Taylor dispersion in heterogeneous porous media: Extended method of moments, theory, and modelling...

PHYSICS OF FLUIDS 26, 022104 (2014)

Taylor dispersion in heterogeneous porous media:Extended method of moments, theory, and modellingwith two-relaxation-times lattice Boltzmann scheme

Alexander Vikhansky1,a) and Irina Ginzburg2,b)

1CD-adapco, Trident Park, Basil Hill Road, Didcot, United Kingdom2Irstea, Antony Regional Centre, HBAN, 1 rue Pierre-Gilles de Gennes CS 10030,92761 Antony Cedex, France

(Received 2 July 2013; accepted 20 January 2014; published online 19 February 2014)

This article describes a generalization of the method of moments, called extended

method of moments (EMM), for dispersion in periodic structures composed of im-

permeable or permeable porous inclusions. Prescribing pre-computed steady state

velocity field in a single periodic cell, the EMM sequentially solves specific lin-

ear stationary advection-diffusion equations and restores any-order moments of the

resident time distribution or the averaged concentration distribution. Like the pio-

neering Brenner’s method, the EMM recovers mean seepage velocity and Taylor

dispersion coefficient as the first two terms of the perturbative expansion. We con-

sider two types of dispersion: spatial dispersion, i.e., spread of initially narrow pulse

of concentration, and temporal dispersion, where different portions of the solute

have different residence times inside the system. While the first (mean velocity)

and the second (Taylor dispersion coefficient) moments coincide for both prob-

lems, the higher moments are different. Our perturbative approach allows to link

them through simple analytical expressions. Although the relative importance of

the higher moments decays downstream, they manifest the non-Gaussian behaviour

of the breakthrough curves, especially if the solute can diffuse into less porous

phase. The EMM quantifies two principal effects of bi-modality, as the appearance

of sharp peaks and elongated tails of the distributions. In addition, the moments can

be used for the numerical reconstruction of the corresponding distribution, avoid-

ing time-consuming computations of solute transition through heterogeneous media.

As illustration, solutions for Taylor dispersion, skewness, and kurtosis in Poiseuille

flow and open/impermeable stratified systems, both in rectangular and cylindrical

channels, power-law duct flows, shallow channels, and Darcy flow in parallel porous

layers are obtained in closed analytical form for the entire range of Peclet num-

bers. The high-order moments and reconstructed profiles are compared to their pre-

dictions from the advection-diffusion equation for averaged concentration, based

on the same averaged seepage velocity and Taylor dispersion coefficient. In paral-

lel, we construct Lattice-Boltzmann equation (LBE) two-relaxation-times scheme

to simulate transport of a passive scalar directly in heterogeneous media specified

by discontinuous porosity distribution. We focus our numerical analysis and as-

sessment on (i) truncation corrections, because of their impact on the moments,

(ii) stability, since we show that stable Darcy velocity amplitude reduces with the

porosity, and (iii) interface accuracy which is found to play the crucial role. The

task is twofold: the LBE supports the EMM predictions, while the EMM provides

non-trivial benchmarks for the numerical schemes. C� 2014 AIP Publishing LLC.

[http://dx.doi.org/10.1063/1.4864631]

a)Electronic mail: [email protected])Corresponding author. Electronic mail: [email protected]

1070-6631/2014/26(2)/022104/52/$30.00 C�2014 AIP Publishing LLC26, 022104-1

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022104-2 A. Vikhansky and I. Ginzburg Phys. Fluids 26, 022104 (2014)

I. INTRODUCTION

The classical works by Taylor,1 Aris,2 Brenner,3 Auriault and Adler4 imply that the transport of a

solute in a porous media is governed by an advection-diffusion equation and provide a framework for

calculation of the effective transport coefficients. This analysis (correct for infinitely long spatially-

periodic systems) is in contradiction with the existing geological data for tracer propagation in

natural soils and laboratory soil columns. The discrepancy stimulated development of alternative

semi-empirical models such as mobile/immobile models (MIM), subordinated Brownian motion,

fractal transport, continuous time random walk (CTRW); for review see Refs. 5 and 6 and references

therein. Although these models can fit virtually any experimental data, they do not predict how the

structure of the porous matrix or its sorptive properties affect the dispersion of the solute. This lack

becomes especially important for the bespoken packings, e.g., in chromatographic columns.7 The

work8 narrows the gap between these two approaches by introducing the non-local effects via a

memory kernel. Since the memory kernel is convoluted with the microscopic concentration field,

this approach requires some additional modelling in order to link the macroscopic flux to the pore-

averaged concentration. It is why upscaling of the convection-diffusion equation is attractive.9, 10

In the present work we suggest an extension of the method of moments that goes beyond the

classical second-order (i.e., advection-diffusion) approximation, and allows (at least in principle)

to calculate moments of any order. The higher moments are generated by the non-uniformity of

the flow field, and/or heterogeneous structure, and transported by advection and diffusion. In fact,

analysis of simple model examples suggests that if the solute can diffuse into the less permeable

porous inclusion, the non-Gaussian effects persist in the system for a very long time. We call the

proposed method extended method of moments (EMM). The EMM extends the previous works11–14

for computing third and fourth moments of the residence time distribution (RTD) to (i) any-order,

(ii) entire range of Peclet (Pe) numbers, (iii) both spatial and temporal dispersions, and (iv) both for

the RTD and averaged concentration distribution. In this work the EMM is presented in the form of

mathematical algorithm.

The idea develops based on the consideration that in a periodic structure solution of the transport

equation can be represented as a product of a long low frequencymonochromatic wave and a spatially

periodic function with period equal to the size of the cell. Similar to the Brenner method3 for

estimation of the dispersion coefficient, the higher moments can be restored sequentially by solving

specific steady-state linear advection-diffusion equations in a single cell. This procedure simplifies

in the limit of infinite Peclet numbers. The interest in knowing the high-order moments is that they

allow to quantify the deviations from the solution predicted by the advection-diffusion equation

(ADE) for averaged concentration, say C(t, x), solving it for the same mean seepage velocity and

Taylor dispersion coefficient. In addition, the RTD distribution P(t, x) and average concentration

profiles C(t, x) can be restored from their predicted moments and compared with the ADE solutions

or experimental breakthrough curves. We adapt reconstruction procedure15 for this purpose.

In fact, since C(t, x) is a function of both time and space, there are two options for the method

of moments. The first one, pioneered by Danckwerts,16 quantifies dispersion via RTD in a system;

in this case one deals with its temporal moments, as µn(x) =�

t tnC(t, x)dt, at a given point x (e.g.,

outlet of a chemical reactor). The second one follows Aris2 and analyses the evolution of the spatial

moments, asµn(t)=�

x xnC(t, x)dx, in time. Hence, the former is referred to as “temporal” dispersion

and the latter as “spatial” dispersion. Although the spatial approach is less sensitive to the boundary

effects, in the past the temporal approach was more popular among experimentalists because it is

much easy to create a step boundary conditions at the inlet than a narrow concentration distribution

inside themedia.Modern nuclear magnetic resonance (NMR)methodsmake the spatial specification

of dispersion in a porous media more feasible.17 Probability distributions of molecular displacements

during a given time interval (so-called propagator) or its spatial moments can be measured with high

precision. The EMM treats temporal and spatial dispersions in similar ways and demonstrates that

the corresponding moments are linked by simple analytical formulae.

In parallel, we aim to confirm the EMM via direct numerical simulations. For this purpose we

construct the Lattice Boltzmann Equation (LBE) scheme for solving transport equation in hetero-

geneous soil characterized by discontinuous porosity distribution. The LBE schemes are routinely



employed for flow computations in packed beds7,18 and reconstructed images of homogeneous soil

samples.19, 20 In fact, already early works21,22 have developed advection-diffusion LBE schemes for

dispersion studies in homogeneous porous media. By showing their validity to predict Taylor-Aris

dispersion in capillary and channel,23 these schemes have been applied for studies of dispersion

in expansion/contraction geometries,23 mixing phenomena in a single intersection24 and collector

efficiency,25 monitoring the breakthrough curves of the nanoparticles in filtrated flow. However,

some real rocks, as carbonate aquifers, manifest multi-scale, fractured/micritic structures where

direct flow and transport computations are difficult. The LBE schemes are attractive for modelling

of flow and transport in heterogeneous porous media because of their simple handling of boundary

and interfaces. So far, within the mesoscopic Stokes-Brinkman26,27 LBE modelling,28–32 the porous

blocks are specified via porosity/permeability distributions where open (Stokes) and porous (Darcy)

flow are coupled via the resistance forcing. In this work, we assume for both EMM and LBE meth-

ods that the steady state periodic (micro, macro or mesoscale) velocity field were resolved with the

Navier-Stokes, Darcy or Stokes-Brinkman equation, respectively. When the porosity is set uniform

and equal to one, the EMM and LBE scheme solve the transport equation in open flow.

Several LBE approaches have been proposed for the modelling of the mesoscopic and macro-

scopic transport. Exact local conservation of mass in soil porosity can be achieved with the ap-

propriate choice of equilibrium distribution. This approach, suitable for any functional relationship

between the conserved and diffusive flux variables, such as φC and C, for concentration C in soil

porosity φ, has been introduced for transport in variably saturated soil,33 extended for modelling

of Richard’s equation,34 saltwater intrusion in density-dependent underground water flow,35 dis-

persive Henry problem in anisotropic heterogeneous porous media,36 and contaminant transport in

variably saturated flow.37 Alternative approaches38,39 apply porosity-dependent collision update for

modelling of flow and transport in bi-modal rock. The work37 also examines another equilibrium

approach where the diffusive and mass variables are the same. However, in current work, because

of the sharp porosity contrasts, continuity relations of the LBE schemes40 dictate a necessary sep-

aration for mass and diffusive flux equilibrium variables. The proposed scheme extends the TRT

models41 from steady state diffusion equation with discontinuous coefficients to transient hetero-

geneous advection-diffusion equation. We aim to examine the capabilities of the LBE scheme to

reproduce the Taylor dispersion and “non-Gaussian” effects in heterogeneous media. In turn, we

compute temporal and spatial moments, simulating temporal and spatial dispersion, respectively,

and compare them, as well as the numerical profiles, to EMM and ADE predictions.

The LBE scheme is designed in the frame of the two-relaxation-times (TRT) collision operator40

and most general equilibrium forms,42 appropriate for all commonly used one-, two-, and three-

dimensional velocity sets. The TRT operator extends available parameter range, especially for

diffusion-dominant regime, where single-relaxation-time model43 loses accuracy.23,37 The TRT

operator may also improve for stability at high Peclet numbers.44–46

The correction of numerical-diffusion and necessary stability conditions44 are extended in this

work for heterogeneous transport equation. In the LBE modelling, these stability conditions play a

very similar role to the well known advection-diffusion stability conditions for forward-time finite-

difference schemes.47 We formulate necessary stability constraints on the velocity magnitude, versus

equilibrium diffusion-scale variable, and validate them by numerical computations. Also, truncation

analysis46 is extended for uniform but porous Darcy flow. The purpose is to derive how the third and

fourth order truncation corrections affect the third and fourth order moments. In the presence of the

interface, they are amended by the effective precision of the interface continuity conditions, which

are implicitly set by the proposed scheme. With a purpose to better understand this coupling, we

analytically construct an approximate solution of the scheme for wave decay in stratified soil. This

solution indicates that the accuracy of the interface conditions dominates the truncated corrections

in bi-modal Darcy flow.

The paper is organized as follows. Section II describes the methodology of the EMM for

temporal and spatial dispersion, expresses the moments through the coefficients of their dispersion

relations, and links the spatial-temporal expansions between them. The reconstruction algorithm

is presented, to restore the distribution shapes from their moments. This section is concluded

with a summary of main findings. Section III applies the above theory for constant Darcy flow



in variable porosity aquifer, constructs the exact and perturbative solutions for stratified channel

and validates them by direct LBE computations. Section IV derives generic governing advection-

diffusion equations of the EMM method for any-order expansion of dispersion relations, illustrates

them for channel flows and provides details of their solutions for four moments in uniform porous

cylindrical flow. Similar solutions for power-law fluid, two-dimensional Poiseuille flow and flow in

shallow channelswith different cross-section shapes conclude this section. SectionV extends channel

solutions to stratified systems and provides analytical solutions for open/impermeable, rectangular,

and cylindrical, systems of two parallel channels, where although no advection happens in the porous

layer, the solute may diffuse there. Numerical results conclude this section. Section VI describes the

TRT LBE scheme for transport in heterogeneous porous aquifers, undertakes its numerical analysis

and validates for the evolution of waves in stratified porous layers. Section VII concludes the paper.

The Appendix provides truncation terms of the proposed TRT scheme in uniform porosity flow,

links them with the corrections for the higher moments, and constructs approximate solution of the

scheme for evolution of waves in stratified Darcy flow.

II. TRANSPORT EQUATION AND METHOD OF MOMENTS

The problem is formulated as following. Consider flow of a viscous incompressible liquid

through a spatially periodic porous media, e.g., similar to that shown in Fig. 1. There is a passive

substance dissolved in the liquid. The particles are microporous and the solute occupies both liquid

and solid phases. The pore scale of the solid particles is small and the diffusive transport inside the

solid phase is dominated by themolecular diffusion, so themechanical dispersion at micro-scales can

be completely neglected. The concentration of the solute is described by the advection-dispersion

equation,48 namely,

∂t (φC) + ∇ ·�

uφC�

− ∇ · (φD0∇C) = 0, uφ = φu, (1)

where C is concentration (in mole of the solute per mole of the solution), φ is porosity of the solid

particles, D0 is molecular diffusivity, and uφ and u are Darcy and seepage velocities, respectively.

Such formulation of the problem allows for the description of the transport through entire the

medium without explicit specifying of interface conditions. The porosity (and possibly–diffusivity)

undergoes a sharp change at the boundary between the liquid phase (where φ = 1) and the solid

phase. Yet, the concentration, Darcy velocity and the diffusive flux −φD0∇C are continuous. Also,

FIG. 1. Schematic view of a periodic porous structure.



we assume that the mean seepage velocity

U =�uφ��φ�

, U = |U | (2)

is directed along one of coordinate axes, where �·� =�

V· dV denotes volume integration over a

single cell. Without loss of generality, we consider its amplitude U as the characteristic velocity in

this work.

In what follows we will assume that the flow has been resolved in the same structure where

Eq. (1) is to be solved, and therefore, the periodic velocity field uφ(r) is known. In particular, we keep

in mind the Brinkman model26 for porous media which provides continuous Darcy velocity field

uφ in block-wise structure, prescribing the permeability for resistance forcing inside each block.

This model allows to couple the open fractures and impermeable zones. One can also compute

the velocity field uφ solving the Darcy equation for pressure distribution. Yet, this approach is

mainly restricted to relatively homogeneous porous obstacles.49 Finally, when the core sample is

well resolved, or provided from synthetic structures, the velocity field can be obtained by solving

the Stokes/Navier-Stokes equations directly in soil porosity. For this microscopic velocity field,

porosity φ is set equal to one in Eq. (1). In this work, we will provide examples for open flow in

Sec. IV B and for simplest heterogeneous configurations in Secs. III andV. In all these cases, exact

velocity profiles are employed to illustrate dispersion coefficients, still the developed mathematical

procedure to predict dispersion relations for Eq. (1) is valid for any steady state periodic velocity

field.

A. Temporal and spatial dispersions

Similarly to the two ways of describing fluid flow in fluid mechanics, i.e., Lagrangian and

Eulerian, there are also two ways of specifying dispersion: in the temporal description the concen-

tration at a point, e.g., outlet of a chemical reactor, is depicted as a function of time; in the spatial

description one considers the shape of the concentration profile at different times.

The temporal specification (sometimes referred to as temporal mixing) is widely used in hy-

drology and chemical engineering and it is formulated as follows. Consider a semi-infinite medium

x ≥ 0, where the mean seepage velocity is directed along the x-axis, and apply the following initial

and boundary conditions:

C|t=0 = 0, C|x=0 = H (t), (3)

where H(t) is Heaviside step function. The concentration averaged over the cross-section at a point

x is called the breakthrough curve

C(t, x) =�

C(t, x)φ(x)dydz�

φ(x)dydz. (4)

Its time derivative is the residence time distribution:16,50

P(t, x) = ∂tC(t, x). (5)

Alternatively, the RTD can be interpreted as the solution of Eq. (1) with boundary condition

C|x=0 = δ(t). (6)

Once the RTD is known it provides solution for more general boundary condition than (6): if the

concentration Cin(t) at x = 0 is a function of time only, the mean concentration at the outlet at

x = L is given by the convolution

Cout (t) =� ∞

0

P(τ, L)Cin(t − τ )dτ. (7)

In the absence of diffusion and temporal dispersion the RTD is just a Dirac delta function and

Cout (t) = Cin(t − |U |−1 x). Quite often instead of direct calculation or measurement of the RTD, its



first few temporal moments are analysed

µn(x) =� ∞

0

tn P(t, x)dt =� ∞

−∞tn P(t, x)dt, n = 0, 1, 2 . . . , (8)

the lower integration limit in the above integral is replaced by −∞ because P(t, x) = 0 when

t < 0.

The spatial specification of the dispersion dating back from the works of Taylor,1 Aris2 and

Brenner3 is more general than the temporal one. Although it allows for consideration of three-

dimensional transport, in the present work we restrict our attention to its one-dimensional version.

Consider a medium infinite in the x-direction; the initial condition for the concentration is

C|t=0 = δ(x), (9)

and our focus is on the evolution of the averaged concentration C(t, x) or its spatial moments:

µn(t) =� ∞

−∞xnC(t, x)dx, n = 0, 1, 2 . . . . (10)

B. Scale separation

While the geometry of the medium and the steady-state flow field u are periodic functions of

x, C(t, x) is not, i.e., as an initially narrow pulse of concentration moves through a porous periodic

structure, its second moment1,3 grows as a linear function of x. In order to make use of the periodic

nature of the problem we introduce the following Ansatz:

C(ω, γ ; t, x) =1

2πP(ω, γ ; x/�) exp [i (γ x − ωt)] , (11)

where P(ω, γ ; x/�) is a spatially periodic function, that is the solution of Eq. (1) is a product of the

oscillating part P(ω, γ ; x/�) with a slowly varying component. The wavenumber (spatial frequency)

γ and temporal frequency ω are unknowns to be found.

In order to underline the scale separation we introduce the small parameter � and the fast

variables x/�, so that, while integrating over a single cell, one can assume that exp [i(γ x − ωt)] is

constant for any f( · ):

� f (x/�) exp [i (γ x − ωt)]� = � f (x/�)� exp [i (γ x − ωt)] . (12)

Without loss of generality it can be assumed that�

Pφ�

�φ�= 1. (13)

Plugging Eq. (11) into Eq. (1), the later becomes

∇ ·�

uφP�

− ∇ ·�

φD0∇P�

= iφ [ω − γ ux ] P

+ iγ

�

φD0

∂P

∂x+

∂φD0P

∂x

�

− γ 2φD0P, (14)

where ux is the x-component of the velocity vector u.

Certainly, ω and γ in Eq. (14) are not independent: if one is fixed, the second has to satisfy the

solvability conditions, i.e., to ensure that the RHS of the equation has zero mean. Solution of (14)

for a fixed ω is given by the pair γ (ω), P(ω, γ (ω); x/�), while solution for a fixed γ reads: ω(γ ),

P(ω(γ ), γ ; x/�). Since we are interested in small frequencies and wavenumbers, these solutions can

be expanded in corresponding Taylor series:

“t-expansion” : γ (ω) = −i

∞�

n=1

γ (n)(iω)n, (15)

“s-expansion” : ω(γ ) = −i

∞�

n=1

ω(n)(iγ )n. (16)



Substitution of Eq. (16) for ω in equation

γ = −i

∞�

n=1

γ (n)(iω)n, (17)

demanding that the coefficient next to γ is equal to 1 and the coefficients next to the higher powers

γ n vanish, yields expressions for the coefficients of “t-expansion” via corresponding coefficients of

“s-expansion”:

γ (1) =1

ω(1), γ (2) = −

ω(2)

ω(1)3, γ (3) = −

−2ω(2)2 + ω(1)ω(3)

ω(1)5,

γ (4) = −5ω(2)3 − 5ω(1)ω(2)ω(3) + ω(1)2ω(4)

ω(1)7, . . . . (18)

Similarly, coefficients of “s-expansion” can be expressed via “t-expansion” coefficients:

ω(1) =1

γ (1), ω(2) = −

γ (2)

γ (1)3, ω(3) = −

−2γ (2)2 + γ (1)γ (3)

γ (1)5,

ω(4) = −5γ (2)3 − 5γ (1)γ (2)γ (3) + γ (1)2γ (4)

γ (1)7, . . . . (19)

Let us demonstrate that these expansions contain the main information about the dispersion.

