Pore-Scale Behavior of Darcy Flow in Static and Dynamic Porous Media

M. Aminpour,1,2,* S. A. Galindo-Torres,1,2,3 A. Scheuermann,1,2 and L. Li21Geotechnical Engineering Center, School of Civil Engineering,The University of Queensland, Brisbane QLD 4072, Australia

2Research Group on Complex Processes in Geo-Systems, School of Civil Engineering,The University of Queensland, Brisbane QLD 4072, Australia

3Department of Civil Engineering and Industrial Design, The University of Liverpool,Liverpool L69 3BX, United Kingdom

(Received 30 June 2016; revised manuscript received 15 March 2018; published 18 June 2018)

Lattice-Boltzmann numerical simulations are conducted to explore the pore-scale flow behavior insidemodeled porous media over the Darcy regime. We use static (fixed) and dynamic (rotating) particles to formthe porous media. The pore flow behavior (tortuosity) is found to be constant in the static medium within theDarcy range. However, the study reveals distinctively different flow structures in the dynamic case dependingon the macroscopic Darcy flow rate and the level of internal energy imposed to the system (via the angularvelocity of particles). With small Darcy flow rates, tortuous flow develops with vortices occupying a largeportion of the pore space but contributing little to the net flow. The formation of the vortices is linked to spatialfluctuations of local pore fluid pressure. As the Darcy flow rate (and, hence, the global fluid pressure gradientacross the medium) increases, the effect of local pressure fluctuations diminishes, and the flow becomes morechannelized. Despite the large variations of the pore-scale flow characteristics in the dynamic porous media,the macroscopic flow satisfies Darcy’s law with an invariant permeability. The applicability of Darcy’s law isproven for an internally disturbed flow through porous media. The results raise questions concerning thegenerality of the models describing the Darcy flow as being channelized with constant (structure-dependent)tortuosity and how the internal sources of energy imposed to the porous media flow are considered.

DOI: 10.1103/PhysRevApplied.9.064025

I. INTRODUCTION

Fluid flow in porous media has been studied extensivelyacross a wide range of disciplines, from mechanical, civil,environmental, chemical, and petroleum engineering, toagricultural, food, material, and biomedical science [1,2].Numerous natural and artificial systems are controlled oraffected by flow in various porous media: for example,seepage flow in soil, multiphase flow of oil, gas, andwater inoil reservoirs, contaminant transport within groundwater,and solute movement through biological tissues [3–7], aswell as ion transport in fuel cells, and fluid flow in buildingmaterials, tablets, textiles, foams, papers, and filters [8–10].The well-known Darcy’s law presents a general method forquantifying the flow in porous media under conditions ofsmall Reynolds numbers (Re), which relates linearly themacroscopic flow velocity V to the fluid pressure gradientacross the medium ∇P,

V ¼ − κμ∇P; ð1Þ

with κ and μ being the permeability of the porous mediumand dynamic viscosity of the fluid, respectively [note that in

Eq. (1), the effect of gravity is not included [11] ]. This law,however, provides no insight into the microscale (pore-scale) flow behavior, which may affect significantly soluteand heat-transport processes in the porous medium. Severalpermeability models have been developed to account for theinfluence of the porous medium’s properties. The classicalKozeny-Carman (KC) equation [12–15] is the most widelyused one, as given below

κ ¼ ϕ3

cð1 − ϕÞ2S2 ; ð2Þ

where c is the KC constant (c ¼ c0T2 with c0 being aconstant and T the flow tortuosity), ϕ is the medium’sporosity, andS is the specific surface area equal to the ratio ofthe total interstitial surface area to the volume of solids [11].The KC equation is derived from the consideration of asimple capillary model, which translates the complexstructure of porous media to a set of capillary tubes witha certain length (equal to the tortuosity times as the directlength of themedium). This equation implies a constant flowtortuosity for a given porous medium (of constant ϕ and S)undergoing flow in the Darcy regime (with constant κ).Consequently, the current conceptualizations dealing with*m.aminpour@uq.edu.au

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the flow behavior at the pore scale have assumed theinvariant flow patterns (hence, the constant tortuosity) whenDarcy’s law applies. Experimental [16], analytical [17],theoretical [18], and numerical [19] models have beendeveloped to relate the tortuosity of flow to the porosityof the porous media (see the review in Refs. [20,21]) but notto the flow rate. However, experimental investigations usingnuclear-magnetic-resonance velocimetry [22–31] and opti-cal imaging techniques [32–38] have demonstrated exten-sive variations of pore-scale fluid velocity in porous media.As important features affecting the transport properties

of porous media, dead-end and closed-pore structures havebeen investigated for several decades [39,40]. Moreover,stagnant zones in porous systems, such as eddies or low-speed regions among the preferential flow paths, have beennumerically observed either at low or high Reynoldsnumbers [41–43]. However, the results have always beenlinked to the medium’s structure. Possible variations ofthese stagnant zones (shape and location) and local flowpatterns have not been thoroughly explored.This study aims to examine systematically the flow

