WATER RESOURCES RESEARCH, VOL. ???, XXXX, DOI:10.1029/,

Moment analysis for spatiotemporal fractional dispersion

Yong ZhangDesert Research Institute, Las Vegas, Nevada, USA

David A. BensonColorado School of Mines, Golden, Colorado, USA

Boris BaeumerUniversity of Otago, Dunedin, New Zealand

Abstract. The evolution of the first five non-negative integer-order spatial moments(corresponding to the mass, mean, variance, skewness, and kurtosis) are investigated sys-tematically for spatiotemporal nonlocal, fractional dispersion. Three commonly used, fractional-order transport equations, including the time fractional advection-dispersion equation(Time-FADE), the fractal mobile-immobile (MIM) equation, and the fully fractional advection-dispersion equation (FFADE), are considered. Analytical solutions verify our numericalresults and reveal the anomalous evolution of the moments. Following Adams and Gel-har’s [1992] work on the classical ADE, we find that a simultaneous analysis of all mo-ments is critical in discriminating between different non-local models. The evolution ofdispersion among the sub- to super-diffusive rates is then further explored numericallyby a non-Markovian random walk particle-tracking method that can be used for any het-erogeneous boundary or initial value problem in 3-D. Both the analytical and the nu-merical results also show the similarity (at the early time) and the difference (at the latetime) of moment growth for solutes in different phases (mobile versus total) describedby the MIM models. Further simulations of the 1-D bromide snapshots measured at theMADE experiments, using all three models with parameters fitted by the observed 0th

to 4th moments, indicate that 1) both the time and space nonlocality strongly affect thesolute transport at the MADE site, 2) all five spatial moments should be considered intransport model selection and calibration because those up to the variance cannot ef-fectively discriminate between nonlocal models, and 3) the log-concentration should beused when evaluating the plume leading edge and the effects of space nonlocality.

1. Introduction

A short-lived injection of conservative tracer that movesthrough field-scale geologic material can exhibit non-Gaussian and non-Fickian growth. Both features are par-ticularly striking in the highly skewed bromide and tri-tium plumes measured in the Macro Dispersion Experi-ment (MADE) at Columbus Air Force Base in Mississippi[Adams and Gelhar, 1992]. The failure of the traditionaladvection-dispersion equation (ADE) model, in either an-alytic or highly discretized numerical forms, to capturethe observed anomalous plumes [Adams and Gelhar, 1992;Zheng and Jiao, 1998; Eggleston and Rojstaczer, 1998], mo-tivates the characterization of fine-scale subsurface hetero-geneity [Zheng and Gorelick, 2003; Liu et al., 2004] andthe development of nonlocal transport models. Three mainnonlocal approaches, including the single/multiple-rate mo-bile/immobile model [Harvey and Gorelick, 2000; Feehley etal., 2000; Julian et al., 2001], the continuous time randomwalk (CTRW) method [Berkowitz and Scher, 1998, 2001;Berkowitz et al., 2006], and the space fractional advection-dispersion equation (fADE) model [Benson et al., 2001;Baeumer et al., 2001; Zhang et al., 2007], have been appliedin place of the 2nd-order ADE to characterize the MADE

Copyright 2007 by the American Geophysical Union.0043-1397 /07/$9.00

site plumes. The MADE site plumes are intriguing becausethey are difficult to render in either the laboratory or thecomputer. They serve as a key to understanding realis-tic transport processes in large-scale, highly heterogeneous,natural porous media. Significant efforts in modeling theMADE site plumes, however, are still needed, including theimproved understanding of subsurface heterogeneity (e.g.,Bowling et al. [2005]) and the establishment of a widely ac-cepted physically-based model (see the discussions by Molzet al. [2006] and Hill et al. [2006]). The latter point is theultimate goal of this study.

The spatial moments, including the mass, mean, variance,skewness, and kurtosis of plumes, were also calculated at theMADE site by Adams and Gelhar [1992] and have been in-terpreted by different physical models. Adams and Gelhar[1992, Figs. 13, 15, and 16] show that analytical solutionsof the ADE, in various forms, cannot simultaneously predictthe evolution of the first several (centered) integer momentsof the bromide plume. Their analysis include an accelerating2-D flow field and a large longitudinal dispersivity (5-10m).Subsequent researchers attribute the discrepancy to insuf-ficient characterization of the hydraulic conductivity (K)field, although many of these studies do not analyze the sta-tistical properties of the simulated plumes (see for example,Barlebo et al. [2004]). Berkowitz and Scher [1998, Figs.10 and 11] fit the MADE bromide snapshots by assuming apower-law waiting time probability density function (pdf) intheir CTRW framework, and then found the similar growthrates of the longitudinal mean and standard deviation of dis-placement of the plume. Berkowitz and Scher [1998, 2001]

1

X - 2 ZHANG ET AL.: MOMENT ANALYSIS FOR SPATIOTEMPORAL FRACTIONAL DISPERSION

concluded that the time nonlocality is the primary factorcausing the anomalous dispersion at the MADE site, and thesubsequent approach of Berkowitz et al. [2002] eliminatesthe space nonlocality that may arise from long-range corre-lation of small structures (e.g., Zheng and Gorelick [2003])that may be present in realistic, regional-scale, natural me-dia. On the other hand, Benson et al. [2001], Baeumeret al. [2001], and Zhang et al. [2007] applied the fractalspatially nonlocal fADE to capture the positively skewedtritium plumes at the MADE site while ignoring the tempo-ral nonlocality that may arise from sequestration of solutein relatively immobile phases. In particular, Benson et al.[2001, Fig. 3] found that the growth rate of the variance ofthe discretely sampled plume supports the space-nonlocalfractal index estimated by the hydraulic conductivity (K)statistics. Schumer et al. [2003] examined the zeroth mo-ment at the MADE site and determined that a fractal mo-bile/immobile (temporally nonlocal) model exactly matchesthe decline in mobile mass, but they did not continue theanalysis to higher moments. In short, efforts to this pointhave looked at a few moments to support either space ortime nonlocality, but have not looked at the effects of both.

This study examines the evolution of the first five spa-tial moments caused by fractal time and space nonlocality,which have been posited as good models of anomalous dis-persion. The rest of the article is organized as follows. Insection 2, the first five moments for solute plumes governedby various nonlocal transport equations are solved analyt-ically using either an integral transform or an asymptoticexpansion method. If the transport model can distinguishthe solutes in different phases (such as the mobile and totalphases) which can be helpful and necessary in field appli-cations, the corresponding moments in each phase are thensolved separately. In section 3, a non-Markovian randomwalk particle-tracking method is developed to approximatethe moment growth, to compensate for insufficiency of theanalytical methods in solving the complete evolution of mo-ments. In section 4, all of the moments data (from the 0th to4th order) observed at the MADE site are then interpretedto assess the dominant nonlocality affecting solute transportthrough typically heterogeneous, alluvial aquifer/aquitardsystem like the MADE site. An analysis of the MADE siteplume moments can illuminate the discrepancies and clarifythe debate mentioned above. We also expect that this studycan provide us a better understanding of anomalous disper-sion and be beneficial to future methodology developmentand model selection.

2. Analytical Solutions of Spatial Moments

The fADE has been used successfully by various hydrolo-gists recently to capture the anomalous transport of conser-vative tracers through unsaturated soils [Pachepsky et al.,2001; Zhang et al., 2005], saturated porous media [Bensonet al., 2000a, 2000b, 2001; Zhou and Selim, 2003; Chang etal., 2005; Huang et al., 2006], streams and rivers [Deng etal., 2004; Zhang et al., 2005; Kim and Kavvas, 2006], andoverland flow [Deng et al., 2006]. The underlying physicalmeanings of fractional dynamics were elucidated clearly inthe review by Metzler and Klafter [2000], and the nonlo-cal property embedded in the fractional derivative and therepresentative aquifer heterogeneity were also introduced byBenson et al. [2000a, 2000b, 2001]. Here we accept theirmain conclusion that the fADE is a potentially reasonablemodel of spatially nonlocal transport, and add the temporalcomponent that is clearly at work at the MADE site. Theinfluence of the space and time nonlocality on spatial mo-ments is analyzed systematically herein. We expect that thisanalysis can help us to decipher the important components

of nonlocality embedded in the measured moment data, aswill be illustrated in Section 4.

Three types of anomalous, fractional-order, transportequations (described in the following subsections) have beenproposed in literature. No systematical analysis has beenmade so far to distinguish their moments or compare theirapplicability to field-measured data. We calculate the spa-tial moments for all these models, with the restriction thatthe transport parameters of velocity and dispersion (V andD) are constant and the dimensionality is 1. The analyti-cal moments are also limited to the Green function solution(i.e., with an instantaneous point source). These restrictionscan be relaxed by using the numerical method discussed inthe next section. Although the analytical methods have in-trinsic limitations, they can provide the most reliable andaccurate solutions. In particular, when a simple functionalform can be obtained for some cases, the analytical solu-tion serves as the best reference to validate, and provide thetheoretical convergence criteria for, the numerical results.

