flow and transport in porous media and fractured rock (from classical methods to modern approaches)...

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341 11 Dispersion in Flow through Porous Media Introduction In Chapters 9 and 10, we described and discussed flow and transport processes that involve only one fluid and one fluid phase. Beginning with this chapter, we take up the flow phenomena that are at the next level of complexity, namely, those that involve at least two fluids and one or more fluid phases. One of the most important of such phenomena is hydrodynamic dispersion that involves two fluids, and is described and studied in the present chapter. Miscible displacements, which represent generalizations of the dispersion processes (in which the viscosities of the two fluids are not equal) will be studied in Chapter 13, while multiphase flows will be described in Chapters 14 and 15. 11.1 The Phenomenon of Dispersion When two miscible fluids are brought into contact with an initially sharp front separating them, a transition zone develops across the initial front, the two fluids slowly diffuse into one another, and after some time, a diffused mixed zone de- velops. If one assumes that the volumes of the two fluids do not change upon the mixing, then the net transport of one of the solute across any arbitrary plane is represented by the Fick’s second law of diffusion @ C @ t D D m r 2 C . (11.1) Here, C is solute the concentration, t is the time, and D m is the molecular diffu- sivity of the solute in the solvent. The mixing of the two fluids is independent of whether or not there is a convective current through the medium. If, however, the solvent is also flowing, then there will be some additional mixing of a different sort – convective mixing – which is caused by a nonuniform velocity field that, in turn, is caused by the morphology of the medium, the fluid flow condition (fast or slow) and the chemical or physical interactions with the solid surface of the medi- Flow and Transport in Porous Media and Fractured Rock, Second Edition. Muhammad Sahimi. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA. Published 2011 by WILEY-VCH Verlag GmbH & Co. KGaA.

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341

11Dispersion in Flow through Porous Media

Introduction

In Chapters 9 and 10, we described and discussed flow and transport processesthat involve only one fluid and one fluid phase. Beginning with this chapter, wetake up the flow phenomena that are at the next level of complexity, namely, thosethat involve at least two fluids and one or more fluid phases. One of the mostimportant of such phenomena is hydrodynamic dispersion that involves two fluids,and is described and studied in the present chapter. Miscible displacements, whichrepresent generalizations of the dispersion processes (in which the viscosities ofthe two fluids are not equal) will be studied in Chapter 13, while multiphase flowswill be described in Chapters 14 and 15.

11.1The Phenomenon of Dispersion

When two miscible fluids are brought into contact with an initially sharp frontseparating them, a transition zone develops across the initial front, the two fluidsslowly diffuse into one another, and after some time, a diffused mixed zone de-velops. If one assumes that the volumes of the two fluids do not change upon themixing, then the net transport of one of the solute across any arbitrary plane isrepresented by the Fick’s second law of diffusion

@C

@tD Dmr2 C . (11.1)

Here, C is solute the concentration, t is the time, and Dm is the molecular diffu-sivity of the solute in the solvent. The mixing of the two fluids is independent ofwhether or not there is a convective current through the medium. If, however, thesolvent is also flowing, then there will be some additional mixing of a differentsort – convective mixing – which is caused by a nonuniform velocity field that, inturn, is caused by the morphology of the medium, the fluid flow condition (fast orslow) and the chemical or physical interactions with the solid surface of the medi-

Flow and Transport in Porous Media and Fractured Rock, Second Edition. Muhammad Sahimi.© 2011 WILEY-VCH Verlag GmbH & Co. KGaA. Published 2011 by WILEY-VCH Verlag GmbH & Co. KGaA.

342 11 Dispersion in Flow through Porous Media

um. The resulting mixing process is called hydrodynamic dispersion, or dispersionfor the sake of brevity.

Dispersion is important to a wide variety of processes, for example, miscible dis-placements in enhanced oil recovery, salt water intrusion in coastal aquifers wherefresh and salt waters mix by dispersion, and in-situ study of the characteristics ofaquifers, where a classical method of determining the characteristics is injectingfluid tracers into the aquifers and measuring their travel times. In addition, pollu-tion of the surface waters due to industrial and nuclear wastes is another impor-tant problem in which dispersion plays a fundamental role. In particular, buryingnuclear wastes deep in the rock has been suggested as a way of sequestering thepollution that they cause, but leakage of the wastes into groundwater aquifers hasbeen a major concern. Dispersion also occurs in flow and reaction in packed-bedchemical reactors and has been studied extensively by chemical engineers for along time; see, for example, Bernard and Wilhelm (1950)1) as one of the earliestpapers on the subject.

11.2Mechanisms of Dispersion Processes

In steady flow through a disordered porous medium, the transit time, or first-passage

time, of a fluid particle or tracer between the entrance and exit planes depends onthe path, or streamline that it follows through the pore space. A population of trac-ers passing the entrance plane at the same instant will arrive at the exit plane bya set of streamlines with a distribution of transit times. Thus, a solute concentra-tion front will spread in the mean-flow direction as it passes through the medium.The resulting first-passage time distribution (FPTD) is a measure of longitudinal

dispersion in a porous medium.Likewise, a population of tracers or particles passing simultaneously through a

restricted area of the entrance plane will not entirely follow the mean flow to theexit plane, but will be dispersed in the transverse directions (perpendicular to thedirection of the mean flow) as well. That is, the population and the set of stream-lines traveled will have a wider distribution of exit locations than of the entranceones. Thus, a concentration front will also spread laterally on the way to the exitplane. The distribution of the first-passage times for crossing the porous mediumat a given transverse plane is a measure of transverse dispersion in the porous medi-um.

Two basic mechanisms drive dispersion in macroscopically homogeneous, mi-croscopically disordered porous media, and arise in the pore-level velocity fieldforced on the flowing fluid by the irregularity of the pore space.

1) Richard Herman Wilhelm (1909–1968), Chairman of Chemical Engineering Department atPrinceton University, made significant contributions to fixed-bed catalytic reactors, fluidizedtransport and separation processes. The Wilhelm Award of the American Institute of ChemicalEngineers has been named in his honor.

11.3 The Convective-Diffusion Equation 343

1. The first mechanism is kinematic: streamtubes divide and rejoin repeatedly atthe junctions of flow passages in the highly interconnected pore space. Theconsequent tangling and divergence of streamlines is accentuated by the widelyvarying orientations of flow passages and coordination numbers of the porespace. The result is a wide variation in the lengths of the streamlines and theirdownstream transverse separations.

2. The second mechanism is dynamic: the speed with which a given flow passageis traversed depends on the flow resistance or hydraulic conductance of thepassage, its orientation, and the local pressure field.

The two mechanisms conspire to produce broad FPTDs between the entrance andexit planes. The two mechanisms also suggest two possible geometrical aspects ofthe dispersion processes defined with respect to the mean-velocity direction, thatis, a longitudinal effect due to the difference between the velocity components inthe direction of mean flow, and a transverse effect due to the differences betweenlocal velocity components orthogonal to the direction of the mean flow.

The two mechanisms of dispersion do not depend on molecular diffusion. Dif-fusion modifies, however, the effects of the two basic mechanisms by moving ma-terial from one streamline to another as well as by the usually weaker streamwisediffusion of material relative to the average velocity. The solid matrix of a porousmedium acts locally, of course, as a separator of the streamlines and, thus, as a bar-rier to diffusion. Therefore, the modification of dispersion by diffusion depends onthe pore space morphology and how it affects local flow and concentration fields.The effect of molecular diffusion is usually important only at the pore level, whereit acts to transfer the tracer particles out of slow or stagnant regions of the porespace.

11.3The Convective-Diffusion Equation

Dispersion processes in microscopically disordered and macroscopically isotrop-ic and homogeneous porous media are usually modeled based on the convective-diffusion (CE) equation:

@C

@tC hvi � r C D DL

@2C

@x2C DTr2

TC , (11.2)

where hvi is the macroscopic mean velocity, C is the mean concentration of the so-lute, and r2

T is the Laplacian in the transverse directions. For the sake of simplicity,we delete h�i and denote the magnitude of the average fluid velocity vector by V

or Vm. Thus, the basic idea is to model dispersion processes as anisotropic diffu-sional spreading of the concentration, with the diffusivities being the longitudinaldispersion coefficient DL and the transverse dispersion coefficient DT. One impor-tant goal of any study of dispersion in porous media is to investigate the conditionsunder which it cannot be represented by the CD equation.

344 11 Dispersion in Flow through Porous Media

Dispersion is said to be diffusive or Gaussian if it is described by the CD equation.If a population of the solute particles is injected into the medium at r0 D (x0, y0, z0)at t D 0 (i.e., C(x0, y0, z0, t) D C0δ(t)), for diffusive dispersion, the probabilitydensity P(r, t) follows the Gaussian distribution

P(r, t) D (8π3DL D2T t)� 3

2 exp�� (x � x0 � Vt)2

4DL t� (y � y0)2

4DT t� (z � z0)2

4DT t

�,

(11.3)

where P(r, t)d r is the probability that a solute particle is in a plane between r andr C d r at time t, and r D (x , y , z). P(r, t) is proportional to C/C0 and, therefore,Eq. (11.3) represents a solution of Eq. (11.2). If one defines Q(� � �0, t)d t as theprobability that a solute particle, beginning in the plane at �0 will cross, for the first

time, a plane at � between t and t C d t. Then, from Eq. (11.3), one can easily obtainthe FPTD in a given direction since Q and P are related:

P(� � �0, t) DtZ

0

P(� � �1, t � τ)Q(�1 � �0, τ)dτ , (11.4)

and, therefore

Q(� � �0, t) D j� � �0j �4πD� t3�� 12 exp

�� (� � �0 � V� t)2

4D� t

�, (11.5)

where D� and v� are the dispersion coefficient and the mean flow velocity in the� -direction, respectively. The moments of Q yield information about the flow fieldand the dispersion processes. For example, the first two moments of the FPTD inthe longitudinal direction are given by

hti D L

V, (11.6)

and

ht2i D hti2�

1 C 2DL

LV

�, (11.7)

where L D � ��0. In general, one can easily show that for large L and to the leadingorder, one has

ht ni � htin , (11.8)

where n > 1 is an integer. Equation (11.8) holds true as long as the descriptionof dispersion by the CD equation is valid. Therefore, one way of showing that aCD equation cannot describe the dispersion process in a given porous medium isto show that ht ni/htin (n > 1) is not a constant (i.e., the ratio depends on t) andone requires more information than just hti to describe the moments of the FPTD.This issue will be discussed later in this chapter.

11.4 The Dispersivity Tensor 345

11.4The Dispersivity Tensor

Because dispersion is an anisotropic phenomenon, and since one must have in-variance under coordinate transformations, the dispersion coefficients constitutea tensor. One may then ask, what is the most general form of the tensor? Thisquestion was addressed by Bear (1961) and Scheidegger (1961). In particular, Bear(1961) noted that the dispersion coefficients should be written as

Di k D ai k l m

vlvm

jVj , (11.9)

where jV j is the magnitude of the fluid velocity. The coefficient ai k l m is called thedispersivity, and represents the typical length scale over which significant disper-sion occurs. Clearly, ai k l m is a fourth-rank tensor with 81 components. However,several symmetry properties significantly reduce the number of independent com-ponents of the tensor. One is that ai k l m D ai k ml. That this symmetry should holdis obvious. If the equality did not hold, the two coefficients could be made to bebecause that would not affect the general form of the equation of motion. Due toOnsager’s principle of microscopic reversibility, one must also have ai k l m D ak i l m.The two symmetries reduce the number of independent components of the disper-sivity tensor a from 81 to 36.

To simplify the notation, it is customary to write a tensor with four indices andonly 36 nonzero components in the form of a 6 � 6 matrix. Suppose that we denotethe matrix by aα� . The connection between the Latin and Greek notations is asfollows. To every pair (xi , x j ) (for i, j D 1, 2, and 3), we attribute a correspondingvalue of α. Let x1 D x , x2 D y , and x3 D z. Then, α D 1, 2, 3, 4, 5, 6 correspondto (x , x ), (y , y ), (z, z), (x , y ), (x , z), (y , z), and similarly for �. Scheidegger (1961)showed that if we require invariance for 90ı rotations around the x and z axes, thetensor [aα� ] reduces to

[aα� ] D

266666664

a11 a12 a12 0 0 0a12 a11 a12 0 0 0a12 a12 a11 0 0 00 0 0 a44 0 00 0 0 0 a44 00 0 0 0 0 a44

377777775

so that [aα� ] is completely symmetric and has only three independent components.For an isotropic porous medium, one can also show that a44 D 1

2 (a11 � a12), so thatthe dispersivity tensor only has two independent components and, therefore, thereare only two dispersion coefficients, the longitudinal and transverse dispersion co-efficients, DL and DT, and the corresponding dispersivities, aL and aT. More re-cently, Salles et al. (1993) studied dispersion in porous media in order to determinethe dispersion tensor.

346 11 Dispersion in Flow through Porous Media

11.5Measurement of the Dispersion Coefficients

Since measurement of DL and DT is not as straightforward as that of the effectivepermeability Ke or the effective diffusivity De, we describe the measurement tech-niques. In principle, there are two distinct ways of observing and measuring thedispersion phenomenon in a porous medium. One may observe the variations ofthe solute concentration either as a function of the distance from the entrance tothe medium at a fixed time, or as a function of the time at a fixed distance from theentrance.

11.5.1Longitudinal Dispersion Coefficient

The longitudinal dispersion coefficient DL can be measured by at least three meth-ods that are all based on the concentration profile of the solute. However, while inone method the profile is measured directly, the profile is inferred from the mea-surement of other quantities in the other two methods.

11.5.1.1 Concentration MeasurementsLaboratory measurements of DL are usually carried out in porous media that areeffectively one dimensional under a constant fluid velocity v. The initial and bound-ary conditions are: C(x � 0, t D 0) D 0, C(x D 0, t > 0) D C0, and C(x ! 1,t � 0) D 0. Then, the solution to the one-dimensional (1D) version of Eq. (11.2)is obtained by the Laplace transform technique. Assuming that the length of theporous medium is L, and defining the dimensionless parameters

α˙ D x ˙ Vt

(4DL t)12

, (11.10)

the solution of the CD equation, subject to the aforementioned initial and boundaryconditions, is given by

C

C0D 1

2erfc(α�) C 1

2exp

�xVDL

�erfc(αC) , (11.11)

where erfc(z) is the complementary error function. Compared with the first term,the second term of Eq. (11.11) is usually very small and, therefore, can be neglectedwithout any significant error, in which case

C

C0D 1

2erfc(α�) D 1p

π

1Zα�

exp(��2)d� D 12

[1 � erf(α�)] . (11.12)

Equation (11.11) is rewritten as

C

C0D 1p

1Zα0

exp�

��2

2

�d� , (11.13)

11.5 Measurement of the Dispersion Coefficients 347

where α0 D 2α�/p

2. Figure 11.1 shows the concentration profiles C/C0 versusxd D x/L and td D R t

0 Vd t/(Lφ) (a dimensionless time), where φ is the porosityof the porous medium. According to Eq. (11.12), at a fixed time, the solution isa normal or Gaussian distribution function 1 � N [(x � hxi)/s] with the averagehxi D Vt and the standard deviation σ D p

2DL t. Two well-known properties of anormal distribution are

N(1) ' 0.84 , N(�1) ' 0.16 , (11.14)

which allow us to measure DL. The width w of the transition or mixing zone, thatis, the zone between a region with pure solute and one with pure solvent, is usu-ally defined as the difference between the values of x at which C/C0 D 0.16 andC/C0 D 0.84. Thus,

w D 2σ D 2p

2DL t D x0.16 � x0.84 , (11.15)

from which we obtain

DL D (x0.16 � x0.84)2

8t. (11.16)

Thus, if we construct a graph of C versus x at a fixed time t, we can determineDL based on Eq. (11.16). Alternatively, we can fit Eq. (11.13) to the data for theconcentration, with DL being the fitting parameter.

In many cases, it is easier to fix x (for example, at the exit of the porous medium)and measure the concentration as a function of time t. If so, Eqs. (11.13) and (11.14)provide an expression for DL:

DL D 18

�x � Vt0.16p

t0.16� x � Vt0.84p

t0.84

�2

, (11.17)

where t0.16 is the time at which C/C0 D 0.16.

Figure 11.1 Normalized concentration profile C/C0 versus the dimensionless distance xd atvarious dimensionless times td.

348 11 Dispersion in Flow through Porous Media

Figure 11.2 A typical solute-composition plot for determining the dispersion coefficient DL onthe arithmetic-probability paper. Circles represent the data (after Brigham et al., 1961).

Measurement of DL in flow through unconsolidated porous medium is usuallyeasier than that for consolidated ones. Thus, we describe methods for measuringDL (and DT) in such porous media, although similar methods may also be usedfor consolidated porous media. One saturates a packed column with one fluid (thesolvent), displaces it with another miscible fluid (the solute), and measures thefluid composition at the exit end of the column as a function of the displacement.Brigham et al. (1961) developed a convenient method for estimating DL from dataof this type. In this method, one plots λp D (v/Vc � 1)/

pv/Vc versus the percent of

the solute on an arithmetic probability paper, where v is the volume of the soluteinjected into the porous medium, and Vc is the column’s volume. Then, DL is givenby

DL D vL

�λ p 90 � λ p10

3.625

�2

, (11.18)

where L is the length of the column, and λp90 is the value of λp when the solventcontains 90% displacing fluid. Equation (11.18) is derived by an argument similarto that for deriving Eq. (11.16), except that the mixing (transition) zone is defined asthe region between the axial positions x at which C/C0 D 0.1 and 0.9. Figure 11.2shows the schematic representation of the method.

11.5.1.2 Resistivity MeasurementsIn this method (see, for example, Odling et al., 2007; Aggelopoulos and Tsakiroglou,2007, for recent references), one measures the resistivity of the porous mediumthat is filled by an electrically-conductive fluid (the solvent) together with the in-jected fluid (the solute). However, the DC resistivity (or conductivity) cannot bemeasured because the ion drift toward each electrode generates a blocking countervoltage. Therefore, the AC measurements are carried out. The typical procedure isas follows (Odling et al., 2007).

11.5 Measurement of the Dispersion Coefficients 349

The sample is first saturated with low-salinity brine and its impedance is mea-sured. Then, a very slow reverse flow of the low-salinity brine is carried out in orderto block any possible upflow into the sample. This is followed by injection from thebottom of the sample of a high-salinity brine at high speed, so that the compositionof fluid at the inflow end of the core is nearly changed instantaneously. The flowrate of the high-salinity solution is then reduced to what is needed for the actualdispersion experiment. The bottom piston drain of the system is then closed, theupper piston drain is opened, hence letting the high-salinity brine to be convectedacross the porous sample. During the entire time, a large number of impedancemeasurements are also carried out. The impedance decreases as the high-salinitysolution displaces and expels the low-salinity solution. When the impedance nolonger changes, the experiment is terminated. In addition, the total fluid flux ismonitored and measured.

At any point x along the sample, the resistance R(x ) of the porous medium isinversely proportional to the fluid saline concentration C(x ). Since each segmentof the sample has a resistance, and the sample is effectively 1D, the effective resis-tance Re of the sample is the sum of all the individual resistances R. Therefore,

Re D c

LZ0

1C(x )

dx , (11.19)

where c is the proportionality factor. If dispersion is Gaussian, which is expected tobe as a laboratory-scale porous sample rarely gives rise to a non-Gaussian disper-sion, then the concentration profile is given by Eq. (11.11). Thus,

Re D c

LZ0

2C0

1

erfc(α�) C exp�

xVDL

�erfc(αC)

dx , (11.20)

where α˙ is given by Eq. (11.10). Thus, the resistance-time response of the poroussample should be fitted to Eq. (11.20), with the fitting parameters being the propor-tionality factor c and the dispersion coefficient DL. As such, the problem is highlynonlinear, and careful numerical integration must be used. As an initial guess forDL, one can use the fact that DL is always larger than the effective molecular diffu-sivity De in the same porous sample. Though somewhat tedious, the method hasproven to be highly accurate.

11.5.1.3 The Acoustic MethodIn an acoustic technique, the concentration profile C(x , t) is determined by mea-suring the variations of the velocity of a sound wave at several cross sections of thesample (Bacri et al., 1984, 1987, 1991; Bacri and Salin, 1986). A calibration curveis first set up that plots the velocity variations as a function of the concentrationof the mixture (of the solvent and solute). Theoretically, the velocity vs of a fastcompressional wave is given by

vs Ds

H

�e, (11.21)

350 11 Dispersion in Flow through Porous Media

where

H D Kb C 43

µs C (Ks � Kb)2Kf

(1 � φ)KsKf C φK2s � Kb Kf

, (11.22)

where Kb and µs are the bulk and shear moduli of the dry solid matrix of the porousmedium, Ks and Kf are the bulk moduli of the porous medium and the fluid, andφ is the porosity. The quantity �e is the effective density of the porous medium Cthe fluid, given by

�e D φ�f C (1 � φ)�s , (11.23)

with �f and �s being the densities of the fluid and solid, respectively. For the caseof an unconsolidated porous medium, one has Kb D µs D 0.

