A Fokker–Planck approach for probability distributions of species concentrations transported in...

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Journal of Computational and Applied Mathematics ( ) –

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A Fokker–Planck approach for probability distributions ofspecies concentrations transported in heterogeneous mediaN. Suciu a,b,∗, F.A. Radu c, S. Attinger d,e, L. Schüler d,e, P. Knabner aa Mathematics Department, Friedrich–Alexander University of Erlangen-Nuremberg, Cauerstraße 11, 91058 Erlangen, Germanyb Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, Fantanele 57, 400320 Cluj-Napoca, Romaniac Department of Mathematics, University of Bergen, Allegaten 41, 5008 Bergen, Norwayd Faculty for Chemistry and Earth sciences, University of Jena, Burgweg 11, 07749 Jena, Germanye Department Computational Hydrosystems, UFZ Centre for Environmental Research, Permoserstraße 15, 04318 Leipzig, Germany

a r t i c l e i n f o

Article history:Received 16 August 2014Received in revised form 5 November 2014

MSC:60J6060G6086A05

Keywords:PDF methodsMixingRandom walkPorous media

a b s t r a c t

We identify sufficient conditions under which evolution equations for probability densityfunctions (PDF) of random concentrations are equivalent to Fokker–Planck equations. Thenovelty of our approach is that it allows consistent PDF approximations by densities of com-putational particles governed by Itô processes in concentration–position spaces. Accuratenumerical solutions are obtained with a global randomwalk (GRW) algorithm, stable, freeof numerical diffusion, and insensitive to the increase of the total number of computationalparticles. The system of Itô equations is specified by drift and diffusion coefficients describ-ing the PDF transport in the physical space, provided by up-scaling procedures, as well asby drift and mixing coefficients describing the PDF transport in concentration spaces. Mix-ing models can be obtained similarly to classical PDF approaches or, alternatively, frommeasured or simulated concentration time series. We compare their performance for aGRW-PDF numerical solution to a problem of contaminant transport in heterogeneousgroundwater systems.

1. Introduction

PDF methods have been developed in turbulence and combustion theory, motivated by the need to close the Reynoldsaveraged transport equations. Beyond achieving the closure (e.g. for convection and reaction terms) these approaches maybe valuable tools in modeling transport in random environments in that they provide the one-dimensional (one-point one-time) PDF of transported quantities, such as turbulent velocity fluctuations and scalars (concentrations, temperature, en-thalpy) [1–3]. In this paper, we only consider scalar PDF methods, with an illustration for the concentration of chemicalspecies transported in saturated groundwater formations. The development of the Fokker–Planck approach and the nu-merical illustration presented in the following are restricted to the case of incompressible flows and constant diffusioncoefficients.

Thus, we consider an array of species concentrations C(x, t) = Cα(x, t) linked through reaction terms S(C) = Sα(C),α = 1, . . . ,Nα , where Nα is the number of chemical species. The diffusion coefficient D is constant and species independent

∗ Corresponding author at: Mathematics Department, Friedrich–Alexander University of Erlangen-Nuremberg, Cauerstraße 11, 91058 Erlangen,Germany.

E-mail address: suciu@math.fau.de (N. Suciu).

2 N. Suciu et al. / Journal of Computational and Applied Mathematics ( ) –

and the chemical components are transported in a statistically homogeneous random velocity field V with divergence-freerealizations according to a system of Nα stochastic balance equations, written in a compact form as

∂tC + V · ∇C = D∇2C + S(C). (1)

The Eulerian PDF f (c; x, t) of the random concentrations C solving (1) satisfies

∂t f + ∇ · (Vf ) =∂2

∂xi∂xj(Dijf ) −

∂cα∂cβ(Mαβ f ) − ∇c · (Sf ) , (2)

whereV = ⟨V⟩+∇·D is the upscaled drift and ⟨V⟩ themean velocity,D = D+D∗ is the diffusion coefficient tensor upscaledthrough the gradient-diffusion closure f ⟨V − ⟨V⟩|C(x, t) = c⟩ = −D∗

∇f of the mean velocity fluctuations conditional onconcentration, ∇c is the gradient in the concentration space, and M = ⟨D∇C∇C|C(x, t) = c⟩ is the conditional dissipationrate tensor which accounts for molecular mixing [2, Section 12.7.4]. The reaction term S occurring in (2) is in a closed form,the same as in the concentration balance equation (1), which is the essential advantage of the PDF methods [3].

The Eulerian PDF f (c; x, t) is usually obtained from the numerical solution of a system of Itô equations describing theevolution of an ensemble of computational particles in physical and concentration spaces [3, Section 6.7]. The system of Itôequations can be written in a general form as

dX(t) = V(X(t), t)dt + dW(X(t), t) (3)dC(t) = M(C(t),X(t), t)dt + S(C(t))dt, (4)

where C(t) = C(X(t), t),W is a Wiener process with EW(X(t), t) = 0 and EW2(X(t), t) = 2 t0 D(X(t), t ′)dt ′

[4, Section 7.1], andM is a mixing model for the conditional dissipation rate M [1–3].In this paper,we identify consistency conditions relating the statistics of the random fieldC(x, t), solution of the systemof

stochastic equations (1), to that of the stochastic process X(t), C(t), describedby the Itô equations (3) and (4),which revealsthe natural relationship between the Eulerian PDF f (c; x, t) and the joint concentration–position PDF p(c, x, t). Further, weestablish sufficient conditions under which p(c, x, t) solves a closed-form equation and, in particular, an equation of theform (2). Such equations are thus Fokker–Planck equations associated to systems of Itô equations (3) and (4). By equivalencebetween Itô and Fokker–Planck representations of a diffusion process, ensured by regularity conditions on coefficients [5]which are assumed fulfilled, p(c, x, t) is approximated numerically by the density of a large ensemble of computationalparticles which evolve in the (x, c) space accordingly to Eqs. (3) and (4) [4].

