Random walks with negative particles for discontinuous diffusion and porosity

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To cite this version:

Oukili, Hamza

and Ababou, Rachid

and Debenest, Gérald

and Noetinger, Benoît Random walks with negative particles

for discontinuous diffusion and porosity.

(2019) Journal of

Computational Physics, 396. 687-701. ISSN 0021-9991.

Official URL:

https://doi.org/10.1016/j.jcp.2019.07.006

Random

Walks

with

negative

particles

for

discontinuous

diffusion

and

porosity

H. Oukili

a

,

∗

,

R. Ababou

a

,

G. Debenest

a

,

B. Noetinger

b

a_{Institut de Mécanique des Fluides de Toulouse, IMFT, Université de Toulouse, CNRS - Toulouse, France} b_{IFP Energies nouvelles, Rueil-Malmaison, France}

a

b

s

t

r

a

c

t

Keywords: Diffusion

RandomWalkParticleTracking(RWPT) Discontinuities

Analyticalsolutions Porousmaterials Negativeparticles

Thisstudydevelopsanew Lagrangianparticlemethodfor modelingflowand transport phenomenaincomplexporousmediawithdiscontinuities.Forinstance,diffusionprocesses can be modeledby Lagrangian Random Walk algorithms. However, discontinuities and heterogeneitiesare difficulttotreat,particularlydiscontinuousdiffusionD(x)orporosity

θ (x).Inthe literatureonparticleRandomWalks, previousmethods used tohandlethis discontinuityproblemcanbecharacterizedintotwomainclassesasfollows:“Interpolation techniques”,and“Partialreflectionmethods”.Oneofthemaindrawbacksofthesemethods isthesmalltimesteprequiredinordertoconvergetotheexpectedsolution,particularly inthepresenceofmanyinterfaces.Theserestrictionsonthetimestep,leadtoinefficient algorithms.The RandomWalk Particle Tracking(RWPT)algorithmproposedhereis,like others intheliterature,discreteintimeand continuousinspace(gridless).We propose a novel approach to partial reflection schemes without restrictions on time step size. The new RWPTalgorithm is based onan adaptive “Stop&Go” time-stepping,combined withpartial reflection/refractionschemes, andextendedwith anew conceptofnegative massparticles.Totestthenew RWPTscheme,wedevelopanalyticalandsemi-analytical solutionsfordiffusioninthepresenceofmultipleinterfaces(discontinuousmulti-layered medium). The results show that the proposed Stop&Go RWPT scheme (with adaptive negative massparticles) fits extremelywell the semi-analytical solutions,even for very high contrasts and in the neighborhood of interfaces. The scheme provides a correct diffusivesolution inonlyafewmacro-timesteps,withaprecisionthatdoesnotdepend ontheirsize.

1. Introduction

Particlemethodshavebeenmuchusedtomodelthetransportofmass,heat,andotherquantitiesthroughsolids,fluids, andfluid-filled porousmedia. The last two cases involveboth diffusive and advective transport phenomena (due to the moving fluid). Hydrodynamicdispersion dueto detailedspatial variations of thevelocity field has alsobeen modeledas aFickiandiffusion-type process,e.g. influid-filledporous structures(see [1]).Other purelydiffuseprocessesincludeheat diffusioninmaterials(Fourier’slaw),andpressurediffusion(compressibleDarcyflowinafluid-filledporousmedium).

*

Correspondingauthor.

Particlemethodsarebasedonadiscreterepresentationofthetransportedquantity(soluteconcentration,fluidpressure, fluidsaturation,heatortemperature)asdiscretepackets(the“particles”),eachcarryingaunitmass,oraunitheat,etc.The advantage ofparticle methodsis thatthey avoidsome ofthe problemsofEulerianmethods basedon PartialDifferential Equations(PDE’s),suchasnumericalinstability,artificialdiffusion,massbalanceerrors,and/oroscillationsthatcouldleadto negativeconcentrationsorsaturations.Varioustypesofparticlemethodshavebeendevised:non-LagrangianParticle-in-Cell methods(PIC);implementingMarkovprocessesinaPICframeworkwithstochastictimes[2];continuous-timeparticleson a grid [3];the time-domain random walk (TDRW)[4,5];andLagrangian particles withdiscrete time-stepsin continuous space (gridless). Such particle methods have been extensively used for modeling advective-diffusive solute transport in poroussoils,aquifers,andreservoirs[6].

In “Lagrangian” methods, space is assumed continuous, and particle positions X

(

t

)

are real numbers (the method is then“gridless”).Inthepresentwork,wefocusonLagrangianparticletrackingtosolvediffusionprocessesbyrandomwalk (Wienerprocess),underthegenericnameRWPT(RandomWalkParticleTracking).

AspecificstudyofthemacroscopicbehaviorofRandomWalkparticlesisnecessarywhendealingwithaheterogeneous ordiscontinuous diffusioncoefficient D

(

x

)

.Thecaseofdiscontinuous diffusionisparticularlytroublesome,andthisisour mainfocus.Suchdiscontinuitiesoccurat“materialinterfaces”,withsuddenchangesofmicrostructure(compositematerials, layeredporousmedia,etc.),andatdiscontinuitiesinphase,forexampletheinterfacebetweensurfacewaterandgrounwater (an importantecological habitatin thehyporheic zone). Forsolute diffusion ina porous medium, an additionalpoint of interest is thecaseofdiscontinuous porosity

θ (

x

)

(if the mediumis water-saturated),or discontinuous volumetricwater content

θ (

x

)

(ifthemediumisunsaturated).

