Fractional ensemble average governing equations of transport by nonstationary stochastic fractional advective velocity and fractional dispersion

$Page 1: Fractional ensemble average governing equations of transport by nonstationary stochastic fractional advective velocity and fractional dispersion$
Published in ASCE Journal of Hydrologic Engineering, 2015

1

FRACTIONAL ENSEMBLE AVERAGE GOVERNING EQUATIONS OF TRANSPORT 1

BY TIME-SPACE NONSTATIONARY STOCHASTIC FRACTIONAL ADVECTIVE 2

VELOCITY AND FRACTIONAL DISPERSION: THEORY 3

M.L.Kavvas1 F.ASCE, S.Kim

2, A. Ercan

3, M.ASCE 4

1 Distinguished Professor, Hydrologic Research Laboratory, and J.Amorocho Hydraulics 5

Laboratory, Department of Civil & Environmental Engineering, University of California, Davis, 6

CA 95616, [email protected] 7

2Associate Professor, Department of Environmental Engineering, Pukyong National University, 8

South Korea, [email protected] 9

3Assistant Project Scientist, J.Amorocho Hydraulics Laboratory, Department of Civil & 10

Environmental Engineering, University of California, Davis, CA 95616, [email protected] 11

12

ABSTRACT 13

In this study, starting from a time-space nonstationary general random walk formulation, the 14

pure advection and advection-dispersion forms of the fractional ensemble average governing 15

equations of solute transport by time-space nonstationary stochastic flow fields were developed. 16

In the case of the purely advective fractional ensemble average equation of transport, the 17

advection coefficient is a fractional ensemble average advective flow velocity in fractional time 18

and space that is dependent on both space and time. As such, in this case the time space-19

nonstationarity of the stochastic advective flow velocity is directly reflected in terms of its mean 20

behavior in the fractional ensemble average transport equation. In fact, the derived purely 21

advective form represents the Lagrangian derivation of the ensemble average mass conservation 22


2

equation for solute transport in fractional time-space. In the case of the fractional ensemble 1

average advection-dispersion transport equation, the moment and cumulant forms of the equation 2

are derived separately. In the moment form of the fractional ensemble average advection-3

dispersion equation of transport, the advection coefficient emerges as a combination of the 4

fractional ensemble average advective flow velocity in fractional time and space with an 5

advective term that is due to dispersion. The fractional dispersion coefficient emerges as a 2- 6

moment of the differential displacement per scaled differential time where and denote, 7

respectively, the fractional orders of the space and time derivatives. Since this fractional 8

dispersion coefficient is in moment form, the corresponding ensemble average equation is a 9

moment form of the fractional ensemble average advection-dispersion equation. The cumulant 10

form of the fractional ensemble average advection-dispersion transport equation is derived for 11

two different cases: (i) when the order of the space fractional derivative of the advective term is 12

the same as that of the time derivative while the order of the space fractional derivative of the 13

dispersion term is twice that of the time derivative, and (ii) when the orders of the time and space 14

fractional derivatives are completely different. In case (i) the advection coefficient emerges as a 15

combination of the fractional ensemble average advective flow velocity with an advective term 16

that is due to dispersion, while the cumulant form of the dispersion coefficient emerges as a 17

time-space-dependent variance of the -moment of the differential displacement per scaled time 18

differential , resulting in a fractional advection-dispersion equation. In case (ii) when the 19

orders of the time and space derivatives are completely different, the cumulant form of the 20

fractional dispersion coefficient emerges as a combination of the moment form of the fractional 21

dispersion coefficient and the square of a fractional ensemble average advective velocity. The 22

derived fractional ensemble average governing equations of transport are rich in structure that 23


3

can accommodate both the non-Fickian and the Fickian behavior of transport. The non-Fickian 1

transport behavior can be modeled by the derived fractional ensemble average transport 2

equations either by means of the long memory in the underlying stochastic flow, or by means of 3

the time-space nonstationarity of the underlying stochastic flow, or by means of the time and 4

space fractional derivatives of the transport equations. 5

Key Words: fractional Ensemble Average Transport Equation, nonstationary stochastic flow 6

7

INTRODUCTION 8

From the works of Taylor (1922; 1954) and Batchelor (1949) the dispersion coefficient (or 9

turbulent mixing coefficient) for the transport of a solute by a time-space stationary stochastic 10

advective velocity field v[xt, t], where xt is the Lagrangian location of a solute particle at time t 11

in one dimension, may be expressed as (see also Kavvas and Karakas, 1996, 1997), 12

13

, (1) 14

where < v > is the constant ensemble mean of v and Var (v) is the finite constant variance of the 15

time-space stationary stochastic advective velocity v ( ), and Cov denotes the covariance function 16

while Rv denotes the autocorrelation function of the stochastic advective velocity v ( ). The 17

integral on the right-hand-side of (1) becomes the correlation time of the stochastic advective 18

velocity for t = . Under the further assumption of finite correlation time (finite memory) for the 19

stochastic advective velocity, one would then end up with a finite constant value for the 20


4

dispersion coefficient for times beyond the correlation time of the advective velocity. That is, for 1

times beyond the correlation time of v( ), 2

, (2) 3

where D is a finite constant in time and space. Within this framework, one can then define a 4

constant dispersion coefficient D for transport of solutes. From Eqns. (1) and (2) it also follows 5

that for times beyond the correlation time of a time-space stationary, finite memory stochastic 6

advective velocity, the variance of the Lagrangian displacement of the solute in a time interval 7