The classical approach to solve the PDF equation (2) is discussed in Section 2. The consistency conditions and the cor-respondence between concentration random fields and stochastic processes in concentration–position spaces, as well asthe derivation of the Fokker–Planck equation for the joint concentration–position PDF, are presented in Section 3. Section 4introduces the new solution approach for the concentration PDF using the GRW algorithm. An application to non-reactivetransport in saturated aquifers is presented in Section 5 and some conclusions are formulated in Section 6.

2. Classical PDF methods

The solution of the system of Itô equations (3) and (4) is the stochastic process Xi(t), Cα(t), Cα(t) = Cα(X(t), t), t ≥ 0,i = 1, 2, 3, α = 1, . . . ,Nα . At given time t , the process takes values (x = X(t), c = C(t)). Here and throughout the paper,we use the convention which denotes random functions by capital letters and their values in the state space by correspond-ing lowercase letters. The state spaceΩ = Ωx ×Ωc is the Cartesian product of the three-dimensional physical spaceΩx andthe Nα-dimensional concentration space Ωc . The one-dimensional (one-time) PDF of this process is the joint PDF of con-centrations and positions p(c, x, t). Classical PDF methods approximate the PDF f (c; x, t) by solving a system of stochasticdifferential equations stochastically equivalent to Eqs. (3) and (4) with coefficients defined by averages conditional on con-centration, which correspond to the coefficients of the PDF equation (2) [1,3,6].

For constant density flows, the one-dimensional Eulerian PDF f (c; x, t) of the random field C(x, t) is assimilated to theconditional PDF of the stochastic process Xi(t), Cα(t), p(c|x, t) = p(c, x, t)/px(x, t), where px(x, t) is the position PDF. Anecessary ingredient to render this approach feasible is to force a uniform position PDF px(x, t), so that f (c; x, t) becomesproportional to the solution p(c, x, t) of the Fokker–Planck equation associated to the system of Itô equations (3) and (4).The position PDF px(x, t) is the solution of the Fokker–Planck equation associated to the position Itô equation (3), whichis Eq. (2) with the last two terms set to zero. The spatially uniform position PDF px(x, t) = const = 1/

dx is also timeindependent (Ωx is a fixed domain). This is a trivial solution of the Fokker–Planck equation in case of space-constant velocityand diffusion coefficients [7] or, in themore general case of space-variable drift and isotropic diffusion coefficient, when thecondition ∇ · V = ∇

2D is fulfilled [2, Section 12.7].The general solution algorithm for variable density flows is based on a discrete representation of the PDF which requires

the equality between the joint concentration–position PDFmultiplied by the total massM of fluid inΩx,Mp(c, x, t), and the‘‘mass density function’’ ρ(c)f (c; x, t), where ρ(c) is the density of the fluid which, for constant temperature, depends onlyon species concentrations [1, Eqs. (3.78) and (3.79)]. For the case of scalar transport considered here, one assumes that thecarrying fluid is also an element of the chemical composition described by the concentration vector C. Equating the integrals

N. Suciu et al. / Journal of Computational and Applied Mathematics ( ) – 3

over the concentration space Ωc of the two densities one finds that Mpx(x, t) =

Ωxρ(c)f (c; x, t)dc = ⟨ρ⟩(x, t). Since for

constant density flows ⟨ρ⟩ is independent of x, the position PDF px(x, t) has to be uniform, irrespective of the properties ofthe coefficientsV andD . We note that while a uniform px(x, t) is physically meaningful in case of homogeneous turbulencewith space-independent statistics [1, p. 140], it is not consistent with a general mechanism of the transport process. Even fordivergence-free upscaled velocity V , the condition ∇

2D = 0 still has to be fulfilled for the existence of a uniform solutionpx(x, t) of the position Fokker–Planck equation.

The implication of a uniform position PDF for the solution algorithm is that the expected particle number density isuniform in physical space at all times. Instead of representing a point in the (3+Nα)-dimensional space Ω which describesthe state of the process Xi(t), Cα(t) at a given time, the ‘‘notional particles’’ introduced by Pope [1] carry a ‘‘composition’’Cα(t), α = 1, . . . ,Nα , of the chemical system in the physical spaceΩx. Concentration evolution equations similar to Eq. (4)are solved for each particle to update its composition and, in a subsequent step, the system of particles is transported in thephysical space. Finally, mean values and PDFs are estimated by counting the number of particleswith different compositionsin cells of the physical space. In this way, one avoids the curse of dimensionality for multi-component reactive transportproblems. The drawback is the linear increase of computational costs with the number of particles and the numericaldiffusion produced by interpolation to particle positions of mean values estimated by averaging over cells [8].

The equivalence between the system of notional particles and the physical system is established by requiring the equal-ity between the joint PDF p(c, x, t) of the system of notional particles and that of a system of fluid particles with the samecomposition Cα(t) (see e.g. [1, Section 4.2] and [3, Section 6.7]). The mathematical definition of the fluid particle usesLagrangian equations for the fluid flow which allow a consistent derivation of the evolution equation for the Eulerian ve-locity PDF [2, Section 12.6]. An intermediate situation is that of reaction–diffusion systems in turbulent flows described byfluid particles with trajectories specified by the upscaled flow velocity, through Eq. (3) with the diffusion term set to zero,and compositions modeled by Eq. (4). The latter accounts for the diffusion process through a mixing term which modelsthe conditional average of the diffusion flux according to the transport balance equation (1) [1,3]. This formulation of theproblem already introduces a disagreement between the Lagrangian description and the diffusion process, which destroysthe individuality of the fluid particles. This is even better seen in the general form of the equations governing the systemof notional particles, where the particle’s position is an Itô diffusion process of type (3) [3,6,7]. Because Eq. (3) describesa diffusion process, there is an infinity of trajectories starting from the same initial state [5] and a fluid particle uniquelyspecified by its initial position cannot be defined. Nevertheless, the heuristic approach based on the equivalence betweennotional and fluid particles carrying compositions in physical space succeeded in designing efficient solution algorithms ina broad field of applications of the PDF methods [9].