IntheliteratureonparticleRandomWalks, thedisplacementschemesusedforhandlingthediscontinuitycanbe char-acterizedintotwoclasses:(1)Interfacecoarsening,interpolation,anddriftvelocityscheme(e.g.[7,8]);(2)Partialreflection schemes (e.g.[9]).Thefirstclass(“interpolationtechniques”)smoothout thediscontinuity[10]:theinterfaceiscoarsened andtheparameters(diffusion,porosity)areconsideredcontinuousthroughthecoarsenedinterface.Thesecondclass (“par-tialreflectionmethods”), introducedby Uffink[9],implementsaprobabilistic reflection/transmission oftheparticles across the discontinuous interface:probabilities are assigned for particle reflection and transmission across the interface. Other similarpartialreflection/transmissionschemeswereinvestigatedby[11–17].

Lejay & Pichot [18] proposed a“two-step algorithm” (theirAlgo.2), equivalent toa Stop&Goprocedure: theparticle is

stopped atthe interface,andthen undergoes a “SkewedBrownianMotion” (SBM)forthenext step, whichmaylead the

particle to cross the interface. Their two-step algorithm was presented fora 1D finite domain with zero flux boundary conditions.Onthe otherhand,they alsopresenteda“one-step algorithm”(theirAlgo.3)where,itseems,they useatype of acceptance-rejectionmethod toobtainthe displacementinthe neighborhoodoftheinterface (this methodisdifferent fromours).Morerecently,Lejay&Pichot[19] testedtheirapproaches[18] byimplementing1Dbenchmarktests,involving comparisonsbetweentheirSBMandtwoRandomWalksapproachesintheliterature [9,15].

Oneofthedrawbacksoftheseapproachesisthatasmalltimestepisrequiredinordertoconvergetothecorrectsolution ofthediscontinuousPDE,evenifthenumberofparticlesisverylarge.Thisisparticularlylimitinginthepresenceofmany interfaces. Thislimitation becomes even more drasticfor very largediffusion contrasts,e.g., two orders of magnitudeor more.Thus,[20,17] showedthattheabovemethodsare onlyvalidforinfinitesimaltimesteps.Asmalltimestepmustbe usedinordertoavoidtheovershootofheterogeneousanddiscontinuoussubregionsofspacebytheparticles.

Inthisstudy,weproposeanovelapproachintheframeworkof“partialreflectionmethods”butwithoutrestrictionson time stepsize.ThenewRWPTalgorithmisdiscreteintimeandcontinuousinspace(gridless),andthenovelaspectshave to dowiththe treatmentof discontinuities.The newalgorithm isbasedon adaptive“Stop&Go” time-stepping,combined withpartialreflection/transmissionschemessimilarto[15–18],andextendedwiththeconceptofnegativemassparticles.

This paper is organized as follows. The next section, 2, presents the theory behind Random Walk Particle Tracking methods (RWPT),andthe correspondingmacroscopicdiffusionPDE. Section3presentsa novelparticle-basedmethodfor solvingheterogeneousanddiscontinuoustransportproblemsusingRWPTwith“negativemassparticles”.Section4compares analyticalsolutionstoourRWPTresults.Section5recapitulatesthemethodanddiscussesextensionsofthiswork.

2. Theory

Thissectionpresentsthetheoryofadvective-diffusivetransport,namely,theconcentration-based,macroscopicPDE’s,and therelatedtheoryofStochasticDifferentialEquations(SDE’s)drivenby whitenoise,governingparticlesatthemicroscopic scale.

2.1. ConcentrationbasedPDE’s 2.1.1. TheGaussianfunction(PDF)

Let usdefine aGaussian PDF, denoted G



μ

,

σ

2

_,

_x



_{, where}

_μ

_{is themean,}

_σ

2 _{thevariance, and x}

₌

_X

₍

_t

₎

_{the particle}

positionatanyfixedtimet:

∀

x



R

;

G



μ

,

σ

2

,x



=

1

σ

√

2

π

exp



−

(x

−

μ

)

2 2

σ

2



(2.1)

A Gaussian random variable (RV) withmean

μ

and variance

σ

2 _{is denoted N}



_μ

_,

_σ

2



and hasthe Probability Den-sity Function (PDF) G



μ

,

σ

2

_,

_x



_{. Letting}

_σ

2

₌

_2D

0t, this Gaussian PDF represents the macroscopic concentration

solu-tion C

(

x

,

t

)

of the diffusion PDE with spatially constant diffusion coefficient D0, for an initial point source condition

C

(

x

,

t

)

=

M0

δ (

x

−

μ

)

,withunitmassM0

=

1,inaninfinitedomain.

2.1.2. Theadvection-diffusiontransportPDEforconcentration

Theequationgoverningthetransportofsolute concentration(C )inaheterogeneousmediumwithvariableparameters

D (diffusioncoefficient),

θ

(porosity)andV (velocity)is(seeforinstance[11]):

∂ (θC

)

∂t

= ∇·(−θ

C V

+

D

·∇ (θ

C

))

= −∇·{θ

C

(

V

+ ∇·

D

+

D

·∇ (

ln

θ ))

} +

1

2

∇·∇·(

2

θ

C D) (2.2)

Notethatwedonotdistinguishbetweenvectorsandsecondranktensors.Thefirstequalitycorrespondstotheconservative (divergence)formofthePDE,whilethe secondequalitycorrespondstoits decomposedform(fromwhichapparent “drift velocity”termsemergeduetospatiallyvariablediffusionandporositycoefficients).

