effective parameter values.
... Hats denote dimensional quantities. Other than for highly idealized geometries (e.g., Kitanidis and Dykaar 1997;Bolster et al. 2009), this equation must be solved numerically, which can be done by any number of methods; see below. Given a solution for the velocity field, the governing equation for transport of the chemical constituents is given by ...
Mixing-driven reactions in porous media are ubiquitous and span natural and engineered environments, yet predicting where and how quickly reactions occur is immensely challenging due to the complex and nonuniform nature of porous media flows. In particular, in many instances, there is an enormous range of spatial and temporal scales over which reactants can mix. This paper aims to review factors that affect mixing-limited reactions in porous media, and approaches used to predict such processes across scales. We focus primarily on the challenges of mixing-driven reactions in porous media at pore scales to provide a concise, but comprehensive picture. We balance our discussion between state-of-the-art experiments, theory and numerical methods, introducing the reader to factors that affect mixing, focusing on the bracketing cases of transverse and longitudinal mixing. We introduce the governing equations for mixing-limited reactions and then summarize several upscaling methods that aim to account for complex pore-scale flow fields. We conclude with perspectives on where the field is going, along with other insights gleaned from this review.
... This phenomenon promotes the formation of fluid recirculations near the walls. Note that these recirculations can also appear under Stokes flow conditions(Kitanidis & Dykaar, 1997;Malevich et al., 2006) but under stronger geometrical constraints. Of FIGURE 10 Variation of the relative error between the local cubic law and Navier-Stokes solutions as a function of the phase shift Δx, with Re = 0.1, = 0.1, and 0 = 0.Variation of the relative error between the local cubic law and Navier-Stokes solutions as a function of Re for the reference geometries(Table 2), for = 0.1 and 0 = 0.2 course, such recirculation zones greatly affect the validity of the LCL. ...
Flow through rough fractures is investigated numerically in order to assess the validity of the local cubic law for different fracture geometries. Two‐dimensional channels with sinusoidal walls having different geometrical properties defined by the aperture, the amplitude and the wavelength of the walls corrugations, the corrugations asymmetry and the phase shift between the two walls are considered to represent different fracture geometries. First, it is analytically shown that the hydraulic aperture clearly deviates from the mean aperture when the walls roughness, the phase shift and/or the asymmetry between the fracture walls are relatively high. The continuity and the Navier‐Stokes equations are then solved by means of the finite element method and the numerical solutions compared to the theoretical predictions of the local cubic law. Reynolds numbers ranging from 0.066 to 66.66 are investigated so as to focus more particularly on the effect of flow inertial effects on the validity of the local cubic law. For low Reynolds number, typically less than 15, the local cubic law properly describes the fracture flow, especially when the fracture walls have small corrugation amplitudes. For Reynolds numbers higher than 15, the local cubic law is valid under the conditions that the fracture presents a low aspect ratio, small corrugation amplitudes, and a moderate phase lag between its walls.
... In this case, a 2D simplification of a representative elementary volume can lead to a sinusoidal channel that accounts for the aperture variation inside the medium. This simplification has been widely applied to study flow and solute dispersion in porous media (Kitanidis and Dikaar [77], Edwards et al. [90], Bolster et al. [91], Bouquain et al. [23]). ...
The aim of the present thesis is to study the transport and deposition of small solid particles in fracture flows. First, single-phase fracture flow is investigated in order to assess the validity of the local cubic law for modeling flow in corrugated fractures. Channels with sinusoidal walls having different geometrical properties are considered to represent different fracture geometries. It is analytically shown that the hydraulic aperture of the fracture clearly deviates from its mean aperture when the walls roughness is relatively high. The finite element method is then used to solve the continuity and the Navier-Stokes equations and to simulate fracture flow in order to compare with the theoretical predictions of the local cubic law for Reynolds numbers Re in the range 0.067-67. The results show that for low Re, typically less than 15, the local cubic law can properly describe the fracture flow, especially when the fracture walls have small corrugation amplitudes. For Re higher than 15, the local cubic law can still be valid under the conditions that the fracture presents a low aspect ratio, small corrugation amplitude, and moderate phase lag between its walls. Second, particle-laden flows are studied. An analytical approach has been developed to show how particles sparsely distributed in steady and laminar fracture flows can be transported for long distances or conversely deposited inside the channel. More precisely, a rather simple particle trajectory equation is established. Based on this equation, it is demonstrated that when particles' inertia is negligible, their behavior is characterized by the fracture geometry and by a dimensionless number W that relates the ratio of the particles sedimentation terminal velocity to the flow mean velocity. The proposed particle trajectory equation is verified by comparing its predictions to particle tracking numerical simulations taking into account particle inertia and resolving the full Navier-Stokes equations. The equation is shown to be valid under the conditions that flow inertial effects are limited. Based on this trajectory equation, regime diagrams that can predict the behavior of particles entering closed channel flows are built. These diagrams enable to forecast if the particles entering the channel will be either deposited or transported till the channel outlet. Finally, an experimental apparatus that was designed to have a practical assessment of the analytical model is presented. Preliminary experimental results tend to verify the analytical model. Overall, the work presented in this thesis give new insights on the behavior of small particles in fracture flows, which may improve our prediction and control of underground contamination, and may have applications in the development of new water filtration and mineral separation techniques
... In solid mechanics, finding the displacement of the bending of elastic plates involves solving the biharmonic equation. In fluid mechanics, the stream function of incompressible Stokes flow in 2D space is the solution of a biharmonic equation as well [44]. Atluri et al. [27] proposed the meshless local Petrov-Galerkin method with the modification of MLS approximation to solve fourth-order problems of thin beams that in this modification, the slope is introduced as an additional independent variable. ...
