C. R. Geoscience 342 (2010) 881–891

Hydrology, environment

A pore-throat model based on grain-size distribution to quantifygravity-dominated DNAPL instabilities in a water-saturatedhomogeneous porous medium

Modele de reseau de pores et de capillaires, base sur la courbe granulometrique pour la

quantification des instabilites gravitaires du deplacement d’un DNAPL dans un milieu

poreux homogene sature en eau

Khalifa Nsir *, Gerhard Schafer

UMR 7517, Centre national de la recherche scientifique (CNRS), laboratoire d’hydrologie et de geochimie de Strasbourg (LHYGES), universite de Strasbourg (UdS), 1,

rue Blessig, 67084 Strasbourg cedex, France

A R T I C L E I N F O

Article history:

Received 21 April 2010

Accepted after revision 13 September 2010

Available online 17 November 2010

Presented by Ghislain de Marsily

Keywords:

Pore-throat model

Instabilities

Two-phase flow

Porous media

DNAPL

Size distribution

Mots cles :

Modele de pores et de capillaires

Instabilites

Ecoulement diphasique

DNAPL

Distribution de la taille

A B S T R A C T

A pore-scale network model based on spherical pore bodies and cylindrical pore throats

was developed to describe the displacement of water by DNAPL. The pore body size and

the pore throat size were given by statistical distributions with user-specified values for

the minimum, mean and maximum sizes. The numerical model was applied to a

laboratory experiment conducted on a sand-filled glass column. The parameters relative to

pore body size and pore throat size that were used in the construction of the equivalent

network were derived from the discrete grain-size distribution of the real porous medium.

The calculated arrival times of the DNAPL front were compared with those measured using

optic fibre sensors placed at different points on the control section of the experimental

device. Furthermore, the model simulated DNAPL pressure measured at the entrance

section of the system. In general, the numerical results obtained with the model were in

good agreement with the actual measurements.

� 2010 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.

R E S U M E

Un modele type reseau de pores et de capillaires a ete developpe pour decrire le

deplacement de l’eau par un DNAPL. Les tailles de pores et de capillaires sont obtenues a

partir d’une distribution statistique basee sur l’introduction d’une taille minimale,

moyenne et maximale, specifiees pour les pores et les capillaires. Le modele numerique

developpe est utilise pour simuler des experiences de laboratoire conduites sur une

colonne en verre. Le milieu poreux utilise est un sable moyen de granulometrie uniforme.

Les parametres relatifs a la taille des pores et des capillaires utilisee pour generer un reseau

equivalent au milieu poreux ont ete derives de la courbe granulometrique discrete du

sable. Les temps d’arrivee du front de DNAPL calcules ont ete compares avec ceux mesures

au moyen des fibres optiques placees a differents points dans une section de controle du

dispositif experimental. De plus, la pression moyenne du DNAPL mesuree pendant

Contents lists available at ScienceDirect

Comptes Rendus Geoscience

www.sciencedi rec t .com

* Corresponding author.

E-mail address: nsir@unistra.fr (K. Nsir).

1631-0713/$ – see front matter � 2010 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.

doi:10.1016/j.crte.2010.09.001

K. Nsir, G. Schafer / C. R. Geoscience 342 (2010) 881–891882

1. Introduction

Two-phase flow is encountered widely in the field ofhydrology because soil and groundwater pollution bydense non-aqueous phase liquids (DNAPLs), such aschlorinated solvents (e.g., trichloroethylene (TCE)), consti-tute a large and serious environmental problem (Banoet al., 2009; Bohy et al., 2004). When infiltrating throughporous media, migration of DNAPLs can result in highlyfingered fluid distributions. The occurrence of fingering iscaused by flow instabilities due to differences in viscosityand density between DNAPL and water (Khataniar andPeters, 1992; Lindner and Wagner, 2009; Riaz andTchelepia, 2006; Tehelepi et al., 1994). These fingerspropagate rapidly, resulting in an early breakthroughcompared to the displacement of a stable front. Generally,two approaches are used to study the displacementprocess through porous media. The first approach usesthe continuum description of the porous medium associ-ated with macroscopic laws (e.g., Muskat’s law) torepresent the volumetric flow vector for each fluid phaseon the REV scale (Barenblatt and Patzek, 2003; Bear, 1972;Galusinski and Saad, 2009). The second approach is basedon the microscopic description of the pore geometry andon the physical laws of flow and transport within the pores(Blunt and King, 1990; Laroche and Vizika, 2005; Lowryand Miller, 1995). One of the common tools that has beenused to study the displacement process is pore-scalenetwork modelling, which uses simple geometric shapes torepresent the pore space. Many network models have beendeveloped to study a wide range of displacementprocesses, including drainage and imbibitions (Hilpertand Miller, 2001; Joekar-Niasar et al., 2009; Nordhauget al., 2003; Patzek, 2000; Singh and Mohenty, 2003). Themost critical aspect of network model construction is theappropriate choice of the pore structure and its capacity torepresent the real physical system (Celia et al., 1995; Raoofand Hassanizadeh, 2009). Generally, an equivalent net-work to the real porous medium is used. This network canbe based on statistical distributions and correlations ofbasic morphologic parameters, such as pore bodies andpore throats. Typical parameters used for building anequivalent network include pore body size distribution,pore throat size distribution and throat length distribution.In the majority of models, the parameters are fictive and donot represent a realistic porous medium. Consequently,these models cannot effectively analyse and reproducephysical experiments.