3. The Fokker–Planck approach

To avoid the above conceptual inconsistency and to overcome the numerical issues of the classical PDF approach,we derive the Fokker–Planck equation for the joint concentration–position PDF, p(c, x, t), consistent with the transportequations (1). Further, we estimate p(c, x, t) from a direct GRW solution of the system of Itô equations (3) and (4) with theinitial condition consisting of a distribution of computational particles in the state space Ω which approximates the initialjoint concentration–position PDF.

3.1. Correspondence between random fields and stochastic processes

The implementation of particle methods for solving PDF equations requires relationships between the Eulerian PDFf (c; x, t)which solves (2) and the joint concentration–position PDF p(c, x, t) associated to the systemof Itô equations (3) and(4). A difficulty, not yet explicitly formulated in the literature, is caused by the use of two quite different stochastic models:random fields and stochastic processes. On one side, the Eulerian PDF f (c; x, t) is the one-point, one-time PDF of the randomconcentration C(x, t), which is a random field indexed by three spatial parameters and time. On the other side, p(c, x, t) isthe one-time PDF of the stochastic process C(t),X(t) described by the system of Itô equations (3) and (4), with a singleparameter, the time t .While in the Eulerian picture x is a parameter of the random concentration,X(t) is a stochastic processwith the one-time PDF px(x, t). The position PDF px(x, t) is the normalized concentration at a given time t of the points X(t)on trajectories of the Itô process (3) [4]. Since (3) describes an upscaled transport process, it follows that px(x, t) representsan ensemble mean concentration of points X(t), irrespective of the species concentrations associated with them via Eq. (4).

Using relations between joint and marginal probabilities, px(x, t) can be expressed as

px(x, t) =

p(c, x, t)dc

Ωc\Ωc1

p(c, x, t)dc2 . . . dcα =

pc1,x(c1, x, t)dc1 = · · · =

ΩcNα

pcNα ,x(cNα , x, t)dcNα

Nαα=1

pcα ,x(cα, x, t)dcα,

that is, as an arithmetic average of position PDFs associated to each species, p(α)x =

pcα ,x(cα, x, t)dcα , where pcα ,x(cα,

x, t) are marginal PDFs of p(c, x, t). Since p(α)x have the meaning of ensemble average concentrations of species α, we con-

jecture that the Eulerian counterpart of the above expression is given in terms of the random field C(x, t) by the stochasticaverage of the arithmetic mean of the species concentrations,

Θ(C(x, t)) =1

C∗Nα

Nαα=1

Cα(x, t), (5)

where C∗ is a normalization constant. This conjecture is true ifΘ is a conserved scalar which verifies Eq. (1) without reactionterm. Then, its average solves Eq. (2) with the last two terms set to zero (e.g. [7]), which is a Fokker–Planck equationassociated to the Itô equation (3). This is always the case if the components of the concentration vector C include all thechemical species of the reaction system. Indeed, since each Cα is a linear combination of chemical elements [10] and the sumof chemical elements in the reaction system is conserved, the sum of species concentrations

Nαα=1 Cα is conserved as well.

Equating the position PDF px(x, t) with the mean ⟨Θ⟩, expressed by the usual change of variables relating integrals overprobability spaces and state spaces [5] as average over the state space Ωc weighted by the Eulerian PDF, one obtains therelations

Θ(c)f (c; x, t)dc = ⟨Θ⟩(x, t)

= px(x, t)

p(c, x, t)dc. (6)

The second equality in (6) allows the determination of the normalization constant C∗. These relations are obviously satisfiedif one chooses

Θ(c)f (c; x, t) = p(c, x, t). (7)The relation (7) establishes a consistent correspondence between the one-dimensional statistics of the random field C(x, t)and the process C(t),X(t). By virtue of (7), p(c, x, t) also solves the evolution equation for F (c; x, t) = Θ(c)f (c; x, t).

As we have seen above, the weighting functionΘ in (7) must be a conserved scalar, but its choice is not unique. It may bedefined by the sum or by the arithmetic mean of the species concentrations, as well as by any other conserved combinationof concentrations, for example the mixture fraction used in diffusion flames approaches [10]. For a given choice of Θ , (7)establishes a particular relation between f (c; x, t) and p(c, x, t). For instance, in combustion theory the weighting factor ischosen as Θ =

Nαα=1 ρα = ρ, where ρα represents mass concentrations of species α and ρ is the fluid density [1,9]. In this

case, both the solutes and the solvent have to be included in the ensemble of species α to close the problem. The relation(7) takes the form

ρ(c)f (c; x, t) = Mp(c, x, t), (8)which by integration over the concentration state space variable c yields

⟨ρ⟩(x, t) = Mpx(x, t), (9)whereM =

Ω⟨ρ⟩(x, t)dx is the total mass of fluid inΩx (see e.g. [1, Section 3.4] and [9, Section 4]). If the relation (8) is used

instead of (7), then F = ρ(c)f (c; x, t) defines the mass density function (e.g. [1, Eq. (3.59)]). For constant density flows, (9)implies a uniform position PDF px(x, t), which, as seen in Section 2, may be inconsistent with the spatial variability of thecoefficients occurring in Eq. (1). The advantage of using (7) with a weighting function Θ different from ρ is that it does notforce a uniform position PDF for constant density and is adequate for the case of dilute solutions, when the solvent is notincluded among the species concentrations.