The sample is saturated by the solvent, the solute is then injected into the porousmedium, and the sound wave velocity is measured at different cross sections (i.e.,various axial positions x). The measurements are then transformed into equivalentsolute concentrations C(x , t). The typical accuracy for the C(x , t) measurementsare better than 1%. The results are then fitted to Eq. (11.12), with the fitting param-eter being DL.

11.5.2Transverse Dispersion Coefficient

Measuring DT is more difficult. If the porous medium is arranged as in Figure 11.3,then a mixed zone develops in the transverse direction. The concentration profilealong a line perpendicular to the direction of the flow (the longitudinal direction)is typically S-shaped (see Figure 11.3). Then, DT is determined by plotting percentcomposition versus the distance from the 50% composition on an arithmetic prob-ability paper, and is estimated by the following equation,

DT D vL

� z0.9 � z0.1

3.625

�2, (11.24)

where z0.9 is the transverse distance between the 90 and 50% compositions. Thereader should be able to derive Eq. (11.24).

Figure 11.3 Schematic of a transverse dispersion experiment. The area between the dashedlines is the dispersion zone (after Harleman and Rumer, 1963).

11.5 Measurement of the Dispersion Coefficients 351

11.5.3Nuclear Magnetic Resonance Method

As described in Chapter 4, nuclear magnetic resonance (NMR) is a powerful toolfor probing the structure of a porous medium. The same method can be used formeasuring the dispersion coefficients as well as the effective diffusion coefficientsin porous media that was already described in Chapter 10. One reason that NMRcan be used for measuring the dispersion coefficients is that, as described above,such measurements typically involve the use of tracer particles of the solute andtheir motion in a flowing fluid in a porous medium. The NMR also provides atracer method in that every single molecule is tagged noninvasively by its localprecession frequency in a nonuniform magnetic field (Callaghan, 1991; Seymourand Callaghan, 1997).

Magnetic resonance imaging makes it possible to measure the distribution ofthe local velocity v(r, t) (Guilfoyle et al., 1992; Müller et al., 1995; Kutsovsky et al.,1996; Lebon et al., 1996; Sederman et al., 1998; Tessier and Packer, 1998). However,as pointed out by Khrapitchev and Callaghan (2003), because the spatial and tem-poral resolutions are constrained by the overall resolution of the instrument, onemust make “trade offs” between the two. For example, if one settles for an ensemble-

averaged signal for the sample as a whole (i.e., coarser spatial resolution), one gainsoptimal temporal resolution in return. The best way of achieving a balance betweenthe two is through pulsed gradient spin echo-NMR (PGSE-NMR) that allows theanalysis of ensemble-averaged mean-square displacements of the solute particlesover a time interval. Recall that the dispersion coefficients are defined in terms ofthe first two moments of the probability distribution functions. In addition, sinceone typically uses a Lagrangian velocity (i.e., one in a moving coordinate system)for describing dispersion (for example, in Taylor–Aris dispersion theory a new co-ordinate, x1 D x � vm t was used; see below), its use with a PGSE-NMR is naturalbecause the method collects signals from moving tracers or particles.

We follow Khrapitchev and Callaghan (2003) and describe the PGSE-NMRmethod. Figure 11.4 shows how two variants of the PGSE-NMR are implementedin which the magnetic-field gradient g is given by g D r Mz , where Mz is thecomponent of the magnetic field M parallel to M0, the homogeneous polarizingfield, with jM0j � jMj. As Figure 11.4 indicates, a sequence of short pulses ofstrength G D jgj are used in order to induce a phase shift (for every nuclear spin)that depends on the positions of its “mother” molecule in the gradient g. The phasefactors are exp(2i πq � r), with r being the position of the host molecule, and q thewavevector, q D (2π)�1γ gδ, where γ is the magnetogyric ratio and δ the durationof the gradient pulse G.

First, consider the single gradient pulse pair sequence, hereafter referred to asS-PGSE-NMR. If r(0) and r(∆) are, respectively, the positions of the spin-bearingmolecule at the time of the first gradient pulse and the time at which a second pulseis applied a time ∆ later, the accumulated spin phase is expf2 i πq � [r(∆) � r(0)]g.

352 11 Dispersion in Flow through Porous Media

Figure 11.4 Implementation of the PGSE-NMR using stimulated echo rf pulse trains for (a) sin-gle PGSE-NMR, and (b) double PGSE-NMR (after Khrapitchev and Callaghan, 2003).

The normalized echo signal ES(q) is given by an ensemble average:

ES(q) D hexpf2 i πq � [r(∆) � r(0)]gi . (11.25)

Since r(∆) � r(0) D R ∆0 v(t)d t, we have

ES(q) D*

exp

0@2 i πq �

∆Z0

v(t)d t

1A+

D exp(2 i πq � v∆)

*exp

0@2 i πq �

∆Z0

u(t)d t

1A+

� 1 � 2π2q2

∆Z0

∆Z0

hu(t)u(t0)id td t0 , (11.26)

where, q D jqj and v D V C u, with V being the ensemble-averaged velocity.Hence, u is the velocity fluctuations around V. The dispersion coefficients D� arethen defined by

D� D 14π2∆

limq!0

@

@q2j ln ES(q)j , (11.27)

where � stands for L or T, and the D� is measured for the velocity component ofu(t) in the direction of q. To measure the dispersion coefficients, the velocity u(t)is decomposed into longitudinal and transverse components. Thus, for example,

DL D 12∆

∆Z0

∆Z0

hu L(t)u L(t0)id td t0 , (11.28)

11.5 Measurement of the Dispersion Coefficients 353

where u L(t) is the longitudinal component of u(t). A similar formula is used forDT based on uT(t).

A similar analysis may be carried out for the second pulse sequence shown inFigure 11.4. Two gradient pairs are applied on the same spin magnetization, butover different time intervals, ∆, separated by a mixing time ∆m. The displacementencoding for the two successive gradient pulse pairs can be applied in either thesame or opposite sense. The experiments are labeled uncompensated and compen-sated, and we refer to the latter as Double PGSE-NMR or D-PGSE-NMR. Then,

ED D*

exp

0@2 i πq �

∆Z0

u(t)d t

1A exp

0B@�2 i πq �

∆mC2∆Z∆mC∆

u(t)d t

1CA+

. (11.29)

If the displacements during the two successive encoding intervals are completelyuncorrelated, then the D-PGSE-NMR will yield twice the stochastic part of the ex-ponent for the S-PGSE-NMR and, thus, ED D jESj2. Then, for the D-PGSE-NMR,

D� D 14∆

264

∆Z0

∆Z0

hu(t)u(t0)id td t0 � 2

∆Z0

∆mC2∆Z∆mC∆

hu(t)u(t0)id td t0

C∆mC2∆mZ∆mC∆

∆mC2∆Z∆C∆m

hu(t)u(t0)id td t0

375 . (11.30)

Clearly, one decomposes u(t) into the longitudinal and transverse components inorder to estimate DL and DT, in a manner similar to Eq. (11.28). If u(t) is stationary,then Eq. (11.30) is simplified to

D� D 12∆

264

∆Z0

∆Z0

hu(t)u(t0)id td t0 �∆Z

0

∆C2∆mZ∆mC∆

hu(t)u(t0)id td t0

375 . (11.31)

Note, however, that one must have hu(t)i D 0. Thus, one can write u(t) Dδu C uf(t), where uf(t) is the fluctuating part of u. Physically, δu can be inter-preted as that part of u that is associated with reversible dispersion. In Taylor–Arisdispersion in a tube (see below), for example, if molecular diffusion is not signif-icant, dispersion will be reversible, that is, by reversing the direction of the flowin the tube, the solute particles will reverse their paths exactly and return to theirorigin. In a D-PGSE-NMR experiment, the phase shift due to δu will be reversed inthe second encoding and, thus, there will be no contribution. Thus, the dispersioncoefficients will still be given by Eq. (11.30), except that u(t) should be replaced byuf(t). That is not, of course, the case for S-PGSE-NMR experiments because in thiscase, the effect of δu is not eliminated. In fact, in this case,

D(S)� D 1

2hδu2i∆ C 1

2∆

∆Z0

∆Z0

huf(t)uf(t0)id td t0 , (11.32)

where the superscript S indicates a S-PGSE-NMR measurement.

354 11 Dispersion in Flow through Porous Media

To use the above formulation for estimating the dispersion coefficients, one musthave detailed data on u(t). As mentioned earlier, PGSE-NMR experiments do pro-vide such data. Then, the data are either used directly in the above information, orone fits a good model to the data, and utilizes the results in the formulation. An ac-curate model for such data is the Ornstein–Uhlenbeck process, which is stationaryand given by

hu(t)u(0)i D hu2i exp�

� t

∆c

�, (11.33)

where ∆c is the correlation time. In principle, ∆c is a distributed quantity. More-over, the distribution of ∆c for the longitudinal correlation times is not necessarilythe same as that of the transverse ones. In addition, one must also consider twodistinct regimes, namely, ∆ ∆c and ∆ � ∆c. After substituting Eq. (11.33) intoEqs. (11.30) and (11.32), and carrying out the integration, one obtains the desiredexpressions for the two dispersion coefficients. For ∆ ∆c, one has

D(S)� � 1

2∆�˝

δu2˛C ˝u2

f

˛�, (11.34)

D(D )� � 1

2

˝u2

f

˛∆�

1 � exp�

� ∆m

∆c

��, (11.35)

whereas for ∆ � ∆c, we obtain

D(S)� � 1

2

˝δu2˛C ˝

u2f

˛∆c , (11.36)

D(D )� � ˝

u2f

˛∆c . (11.37)

Seymour and Callaghan (1997), Stapf and Packer (1998), Khrapitchev andCallaghan (2003), and Hunter and Callaghan (2007) used the above and similarideas to measure the dispersion coefficients by the PGSE-NMR method. In gen-eral, the measured dispersion coefficients are consistent with those measured byother methods (see Figures 11.5 and 11.6), but typically somewhat lower. Damionet al. (2000) carried out pore network simulations of dispersion (see below), andcomputed the important quantities in a NMR experiments using the model.

11.6Dispersion in Systems with Simple Geometry

Dispersion is a general mass transfer phenomenon and can occur in a wide vari-ety of systems. At the same time, dispersion is sensitive to the microstructure ofthe system in which it is occurring. Thus, dispersion has been studied in a largenumber of systems with a wide variety of morphologies. The simplest system inwhich dispersion can occur is in laminar flow through a single capillary tube, orin a channel between two parallel flat plates. The problem has been studied exten-sively, and in what follows, we summarize the main results. As we describe later

11.6 Dispersion in Systems with Simple Geometry 355

Figure 11.5 Dependence of the longitudinal dispersion coefficient DL on the Péclet numberPe, and the various dispersion regimes. Dm D D is the molecular diffusivity (after Fried andCombarnous, 1971).

Figure 11.6 Dependence of the transverse dispersion coefficient DT on the Péclet number Pe.For comparison, the DL � Pe curve (dashed) is also sketched (after Fried and Combarnous,1971).

in this chapter, some of the results for dispersion in a single capillary tube aresurprisingly similar and applicable to those in porous media with complex mor-phologies.

356 11 Dispersion in Flow through Porous Media

11.6.1Dispersion in a Capillary Tube: The Taylor–Aris Theory

Historically, Griffiths (1911) was the first to report some experimental data thatdemonstrated the essence of the dispersion process in a tube with the effect ofmolecular diffusion being present. Griffiths did not, however, analyze the problemmathematically. He observed that a tracer solute injected into a tube in which wateris flowing spreads out symmetrically about a plane in the cross section that moveswith the speed of flow. He commented that, “It is obvious that the movement of the

center of the column of the tracers must measure the mean speed of flow.” It turned outthat this was not as obvious as Griffiths had thought!

Forty two years later, Taylor (1953)2) pointed out that Griffiths’ observation is arather startling result for two reasons. First, because the water at the center of thetube moves with twice the mean speed of the flow (the Hagen–Poiseuille flow), thewater at (or near) the center must approach the column of the tracer, absorb thetracer as it passes through the column, and then reject it as it leaves on the otherside of the column. Secondly, although the velocity is asymmetrical about the planemoving at the mean speed, the column of tracer spreads out symmetrically.

Taylor was known to have excellent intuition about the most important physicalfactors in any problem that he studied. He analyzed dispersion in a capillary tubeapproximately, but his analysis turned out later to be correct. Aris (1956)3) studiedthe same problem without making any of the approximations that Taylor had made.They both studied dispersion in a cylindrical capillary tube of radius R. Startingfrom a CD equation for a tube

@C

@tC 2Vm

�1 �

� r

R

�2�

@C

@xD Dm

�@2C

@r2C 1

r

@C

@rC @2C

@x2

�, (11.38)

where Vm is the mean flow velocity in the tube, and defining a mean concentrationCm by

Cm D

2πR0

RR0

C(r, x )r d r dθ

2πR0

RR0

r d r dθD 2

R2

RZ0

Cr dr , (11.39)

2) Sir Geoffrey Ingram Taylor (1886–1975) wasa British physicist and mathematician whomade seminal contributions to the theory ofturbulence and fluid dynamics, wave theory,solid mechanics, and dispersion amongother things. He has been described as “oneof the greatest physical scientists of the 20thcentury”.

3) Rutherford “Gus” Aris (1929–2005) was theRegents Professor of Chemical Engineeringand Materials Science at the Universityof Minnesota. He finished a mathematics

degree from the University of London atthe age of 16, and received a doctorateby correspondence, writing his thesis insix weeks. A member of the NationalAcademy of Engineering, he made seminalcontributions to the field of control theory aswell as reaction engineering and catalysis.One of his best known quotes, the basis ofdynamic programming is, “If you don’t dothe best you can with what you happen tohave, you’ll never do the best you might havedone with what you should have had.”

11.6 Dispersion in Systems with Simple Geometry 357

Taylor and Aris showed that in the limit of long times,

@Cm

@tD DL

@2 Cm

@x21

, (11.40)

where x1 D x � Vm t is the moving coordinate with respect to the mean-flowvelocity, and

DL D Da C R2V 2m

48Dr, (11.41)

with the subscripts a and r signifying the fact that Da and Dr are the contribu-tions of the axial and radial molecular diffusion, respectively. That is, if we delete@2 C/@x2 from Eq. (11.38) (i.e., neglect axial diffusion), Da will also be deleted fromEq. (11.41). Numerically, of course, Da D Dr D Dm. Note that in the Taylor–Arisdispersion theory, DL depends quadratically on Vm.

Equation (11.40) has a deep physical meaning: The average solute concentrationspread out by a dispersion process in the solvent – by a combination of radial dif-fusion and axial convection – follows a diffusion equation and, hence, a Gaussiandistribution, except that the effective diffusivity is not the molecular diffusivity Dm,but the dispersion coefficient that may be viewed as an effective diffusivity givenby Eq. (11.41).

We define the Péclet number4) Pe by Pe D RVm/Dm D tr/ tc, where tr D R2/Dm

is a time scale for significant radial diffusion, and tc D R/Vm is the convection timescale. Thus, Pe is simply a measure of the competition between radial diffusion andconvection. Then, Eq. (11.41) is rewritten as

DL

DmD 1 C 1

48Pe2 . (11.42)

Aris (1956) also showed that for a tube with a cross section of any shape, one has

DL D Dm C δsl2s V 2

m

Dm, (11.43)

where ls is a characteristic length scale of the tube, and δs is a numerical factorthat depends on the shape of the tube’s cross section. For example, for an ellipticalcross section with the major and minor semi-axes a and b, one has ls D a and

δs D 148

24 � 24e2 C 5e4

24 � 12e2 , (11.44)

with e D p1 � b2/a2. For a circular cross section, b D a, e D 0 and δ D 1/48, as

expected. For dispersion in a slit pore with fully-developed laminar flow, ls D h andδs D 2/105, where h is the half-width of the pore. Thus, the quadratic dependenceof DL on Vm is independent of the shape of the cross section. It is, in fact, the resultof the competition between radial molecular diffusion and axial convection.

4) Jean Claude Eugene Péclet (1793–1857) published several books, including one on heat conduction.

358 11 Dispersion in Flow through Porous Media

Aris (1956) conjectured that any initial distribution of the solute concentrationwill ultimately approach a Gaussian distribution. Chatwin, in a series of papers(see Chatwin, 1977, for earlier references to his work), proved Aris’ conjecture.The range of validity of the Taylor–Aris dispersion has been studied thoroughlyby finite-difference calculations (Annanthakrishnan et al., 1965) and by orthogonalcollocation (Wang and Stewart, 1983).

Further important work on dispersion in tubes was carried out by Horn (1971)and Brenner (1980). In particular, Brenner (1980) generalized the Taylor–Aris dis-persion theory very significantly by employing a formulation in which both localand global spaces are utilized. For example, in the problem of dispersion in a cap-illary tube, r represents the local space, whereas x is the global one. Brenner alsoassumed that the solute particles may have a finite size. One can also exploit theequivalence between Langevin and Fokker–Planck equations in order to derive theTaylor–Aris results (van den Broeck, 1982).

Experiments by Koutsky and Adler (1964) demonstrated that coiled tubes yieldreduced longitudinal dispersion, which is important in chemical reactor design aswell as in diffusivity measurements. Hoagland and Prud’homme (1985) analyzedlongitudinal dispersion in tubes of sinusoidally-varying radius, R(z) D R0[1 Ca sin(2πz/ω)], in order to model dispersion in packed-bed processes. Surprisingly,their analysis indicated that the longitudinal dispersion coefficient DL varies linear-

ly with the Péclet number, rather than the second power as in the classical Taylor–Aris dispersion.

11.6.2Dispersion in Spatially-Periodic Models of Porous Media

At the next level of complexity are spatially-periodic models of porous media. Wealready described the geometrical structure of such models in Chapter 5. Diffu-sion, electrical conduction, and flow in such models of porous media were stud-ied in Chapter 9. Thus, it suffices to mention that Brenner (1980), Brenner andAdler (1982), Eidsath et al. (1983), Koch et al. (1989), and Salles et al. (1993) exam-ined theoretically dispersion in spatially-periodic 3D porous media, while Edwardset al. (1991) studied it within the 2D model. Edwards (1995) studied charge trans-port together with convective dispersion in spatially-periodic models. Dorfman andBrenner (2002) studied Taylor–Aris dispersion for Brownian particles of nonze-ro sizes in a spatially-periodic network of capillary tubes. Quintard and Whitak-er (1993) developed a method for studying transport and dispersion in spatially-disordered porous media with or without disorder. Brenner and Edwards (1993)presented analyses of convective dispersion and reaction in various geometries, in-cluding tubes and spatially-periodic packed beds. On the experimental side, Gunnand Pryce (1969) reported measurements of the longitudinal dispersion coefficientDL for flow parallel to one of the axes of a simple-cubic lattice of spherical particles.

Such models are not realistic representations of real porous media that are nei-ther periodic, nor ordered the way such models are. In fact, because dispersion isvery sensitive to the microstructure of the system in which it occurs, a spatially-

11.7 Classification of Dispersion Regimes in Porous Media 359

periodic model of porous media cannot be expected to yield useful predictions fordisordered porous media. Indeed, Koch et al. (1989) showed, for example, that fora square array of cylinders or a cubic array of spheres and in the limit Pe ! 1, DL

depends quadratically on Pe, and that DT approaches a constant value. Their predic-tions are in contradiction with dispersion in disordered porous media for which DL

has a much weaker dependence on Pe, and DT does not reach a constant value (seebelow).

11.7Classification of Dispersion Regimes in Porous Media

Over the past several decades, dispersion in porous media has been studied exper-imentally, particularly in beadpacks, unconsolidated sandpacks, and sandstones.Some of the reported data, mainly for unconsolidated sands, were compiled byFried and Combarnous (1971). Data on sandstones, compiled by Perkins and John-ston (1963) and Legaski and Katz (1967) show, however, that dispersion in consoli-dated porous media is similar to that in unconsolidated media. Figure 11.5 collectsexperimental data for DL/Dm for sandpacks, which show that there are five dis-cernible regimes of dispersion; see Bijeljic et al. (2004) for a collection of the morerecent references on the experimental data for DL and its dependence on Pe.

Figure 11.6 presents the typical experimental data for DT/Dm. For unconsoli-dated porous media, the Péclet number is defined as Pe D dgV/Dm, where dg isfrequently taken to be the average diameter of a grain or bead. Bijeljic and Blunt(2007) provide a more recent list of references for the experimental measurementsof DT and its dependence on Pe.

The five dispersion regimes, indicated by Figure 11.5, are as follows.