structures at the pore scale over the Darcy regime with afocus on the flow tortuosity. A static porous mediumcomposed of fixed spherical particles (with no-slip boun-daries) is employed. We also investigate the flow behaviorin dynamic porous media. In this case, we are interested inactive porous structures in which the porosity remainsconstant (e.g., exothermic porous media [44]), eliminatingthe effects of porosity variation on the permeability. Suchan active condition is obtained by using the particlesconstantly spinning with different angular velocities aroundfixed axes.Fluid flow due to rotating spheres is of intrinsic interest

in fluid dynamics, meteorology, astrophysics, and manyother fields. It has received much attention theoretically,numerically, and experimentally [45–47]. Here, we inves-tigate pore fluid structures in a pack of rotating spheres thatalso undergo a general flow crossing the medium. Therotation of spheres may appear in certain conditions, forexample, under Quincke effects [48]. The rotating spherepack is generally considered an example of dynamic media.Fluid flow in dynamic porous media can have prospectiveapplications in physics and industry, for example, dynamiccontaminant mobilization in soils [49] or groundwater flowthrough dynamically loaded sites, e.g., via vehicles, pilingconstruction, or machinery foundations.Apart from that, we examine the validity of Darcy’s law

where the fluid is subject to internal sources of energy fromthe medium (as a disturbance energy), and yet the net flowis driven by the macroscopic pressure gradient (or bodyforce). The macroscopic linearity as per Darcy’s law istested in the presence of highly variable microscopic flowbehavior. The findings have implications for the “univer-sality” of the concept of constant (structure-dependent)flow tortuosity within the Darcy regime, which is the

implicit assumption lying at the root of the models thatdescribe flow (and transport) in porous media. We areinterested in the portion of pore space that contributes tothe flow of the passing-through fluid (the Darcy flux). Themechanism underlying the pore-scale flow behavior is alsoinvestigated.

II. METHODOLOGY

We conduct numerical simulations using a computermodel based on the lattice-Boltzmann method (LBM) andthe discrete-element method (DEM) [50]. The simulatedporous medium is made of 500 spherical particles, each of16 lattice units (l.u.) in diameter, randomly deposited into abox with a cross section of 182 × 182 l.u. In the LBM, thevariables are defined in lattice units and time steps but canbe converted to physical units based on dimensionlessnumbers, e.g., Reynolds numbers [51]. The model dimen-sions and the particle diameter are convertible to physicalunits of 9 mm for the sphere diameter and 10 × 10 cm forthe cross section of the box domain with water being thefluid. The packing of particles is obtained with thesimulation of a natural deposition through a free-fallprocess using the discrete-element method resulting in amedium with the porosity of ϕ ¼ 0.42. This medium isthen imported into the LBM model where the fluid flowthrough the pore space is simulated. When the spheressettle into the box, they spin (with different angularvelocities) in their location for some time depending onthe surface friction specified. To generate static porousmedia, the angular velocities of the spheres are set to zeroafter deposition is completed (zero linear velocities). Incontrast, the spheres in dynamic porous media are fixed intheir locations (with zero linear velocity) but keep rotatingat constant angular velocities caused by the depositionprocess. In the dynamic porous medium, each sphere isspinning at a different but constant angular velocity overtime [Fig. 1(a)] around a fixed axis [with a differentorientation from others; Fig. 1(b)]. While the angularvelocity distribution depicted in Figs. 1(a) and 1(b) isreferred to as the ω0 dynamic case, other dynamic con-ditions are also simulated by multiplying the angularvelocity of each sphere by a constant factor (0.5 and 2)but maintaining the same axes of rotations.The process of particle deposition by which the initial

rotations of the spheres are induced is a natural examplethat makes the preferential direction of rotations: theparticles in contact with others will rotate in directionsand speeds correlated by the relative movement of others.The rotation of particles in our model is then not a randombut a correlated distribution. By maintaining the rotations indynamic media, we suggest that a natural condition inwhich preferential movements of particles are induced(e.g., rearrangement of packings by continuous shearstrains) can be mimicked.

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A constant velocity and pressure boundary condition[52,53] are set at the bottom and top of the model domain,respectively, to simulate a constant discharge of fluidthrough the medium with a constant fluid pressure main-tained at the top boundary. Periodic boundary conditions(PBCs) are implemented for the lateral boundaries repre-senting an infinite domain in the lateral directions. Thesimulations are conducted with different discharge rates tocover the whole Darcy regime (with the post-Darcy con-ditions also simulated). Each simulation is run until thesteady state in flow characteristics (e.g., distributionof velocity and fluid pressure) is reached (normally after3 × 104 simulation time steps).More simulations are also conducted to analyze the