3.2. Derivation of the Fokker–Planck equation

In studies of turbulent reacting flows [1,3], evolution equations for f (c; x, t) or for the weighted concentration PDFF (c, x, t) are derived by integrating over the velocity space evolution equations for joint velocity–concentration PDFs. Inthe following, we derive evolution equations for F (c, x, t) in the form of Fokker–Planck equations which, according to (7),also govern the evolution of the joint concentration–position PDF p(c, x, t) of the Itô process (3)–(4).

Proposition 1. Assume that the velocity fieldV is statistically homogeneouswith divergence-free samples, the diffusion coefficientis constant and species independent, the system of reacting species is governed by the system of equations (1), and the weightingfunction Θ verifies the continuity equation

∂tΘ + V · ∇Θ = 0. (10)

Then, the weighted concentration PDF F (c, x, t) satisfies the evolution equation

∂tF + ⟨V⟩ · ∇F = −∇ · [⟨U|C = c⟩F ] − ∇c ·D∇

2C|C = cF

− ∇c · (SF ) . (11)

Proof. Let us consider the differential operator

A = ∂t + V · ∇ (12)

applied to an arbitrary differentiable function Q which depends on space–time variables only through the random concen-tration vector, Q (x, t) = Q (C(x, t)), with compact support in Ω0

c (the interior of Ωc). Similarly to the Pope derivation ofthe velocity-composition PDF equation [1], we derive Eq. (11) by equating two independent expressions for the ensembleaverage of the product between Θ and the operator (12) applied to Q .

The first expression is given by

⟨ΘAQ ⟩ = ⟨Θ∂tQ + ΘV · ∇Q ⟩ = ⟨∂t(ΘQ ) + ∇ · (VΘQ )⟩ = ∂t⟨ΘQ ⟩ + ⟨V⟩ · ∇⟨ΘQ ⟩ + ∇ · ⟨UΘQ ⟩, (13)

where we used the continuity equation (10), the incompressibility condition ∇ · V = 0, and by U = V− ⟨V⟩ we defined thefluctuation about the constant mean ⟨V⟩ of the statistical homogeneous velocity field. To compute the ensemble average(13) we also need information not included in the one-point PDF f (c; x, t), in this case, the statistics of the velocity fluctu-ations. Following Fox [3, Section 6.2], we lump the unknown statistics in a random vector Z and consider the joint EulerianPDF f (c, z; x, t). The ensemble average of a function F(C, Z) = F1(C)F2(Z) will then be computed as

⟨F(C(x, t), Z(x, t))⟩ =

F1(c)F2(z)f (c, z; x, t)dcdz =

F1(c)⟨F2|C = c⟩f (c; x, t)dc, (14)

where the conditional average is performed with respect to the conditional PDF f (z|c; x, t) = f (c, z; x, t)/f (c; x, t),

⟨F2|C = c⟩ =

F2(z)f (z|c; x, t)dz. (15)

With (14) and (15), the ensemble average (13) of ΘAQ becomes

⟨ΘAQ ⟩ =

Q (c)Θ(c) ∂t f (c; x, t) + ⟨V⟩ · ∇f (c; x, t) + ∇ · [⟨U|C = c⟩f (c; x, t)] dc. (16)

The second expression of ⟨ΘAQ ⟩ follows from the fact that Q depends on time–space variables through the randomconcentration C(x, t) and AC is given by the right hand side of the transport equation (1):

⟨ΘAQ ⟩ = ⟨Θ∇cQ · AC⟩ =Θ∇cQ ·

2C + S(C)

Θ(c)∇cQ (c) ·D∇

2C|C = c+ S(c)

f (c; x, t)dc. (17)

To obtain (17) we used the chain rule of calculus and the averaging procedure (14). Integration by parts yields

⟨ΘAQ ⟩ =

∂Ωc

Q (c)Θ(c)

2C|C = c+ S(c)

f (c; x, t)

· ndΓc

Q (c)∇c ·Θ(c)

2C|C = c,+ S(c)

f (c; x, t)

where n is the outward unit normal to the boundary ∂Ωc of the concentration space Ωc and dΓc is the surface element.Since the function Q (c) has compact support inΩc it vanishes on ∂Ωc and cancels the first integral, so that we finally obtain

⟨ΘAQ ⟩ = −

Q (c)∇c ·Θ(c)

2C|C = c,X = x, t+ S(c)

f (c; x, t)

dc. (18)

Since the expressions (16) and (18) should give the same result for any function Q with compact support, equating themone obtains the evolution equation (11) for F (c; x, t) = Θ(c)f (c; x, t).

Eq. (11) is not yet in a closed form and the conditional average of the velocity fluctuations and that of the diffusive fluxfrom the right hand side of (11) require modeling. Often used in turbulence [1,3,9] (with some caution in case of variabledensity flows) is a gradient-diffusion closure of the first conditional average in (11),

⟨U|C = c⟩f (c; x, t) = −D∗∇f (c; x, t). (19)

Introduced in (11), the closure (19) gives the diffusion term ∂∂xi

i,j∂

∂xjF . To put it in a Fokker–Planck formwe add ∂

∂xiF ∂

∂xjD∗

in both sides of Eq. (11), which also implies defining the new drift coefficients

Vi = ⟨Vi⟩ +∂

∂xjD∗

i,j. (20)

Proposition 1 ensures the existence of the evolution equation (11) only if the weighting factor verifies the continuityequation (10). This is the case when Θ = ρ and Eq. (11) solves for the mass density function F (c; x, t) = ρ(c)f (c; x, t). In

case of dilute solutions, when the solvent is not included in the reaction system, an evolution equation for F can still be de-rived for advection–reaction problems. Then,D = 0, all conserved scalarsΘ verify (10), andwe have the following corollary.