Different approaches exist for identifying pore struc-tures (Saad et al., 2010). One of the common approaches isthe inverse method. Pressure-saturation curves measuredin the laboratory for a given two-phase fluid system in aporous medium are fitted using theoretical relationships to

obtain the equivalent geometric parameters of thenetwork model. Other experimental approaches are directmethods, such as the use of mercury porosimeters toassess the pore-size distribution of the porous medium.Because direct measurements of pore space structure aregenerally costly and time consuming, recent conceptualapproaches have been developed to derive the void sizedistribution (VSD) from the grain size distribution (GSD).The GSD is widely available for many soil types and can beroutinely determined in the laboratory. A novel approachbased on the geometric proprieties of the packing ofspheres has been elaborated (Dodds, 1980; Geoffrey andMellor, 1995). This approach computes the VSD resultingfrom a specific packing of spheres. The shape and size-distribution of the voids within a random packing of anarray of spheres and the effect of GSD on the packing havereceived considerable attention (Rouault and Assouline,1998).

In this article, we present a novel network model forsimulating drainage processes in a homogeneous porousmedium. The developed model is based on the same basicflow characteristics as the model introduced by Nordhauget al. (2003). However, our model considers capillaryeffects at water-DNAPL interfaces. Capillary pressure isassumed to be located at the boundary between a porebody and a pore throat. To quantify the flow rate in a porethroat, which is connected to a pore body separated by awater-DNAPL interface, we used the Washburn equation.Furthermore, to obtain a predictive network model, themodel parameters used to generate both a pore-body sizedistribution and pore-throat size distribution were quan-tified using a geometrical model of an array of spheresbased on the grain-size distribution combined with aprobabilistic approach. The numerical model was thenapplied to a laboratory drainage experiment (DNAPL/water) that was conducted on a sand-filled column understable and unstable flow conditions. The pressure at thecolumn entrance section and the arrival times distributionof the DNAPL front in a cross-section, measured during thedisplacement process, were compared to the numericalresults. A sensitivity study of the size of pore throats andthe size of the lattice was used to analyse the influence ofthese model parameters on the displacement of water byDNAPL.

2. Network model

Pore-scale network models typically represent the porespace of the medium using simplified geometries. Withinthese geometric representations, the models solve equa-tions to explicitly track the location of all fluid–fluidinterfaces within the network. We developed a pore-throatnetwork model to simulate the drainage process insaturated porous media. Drainage occurs during an

l’experience dans la section d’entree de la colonne a ete confrontee a la pression

transitoire predite par le modele developpe. Les resultats numeriques sont globalement

en bon accord avec les observations experimentales.

� 2010 Academie des sciences. Publie par Elsevier Masson SAS. Tous droits reserves.

K. Nsir, G. Schafer / C. R. Geoscience 342 (2010) 881–891 883

immiscible displacement of a wetting fluid (water) by anonwetting fluid, such as trichloroethylene.

2.1. Geometry of the network

The pore network that is simulated in the model is acubic lattice with constant node spacing. The pore bodiesare represented by spheres. Similarly, pore throats arecylindrical with a circular cross-section (Fig. 1). Thelocations of fluid-fluid interfaces are restricted to theconnections between the pore bodies and pore throats. Thepore space is constructed by assigning radii to the latticeelements using a Weibull distribution (Weibull, 1951),which is frequently used to construct network analogs ofporous media. The distribution is attractive due to itsflexibility and to the fact that it is based on fixedparameters that describe the characteristics of the sizesof pore bodies and pore throats of the network. Aminimum, mean and maximum radius (rmin, rmean andrmax, respectively) specified for both pore bodies and porethroats are therefore required to randomly generate thepore body-size distribution and pore-throat size distribu-tion in the network. The probability distribution that wasused is given by:

g xð Þ ¼x

x22

exp � x2

x22

!for x � x3

0 for x > x3

8><>: (1)

[()TD$FIG]

Fig. 1. 3D view of a pore-throat network with randomly distributed

spherical pores and uniformly sized cylindrical throats.

Fig. 1. Illustration 3D du reseau de pores et de capillaires avec une

distribution aleatoire de la taille de pores spheriques et des capillaires

cylindriques de taille uniforme.

where x = r� rmin, x2 = rmean� rmin, and x3 = rmax� rmin.Pore body radius and pore throat radius are representedby r. For a throat connecting pores i and j, the throat lengthis defined as Lij = L� (rpore, i + rpore, j), where L is the distancebetween the centres of two adjacent pore bodies. L isconstant in the chosen network. Pore body and pore throatradii are generated independently of each other. Therefore,there is no relationship between the radii of throats andthe neighbouring radii of pores. However, radii of porebodies are larger than those of pore throats to ensure theconverging–diverging nature of pores. In section 3, weexplain the model concept to quantify the ensemble of thepore body radius and the pore throat radius (rmin, rmean,rmax), as well as the node spacing, L, needed to construct anequivalent network of a real porous medium.

The porosity of the equivalent porous medium isassumed to be equal to the areal porosity of a horizontaltransect of the network model. The reason for using arealporosity is that volumes do not scale as the third power ofthe radius, whereas areas scale as second power of theradius. Assuming that the pore throat volumes arenegligible compared to the total volume of pore bodies,the porosity ø[-]of the pore-throat network is given by:

f ¼P

iApore;i

nxnynzL2(2)

where Apore, i [L2] is the area of pore body, and nx, ny and nz

are the number of pores in the x-, y- and z-directions,respectively. The index i runs over all the pore bodies of thenetwork.