C. Moments of the RTD

Inspection of Eq. (11) reveals that P(ω; x/�)eiγ (ω)x is the time Fourier transform of the concen-

tration:

P(ω; x/�) exp (iγ (ω)x) =� ∞

−∞C(t, x) exp(iωt)dt. (20)

Hence, multiplication by φ and averaging over a cell yields

exp (iγ (ω)x) =� ∞

−∞P(t, x) exp(iωt)dt, P(t, x) =

�Cφ��φ�

, (21)

where we use the fact that�

Pφ�

/ �φ� = 1 (see Eq. (13)). The analysis below shows that P(t, x)

approximates the RTD at a distance x from the inlet in case of boundary condition (6). It follows

that P(t, x) = ∂ tC(t, x) approximates the RTD for boundary condition (3).

Let µn(x) denote the nth raw moment (8) of P(t, x) across a single cell. Since

dn

dωn

� ∞

−∞P(t, x) exp(iωt)dt

�

�

�

�

ω=0

=� ∞

−∞(i t)n P(t, x)dt, (22)

the raw temporal moments of P(t, x) can be expressed via the coefficients of the “t-expansion”:

µn(x) = (−i)ndn

dωnexp(iγ (ω)x)

�

�

�

�

ω=0

, n = 1, 2, . . . , , with µ0 = 1. (23)

The centroid of P(t, x) is at

µ1(x) = −id

dωexp [iγ (ω)x]

�

�

�

�

ω=0

= γ (1)x, (24)

and formulæ for the central moments read

µ�n(x) = (−i)n

dn

dωnexp

�

i(γ (ω) − γ (1)ω)x�

�

�

�

�

ω=0

, n = 1, 2, . . . , (25)



in particular,

µ�2(x) = 2γ (2)x, µ�

3(x) = 6γ (3)x, µ�4(x) = 12x

�

2γ (4) + γ (2)2x�

. (26)

Since µ0 = 1 and µn(0) = 0 for n > 0, P(t, x) satisfies boundary condition (6). Certainly, the

concentration itself does not satisfy Eq. (6): C(t, x) = P(t, x) + C �(t, x), but �C �φ� = 0 by definition.

According to our numerical analysis, the effect of the inlet boundary condition decays rather quickly,

whileP(t, x) growswith x. Therefore,P(t, x) can be regarded as an approximation of the cell-averaged

RTD at point x.

D. The spatial moments

Ansatz (11) can be interpreted in another way, i.e., one can say that P(ω; x/�)e−iω(γ )t is the

space Fourier transform of the concentration:

P(ω; x/�) exp (−iω(γ )t) =� ∞

−∞C(t, x) exp(−iγ x)dx . (27)

Let µn(t) denotes the nth raw spatial moment (10) of the C(t, x) across a single cell. Integration over

a cell and differentiation yield

dn

dγ n

� ∞

∞�C(t, x)� exp(−iγ x)dx

�

�

�

�

γ=0

=� ∞

−∞(−i x)n �C(t, x)� dx, (28)

and the raw and central spatial moments of C(t, x) are, respectively,

µn(t) = (i)ndn

dγ nexp [−iω(γ )t]

�

�

�

�

γ=0

, n = 1, 2, . . . , with µ0 = 1, and

µ�n(t) = (i)n

dn

dγ nexp

�

−i(ω(γ ) − ω(1)γ )t�

�

�

�

�

γ=0

, n = 1, 2, . . . , (29)

in particular,

µ1(t) = ω(1)t, µ�2(t) = −2ω(2)t, µ�

3(t) = 6ω(3)t, µ�4(t) = 12t

�

−2ω(4) + ω(2)2t�

. (30)

Since µ0 = 1 and µn(0) = 0 for n > 0, C(t, x) satisfies the initial condition (9) and therefore Eq. (30)

describe a spread of initially narrow concentration profile in an infinite (in x-direction) medium.

E. Spatial and temporal moments

The spatial and temporal moments are inter-related through Eqs. (18) and (19). Namely, giving

ω(n) via µ1 = µ1(x) and µ�n(x) in Eq. (19), one expresses the spatial moments (30) via the temporal

moments (26):

µ1(t) =t x

µ1(x), µ�

2(t) =t x2µ�

2

µ31

, µ�3(t) =

t x3(3µ�22 − µ�

3µ1)

µ51

,

µ�4(t) =

t x4(15µ�23 + (µ�

4 − 3µ�22)µ2

1 + (3µ�2t − 10µ�

3)µ�2µ1

µ71

. (31)

Conversely, the temporal moments (26) read via the spatial moments (30)

µ1(x) =xt

µ1(t), µ�

2(x) =xt2µ�

2

µ31

, µ�3(x) =

xt3(3µ�22 − µ�

3µ1)

µ51

,

µ�4(x) =

xt4(15µ�23 + (µ�

4 − 3µ�22)µ2

1 + (3µ�2x − 10µ�

3)µ�2µ1

µ71

. (32)

Notice, the two results are exchanged replacing t by x and x by t.



F. Advection-diffusion equation

In order to illustrate our approach consider a one-dimensional ADE for averaged concentration:

∂C

∂t+ U

∂C

∂x= D

∂C2

∂x2, U = |U | , (33)

where U is the mean seepage velocity (cf. Eq. (2)) and D is the effective diffusion coefficient, which

we present as a sum of molecular diffusion D0 and the Taylor dispersion coefficient DT:

D = D0 + DT . (34)

The Taylor dispersion coefficientDT will be prescribed by the EMM solution for the secondmoment,

depending on the individual velocity field and porous structure (see after Eq. (43)). Substitution of

C = exp [i(γ x − ωt)] in Eq. (33) yields the following “s-expansion”:

ω = Uγ − iγ 2D. (35)

Spatial dispersion. Combining (35) with Eq. (30) one obtains the formulæ for the first and

non-zero central spatial moments:

µ1 = U t, µ�2 = 2Dt, µ�

4 = 12 (Dt)2 , . . . , (36)

which are the moments of the fundamental solution

C(t, x) =1

√4πDt

exp [(x − U t)2

4Dt]. (37)

Temporal dispersion. Expanding (35) in Taylor series by powers of ω and keeping the first four

terms of the expansion we obtain the following formulæ for the coefficients of the “t-expansion”:

γ(1)G = U−1, γ

(2)G = U−3D, γ

(3)G = 2U−5D2, γ

(4)G = 5U−7D3. (38)

Correspondingly, the first and higher central temporal moments read

µ1 =1

U, µ�

2 =2Dx

U3, µ�

3 =12D2x

U5, µ�

4 =120D3x

U7+

12D2x2

U6. (39)

The solution of Eq. (33) for x ≥ 0 with initial condition: C(t = 0, x) = 0, and boundary condition:

C(t, x = 0) = 1, hereafter called CG(t, x) reads

CG(t, x) =1

2Erfc[

x − U t

2√Dt

] +1

2Erfc[

x + U t

2√Dt

] exp[Ux

D]. (40)

The corresponding RTD is obtained by differentiating CG(t, x) over t:

PG(t, x) = ∂tCG(t, x) =x exp[− (x−U t)2

4Dt]

2t√Dt

. (41)

One can easily verify that the moments of the RTD,�∞0

PG(t, x)(t − t)ndt with the centroid t = xU

are given by Eq. (39).

This example demonstrates that the “s-expansion” carries the information about the spatial

dispersion, while the “t-expansion” describes the temporal dispersion, which is provided with the

moments of the RTD via Eqs. (24)–(26).

G. Physical interpretation

The coefficients of expansion introduced in the above sections have clear physical meanings as

exposed in the following. The mean residence time in a channel of length L is t = L/U where Uis the mean effective velocity. The same velocity transports a concentration centroid in an infinite

medium. Therefore, from Eqs. (24)–(30) imply that

U = ω(1) =1

γ (1). (42)



According to work of Taylor1 the second moment of a pulse of concentration grows with time as

2Dt, where D is effective diffusion coefficient. It is also known50 that the second moment of the

RTD at point x is 2�

D/U3�

x . Consequently, it follows from Eqs. (26)–(30) that

D = −ω(2) = U3γ (2). (43)

In what follows, we apply this relation to derive the effective diffusion coefficient D from γ (2) or

ω(2). We refer to the difference D − D0 as the Taylor dispersion coefficient DT, by analogy with

Eq. (34). A simplified procedure, valid in the limit of infinite Peclet numbers, will provide the

coefficient DT directly.

The effect of the higher moments on the shape of a distribution is characterised by two dimen-

sionless parameters, namely, skewness (Sk) and kurtosis (Ku):

Sk =µ�3

�

µ�2

�3/2, Ku =

µ�4

�

µ�2

�2− 3. (44)

The Sk characterises the asymmetry of the distribution; positive Sk means that the distribution

has an elongated tail to its right. Ku is a measure of the “peakiness” of the distribution; Ku > 0

means that the distribution has a higher peak at the centre and longer tails at the periphery than a

Gaussian with the same standard deviation. Hence, for the spatial and temporal dispersions given by

Eqs. (25) and (26), one has, respectively,

Sk(t) =3ω(3)

√2�

�ω(2)�

�

3/2t−1/2, Ku(t) = −

6ω(4)

�

ω(2)�2t−1, (45)

Sk(x) =3γ (3)

√2�

γ (2)�3/2

x−1/2, Ku(x) =6γ (4)

�

γ (2)�2x−1. (46)

Notice, the ADE solution (36) yields SkG = KuG = 0 for the spatial dispersion. At the same time,

the ADE solution (39) gives for the “temporal” dispersion:

SkG(x) = 3√2

�

D

Ux, KuG(x) = 30

D

Ux. (47)

For long enough times or distances, Sk and Ku tend to zero and the distribution is completely

characterised by its first two moments, that is advection-diffusion equation becomes exact model

of the process, irrespective to the details of the flow. However, the analysis below shows that the

deviations from the ADE solution may persist for long time and therefore they cannot be neglected

in finite-length systems.

H. Dimensionless expansions

Using a characteristic length L, characteristic molecular diffusivity D (e.g., liquid phase),

characteristic velocity U and characteristic time T = L/U , one can rewrite Eq. (14) in dimensionless

form

∇� ·�

u�

φP�

− Pe−1∇

� ·�

φD�∇

�P�

= iφ�

ω� − γ �u�x

�

P

+iγ �Pe−1

�

φD� ∂P

∂x � +∂φD�P

∂x �

�

− Pe−1γ �2φD�P, ∇� =∇L

, (48)

where Peclet number and dimensionless frequency, wavenumber, velocity, and diffusivity are given

as

Pe =UL

D, ω� =

L

Uω, γ � = Lγ, u

�

φ =uφ

U, D� =

D0

D(49)



and primes denote dimensionless variables. Dimensionless coefficients of “t-expansion” and

“s-expansion” are

γ �(n) =γ (n)Un

L(n−1), ω�(n) =

ω(n)

UL(n−1), n = 1, 2, . . . , . (50)

They are related via Eqs. (18) and (19). As one example, solution (38) takes dimensionless

form:

γ �(1)G = 1, γ �(2)

G =D�

Pe, γ �(3)

G =2D�2

Pe2, γ �(4)

G =5D�3

Pe3, with Pe =

UL

D0

, D� =D

D0

. (51)

Note that in a non-uniform flow field, e.g., flow in a pipe,1 dispersivity increases with Pe.

In all examples considered in the present study D� ∼ Pe2 at high Pe. Therefore, in Eq. (51) γ �(n)

∼ ω�(n) ∼ Pen−1, i.e., Sk ∼√Pe/x �, Ku ∼ Pe/x�. The same scaling (with different coefficients of

proportionality) holds for Sk and Ku obtained for other systems, that is, the non-Gaussian effects

cannot be neglected at high Pe and the distribution becomes Gaussian at a distance x� ∼ Pe from

the inlet. It is why we redefine length and time variables as x � = x/(LPe) and t � = tU/(LPe). Therelations (45) and (46) yield characteristic values Sk � = Sk

√x �, Ku� = x�Ku, in case of temporal

dispersion, and Sk � = Sk√t �, Ku� = t�Ku, in case of spatial dispersion, respectively, as

“t-expansion” : Sk � =3γ �(3)

√2�

γ �(2)�3/2

Pe−1/2, Ku� =6γ �(4)

�

γ �(2)�2

Pe−1, (52)

“s-expansion” : Sk � =3ω�(3)

√2�

�ω�(2)�

�

3/2Pe−1/2, Ku� = −

6ω�(4)

�

ω�(2)�2

Pe−1. (53)

As one can see later, for the systems considered in the present study Sk� ∼ Ku� ∼ 1, while Eq. (47)

reads with Sk �G = SkG

√x � and Ku�

G = KuGx�,

Sk �G = 3

√2D�Pe−1, Ku�

G = 30D�Pe−2 =5

3Sk �

G

2, Pe =

UL

D0

, D� =D

D0

. (54)

I. Reconstruction of a distribution from its moments

Following the framework,15 we develop the reconstruction procedure to restore the distribution

profile from its moments µn. For a distribution p(x) we require that its entropy

S(x) = −�

p(x) (ln p(x) − 1) dx (55)

attains its maximum value under constraints�

p(x)xndx − µn = 0. (56)

Multiplication of (56) by corresponding Lagrange multipliers and substitution into (55) yields

L(p; λn) = −�

p(x) (ln p(x) − 1) dx +�

n

�

λn

��

p(x)xndx − µn

��

→pmax . (57)

Euler conditions for the above problem δpL = 0, where δp is variation with respect to p,

read

ln p(x) =�

n

λnxn (58)

and therefore

p(x) = exp

�

�

n

λnxn

�

. (59)



5 5Τ

0.1

0.2

0.3

0.4

P Τ

5 5Τ

0.1

0.2

0.3

0.4

0.5

0.6

P Τ

FIG. 2. This figure demonstrates reconstruction of the RTD from the ADE moments (54) in two-layered Darcy system of

porosity contrast Rφ , with Rφ = 10−1 (left diagram) and Rφ = 10−2 (right diagram). The two diagrams plot P(τ ) = σP(t),

σ =�

µ�2, versus the centered time τ = (t − τ )/σ , τ = x/U . Exact solution (41) is plotted by dotted (red) line, reconstruction

from 5 moments by dotted-dashed (blue) line, and reconstruction from 7 moments by dashed (black) line. The prescribed

sets of 7 moments are, approximately, {1, 0, 1, 0.53, 3.46, 5.82, 25.57} (left) and {1, 0, 1, 1.91, 9.08, 46.2, 298} (right).

Substitution of (59) into (57) yields after some algebra:

L(λn) = −�

exp

�

�

n

λnxn

�

dx −�

n

λnµn →λn

min . (60)

The unconstrained minimisation problem (60) is convex and can be solved by a pseudo-Newtonian

method.We apply this procedure throughout the paper to reconstruct a distribution from its predicted

central moments.

Central moments of the distributions are given by Eq. (25) for “temporal” dispersion and by

Eq. (29) for “spatial” dispersion, with µ�0 = 1. In some cases, we extend these relations up to

nine moments which are functions of γ (1) − γ (8) or ω(1) − ω(8), respectively. In particular, for

the ADE solution (38): γ (5) = 14D4/U9, γ (6) = 42D5/U11, γ (7) = 132D6/U13, γ (8) = 429D7/U15.

The reconstruction is then performed from the normalized set of the central moments, such as

{1, 0, µ�2/σ

2 = 1, µ�3/σ

3, µ�4/σ

4, . . .} with σ =�

µ�2, where we substitute either x = x�HPe for the

RTD or t = t �HPe/U for the averaged concentration. The set of the first five moments reads:

{1, 0, 1, Sk �√x � ,

Ku�

x � }, with Sk� and Ku� prescribed by Eqs. (52) and (53). We plot the centered RTD

profiles P(τ , x�) = σP(t, x), with τ = (t − τ )/σ , τ = γ (1)x = xU. Similarly, the reconstructed profile

of the averaged concentration are plotted for centered distributions C(t�, x�), with x � = (x − x)/σ ,

x = ω(1)t = U t .Two-layered periodic porous system of porosity ratio is discussed in Sec. III. As an example,

Fig. 2 shows the RTD profiles reconstructed from the ADE moments (54) in this system when

Pe = 16. They are plotted together with the exact ADE solution (41). The moments are computed

with the averaged velocity U and D = D0 + DT (see Eqs. (68) and (70) below), for Rφ = 10−1

(left diagram) and Rφ = 10−2 (right diagram). The RTD is monitored at the distance x� = x/(HPe)

= 3 from the inlet. When Rφ = 10−1, the exact profile is restored accurately. When Rφ = 10−2, the

five-moments based approximation is deficient but it improves for seven moments.

J. Summary

In this section we postulate the solution of Eq. (1) in form (11), assuming any steady-state

periodic velocity field. The two parameters of this solution, namely, the wavenumber γ and temporal

frequency ω, are not independent but related via Eqs. (18) and (19), by expanding γ (ω) and ω(γ )

into Taylor series. We show that the temporal central moments µ�n(x) can be restored from the n first

coefficients γ (n), as given by Eq. (26), while the central spatial moment µ�n(t) can be restored from

the n first coefficients ω(n), as given by Eq. (30). Since the coefficients of the two expansions are

related, the temporal and spatial moments are shown to be linked, as given by Eqs. (31) and (32).



Our construction of dispersion relations has been validated for the particular case of advection-

diffusion equation for averaged concentration C(t, x). First solution (spatial dispersion) addresses

the evolution of the initial pulse, where the proposed construction of moments from “s-expansion”

provides their fundamental solution. Second solution (temporal dispersion) applies for inlet concen-

tration step, where the RTD is obtained by differentiation in time of the exact concentration profile.

Its moments coincide with those predicted by Eq. (39) from “t-expansion.”

The first two moments are related to the mean seepage velocity amplitude U and the effective

dispersion coefficient D, they are predicted to be the same for the governing equation (1) and the

ADE equation for averaged concentration. It follows that the solution for γ (2) or ω(2) will allow to

compute the effective dispersion coefficient in a heterogeneous system. Next, γ (3) or ω(3), and then

γ (4) or ω(4), will allow to estimate the skewness (Sk) and kurtosis (Ku) via Eq. (45). The procedure

can be continued up to any order. In particular, Eq. (30) may allow for upscaling of the experimental

data, that is, if the moments of a propagator are measured for a given time interval t, solving (30)

with respect to ω(n) allows estimating the moments for higher times. The dimensionless form (50)

indicates the scaling for coefficients of two expansions with the characteristic length and velocity.

This can be useful for the re-scaling of the high-order moments and estimation of their respective

significance.

Finally, the numerical procedure described in Sec. II I allows to reconstruct the concentration

profiles or the RTD from their moments. This procedure is independent of the physical nature

of the problem and it can be efficiently implemented for normalized moments, e.g., {1, 0, µ�2/σ

2

= 1, µ�3/σ

3, µ�4/σ

4, . . .}. Overall, we will find that the reconstruction with the five EMM moments

suffices to capture for the principal deviations of the solution from its ADE predictions. Increasing

their number mainly increases the sharpness of the distribution. However, the reconstruction method

should be applied with care for high Peclet numbers, high porosity contrast and, especially, near the

inlet or where the initial effects are still strong.

To conclude, solution for n coefficients γ (n) or ω(n) will allow us to predict the breakthrough

curves, on the one side, and to verify the numerical solutions of Eq. (1), on the other. In Sec. III we

will demonstrate how to construct these coefficients for constant Darcy velocity in heterogeneous

stratified system. In Sec. IV we will construct the subsequent sets of steady state advection-diffusion

equations which allow to compute γ (n) for any soil porosity in pre-computed periodic velocity

field.

III. STRATIFIED DARCY FLOW

In this example we consider a system where, at least in principle, all moments of the distribution

can be obtained analytically. We provide the formulæ up to fourth-order, but, using symbolic

computations, we could attain the eighth order easily. The considered periodic system consists of

two interchangeably placed infinite parallel porous layers with thicknesses h1 and h2. The pressure

gradient and velocity of the fluid are directed parallel along the layers (see Fig. 3). It is assumed

that the layers have the same permeability k, but different porosity φ1 and φ2, respectively (see

schematic view below). Although this example is chosen for its simplicity, it has the methodological

and practical utility. For instance, it can be realised in an experiment featuring loosely packed fine

sand and a more dense rock. While porosity of the sand is higher, the pore diameter is smaller than in

the rock, this makes the permeability in both layers equal and the Darcy velocity in the entire system

is constant. Despite the simplicity of the flow field, the system has non-trivial dispersive features as

shown below.