We are going to study a simple and effective method for the numerical solution of the closed interface boundary value problem with both discontinuities in the solution and its derivatives. It uses a strong-form meshfree method based on the moving least squares (MLS) approximation. In this method, for the solution of elliptic equation, the second-order derivatives of the shape functions are needed in constructing the global stiffness matrix. It is well-known that the calculation of full derivatives of the MLS approximation, especially in high dimensions, is quite costly. In the current work, we apply the diffuse derivatives using an efficient technique. In this technique, we calculate the higher-order derivatives using the approximation of lower-order derivatives, instead of calculating directly derivatives. This technique can improve the accuracy of meshfree point collocation method for interface problems with nonhomogeneous jump conditions and can efficiently estimate diffuse derivatives of second- and higher-orders using only linear basis functions. To introduce the appropriate discontinuous shape functions in the vicinity of interface, we choose the visibility criterion method that modifies the support of weight function in MLS approximation and leads to an efficient computational procedure for the solution of closed interface problems. The proposed method is applied for elliptic and biharmonic interface problems. For the biharmonic equation, we use a mixed scheme, which replaces this equation by a coupled elliptic system. Also the application of the present method to elasticity equation with discontinuities in the coefficients across a closed interface has been provided. Representative numerical examples demonstrate the accuracy and robustness of the proposed methodology for the closed interface problems. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2014
... Nevertheless , under a number of simplifying assumptions , analytical solutions can be found by perturbative methods . Kitanidis and Dykaar ( 1997 ) and Di Federico et al . ( 2002 ) developed an analytic method to determine the solution of Stokes ' problem for a 2D , axis symmetric , periodic , waving channel , in terms of a power series expansion , under the assumption that the width is much smaller than the wavelength . ...
The role of the microvascular network geometry on transport phenomena in solid tumors and its interplay with the leakage and pressure drop across the vessels is qualitatively and quantitatively discussed. Our starting point is a multiscale homogenization, suggested by the sharp length scale separation that exists between the characteristic vessels and tumor tissue spatial scales, referred to as the microscale and the macroscale, respectively. The coupling between interstitial and capillary compartment is described by a double Darcy model on the macroscale, whereas the geometric information on the microvascular structure is encoded in the effective hydraulic conductivities, which are numerically computed solving classical differential problems on the microscale representative cell. Then, microscale information is injected into the macroscopic model, which is analytically solved in a prototypical geometry and compared with previous experimentally validated, phenomenological models. In this way, we are able to capture the role of the standard blood flow determinants in the tumor, such as the tumor radius, tissue hydraulic conductivity and vessels permeability, as well as the influence of the vascular tortuosity on fluid convection. The results quantitatively confirm that transport of blood (and, as a consequence, of any advected anti-cancer drug) can be dramatically impaired by increasing the geometrical complexity of the microvasculature. Hence, our quantitative analysis supports the argument that geometric regularization of the capillary network improve blood transport and drug delivery in the tumor mass.
... Additionally, transport through periodic porous flow becomes Fickian after approximately 12 pores (Cardenas, 2008(Cardenas, , 2009Bolster et al., 2009). This approach of scaling a single grain to Darcy scales has been followed in many pore-scale examinations of permeability, e.g., Kitanidis and Dykaar (1997), Balhoff and Wheeler (2009), Chaudhary et al. (2011. This simplification also allows for the sulphur kinetics described in Eqs. ...
It has recently been hypothesized that bulk denitrification rates in carbonate sands may be enhanced by reactions occurring in the intra-granular pores, cracks and crevices. We tested this hypothesis using a series of flow and reactive transport models spanning from the pore-scale (∼mm) to the continuum scales (∼10 cm bedforms). Pore-scale simulations solved the coupled Navier–Stokes and Brinkman equations and represented flow-through reactor experiments previously performed on coral reef sands. The results revealed that intra-granular transport and reactions can explain over-all denitrification enhancement. A sensitivity study with a single grain diffusive transport model showed that in the majority of cases, the resultant increase in denitrification was not coupled to nitrification within a single grain. Only for large grain diameters of 2 and 4 mm was coupled nitrification–denitrification important. In most cases, coupled nitrification–denitrification instead arose as conditions became more reducing along a flow path, as is the case in quartz sands without intragranular pores. An intra-granular reaction rate based on a single grain model was incorporated into a continuum-scale Darcy flow and reactive transport model for a rippled sand bed, where porewater flow is driven by the turbulent current over the ripple. The results of the Darcy-scale model suggest that intra-granular pores increase the amount of slow-flowing areas within which coupled nitrification–denitrification can occur. We conclude that the complex advective flow field does not strongly inhibit denitrification enhancement by carbonate sand grains, as it does in silica sands. Thus, intra-granular reactions may enhance bulk denitrification in carbonate sediment with porous grains under natural advective conditions.
... In particular, as inertial effects increase, the presence of recirculation zones also increases. While such recirculation zones can and do occur for flows at low (and even vanishing) Reynolds number [40,35,10] 4 they become more pronounced as inertial effects become increasingly important; ...
We study solute transport in a periodic channel with a sinusoidal wavy boundary when inertial flow effects are sufficiently large to be important, but do not give rise to turbulence. This configuration and setup are known to result in large recirculation zones that can act as traps for solutes; these traps can significantly affect dispersion of the solute as it moves through the domain. Previous studies have considered the effect of inertia on asymptotic dispersion in such geometries. Here we develop an effective spatial Markov model that aims to describe transport all the way from preasymptotic to asymptotic times. In particular we demonstrate that correlation effects must be included in such an effective model when Péclet numbers are larger than O(100) in order to reliably predict observed breakthrough curves and the temporal evolution of second centered moments. For smaller Péclet numbers correlation effects, while present, are weak and do not appear to play a significant role. For many systems of practical interest, if Reynolds numbers are large, it may be typical that Péclet numbers are large also given that Schmidt numbers for typical fluids and solutes can vary between 1-500. This suggests that when Reynolds numbers are large, any effective theories of transport should incorporate correlation as part of the upscaling procedure, which many conventional approaches currently do not do. We define a novel parameter to quantify the importance of this correlation. Next, using the theory of CTRWs we explain a to date unexplained phenomenon of why dispersion coefficients for a fixed Péclet number increase with increasing Reynolds number, but saturate above a certain value. Finally we also demonstrate that effective preasymptotic models that do not adequately account for velocity correlations will also not predict asymptotic dispersion coefficients correctly.
... It should be noted that Kitanidis and Dykaar [1997] computed the perturbation solution to the Stokes equations for flow between two out-of-phase (i.e., un-mated) sinusoidal walls, the aperture of which is still represented by eq. (15), and found precisely the same expression for the ratio T Stokes /T Reynolds . ...