2.1. Model 1: The Time-FADE Model

The time fractional advection-dispersion equation (Time-FADE) is of the form (see Metzler and Klafter [2000], eq.(44); and the numerous references cited therein)

∂γC

∂tγ=

(−V

∂

∂x+ D

∂2

∂x2

)C, (1)

where γ (0 < γ < 1) is the scaling index in time describedby a Caputo fractional derivative, C = C(x, t) is the soluteconcentration, V is the velocity, and D is the dispersion co-efficient. When γ = 1, (1) reduces to the traditional ADE.The Time-FADE (1) is a subset of the CTRW [Berkowitz etal., 2002], and a functionally similar form has been put for-ward as a reasonable candidate to capture the moments ob-served at the MADE site [Berkowitz and Scher, 1998, 2001].As derived in Appendix A, we get the centered momentsgoverned by (1) using an integral transform method

M(t) = 1 (2a)

E(t) = a1V tγ (2b)

σ2(t) = a2Dtγ + a3V2t2γ (2c)

S(t) =a4V

3t3γ + a5V Dt2γ

(a2Dtγ + a3V 2t2γ)3/2(2d)

K(t) =a6V

4t4γ + a7V2Dt3γ + a8D

2t2γ

(a2Dtγ + a3V 2t2γ)2− 3 (2e)

where M, E, σ2, S and K denote the mass, mean displace-ment, variance, skewness and kurtosis, respectively. Theparameters a1 ∼ a8 used in (2) depend on γ only:

a1 = 1/Γ1 (3a)

a2 = 2/Γ1 (3b)

a3 = 2/Γ2 − 1/(Γ1)2 (3c)

a4 = 6/Γ3 − 6/(Γ2Γ1) + 2/(Γ1)3 (3d)

a5 = 12/Γ2 − 6/(Γ1)2 (3e)

a6 = 24/Γ4 − 24/(Γ3Γ1) + 12/[Γ2(Γ1)2]− 3/(Γ1)

4 (3f)

a7 = 72/Γ3 − 48/(Γ2Γ1) + 12/(Γ1)3 (3g)

a8 = 24/Γ2 (3h)

where Γi = Γ(iγ + 1) is the Gamma function.The analytical solution (2) reveals the evolution of mo-

ments at all time scales. First, the solute mass M(t) isconstant in time and equal to the initial mass (which is nor-malized to be 1 here). In other words, the Time-FADE (1)does not distinguish the solute in different phases. Second,

ZHANG ET AL.: MOMENT ANALYSIS FOR SPATIOTEMPORAL FRACTIONAL DISPERSION X - 3

the mean displacement E(t) increases slower than linearlywith time. This is expected because solutes are trapped forlonger periods of time as time increases, which further de-lays the mean shift of the whole plume. Third, the varianceσ2(t) can grow either faster or slower than Fickian diffusion.For 0 < γ < 0.5, the time fADE (1) induces subdiffusion be-cause the variance always grows slower than the Gaussiancase. For 0.5 < γ < 1, however, superdiffusion will appear ifthe second term on the right-hand side (RHS) of (2c) dom-inates. In addition, the skewness and kurtosis grow nonlin-early (and may not be monotone) with time. We explorethese moments further using various examples in the nextsection.

When γ = 1, the moments given by (2) reduce toM(t) = 1, E(t) = V t, σ2(t) = 2Dt, S(t) = 0, and K(t) = 0,respectively, which are the results of the classical 2nd-orderADE. So the Time-FADE (1) and its analytical moments(2) still hold for the case of γ = 1, and we can regard themas the generalized ADE and moments. Also note that themean displacement depends on the velocity V and the scaleindex γ. The variance depends on V , γ and the dispersioncoefficient D. Therefore, given the measured first three mo-ments, all model parameters required by the Time-FADE (1)can be calibrated. This provides an indirect way to checkthe applicability of the Time-FADE (1), because one caneasily compare the resultant skewness and kurtosis to themeasured ones if they are available.

2.2. Model 2: The Mobile/Immobile (MIM) Model

The fractal mobile/immobile (MIM) model (see (4) and(10)) extends the Time-FADE (1) by distinguishing the so-lutes in different phases, and thus we need to solve the spa-tial moments for each phase separately. Since the soluteconcentration in the immobile phase is simply the differenceof concentration in the total phase and the mobile phase (forexample, see eq. (9) in Schumer et al. [2003]), we ignorethe immobile phase in this study.

2.2.1. The Total PhaseThe fractal MIM model for conservative tracer in the to-

tal domain is [Schumer et al., 2003; Baeumer et al., 2005]

(∂

∂t+ β

∂γ

∂tγ

)Ctot =

(−V

∂

∂x+ D

∂2

∂x2

)Ctot, (4)

with the initial condition Ctot(x, t = 0) = θmCm(x, t =0)/θtot, where θtot and θm are the total and mobile zoneporosities, respectively, Cm(x, t = 0) is the initial concen-tration in the mobile zone, and β is the capacity coefficient.Here the solute is assumed to be placed into the mobilezone and the immobile zone is initially clean, representinga typical tracer test (see also Schumer et al. [2003]). Whenγ = 1, (4) reduces to the 2nd-order ADE with a retardationfactor 1 + β. Compared to the Time-FADE (1), the MIMmodel (4) has an additional term (∂C/∂t) on the left-handside, representing the linear drift of time when particles aremobile. This term can help us to distinguish the solute par-ticles in various phases (motion or waiting) [for details, seeBenson et al., submitted], and thus the MIM model may besuperior to the Time-FADE (1) in capturing the real plumesmeasured in the field, where the sampling process tends tocollect solutes preferentially from mobile zones.

The integral transform method shown in Appendix Bshows that the total mass (mobile + immobile) is constant(see (B2)):

Mtot(t) = θm/θtot, (5)

where the subscript “tot” denotes the total domain.The higher moments cannot be solved using the integral

transform method. There are no exact analytical solutions

for these moments, except for asymptotic approximations.We used the asymptotic expansion to derive the early time(denoted as tearly) and the late time (tlate) approximationsof these moments. The main steps of the expansion and theresultant moments are shown in Appendix B. Here we listthe approximation of each moment at both small and largetime, to assess the type of growth which can be importantto understand the anomalous transport.

At very early or late time, the first term (with n = 0) onthe RHS of (B8) and (B9) dominates, and thus we get thesimple approximation revealing the type of growth of meandisplacement Etot:

Etot(tearly) ≈ V t , (6a)

Etot(tlate) ≈ V β−1tγ/Γ1 . (6b)

So the mean displacement for the MIM model (4) differsfrom the Time-FADE (1) (see (2b)) due to its time-scaledependency: at very early time, the mean displacement ofsolutes grows linearly, while it grows slower than the Gaus-sian case at very large t when more mass becomes trapped.

Similarly, based on (B14) and (B15), the variance σ2tot

grows at very small and large t as:

σ2tot(tearly) ≈ 2Dt , (7a)

σ2tot(tlate) ≈ V 2β−2t2γ

[2

Γ2− 1

(Γ1)2

]+ Dβ−1tγ 2

Γ1. (7b)

It is possible that the solute transport may experience threedifferent types of dispersion. The plume can spread at aFickian rate at early time (σ2

tot = 2Dt), then turn to sub-diffusive growth (σ2

tot ∝ tγ), and finish with superdiffusion(σ2

tot ∝ t2γ with γ > 0.5). The possible superdiffusive pro-cess is not caused by the dispersion D, but by V 2. We willcheck this conclusion extensively using examples in section3. This unexpected evolution has not been derived previ-ously. This time-dependent transport process, however, maybe critical in real applications.

Based on (B19) and (B20), the skewness Stot at very smalland large t behaves like:

Stot(tearly) ≈ 0

[σtot(tearly)]3= 0 , (8a)

Stot(tlate) ≈ b1V3β−3t3γ + b2DV β−2t2γ

[σtot(tlate)]3, (8b)

where the parameters b1 = 6/Γ3 − 6/(Γ2Γ1) + 2/(Γ1)3 and

b2 = 12/Γ2−6/(Γ1)2. The zero skewness confirms the Gaus-

sian diffusion at early time.Finally, based on (B24) and (B25), the kurtosis Ktot at

very small and large t behaviors like:

Ktot(tearly) ≈ 12D2t2

[σtot(tearly)]4− 3 = 0 , (9a)

Ktot(tlate) ≈ c1V4β−4t4γ + c2DV 2β−3t3γ + c3D

2β−2t2γ

[σtot(tlate)]4− 3 ,

(9b)

where the parameters c1 = 24/Γ4 − 24/(Γ3Γ1) +12/[Γ2(Γ1)

2]− 3/(Γ1)4, c2 = 72/Γ3− 48/(Γ2Γ1)+ 12/(Γ1)

3,and c3 = 24/Γ2. The zero kurtosis confirms further theGaussian diffusion at early time.

We will validate the above small/large time behavior ofmoments in section 3. More importantly, we will evaluateempirically the effective ranges of the “early” and “late”times discussed above.

2.2.2. The Mobile Phase

X - 4 ZHANG ET AL.: MOMENT ANALYSIS FOR SPATIOTEMPORAL FRACTIONAL DISPERSION

The MIM model for solute in the mobile domain is[Schumer et al., 2003, eq. (27)]

(∂

∂t+ β

∂γ

∂tγ

)Cm =

(−V

∂

∂x+ D

∂2

∂x2

)Cm

− βCm(x, t = 0)t−γ

Γ(1− γ),

(10)

where the last term on the RHS of (10) denotes the power-law release of solute from the source (which will be assumedto be a Dirac delta function here). The MIM model (10)has been used successfully by Schumer et al. [2003] to fitthe breakthrough of tracer in a mountain stream [see alsoHaggerty et al., 2002] and match the mobile mass decline atthe MADE site with γ = 0.33.

We solve the spatial moments for (10) using the asymp-totic expansion in Appendix C. Similar to the last subsec-tion, here we only show the type of moment growth at smalland large times. First, the mass decays as

Mm(tearly) ≈ 1− βt1−γ/Γ(2− γ) , (11a)

Mm(tlate) ≈ β−1tγ−1/Γ(γ) , (11b)

where the subscript “m” denotes the mobile phase. Thelater-time mass decay (11b) is consistent with the approxi-mation given by Schumer et al. [2003, eq. (26)] using a dif-ferent method. The formula (11b) can be practically helpfulbecause the log-log plot of the measured mass versus timedirectly provides the values of γ and β, if (10) is the appro-priate governing equation.

Second, the mean displacement grows like

Em(tearly) ≈ V t , (12a)

Em(tlate) ≈ V β−1tγΓ(γ)/Γ(2γ) , (12b)

and the variance is

σ2m(tearly) ≈ 2Dt , (13a)

σ2m(tlate) ≈ V 2β−2t2γd1 + Dβ−1tγd2 , (13b)

where d1 = 2Γ(γ)/Γ(3γ) − [Γ(γ)/Γ(2γ)]2 and d2 =2Γ(γ)/Γ(2γ).