1. Pe < 0.3 defines the diffusion regime in which convection is so slow that molec-ular diffusion controls the mixing almost completely. In this regime, we haveisotropic dispersion such that (Brigham et al., 1961)

DL

DmD DT

DmD 1

F φ, (11.45)

where F is the formation factor and φ is the porosity of the porous medium.The quantity 1/(F φ) varies commonly between 0.15 and 0.7, depending onthe type of porous media. Due to the isotropy, a concentrated sphere of so-lute will remain a sphere (rather than developing into an ellipsoid as indicat-ed by Eq. (11.3)), but will increase in size as dispersion progresses. AlthoughEq. (11.45) is quoted widely in the literature, a proof of it, even by a heuristicargument, is not usually given. In particular, the presence of φ is not obvious.However, if we consider the limit Pe D 0, then we derive Eq. (11.45) by thefollowing argument.As mentioned in Chapter 3, Einstein’s relation relates the electrical conductivi-ty gf of a fluid to the molecular diffusivity Dm by gf D ne2Dm/(kB T ), where n

360 11 Dispersion in Flow through Porous Media

is the density of the charge carriers, e is the charge, kB is the Boltzmann’s con-stant, and T is the temperature. For a porous medium, the same equation maybe used, except that gf should be replaced by ge, the effective electrical conduc-tivity of the fluid-saturated medium, Dm by De, the effective diffusivity in theporous medium, and n by nφ, the density of charge carriers in the pore spaceof the medium and, thus, ge D nφe2De/(kB T ). Thus, De/Dm D ge/(φgf) D1/(F φ), with F D gf/ge, which is equivalent to Eq. (11.45), because in the limitPe ! 0, we have DL D DT D De.

2. The regime 0.3 < Pe < 5 defines the transition zone in which convection con-tributes to dispersion, but the effect of diffusion is still quite strong. DL/Dm

appears to increase with Pe, although it is difficult to say how.3. The interval 5 < Pe < 300 defines the power-law regime. Convection dominates

dispersion, but the effect of molecular diffusion cannot be neglected, and onewrites

DL

DmD 1

F φC aLPe�L , (11.46)

DT

DmD 1

F φC aTPe�T . (11.47)

The average values of �L and �T from all the available experimental data are�L ' 1.2 and �T ' 0.9. We call this regime the boundary-layer dispersionafter Koch and Brady (1985) since, as we show below, it is consistent with theexistence of a diffusive boundary layer near the solid surface of the pores, firstidentified by Saffman (1959), where molecular diffusion transfers the solutefrom the very slow regions near the solid walls to faster streamlines. In practicalapplications, �L and �T often are taken to be unity. The coefficients aL and aT

depend on the heterogeneities of the pore space, and their typical values areaL ' 0.5 and aT ' 0.01�0.05.

4. The purely convective regime is defined by 300 < Pe < 105. Simple dimensionalanalysis indicates that (ignoring the 1/(F φ) term, which is small)

DL

Dm� Pe , (11.48)

DT

Dm� Pe . (11.49)

This type of dispersion is also called mechanical dispersion. In this case, dis-persion is simply the result of a stochastic fluid velocity field induced by therandomly distributed pore boundaries.

5. For Pe > 105, dispersion is in the turbulent regime. The Péclet number is nolonger the only correlating parameter, as the Reynolds number should also beconsidered. For flow through porous media, however, this regime is not of in-terest.

6. There is a sixth dispersion regime that is not evident in Figure 11.5. This isthe so-called holdup dispersion (Koch and Brady, 1985) first studied by Carberry

11.8 Continuum Models of Dispersion in Porous Media 361

and Bretton (1958), Turner (1959), and Aris (1959). In this regime, the soluteis trapped in the dead-end pores or inside the solid grains, from which it canescape only by molecular diffusion. One has (again, ignoring the 1/(F φ) term)

DL

Dm� Pe2 , (11.50)

DT

Dm� Pe2 , (11.51)

which indicate rather strong dependence of DL and DT on Pe. In low-porosityporous media that are barely connected with their porosity being close tothe percolation threshold, there are many dead-end pores and, therefore, thisregime is relevant to dispersion in such porous media.

Bacri et al. (1987) used an acoustic technique (see Section 11.5.1.3) to measure DL

in flow through three different porous media, namely, an unconsolidated pack ofglass beads, a fireproof brick, and a mill sandstone. Their data indicated that pore-level disorder strongly affects DL and its dependence on Pe. They also reported thattheir data follow power laws (11.46)–(11.49), depending on the breadth of the poresize distribution and the connectivity of the pore space.

11.8Continuum Models of Dispersion in Porous Media

Having gained a qualitative understanding of dispersion in porous media and whatwe may expect, let us now describe and discuss various models of dispersion inporous media. We begin by describing the continuum approach.

11.8.1The Volume-Averaging Method

Chapter 9 presented the continuum approach for modeling of flow and transport inporous media. The same approach may be used for studying dispersion in porousmedia. The works of Whitaker (1967), Bachmat (1969), Gray (1975), Carbonell andWhitaker (1983), Eidsath et al. (1983), and Plumb and Whitaker (1988a) fall inthis class of methods. For example, Plumb and Whitaker (1988a) used a volume-averaging technique (see Whitaker, 1999, for details of the method) in order toaverage the microscopic CD equation, where the averaging is over the disorder,in order to arrive at the macroscopic averaged equation. The final result for thevolume-averaged concentration hCi of the solute is given by

φ@hCi@t

C r � (φhvihCi) D r � (φD� � rhCi) . (11.52)

362 11 Dispersion in Flow through Porous Media

Here, D� is the dispersion coefficient tensor given by

D� D Dm

24I C 1

V

ZA

n � f dA

35 � hQv f i , (11.53)

where f is a trial solution, V is the pore volume, A is the interfacial area containedbetween the averaging volume, and n is the unit outwardly-directed normal vectorfor the pores. Here, Qv is the fluctuations in the velocity v (i.e., v D hvi C Qv).

The volume-averaging analysis indicates that on a large enough length scale overwhich the porous medium is homogeneous, a CD equation describes the average

solute concentration. The tensor D� contains two terms. One is the contribution ofmolecular diffusion, while the second one is due to hydrodynamic transport. How-ever, the contribution of molecular diffusion appears only as Dm, not as Dm(F φ)�1,as the method of volume-averaging cannot take into account the effect of the tortu-osity of a porous medium.

The unknown function f must be determined in order for D� to be computed. Inpractice, f is the solution of the following boundary value problem

Qv C v � r f D Dmr2 f , (11.54)

�n � r f D n at A , (11.55)

provided that Dm t/ l2 � 1 (where l is the length scale associated with the pores)and, therefore, f is determined if a model of the pore space is specified. If, how-ever, the pore space is disordered, then the numerical calculation of f is no easierthan the numerical simulation of the CD equation itself within the domain of theproblem.

11.8.2The Ensemble-Averaging Method

Koch and Brady (1985) used an ensemble-averaging technique (see also Chapter 9)in order to study dispersion in a packed bed in which the volume fraction of theporous particles was φp. They first formally related the average concentration fieldto the probability distribution of the solid material, and then derived the effectivedispersion coefficients in the high porosity limit (low solid volume fraction φp), in-cluding all the relevant proportionality constants. Koch and Brady (1985) showedthat the macroscopic equation of mass conservation (i.e., the CD equation) in thelong-time limit takes the form of a macroscopic Fick’s (second) law with a constanteffective diffusivity, or dispersion, tensor. They also carried out an asymptotic anal-ysis in low volume fraction of the dispersion coefficients in a bed of fixed sphericalparticles for all values of the Péclet number. Koch and Brady (1985) identified sev-eral physical mechanisms that cause dispersion:

1. The stochastic velocity fluctuations induced in the fluid by the randomly po-sitioned bed particles give rise to a convectively-driven contribution to disper-

11.9 Fluid-Mechanical Models 363

sion. At high Péclet numbers, the convective dispersion mechanism is pure-ly mechanical, with the resulting dispersion coefficients being independent ofmolecular diffusion and grow linearly with Pe.

2. The region of zero velocity in and near the particles gives rise to nonmechanicaldispersion mechanisms that dominate the longitudinal dispersion coefficient atvery high Péclet numbers. One such mechanism involves the retention of thediffusing solute particles in permeable particles, from which they can escapeonly by molecular diffusion, leading to a dispersion coefficient that grows asPe2 (see the earlier discussion of holdup dispersion)

3. Even if the particles are impermeable, nonmechanical contributions that growas Pe ln Pe and Pe2 at high Pe arise from a diffusive boundary layer near thepores’ surfaces and from regions of closed streamlines (the dead-end pores),respectively.

These results are all in agreement with the experimental observations describedin the last section. As an example, the predictions of Koch and Brady (1985) forrelatively high Péclet numbers are given by

DL

DmD 1 C 3

4Pe C π2

6φpPe ln Pe C Dm(1 C )2

15�DmpφpPe2 , (11.56)

DT

DmD 1 C 63

p2

320

pφpPe . (11.57)

Here, � is the partition coefficient for the solute (i.e., partitioning of the solutebetween the fluid flowing through the bed and that in the void space inside theparticles), and Dmp is the molecular diffusivity of the solute inside the particles. is defined by

D φp(� � 1)� � φp(� � 1)

. (11.58)

An appealing aspect of Koch and Brady’s work was that the fluid mechanicalaspects of the problem were treated without any approximations. The ensemble-averaging technique may also be extended to the case where dispersion is notGaussian and cannot be described by a CD equation (see below).

11.9Fluid-Mechanical Models

The fluid-mechanical models are based on three basic attributes:

1. a Lagrangian description of the motion of solute-containing fluid through a sin-gle pore. As mentioned earlier, in a Lagrangian approach, the motion of soluteparticles is followed, and the average velocity and the dispersion coefficients aredefined as the time-rate of change of the mean and mean-square positions ofthe particles, respectively;

364 11 Dispersion in Flow through Porous Media

2. specific assumptions about the medium, for example, homogeneity and iso-tropy, and

3. calculation of the quantities of interest as statistical averages.

The works of Scheidegger (1954), Day (1956), de Josselin de Jong (1958), Saffman(1959)5), Saffman (1960), Haring and Greenkorn (1970), and Bear (1972) are in thisgroup. Saffman’s work is the most general of these and, hence, is described here.His model consisted of a network of randomly oriented and distributed straightcapillaries, in each one of which the flow was uniform. The path of solute particleswas regarded as a random walk (see Chapters 9 and 10) in which the length, direc-tion, and duration of each step were random variables. However, the only morpho-logical disorder allowed in the model was the random orientations of the capillarytubes. Saffman was careful to introduce a dynamical basis for his model found-ed explicitly on fluid-mechanical aspects. He assumed that all the pores have anequal circular cross section of radius R, that flow was laminar in all the tubes, anddistinguished five cases in his first paper Saffman (1959):

1. tc tr, where tc is the convective time spent by a solute particle in a pore andtr the time required for appreciable radial diffusion of the same particle, tr DR2/(8Dm) (i.e., the time that the solute particle spends to jump a distance R/2from one streamline to another). Thus, radial diffusion is negligible and theduration of a step is t D tc D l/Vm, where l is the pores’ length.

2. The solute particle is on a streamline close to a pore wall, that is, its speedis small and molecular diffusion is important. The duration of a step is thent D tr C l/Vm, that is, the particle makes one jump from a streamline close tothe pore wall to another one with speed Vm, and then is convected out of thepore.

3. tr < t ta, where ta is the time for appreciable axial diffusion with ta Dl2/(2Dm). The effect of the axial diffusion is negligible and t D tr C l/Vm.

4. tr < t ta, which means that the pore is very narrow and thus, t D l/Vm.5. ta t. Therefore, the duration of a step is t D ta.

Saffman found that in all the cases, DT is given by

DT D 316

lV . (11.59)

DL was found, however, to depend on the dispersion regime considered. If we let

DL D 12

V l s2 , (11.60)

5) Philip Geoffrey Saffman (1931–2008), theTheodore von Karman Professor of AppliedMathematics and Aeronautics at CaliforniaInstitute of Technology, was a leading figurein fluid mechanics. In addition to his workon dispersion and the Saffman–Taylor

instability (see Chapter 13), he developed,together with his colleague Max Delbrück,the Saffman–Delbrück model of proteindiffusion, and made significant contributionsto the theory of fluid vorticity arising fromthe motion of ships and aircrafts.

11.9 Fluid-Mechanical Models 365

then

s2 D 13

ln�

3V ta

l

�C 1

12

�ln�

6V tr

l

��2

� 14

ln�

6V tr d

l

�C 19

24, (11.61)

if

V talq

ns ln� 3V ta

l

� 1 I (11.62)

and

s2 D 16

ln�

27V tm

2l

�C 1

12

�ln�

6V tr

l

��2

� 14

ln�

6V tr

l

�C 19

24, (11.63)

if

ln ns � 2 ,3V tr

l

n12s

ln n12s 1 ,

3V tal

n12s

ln n12s � 1 ,

and

s2 D 148

�ln�

54V tm

l

��2

, (11.64)

if

ln ns � 2 ,4V tr

l

n12s

ln n12s � 1 ,

4V tal

n12s

ln n12s � 1 ,

where ns is the mean number of steps taken by the fluid particles after a largetime Tm at which DL and DT are measured, given by Tm D 3hxi/(2l) with hxibeing the mean longitudinal position of the particles at time Tm. We may viewthe constant 19/24 in Eqs. (11.61) and (11.63) as the prediction of the Saffman’stheory for (F φ)�1. Ignoring the constant that is usually much smaller than theother terms in the above equations, Saffman’s results for DL may be summarizedas

DL

Dm� Pe(ln Pe)α , (11.65)

where α D 1 or 2. Equation (11.65) may now be compared with Eq. (11.46). Asfirst pointed out by Sahimi (1984), if we take α D 1 and fit the experimental datato Eq. (11.65), the resulting fit would be as accurate as that provided by Eq. (11.46)if �L ' 1.25. On the other hand, if we take α D 2, the accuracy of the resultingfit would be compatible with that of Eq. (11.46) if �L ' 1.15. Thus, two importantfeatures of all the experimental data are explained:

366 11 Dispersion in Flow through Porous Media

1. The data indicate that �L is either about 1.13–1.16 (obtained by Legaski andKatz (1967) for Bandera sandstone, by Salter and Mohanty (1982) for Bereasandstone, and by Blackwell et al. (1959) for unconsolidated packed sands), orabout 1.24–1.30 (as reported by Brigham et al. (1961) and Pakula and Greenko-rn (1971) for glass beads, and by Legaski and Katz (1967) for Boise and Nor-dosaria sandstones and for Dolomites), with an overall average of about 1.2, asmentioned above.

2. �L is probably not universal; it depends on the strength of the competition be-tween molecular diffusion and convection, which in turn depends on the poreshapes and sizes. On the other hand, Eq. (11.59) is not completely compatiblewith Eq. (11.47) because most of the data (see, for example, Blackwell, 1962)indicate that �T ' 0.9, as mentioned above. Although, in the numerical simu-lations, �T is usually taken to be unity. Similar to �L, it is likely that �T is not

universal.

Saffman also found that dispersion cannot be described by a CD equation unlessthe characteristic time Tm is sufficiently large. His analysis clearly points to the sig-nificance of molecular diffusion to dispersion in microscopically-disordered porousmedia, no matter how small it may be, as long as it is not exactly zero. In fact, thepresence of molecular diffusion in such porous media is essential. In its absence,a solute particle that is traveling along a streamline very close to a pore wall willneed a huge amount of time to escape from that region. Otherwise, DL D 0. It ismolecular diffusion that intervenes and transfers the solute to a faster streamlineand prevents DL from vanishing. The logarithmic terms in Eqs. (11.61)–(11.64) areprecisely due to the diffusion time scale for transferring the solute from a very slowstreamline to a faster one.

Saffman’s results are presumably valid if Pe is large but finite. In his secondpaper (Saffman, 1960), he considered the case where Pe is “less than some largevalue”, and found that both DL and DT depend quadratically on Pe. The agreementbetween Saffman’s results and the various experimental data ranges from reason-able to good. Saffman’s work is the most detailed and carefully analytical analysis ofdispersion in microscopically disordered porous media. It has not, however, beenfully appreciated.

Saffman did not allow the possibility that the pores’ sizes are distributed accord-ing to a pore size distribution (all the pores were assumed to have the same radius).Haring and Greenkorn (1970) rederived some of Saffman’s results, assuming thatthe pores’ sizes follow a statistical distribution. Moreover, in Saffman’s work, theflow field is represented by a sort of a mean-field approximation. There are no cor-relations between the successive steps of the particle. This restriction can also beremoved by Monte Carlo calculations using pore network models of pore space,as was first done by Sahimi et al. (1982), which is described below. Finally, it isworth mentioning that the logarithmic singularities derived by Saffman were re-discovered by Aronovitz and Nelson (1984) in what they called “diffusion in steadyflow” through a porous medium, which is nothing but hydrodynamic dispersiondescribed herein.

11.10 Pore Network Models 367

11.10Pore Network Models

These models belong to the class of fluid-mechanical models that we already de-scribed, except that the mean-field nature of the flow field and the absence of theheterogeneity are explicitly deleted. As already mentioned in Chapter 3, Torelli andScheidegger (1972) appear to be the first to propose a random network model forstudying dispersion processes in porous media (and invoked percolation theory toexplain the phenomenon), although they did not report any result. Torelli (1972)did simulate dispersion processes in flow through a random network, but his re-sults pertain to a type of non-Gaussian transport process not closely related to whatwe are interested in here.

11.10.1First-Passage Time and Random Walk Simulation

Sahimi et al. (1982) were the first to use pore network models of porous media tosimulate dispersion in flow through the networks. In their model, one first deter-mines the flow field in the network by the method described in Chapter 10, wherewe described pore network models for calculating the effective permeability of aporous medium. Then, solute particles are injected into the network at random atthe upstream plane at x D 0. Each particle selects a streamline at random. Theconvective travel time for a given pore is given by t D l/vp, where l is the pore’slength, and vp is the flow velocity in the pore. Complete mixing at the nodes wasassumed and, therefore, the probability that a pore is selected once a particle hasarrived at a node is proportional to the flow rate in that pore. The FPTD Q for theparticles are computed by fixing the longitudinal or lateral positions and measur-ing the time at which the particles arrive at the fixed positions for the first time.It is not difficult to show that D� , the dispersion coefficient in the � -direction, isgiven by

D� D1Z0

Q(� � �0, t)s2

2td t , (11.66)

where �0(x0) is the starting position of the particles, s2x D (x � x0 � V t)2, and,

s2� D (� � �0)2 for � D y and z.However, such a model is appropriate for purely mechanical dispersion since it

ignores the pore-level molecular diffusion. It only considers the effect of a stochas-tic velocity field throughout the network. To include the effect of molecular diffu-sion and simulate the boundary-layer dispersion, the following method was adopt-ed (Sahimi and Imdakm, 1988). The convective time tc for traveling along a stream-line in a pore is calculated. If tc � tr, where tr is the time scale for significant radialdiffusion that was described above, then one sets t D tc C tr since the tracer hasenough time to diffuse to a faster streamline. Sorbie and Clifford (1991) suggest-ed a somewhat different method for including the effect of molecular diffusion

368 11 Dispersion in Flow through Porous Media

on the dispersion process in the pore network. To simulate the holdup dispersion(i.e., dispersion in which the tracer may diffuse into the stagnant region), the tracerparticles are allowed to diffuse into the dead-end pores of the network. Transportin such pores is only by molecular diffusion. In a series of papers, Sahimi et al.

(1982, 1983a, 1986a,b) and Sahimi and Imdakm (1988) demonstrated that suchpore network models reproduce and simulate all the regimes of dispersion de-scribed above. In particular, Eqs. (11.33)–(11.38) are all correctly predicted by suchFPT simulations and pore network models.

A more refined pore network model of dispersion using random walk simula-tions was developed by Bijeljic et al. (2004). They mapped a Berea sandstone ontoan equivalent pore network with square cross sections, and utilized the randomwalk simulation similar to that developed by Sahimi and co-workers. To includethe effect of molecular diffusion, the particles are moved randomly, as defined bythe coordinates of a spherical coordinate system. In a time step ∆ t, the diffusiondistance is rd D p

2Dm ∆ t. Thus, the coordinates of the particle are given by theusual relation, x D rd cos θ sin ', y D rd sin θ sin ', and z D rd cos '. The transi-tion probabilities for moving the particles from one pore to another depended onthe last step of the particles. If they arrived at a node of a network as a result ofconvection, then the next pore was, similar to the work of Sahimi and co-workers,selected with a probability proportional to the volume flow rate in that pore. Ifthe particles arrived at a node as a result of a diffusion step, then the probabil-ity of selecting the next pore was taken to be proportional to the cross-sectionalarea of that pore. The longitudinal dispersion coefficient DL was then computed bythe method that Sahimi and co-workers used. Bijeljic et al.’s simulations indicatedthat their network model correctly reproduces the dependence of DL on the Pécletnumber Pe, as shown in Figure 11.5. A somewhat similar model was also usedby Bruderer and Bernabé (2001) in their studies of the transition from Taylor–Arisdispersion to purely mechanical dispersion in very heterogeneous porous media;see also Acharya et al. (2007).