A. Exact solution

Assume that the Darcy velocity is along x axis, and evolution of concentration Ci (t, x, z) in eachlayer is described by Eq. (1) with Darcy’s velocity uφ, i:

φi∂tCi + uφ,i∂xCi = ∇ · φi D0∇Ci , uφ,x = {uφ,i }, i = 1, 2. (61)



FIG. 3. Schematic view of periodic stratified media subject constant Darcy’s flow.

We are looking for the solution in the form: Ci (t, x, z) = Ci (z) exp[i(γ x − ωt)]. Substitution into

the above equation yields

d2Ci

dz2= K 2

i Ci , with K 2i (ω, γ ) =

−iω + iγ ui + D0γ2

D0

, ui =uφ,i

φi

. (62)

When uφ,i is the same for all layers, then ui is constant per layer, solution reads Ci (z) = Ai exp(Ki z)

+ Bi exp(−Ki z). The two constants per layer, {Ai, Bi} should satisfy continuity conditions for Ci (z)and diffusive flux −φi D0∂z Ci (z) at the interface z = z0, as

A1 + B1 = A2 + B2, φ1K1 (A1 − B1) = φ2K2 (A2 − B2) , if z0 = 0. (63)

Two symmetry conditions prescribe zero flux conditions at the centres of the layers:

A1 exp�

− K1

h1

2

�

− B1 exp�

K1

h1

2

�

= 0,

A2 exp(K2

h2

2) − B2 exp

�

− K2

h2

2

�

= 0. (64)

Equations (63) and (64) have the non-trivial solution only if the determinant of the linear system is

zero. This gives us exact solvability condition

K1φ1 tanh�

K1

h1

2

�

= −K2φ2 tanh�

− K2

h2

2

�

. (65)

Solving this equation for fixed γ , the real part of ω provides decay rate of a harmonic wave with

wavenumber γ and its imaginary part provides its effective velocity. Solution of (65) is not unique;

we are interested in that branch of the solution which can be continuously prolonged from γ = 0.

B. The “temporal” dispersion

We substitute expansion (15) into Eq. (62) for K1 and K2 and plug the obtained expressions into

Eq. (65), then sequentially evaluate the coefficients of the obtained series in ω. Equating them to

zero we obtain formulæ for {γ (n)}. The solution is decomposed into a sum of two components:

γ (n) = γ(n)0 + δγ (n), and γ �(n) = γ �(n)

0 + δγ �(n), (66)

where γ(n)0 is the solution for Pe → ∞, i.e. when K 2

i (ω, γ ) = −iω+iγ uiD0

. The dimensionless solution

γ �(n) is obtained with Eq. (50) where we set Pe = UH/D0 for channel width H = h1 + h2. Using

the auxiliary dimensionless variables:

r1 = 1 + Rh, r2 = Rφ − 1, r3 = 1 + Rh Rφ, r4 = Rh + Rφ, Rh =h1

h2, Rφ =

φ1

φ2

, (67)



0.2 0.4 0.6 0.8 1.0Rh

0.02

0.04

0.06

0.08

0.10

0.12

0.14

kT Φ2

2

20 40 60 80 100Rh

105

104

0.001

0.01

0.1

kT Φ2

2

FIG. 4. The two diagrams plot the Darcy flow solution (70) in two-layered system for kT φ22 versus aspect ratio Rh when

porosity ratio Rφ = { 23, 12, 2 × 10−1, 10−1} (solid, dashed, dotted, dashed-dotted lines). Left diagram: Rh ∈ [0, 1]. Right

diagram: Rh ∈ [1, 100] (log-scale).

the averaged velocity reads

U =uφH

�φ�=

uφRφr1

φ1r3, �φ� = h1φ1 + h2φ2 =

Hφ1r3

r1Rφ

. (68)

The solution for the first four coefficients of the dimensionless expansion reads

γ �(1)0 = 1,

γ �(2)0

Pe=

R2hr

22r4

12Rφr41r3

,γ �(3)

0

Pe2=

R3hr

32r4(2Rφ − 3Rhr2 − 2R2

h)

360R2φr

81r3

,

γ �(4)0

Pe3=

R4hr

42r4

60480R3φr

121 r3

(16R2φ + 4Rh Rφ(17 − 27Rφ) + R2

h(51(1 + Rφ) − 214Rφ)

+ 4R3h(17Rφ − 27) + 16R4

h). (69)

Then, solution for the Taylor dispersion DT and the dimensionless coefficient kT read

DT = γ(2)0 U3 = γ �(2)

0 PeD0 =R2hr

22r4

12Rφr41r3

Pe2D0, kT =DT D0

U2�φ�2=

R2hr4r

22

12φ22Rφr

21r

33

. (70)

This solution is illustrated in Fig. 4 for kTφ22 versus Rh, when Rφ decreases from 2/3 to 1/10.

This shows that kTφ22 increases when Rφ decreases but its behaviour with Rh is non-monotonic, in

particular kTφ22 increases with Rh when the less porous layer is finer, Rh ≤ 1, but decreases when

Rh → ∞. Solution for γ �(n) is valid for any Peclet number with the correction δγ �(1):

δγ �(1) = 0,δγ �(2)

Pe=

1

Pe2,

δγ �(3)

Pe2=

2

Pe4+

R2hr

22r4

3Pe2Rφr41r3

,

δγ �(4)

Pe3=

5

Pe6+

c1Pe2 + c2Pe

4

144Pe6R2φr

81r

23

, with c1 = 180R2h Rφr

41r

22r3r4,

c2 = R3hr

32r4(4Rφ(1 − R3

h) + Rh(Rφ(9Rφ − 11) + 6) + R2h(Rφ(11 − 6Rφ) − 9)). (71)

When there is no porosity contrast (Rφ = 1, r2 = 0), then component (69) vanishes for n ≥ 2 and

γ �(n) reduces to ADE solution (51) with D� = 1. The solution above for γ �(n) allows to compute Sk�

and Ku� with Eq. (52) and to compare them with the ADE solution (54). The latter becomes using

Eq. (70):

Sk �G =

�

3

2

�

12

Pe2+

R2hr

22r4

Rφr41r3

, Ku�G =

5

3Sk �

G

2. (72)

Figures 5 and 6 plot the ADE and EMM predictions for Sk� and Ku�, respectively, and their ratios,

Sk �/Sk �G and Ku�/Ku�

G , versus Peclet number. The aspect ratio increases from Rh = 10−1 (less



20 40 60 80 100Pe

0.2

0.4

0.6

0.8

1.0

1.2

Sk'G

20 40 60 80 100Pe

0.2

0.4

0.6

0.8

1.0

1.2

Sk'G

20 40 60 80 100Pe

0.2

0.4

0.6

0.8

1.0

1.2

Sk'G

20 40 60 80 100Pe

0.5

0.5

1.0

Sk'

20 40 60 80 100Pe

0.5

0.5

1.0

Sk'

20 40 60 80 100Pe

0.5

0.5

1.0

Sk'

20 40 60 80 100Pe

4

3

2

1

1

Sk SkG

20 40 60 80 100Pe

0.2

0.2

0.4

0.6

0.8

1.0

Sk SkG

20 40 60 80 100Pe

1.0

1.5

2.0

2.5

3.0

Sk SkG

FIG. 5. The Darcy flow, “temporal” dispersion. These figures plot the Gaussian solution (72) for Sk�G (top row), the EMM

solution (52) with Eqs. (66)–(71) for Sk� (middle row), and their ratio Sk�/Sk�G (bottom row), versus Peclet number, when

porosity ratio Rφ = { 23, 12, 2 × 10−1, 10−1} (solid, dashed, dotted, dotted-dashed lines). The aspect ratio Rh = {10−1, 1, 10}

from the left to right panel.

porous block is finer, left panel) to Rh = 1 (middle panel) and Rh = 10 (right panel). The two limits

are Rφ = 1 where Sk �/Sk �G = Ku�/Ku�

G = 1, and Rφ = 0 where

limRφ→0

Sk �

Sk �G

=2

5−

3

5Rh

, limRφ→0

Ku�

Ku�G

=51 − 108Rh + 16R2

h

175R2h

. (73)

20 40 60 80 100Pe

0.2

0.4

0.6

0.8

1.0

1.2

Ku'

G

20 40 60 80 100Pe

0.5

1.0

1.5

2.0

2.5

Ku'

G

20 40 60 80 100Pe

0.5

1.0

1.5

Ku'

G

20 40 60 80 100Pe

0.5

0.5

1.0

1.5

Ku'

20 40 60 80 100Pe

0.5

0.5

1.0

1.5

Ku'

20 40 60 80 100Pe

0.5

0.5

1.0

1.5

Ku'

20 40 60 80 100Pe

2

4

6

8

10

Ku KuG

20 40 60 80 100Pe

4

3

2

1

1

Ku KuG

20 40 60 80 100Pe

1

2

3

4

5

6

7

Ku KuG

FIG. 6. Similar as Fig. 5 but for the kurtosis Ku�.



These figures show that at high Pe, an important relative difference with the Gaussian solution may

appear even for small porosity contrast, as Rφ = 12or Rφ = 2

3where the ADE has small values Sk �

G

and Ku�G . Moreover, unlike the ADE solution, both Sk� and Ku� may become negative. These figures

also show an important role of the aspect ratio Rh for this deviation.

C. The “spatial” dispersion

Substitution of Eq. (16) into Eq. (62) and then Eq. (65), followed by the expansion into series

in γ and sequential evaluation of the coefficients of the expansion yield the solution for ω(n). Using

the auxiliary variables (67), the first four coefficients can be expressed as

ω�(1) = 1,ω�(2)

Pe=

1

Pe2+

R2hr4r

22

12Rφr41r3

,ω�(3)

Pe2=

R3hr

32r4(−2Rφ − 3Rh(1 − R2

φ) + 2R2h Rφ)

360R2φr

81r3

,

ω�(4)

Pe3= −

R4hr

42r4

60480R3φr

121 r3

(16R2φ + 4Rh Rφ(17 − 27R2

φ)

+ R2h(51(1 + R4

φ) − 214R2φ) + 4R3

h Rφ(17R2φ − 27) + 16R4

h R2φ). (74)

As predicted, ω(2) = −(D0 + DT) where the Taylor dispersion coefficient DT is the same as that

obtained from the temporal dispersion coefficient γ(2)0 in Eq. (70). The coefficients of two expansions

are related via Eqs. (18) and (19). Note that without the diffusion component D0γ2 in Eq. (62), the

solution (74) is the same except for ω(2) = −DT. As one example, solution (53) reads for Rh = 1:

Sk � =3

�

32Pe3r42

10�

Rφ(192Rφ + Pe2r22 )32

, Ku� =Pe4r42 (51 + Rφ(51Rφ − 142))

280Rφ(192Rφ + Pe2r22 )2

. (75)

This solution is illustrated in Fig. 7 for Sk� and Ku� versus Peclet. The porosity ratio Rφ and aspect

ratio Rh are the same as in two previous figures. These functions take zero values for Pe = 0 in

“spatial” dispersion. When Rh = 1, Sk� is positive and its amplitude increases when Rφ decreases.

In turn, Ku� may become negative. Again, the aspect ratio Rh noticeably modifies these distributions

when Rh ≥ 1 (right panel). On the other hand, we should emphasize that the account of the next

20 40 60 80 100Pe

0.1

0.2

0.3

0.4

0.5

0.6

Sk'

20 40 60 80 100Pe

0.2

0.4

0.6

0.8

1.0

Sk'

20 40 60 80 100Pe

0.2

0.1

0.1

Sk'

20 40 60 80 100Pe

0.1

0.2

0.3

0.4

0.5

0.6

Ku'

20 40 60 80 100Pe

0.2

0.4

0.6

0.8

1.0

1.2

Ku'

20 40 60 80 100Pe

0.25

0.20

0.15

0.10

0.05

Ku'

FIG. 7. The Darcy flow, “spatial” dispersion. This figure plots the EMM solution (53) with Eq. (74) for Sk� (top row) and

Ku� (bottom row) versus Peclet number when Rφ = { 23, 12, 2 × 10−1, 10−1} (solid, dashed, dotted, dotted-dashed lines). The

aspect ratio Rh = {10−1, 1, 10} from the left to the right panel. The Gaussian predicts Sk� = Ku� = 0.



moments may become important when the porosity contrast and/or Peclet number increase. The

examples of that can be found in Secs. III E 3 andVI C 1.

D. Physical interpretation

Dependence of the non-Gaussian effects on porosity and thicknesses of the layers can be

discussed using the results in Fig. 7 as an illustration. Since the Darcy velocity is constant, the

concentration propagates along the less porous layer with higher speed. The total capacitance of ith

layer, i.e., the amount of solute which can be absorbed if two layers are in equilibrium, is proportional

to hiφi. If the less porous “fast” layer is thinner, h1φ1 < h2φ2 and most of the solute is transported

by the low velocity uφ ,2. The smallest part moving with uφ ,1 contributes by a small mass tail of the

concentration profile. Since the profile is skewed to its left, Sk� > 0 (left panel). In the right panel

Sk� < 0 for all porosities except the smallest one. If h1φ1 > h2φ2, most of the solute moves with high

velocity uφ ,1 and the slow layer contributes to a small tail to the profiles’ left. The obvious exception

φ1/φ2 = 0.1 is because in that case h1φ1 is not greater than h2φ2.

In order to explain the behaviour of the kurtosis let us remind that Ku is positive, when the

distribution has long tails and therefore a higher peak; on the other hand a two-peak distribution has

negative Ku. Two peak distribution is possible, when both layers have comparable capacitance; it

can be achieved either when both φ1 ≈ φ2 and h1 ≈ h2 (central panel, solid line), or when φ1 � φ2

and h1 � h2 (right panel, dotted line).

E. Numerical tests and truncation corrections

The procedure described in this section to derive {γ (n)} is much simpler than the EMM in its

general form (see Sec. IV). Moreover, since the number of required moments increases with the

Peclet number and porosity contrast, alternative, direct solving of Eq. (65) provides the worthwhile

benchmark solution for numerical modelling. We have to emphasize that truncation corrections of

numerical scheme impact the moments. To explain this point, assume first that the numerical scheme

models Eq. (61) with two truncation corrections, φi ai∂3Ci∂x3

and φi bi∂4Ci∂x4

:

φi

∂Ci

∂t+ uφ,i

∂Ci

∂x= φi

�

D0

∂2Ci

∂x2+ ai

∂3Ci

∂x3+ bi

∂4Ci

∂x4

�

. (76)

Then solution Ci (t, x, z) = Ci (z) exp[i(γ x − ωt)] satisfies this scheme as (cf. Eq. (62)):

d2Ci

dz2= K 2

i Ci ,

K 2i (ω, γ ) =

−iω + iγ ui + D0γ2 + iaiγ

3 − biγ4

D0

, ui =uφ,i

φi

. (77)

Therefore, the resulting third moment will depend on ai, and the fourth moment will depend on both

ai and bi. For instance, in homogeneous flow φi = 1, the scheme (76) will modify the RTD ADE

moments (39) as (replacing uφ,i by U , D0 by D, ai by a, bi by b):

µ�3(a, b) =

6(2D2 + aU )x

U5,

µ�4(a, b) =

12x(10D3 + 10aDU + 2bU2 + D2Ux)

U7. (78)

It follows that trying to validate predictions of the high-order moments, one should guarantee that

the truncation numerical errors are relatively small. The parameter choice for the TRT scheme is

verified in Sec. VI C 1 with respect to exact solution (65) for evolution of concentration waves in

stratified Darcy flow. In Secs. III E 1 and VI C 3 one may find the comparison of numerical and

predicted EMM moments.



TABLE I. Characteristic values of the TRT computations for “temporal”

dispersion in constant Darcy flow. The Sk� and Ku� are defined by Eq. (52)

for EMM predictions (66) with Eqs. (69) and (71). The Sk�G and Ku�

G are

given by ADE solution (54).

Rφ 2 × 10−1 10−1 4 × 10−2 2 × 10−2

U/uφ 1.67 1.82 1.92 1.96

DT/D0 1.15 3.47 11.04 23.93

Pe(U ) 8.31 9.07 9.59 9.78

Sk� 0.49 0.27 − 0.014 − 0.23

Ku� 0.42 0.076 − 0.59 − 1.54

Sk�G 0.75 0.99 1.54 2.17

Ku�G 0.93 1.63 3.93 7.82

1. The “temporal” dispersion in Darcy flow

The model configuration and the domain size are the same as in Sec. VI C 1. The box is periodic

in the transverse direction, for z-axis, but the Dirichlet boundary condition Ci (t, x = 0, z) = 1 is

implemented at the inlet via the anti-bounce-back rule (142), while the Dankwerts (zero-flux)

condition ∂xCi (t, x = L) = 0 is set at the outlet with the bounce-back rule (142). As above, �−φ

= φ/√12 and then, D0 = cφ/

√12. We mainly discuss the results when cφ = 10−3, uφ = 1.2

× 10−4, and �φ = 112. In these simulations, Pe(U ) = UH

D0belongs to the interval [8, 10] when

porosity ratio Rφ decreases from 2 × 10−1 to 2 × 10−2. The TRT scheme provides solution

Ci (t, x, z) for two layers, i= 1, 2.We compute the averaged concentrationC(t, x �) =�

i Ci (t,x�,z)φi

�φ� , �φ�=�

ihiφi and the RTD distribution, P(t, x�) ≈ (C(t + �t, x�) − C(t − �t, x�))/(2�t) at given set of

the grid points {x� = x/(HPe}. At the same time, we accumulate the raw moments of the RTD, µn(x�)

=�

tP(t, x�)tn�t, until P(t, x�) ≡ 0. We set �t = 1 for all computations but this integration time

step could be increased by factor 10–100, at least, without noticeable loss of accuracy. The effective

numerical value of the average velocity, Unum(x �) = x �/µ1(x�) is then used to restore sequentially the

central moments µ�n(x

�) from their raw values. The EMM predictions for dimensionless moments

are computed with Eq. (26) where we use relations (50) to restore γ (n) from its dimensionless

solution γ �(n), which as given by Eq. (66) with Eqs. (69) and (71). The predicted values for Sk� and

Ku� are computed with Eq. (52) and given in Table I, together with their ADE values Sk �G and

Ku�G given by Eq. (54). The numerical moments are compared to their EMM and ADE solutions in

Fig. 8 where their relative differences are plotted for Rφ = 2 × 10−1 and Rφ = 2 × 10−2. Clearly,

the TRT solutions for Sk and Ku are much closer to the EMM predictions than to the ADE solution.

10 20 30 40 50x H

0.6

0.5

0.4

0.3

0.2

0.1

0.1

err m om ents

10 20 30 40 50x H

1.2

1.0

0.8

0.6

0.4

0.2

err m om ents

FIG. 8. The Darcy flow, “temporal” dispersion, Rφ = 2 × 10−1, Pe = 8.31 (left diagram) and Rφ = 2 × 10−2, Pe = 9.78

(right diagram). Data in Table I. The figure plots the relative differences of the TRT numerical moments with respect to the

EMM predictions versus x/H for: (a) U (thin solid line, U ≈ 0 on the left diagram), (b) D = D0 + DT (thin dashed line), (c)

Sk (thick solid line), and (d) Ku (thick dashed line). The two lowest lines plot the TRT errors with respect the ADE for Sk

(thick dotted line) and Ku (thick dotted-dashed line).



20 40 60 80 100t'

0.05

0.10

0.15

P t'

6 4 2 2 4 6Τ

0.1

0.2

0.3

0.4

P Τ

FIG. 9. The “temporal” dispersion in Darcy flow when Rφ = 10−1, Pe ≈ 9.07. Left diagram plots together solution P(t�)= σ �P(t), σ � = H/U , versus time t� = t/σ � for the ADE (red-dotted line) and EMM (black-dashed line), at three points x�

= x/(HPe) ≈ {0.91, 1.64, 5.5}. Right diagram reports these results for P(τ ) = σP(t), σ =�

µ�2(x), versus the centered time

τ = (t − τ )/σ , τ = x/U . Highest amplitude ADE profile corresponds to x� ≈ 0.91.

Further reduction of velocity (or its increasing, for example by factor 5 when Rφ = 10−1) does not

noticeably impact these results. In turn, refinement of the grid from H = 6 to H = 12 significantly

improves the accuracy. We specially perform these computations on the relatively coarse grids in the

transverse direction, to mimic the realistic porous media applications. Finally, we emphasize that

the numerical values for Sk� and Ku� strongly depend on the obtained Taylor dispersion coefficient.