The mathematical analysis of the flow of a single-phase Newtonian fluid
through a rough-walled rock fracture is reviewed, starting with the
Navier-Stokes equations. By a combination of order-of-magnitude
analysis, appeal to available analytical solutions, and reanalysis of
some data from the literature, it is shown that the Navier-Stokes
equations can be linearized if the Reynolds number is less than about
10. Further analysis shows that the linear Stokes equations can be
replaced by the simpler Reynolds lubrication equation if the wavelength
of the dominant aperture variations is about three times greater than
the mean aperture. However, this criterion does not seem to be strongly
obeyed by all fractures. The Reynolds equation (i.e., the local cubic
law) may therefore suffice in estimating fracture permeabilities to
within a factor of about 2, but more accurate estimates will require
solution of the Stokes equations. Similarly, estimates of mean aperture
based on inverting transmissivity data may have errors of a factor of
two if any version of the local cubic law is used to relate
transmissivity to mean aperture.
... [4] Grains comprising geologic porous media can be very angular to round in shape, which results in flow channels having a diverse range of diverging-converging pore to pore-throat geometries. Few studies have investigated the fluid flow fields in idealized diverging-converging pores, but with simplistic pore geometries ; for example, pore walls with sinusoidal curves [Bolster et al., 2009;Bouquain et al., 2012;Dykaar and Kitanidis, 1996;Kitanidis and Dykaar, 1997;Malevich et al., 2006;Pozrikidis, 1987;Sisavath et al., 2001], ellipses [McClure et al., 2010], a box shape [Cao and Kitanidis, 1998;Ruth, 1993, 1994;Meleshko, 1996;Panfilov and Fourar, 2006], tortuous pores [Cardenas, 2008;Cardenas et al., 2007;Chaudhary et al., 2011;Fourar et al., 2004], or periodic porous media [Brenner and Adler, 1982]. Moreover, most of the above-mentioned studies use Stokes flow or viscous flow, and only few inspect the flow fields in detail at increasing inertial flow regimes [Chaudhary et al., 2011;Fourar et al., 2004;Leneweit and Auerbach, 1999;Ma and Ruth, 1993;Meleshko, 1996;Panfilov and Fourar, 2006]. ...
[1] In this article, the effects of different diverging-converging pore geometries were investigated, and the microscale fluid flow and effective hydraulic properties from these pores were compared with that of a pipe from viscous to inertial laminar flow regimes. The flow fields are obtained using computational fluid dynamics, and the comparative analysis is based on a new dimensionless hydraulic shape factor β, which is the “specific surface” scaled by the length of pores. Results from all diverging-converging pores show an inverse pattern in velocity and vorticity distributions relative to the pipe flow. The hydraulic conductivity K of all pores is dependent on and can be predicted from β with a power function with an exponent of 3/2. The differences in K are due to the differences in distribution of local friction drag on the pore walls. At Reynolds number (Re) ∼ 0 flows, viscous eddies are found to exist almost in all pores in different sizes, but not in the pipe. Eddies grow when Re → 1 and leads to the failure of Darcy's law. During non-Darcy or Forchheimer flows, the apparent hydraulic conductivity Ka decreases due to the growth of eddies, which constricts the bulk flow region. At Re > 1, the rate of decrease in Ka increases, and at Re >> 1, it decreases to where the change in Ka ≈ 0, and flows once again exhibits a Darcy-type relationship. The degree of nonlinearity during non-Darcy flow decreases for pores with increasing β. The nonlinear flow behavior becomes weaker as β increases to its maximum value in the pipe, which shows no nonlinearity in the flow; in essence, Darcy's law stays valid in the pipe at all laminar flow conditions. The diverging-converging geometry in pores plays a critical role in modifying the intrapore fluid flow, implying that this property should be incorporated in effective larger-scale models, e.g., pore-network models.
... The terms hЈ and hЉ are the first and second derivatives of h with respect to Z . This final result is similar to the result obtained by Kitanidis and Dykaar, 16 who developed a series solution for a two-dimensional sinusoidal channel. ...
Creeping flow of a Newtonian fluid through tubes of varying radius is studied. Using an asymptotic series solution for low Reynolds number flow, velocity profiles and streamlines are obtained for constricted tubes, for various values of constriction wavelength and amplitude. A closed-form expression is derived to estimate the pressure drop through this type of tube. The results obtained with this new expression are compared to data from previous experimental and numerical studies for sinusoidally constricted tubes. Good agreement is found in the creeping flow regime for the pressure drop versus flow rate relationship. Our method offers an improvement over the integrated form of the Hagen-Poiseuille equation (i.e., lubrication approximation), which does not account for the wavelength of the constrictions.
Cette thèse a pour objet l'étude de la dispersion, c'est à dire la vitesse d'étalement de particules traceurs, dans des milieux hétérogènes, présentant une périodicité spatiale. Ce processus se quantifie par un coefficient de diffusivité effective. Il est bien connu que la présence de parois ou d'obstacles diminue la dispersion, tandis qu'un écoulement de cisaillement a pour effet de l'augmenter. Cependant, la plupart des études considèrent des parois parfaitement réfléchissantes. Dans un premier chapitre, nous dérivons une expression générale de la diffusivité effective pour un modèle de diffusion médiée par la surface qui inclut la possibilité pour les particules d'adhérer et de diffuser sur les parois. En appliquant cette formule à deux systèmes (un réseau d'obstacles sphériques et un canal périodique), nous montrons que ce mécanisme d'adhésion permet d'augmenter la dispersion par rapport au cas de parois réfléchissantes, et ce même si la diffusion à la surface est plus lente que dans le volume. Dans un deuxième temps, nous examinons les effets combinés d'un écoulement visqueux et du confinement sur la dispersion dans un canal ondulé. Nous mettons en évidence à l'aide d'expressions analytiques la compétition entre ces deux effets antagonistes dans la limite de faibles ondulations des parois. Dans la limite opposée, les fortes ondulations des parois ont pour effet de piéger les particules traceurs. Nous mettons en évidence à l'aide de l'expression de la diffusivité une équivalence avec un problème de dispersion de Taylor dans un canal uniforme dont les parois sont partiellement adsorbantes. Dans un troisième temps, nous prenons en compte simultanément l'ondulation des parois, l'écoulement visqueux ainsi que la possibilité d'adhésion aux surfaces. En dérivant un formalisme général, nous montrons que la combinaison de ces trois paramètres induit des effets non-triviaux sur la dispersion. Dans un dernier chapitre, nous nous intéressons à la dynamique d'une particule Brownienne confinée dans un canal plat. Nous dérivons une expression généralisée de la diffusivité de Taylor en exprimant le deuxième cumulant. De même, nous quantifions la non-gaussianité de la distribution en position suivant l'axe du canal en étudiant analytiquement les cumulants d'ordres supérieurs.