A similar model was used (Bijeljic and Blunt, 2007) for computing the transversedispersion coefficient DT. As in the case of DL, the computed dependence of DT onthe Péclet number Pe was found to be similar to what is presented in Figure 11.6.

11.10.2Probability Propagation Algorithm

De Arcangelis et al. (1986) proposed another pore network model that they calledthe probability propagation algorithm. In their model, a 1D CD equation is assumedto describe mass transfer in each pore of the network, that is,

@C

@tC Vm

@C

@xD Dm

@2C

@x2. (11.67)

Consider a network of capillary tubes fi j g. The concentration Ci j in each tube fol-lows Eq. (11.67), with the initial condition that Ci j (xi j , 0) D 0, and three boundaryconditions:

11.10 Pore Network Models 369

1. a unit pulse of input flux at node i at time t D 0,

Xf j g

Si j

�vmi j Ci j � Dm

@Ci j

@xi j

�xi j D0

D δ(t) , (11.68)

where Si j is the cross-sectional area of tube i j , and vmi j is the mean-flowvelocity in that tube;

2. a common concentration Ci (t) at the starting junction, Ci j (0, t) D Ci (t) for allj, and

3. a sink at each tube end, Ci j (l, t) D 0, for all j, corresponding to the fact thata tracer reaching the end acts as a source for the junction problem at the newnode.

The FPT probability is given by qi j (t) D �Si j Dm@Ci j (l, t)/@xi j . Equation (11.67)is solved in the Laplace transform space. The solution is given by

QCi j (x , λ) D A i j exp(α i j x ) C Bi j exp(� i j x ) , (11.69)

with

α i j , � i j Dvi j ˙

qv2

i j C 4Dmλ

2Dm, (11.70)

where A i j and Bi j are determined from the specified boundary conditions, and λis the Laplace transform variable conjugate to t. We then obtain

Qqi j (λ) D QCi j (λ)Si j

α i j � � i j

exp(�� i j l) � exp(�α i j l). (11.71)

Having determined Qqi j (λ), one obtains the FPTD QQ(L, λ) for the entire network,

QQ(L, λ) DX

Γ

Yi, j 2Γ

Qqi j (λ) , (11.72)

where the sum is over all the paths Γ from the inlet to the outlet of the network.To efficiently compute the sum in Eq. (11.72), de Arcangelis et al. (1986) ordered

the nodes of the network in decreasing pressure, starting with the inlet and finish-ing with the outlet. At each node i, a quantity QQi (λ) was introduced that representeda partial sum in Eq. (11.72), over paths running from the inlet to site i. For a delta-function input of the tracer, one initially has QI D 1 at the inlet I and Qi D 0elsewhere. One then proceeds recursively through the pressure-ordered node list,propagating the quantity Qi from each node i to its network neighbors j accordingto the rule that QQ j (λ) ! QQ j (λ) C QQi (λ) Qqi j (λ), QQi (λ) ! 0. After all the inter-nal nodes have been propagated once in this way, the quantity QQ0(λ) at the outletcontains all terms of the sum in Eq. (11.72) corresponding to purely downstreampaths.

370 11 Dispersion in Flow through Porous Media

However, because molecular diffusion is present, the solute motion includesupstream paths as well. Hence, after one sweep through the network, one hasQQn ¤ 0 for the internal nodes n. By repeated sweeps through the network, the

contributions of the paths with progressively more upstream steps are included.Once QQ(L, λ) is determined, it is inverted to the time domain and DL is calculatedusing Eq. (11.66). Note that this model, in the mechanical dispersion regime (i.e.,with no boundary-layer diffusion), is equivalent to the FPT model of Sahimi andco-workers. De Arcangelis et al. (1986) showed that their method reproduces theresults for both mechanical and boundary-layer dispersion. The method is very ef-ficient as long as the network is well connected. For percolation networks near thepercolation threshold, the method is very inefficient because calculating the sumin Eq. (11.72) becomes very time consuming.

11.10.3Deterministic Models

Koplik et al. (1988b) developed another method for studying dispersion in porenetworks. In their model, one first calculates the flow field throughout the porenetwork by the method described in Chapter 10. Assuming that dispersion in eachpore is described by a CD equation, Eq. (11.68) with its right-hand side being zero(which is simply a statement of the continuity of mass at each node), is writtenfor all the interior nodes of the pore network. The resulting set of linear equationsfor nodal concentrations is solved (in the Laplace transform space), from which DL

is calculated. Sahimi and Jue (1989) and Sahimi (1992a) used a similar method tostudy generalized Taylor–Aris dispersion of finite-size molecules in porous media,that is, one in which the hydrodynamic radius of the molecules is comparable tothe pore sizes.

Roux et al. (1986) used the same method, except that they used a transfer-matrixmethod. In this method, the network is not constructed all at once, but by a step-by-step procedure by adding rows (or planes in 3D) of nodes to the network, startingfrom the inlet row (or plane). Alvarado et al. (1997) used a pore network modelsimilar to that of Koplik et al. (1988b), except that they also considered the effect ofa first-order reaction and linear adsorption on the dispersion process. Their simu-lations indicated that if the kinetic coefficients of the reaction and adsorption is thesame everywhere, then the dispersion coefficient is the same as that of the nonre-active case. The same is true when the Damköhler number Da is much larger thanone. However, when Da ! 0, the description of the dispersion process by a CDequation breaks down.

11.11Long-Time Tails: Dead-End Pores versus Disorder

As already discussed, molecular diffusion transfers the solute into and out of stag-nant, dead-end or low-velocity regions of a pore space. Many measurements of the

11.11 Long-Time Tails: Dead-End Pores versus Disorder 371

concentration profiles of solutes indicate the presence of a long-time tail in theprofiles. Diffusion into and out of the stagnant regions is often invoked to explainsuch long-time tails, which have been of great interest for a long time. Carberry andBretton (1958), Aris (1959), and Turner (1959) were probably the first who studieddispersion in systems with stagnant regions. In particular, Aris (1959) showed thatDL/Dm � Pe2, a result that was rediscovered by Koch and Brady (1985).

In the early 1960s, there were several studies of the relation between the long-time tails and the effect of dead-end pores. Deans (1963) and Coats and Smith(1964) attributed the long-time tails to the presence of dead-end pores that cancause long delays in the travel times of a solute and, hence long tails in its concen-tration profile. They developed a semi-empirical model to account for the effect,which is described below. Brigham (1974) and Baker (1977) found that trappingin dead-end pores is needed to describe dispersion in carbonate rocks, but not insandstones. They proposed that the origin of stagnant regions in carbonate rocksis either regular or bimodal porosity. Their proposal was disputed by Gist et al.

(1990), who measured dispersion coefficients in a variety of sandstones and car-bonate porous media. Their mercury capillary-pressure data for the Austin Chalkand Indiana limestone indicated the presence of bimodal porosity, yet no long-timetails were observed in the measured concentration profiles.

Deans (1963), Coats and Smith (1964), Passioura (1971), Baker (1977), Rao et al.

(1980), and Salter and Mohanty (1982) all investigated the effect of the long-timetails and dead-end pores. In Baker’s model, which is the most sophisticated of suchworks, it is assumed that a fraction φf of the pore volume is available for flow, while1�φf is the fraction of the stagnant or dead-end fraction. A 1D CD equation is used,modified to account for the effect of the stagnant regions:

φf@Cf

@tC (1 � φf)

@Cs

@tC V

@Cf

@xD DL

@2 Cf

@x2, (11.73)

where Cf and Cs are, respectively, the concentrations of the solute in the flowingand stagnant regions. Equation (11.73) is augmented by a mass balance betweenthe stagnant and flowing fluids:

(1 � φf)@Cs

@tD kc(Cf � Cs) , (11.74)

where kc is the mass transfer coefficient such that k�1c is the time the solute spends

in the stagnant regions. Equations (11.73) and (11.74), with the appropriate initialand boundary conditions, are then solved. Normally, φf and kc are not known a

priori and are treated as adjustable parameters.Bacri et al. (1990a) who used an acoustic technique to measure the concentration

and velocity profiles during dispersion in unsaturated porous media (i.e., disper-sion in one fluid phase in a presence of another immiscible fluid), Charlaix et al.

(1987b) who measured the dispersion coefficients and concentration profiles insintered-glass bead packs, and Gist et al. (1990) who did the same in a variety ofsandstones and carbonate rocks, all used the Coats–Smith–Baker model to fit their

372 11 Dispersion in Flow through Porous Media

Figure 11.7 Unsaturated concentration profiles C at several cross sections of the porous medi-um with mean flow velocity V D 3.6 cm/h (a) and V D 0.9 cm/h (b). The dashed curvescorrespond to Gaussian profiles (after Bacri et al., 1990a).

data and obtained very accurate fits. However, whereas Bacri et al. (1990a) attribut-ed the long-time tails in their data to the length of their porous medium as beingtoo short to allow for the development of the Gaussian dispersion (see Figure 11.7),Charlaix et al. (1987b) and Gist et al. (1990) attributed them to the heterogeneity oftheir porous media. Thus, it is important to understand why the Coats–Smith–Baker model is able to provide such good fits to the data.

Based on their studies of dispersion in consolidated porous media, Gist et al.

(1990) identified two cases in which the long-time tails can occur. The first is thatof a heterogeneous porous medium in which the permeability contrast betweenvarious regions is strong enough. The heterogeneity gives rise to a long-time tail inthe concentration profile. The second case is that of a narrow pore-size distributionin which the permeability heterogeneities are due to defects in the packing density.If the long-time tails are in fact due to the permeability heterogeneities, the impli-cations for the upscaling of laboratory results to field conditions are important.

For example, the Coats–Smith–Baker model predicts that the long-time tails willdisappear if k�1

c is much smaller than the total travel time of the solute particles(this is easily seen by inspecting Eqs. (11.73) and (11.74)), whereas the long-timetails will persist if there are strong permeability heterogeneities at any length scale.This is also consistent with the studies and measurements of tracer dispersion ingroundwater flow in heterogeneous aquifers (Pickens and Grisak, 1981). However,before we go on and explain this complex phenomenon, let us first study dispersionin short porous media, a closely related subject.

11.12Dispersion in Short Porous Media

We already mentioned that Bacri et al. (1990a) attributed the long-time tails in theconcentration profiles to the small size of their porous medium. Thus, dispersion

11.12 Dispersion in Short Porous Media 373

in short porous media can be important because, then, the mixing zone will belarge compared with the medium’s length. Brenner (1962) and Brigham (1974)were among the first to study the issue, with Brigham (1974) making a compre-hensive and definitive analysis of this problem.

We already presented the solution of a CD equation for dispersion in a 1D porousmedium of length L and a step change in the inlet concentration at time t D 0 (seeEqs. (11.10)) and (11.11). We define

�˙ DL ˙ V

�t � Vinj

Q

�r

4DL

�t � Vinj

Q

� , (11.75)

where Q is the volume flow rate, and Vinj the total volume of the injected tracersolution. Then, the solution for a pulse input of total volume Vinj is obtained bysuperimposing two solutions for step changes, in which case the outlet solute con-centration is given by

C

C0D 1

2erfc(α�) C 1

2exp

�LV

DL

�erfc(αC) � 1

2erfc(��)

� 12

exp�

LV

DL

�erfc(�C) , (11.76)

where α˙ are defined by Eq. (11.10). If one is only interested in observation timest � Vinj/Q, then Eq. (11.76) is simplified to

C

C0D�

L

�(�DL

�t � Vinj

Q

��� 12

� (DL t)� 12

) exp(��2

�) C exp(�α2�)

.

(11.77)

Gist et al. (1990) used Eq. (11.77) to fit the concentration profiles that they hadmeasured, and found that the resulting fits are as accurate as those provided by theCoats–Smith–Baker model.

Brigham (1974) showed that if the Coats–Smith–Baker model is adjusted at theeffluent boundary to account for the difference between the in-situ and flowingconcentrations, then Eqs. (11.11), (11.76) and (11.77) and the Coats–Smith–Bakermodel will essentially provide identical fits to the data. This explains why Bacri et

al. (1990a) could fit their data for dispersion in a short porous medium with theCoats–Smith–Baker model.

Koch and Brady (1987) also considered dispersion in porous media of short tomoderate lengths. They derived an expression for the Fourier transform of theconcentration and the effective dispersion coefficients, and showed that the char-acteristic time τKB for reaching a diffusive transport described by a CD equation isrelated to a Péclet number Pe1 by

τKB � Pe� 2

31 , (11.78)

374 11 Dispersion in Flow through Porous Media

where Pe1 D dV/Dm, with d being the typical grain size before the grains are fusedto produce a consolidated porous medium. Koch and Brady (1987) also found qual-itative agreement between Eq. (11.78) and the data of Charlaix et al. (1987b). Bacriet al. (1990a) also used the Koch–Brady expression for the concentration profile, butfound only qualitative agreement between the predictions and their data, whereasthe Coats–Smith–Baker model provided an accurate fit to their data. The reasonfor the discrepancy is perhaps that the Koch–Brady results are valid in the limit ofhigh porosities, whereas the data of Bacri et al. (1990a) and Charlaix et al. (1987b)were both for porosities that are beyond the region of validity of the Koch–Bradyresults. Koch and Brady (1987) also proposed that τKB is the same as k�1

c in theCoats–Smith–Baker model.

11.13Dispersion in Porous Media with Percolation Disorder

In this section, we describe dispersion in porous media with percolation disor-der, modeled by pore networks in which a fraction of the pores (bonds) are cutat random. As Katz and Thompson (1986, 1987) showed (see Chapter 10), flowin a porous medium with a broad pore size or permeability distribution may bemapped onto an equivalent percolation problem. The same must be true aboutdispersion in such porous media, as a broad pore size distribution gives rise toa broad distribution of pore flow velocities that, in turn, affects dispersion. Thereare two features of percolation pore networks that influence dispersion. One is thefact that such pore networks have a large number of dead-end pores (bonds) nearthe percolation threshold pc (see Chapter 3) and, thus, holdup dispersion may beimportant. The second feature is that for length scales shorter than the percolationcorrelation length �p, the sample-spanning cluster and its backbone are fractal ob-jects (see Chapter 3) and, thus, dispersion is not expected to be described by a CDequation. We call this regime fractal dispersion.

To study dispersion in percolation pore networks, two important characteristicquantities are essential: the dispersivities, αL D DL/V , and αT D DT/V , which areproportional to each other, but αL is usually larger than αT. Physically, the disper-sivities represent the length scale over which a CD equation can describe dispersionand, thus, in some sense, they are similar to the percolation correlation length �p

(see Chapter 3) because dispersion in a percolation network can be described by aCD equation if the dominant length scale of the system is larger than �p.

11.13.1Theoretical Developments

In their simulations of dispersion in percolation networks, Sahimi et al. (1982,1983a, 1986a,b) found that as pc is approached, the dispersivities and dispersioncoefficients also increase. The increase may be attributed to the fact that near pc,the transport paths are very tortuous, resulting in broad first-passage time distri-

11.13 Dispersion in Porous Media with Percolation Disorder 375

butions (FPTDs) and, hence, large dispersive mixing of the two fluids. Figure 11.8shows their results for dispersion in a percolating square network.

We should mention here the work of de Gennes (1983a). He studied dispersionnear pc. After a rather long analysis, he showed that in calculating DL, the averageflow velocity must be based on the total travel time of the solute particles in thesample-spanning cluster, rather than the travel time along the backbone alone. In-tuitively, this is clear. Indeed, in experimental measurements of the concentrationprofiles and DL, there is no way of measuring the travel times along the backbonealone. Instead, what is routinely measured is the total travel or transit time.

In Chapter 3, we described universal power laws for various properties of perco-lation networks near the percolation threshold. What are the power laws for DL andDT near pc? Our discussions so far must have made it clear that DL and DT are sen-sitive to the structure of a porous medium. Similar to fractal diffusion described inChapter 10, we may define a crossover time τco such that for t � τco, dispersion isGaussian or diffusive and follows a CD equation, whereas for t τco, dispersionis non-diffusive, with the crossover between the two regimes taking place at aboutt ' τco. For dispersion near pc, this time scale is estimated from

τco � � 2p

DL(11.79)

Figure 11.8 Longitudinal (circles) and trans-verse (triangles) dispersivities in a percolatingsquare network in which a fraction p of thebonds carry flow. The dispersivities are mea-

sured in units of the length of a bond (pore).At p D 1, there is no disorder, and thus thedispersion coefficients and the dispersivitiesare both zero (after Sahimi et al., 1986b).

376 11 Dispersion in Flow through Porous Media

since the dominant length scale in the system is �p. To derive the power laws forDL and DT, we must separately consider the various dispersion regimes describedin Section 11.7. Let us first introduce two random-walk fractal dimensions by

h∆x2i � t2

D lw , (11.80)

hy 2i � hz2i � t2

D tw , (11.81)

where, h∆x2i D h(x � hxi)2i D hx2i � hxi2. Equations (11.80) and (11.81) aredefined for length scales L �p. Two average flow velocities may also be defined.One is an average velocity Vc, defined in terms of the travel time in the sample-spanning cluster. Then, Vc � Ke/X A, where Ke is the effective permeability ofthe pore network (porous medium), and X A is the percolation accessible fractiondescribed in Chapter 3. Thus, near pc,

Vc � (p � pc)e�� � � �θcp , (11.82)

where θc D (e � �)/ν, and e, ν and � are, respectively, the critical exponents of thepermeability, the correlation length, and the accessible fraction that were definedin Chapter 3. On the other hand, if an average particle velocity VB is defined interms of the travel times along the backbone, then VB � Ke/X B, where X B is thepercolation backbone fraction defined in Chapter 3. Thus, near pc,

VB � (p � pc)e��B � � �θbbp , (11.83)

where θB D (e��bb)/ν, and �bb is the critical exponent of X B defined in Chapter 3.For length scales L �p, we should replace �p in Eqs. (11.82) and (11.83) by L and,therefore, Vc � L�θc and VB � L�θbb , respectively. We also define a macroscopic

Péclet number by

PeM D V �p

De, (11.84)

where De is the effective diffusivity of the system, and V is either Vc or VB. ForL �p, we replace �p in Eq. (11.84) by L. Having defined the essential quantities,we now investigate the power laws that the dispersion coefficients and τco follownear pc.