The skewness is:

Sm(tearly) ≈ 0

[σm(tearly)]3= 0 , (14a)

Sm(tlate) ≈ e1V3β−3t3γ + e2DV β−2t2γ

[σm(tlate)]3, (14b)

where e1 = 6Γ(γ)/Γ(4γ) − 6[Γ(γ)]2/[Γ(3γ)Γ(2γ)] +2[Γ(γ)/Γ(2γ)]3 and e2 = 12Γ(γ)/Γ(3γ)− 6[Γ(γ)/Γ(2γ)]2.

Finally, the kurtosis grows as:

Km(tearly) ≈ 12D2t2

[σm(tearly)]4− 3 = 0 , (15a)

Km(tlate) ≈ f1V4β−4t4γ + f2DV 2β−3t3γ + f3D

2β−2t2γ

[σm(tlate)]4− 3 ,

(15b)

where f1 = 24Γ(γ)/Γ(5γ) − 24[Γ(γ)]2/[Γ(4γ)Γ(2γ)] +12[Γ(γ)]3/[Γ(3γ)(Γ(2γ))2]−3[Γ(γ)/Γ(2γ)]4, f2 = 72Γ(γ)/Γ(4γ)−48[Γ(γ)]/[Γ(3γ)Γ(2γ)]+12[Γ(γ)/Γ(2γ)]3, and f3 = 24Γ(γ)/Γ(3γ).

At early time, the mean, variance, skewness, and kurtosisin the mobile phase are the same as the corresponding mo-ments for the total phase. At late time, these moments growat the same rate as those for the total phase, but the mag-nitude is different (for example, see the difference betweenthe parameters b in (8b) and e in (14b)). Interestingly, this

difference depends only on γ (or more specifically, the dif-ference increases with a decrease of γ).

2.3. Model 3: The Fully Fractional Advection-Dispersion Equation (FFADE) Model

The fully fractional advection-dispersion equation (FFADE)is the same as the fractal MIM (model 2) with the additionof Levy motion instead of Brownian motion as the model ofsolute transport in the mobile phase. This model containsa nonlocal fractional derivative term in space [Zhang et al.,2006c]:

(∂

∂t+ β

∂γ

∂tγ

)Ctot =

(−V

∂

∂x+ D

∂α

∂xα

)Ctot, (16)

where α (1 < α < 2) is the scaling index in space describedby a Riemann-Liouville fractional derivative. The Riemann-Liouville type of fractional derivative is equal to its Caputocounterpart, if the upstream boundary remains clean or itsconcentration eventually falls to zero. It is noteworthy thatthe current applications of the fADE (mentioned above) aremainly limited to be the space fADE (with β = 0 in (16)).

For brevity, we only show the governing equation for thetotal phase. The integral transform reveals that the spacenonlocality does not change the mass or the mean displace-ment, and the mass and mean displacement for (16) are thesame as (5) and (6), respectively (Appendix D). The spa-tial moments higher than the first diverge for the FFADE(16) with scale index 1 < α < 2, if the whole plume canbe observed. However, the detection limit and/or finite wellfields cut the leading edge of plume artificially, and the spa-tial moments are finite and measurable (as demonstratedby Baeumer et al. [2001] for the mean and variance of asubordinated Brownian process, with a truncation of de-tection length). The truncation of low concentrations orconcentrations downstream, however, can decrease the totalmeasurable mass gradually, and the calculation of the corre-sponding analytical spatial moments is not trivial. Bensonet al. [2001] found that the variance of the discretely sam-pled plume should grow proportionally with t2/α, becausethe density of an α-stable plume is scale invariant with t1/α.This conclusion, however, is based on the assumption thatthe mass will not drop. Here we solve the analytical mo-ments for a reduced version of (16) with β = 0, which isthe space fADE, as shown in Appendix D. For a completeFFADE model, we rely on the following numerical methoddue to its simplicity.

3. Random Walk Approximations andNumerical Examples of Spatial Moments

A non-Markovian random walk (RW) particle-trackingalgorithm is developed to approximate the spatial mo-ments for all three models discussed above. The numericalmethod is necessary for obtaining a complete growth of mo-ments, which can not be derived analytically for the MIMand FFADE models. For non-ideal cases involving space-dependent transport parameters, non-ideal boundary andinitial conditions, and/or high-dimensional transport, themoments for the Green function solutions discussed aboveare not tractable and the numerical method is required.

The RW method is selected here due to its potentialflexibility and efficiency (as demonstrated partially by theMarkovian RW method developed by Zhang et al. [2006a,2006b] for the space fractional ADE). Particle dynamics andthe main steps of the RW method for each nonlocal modelare described in Appendix E.

ZHANG ET AL.: MOMENT ANALYSIS FOR SPATIOTEMPORAL FRACTIONAL DISPERSION X - 5

Time

Mea

n D

ispl

acem

ent

Time

Var

ianc

e

Time

-1

0

1

2

Ske

wne

ss

Time

-1

0

1

2

Kur

tosi

s

10-2 10-1 100 101 102 103

103

102

101

100

10-1

10-2

105

104

103

102

101

100

10-1

RW vs. Analyticγ = 0.9γ = 0.5γ = 0.1

10-2 10-1 100 101 102 103

10-2 10-1 100 101 102 103 10-2 10-1 100 101 102 103

Figure 1. Random walk solutions (symbols) versus the analytical solutions (lines), showing the influenceof the time scale index γ on spatial moments, where the transport equation is the Time-FADE (1). Theparameters are: β = 1, V = 1.3, and D = 2.8.

3.1. The Time-FADE Model (1)

The simulated moments using the the RW method match

the analytical solutions (Figure 1), validating the RW tech-

nique. For all examples considered here, the skewness and

the kurtosis keep relatively stable.

3.2. The MIM Model (4) and (10)

RW approximations of moments for the MIM model (4)match the approximations of moments at early and late timederived above (Figure 2). We can now empirically define theeffective range of these early and late times.

At early time when the contribution of the time-advectionterm ∂C/∂t is much larger than the time-dispersion termβ∂γC/∂tγ (i.e., t << (1/β)1/(1−γ)), the MIM model (4) re-

Time

Var

ianc

e

Time

-1

0

1

2

Ske

wne

ss

Time

-1

0

1

2

3

Kur

tosi

s

10-2 10-1 100 101 102 103 104

106

104

102

100

10-2

104

103

102

101

100

10-1

10-2

Time

Mea

n D

ispl

acem

ent

10-2 10-1 100 101 102 103 104

10-2 10-1 100 101 102 103 104 10-2 10-1 100 101 102 103 104

RW vs. Analyticγ = 0.9γ = 0.5γ = 0.1

Figure 2. Random walk (RW) solutions (symbols) versus the analytical solutions (lines) of momentscaused by different time scale index γ, where the transport equation is the MIM model for the totalphase (4). The model parameters are: β = 1, V = 1.3, and D = 2.8. The noise of the kurtosis simulatedby the RW method is due to the relatively large time step at late time. The dashed line is the analyticalsolution at early time.

X - 6 ZHANG ET AL.: MOMENT ANALYSIS FOR SPATIOTEMPORAL FRACTIONAL DISPERSION

Time

Var

ianc

e

Time

-0.5

0

0.5

1

Ske

wne

ss

Time

-1

0

1

2

3

Kur

tosi

s

Time

Mea

n D

ispl

acem

ent

Time

Mas

s102

101

100

10-1

10-2

101

100

10-1

10-2

10-3

10-4

10-5

101

100

10-1

10-2

10-2 10-1 100 101 102 103 104 10-2 10-1 100 101 102 103 104 10-2 10-1 100 101 102 103 104

10-2 10-1 100 101 102 103 10410-2 10-1 100 101 102 103 104

Mobile phaseTotal phaseEarly-time analytic solutionLate-time analytic solution

Figure 3. RW solutions (symbols) versus the analytical solutions (lines) of moments for the MIM modelsfor solute in the total and mobile phases. The model parameters are: γ = 0.1, β = 1, V = 1.3, andD = 2.8. The lines represent the moments calculated by the very small or large time approximations(including (11)∼(15)). The mobile mass represented by the green and blue lines in the top left figure iscalculated by using (C2) with the first four terms.

duces to the traditional ADE

∂Ctot(x, t)

∂t= −V

∂Ctot(x, t)

∂x+ D

∂2Ctot(x, t)

∂x2,

and thus all moments behave like a Fickian model (see Fig-

ure 2). The empirical time is

t < tearly = (0.1/β)1/(1−γ) . (17)

At late time when t >> (1/β)1/(1−γ), the MIM model (4)

reduces to

β∂γCtot(x, t)

∂tγ= −V

∂Ctot(x, t)

∂x+ D

∂2Ctot(x, t)

∂x2, (18)

which is similar to the Time-FADE (1) except for the addi-

tional factor β. The empirical time is

t > tlate = (3/β)1/(1−γ), (19)

which is consistent with the empirical value found bySchumer et al. [2003].

Therefore, the Fickian-type moments appear at earlytime and then transfer to subdiffusion for 0 < γ < 0.5 andsuperdiffusion for 0.5 < γ < 1 at late time. However, theGaussian moments at early time may sometimes be difficultto capture due to the short duration. Once an appreciableamount of mobile mass is lost, the effects of trapping are feltby all moments. Note that for smaller γ and/or a larger β,tlate in (19) is relatively smaller, and that is why the later-time approximations derived in section 2 require less timeto converge to the real moments for a smaller γ and/or alarger β. Also note that the similarity between (18) and (1)reveals that the later-time moments for the MIM model (4)with β = 1 should be the same as those for the Time-FADE(1). This can be verified by comparing the correspondingmoment formula shown in section 2. For example, the vari-ance (7b) with β = 1 is equal to (2c). The same is true forall other moments. In other words, by replacing V with V/βand D with D/β in the moments for the MIM model (4),we should get the same later-time formulas as those for theTime-FADE (1).

