Journal of Hydrology 519 (2014) 1792–1803
Contents lists available at ScienceDirect
Journal of Hydrology
journal homepage: www.elsevier.com/locate/jhydrol
Mass-time and space-time fractional partial differential equations
of water movement in soils: Theoretical framework and application
to infiltration
Ninghu Su ⇑
School of Mathematics and System Science, Shenyang Normal University, Shenyang, Liaoning 110034, China
College of Science, Technology and Engineering, James Cook University, Cairns, Queensland 4870, Australia
a r t i c l e
i n f o
Article history:
Received 3 February 2014
Received in revised form 3 September 2014
Accepted 9 September 2014
Available online 18 September 2014
This manuscript was handled by
Konstantine P. Georgakakos, Editor-in-Chief,
with the assistance of Ana P. Barros,
Associate Editor
Keywords:
Water movement in soils
Mass-time and space-time fractional partial
differential equations
Swelling and non-swelling soils
Infiltration
Mobile and immobile zones
Transport exponent
s u m m a r y
This paper presents mass-time fractional partial differential equations (fPDEs) formulated in a material
coordinate for swelling–shrinking soils, and space-time fPDEs formulated in Cartesian coordinates for
non-swelling soils. The fPDEs are capable of incorporating mobile and immobile zones or without immobile zones. As an example of the applications, the solutions of the fPDEs are derived and used to construct
equations of infiltration. The new equation of cumulative infiltration into soils with mobile and immobile
b þb 1=ð2k1Þ
, where A is the final infiltration rate, S is the sorptivity which differs
zones is IðtÞ ¼ At þ S S ttb22þS1 tb1
1
infiltration without an immobile zone is IðtÞ ¼ At þ St b=ð2k1Þ , where b is the order of temporal fractional
derivatives. Published data are used to demonstrate the use of the new equations and derive the parameters. The transport exponent for soils with mobile and immobile zones is l ¼ 2ðb2 þ b1 Þ=k, and l ¼ 2b=k
for soils without an immobile zone. The transport exponent is the criteria for defining flow patterns: for
l < 1, the flow process is sub-diffusion as compared to l = 1 for classic diffusion and l > 1 for
super-diffusion.
Ó 2014 Elsevier B.V. All rights reserved.
1. Introduction
In this paper we present mass-time fractional partial differential equations (fPDEs) of water movement formulated in a material
coordinate for swelling–shrinking soils, and space-time fPDEs of
water movement formulated in Cartesian coordinates in
non-swelling soils. We demonstrate the connection between the
mass-time and space-time fPDEs and the continuous-time random
walk (CTRW) theory for water flow in soils. We also investigate the
interrelationships between the parameters in the fPDEs based on
material balance, the CTRW theory and the fPDEs derived based
on the mass-time (or space-time) scaling laws (Zaslavsky, 2002).
Following a detailed discussion of the mechanisms represented
⇑ Address: College of Science, Technology and Engineering, James Cook University, Cairns, Queensland 4870, Australia. Tel.: +61 7 4232 1551; fax: +61 7 4232
1284.
E-mail address: [email protected]
http://dx.doi.org/10.1016/j.jhydrol.2014.09.021
0022-1694/Ó 2014 Elsevier B.V. All rights reserved.
2
between swelling and non-swelling soils, b2 and b1 are orders of temporal fractional derivatives for
mobile and immobile zones, respectively, k is the order of spatial fractional derivatives, and S2 and S1
are parameters incorporating relative porosities and b2 and b1, respectively. The equation of cumulative
by the fractional models, we develop solutions of the fPDEs, and
use them to construct new equations of infiltration into swelling
and non-swelling soils.
The current presentation is a step forward from the previous
findings of fPDEs for water movement formulated in a material
coordinate in swelling porous media such as absorption (Su,
2009a) and infiltration (Su, 2010, 2012). The previous models were
formulated for both single porosity media for one-phase flow (Su,
2010) and two-phase flow (Su, 2009b), and dual porous media with
mobile and immobile zones (Su, 2012) where a constant diffusivity, D, and a linear hydraulic conductivity, K, were used. For infiltration and absorption into a saturated surface, a constant diffusivity
represents reasonable approximations to the process with the
boundary condition modelling sudden saturation. Those equations
presented earlier provide new insights into the physics of fluid
flow in swelling soils with or without an immobile zone.