Corollary 1. AssumeD = 0 and the closure (19). Then theweighted concentration PDF F (c, x, t) satisfies the evolution equation

∂tF + ∇ · (Vf ) =∂2

∂xi∂xj(D∗

ijF ) − ∇c · (SF ) . (21)

According to Eq. (7), F (c; x, t) = p(c, x, t). Hence, (11) is the general form of the Fokker–Planck equation associatedto the system of Itô equations (3) and (4). Expressing p(c, x, t) as a product of conditional PDF p(c|x, t) and position PDFpx(x, t), p(c, x, t) = p(c|x, t)px(x, t), and using (6) and (7) one obtains

p(c|x, t) = Θ(c)f (c; x, t)/⟨Θ⟩(x, t). (22)

Eq. (22) shows that the concentration PDF conditional on position, approximated by numerical solutions of the Itô equations(3) and (4), is a numerical solution for the one-dimensional Eulerian concentration PDFweighted byΘ/⟨Θ⟩. In classical PDFmethods Θ = ρ, Eq. (11) describes the evolution of the mass density function F (c; x, t) = ρ(c)f (c; x, t), and the condi-tional PDF p(c|x, t) of the system of computational particles approximates, according to (8) and (9), the density weightedEulerian PDF,

p(c|x, t) = ρ(c)f (c; x, t)/⟨ρ⟩. (23)

The solution of the classical ‘‘stochastic particles algorithm’’ of Pope is based on Eq. (23), which allows approximating den-sity weighted averages by ensemble averages over the system of computational particles (see e.g. [1, Eqs. (3.84) and (3.89)]).But, even if in this case (11) is a Fokker–Planck equation, the Pope particles algorithm, shortly described in Section 2, doesnot solve the system of Itô equations (3) and (4).

The numerical solution for the weighted PDF based on Eq. (22) which does not require including the solvent among theensemble of species concentrations has the advantage that it does not force a uniform position PDF for constant densityflows. As shown by Corollary 1, this approach is valid for advection–reaction problems, which also may be thought as a con-sistent first-order approximation of the full, advection–diffusion–reaction problem [4]. Another situation when a weightingfunctionΘ = ρ could be used is when the approximationΘ ≃ ⟨Θ⟩(x, t)may be adopted, as in case of ‘‘ergodic behavior’’ oftransport in groundwater [4,11]. Then the Eulerian PDF f (c; x, t) can be approximated by the conditional PDF p(c|x, t), evenif px(x, t) = ⟨Θ⟩(x, t) is not uniform. Within this ergodic approximation, Eq. (2) is the Fokker–Planck equation describingthe evolution of p(c|x, t).

3.3. The treatment of the conditional diffusion term in Eq. (11)

The conditional averages in (11) depend not only on independent state-space variables c and x but also on space–timedependent variables describing the random field C(x, t). A way to relate the two types of variables is to use probabilitydensities defined by averages of Dirac-δ functions [1,9]. Even though such densities are singular, their integration withrespect to the state space variables yields well-defined statistics [4, Appendix A.1].

In this frame, a useful expression for the conditional average (15) is obtained by using the definition of the joint PDF inthe (c, z) space, f (c, z; x, t) = ⟨δ(Z(x, t) − z)δ(C(x, t) − c)⟩, as follows:

⟨F |C = c⟩ (x, t) =

F(z)f (z|c; x, t)dz =

F(z)f (c, z; x, t)f (c; x, t)

f (c; x, t)

F(z) ⟨δ(Z(x, t) − z)δ(C(x, t) − c)⟩ dz

f (c; x, t)

δ(C(x, t) − c)

F(z)δ(Z(x, t) − z)dz

f (c; x, t)⟨F(Z(x, t))δ(C(x, t) − c)⟩ . (24)

In the above calculations we used multi-dimensional δ functions, δ(y0 − y) =i=n

i=1 δ(y0,i − yi), and the obvious propertyΩf (y)δ(y0 − y)dy =

Rn IΩ(y)f (y)δ(y0 − y)dy = f (y0) for all y ∈ Ω ⊂ Rn, where IΩ is the indicator function of Ω .

Expressions of conditional averages similar to (24) are often used in classical PDF methods [1,2].Another ingredient towork out the conditional averages in (11) is the derivative of the δ function. This is properly defined

in terms of Dirac distributions by∞

−∞

δ′(y0 − y)ϕ(y)dy = −

−∞

δ(y0 − y)ϕ′(y)dy, (25)

for any smooth function ϕ with compact support in R. The usual notation for the distributional derivative is δ′[ϕ] = −δ[ϕ′

= −ϕ′(y0). Using (25) for a composite function ϕ(g(x)) we have

−∞

δ(g(x) − y)ϕ(y)dy =ddx

ϕ(g(x)) = ϕ′(g(x))dg(x)dx

=dg(x)dx

−∞

δ(g(x) − y)ϕ′(y)dy

= −dg(x)dx

−∞

δ′(g(x) − y)ϕ(y)dy,

or, in compact notation,

δ(g(x) − y)[ϕ] = −dg(x)dx

δ′(g(x) − y)[ϕ]. (26)

Bearing in mind its interpretation as distributional derivative, the relation (26) is formally written in applications as [1,2]

δ(g(x) − y) = −dg(x)dx

δ(g(x) − y). (27)

