journal of contaminant hydrology · 2018. 7. 18. · and engineering. speciﬁc examples include...

Journal of Contaminant Hydrology 124 (2011) 68–81

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Journal of Contaminant Hydrology

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Pore-scale modeling of dispersion in disordered porous media

Saeed Ovaysi⁎, Mohammad PiriDepartment of Chemical and Petroleum Engineering, University of Wyoming, Laramie, WY 82071–2000, USA

a r t i c l e i n f o

⁎ Corresponding author. Tel.: +1 307 766 4923; faE-mail addresses: [email protected] (S. Ovaysi), mp

0169-7722/$ – see front matter © 2011 Elsevier B.V.doi:10.1016/j.jconhyd.2011.02.004

a b s t r a c t

Article history:Received 7 September 2010Received in revised form 3 February 2011Accepted 3 February 2011Available online 15 February 2011

We employ a direct pore-level model of incompressible flow that uses the modified movingparticle semi-implicit (MMPS) method. The model is capable of simulating both unsteady- andsteady-state flow directly in microtomography images of naturally-occurring porous media.We further develop this model to simulate solute transport in disordered porous media. Thegoverning equations of flow and transport at the pore level, i.e., Navier–Stokes and convection–diffusion, are solved directly in the pore space mapped by microtomography techniques. Threenaturally-occurring sandstones are studied in this work.We verify the accuracy of themodel bycomparing the computed longitudinal dispersion coefficients against the experimental data fora wide range of Peclet numbers, i.e., 5×10−2bPeb1×106. Solutions of full Navier–Stokesenable us to examine the impact of inertial forces at the very high Peclet numbers. We showthat inclusion of the inertial forces improves the agreement between the computed dispersioncoefficients with their experimental counterparts. We then investigate the impact of pore-space topology on the pre-asymptotic and asymptotic dispersion regimes by comparing solutedispersion in the three sandstones that possess different topological features. We illustrate howgrain size and homogeneity of the two sandstones dictate the threshold and magnitude of theasymptotic regime.

© 2011 Elsevier B.V. All rights reserved.

Keywords:Moving particle semi-implicitDirect pore-level modelingPorous mediaLongitudinal DispersionPore space topology

1. Introduction

Dispersive transport of various solutes in disorderedporous media is of great importance in many areas of scienceand engineering. Specific examples include tracer studies inoil recovery, subsurface contaminant transport, chemicaltransport in packed bed reactors, water filtration, fuel cells,and development of modern chemical catalysts.

The conventional practice in modeling field-scale disper-sion is to use the advection–dispersion equation with thedispersion tensor obtained from column experiments orvarious simulation techniques such as pore-scale modeling.The injected plume of solute will initially start spreadingthroughout the system in what is called a pre-asymptoticregime. Once the pore space heterogeneities at differentscales are scanned, the plume of solute follows an asymptoticspreading regime. Dispersion in porous systems with multi-

x: +1 307 766 [email protected] (M. Piri)

All rights reserved.

.

scale heterogeneities like fractal media, however, will neverreach an asymptotic regime (Cushman, 1997).

Furthermore, asymptotic dispersion coefficient stronglydepends on Peclet number, Pe = UaDm, where U is the averageflow velocity, a is the characteristic length, and Dm is themolecular diffusion coefficient, which scales the rate ofadvection to diffusion. The asymptotic behavior of thelongitudinal dispersion coefficient, DL, for solute transportin consolidated and unconsolidated porous media has beenstudied extensively using experimental techniques (Blackwellet al., 1959; Carberry and Bretton, 1958; Dullien, 1992;Ebach and White, 1958; Edwards and Richardson, 1968;Frosch et al., 2000; Gist et al., 1990; Kandhai et al., 2002;Khrapitchev and Callaghan, 2003; Legatski and Katz, 1967;Pfannkuch, 1963; Rifai et al., 1956; Seymour and Callaghan,1997; Stöhr, 2003). Bear (1972) identifies five distinct zonesfor variation of the longitudinal dispersion coefficient withPeclet number. A pure molecular diffusion zone for Peb0.4where the longitudinal dispersion coefficient is independentof Peclet number and less than molecular diffusion coefficient

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69S. Ovaysi, M. Piri / Journal of Contaminant Hydrology 124 (2011) 68–81

due to the tortuosity of the porous system. Zone II is atransition zone and corresponds to 0.4bPeb5 where theeffects of both the molecular diffusion and mechanicaldispersion should be added as they are in the same order ofmagnitude. In zone III, transversal molecular diffusion reducesthe effect ofmechanical dispersion. The longitudinal dispersioncoefficient follows a power law function of Peclet numberthrough DL∼Peδ where 1bδb1.2. This zone is also referred toas boundary layer by Sahimi (1995). Mechanical dispersion isfully dominant in zone IV and the dispersion coefficientbecomes proportional to Peclet number, i.e., DL∼Pe. In zoneV, as the fluid velocity and hence Peclet number increase, theinertia and turbulence gain considerable impact on thedispersion coefficient leading to a sub-linear dependence ofDL on Pe.