2.2. Governing flow equations of the pore-throat model

The developed network model makes the followingassumptions: (i) the fluids used are Newtonian, incom-pressible and immiscible; (ii) the flow is laminar; and (iii)the local capillary pressure in the pore bodies is negligible,resulting in only one mean pressure within a pore body,independent of the phase saturation of that pore body. Themass balance equation for each pore is therefore written asfollows:X

j

Q ij ¼ Qei (3)

where Qij [L3T�1] denotes the flow rate through a porethroat connecting pore node i and pore j. Qe

i [L3T�1] is theexternal flow rate into pore i, which is zero unless the poreis located at the upstream flow boundary. The sum refers toall neighbouring pores of pore body i. The expression usedfor the flow rate Qij depends on the fluid present in the porethroat connecting pores i and j and on the presence of ameniscus at the boundaries between a pore body and apore throat.

If no meniscus is present at pore throat entry, the flowrate Qij through the pore throat is given by Poiseuille’s Law

Qi j ¼pr4

i j

8LmaPi � Pj � zi � z j

� �rag

� �(4)

where ma [ML�1T�1] and ra [ML�3] are the dynamicviscosity and the density of the fluid moving in the pore

K. Nsir, G. Schafer / C. R. Geoscience 342 (2010) 881–891884

throat, respectively. Pi and Pj are the fluid pressures inpores i and j at vertical positions zi and zj, respectively.

If a meniscus is present at pore throat entry, the flowrate Qij through the pore-throat is expressed by theWashburn equation (Washburn, 1921):

Qi j ¼ H Pi � P j � zi � z j

� �rag � Pc ri j

� �� � pr4i j

8Lm

� Pi � P j � zi � z j

� �rg

� �(5)

Pc(rij) [ML�1T�2] used in Eq. (5) denotes the capillaryentry pressure assigned to pore throat connecting pore i

and j that needs to be exceeded by the DNAPL to enter thewater filled pore throat. The capillary entry pressure isgiven by Young Laplace’s Law as follows:

Pc rij

� �¼ 2scosu

ri j(6)

where u is the contact angle (assumed zero in our case),and rij denotes the radius of the pore throat connectingpore i and pore j.

H Xh i is a Heaviside function that stands for thedisplacement condition of the interface in a pore throat.The throat can be blocked by capillarity if the capillaryentry pressure (Pc (rij)) has not yet been reached. TheHeaviside function is defined as:

H Xh i ¼ 1 for X>00 for X � 0

�(7)

In situations where a meniscus is present at boundariesbetween a pore body and a pore throat, the identity of thefluid flowing through the pore throat (DNAPL or water, orcapillary blocked) can only be determined after computa-tion of the pressure value of the whole network. Therefore,we use an ‘‘effective’’ viscosity m and an ‘‘effective’’ densityr for the displacing fluid in Eq. (5), which are defined by:

m ¼ mnwSij þ 1� Sij

� �mw

r ¼ rnwSij þ 1� Sij

� �rw

Sij ¼ Si þ S j

� �=2

(8)

where the suffixes w and nw refer to the wetting andnonwetting fluid, respectively, and Sij [-] is an averagephase saturation between pore bodies i and j. This factorevolves as function of time and indirectly takes interfacemovement inside a pore throat into account.

Introducing the appropriate expression of flow rate Qij

in Eq. (3) leads to a set of algebraic equations with the porebody pressures as unknowns. This set of equations iswritten in matrix form as follows:

AP ¼ B (9)

where A is the conductance matrix with elements thatdepend on the connection between different pore throats,P denotes the pressure vector, and the B vector containsthe pressures and the flow rates at the boundaries and thecapillary pressure term if a meniscus is present in suchpore throat entry. For a given time step, the flow isassumed to be in steady state, with constant fluidproperties. To control the nonlinearities under suchdisplacement conditions, the system of Eq. (9) is solved

for pressure P using the value of the saturation term of theprevious time step (Singh and Mohenty, 2003).

The saturation field is updated after computation of thepressure terms by quantifying a characteristic minimumtime (hereafter called Tmin), which corresponds to aminimum filling time for each of the pore bodies. Thecalculated pressure field is used to quantify fluid fluxesthrough the pore throats. The ratio of free volume in a porebody (not invaded by displacing fluid) to the flow rate at agiven interface location gives the filling time needed forthat pore body to be invaded. The saturations in each porebody are then updated using:

Snþ1i ¼ Sn

i þQi � Tmin

Vi(10)

where Sin and Si

n+1 denote the saturation at the previoustime (n) and calculated at the new time (n + 1), respective-ly, Qi [L3T�1] represents the sum of flow rates in all porethroats connected to pore body i and contains a movinginterface, Vi is the volume of pore body i, and Tmin is thetime step as defined above.

During one time step, only one pore body reaches fullnonwetting phase saturation. This pore body generates theadditional interfaces, which are located at all connectingpore throats that are filled with wetting fluid. Newlycreated interfaces are tested for stability, and appropriateexpressions for flow rate Qij are then identified. This leadsto a modified matrix of conductance terms that isreformulated for each time step in the set of equations(Eq. (9)). The field pressure is computed and the procedurefor updating saturation in pore bodies is repeated; thisprovides a transient response for pressure and saturationfields.