Fora1Dproblemwithscalardiffusion D, thetransportPDEforaninitialsourceatx

=

x0 ina homogeneousmedium

withconstantparameters D,

θ

andV is:

⎧

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎩

∀

t

>

0

; ∀

x



R;

∂

C

∂t

(x,t)

= −

V

∂

C

∂

x

(x,t

)

+

D

∂

2C

∂

x2

(x,t)

∀

t

>

0

;

lim x→±∞C

(x,t)

=

0

∀

x



R

;

C

(x,

0

)

=

M0

θ

δ (x

−

x0

)

(2.3)

The last equation represents an initial point source located at x

=

x0 with mass M0, and

δ (

x

)

represents the Dirac

pseudo-function(

δ

distribution)(e.g.Schwartz[21]).

ThisPDEwillbelaterformulatedforpurelydiffusivediscontinuousdiffusionandporositycoefficientsinsection2.2. TheanalyticalsolutionofEq. (2.3) is:

∀t

>

0

; ∀x



R;

C

(x,t)

=

M0

θ

G

(x

0

+

V t

,

2Dt

,

x) (2.4)

whereG istheGaussianPDFdefinedinEq. (2.1).

2.1.3. Fromconcentrationtoparticlepositions

LetusconsidernowaparticlebasedmethodtosolveEq. (2.3).Theconcentrationcanbeexpressedasfollows[22,23]:

C

(x,

t)

=



R C

(X

t

,t

) δ (X

t

−

x)d Xt

=



R

δ (X

t

−

x)dmt (2.5)

where Xt anddmt represent,respectively,thepositionandmassofaninfinitesimalconcentrationpacket(tobediscretized

asa“particle”).

ThePDFofparticlespositionsatanyfixedtimet shouldfollowthedistributionG

(

x0

+

V t

,

2Dt

,

x

)

.Thus,the

correspond-ingparticlepositionscanbegeneratedusingaGaussianRV:

Xt

=

N

(x

0

+

V t

,

2Dt

)

=

x0

+

V t

+

√

2 D t N

(

0

,

1

)

(2.6)

whereN

(

0

,

1

)

designatesanormalizedGaussianRV(zeromeanandunitvariance).ForV

=

0,

(

Xt

)

istheWienerprocess.

Ascanbeseen,thePDFof

(

Xt

)

isidenticaltotheconcentrationsolutioninEq. (2.4) dividedby M0.

InthecaseofspatiallyvariablebutdifferentiablecoefficientsD

(

x

)

and

θ (

x

)

,thecorrespondingSDEbecomes[22,11,24]:

Xt+dt

=

Xt

+

2D

(

Xt

)

dt N

(

0

,

1

)

+ {

V

(

Xt

)

+ ∇ (

D

(

Xt

))

+

D

(

Xt

)

∇ (

ln

θ (X

t

))}

dt (2.7)

whichgovernstheGaussianMarkovianprocess

(

Xt

)

.After“explicit”“discretization”,dt is replacedwiththefinite



t step

[7–9,12,25,13–17].

2.2. Prototypeproblem:1Ddiffusionwithasinglesourceandasinglediscontinuity

In thissection,we define a purelydiffusive problem, withan initialDiracsource, inan infiniteporousmedium com-prising twosubdomains

1 and

2 separatedbyasingle discontinuity,or“materialinterface”.The interfaceislocatedat

x1−2

=

0,andtheinitial point sourceis locatedat XSource

=

x0

<

0.The subdomains

1 and

2 havedifferentdiffusion

coefficients D1 andD2,anddifferentporosities

θ

1 and

θ

2.Suchdiscontinuities canbe foundinsoils,fracturedrocks,and

manyother porousmaterials. Forinstance,[8] studiedoxygendiffusionthrough adiscontinuous grains/jointssystemin a submicronlayerofNickelOxide.

Here,toillustrateourRWPTmethod(asinLabolle’sanalysis[11]),wefocuson1Dsolutediffusioninaporousmedium withasingleinterface,wherebothD

(

x

)

and

θ (

x

)

arediscontinuous.ThePDEsystemforthediscontinuousproblemis,for

thedomain

= 1

∪ 2

:

∀t

>

0

; ∀x



i

;

∂ (θ

iCi

)

∂t

=

∂

∂x



θ

iDi

∂

Ci

∂

x



(2.8a)

∀

t

≥

0

;

lim x→±∞Ci

(x,

t)

=

0 (2.8b)

⎧

⎨

⎩

∀t

≥

0

;

C1

(x

1−2

,

t)

=

C2

(x

1−2

,t)

∀

t

≥

0

; −θ

1D1

∂

C1

∂

x

(x

1−2

,t)

= −θ

2D2

∂

C2

∂

x

(x

1−2

,t)

(2.8c)

∀

x



i

;

Ci

(x,

0

)

=

M0

θ

i

δ (x

−

x0

)

(2.8d)

InEq.(2.8a),each PDErepresentsamassconservationequation forthesolute ineachsubdomain.Inthecaseathand, porosities

θ

1 and

θ

2 are constantin eachsubdomain andcan befactored out fromeach PDE.Thesystem(2.8c) enforces

thecontinuityofsolute concentration(masspervolumeofsolvent)andofarealsolutefluxdensity.Inalltheseequations, Fick’slawisusedforthediffusiveflux.