Fundamental understanding of dispersion in porous mediarequires development of physically-based predictive modelsthat can be used to study the impact of various phenomena onthe dispersive transport of solutes. Pore-level transport modelsare among appealing approaches that have been used over theyears to study dispersion in porous media. The success of suchmodels depends on detailed understanding of pore-levelphenomena and utilization of faithful representations ofrandompore space aswell as robust computational algorithms.One may categorize these models into two main groups:1) pore-scale networkmodels and 2) direct pore-level models.Here we include a short review of examples from each group.

In the first group of pore-level models, the pore space isdescribed as a two- or three-dimensional network of poresconnected by throats with some idealized geometry. Then aseries of transport (or displacement) rules in each pore orthroat are combined to simulate various processes. Recentadvances in pore scale network modeling have been reviewedbyCelia et al. (1995) and Blunt et al. (2002). Bijeljic et al. (2004)used a two-dimensional networkmodel alongwith continuoustime random walk to study dispersion in porous media. Thepore-size distribution of a three-dimensional Berea sandstonenetwork was adopted to build the two-dimensional networkused in their work. Throats had square cross-section and ananalytical solution for steady-state laminar flow was used toobtain the velocity field in each capillary element. Thecalculated longitudinal dispersion coefficients were in goodagreement with the experimental data for Peb4000. ForPeN4000 the results differed up to 50% from their experimentalcounterpart. The authors ascribed this difference to the inertialeffects in pores with high velocities. Jha et al. (2008) used anetwork model representing a random packing of spheresalong with a set of deterministic rules for particle tracking tocalculate the dispersion coefficient for Peb1×105. The resultscompared well with the experimental data when a parabolicvelocity field was employed in the throats. Acharya et al.(2007a) employed the mixing cell model to calculate disper-sivity in three dimensional regular pore-networks and com-pared the results against those obtained utilizing the Brownianparticle tracking model.

In the second group of pore-level models, the governingequations of flow are directly solved in the pore spaceaccompanied by a solute transport model. Maier et al. (2000)used Lattice–Boltzmann method to calculate the flow fieldand random walk particle tracking method to model solutetransport in a random packing of spheres for 1bPeb5000 and

obtained a good agreement with the experimental data.Acharya et al. (2007b) used a hybrid model, i.e., Lattice–Boltzmann for steady-state flow field and finite volume forreactive transport, to study dispersion and reaction along thetransverse mixing zone in two-dimensional porous media.They concluded that, regardless of the geometrical propertiesof the media, product formation can be predicted usingnonreactive transverse dispersion coefficients. Coelho et al.(1997) applied finite-difference to solve the Stokes equationfor flow field and the so called B-equation for solute transportto study dispersion in random packing of spherical and non-spherical particles. The calculated dispersion coefficientswere in good agreement with the experimental data forPeb1000. The same approach for the flow field had been usedby Salles et al. (1993) and Yao et al. (1997) in which solutetransport was modeled using a Monte Carlo approach. Theaccuracy of the predicted results, however, was not satisfac-tory. Garmeh et al. (2009) used the commercially availablefinite-element code COMSOL to model dispersion in twodimensional artificial packed beds. The steady-state incom-pressible Navier–Stokes equation along with the convection–diffusion equation was solved to obtain an interestinganalysis of the effect of porosity and heterogeneity ondispersion coefficient. Mostaghimi et al. (2010) used a finitedifference solver to obtain a solution to the Stokes equation ina variety of natural sandstones. A streamline-based algorithmwas then employed along with a random walk method tostudy solute dispersion in naturally-occurring sandstones for0.01bPeb105. The simulation results at high Peclet numbersunderestimated the experimental data. Recently, Zaretskiyet al. (2010) used a hybrid of finite element and finitedifference to solve the Stokes equation in a microtomographyimage of Fontainebleau sandstone. The computed velocitystreamlines were used in the advection–diffusion equation tostudy dispersion in this sandstone for 0.1bPeb104. Theresults were consistent with the experimental data in theliterature. Smoothed Particle Hydrodynamics (SPH), which isa particle-based Lagrangian method, has also been employedto solve the governing equations for both the flow field andsolute transport. Unlike Eulerian methods, the numericaldispersion due to the convective term in the convection–diffusion equation is eliminated in Lagrangian methods. Zhuand Fox (2002) used this technique to model dispersion intwo dimensional artificial periodic porous media. Further-more, Tartakovsky and Meakin (2006) used the sametechnique to investigate dispersion in artificial 2D models ofporous media. The results from these two studies, however,were not compared with the experimental data.