(0,t) becomes proportional to time t. That is, 8

2Dt (3) 9

The behavior of solute transport, as described by Eqns. (2) and (3) is known as “Fickian 10

transport”. This Fickian framework provided the basis for the advection-dispersion equation with 11

a constant dispersion coefficient in transport modeling studies in 1970s and 1980s (Fischer et 12

al.1979; Orlob, 1983 among others). 13

However, various studies of the field data for solute transport in rivers ( Nordin and Sabol, 1974; 14

Nordin and Troutman, 1980; Johnson, 2001) have shown significant deviations from the Fickian 15

behavior. Various laboratory (Silliman and Simpson, 1987; Levy and Berkowitz, 2003) and field 16

studies (Peaudecerf and Sauty, 1978; Sudicky et al., 1983; Sidle et al., 1998) of transport in 17

subsurface porous media have similarly shown significant non-Fickian behavior. As one 18

approach to the modeling of the generally non-Fickian behavior of transport, Meerschaert, 19

Benson, Baumer, Schumer, Zhang and their co-workers (Meerschaert et al. 1999, 2002, 2006; 20

Benson et al. 2000a,b; Baumer et al. 2005, 2007; Schumer et al. 2001, 2009; Zhang et al. 2007, 21


5

2008 and 2009) have introduced the fractional advection-dispersion equation (fADE) as a model 1

for transport in heterogeneous subsurface media, while Deng, Singh and their coworkers (Deng 2

et al. 2004; 2005; 2006a,b) and Kim and Kavvas (2006) have introduced the fADE as a model 3

for turbulent transport in surface waters. By theoretical, numerical and field studies the above 4

authors have shown that fADE has a nonlocal structure that can model well the heavy tailed non-5

Fickian dispersion both in subsurface media as well as in rivers and overland flows, mainly by 6

means of a fractional spatial derivative in the dispersion term of the equation. Meanwhile, they 7

have also shown that fADE, with a fractional time derivative, can also model well the long 8

particle waiting times in transport in both surface and subsurface environments. 9

Berkowitz et al. (2006) questioned the fADE model, mentioning that fADE places the 10

mechanism for non-Fickian behavior entirely on the fractional powers of the space and time 11

derivatives in the fADE. They also questioned fADE model’s ability to account for the evolution 12

to a Fickian regime. Neuman and Tartakovsky(2009) mentioned a lack of specific theoretical 13

framework that would lead to fADE equations, and questioned the physical meaning of the 14

fractional parameters in the fADE. They also questioned the flow field stationarity and no-15

correlation assumptions underlying the Continuous Time Random Walk (CTRW) model. 16

This study shall specifically address the above concerns of Berkowitz et al. (2006) and Neuman 17

and Tartakovsky (2009) on fADE by developing the moment and cumulant forms of the 18

ensemble average equations of transport by time-space nonstationary stochastic flow in 19

fractional time and space , starting from a general time-space nonstationary random walk 20

formulation, and deriving the fractional drift and fractional dispersion parameters of transport in 21

terms of the statistical properties of the underlying fractional stochastic flow. 22


6

Comprehensive derivations of the ensemble average advection-dispersion equation for 1

conservative and reactive solute transport by time-space nonstationary flow random fields in 2

integer time-space have been accomplished recently (Kavvas and Karakas, 1996 for conservative 3

transport; Kavvas, 2001 for reactive transport). In these derivations, the advective flow velocity 4

random field is nonstationary both in time and space, and there are no restrictions placed on the 5

distribution or the magnitude of variability of the flow velocity field. Not only Kavvas and 6

Karakas (1996) have obtained the ensemble average form of the stochastic advection-dispersion 7

transport equation in second order cumulant form but also the n-point moment equations of 8

conservative transport by time-space nonstationary flow random fields in second order cumulant 9

form. These equations are non-local, having their advection and dispersion coefficients given 10

explicitly in terms of the time-space mean and chronologically-ordered time-space covariance 11

functions of the advective flow velocity random field. The only restriction placed on these 12

derivations is the finite correlation time for the advective flow velocity random field. 13

Accordingly, one can express the ensemble average advection-dispersion equation for 14

multidimensional transport by time-space nonstationary flow random fields by (Kavvas and 15

Karakas, 1996; Kavvas, 2001); 16

17

(4) 18

to the order of the covariance time of the advective flow velocity random field v (second-order 19

cumulant closure). In Eqn. (4) < . > denotes the ensemble averaging operation, c denotes the 20

solute concentration at the three-dimensional spatial location xt at time t, Covo is the time-21


7

ordered covariance function of the multidimensional advective velocity random field, and xt-s is 1

the Lagrangian location of particle motion at time t-s. It is formulated as (Kavvas and Karakas, 2