In the case of a single-phase flow, the intrinsicpermeability k [L2] of the equivalent porous medium canbe furthermore computed using Darcy’s Law as follows:

k ¼maQe

i;totalh

A Fup �Fdown

� � (11)

where Qei,total [L3T�1] is the sum of external flow rate into

pores i, located at the upstream boundary of the network,A [L2] and h [L] are the cross-sectional area and the totalheight of the network, respectively. Fup [ML�1T�2] andFdown [ML�1T�2] denote the phase potential at theupstream and downstream boundary of the network,respectively. Here, the phase potential is defined asF = P + ragz, where P [ML�1T�2] is the fluid pressurecomputed by the pore-throat model.

3. An approach based on grain-size distribution to derivethe radii of pore bodies and pore throats

3.1. Theory

To generate a distribution of pore-body sizes and porethroat-sizes for the network, we used a statisticaldistribution that specifies values of the minimum, meanand maximum size (Section 2.1). Assessment of thesenetwork parameters is essential to deliver a quantitativemodel and to obtain reliable simulations of laboratory

[()TD$FIG]

Fig. 2. A tetrahedral subunit of four touching spheres with given radii r1, r2, r3 and r4 and its pore space approximation as four pore throats with computed

radii rc1, rc2, rc3 and rc4 meeting at a cavity pore body with computed radius rp.

Fig. 2. Illustration d’un tetraedre unite, forme par le chevauchement de quatre spheres de rayons r1, r2, r3, et r4 et approximation de l’espace poral, donnee

par quatre capillaires de rayons calcules rc1, rc2, rc3 rc4 et un pore central de rayon calcule rp.

K. Nsir, G. Schafer / C. R. Geoscience 342 (2010) 881–891 885

experiments. Our modelling concept is based on geometricproperties of packing spheres, which are combined with aprobability approach proposed by Rouault and Assouline(1998). Models using packing spheres represent the porousmedium as a geometric system that can be described bytetrahedral subunits formed by groups of four touchingspheres. The probability of occurrence of each subunit isexpressed in terms of the volume of the spheres, theirprobability of being present and the probability thatspecific pairs of spheres in contact may exist. The porespace in a subunit can be approximated by a narrow throatin each of the four faces of the tetrahedral subunit thatmeet at a pore body, which occupies the central cavity ofthe tetrahedron. Fig. 2 shows an example of a tetrahedralsubunit formed by four touching spheres and an approxi-mation of the pore space structure. The size of the porethroat is evaluated by a cylinder with a diameter given bythe sphere in between the three spheres in the face. Thepore body-size is determined by the sphere created inbetween the four touching spheres. Formulas given in theRouault and Assouline (1998) work for computing porebody size (rp), pore throat size (rci) and their relativeprobability densities were used.

The input data for sphere packing are in the form of adiscrete sphere-size distribution, which can be derivedfrom the GSD of the porous medium. We describe the GSDof a sand used in our laboratory experiment by a normaldistribution, which is commonly used for an analyticaldescription of grain-size distribution and can easily beapplied in practice. The density probability function f(d) ofthis distribution is given by:

f dð Þ ¼ 1ffiffiffiffiffiffiffiffiffiffiffiffi2psg

p exp �1

2

d� d50

sg

� 2 !

(12)

where d is the grain-size diameter, d50 is the mean grainsize diameter, and sg is the standard deviation of the grain-size distribution deduced by graphical analysis.

For each set of four radii, the probability density of thetetrahedron subunits and the corresponding radius of the‘‘kissing’’ sphere are obtained. The final result is one pore-size distribution curve and four throat-size distribution

curves. Assuming that the generated pore body-sizes andpore throat-sizes are normally distributed, which isgenerally assumed for a large sampling number, theassessment of the mean value and the standard deviationof corresponding curves allows for calculation of variousranges of pore body radii and pore throat radii. Theseparameters are geometric input data of the pore-throatmodel.

3.2. Application to a medium-size sand

In this section, we use the conceptual approachdescribed in Section 3.1 to parameterise the equivalentnetwork of the H2F sand that was used in the physicalexperiment. H2F is a medium-size quartz sand witha hydraulic conductivity of about 8� 10�4 m/s(k = 8� 10�11 m2), a total porosity of 40% and a lowfraction of organic content (foc = 0.09%, based on NFT 31–109) (Bohy et al., 2006).

Fig. 3 shows the grain-size distribution curve of the H2Fsand used in the experiment and its corresponding derivedsphere-size distribution considered in the model of spherepacking. The mean value and the standard deviation of thesphere-size distribution allow positive values at eachdistance even at three standard deviations away from themean value. Twenty-one spheres of different sizes wereconsidered in the packing. This relatively high number ofspheres ensured the reliability of statistics on the evaluationparameters as C4

21 (5985) possible sets of four radii might beformed. The radii of chosen spheres are equally spaced; thismeans that the radii are arithmetically related. The sizes ofthe radii vary between a minimum value of 0.05 mm and amaximum value of 0.34 mm. The ratio between themaximum and minimum sizes satisfies the packing condi-tion mentioned in the work of Dodds (1980).

The generated pore body and pore throat-size distribu-tions are shown in Fig. 4. In contrast to the pore body-sizedistribution, the packing model allows us to compute fourdifferent distributions that can be used to describe thepore-throat size. The pore body-size distribution has ashape similar to the sphere-size distribution. The resultsagree well with the findings of Rouault and Assouline

[()TD$FIG]

(a) (b)

0102030405060708090100

10.10.01

d (mm)

cumulativefrequency

(%)

0

0.3

0.6

0.9

1.2

0.80.60.40.20

d(mm)

f(d)

Fig. 3. Medium-size H2F sand: (a) grain-size distribution; (b) derived sphere-size distribution.