In this work, we extend the direct pore-level model ofincompressible fluid flow in porous media developed byOvaysi and Piri (2010) to model dispersive transport of solutein porous media. We first introduce high-resolution X-rayimages of the three sandstones used in this work. A briefdescription of the Modified Moving Particle Semi-implicit(MMPS) method is given next that is then followed by adetailed formulation of the solute transport model. We thenanalyze the modeled pore-scale flow fields as well as solutetransport in the samples studied here and compare thecalculated longitudinal dispersion coefficients against theexperimental data available in literature. The impact of theinertial forces on dispersion coefficient as well as the

70 S. Ovaysi, M. Piri / Journal of Contaminant Hydrology 124 (2011) 68–81

relevance of the pore-space topology in the pre-asymptoticregime is discussed in the closing section.

2. Pore space representation

We use high-resolution X-ray images of three differentsandstones, taken from Dong (2007), in this work. As shown inFig. 1, these sandstones have different pore-space topologies.The Berea sandstone has a porosity of 19.6% and an averagecomputed permeability of 1.3 D. S9 is a reservoir sandstonewith 22.2% porosity and a computed permeability of 2.2 D.Sandstone S1 has a porosity of 14.1% and an average computedpermeability of 1.7 D. Pore-size distribution graphs for both S9and S1 samples show broad distributions with pores as large as100 μm. Whereas, Berea sandstone has narrower pore-sizedistribution, i.e., 3–73 μm. Resolution of the microtomographyimage of the Berea sample is 5.345 μm with 400×400×400voxels in x, y, and z directions whereas both S9 and S1 are300×300×300 voxels but with different resolutions of3.398 μm and 8.683 μm, respectively. This makes the Bereaand S1 images more than twice the size of S9 image in everydirection. The statistics reportedaboveare extracted fromDong

Fig. 1. High-resolution X-ray images of the three sandstones studied in this worand resolution of 5.345 μm. (b) Sandstone S9 with 1.019 mm×1.019 mm×1.02.605 mm×2.605 mm×2.605 mm dimensions and resolution of 8.683 μm.

(2007). In Section 4 we elaborate on the samples selected fromthese sandstones to carry out the simulations.

3. Model description

The convective–diffusive transport of a nonreactive solutein an incompressible Newtonian flow field is governed by thecombination of the incompressible Navier–Stokes equationsand the convection–diffusion equation, i.e.,

DvDt

= −1ρ∇P + μ

ρ∇2v + g ð1Þ

∇:v = 0 ð2Þ

and

DCDt

= Dm∇2C ð3Þ

where C is the volume concentration of solute, Dm is themolecular diffusion coefficient, v is the velocity vector, ρ is thedensity, μ is the viscosity, g is the gravity vector, and P is

k. (a) Berea sandstone with 2.138 mm×2.138 mm×2.138 mm dimensions19 mm dimensions and resolution of 3.398 μm. (c) Sandstone S1 with

Table 1Some of the details of the three samples taken from the sandstones Berea, S9, and S1 studied in this work. For additional information about these sandstones seeDong (2007).

Item Sample A Sample B Sample C

Sandstone Berea S9 S1

Number of voxels in x direction 42 66 52Number of voxels in y direction 42 66 52Number of voxels in z direction 190 298 234Resolution (μm) 10.69 6.796 8.683Sample size in x direction (mm) 0.448980 0.448536 0.4515Sample size in y direction (mm) 0.448980 0.448536 0.4515Sample size in z direction (mm) 2.0311 2.025208 2.031822Sample volume (mm3) 0.40944 0.40744 0.41422Total number of main domain particles (solid and fluid) 335,160 1,298,088 632,736Number of main domain fluid particles (including stabilizer and upstream and downstream) 110,175 412,636 169,445Total number of particles (solid and fluid) including shell, stabilizer, and upstream and downstream 629,856 1,971,216 1,143,072Kernel size (μm) 32.07 20.388 26.049


pressure.Note, DDt

representsmaterial derivativewhich is used todescribe the Lagrangian movement.