1996; Kavvas, 2001); 3

(5) 4

in terms of the time-ordered exponential which is basically a displacement operator (Lie 5

operator). To first order (Kavvas and Karakas, 1996); 6

. (6) 7

Within the framework of Eqns. (4), (5) and (6), the macrodispersion tensor parameter of the 8

ensemble average advection-dispersion equation (4) for transport by a time-space nonstationary 9

advective flow velocity random field is 10

(7) 11

while the local-scale dispersion tensor is Dji . From Eqn. (7) it follows that due to the finite 12

correlation time assumption for the advective velocity in order to be able to derive Eqn. (4)), the 13

integral in Eqn. (7) (representing the ij element of the macrodispersion tensor) will take finite 14

values at every specified time t as t goes to infinity for every spatial location . However, due 15

to the time-space nonstationary nature of the advective flow velocity random field the time-16

ordered covariance function Covo of the advective velocity will vary with every location and 17

time t. As such, the integral in Eqn. (7) will converge to different constant values at different 18

(xt, t) locations as the process evolves in time and space, as opposed to the dispersion case of 19

time-space stationary advective flow velocity with a constant dispersion parameter for times 20


8

beyond the correlation time of the stationary velocity field, as described by Eqns. (1) and (2). 1

Hence, in the case of stochastic transport by time-space nonstationary flow random fields, 2

although the advective flow velocity may have finite correlation time (thin-tailed correlation 3

structure, or short memory), the macrodispersion tensor will not be a constant, and, hence, the 4

stochastic transport will be non-Fickian. This was shown to be the case by detailed Monte Carlo 5

experiments on transport by time-space nonstationary but short memory Saint Venant’s 6

stochastic open channel flow by Liang and Kavvas (2008). This situation is very different from 7

the case of the underlying flow fields with long-memory (with heavy-tailed correlation 8

functions) but are stationary in time and space, as in the case of CTRW (Berkowitz et al. 2006). 9

It is also very different from the case of fractional advection-dispersion equation (fADE) with 10

constant drift and dispersion coefficients with fractional time and space derivatives that yield 11

long memory for solute concentrations. All these cases yield non-Fickian transport behavior but 12

for different reasons. 13

Within the above framework, what seems to be a desirable model of stochastic transport is an 14

ensemble average fractional advection-dispersion equation (fADE) which can yield the 15

appropriate memory structures both in time and space (by means of both fractional time and 16

fractional space derivatives), and which is also based on an underlying time-space nonstationary 17

stochastic flow field in fractional time-space which can take a long memory or short memory 18

structure, to be able to reproduce both the non-Fickian and Fickian transport behavior. In the 19

following, this study will attempt to develop such a model. 20

21


9

DERIVATION OF FRACTIONAL ENSEMBLE AVERAGE EQUATION OF TRANSPORT 1

BY TIME-SPACE NONSTATIONARY STOCHASTIC FLOW IN ADVECTIVE FORM 2

Over a 1-dimensional spatial lattice with increments x, let us consider a generalized random 3

walk of a solute particle in time with incremental steps of size t. Let the motion of this particle 4

start at time t=0 at the spatial location x(0) = x0. Let the spatial location of this particle at time 5

t+t be x(t+t). Let the concentration at time (t+t) of the ensemble of solute particles that start 6

their motions at a generic space-time origin (x0, 0), be denoted by 7

C(x(t+t), t+t | x0, 0). Let the probability density of finding the transported particle at some 8

spatial location at time t, given that it started its motion at the generic space-time origin (x0, 0), 9

be denoted by P(,t | x0, 0). Then for an initial solute mass of unity, this probability density is 10

equivalent to the ensemble average solute concentration at time-space location (,t) (Csanady, 11

1980). That is, 12

(8) 13

where the symbol < . > denotes the ensemble averaging operation. Hence, the goal of this study 14

is to develop evolution equations for the ensemble average solute concentration <C(,t | xo,0)> in 15

fractional time-space by means of the relationship (8) between the displacement probability 16

density P(, t | xo, 0) and the ensemble average solute concentration <C(, t | xo, 0)>. 17

For a generalized transport process, a particle can make a displacement over the 1-dimensional 18

lattice during a time interval (t, t+Δt) from a location x(t) = x(t+t) – kΔx, k = -…-1,0,+1,…+∞ 19

at time t =nΔt (assuming that it takes n time steps, each of size t, for the particle to reach time t 20

from the time origin t=0) , in order to reach a spatial location x(t+Δt) at time (t+Δt) = (n+1)Δt, 21


10

given the whole transport process has started at some time-space location (xo, 0). In other words, 1

during any time interval (t, t+t) the discrete particle displacement x(t+t) – x(t) = kx can start 2

from any location x(t) = x(t+t) – kx, k = - ,…-1,0,+1,…+∞ in order to reach x(t+t) at time 3

t+t. Accordingly, it is possible that the particle can make a very large displacement during the 4

time interval (t, t+t) in either of two directions. However, it is also possible that it can make a 5

very small displacement during this time interval. 6

Within this framework, denoting the probability of finding a particle over the discrete lattice at 7

location x(t+t) given that it started its motion at some space-time origin (xo, 0), by 8

9

the evolution equation for the displacement probability may be written as, 10

(9) 11

where is the probability that the particle will make a 12

displacement of size kΔx = x(t+t) – x(t) during interval (t, t+Δt) given that it was at location 13

x(t) = at time t. As such this probability varies not only with the displacement 14

size and direction, but also with the space-time origin of the displacement. As such, the random 15

walk, described by Eqn. (9) is non-stationary both in space and time. Also in Eqn. (9), 16

is the probability that the particle is at location x(t) = 17

at time t, given that the particle motion has started at ( . 18

Let y= x(t+t) so that Eqn. (9) may be re-written as, 19


11

1

(10) 2

+ (11) 3

Since we are considering transport in fractional time-space, Equation (11) can be expanded in 4

fractional time-space increments in terms of generalized Taylor’s series (Odibat and Shawagfeh, 5