1. Let us first consider the small Péclet number regime described earlier. In thiscase, convection has no effect and DL � DT � D � (p � pc)µp�� , as derived inChapter 3, where µp is the critical exponent of the electrical conductivity of the(fluid-saturated) system. For L �p, we have fractal diffusion. Based on thediscussions in Chapters 9 and 10, we can immediately write

D lw D D t

w D Dw D 2 C θ , (11.85)

where θ D (µp � �)/ν and Dw is the fractal dimension of the random walkdefined in Chapter 10. Moreover, according to Eq. (11.79),

τco � (p � pc)�µp�2νC� � � 2Cθp , (11.86)

11.13 Dispersion in Porous Media with Percolation Disorder 377

so that τco � L2Cθp for L �p. Equations (11.85) are valid for dispersionin the entire sample-spanning cluster in the limit PeM ! 0. For dispersionin the backbone, θ in Eqs. (11.85) and (11.86) should be replaced with θB D(µp � �bb)/ν. Moreover, for the nth moment of the FPTD, we have ht ni � htin ,where n > 1 is an integer and, therefore, for L �p, we have ht ni � (L2/De)n .Therefore,

ht ni � Ln(2Cθp) . (11.87)

2. Now, suppose that dispersion only takes place in flow through the backbone ofthe pore network, and that PeM is relatively large. Although any porous mediumhas a large number of dead-end pores near its percolation threshold (criticalporosity) pc, as the experiments of Charlaix et al. (1988a) indicated, the porousmedium must be extremely close to pc if the effect of the dead-end pores is tobe seen, so that dispersion along the backbone has practical importance. Fordispersion in the backbone, we have DL/De � PeM, and DT/De � PeM (i.e.,mechanical dispersion) since the logarithmic correction indicated by Eq. (11.65)is neglected in scaling analyses (as ln x grows with x slower than any power ofx and, therefore, it is equivalent to a zero critical exponent). Therefore, DL �DT � �pVB � � 1�θB

p � (p � pc)e��bb�ν .Using the numerical values of e, �bb and ν given in Table 3.2, we obtain DL �DT � (p � pc)�0.56 in 2D, and DL � DT � (p � pc)0.04. Thus, DL and DT diverge

in 2D, but vanish very weakly in 3D. The difference demonstrates the strongeffect of the backbone structure on dispersion processes. As described in Chap-ter 3, the backbone is approximated by nodes, links, and blobs. Links are thebonds or pores that connect the blobs and the remaining multiply-connectedbonds aggregate together in the blobs. The blobs are very dense in 2D, pro-viding a wide variety of paths for the solute particles with broad FPTDs. As aresult, DL and DT both diverge as pc is approached. On the other hand, theblobs are not dense in 3D, which means that the FPTDs are not broad enoughto give rise to divergent DL and DT. In a hypothetical porous medium similar toa Bethe lattice (for which e D 3, �bb D 1 and ν D 1/2), DL � DT � (p � pc)1/2,indicating the strong effect of the closed loops of pores (which are absent in theBethe lattices) on DL and DT.For L �p, we replace �p with L and, thus,

DL � L1�θB (11.88)

with a similar power law for DT. As all the length scales of the system mustbe proportional to each other (and to L), Eq. (11.88) may be rewritten as DL �h∆x2i(1�θB)/2. Using a fundamental property of random-walk processes, DL �dh∆x2i/d t, we obtain dh∆x2i/d t � h∆x2i(1�θB)/2, which, after integration,yields

h∆x2i � t2

1CθB , (11.89)

378 11 Dispersion in Flow through Porous Media

with similar equations for hy 2i and hz2i. Equation (11.89) implies that

DL � t1�θB1CθB , (11.90)

and that

D lw D D t

w D 1 C θB . (11.91)

Equation (11.89) implies that in 2D, h∆x2i � hy 2i � t1.26, and in 3D, h∆x2i �hy 2i � hz2i � t0.97. That is, dispersion is superdiffusive in 2D, so that the mean-square displacements of the solute particles grow with time faster than linearly,whereas it is fractal or subdiffusive in 3D, so that the mean-square displacementsgrow slower than linearly with time.On the other hand, according to Eq. (11.83) for L �p, we have VB � L�θB �hxi�θB , and since VB � dhxi/d t, we obtain, after integration,

VB � t�

θB1CθB , (11.92)

which is in sharp contrast with diffusive dispersion for which VB is constant.Finally, since αL D DL/VB, we obtain

αL � t1

1CθB , (11.93)

which means that for nondiffusive dispersion, the dispersivity depends on t.The time scale τco is given by

τco � (p � pc)�eC�bb�ν � � 1CθBp , (11.94)

and for L �p, we have τco � L1CθB . Equation (11.94) should be comparedwith Eq. (11.86). It is not too difficult to show that ht ni � (p � pc)�ν�n(e��bb) �� 1CnθB

p , and hti � � 1CθB . Thus, ht ni/htin � � 1�np . That is, from the scaling of

hti alone, one cannot obtain the scaling of ht ni for n > 1. For �p � L, we have

ht ni � L1CnθB , (11.95)

which should be compared with Eq. (11.87) if we replace θ by θB.3. Consider the holdup dispersion described earlier. We have DL � (Vc�p)2/De,

which is the same as Eq. (11.50) in which the length scale is �p and the molec-ular diffusivity Dm has been replaced by the effective diffusivity De in theporous medium, as suggested by de Gennes (1983b). Therefore, we can writeDL � � 2�2θcCθ

p � (p � pc)�2�p��C2e, with a similar result for DT and, thus,DL � (p � pc)�1.5 and DL � (p � pc)�0.17 in 2D and 3D, respectively. That is,as pc is approached, the dispersion coefficients diverge, which undoubtedly isdue to the contribution of the dead-end pores and the long times that the soluteparticles spend there. For L �p, we have

DL � L2�2θcCθ . (11.96)

11.13 Dispersion in Porous Media with Percolation Disorder 379

Using the same type of reasoning as before, we find that

h∆x2i � t2

2θc�θ , (11.97)

with similar scalings with the time t for hy 2i and hz2i. Thus,

DL � t2�2θcCθ

2θc�θ , (11.98)

and

D lw D D t

w D 2θc � θ , (11.99)

and, therefore, h∆x2i � t2.3 and h∆x2i � t1.1 in 2D and 3D, respectively.That is, dispersion is always superdiffusive when the holdup dispersion, that is,trapping of the solute particles in the stagnant regions of the pore space, isdominant. It is then straightforward to show that

Vc � t�θc

1Cθc , (11.100)

and, therefore,

αL � t2Cθ

(2θc�θ )(1Cθc) , (11.101)

which is more complex than Eq. (11.93). The time scale τco is given by

τco � (p � pc)�2eCµpC�p � � 2θc�θp , (11.102)

and τco � L2θc�θ for L �p. Therefore, we obtain ht ni � (p�pc)�nν(θC2)Cν �� n(θpC2)�1

p . Thus, ht ni/htin � � n�1p and, therefore, the scaling of hti alone is

not enough for obtaining the scaling of the higher moments ht ni of the FPTDfor any n > 1. In the L �p regime, we have

ht ni � Ln(θC2)�1 . (11.103)

Aside from Eqs. (11.85)–(11.87), all the above results were derived by Sahimi(1987) and were confirmed by the pore network simulations of Sahimi and Im-dakm (1988) and Koplik et al. (1988b). The fact that ht ni/htin depends on n meansthat there is no unique time scale for characterizing non-Gaussian dispersion in porousmedia with percolation disorder. Koplik et al. (1988b) also proposed a generalscaling equation for the FPTD given by

Q(t) D 1td

Fs

�t

td,

tc

td

�, (11.104)

where td and tc are the diffusion and convective time scales, respectively, and td �� 2Cθ

p , as Eq. (11.86) indicates. td is also the largest time that the solute particles

380 11 Dispersion in Flow through Porous Media

spend in the dead-end pores since the length of the longest dead-end branches isof the order �p. The scaling function Fs has the following limiting behavior

Fs(x , y ) !(

F1(x ) as y ! 1 ,

y F2(x ) as y ! 0 .(11.105)

The limit of pure diffusion corresponds to y D tc/ td ! 1, whereas the convectivelimit corresponds to y ! 0. Numerical simulations supported Eq. (11.104).

Related pore network simulations were carried out by Andrade and co-workers(Lee et al., 1999; Makse et al., 2000; Andrade et al., 2000). They utilized pore net-works at the percolation threshold and computed several important properties, in-cluding the minimum traveling time tmin and the most probable time t� betweentwo points separated by a distance r, the length `min of the path corresponding tothe time tmin, and `� corresponding to t�. For 2D pore networks at the percolationthreshold, they found that tmin � r1.33, t� � r Dbb � r1.64, `min � r Dmin � r1.13,and `� � r Dop � r1.22. Here, Dbb is the fractal dimension of the backbone (theflow-carrying part of the network; see Chapter 3), Dmin is the fractal dimension ofthe shortest path between the two points, and Dop is the fractal dimension of theoptimal path between the two points.

11.13.2Experimental Measurements

The increase in the dispersivities and the dispersion coefficients near pc that isshown in Figure 11.8 was confirmed by the experiments of Charlaix et al. (1987b,1988a) and Hulin et al. (1988a), who studied dispersion in model porous mediaand measured DL. Charlaix et al. (1988a) constructed 2D hexagonal networks ofpores with diameters that were of the order of millimeters. They reported that asthe fraction of the open pores decreased, DL increased sharply, and that Eq. (11.46)seemed to explain the data. However, even when the dispersion coefficients weremeasured quite close to pc, the quadratic dependence of DL/Dm on Pe, Eq. (11.50),was not observed (although the fraction of the dead-end pores is quite large nearpc), presumably because the exchange time between the flowing fluids and thedead-end regions was so long that it could not be detected during the experiment.

Hulin et al. (1988a) measured DL in bidispersed sintered glass materials pre-pared from mixtures of two sizes of beads. They reported that when the porositywas decreased from 30 to 12% DL increased by a factor of 30. The results of thetwo studies also indicated that dispersion is more sensitive to large-scale inhomo-geneities of a porous medium than to its detailed local structure. Somewhat similarresults were obtained by Charlaix et al. (1987b).

Gist et al. (1990) studied dispersion in a variety of sandstones and carbonaterocks, and used the percolation model of Katz and Thompson (1986, 1987), de-scribed in Chapter 10, to quantify their results. Following Sahimi et al. (1982, 1983a,1986a,b), they argued that the fundamental quantity to be considered is the ratio�p/dg, where dg is the mean grain size. Since �p/d � (X A)�ν/�, and as ν/� ' 2

11.13 Dispersion in Porous Media with Percolation Disorder 381

(see Table 3.2), one can write �p/dg � (X A)�2. Because X A is roughly proportionalto the fluid saturation S, we obtain

�p

dg� S�2 . (11.106)

Gist et al. (1990) derived a relation between αr D αL/dg and �p/dg using the perco-lation model of Sahimi et al. Their final result is

αr ��

�p

dg

�2.2

� S�4.4 . (11.107)

Gist et al.’s data for sandstones, epoxies and carbonates supported the validity ofEq. (11.107) (see Figure 11.9), hence confirming the relevance of percolation-typedisorder to dispersion in a heterogeneous porous medium – even when the porousmedium does seem to contain such disorder – and the power laws derived above.

The last question to be addressed in this section is: What is the equation for theprobability density function P(r, t) or, equivalently, the (normalized) solute con-centration – if dispersion is non-Gaussian? For example, as the discussion of dis-persion in percolation networks near pc indicates, dispersion in a fractal porousmedium is not expected to be Gaussian and, thus, P(r, t) given by Eq. (11.3) is notexpected to be valid. In Chapter 10, we discussed the appropriate form of P(r, t) fordiffusion in fractal porous media. For non-Gaussian dispersion in a porous medi-um with a fractal dimension Df, Sahimi (1987) proposed the following equation

Figure 11.9 Dependence of the reduced lon-gitudinal dispersivity αL/dg on the porosityφ and the formation factor F in sandstones(squares), epoxies (triangles), and carbonates

(diamonds). Solid and dashed lines representthe best fit of the sandstones and carbonatesdata, respectively (after Gist et al., 1990).

382 11 Dispersion in Flow through Porous Media

for P(r, t), in the limit of long times

P(r, t) � t�ds exp

"�α1

jx � hxij

t1

Dw l

!νl

� α2

jy j

t1

Dwt

!νt

� α3

jzj

t1

Dwt

!νt#,

(11.108)

where α1, α2 and α3 are constant and, as we already showed, D lw D D t

w for mostcases. Here ds D Df/D l

w, νl D D lw(D l

w � 1)�1, and νt D D tw(D t

w � 1)�1. Equa-tion (11.108) reduces to Eq. (11.3) when Df D d and D l

w D D tw D 2. Pore net-

work simulations of Sahimi and Imdakm (1988) appeared to support the validityof Eq. (11.108), but no rigorous derivation of Eq. (11.108) is yet available, and thematter is still an open question.

11.14Dispersion in Field-Scale Porous Media

Dispersion in field-scale (FS) porous media has attracted considerable attention byhydrologists, petroleum, chemical and environmental engineers, and even politi-

cians over the past three decades. The main cause for the attention is the growingconcerns about pollution and water quality. Due to the intensifying exploitation ofgroundwater and the increase in solute concentrations in aquifers caused by salt-water intrusion, leaking repositories, and use of fertilizers, dispersion in the FSporous media has been a main topic of research. Moreover, dispersion in misci-ble displacement processes is an important phenomenon during oil recovery, anddepending on the magnitudes of the dispersion coefficients DL and DT and otherphysical parameters of the process, dispersion may help or hurt the displacementprocesses and their efficiency.

As discussed in Chapter 1, in studying transport in heterogeneous porous media,one should define precisely what is meant by heterogeneous. Chapter 1 describedfour important scales of heterogeneities, that is, microscopic, macroscopic, megas-copic, and gigascopic scales. Bhattacharya and Gupta (1983) described a variety oflength scales, ranging from the kinetic and Taylorian to the Darcy scales, while Da-gan (1986) considered length scales, ranging from pore to laboratory to formationto regional levels. Cushman (1984) provided a brief review of the general problemof the development of N-scale transport equations. A complete treatment of theproblem at all the relevant length scales is still an open question.

We remind the reader that dispersion in a FS porous medium is purely mechan-ical, arising as a result of large-scale spatial variations of the permeability of themedium and the resulting random velocity field. Thus, Eqs. (11.48) and (11.49) aregenerally expected to hold. The length scale used to define the Péclet number Pein Eqs. (11.48) and (11.49) may be the permeability correlation length �k. In thissense, dispersion in a FS porous medium might seem to be somewhat simpler thanwhat is studied so far. One of the main problems to be solved for dispersion in aFS porous medium is the relation between the implied prefactors in Eqs. (11.48)

11.14 Dispersion in Field-Scale Porous Media 383

and (11.49) and the distribution of large-scale heterogeneities of the medium. Inaddition, it turns out that the dispersion coefficients in the FS porous media maydepend on the length scale of the observations and, therefore, the precise scalingof DL and DT with the characteristics length(s) is also important.

In the early years of investigating dispersion in the FS porous media with large-scale permeability and porosity variations, a CD equation served as the startingpoint for analyzing the field data, while the analysis of the problem was based oncompletely deterministic methods. A considerable amount of data has, however,indicated unequivocally that DL and DT measured in a field are larger by severalorders of magnitude than those measured in a laboratory, and an entirely deter-ministic approach cannot provide a completely satisfactory explanation for suchdata. Moreover, field experiments of Sudicky and Cherry (1979), Pickens and Grisak(1981), Sudicky and Frind (1982), Sudicky et al. (1985), and Molz et al. (1983) (forreviews see, for example, Gelhar et al., 1992; Vanderborght and Vereecken, 2007)indicate that dispersion coefficients and dispersivities are often scale-dependent,and that the apparent dispersivities αL and αT seem to increase with the transittimes of the solute particles, similar to those in percolation networks for lengthscales L �p. Figure 11.10, adapted after Arya et al. (1988), demonstrates thisphenomenon very clearly. As the distance from the source increases, so does thedispersivity with no asymptotic limit in which αL is constantly being apparent.

Warren and Price (1961) seem to be the first to investigate dispersion in theFS porous media, taking into account the effect of the permeability distribution.They used a Monte Carlo method that will be described later in this chapter. De-

Figure 11.10 Field-scale longitudinal dispersivity as a function of the distance from the source(both are in meters). Solid line represents the best fit to about 75% of the data, while thedashed line represents Gaussian dispersion (after Arya et al., 1988).

384 11 Dispersion in Flow through Porous Media

spite their many shortcomings, deterministic approaches are still used by many.Although they provide valuable insights into the problem, deterministic approach-es fail to provide accurate quantitative predictions for dispersion coefficients andthe solute concentration profile. Typical of such approaches is the large-scale vol-ume averaging method of Plumb and Whitaker (1988a,b), which we now describebriefly.

11.14.1Large-Scale Volume Averaging

Plumb and Whitaker (1988a,b) (see also Tompson and Gray, 1986) considered atwo-scale problem and developed a large-scale averaging technique for determiningthe averaged transport equation. The starting point of their analysis was Eq. (11.52),which must be averaged over the regions in which the permeability and porosityvary spatially. To do this, one writes

φ D fφg C Qφ , (11.109)

hvi D fhvig C Qv , (11.110)

hCi D fhCig C QC , (11.111)

D� D fD�g C QD , (11.112)

where f�g denotes the large-scale quantity. By substituting Eqs. (11.109)–(11.112)into Eq. (11.52) and averaging the results over regions in which the permeabilityand porosity vary appreciably, Plumb and Whitaker (1988a,b) derived a large-scaleaveraged equation for the average solute concentration that contained such termsas rrrfhCig and rr@fhCig/@t, indicating that dispersion in the FS porous mediacannot, in general, be described by a CD equation.

As our discussion of dispersion in percolation systems in the non-Gaussianregime indicated, when dispersion cannot be described by a CD equation, one mustdeal with time- and scale-dependent dispersion coefficients. The work of Plumband Whitaker is valuable in that it clearly demonstrates the deviations of dispersionin the FS porous media from a conventional description by the CD equation. Themain difficulty with the approach of Plumb and Whitaker (1988a,b) is that exceptfor very simplified models of a pore space, the numerical solution of their averagedequation is difficult to obtain. The problem must first be solved at the local level inorder to use the solution as the starting point for determining the solution for thelarge-scale averaged equation. Moreover, as discussed earlier, the main contribu-tion to dispersion in a FS porous medium is made by large-scale variations of thepermeability and porosity of the medium and local phenomena, such as, diffusion,do not play an important role.

11.14 Dispersion in Field-Scale Porous Media 385

11.14.2Ensemble Averaging

Koch and Brady (1988) also studied dispersion in the FS porous media, throughuse of the ensemble-averaging technique described in Chapter 9 and earlier in thischapter. They were able to show that if the correlation length �k for the permeabil-ity fluctuations is finite, then dispersion is diffusive and follows a CD equation.If, however, �k is divergent, then superdiffusive dispersion occurs in which themean-square displacements of the solute particles grow with time faster than lin-early, qualitatively similar to superdiffusive dispersion near and at the percolationthreshold for length scales L �p. In this sense, �k plays a role similar to thepercolation correlation length �p. Moreover, Koch and Brady (1988) showed that inthe superdiffusive dispersion regime, the space-time evolution of the solute con-centration is universal and uniquely related to the covariance of the permeabilityfield. This is again similar to the superdiffusive dispersion in percolation systemsat length scales L �p.

11.14.3Stochastic Spectral Method

Stochastic spectral method has been popular with geologists and hydrologists,and has been used extensively. The main motivation for developing the methodis that the complex geohydrological structure of aquifers, the nonuniformity andunsteadiness of flow, and other influencing factors make dispersion in a FS porousmedium a very complex phenomenon. Field measurements of the dispersivitiesare often costly and time consuming. One needs to drill many observation wells tomonitor the spread of the solute, and the spreading itself is often so slow that a fewyears may be needed for completing the investigations. The level of uncertaintiesin all the operations and measurements is quite high and, therefore, stochasticmethods have been advocated so that the concepts of randomness, uncertainty anderrors can be introduced into the models and analyses.

Early analytical studies of this problem using stochastic concepts were carriedout by Mercado (1967) and Buyevich et al. (1969), but not all the ingredients wereknown at that time. Only since the early 1980s has there been a more compre-hensive analysis of the problem. The assumption of ergodicity is implicit in thestochastic approaches. That is, one assumes that dispersion of a solute in an en-semble of porous media with the given statistical properties mimics the situationin a real field that is a single realization of a FS porous medium with large-scalevariations of the permeability, porosity, and other properties. Such an assumptionis valid if the scale of the flow system is large compared with the correlation lengthof the system. Thus, if the permeability, porosity, or other properties are, for ex-ample, fractally distributed (see Chapter 5), that implies that the correlation lengthmay be as large as the linear size of the porous medium, and the stochastic spectralapproach may break down.

386 11 Dispersion in Flow through Porous Media

To give the reader some ideas about stochastic spectral model of dispersion ina FS porous medium, let us describe the work of Gelhar et al. (1979), Gelhar andAxness (1983), and Gelhar (1986), which is representative of this class of models(see also Dagan, 1987, and references therein). The starting point is a CD equationat the local level. Assuming, for example, a 2D porous medium (the extension to3D is described by Gelhar and Axness, 1983), one starts with

@C

@tC @

@x(v C ) D @

@x

�DL

@C

@x

�C @

@z

�DT

@C

@z

�, (11.113)

where it has been assumed that C, DL and DT, which are local properties, are ran-dom processes with

C(x , z, t) D Cm(x , t) C c(x , z, t) , (11.114)

v D Vm C u , (11.115)

DL D DLm C dL , (11.116)

DT D DTm C dT , (11.117)

where subscript m denotes a mean value, for example, Cm(x , t) D hC(x , z, t)i, withthe averaging being taken with respect to the vertical depth z. Here, c(x , z, t), dL

and dT represent the fluctuations with zero averages. If we substitute Eqs. (11.114)–(11.117) into Eq. (11.113) and take the average of both sides, we find that

@Cm

@tC @

@x(VmCm)C @

@xhuci D @

@x

�DLm

@Cm

@x

�C @

@x

�dL

@c

@x

�C @

@x

�dT

@c

@z

�,

(11.118)

where we have used the fact that the averages of the quantities are independentof z. Equation (11.118) is now subtracted from Eq. (11.113), and the coordinate� D x � Vm t is used to obtain

@C

@t1

C u@Cm

@�2

C @

@�(uc � huci)

3

D DTm@2c

@z24

C @

@�

�dL

@c

@�

�5

C DLm@2c

@�2

6

C @

@z

0@dT

@c

@z��dT

@c

@z

�7

1AC @

@�

0@dL

@c

@���dT

@c

@�

�8

1A . (11.119)

If the perturbation u is small, then the second-order terms (numbered 3, 7 and 8)can be neglected, and one obtains an approximate equation of the form

@c

@tC u

@Cm

@�D DTm

@2 c

@z2 C dL@2c

@�2 C DLm@2 c

@�2 , (11.120)

so that even at this level of approximation, one already has the additional termdL@2c/@�2.