Eqs. (1) and (2) are solved using the Modified MovingParticle Semi-implicit (MMPS) method, cf. (Ovaysi and Piri,2010). MMPS is an adaptive particle-based method that iscapable of simulating flow of an incompressible fluid in highly

Fig. 2. Static pressure profile for the three samples studied here,the upstream and downstream sections). (a) Sample A from the Berea sandstoneresolution. (c) Sample C from sandstone S1 with 8.683 μm resolution. Note: Darmicrotomography images. Flow is from left to right (z direction).

disorderedporousmedia, e.g., naturally-occurring rocks. Eq. (3)is also solved using the same particle-based techniquedescribed by Ovaysi and Piri (2010). The convection–diffusionequationmust be coupledwith the imperviousness of the soliddomain which implies n.∇C=0, where n is the outwardnormal vector to the fluid domain. In MMPS, this boundary

all with dimensions of 0.45 mm×0.45 mm×2 mm (excluding stabilizers andwith 10.69 μm resolution. (b) Sample B from sandstone S9 with 6.796 μm

k blue isolated particle clusters represent isolated pores from the original

Unlabelled image


condition is satisfied by assigning a virtual concentration equalto Ci to the solid particles in the neighborhood of i. More detailscan be found in (Ovaysi, 2010). Furthermore, the dispersiontensor is computed using themethod ofmomentswhich is alsoused in (Zhu and Fox, 2002).

Fig. 3. Concentration maps in sample A for Pe=50 at dishown. (a) t*=0.002. (b) t*=0.05. (c) t*=0.1. (d) t*=0.15. (e) t*=0.2. (f) t*=0

4. Results and discussions

In this sectionwe present the results of our simulations.Weuse three samples taken from sandstones Berea, S9, and S1,introduced in Section 2, with approximately equal dimensions

fferent times during the simulation. Only the particles with CN3×10−5 are.25.

Unlabelled image


of 0.45 mm×0.45 mm×2 mm in x, y, and z directions,respectively, see Table 1 for more details. Fig. 2 shows avisualization of the pore space in these samples along with thestatic pressure profile when each sample is under 5 Pa pressuredifference across the z direction. We study the behavior of thelongitudinal dispersion coefficient along the longest span of thesample, i.e., z direction. This is to provide the plume with alonger medium to establish the asymptotic behavior.

Sample A, selected from the Berea sandstone, is coarsenedin all directions by a factor of 2 resulting in a resolution of10.69 μm. Given the pore-size distribution of 3–73 μm forBerea sandstone and the fact that only the pores smaller than10.69 μm are lost by this coarsening, majority of the pore-level features of this sandstone are still preserved in sample A.Sample B was taken from a micro-CT image of the reservoirsandstone S9. But since the original image did not have thesame length as the Berea sandstone, we mirrored the X-rayimage of S9 in the z direction. The entire image was thencoarsened by a factor of 2 (to reduce the computational cost)and then sample B was taken from it with the samedimensions as those of sample A. Sample C is directly takenfrom sandstone S1 with the original resolution of 8.683 μm.

We study both pre-asymptotic and asymptotic regimes ofthe longitudinal dispersion coefficient in the above-men-tioned samples. We use simulations in sample A to compareour results with the experimental data available in theliterature. We then present an analysis of the inertial effectsat high Pe flows in the same sample. Finally, we compare theresults of simulations in samples A, B, and C to investigate theimpact of pore-space topology on solute transport at the porelevel under similar flow conditions.

4.1. Pre-asymptotic and asymptotic dispersion regimes

Sample A is chosen to study the pre-asymptotic andasymptotic behavior of the longitudinal dispersion coefficientunder awide range of flow conditions. To do that, we carry out aset of simulations targeting the range 5×10−2bPeb1×106.Water at 25 °C with density ρ=997 Kg m−3 and viscosity μ=

400

350

250

350

0 0.05 00.1

150

300

200

100

0

DL/

Dm

Fig. 4. Progress of the longitudinal dispersion coefficient divided by molecular diffusregimes.