sensitivity of the results to the number of spheres in the

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FIG. 1. Dynamic porous media withperiodic boundary conditions. (a) Pro-bability density function (PDF) ofthe magnitudes of angular velocities[jωjðrad= secÞ] imposed to the spheresto produce dynamic porous media (re-sulting from the natural deposition proc-ess as we describe in Sec. II). Eachparticle of such media is rotating at adifferent but constant ω ¼ ðωx;ωy;ωzÞaround a fixed axis also different inorientation from the others. The meanof the rotation speed magnitudes is equalto jω̄j ¼ 0.026 rad= sec. (b) The orien-tation of rotations displayed as angularvelocity components. (c) Streamlinesfor the dynamic porous medium. Here,the boundary conditions for all sides ofthe model are periodic (no pressure orvelocity boundary conditions); thus, theresulting flow (downward) is caused bythe rotating spheres only. (d) The veloc-ity vectors shown on an internal crosssection of this model, indicating themain direction of flow (−z, negativelongitudinal) indicated by the rotatingsphere pack. (e) Pore fluid velocitydistributions of the rotating sphere packfor longitudinal (VL) and transverse(VT) velocities and velocity magnitudes(jVj). (f) The macroscopic velocity ofoutflow caused by the rotating spheresystem (the component in the −z direc-tion) versus jω̄jr where r is the sphereradius. The equation shows a linearfitting. The models of rotating spheresare subsequently exposed to additionalvelocity (pressure) boundary conditionsto examine a net Darcy flow (in the þzdirection) passing through the dynamicporous medium.

FIG. 2. Monodispersed particle packing achieved by free-falldeposition of 8400 spheres into a box using DEM. The resultingporous medium is then used for LBM simulations.

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porous medium (i.e., porous medium dimensions) and thenumber of lattices per sphere diameter (tested for up to70 cells=diameter), as well as the lateral boundary con-ditions (PBC vs no-slip walls). The results reported arefound to be unaffected by these factors. For instance, thesimulations with 8400 particles with a medium of 2003 l.u.(PBC for the lateral boundaries, and velocity and densityBC for inlet and outlet boundaries, respectively) confirmthe consistency of the results (Fig. 2).

III. FLOW REGIME IDENTIFICATION(V −∇P RELATIONSHIP)

Based on the simulated (linear) relationship between thepressure gradient across the medium (∇P) and macroscopicflow velocity (V), we determine the Darcy regime (Fig. 3)given by Re ⪅ 1.8, where Re is defined as

Re ¼ ρVdpμ

; ð3Þ

with ρ and dp being the fluid density and particle diameter,respectively. Within this regime, Darcy’s law is indeedfound to apply with a constant permeability for thesimulated porous media (both static and dynamic).The straight line fitted to the V versus ∇P relationship

for the static porous media passes the origin. However,fitted V versus ∇P lines for dynamic porous media exhibit

intercepts on the positive part of the∇P axis. With a highermean angular velocity for the particles (jωj), the value ofthe intercept increases. This behavior can be explained byconsidering the resulting flow induced by the rotation ofspheres. Each spinning sphere can produce a flow aroundits surface from the two poles (with respect to the axis ofrotation) toward the equator and then radially away from it[45]. The resulting pore fluid flow structures appear in theform of vortices. Resulting from the integration of flowgenerated by the individual spheres, a net flow is inducedoutward from the medium. The average velocity magnitudeof the outflow is presumably proportional to the mean ofthe tangential velocity of particles v⊤ ¼ jωjr (with r beingthe particle radius). The mean of the angular velocitiesfor the dynamic base case is jω̄0j ¼ 0.026 rad= sec.To examine the generation of a net outflow from the

system of dynamic spheres, the pack of rotating sphereswith the PBC for all boundaries (no velocity boundaryconditions) is analyzed (see Fig. 1). In this case, the flow isinduced only by the rotating spheres. As an engine made ofrotating spheres with different angular velocities arounddifferent axes, the system generates a net flow exiting themedium with the longitudinal velocity component in thedirection of −z (negative longitudinal). This outflow alsoresults in a gradient of pressure. When we assign themacroscopic flow with velocity boundary conditions in theþz direction (in further simulations for Darcy flow inves-tigation), the net flow overcomes the background flowgenerated by the rotating sphere system. This combinedflow induces a pressure gradient which is the sum of thegradient required to overcome the backward flow and thegradient due to the main flow. With a higher angularvelocity of spheres, the net outflow (in the −z direction dueto the rotating spheres) is larger. A larger outflow induces alarger background hydraulic gradient across the medium.Thus, for the net macroscopic flow (developed by outerboundary conditions in þz), a larger pressure intercept isobserved in the V versus ∇P graph (see Fig. 3).To better understand themechanism involved in creating a

net flow out of the system of rotating spheres, we set up asimple model (Fig. 4). While a rotating sphere produces asymmetrical recirculation zone around it with no net flow inany direction, the symmetry breaking can cause the nonzeronet flow in a direction depending on the obstacle location(and the free void space available). A single fixed (non-rotating) sphere next to the same rotating sphere [Fig. 4(b)]accommodates the asymmetric fluid movement that resultsin an outflow. Two spheres both rotatingwith the same speedmay also produce zero or nonzero net flow depending onthe orientations of the rotations. For the same orientationof rotations, the fluid recirculates with no net outflow[Fig. 4(c)]. With opposite orientations of rotation, net flowis produced [Fig. 4(d)]. A similar phenomenon can happenfor the medium studied here as the dynamic porous mediumwhich is comprised of spheres rotating individually with

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FIG. 3. Pressure gradient versus macroscopic flow velocityfor static and dynamic porous media, with the Darcy regimeindicated by the linear range, corresponding to Re ⪅ 1.8 (theequations on the figure show the fitting to the Darcy range ofthe curves). jω̄j is the mean angular velocity of the spheres in thedynamic porous media. The permeability of the media within theDarcy regime is found to be the same (κ ¼ 2.78 × 10−7 m2) forall cases. The top left inset shows a close-up near the origin, andthe top right inset displays the other end of the graph on a logscale, as the transition to the post-Darcy regime. It also shows theForchheimer fitting to this part of the curve.