Los procesos de disolución pueden afectar las propiedades macroscópicas de las rocas y en consecuencia modificar los patrones de flujo a largo plazo. En este trabajo se presenta un modelo teórico para describir la evolución temporal de la porosidad y la conductividad hidráulica de una roca sometida a procesos de disolución. Para derivar el modelo, la porosidad de la roca se representa mediante un conjunto de poros cilíndricos de radio variable cuya distribución obedece a una ley fractal. En base a esta descripción y propiedades físicas conocidas se obtienen expresiones analíticas para la porosidad y la conectividad hidráulica en función del radio máximo y la dimensión fractal. Estas expresiones pueden combinarse para obtener una fórmula similar a la ecuación de Kozeny-Carman. Finalmente, asumiendo una disolución constante es posible derivar expresiones analíticas cerradas para la porosidad y la permeabilidad que dependen explícitamente del tiempo. Para validar el modelo propuesto se comparan las expresiones analíticas obtenidas con un ensayo de laboratorio realizado sobre una muestra de arenisca de baja porosidad
Heterogeneity across a broad range of scales in geologic porous media often manifests in observations of non-Fickian or anomalous transport. While traditional anomalous transport models can successfully make predictions in certain geological systems, increasing evidence suggests that assumptions relating to independent and identically distributed increments constrain where and when they can be reliably applied. A relatively novel model, the Spatial Markov model (SMM), relaxes the assumption of independence. The SMM belongs to the family of correlated continuous time random walks and has shown promise across a wide range of transport problems relevant to natural porous media. It has been successfully used to model conservative as well as more recently reactive transport in highly complex flows ranging from pore scales to much larger scales of interest in geology and subsurface hydrology. In this review paper we summarize its original development and provide a comprehensive review of its advances and applications as well as lay out a vision for its future development.
The effective diffusivity of a Brownian tracer in unidirectional flow is well known to be enhanced due to shear by the classic phenomenon of Taylor dispersion. At long times, the average concentration of the tracer follows a simplified advection–diffusion equation with an effective shear-dependent dispersivity. In this work, we make use of the generalized Taylor dispersion theory for periodic domains to analyze tracer dispersion by peristaltic pumping. In channels with small aspect ratios, asymptotic expansions in the lubrication limit are employed to obtain analytical expressions for the dispersion coefficient at both small and high Péclet numbers. Channels of arbitrary aspect ratios are also considered using a boundary integral formulation for the fluid flow coupled to a conservation equation for the effective dispersivity, which is solved using the finite-volume method. Our theoretical calculations, which compare well with results from Brownian dynamics simulations, elucidate the effects of channel geometry and pumping strength on shear-induced dispersion. We further discuss the connection between the present problem and dispersion due to Taylor’s swimming sheet and interpret our results in the purely diffusive regime in the context of Fick–Jacobs theory. Our results provide the theoretical basis for understanding passive scalar transport in peristaltic flow, for instance, in the ureter or in microfluidic peristaltic pumps.
Diffusion of particles carried by Poiseuille flow of the surrounding solvent in a three-dimensional (3D) tube of varying diameter is considered. We revisit our mapping technique [F. Slanina and P. Kalinay, Phys. Rev. E 100, 032606 (2019)], projecting the corresponding 3D advection-diffusion equation onto the longitudinal coordinate and generating an effective one-dimensional modified Fick-Jacobs (or Smoluchowski) equation. A different scaling of the transverse forces by a small auxiliary parameter ε is used here. It results in a recurrence scheme enabling us to derive the corrections of the effective diffusion coefficient and the averaged driving force up to higher orders in ε. The new scaling also preserves symmetries of the stationary solution in any order of ε. Finally we show that Reguera-Rubí's formula, widely applied for description of diffusion in corrugated tubes, can be systematically corrected by the strength of the flow Q; we give here the first two terms in the form of closed analytic formulas.
We investigate diffusion of colloidal particles carried by flow in tubes of variable diameter and under the influence of an external field. We generalize the method mapping the three-dimensional confined diffusion onto an effective one-dimensional problem to the case of nonconservative forces and use this mapping for the problem in question. We show that in the presence of hydrodynamic drag, the lowest approximation (the Fick-Jacobs approximation) may be insufficient, and inclusion of at least the first-order correction is desirable to obtain more reliable results. As a practical application, we use the method for investigation of separation of colloidal particles carried by a fluid flow according to their size, using flotation and centrifugation.
The Spatial Markov Model (SMM) is an upscaled model with a strong track record in predicting upscaled behavior of conservative solute transport across hydrologic systems. Here we propose an SMM that can account for reactive linear adsorption and desorption processes and test it on a simple benchmark problem: flow and transport through an idealized periodic wavy channel. The methodology is built using trajectories that are obtained from a single high resolution random walk simulation of conservative transport across one periodic element. Our approach encodes information about where a particle starts at the inlet, where it leaves at the outlet, how long it takes to cross the domain and one additional piece of information, the number of times a particle strikes the boundary, with the objective of predicting large scale transport with arbitrary linear adsorption and desorption rates. Our benchmark problem demonstrates that predictions made with our proposed SMM agree favorably with results from direct numerical simulations, which resolve the full transport problem.
We provide analytical formulas for the movement of spherical particles in a corrugated tube, in the approximation of small amplitude of the tube diameter variation. We calculate how the particle is pushed toward the wall at some places and pulled off the wall at others. We show that this effect causes rectification of the particle movement, when the direction of the fluid flow is alternated, thus leading to the hydrodynamic ratchet effect. We propose such scheme as a particle-separation device.