Time

Var

ianc

e

Time

Mas

s

10-3 10-2 10-1 100 101 102 103 104

100

10-1

10-2

107

106

105

104

103

102

101

100

10-1

10-2

RW vs. Analyticβ = 0.01β = 0.1β = 1

10-3 10-2 10-1 100 101 102 103 104

Mass in the Mobile phase Variance in the Total phase

Figure 4. Random walk solutions (symbols) versus the analytical solutions (lines) of moments causedby different capacity coefficient β, where the transport equation is the fractal MIM model. The modelparameters are: γ = 0.5, V = 1.3, and D = 2.8. The dashed line is the analytical solution at early time.

ZHANG ET AL.: MOMENT ANALYSIS FOR SPATIOTEMPORAL FRACTIONAL DISPERSION X - 7

The RW solutions also demonstrate that the mobile solutehas similar moments at early time as the total solute, whiletheir later-time moments have the same trend in a log-logplot with different magnitudes (Figure 4), just as concludedabove. The power-law decay of the mobile mass implies thenecessity of distinguishing the origin of sampled water in thefield. In addition, using more terms in the power series ofmoment formula (such as (C2)), increases the effective rangeof the moment formula (see the top left picture in Figure 4).

A smaller β causes less solute to be trapped in the im-mobile phase, and thus the mobile mass decays slower, andboth the mean displacement and the plume variance arelarger (Figure 3). Also notable is our confirmation of theobservation by Schumer et al. [2003] that skewness will in-crease from negative to positive with an increase of β, andthat large values of β are needed for this transition (notshown here). The transition time between tearly and tlate in-creases with a decrease of β (Figure 3). The plume varianceduring this transition zone can behave quite different (i.e.,with a non-linear growth rate) compared to the late timevariance (with a stable growth rate), resulting in additionaluncertainty in the interpretation of measured moments data.For example, the time-dependence of variance can be mis-interpreted as measurement noise, and the transition zonecan thus be shortened significantly or even ignored by as-signing a wrong γ and/or β. Caution must be taken whenestimating parameters based solely on late-time behavior ofmoments.

3.3. The FFADE Model (16)

As expected, the numerical solutions (Figure 5) show thatthe space nonlocality enhances the plume variance, skew-ness, and kurtosis significantly. Here we have imposed acutoff of concentration due to the detection limit for thesemoments to exist. The mean displacement and the massdecay rate does not change significantly. The slight delayof the mean displacement for small α can be expected dueto the cutoff of large jumps, but the discrepancy of meandisplacement for α = 1.2 shown in Figure 5 may mainly dueto the numerical noise (more particles are needed to cap-ture the fast motion for the very small fraction of particlesmoving large distances when α is small).

In the model used to generate the results shown in Fig-ure 5, 1,000,000 particles were released from the source andthus the detection limit (normalized concentration) is onthe order of 10−6. In real-world cases, the finite distanceof observation wells may impose a more significant limit ondetection (such as the MADE test discussed below), andthus the moments higher than 1st order will be lower thanthose shown in Figure 5. This will cause some uncertaintyin model selection. Specifically, a space nonlocal model witha significant cutoff of large jumps (by detection limits or fi-nite well fields) may have plume variance the same order asa time nonlocal model with different dispersion coefficientD. This can be found at the MADE site discussed below.

The apparent skewness and kurtosis for 1 < α < 2 are sig-nificantly larger than the other two models discussed above.The skewness and kurtosis for 1 < α < 2 also remain rela-tively stable at late times. These discrepancies may help usto interpret the measured moments data using the appropri-ate model, especially to allow for the effect of plume lengthcutoffs on measured variances. Also note that changes in αwill not affect the mass or mean, but will increase the otherspatial moments.

4. The MADE Test Site Revisited: SpatialMoments and Dominant Nonlocality

The MADE site moments data (up to 4th order) are in-terpreted by applying all three nonlocal models discussedabove. To evaluate further the nonlocal models, the best-fittransport parameters (including the time scale index γ, ca-pacity coefficient β, velocity V , and dispersion coefficient D)are then used to simulate the concentration distribution atgiven observation times, which can be compared to the snap-shots observed at the MADE site. To compare directly withprevious 1-D models built by Berkowitz and Scher [1998,2001] and Benson et al. [2001], we check also the longitu-dinal moments and the 1-D projected concentrations. All7 sampling cycles from 9 to 503 days are considered, eventhough not all snapshots have the same density or range ofsampling. The spatial moments in each model are calcu-lated using the space cutoff corresponding to the furthestobservation well at each measurement cycle.

4.1. Moments Fitted by the Time-FADE Model (1)

Berkowitz and Scher’s [1998] CTRW model for fitting theMADE plume corresponds to the Time-FADE (1). By 1)assuming waiting times independent of subsequent motionsize, 2) assigning a long-tailed waiting time pdf with theasymptotic behavior w(t) ∼ t−1−γ (where 0 < γ < 1 is thesame as the time scale index in the Time-FADE (1)), and3) placing w(t) into the CTRW solution or the generalizedmaster equation, the following fractional derivative equa-tion results (see also Metzler and Klafter [2000], eq. (44) inBerkowitz et al. [2002], and eq. (107) in Berkowitz et al.[2006]):

∂C(x, t)

∂t=

∂1−γ

∂t1−γ

[−V

∂C(x, t)

∂x+ D

∂2C(x, t)

∂x2

], (20)

This equation is the same as the Time-FADE (1) if the up-stream boundary remains clean.

Here we use the time scale index fitted by Berkowitz andScher [1998], which is γ = 0.5, and calibrate V = 0.78 m/dand D = 4.60m2/d to fit simultaneously the measured meanand variance. The final mean and variance of plumes (shownin Figure 6 with dots and named as “Time-FADE”) matchwell the measured ones. The resultant skewness and kurto-sis, however, obviously underestimate the measured values.Moreover, because the Time-FADE (1) cannot distinguishthe particles in different phases, it cannot capture the ob-served mass decay (Figure 6(a)).

4.2. Moments Fitted by the MIM Model (10)

The failure of the Time-FADE model (1) in capturing theobserved mass decay motivates the application of the fractalMIM model (10). The simultaneous fit of the mass, mean,and variance of plumes using the fractal MIM model (10) forthe mobile phase (Figures 6) gives us all of the model pa-rameters, including γ = 0.35, β = 0.08d−0.65, V = 0.13m/dand D = 0.9 m2/d. As shown by Figure 6(a), the slope ofthe later-time mobile mass versus time in a Log-Log plot isaround −0.65, which equals to −(1−γ) and thus reveals thetime scale index γ (0.35) directly. The best-fit γ and β areclose to or the same as those fitted by Schumer et al. [2003](which are 0.33 and 0.08 d−0.67, respectively). To distin-guish it from another MIM model built below, we name itthe “MIM-1” model.

Here we have tearly = (0.1/β)1/(1−γ) = 1.4 days andtlate = (3/β)1/(1−γ) = 264 days, and thus most of the mo-ments measured from 9 to 503 days are within the transitionzone time discussed in Section 3. As shown by Figures 6(d)and 6(e), the skewness and kurtosis predicted by the MIM-1model significantly underestimate the measured values dur-ing the most period. This mismatch is due to the underlyingassumption of Brownian motion for mobile particles.

X - 8 ZHANG ET AL.: MOMENT ANALYSIS FOR SPATIOTEMPORAL FRACTIONAL DISPERSION

Time

Var

ianc

e

Time

Ske

wne

ss

Time

Kur

tosi

s

Time

Mea

n D

ispl

acem

ent

Time

Mas

s

103

102

101

100

10-1

10-2

10-3

α = 2, γ = 0.5α = 1.9, γ = 0.5α = 1.5, γ = 0.5α = 1.2, γ = 0.5

100

10-1

10-2

105

104

103

102

101

100

10-1

10-2

106

105

104

103

102

101

100

10-1

10-2

10-2 10-1 100 101 102 103 10-2 10-1 100 101 102 103

10-2 10-1 100 101 102 103

102

101

100

10-1

10-2

10-2 10-1 100 101 102 10310-2 10-1 100 101 102 103

Mass in the Mobile phase Mean in the Total phase Variance in the Total phase

Skewness in the Total phase Kurtosis in the Total phase

Figure 5. Random walk solutions of moments caused by different space scale index α, where the trans-port equation is the FFADE model (16). The model parameters are: β = 1, V = 1.3, and D = 2.8. Themass (and also the mean) for different α are almost identical.

4.3. Moments Fitted by the FFADE model

Here we interpret further the moments data using theFFADE model (16) (note it is modified for the mobile phaseby adding the source term on the RHS, which is simi-lar to (10)). The time-nonlocal parameters of γ = 0.35,β = 0.08 d−0.65 come from the mass decline. We visuallyfit V = 0.28 m/d, and D = 0.28 mα/d based on the meanand variance growth. The space scale index α (= 1.1) is not

fitted but estimated a priori based on the K statistics (seeBenson et al. [2001] for details). The best-fit velocity anddispersion coefficient are still consistent with the analysisof the K statistics by Benson et al. [2001]. The FFADEmodel does not significantly improve the fit of either themean or the variance over the MIM model with Brownianmotion (10), except that the slopes are steeper and bettermatched by the FFADE model. Benson et al. [2001] discussthe very Gaussian-like early plume that may have inherited

1 10 100 1000Time (day)

0.1

1

10

Mas

s F

ract

ion

100Time (day)

1

10

100

Mea

n D

ispl

acem

ent

(m)

0 200 400 600Time (day)

-1

0

1

2

3

Ske

wne

ss

0 200 400 600Time (day)

-2

0

2

4

6

8

Kur

tosi

s

a) b)

d) e)

Slope = -0.65

40 600 100Time (day)

10

100

1000

10000

Var

ianc

e (m

2/d

ay)

c)

40 600

Legend

Measured dataTime-FADEMIM-1MIM-2FFADE

Figure 6. The fitted (lines) versus measured (circles) centered moments at the MADE site (MADE-1bromide). a) mass, b) mean displacement, c) variance, d) skewness, and e) kurtosis. In a), the massfraction for the MIM-1, MIM-2, and the FFADE models are almost identical. In d) and e), the momentsfor MIM-1 and MIM-2 are very close.