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various speeds in different orientations. In a packingwith thepore space randomly variable, the voids surrounding theparticles are not symmetrical, leading to preferential flowdirections depending on the relative rotation velocities andthe void space available. Spheres may play an obstacle rolefor neighbors depending on their relative rotations. Thecomplex integration of fluid recirculations and preferentialflow streams within the pores provides a net flow out of themedium in a certain direction.As indicated by Fig. 1(e), the entirely circulating fluid

in pore space leads to similar distributions for velocitycomponents in different directions. However, a small netflow is generated in a certain direction out of the medium asdiscussed earlier. The net flow rate is shown to be morethan 2 orders of magnitude smaller than the average of thefluid recirculation velocity, i.e., jω̄jr [see Fig. 1(f)].The intercept on the ∇P axis discussed here is similar to

what is well reported as the initial (minimum) pressuregradient (∇P0) required for the Darcy flow in porous

media [11]; that is, Darcy’s law applies only for the pressuregradients greater than ∇P0, with the new form ofV ¼ −ðκ=μÞð∇P −∇P0Þ. Although the static porous mediasimulation (no-slip boundary condition for solids) providesno evidence of such an initial pressure gradient, with thedynamic conditions simulated here, the initial pressuregradient is required to generate the net flow through themedia. The mechanisms of the initial pressure gradient innature (the electrical streaming potential, non-Newtonianviscosity, or immobile water layers [54]) and the behaviorreported here are different. However, the simulations can bea step toward better understanding of the phenomenon.The permeability of the porous medium is found to be

constant in all conditions (static or dynamic), equal to2.78 × 10−7 m2. This value is comparable to the permeabil-ity of well-sorted gravels. The medium is made of sphereswith a diameter of d ¼ 9 mm. Using the Kozeny-Carmanequation (2) with ϕ ¼ 0.42, S ¼ 6=d (as for spheres), T ¼1.168 (as calculated in our simulations), and c0 ¼ 2.5 assuggested by Ref. [14], the permeability estimation is equalto 1.45 × 10−7 m2 which is a close estimation to that of thesimulations.The upper end of the V −∇P graph deviates from the

linear regime (for Re⪆1.8). As shown in Fig. 3, top rightinset, the non-Darcy behavior can be fitted with aForchheimer equation in the form of [55]

∇P ¼ μκV þ ρCfffiffiffi

κp V2; ð4Þ

with κ being the permeability of the medium obtained fromthe linear regime, and Cf being the form coefficient takento be 0.9 here.

IV. STATIC POROUS MEDIA

A. Pore-scale flow variations

Pore fluid velocities obtained from the numerical modelare found to vary largely in the longitudinal (L) andtransverse (T) directions, as shown in Fig. 5. As expected,the transverse velocity distributions are symmetric aboutzero indicating zero mean velocity in the transversedirection [Fig. 5(b)]. The longitudinal (L) velocities aremostly distributed in the positive range with a smallnegative tail [Fig. 5(b)]. Further comparisons betweenour numerical results and experimental and numerical datafrom the literature are presented in Fig. 5, where asatisfying agreement is observed.

B. Streamlines

Streamlines are computed using the numerical integra-tion [58] on the 3D flow fields obtained from the simu-lations. As compared in Fig. 6, streamlines in static porousmedia with different Darcy flow rates show channelizedflow mostly directed along the general flow direction.

(a)

(c) (d)

(b)

FIG. 4. Simple models to explore the mechanism of thegeneration of net flow out of the rotating sphere models. Themodel boundaries are all periodic. (a) A single rotating sphere(jωj ¼ 1.0 rad= sec) produces no net flow with the symmetricalrecirculation around the rotating sphere. (b) The same rotatingsphere when next to a fixed sphere generates a net flow out of thesystem due to the symmetry breaking for circulating fluid.(c) Two adjacent spheres when rotating with the same orienta-tions produce only local rotational flow but no net flow. (d) Twospheres rotating with opposite orientations generate a net outflow.The combination of differently rotating spheres (different inspeed and orientation of rotation) in our porous medium (Fig. 1)can produce complex fluid structures resulting in a net outflow ina similar way as presented here.

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Alterations in the shape and direction of streamlinesassociated with different flow rates within the Darcy regimeare negligible.

C. Tortuosity

The flow tortuosity can be calculated based on the flowvelocities [59]

T ¼P

vP

vL; ð5Þ

where v and vL denote the pore fluid velocity and itscomponent in the direction of macroscopic flow (longi-tudinal direction), respectively. The method has beenused in several studies to investigate the flow tortuosity(e.g., Refs. [19,60,61]). We use Eq. (5) to calculate thetortuosity of the flow in porous media. A cubic sectionfrom the middle of the porous media with all sides at adistance from the boundaries is selected as the represen-tative volume for the velocity fields. We use Eq. (5) tocalculate the tortuosity of the flow within this cubicsection. The results are shown in Fig. 7 where a constanttortuosity (T ¼ 1.168) is observed within the Darcyrange. With the flow rate exceeding the upper limit ofthe Darcy range, tortuosity slightly decreases up to T ¼1.156 for Re ¼ 18. The decrease in tortuosity can beattributed to the effects of the transition to the non-Darcyregime as also indicated by some researchers(see Ref. [62]).