Theanalyticalsolutionofproblem2.8isgivenbyC1 andC2(

∀

t

≥

0):

(

1

)

: ∀

x

≤

x1−2

;

C1

(x,t

)

=

C1S

(x,

t)

+

C1R

(x,t)

(2.9a) C₁S

(x,t)

=

M0

θ

1 G

(x

1−2

+ (

x0

−

x1−2

) ,

2D1t,x) (2.9b) C₁R

(x,t)

=

M0

θ

1 R1−2G

(x

1−2

− (

x0

−

x1−2

) ,

2D1t,x) (2.9c)

(

2

)

: ∀

x

≥

x1−2

;

C2

(x,t

)

=

M0

θ

2

(

1

−

R1−2

)

×

G

(x

1−2

+ β

1−2

(x

0

−

x1−2

) ,

2D2t

,x)

(2.9d) R1−2

=

θ

1

√

D1

− θ

2

√

D2

θ

1

√

D1

+ θ

2

√

D2 and

β

1−2

=

√

D2

√

D1 (2.9e)

•

C₁S (“S”for“Source”)isthesolutionofthisdiffusionproblemwithoutinterface(nodiscontinuities).

•

C₁R isthesymmetricofC₁S relativetotheinterfacex

=

x1−2,multipliedbycoefficient R1−2 (R forReflection).

•

C2 isthesolutionofadiffusionproblemwithinitialmassM0

(

1

−

R1−2

)

locatedatx1−2

+ β1

−2

(

x0

−

x1−2

)

.

Eqs.(2.9) extendpreviousanalyticalsolutionsgivenby[26,7,27].

3. Methodsandalgorithms

In this section, we start by explaining the need for a new algorithm to deal with discontinuities. Then we discuss previous methodsproposed intheliterature.Finally,wepresentournewmethodanddiscussits advantagescompared to thepreviousones.

3.1. DiscontinuityproblemforRandomWalk

ThemoststraightforwardtestofanalgorithmfordiffusionwithdiscontinuousD

(

x

)

isthe“uniformconcentrationtest”, wheretheexactsolutionofthediffusionPDEisconstantconcentrationC

(

x

,

t

)

=

C0atalltimes(t)andallpositions(x).This

isobtainedbyimposingconstantinitialconcentration C

(

x

,

0

)

=

C0,andimposingeitherzerofluxconditions

∂

C

/∂

x

=

0,or

else DirichletconditionsC

=

C0,atbothboundaries.

Therandomwalkequationcanbewrittenasfollowsfor1DdiffusionwithvariableD

(

x

)

:

X(p)

(t

n

+ 

tn

)

=

X(p)

(t

n

)

+ 

tn

∂

D

∂x



X(p)

(t

n

)



+



2D



X(p)

_(t

n

)



t

nZ(p)

(t

n

)

(3.1)

However,thisequationislimitedtothecaseofcontinuouslyvariablediffusioncoefficient.Letusnowfocusonthecase ofdiscontinuous D

(

x

)

.Ifwenaïvelyinsertthediscontinuous D

(

x

)

intheaboveequation,adeficitofconcentrationappears near the interface of discontinuity, where D1

<

D2. We can deal with this deficit of concentration using two different

schemes(reflection,smoothing).Forthispurpose,aStop&Goalgorithmisnecessary;itisdescribedinthefollowingsection.

3.2. Partialreflectionandextensions(algorithm) 3.2.1. PartialreflectionschemeforthecaseR1−2

≥

0

Thegeneralprincipleofthispartialreflectionscheme,sofar,issimilartotheonepreviouslyusedinliterature[15–17]. The fractions

|

R1−2|and

(

1

− |

R1−2|)are interpretedasprobabilities.The issueof“negativeprobabilities”will betackled

laterbelowandinsection3.2.2.Here,wefocusonR1−2

≥

0.

Thus,thedisplacementalgorithmfor X

(

t

)

becomes:

⎧

⎪

⎪

⎨

⎪

⎪

⎩

JR

1−2

K =

1

;

X

(t)

=

x1−2

−

N

(x

0

−

x1−2

,

2D1t

)

:



XRlreflected



JR

1−2

K =

0

;

X

(t)

=

x1−2

+



D2 D1 N

(x

0

−

x1−2

,

2D1t

)

:



XRrrefracted



(3.2)

where

J

R1−2K designates a Bernoulli RV that is equal to 1 with probability

|

R1−2| and equal to 0 with probability

(

1

− |

R1−2|),x0 istheinitialparticleposition X

(

0

)

,andx1−2istheinterfaceposition.

However,untilnowwehaveconsideredonlytheabsolutevalueofR1−2.Thus,thepreviousalgorithmissufficientonly

incaseswhereR1−2 ispositive.

3.2.2. PartialreflectionschemeforthecaseR1−2

<

0

Negative partialreflection probability R1−2 corresponds to a subtractionin theanalytical solution

(

C

(

x

,

t

))

. However,

itis difficultto“substract” particlesata specificlocation (position)withRWPTcomparedto addingparticles.The reason isthat,whenattempting tosubstractaparticleata specifiedlocation,onehastoact indirectlybysearchingforparticles in a neighborhood ofthe desired location, whileadding a particle ata specific location isalways possibledirectly. This subsectiondiscussesanewmethodtodealwithnegativeR1−2 forasingleinterface(wehavealsoextendedthemethodto

multipleinterfacesinthenextsubsection).