11.14 Dispersion in Field-Scale Porous Media 387

Equation (11.120) is solved by assuming that the permeability is a statistically-homogeneous random process, and introducing the spectral representation of therandom variables (see Chapter 5). If the field permeability K is written as K DKm C k, where Km D hKi, and hki D 0, then

k D1Z

�1

e i ωz dZk (ω) , (11.121)

where ω is the wave number and Zk (ω) is a complex stochastic process with or-thogonal increments. The random processes u, c, dL, and dT also have similar spec-tral representations. Based on the experimental results of Harleman and Rumer(1963) that indicated that αL seems to be proportional to

pK (this may be in-

tuitively clear, asp

K is a length scale, as is αL), it is not difficult to show thatdL/DLm D 3k/(2Km). If we introduce a spectral representation for c,

c D1Z

�1

e i ωz dZc(ω) , (11.122)

then Eq. (11.120) becomes

@y

@tC aTVmω2 y � DLm

@2 y

@�2 D Vs(� , t) (11.123)

with y D dZc, and

Vs D VmdZk

kmG , G D �@Cm

@�C 3

2aL

@2Cm

@�2,

and aL D DLm/Vm, and aT D DTm/Vm. The cross spectrum of u and c, Suc(ω),and the spectrum of k, Sk k(ω), are represented by (see, for example, Lumley andPanofsky, 1974)

Suc(ω) D E(dZu dZ�c ) , (11.124)

Sk k(ω) D E(dZk dZ�k ) , (11.125)

where E denotes the expected value, and � denotes the complex conjugate. SincedZu/Vm D dZk / k, we obtain

Suc (ω) D Sk k(ω)2

k2m

(Vm G

1 � e�� t

aT ω2 ��

@G

@t� DLm

@2 G

@�2

�"1 � (1 C � t)e�� t

a2T ω4

#),

(11.126)

where � D aTVmω2. Similarly, huci is given by

huci D AVmG � B

�@G

@t� DLm

@2G

@�2

�, (11.127)

388 11 Dispersion in Flow through Porous Media

and

A D1Z

�1

Sk k(ω)k2

m

1 � e�� t

aTω2dω , B D

1Z�1

Sk k(ω)k2

m

1 � e�� t

a2T ω4

dω .

Using all the results, Eq. (11.119) is rewritten as

@Cm

@tD (A C aL)Vm

@2 Cm

@�2� B

@3Cm

@�2@t� 3aL AVm

@3 Cm

@�3C 3aL B

@4Cm

@�3@t

C�

aL BVm C 94

a2L AVm

�@4Cm

@�4 C � � � (11.128)

Equation (11.128) indicates that the average concentration Cm does not follow a CDequation, a result similar to that obtained by Plumb and Whitaker (1988b). The restof the analysis is clear: a spectrum Sk k(ω) is assumed and the quantities A and B

are calculated. Having determined A and B, one proceeds to analyze Eq. (11.128).Gelhar and Axness (1983) extended the above analysis to 3D heterogeneous mediaand showed that the dispersion coefficients depend linearly on the average velocity,which is not surprising. The results of the stochastic model have been used withsome success for predicting the breakthrough of solutes at field scales (see, forexample, Kapoor and Gelhar, 1994, and references therein).

The analysis so far assumes that all the randomness in a porous medium is dueto the variations of the permeability. An alternative approach relies on a stochasticrepresentation of the velocity (Tang et al., 1982) and develops an ensemble-averagedequation containing coupling between the velocity and the concentration fluctua-tions (which is similar to that found by Gelhar et al., 1979) that leads to a coefficientin the stochastic transport equation similar to DL in a conventional CD equation.This term, the ensemble dispersion coefficient, depends upon the variance-covariancestructure of the velocity field. If the neighboring velocities are uncorrelated, the en-semble dispersion coefficients increase as a function of the travel distance from thesource. If the covariance of the velocity field is an exponentially-decaying function,then the ensemble dispersion coefficients reach a constant value. Thus, this type ofanalysis clearly demonstrates the effect of the correlations in the stochastic analysisof dispersion.

The stochastic models of the type that we described here are useful if an adequaterepresentation of the velocity or permeability fields is available. Such representa-tions were described in Chapter 5. The interested reader should consult Dagan(1986, 1987) and Haldorsen and Damsleth (1990) for more details and referenceson stochastic modeling of transport in the FS porous media.

11.14.4Continuous-Time Random Walk Approach

In addition to the various theoretical developments described in the last three sub-sections, there is also considerable experimental evidence that the CD equation

11.14 Dispersion in Field-Scale Porous Media 389

cannot describe dispersion in flow through a large class of heterogeneous porousmedia. In fact, over 50 years ago, Aronofsky and Heller (1957) had already reportedon the deviations of their data for dispersion from the description by a CD equationand noted that the deviations were systematic. Scheidegger (1959) also reportedon some careful experiments on dispersion in columns. He noted that the break-through solute concentration profile (when the solute first exits the column), whenfitted to the CD equation, significantly deviates from his data and commented that,“The deviations are systematic which appears to point toward an additional, hith-erto unknown, effect.” Silliman and Simpson (1987) reported convincing data fordispersion in laboratory experiments that indicated scale-dependence of the disper-sivity, the hallmark of non-Fickian dispersion.

Such deviations have motivated the development of new models for describingdispersion in flow through heterogeneous porous media. One such model is basedon continuous-time random walks (CTRW), an idea whose origin goes back to animportant paper of Montroll6) and Weiss (1965), and further developed by Montrolland Scher (1973) and Scher and Montroll (1975). The application of the CTRWsto describing dispersion in flow through heterogeneous porous media has beendeveloped by Berkowitz, Cortis, Scher and their co-workers. In this section we de-scribe the basic ideas of the model by closely following the comprehensive reviewof Berkowitz et al. (2006); see also Berkowitz et al. (2008)

The starting point of the model is the master equation (ME), already used in Chap-ter 10 for describing diffusion and conduction in porous media. Recall that the MEis given by

@C(s, t)@t

DX

s0

W(s, s0)C(s0, t) �X

s0

W(s0, s)C(s, t) , (11.129)

where, as in Chapter 10, C(s, t) is the (solute) concentration, or the probability thata solute particle is at s at time t (if the concentration is suitably normalized), andW(s, s0) is the transition rate, or the probability of moving from s to s0. The transi-tion rates describe the effect of the flow velocity on the motion of the solute. Notetwo differences with the formulation presented in Chapter 10. One is that, unlikethe analysis in Chapter 10, the transition rates in Eq. (11.129) are not necessarilysymmetric. The second difference is that the points s and s0 are not necessarily thesites of a lattice. Note also that the ME does not separate the effects of convectionand diffusion into distinct parts.

The crucial aspect of the formulation is specification of the transition rates,which entails detailed knowledge of the heterogeneities of porous media. As thediscussions throughout this book should have made it clear thus far, below a lengthscale `c, the heterogeneities are unresolved. Thus, over such scales, one must resortto a statistical description of the set fW(s, s0)g through distribution functions. Todo so, one resorts to the ensemble-averaged version of Eq. (11.129) which is given

6) Elliott Waters Montroll (1916–1983), Albert Einstein Professor of Physics at the University ofRochester, was a mathematician who made very significant contributions to random walks, phasetransitions and traffic flow.

390 11 Dispersion in Flow through Porous Media

by

@Cm(s, t)@t

DX

s0

tZ0

'(s � s0, t � t0)Cm(s0, t0)d t0

�X

s0

tZ0

'(s0 � s, t � t0)Cm(s, t0)d t0 , (11.130)

where Cm(s, t) is the ensemble-averaged concentration, and the memory func-tion '(s, t) will be specified shortly. Note that Eq. (11.130) is fully compatiblewith Eq. (10.16), except that symmetry of the transition rates was assumed in theformulation of Chapter 10. As mentioned in Chapter 10, Eq. (11.130) is knownas the generalized master equation (GME). Moreover, as discussed there, goingfrom Eq. (11.129) to Eq. (11.130) forces the governing equation for the ensemble-averaged concentration to be nonlocal and contain memory. Use of nonlocal trans-port equations with memory goes back to the work of Zwanzig (1960). Note thatalthough the function ' depends on time, it is still stationary in space, only de-pending on the difference s � s0.

A CTRW is described by

P(s, t) DX

s0

tZ0

ψ(s � s0, t � t0)P(s0, t0)d t0 , (11.131)

where P(s, t) is the probability per time that a random walker has just arrived atposition s and time t, and ψ(s, t) is the probability per time for a displacement swith a difference of arrival times of t. The initial condition for P(s, t) is that thewalker is at the origin at time zero. Equation (11.131), which was first proposedby Montroll and Weiss (1965), is Markovian in space (no step of the random walkdepends on the previous steps), but not in time (the equation contains memory).Its first application to a physical system was suggested by Scher and Lax (1973a).As pointed out in Chapter 10, Kenkre et al. (1973) and Shlesinger (1974) showedthat the GME is completely equivalent to a CTRW. The correspondence betweenEqs. (11.130) and (11.131) is given by

Cm(s, t) DtZ

0

Ψ (t � t0)P(s, t0)d t0 . (11.132)

Here, Ψ (t) is given by

Ψ (t) D 1 �tZ

0

ψ(t0)d t0 , (11.133)

and is the probability for the walker to remain – to wait – at a point s,

ψ(t) DX

s

ψ(s, t) , (11.134)

11.14 Dispersion in Field-Scale Porous Media 391

and the function ' is related to the ensemble-averaged concentration in the Laplacetransform space by

Q'(s, λ) D λ Qψ(s, λ)1 � Qψ(λ)

, (11.135)

where λ denotes the Laplace transform variable conjugate to t.Equations (11.131)–(11.133) are solved by using Fourier transform (Scher and

Lax, 1973a). The solution is

C(ω, λ) D 1 � Qψ(λ)λ

11 � Λ(ω, λ)

, (11.136)

where C(ω, λ) and Λ(ω, λ) are, respectively, the Fourier transforms of QCm(s, λ) andQψ(s, λ). Equation (11.136) is valid for a lattice of N sites with periodic boundary

conditions, in which the sites’ positions are given by s D P3j D1 s j a j with s j D

1, 2, . . . , N , and a j being the lattice constants. The components ω i of ω are givenby ω i D 2π mi/N with �(N � 1)/2 mi (N � 1)/2 for odd N. If N is very large,the lattice constant can be arbitrarily small, approaching the continuum limit.

Another important quantity, which has been already mentioned and its use de-scribed, is the first passage-time distribution Q(s, t), the probability density of awalker arriving at s at time t for the first time. Introducing the FPTD into the CTRWformulation is necessitated by the fact that Eq. (11.136) gives the solution for a sys-tem with periodic boundary conditions, whereas such conditions do not necessari-ly exist in actual experiments. In addition, as mentioned earlier, the breakthroughcurve is equivalent to Q(s, t). Thus, we write Eq. (11.4) in a slightly more generalform and in the notation adopted for the CTRW:

P(s, t) D δ s,0δ(t � 0C) CtZ

0

Q(s, t0)P(0, t � t0)d t0 , (11.137)

where the first term on the right side indicates the starting point of the randomwalker. The solution of Eq. (11.137) in the Laplace transform yields

QQ(s, λ) DQP(s, λ) � δ s,0

QP(0, λ). (11.138)

The breakthrough curve fBT at, say s1 D L, is obtained by summing over all thedirections s2 and s3:

fBT DX

s2

Xs3

Q(s1 D L, s2, s3) , (11.139)

which is similar to Eq. (11.72). The inverse Laplace and/or Fourier transforms ofall the solutions presented so far must be computed numerically, which can bedone in a number of ways. Cortis and Berkowitz (2005) developed a collection ofeasy-to-use MATLAB programs and functions to calculate the temporal and spatialfunction numerically.

392 11 Dispersion in Flow through Porous Media

11.14.4.1 Relation between the Transition Rates and the Waiting-Time DistributionAs far as the application of the CTRW to describing solute dispersion in hetero-geneous porous media is concerned, identifying the distribution ψ(s, t) and itsrelation with the transition rates W(s, s0) are most crucial. Naturally, ψ(s, t) shouldreflect the heterogeneity of porous media. In the CTRW formulation, it is assumedthat the heterogeneities are reflected in the distribution of the flow velocities that, inturn, represent the heterogeneity in the spatial distribution of the permeabilities.One first writes a ME as a random walk equation in order to obtain a transitionlength and time distribution

ψs,s0 (t) D W(s0, s) exp

"�tX

s00

W(s00, s)

#. (11.140)

Then, summing over all s0, one obtains

ψs(t) DX

s0

W(s0, s) exp

"�tX

s0

W(s0, t)

#D � dQs

d t, (11.141)

with

Qs � exp

"�tX

s0

W(s0, s)

#, (11.142)

ψ(t) D � d

d t[[Qs]] , (11.143)

where [[�]] denotes an ensemble average. Calculating the ensemble averageEq. (11.143) in its most general form is, however, difficult.

Hence, consider, first, the case of an ordered system with W(s0, s) D W(s0 �s), and Ws D P

s0 W(s0, s) D constant. Then, ψ(t) D Ws exp(�Ws t), Qψ(λ) DWs/(Ws C λ), and

Q'(s, λ) D W(s) , (11.144)

which is similar to Eq. (10.16). Observe that, as the discussions in Chapter 10 in-dicated, even if the porous medium is not ordered, but an effective (and uniforms)transition rate We (representing the porous medium) can be determined (Chap-ter 10 described how We is determined by an effective-medium approximation),then Eq. (11.144) is still valid. In the more general case of a disordered medium,one can maintain a fixed value W(s0 � s), but have a random spatial distribution ofthe sites s. Thus, if the site density is Ns (N sites per unit volume), one can write(Scher and Lax, 1973b)

[[Qs]] D exp �Ns

Zf1 � exp[�W(s)t]gd3 s

�. (11.145)

A standard transition rate is given by

W(s0 � s) D Wm exp�

�js0 � sjr0

�, (11.146)

11.14 Dispersion in Field-Scale Porous Media 393

with r0 and Wm being the range and maximum value of the transition rates. Equa-tion (11.144) can then be evaluated, (Cortis et al., 2004a)

ψ(τ)Wm

D � 3F3

�1, 1, 1,2, 2, 2,

I �τ�

exp(�� τ4F4) . (11.147)

τ � Wm t, � � 4πNsr30 , and p Fq is the generalized hypergeometric function7)

(Abramowitz and Stegun, 1970). In Eq. (11.147), the arguments of 4F4 in the expo-nential term is the same as that of 3F3 given above. A plot of ψ(τ)/ Wm versus τindicates that for low values of τ, ψ(τ) decreases with τ slowly, as in a power law.Thus, for such values of τ, one has

ψ(τ) � τ�1�α , (11.148)

which has been used frequently in the application of the CTRW.The range of the transition rates that are given by Eq. (11.146) depends on the

interplay between the spatial extent r0 and the average separation distance rN be-tween the sites, where r�3

N � 4πNs/3. Note that � D 3(r0/rN )3 for r0 rN . Forevery � , there exists a τc such that α > 2 for τ > τc, and dispersion evolves to-ward the Gaussian (Fickian) regime. As such, α and its value may be viewed asdescribing the asymptotic behavior of ψ(t) over a time range that corresponds tothe duration of the measurements or observations. Scher and Lax (1973b) gave thefollowing expression for the large τ limit:

ψ(τ)Wm

D �(ln a1τ)2 C a2

C τ�1 exp

� 1

3�(ln a1τ)3 C 3a2 ln(a1τ) C 2a3

�, (11.149)

with a1, a2, and a3 being three constants.

11.14.4.2 Continuum Limit of the CTRWEngineers are most familiar with the continuum mechanics and partial differentialequations, the best known example of which are the Navier–Stokes and the CDequations. Thus, it is of importance to recognize that the CTRW may be extended tothe continuum limit. If the moments of ψ(s, t) exist, then one can expand Cm(s0, y )as a Taylor expansion over the finite range of the transition rates:

Cm(s0, t) � Cm(s, t) C (s0 � s) � r Cm(s, t) C 12

(s0 � s)(s0 � s) W rr Cm(s, t) ,

(11.150)

7) The generalized hypergeometric functionp Fq is a hypergeometric series,

P1kD0 ck x k , for

which

ckC1

ck

D (k C a1)(k C a2) � � � (k C a p )(k C b1)(k C b2) � � � (k C bq )(k C 1)

.

Then, the function is given by

p Fq

�a1, a2, � � � , a p ,b1, b2, � � � , bq , I �x

D1X

kD0

(a1)k (a2)k � � � (a p )k

(b1)k (b2)k � � � (bq )k

x k

k!

with (a)k D Γ (ak )/Γ (a).

394 11 Dispersion in Flow through Porous Media

where : indicates a tensor product. Inserting Eq. (11.150) into Eq. (11.130) yields(Berkowitz et al., 2002; Dentz et al., 2004) the following equation in the Laplacetransform space

λ QCm(s, λ) � Cm(s, 0) D �v�(λ) � r QCm(s, λ) C D�(λ) W rr QCm(s, λ) , (11.151)

where the convective part of the equation is identified with an effective flow velocity,

v�(λ) �Z

s Q'(s, λ)d d s , (11.152)

whereas the effective dispersion (diffusion) tensor is given by

D�(λ) �Z

12

ss Q'(s, t)d d s . (11.153)

The fact that both v�(λ) and D�(λ) depend on λ (and, hence, the time t) is a man-ifestation of the nonlocality of the governing equation (which was also empha-sized in Chapter 10). An expansion similar to Eq. (11.150) for a single realization

(not ensemble-averaged), together with a Taylor expansion of the transition ratesW(s, s0), yields

@C(s, t)@t

C v(s) � r C(s, t) D r � r [D(s)C(s, t)] , (11.154)

which represents a slight generalization of the CD equation.In some cases, one can write ψ(s, t) D p (s)ψ(t), where p (s) and ψ(t) are the

probability distributions for the length of the jump of the random walker and thewaiting time before the jump is made, respectively. The decoupling is justified ifthe flow velocity correlates very weakly (or not at all) with jsj. In many situations,that is indeed the case (see Berkowitz et al., 2006 for more discussions). In this case(Berkowitz et al., 2002; Cortis et al., 2004b; Dentz et al., 2004),

v�(λ) D QM (λ)vψ , (11.155)

D�(λ) D QM (λ)Dψ , (11.156)

with a memory function

QM � tcλQψ(λ)

1 � Qψ(λ), (11.157)

and

vψ D 1tc

Zs p (s)d d s , (11.158)

Dψ D 1tc

Z12

ssp (s)d d s , (11.159)

with tc being a characteristic time. One may also write

Dψ D aψ jvψ j , (11.160)

11.14 Dispersion in Field-Scale Porous Media 395

where the dispersivity-like quantity aψ is given by

aψ DR 1

2 ssp (s)d d sˇ̌Rs p (s)d d s

ˇ̌ . (11.161)

The working transport equation for the ensemble-averaged concentration is thengiven by

λ QCm(s, λ) � Cm(s, 0) D � QM(λ)vψ � r QCm(s, λ) � Dψ W rr QCm(s, λ)

.

(11.162)

Note that the transport velocity vψ is not the same as the average fluid velocity v,and similarly for the dispersion tensor Dψ . The mass flux J, defined through theusual relation, @Cm/@t D �r � J, is then given by

QJ(s, λ) � QM (λ)vψ QCm(s, λ) � Dψ � r QCm(s, λ)

. (11.163)

Berkowitz et al. (2006) presented some asymptotic solution of the governing trans-port equation for the ensemble-averaged solute concentration. We note that Bijeljicand Blunt (2006) showed that pore-network simulation of dispersion can produceresults that are consistent with the CTRW predictions.

11.14.4.3 Application to Laboratory ExperimentsBerkowitz et al. (2006) reviewed many applications of the CTRW approach to mod-eling dispersion in laboratory-scale porous media. Here, we consider two of suchapplications. One is to the classical experiments of Scheidegger (1959) mentionedat the beginning of this section, reported by Cortis and Berkowitz (2004). Schei-degger’s experiments were carried out in Berea sandstone cores 30 in long with adiameter of 2 in, and porosity of 0.204. The flux was 1.73 cm3/min. The cores werefirst fully saturated with the solute tracers and subsequently flushed with clear liq-uid. Schedegger measured the breakthrough curves. Figure 11.11 presents the fitof the data by both a CD equation and the CTRW formulation. Equation (11.148)was used and the most accurate fit was obtained with α D 1.59. As Figure 11.11indicates, the CTRW formulation provides an excellent fit of the data, whereas theCD equation systematically fails to capture the trends in the data.