Fig. 3. Sable moyen H2F : (a) courbe granulometrique ; (b) distribution calculee de la taille des spheres.[()TD$FIG]

(a) (b)

0

0.04

0.08

0.12

0.16

0.2

0.050.040.030.020.010

rci(mm)

pro

bab

ility

den

sity

fu

nct

ion

g(r

ci)

rc1rc2rc3rc4

0

0.06

0.12

0.18

0.080.060.040.020.00

rp(mm)

pro

bab

ility

den

sity

fu

nct

ion

g(r

p)

Fig. 4. Medium-size H2F sand: (a) calculated pore-body size (rp) distribution; (b) pore-throat size (rci) distributions.

Fig. 4. Sable moyen H2F : (a) distribution calculee de la taille des pores (rp) ; (b) distribution calculee de la taille des capillaires (rci).

K. Nsir, G. Schafer / C. R. Geoscience 342 (2010) 881–891886

(1998). The forms of throat-size distributions vary andoccupy a large range of radii. Based on the assumption thatthe volume of pore throats is negligible compared to that ofpore bodies, we limited our analysis to the pore-throatdistribution that presents the smallest mean radius (rc1).

Table 1 shows various values derived for both pore-body radii and pore-throat radii using the mean value andthe standard deviation of the corresponding generated sizedistributions. For both size distributions, Case 1, Case 2 andCase 3 refer to the minimum and maximum radii derivedfrom a size distribution as one time, two times and threetimes the standard deviation of the size distribution,respectively. For fixed distribution parameters of porebody radius and by combing the pore body radii with thepore throat radii given in Case 1, Case 2 and Case 3, weobtained at least nine possible realisations of the pore

Table 1

Minimum, mean and maximum radii of pore-bodies and pore-throats derived

Tableau 1

Rayons minimal, moyen et maximal de pores et de capillaires, calcules par le m

Pore body radius

rmin (mm) rmean (mm) rmax (mm)

Case 1 0.042 0.048 0.057

Case 2 0.036 0.048 0.061

Case 3 0.03 0.048 0.067

space geometry of our network model. To reduce thenumber of possible realisations of the geometry of thenetwork model, we used the macroscopic properties ofmedium-size H2F sand, such as the intrinsic permeability(k = 8� 10�11 m2) and porosity (ø = 40%). For each of thenine possible combinations, we fixed the distance betweenpore-body centres at L = 0.14 mm, which is greater thantwo times the maximum radii of pore bodies. A single-phase water flow was simulated using the generatednetwork model with given pressure boundaries. Theintrinsic permeability and the porosity of the network,calculated with Eqs. (2) and (11), respectively, were thencompared to the measured values of the sand used in thelaboratory experiment (Table 2). The combination of Case 1of the pore-body radii with Case 2 of the pore throat radiigave the best fit for the macroscopic properties of the sand.

from the sphere packing model.

odele d’empilement de spheres.

Pore throat radius

rmin (mm) rmean (mm) rmax (mm)

0.026 0.03 0.036

0.022 0.03 0.04

0.017 0.03 0.046

Table 2

Calculated macroscopic properties of the network of each of the nine realizations compared to the measured intrinsic permeability and porosity of the

medium-size H2F sand.

Tableau 2

Proprietes macroscopiques du reseau de pores et de capillaires (calculees pour neuf realisations), comparees a la permeabilite intrinseque et a la porosite du

sable moyen H2F.

Pore body radius Pore throat radius Macroscopic proprieties

rmin (mm) rmean (mm) rmax (mm) rmin (mm) rmean (mm) rmax (mm) ø (%) K (10�11 m2)

0.042 0.048 0.057 0.026 0.03 0.036 47 12.87

0.042 0.048 0.057 0.022 0.03 0.040 47 8.6

0.042 0.048 0.057 0.017 0.03 0.046 47 13.2

0.036 0.048 0.061 0.026 0.03 0.036 49 12.84

0.036 0.048 0.061 0.022 0.03 0.040 49 7.64

0.36 0.048 0.061 0.017 0.03 0.046 49 13.42

0.30 0.048 0.067 0.026 0.03 0.036 53 11.73

0.30 0.048 0.067 0.022 0.03 0.040 53 11.83

0.30 0.048 0.067 0.017 0.03 0.046 53 12.05

K. Nsir, G. Schafer / C. R. Geoscience 342 (2010) 881–891 887

While the calculated permeability of 8.6� 10�11 m2 wasquite close to the experimental value, the porosity of ournetwork model (ø = 47%) overestimated the porosity of oursand. The porosity might be better estimated by increasingthe distance L between pore body centres. However, thiswould cause a decrease in the equivalent macroscopicpermeability.

4. Numerical simulation of a laboratory experiment

4.1. Model parameters

One of the objectives of our study was to useexperimental results (Nsir, 2009) for verification of thenetwork model developed. Experiments were conducted ina sand-filled glass column with a length of 70 cm and innerdiameter of 10 cm. The DNAPL used was TCE, which wasinjected at a constant flow rate of 40 mL/min. Thedisplacement was performed from bottom to top, whichreproduced a stable displacement condition, and from topto bottom, which reproduced an unstable gravity condi-tion. The experimental device was equipped with pressuretransducers for monitoring pressure at the inlet and outletsections of the glass column. The experimental programmealso allowed the measurement of the arrival time of theDNAPL front at eight points on a control section equippedwith optical fibre sensors (Nsir, 2009).