Fig. 3.1. Partialreflectionschemewithnegativemassparticles,foradomainwithdiscontinuousdiffusionandporosity(e.g.,here,

D

2<D1).Eachgiven

particlearrivingattheinterfacewithnegative

R

1−2istransmittedwithprobability1.Inaddition,twoparticleswithoppositemassespopupwith|R1−2|

probability.

Forthe caseof negative R, we propose thefollowing algorithm. First,the particle isalways refracted. Secondly, with probability

|

R1−2|, two newparticles are created: one is refracted and has the same mass asthe original particle,and

theother isreflected withamassofopposite sign(Fig. 3.1). Thisalgorithmallows ustomodeltheexact solutiontothe discontinuousdiffusionproblem.Thus,thedisplacementalgorithmforeachparticle Xk

(

t

)

becomes:

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎩

Xk

(t

)

=

x1−2

+



D2 D1 N

(x

0

−

x1−2

,

2D1t)

{

refracted

}

J

R1−2

K =

1

;

Xkpos

(t)

=

x1−2

+



D2 D1 N

(x

0

−

x1−2

,

2D1t)

{

refracted

}

J

R1−2

K =

1

;

Xkneg

(t)

=

x1−2

−

N

(x

0

−

x1−2

,

2D1t

)

{

reflected

}

(3.3)

3.3. Multipleinterfaces

Aftercrossing oneinterface, aparticlecould eventually crossasecond interface. Thealgorithm shouldbe abletodeal withanynumberofinterfacescrossedbyagivenparticle,inasingletimestep.

3.3.1. Semi-analyticalsolutionforadiffusionproblemwithN

≥

2 interfaces

ToobtaintheRWPTalgorithmthatdealswithmultipleinterfaces,letusfirstgeneralizethesolutionEq. (2.9) ofEq. (2.8) forN-interfacesand(N

+

1)layers.ThegeneralizationofEq. (2.8) fori

J

1

;

N

+

1

K

is:

∀t

>

0

; ∀x



i

;

∂ (θ

iCi

)

∂t

=

∂

∂x



θ

iDi

∂

Ci

∂

x



(3.4a)

⎧

⎨

⎩

∀

t

≥

0

;

lim x→−∞C1

(x,t)

=

0

∀t

≥

0

;

lim x→+∞CN+1

(x,t

)

=

0 (3.4b)

⎧

⎨

⎩

∀t

≥

0

; ∀i



J

1

;

NK Ci



xi,i+1

,t



=

Ci+1



xi,i+1

,t



∀

t

≥

0

; ∀

i



J

1

;

N

K −θ

iDi

∂

Ci

∂

x



xi,i+1

,t



= −θ

i+1Di+1

∂

Ci+1

∂

x



xi,i+1

,t



(3.4c)

∀x



i

;

Ci

(x,

0

)

=

M0

θ

i

δ (x

−

x0

)

(3.4d)

The solutionEq. (2.9) iscomposed ofthreegaussians,one ofwhich(C₁S

(

x

,

t

)

) isthesolutionofa purediffusiveproblem, andtheothertwodependonCS

1 andontheinterfaceposition.LetusdefinetwolinearoperatorsLi j andL∗i j:

Li j

(G

(x

0

,

2Dit,x))

=

Ri jG



2xi−j

−

x0

,

2Dit,x



(3.5) L∗_{i j}

(G

(x

0

,

2Dit,x))

=



1

−

Ri j



G



xi j

+



Dj Di



x0

−

xi−j



,

2Djt,x



(3.6) ThusthesolutionEq. (2.9) couldbewrittenasfollows:

⎧

⎨

⎩

∀

t

>

0

; ∀

x

≤

x1−2

;

C1

(x,t)

=

C1S

(x,t)

+

L12



C₁S

(x,

t)



∀

t

>

0

; ∀

x

≥

x1−2

;

C2

(x,t)

=

L∗12



C₁S

(x,t)



(3.7)

Thedecomposition(Eq. (3.7))canbefurthergeneralized:foreachindividualinterface,newgaussiansaregenerated,with parameters chosen to fitthe solution.Hence, each time agaussian function g initiallyina subdomain

(

i

)

encountersan interfaceatpositionxi−j,weaddLi,j

(

g

)

tothesolutioninsubdomain

(

i

)

andL∗_i,j

(

g

)

tothesolutioninsubdomain

(

j

)

.See

Algorithm1.

Algorithm1Semi-analyticalsolutionfordiffusionwithN

≥

2 interfaces.

1. Ckconcentrationinsubdomain

k

2. Initialize

C ik

=CSwith

CS

thesolutionofadiffusionproblemwithnodiscontinuity. 3. uk= ±1 thedirectiontowardstheinterfaceslimitingsubdomain(k).