The second application is to the data of Nielsen and Biggar (1962), who report-ed breakthrough curves for both saturated and unsaturated porous media. Thecolumns were filled with Aiken clay loam 0.23–0.5 mm aggregates in saturatedconditions. Two breakthrough curves for the saturated case were measured withimposed fluid velocity of 3.4 and 0.058 cm/h. Cortis and Berkowitz (2004) reana-lyzed the data and fitted them with the CTRW model. Figure 11.12 presents thefit of the data for one of the breakthrough curves using the CTRW formulation,and compares the results with those obtained with the CD equation. Once again,Eq. (11.148) was utilized, and the most accurate fit was obtained with α D 1.29.Clearly, the parameter α is not, and cannot be, universal and must be evaluated foreach porous medium.

396 11 Dispersion in Flow through Porous Media

Figure 11.11 The breakthrough data of Schei-degger (1959) in terms of the fluy j versus theinjected pore volume, and their fit with theconvective (advective)-diffusion equation and

the CTRW model. Parts (a) and (b) depict twodifferent plots of the same data (after Cortisand Berkowitz, 2004; courtesy of ProfessorBrian Berkowitz).

Berkowitz and Scher (2009) discussed more recent developments, including theuse of the truncated waiting time distribution,

ψ(t) � (t1 C t)�1�α exp�

� t

t2

�, (11.164)

where, in addition to α, t1 and t2 are other adjustable parameters that are evaluatedby fitting the breakthrough curves to the model. Modeling of dispersion in corre-lated velocity fields based on the CTRW formulation was studied by Berkowitz andScher (2010). Edery et al. (2009) extended the CTRW formulation to the case inwhich, in addition to the transport, a reaction of the type, A C B ! C , also occursin porous media.

11.14.4.4 Application to Field-Scale ExperimentsThe CTRW formulation has also been used to model the data for dispersion of so-lute in a field-scale (FS) porous medium. Berkowitz and Scher (1998) employedthe model in order to explain the data for solute transport in a heterogeneousalluvial aquifer at the Columbus Air Force Base (in Mississippi). In the experi-ments (which were costly and time consuming), a pulse of bromide was injected

11.14 Dispersion in Field-Scale Porous Media 397

Figure 11.12 The breakthrough data ofNielsen and Biggar (1962) in terms of theflux j versus time t and their fit with the con-vective (advective)-diffusion equation and the

CTRW model. Parts (a) and (b) depict two dif-ferent plots of the same data (after Cortis andBerkowitz, 2004; courtesy of Professor BrianBerkowitz).

and traced over a 20 month period. The sampling was done using an extensive3D well network. The measured tracer plume turned to be completely asymmet-ric and, hence non-Gaussian. Therefore, the plume could not be described by aCD equation. Berkowitz and Scher utilized the CTRW model to fit the data. Theplume shapes obtained by the model displayed the same essential characteristics.In addition, the CTRW model could provide quantitative predictions for the time-dependence of the correlation between the mean and standard deviation of the fieldplumes and their shape.

In summary, the CTRW formulation is elegant and has yielded considerableinsights into non-Fickian transport in heterogeneous porous media. It can al-so provide quantitative information on many aspects of the transport processes.On the other hand, its parameters, and, in particular, the exponent α defined byEq. (11.148), are, at this point, purely phenomelogical and must be fitted to eachand every porous medium. It is not yet clear how to relate the parameters to thefundamental properties of a porous medium and, in particular, its porosity andpermeability distribution in the case of the FS porous media, and to the pore sizedistribution in the case of laboratory-scale porous media.

398 11 Dispersion in Flow through Porous Media

On the other hand, one may argue that the exponent α is a dynamical property ofa porous medium and, thus, it should be related to a directly-measurable dynamicalproperty of the medium. In that case, the leading candidates could be the FPTDand the residence time distribution of the solute particles in a porous medium.However, the problem remains open.

An alternative way is to try to determine an ensemble averaged distribution ofthe transition rates as defined by Eq. (11.129). If such a distribution can be deter-mined, it will directly lead to the function ψ(r , t) and '(r , t) and, hence, provideclues about the type of functions that one might expect. As the same time, theensemble-averaged distribution of the transition rates will enable one to directlyuse the ensemble-averaged master equation to model dispersion in flow throughporous media. Progress in this direction has been made; see Cortis et al. (2004a,b).

11.14.5Fractional Convective-Diffusion Equation

Another attempt to model non-Fickian dispersion in flow through heterogeneousporous media has been based on the fractional convective-diffusion (FCD) equa-tion. In Chapter 10, we already briefly described the fractional diffusion (FD) equa-tion for modeling of anomalous diffusion in porous media. In the same spirit, at-tempts have been made to extend the FD equation to model anomalous dispersion(see, for example, Meerschaert et al., 1999; Benson et al., 2000a,b; Pachepsky et al.,2000; Zhou and Selim, 2003; Deng et al., 2004; Sanchez et al., 2005; Zhang et al.,2005, 2007; Huang et al., 2006; Krepysheva et al., 2006). The main motivation fordeveloping this approach has been to develop a formulation that includes two fea-tures of non-Fickian dispersion, namely, scale-dependent dispersion coefficients,and the nonlocal nature of the process.

To explain the approach, we consider a 1D porous medium. The generalizationsto higher-dimensional porous media have also been developed (Meerschaert et al.,1999). The simplest FCD equation is given by

@C

@tC V

@C

@xD DL

@γ C

@x γ , (11.165)

where the derivative on the right side represents a fractional derivative. Such aderivative can be defined in several ways (see, for example, Samko et al., 1993).One is the Riemann–Liousville definition, given by

@γ C

@x γD 1

Γ (m � γ )@m

@x m

xZ0

C(y , t)(x � y )γ�mC1

d y , (11.166)

which is consistent with, but slightly more general than Eq. (10.60). Here, m isthe smallest integer larger than γ , and Γ represents the gamma function. Thus, if1 < γ 2, then m D 2. If γ < 1, then m D 1, in which case we recover Eq. (10.60).The limit γ D 2 reproduces the conventional CD equation. Equation (11.166) doeshave some troubling aspects, such as the fact that it yields a nonzero fractional

11.14 Dispersion in Field-Scale Porous Media 399

derivative for a constant value. An alternative definition is the so-called Caputofractional derivative, defined by

@γ C

@x γ D 1Γ (2 � γ )

xZ0

1(x � y )γ�1

@2 C

@y 2 d y , (11.167)

but Eq. (11.166) is often used because it is consistent with the mass conservationequation (see below). Equation (11.166) is also written in an equivalent form (withm D 2):

@γ C

@x γ D @

@x

24 1

Γ (2 � γ )@

@x

xZ0

C(y , t)(x � y )γ�1 d y

35 , (11.168)

from which it follows that

@γ C

@(�x )γD @

@x

24 1

Γ (2 � γ )@

@x

LZx

C(y , t)(y � x )γ�1

d y

35 , (11.169)

where L is the length of the 1D system. However, regardless of what definition ofa fractional derivative one utilizes, one aspect is clear: any FCD equation is nonlo-cal because the fractional derivative involves convolution integrals. The same is ofcourse true about the CTRW formulation described in the last section.

Equation (11.165) assumes that the dispersion coefficient DL does not dependon any length scale (i.e., DL does not depend on x). However, as discussed earlier,in many cases (depending on the degree of the heterogeneities), scale-dependenceof the dispersion coefficient in flow through heterogeneous porous media, even atlaboratory scale, has been convincingly established by experiments (Silliman andSimpson, 1987) as well as by pore-scale simulations (Zhang and Lv, 2007). Hence,to include the scale-dependence of DL in the FCD equation, Eq. (11.165) was gen-eralized, first to (Lu et al., 2002),

@C

@tC @

@x[V(x )C ] D @

@x

�DL(x )

@γ�1C

@x γ�1

�, (11.170)

and then to (Zhang et al., 2006),

@C

@tC @

@x[V(x )C ] D @γ�1

@x γ�1

�DL(x )

@C

@x

�. (11.171)

What is the form of the dispersive flux j if a FCD equation governs dispersion?Writing down an expression for j entails generalizing Fick’s first law. Paradisi et al.

(2001) suggested that

j D � 12

DL

�(1 C s)

@γ�1C

@x γ�1C (1 � s)

@γ�1 C

@(�x )γ�1

�, (11.172)

400 11 Dispersion in Flow through Porous Media

where 1 s 1 is the skewness. Equation (11.172) reduces to Fick’s first lawwhen γ D 2. As described earlier in this chapter, one may have a superdiffusivedispersion regime in which some of the solute particles are far ahead of the averagemotion. This limit corresponds to s D 1. In a FS porous medium, superdiffusivedispersion may occur if the spatial distribution of the permeabilities is very broadand contains long-range correlations.

If Eq. (11.172) is used in conjunction with the mass conservation equation,Eq. (11.170) is obtained. On the other hand, if we set s D 1 in Eq. (11.172) andreplace the first term on its right side by

� @γ�2

@x γ�2

�DL(x )

@C

@x

�,

we recover Eq. (11.171). Zhang et al. (2007) proposed yet another FCD equation,

@C

@tC @γ�1

@x γ�1 [V(x )C ] D @γ�1

@x γ�1

�DL(x )

@C

@x

�, (11.173)

which implies that the total flux J (convective plus dispersive) is given by

J D @γ�2

@x γ�2

�V(x )C � DL(x )

@C

@x

�. (11.174)

Thus, in this formulation, the total flux is also given by a nonlocal equation.In any case, use of the FCD equation in the modeling of dispersion in FS porous

media has had some success; see, for example, Zhang et al. (2005). On the otherhand, as the discussion makes it clear, there is still some debate as to what thegeneral form of the generalized Fick’s law is, and how the nonlocality should beincluded by the fractional derivatives. The problem is still open.

11.14.6The Critical Path Analysis

In Chapter 9, we described the critical path analysis (CPA) that has been used forderiving highly accurate expressions for the effective permeability and electricalconductivity of porous media. The CPA was extended by Hunt and Skinner (2008,2010) in order to derive expressions for the dispersion coefficient DL and the dis-tribution of the travel times of the solute particles.

Recall that in CPA, transport in a highly heterogeneous porous medium ismapped onto a percolation system at or very near the percolation threshold pc.Thus, Hunt and Skinner first defined the tortuosity τp of the percolation cluster,and in particular that of the backbone – the flow-carrying part of the cluster. Nearpc the length of the shortest path �min on the backbone follows a power law,

�min � jp � pcj�η , (11.175)

where η is a universal exponent. The tortuosity was then defined by Hunt andSkineer as (others have defined τp differently) τp D �min/�p, where �p is the corre-lation length of percolation. As described in Chapter 3, near pc, �p � jp � pcj�ν ,

11.14 Dispersion in Field-Scale Porous Media 401

where ν is the associated universal exponent. Hence,

τp � jp � pcjν�η . (11.176)

On the other hand, as mentioned earlier, Lee et al. (1999) studied the statistics ofthe travel times of solute particles in the percolation clusters near pc. They showedthat the time t� for traveling along the most probable path between two pointsseparated by a distance x scales as

t� � x Dbb , (11.177)

where Dbb is the fractal dimension of the backbone (see Chapter 3). To derive theirresults, Hunt and Skinner assumed a particular form for the pore size distributionfp(r) of the pore space, namely,

fp(r) / r�Dp�1 , r0 < r < rM , (11.178)

where Dp is the fractal dimension of the pore space that, as described in Chapters 4and 9, is not necessarily the same as the fractal dimension of the sample-spanningpercolation cluster. Here, r0 and rM are, respectively, the minimum and maximumpore sizes. The porosity of the porous medium is then given by

φ D 3 � Dp

r3�DpM

rMZr0

r3 fp(r)dr D 1 ��

r0

rM

�3�Dp

. (11.179)

The volume fraction Vp of the pores with sizes r0 < r < rM is given by Vp DR rMr r3 fp(r)dr . As described in Chapter 9, in the CPA, one identifies a critical pore

size rc as the smallest pore on the sample-spanning cluster that has the largestpossible value of the smallest pore. Such a pore controls the flow on the backboneby acting as a bottleneck. Thus, one may define a critical volume fraction Vc cor-responding to rc by Vc D R rM

rcr3 fp(r)dr . In laminar flow through a pore, the flow

conductance g is proportional to r4/ lp, where lp is the pore’s length that is usuallytaken to be lp � 1/r . Hence, g � r3 and, therefore,

p � pc / g1�

Dp3 � g

1�Dp3

c

g1�

Dp3

c

,

where gc is the critical conductance corresponding to rc. Hunt and Skinner thenused the cluster size distribution ns of the percolation clusters. As described inChapter 3, ns is the number of clusters of size s (normalized by the volume Ld

of the system, with d being the dimensionality of the system). Then, consider apath of N resistances along a quasi-1D path of the fluid on the backbone. One hasns d s D nN dN , where nN is the number of paths of N resistances. Recall fromChapter 3 that ns follows a universal scaling equation. More explicitly,

ns � s�τ exp�s2σ(p � pc)2 , (11.180)

402 11 Dispersion in Flow through Porous Media

where τ and σ are two percolation exponents (see Table 3.2) such that τ �1 D σνd.Thus, one obtains

nN D N�d�1 exp

8̂<:̂�

24�N l

L

� 1ν

ˇ̌̌ˇ̌̌1 �

�g

gc

�1�Dp3

ˇ̌̌ˇ̌̌35

29>=>; . (11.181)

The probability that a given porous medium of Euclidean length N l is spanned bya cluster that contains a controlling conductance g is proportional to the integralof N d nN over clusters of all sizes larger than or equal to the volume in question.Then, the conductance distribution f c(g) is given by

f c(g) / 2ν�1 E i

264ˇ̌̌ˇ̌̌1 �

�g

gc

�1�Dp3

ˇ̌̌ˇ̌̌2 � x

L

� 2ν

375 , (11.182)

where x is the linear extent of the porous medium, L now represents the linearsize of the representative elementary volume, and E i is the exponential integral,E i(y ) D R1

y z�1 exp(�z)dz. An accurate approximation to Eq. (11.182) is

f c(g) / ln

264� L

l C x

� 1ν

ˇ̌̌ˇ̌̌1 �

�g

gc

�1�Dp3

ˇ̌̌ˇ̌̌�1375 . (11.183)

Hunt and Skinner argued that Q(t)d t D g f c(g)dg since the fluid flux is pro-portional to g f c(g). Here, Q(t) is the distribution of the arrival times, or the first-passage time distribution already described. They then assumed that the flow pathis quasi-1D, but corrected by the tortuosity τp. The time that a solute particle takesto travel a pore is t / r A/q, where A is the cross-sectional area of the pore and q isthe volume flow rate. All the pores along the critical percolation path have the sameconductance (see Chapter 9), and q / r3 t0, where t0 is a pore time scale for flow.Thus, the passage time (without considering the correction due to the tortuosity) isgiven by

t(r) / t0

rMZr

r 03

q

fp(r)r�Dp

dr 0

D t0

24 rcZ

r

r 03

q

(r 0)�Dp�1

r�Dpdr 0 C

rMZrc

r 03

q

(r 0)�Dp�1

r�Dpdr 0

35 . (11.184)

The tortuosity makes the path longer. Equation (11.176) is rewritten as

τp � jVp � Vcjν�η D jVp � Vcjν�νDbb , (11.185)

where the relation η D νDbb was employed. Moreover, Eq. (11.177) also entersthe derivation, as it provides t with the explicit dependence on x. Thus, putting

11.15 Numerical Simulation 403

everything together and carrying out the integrations in Eq. (11.184), one finallyobtains

t D t0

� x

L

�Dbb 13 � Dp

gc

g

1(1 � Vc)η�ν

24 1

1 � Vc

�gc

g

�1�Dp3 � 1

35

�ˇ̌̌ˇ̌̌� g

gc

�1�Dp3 � 1

ˇ̌̌ˇ̌̌

1�Dbbν

� tg

� x

L

�Dbb. (11.186)

One may also determine the distribution P(x I t) of clusters of size of at least x

that are dominated by minimum conductances g at a fixed time t. Note that x rep-resents both the size of a cluster of connected pores as well as the distance. Then,the dispersion coefficient DL is simply DL D (hx2i � hxi2)/(2t) (for an effectively1D system). From Eq. (11.177), we obtain

x D L

�t

tg

� 1Dbb

, (11.187)

where tg is defined by Eq. (11.186). Using Eq. (11.183), one obtains

f c(g) / ln

8̂̂̂<̂ˆ̂̂̂:

2664 L

l C L�

ttg

� 1Dbb

3775

1ν ˇ̌̌ˇ̌1 �

�g

gc

�1�Dpˇ̌̌ˇ̌�1

9>>>>=>>>>;

. (11.188)

The distribution P(x I t) is then obtained from P(x I t)dx D f c(g)dg, where f c(g)is given by Eq. (11.188), and x and t are related through Eq. (11.187), with tg givenby Eq. (11.186).

A comparison of the CPA predictions with the experimental data of Nielsen andBiggar (1962), as analyzed by Cortis and Berkowitz (2004), indicated that the pre-dictions are in rough agreement with the data. Thus, clearly, the CPA does have thepotential of being able to provide accurate predictions for dispersion in heteroge-neous porous media, although it must still be refined.

11.15Numerical Simulation

Section 11.10 described various models and computer simulation methods for sim-ulating dispersion in laboratory-scale porous media. Before we begin to describenumerical simulation of dispersion in larger-scale, and in particular field-scale (FS)porous media, we should mention that the lattice Boltzmann (LB) method, whichwas described in Chapter 9 for simulating fluid flow in porous media, has also beenextended for simulating dispersion in porous media. Hence, let us first describe themethod.

404 11 Dispersion in Flow through Porous Media

11.15.1Lattice-Boltzmann Method

Flekkøy (1993) extended the standard LB model based on the Bhatnagar–Gross–Krook (BGK) (Bhatnagar et al., 1954) collision rules (see Chapter 9) in order tosimulate diffusion of one component, say B, in another component, say A. Flekkøyet al. (1995) extended the model further in order to simulate dispersion in a flowingfluid. Maier et al. (1998, 2000) utilized the LB method to simulate dispersion inpacked beds, while Jimenez-Hornero et al. (2005) used the LB approach to modeldispersion in soil.

We assume that component B is present only in trace amounts so that the A–Band B–A collisions can be neglected. Component A has the same equilibrium dis-tribution that was described in Chapter 9, but component B – the solute – evolvestowards a new equilibrium as expressed by its own equilibrium distribution func-tion. Since B is present only in small amounts, its equilibrium distribution functioncontains only up to the first-order term in fluid velocity:

neqBi (x ) D wi�B(x)(c2

s C vi � vA) , (11.189)

where wi is the weight, �B is the density of B, cs is the speed of sound, vA is thevelocity vector of A, and vi is the microscopic velocities. The weights depend onthe structure of the grid used in the simulations. For example, with a square gridwith diagonal connections (so that each grid point is connected to eight others), theweights are wi D 1/3 for i D 1�4 (the four main directions leading to a grid pointalong the Cartesian axes), wi D 1/12 for i D 5�8 (for diagonal links), and w0 D 4/9for a particle at rest. The density B is computed by the same equation that we usefor component A, Eq. (9.141). Note that because B is present in trace amounts, itsvelocity is assumed to be the same as that of A, vA. The molecular diffusivity Dm isgiven by Flekkøy (1993)

Dm D c2s

�τD � 1

2

�, (11.190)

where τD is a relaxation time for diffusion. The dispersion coefficient DL is stillcomputed from the mean-square displacements of the B particles, namely, DL D(hx2i � hxi2)/(2t), where the position of the B particles is evolved according to theLB simulation. The choice of the time step ∆ t is important, as it must minimize thenumber of displacements of the particles, but also have enough accuracy. Supposethat l is the distance between two neighboring nodes. Then, Maier et al. (2000)suggested that ∆ t must be selected such that

vmax ∆ t C 2p

Dm ∆ t 12

l , (11.191)

where vmax is the maximum velocity.The LB method as described only yields one dispersion coefficient, DL. However,

as we have emphasized throughout this chapter, the solute spreads anisotropically,

11.15 Numerical Simulation 405

with the anisotropy being characterized by the dispersion coefficients DL and DT.Therefore, it is essential to develop an “anisotropic” LB (ALB) model that can alsoyield DT. Zhang et al. (2002) and Ginzburg (2005) developed such ALB methods.For example, in their 2D simulations using a LB model with four velocities in ninedirections (see Chapter 9), Zhang et al. (2002) introduced nine relaxation timesτ i , but with the constraint that τ i D τ iC4 along with τ0. Then, the dispersioncoefficients are given by

Dx x D ∆x2

18∆ t(4µ1 C µ2 C µ3 � 3) , (11.192)

Dy y D ∆y 2

18∆ t(4µ4 C µ2 C µ3 � 3) , (11.193)

Dx y D ∆x∆y

18∆ t(µ2 � µ3) , (11.194)

where, Dx x D DL, Dy y D DT, Dx y D Dy x is the off-diagonal dispersion coefficientin the dispersion coefficient tensor, and µ1 D τ1/∆ t, µ2 D τ2/∆ t, µ3 D τ4/∆ t,and µ4 D τ3/∆ t. The relaxation parameters are selected such that they are closeto each other, but not too close to 0.5, in order to ensure stable solutions. If thegoal of the study is to determine the evolution with the time and length scale of thesolute concentration, the dispersion coefficients are set, and Eqs. (11.192)–(11.194)are used to estimate the relaxation parameters. Since there are three dispersioncoefficients but four relaxation parameters, one of the parameters must be set in-dependent of the dispersion coefficient.