V. DYNAMIC POROUS MEDIA

In this section, we use the dynamic sphere pack (as wedescribe in Sec. II) to conduct the flow simulations throughthe dynamic porous media. The discussion focuses on thecase of ω0 unless mentioned otherwise.

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FIG. 6. Static porous media. Flow streamlines for (a) low and(b) high flow rates within the Darcy range.

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A. Pore-scale flow variations

PDFs of normalized transverse [Fig. 8(a)] andlongitudinal [Fig. 8(b)] velocities and velocity magnitudes[Fig. 8(c)] all show wider distributions in the lower Darcyrange with smaller macroscopic flow rates (Re). In thisrange, the countercurrents (VL < 0) are alsomore profound.The normalized longitudinal flow velocities through thepores display different patterns of flow activities (Fig. 9).Under highmacroscopic flow rates, active flow zones appearto be well connected and channelized in the Darcy flowdirection [Fig. 9(b)]. In contrast, flow streams under lowflow rates are more tortuous with local vortices, separatingthe pore flow zones into streams directing toward or oppositethe main flow direction (countercurrents) [Fig. 9(a)].With the higher flow rates, the pore flow velocity distri-

butions coincidewith those of static medium. This is becausethe local flow rotational velocities due to dynamic effects arenot relatively strong enough to disturb the main flow streams(see Sec. V F for the associated threshold). The transition tounaffected streams is indicated by the decrease of thetortuosity to that of the static conditions (discussed later).

B. Streamlines

The results show different pore-scale flow patternsdepending on the macroscopic flow rate or Re (Fig. 10).At low macroscopic flow rates, the fluid moves throughvery tortuous streamlines, many of which do not even endat the exit boundary and instead form circulation cellsinside the medium [Fig. 10(a)]. As the macroscopic flowrate increases, the streamlines become straighter, like thosein the static media.Circulating streamlines under the condition of lowmacro-

scopic flow rates in dynamic porous media appear in theform of vortices. The nature of these vortices which areformed in an ideally laminar flow (Darcy regime) [63,64] isdifferent from that of turbulent eddies. Considering thatthe permeability of the medium is found the same in all ofthe static and dynamic cases investigated here, we further

examine themechanismof the flow through dynamic porousmedia with identification of the conducting flow channelsand recirculating fluid capsules (no net flow).

C. Vorticity

We compute the vorticity based on the simulated flowvelocity, i.e., the curl of the velocity vector. The variation of

(a) (b) (c)

FIG. 8. Dynamic porous media. PDFs of normalized transverse VT (a) and longitudinal VL (b) velocity components, and velocitymagnitudes (c) obtained from numerical simulations (all within the Darcy range). Under a lower Darcy flow rate, all distributions appearto be wider, with a larger negative portion for VL; (ðV=jω̄jrÞ ¼ 0.95Re).

1

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(a)

(b)

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–1

FIG. 9. Dynamic porous media. Normalized longitudinalvelocity (ðVL=hVLiÞ) in a cross section at the middle of the3D model, for (a) Re ¼ 0.025 (ðV=jω̄jrÞ ¼ 0.024) and (b) Re ¼ 2(ðV=jω̄jrÞ ¼ 1.9). The color scale is adjusted so that the normal-ized velocities greater (or smaller) than 1 (or −1) are shown in thesame color. Thus, the color map mainly differentiates the positiveand negative longitudinal velocities (countercurrents). Smallzones at the middle indicated with squares on the left aremagnified at right panels, also showing the streamlines. Themacroscopic Darcy flow direction is upwards.

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the curl magnitude within the pore space is investigated.For the purpose of comparing the flow structures ofdifferent Darcy flow rates, we normalize the curl magnitudeby the macroscopic flow velocity. That is, the curl isfirst calculated over normalized velocity fields and thendivided by local normalized velocity magnitude, i.e.,j∇ × ðv=hviÞj=jv=hvij. As shown in Fig. 10(a), the zonesof curl magnitudes with converged gradients can beidentified as vortices and linked to circulating streamlinesunder the condition of low macroscopic flow rates indynamic porous media. While some vortices appear tocirculate solid particles, smaller vortices also exist entirelywithin the pore space. The vortex flow structures, however,disappear in cases with larger macroscopic flow rates (Re)within the Darcy regime [Fig. 10(b)].Combining the results of streamline and curl magnitude,

we determine a critical value of the normalized curlmagnitude (Ω0) to separate the vortical zone (pore spaceoccupied by vortices) and the passing-through flow zone(pore space with streamlines passing through the wholemedium). When the macroscopic flow rate is low (Re → 0),the vortical zone caused by spinning particles dominates,but such a dominance diminishes as the macroscopic flow

rate increases (Re⪆ 1), and the whole pore space contrib-utes to the passing-through flow toward the upper end ofthe Darcy regime (Re ≈ 1.8) [Fig. 13(c)]. The volume ofpore space contributing to the passing-through flow[denoted as net flow streams in Fig. 13(c)] increases withthe Darcy flow rate reaching the maximum value (entirepores) at high flow rates.The generation of the vortices can be characterized

by the value of ðV=jω̄jrÞ (see Fig. 11). For values lessthan 1, the main flow rate is weaker than the rotationalstreams, resulting in evolving vortices. As this ratioincreases, the main flow gradually overwhelms the rota-tional zones, making the vortices progressively shrink andtransform to channelized flow streams for ðV=jω̄jrÞ > 1.