Simulation results reported here were carried out on anetwork of 7�7�4858 pore bodies. The spatial discretisa-tion used reproduces the total height of the physical modelbut only a small fraction of the column section. Weassumed that the chosen network is sufficiently large to bea representative elementary volume (REV) which mayreproduce the macroscopic properties of the sand columnsuch as intrinsic permeability and porosity. The pore-bodyand pore-throat radii considered in the simulationscorrespond to Case 1 for pore bodies and Case 2 for porethroats, as given in Table 1. The distance between pore-body centres was fixed at L = 0.14 mm.

The network model is initially water-saturated, and aconstant flow rate of 29.6� 10�4 mL/min of TCE wasapplied at the inlet section of the model. The injection rate

was calculated by accounting for the ratio between themodelled section and the total section of the glass column.At the outlet section of the model, the observed water headwas specified as a constant pressure boundary. Thesimulation stopped when the duration of the DNAPLinjection that occurred in the experiment was reached.

4.2. Stable case of water displacement by DNAPL

To illustrate the case of stable displacement, thedisplacing fluid (TCE) was injected at the bottom of thenetwork. The boundary pressure applied to the outletsection corresponded to a water height of 18 cm. Fig. 5ashows a comparison of calculated and measured inletpressures as a function of time. According to theexperimental observations, the value of the pressure inthe whole domain increased with time. The water/DNAPLinterface was stabilized by the gravity effect; thepotentially destabilizing influence of the viscosity ratioon the front behaviour was not significant. The increase inpressure was therefore caused by the high density of theinvading fluid, which overrides the decrease of viscouspressure during the displacement of the less viscousDNAPL.

Even though the increasing pressure behaviour ob-served in the experiment was adequately reproduced, themeasured pressure increases were stronger than what wascalculated experimentally. This can be explained by thechosen arrangement of pore and throat sizes in the pore-throat model. The very narrow throats that exist in the realporous medium were not considered in the network andmay require a higher displacement pressure to be invadedby the displacing fluid. Using a wider pore body-sizedistribution might also reduce this misfit of inlet pressure.

The curves representing the distributions of dimen-sionless arrival times of the water/DNAPL front in theexperiment and the simulation were similar (Fig. 5b). Thecalculated standard deviations were very close to themeasured values (14 s calculated compared to 9 s mea-sured in the experiment). According to our experimentalobservations, the distribution of front arrival times at thecontrol section is almost uniform. Due to the stabilizingeffect of gravity, a flat piston-like front was observed

[()TD$FIG]

(b) (a)

0

40

80

120

160

25002000150010005000

time (s)

pre

ssu

re h

ead

(cm

) calculated pressuremeasured pressureboundary pressure

0

0.02

0.04

0.06

1.0210.980.96t/t mean

frequency(-)

numericalexperiment

Fig. 5. Stable displacement of water by DNAPL: (a) inlet pressure measured and simulated as functions of time; (b) distribution of calculated and measured

dimensionless arrival times of the water/DNAPL front.

Fig. 5. Deplacement stable de l’eau par du DNAPL : (a) pression mesuree a l’entree de la colonne et simulee en fonction du temps ; (b) distribution des temps

d’arrivee adimensionnels du front d’eau/DNAPL.

K. Nsir, G. Schafer / C. R. Geoscience 342 (2010) 881–891888

moving through the network. Consequently, very littlewetting fluid was left behind and a compact pattern of theinvading fluid occurred. The network model adequatelyreproduced the stability mechanisms of displacementcaused by the positive contribution of gravity.

4.3. Unstable case of water displacement by DNAPL

In the case of unstable water displacement, the DNAPLwas injected at the top of the network using the sameboundary and initial conditions as described in section 4.1.For this displacement, we expected a much more irregularfront with possible gravity fingering. Fig. 6a shows thevariation of displacement pressure measured and calcu-lated by the model. Due to the constant injection rate, thepressure value in the domain decreased as the less viscousand denser fluid invaded the system. The sudden increaseof pressure shown in Fig. 6a corresponds to the break-through of the invading fluid at the outlet section. Underthese displacement conditions, gravity forces are destabi-lizing and dominate the movement, resulting in typicalDNAPL fingers in the displaced fluid. The macroscopicwater/DNAPL interface was no longer horizontal and manythroats were blocked by the capillary effect. The number ofactive invasion paths was therefore low, and the flow[()TD$FIG]

(a)

0

20

40

60

80

100

25002000150010005000

time (s)

pre

ssu

re h

ead

(cm

)

calculated pressuremeasured pressureboundary pressure

frequency(-)

Fig. 6. Unstable displacement of water by DNAPL: (a) inlet pressure measure

measured dimensionless arrival times of the DNAPL front.

Fig. 6. Deplacement instable de l’eau par du DNAPL : (a) pression mesuree a l’e

temps d’arrivee adimensionnels du front d’eau/DNAPL.

section of DNAPL was reduced. The contribution of viscouspressure was also limited and was dominated by thecontribution of the higher density of the invading fluid.This resulted in a decrease of the inlet pressure. The inletpressure observed was reproduced by the numericalmodel, but the rate at which it decreased in the modelwas different from the measured one, the calculatedpressure decrease was more substantial than that ob-served. In addition, we recorded an earlier breakthroughtime than we observed. This result can be explained by thenumber of active invasion paths that were lower in thenetwork model than in the real porous medium. This effectmay be due to the narrow generated pore-throat sizedistribution that contains many large pore throats.