2007; Osler, 1972). The generalized Taylor’s series for some function F(x) around z can be 6

expressed as (Odibat and Shawagfeh, 2007), 7

(12) 8

where (.) is the gamma function and the fractional derivative

that appears in Eqn. (12) is 9

a Caputo derivative ( Odibat and Shawagfeh, 2007; Podlubny, 1999). 10

Using Eqn. (12) one can expand Eqn. (11) around (y, t) to obtain, 11

12

13

(13) 14

15


12

where for k x(t+t) < x(t), and the solute particle displacement kΔx = x(t+t) - x(t) < 0 is in 1

the direction opposite to the main direction of the motion. Similarly, for k > 0, x(t+t) > x(t), and 2

the solute particle displacement kΔx = x(t+t) - x(t) > 0 is in the main direction of motion. The 3

right-sided Caputo fractional derivative

has been discussed in some detail in Podlubny 4

(1999). Essentially, the direction of the right-sided fractional derivative is opposite to the 5

direction of the standard derivative while the direction of the left-sided fractional derivative 6

is in the same direction as of the standard derivative. In Eqn. (13) k can take any integer 7

value randomly within (-∞ , + ∞). Hence, it is possible for the transported particle to make a 8

wide range of displacements, ranging from very large displacements to very small displacements 9

in two possible directions. 10

Hence, one can express Eqn. (13) as 11

12

13

(14) 14

where the terms in brackets <·> in Eqn. (14), raised to the power jβ, are the jβ-moments of the 15

particle displacement during (t, t+Δt) = (nΔt, (n+1)Δt). 16

In order to obtain transport equations that can be utilized in practice it is necessary to 17

approximate the Eqn. (14) which represents the fractional ensemble average transport in the 18

general setting, to some special cases. In the following, the fractional analogs of the purely 19


13

advective and advective-dispersive ensemble average transport equations shall be developed by 1

specializing the general and expansions in Eqn. (14) to the appropriately fixed and 2

powers in time and space. 3

In order to obtain the purely advective form of the fractional ensemble average transport 4

equation it is necessary to approximate Eqn. (14) to just -order in time and β-order in space. 5

Accordingly, retaining the leading terms on both sides of Eqn. (14) (approximating the 6

expansions to j=1), one obtains 7

8

(15) 9

10

+

(16) 11

Now recognizing, 12

, 13

and taking x(t) = x, one can define a time-space ensemble average fractional advective flow 14

velocity , considered only in the direction opposite to the main direction of flow 15

(reverse direction), by 16

, where x(t+t) < x(t), (17) 17


14

where may be interpreted as a “scaled” ensemble average advective flow velocity in 1

the sense that it is the expected scaled differential displacement per scaled differential time only 2

in the direction opposite to the main direction of flow. Similarly, one can define a time-space 3

ensemble average fractional advective flow velocity , considered only in the main 4

direction of flow, by 5

, where x(t+t) > x(t) . (18) 6

Now taking the limit as t on Eqn. (16) yields, 7

8

. (19) 9

Since the probability density P[x,t|x0,0] of solute particle displacement (given that it started at 10

some space-time location (x0,0)) to a location x at time t is the ensemble average concentration 11

of the solute <C(x,t|x0,0)> at space-time location (x,t) (Eqn.(8)) (Csanady, 1980), and since the 12

space-time origin (x0,0) is generic so that <C(x,t|x0,0)> may be represented as <C(x,t)> for 13

brevity, the Eqn.(19) above may be re-written in terms of the ensemble average concentration 14

as, 15

16

(20) 17


15

as the pure advection form of the fractional ensemble average equation of transport (fEATE) by 1

time-space nonstationary stochastic fractional advective flow in fractional time-space. This 2

equation is both the first-order moment and the first order cumulant expansion in terms of 3

since

are both the first moments and the first cumulants of the time-4

space nonstationary stochastic fractional advective flow velocity in the main flow direction and 5

in the reverse direction to flow. In Eqn. (20), the fractional derivatives are Caputo derivatives 6

(Odibat and Shawagfeh, 2007; Podlubny, 1999). 7

It is important to note that at any one location the fluid element is not moving in both the 8

upstream and the downstream directions. Instead, there is a certain probability for the element to 9

move upstream and certain probability to move downstream at any given time and space 10

location, as formulated in Equations (9) and (11) above. The case of flow in two directions at 11

two different sections of a river may happen in a river that is flowing over very mild slopes 12

toward a sea. During a high tide, the sea level may be significantly higher than the water surface 13

at the downstream end of the river, and the sea water may start flowing upstream within the 14

river. This water flow will meet the river flow that is coming from the upstream of the river, 15

somewhere along the river, forming a transition section in the river. Downstream-to-upstream 16

reverse open channel flow may also happen at the river junctions where the hydraulic head in the 17

main river may be higher than that of a joining branch. In such a situation, flow may go from the 18

junction location upstream within the river branch, to meet the flow that is coming from the 19

upstream section of the branch in the downstream direction, somewhere along the branch. These 20

situations correspond to the flow velocity components being both in the direction against the 21

main upstream-to-downstream direction of the river flow and in the direction of the main river 22

flow at two separate reaches of the same river. The defined fractional flow velocity in Eqn. (17) 23