0.001 Kg m−1 s−1 is usedas solvent.We inject a 6×10−5 mm3

pulse of a nonreactive solute with the volume-based dimen-sionless concentration of C=0.1. That is 0.004% of the total porevolume of sample A.We assume that the injection of solute doesnot impact the overall fluid's viscosity and density. Injectionstarts at t=0 and hence the pre-asymptotic regime includes theunsteady-state flow field. Also, to ensure that only the porespace is scanned by the solute plume, the pulse is injectedimmediately after the upstream stabilizer and is withdrawnright before the downstream stabilizer.

Fig. 3 presents visualization of solute concentrationdistribution for Pe=50 at different dimensionless timest� = QtVp, where Q is the steady-state flow rate, t is time, andVp is the pore volume. The development of solute plumeshown in this figure underlines the role of the random porespace on the dynamic behavior of solute dispersion indisordered porous media. As time progresses, the initialhigh concentration plume spreads throughout the pore space.The convective and diffusive forces are both important at thisPeclet number. The plume is mainly transported by convec-tion through the most active areas of the pore space while theedges of the plume experience the maximum diffusion due tothe highest concentration gradient with the bulk of fluid.Parts of the pore space that are less conductive, or dead ends,receive the solute mass only through diffusion. This diffusedmass is then released back to the bulk of fluid after the soluteplume passes the region, which leads to a further stretching ofthe plume. This process continues to the extent that all of thefeatures of the pore space have played a role in spreading thesolute plume. When this happens, the plume's rate ofexpansion, i.e., dispersion coefficient, becomes constantwhich marks the threshold of the asymptotic regime. Usingthe concentration map at each time step and the method ofmoments, the pre-asymptotic and asymptotic regimes of thedispersion coefficient can be plotted as a function of time. Theresults obtained from the simulations enable us to computethe entire dispersion tensor. However, the samples we haveselected were not, to limit the computational cost, longenough in the transverse directions of x and y to establish an

.15

t*0.2 0.25 0.3

ion coefficient in sample A at Pe=50 for the pre-asymptotic and asymptotic

image of Fig.�4


asymptotic regime in those directions. Therefore, we limit thediscussion to only longitudinal dispersion coefficient in the zdirection. Shown in Fig. 4 is the variation of DL, thelongitudinal dispersion coefficient, normalized by Dm, molec-ular diffusion coefficient, with time for Pe=50. The DL/Dmexhibits a pre-asymptotic trend that ultimately stabilizes atthe asymptotic value. This transition is affected by numerousfactors taken into account by the model, e.g., pore-space

Fig. 5. Concentration maps in sample A at t*=0.11 for(a) Pe=0.5. (b) Pe=50. (c) Pe=500. (d) Pe=5000. (e) Pe=5×104. (f) Pe=5×1

topology, local Pe, pore-level molecular diffusion, and pore-level mixing induced by tortuosity.

Fig. 5 illustrates the plume's progress at t*=0.11 for sixdifferent Peclet numbers covering different dispersionregimes. At low Peclet numbers molecular diffusion isexpectedly the dominant process to transport the plume.This mass transfer, however, takes place with little influencefrom pore conductivities, giving equal importance to the highly

different Peclet numbers. Only the particles with CN3×10−5 are shown05.

.


conductive and less conductive pores in solute transport. As thePeclet number increases, convective transport gains dominanceand the plume becomes more selective in the route it takes bypassing through the most conductive pores. By increasing thePeclet number, faster movement of the solute particlesgradually leaves no time for diffusion to take place. Eventuallyat the very high Peclet numbers, i.e., PeN5000, solute particlespass through the system with negligible diffusion taking place.

0.016

0.014

0.012

0.010

0.008

0 500 1,000 1,500 2,000 2,500

0 500 1,000 1,500 2,000 2,500

0 500 1,000 1,500 2,000 2,500

t*=0.002t*=0.05t*=0.1

t*=0.002t*=0.05t*=0.2

t*=0.002

t*=0.05

t*=0.25

0.006

0.004

C/C

0C

/C0

C/C

0

L (microns)

L (microns)

L (microns)

0.002

0.000

0.018

0.016

0.014

0.012

0.008

0.01

0.006

0.004

0.002

0

0.018

0.016

0.014

0.012

0.008

0.010

0.006

0.004

0.002

0.000

a

c

eFig. 6. Progress of the normalized concentration profiles at different Peclet numbers i(a) Pe=0.5. (b) Pe=50. (c) Pe=500. (d) Pe=5000. (e) Pe=5×104. (f) Pe=5×1