Fig. 6b shows a comparison between the distribution ofcalculated arrival times of the front and those observed atthe control section. The statistical data of both distribu-tions are similar. They both illustrate highly non-uniformdistributions of arrival times with significant standarddeviations. The calculated standard deviation was 75 s,compared to 70 s measured in the laboratory experiment.Thus, gravity fingering was numerically confirmed by thedistribution of arrival times of the front at the chosencross-section of the network. However, the mean arrivaltime of the water/DNAPL front calculated at the control

(b)

0

0.04

0.08

1.31.21.110.90.80.7t/t mean

numericalexperiment

d and simulated as functions of time; (b) distribution of calculated and

ntree de la colonne et simulee en fonction du temps ; (b) distribution des

K. Nsir, G. Schafer / C. R. Geoscience 342 (2010) 881–891 889

section was approximately 633 s, compared to theobserved time of approximately 747 s. This finding directlyconfirms the earlier computed breakthrough of TCE at thedownstream section (Fig. 6a).

4.4. Sensitivity study

A sensitivity study of specific model parameters, suchas the size of the lattice and the size of pore throats, wasundertaken to analyse the influence of these parameters onthe displacement of water by DNAPL. Furthermore, weassessed whether a reasonable modification of their valuesmight explain the differences between the simulated andmeasured DNAPL pressures at the inlet section, as well asthe simulated and measured arrival times of the DNAPLfront. Specifically, we analysed the unstable case of waterdisplacement by DNAPL.

4.4.1. Effect of the lattice size

Using a larger section of network may be more valid forthe appropriate representation of the real porous medium.However, larger networks suffer from computation timesthat can become, in some cases, very high, even withpowerful computers. Fig. 7a shows the calculated inletpressure obtained from simulations carried out on net-works of two lattice sizes (5�5�4858 and 6�6�4858 porebodies) that were smaller than the reference case and onelattice size (9�9�4858) that was larger than the referencecase discussed in Section 4.3. The size of the reference casewas 7�7�4858. On a Linux-computer with 3956 MB RAMand a Celeron processor running at 1998 MHz, thecorresponding Central Processing Unit (CPU) times werevery different; the times were one day, four days, thirteendays and twenty days for the two small lattices, themedium lattice and the large lattice, respectively. In allcases, the modelled inlet pressure behaviour was similar; itdecreased with increasing time, particularly for thereference case and the case of the bigger lattice (thepressure profiles for those cases were almost the same).However, the breakthrough times calculated for the biggerlattice and the reference lattice were shorter than thetimes calculated for the two smaller lattice sizes. Thesmaller lattices were closer to the observed breakthrough

[()TD$FIG]

(a)

0

10

20

30

40

25002000150010005000

time (s)

pre

ssu

re h

ead

(cm

) lattice (5x5x4858)lattice (6x6x4858)lattice (7x7x4858)lattice (9x9x4858)experiment

Fig. 7. Influence of the lattice size on the unstable DNAPL displacement: (a) inlet

of calculated and measured dimensionless arrival times of the water/DNAPL fr

Fig. 7. Influence de la taille de reseau sur le deplacement instable de l’eau par

distribution des temps d’arrivee, mesures et calcules du front d’eau/DNAPL.

time. Thus, developed displacement instabilities are moresignificant in bigger lattices than in smaller lattices. Thearrival time distribution of the front water/DNAPLcalculated and measured (Fig. 7b) in the same controlsection confirms this conclusion. The corresponding curvesfor both bigger lattices were almost the same and weremore widely distributed than those computed using thesmaller lattices. The standard deviation of the distributionof arrival times was 75 s and 79 s for lattice sizes7�7�4858 and 9�9�4858, respectively. These valuesare very close to the measured standard deviation of 72 s.The standard deviation arrival times for the smaller latticessizes (5�5�4858 and 6�6�4858) were 59 s and 64 s,respectively. A large lattice size leads to a spatial pore bodysize distribution and pore throat size distribution that ismore heterogeneous than for a small lattice. Consequently,preferential pathways are more likely to occur duringdisplacement. As a result, the water/DNAPL front arrivesearlier at the downstream section of the model in the caseof bigger lattice sizes. Thus, the distribution of arrival timesof the water/DNAPL front is not uniform, as shown inFig. 7b. In contrast, based on the numerical resultsobtained from the 7�7�4858 and 9�9�4858 lattices,which were quite similar, we concluded that a lattice sizeof 7�7�4858 adequately reproduces the unstable dis-placement process in the given porous medium.

4.4.2. Effect of the variance in pore-throat size

In this study, we used the three distribution parametersfor pore throat radius as given in Table 1 (Case 1, Case 2 andCase 3) to analyse the influence of the variance in the porethroat size on the unstable displacement of water byDNAPL. Case 1 has a pore throat size variation rangesmaller than Case 2, which is hereafter referred to as thereference case. Case 3 has pore throat size variation rangelarger than the reference case. The size of the networkconsidered in our simulations was the same for each porethroat size distribution (with identical pore body sizedistribution). We chose the smallest lattice size(5�5�4858) because it required less computational timeand memory.