16

accounts for such downstream-to-upstream flow conditions, while the fractional flow velocity in 1

Eqn. (18) accounts for the upstream-to-downstream flow conditions. However, in most real-life 2

cases the river flow will be only in the upstream-to-downstream main flow direction, and 3

will be zero, reducing the pure advection form of the fEATE to 4

. (21) 5

In order to gain more physical insight into Eqn. (20), let us consider this equation for the 6

standard integer case = = 1 of the pure advective transport. In this case Eqn. (20) takes the 7

form 8

9

, = 1 (22) 10

11

(23) 12

However, as mentioned earlier, is the ensemble average fractional advective 13

velocity only for the motion in the reverse direction to flow while is the 14

ensemble average fractional advective velocity only for the motion in the main direction of flow. 15

As such, if < (x,t)> represents the ensemble average fractional advective velocity for the 16

motion in both possible directions, then 17

< (x,t)> = -

(24) 18


17

or 1

< (x,t)> = -

(25) 2

But it also follows from Eqn. (23) that 3

= -

(26) 4

Then from Eqns. (22), (23), (24), (25) and (26) it follows that for the case of = 1 the form 5

of the purely advective transport becomes 6

(27) 7

where is the advective flow velocity in the standard integer power case. 8

Hence, Eqn. (20) is a physically realistic pure advection form of the fractional ensemble average 9

equation of transport by time-space nonstationary stochastic flow in fractional time-space, and 10

becomes identical in form to the classical advective transport equation when When 11

becomes zero for flow only in the upstream-to-downstream direction, arguments 12

similar to those above lead Eqn. (21) again to Eqn. (27) for 13

It is important to note that Equation (20) for two flow directions and Equation (21) for only the 14

upstream-to-downstream main flow direction are also the ensemble average mass conservation 15

equations for stochastic solute transport in fractional time-space. The above derivation is the 16

Lagrangian derivation of the ensemble average mass conservation equation for stochastic solute 17

transport in fractional time-space. 18

In Eqn. (20) if , then 19


18

, (28) 1

and 2

(29) 3

where is the reverse direction component of the standard advective flow velocity 4

of the integer case while is the main direction component of the advective 5

flow velocity of the integer case. Therefore, the right-hand-side of Eqn. (28) is the - 6

moment of the reverse direction component of the standard advective flow velocity while the 7

right-hand-side of Eqn. (29) is the - moment of the main flow direction component of the 8

standard advective flow velocity. This is an interesting result which gives the fractional 9

advection term a different meaning than the standard advection term of the integer time-space. 10

While in the integer time-space the ensemble average advection term for transport by time-space 11

nonstationary stochastic flow is in terms of the ensemble average of standard advective velocity 12

as seen from Eqn. (27) above, in the case of fractional time-space where the time and spatial 13

derivatives are of the same order (), the fractional ensemble average advection term for 14

transport is in terms of the -moments of the two possible directional components of the 15

standard advective velocity of integer time-space. 16

Hence, in the case , one may express the pure advection form of the fractional ensemble 17

average equation of transport (fEATE) by time-space nonstationary stochastic flow in fractional 18

time-space as, 19

20


19

(30) 1

in terms of the - moment of the reverse direction component of the standard advective flow 2

velocity and the - moment of the main flow direction component of the standard advective flow 3

velocity. In the mostly-encountered situation of upstream-to-downstream single flow direction, 4

the Eqn. (30) will reduce to 5

(31) 6

As shall be shown in the companion paper by Kim et al. (2014), unlike the standard integer form 7

of the advective transport equation where the solute displacement takes place as the displacement 8

of a distinct front (the piston movement), in the fractional form of the pure advective ensemble 9

average transport equation, due to the above explanation the solute may be spread out around the 10

moving solute front. 11

12

DERIVATION OF FRACTIONAL ENSEMBLE AVERAGE GOVERNING EQUATION OF 13

TRANSPORT BY TIME-SPACE NONSTATIONARY STOCHASTIC FLOW IN 14

ADVECTION-DISPERSION FORM: THE MOMENT FORM 15

In order to obtain a practical advection-dispersion form of the fractional ensemble average 16

transport equation that can be utilized in practice, it is necessary to approximate Equation (14) to 17

-order in time and 2β-order in space. Retaining the first term on the left-hand-side (LHS) and 18

first two terms on the right-hand side (RHS) of Eqn. (14) leads to 19

20


20

1

2

, (32) 3

or, 4

5

+

6

7

+

(33) 8

Then proceeding in the same way as in the above pure advection case, but in addition to defining 9

, also defining 10

(34) 11

where represents the dispersion for the fractional movements only in the reverse flow 12

direction, and 13

(35) 14


21

represents the dispersion for the fractional movements only in the main flow direction, and then 1

taking the limit as on Eqn. (33) yields a fractional Fokker-Planck-Kolmogorov equation 2

(FFPKE), 3

4

5

6

. (36) 7

Since the probability density P[x,t|x0,0] of solute particle displacement (given that it started at 8

some space-time location (x0,0)) to a particle location (x,t), is the ensemble average 9

concentration of the solute <C(x,t|x0,0)> at space-time location (x,t) (Eqn.(8)) (Csanady, 1980), 10

and since the space-time origin (x0,0) is generic so that <C(x,t|x0,0)> may be represented as 11