We note that in similar parameter range, DT is matched less accurately than we could expect from

the above simulations in waves. Most likely, this is related to the time-dependent character of this

example. Figures 9–11 compare the RTD profiles for the EMM, ADE, and TRT when Rφ = 10−1

and Rφ = 4 × 10−2. The TRT profiles agree very well with the EMM predictions, and they clearly

differ from the ADE solutions when the porosity contrast increases. We recall that �φ = 112

was

selected for its advanced advection accuracy in waves. However, we do not detect any noticeable

difference for high moments with �φ = 16or �φ = 1

4in these simulations. These larger �φ values

allow for increasing of stable velocity amplitude.

2. The “temporal” dispersion

The concentration value Ci (t, x = 0) = 1 is set at the inlet. Section VI C 2 describes the

numerical setup of the TRT scheme. The parameters and predicted values are gathered in Table I.

The numerical moments are compared to their EMM and ADE solutions in Fig. 8 where their relative

differences are plotted for porosity ratio Rφ = φ1/φ2 = 2 × 10−1 and Rφ = 2 × 10−2. Clearly, the

TRT solutions for Sk and Ku are much closer to the EMM predictions than to the ADE solution.

Figure 9 compares the RTD profiles for the EMM and ADE, and Fig. 10 plots them for the EMM,

ADE, and TRT, when Rφ = 10−1. Notice, Fig. 10 plots the rescaled distributions P(τ ) = σP(t),

FIG. 10. The “temporal” dispersion in Darcy flow when Rφ = 10−1. Similar as in the right diagram in Fig. 9, but for the

ADE (dotted line, red), EMM (dashed line, black), and TRT (solid line, blue) profiles are plotted together at three points x�

= x/(HPe) ≈ {0.91, 1.64, 5.5}. The TRT and EMM practically coincide.



20 20 40 60t'

0.02

0.04

0.06

0.08

P t'

6 4 2 2 4 6Τ

0.1

0.2

0.3

0.4

P Τ

6 4 2 2 4 6Τ

0.1

0.2

0.3

0.4

P Τ

FIG. 11. The Darcy flow, “temporal” dispersion, similar as in Figs. 9 and 10 but Rφ = 4 × 10−2, Pe ≈ 9.59, and the RTD

is plotted at x� ≈ {1.55, 1.9, 2.25} in first two diagrams. Last diagram plots solution profiles for the ADE (dotted line, red),

EMM (dashed line, black), and TRT (solid line, blue) when x� = 1.9.

with the scale factor σ =�

2γ (2)x , versus τ = (t − τ )/σ , τ = x/U ; the TRT results being rescaled

with their effective values γ (1) and γ (2). The TRT results practically coincide with the reconstructed

EMM profiles, even relatively close to the inlet (see results for x� = x/(HPe) = 0.9). Figure 11 shows

similar results for Rφ = 4× 10−2. Note that closer to the inlet, the reconstruction procedure may lose

feasibility, especially for larger porosity contrasts. On the whole, the TRT results clearly confirm the

EMM predictions for all cases in Table I.

3. The “spatial” dispersion

The initial concentration plume Ci (t, x0) = 1 is released in the periodic domain. The numerical

setup is described in Sec. VI C 3 where the numerical and EMM moments are compared. The

parameters and predicted values are gathered in Table II. Figure 12 illustrates evolution of the

averaged concentration C(t, x) for porosity ratios Rφ = 4 × 10−2 and Rφ = 2 × 10−2. In this figure,

the TRT and EMM profiles practically coincide, while the Gaussian solution is quite different,

especially for a short time after release. Figure 13 plots these TRT profiles C(t, x�) = σC(t, x)

in centered coordinates x � = (x − U t)/σ , σ =�

µ�2(t). Note that the ADE solutions C(t, x�) then

coincide for all t. We emphasize that the EMM results noticeably differ for five, seven and nine

moments when Rφ decreases and t� is relatively small. This is illustrated for Rφ = 2 × 10−2 in

Figures 14 and 15, where five or nine moments are used, respectively. We observe that the larger

number of moments increases the peak amplitude. The TRT results then agree much better with the

nine moments EMM profiles, as depicted in Fig. 15.

F. Concluding remarks

The algorithm proposed in this section is limited to a very simple flow field. It allows for

analytical calculations of high-order moments in bi-modal systems. They exhibit persistent non-

Gaussian behaviour due to the high porosity contrast, which is validated by direct numerical compu-

tations. Although the first five moments of the distribution suffice for capturing the most prominent

non-Gaussian effects, a larger number may be required for more accurate reconstruction of the

distribution. Here, it was assumed that the transport across the layers is only due to molecular

TABLE II. The characteristic values of the TRT computations for “spatial”

dispersion in constant Darcy flow when Pe ≈ 9.07. The Sk� and Ku� aredefined by Eq. (53) with EMM predictions (74).

Rφ 4 × 10−2 2 × 10−2

U/uφ 1.92 1.96

DT/D0 9.87 20.57

Sk� 1.53 2.37

Ku� 3.34 7.82



20 20 40 60 80 100 120x H

0.005

0.010

0.015

0.020

C

20 20 40 60 80 100 120x H

0.002

0.004

0.006

0.008

0.010

0.012

C

FIG. 12. The “spatial” dispersion in Darcy flow when Rφ = 4 × 10−2 (left) and Rφ = 2 × 10−2 (right). Evolution of the

averaged concentration C(t, x) is shown with the ADE (dotted line, red), EMM (dashed line, black), and TRT (dotted (first

profile) and solid line, blue) solutions. The plots correspond to t = t�T, T = HPe/U , with t � ≈ { 12, 1, 2, 3, 4, 6} for Rφ = 4

× 10−2 and t� ≈ {1, 2, 3, 4, 5, 6} with Rφ = 2 × 10−2. The EMM solution is reconstructed with 9 moments.

6 4 2 0 2 4 6x'

0.2

0.4

0.6

0.8

1.0

C x'

6 4 2 2 4 6x'

0.2

0.4

0.6

0.8

1.0

C x'

FIG. 13. The Darcy flow, “spatial” dispersion, Rφ = 4 × 10−2 (left) and Rφ = 2 × 10−2 (right). The ADE (dotted line, red)

and TRT (dotted (first profile) and solid line, blue) results for C(t, x) from Fig. 12 are plotted together as C(t�, x�) = σC(t, x),

x � = (x − U t)/σ , σ =�

µ�2(t).

6 4 2 0 2 4 6x'

0.2

0.4

0.6

0.8

1.0

C x'

6 4 2 0 2 4 6x'

0.2

0.4

0.6

0.8

1.0

C x'

6 4 2 0 2 4 6x'

0.2

0.4

0.6

0.8

1.0

C x'

FIG. 14. The Darcy flow, “spatial” dispersion, Rφ = 2 × 10−2. The same parameters as in Fig. 12. The ADE (dotted line,

red), EMM (dashed line, black), and TRT (solid line, blue) normalized profiles are plotted in centered coordinates when

t� = {1, 2, 3}. The EMM profiles are reconstructed with the five moments.

6 4 2 0 2 4 6x'

0.2

0.4

0.6

0.8

1.0

C x'

6 4 2 0 2 4 6x'

0.2

0.4

0.6

0.8

1.0

C x'

6 4 2 0 2 4 6x'

0.2

0.4

0.6

0.8

1.0

C x'

FIG. 15. The Darcy flow, “spatial” dispersion, Rφ = 2 × 10−2. Similar as in Fig. 14 but the EMM profiles (dashed line,

black) are reconstructed with the nine moments. The TRT and the EMM profiles then agree well.



diffusion, future research is needed to include the role of mechanical dispersion inside individual

porous layers.

IV. EXTENDED B-FIELDS

In Sec. IV A, governing advection-diffusion equations of the EMM are derived for any order.

They are worked-out in details for flat and axisymmetric channel flow in Sec. IV B. In last section,

the B-field equations are analytically solved for coefficients γ �(1) − γ �(4) for Newtonian and power-

law flow in cylindrical pipes, Poiseuille flow between two parallel plates, flow in shallow channels

with parabolic, triangular and elliptic cross-sections. Numerical validation is left to Sec. V which

covers for channel flow as particular case. We recall that the coefficients ω(n) in “s-expansion” can

be obtained with the help of Eq. (19) from solution derived for γ (n). The same relations link the

dimensionless coefficients ω�(n) and γ �(n) defined by Eq. (50).

A. Generic equations

In Sec. III solution for coefficients of the expansions (15) and (16) was constructed for constant

Darcy velocity filed uφ . In this section, we extend the idea of the B-fields3 for high moments of the

RTD in any velocity field. Our starting point is Eq. (14) in its dimensionless form given by Eq. (48).

The idea is to couple the dimensionless expansion (15) with the Taylor series for P(ω�), as

γ �(ω�) = −i

∞�

n=1

γ �(n)(iω�)n, (79)

P(ω�) =∞�

n=0

B(n)(iω�)n. (80)

Substitution of these two expansions into Eq. (48) yields at zero order:

∇� ·�

u�

φB(0)�

− Pe−1∇

� ·�

φD�∇

�B(0)�

= 0. (81)

The obvious solution of (81) isB(0) = const, wherewe assume for convenience that�

φB(0)�

/ �φ� = 1,

then B(0) = 1. The next order of expansion gives for ω�:

∇� ·�

u�

φB(1)�

− Pe−1∇

� ·�

φD�∇

�B(1)�

= φ�

1 − γ �(1)u�x

�

B(0). (82)

The solvability conditions imply that in order to have a spatially periodic solution, the mean value

of the RHS of Eq. (82) must be zero:�

φ�

1 − γ �(1)u�x

�

B(0)�

= 0. Therefore

γ �(1) =�φ��

φu�x

� . (83)

That is, the effective velocity U � = 1/γ �(1) is given by porosity-weighted averaging of the seepage

velocity across single cell. For any solution B(1) of Eq. (82), B(1) + constant also satisfies the

equation and therefore, without loss of generality we assume that�

φB(1)�

= 0. At nth order of

expansion one has for ω�n:

∇� ·�

u�

φB(n)�

− Pe−1∇

� ·�

φD�∇

�B(n)�

= φ

n�

k=1

�

δ1k − γ �(k)u�x

�

B(n−k) + Pe−1S (n),

(84)

where δ1k is Kronecker delta and

S (n) =n�

k=1

γ �(k)�

φD� ∂B(n−k)

∂x � +∂φD�B(n−k)

∂x �

�

+ φD�n�

k=1

�

k�

m=1

γ �(m)γ �(k−m)

�

B(n−k). (85)



Let us rewrite Eq. (84) as

B(0) ≡ 1, ∇� ·�

u�

φB(n)�

− Pe−1∇

� ·�

φD�∇

�B(n)�

= M(n) + Pe−1S(n), n ≥ 1, (86)

where the RHS is given as (with γ �(0) = 0):

M(n) = M (n) − γ �(n)u�φ , n ≥ 1, with u�

φ = φu�x ,

M (1) = φ, S(1) = 0,

M (n) =n−1�

k=1

�

φδ1k − γ �(k)u�φ

�

B(n−k) + φD�Pe−1

n�

k=1

�

k�

m=1

γ �(m)γ �(k−m)

�

B(n−k), n ≥ 2,

S(n) =n�

k=1

γ �(k)�

φD� ∂B(n−k)

∂x � +∂φD�B(n−k)

∂x �

�

, n ≥ 2. (87)

The solvability condition �RHS� = 0 yields

γ �(n) =�

M (n) + Pe−1S(n)�

�

u�φ

� , u�φ = φu�

x , n ≥ 1. (88)

One can see that Eq. (83) is a particular form of (88) for n = 1. In a general case Eq. (84) has to

be solved numerically. At each order of expansion the calculations proceed in the following order:

(i) γ �(n) with Eq. (88); (ii) M (n) and S(n) with Eq. (87), (iii) B(n) with Eq. (86), subject then to

normalization condition �φB(n)� = 0. Note that at the last order of expansion calculation of γ �(n)

may suffice and solution of the differential equation can be omitted. If one restricts the calculations

to second order, i.e., the effective velocity and dispersion coefficient, Eq. (82) is equivalent to that

used in work.3 In the limit of infinite Peclet number , the method simplifies noticeably since the last

term in M (n) and the term Pe−1S(n) vanish when Pe−1 → 0. In flat or axisymmetric rectilinear flows

the equations allow for analytical solutions. We first present them in Sec. IV B for channel flows,

and then extend in Sec. V for stratified systems.

B. Channel flow

Consider a parallel flow, where D0, φ and velocity are functions of a transverse coordinate r� ∈[0, 1] only. Then the first term in S(n) vanishes and Eq. (84) for the B-field of order n ≥ 1 reads

1

h(r �)

∂

∂r �

�

φD�h(r �)∂B(n)

∂r �

�

= −PeM(n)(r �),∂B(n)

∂r �

�

�

�

�

r �=0

=∂B(n)

∂r �

�

�

�

�

r �=1

= 0, (89)

where B(0) = 1, h( · ) is scale factor, e.g., h(r�) = 1 for a two-dimensional channel with r� = 0 at

the bottom, and h(r�) = r� for a pipe with r� = 0 at the center. The source termM(n) is specified by

Eq. (87). In particular, for first, second, third, and fourth orders of expansion it has the following

form:

M (1) = φ, M (2)(r �) = (φ − γ �(1)u�φ )B

(1)(r �) + Pe−1φD�γ �(1)2,

M (3)(r �) = (φ − γ �(1)u�φ )B

(2)(r �) − γ �(2)u�φB

(1)(r �) + Pe−1φD��

2γ �(1)γ �(2) + γ �(1)2B(1)(r �)�

,

M (4)(r �) = (φ − γ �(1)u�φ )B

(3)(r �) − γ �(2)u�φB

(2)(r �) − γ �(3)u�φB

(1)(r �)

+ Pe−1φD��

γ �(2)2 + 2γ �(1)γ �(3) + 2γ �(1)γ �(2)B(1)(r �) + γ �(1)2B(2)(r �)�

. (90)

Solvability condition (88) yields following solution for coefficients γ �(n), n ≥ 1:

γ �(n) =1�

u�φ

�

� 1

0

M (n)h(r �)dr �,�

u�φ

�

=� 1

0

u�φh(r

�)dr �. (91)



In order to write solution of Eqs. (89) and (90) in quadratures we introduce the transverse flux of the

B-field:

Q(n)(r �) = φD�h(r �)∂B(n)

∂r � , n ≥ 1. (92)

Solution for Q(n) is then given by integration:

Q(n)(r �) = −Pe

� r �

0

M(n)(r )h(r )dr. (93)

As one can see, the zero flux boundary conditions hold for r� = 0 and at r� = 1. The latter one is

satisfied automatically since the source term has zero mean. Second integration of Eq. (89) yields

B(n)∗ (r �) =

� r �

0

Q(n)(r )

φD�h(r )dr. (94)

Finally, the solution reads

B(n)(r �) = B(n)∗ (r �) −K(n)

� , (95)

where the constant K(n)� is to satisfy the normalisation conditions

�

φB(n)�

= 0:

K(n)� =

� 1

0B(n)

∗ (r �) h(r �)φ(r �)dr �

� 1

0h(r �)φ(r �)dr �

, n ≥ 1. (96)

Thus, at each order of expansion the calculations proceed in the following order: (i) γ �(n) with

Eq. (91); (ii) Q(n)(r �) with Eq. (93); (iii) B(n)∗ (r �) with Eq. (94); (iv) K(n)

� with Eq. (96); (v) B(n)(r �)

with Eq. (95). Finally, solution for γ �(n)0 = γ �(n)|Pe→∞, n ≥ 2, can be obtained dropping the term

with the prefactor Pe−1 in Eq. (90).

C. Flow in a pipe

1. Poiseuille flow in cylindrical pipe

In order to illustrate our approach consider Poiseuille flow in a cylindrical pipe of radius R1.

Let us omit the primes for r� in the rest of this section (we use r for r� = r/R1). The porosity and

dimensionless diffusivity are both equal to 1, uφ(r ) = 2U (1 − r2), u�φ (r ) = 2(1 − r2). Equation (89)

reads

1

r

∂

∂r

�

r∂B(n)

∂r

�

= −PeM(n)(r ), n ≥ 1, (97)

where Peclet number Pe = UR1/D0, U = U . First, Eqs. (87) and (91) suggest that M (1) = 1, γ �(1)

= 1,M(1) = 2r2 − 1. Then Eqs. (93)–(96) yield

B(1)(r ) = Pe

�

−1

12+

r2

4−

r4

8

�

, (98)

what can be used to calculate γ �(2) via Eq. (91):

γ �(2) =1

Pe+

Pe

48, DT =

Pe2

48D0, kT =

DT R1

U2�φ�2=

1

12. (99)

This is famous solution1 for dispersion coefficient. Knowing γ �(2) one obtains

M(2)(r ) =24 − 48r2 + Pe2

�

−1 + 9r2 − 15r4 + 6r6�

24Pe(100)

and B(2) is given by integration (93)–(96):

B(2)(r ) =−480

�

2 − 6r2 + 3r4�

+ Pe2�

11 + 5r2�

−24 + 54r2 − 40r4 + 9r6��

11520. (101)



Again, Eq. (91) yields

γ �(3) =2

Pe2+

1

12+

Pe2

1920. (102)

Repeating the same procedure for the forth-order terms of expansion one obtains

M(3)(r ) =2 − 4r2

Pe2+

1

24

�

−1 + 16r2 − 30r4 + 12r6�

+Pe2

�

17 − 6r2 − 330r4 + 680r6 − 445r8 + 90r10�

11520,

B(3)(r ) =−480

�

2 − 6r2 + 3r4�

+ Pe2�

−9 + 5r2�

−12 + 48r2 − 40r4 + 9r6��

5760Pe

+Pe3

�

992 − 7r2�

510 − 45r2 − 1100r4 + 1275r6 − 534r8 + 75r10��

9676800,

γ �(4) =5

Pe3+

5

16Pe+

11Pe

2304−

53Pe3

7741440. (103)

This solution is then used to compute the moments of the RTD via relations (26) with (50), and

Sk�, Ku� with Eq. (52). The relative differences with the ADE solution (54) then becomes

Sk

SkG=

Sk �

Sk �G

=γ �(3)

γ �(3)G

=3(3840 + Pe4 + 160Pe2)

5(48 + Pe2)2,

Ku

KuG

=Ku�

Ku�G

=γ �(4)

γ �(4)G

=38707200 + 36960Pe4 − 53Pe6 + 2419200Pe2

350(48 + Pe2)3,

limPe→∞

Sk

SkG=

3

5, lim

Pe→∞

Ku

KuG

= −53

350. (104)

Note that Sk = SkG and Ku = KuG when Pe = 0, but Sk < SkG and Ku < KuG for any non-zero Peclet

number; the distribution has shorter tails than corresponding Gaussian, i.e., the solute is washed out

of the pipe faster than it is predicted by the ADE. Solution for ω�(n) in “spatial” dispersion can be

obtained with the help of Eq. (19) from the above solution for γ �(n).

2. Power-law flow in cylindrical pipe

Following the same pattern we can consider flow of a power-law fluid:51

u�φ (r ) =

(1 + 3n)�

1 − r (1+1n )�

1 + n, r ∈ [0, 1], (105)

where n is rheological index. After a bulky but straightforward algebra one obtains

γ �(1) = 1, γ �(2) =1

Pe+

n2Pe

2 + 16n + 30n2,

γ �(3) =2

Pe2+

2n2

1 + 8n + 15n2+

n3(1 + n(7 + 16n))Pe2

12(1 + 3n)2(1 + 4n)(1 + 5n)(1 + 7n),

γ �(4) =5

Pe3+

15n2

2Pe + 16nPe + 30n2Pe+

5n3(1 + n(15 + 4n(21 + 41n)))Pe

12(1 + 3n)2(1 + 4n)(1 + 5n)2(1 + 7n)

−n4(−3 + n(−35 + n(86 + n(2994 + n(17321 + 23n(1847 + 1692n))))))Pe3

192(1 + 3n)3(1 + 4n)(1 + 5n)3(1 + 7n)(1 + 9n)(3 + 11n).

(106)



At high Pe, Eq. (106) and ADE solution (50) and (38) differ significantly, e.g., Eq. (38) predicts that

the leading term in γ �(4) is positive when Pe → ∞, while it is negative in Eq. (106), similar to result

(104) for Newtonian fluid.