A regular perturbation method in which the small parameter is taken as the characteristic frequency of variation of a two-dimensional slit wall height is applied to the creeping flow of a viscoelastic fluid. The Upper-Convected Maxwell model has been chosen as the constitutive equation. The equations of motion are expressed in terms of the stream function as the only unknown. The stream function is expressed then as a regular perturbation expansion up to the fourth order in the small parameter. When these expansions are written for slits with slowly varying walls of any shape, curvature and higher-order derivatives appear in the expansion solution. From the stream function solution, we distinguish two different types of viscoelastic higher-order effects: one is symmetric and the other is a loss of fore-and-aft flow symmetry due to the convected derivatives appearing in the constitutive equation. The expansion is formally validated with a ratio between two consecutive higher-order terms. It shows that the accuracy of the perturbation will depend on four factors: the small parameter ϵ, the Weissenberg number We0, the local depth h and the shape of the slit. In some instances, Padé approximants can restore a diverging perturbation expansion and extend the range of viscoelastic levels for which the expansion is usable. Results for diverging and for converging flows in a wedge, as well as flows in corrugated slits, are checked against Finite Elements Method calculations. At the same level of viscoelasticity, the stress profiles are completely different in a diverging flow from a converging flow. In particular, the shear stress component of the stress tensor becomes non-monotonic in divergent flows.
We investigate analytically a microfluidic device consisting of a tube with a nonuniform but spatially periodic diameter, where a fluid driven back and forth by a pump carries colloidal particles. Although the net flow of the fluid is zero, the particles move preferentially in one direction due to the ratchet mechanism, which occurs due to the simultaneous effect of inertial hydrodynamics and Brownian motion. We show that the average current is strongly sensitive to particle size, thus facilitating colloidal particle sorting.
This chapter provides an overview of the simplest transport properties of single fractures, namely permeability, conductivity and solute transport.
This chapter is concerned with dispersion of nonreactive solutes in saturated porous media. It reviews velocity‐based solute transport models, and defines their parameters. The chapter presents the different mechanisms contributing to the spreading of a conservative nonreactive solute in saturated soil. It also defines pore characteristics relevant to solute transport and discusses methods of measuring them. The chapter also reviews experimental studies relating pore characteristics to solute dispersion. It is then devoted to theoretical models for solute transport based on various pore characteristics. These models are presented in order of increasing complexity in their representation of the pore space geometry. Emerging areas of research are identified. Further research into the feasibility of predicting dispersion at the reservoir scale by renormalizing core‐scale dispersivity measurements may prove fruitful. Finally, the chapter summarizes relationships between solute spreading and pore characteristics, and discusses the predictive potential of pore‐based transport models.
A coupling analysis was proposed to study the hydro-mechanical response for the fluid flow inside fractured rock mass with the method of discontinuous deformation analysis (DDA). Detailed governing equations, including basic assumptions and related algorithms were introduced to study the interaction between the fluid flow along the fractures and the movement of the blocks. Residual flow method was applied to determine the location of free surfaces for the fluid flow problems, which was then coupled with the mechanical motion equations expressed in DDA. A simple validation example was used to test the accuracy of the coupled model. The study helped understand the influence in the fractured rock structures under the condition of fluid flow. Copyright © 2013 by The Society for Rock Mechanics & Engineering Geology.
In this study we extend the Spatial Markov model, which has been successfully used to upscale conservative transport across a diverse range of porous media flows, to test if it can accurately upscale reactive transport, defined by a spatially heterogeneous first order degradation rate. We test the model in a well known highly simplified geometry, commonly considered as an idealized pore or fracture structure, a periodic channel with wavy boundaries. The edges of the flow domain have a layer through which there is no flow, but in which diffusion of a solute still occurs. Reactions are confined to this region. We demonstrate that the Spatial Markov model, an upscaled random walk model that enforces correlation between successive jumps, can reproduce breakthrough curves measured from microscale simulations that explicitly resolve all pertinent processes. We also demonstrate that a similar random walk model that does not enforce successive correlations is unable to reproduce all features of the measured breakthrough curves.
Copyright © 2015 Elsevier B.V. All rights reserved.
During composites processing, thermoset polymer resin is injected into network of densely packed continuous fibers with the goal of complete saturation. The formation and entrapment of gas bubbles, due to the presence of air or volatiles during processing, will create voids in the cured composite. Voids can degrade the mechanical properties and increase design risks and costs. Thus, there is a need to understand the two phase flow of resin and bubbles through channels within fibrous porous media. A two-phase flow model of a channel containing resin and gas bubbles is presented. The boundaries of the channel are porous media with sinusoidal wavy or corrugated walls, which represents the wavy nature of the porous media. This causes the change in bubble movement dynamics, due to the non-uniform pressure gradient induced by non-rectilinear walls. Parameters such as porous media permeability, channel waviness, and channel width are studied to investigate the influence of wavy porous wall effects on the two-phase flow and how these parameters may influence the likelihood of bubble entrapment. By maximizing the bubble mobility, which is the ratio of average bubble velocity to average resin velocity, one can remove the bubbles from the system before the resin cures.
In this standard reference of the field, theoretical and experimental approaches to flow, hydrodynamic dispersion, and miscible displacements in porous media and fractured rock are considered. Two different approaches are discussed and contrasted with each other. The first approach is based on the classical equations of flow and transport, called 'continuum models'. The second approach is based on modern methods of statistical physics of disordered media; that is, on 'discrete models', which have become increasingly popular over the past 15 years. The book is unique in its scope, since (1) there is currently no book that compares the two approaches, and covers all important aspects of porous media problems; and (2) includes discussion of fractured rocks, which so far has been treated as a separate subject. Portions of the book would be suitable for an advanced undergraduate course. The book will be ideal for graduate courses on the subject, and can be used by chemical, petroleum, civil, environmental engineers, and geologists, as well as physicists, applied physicist and allied scientists that deal with various porous media problems.
Transport of overdamped Brownian particles in a periodic hydrodynamical channel is investigated in the presence of an asymmetric unbiased force, a transverse gravitational force, and a pressure-driven flow. With the help of the generalized Fick–Jacobs approach, we obtain an analytical expression for the directed current and the generalized potential of mean force. It is found that, when the transverse gravitational force is larger than a certain value, the current is suppressed. Moreover, when the temporal asymmetry parameter of the unbiased force is negative, the current is always negative. However, when the temporal asymmetry parameter is positive, the transverse gravitational force and the pressure drop not only determine the direction of the current but also affect its amplitude. In particular, the competition between the asymmetric unbiased force and the pressure drop can result in multiple current reversals.