In Fig. 6, we show the concentration profiles at different Pecletnumbers until Gaussian distribution is reached. The concentra-tions are averaged over the entire transverse cross-section.Since the injected solute pulse does not cover the entiremedium in the transverse directions, the initial normalizedconcentration is smaller than unity. Noteworthy, at high Pecletnumbers the high concentration individual solute particlescontribute tonon-Gaussiandistributionswhichare also evident

0 500 1,000 1,500 2,000 2,500

t*=0.002t*=0.05t*=0.25

t*=0.002

t*=0.05t*=0.25

t*=0.002t*=0.05t*=0.25

0 500 1,000 1,500 2,000 2,500

0 500 1,000 1,500 2,000 2,500

C/C

0C

/C0

C/C

0

L (microns)

L (microns)

L (microns)

0.018

0.016

0.014

0.012

0.008

0.01

0.006

0.004

0.002

0

0.018

0.016

0.014

0.012

0.008

0.01

0.006

0.004

0.002

0

0.018

0.016

0.014

0.012

0.008

0.010

0.006

0.004

0.002

0.000

b

d

fn sample A. C0=0.1 is the initial concentration of the injected solute particles.05.

image of Fig.�6


from Fig. 5. However,we believe a bigger solute pulse, i.e.,moresolute particles, along with a longer sample length in thelongitudinal direction z would have allowed for Gaussiandistribution to establish in this high Peclet flows.

4.2. Comparison with experimental data

The asymptotic longitudinal dispersion coefficientsobtained from the simulations in sample A are comparedagainst the experimental data available in the literature inFig. 7. The reportedPeclet numbers are based on the80–330 μmgrain size range reported by Øren and Bakke (2003) for Bereasandstone. We use the average value of 200 μm for a in thiswork. For Peb1×105, we kept the pressure difference acrossthe system fixed at 5 Pa while changing Dm from 1×10−7 to1×10−13 m2 s−1. Whereas for PeN1×105, we used the twoapproaches discussed in the next section. The predicted longi-tudinal dispersion coefficients showanencouraging agreementwith their experimental counterparts for 5×10−2bPeb1×105.For PeN1×105, we slightly overestimate the longitudinaldispersion coefficients, which in the next section we showthat it can be improved by inclusion of inertial forces.

4.2.1. Inertial effectsTo investigate the impact of inertial forces on the longitu-

dinal dispersion coefficient, simulations at 1×105bPeb1×106

were carried out in two ways: 1) by decreasing the moleculardiffusion coefficient, Dm and 2) by increasing the appliedpressure difference and hence velocity, U. In the first approach,wefixed the pressure difference across the length of the systemat 5 Pa and lowered the molecular diffusion coefficient from1×10−13 to 5×10−14 m2 s−1. This corresponds to a fixedReynolds number of Re = ρUaμ = 0:005. As shown in Fig. 7, thecomputed dispersion coefficients follow the same trend as the

DL/

Dm

P

107

106

105

104

103

102

1

101

101

100

10010-1

10-3 10-2 10-1

Fig. 7. The computed longitudinal dispersion coefficient divided by molecular diffusfrom Ebach and White (1958), Carberry and Bretton (1958), Edwards and Richards(1997), Kandhai et al. (2002), Khrapitchev and Callaghan (2003), Stöhr (2003), and(2004), and Pfannkuch (1963).

mechanical dispersion regime leading to overestimation of thedispersion coefficient at very high Peclet numbers. In thesecond approach, however, we used a fixedmolecular diffusioncoefficient of 1×10−12 m2 s−1 and increased the pressuregradient from 5 to 1000 Pawhich corresponds to 0.005bReb1.As shown in Fig. 7, this leads to an improved agreement withthe experimental data, which highlights the role of inertialforces at PeN1×105. In line with the concluding remarks byWood (2007), we believe the correct measure for the inertialzone of dispersion is both the Peclet and Reynolds numbers.

4.3. Impact of pore-space topology

Samples B and C are used to investigate the impact ofpore-space topology on solute transport at the pore level.Similar to sample A, we inject a pulse of 6×10−5 mm3 withC=0.1 and calculate the longitudinal dispersion coefficient atdifferent times. The absolute permeability (K) of sandstonesS9, S1, and Berea, reported in Section 2, is used to calculate thegrain size for samples B and C through K∝am, whereShepherd (1989) suggests mb1.5 for consolidated sand-stones. We use m=1.4, which results in a=300 μm forsample B and a=270 μm for sample C. Both pre-asymptoticand asymptotic longitudinal dispersion coefficients are thencomputed. Comparison of the latter with the experimentaldata is shown in Fig. 7 while Fig. 8 shows a comparison ofboth with those of sample A for the same Peclet number of 50.