3. Poiseuille channel flow

For the sake of completeness, we also give the results for Poiseuille flow in a flat channel of

width H, with the averaged velocity value U , Pe = UHD0

:

γ �(1) = 1,γ �(2)

Pe=

1

Pe2+

1

210,

γ �(3)

Pe2=

2

Pe4+

2

105Pe2+

29

485100, DT =

Pe2

210D0,

γ �(4)

Pe3=

5

Pe6+

1

14Pe4+

2

4851Pe2+

97

220720500. (107)

Again, we recover the well known coefficient DT at the second order. The relative differences with

the ADE solution (47) then becomes

Sk

SkG=

Sk �

Sk �G

=γ �(3)

γ �(3)G

=970200 + 9240Pe2 + 29Pe4

22(210 + Pe2)2,

Ku

KuG

=Ku�

Ku�G

=γ �(4)

γ �(4)G

=6(1103602500 + 15765750Pe2 + 91000Pe4 + 97Pe6)

715(210 + Pe2)3,

limPe→∞

Sk

SkG=

29

22, lim

Pe→∞

Ku

KuG

=582

715. (108)

Note that Sk > SkG for all Peclet numbers, unlike for circular pipe. Also, Ku is positive

in the limit Pe → ∞, namely, Ku ∈ [0, KuG] when Pe ≥ 5

�

54619, while for the circular pipe,

Ku

< KuG, ∀Pe, and Ku is negative when Pe → ∞ (cf. Eq. (108)). While compared to the

ADE solution, concentration in the flat channel has higher peak and longer tail to its right, i.e., most

of the material is washed out faster, while there is a portion of the solute, which remains in the

channel longer than the ADE predicts.

D. Dispersion in a shallow channel

Consider an incompressible liquid with viscosity µ driven by a pressure gradient pz through

a straight channel with cross-section y = �h(x) (h(−1) = h(1) = 0, h(0) = 1) as it is shown

schematically in Fig. 16. We assume that the flow field is steady and laminar, thus the velocity field

does not depend on the axial coordinate z: u(x, y, z) = (0, 0, u(x, y)), where u(x, y) is the solution

of the Stokes equation

µ∇2u = pz . (109)

x

y

FIG. 16. Schematic view of the cross-section of the channel.



TABLE III. Table shows γ �(i) for different cross-section shapes in shallow channel (γ �(1) = 1).

Shape γ �(2) γ �(3) γ �(4)

Parabolic 1Pe

+ 3347Pe270270

2

Pe2+ 6694

1351355

Pe3+ 3347

18018Pe

+ 18221834Pe2

33705452781+ 16373661907Pe

4718763389340+ 99753278851Pe3

4988575538851905

Triangular 1Pe

+ Pe48

2

Pe2+ 1

12+ 7Pe2

57605

Pe3+ 5

16Pe+ 19Pe

2304+ 169Pe3

2580480

Elliptic 1Pe

+ 5Pe576

2

Pe2+ 5

144+ 49Pe2

1658885

Pe3+ 25

192Pe+ 205Pe

110592+ 5333Pe3

764411904

The channel is shallow, i.e., � � 1 and Eq. (109) reads after transformation of variables y = �y�:

(�2∂xx + ∂y� y� )u = µ−1�2 pz . (110)

We are looking for a solution of Eq. (110) in the following form:

u(x, y�) = 6u(x)

�

y�

h−�

y�

h

�2�

. (111)

Substitution of the above equation into Eq. (110) and averaging over the depth of the channel yields

the following equation for the depth-averaged velocity:

u(x) = �2h2(x)

12µpz +O(�4). (112)

The expression for mean velocity reads

U =� 1

−1h(x)u(x) dx� 1

−1h(x) dx

= �2I3 pz

12µI1, where In =

� 1

−1

hn(x) dx .

Finally, u(x) = U I1I3h2(x) and the depth-averaged version of Eq. (14) reads

−Pe−1∂x�

h(x)∂x P�

= ih(x) [ω − γ (ω)u(x)] P − Pe−1h(x)γ 2(ω)P . (113)

There are several shapes of the channel allowing for analytical solution of Eq. (113); following

Refs. 52 and 53 we consider parabolic (h(x) = 1 − x2), triangular (h(x) = 1 − |x|) and elliptic

(h(x) =√1 − x2) shapes. Consecutive evaluation of the B-fields and solvability conditions yields

after some algebra dimensionless coefficients of γ -expansion shown in Table III.

V. RECTILINEAR STRATIFIED FLOWS

It was shown in Sec. III that a rectilinear flow in a two-layer systems might have non-trivial

dispersive properties. Such flows are important for environmental and physiological applications,

e.g., channel flow over a porous bed or capillary transport in tissues;14 later we will see that they

also constitute a non-trivial benchmark for the numerical schemes. In this section we extend the

B-field equations in pipes for multilayer flows with interface continuity conditions. The coefficients

γ �(1) − γ �(4) are then obtained for flat and axisymmetric geometries where an open channel adjoins

with a zero-permeability porous bed, referred to as open/impermeable system hereafter. The derived

moments, the RTD and concentration profiles are confronted to ADE and numerical TRT solutions

for temporal and spatial dispersion.

A. B-field equations with interface conditions

Assume now that the channel is composed of N layers of porosity φi. Each layer belongs to the

interval r � ∈ [R�i−1, R

�i ], with R�

0 = 0, and integration of velocity and porosity across the channel



reads

�

u�φ

�

=N�

i=1

� R�i

R�i−1

u�φ ih(r �)dr �, �φ� =

N�

i=1

� R�i

R�i−1

φi h(r�)dr �, (114)

with scale factor h( · ) (see after Eq. (89)). The B-field equations (89) become for B(n)i (r �),

r �

∈ [R�i−1, R

�i ]:

B(0)i = 1,

1

h(r �)

∂

∂r �

�

φi D�h(r �)

∂B(n)i

∂r �

�

= −PeM(n)i (r �), n ≥ 1, i = 1, . . . , N , (115)

where M(n)i is given by Eqs. (87) and (90) replacing M(n) → M(n)

i , M (n) → M(n)i , φ → φi,

u�φ → u�

φ iand B(n) → B(n)

i . Solvability condition (88) yields following solution for coefficients

γ �(n):

γ �(n) =1�

u�φ

�

N�

i=1

� R�i

R�i−1

M(n)i (r �)h(r �)dr �, n ≥ 1. (116)

At the interfaces r � = R�i , solution has to satisfy continuity relations for B-field and its diffusion flux:

[|B(n)i |]r �=R�

i= 0, [|Q(n)

i |]r �=R�i= 0, with Q(n)

i (r �) = −φi D�h(r �)

∂B(n)i

∂r � , (117)

[|� (n)i |] = �

(n)i+1 − �

(n)i , ∀ �

(n)i . This can be achieved with the help of two constants per layer, say

K(n)i for B(n)

i and F (n)i for Q(n)

i , and one global constant K(n)� for the entire system:

B(n)i (r �) = B(n)

∗i (r�) +K(n)

i −K(n)� ,

Q(n)i (r �) = −Pe

� r �

R�i−1

M(n)i (r ) h(r )dr − F (n)

i , (118)

where we set

B(n)∗i (r

�) =� r �

R�i−1

Q(n)i (r )

φi D�h(r )dr, i = 1, . . . , N . (119)

Note that K(n)� vanishes in interface relations. The solution for 2 × N constants K(n)

i and F (n)i is then

set by 2 × (N − 1) linear continuity equations (117) complemented by two boundary conditions for

r� = 0 and r � = R�N , for each expansion order n > 1. Finally, the constants K(n)

� are determined from

normalisation conditions�N

i=1�φiB(n)i � = 0:

K(n)� =

�Ni=1

� R�i

R�i−1(B(n)

∗i (r�) +K(n)

i )h(r �)φi (r�)dr �

�φ�, n ≥ 1. (120)

1. Example: Two layered channel flow

Below we will specify solution for straight and radial systems of two parallel layers, open with

porosity 1 and impermeable with porosity φ, with the interface at r � = R�1. No-flux condition takes

place for r� = 0 and r� = 1.Without loss of generality, one can putK(n)1 = 0,F (n)

1 = 0, then continuity

conditions (117) give

K(n)2 = B(n)

∗1 (R�1), F

(n)2 = Pe

� R�1

0

M(n)1 (r ) h(r )dr, n ≥ 1. (121)



2. Example: Two-layered Darcy flow

Two-layered periodic Darcy flow has been considered in Sec. III. Solution for γ �(n) is constructed

in Sec. III B plugging expansion (15) into exact solvability condition (65). The solution for γ �(1)

− γ �(4) is given by Eq. (66) with Eqs. (69) and (71). The same solution can be obtained from the

generic procedure described above. The only difference with the two-layered bounded channel is that

the zero flux (symmetry) conditions have to be imposed at the middle of two layers, at z� = h�1/2 and

z� = h�2/2 + h�

1, h�1, and h

�2 being respective dimensionless width for two channels, with h�

1 + h�2 = 1.

All interface conditions then become equivalent and solution reads with h(r�) ≡ 1:

B(n)i (z�) = B(n)

∗i (z�) +K(n)

i −K(n)� , i = 1, 2, n ≥ 1. (122)

where

B(n)∗1 (z

�) =� z�

h�1/2

Q(n)1 (z)

φ1D� dz, Q(n)1 (z�) = −Pe

� z�

h�1/2

M(n)1 (z)dz,

B(n)∗2 (z

�) =� z�

h�1

Q(n)2 (z)

φ2D� dz, Q(n)2 (z�) = −Pe

� z�

h�1

M(n)2 (z)dz − F (n)

2 ,

K(n)1 = 0, K(n)

2 = B(n)∗1 |z�=h�

1, F (n)

1 = 0, F (n)2 = Pe

� h�1

h�1/2

M(n)1 (z�) dz�,

K(n)� =

� h�1

0B(n)

∗1 dz� +� 1

h�1(B(n)

∗2 +K(n)2 )dz�

�φ�, γ �(n) =

� h�1

0M

(n)1 dz� +

� 1

h�1M

(n)2 dz�

�

u�φ

� . (123)

This solution coincides with our direct symbolic computations in Sec. III B.

B. Open/porous Poiseuille flow in straight channel

We consider a system consisting of two interchangeably placed infinite parallel layers with

thicknesses h1 and h2. The system is bounded by impermeable boundaries at z = 0 and z = H, H

= h1 + h2. The pressure gradient and velocity of the fluid are directed along the layers. It is assumed

that the first (impermeable) layer has porosity φ and k1 = 0, and then no advection happens there

but the diffusion process can take place in it. In turn, the second layer is open and hence, the velocity

profile is parabolic there,

uφ(x) = 0, if x ∈ [0, h1] and

uφ(x) = −U0

H 2

(1 − x/H )(Rh − (1 + Rh)x/H )

2(1 + Rh), x � = x/H, z ∈ [h1, H ], Rh =

h1

h2, (124)

with scale factor U0, boundary conditions uφ(h1) = 0 and uφ(H) = 0. In this section we will use the

auxiliary variables (67) with porosity ratio Rφ = φ, then

r1 = 1 + Rh, r2 = −1 + φ, r3 = 1 + Rhφ, r4 = Rh + φ. (125)

The limit case Rh = 0 corresponds to single channel considered in Sec. IV C 3, with H = h2.

1. The “temporal” dispersion

The B-field procedure to built solution for coefficients γ �(n) of expansion (79) is described

in Sec. V A and specified for two layered channel by Eq. (121) with h( · ) = 1. Similar as in

Sec. III B, the solution for γ �(n) is decomposed into two components: γ �(n) = γ �(n)0 + δγ �(n), where



γ �(n)0 is solution in the limit Pe → ∞:

γ(1)0 =

1

U, U =

�uφ��φ�

=H 2U0

12r21r3, �φ� = �φ(x)� =

Hr3

r1, Pe =

UH

D0

,

γ �(1)0 = 1,

γ �(2)0

Pe=

1 + φRh(9 + 78φRh + 70R2h)

210r21r3,

γ �(3)0

Pe2=

29 + 472φRh + 12176φ2R2h + (6930φ + 60243φ3)R3

h + 120120φ2R4h

485100r41r3,

γ �(4)0

Pe3=

1

1324323000r61r3(582 + 16713φRh + 1296441φ2R2

h

+ 80φ(5369 + 176070φ2)R3h + 160φ2(138502 + 304695φ2)R4

h

+ 8190φ(924 + 20081φ2)R5h + 185825640φ2R6

h + 71471400φR7h). (126)

Then solution for Taylor dispersion coefficient DT and dimensionless coefficient kT read

DT = γ(2)0 U3 = γ �(2)

0 PeD0, DT |Rh=0 =Pe2D0

210,

kT =D0DT

U2�φ�2=

1 + φRh(9 + 78φRh + 70R2h)

210r33, kT ≥ kT |Rh=0 =

1

210, ∀φ, ∀Rh . (127)

As expected, this solution reduces to solution (107) for single channel when Rh = 0, then, in the

presence of impermeable layer, kT(φ) increases as illustrated in Fig. 18 for several values of the

aspect ratio Rh, the corresponding velocity profiles are schematized in Fig. 17. In the left diagram,

Rh ∈ [0, 1] and the limit case is single channel, kT |Rh→0 = 1210

. In the right diagram, Rh ≥ 1

and the asymptotic limit case is kT |Rh→∞ = 13φ2 . The solution is valid for any Peclet number with

0.0 0.2 0.4 0.6 0.8 1.0z '

1

2

1

3

2

u Φ

0.2 0.4 0.6 0.8 1.0z '

1

6

1

3

u Φ

FIG. 17. The two diagrams plot velocity profiles (124) normalized by factor U0H2/12 for results in Fig. 18, when

Rh = {1, 12, 10−1, 10−2} (left diagram) and Rh = {1, 2, 10} (right diagram), (solid, dashed, dotted, dashed-dotted lines,

respectively). Impermeable layer is x � ∈ [0,Rh

1+Rh] and the open layer is x � ∈ [

Rh1+Rh

, 1]. Poiseuille solution in open channel

is approached when Rh → 0 (left diagram).



0.2 0.4 0.6 0.8 1.0Φ

1

25

3

50

2

25

kT

0.2 0.4 0.6 0.8 1.0Φ

0.1

1

10

100

1000

kT

FIG. 18. The two diagrams plot solution (127) for kT(φ) in open/impermeable system with porous layer of porosity φ,

when Rh = {1, 12, 10−1, 10−2} (left diagram) and Rh = {1, 2, 10, 103} (right diagram, log-scale) (solid, dashed, dotted,

dashed-dotted lines, respectively). The corresponding velocity profiles are plotted in Fig. 17.

correction δγ �(n):

δγ �(1) = 0, δγ �(2) =1

Pe,

δγ �(3)

Pe2=

2

105Pe4+

2(1 + φRh(9 + 78φRh + 70R2h))

105Pe2r21r3,

δγ �(4)

Pe3=

5

Pe6+

c1Pe2 + c2Pe

4

97020Pe6r41r23

, c1 = 6930r21r3(1 + φRh(9 + 78φRh + 70R2h)),

c2 = 40 + 699φRh + 15255φ2R2h + φ(8470 + 87863φ2)R3

h

+ 3φ2(46970 + 42389φ2)R4h + 9240φ(7 + 26φ2)R5

h + 118580φ2R6h . (128)

When Rh = 0, {γ �(n)} reduces to the single channel solution (107). The obtained solution allows forthe computation of Sk� and Ku� via Eq. (52), and then to compare them with the Gaussian solution

(47) whereD=D0 +DT andDT is given in Eq. (127). Finally, the coefficients ω�(n) of “s-expansion”

are obtained via relations (19) from solution for γ �(n).

2. Numerical tests: “Temporal” dispersion in open/impermeable channel

This test is similar to the one in Sec. III E 2 but the velocity profile (124) is prescribed for

two channels. The details on the TRT scheme are gathered in Sec. VI C 4. We illustrate the results

for porosity φ = 10−1 of impermeable layer, when the two layers have equal width: h1 = h2 = 6

(grid nodes), and Pe = UHD0

≈ 13.6. The EMM predictions (126)–(128) yield: DT ≈ 1.4 × 10−3,

DT/D0 ≈ 1.94, Sk� ≈ 1.37, Ku� ≈ 3.46. The ADE solution yields much smaller values: Sk �G ≈ 0.53,

Ku�G ≈ 0.48. The TRT results for Sk� and Ku� in Fig. 19 agree with the EMM predictions within 5%

5 10 15 20 25 30 35 40x H

0.05

0.05

0.10

err m om ents to EMM

5 10 15 20 25 30 35 40x H

1

2

3

4

5

6

err m om ents to ADE

FIG. 19. The open/impermeable system, “temporal” dispersion, Rφ = 10−1. The left diagram plots the relative difference

for four first TRT moments with respect to EMM: (a) U (thin solid line), (b) D = D0 + DT (thin dashed line), (c) Sk� (thicksolid line), and (d) Ku� (thick dashed line). The right diagram plots the relative differences with respect the ADE solution

(47) for Sk� (dotted line) and Ku� (dotted-dashed line).



10 20 30 40 50 60t'

0.1

0.2

0.3

0.4

0.5

P t'

6 4 2 2 4 6Τ

0.1

0.2

0.3

0.4

0.5

P Τ

FIG. 20. Left diagram plots together the RTD profile P(t�) = σ �P(t) in open/impermeable channel when Rφ = 10−1 for the

ADE (dotted line, red) and the EMM (dashed line) versus time t� = t/σ �, σ � = H/U at three points x� = x/(HPe) = {0.3,

0.98, 2.51}. The right diagram re-plots these results for P(τ ) = σP(t), σ =�

µ�2(x) versus the centered time τ = (t − τ )/σ ,

τ = xU .

(left diagram) and thus, they exceed very noticeably the ADE prediction (see right diagram).

Figure 20 compares the RTD profiles for the EMM and ADE. Figure 21 plots the centered pro-

files for EMM, ADE, and TRT together. The TRT and EMM profiles agree well, some deviation

is however observed for first point x� = x/(HPe) = 0.3. This can be explained by several factors,

namely (i) the TRT truncation errors may increase when the velocity field varies; (ii) the modeled

Peclet number is higher; and (iii) x� = 0.3 is closer to the inlet than in results reported above for

Darcy law. Finally, as for the sharp initial profiles in Darcy flow, the RTD reconstruction based on

first five moments could become not accurate enough near the inlet. However, unlike for Darcy flow,

we are not yet aware of the analytical solution for the next moments in this example.

3. Numerical test: “Spatial” dispersion in open/impermeable channel

Similar as in Sec. III E 3, an initial narrow impulse is released in the periodic in the streamwise

direction channel where velocity profile (124) is prescribed. The details for the TRT scheme can be

found in Sec. VI C 4. The coefficients ω�(n) of “s-expansion” are computed with the help of relations

(19) from the RTD solution (126)–(128). The U and DT are the same as for “temporal” dispersion

and they are predicted by Eqs. (126) and (127). The EMM profiles for averaged concentration

are restored from the obtained five moments. The predicted characteristic values are gathered in

Table IV for Poiseuille flow in single channel (Pe = 59.86) and when the impermeable layer has

porosity φ = 10−1. The numerical values for Sk� and Ku� are compared to their EMM solution in

Fig. 22 (recall that the ADE solution yields zero values). We note that the difference in moments

is larger at the initial stage for open/impermeable system than for Darcy flow. In single channel,

Sk� and Ku� agree with the predictions within about 10% soon after the contaminant release. In

open/impermeable system this is achieved when the dimensionless time t� = t/T> 1, T=HPe/U. The

profiles of mean concentration are illustrated in Figs. 23 and 24 for single channel and open/porous

porosity ratio φ = 10−1, respectively. The profiles C(t�, x�) = σC(t, x) are plotted again in centered

6 4 2 0 2 4 6Τ

0.1

0.2

0.3

0.4

0.5

0.6

0.7

P Τ

6 4 2 0 2 4 6Τ

0.1

0.2

0.3

0.4

0.5

0.6

0.7

P Τ

6 4 2 0 2 4 6Τ

0.1

0.2

0.3

0.4

0.5

0.6

0.7

P Τ

FIG. 21. Similar as in the right diagram in Fig. 20, the three diagrams plot distributions P(τ ) = σP(t), for the ADE (dotted

line, red), EMM (dashed line, black), and TRT (sold line, blue) at points x� = {0.3, 0.98, 2.51}.



TABLE IV. Characteristic values of the TRT computations for “spatial”

dispersion in single channel (Rh = 0, U = U0H2

12) and open/porous channel

(Rh = 1, φ = 10−1), using the scale factor U0 = 3 × 10−4 and H = 12 in

Eq. (124). The Sk� andKu� are predicted by the EMM; and Sk�G = Ku�

G = 0

for the Gaussian.