[1] Diffusive mass transfer in rock fractures is strongly affected by fluid flow in addition to material properties. The flow-dependence of matrix diffusion is quantified by a random variable (“transport resistance”) denoted as β [T/L] and computed from the flow field by following advection trajectories. The numerical methodology for simulating fluid flow is mesh-free, using Fup basis functions. A generic statistical model is used for the transmissivity field, featuring three correlation structures: (i) highly connected non-multi-Gaussian; (ii) poorly connected (or disconnected) non-multi-Gaussian; and (iii) multi-Gaussian. The moments of β are shown to be linear with distance, irrespective of the structure, after approximately 10 integral scales of ln T. Percentiles of β are found to be linear with the mean β when considering all three structures. Taking advantage of this property, a potentially useful relationship is presented between β percentiles and the fracture mean water residence time that integrates all structures with high variability; it can be used in discrete fracture network simulations where T statistical data on individual fractures are not available.
Dispersive transport in porous media is usually described through a Fickian model, in which the flux is the product of a dispersion tensor times the concentration gradient. This model is based on certain implicit assumptions, including slowly varying conditions. About fifty years ago, it was first suggested that the parameterization of the second-order dispersion tensor for anisotropic porous media involves a fourth-order dispersivity tensor. However, the properties of the dispersivity tensor have not been adequately studied. This work contributes to achieving a better grasp of dispersion in anisotropic porous media through a number of ways. First, with clearly stated assumptions and from first principles, we use the method of moments to derive a mathematical formula for the fourth-order dispersivity tensor, and show that it is a function of pore geometry, fluid velocity, and pore diffusion. Second, by using pore-scale flow and transport simulations through orderly and randomly packed 2-D and 3-D porous media, we evaluate the effects of the three factors on dispersivity. Different relationships with the Peclét number are observed for the longitudinal and transverse dispersivities and for orderly and randomly packed media. Third, we discuss the limitations of 2-D periodic media with simple structures in computing transverse dispersivity, which is more accurately predicted in the 3-D periodic media and 2-D randomly packed media. Fourth, we exhibit through numerical simulations that the method of moments can, computational limitations notwithstanding, be extended to stationary porous media.
We investigate the upscaling of dispersion from a pore-scale analysis of
Lagrangian velocities. A key challenge in the upscaling procedure is to
relate the temporal evolution of spreading to the pore-scale velocity
field properties. We test the hypothesis that one can represent
Lagrangian velocities at the pore scale as a Markov process in space.
The resulting effective transport model is a continuous time random walk
(CTRW) characterized by a correlated random time increment, here denoted
as correlated CTRW. We consider a simplified sinusoidal wavy channel
model as well as a more complex heterogeneous pore space. For both
systems, the predictions of the correlated CTRW model, with parameters
defined from the velocity field properties (both distribution and
correlation), are found to be in good agreement with results from direct
pore-scale simulations over preasymptotic and asymptotic times. In this
framework, the nontrivial dependence of dispersion on the pore boundary
fluctuations is shown to be related to the competition between
distribution and correlation effects. In particular, explicit inclusion
of spatial velocity correlation in the effective CTRW model is found to
be important to represent incomplete mixing in the pore throats.
We present the development of an improved 2-D flow equation for rough-walled fractures. Our improved equation accounts for the influence of midsurface tortuosity and the fact that the aperture normal to the midsurface is in general smaller than the vertical aperture. It thus improves upon the well-known Reynolds equation that is widely used for modeling flow in fractures. Unlike the Reynolds equation, our approach begins from the lubrication approximation applied in an inclined local coordinate system tangential to the fracture midsurface. The local flow equation thus obtained is rigorously transformed to an arbitrary global Cartesian coordinate system, invoking the concepts of covariant and contravariant transformations for vectors defined on surfaces. Unlike previously proposed improvements to the Reynolds equation, our improved flow equation accounts for tortuosity both along and perpendicular to a flow path. Our approach also leads to a well-defined anisotropic local transmissivity tensor relating the representations of the flux and head gradient vectors in a global Cartesian coordinate system. We show that the principal components of the transmissivity tensor and the orientation of its principal axes depend on the directional local midsurface slopes. In rough-walled fractures, the orientations of the principal axes of the local transmissivity tensor will vary from point to point. The local transmissivity tensor also incorporates the influence of the local normal aperture, which is uniquely defined at each point in the fracture. Our improved flow equation is a rigorous statement of mass conservation in any global Cartesian coordinate system. We present three examples of simple geometries to compare our flow equation to analytical solutions obtained using the exact Stokes equations: an inclined parallel plate, and circumferential and axial flows in an incomplete annulus. The effective transmissivities predicted by our flow equation agree very well with values obtained using the exact Stokes equations in all these cases. We discuss potential limitations of our depth-integrated equation, which include the neglect of convergence/divergence and the inaccuracies implicit in any depth-averaging process near sharp corners where the wall and midsurface curvatures are large.
Predicting the dissolution rate of nonaqueous phase liquids (NAPLs) in groundwater is difficult, as the effects of variable pore and NAPL blob geometry are poorly understood. To elucidate these effects, fluorescence microscopy and digital image analysis were used to quantify the size and location of variably distributed NAPL blobs during dissolution in homogeneous and heterogeneous pore networks etched into silicon wafers. Results show that the dissolution rate constant (expressed as the Sherwood number, Sh) is relatively constant regardless of pore and NAPL blob geometry when the average mass transfer length scale remains constant during dissolution. Results also show that Sh increases with Peclet (Pe) between 2 and 26 and then levels off. The limiting value of Sh reached depends on the average diffusion length scale; this length scale was directly calculated and found to vary depending on the pore and NAPL blob geometry. For example, the average diffusion length scale decreases (and Sh increases) as the pore throat width to grain diameter increases. Last, results show that the volumetric NAPL content (thetan) is linearly related to the specific NAPL-water interfacial area (ait) over much of the dissolution process. However, this relationship depends on the pore and blob size distribution. For example, when multipore blobs control dissolution, the relationship between these parameters will change as smaller blobs dominate dissolution at low thetan. These results are important because existing mass transfer correlations do not account for limiting values of Sh that can be obtained at high Pe for the effect of blob or pore geometry on the average diffusion length scale (and therefore on Sh) or for the effect of pore geometry and transient blob size distribution on the relationship between ait and thetan.