D. Tortuosity

1. Tortuosity calculated using streamlines

The results that we present above for dynamic porousmedia demonstrate large variations of pore-scale flowbehavior with different characteristics over the same macro-scopic Darcy flow regime (with invariant permeability). Toquantify such variations, we compute flow tortuosity, a

(a)

(b)

FIG. 10. Dynamic porous media. Flow streamlines shown in 3D (left) and in 2D (middle) at a vertical section of the modeled porousmedium for a low (a) and high (b) flow rate both within the Darcy range. A single-pore zone from the selected section is enlarged at theright-hand side with color images showing the normalized vorticity.

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quantity often treated as a function of only porosity inporous media flow studies [19,60,65–69]. With the stream-lines already computed, the tortuosity of the streamlinescan be calculated as the ratio of l=L, where l is the actuallength of the streamline, and L is its direct length in themacroscopic flow direction. The PDF of the tortuosityvalues calculated for all individual streamlines (Fig. 12)

indicate a highly variable behavior of the flow in dynamicporous media: at low flow rates, streamlines have a widerange of tortuosities including a portion of highly tortuousstreamlines. At high flow rates, the tortuosity values span anarrow range, mostly less than 2. In all cases, there are anumber of streamlines with small tortuosities (T < 2),which are the passing-through streamlines, but their pro-portions to all streamlines vary largely with the Darcyflow rate.The average of the flow tortuosity is also first calculated

directly from all the streamlines, i.e., the ratio of the sum ofthe streamlines’ actual lengths (

Pli) to the sum of their

length in the macroscopic flow direction (P

Li). Given thatsome streamlines belong to vortical regions with no sharein the net flow, this method provides a tortuosity valueassociated with all pore fluid movements regardless ofcontributions to the net flow. A decreasing trend in thecalculated tortuosity from large values of the order of 102 atlow Darcy flow rates to small values less than 2 at the upperlimit of the Darcy range is observed [Fig. 13(b), allstreamlines]. While the tortuosity variation reflects thechanges of pore fluid flow structures over the Darcyregime, the calculated tortuosity cannot be linked directlyto the macroscopic passing-through Darcy flow. Therefore,

(a) (b) (c)

(d) (e) (f)

FIG. 11. Dynamicporous media. Vortex pro-gressive dissipation. Thevortices are generated bythe effects of rotatingspheres. As the Darcyflow becomes relativelystronger, indicated by alarger ðV=jω̄jrÞ, the vorti-ces progressively disap-pear and the flow becomeschannelized.

100 101 102 103

Tortuosity

10–4

10–3

10–2

10–1

10 0

PD

F

Re = 0.01

Re = 0.09

Re = 0.8

Re = 1.8

FIG. 12. Dynamic porous media. PDF of the tortuosity valuesof all streamlines for different flow rates, all within the Darcyrange; (V=jω̄jr ¼ 0.95Re).

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we recalculate the tortuosity based on only passing-throughstreamlines, those that span the medium from thebottom (net flow entrance) to top (net flow exit) boundary[Fig. 13(b), complete streamlines]. Another criterion is alsoapplied to select additional streamlines that are likely tobelong to the passing-through group but do not appear toexit the medium. The criterion is that streamlines whoseindividual tortuosity is less than 4 with an ending pointlocated at the highest elevation of the streamline path arealso passing-through streamlines [Fig. 13(b), selectedstreamlines]. The threshold of 4 is chosen according tothe outcomes of the other methods we explain below (i.e.,T1), which show that the passing-through flow has amaximum tortuosity less than this value. A sensitivityanalysis confirms the rationale of this threshold.

2. Tortuosity calculated over velocity fields

While being popular as a preferredmethod to calculate thetortuosity, Eq. (5) has some limitations [19] that clearly existwhen applied to complex flows as encountered here. Thus,modifications are needed to exclude the effects of counter-currents and circulations that make no contribution to themacroscopic flow through the whole porous medium. Thefollowing modified tortuosity equations are introduced,

T1 ¼P

vP

vLðvL>0Þ; ð6Þ

T2 ¼P

vðvL>0ÞPvLðvL>0Þ

; ð7Þ

T3 ¼P

vðΩ0ÞPvLðΩ0Þ

; ð8Þ

with the parameters defined as follows: vLðvL>0Þ are thelongitudinal-component velocities for those velocity vectorswith positive L components (directing in the macroscopicflow direction, i.e., þL), vðvL>0Þ are the velocities withpositive L components, vðΩ0Þ and vLðΩ0Þare velocities and L-component velocities for those vectorswith positiveL components and the curlmagnitudes less thanthe critical value ofΩ0. Thesemodified equations [Fig. 13(b)]give similar results to those calculated based on passing-through streamlines. In summary, the calculated tortuositybased on all streamlines for the dynamic porous mediumvaries by 2 orders of magnitude over the Darcy regime,consistent with the large changes of pore-scale flow struc-tures. With the vortical zone excluded, the tortuosity of theconducting channels given consistently by different methodsremains variable but to a much less extent (by a factor of 2).