Rh 0 1

φ . . . 10−1

Pe(U ) 59.86 13.6U

U |Rh=01 0.23

DTD0

17.06 1.94

Sk� −0.086 −0.84

Ku� −0.1 1.49

0.2 0.4 0.6 0.8 1.0t'

0.10

0.08

0.06

0.04

0.02

0.00

0.02

0.04

rel . err

0.5 1.0 1.5 2.0 2.5 3.0t

0.01

0.02

0.03

0.04

0.05

rel . err

FIG. 22. The “spatial” dispersion is addressed for Poiseuille flow in single channel (left diagram, Pe = 59.86) and for

open/impermeable system with Rφ = 10−1 (right diagram, Pe = 13.6). The figure plots the relative difference of the TRT

numerical moments to EMM predictions versus t� = t/T, T = HPe/U : (a) U (solid line), (b) D = D0 + DT (dashed line,

red), (c) Sk� (dotted line, blue), and (d) Ku� (dotted-dashed line, magenta). Data are given in Table IV.

20 40 60x H

0.002

0.004

0.006

0.008

0.010

C

6 4 2 0 2 4 6x'

0.1

0.2

0.3

0.4

C x'

6 4 2 0 2 4 6x'

0.1

0.2

0.3

0.4

C x'

FIG. 23. The “spatial” dispersion in Poiseuille flow in single channel. Left diagram shows evolution of the averaged

concentration C(t�, x�) with the EMM (dashed line, black) and TRT (solid line, blue) when t = t�T, T = HPe/U , with

t � ≈ { 14, 12, 34}. Next two diagrams plot the centered profiles for the ADE (dotted line, red), EMM (dashed line, black), and

TRT (solid line, blue) when t � ≈ 14(middle) and t � ≈ 1

2(right). Data are given is Table IV.

10 10 20 30 40 50 60x H

0.005

0.010

0.015

0.020

0.025

0.030

0.035

C

6 4 2 0 2 4 6x'

0.1

0.2

0.3

0.4

0.5

C x'

6 4 2 0 2 4 6x'

0.1

0.2

0.3

0.4

0.5

C x'

FIG. 24. The “spatial” dispersion in open/impermeable systemwith φ = 10−1 when Pe= 13.6. Left diagram shows evolution

of the averaged concentrationC(t, x) with the EMM (dashed line, black) and TRT (solid line, blue) when t= t�T, T = HPe/U ,

with t � ≈ { 14, 1, 3}. Next two diagrams plot the centered profiles of the ADE (dotted line, red), EMM (dashed line, black), and

TRT (solid line, blue) when t � ≈ 14(middle) and t� ≈ 1 (right). The EMM profiles are reconstructed with the five moments.

Data are given is Table IV.



coordinates x � = (x − U t)/σ , σ =�

µ�2(t). They confirm good agreement of the TRTwith the EMM

when t � ≥≈ 14in single channel, and t � ≥≈ 1

2in bi-modal system. The results for Rφ = 10−2 are

very similar to those for Rφ = 10−1 (data are not shown). In fact, the difference between the ADE

and EMM is less significant in these example than in Darcy constant flow.

C. Open/porous flow in cylindrical pipe

We consider a system consisting of two circular layers. The internal open layer has radius

r = R1. The porous layer is between r = R1 and r = R, and it has porosity φ. It is assumed that this

layer is impermeable and then no advection happens there, but the diffusion process can take place

in it. The system is bounded by impermeable circle boundary at r = R. The pressure gradient and

velocity of the fluid are directed along the x-axis. The velocity profile is parabolic in internal open

layer: uφ,x (r ) = 2U0(1 − r2

R21

), with scale factor U0. In this section we will use following auxiliary

variables:

r1 = Rr − 1, r2 = φ − 1, r3 = R2r − 1, r4 = R2

r − φr3, c = ln [Rr ], Rr =R1

R. (129)

The B-field procedure to built solution for coefficients γ �(n) of expansion (79) is described in

Sec. V A and specified for two layered system by Eq. (121). We apply them for cylindrical capillary

with h(r�) ≡ r�. As predicted, γ(1)0 = γ (1) = 1

U, with U = �uφ,x �

�φ� = R2r U0

r4, �φ� = R2

r r42. Similar as in

Secs. III B and V B 1, the solution for γ �(n) is decomposed into two components: γ �(n) = γ �(n)0

+ δγ �(n), where first component γ �(n)0 is solution in the limit Pe → ∞. Applying the B-field

procedure above, solution for Taylor dispersion coefficient DT and dimensionless coefficient kTbecomes

DT = γ �(2)0 PeD0, γ �(2)

0 =R4r − 24cφ − 6φ(3 − 5R2

r + 2R4r ) + 11r23φ

2

48r4Pe,

kT =DT D0R

2

U2�φ�2=

R4r − 6φ(3 + 4c + 2R4

r − 5R2r ) + 11φ2r23

12r34, Pe =

UR

D0

. (130)

When Rr = 1, this solution reduces to Eq. (99): DT = Pe2

48D0. A typical behaviour for kT(φ) is

illustrated in Fig. 25 for several values of aspect ratio Rr. In the left diagram, Rr ∈ [ 12, 1] and the

0.2 0.4 0.6 0.8 1.0Φ

1.00

0.50

0.20

0.30

0.15

1.50

0.70

kT

0.2 0.4 0.6 0.8 1.0Φ

10

100

1000

104

105

kT

FIG. 25. The two diagrams plot solution (130) for kT(φ) in open/impermeable cylindrical system with porous layer of

porosity φ, when Rr = { 12, 34, 910

, 99100

} (left diagram) and Rr = { 12, 14, 10−1, 10−3} (right diagram, log-scale) (solid, dashed,

dotted, dashed-dotted lines, respectively).



limit case is the single pipe solution (99), kT |Rr→1 = 112. Next, solution for γ �(3)

0 and γ �(4)0 reads

γ �(3)0

Pe2=

1

5760(r2R2r − φ)

(−3R6r + 251φ3r33 + r3φ(1020 − 1050R2

r + 247R4r )

− 120cφ(18 + 12c + 11r3φ − 15R2r ) − 495r23φ

2(−2 + R2r )),

γ �(4)0

Pe3=

1

7741440(r2R2r − φ)

(53R8r + 252φ3r33 (−1506 + 635R2

r )

+ 12r3φ(−50330 − 35924R4r + 4525R6

r + 81550R2r )

− 50427r43φ4 − 2r23φ

2(438900 + 81973R4r − 406560R2

r )

+ 672cφ(2850 + 753r23φ2 + 1297R4

r − 4140R2r − 990φ(3 + 2R4

r − 5R2r )

+ 60c(54 + 24c − 30R2r + 11φ(−3 + 2R2

r )))), and

γ �(3)0 |Rr=1 =

Pe2

1920, γ �(4)

0 |Rr=1 = −53Pe3

7741440. (131)

In turn, solution for δγ �(2) − δγ �(4) becomes

δγ �(2) =1

Pe,

δγ �(3)

Pe2=

2

Pe4+

R4r − 24cφ − 12φR2

r r3 + φr3(18 + 11φr3)

12Pe2r4,

δγ �(4)

Pe3=

5

Pe6+

c1Pe2 + c2Pe

4

2304Pe6r24,

c1 = 720(R2r − φr3)(−24cφ + R4

r − 12φR2r r3 + φr3(18 + 11φr3)),

c2 = 11R8r − 2880c2φ(−2φ + (−1 + φ)R2

r )

− 620φR6r r3 + 2φR4

r r3(1140 + 1157φr3)

− 480cφ(8R4r + φr3(18 + 11φr3) − R2

r (9 + 19φr3))

− 4φR2r r3(510 + φr3(1560 + 703φr3))

+ 3φ2r23 (1220 + 3φr3(440 + 123φr3)), and

δγ �(3)|Rr=1 =2

Pe2+

1

12, δγ �(4)|Rr=1 =

5

Pe3+

5

16Pe+

11Pe

2304. (132)

When Rr = 1, γ �(n) reduces to solution (102)–(103) for open pipe. The obtained solution for γ �(n)

allows to compute Sk� and Ku� via Eq. (52), and then to compare them with the Gaussian solution

(54) whereD=D0 +DT andDT is given by Eq. (130). Finally, the coefficientsω�(n) of “s-expansion”

can be obtained via relations (19) from solution for γ �(n).

VI. THE TRT TRANSPORT SCHEME FOR HETEROGENEOUS POROUS FLOW

A. Numerical algorithm

We extend advection-diffusion TRT schemes46 for transport equation (1) from open to het-

erogeneous porous flow. In this section, we keep symbols u (seepage velocity) and D0 (molecular

diffusion) for their numerical values. The physical and modelled equations yield the same Peclet

numbers, porosity distribution and aspect ratios. The discrete, d-dimensional velocity vectors consist

of zero vector c0 and Qm = Q − 1 vectors cq connecting grid nodes r. This velocity set is anti-

symmetric, that is each vector cq has the opposite one cq , hereafter cq = −cq . Two local equilibrium

values e±q (r, t) are prescribed for each couple of the opposite velocities. The primary variable of the

scheme is the vector { fq (r, t)} composed of Q “populations.” The TRT scheme40 updates them with



the help of two relaxation parameters s±(r, t) restricted to the linear stability interval ]0, 2[:

fq (r + cq , t + 1) = fq (r, t) + g+q + g−

q , q = 0, . . . ,Qm

2,

fq (r − cq , t + 1) = fq (r, t) + g+q − g−

q , q = 1, . . . ,Qm

2, with

g±q = −s±( f ±

q − e±q ), f ±

q =fq ± fq

2, q = 1, . . . ,

Qm

2,

g+0 = −2

Qm/2�

q=1

g+q , g−

0 = 0. (133)

The local mass (sum) of populations defines the conserved quantity φC:

φC(r, t) =Q−1�

q=0

fq = f0 + 2

Qm/2�

q=1

f +q . (134)

The solution C(r, t) is set equal to�Q−1

q=0 fq/φ. The equilibrium distribution e±q (r, t) = E±

q C is

prescribed in such a way that it contains the local mass:�Q−1

q=0 eq = φC, then e0 = e+0 =

(φ − 2�

Qm2

q=1 E+q )C, with

E+q = t (m)

q cφ + InφE(u)q (u), E−

q = t (c)q (uφ · cq ), uφ = φu, q = 1, . . . ,Qm

2, (135)

E (u)q (u) = t (u)q u2 + w(u)

q �cq�2d�

α

(u2α − u2)c2qα +�

α �=β

uαuβcqαcqβ

2�

Qm2

j=1 c2jαc

2jβ

,

with u2 =u2

d, u2 =

d�

α

u2α,

Q−1�

q=1

t (·)q cqαcqβ =Q−1�

q=1

w(·)q cqαcqβ = δαβ. (136)

The free-tunable positive parameter cφ will determine the modeled diffusion coefficient. However, it

is restricted by stability conditions as discussed below. The TRT scheme (133)–(135) copes with the

“minimal” velocity sets dDQ(2D + 1), as d1Q3, d2Q5, and d3Q7, where t (m)q = t (c)q = t (u)q = w(u)

q

= 12, ∀ q, but also with the “full” (hydrodynamic) sets.43 Stability dependence on the equilibrium

weights has been first analyzed for d2Q9 and d3Q15 sets.44, 45 This analysis is extended for d3Q13

and d3Q19 sets,46 including the anisotropy and role of the weights for truncation corrections in

uniform flow. The minimal schemes yield

E+q =

1

2(cφ + Inφu

2αc

2qα), E

−q =

1

2φuαcqα, q = 1, . . . ,

Qm

2, cqα �= 0. (137)

The d2Q9 equilibrium for open flow is presented in more detail in work.42 Applying the stan-

dard second-order Chapman-Enskog analysis,42 the concentration C obeys the following effective

equation of the model:

∂tφC + ∇ · uφC =�

α,β

∂α�−φ ∂β(cφδαβ + (In − 1)φuαuβ)C. (138)

The two positive functions of relaxation rates,�±φ = 1

s± − 12, are given below in Eqs. (139) and (140).

When In = 0, Eq. (138) incorporates the anisotropic tensor of numerical diffusion, with the entries

{−φ�−φ uαuβ} in uniform porosity block. When In = 1, the full velocity sets remove it completely

while the minimal sets only cancel its diagonal components. However, when the velocity vector is

along the coordinate axis, as in numerical examples in this work, the off-diagonal entries vanish and

the minimal models are consistent as well. An extension for modelling of the anisotropic diffusion

tensor is straightforward by adding the anisotropic,mass conserving equilibrium correction E (a)q (cφ)C



to e+q (e.g., E (a)

q (ce) is given by Eq. (26) in work42). In principle, all model parameters, such as uφ ,

cφ , and �±φ , may depend on space and time. In this work, they are time-independent. Also, all

equilibrium weights are space-independent, i.e., the same in all grid nodes. When D0 and cφ(r) are

prescribed, Eq. (138) fits Eq. (1) if �−φ (r) obeys

�−φ (r) =

φ(r)D0

cφ(r)and �−

φ (r) =1

s−(r)−

1

2. (139)

This relation sets a numerical value for s−(r). The relaxation parameter s+(r) is free-tunable and it

is set giving the positive value �φ :

�+φ (r) =

�φ(r)

�−φ (r)

and �+φ (r) =

1

s+(r)−

1

2. (140)

Further parameter choice is dictated by the continuity conditions40 at the interface between two

blocks. So far, the continuous pre-factors E+q (r) assure leading-order continuity for C midway the

interface-cutted velocity link. When In = 0, they are continuous if cφ is the same for all points, and

hence, �−φ (r) varies with φ(r):

�−φ (φ) = φ(r)�−

1 ,�−1 = �−

φ (φ = 1) =D0

cφ

. (141)

When In = 1, since the Darcy velocity uφ is continuous, the correction term φE (u)q (u) = E (u)q (uφ )

φ

undergoes a jump when φ changes abruptly, except if the interface is parallel to the velocity vector

uφ and then E (u)q (uφ) vanishes for interface-cutted links, as in stratified media. It follows that in

general, this term has to be relatively small with respect to cφ . The stability conditions will support

such a choice as well.

We note that when φ(r) is uniform, solution of Eq. (1) for C(r, t) is the same for any φ

when u = uφ

φis φ−independent. However, numerical solutions yield this property only if the same

relaxation rates s± are applied for all φ, and then cφ(r) = ceφ(r). When In = 0, one easily observes

this in Eq. (138) (dividing it by φ) since the numerical diffusion tensors are not the same when

�−φ differs. In general, and when In = 1, the same relaxation rates are dictated by the third and

high-order truncation errors (see in Subsection A 1 of the Appendix). Therefore, applying strategy

(141) where �−φ varies with φ, the modeled second-order equations (138) are equivalent for any

uniform porosity value when In = 1, but the obtained concentration profiles will differ for them

because of distinguished truncation corrections.

Finally, the simplest and local boundary conditions are the “bounce-back” (BB), to mimic

the zero-flux condition, and the “anti-bounce-back” (ABB), to prescribe the concentration value

Cb. Namely, if the point rb = r + cq lies outside the computational domain, these rules prescribe

incoming population fq (r, t + 1), as

BB : fq (r, t + 1) = [ fq + g+q + g−

q ](r, t),

ABB : fq (r, t + 1) = −[ fq + g+q + g−

q ](r, t) + 2E+q (r)Cb

�

r + rb

2

�

. (142)

Note that the two rules locate boundary point midway the link (r, rb) with the first-order accuracy,

and hence, exactly only for linear concentration profile and straight walls. The second-order accurate

location of boundary and the interface is controlled by the free parameter�φ , at steady state at least.40

Several other choices for �φ , as dictated by the wish for reduction of the truncation errors and for

the improvement of stability, may be taken as discussed below.

B. Necessary stability conditions

Here, we extend both the stability and non-negativity conditions of the TRT and BGK schemes,

from open to porous flow, assuming porous blocks of uniform porosity φ. For this purpose, let us



rewrite Eqs. (135) and (138) for evolution of ρ = φC in velocity field u = uφ/φ:

e+q = E+

q ρ, e−q = E−

q ρ, e0 = E0ρ,

E+q = t (m)

q ce + InE(u)q (u), E−

q = t (c)q (u · cq ), E0 = 1 − 2

Qm2�

q=1

E+q . (143)

This form is equivalent to Eq. (135) with

ρ = φC, ce =cφ

φ, u =

uφ

φ. (144)

The modelled equation (138) then reads

∂tρ + ∇ · uρ =�

α,β

∂α�−φ ∂β(ceδαβ + (In − 1)uαuβ)ρ, ce�

−φ = D0. (145)

The necessary and sufficient stability conditions have been examined in works.44,45 Replacing ce by

cφ /φ and u2 = �u�2 by u2φ/φ2, they give for the minimal TRT schemes (137):

In = 0 : u2φ ≤ φcφ, cφ ∈�

0,φ

d

�

,

In = 1 : u2φ ≤ min{φ(φ − dcφ),d

d − 1φcφ}, cφ ∈

�

0,φ

d

�

. (146)

These conditions reduce to open flow for φ = 1. Similarly, the necessary stability conditions44,46

for modelling of advection-diffusion equation in open flow with “full” velocity schemes can be

re-interpreted for porous flow. We stress that cφ reduces with φ for all schemes. When cφ is

set, the stable Darcy velocity amplitude �uφ� reduces with porosity as dictated by the least

porous block. Following exact solutions for open flow, the constraints (146) are expected to be

sufficient for any velocity direction only on the so-called “optimal” TRT subclass45,46 where

�φ ≥�

1 − 8(�−φ )

2 +�

64(�−φ )

4 + 1�

/8. The stable strategy consists of increasing �φ from 16

to 14when (�−

φ )2 reduces from 1

6towards zero. Beyond the optimal subclass, the effective stability

bound u2φ(cφ) depends on the two relaxation rates, as predicted by the exact45 or the approximate37

stability curves. In open flow, the optimal TRT subclass �φ = 14may remove the left (advection)

stability branch, as u2φ ≤ dd−1

φcφ in Eq. (146), for specific equilibrium weights of velocity sets

with the diagonal links.44–46 In principle, these models could then reach the best possible stability

condition as u2φ ≤ φ2 when cφ → 0 (i.e. u2 ≤ 1 for infinite Peclet number). The “hydrodynamic”

velocity sets may also increase the stable cφ interval to cφ ∈ [0, φ], depending on weights t (m)q .

The BGK subclass of the TRT model yields �−φ = �+

φ =�

�φ . Its sufficient stability bounds

for equilibrium (143) are set by the non-negativity conditions:44 {E+q ± E−

q ≥ 0,∀ q = 0, . . . , Qm

2}.

In particular, the minimal schemes (137) yield

In = 0 : u2φ ≤ c2φ, cφ ∈�

0,φ

d

�

,

In = 1 : u2φ ≤ min

�

φ(φ − dcφ),

�

φ −�

φ(φ − 4cφ)�2

4

�

, cφ ∈�

0,φ

d

�

. (147)

The non-negativity conditions become necessary for the BGK when �−φ decreases towards zero.

Note that the non-negativity conditions reduce the velocity amplitude below its optimal bound (146).

C. Numerical assessment

In this work we apply the TRT scheme (133) with equilibrium (135) for the d2Q5 velocity set, in

a stratified two-dimensional two-layered domain [L × H] where the interface is placed in the middle

z = H2. The porosity φ1 of lower layer varies between 10−2 and 1. The upper layer is set empty,



φ2 = 1. The relaxation rates are selected with Eqs. (139) and (140) where we set �−φ = φ/

√12

because of the particular accuracy of this choice (see in Subsection A 1 of the Appendix). Unless oth-

erwise indicated, we adopt In = 1 in equilibrium (137) which removes numerical diffusion. The equi-

librium parameter cφ is the same for all points and it is restricted to the stability interval ]0, φ1/2] (cf.

Eq. (146)). We compare results for �φ = { 112

, 16, 14} and the BGK choice �φ = (�−

φ )2 = φ2/12.