Apart from different pre-asymptotic regimes, Fig. 8 showslower asymptotic longitudinal dispersion coefficients forsamples B and C compared to that of sample A, while all thethree points are still in the range of the experimental data inFig. 7. This difference can be attributed to different pore-spacetopologies in these three sandstones. Sample A has a finer grainsizewith a homogeneous distribution of the pore space. Sample

e10710610510410302

ion coefficient versus Peclet number compared against the experimental dataon (1968), Blackwell et al. (1959), Rifai et al. (1956), Seymour and CallaghanPfannkuch (1963), which were extracted fromWhitaker (1999), Bijeljic et al

.

0

50

100

150

200

250

300

350

400

0 0.05 0.15 0.250.1 0.2 0.3

t*

SampleA

SampleB

SampleC

DL/

Dm

Fig. 8. Comparison of the pre-asymptotic and asymptotic regimes of the longitudinal dispersion coefficients in samples A, B, and C at Pe=50.


B, on the other hand, is made of coarser grains with a relativelyless homogeneous distribution of the pore space. Grain sizedistribution and tortuosity of Sample C lies between those ofsamples A and B. These different topologies are manifested invelocity distributions shown in Fig. 9. The homogeneousdistribution of fine grains in sample A has led to a well-connected active flow field creating high local Peclet numbersthroughout the system. In sample B, however, the high velocityareas are not well-connected which leaves most of the systemwith local Peclet of less than 50. The symmetric velocitymap in

Fig. 9. Velocity profile at Pe=50 in (a) sample AwithU=2with U=1.65×10−5 ms−1. Only the particles with vN12U are shown, where U=individual particle.

Fig. 9 is expected as sample B is mirrored in the longitudinaldirection. Velocity distribution in sample C reveals a moder-ately well-connected pore space. It is also evident from Fig. 9that pores in sample C are larger than those in samples A and B.This is inlinewith the pore-size distribution graphs reported forthe Berea sandstone and sandstones S9 and S1 in Dong (2007).Unlike sample B, high velocity areas in sample C are distributedthroughout the system. This distribution, however, is not asuniform as in sample A. The overall impact of the distribution oflocal Peclet numbers is a higher dispersion coefficient in sample

.63×10−5 ms−1, (b) sample BwithU=1.68×10−5 ms−1, and (c) sample CQ/A is the average velocity in the entire system and v is the velocity of each

Fig. 10. Concentration maps in sample B for Pe=50 at different times during the simulation. Only the particles with CN3×10−5 areshown. (a) t*=0.002. (b) t*=0.05. (c) t*=0.1. (d) t*=0.15. (e) t*=0.2. (f) t*=0.25.

Fig. 11. Concentration maps in sample C for Pe=50 at different times during the simulation. Only the particles with CN3×10−5 areshown. (a) t*=0.02 corresponding to point A in Fig. 12. (b) t*=0.028 corresponding to point B in Fig. 12. (c) t*=0.037 corresponding to point C in Fig. 12(d) t*=0.053 corresponding to point D in Fig. 12. (e) t*=0.1 corresponding to point E in Fig. 12. (f) t*=0.13 corresponding to point F in Fig. 12. (g) t*=0.165corresponding to point G in Fig. 12. (h) t*=0.236 corresponding to point H in Fig. 12.


.

Unlabelled imageUnlabelled image


image of Fig.�11

0

50

100

150

200

250

300

0 0.05 0.1 0.2 0.30.15 0.25

t (s)

A

B

CD

E

FG

HDL/

Dm

Fig. 12. Dynamic behavior of the longitudinal dispersion coefficient for sample C at Pe=50.


A compared to those in samples B and C, as shown in Fig. 8. Dueto its superior connectivity and tortuosity, sample C, yields ahigher asymptotic dispersion coefficient than sample B. Thisphenomenon is also evident from Figs. 10 and 11, where wevisualize theprogress of the solute plume in samples B andC forPe=50. The plume of solute in samples B and C show a lowertendency to disperse in Figs. 10 and 11 compared to theprogress in sample A, see Fig. 3, leading to lower asymptoticdispersion coefficients in samples B and C.