4. while(Ck)notconverged do

(a) Ck=Ck+Lk,k+uk(C ik)and

Ck

+uk=Ck+uk+L∗k,k+uk  Ci k  (b) uk+uk=ukand

uk

= −uk (c) C ik=Lk,k+uk(C ik)and

C ik

+uk=C ik+uk+L∗k,k+uk(C ik) 5. end

For the two-interface problem (N

=

2), the previous algorithm leads to an analytical solution for diffusion with an initial source,withdiscontinuousdiffusioncoefficientsandporositieshavingthreedifferentvalues(threelayers).Thus the analytical

_⎧

solutionforasourcelocatedatthepositionx0 insubdomain

(

i

=

1

)

,

∀

t

≥

0:

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎩

∀

x

≤

x1;2

;

θ

1 M0 C1

(x,

t)

=

id

+

L12

+

+∞



i=0 L∗₂₁

(L

23L21

)

iL23L∗12

∀

x1;2

≤

x

≤

x2;3

;

θ

2 M0 C2

(x,

t)

=

+∞



i=0

(id

+

L23

) (L

21L23

)

iL∗12

∀

x

≥

x2;3

;

θ

3 M0 C3

(x,

t)

=

L∗23 +∞



i=0

(L

21L23

)

iL∗12 (3.8)

Theanalyticalsolutionforasourcelocatedatthepositionx0insubdomain

(

i

=

2

)

,

∀

t

≥

0:

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎩

∀

x

≤

x1;2

;

θ

1 M0 C1

(x,t)

=

L∗21 +∞



i=0

(L

23L21

)

i

(id

+

L23

)

∀

x1;2

≤

x

≤

x2;3

;

θ

2 M0 C2

(x,t)

=

+∞



i=0

(id

+

L21

) (L

23L21

)

i

(id

+

L23

)

∀

x

≥

x2;3

;

θ

3 M0 C3

(x,t)

=

L∗23 +∞



i=0

(L

21L23

)

i

(id

+

L21

)

(3.9)

withtherightsideofEq. (3.8) isappliedtoG

(

x0

,

2D2t

,

x

)

.

ThissolutionisdetailedinAppendixA,andithasbeenverifiedbysubstitutionintothegoverningPDE’s.

3.3.2. GeneralizationoftheRWPTalgorithmforN

≥

2 interfaces

Thesameideaofgeneralization oftheanalyticalsolutioninsubsection3.3.1 (fromoneinterfaceintoamulti-interface) hasbeenappliedtotheRWPTmethodforaproblemwithN interfaces.Ifaparticlecrossesan interface,then(step 1)its positionis alteredaccordingtoprevious algorithms that dealwithdiscontinuities (see subsection3.2,Eq. (3.2) for R

≥

0 andEq. (3.3) for R

<

0).After this(step 2), ifthenew particleposition doesnot belong toits initial subdomain, norto the adjacent subdomains, then go back to “step 1”. Thereafter, the particle continues undergoing this algorithm within a conditional loop, until the particle doesnot cross an interface (it then reaches its final position within the loop). See Algorithm2.

Algorithm2RWPTalgorithmforN

≥

2 interfaces.

1. Considerparticle(k)withmass

mk

andposition

Xk

. 2. while particle(k)crossesinterfaces do

(a) If R1,2≥0 then

i. IfJR1,2K =0; then theparticle(k)is

refracted to

theposition

X

kRrasinEq. (3.2) endif ii. IfJR1,2K =1; then theparticle(k)is

refracted to

theposition

X

kRlasinEq. (3.2) endif (b) Else (case

R

1,2<0)

i. Theparticle(k)is

refracted to

theposition

X

Rr k . ii. IfJR1,2K =1; then Createtwoparticles“A”and“B”:

A. withmass

mk

andatthe

refracted position X

Rr k . B. withmass−mkandatthe

refracted position X

Rl

k. iii. endif

(c) end 3. end

Thisalgorithmwillbe testedinsection4,withtheanalyticalsolutiondefinedinsubsection2.2.Then,itwillbe com-paredwithageneralizedanalyticalsolutionforapurediffusionproblemwithtwointerfacesandthreedifferentdiffusion coefficientsandwatercontents:thedetailedanalyticalsolutionforthis3-layercaseispresentedinEq. (3.8).Andfinally,it willbevalidatedwithanevenmoregeneralizedsemi-analyticalsolutionwhichalgorithmhasbeendetailedintheprevious subsection3.3.1.

3.4. Postprocessing:fromparticlestoconcentrations

Post-processinginRandomWalkmethodisessentialsincetheprimaryobjectiveofthesimulationistogetthe concen-tration(temperatureorpressure)field.AspecialattentionshouldbegiventoNegativeunitmassandAdaptivemassparticle methodsinparticular,sincetheyareverydifferentfromtheclassicalRandomWalksimulation.Here,massconservationis stillmaintained,sinceeachtimewecreateanegativemass,wecreatealsoapositiveone.

Themacroscopicconcentrationisdeterminedfromthedistributionofparticlepositions Xt weightedbytheirrespective

masses,dmt

=

C

(

Xt

,

t

)

d Xt.Thiscanbeexpressedformallyas1:

C

(x,

t)

=



R C

(X

t

,t

) δ (X

t

−

x)d Xt

=



R

δ (X

t

−

x)dmt (3.10)

1 _{Moreprecisely,inagivendomain ,thelocalconcentration}

_C

(x,t)isrelatedtothePDFofparticlepositions fXt(x;t)by

C

(x,t)=M (t)fXt(x;t) where

M

(t)isthetotalmassoftheparticlesinsidethedomain attime

t.

The solutionisdescribed fortwo casesof“initial”point sources C

(

x

,

0

)

=

M0

.δ (

x

−

x0

)

locatedatx0

<

x1:2

<

x2:3 and

C

(

x

,

0

)

=

M0

.δ (

x

−

x1

)

locatedatx1:2

<

x1

<

x2:3.Inthefirst case,thesourceisintheleft layer,butsimilarsolutionsare

obtainedforanysourceposition.ThesesolutionsareusedinthetextinrelationtothegeneralizationofourRandomWalk algorithm,andforcomparison/validationofresults(Section4,Fig.4.2).