1. Evolution of waves in Darcy flow

a. Effective dispersion relations. Exact solution for evolution of waves in constant two-

layered Darcy flow is set by solvability condition (65). In this section, accuracy and stability of

the TRT scheme is examined in this configuration. Suitable parameter choice is then adopted for

simulations of the temporal and spatial dispersion with large porosity contrasts. In this system, the

periodic boundary conditions are applied at the ends of the computational domain and the solu-

tion is searched as: Ci (t, x, z) = Ci (z) cos(k · (x − �imt)) exp(−�r t) with wavevector k = {k, 0},k = 2π /L, then K 2

i (�, k) in Eq. (65) reads with

K 2i (�, k) =

−� + ikui + D0k2

D0

, � = �r + i�im, ui =uφ

φi

. (148)

When the porosity is the same in two layers, solution reduces to �r = D0k2 and �im = uφ

φk. In two

different porosity layers, the averaged velocity U and Taylor dispersion coefficient DT are specified

by Eqs. (68) and (70), respectively. However, the solution for a finite wavenumber may strongly

deviate from the zero-k limit: �r = (D0 + DT)k2 and �im = Uk; this difference increases with the

porosity contrast and/or Peclet number, as illustrated in Table V. Numerical solution of Eq. (65)

with Eq. (148) is used as a benchmark. Corresponding values of �(num)r and �

(num)im for �(num), where

�(num) = �(num)r + i�

(num)im , are estimated from the TRT solution as follows:

�(num)r =

1

ta(log[I (ta)] − log[I (t + ta)]), I 2(t) = I 2s (t) + I 2c (t),

k�(num)im =

1

ta

�

arctan

�

Is(t + ta)

Ic(t + ta)

�

− arctan

�

Is(t)

Ic(t)

��

, with

Is(t) =1

HL

� H

0

� L

0

C(t, x, z) sin(kx)dxdz,

Ic(t) =1

HL

� H

0

� L

0

C(t, x, z) cos(kx)dxdz. (149)

b. Homogeneous porous flow. In constant porosity field, we first verify that numerical simu-

lations produce the identical results for different φ using the same eigenvalues and keeping the same

values for ratios cφ /φ and uφ /φ. In what follows we apply the alternative parameter choice (141),

most suitable for heterogeneous layers. The longitudinal truncation corrections impact the solution

TABLE V. This table shows the relative differences { �r

(D0+DT )k2 − 1,

�imkU − 1} between exact solution given by Eq. (65)

with Eq. (148) and its leading order approximation (D0 + DT )k2 + iUk in two layered Darcy flow, versus Peclet number

Pe(uφ) = uφ H

D0, D0 = cφ√

12. Parameter range corresponds to computations below, with φ2 = 1, k = 2π

Lin finite box L × H

= 800 × 12.

Pe(uφ ) φ1 = 2 × 10−1 φ1 = 2 × 10−2

4.16 {−2.32 × 10−4, −2.84 × 10−4} {−1.66 × 10−1, −7.32 × 10−2}

8.31 {−1.59 × 10−3, −1.14 × 10−3} {−4.56 × 10−1, −1.91 × 10−1}

41.57 {−5.29 × 10−2, −2.78 × 10−2} {−9.46 × 10−1, −3.93 × 10−1}

83.14 {−2.15 × 10−1, −9.96 × 10−2} {−9.81 × 10−1, −4.21 × 10−1}



TABLE VI. Table shows the relative differences err(�) = {err(�r), err(�im)} and err (�(tr )) = {err (�(tr )r , err (�

(tr )im )} for

Darcy flow in homogeneous media of porosity φ, using cφ = φ4, uφ = φ

2, �−

φ = φ√12

(except for (�−φ )

2 = 112). The error

estimate in right column accounts for truncation errors via Eq. (76) with Eq. (A10).

�φ err(�) err(�(tr))

φ = 1112

{−1.91 × 10−11, −3.75 × 10−14} {−1.91 × 10−11, −3.75 × 10−14}16

{ 2.57 × 10−6 , −2.57 × 10−6} { 7.05 × 10−11, −2.05 × 10−11}14

{ 1.03 × 10−5 , −5.14 × 10−6} { 6.21 × 10−11, −1.1 × 10−11}

φ = 2 × 10−2

(�−φ )

2 = 112

{≈10−11, ≈10−14} {≈10−11, ≈10−14}

φ2

12{−1.29 × 10−6, −2.22 × 10−16} {−1.6 × 10−9, −4.44 × 10−16}

16

{−1.27 × 10−2, −5.17 × 10−6} { 1.65 × 10−4, −3.26 × 10−8}14

{ 1.41 × 10−5, −7.71 × 10−6} {−1.6 × 10−9, −4.44 × 10−16}

in each layer as indicated by Eq. (76) with Eq. (77). One then expects for the TRT scheme:

�(num)r ≈ �(tr )

r = D0k2 − bk4 and k�

(num)im ≈ k�

(tr )im = uφ

φk + ak3, where a(

cφ

φ,

uφ

φ,�−

φ ,�φ) and

b(cφ

φ,

uφ

φ,�−

φ ,�φ) are given by Eq. (A10). For comparison, we provide the two estimates, err(�)

and err(�(tr)) computed as the two relative differences of the solution (149) with � = D0k2 + i

uφ

φk

and �(tr), respectively. Table VI shows them for φ = 1 and φ = 2 × 10−2, in finite box 800 × 12.

The results confirm that for the very particular choice (A11): cφ = φ

4and uφ = φ

2, as given by

Eq. (A11), the truncation corrections vanish for �im on the BGK subclass �φ = (�−φ )

2, and then

they vanish for both �r and �im when �φ = (�−φ )

2 = 112. Furthermore, since err(�(tr)) is several

orders of magnitude smaller then err(�), the third and fourth order truncation errors are mainly

responsible for the deviation from the exact solution in homogeneous case.

These computations employ In = 1 and they are stable in the limits prescribed by Eq. (146) for

�φ = 16and�φ = 1

4. When In = 0, the simulations remain stable on the stability line u2φ = φcφ only

for �φ = 14(cf. Eq. (146)). This is in agreement with the exact stability results since (�−

φ )2 < 1

6in

these computations (see in Sec. VI B). Moreover, when In = 0, the effective diffusion value derived

with Eq. (149) reduces from D0 = cφ�−φ to D0

(num) ≈ 10−8 − 10−7 for cφ = φ/4 and uφ = φ/2, in

agreement with the prediction: D0(num) = �−

φ (cφ − u2φφ) ≈ 0. This confirms that the optional choice

with In = 0 should be restricted to u2φ � φcφ .

c. Heterogeneous porous flow. In heterogeneous porosity, we examine two cases: φ1 = 2 ×10−1 and φ1 = 2 × 10−2 while φ2 = 1. Unless indicated, cφ = 5 × 10−3. Stability condition (146)

restricts uφ|cφ= φ1

4

to√22

φ1. Like in homogeneous media, the simulations remain stable to this limit

when �φ = { 16, 14}. We then examine four velocity values, when uφ increases from uφ1 = 5 × 10−4

to uφ4 = 10−2. The results for �φ = 16are gathered in Tables VII and VIII, and in Table IX for the

BGK model. Table VII addresses two values, cφ = 5 × 10−3 and cφ = 5 × 10−2, applying the same

TABLE VII. The relative difference err(�) = {err(�r), err(�im)} for Darcy flow in heterogeneous media when φ1 = 2

× 10−1, φ2 = 1, �φ = 16, for two configurations with the same Peclet numbers: cφ = 5 × 10−3 and cφ = 5 × 10−2.

uφ cφ = 5 × 10−3 uφ cφ = 5 × 10−2

uφ1 5 × 10−4 {1.25 × 10−2, −1.22 × 10−5} uφ1 × 10 {1.44 × 10−2, −1.38 × 10−5}

uφ2 10−3 {2.15 × 10−2, −3.33 × 10−5} uφ2 × 10 {2.47 × 10−2, −3.77 × 10−5}

uφ3 5 × 10−3 {2.85 × 10−2, −7.55 × 10−4} uφ3 × 10 {3.29 × 10−2, −8.63 × 10−4}

uφ4 10−2 {2.94 × 10−2, −3.2 × 10−3} uφ4 × 10 {3.35 × 10−2, −3.72 × 10−3}



TABLE VIII. Table shows the TRT relative errors err(�) in Darcy heterogeneous flow when φ1 = 2 × 10−2, φ2 = 1, in

finite boxes L1 × 12 and L2 × 12.

Darcy velocity Box length

uφ L1 = 800 L2 = uφ nuφ n−1

L1

uφ1 5 × 10−4 {2.72 × 10−2, −1.99 × 10−3}

uφ2 10−3 {3.07 × 10−2, −5.99 × 10−3} {2.87 × 10−2, −1.98 × 10−3}

uφ3 5 × 10−3 {8.15 × 10−2, −2.12 × 10−2} {3.15 × 10−2, −5.99 × 10−3}

uφ4 10−2 {1.19 × 10−1, −3.21 × 10−2} {8.21 × 10−2, −2.12 × 10−2}

eigenvalues but keeping the same Peclet numbers via velocity increase, by factor of 10. As could be

expected from the truncation analysis in Subsection A 1 of the Appendix, the results are very similar

but not identical: err(�) slightly increases with cφ . The results in these three tables for particular

choice (A11): cφ = φ1

4, with uφ4 when cφ = 2 × 10−2 and uφ4 × 10 when cφ = 2 × 10−1, show that

the BGK model remains most accurate, similar as in the homogeneous case. We note that �φ = 112

is also more accurate than �φ = { 16, 14}, e.g., err (�)|uφ=uφ1

=�

7.83 × 10−3, 1.16 × 10−3�

when

φ1 = 2 × 10−2. However, this choice becomes unstable for larger velocities in agreement with its

exact stability analysis.45 At the same time, the BGK choice �φ = (�−φ )

2 = φ2

12remains stable on

the non-negativity boundary (147), which is however below the optimal choice (146).

The second column in Table VIII shows that the err(�) scales as U/L when φ1 = 2 × 10−2 (and

very similar for φ = 2 × 10−1). These observations suggest that err(�) is set by uφk when all other

parameters are fixed. In turn, increasing twice the channel widthH, and thus keeping the same Peclet

number for {uφ1,H= 24} and {uφ2,H= 12}, we obtain err (�)|uφ 1=�

1.01 × 10−2,−2.93 × 10−3�

when φ1 = 2 × 10−2. This result is about three times more accurate than err |uφ2(�r ) and twice

as accurate as err |uφ2(�im) in Table VIII. However, increasing H by a factor of 2 for uφ3 and uφ4,

err(�im) reduces, again, by a factor of 2–3 but the reduction for err(�r) is much weaker.

d. Account of interface conditions. In fact, in contrast to the homogeneous case, the longitu-

dinal truncation corrections alone do not quantify the err(�), because of (a) transverse discretisation

errors and (b), inexactness of the interface conditions. To clarify the impact of these two factors,

we first partly accounted for the transverse truncation corrections computing K 2i with Eq. (A12).

However, this remains insufficient to match err(�r). We then construct in Subsection A 3 of the

Appendix the effective solvability conditions of the TRT scheme based on its effective interface

conditions (cf. (A16) with (A17)).

In this solution, the non-equilibrium component is only partly restored, at best up to fourth-order

when�φ = 14. This approximate solution provides us solution for�(tr).We then compute err(�(tr)) as

the relative difference between the numerical and approximate solutions,�num and�(tr), respectively.

The results given in Tables X and XI for �φ = 16and �φ = 1

4, respectively, confirm that err(�(tr))

TABLE IX. Table shows err(�) for the BGK model in finite box 800 × 12. The two last lines address the non-negativity

limit uφ |cφ= φ1

4

= φ12where the third-order longitudinal truncation errors vanish in φ1 (porous) layer.

uφ φ1 = 2 × 10−1 φ1 = 2 × 10−2

cφ = 5 × 10−3

uφ1 5 × 10−4 {−9.87 × 10−3, 1.95 × 10−5} {−1.16 × 10−2, 4.2 × 10−3}

uφ2 10−3 {−1.68 × 10−2, 7.07 × 10−5} { 1.08 × 10−2, 6.51 × 10−3}

uφ3 5 × 10−3 {−1.72 × 10−2, 1.63 × 10−3} {−4.72 × 10−3, −3.65 × 10−3}

uφ4 10−2 {−8.43 × 10−3, 1.11 × 10−3}

cφ = 5 × 10−2

uφ4 × 10 10−1 {3.07 × 10−2, −5.99 × 10−3}



TABLEX. Table shows err(�) with respect the benchmark solution and err(�(tr)) with respect the TRT approximate solution

when �φ = 16. The TRT approximation is built is Subsection A 3 of the Appendix.

φ1 = 2 × 10−1, cφ = 5 × 10−2

Pe(uφ ) err(�) err(�(tr))

4.16 {2.15 × 10−2, −3.33 × 10−5} {−3.71 × 10−4, 10−5}

83.14 {3.35 × 10−2, −3.72 × 10−3} {7.55 × 10−3, 7.89 × 10−4}

φ1 = 2 × 10−2, cφ = 5 × 10−3


4.16 {2.72 × 10−2, −1.99 × 10−3} {4.38 × 10−3, 6.07 × 10−4}

83.14 {1.19 × 10−1, −3.21 × 10−2} {−3.22 × 10−2, −2.18 × 10−3}

is about one order of magnitude smaller than err(�) for �φ = 16, and err(�(tr)) almost vanishes for

�φ = 14. This indicates that account of effective interface conditions is imperative for numerical

analysis of the solution of the LBE schemes, while the bulk truncation corrections alone are not

sufficient in heterogeneous limit.

e. Resume. The numerical simulation of waves confirm the validity of our analysis for nu-

merical diffusion, stability, and high order truncation corrections. Namely, the observed numerical

diffusion for the equilibrium with In = 0 in the wave decay agrees with the predictions. In hetero-

geneous media, the minimal stable velocity amplitude has to be respected. When the equilibrium

parameter cφ(r) is set the same for all points, the least porous block dictates the stable velocity range

for uφ . The optimal TRT choice �φ = 14remains the most stable in heterogeneous case as these

necessary conditions suffice. When numerical diffusion is removed, �φ can be reduced towards

�φ = 16. In turn, the BGK subclass remains controlled by the minimal velocity amplitude as set by

the non-negativity conditions for all porous blocks. Further work is necessary to check if the optimal

TRT subclass keeps its advanced stability for models with the diagonal links, such as d2Q9 and

d3Q19, where the advection line dd−1

φcφ in Eq. (146) vanishes.44,46

Concerning the precision of the scheme, the BGK model with �−φ = φ/

√12 is particularly

accurate for waves using very specific equilibrium parameters where truncation corrections vanish.

However, the TRTmodel is preferred in general, and especially for large diffusion coefficients and/or

in the presence of boundaries.37 The deviation of the numerical solutions from the theoretical predic-

tions in heterogeneous media for coarse resolutions has been explained by taking into consideration

the effective interface conditions, along with the effective longitudinal and transverse discretisations,

as solvability conditions in Darcy flow.

TABLE XI. Same as in Table X but for �φ = 14.

φ1 = 2 × 10−1, cφ = 5 × 10−2


4.16 {2.77 × 10−2, −3.44 × 10−5} {4.5 × 10−4, − 2.87 × 10−7}

83.14 {5.07 × 10−2, −8.57 × 10−3} {8.67 × 10−4, −6.89 × 10−5}

φ1 = 2 × 10−2, cφ = 5 × 10−3


4.16 {4.61 × 10−2, −5.17 × 10−3} {5.22 × 10−6, −4.14 × 10−7}

83.14 {1.21 × 10−1, −4.76 × 10−2} {1.41 × 10−4, −5.39 × 10−8}



2. The “temporal” dispersion in Darcy flow

The model configuration and the domain size are the same as in Sec. VI C 1. The box is periodic

in the transverse direction, for z-axis, but the Dirichlet boundary condition Ci (t, x = 0, z) = 1 is

implemented at the inlet via the anti-bounce-back rule (142), while the Dankwerts (zero-flux)

condition ∂xCi (t, x = L) = 0 is set at the outlet with the bounce-back rule (142). As above, �−φ

= φ/√12 and then, D0 = cφ/

√12. We mainly discuss the results when cφ = 10−3, uφ = 1.2 ×

10−4 and �φ = 112. In these simulations, Pe(U ) = UH

D0belongs to the interval [8, 10] when porosity

ratio Rφ decreases from 2 × 10−1 to 2 × 10−2. The TRT scheme provides solution Ci (t, x, z) for two

layers, i = 1, 2. We compute the averaged concentration C(t, x �) =�

i Ci (t,x�,z)φi

�φ� , �φ� =�

ihiφi and

the RTD distribution, P(t, x�)≈ (C(t+ �t, x�)−C(t− �t, x�))/(2�t) at given set of the grid points {x�

= x/(HPe}. At the same time, we accumulate the raw moments of the RTD, µn(x�) =

�

tP(t, x�)tn�t,

until P(t, x�)≡ 0. We set�t= 1 for all computations but this integration time step could be increased

by factor 10–100, at least, without noticeable loss of accuracy. The effective numerical value of

the average velocity, Unum(x �) = x �/µ1(x�) is then used to restore sequentially the central moments

µ�n(x

�) from their raw values. The EMM predictions for dimensionless moments are computed with

Eq. (26) where we use relations (50) to restore γ (n) from its dimensionless solution γ �(n), which as

given by Eq. (66) with Eqs. (69) and (71). The predicted values for Sk� and Ku� are computed with

Eq. (52) and given in Table I, together with their ADE values Sk �G and Ku�

G given by Eq. (54). The

numerical moments are compared to their EMM and ADE solutions in Fig. 8 where their relative

differences are plotted for Rφ = 2 × 10−1 and Rφ = 2 × 10−2. Clearly, the TRT solutions for Sk and

Ku are much closer to the EMM predictions than to the ADE solution. Further reduction of velocity

(or its increasing, for example, by factor 5 when Rφ = 10−1) does not noticeably impact these

results. In turn, refinement of the grid from H = 6 to H = 12 significantly improves the accuracy.

We specially perform these computations on the relatively coarse grids in the transverse direction, to

mimic the realistic porous media applications. Finally, we emphasize that the numerical values for

Sk� and Ku� strongly depend on the obtained Taylor dispersion coefficient. We note that in similar

parameter range, DT is matched less accurately than we could expect from the above simulations

in waves. Most likely, this is related to the time-dependent character of this example. Figures 9–11

compare the RTD profiles for the EMM, ADE, and TRT when Rφ = 10−1 and Rφ = 4 × 10−2.

The TRT profiles agree very well with the EMM predictions, and they clearly differ from the ADE

solutions when the porosity contrast increases. We recall that �φ = 112

was selected for its advanced

advection accuracy in waves. However, we do not detect any noticeable difference for high moments

with �φ = 16or �φ = 1

4in these simulations. These larger �φ values allow for increasing of stable

velocity amplitude.

3. The “spatial” dispersion in Darcy flow

The TRT scheme is similar to that applied for “temporal” dispersion but the channel is longer:

L × H = 2400 × 12. This allows us to use the periodic boundary conditions for x-ends during

the observation time for release of the initial plume: Ci (t = 0, x = x0, z) = 1, Ci (t = 0, x, z) = 0

if x �= x0. The ADE solution for averaged concentration is given by the Gaussian distribution,

C(x, t) = exp(−(x−x0−U t)2/4Dt)

2√

πDtwith D = D0 + DT. The first two moments are the same for the ADE

and EMM. However, all next central moments vanish for the ADE while they are non-zero for the

EMM, unless when φi is the same for two layers and DT reduces to zero. We discuss the results

for Rφ = 4 × 10−2 and Rφ = 2 × 10−2, when Pe(U ) ≈ 9.07 in the two cases. The TRT scheme is

applied with cφ = 5 × 10−3, �φ = 16, uφ(Rφ = 4 × 10−2) ≈ 5.67 × 10−4 and uφ(Rφ = 2 × 10−2)

≈ 5.56 × 10−4. The predicted characteristic values are gathered in Table II. We first compute the

spatial raw moments of the averaged concentration: µ(n)(t)=�

x(x− x0)nC(t�, x)dx for several values

of the dimensionless time t� = t/T, T = HPe/U . The central moments µ�(n)(t) and coefficients

ω�(n) are sequentially restored using numerical values Unum(t) = µ1(t)

tand previous order moments

(cf. Eq. (30)). The numerical values for Sk�(t) and Ku�(t) are then computed from Eq. (53). Figure 26

plots the absolute values for err(D0 + DT), err(Sk), and err(Ku), which are the relative differences

with the EMM predictions for three configurations: the BGK with �φ = (�−φ )

2 and TRT with



1 2 3 4 5 6t'

0.05

0.00

0.05

rel . err

1 2 3 4 5 6t'

0.05

0.00

0.05

rel . err

1 2 3 4 5 6t'

0.05

0.00

0.05

rel . err

FIG. 26. The Darcy flow, “spatial” dispersion, Rφ = 4 × 10−2 with �φ = (�−φ )

2 (first diagram), �φ = 16(second diagram),

and �φ = 14(last diagram). Data in Table II. The figure plots the relative difference of the TRT numerical moments to EMM

predictions: (a) U (solid line, black), (b) D = D0 + DT (dotted line, red), (c) Sk (dashed line, blue), and (d) Ku (dotted-dashed

line, magenta).

�φ = { 16, 14}. The relative velocity error err (U ) is small in these computations, its smaller than

10−13 up to t� ≈ 3 and then increases to 10−8–10−7 when t� ∈ [3, 6]. In agreement with the truncation

analysis for waves, the BGK matches slightly more accurately the averaged velocity, and �φ = 16is

more accurate than �φ = 14for DT. However, the velocity and dispersion errors can mutually cancel

or amplify for Sk(t) and Ku(t), such that they best agree with predictions for �φ = 14in this example.