<C(x,t)> for brevity, the Eqn.(36) above may be re-written in terms of the ensemble average 12

concentration as, 13

14

15

, (37) 16

which is the moment form of the fractional ensemble average advection-dispersion equation 17

(fEAADE) of transport by time-space nonstationary stochastic flow in fractional time-space. In 18


22

Eqn. (37), there are both left-sided and right-sided fractional derivatives that are Caputo 1

derivatives (Odibat and Shawagfeh, 2007; Podlubny, 1999).It is important to note that in Eqn. 2

(37) the dispersion coefficients for the two possible movement directions (main and reverse 3

directions) are defined in terms of the second moment of the fractional differential displacement 4

(dx) per fractional differential time . As such, Eqn. (37) is the moment form of the 5

fractional ensemble average advection-dispersion equation. It is also important to note in Eqn. 6

(37) that when , then 2 As such for 0< the fractional dispersion term will 7

act as a fractional advection term. 8

In the mostly-encountered real-life situation of upstream-to-downstream single flow direction in 9

a river Eqn. (37) reduces to 10

11

, 12

13

, . (38) 14

as the fractional ensemble average advection-dispersion equation for nonstationary stochastic 15

flow only in the main flow direction. From the fractional advective term on the RHS of Eqn. (38) 16

it is seen that the ensemble average advective solute transport velocity is different from the 17

ensemble average advective flow velocity by

due to the effect 18

of dispersion. 19


23

For the special case of , Eqn. (37) takes the form, 1

2

, (39) 3

and Eqn. (38) takes the form 4

5

, (40) 6

7

For the special case when = 1, the moment form of the fractional ensemble average 8

advection-dispersion equation, Eqn. (37), becomes 9

10

+

(41) 11

From the definitions of and

in Eqns. (34) and (35) it follows that 12

=

=

=

(42) 13

However, is the dispersion only for the backward movements while

is 14

the dispersion only for the forward movements. As such, if represents the dispersion of 15

the totality of both forward and backward movements, then 16


24

= +

(43) 1

Combining Eqn. (43) with Eqns. (24), (25), (26) and (41) yields for = = 1, 2

(44) 3

as the ensemble average advection-dispersion equation of transport by nonstationary stochastic 4

flow for the integer power case. 5

In the mostly-encountered real-life situation of upstream-to-downstream single direction river 6

flow becomes zero. Accordingly, in this case 7

= , (45) 8

and one obtains from Eqn. (40) 9

(46) 10

as the ensemble average advection-dispersion equation of transport by nonstationary stochastic 11

flow in only the main flow direction for the integer power case. From the advective term on the 12

RHS of Eqn. (46) it is seen that the ensemble average advective solute transport velocity is 13

different from the ensemble average advective flow velocity due to the effect of dispersion. 14

As mentioned above, Eqns. (37) and (38) are the moment forms of the fractional ensemble 15

average advection-dispersion equation. As was recognized by Taylor as early as 1954, (Taylor, 16

1954), the cumulant form of the dispersion coefficient is the more appropriate form for a 17

transport equation since cumulants decrease in magnitude with increasing order. Meanwhile, in 18

the moment form of the transport equation, the moments increase in magnitude with the order 19


25

(Van Kampen, 1981). Therefore, in the following, the cumulant form of the fractional ensemble 1

average advection-dispersion equation of transport by time-space nonstationary stochastic flow 2

in fractional time-space will be explored. 3

4

DERIVATION OF FRACTIONAL ENSEMBLE AVERAGE EQUATION OF TRANSPORT 5

BY TIME-SPACE NONSTATIONARY STOCHASTIC FLOW IN ADVECTION-6

DISPERSION FORM: CUMULANT FORMS 7

Let us define: 8

(47) 9

and 10

(48) 11

Combining Eqns. (47) and (48) with Eqn. (14) yields, 12

13

(49) 14

In order to obtain a cumulant form of the advection-dispersion equation for the fractional 15

ensemble average transport that can be utilized in practice, it is necessary to approximate 16

Equation (14) to 2 -order in time and 2β-order in space. To 2 -order in time and 2 -order in 17

space Eqn. (49) reduces to, 18


26

1

β

β

. (50) 2

In the general case when , using the Leibniz rule for the Caputo derivative (Ishteva et al. 3

2007) and order analysis on Eqn. (50), as explained in Appendix I, one obtains a cumulant form 4

for the fractional Fokker-Planck-Kolmogorov equation (FFPKE) (Eqn.(I-4) in Appendix I), 5

6

7

8

(51) 9

for Similar to the previous cases, the probability density P[x,t|x0,0] 10

of solute particle displacement may be represented by the ensemble average concentration of the 11

solute <C(x,t)>. Accordingly, the Eqn. (51) above may be re-written in terms of the ensemble 12

average concentration as, 13

14

15


27

(52) 1

for and , as the cumulant form of the fractional ensemble average 2

advection-dispersion equation (fEAADE) of transport by time-space nonstationary stochastic 3

flow in fractional time-space for the general case . 4

In the mostly-encountered real-life situation of upstream-to-downstream single flow direction 5