In this paper, we consider the following major issues in the new
forms of the fPDEs for soil water movement:
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N. Su / Journal of Hydrology 519 (2014) 1792–1803
(1) Different forms of fPDEs are presented. A mass-time fPDE is
presented for swelling soils and a space-time fPDE for nonswelling soils. In the non-swelling case, the fPDE can be called
the fractional Richards’ equation (fRE), compared to the
diffusion component of a fRE presented by Sun et al. (2013).
(2) Power functions for the diffusivity and hydraulic conductivity are incorporated in the fPDEs. These inclusions allow the
diffusivity and hydraulic conductivity to vary with the moisture ratio. The swelling–shrinking properties of the porous
media are based on the material coordinate (Smiles and
Rosenthal, 1968; Smiles, 1974; Philip, 1969a, 1986, 1992a;
Smiles and Raats, 2005; Su, 2009a, 2009b, 2010, 2012).
(3) The mobile and immobile zones in soils are optional. The
mobile–immobile concept, which was first proposed by
Barenblatt et al. (1960), (see Gao et al., 2010 for a review),
is taken into account in both the mass-time and space-time
fPDEs. The fPDEs are also formulated for soils without
considering immobile zones as an optional simplification.
(4) The connection between the mass-time fPDE and CTRW
theory for soils is established. The mass-time fPDE is an
extension to the space-time fPDE. The fPDE is the consequence of the long-time limit or asymptotic result of the
two probability density functions of the CTRW model with
power-law waiting times and power-law jumps (Zaslavsky,
2002; Gorenflo and Mainardi, 2005; Gorenflo et al., 2007;
Meerschaert, 2011). These distributed-order fPDEs, with a
drift term due to convection added, form the mass-time fractional diffusion and advection equation (fDAE) and the
space-time fDAE. This connection offers another option for
deriving the fPDE-based models without resorting to the traditional mass balance method for their derivation.
(5) The fPDE and the two fractional derivatives are discussed in
the context of mass-time and space-time scaling laws.
Detailed discussions of the connection between the CTRW and
the fPDE can be found in the literature (Gorenflo and Mainardi,
2005, 2009, 2011; Gorenflo et al., 2007; Meerschaert, 2011;
Tejedor and Metzler, 2010; Uchaikin and Saenko, 2003;
Zaslavsky, 2002), here we only briefly discuss the connection
between the distributed-order fPDE and CTRW for the benefit of
readers in hydrology and soil physics.
The CTRW concept interprets the motion of a particle in a
sequence of two states, the jump and the waiting preceding the
next jump. Each jump length and the waiting time are independent
random variables, and each probability is independently identically distributed (iid) (Gorenflo et al., 2007; Tejedor and Metzler,
2010). The iid positive waiting times are denoted by T1, T2, T3, . . .,
each having the same probability density function (pdf), /
(t), t > 0, and the iid random jumps are denoted by X1, X2, X3, . . .
in a real domain, R, each having the same pdf w(x), x e R. With
these definitions, the probability density of the particle (or water
parcel) movement in the soil is p(x, t), which is represented by
the following series (Gorenflo et al., 2007, p. 89–90),
pðx; tÞ ¼ WðtÞdðxÞ þ
1
X
v n ðtÞwn ðxÞ
ð1Þ
n¼1
where d(x) is the Dirac delta function, and W(t) is the survival function given by
WðtÞ ¼
Z
1
/ðt0 Þdt
0
ð2Þ
t
with vn(t) and wn(x) being repeated convolutions in time and space,
respectively, i.e., /(t) vn(t) = (W ⁄ /⁄n)(t), and wn(x) = (w⁄n)(t).