Figures 12–15 illustrate ADE, EMM, and TRT solutions. Last two figures clearly show that the TRT

better agrees with the EMM shapes reconstructed for larger number of moments. This validates the

two methods. Altogether, the TRT difference with the EMM results is smaller in these simulations

than for “temporal” dispersion. Future work needs to examine whether this feature is because of

low-order (ABB) inlet boundary conditions used for “temporal” dispersion.

4. Open/porous channel

The numerical configuration for “temporal” and “spatial” dispersion in open/impermeable

system is the same as in Darcy flow, except that the constant Darcy velocity in Eq. (135) is replaced

by the prescribed parabolic profile for the open layer, with the zero velocity for the impermeable layer

(cf. Eq. (124)). The bounce-back condition (142) is applied for the two horizontal boundaries. The

tests were run in L×H= 600× 12 channel for “temporal” dispersion, with�φ = 112, and in a twice

longer channel for “spatial” dispersion, with �φ = 16, applying cφ = 2.5 × 10−3, �−

φ =√

φ/12, U0

= 3 × 10−4, D0 ≈ 7.22 × 10−4 in both cases. The results are discussed in Secs. V B 2 andV B 3.

For spatial dispersion, the data is given in Table IV for (a) single channel (Rh = 0, Pe ≈ 59.86) and

(b) porosity contrast Rφ = 10−1, when the interface is in the middle, Rh = 1, Pe ≈ 13.6. Further

reduction of cφ and uφ by factor 10 does not noticeably modify these results. Also, using of the

BGK model with �φ = �−φ = φ/12 does not improve the effective averaged velocity and results in

slightly less accurate moments compared to TRT solution as depicted in Fig. 22. Future work needs

to clarify the role of the truncation corrections of the TRT model in variable velocity field.

VII. CONCLUSION

This paper proposes a mathematical algorithm that allows for the calculation of the sequence

of moments for concentration or residence time distributions in periodic steady velocity fields,

in homogeneous or heterogeneous media. The procedure is based on the successive solution of

one-dimensional linear stationary advection-diffusion equations with heterogeneous sources in a

single periodic porous cell. The advantage of the EMM is that it avoids the computationally in-

volved transient modelling of the contaminant release in the entire domain. The scalar variable of

these equations is called B-field since the EMM method can be regarded as an extension of the

pioneering work by Brenner,3 from the second to higher moments, on the one side, and from the

“spatial” to “temporal” dispersion, on another. This work derives generic relations between the

spatial and temporal moments as it is specified by Eqs. (31) and (32). Thus, when solving the B-field

equations (86)–(88) for the RTD, one also gets the solution for the spatial moments. In principle,

this methodology could be reversed, starting from the B-field equations for the spatial dispersion.



Proceeding this way, the EMM could be straightforwardly extended for arbitrarily directed seepage

velocity and/or anisotropic transport of solute.

The first two moments are related to the effective velocity and dispersion coefficient, while the

higher moments account for the “non-Gaussian” effects, similar to those in breakthrough curves

in experimental tracer tests. Although the EMM method does not cover the whole spectrum of

possible reasons to justify this deviation, our results undoubtedly show that the heterogeneity of

the porosity alone may become responsible for the non-Fickian behaviour. In order to illustrate

our approach, the analytical predictions have been constructed for Newtonian and power-law flow

in cylindrical capillary, as well as the Poiseuille and shallow water flows in channel. Then we

analytically addressed the two principal limits of interface configuration: (1) Darcy constant flow in

different porosity parallel channels, and (2) open/impermeable, rectangular, and cylindrical systems.

The derived solutions for Taylor dispersion coefficient, skewness, and kurtosis depend on three

parameters: Peclet number, porosity, and aspect ratios. In fact, the “non-Gaussian” effects manifest

themselves in constant Darcy flow uniquely because of the porosity contrast. Even in open Poiseulle

flow and shallow channels, the deviation from the ADE solution is predicted for high-order moments.

In open/impermeable flow it is amended by heterogeneity of permeability. We believe that the

constructed solutions andmethodology are helpful to get insight into their respective role in bi-modal

porous systems and, perhaps, for better understanding of environmental dispersion, in particular, in

wetlands with submerged vegetation represented by regular porous blocks.54

The obtained results hold for the entire range of Peclet numbers. For the rectilinear flows

considered in the present work dimensionless nth moment of the distribution scales as Pe(n−1) for Pe

→ ∞ and as Pe−(n−1) for Pe → 0. For other types of flow these scalings should change, however,

it is important to note that (i) characteristics of the distribution are non-trivial functions of Pe and

(ii) in many practically important cases the characteristic values of Pe belong to an intermediate

range, far from the above-mentioned asymptotæ. In the present study, the two next moments,

characterised by the dimensionless parameters as skewness and kurtosis, were sufficient for one-two

order magnitude porosity contrasts, except for very sharp peaks in bi-modal soils. Future work needs

to establish the problem-dependent criteria to forecast the number of required moments for reliable

distribution reconstruction. Besides the physical aspects, the proposed solutions present the valuable

benchmark tests for numerical methods. In particular, we show that in stratified Darcy flow, solution

for a finite wavenumber may strongly deviate from the zero-k (advection-diffusion)) limit (see

Table V).

In this paper, the two-relaxation-times Lattice Boltzmann schemewas constructed for modelling

of transport phenomena in heterogeneous media. We specify how to numerically compute the

temporal and spatial dispersions, the RTD, the spatial and temporal moments, and how to derive

effective dispersion relations for waves. We prefer this numerical method because of (i) its implicit

interface treatment, which allows for simple incorporation of sharp porosity distributions, (ii) its

deterministic character, which allows for estimation of the truncation errors impact on the moments,

and (iii) its versatility in solving velocity field and transport. The presented TRT scheme is supported

by the necessary stability criteria. They show that the less porous block mainly dictates the stable

parameter map. The third- and fourth-order truncation corrections of the scheme are derived for

uniform porous Darcy flow and related to the obtained moments, where they explain small deviations

from the predicted solution which vanish for particular parameter choice, as given by Eq. (A11). We

show that in stratified Darcy flow, the effective accuracy of the interface continuity conditions mainly

determine the accuracy of the scheme. This example provides the methodology for construction of

time-space-dependent approximate solutions of the LBE scheme in the presence of sharp interfaces

and truncation corrections, extending their previous steady state analysis40 from polynomial to

harmonic solutions.

To conclude, we emphasize that the non-Gaussian effects predicted by the EMM and TRT

schemes are in a good agreement. Although in the present work the two methods were only applied

with the analytical velocity profiles, no modification is needed for any heterogeneous flow, e.g.,

obtained by solving Navier-Stokes or Stokes-Brinkman equation. Furthermore, by incorporating the

heterogeneous source terms, the scheme can be applied for solving the B-field equations themselves.

This would offer a promising alternative to the random walk and direct discretisation schemes,



previously applied for solving B-field equations for Taylor dispersion in homogeneous velocity

fields.55

ACKNOWLEDGMENTS

A. Vikhansky thanks the Irstea for hospitality and financial support during his visits. I. Ginzburg

is grateful to ANR for funding the project LaboCothep ANR-12-MONU0011. The authors thank Dr.

G. Silva for critical reading of the manuscript and D. Bauer, M. Fleury, and L. Talon for interesting

discussions.

APPENDIX: TRUNCATION CORRECTIONS

In Subsection A 1 of the Appendix, we provide the fourth-order truncation errors of TRT scheme

(133)–(135) for constant Darcy porous flow, valid for any considered velocity set from one to three

dimensions. In Subsection A 2 of the Appendix, we write down the exact form of the longitudinal

truncation corrections for third and fourth order moments in one-dimensional flow, and then partly

account for the transverse direction. In Subsection A 3 of the Appendix, we construct approximate

time-dependent two-dimensional solution of the d2Q5 scheme, used in our numerical computations,

in stratified layers of different porosity.

1. Fourth-order results

We extend the fourth-order solution46 for constant Darcy porous flow assuming that inside

each porous block of porosity φ, the relaxation functions �±φ are space and time independent. The

post-collision non-equilibrium components {g±q } in Eq. (133) can then be written as56

g±q (r, t) = �t e

±q + �qe

∓q − �∓

φ (�2q − �2

t )e±q + δg±

q , (A1)

δg±q (r, t) = −

1

2�2

t g±q − (�±

φ + �∓φ )�t g

±q +

�

�φ −1

4)(�2

q − �2t

�

g±q , (A2)

with the help of central time-differences: �tψ(r, t) = 12(ψ(r, t + 1) − ψ(r, t − 1))

and �2t ψ(r, t) = ψ(r, t + 1) − 2ψ(r, t) + ψ(r, t − 1), and the directional differ-

ence operators in space: �qψ(r, t) = 12(ψ(r + cq , t) − ψ(r − cq , t)) and �2

qψ(r, t)

= ψ(r + cq , t) − 2ψ(r, t) + ψ(r − cq , t), ∀ψ = {e±q , g±

q }. Exact mass conservation equation

reads�Q−1

q=0 g+q (r, t) = 0. Since

�Q−1q=0 e+

q =�Q−1

q=0 fq = φC, Eqs. (A1) and (A2) yield mass

conservation equation in the form

[�t + �−φ �2

t ]φC +Q−1�

q=1

�qe−q − �−

φ

Q−1�

q=1

�2qe

+q = −

�

�φ −1

4

� Q−1�

q=1

�2qg

+q . (A3)

In derivation of Eq. (A3) we have taken into account that the differentiation in time vanishes for�Q−1

q=0 g+q (r, t) = 0. Plugging Eqs. (A1) and (A2) into Eq. (A3), and expanding it into Taylor’s series,

the fourth-order accurate approximation of this equation takes the form

∂tφC = [Rφ,1 + Rφ,2 + Rφ,3 + Rφ,4]C, Rφ,k = φRk

�

cφ

φ,

uφ

φ,�−

φ ,�φ

�

, (A4)

where the four operators Rk(ce, u,�−,�) have been derived in work46 for the isotropic and

anisotropic TRT advection-diffusion scheme in open flow. These operators are here reproduced

in Eqs. (A6)–(A8) with the help of the following operators, for k ≥ 1:

S2k = 2

Qm2�

q=1

E+q (ce, u)∂2k

q , S2k−1 = 2

Qm2�

q=1

E−q (u)∂

2k−1q , ∂q = ∇ · cq . (A5)



Notice that these operators do not depend on the relaxation rates. At the second-order, Eq. (A4)

reads

∂tφC + φS1C = φ�−φ D2C, R1 = −S1, R2 = �−

φ D2, D2 = S2 − S21 . (A6)

Hereafter, Snk C means that the operator Sk is applied n − times to solution C. Thus, the discrete

convective term is φS1(uφ

φ)C = ∇ · uφC, and the effective diffusion term is a sum of the modeled

diffusion form φ�−φ S2(

cφ

φ,

uφ

φ)C and the second-order numerical diffusion form −φ�−

φ S21 (

uφ

φ)C. In

case of equilibrium (135), Eq. (A6) reduces to Eqs. (138) and (145). The third-order truncated term

φR3(cφ

φ,

uφ

φ,�−

φ ,�φ)C in Eq. (A4) reads as

R3(ce, u,�−,�) = c3,1R3,1 + c3,2R3,2, where

R3,1(ce, u) = D2S1, R3,2(u) = S31 − S3, and

c3,1(�−,�) = 2(�−)2 + � −

1

4, c3,2(�) = � −

1

12. (A7)

The two coefficients c3,1 and c3,2 vanish for the BGK model with (�−)2 = � = 112

(so-called

optimal-advection choice46). The fourth-order truncation term φR4(cφ

φ,

uφ

φ,�−

φ ,�φ)C is read as

R4 = c4,1D22 + c4,2D2S

21 + c4,3R4,3 + c4,4R4,4,

R4,3 = S4 − S1S3, R4,4 = S41 − S1S3, and

c4,1(�−,�) = −�−

�

(�−)2 + � −1

4

�

,

c4,2(�−,�) = �−

�

4(�−)2 + � −3

4+

�(4� − 1)

4(�−)2

�

,

c4,3(�−,�) = �−

�

� −1

6

�

,

c4,4(�−,�) =

�−

4

�

8� − 1 +�(4� − 1)

(�−)2

�

. (A8)

It has been recognized that there is no solution for �± where the four coefficients c4,1 − c4,4 vanish.

In pure diffusion equation, the fourth-order truncation correction vanishes when c4, 1 = 0, i.e., for

TRT operator with (�−)2 = 112

and � = 16. This is the optimal-diffusion choice.46

2. Fourth-order one-dimensional equation

When the velocity vector u = {u, 0, 0} is parallel to the x −axis, and the porosity is uniform,

all velocity sets will produce the same one-dimensional solution C(t, x). Plugging Eq. (143) into

Eq. (A5), the discrete operators become

SkC = u∂kC

∂xk, S2kC = (ce + Inu

2)∂2kC

∂x2k, k = 1, 2, . . . . (A9)

Plugging them into Eqs. (A6)–(A8), the truncation terms R3 and R4 give the coefficients

a(cφ

φ,

uφ

φ,�−

φ ,�φ) and b(cφ

φ,

uφ

φ,�−

φ ,�φ) in Eq. (76), as

a(ce, u,�−,�) = (c3,1ce − c3,2)u + (c3,2 + c3,1(In − 1))u3,

b(ce, u,�−,�) = ce(cec4,1 + c4,3) + (cec4,2 − c4,4)u2 + c4,4u

4

+ (In − 1)((2c4,1ce + c4,3)u2 + (c4,2 + c4,1(In − 1))u4). (A10)

Hence, the Rφ,3 and the a, vanish for any φ, cφ , and uφ only for the specific BGK choice (�−φ )

2

= �φ = 112

where c3,1 = 0 and c3,2 = 0. With this relaxation choice, the Rφ,4 and the bi vanish if



cφ = φ

4and uφ = φ

2:

(�−φ )

2 = �φ =1

12, cφ =

φ

4, uφ =

φ

2: a = b = 0. (A11)

These particular parameters belong to advection part of the non-negativity boundaries (147) (see

second equation there), and therefore, they are found on the stability line of the BGK model.

Furthermore, solution (A11) nullifies Rφ,3 and ai for BGK subclass �φ = (�−φ )

2, ∀ �−φ , i.e., for

any relaxation rate. The numerical assessment of the wave evolution in homogeneous porosity in

Table VI confirms that the BGK scheme (A11) yields exact velocity (see results for �φ = φ2/12,

any relaxation rate �−φ can be used), and then it produces nearly exact solution when (�−

φ )2 = 1

12.

Otherwise, the truncation correction principally match the difference between the TRT and reference

solution for uniform porosity (cf. err(�) and err(�(tr))).

When the two layers have different porosity then ce = cφ

φiand ui = uφ,i

φi, Eq. (148) reads with

K 2i (�, k) ≈

−� + iui k + D0k2 + iai k

3 − bik4

D0 + cekui (ic3,1(�−φ,i ,�φ,i ) − c4,2(�

−φ,i ,�φ,i )kui )

. (A12)

This estimate, however, neglects the fourth-order transverse derivatives from terms c4,1D22 and

c4,3R4,3 in Eq. (A8). We find that Eq. (A12) is not sufficient to match the obtained numerical

solutions, unless the effective interface conditions are accounted for. Such a solution is constructed

in Subsection A 3 of the Appendix.

3. Approximate time-space dependent TRT solution in stratified channel

We assume that the d2Q5 set is employed for modelling of the constant Darcy flow in two layers

separated by flat interface z = h1 parallel to the x−axis. In this case, only the vertical links {cqx =0, cqz = ±1} are cut by the interface.

Interface conditions are implicitly set by population update (133) in the form of two relations

per each cutted link (a pair of two opposite velocities):

(e+q − e−

q − ( 12

+ �+φ )g

+q + ( 1

2+ �−

φ )g−q )|z=h1+ 1

2,t+1

= (e+q + e−

q + ( 12

− �+φ )g

+q + ( 1

2− �−

φ )g−q )|z=h1− 1

2,t ,

(e+q + e−

q + ( 12

− �+φ )g

+q + ( 1

2− �−

φ )g−q )|z=h1+ 1

2,t

= (e+q − e−

q − ( 12

+ �+φ )g

+q + ( 1

2+ �−

φ )g−q )|z=h1− 1

2,t+1.

(A13)

We take into account that the non-equilibrium components f ±q − e±

q are equal to −g±q /s± = −( 1

2

+ �±φ )g

±q . In case of the d2Q5 set, these equations involve the equilibrium and non-equilibrium

components for the same coordinate x. In general, �±φ differ for two sides of the interface. The sum

and the difference of the two interface relations yield two effective interface continuity conditions

of the TRT model, one for e+q (and hence, for the concentration), and another one for the sum of

e−q and −�−

φ g−q (and hence, for the advection-diffusion flux). At the steady state, they reduce to the

following relation:40

�e±q ±

1

2g∓q − �±

φ g±q �z=h1∓ 1

2= 0. (A14)

These relations can be used as the first approximation, e.g., when Darcy velocity uφ is small. Below,

we use the time-dependent sum and the difference of two interface relations (A13), where Eq. (A1)

is used to express g±q . In more detail, we first put δg±

q = 0 in Eq. (A1) and express g±q via the exact

central differences of e±q , in time and space, where we take into account that e−

q and its differences

all vanish for the vertical links (since the velocity is along the x−axis). Equation (A1) gives for

e+q = e+

q (t, x, z):

g+q (t, x, z) ≈ �t e

+q − �−

φ (�2q − �2

t )e+q , g−

q (t, x, z) = �qe+q . (A15)



The central differences �qe+q and �2

qe+q reduce for the interface cutted links to (e+

q (z + cqz)

− e+q (z − cqz))/2 and e+

q (z + cqz) − 2e+q (z) + e+

q (z − cqz), respectively. Next, plugging Eq. (A15)

into Eq. (A2), and neglecting the term with prefactor �φ − 14there, we express the correction

δg±q = −[ 1

2�2

t + (�±φ + �∓

φ )�t ]g±q . We substitute then g±

q , as given by Eq. (A15), plus δg±q into

effective interface conditions (A13). Up to this point, the construction is valid for any e+q .

Inside each layer, e+q is prescribed in the form (137) with Ci (t, x, z) = Ci (z) exp(ikx − �t).

Plugging Ci (t, x, z) into Eq. (A3) (where we omit the term with prefactor �φ − 14on the RHS, to be

consistent with the approximation above), Ci (z) obeys the following discrete equation:

�2z Ci (z) = K 2

i Ci (z), with �2z Ci (z) = Ci (z + 1) − 2Ci (z) + Ci (z − 1), i = 1, 2, (A16)

where

K 2i =

[�t + �−φ,i�

2t ]φCi + uφ,i�xCi − �−

φ,i cφ,i�2xCi

φi D0Ci, then

K 2i =

−2�−φ,i (φi + (cφ,i cos [k] − 1) + φi (2�

−φ,i cosh [�] − sinh [�]) + iuφ,i sin [k]

φi D0

, (A17)

Solution to Eq. (A16) takes the form

Ci (z) = Aiξzi + Biξ

−zi , ξi =

2 + K 2i + Ki

�

4 + K 2i

2. (A18)

Asymptotic solution for k → 0, � → 0, reads

K 2i ≈

−�φi + iuφ,i k + cφ,i�−φ,i k

2

D0φi

+�−

φ,i

D0

�2 + O(�3) + O(k3), cφ,i�−φ,i = φi D0. (A19)

This confirms that the discrete solution asymptotically converges towards the exact solution (148).

Four constants A1, B1 and A2, B2 are derived from two interface conditions and two symmetry

conditions in themiddle of two layers, say C1|z= h12

+ 12

= C1|z= h12

− 12

and C2|z=h1+h22

+ 12

= C2|z=h2+h12

− 12

,

when the two layers have width h1 and h2, respectively.

a. Coupling of bulk solution with closure conditions

Solution Ci (t, x, z) = Ci (z) exp(ikx − �t), where Ci (z) is given by Eq. (A18) with Eq. (A17),

is substituted into the previously obtained interface conditions, where we apply the spatial dis-

crete differential operators separately on the solution of each layer. The symmetry conditions

close the system. Together, they give us four linear equations with respect to four constants in

Eq. (A18). Solvability condition guarantees zero determinant of this system, thus giving the approx-

imate solution�(tr) of the scheme. This approximate solution is compared to the numerical solution in

Tables X and XI. As expected, the obtained estimate �(tr) matches �(num) at the best when �φ = 14.

Further works need to account for corrections with the prefactor (�φ − 14)(�2

q − �2t )g

±q where

Eqs. (A16) and (A17) have to include the fourth-order corrections in transverse direction.

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