Forasourcelocatedatleft

(

x0

<

x1:2

)

theinitialcondition forthediffusionproblemofEq. (A.1) is:

⎧

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎩

∀

x

≤

x1:2

;

C1

(x,

0

)

=

M0

θ

1

δ (x

−

x0

)

∀

x1:2

≤

x

≤

x2:3

;

C2

(x,

0

)

=

0

∀

x

≥

x2:3

;

C3

(x,

0

)

=

0 (A.2)

Thesolutionofthediscontinuousdiffusionproblem(Eq. (A.1))withinitialcondition(A.2) is:

∀

t

≥

0

; ∀

x

≤

x1:2

;

θ

1 M0 C1

(x,

t)

=

G

(x

0

,

2D1t,x)

+

R1:2G

(−

x0

+

2x1:2

,

2D1t

,x)

+

+∞



i=0

(

1

−

R1:2

)

R2:3

(R

2:3R2:1

)

i

(

1

−

R2:1

)

×

G



−

x0

+

2x1:2

+

2

(i

+

1

)

√

D1

√

D2

(x

2:3

−

x1:2

) ,

2D1t,x



(A.3a)

∀t

≥

0

; ∀x

1:2

≤

x

≤

x2:3

;

θ

2 M0 C2

(x,t

)

=

+∞



i=0

(

1

−

R1:2

) (R

2:3R2:1

)

i

×



G

√

D2

√

D1

(x

0

−

x1:2

)

− (

2i

+

1

) (x

2:3

−

x1:2

)

+

x2:3

,

2D2t

,x



+

R2:3

×

G



−

√

D2

√

D1

(x

0

−

x1:2

)

+ (

2i

+

1

) (x

2:3

−

x1:2

)

+

x2:3

,

2D2t,x



(A.3b)

∀

t

≥

0

; ∀

x

≥

x2:3

;

θ

3 M0 C3

(x,

t)

=

+∞



i=0

(

1

−

R1:2

) (R

2:3R2:1

)

i

(

1

−

R2:3

)

×

G

√

D3

√

D1

(x

0

−

x1:2

)

+ (

2i

+

1

)

√

D3

√

D2

(x

1:2

−

x2:3

)

+

x2:3

,

2D3t

,x



(A.3c) AndthesolutionofEq. (A.1) withtheinitialcondition

⎧

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎩

∀x

≤

x1:2

;

C1

(x,

0

)

=

0

∀

x1:2

≤

x

≤

x2:3

;

C2

(x,

0

)

=

M0

θ

1

δ (x

−

x1

)

∀

x

≥

x2:3

;

C3

(x,

0

)

=

0 (A.4) is:

∀t

≥

0

; ∀x

≤

x1:2

;

θ

1 M0 C1

(x,

t)

= (

1

−

R2:1

)

+∞



i=0

(R

2:3R2:1

)

i

×



G



x1:2

+



D1 D2

(

2i

(x

2:3

−

x2:1

)

+

x0

−

x1:2

) ,

2D1t

,x



+

R2:3G



x1:2

+



D1 D2

(

2i

(x

2:3

−

x2:1

)

+

2x2:3

−

x0

−

x1:2

) ,

2D1t

,

x



(A.5a)

∀

t

≥

0

; ∀

x1:2

≤

x

≤

x2:3

;

θ

2 M0 C2

(x,t

)

= −

G

(x

0

,

2D2t,x)

+

+∞



i=0

(R

2:3R2:1

)

i

× (

G

(

2i

(x

2:3

−

x1:2

)

+

x0

,

2D2t

,x)

+

G

(

2i

(x

1:2

−

x2:3

)

+

x0

,

2D2t,x)

+

R2:3G

(

2i

(x

2:3

−

x1:2

)

+

2x2:3

−

x0

−

x1:2

,

2D2t,x)

+

R2:1G

(

2i

(x

1:2

−

x2:3

)

+

2x1:2

−

x0

−

x2:3

,

2D2t,x)) (A.5b)

∀

t

≥

0

; ∀

x

≥

x2:3

;

θ

3 M0 C3

(x,t)

= (

1

−

R2:3

)

+∞



i=0

(R

2:1R2:3

)

i

×



G



x2:3

+



D3 D2

(

2i

(x

1:2

−

x2:3

)

+

x0

−

x2:3

) ,

2D3t,x



+

R2:1G



x2:3

+



D3 D2

(

2i

(x

1:2

−

x2:3

)

+

2x1:2

−

x0

−

x2:3

) ,

2D3t,x



(A.5c) The above solutions have been checked by direct substitution in Eq. (A.1a) and the initial and boundary conditions Eq. (A.1b),Eq. (A.1c) andEq. (A.2).

Theinfiniteseriesobtainedinthissectionforconcentration C

(

x

,

t

)

havetheformofafunctionh

(

x

,

t

)

definedas:

h

(x,t)

=

+∞



n=0 A

√

tz n_exp



−

(x

+

Cn

+

D)2 Bt



(A.6) withB

,

t

>

0; x

,

A

,

C

,

D

R

andz

=

R2:1R2:3 (productofpartialreflectioncoefficients).Theirconvergenceisstudiedinthe

nextappendix(AppendixB).