Also noteworthy in Fig. 9, most of the active (highvelocity) pore space in sample B is located at the beginningof the system. Knowing that scanning the active pore space bythe plume is a necessity to reach the asymptotic regime, thesolute plume in sample B does not need to travel far in orderto reach the asymptotic regime. Consequently, threshold ofthe asymptotic regime in this sample has been reached evenbefore the augmented replica starts as it is clearly seen inFigs. 8 and 10. In samples A and C, on the other hand, theactive pore space is scattered throughout the system whichrequires the plume to scan the entire system to reach theasymptotic regime. This explains the relatively short asymp-totic threshold observed for sample B in Fig. 8 compared tothose for samples A and C.

Lastly, we use Figs. 11 and 12 to further elaborate on theimpact of pore-space topology on the dynamic behavior ofdispersion coefficient in sample C. Starting from point A, theplume of solute is confined to only two fluid channels. Frompoint A to B, the front of this plume stretches to new junctionswhich causes the plume to mix. This mixing intensifies wheneven more junctions are met at point C. During this period,dispersion coefficient incrementally increases until point C.This is then followed by a relatively sharp decrease indispersion coefficient. The reason for this decrease is clearlyillustrated in Fig. 11d which corresponds to point D. At thispoint, the plume of solute does not encounter new junctions.Therefore,minimummixing occurs and solute dispersion slowsdown, leading to a decrease in dispersion coefficient. Thiscontinues until point E, where new junctions are faced by theplume.However, compared to the junctionsmet at points B andC, the junctions at point E cannot give a sharp increase in the

dispersion coefficient. This is mainly due to the fact that theplume of solute has previously experienced a strong mixingfromB to C. Themixing continues until point F is reached. FromF to G the plume does not meet new junctions and simply fillsup the fluid channels. This cycle of scanning new junctionsand filling up the pore bodies causes a fluctuating trend in thedynamic behavior of dispersion coefficient. These fluctuations,however, dampen quickly after the porous system does notoffer stronger mixing environments. Even though the plumepasses through new junctions from G to H, the new junctionscannot offer a highly superior mixing thanwhat has previouslybeen experienced. Consequently, the dispersion coefficientonly increases slightly and eventually stabilizes at point H,where the asymptotic behavior is established. The processexplained above further supports the statement made earlierthat active pore space, where plume is mixed best, determinesthe onset of asymptotic behavior which is consistent with thefindings of Maier et al. (2000).

5. Conclusions

A direct particle-based pore-level model of solute trans-port in random porous media was developed. Themodel is anextension of the work presented by Ovaysi and Piri (2010)and is able to handle both unsteady- and steady-stateconvective–diffusive transport of a nonreactive solute in anincompressible Newtonian solvent through disordered po-rous media. A highly scalable parallel implementation of thismodel was used to carry out direct pore-level simulations ofsolute transport in high-resolution microtomography imagesof three sandstone samples A, B, and C. Simulations in sampleA were performed over a wide range of Peclet numbers(5×10−2bPeb1×106) to predict both pre-asymptotic andasymptotic longitudinal dispersion coefficients. The asymp-totic values were then successfully compared with theirexperimental counterparts. For 1×105bPeb1×106 an inves-tigation of the inertial effects was presented throughsimulations with higher Reynolds number. This resulted inan improved comparison of the predicted longitudinal

image of Fig.�12


dispersion coefficients with the experimental data in thisrange of Peclet numbers.

Finally, simulations in sample A, B, and C were comparedfor a given Peclet number. The differences observed in thepredicted pre-asymptotic and asymptotic dispersion in thesesamples were explained using the differences in pore-spacetopologies and local Pe. Sample C was further used to shedlight on the physics of solute dispersion at the pore level.

To the best of our knowledge, this is the first directLagrangian pore-level model of solute transport in micro-tomography images of naturally-occurring porous systems.The model can be used to study a wide range of pore-leveltransport problems in porous media including reactivetransport, multi-component mass transfer, and ion transport.

Acknowledgments

We gratefully acknowledge financial support of EnCanaand the School of Energy Resources and the Enhanced OilRecovery Institute at the University of Wyoming. Hu Dong(iRock Technologies) is gratefully thanked for sharing hismicro-CT data with us.

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Pore-scale modeling of dispersion in disordered porous mediaIntroductionPore space representationModel descriptionResults and discussionsPre-asymptotic and asymptotic dispersion regimesComparison with experimental dataInertial effects

Impact of pore-space topology

ConclusionsAcknowledgmentsReferences

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