This work derives the fracture flow equation from the two-dimensional
steady form of the Navier-Stokes equation. Asymptotic solutions are
obtained whereby the perturbation parameter is the ratio of the mean
width over the length of the fracture segment. The perturbation
expansion can handle arbitrary variation of the fracture walls as long
as the dominant velocity is in the longitudinal direction. The effect of
the matrix-fracture interaction is also taken into account by allowing
leakage through the fracture walls. The perturbation solution is used to
obtain an estimate of the flow rate and the fracture transmissivity as
well as the velocity and the pressure distribution in fractures of
various geometries. The analysis covers eight different configurations
of fracture geometry including linear and curvilinear variation as well
as sinusoidal variation in the top and bottom walls with varying
horizontal alignment and roughness wavelengths. The zero-order solution
yields the Reynolds lubrication approximation, and the higher-order
equations provide a correction term to the flow rate in terms of the
roughness frequency and the Reynolds number. For sinusoidal and linear
walls, the mathematical analysis shows that the zero-order flow rate
could be expressed in terms of the maximum to minimum width ratio. For
equal widths at both ends of the fracture, the first-order correction is
zero. For sinusoidal fractures, the flow rate decreases with increasing
Reynolds number and with increasing roughness amplitude and frequency.
The effect of leakage is to create a nonuniform flow distribution in the
fracture that deviates significantly from the flow rate estimate for
impermeable walls. The derived flow expressions can provide a more
reliable tool for flow and transport predictions in fractured domain.
Flow of non-Newtonian fluids between rough walls is of interest in several geophysical and industrial applications. In this work (mainly geared toward fractured media) a governing equation for creeping flow of a purely viscous power law fluid of flow behavior index n in a rough-walled fracture is obtained, generalizing past results for a Newtonian fluid. An equivalent fracture aperture is defined, in analogy to the well-known hydraulic aperture valid for n = 1. Tortuosity is introduced as a vectorial quantity, thereby distinguishing between true and apparent fracture aperture. Examples are provided to illustrate the utility of the proposed approach. It is demonstrated that tortuosity effects significantly decrease the equivalent fracture permeability. Depending on the specific geometry considered, the flow behavior index may or may not have a significant impact on the equivalent fracture permeability. When it does, the reduction effect due to tortuosity is enhanced as the flow behavior index decreases.
A steady viscous flow through a two-dimensional, infinite channel consisting of an uneven wall and a plane wall is theoretically investigated under a given pressure gradient. The profile of the uneven wall is assumed to be periodic in the direction of the mean flow. A systematic expansion procedure is developed for the case where the ratio of the mean channel width to the period of the uneven wall, k, is small. An approximate solution is obtained up to the order of k2 . As a result, the flow rate is explicitly found as a function of the profile of the uneven wall, k and the Reynolds number. The unevenness of the wall always decreases the flow rate in proportion to k2 . The inertia effect on the flow rate first arises from the second order of k. The flow rate and the stream line are concretely obtained for the case of a sinusoidal wall. The approximate solutions show in good agreement with the solutions obtained by a numerical calculation.
Perturbation methods are one of the fundamental tools used by all applied mathematicians and theoretical physicists. In this book, the author has managed to present the theory and techniques underlying such methods in a manner which will give the text wide appeal to students from a broad range of disciplines. Asymptotic expansions, strained coordinates and multiple scales are illustrated by copious use of examples drawn from all areas of applied mathematics and theoretical physics. The philosophy adopted is that there is no single or best method for such problems, but that one may exploit the small parameter given some experience and understanding of similar perturbation problems. The author does not look to perturbation methods to give quantitative answers but rather to give a physical understanding of the subtle balances in a complex problem.
Physical and numerical fracture network models have been used to analyze the transport of conservative solutes in systems of parallel-sided fractures. The processes controlling dispersion in fracture systems that are explicitly simulated by the numerical model are (1) development of a velocity profile within individual fractures, (2) transverse molecular diffusion between streamlines, both within fractures and at fracture junctions, and (3) advection with the bulk fluid through a system of fractures with a range of hydraulic gradients and apertures. The first two processes, referred to as microdispersion, are often assumed to be secondary to the third, referred to as macrodispersion. The validity of this assumption is, however, highly dependent on the hydrodynamics of the system under consideration. Data collected from a physical model of a fracture network are used to validate a numerical model that explicitly simulates all three transport processes. The numerical model is then used to evaluate the relevance of microdispersion processes in a system where macrodispersion is significant.
This book presents a coherent introduction to boundary integral, boundary element and singularity methods for steady and unsteady flow at zero Reynolds number. The focus of the discussion is not only on the theoretical foundation, but also on the practical application and computer implementation. The text is supplemented with a number of examples and unsolved problems, many drawn from the field of particulate creeping flows. The material is selected so that the book may serve both as a reference monograph and as a textbook in a graduate course on fluid mechanics or computational fluid mechanics.
Flow fields within spatially periodic arrays of cylinders arranged in square and hexagonal lattices are calculated, with microscale Reynolds number ranging between zero and 200, employing a finite element numerical scheme. The terminology of an ‘‘apparent permeability’’ is introduced to establish a relationship existing between mean velocity and macroscopic pressure gradient characterized by a finite Reynolds number flow. In contrast with the low Reynolds number ‘‘true ’’ permeability, the apparent permeability is shown here to generally depend upon the direction of the applied pressure gradient, owing to nonlinearities existing within the local fluid motion. The orientation-dependent permeabilities of both square and hexagonal monodisperse arrays are observed to diminish with increasing Reynolds number. Similar behavior is also observed for a bidisperse square array, though the apparent permeability of the latter is shown less sensitive to Darcy velocity orientation at large Reynolds numbers in comparison to the corresponding monodisperse square array, for all cylinder concentrations examined.
The results and interpretation of five induced-gradient tracer tests performed at five different average interborehole fluid velocities in a single fracture in monzonitic gneiss are described. The experiments were conducted using radioactive 82Br and a fluorescent dye as conservative tracers where the tracers were pulse injected into radial convergent and injection-withdrawal flow fields. The flow fields were established between straddle packers isolating the fracture in three boreholes over distances of 12.7–29.8 m. The tracer breakthrough curves were determined from samples of the withdrawn groundwater and were interpreted using residence time distribution (RTD) theory and two deterministic simulation models. The RTD curves of the tracer experiments were interpreted by fitting to the field data a simple advection-dispersion model and an advection-dispersion model with transient solute storage in immobile fluid zones. Both models consider the different flow field geometries associated with injection-withdrawal and radial convergent tests. Comparison of the fits obtained by the simulation models suggest that the initial period of solute transport in single fractures is advection dominated and with increasing tracer residence time or decreasing fluid velocity, transport progresses toward more Fickian-like behavior. During the advective-dominated period, the transient solute storage model is shown to adequately describe the asymmetries and long tails characteristic of the fracture RTDs. Interpretation of the tracer experiments using both simulation models further suggests that induced-gradient tracer experiments are likely to underestimate the dispersive characteristics of single fractures under natural flow conditions.