ZHANG ET AL.: MOMENT ANALYSIS FOR SPATIOTEMPORAL FRACTIONAL DISPERSION X - 9

-25 0 25 50 75Distance from source (meter)

0

400

800

1200

1600B

rom

ide

Con

cent

atio

n (mg/L

)

-25 0 25 50 75Distance from source (meter)

0

200

400

600

-25 25 75 125 175Distance from source (meter)

0

50

100

150

200

250

Bro

mid

e C

once

ntat

ion

(mg/L

)

-25 25 75 125 175Distance from source (meter)

0

50

100

150

200

250

-25 25 75 125 175Distance from source (meter)

0

50

100

150

200

250

Snapshot 2 (day 49)

Snapshot 3 (day 126)

Snapshot 4 (day 202)

Snapshot 7 (day 530)

Snapshot 6 (day 370)

Sanpshot 5 (day 279)

-25 0 25 50 75Distance from source (meter)

0

200

400

600

Time-FADEMIM-1MIM-2FFADE

Figure 7. Simulated (lines) and measured (circles) bromide concentrations from the MADE-1 test usinglinear axes. Note different y-axis scales used to show more clearly the differences between models.

some symmetry from the large injection volume. However,the FFADE model can obviously improve the fitting of theskewness and kurtosis at most times (Figure 6), except forthe first and last snapshots which are discussed in the nextsubsection.

4.4. Comparison of Snapshots Simulated by DifferentModels

In addition to the moments, the “gold standard” of modelfitness is how well it reproduces the concentrations in the

plume profiles over time. All three models built above areused further to calculate the space distributions of bromide.

The Time-FADE model, with V and D best fit using thefirst and second moments, underestimates the peak concen-tration and the measured mass at the early time, and thenoverestimates the mass at late time (Figure 7), due to thefailure of this model in distinguishing the solute in differentphases. To capture the fast movement of the plume frontby using the Gaussian (symmetric) diffusion with a con-stant dispersion coefficient, the Time-FADE model requires

-25 25 75 125 175Distance from source (meter)

Bro

mid

e C

once

ntat

ion

(mg/L

)

-25 25 75 125 175Distance from source (meter)

-25 25 75 125 175Distance from source (meter)

Bro

mid

e C

once

ntat

ion

(mg/L

)

-25 25 75 125 175Distance from source (meter)

-25 25 75 125 175Distance from source (meter)

Snapshot 2 (day 49)

Snapshot 3 (day 126)

Snapshot 4 (day 202)

Snapshot 7 (day 530)

Snapshot 6 (day 370)

Sanpshot 5 (day 279)

-25 25 75 125 175Distance from source (meter)

104

103

102

101

100

10-1

104

103

102

101

100

10-1

104

103

102

101

100

10-1

104

103

102

101

100

10-1

104

103

102

101

100

10-1

104

103

102

101

100

10-1

Time-FADEMIM-1MIM-2FFADE

Figure 8. Semi-log version of Figure 7: the simulated (lines) versus measured (circles) bromide concen-trations at the MADE-1 test.

X - 10 ZHANG ET AL.: MOMENT ANALYSIS FOR SPATIOTEMPORAL FRACTIONAL DISPERSION

an extremely large dispersivity (αL = D/V ≈ 6m) and thusproduces unrealistic strong upstream diffusion.

In the other two alternative physical models (i.e., theMIM and FFADE models), the values of γ and β are es-sentially fixed by the decline of mobile mass. The concen-tration distributions simulated by the MIM-1 model under-estimate significantly the peak value (Figure 7) and createsthe requisite variance via an unrealistically large dispersiv-ity (similar to the Time-FADE model). The failure of theMIM-1 model motivates us to build an alternative time non-local model with Gaussian transport (named “MIM-2”) tocapture the peak concentration. After fitting the snapshotsdirectly to get the parameters V and D, we obtain best-fitV = 0.05m/d, and D = 0.18m2/d, with the same γ and β asthe MIM-1 model. The MIM-2 model does model the peakconcentrations, but with a price of underestimating all thespatial moments higher than the zeroth order (see Figure6).

As shown above, the skewness and kurtosis in the caseof the FFADE model are much larger than those predictedby the Time-FADE and the fractal MIM models. The largeskewness at the early time calculated by the FFADE modelis due to the biased (and large) dispersive motion in thedownstream direction modeled by the direction-dependentfractional space derivative. At early time (t = 9 and 49days), the measured plume looks symmetric, deviating sig-nificantly from the positively skewed plumes at the remain-ing sampling cycles (see also Benson et al. [2001]). Thusthe FFADE model overestimates the skewness and kurto-sis at the first two sampling cycles. The overestimation ofskewness and kurtosis using the FFADE model at late time(t = 503 days) shown by Figure 6, however, may not in-dicate the failure of the model itself. We suspect that theskewness at t = 503 days calculated by Adams and Gelhar[1992] may be too small (0.58) due to a truncated samplinground. A complete analysis of the 3-D sample data would beneeded to confirm this. Furthermore, the symmetric distri-bution of the plumes at the earliest time step (9 days) maybe due to the radial flow caused by the 48-hour injectionof source water. The first time step data shows essentiallyzero skewness and kurtosis. The late-time plumes, whichhave less influence of the initial injection and are more re-liable and complete due to the relatively large observationranges, support the applicability of the FFADE model.

If we ignore the moment data, then the MIM-2 modelseems to be a reasonable one for capturing the plume snap-shots (as demonstrated by Figure 7). The simulated con-centrations using the MIM-2 and the FFADE models (withdifferent V and D) are very similar to each other in linear-linear plots, and they both match generally the measuredsnapshots (Figure 7). However, semi-log plots of concen-tration distributions (Figure 8) show clearly the discrep-ancy between these two models. The MIM-2 model fits thefirst two snapshots (t = 9 and 49 days) much better thanthe FFADE model, because the MIM model (10) results inGaussian-type moments at early time. However, the MIM-2model with Brownian motion underestimates the fast mov-ing leading-edge for time t ≥ 126 days (Figure 8). The sameunderestimation can be found for the MIM-1 model, whichis the best-fit model for the measured mass, mean, and vari-ance of solute plumes. As shown by Schumer et al. [2003],the fractal mobile/immobile model with motion describedby the classical 2nd-order ADE predicts a Gaussian lead-ing edge only. On the contrary, the leading edge describedby the FFADE model decays as a power-law, matching theMADE-1 bromide data. The leading edge is delivered byhighly correlated yet fine-scale features such as gravel fillin the bottom of paleochannels [Zheng and Gorelick, 2003].This motion is highly non-Gaussian and requires a spatiallynonlocal operator to account for the dependence on far up-stream concentrations.

When 1 < γ < 2 in the fractal MIM model [see Baeumeret al., 2005], particles can jump faster than the case of

0 < γ < 1 and thus form a stronger downstream front.However, the leading edge still grows slower than the powerlaw (not shown here). Therefore, the space nonlocality isthe dominant mechanism that contributes to large motionsin the leading edge of a plume. In geologic material withstrong space nonlocality that arises from long-range correla-tion of high-K material (even when the material is presentin very low volumes) the space nonlocality should not be ig-nored. This would apply also to K-structures that span the“grid scale” in a numerical model. The fractal space nonlo-cality can be represented with any degree of conditioning bymeasured parameters [Zhang et al., 2007]. Berkowitz andScher [1998, 2001] and Berkowitz et al. [2006] do not ac-count for space nonlocality and do not directly account forthe mobile versus immobile porosity, so their CTRW modelmisses both the mobile mass decay and the heavy leadingedge. These features are not obvious when normalized con-centrations and double linear plots are used to representmodel fit.

5. Conclusions

1. Analytic expressions give a clear picture of the anoma-lous evolution of some commonly measured moments. Thefractal MIM model produces classical ADE-type momentsbefore time tearly ≈ (0.1/β)1/(1−γ), and then either subdif-fusive growth for 0 < γ < 0.5 or superdiffusive growth for0.5 < γ < 1 after time tlate ≈ (3/β)1/(1−γ). On the otherhand, the time (between tearly and tlate) forms a transitionzone where the growth rate of moments (higher than the ze-roth order) can change. It would be misleading to assess thetype of diffusion based on the measured moments withoutconsidering the time scale of observation. At the MADEsite, this transition period is calculated to be from a fewdays to several hundred days.

2. A non-Markovian random walk model can simulate allaspects of fractal time and space nonlocality, and allows usto calculate moments under conditions more complex thanthe Green function of a homogeneous transport equation.

3. Classical CRTW and the Time-FADE do not dis-tinguish between particles in mobile or relatively immobilestates. The 0th moment is constant and any observed dropin mobile mass is not accounted for, resulting in serious mis-match of the declining 0th moment at the MADE site.

4. A fractal MIM model with classical advection and dis-persion matches the 0th, 1st, and 2nd moments at the MADEsite, but at the expense of accurately representing the plumesnapshots. This is due to the classical local space operatorimparting Brownian motion to moving particles, which can-not match the skewed and heavy-tailed leading edge.

5. A fractional dispersion derivative can model the skew-ness and heavy leading edge, but when integrated to infinity,has diverging higher order moments. These moments are fi-nite when the concentration drops below detection limits ormoves beyond some finite well field.

6. Analysis of the 0th through 4th moments, along withsnapshots of the MADE site plumes, shows that space non-locality is a dominant control on plume evolution. Specifi-cally, the time nonlocality primarily leads to the mass decay(power-law decay at late time), and the space nonlocalityleads to the heavy leading edge, the resultant relatively largeskewness and kurtosis. This behavior is expected due to thepresence of long-range paleo-channel-bottom gravel from thebraided stream depositional environment.

7. All of the first five spatial moments should be consid-ered in building the conceptual/physical model and/or cal-ibrating transport parameters, because the first three onescannot capture all of the nonlocalities. The plume skewnessis especially helpful and critical in model selection.