Eqn. (52) reduces to 6

7

(53) 8

for and , as the cumulant form of the fractional ensemble average 9

advection-dispersion equation (fEAADE) of transport by time-space nonstationary stochastic 10

flow in fractional time-space for the general case . 11

For the case of , the Eqn. (52) reduces to, 12

13

(54) 14

for , as the cumulant form of the fractional ensemble average advection-dispersion 15

equation of transport by time-space nonstationary stochastic flow in fractional time-space when 16

. 17


28

In the mostly-encountered real-life situation of upstream-to-downstream single flow direction, 1

for Eqn. (52) reduces to 2

, 3

=

4

, (55) 5

as the cumulant form of the fractional ensemble average advection-dispersion equation of 6

transport by time-space nonstationary stochastic flow in the main flow direction in fractional 7

time-space when . From the fractional advective term on the RHS of Eqn. (55) it is seen 8

that the ensemble average advective solute transport velocity is a combination of the ensemble 9

average advective flow velocity with a term that is due to the effect of dispersion. 10

In Eqn. (54) 11

, (56) 12

and is the variance of the fractional differential displacements per fractional differential time in 13

the reverse direction to the main flow direction, while in Eqns. (54) and (55) 14

(57) 15

is the variance of the fractional differential displacements per fractional differential time in the 16

main flow direction. Accordingly, if one defines 17


29

as the variance of the fractional differential displacements per fractional differential time in both 1

of the possible flow directions, then 2

(58) 3

For 1, 4

5

(59) 6

Combining Eqns. (54), (24), (25) and (26) with Eqn. (59) yields for 1, 7

(60) 8

which is in a form similar to that of an ensemble average advection-dispersion equation for 9

transport by nonstationary flow in the integer power case. In the mostly-encountered real-life 10

situation of upstream-to-downstream single flow direction, 11

, (61) 12

and from Eqn. (55) one obtains 13

(62) 14

15


30

as the cumulant form of the ensemble average advection-dispersion equation for transport by 1

nonstationary stochastic flow in the main flow direction in the integer power case. 2

3

DISCUSSION AND CONCLUSIONS 4

In this study, starting from a general formulation of the time-space nonstationary random walk of 5

a solute particle, the fractional ensemble average governing equations of transport by time-space 6

nonstationary stochastic flow in fractional time-space were developed in terms of their moment 7

and cumulant forms. First, the purely advective form of the fractional ensemble average equation 8

of transport by time-space nonstationary stochastic advective flow velocity was developed, and 9

is given in Eqn.(20) (for flows in two possible directions) and Eqn.(21) (for flows only in the 10

main flow direction) for the general case when the orders of the time and space fractional 11

derivatives are different ( . From Eqns. (20) and (21) it is seen that the advection 12

coefficient is an ensemble average fractional advective velocity which is an explicit function of 13

space and time, and which is defined in Eqns. (17) and (18) as the expected fractional differential 14

displacement <(dx)> per fractional differential time at a particular space-time position 15

(x,t). <(dx)> may also be interpreted as the -moment of the differential displacement dx. As 16

such, this ensemble average fractional advective velocity varies in space and time and signifies 17

the time-space nonstationary nature of the stochastic advective velocity that transports the solute 18

particles. In the special case when the orders of the time and space fractional derivatives are the 19

same (then the ensemble average fractional advective velocity is expressed as the -20

moment of the corresponding standard advective velocity for the integer time-space, as shown in 21

Eqns. (30) and (31), respectively for two-directional and one-directional flows. This is an 22


31

interesting result which gives the fractional advection term a different meaning than the standard 1

advection term in the case of integer time-space. While in the integer time-space the ensemble 2

average advection term for transport by time-space nonstationary stochastic flow is in terms of 3

the ensemble average of standard advective velocity, as seen from Eqn. (4) above, in the case of 4

fractional time-space where the time and spatial derivatives are of the same order (), the 5

fractional ensemble average advection term for transport is in terms of the -moment of the 6

standard advective velocity of integer time-space. As such, while the ensemble average 7

advection term essentially describes a frontal behavior in integer time-space, the corresponding 8

ensemble average advection term in fractional time-space may describe a spreading of the front. 9

This behavior may have significant implications in the application of the fractional ensemble 10

average purely advective transport model to practical problems. As is shown by Kim et al. 11

(2014) in the companion paper to this paper, for transport cases of small dispersion the purely 12

advective fractional transport equation becomes a viable model of transport for both stationary 13

and nonstationary stochastic flows with fractional powers of the transport equation being less 14

than unity. 15

Next, the moment form of the fractional ensemble average advective-dispersive governing 16

equation of transport by time-space nonstationary stochastic flow in fractional time-space was 17

developed, and is given in Eqns. (37) and (38), respectively for two-directional and one-18

directional flows. The two coefficients that emerge in this equation are the ensemble average 19

fractional advective flow velocity which is an explicit function of space and time, and a 20

fractional dispersion coefficient which is also an explicit function of space and time 21

and is defined in terms of the second moment of the fractional differential displacement (dx)) 22

per fractional differential time . As such, Eqns. (37) and (38) are the moment forms of the 23


32

fractional ensemble average advection-dispersion equation, respectively for two-directional and 1

one-directional stochastic nonstationary flows. It can be seen from Eqn. (46) that for , the 2

derived moment form of the fractional ensemble average advection-dispersion equation (Eqn. 3