E. Pressure fluctuation

We also analyze local pore fluid pressure fluctuations(spatial variations) relative to the global pressure drop

10–2 10–1 100 101

Re

10–2

100

102

LBMLinear fitting

Darcy range

(a)

10–2 10–1 100 101

Re

Tor

tuos

ity

Selected streamlines(T

streamline

across the medium. A dimensionless number is intro-duced, i.e.,

M ¼Mean of local fluid pressure fluctuations

Specific length

Global pressure gradient: ð9Þ

For the mean of local fluid pressure fluctuations, thestandard deviation of fluid pressure σP is evaluated atthe lower boundary of the porous medium, while the speci-fic length is chosen as the particle diameter. Over the Darcyregime, this dimensionless number is found to vary in asimilar manner to that of the tortuosity [Fig. 13(d)]. Wesuggest that the similarity is linked to the mechanism of thepore-scale flow behavior. Local pressure fluctuations canpotentially induce vortical flow (or be caused by such aflow), depending on the intensity of the net (global) flow.Large local pressure fluctuations under a weak net flowcondition (small global pressure drop) as indicated by alarge M number tend to produce vortical, tortuous pore-scale flow structures.

F. General range of tortuosity variation

With different speeds of rotations for the spheres in theporous medium, the velocity of disturbed (vortical) porefluid changes. We examine the effects of rotation speeds bychanging the angular velocity of spheres (nω0). The resultsshow that the increase of rotation speeds causes a largertortuosity of the flow (Fig. 14), and yet the permeabilityremains constant (see Fig. 3).

A unique relationship is found between the tortuosity offlow and the dimensionless number of V=jω̄jr (Fig. 14,inset). With the net flow velocity (V) exceeding the meantangential velocity of spheres (jω̄jr), the tortuosity drops tothe minimum possible value that coincides with that of thestatic medium. Thus, V ¼ jω̄jr can be taken as the thresh-old of dynamic porous medium effects on local pore fluidvelocity distributions (tortuosity).

G. Effects of the medium’s characteristic length

We examine the effect of variations in particle diameteron the flow behavior under the dynamic porous mediacircumstances. The simulations are performed for themedia comprised of particles with diameters of 1.0 to10−4 in the unit of length (with water as the fluid and sizes

10–10 10–5 100 105

Re

d = 1.0

d = 10–1

d = 10–2

d = 10–310–5 100 1051

1.5

2

2.5

3

3.5

4

1

1.5

2

2.5

3

3.5

4

FIG. 15. Dynamic porous media. Tortuosity vs Reynoldsnumber for dynamic porous media of spheres with differentsizes. dp is the spherical particle size (in centimeters). Thespheres rotate on average with angular velocity of 4.3jω̄0j. Theinset shows the tortuosity vs V=jω̄jr where all curves collapseinto a single relationship. Here, V is the flow macroscopicvelocity, and jω̄j is the mean of the angular velocity of spheres.

10–6 10–4 10 –2

10 –5 100 1051

2

3

4

1

1.5

2

2.5

3

3.5

FIG. 14. Dynamic porous media. Tortuosity vs flow macro-scopic velocity (V) for porous media of rotating particles withdifferent mean angular velocities (jω̄j). Here, V is the velocity ofnet flow passing through the dynamic medium which is con-trolled by external boundary conditions. The inset shows thetortuosity vs V=jω̄jr where r is the particle radius. The uniquerelationship indicates that when the mean flow velocity (V)reaches the mean velocity of disturbed pore fluid motions due torotating spheres (jω̄jr), the effects of the dynamic medium onflow diminish, and the flow retrieves the minimum tortuosityequivalent to that of static media.

10–1 100 101 102

M

dp

= 1.0

dp

= 10–1

dp

= 10–2

dp

= 10–3

1

1.5

2

2.5

3

3.5

4

FIG. 16. Dynamic porous media. Tortuosity vs M number(as defined for Fig. 13). dp is the spherical particle size (incentimeters).

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in units of centimeters). The angular velocity distribution isdetermined in the same way as shown in Fig. 1(a) butmultiplied by a constant factor of 4.3 for all spheres.The tortuosity varies with the flow rate and the character-

istic length (Fig. 15). As expected, the flow disturbanceresulting from the rotation of particles can alter thetortuosity, while the range of this effect varies with theparticle diameter. The velocity of disturbed (circulating)fluid is a function of the tangential velocity of particles(v⊤ ¼ jωjr). Hence, with a decrease in particle diameter,the range of Reynolds numbers in which the tortuosityvaries is reduced to the same orders as that of r2.A unique relationship is found between the tortuosity

and the M number (Fig. 16). This indicates a similarrelative local pressure fluctuation with respect to the flowtortuosity, regardless of the porous medium characteristiclengths. The uniqueness of the T versus M relationshipprovides a measure to integrate the flow characteristics inpore scale. With the interrelationship between flow tor-tuosity and the local pressure fluctuations, the circum-stances where one may cause the other can be furtheridentified.