8. A log-concentration plot (instead of a linear one)should be used to evaluate the plume leading edge and thespace nonlocality. The functional form of the leading edgeis an indication of the appropriateness of a fractional spacederivative.

ZHANG ET AL.: MOMENT ANALYSIS FOR SPATIOTEMPORAL FRACTIONAL DISPERSION X - 11

Appendix A: The Integral Transform Methodto Solve the Spatial Moments of PlumesGoverned by the Time-FADE (1)

Taking the Fourier and Laplace transforms (x → k, t → s)

of (1), and after re-arrangement, we get the solution of C in

Fourier and Laplace spaces

C(k, s) =sγ−1

sγ + V ik + Dk2C0(k) , (A1)

where C0(k) is the initial concentration, and we use the nor-

malized concentration here (so C0(k) = 1).

The nth moment about the origin (denoted as νn) for

solutes relates to the nth-order derivative of its concentra-

tion in Fourier space (because the image function is also the

characteristic function)

νn(s) =1

(−i)n

dnC(k, s)

dkn

∣∣∣∣k=0

. (A2)

Placing (A1) into (A2) and assuming n = 0, we get the

zeroth moment

ν0(s) =1

(−i)0d0C(k, s)

dk0

∣∣∣∣k=0

=1

s. (A3)

Taking the Laplace reverse transform of (A3), we get the

moment in real time

ν0(t) = 1 . (A4)

Similarly, we get the 1st- to 4th-order moments about the

origin

ν1(t) =V tγ

Γ(γ + 1), (A5a)

ν2(t) =2V 2t2γ

Γ(2γ + 1)+

2Dtγ

Γ(γ + 1), (A5b)

ν3(t) =6V 3t3γ

Γ(3γ + 1)+

12V Dt2γ

Γ(2γ + 1), (A5c)

ν4(t) =24V 4t4γ

Γ(4γ + 1)+

72V 2Dt3γ

Γ(3γ + 1)+

24D2t2γ

Γ(2γ + 1). (A5d)

The spatial moments are functions of the moments about

the origin:

M(t) = ν0(t) , (A6a)

E(t) =ν1(t)

ν0(t), (A6b)

σ2(t) =ν2(t)

ν0(t)− [ν1(t)]

2

[ν0(t)]2, (A6c)

S(t) =

ν3(t)ν0(t)

− 3 ν2(t)ν0(t)

ν1(t)ν0(t)

+ 2 [ν1(t)]3

[ν0(t)]3

σ3(t), (A6d)

K(t) =

ν4(t)ν0(t)

− 4 ν3(t)ν1(t)

[ν0(t)]2+ 6 ν2(t)[ν1(t)]2

[ν0(t)]3− 3 [ν1(t)]4

[ν0(t)]4

σ4(t)− 3 .

(A6e)

Leading (A4) and (A5) into (A6), we get the spatial mo-

ments shown by (2).

Appendix B: The Asymptotic ExpansionMethod to Solve the Spatial Moments ofPlumes in the Total Phase Governed by theFractal MIM Model (4)

The solution of (4) in Fourier and Laplace spaces is

Ctot(k, s) =1 + βsγ−1

s + βsγ + V ik + Dk2Ctot,0 . (B1)

B1. Mass

Different from the asymptotic expansion method dis-cussed in the following subsections, the integral transformmethod can be used here to derive the mass growth. Similarto (A4), we get the mass growth

Mtot(t) = ν0tot(t) = θm/θtot = ϑ , (B2)

where ν0tot(t) denotes the zeroth moment about the origin

(where the suffix “0” denoting the order of moment, andthe same notation will be used in the following). By assum-ing that the contaminant is initially located in the mobilezone only, we have the initial concentration Cm(t = 0) = 1(after normalization) and Ctot(t = 0) = θm/θtot.

B2. Mean Displacement

Based on (A2) and (B1), we get the first moment aboutthe origin

ν1tot(s) =

V ϑ

s(s + βsγ). (B3)

We then expand the above Laplace transform both at zeroand at infinity in order to derive ν1

tot in real time t. Firstly,for |βsγ−1| < 1 (or |s1−γ | > β, representing the small realtime t), we can expand (B3) to

ν1tot(s) =

V ϑ

s2(1 + βsγ−1)=

V ϑ

s2

∞∑n=0

(−1)n(βsγ−1)n , (B4)

which has the following Laplace reverse transform

ν1tot(tearly) =

∞∑n=0

(−1)nϑV βn t(1−γ)n+1

Γ[(1− γ)n + 2]. (B5)

Secondly, for |β−1s1−γ | < 1 (or |s1−γ | < β, representingthe large real time t), we can expand (B3) to

ν1tot(s) =

V ϑ

βsγ+1(s1−γ/β + 1)=

V ϑ

βsγ+1

∞∑n=0

(−1)nβ−ns(1−γ)n

(B6)which has the following Laplace reverse transform

ν1tot(tlate) =

∞∑n=0

(−1)nϑV β−n−1 t(γ−1)n+γ

Γ[(γ − 1)n + γ + 1]. (B7)

By combining (B5) and (B2), we get the mean displace-ment at early time

Etot(tearly) =ν1

tot(tearly)

ν0tot(t)

= V

∞∑n=0

(−1)nβn t(1−γ)n+1

Γ[(1− γ)n + 2].

(B8)

X - 12 ZHANG ET AL.: MOMENT ANALYSIS FOR SPATIOTEMPORAL FRACTIONAL DISPERSION

Then the combination of (B7) and (B2) results in the later

time mean displacement

Etot(tlate) =ν1

tot(tlate)

ν0tot(t)

= V

∞∑n=0

(−1)nβ−n−1 t(γ−1)n+γ

Γ[(γ − 1)n + γ + 1].

(B9)

B3. Variance

Based on (A2) and (B1), we get the second moment about

the origin

ν2tot(s) =

2V 2ϑ

s(s + βsγ)2+

2Dϑ

s(s + βsγ)= f1(s)+ f2(s) . (B10)

For |βsγ−1| < 1, the first term on the RHS of (B10) can

be expanded as [Baeumer and Meerschaert, 2007]

f1(s) =2V 2ϑ

s(s + βsγ)2=

2V 2ϑ

s3

1

(1 + βsγ−1)2

≈ 2V 2ϑ(N + 1)(βsγ−1)N+1

s3(1 + βsγ−1)2

+2V 2ϑ

s3

N∑n=0

(−1)n(n + 1)(βsγ−1)n ,

(B11)

which has the following Laplace reverse transform (note the

first term on the RHS of (B11) is analytic in a sectorial re-

gion and its produce with s is bounded, see also Baeumer

and Meerschaert [2007])

f1(tearly) ≈ 2ϑV 2N∑

n=0

(−1)n(n + 1)βn t(1−γ)n+2

Γ[(1− γ)n + 3].

(B12)

Similar to (B5), for |βsγ−1| < 1, we can also expand the

second term on the RHS of (B10), take the reverse Laplace

transform, and finally get

f2(tearly) = 2ϑD

∞∑n=0

(−1)nβn t(1−γ)n+1

Γ[(1− γ)n + 2]. (B13)

Then by combining (B12), (B13), (B5), and (B2), we can

get the variance at early time

σ2tot(tearly) =

ν2tot(tearly)

ν0tot(tearly)

−[

ν1tot(tearly)

ν0tot(tearly)

]2

≈ 2V 2N∑

n=0

(−1)n(n + 1)βn t(1−γ)n+2

Γ[(1− γ)n + 3]

+ 2D

∞∑n=0

(−1)nβn t(1−γ)n+1

Γ[(1− γ)n + 2]

−{

V

∞∑n=0

(−1)nβn t(1−γ)n+1

Γ[(1− γ)n + 2]

}2

.

(B14)

Following the same procedure by expanding f1 and f2 in

(B10) at late time, we get the variance at late time

σ2tot(tlate) =

ν2tot(tlate)

ν0tot(tlate)

−[

ν1tot(tlate)

ν0tot(tlate)

]2

≈ 2V 2N∑

n=0

(−1)n(n + 1)βn t(1−γ)n+2

Γ[(1− γ)n + 3]

+ 2D

∞∑n=0

(−1)nβn t(1−γ)n+1

Γ[(1− γ)n + 2]

−{

V

∞∑n=0

(−1)nβn t(1−γ)n+1

Γ[(1− γ)n + 2]

}2

.

(B15)

B4. Skewness

Based on (A2) and (B1), we get the third moment about

the origin

ν3tot(s) =

6V 3ϑ

s(s + βsγ)3+

12DV ϑ

s(s + βsγ)2. (B16)

We can expand the two terms on the RHS of (B16) at

early and late time, respectively, by following the same pro-

cedure discussed in the above subsection. The resultant

third moment about the origin at early time is of the form

ν3tot(tearly) ≈ 6V 3ϑ

N∑n=0

(−1)n(n + 1)βn t(1−γ)n+3

Γ[(1− γ)n + 4]

+ 12DV ϑ

N∑n=0

(−1)n(n + 1)βn t(1−γ)n+2

Γ[(1− γ)n + 3],

(B17)

and the later-time form is

ν3tot(tlate) ≈ 6V 3ϑ

N∑n=0

(−1)n(n + 1)β−n−3 t(γ−1)n+3γ

Γ[(γ − 1)n + 3γ + 1]

+ 12DV ϑ

N∑n=0

(−1)n(n + 1)β−n−2 t(γ−1)n+2γ

Γ[(γ − 1)n + 2γ + 1].

(B18)

Then by combining (B2), (B5), (B14), (B17), and the

Laplace reverse transform of (B10), we can get the skewness

at early time immediately by using the following relationship

Stot(tearly) =

{ν3

tot(tearly)

ν0tot(tearly)

− 3ν2

tot(tearly)ν1tot(tearly)

[ν0tot(tearly)]

2

+ 2

[ν1

tot(tearly)

ν0tot(tearly)

]3}/σ3

tot(tearly) .