For the vectorial function g = C(x, t), (27) can be generalized to the following expression of the partial derivative withrespect to the space variable,

∂xiδ(C(x, t) − c) = −

∂Cα

∂xi(x, t)

∂cαδ(C(x, t) − c). (28)

Proposition 2. If the weighting factor Θ ≡ 1 the conditional diffusion term in Eq. (11) can be decomposed as sum of diffusionterms in physical and concentration spaces,

−∂

∂cα

∂2Cα

∂xi∂xi

c f (c; x, t) = D∂2

∂xi∂xif (c; x, t) −

∂cα∂cβ

∂Cα

∂Cβ

c f (c; x, t) . (29)

Proof. Θ ≡ 1 implies F (c; x, t) ≡ f (c; x, t). Using (24), (28), and the definition ⟨δ(C(x, t) − c)⟩ = f (c; x, t) for the one-dimensional Eulerian PDF, the second conditional average from (11) can be computed as follows:

−∂

∂cα

∂2Cα

∂xi∂xi

c f (c; x, t) = −∂

∂cα

∂2Cα

∂xi∂xiδ(C(x, t) − c)

= −D

∂cα

∂Cα

∂xiδ(C(x, t) − c)

∂Cα

= −D

∂Cα

∂cαδ(C(x, t) − c)

∂cα

∂Cα

∂xi∂xi⟨δ(C(x, t) − c)⟩ −

∂cα

∂Cα

∂Cβ

∂cβδ(C(x, t) − c)

∂xi∂xif (c; x, t) −

∂cα∂cβ

∂Cα

∂Cβ

c f (c; x, t) .

Inserting (29) into (11) and using the gradient-diffusion closure (19) and the definition (20) of the drift coefficients oneobtains the evolution equation (2) for the Eulerian PDF f (c; x, t), with upscaled diffusion coefficients defined by

Dij = D + D∗

i,j (30)

and conditional dissipation tensor defined, according to the second term of the decomposition (29), by

Mαβ =

∂Cα

∂Cβ

c . (31)

We have thus the following corollary of Propositions 1 and 2.

Corollary 2. Under the assumptions of Proposition 1 and for weighting factor Θ ≡ 1, the Eulerian PDF f (c; x, t) verifiesEq. (2).

The PDF evolution equation (2) has the form of a Fokker–Planck equation and, according to (22), it is verified by theconditional PDF p(c|x, t) of the system of Itô equations (3)–(4) for uniform position PDF px(x, t). Corollary 2 also applies ifas weighting factor Θ one chooses ρ = const , which corresponds to the case of homogeneous turbulence in classical PDFmethods [1, p. 140].

For variable Θ and F (c; x, t) = Θ(c)f (c; x, t), one obtains a Fokker–Planck equation of form (2) but with additionalsource terms. A computation similar to the proof of Proposition 2 shows that these terms depend on partial derivatives ofΘ with respect to cα . The equation is thus unclosed and it is no longer equivalent to the system of Itô equations (3) and(4). However, in classical PDF methods Eq. (2), with diffusion coefficients (30) andmixing coefficients (31), is assumed validin case of variable density, without further justification [12,13], or in some cases, e.g. [14], by referring to its derivation forconstant density flows [15].

Summarizing this section, we note that the exact Fokker–Planck equation has the from (11) and has been derived forweighting functionsΘ which verify the continuity equation (10). This condition can be relaxed for advection–reaction prob-lems when, by Corollary 1, one obtains the Fokker–Planck equation (21), without mixing term. With the gradient-diffusionclosure (19), diffusion in physical space may be introduced in both equations, yet with incomplete upscaled diffusion coef-ficient D , which does not contain themolecular diffusion coefficient D. The general form of themixing coefficients in (11) isthat given by the conditional average of the diffusion flux from the transport equation (1). The evolution equation (2) for theEulerian PDF f (c; x, t), with diffusion in physical space described by the full upscaled coefficients (30) and mixing modeledas a diffusion in concentration space with coefficients (31) given by the conditional dissipation rate, can be derived only incase of constant Θ , e.g., constant density ρ as in turbulence studies, or as an approximation under ergodicity assumptions.

4. GRW solutions to PDF equations

The solutions of the PDF evolution equations, defined in the state space Ω = Ωx × Ωc , depend on 3 + Nα independentvariables plus the time variable. Numerical methods for PDF equations are thus confrontedwith the curse of dimensionality.Therefore, instead of using finite difference/element schemes, solution methods for PDF equations are based on equivalentrepresentations as systems of computational particles. Particle methods also avoid the increase of computational errorsby numerical diffusion [16], unavoidable in classical schemes for PDF equations, which usually result in strongly advection-dominated problems. For either Pope’s particles approach or particle tracking solutions to the systemof Itô equations (3)–(4),the computational cost increases linearly with the number of particles. The GRW algorithm [17] described in the followingis insensitive to the increase of the number of particles. For advection–diffusion problems with constant coefficients GRWis equivalent to a finite difference scheme [4] but in case of variable coefficients it is faster and free of numerical diffusion.For instance, when solving flow and transport problems in highly heterogeneous media by finite element schemes, if thetransport solver is replaced by a GRW scheme the numerical diffusion is completely removed and the total computationtime is about 20 times smaller [18]. As compared with sequential procedures, we note that for a grid-free particle trackingscheme the computing time is of the order of the number of particles and for a GRW solution to the same problem and thesame number of time steps it is of the order of grid points occupied by particles. For the solution of the PDF problem solvedin the next section, using ∼1024 particles which move on a lattice with ∼105 points, the GRW computing time is about 0.5 swhile a sequential particle tracking would require a computing time 1019 times larger.