Starting from Eq. (1), Gorenflo et al. (2007, Eq. (5.18)) and
Gorenflo and Mainardi (2011, Eq. (71)) show that the Laplace–
Fourier transform of Eq. (1) results in the following expression,
^~ ðj; sÞ ¼
u
Z
1
h
x
exp t jjjc i
sign
j
i
sb1 exp t sb dt ð3Þ
0
New equations of infiltration into swelling and non-swelling
soils are presented based on the solutions of the new fPDEs with
examples illustrated using published data collected in the field by
Talsma and van der Lelij (1976). The equations of infiltration without immobile zones are presented as simpler options. These equations of infiltration derived here are examples for the use of the
new fPDEs. The new fPDEs can be applied to investigate other flow
processes such as redistribution and water movement at different
times and locations with different initial and boundary conditions.
The new formulations explain different mechanisms of anomalous fluid dynamics in natural porous media by including masstime fractional derivatives in the fPDE, and these new models have
more realistic features and properties of the flow in soils, particularly when the soils are treated as swelling–shrinking porous media.
2. The CTRW concept and its connection with the fPDEs for
water flow in soils
The connection between the CTRW concept and anomalous
transport process has been extensively investigated (Zaslavsky,
2002; Gorenflo and Mainardi, 2005; Gorenflo et al., 2007), and is
understood in the framework of the classical renewal theory (Cox,
1967). The CTRW theory models particle motion with two probabilities for the two stages of random particle movements: one probability relates to the motion length and the second to the waiting
time of the particles before the next movement. When the motion
of water-driven solute particles in mobile–immobile zones of
porous media is explained successfully using the CTRW concept
(Berkowitz et al., 2006; Dentz and Berkowitz, 2003; Schumer
et al., 2003), the movement of water itself should be logically
investigated using the CTRW concept because water is the original
driving force for the motion of solute particles in the subsurface.
where j and s are the Fourier and Laplace transform variables,
respectively; b is the exponent for the probability of the waiting
time intervals between two consecutive steps; c is the exponent
for the probability of the length of steps for the random walks
(Zaslavsky, 2002, p. 497–500), and x is the skewness acting on
the space variable, |x| 6 min {c, 2 c}. Eq. (3) can be written
Gorenflo and Mainardi (2005)
^~ ðj; sÞ ¼
u
sb1
sb
ð4Þ
x sign j
þ jjjc i
For the symmetrical case of x = 0, Gorenflo and Mainardi (2005,
2009) show that Eq. (4) is the Laplace–Fourier transform of the following fractional diffusion-wave equation (fDWE)
@ b uðx; tÞ @ c uðx; tÞ
¼
;
@xc
@tb
uðx; 0Þ ¼ dðxÞ
ð5Þ
where b is here called the order of the time (or temporal) fractional
derivatives, and c the order of the space (or spatial) fractional
derivatives. The Delta function in the initial condition of Eq. (5)
can be incorporated in the fDWE (Zaslavsky, 2002) resulting in
the following form of the fDWE,
@ b uðx; tÞ @ c uðx; tÞ
tb
¼
þ
dðxÞ
b
@xc
Cð1 bÞ
@t
ð6Þ
The fDWE in Eq. (5) results from the asymptotic or long-time
approximation of the CTRW model with the two transitional probability distribution functions for the length of jumps, P(X > x), and
waiting time intervals, P(J > t), obeying power laws, i.e.,
P(X > x) xc, and P(J > t) tb (Meerschaert, 2011, p. 273–274).
Eq. (6) can be written in a different form
b
t D uðx; tÞ
¼ x Dcx uðx; tÞ;
uðx; 0Þ ¼ dðxÞ
ð7Þ
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N. Su / Journal of Hydrology 519 (2014) 1792–1803
where the left hand side of Eq. (7) is the Caputo fractional derivative
with respect to time, t, while the right hand side is the Riesz–Feller
fractional derivative with respect to space, x. Note that the Riesz–
Feller fractional derivative becomes the Liouville fractional derivative for x = ±c with the positive sign for the forward fractional
derivative and the negative sign for the backward fractional derivative (Ortigueira and Trujillo, 2012, p. 5155–5156).