(B19)

And combining (B2), (B7), (B15), (B18), and the Laplace

reverse transform of (B10), we can get the skewness at late

time immediately by using

Stot(tlate) =

{ν3

tot(tlate)

ν0tot(tlate)

− 3ν2

tot(tlate)ν1tot(tlate)

[ν0tot(tlate)]

2

+ 2

[ν1

tot(tlate)

ν0tot(tlate)

]3}/σ3

tot(tlate) .

(B20)

ZHANG ET AL.: MOMENT ANALYSIS FOR SPATIOTEMPORAL FRACTIONAL DISPERSION X - 13

B5. Kurtosis

Based on (A2) and (B1), we get the forth moment about

the origin

ν4tot(s) =

24V 4ϑ

s(s + βsγ)4+

72DV 2ϑ

s(s + βsγ)3+

24D2ϑ

s(s + βsγ)2. (B21)

Comparing to (B16), the above equation has one more term

on the RHS. Similarly, we can expand all three terms at

early and late time, and the resultant moment about the

origin at early time is of the form

ν4tot(tearly) ≈ 24V 4ϑ

N∑n=0

(−1)n(n + 1)βn t(1−γ)n+4

Γ[(1− γ)n + 5]

+ 72DV 2ϑ

N∑n=0

(−1)n(n + 1)βn t(1−γ)n+3

Γ[(1− γ)n + 4]

+ 24D2ϑ

N∑n=0

(−1)n(n + 1)βn t(1−γ)n+2

Γ[(1− γ)n + 3],

(B22)

and the later-time form is

ν4tot(tlate) ≈ 24V 4ϑ

N∑n=0

(−1)n(n + 1)β−n−4 t(γ−1)n+4γ

Γ[(γ − 1)n + 4γ + 1]

+ 72DV 2ϑ

N∑n=0

(−1)n(n + 1)β−n−3 t(γ−1)n+3γ

Γ[(γ − 1)n + 3γ + 1]

+ 24D2ϑ

N∑n=0

(−1)n(n + 1)β−n−2 t(γ−1)n+2γ

Γ[(γ − 1)n + 2γ + 1].

(B23)

Then by combining (B2), (B5), (B14), (B17), (B22), and

the Laplace reverse transform of (B10), we can get the skew-

ness at early time immediately by using the following rela-

tionship

Ktot(tearly) =

{ν4

tot(tearly)

ν0tot(tearly)

− 4ν3

tot(tearly)ν1tot(tearly)

[ν0tot(tearly)]

2

6ν2

tot(tearly)[ν1tot(tearly)]

2

[ν0tot(tearly)]

3− 3

[ν1

tot(tearly)

ν0tot(tearly)

]4}

/σ4

tot(tearly)− 3 .

(B24)

And combining (B2), (B7), (B15), (B18), (B23), and the

Laplace reverse transform of (B10), we can get the skew-

ness at late time immediately by using

Ktot(tlate) =

{ν4

tot(tlate)

ν0tot(tlate)

− 4ν3

tot(tlate)ν1tot(tlate)

[ν0tot(tlate)]

2

6ν2

tot(tlate)[ν1tot(tlate)]

2

[ν0tot(tlate)]

3− 3

[ν1

tot(tlate)

ν0tot(tlate)

]4}

/σ4

tot(tlate)− 3 .

(B25)

Appendix C: The Asymptotic ExpansionMethod to Solve the Spatial Moments ofPlumes in the Mobile Phase Governed bythe Fractal MIM Model (10)

The solution of (10) in Fourier and Laplace spaces is

Cm(k, s) =1

s + βsγ + V ik + Dk2Cm,0 . (C1)

The same asymptotic expansion used to solve the moments

for solutes in the total phase discussed in the previous ap-

pendix can be used here. For brevity, here we list only the

final moments about the origin.

The zeroth moment about the origin (which is also the

mass Mm(tearly) and Mm(tlate) ) decays like

ν0m(tearly) =

∞∑n=0

(−1)nβn t(1−γ)n

Γ[(1− γ)n + 1], (C2a)

ν0m(tlate) =

∞∑n=0

(−1)nβ−n−1 t(γ−1)n+γ−1

Γ[(γ − 1)n + γ], (C2b)

which leads to the approximation (11) directly.

The first moment about the origin is:

ν1m(tearly) ≈ V

N∑n=0

(−1)n(n + 1)βn t(1−γ)n+1

Γ[(1− γ)n + 2], (C3a)

ν1m(tlate) ≈ V

N∑n=0

(−1)n(n + 1)β−n−2 t(γ−1)n+2γ−1

Γ[(γ − 1)n + 2γ].

(C3b)

The mean displacement can then be calculated by dividing

(C3) by (C2) (see also (A6b)), leading to the approximation

(12).

The second moment about the origin at early time is

ν2m(tearly) ≈ 2V 2

N∑n=0

(−1)n(n + 1)βn t(1−γ)n+2

Γ[(1− γ)n + 3]

+ 2D

N∑n=0

(−1)n(n + 1)βn t(1−γ)n+1

Γ[(1− γ)n + 2],

(C4)

and the later-time second moment is

ν2m(tlate) ≈ 2V 2

N∑n=0

(−1)n(n + 1)β−n−3 t(γ−1)n+3γ−1

Γ[(γ − 1)n + 3γ]

+ 2D

N∑n=0

(−1)n(n + 1)β−n−2 t(γ−1)n+2γ−1

Γ[(γ − 1)n + 2γ],

(C5)

The variance can then be calculated by combining (C4) (or

(C5)) with (C3) and (C2) (see also (A6c)), leading to the

approximation (13).

Similarly, we get the third moment about the origin at

early time

ν3m(tearly) ≈ 6V 3

N∑n=0

(−1)n(n + 1)βn t(1−γ)n+3

Γ[(1− γ)n + 4]

+ 12DV

N∑n=0

(−1)n(n + 1)βn t(1−γ)n+2

Γ[(1− γ)n + 3],

(C6)

X - 14 ZHANG ET AL.: MOMENT ANALYSIS FOR SPATIOTEMPORAL FRACTIONAL DISPERSION

and late time

ν3m(tlate) ≈ 6V 3

N∑n=0

(−1)n(n + 1)β−n−4 t(γ−1)n+4γ−1

Γ[(γ − 1)n + 4γ]

+ 12DV ϑ

N∑n=0

(−1)n(n + 1)β−n−3 t(γ−1)n+3γ−1

Γ[(γ − 1)n + 3γ].

(C7)

The skewness can then be calculated by combining (C6) (or(C7)) with (C4) (or (C5)), (C3), and (C2) (see also (A6d)),leading to the approximation (14).

Finally, we get the fourth moment about the origin atearly time

ν4m(tearly) ≈ 24V 4

N∑n=0

(−1)n(n + 1)βn t(1−γ)n+4

Γ[(1− γ)n + 5]

+ 72DV 2N∑

n=0

(−1)n(n + 1)βn t(1−γ)n+3

Γ[(1− γ)n + 4]

+ 24D2N∑

n=0

(−1)n(n + 1)βn t(1−γ)n+2

Γ[(1− γ)n + 3],

(C8)

and late time

ν4m(tlate) ≈ 24V 4

N∑n=0

(−1)n(n + 1)β−n−5 t(γ−1)n+5γ−1

Γ[(γ − 1)n + 5γ]

+ 72DV 2N∑

n=0

(−1)n(n + 1)β−n−4 t(γ−1)n+4γ−1

Γ[(γ − 1)n + 4γ]

+ 24D2N∑

n=0

(−1)n(n + 1)β−n−3 t(γ−1)n+3γ−1

Γ[(γ − 1)n + 3γ].

(C9)

The skewness can then be calculated by combining (C8) (or(C9)) with (C6) (or (C7)), (C4) (or (C5)), (C3), and (C2)(see also (A6e)), leading to the approximation (15).

Appendix D: Spatial Moments of PlumesGoverned by the FFADE Model (16)

The solution of the FFADE model (16) in Fourier andLaplace spaces is

C(k, s) =1 + βsγ−1

s + βsγ + V ik + D(ik)αC0 . (D1)

Using (A2), we have the zeroth moment about the origin

ν0(s) =C0

s, (D2)

and thus the mass is the same as (5).The first moment about the origin is

ν1(s) =V C0

s(s + βsγ). (D3)

which is similar to (B3). So the mean displacement for (16)is the same as (6).

The solution of the FFADE model (16) has a heavy lead-ing tail, so heavy that the moments higher than 1 are in-finite for 1 < α < 2. However, given a truncation ofdetection length L or a truncation of concentrations (dueto the detection limit), the apparent moments will still be

finite (as demonstrated by the numerical solutions in thetext) although they can not be solved easily using analyt-ical method. To further demonstrate this analytically, weremove the mass partition and keep the heavy leading tailin the FFADE model (16), and then we get the followingspace-only fADE

∂

∂tC(x, t) = −V

∂

∂xC(x, t) + D

∂α

∂xαC(x, t) . (D4)

Here we assume that the concentration was measuredaccurately over a finite domain [−L, L]; i.e. νL

0 (t) =∫ L

−LC(x, t) dx, νL

1 (t) =∫ L

−LxC(x, t) dx, etc., with L being

large enough for the concentration to exhibit a strong power-

law tail. In particular, as C(x, t) = 1

(Dt)1/α gα

(x−V t

(Dt)1/α

)

with Fourier transform gα(k) = exp((ik)α) [Meerschaert andScheffler, 2001], we assume that L >> V t is large enoughso that

C(L, t) ≈ 1

(Dt)1/α

(L− V t

(Dt)1/α

)−α−1 /Γ(−α)

=Dt

Γ(−α)(L− V t)−α−1

(D5)

using the fact that gα(x) ≈ x−α−1/Γ(−α) for x large [Meer-schaert and Scheffler, 2001]. As ν0 = 1 we can approximateνL0 via

νL0 (t) ≈1− Dt

Γ(−α)

∫ ∞

L

(x− V t)−α−1 dx

=1− Dt(L− V t)−α

αΓ(−α).

(D6)

where the suffix “L” denotes the apparent moments due tothe truncation of plumes with a maximum detection length.