Fig. 8a shows the calculated pressure for differentnetwork constructions that were compared to the

(b)

0

0.04

0.08

1.31.21.110.90.80.7

t/tmean

freq

uen

cy (

-)

lattice (5x5x4858)lattice (6x6x4858)lattice (7x7x4858)lattice (9x9x4858)experiment

pressure measured and simulated as functions of time; (b) the distribution

ont.

du DNAPL : (a) pression mesuree et simulee en fonction du temps ; (b)

[()TD$FIG]

(a) (b)

0

10

20

30

40

25002000150010005000

time (s)

pre

ssu

re h

ead

(cm

) case 1reference casecase 3experiment

0

0.02

0.04

0.06

0.08

1.31.21.110.90.8

t/tmean

freq

uen

cy (

-)

case 1reference casecase 3experiment

Fig. 8. Influence of the variance of the pore-throat size on the unstable DNAPL displacement: (a) inlet pressure measured and simulated as functions of

time; (b) distribution of calculated and measured dimensionless arrival times of the water/DNAPL front.

Fig. 8. Influence de la variance de la taille de capillaires sur le deplacement instable de l’eau par du DNAPL : (a) pression mesuree et simulee en fonction du

temps ; (b) distribution des temps d’arrivee, mesures et calcules du front d’eau/DNAPL.

K. Nsir, G. Schafer / C. R. Geoscience 342 (2010) 881–891890

measured pressure. We note that the inlet pressureincreased with increasing variance of the pore throatdistribution and was accompanied by a later breakthroughof the water/DNAPL front. Apparently, the pore-throatradius controls the dynamic displacement of the water/DNAPL interface in the network. Case 3 had the smallestpore-throat sizes. This resulted in an increase in pressurein the pore bodies at the upstream boundary of the model,which was required to maintain the prescribed constantmass flow rate for DNAPL. A higher local pressure fieldpermitted the DNAPL to overcome the capillary entrypressure in the throats and therefore resulted in a moreuniform invasion of the water-saturated pore bodies byDNAPL in the model transect. However, the pore-throatsize distributions of Case 1 and Case 2 had much largerthroats, resulting in preferential pathways that were morelikely to occur. This made the overall invasion of thesection harder. As a consequence, the model predictedearlier breakthrough times, and the pressure field de-creased rapidly. This may explain why the computedbreakthrough of the displacement front in Case 3 was laterthan in Cases 1 and 2. Furthermore, in analysing thecalculated DNAPL saturation field (not presented here), theDNAPL front in Case 3 was characterised by severalgravity-dominated fingers, whereas in Case 1, fewerfingers were formed.

Fig. 8b shows a comparison between distributions ofdimensionless arrival times of the water/DNAPL frontcalculated and measured at the control section. Theprediction with a pore-throat size distribution as in Case3 was quite close to the experimental data. The corre-sponding curve was more widely spread than thoseobtained with the narrow pore-throat size distributionsof Case 1 and Case 2. The calculated standard deviationrelative to the pore-throat distribution of Case 3 wasapproximately 69 s, which is quite close to the experimen-tally determined value of 72 s. However, the calculatedstandard deviation was only 49 s and 59 s for Case 1 andthe reference case, respectively. Case 3 provides a widerpore-throat radius distribution. This may allow muchinvasion pathways to be formed, leading to a betterreproduction of the displacement front observed in thelaboratory experiment. Because the pore-throat size

distribution variability was lower in Case 1 and Case 2,one may think that the number of accessible pore throatsfor the displacing fluid was reduced and therefore only afew fingers resulted.

5. Conclusions

We developed a pore-throat network model thatadequately describes the dynamic drainage process. Ageometric model of packing spheres combined with aprobabilistic approach was used to obtain realistic poresizes and throat sizes based on grain-size distribution. Thepore-scale network model was tested using experimentaldata. The prediction of the temporal evolution of pressureat the inlet section and the distribution of arrival times ofthe water/DNAPL front in a given cross-section matchedwell with laboratory measurements. An important featureof the model is its capability to reproduce the observedpressure behaviour for stable and unstable displacementregimes. The sensitivity study indicated that the lowerpore-throat size strongly influenced the instability ofDNAPL displacement. However, additional numericalstudies may be necessary to analyse the influence of porebody size on unstable DNAPL displacement. Furthermore,application of the developed numerical model to adrainage experiment recently conducted in a homoge-neous fine sand may allow an additional verification stepof the model to derive pore body sizes and pore throat sizesin the case of a low permeability porous medium.

The network model used in the present work considersthe converging–diverging nature of pores and is based onthe pore body size distribution and pore-throat sizedistribution. However, no correlation exists between porethroat sizes and neighbouring pore bodies. Besides, porespace asperities and roughness were not considered.Additional quantitative information about the degree ofcorrelation between pore-body size and pore-throat sizeare needed to take into account more features of the porespace geometry of the porous medium. The lattice sizesused resulted in time consuming simulations. Thisprohibits application of the model to a larger part of thecross-section of the porous medium. One possible way to

K. Nsir, G. Schafer / C. R. Geoscience 342 (2010) 881–891 891

overcome this problem might be to develop a moreefficient algorithm to solve the large set of equations.

Acknowledgements

Financial support for this research was received fromthe programme REALISE (REseau Alsace de Laboratoires enIngenierie et Sciences pour l’Environnement). The RegionAlsace, the GDR ‘‘Hydrodynamique et Transferts dans lesHydrosystemes Souterrains’’ (INSU-CNRS), and the ConseilScientifique de l’Universite Louis-Pasteur Strasbourg aregratefully acknowledged. Furthermore, the authors wouldlike to thank Professor Insa Neuweiler and the two otheranonymous reviewers for their valuable comments andsuggestions, which helped to improve the article signifi-cantly.

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