To illustrate the GRW-PDF approach we consider a two dimensional PDF problem for joint concentration–position PDFp(c, x, t), solution of the particular form of (2)

∂tp + V∂p∂x

+ Vc∂p∂c

= D∂2p∂x2

+ Dc∂2p∂c2

. (32)

The solution of the Fokker–Planck equation (32) is approximated by the point-density at lattice sites of a large numberof computational particles evolving according to Itô equations of the form (3)–(4). At a given time step, the computationalparticles from a lattice site (blue column in Fig. 1) are globally scattered in groups of particles which remain at the positiondetermined by the drift coefficients and particles undergoing diffusive jumps (red columns). The numbers of particles ineach group are binomial random variables with parameters determined by the coefficients of the Itô equations, the timestep, and the lattice constants. The GRW algorithm is thus a superposition of many weak Euler schemes for systems of Itôequations projected on a regular lattice [4,17].

The two-dimensional GRW illustrated in Fig. 1 is defined by the relations

n(l,m, k + 1) = δn(l,m, k) +

i=l,j=m

δn(l,m|i, j, k), (34)

where n(i, j, k) is the number of particles at the site (x, c) = (iδx, jδc) at the time t = kδt and the δn are binomial randomvariables describing the spread of n(i, j, k). To the drift and diffusion coefficients of the transport problem, V(iδx, kδt),D(iδx, kδt), Vc(iδx, jδc, kδt), and Dc(iδx, jδc, kδt), one associates dimensionless parameters

vi = Vδtδx

, vj = Vcδtδc

, ri = D2δt

(diδx)2, rj = Dc

2δt(djδc)2

, (35)

where di and dj are integers describing the amplitude of the diffusive jumps.

Fig. 1. Two-dimensional GRW algorithm in (x, c) space. (For interpretation of the references to colour in this figure, the reader is referred to the webversion of this article.)

The average over GRW runs of the terms in (33) are related by

δn(i + vi, j + vj|i, j, k) = (1 − ri − rj) n(i, j, k),

δn(i + vi ± di, j + vj|i, j, k) =ri2

n(i, j, k),

δn(i + vi, j + vj ± dj|i, j, k) =rj2

n(i, j, k). (36)

Themajor feature of the GRW algorithm is its self-averaging property [17, Fig. 5]. That means that for large enough numbersof computational particles the averages over runs are well approximated by the corresponding quantities computed in asingle GRW run, e.g. n ≃ n and δn ≃ δn.

An efficient implementation of the GRWalgorithm is obtainedwhen the binomial random variables δn are approximatedby

δn(i + vi − d, j + vj|i, j, k) =

n/2 if n is even

[n/2] + θ if n is odd, (37)

where n = ri n(i, j, k) and θ is a random variable taking the values 0 and 1 with probability 1/2. The number of particlesjumping in the opposite direction is simply given by δn(i + vi + d, j + vj|i, j, k) = n − δn(i + vi − d, j + vj|i, j, k). Thesame procedure is further applied for the jumps on the j-axis. In practice, (37) is implemented by summing up remindersof division by 2 and multiplication of n(i, j, k) by ri and rj and by assigning a particle to the lattice site where the sum ofreminders reaches the unity. In this way, one avoids the need to use a random number generator to compute θ [18].

The GRW algorithm is free of numerical diffusion, because the diffusive jumps and the nominal diffusion coefficientsare related by the last two equations (35), and, when implemented by (37), it is practically insensitive to the increase ofthe number of particles (see [4,17,18] for implementation details and convergence tests). Since the particles move betweenlattice sites on which mean values are also defined through particle densities, the GRW-PDF solution presented in the nextsection avoids the artificial diffusion generated in classical PDF methods by interpolation to particle positions of the meanscomputed by averaging over computational cells.

5. Application to transport in groundwater

Monte Carlo simulations of two-dimensional passive transport of a single chemical species in saturated aquifers wereused to estimate the PDF f (c; x, t) of the cross-section space-average concentration C(x, t) =

Ly0 c(x, y, t)dy, where Ly is the

transverse dimension of the computational domain, estimated at the x-coordinate of the plume center ofmass, x = ⟨V ⟩t [11].The upscaled drift coefficient V in Eq. (32) is the ensemble mean velocity, equal to the velocity of the center of mass

⟨V ⟩ = 1 m/day. The upscaled diffusion coefficient D is the longitudinal component of the time dependent ‘‘ensembledispersion coefficient’’. The latter is a self-averaging quantity for transport in random velocity fields with finite correlationrange considered here. Hence,D(t)was efficiently determined by using a single trajectory of diffusion in a single realizationof the random velocity field [4, Section 7.2].

Two different mixing models M were considered in the concentration Itô equation (4). The first one is a general mixingmodel written as a sum of the interaction by exchange with the meanmodel (IEM) [1], a drift term given by the attenuationof themean concentration due to the local diffusion [19], and an additional noise termmodeled as aWiener process, neededto control the shape of the PDF [3], written on the right hand side of Eq. (38),

dC(t) = −a(t)[C(t) − ⟨C(t)⟩]dt + ∆⟨C(t)⟩ + bdW (t). (38)

The coefficient a(t) was estimated, similarly to turbulence problems [1,12], by the inverse of diffusion time scale D(t)/λ2,with λ = 1 m, the characteristic correlation scale of the transport problem [11]. ∆⟨C(t)⟩ was estimated from a fractional

Fig. 2. Distribution of particles in the (x, c) GRW lattice at successive times, n(x, c; t = 0, 10, 50, 100).