(38)) is in a similar form to the classical advection-dispersion transport equation. 4

As was recognized by Taylor as early as 1954, (Taylor, 1954), the cumulant form of the 5

dispersion coefficient is the more appropriate form for a transport equation since it is the exact 6

second order closure, not needing any information from the higher order cumulants that 7

generally decrease in magnitude with increasing order (Van Kampen, 1981). Therefore, next the 8

cumulant form of the fractional ensemble average advection-dispersion equation of transport by 9

time-space nonstationary stochastic flow was developed for two different cases: (i) when the 10

order of the fractional space derivative of the advective term is the same as that of the time 11

derivative while the order of the fractional space derivative of the dispersion term is twice that of 12

the time derivative, and (ii) when the orders of the time and space fractional derivatives are 13

completely different. In case (i) while the advection coefficient is essentially the same as in the 14

previous cases, the cumulant form of the dispersion coefficient, as given in Eqns.(54) and (55), 15

emerges as a time-space-dependent variance of the fractional differential displacement (dx) per 16

fractional differential time , as defined in Eqns. (56) and (57). Meanwhile, the dispersion 17

coefficient of the moment form of the fractional advection-dispersion equation, defined in Eqns. 18

(34) and (35), is the time-space dependent second moment of the differential displacement (dx) 19

per fractional differential time when = . The basic difference between the two 20

dispersion coefficients is that the one for the cumulant form is the second cumulant (variance) of 21

the fractional differential displacement per fractional differential time while the moment form is 22

the second moment of the fractional differential displacement per fractional differential time. As 23


33

will be seen in the accompanying paper by Kim et al. (2014) these forms can render very 1

different modeling results, with the cumulant form being generally superior. The cumulant form 2

of the fractional ensemble average advection-dispersion equation is in a form similar to that of 3

the standard advection-dispersion equation of transport. In case (ii) when the orders of the time 4

and space derivatives are completely different, the cumulant form of the fractional dispersion 5

coefficient emerges as a combination of the moment form of the fractional dispersion coefficient 6

and the square of an ensemble average fractional advective flow velocity. Hence, when 7

compared to the dispersion coefficient of the moment form, the dispersion coefficient of the 8

cumulant form, in the general case when the orders of the time and space derivatives are 9

completely different, contains an extra term accounting for the effect of the ensemble average 10

fractional advective flow velocity on dispersion. 11

In the accompanying paper by Kim et al. (2014), the numerical simulations of transport by 12

stationary and nonstationary flows under various memory lengths show the general superiority of 13

the cumulant form of the fractional transport equation over the moment form when the –14

fractional powers of the transport equation are less than unity. 15

As may be seen from their respective equations, the derived pure advection, and advection-16

dispersion forms of the fractional ensemble average governing equations of transport by time-17

space nonstationary stochastic flow in fractional time-space are rich in structure that can 18

accommodate both the non-Fickian and the Fickian behavior of transport under various memory 19

structures for both the underlying flow fields and the transport. The non-Fickian transport 20

behavior can be described by the fractional ensemble average transport equations, derived in this 21

study, either by means of the long memory in the underlying stochastic flow, or by means of the 22

time-space nonstationarity of the underlying stochastic flow, or by means of the time and space 23


34

fractional derivatives of the transport equations. These issues and the performance of the various 1

forms of the fractional ensemble average transport equations (fEATEs), developed here, are 2

explored in the accompanying paper by Kim et al. (2014). 3

4

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10

APPENDIX I 11

Differentiating both sides of Eqn. (50) with respect to , introducing the generalized 12

Leibniz rule for the Caputo derivative (Ishteva et al. 2007), and assuming Bj J=1,2, and P are 13

real, analytic functions with (Gardiner, 1985), 14

results in the equation, 15


40

(I-1) 1

2

Multiplying both sides of Equation (I-1) by –

, and then substituting into the 3

resulting equation the expression for

from the Eqn. (50), yields, 4

– Δ

Δ

β

β

β

β

β

β

β

β

5

Δ

β

β

β

β

β

β

β

β

6

Δ

β

β

β

β

-

Δ

β

β

β

β

7

Δ

β

β

β

β

β

β

β

β

Δ

β

β

β

β

β

β

β

β

Δ

β

β

β

β

Δ

β

β

β

β

. 8

(I-2) 9


41

Again using the Leibniz rule for the Caputo derivative (Ishteva et al. 2007) on Equation (I-2), 1

along with the definitions (47) and (48) for for j=1,2, and taking the limit as t goes to 2

zero yields 3

– Δ

β

β

4

β

β

5

β

β

β

β

β

β

β

β

β

β β

β β

β β

β (I-3) 6

Combining Equation (I-3) with Equation (50) of the main text while taking the limit as t goes to 7

zero, and introducing the definitions of the fractional ensemble average advective velocity 8

and the fractional dispersion coefficient

of the main text yields the cumulant 9

form of the fractional Fokker-Planck-Kolmogorov Equation (FFPKE) for transport by 10

nonstationary stochastic flow in fractional time-space as 11

12

13

14

(I-4) 15


42

for Equation (I-4) is the Equation (51) in the main text. 1

2

3

4

5

6

7

Fractional ensemble average governing equations of transport by nonstationary stochastic fractional advective velocity and fractional dispersion

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