Thus, the method cannot yield information on the dispersion coefficients, butmay be used for studying the evolution of the solute concentration. Ginzburg(2005)’s method is more rigorous than that of Zhang et al., but it is also muchmore involved.

11.15.2Particle-Tracking Method

The LB method is not efficient enough for simulating dispersion in the FS porousmedia. In addition, a FS porous medium is characterized by spatial distributionsof the permeability and porosity, rather than a pore size distribution. In its presentform, the LB model cannot be extended to include the spatial distribution of thepermeabilities.

We already described in Chapter 5 models of the FS porous media that were de-veloped by Warren and Price (1961), Warren and Skiba (1964), and Heller (1972),and their improvement by Smith and Freeze (1979) and Smith and Schwartz(1980, 1981a,b), who incorporated short-range correlations between the neighbor-ing blocks. Smith and Schwartz used the model for studying dispersion in the FSporous media. They used an algorithm for the motion of the solute particles thatincluded both deterministic and random displacements. The algorithm is popular-ly known as the particle-tracking method and is a combination of a random walkand a deterministic component.

406 11 Dispersion in Flow through Porous Media

In the simulations of Smith and Schwartz, a solute particle is released in the flowfield. For each time step ∆ t, a fluid velocity at any point is calculated by linearlyinterpolating the four surrounding values (in a 2D system). The particle is thenmoved a distance that is fixed by the magnitude of the time step and the fluidvelocity. This step represents the deterministic portion of the displacement, whichis due to convection. Random relocation from the deterministic position, due todispersion, is accomplished first by moving the particle a distance dx in a directionthat coincides with the flow vector, and second a distance dy in a direction normalto it. The random displacements dx and dy are calculated from

dx D (0.5 � [R ])q

24D(l)L ∆ t , (11.195)

dy D (0.5 � [R ])q

24D(l)T ∆ t , (11.196)

where [R ] a random number uniformly distributed in (0, 1), and D(l)L and D

(l)T are

the local dispersion coefficients, that is, the dispersion coefficients on the scaleof the size of the grid blocks. Using the model, Smith and Schwartz investigat-ed many aspects of dispersion in heterogeneous porous media and showed thatstrong permeability heterogeneity gives rise to non-Gaussian dispersion, and thatwhen the permeability correlation length �k is of the order of the system length, aunique dispersion coefficient may not be possible to define, in agreement with theresults described earlier. The simulations by Smith and Schwartz represent someof the earliest computational evidence for non-Gaussian dispersion in heteroge-neous porous media.

The particle-tracking method is now the standard approach for simulating dis-persion in heterogeneous porous media.

11.15.3Fractal Models

In our discussions of the various variograms in Chapter 5, we mentioned the workof Hewett (1986) (and those of several others) who analyzed the porosity logs ofthe FS porous media and found that the porosity distributions often follow fractalstatistics. More precisely, the vertical porosity logs seem to follow the statistics ofthe fractional Guassian noise (FGN), while the lateral logs seemed to be describedby the fractional Brownian motion (FBM). Vertical porosity logs analyzed by Hewett(1986) produced values of the Hurst exponent H ' 0.7�0.8, indicating long-rangepositive correlations.

Arya et al. (1988) analyzed over 130 greatly-varying dispersivities, collected onlength scales up to 100 km. The data collected by them, which are shown in Fig-ure 11.10, exhibit large scatter, but their analysis indicated that about 75% of themfollow the following scaling law, that is,

αL � Lδ , (11.197)

11.15 Numerical Simulation 407

where L is the length scale of measurements, or the distance from the source wherethe solute had been injected into the solvent in the porous medium. Arya et al.

(1988) suggested that δ ' 0.75. Neuman (1990) presented a different analysis ofthese and other data, and proposed that there are in fact two distinct regimes. Oneis for L 100 m, for which δ ' 1.5, whereas the second regime is for L � 100 m,for which δ was found to be close to unity. Equation (11.197) is reminiscent ofdispersion in percolation networks described earlier in this chapter. As the discus-sions indicated, scale-dependence of αL implies that it is time dependent as well.Thus,

αL � tυ , (11.198)

which is again similar to the results for a percolation system at length scales L �p described earlier. A non-universal υ has been found to provide reasonably ac-curate fits of various field data, including those shown in Figure 11.10 with υ '0.5�0.6. Of course, scale- and time-dependence of αL implies the same for thefluid velocity V and DL.

In Hewett’s work, the dispersivity αL was implicitly assumed to followEq. (11.198) with

υ D 2H � 1 . (11.199)

Therefore, with H ' 0.75, one obtains αL � t0.5. A similar result was obtainedby Philip (1986) and Ababou and Gelhar (1990), and was also implicitly assumedby Arya et al. (1988). Philip’s work also predicted that at short times, αL � t, andthe constraint 0.5 < H < 1 was also proposed, consistent with Hewett’s analysisof the porosity logs. In a series of paper by the Chevron group, Eq. (11.198) wasused in the numerical simulation of flow phenomena in oil reservoirs (Mathewset al., 1989; Emanuel et al., 1989; Hewett and Behrens, 1990). The phenomenathat were studied included dispersion, miscible displacements (to be described inChapter 13), and a waterflood (see Chapter 14).

How are such numerical simulations with fractal distributions of the perme-ability and/or porosity carried out? Consider, for example, simulation of disper-sion. One first generates the permeability and porosity fields using the FBM andFGN distributions. Methods for synthetic generations of the FBM and FGN weredescribed in Chapter 5. The simulations are conditional (see Chapter 5) becausethe permeability and porosity fields must honor the actual data collected at certainplaces in the field. The flow field is calculated using Darcy’s law, which reducesthe problem to calculation of the pressure field. Often, a finite-difference approx-imation with a rectangular grid is used. The CD equation is then discretized andsolved with the resulting flow field. Thus, the implicit assumption is that at thescale of the grid blocks, the CD equation governs dispersion. The assumption isjustified, as each block is assumed to be homogeneous. In practice, the discretizedequations must be solved for a finely-structured grid so that the effect of the per-meability and porosity distributions and their long-range correlations are captured.Comparison of the simulation results with the field data indicates that such condi-tional simulations with fractal distributions of the permeability and porosity are far

408 11 Dispersion in Flow through Porous Media

more realistic than the conventional simulations without such distributions. Theyalso demonstrate most definitively the relevance of fractal statistics to modeling theFS porous media and transport processes in them.

11.15.4Long-Range Correlated Percolation Model

The percolation models that we have described and used so far to model and ex-plain various phenomena in porous media represent random percolation models.Such models provide us with a relatively simple theoretical foundation for the exis-tence of scale- and time-dependent dispersion coefficients and dispersivities in theFS porous media. Random percolation models, however, predict universal scalingof αL and DL with L and t, whereas, as discussed above, field measurements andobservations indicate nonuniversal scalings. We now describe a correlated percola-tion model based on a FBM distribution of the permeabilities, which we utilize toexplain the scale-dependence of the dispersivities and the dispersion coefficients ofthe FS porous media described above. For more details, see Sahimi (1994b).

To each bond of a network, we assign a number selected from a FBM, and inter-pret it as the effective permeability of a portion of the porous medium over whichit is homogeneous. The permeabilities are, therefore, infinitely correlated if theHurst exponent H > 0.5 (see Chapter 5), or anticorrelated if H < 0.5. In the un-correlated percolation, the network’s bonds are removed randomly. To preserve thecorrelations between the permeabilities, however, we remove those bonds that havebeen assigned the smallest permeabilities. The idea is that in a porous medium witha broad distribution of the permeabilities, a finite fraction of the medium’s zoneshave very small permeabilities and, therefore, their contribution to the overall flowand transport properties of the medium would be negligible. Note that since forH > 0.5, the correlations are positive, most bonds with large or small permeabili-ties are clustered together. As a result, removal of the low-permeability bonds doesnot generate a tortuous cluster in the system. Moreover, precisely for the samereason, the percolation cluster generated by such a model does not have manydead-end bonds and is close to its backbone. This assertion is confirmed by the nu-merical results described below. On the other hand, if we consider the percolationcluster for H < 0.5, it is more tortuous and chaotic because the permeabilities arenegatively correlated.

Several properties of the percolation model with long-range correlations were in-vestigated using large-scale simulations and finite-size scaling (see Chapter 3). Inparticular, pc, ν (the correlation length exponent), Oe D e/ν, Oµ D µp/ν, and the frac-tal dimensions Df and Dbb of the sample-spanning cluster and its backbone werecalculated. Here, e and µp are the critical exponents associated with the power-lawbehavior of the permeability and conductivity near the percolation threshold (seeChapter 3). The results indicated that for 1/2 < H < 1 – the range of interesthere – the percolation threshold pc decreases with increasing H, whereas the re-verse is true for 0 < H < 1/2. Moreover, ν and Df essentially retain their valuefor random percolation, except when H ' 1, where Df ! 2 (in 2D). It was also

11.15 Numerical Simulation 409

found that for H > 1/2, Dbb increases with H and that Dbb ! 2 as H ! 1, thatis, the cluster becomes compact. Moreover, Df ' Dbb, confirming the assertionthat for H > 1/2, the cluster and its backbone are similar. Table 11.1 presents theresults for 1/2 < H < 1, which, unlike random percolation for which the criticalexponents are universal, indicates a smooth dependence of the exponents on H.

The correlated percolation model provides a rational explanation for the field-scale data for dispersion in the FS porous media. As emphasized earlier, if the per-meabilities are distributed according to a FBM, then the pore space must containzones of very low permeabilities, the elimination of which gives rise to a correlatedpercolation cluster with a backbone very close to the sample-spanning cluster. Infact, the backbone becomes identical with the cluster for H ! 1 (see Table 11.1).Therefore, since the fraction of the dead-end pores or stagnant regions in the sys-tem is very small, molecular diffusion that transfers the solute into and out of suchregions plays no significant role. This means that dispersion is dominated by thestochastic velocity field imposed on the medium by the permeability distributionand, consistent with the field data, DL depends on the average flow velocity V as

DL � � V , (11.200)

where � is some appropriate length scale. Under such conditions, the role of dif-fusion is to transfer the solute out of the slow boundary layer zones near the poresurfaces, and its effect only appears as a logarithmic correction to Eq. (11.200) (seeEq. (11.65)). Note that had we not removed the low permeability zones, diffusioninto and out of such zones would have been important, and Eq. (11.50) would haveimplied that DL � V 2, contradicting the field data.

Because the flow only takes place in the backbone of a percolation cluster, weshould consider dispersion in the backbone of a correlated percolation cluster.Since the permeabilities are infinitely correlated, their correlation length is largerthan any other relevant length scale of the system and, therefore, the only relevantlength scale of the system is its linear size L, implying that the system is a backbonefractal for any L. Thus, Eqs. (11.90) and (11.93) should be used, implying that theexponent υ of Eq. (11.198) is given by υ D 1/(1 C θB), except that when calculatingθB, we must use the numerical values of the relevant exponents given in Table 11.1.

Table 11.1 Values of the critical exponents for 2D percolation with long-range correlation as afunction of the Hurst exponent H.

H Oe Oµ Dbb

0.50 0.98 0.98 1.64

0.60 0.91 0.95 1.82

0.75 0.86 0.80 1.850.90 0.82 0.50 1.89

0.98 0.76 0.32 1.96

410 11 Dispersion in Flow through Porous Media

Another aspect of the model is that it is the 2D correlated percolation that isrelevant to the field-scale data since such data are obtained at large distances fromthe source (up to several tens of kilometers), whereas the thickness of such porousmedia is at most a few hundred meters and, therefore, such porous media are longand thin and, thus, are essentially two-dimensional. Since the analysis of variousfield-scale permeability data by Hewett (1986) and others have indicated that H >

1/2 (and mostly 0.7 < H < 0.9; see Chapter 5), from Table 11.1, we obtain anonuniversal υ D 1/(1 C θB) ' 0.53�0.62 for this range, consistent with the fielddata described earlier. Note that if we do not remove the low permeability zones, wewould have to use Eq. (11.101), which would yield estimates of υ that are not in therange of the field data. For example, we would obtain υ(H D 0.6) ' 2, completelyinconsistent with the field data.

Thus, percolation with long-range correlation is relevant to flow phenomena infield-scale porous media, for example, oil reservoirs and groundwater aquifers, as itprovides a sound explanation for the field-scale data on the dispersion coefficientsand dispersivities.

11.16Dispersion in Unconsolidated Porous Media

Dispersion in flow through consolidated porous media is similar to that in uncon-solidated ones and, therefore, all theories of dispersion described earlier are equallyapplicable to what we discuss in this section. Over the years, there have been manyexperimental studies of dispersion in flow through a packed bed. Some of the oldestworks that the author is aware of are those of Ebach and White (1958), Carberry andBretton (1958), Blackwell et al. (1959), Grane and Gardner (1961), Blackwell (1962),Harleman and Rumer (1963), Pfannkuch (1963), Edwards and Richardson (1968),Hassinger and von Rosenberg (1968), and Gunn and Pryce (1969). These works re-ported extensive experimental data for the longitudinal and transverse dispersioncoefficients in a variety of packed beds using a wide variety of fluids, and over sev-eral orders of magnitude in the Péclet number Pe. Their results for both dispersioncoefficients are completely similar to those shown in Figures 11.5 and 11.6.

Figure 11.13 The effect of particle size distribution on thelongitudinal dispersion coefficient DL in packed beds. Dm isthe molecular diffusivity and Pep is the particle Péclet number(after Han et al., 1985).

11.16 Dispersion in Unconsolidated Porous Media 411

As pointed out earlier, a crucial question about dispersion in flow through anykind of porous media is the condition(s) under which the dispersion coefficientsbecome independent of time. This issue was studied by Han et al. (1985) for dis-persion in flow through a packed bed. If a particle Péclet number is defined by

Pep D Vsdp

Dm

φ1 � φ

, (11.201)

where Dm is the molecular diffusivity, then Han et al. (1985) showed that constantdispersion coefficients are obtained when a dimensionless time td

td D 1Pep

L

dp

1 � φp

φp� 0.3 , (11.202)

where L is the length of the bed, and φp is the particles’ volume fraction. Han et

al. (1985) also studied the effect of a particle size distribution on the dispersioncoefficients. Figure 11.13 presents their results for DL for three types of beds. Onewas a uniform bed in which all the spherical particles had the same size, while theother two contained a range of particle sizes, but with the same average particlesize as the uniform porous medium. The porosities of all the three porous mediaare also the same. As Figure 11.13 indicates, a broader particle size distributiongives rise to larger values of DL, which is expected because a broader distributiongenerates more tortuous transport paths, and thus larger DL. On the other hand,the results of Han et al. (1985) indicate no appreciable effect of the particle sizedistribution on the transverse dispersion coefficient.

Unlike the problem of flow and conduction through packed beds described inChapter 9, theoretical investigation of dispersion through packed beds has not re-ceived a lot of attention. This is partly because theoretical studies of dispersion inrandom packings of particles is too complex and, as far as dispersion phenomenaare concerned, regular or periodic arrays of particles do not provide realistic modelsof unconsolidated porous media. Such porous media do not have the type of tortu-ous flow paths that random packings have and, as emphasized earlier, dispersionis sensitive to the microstructure of porous medium. A spatially-periodic array ofparticles does not allow any disorder or heterogeneities.

Brenner (1980), Brenner and Adler (1982), Carbonell and Whitaker (1983), Eid-sath et al. (1983), Koch et al. (1989), and Salles et al. (1993) studied dispersion inspatially-periodic systems. For example, Koch et al. (1989) showed that at high Pé-clet numbers, the mechanical dispersion that is caused by a stochastic velocity fieldis absent because flow in a spatially-periodic medium is completely deterministic,and at high values of Pe, molecular diffusion that can generate some microscop-ic stochasticity in the solute paths is not important. Thus, both DL and DT wereshown to depend quadratically on the Péclet number, in contradiction with theexperimental data for dispersion in disordered porous media – including random

packings of particles – that indicate a much weaker dependence of the coefficientson the Péclet number.

Eidsath et al. (1983) carried out numerical simulations of dispersion in a squarearray of parallel cylinders in which the flow was parallel to the axis of the cylinders.

412 11 Dispersion in Flow through Porous Media

Using a finite-element method, they first solved for the flow field, and then solvedthe convective-diffusion equation. From their numerical simulations, Eidsath et al.

found that DL � Pe1.7, which is closer to the experimental data described earlierin this chapter. From a theoretical view, however, their result cannot be explainedbecause at high Pe, DL must depend on Pe quadratically. Moreover, at very low Pe(i.e., when the flow field essentially has vanished), the results should be compati-ble with the conduction calculations described in Chapter 9. However, the resultsreported by Eidsath et al. were also not compatible with the limit of very low Pe.It is not known whether these discrepancies are due to numerical inaccuracies, tonot having achieved the high-Pe limit, or reflect some other form of error.

11.17Dispersion in Stratified Porous Media

The last topic to be briefly discussed is dispersion in stratified porous media. Field-scale porous media are usually stratified, consisting of many layers. The structural,and the flow and transport properties may vary greatly from stratum to stratum. Forthis reason, dispersion in stratified porous media has always been of interest. Theearliest studies on transport in stratified porous media appear to be those of Koonceand Blackwell (1965) and Goddin et al. (1966), who studied the displacement of oilby water or a solvent in a stratified porous medium. Such works will be describedin Chapter 14.

Marle et al. (1967) and Güven et al. (1984, 1985) applied the Taylor–Aris disper-sion theory described above to a system of N strata that communicate with oneanother. Marle et al. (1967) obtained an integral expression for the longitudinaldispersion coefficient involving the porosity, fluid velocity, and the local transversedispersion coefficient that were all functions of the distance perpendicular to thestrata. Lake and Hirasaki (1981) and van den Broeck and Mazo (1983, 1984) alsoconsidered the Taylor–Aris dispersion in a stratified medium. In particular, Vanden Broeck and Mazo derived several interesting results, including the FPTD andthe longitudinal dispersion coefficient. Gelhar et al. (1979)’s model described earli-er is, in fact, a method for studying dispersion in 2D stratified porous media sincetheir equations represent averages over the vertical distance z, and the permeabilityfield is assumed to depend on the distance perpendicular to the strata. Plumb andWhitaker (1988a,b) used their large-scale volume-averaging method mentioned ear-lier to study dispersion in stratified porous media.

In a seminal paper, Matheron and de Marsily (1980) studied dispersion analyt-ically in a 2D stratified porous medium using analytical methods and asymptoticexpansions. The direction of the flow velocity was assumed to be parallel to the bed-ding and constant for a given stratum. It was further assumed that the componentof the velocity along the direction of macroscopic flow field is a weakly-stationarystochastic process. The permeability was assumed to be an isotropic stochastic pro-cess and the medium was of infinite extent in both directions. Matheron and deMarsily (1980) showed that under such conditions, dispersion is never Guassian.

11.17 Dispersion in Stratified Porous Media 413

The reason is that because the porous medium is infinitely large and heteroge-neous, a traveling solute particle always samples new regions and strata with newheterogeneities. As a result, a Guassian dispersion regime can never be reached.Matheron and de Marsily also showed that if dispersion is to be Gaussian, thenthe integral of the covariance of the velocity (or permeability) must be zero whichis, however, almost never satisfied for most realistic situations. On the other hand,if the macroscopic flow is not strictly parallel to the stratification (i.e., a small butfinite perpendicular flow component is added), then dispersion will asymptoticallybe Gaussian if the integral is finite.

Bouchaud, J.P. et al. (1990) extended Matheron and de Marsily’s work by studyinga random walk in a 2D stratified medium containing a random velocity field. If thevelocities in the x-direction – the macroscopic direction of the flow – are a functionof the vertical distance, then Bouchaud et al. showed that

h∆x2i � t32 , (11.203)

that is, dispersion is superdiffusive, as DL � h∆x2i/ t � t1/2 and grows with thetime. They also showed that there are large sample-to-sample fluctuations. Theprobability density P(x , t), when averaged over various environments (realizationsof the medium), was found to be non-Gaussian. It is approximately given by

hP(x , t)i � t� 34 f

�x

t34

�, (11.204)

where f (u) is a scaling function with the properties that f (u) � exp(�uδ) foru � 1, with δ D 4/3. These results once again demonstrate the non-Gaussiannature of dispersion in the FS and stratified heterogeneous porous media, and theinadequacy of the CD equation for describing it.