To derive Eqs. (5) or (6), many reports present the case for
0 < b 6 1 and/or 0 < c 6 2. However, Barkai (2002) and Zaslavsky
(2002, p. 499–511) show that the restriction of 0 < b 6 1 is unnecessary. For example, the case of 1 < b 6 2 and 0 < c 6 2 corresponds
to anomalous transport with two-time scales (Becker-Kern et al.,
2004; Baeumer and Meerschaert, 2007) and the scaling limit of a
decoupled CTRW (Caceres, 1986; Meerschaert et al., 2010), which
is an important property relevant to the two-term distributedorder fPDE for flow in soils to be discussed below.
The above presentation means that in the Laplace–Fourier
domain the CTRW model is identical to the fDE in Eq. (5). In other
words as shown by Zaslavsky (2002, p. 497–501), Baeumer et al.
(2005), and Gorenflo et al. (2007, p. 88–92), under the asymptotic
or long-time limit condition the process modelled by a CTRW converges to a simpler function whose probability density solves the
fDWE. This connection provides an alternative approach for deriving the fDWE without resorting to the classic mass balance method
for its derivation in addition to the renormalization group of kinetics (Zaslavsky, 2002, p. 507–508) which also yields the fDWE.
The CTRW concept has several versions under different assumptions regarding the dependence or correlation between the two
successive jumps and waits as well as the relationships between
jumps and waits. There are CTRW models with correlated waiting
times or correlated jump lengths (Chechkin et al., 2009; Tejedor
and Metzler, 2010), and coupled CTRW with the sum of random
jump lengths dependent on the random waiting times immediately preceding each jump (Schumer et al., 2011) or the delay
between particle jumps affecting the subsequent jump magnitude
(Meerschaert et al., 2002) etc. For more background on CTRW models and their applications, the readers are referred to Klafter et al.
(1987), Balescu (1995), Zaslavsky (2002), Berkowitz et al. (2006),
Gorenflo and Mainardi (2005, 2009, 2011), Gorenflo et al. (2007)
and Meerschaert (2011).
Now it is seen that the two parameters, b and k, have three meanings: they are (1) the orders of temporal fractional derivatives and
spatial fractional derivatives, respectively, in the fPDE; (2) the exponents for the two probability density functions in the CTRW model,
and (3) the critical exponents characterising the fractal structures of
space-time fractals (Zaslavsky, 2002, p. 507; Meerschaert, 2011). As
such these two parameters span and connect the three distinctive
fields of fractional calculus, stochastic/probability, and fractal
geometry. This connection is very valuable for evaluating these
parameters using different methods in these fields.
This paper presents the fPDE and CTRW concepts to explain the
stochastic process of water movement resulting from fluctuations
during its movement (Timashev et al., 2010) in soils and derive
solutions of the fPDEs and demonstrate their applications to infiltration as examples.
defined as h = hl/hs, with hl and hs being the volume fractions of
liquid and solid, respectively.
With the transformation in Eq. (8), and the previous analysis
(Su, 2010, 2012), the physical coordinate x in Eq. (7) is replaced
by the material coordinate defined by Eq. (8) and u replaced by h
to yield
b
t D hðm; tÞ
¼
c
m Dx hðm; tÞ;
hðm; 0Þ ¼ dðmÞ
ð9Þ
which results from the analogue to the long-time result of the
continuous time random walks on non-swelling media. The classic
CTRW model does not consider the change of the shapes (or mass)
when a liquid is introduced in the media. The connection between
the CTRW concept through Eqs. (3) and (5) and further with Eq. (9)
means that the mass-time fPDE in Eq. (9) can now be interpreted
as the long-time limit of the continuous time random walks in
swelling soils.
Let us further discuss the distributed-order time fPDE for water
movement in swelling–shrinking soils with mobile and immobile
zones (Su, 2012, Eq. 10), which is a counterpart for the motion
of water-driven solutes in mobile–immobile zones of porous
media as explained using the CTRW concept by Schumer et al.