Creeping flow in two-dimensional periodic channels of arbitrary geometry is considered. The problem is formulated using the boundary-integral method for Stokes flow, presently adapted for periodic flows with special geometrical characteristics. Numerical calculations for steady flow in channels constricted by a plane and a sinusoidal wall are performed. Detailed streamline patterns are presented and criteria for flow reversal are established. It is shown that for narrow channels the mechanism driving the flow has a strong effect on the structure of the flow. The results are discussed with reference to lubrication, coating and molecular-convective processes.
We develop a model for simulating the growth of a biofilm in a tortuous tube. The solutions to the Navier-Stokes equations and the advection-diffusion equation are calculated numerically using finite differences. These solutions are then coupled with a biofilm growth model. © 1994 John Wiley & Sons, Inc.
A physically based model is developed to study the transport of a solute utilized by microorganisms forming a biofilm coating on soil grains in a porous medium. A wavy-walled channel is used as a geometrical model of a porous medium and a biofilm is attached to the channel wall. Within the biofilm the solute is consumed according to a first-order volumetric rate. A numerical study is performed to obtain the dependence of the macrotransport coefficients on the Peclet number and Damkohler number. It is found that in some cases of practical importance the pore fluid is not well mixed, and mass transport limitations can control macroreaction rates. For diffusion-limited cases (large Damkohler numbers) increased solvent velocity can enhance the macroreaction rate by a factor of almost 3. Mean solute and mean solvent velocities are, in general, not equal, and mean solute velocities can exceed mean solvent velocities by 60% at high Damkohler numbers. These results agree qualitatively with those of a previous numerical study by Edwards et al. [1993]. The results also suggest that due to the spatially variable pore geometry, the biomass nearest the pore throat is more effective at consuming the solute than biomass in the pore chamber. A comparison is made between mass transfer correlations and the results determined for the macroreaction rate coefficient. We find that over a limited range of Peclet numbers a macroscale Sherwood number follows the Pe l/3 behavior determined from experimental mass transfer correlations and predicted by boundary layer theory.
Taylor dispersion of a passive solute within a fluid flowing through a porous medium is characterized by an effective or Darcy scale, transversely isotropic dispersitivity
[`(D)]*\bar D^*
, which depends upon the geometrical microstructure, mean fluid velocity, and physicochemical properties of the system. The longitudinal,
[`(D)]|| *\bar D_\parallel ^{*}
and lateral,
[`(D)]^ *\bar D_ \bot ^{*}
dispersivity components for two-dimensional, spatially periodic arrays of circular cylinders are here calculated by finite element techniques. The effects of bed voidage, packing arrangement, and microscale Pclet and Reynolds numbers upon these dispersivities are systematically investigated.The longitudinal dispersivity component is found to increase with the microscale Pclet number at a rate less than Pe2. This accords with previous calculations by Eidsath et al. (1983), although the latter calculations were found to yield significantly lower longitudinal dispersivities than those obtained with the present numerical scheme. With increasing Pclet number, a Pe2 dependence is, however, approached asymptotically, particularly for square cylindrical arrays - owing to the creation of a linear streamline zone between cylinders.Increasing tortuosity of the intercellular flow pattern reduces the longitudinal dispersivity component and enhances the lateral component. Longitudinal dispersivities for square and hexagonal arrays are found to be quite similar at high porosities; yet they diverge dramatically from one another with decreasing porosity. The longitudinal dispersivity is found to increase markedly with increasing Reynolds number. Comparison of this longitudinal dispersivity with available experimental results shows that
[`(D)]|| *\bar D_\parallel ^{*}
experimentally measured for three-dimensional arrays of spheres may be correlated by the present two-dimensional model by an appropriate choice of the array's packing arrangement. In general, the calculated dispersivities were found to be sensitive to the bed packing arrangement and apparently no rationale exists for choosing any one particular geometric microstructure over another for a comparison with existing experimental data. It is thus concluded that existing experimental data pertaining to three-dimensional beds of spherical particles cannot rationally provide a basis for verification of two-dimensional, circular cylindrical dispersion models.The finite-element scheme employed in this work was tested in the purely diffusive, nonflow limit by calculating the composite diffusivities of square cylindrical arrays for different volume fractions and various dispersed solid-continuous phase diffusivity ratios, subsequently comparing these with existing analytical results. An additional test was provided by comparing calculated with analytical axial dispersivities for transport of a dissolved solute in a Poiseuille flow between two parallel plates.
Lubrication theory is used to study the permeability of rough-walled rock fractures. In this approximation, which is valid for low Reynolds numbers and under certain restrictions on the magnitude of the roughness, the Navier-Stokes equations that govern fluid flow are reduced to the more tractable Reynolds equation. An idealized model of a fracture, in which the roughness follows a sinusoidal variation, is studied in detail. This fracture is considered to consist of a random mixture of elements in which the fluid flows either parallel or transverse to the sinusoidal bumps. The overall permeability is then found by a suitable averaging procedure. The results are similar to those found by other researchers from numerical analysis of the Reynolds equation, in that the ratio of the hydraulic aperture to the mean aperture correlates well with the ratio of the mean aperture to the standard deviation of the aperture. Higher-order approximations to the Navier-Stokes equations for flow between sinusoidal walls are then studied, and it is concluded that in order for the lubrication approximation to be valid, the fracture walls must be smooth over lengths on the order of one standard deviation of the aperture, which is much less restrictive a condition than had previously been thought to apply.
Thesis (Ph. D.)--Stanford University, 1994. Submitted to the Department of Civil Engineering. Copyright by the author. Photocopy.
Two-dimensional modeling of microscale transport and biotransformation inporous media, Numer. Methods Partial Differential Equations 10 On steady flow through a channel consisting of an uneven wall and a plane wall. Part 1. Case of no relative motion in two walls, Bull
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- J E Gale
- A Rouleau
- L C Atkinson