Fig. 3. ⟨C⟩(x) at fixed t = 10, 50, and 100 days (peaks), and ⟨C⟩(t) at the plume center of mass x = Vt (monotone curves) compared with Monte Carloresults. (For interpretation of the references to colour in this figure, the reader is referred to the web version of this article.)

step, performed at each time step of the GRW-PDF simulation, consisting of a GRW simulation of the one-dimensionaldiffusion with the constant local diffusion coefficient D = 0.01 m/day considered in the Monte Carlo simulations. Theamplitude b ≈ 10−6 day−1 of theWiener processW (t)was adjusted by comparisonswith theMonte Carlo inferred EulerianPDF f (c; x, t). The results obtained by this extended IEM model are represented by green curves in Figs. 3, 4, and 5.

The second model consisted of a drift term equal to the rate of decay of the mean concentration at the center of mass,⟨C(x, t)⟩, determined from the ensemble of Monte Carlo simulations [11], and a small diffusion in the concentration space.The corresponding diffusion coefficient starts with an initial value adjusted in the same way as the noise term in (38) anddecays exponentially in time, as suggested by a preliminary analysis of concentration time series generated during theMonteCarlo simulations, carried out with an automatic algorithm [20].

The Eulerian concentration PDF f (c; x, t) was simulated by the GRW algorithm and compared with the one obtained byMonte Carlo simulations. Since the cross-section concentration C(x, t) is ergodic with a good approximation [11], C(x, t) ≃

⟨C(x, t)⟩ and f (c; x, t) is well approximated by the conditional PDF p(c|x, t), according to (22). The latter was determined bythe concentration–position PDF p(c, x, t), approximated by the GRWsolution of the Fokker–Planck equation (32), divided bythe position PDF px(x, t). The initial distribution of particles in the (x, c) plane was approximated by multiplying the MonteCarlo PDF at t = 1day by 1024 particles. Fig. 2 shows the evolution of the computational particles during theGRWsimulation.

The position PDF px(x, t), obtained by integrating over the c-coordinate the joint PDF p(c, x, t), defines the ensembleaverage concentration ⟨C⟩(x, t), according to (6). Fig. 3 shows a good agreement, for both mixing models, between MonteCarlo results and GRW-PDF simulations. This result is already expected because ⟨C⟩(x, t) is the probability density of thecomputational particles governed by the Itô equation (3), which is independent of the concentration Itô equation (4) andof the mixing model. The comparison presented in Figs. 4 and 5 shows that the mixing model M based on the rate of decayof ‘‘measured’’ (i.e. simulated) concentration resembles quite well the Monte Carlo results. The extended IEM model (38)instead fails to capture the transport of the PDF in physical and concentration spaces. The discrepancy may be attributedto structural differences between groundwater and turbulent flows. Such models are better suited for homogeneous sys-tems [1,19], the IEM model being exact in case of statistically homogeneous Gaussian concentrations [2, p. 551].

Fig. 4. Transported PDF f (c; x, t) at the plume center of mass x = Vt . (For interpretation of the references to colour in this figure, the reader is referredto the web version of this article.)

Fig. 5. GRWandMonte Carlo cumulative distribution functions cdf (c; x, t), x = Vt, t = 0, 10, 30, 50, and 100 days (from right to left). (For interpretationof the references to colour in this figure, the reader is referred to the web version of this article.)

6. Conclusions

We establish relations between Eulerian random fields describing concentrations of chemical species transported in ran-dom environments and Itô processes in physical and concentration spaces by equating the PDF of the position Itô process tothe ensemble average of a conserved scalar specified by a combination species concentrations. Based on this correspondencewe show that the Fokker–Planck equation for the concentration–position PDF of the Itô process coincides with the Eulerianevolution equation of the concentration PDF weighted by the conserved scalar, provided that the latter obeys a continuityequation. For a constant weighting factor or under the ergodic assumption of small randomness of the conserved scalar, theFokker–Planck equation takes the form of a diffusion in the (x, c) space.

In classical PDF methods for turbulent reacting flows the weighting function is the fluid density. This implies a uniformposition PDF for constant density flows which may be inconsistent with the physical problem. To overcome this difficulty,numerical solutions to PDF equations are obtained by tracking in the physical space notional particles with an associatedcomposition of species concentrations, which is updated by solving concentration evolution equations for each particle. Thisapproach, mainly useful in problems for large reaction systems, is affected by the increase of the computational cost withthe number of particles and by the numerical diffusion induced by interpolations of cell-averages to particle positions.

The alternative GRW-PDF approach proposed in this paper provides solutions to Fokker–Planck equations for realisticinitial conditions represented byparticles distributions in the (x, c) space. The Eulerian PDFweighted by a conserved scalar isestimated by dividing the joint concentration–position PDF by the position PDF of the systemof computational particles. Theprocedure is insensitive to the increase of the number of particles and is strictly free of numerical diffusion because it doesnot involve cell-averages and interpolation procedures. The computational effort in PDF simulations for multi-componentreactive transport could be reduced by parallelization with respect to the number of reactants.

Our numerical illustration for transport in groundwater shows that the mixing model consisting of the rate of decay ofthe mean concentration and an additive noise performs much better than the IEM model used in turbulence studies. Sometests done with mixing models inferred from single-realizations of Monte Carlo simulated concentrations also yield fairlygood GRW-PDF solutions. Therefore, detrending time series of measured concentrations by efficient methods [21,20] seemsto be a promising approach to model mixing in practical applications.

Acknowledgments

This work was supported by the Deutsche Forschungsgemeinschaft-Germany under Grants AT 102/7-1, KN 229/15-1, SU415/2-1 and by the Jülich Supercomputing Centre-Germany in the frame of the Project JICG41.

References

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A Fokker–Planck approach for probability distributions of species concentrations transported in...

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