VI. PRESSURE FLUCTUATION-TORTUOSITYRELATIONSHIP

To further explore the mechanism behind the tortuosityvariations, we set up a simple model with only two spheresrepresenting the solid phase and confining the pore spacewhere the fluid flows (see Fig. 17). This simple medium issubjected to different pressures at the lower and upperboundaries. While the upper boundary pressure is set to bespatially uniform (P0), the lower boundary pressure oscil-lates spatially in a sinusoidal form (P ¼ P0 þ ΔPþ Ap×sinðπ=5 × iÞ, where AP is the pressure fluctuation ampli-tude, and i is the lateral coordinate in lattice units for thespheres with diameter of 8 l.u. The simulations areconducted under various conditions with different averagedpressure drop over the medium and different amplitudesand wavelengths of the sinusoidal pressure oscillationsat the lower boundary. For this model, the M number isgiven by M ¼ AP=ΔP, where ΔP is the average pressuredifference between the boundaries. The results clearlydemonstrate that local fluid pressure fluctuations incounterbalance with the global (averaged) pressure gradientcause variations of pore-scale flow behavior over the Darcyregime (Fig. 18). In particular, large pressure fluctuationscombined with low global pressure gradients lead tovortical flows in the porous medium. The results also showa strong relationship betweenM and flow tortuosity, whichappears not to be directly affected by the value of the globalpressure drop.

VII. CONCLUSIONS

This study provides insights into the pore-scale flowbehavior and reveals the vastly different pore-scale flowstructures within the Darcy regime in porous media that aresubjected to internal energy (here, via spinning particles).The static (ordinary) porous media result in invariantchannelized pore-scale flow structures (constant tortuos-ity). In dynamic porous media, for the lower Darcy ranges(where ðV=jω̄jrÞ < 1), tortuous flow is shown to produce

(a) (b)

(c)

×

FIG. 17. Velocity vectors for a simpledouble-sphere model. In this model, pres-sure boundary conditions are imposed tothe upper (outlet) and lower (inlet) boun-daries as constant and sine waves, re-spectively. The overall ΔP is the samein both cases, while the M number(M ¼ ðAmplitude of pressure sine wave=Macroscopic pressure differenceÞ ¼ AP=ΔP) is different [M¼10−3 (a) and 0.7 (b)].The pressure applied at the lower boundaryis equal to P0 þ ΔPþ Pwave, where P0 isthe pressure at the upper boundary, andPwave is a function of the lateral coordinatein the form of (c).

10–4 10–3 10–2 10–1 100

Tor

tuos

ity

ΔP = 10–3 Pa

ΔP = 10–4 Pa

ΔP = 10–5 Pa

1.15

1.2

1.25

1.3

1.35

FIG. 18. Variations of the M number (M ¼ AP=ΔP) andflow tortuosity (T1) under conditions of different macroscopicpressure gradients obtained from the simple double-sphere model(Fig. 17).

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vortices with little contribution to the net flux. As the Darcyflow rate increases, vortices shrink in volumes and thechannelized flow zones expand. Despite the variations,particularly in flow tortuosity, the macroscopic flowthrough the medium in the simulated cases satisfiesDarcy’s law with an invariant permeability for all cases(static and dynamic). The microscopic flow behavior isshown to be linked to the local fluctuations of the pressurecompared to the global pressure gradient.With the variable tortuosity demonstrated here, the

findings challenge the universality of the existing porousmedia models (e.g., the KC model) that relate the medium’spermeability to a constant tortuosity. The argument can bemore significant in cases where there are considerableinternal sources of energy from solid material interactingwith the fluid (e.g., geothermal, chemical, or mechanicalenergies). However, how the flow tortuosity variations thatwe reveal here are related to an invariant permeabilityremains a question.Given that a large portion of pore space occupied by

vortices is virtually immobile under relatively low Darcyfluxes for a dynamic medium, how is the solute transport insuch a medium? The complex transport mechanism callednon-Fickian transport [70,71] has been shown to be aninherent characteristic of many types of porous media(packed and granular media, fractured and open networks),whether homogeneous or heterogeneous [72]. To someextent, can it be related to the pore fluid structures thatdiffer from the ordinary channelized flow? The immobilevortical zones may provide transient storage for solute viaexchange with the main solute transport path along thepassing-through flow. When examined in realistic porousmedia conditions, this argument may help us to betterunderstand complex transport mechanisms in porousmedia. The solute transport mechanisms with respect tothe variable pore flow behaviors, thus, require furtherinvestigations.

ACKNOWLEDGMENTS

This research is funded by the Australian ResearchCouncil via the Discovery Projects (No. DPI120102188:Hydraulic erosion of granular structures: experimentsand computational simulations and No. DP140100490).The authors also acknowledge support from theNational Natural Science Foundation of China (GrantNo. 51421006).

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