(2003),
b1
@ b1 h
@ b2 h
@
@h
dKm ðhÞ @h
ðcn a 1Þ
Dm ðhÞ
þ b2 b2 ¼
b1
@m
@m
dh @m
@t
@t
ð10Þ
where m is the material coordinate defined in Eq. (8); t is time; b1
and b2 are the relative porosities in immobile and mobile zones,
respectively, i.e., b1 ¼ //im and b2 ¼ //m with /im, /m and / being the
porosities in the immobile and mobile zones, and total porosity,
respectively; b1 and b2 are the orders of fractional derivatives for
immobile and mobile zones, respectively; cn is the particle specific
gravity; a is the gradient (or slope) of the shrinkage curve, which is
a ratio on the graph of the specific volume, v, versus water content
or moisture ratio, h; Dm(h) is the material diffusivity given by
Dm ðhÞ ¼
K m ðhÞ dU
1 þ h dh
ð11Þ
with U being the unloaded matrix potential, and Km(h) being the
unsaturated material hydraulic conductivity defined as (Smiles
and Raats, 2005, Eq. (29), for a negative sign in their Eq. (28))
K m ðhÞ ¼
km
ðc a 1Þ
hs n
ð12Þ
with km being the saturated material hydraulic conductivity (Smiles
and Raats, 2005) given as
km ¼ KðhÞhs
ð13Þ
where K(h) is the conventional unsaturated hydraulic conductivity.
While the diffusivity and hydraulic conductivity are defined
above, in practice, empirical formulae are often used to relate the
diffusivity and hydraulic conductivity to the moisture ratio. In
particular, power functions are often used for the diffusivity
(Philip, 1992b),
Dm ðhÞ ¼ D0 hb
ð14Þ
and the hydraulic conductivity
3. The mass-time fPDEs for water movement in swelling soils
K m ðhÞ ¼ K 0 hk
We start with the material coordinate, m, which is defined by
(Smiles and Rosenthal, 1968; Philip, 1969a)
where D0, b, K0 and k are constants, which need to be determined
experimentally. With Eqs. (8), (14) and (15), Eq. (10) is written in
the following form by introducing the mass fractional derivative,
m¼
Z
z
ð1 þ eÞ1 dz1
ð8Þ
0
where z is the conventional space coordinate in a vertical direction,
and e is the void ratio given by e = h/(1 h); h is the moisture ratio
ð15Þ
"
#
@ b1 h
@ b2 h
D0
@ @ g hbþ1
@h
ðcn a 1ÞK 0 khk1
b1 b þ b2 b ¼
@m
b þ 1 @m @mg
@t 1
@t 2
ð16Þ
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N. Su / Journal of Hydrology 519 (2014) 1792–1803
where 0 < g 6 1, which is the mass-time distributed-order fractional advection–dispersion equation (fADE) or multi-term fADE
(Liu et al., 2013) for water movement in swelling soils with the
material coordinate being dependent on the moisture ratio.
Eq. (16) is based on the CTRW theory with a distributed-order
time fractional derivative due to the two time-scaling property
and convection due to a shift jump size distribution in the CTRW
theory (Zhang et al., 2009, p. 577). The connection between CTRW
and the distributed-order fDWE has been established by Chechkin
et al. (2002), which is also applicable to Eq. (16) – this is an extension of the CTRW theory and fPDE to swelling soils.
Note that different terminologies appear in the literature for the
flow patterns when the orders of fractional derivatives, b1 and b2,
in Eq. (10) take different values. Our case here with 0 < b1 6 2
and 0 < b2 6 2 is similar to the concept of a decoupled CTRW
(Meerschaert et al., 2010) in the two time-scaling case, and also
similar to the alternative derivation by Zaslavsky (2002) where
b1 and b2 are not restricted. In the above formulation the relationships b1 < b2 and b1 < b2 hold because for b1 = b2 there is no need to
distinguish between the mobile and immobile zones. For 0 < b1 6 2
and 0 < b2 6 2 the fPDE can be solved using integral transform
methods (Debnath, 2003; Mainardi et al., 2001) in fluid mechanics
in addition to numerical methods.
The two-term distributed-order fPDEs are special forms of the
variable-order fPDEs. According to Gorenflo and Mainardi (2005),
it was Caputo (1969) who developed the idea of distributed-order
differential equations with a distribution function for b being used.