Similarly, using ν1(t) = V t, we approximate

νL1 (t) ≈ V t− Dt(L− V t)1−α

(α− 1)Γ(−α).

As the second and higher moments are infinite, the inte-grals are dominated by the behavior of C(x, t) near L; i.e.

∫ L

−L

x2C(x, t) dx = ε +

∫x2 Dt

Γ(−α)(x− V t)−α−1 dx

∣∣∣∣x=L

.

Centering about the mean yields an apparent variance of

σ2,L(t) ≈ Dt(L− V t)2−α

(2− α)Γ(−α).

Note the illusion of Fickian behavior with alluded dis-persion coefficient of σ2,L(t)/2t ≈ DL2−α

2(2−α)Γ(−α)growing with

scale. Also note the consistency with the Fickian (α = 2)case as limα→2−(2−α)Γ(−α) = 1/2. Similarly we computeapparent skewness of

SL(t) ≈ Dt(L− V t)3−α

(3− α)Γ(−α)

/[σ2,L(t)]1.5, (D7)

and apparent kurtosis of

KL(t) ≈ Dt(L− V t)4−α

(4− α)Γ(−α)

/[σ2,L(t)]2 − 3. (D8)

More often than not, however, the concentration will bemeasured as long as the concentration is above a minimummeasurable concentration cmin. Assume the minimum mea-

ZHANG ET AL.: MOMENT ANALYSIS FOR SPATIOTEMPORAL FRACTIONAL DISPERSION X - 15

surable concentration is such that a power-law tail is appar-ent. Then cmin = Dt(L− V t)−α−1

/Γ(−α) or

L(t) = V t +

(Dt

cminΓ(−α)

) 1α+1

. (D9)

Inserting into above formulas yields the apparent moments

νmin0 (t) = 1− cmin

α

(Dt

Γ(−α)cmin

)1/(1+α)

, (D10)

νmin1 (t) = V t− cmin

α− 1

(Dt

Γ(−α)cmin

)2/(1+α)

, (D11)

σ2,min(t) =cmin

2− α

(Dt

Γ(−α)cmin

)3/(1+α)

, (D12)

Smin(t) =(2− α)1.5

c0.5min(3− α)

(Dt

Γ(−α)cmin

)− 12(1+α)

, (D13)

and

Kmin(t) =(2− α)2

cmin(4− α)

(Dt

Γ(−α)cmin

)−1/(1+α)

− 3, (D14)

where the suffix “min” denotes the apparent moments dueto the truncation of plumes with a minimum concentrationdetection limit.

Extending to the FFADE (16) is not trivial, where thesubordination technique is required to capture the influenceof additional temporal fractional derivatives on spatial mo-ments (for example, see Baeumer et al. [2005] for the ap-plication of subordination technique). We will pursue thisin a future study. Here the random walk method discussedin the text is recommended to approximate the momentshigher than 1 for the FFADE model (16). As a preliminaryverification of the random walk method, we solved the spa-tial moments for the space fADE (D4) using random walks,as shown in Figure 9 for a specific example with a trunca-

Time

Spa

tial

Mom

ents

Mass

MeanVariance

Skewness

Kurtosis104

102

100

10-2

10-4

10-2 10-1 100 101 102

Figure 9. RW solutions (symbols) versus the ana-lytical approximations (lines) of moments for the spacefADE model (D4). The model parameters are: α = 1.3,V = 0.3, D = 1, and L = 100.

tion of detection length (representing the furthest observa-tion well). The random walk solutions match the analyticalapproximations well for t ¿ L/V . Within this period, be-cause the fastest leading edge is truncated, the variance ofthe truncated plumes is no longer infinite but increases al-most linearly with time (which can be confused with theGaussian model). Meanwhile, the skewness decreases liket0.5 and the kurtosis decreases linearly with time. All ofthese apparent moments are finite and measurable in thefield.

Appendix E: Non-Markovian Random WalkApproximation of the Spatial Moments

To approximate the spatial moments using the randomwalk method, we first track the walker (or particle) whosedynamics satisfies the Lagrangian version of the nonlocaltransport equation. After knowing each particle’s position(and status) at the observation time t, we can calculate thenth sample central-moment µn(t) at t by using

µn(t) =

J∑i=1

[Xi(t)− E(t)]nCi(t) , (E1)

where i denotes the ith particle, Xi(t) represents the par-ticle location at time t, J is the total number of particles(should be sufficiently large), and Ci(t) represents the par-ticle’s concentration. For nonreactive tracer and constantporosity, the normalized concentration Ci(t) can be simpli-fied to 1/J if every particle contains the same mass. Themean displacement E(t) in (E1) can be calculated simplyusing E(t) =

∑Ji=1 Xi(t)/J . In the following we explain

how to make the particle to move and rest.

E1. The Time-FADE Model (1)

We use the following two-step RW algorithm to approxi-mate the Time-FADE (1):

tnext = tnow + W , (E2a)

X(tnext) = X(tnow) + V [X(tnow)]dt + ω√

2Ddt , (E2b)

where tnow denotes the current time, W is a γ-stable randomvariable that represents the real (clock) time correspondingto the operational time (dt) spent during the next jump, dtis the pre-defined, small enough (motion) time step, and ωis independent normally distributed random variables withzero mean and unit variance. The random time W can begenerated numerically by using the formula

W =

[∣∣∣∣cos

(πγ

2

)∣∣∣∣dtβ

]1/γ

Sγ(β∗ = 1, σ = 1, ψ = 0) , (E3)

where Sγ(β∗ = 1, σ = 1, ψ = 0) denotes a maxi-mally skewed, standard, zero-mean stable variable (see alsoSamorodnitsky and Taqqu [1994] for the property of stan-dard stable). If W = dt, then the scheme (E2) reduces tothe RW approximation of solution for the 2nd-order ADE(for instance, see LaBolle at al. [1996, 1998]). If the disper-sion coefficient D varies in space, then an additional term,−[∂D(x)/∂x]dt, should be added to the RHS of (E2b) tocapture the additional drift (but not dispersion) caused bythe space-gradient of D(x) (see also LaBolle at al. [1996],among many others, for the RW scheme for the 2nd-orderADE with variable D).

Note equation (E3) converts the operational time dt tothe real time W . We emphasize here that the sequenceof this conversion is opposite to the process developed by

X - 16 ZHANG ET AL.: MOMENT ANALYSIS FOR SPATIOTEMPORAL FRACTIONAL DISPERSION

Benson et al. [submitted] and Magdziarz et al. [2007]. Theopposite conversion is selected here due to its computationalefficiency for the specific governing equation considered inthis study, while the process in Benson et al. [submitted] is amore generalized, “standard” one for various time-nonlocaltransport equations. More details abut the comparison be-tween different conversions will be discussed in a future pa-per. Also note that all particles are in motion during thewhole real time W , which is different from other particledynamics discussed below.

E2. The Fractal MIM Model (4) and (10)

The fractal MIM model has different governing equationsfor plumes in different phases, and the analytical methodhas to be applied separately for different equations to getthe moments, as demonstrated above. The RW method,however, can distinguish the particle’s status (in motion ornot) easily, and there is no need to run the RW scheme morethan once.

We use the following algorithm to approximate the MIMmodel for all phases (including (4) and (10)):

tnext = tnow + dt + W , (E4a)

X(tnow + dt) = X(tnow) + V [X(tnow)]dt + ω√

2Ddt .(E4b)

This scheme requires less CPU time to finish a model than(E2), because the tnext increases faster due to the additionalmotion time dt in (E4a). The dt term in (E4a) is used to cap-ture the drift of time expressed by ∂/∂t in the MIM model(4) and (10).

The particle is in motion from tnow to tnow + dt, andthen stays stagnant in the immobile zone from tnow + dt totnext = tnow + dt + W . So the status of each particle atany observation time can be determined directly using thescheme (E4).

E3. The FFADE model (16)

The FFADE model (16) has the same temporal derivativeterms (representing the drift and dispersion in time) as thefractal MIM model (4) and (10). The only difference is thatthe FFADE has fractional dispersion (instead of the Gaus-sian type) in space. So its RW scheme is similar to (E4)except for the dispersive length in space

tnext = tnow + dt + W , (E5a)

X(tnow + dt) = X(tnow) + V [X(tnow)]dt + D1/αdLα(t) ,(E5b)

where dLα(t) denotes random noise underlying an α-orderLevy motion. Here the variable dLα(t) can be generatednumerically using:

dLα =

[∣∣∣∣cos

(πα

2

)∣∣∣∣dt

]1/α

Sα(β∗ = 1, σ = 1, ψ = 0) . (E6)

If W = 0 in (E5a), then this non-Markovian RW schemereduces to the Markovian RW scheme proposed by Zhang etal. [2006a] to approximate the space fractional ADE. More-over, if D is space dependent, then an additional term shouldbe added in (E5b) to compensate the influence of D(x) onsolute drift (see Zhang et al. [2006a] for details).

All of the RW schemes can be extended to high dimen-sions. Because the time is dimensionless, the first step (ac-counting for the real time and the waiting time) in the RWscheme remains unchanged, and only the second step (calcu-lating the jump) needs to be extended. However, this exten-sion is not straightforward, because the simultaneous jumps

along each direction can be dependent and non-orthogonal.The Markovian RW scheme derived by Zhang et al. [2006b]for approximating the multiscaling, space-fractional ADEcan be used for this extension.

Acknowledgments. This work was supported by the Na-tional Science Foundation under DMS–0417869, DMS–0417972,DMS–0139927 and DMS–0139943, and the grant DE-FG02-07ER15841 from the Chemical Sciences, Geosciences, and Bio-sciences Division, Office of Basic Energy Sciences, Office of Sci-ence, U.S. Department of Energy. Y.Z. was also partially sup-ported by the Desert Research Institute. Any opinions, findings,conclusions or recommendations do not necessary reflect the viewsof the NSF, DOE or DRI.

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Boris Baeumer, Department of Mathematics and Statistics,University of Otago, Dunedin, New Zealand. (bbaeumer@maths.otago.ac.nz)