Appendix B. Studyoftheconvergenceandcontinuityoftheseriesh

Inthisappendix,westudytheconvergenceoftheinfiniteseriesh

(

x

,

t

)

(A.6) whichhastheformofthesolutionfoundin AppendixA.Thissolution(concentrationC

(

x

,

t

)

)wasusedinFig.4.2tovalidatetheRandomWalkParticleTrackingmodel proposedinthispaperfordiscontinuousdiffusion.

B.1. Pointwiseconvergenceusingd’Alembertcriteria Theorem1(d’AlembertRatiotest[29]).Let



an

,

n

N



beasequenceofcomplex numberssuchthatL

=

limn→+∞|an_|an+1||exists.

IfL

<

1,thentheseries



+∞_n=0anconvergesabsolutely.Thus,



+∞n=0anispointwiseconvergent.

IfL

>

1,thentheseries



+∞_n=0anisdivergent.

IfL

=

1,thenthecaseisundecided.

Using Theorem 1, we now show that our series h

(

x

,

t

)

is pointwise convergent. This series is of the form h

(

x

,

t

)

=



_+∞ n=0Un

(

x

,

t

)

,where: Un

=

A

√

tz n_exp



−

(x

+

Cn

+

D)2 Bt



(B.1)

|

Un+1

|

|

Un

|

= |

z

|

exp



−

(x

+

C

(n

+

1

)

+

D)2

− (

x

+

Cn

+

D)2 Bt



(B.2)

|

Un+1

|

|U

n

|

= |

z| exp



−

2xC

+

C2

+

2DC Bt



exp



−

2C2n Bt



(B.3) IfC

=

0,thentheseriesconvergesfor

|

z

| <

1 and B

,

t

>

0;x

,

A

,

D

R

,h

(

t

,

x

)

=

√A

t 1 1−zexp



−

(x+D)2 Bt



.Thiscaseoccursif thedistancebetweentwointerfaces

(

|

x1:2

−

x2:3|)goestozero(wemaydismissthiscasehere).

Otherwise,ifC

=

0,then |Un_|Un+1_||

→

n→+∞0.Bythe d’AlembertRatio test,the seriesh converges

∀

z

C

;

∀

B

>

0

,

∀

t

>

0;

∀

x

,

A

,

C

,

D

R

.

Inconclusion,theseriesh

(

x

,

t

)

andtheanalyticalconcentrationssolutions(Eqs. (3.8) and(3.9))convergepointwisein allcasesofinterest.

B.2. Uniformconvergence

Theorem2([29]).Assume

(

fn

)

isasequenceoffunctionsdefinedonE,andassume

|

fn

(

x

)

| ≤

Mn

(

xE

,

n

=

1

,

2

,

3

, . . .)

.Then



fn

convergesuniformlyonE if



Mnconverges.

Theinfiniteseriesh

(

x

,

t

)

isoftheform:

h

(x,t)

=

+∞



n=0 Aznexp



−

Bt ln(t

)

+

2

(x

+

Cn

+

D)2 2Bt



(B.4)

Letusdefine gn

(

x

,

t

)

as:

gn

(x,t

)

= −

Bt ln

(t)

+

2

(x

+

Cn

+

D)2

2Bt (B.5)

Fort

≥

1 wehave gn

≤

0.Thus,







Aznexp



−

Bt ln

(t)

+

2

(x

+

Cn

+

D)2 2Bt







≤ |

A

| |

z

|

n (B.6)

Sincetheseries



_n+∞₌₀

|

A

| |

z

|

n convergesfor

|

z

| <

1,h isuniformlyconvergenton

R

×

[1

; +∞

[. For0

<

t

≤

1 wehave −_eB

≤

Bt ln

(

t

)

≤

0.

•

First,ifC

>

0 andforafixed x0



R

;

∃

n1

/

∀

n

≥

n1;

∀

x

≥

x0;Bt ln

(

t

)

+

2

(

x

+

Cn

+

D

)

2

≥

Bt ln

(

t

)

+

2

(

x0

+

Cn

+

D

)

2

≥

0.

Hence,theseriesh convergesuniformlyon[x0; +∞[

×

]0

;

1].

•

Secondly,ifC

<

0 andforafixedx0



R

;

∃

n2

/

∀

n

≥

n2;

∀

x

≤

x0;Bt ln

(

t

)

+

2

(

−

x

−

Cn

−

D

)

2

≥

Bt ln

(

t

)

+

2

(

−

x0

−

Cn

−

D

)

2

≥

0.Theseriesh convergesuniformlyon]0

;

1]

×

]

−∞;

x0].

Theorem3([29]).If

(

fn

)

isasequenceofcontinuousfunctionsonE,andif fn

→

f uniformlyonE,thenf iscontinuousonE.

Consequence Forz

C

suchthat

|

z

| <

1,h iscontinuouswithrespectto

(

x

,

t

)

on

R

_×

]0

; +∞

[.

Conclusion Inallcasesofinterest,theanalyticalinfiniteseriesconcentrationsolutions(Eqs. (3.8) and(3.9))arepointwise convergentandcontinuouswithrespectto

(

x

,

t

)

on

R

×

]0

; +∞

[.

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[2]M.Spiller,R.Ababou,J.Koengeter,Alternativeapproachtosimulatetransportbasedonthemasterequation,in:TracersandModellinginHydrogeology, ProceedingsoftheTraM’2000ConferenceheldatLiège,Belgium,May2000,in:IAHSPubl.,vol. 262,2000.

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