Algebraic functions are also reported for b (Jacob and Leopold,
1993; Lorenzo and Hartley, 2002). Here we use the two-term distributed-order fPDE (Gorenflo and Mainardi, 2005) to model different flow patterns in different pores or different levels of flow
patterns in the same types of pores. Another option is to use different values for the fractional orders (Hahn and Umarov, 2011) at
different stages for changing diffusion patterns from one stage
described by b1 to the next stage by b2.
As to the forms for the diffusivity and hydraulic conductivity,
Philip (1960a, 1960b) reported a very large set of diffusivity
functions that yield exact solutions of concentration-dependent
diffusion. With those different functions for the diffusivity, different solutions can be derived for Eqs. (10) and (16). For example,
Gerolymatou et al. (2006) used an exponential function in the fractional diffusion equation to model water absorption. We choose Eq.
(14) in particular because the diffusivity as a power function
bridges the diffusion equation and the nonlinear porous media
equation (Vazquez, 2007), which is characterised by the term
h bþ1 i
@
@h
. Due to the functional relationship in Eq. (11) between
@m
@m
the diffusivity and hydraulic conductivity, a power function is also
used for the hydraulic conductivity.
As a counterpart of the space fractional derivative in terms of
Riemann–Liouville fractional derivatives (Voller, 2014, p. 272),
the following identity (Podlubny, 1999, p. 59)
@ @ g hbþ1
@m @mg
!
¼
@ gþ1 hbþ1
@mgþ1
ð17Þ
@ b1 h
@ b2 h
D0 @ k hbþ1
@
þ b 2 b2 ¼
ðcn a 1Þ
ðK 0 hk Þ
b1
@m
b þ 1 @mk
@t
@t
9
0 < b1 6 2; 0 < b2 6 2 >
=
0 < k 6 2; k ¼ g þ 1
>
;
b1 þ b2 ¼ 1
ð20Þ
0 < b 6 2;
ð21Þ
0<k62
The parameters and the material coordinate in Eqs. (18) and
(20) characterise the different flow patterns in swelling soils with
mobile and immobile zones which have a variable material diffusivity, Dm(h), and hydraulic conductivity, Km(h) as functions of the
moisture ratio. The parameters in the two functions, namely, D0,
b, K0 and k, may also be functions of mass or directions when the
soils are heterogeneous and/or anisotropic.
4. The space-time fPDEs for water movement in non-swelling
soils
With power functions for the diffusivity, D(h), and hydraulic
conductivity, K(h), as in Eqs. (14) and (15), the fPDE for nonswelling soils can be derived as
b1
@ b1 h
@ b2 h
D0 @ k hbþ1 @
þ b2 b ¼
ðK 0 hk Þ
b1
2
@z
b
þ 1 @zk
@t
@t
0 < b1 6 2;
0 < b2 6 2
ð22Þ
ð23Þ
b1 þ b2 ¼ 1; 0 < k 6 2
where z is the usual physical coordinate. Eq. (22) can be extended to
two and three dimensions for non-swelling soils,
b1
@ b1 h
@ b2 h
@
g bþ1
þ b2 b ¼ @x@ D0 @ @xh g ðK 0 hk Þ
b1
i
@z
i
@t
@t 2
0 < b1 6 2;
0 < b2 6 2
b1 þ b2 ¼ 1;
0<g61
ð24Þ
ð25Þ
where i = 1, 2, 3 is the number of dimensions, and D0, b, K0 and k
may vary with space or directions in different dimensions.
Similar to Eq. (20) for swelling soils without immobile zones,
Eq. (24) with b1 = 0 and b2 = b can be simplified for non-swelling
soils without immobile zones,
@bh
@
@
@ g hbþ1
ðK 0 hk Þ
¼
D
g
0 @x
@z
i
@t b @xi
0 < b 6 2; 0 < g 6 1
ð26Þ
ð27Þ
Compared to the classic advection–diffusion equation (ADE),
the space- and time-fractional models are shown to better represent the first and second moments of solute transport at early
times and the tails of tracer plumes (Zhang et al., 2009, p. 573).
As water is the carrier of the solutes, the fractional model is then
expected to perform better than its classic counterpart for water
flow including infiltration into soils, particularly at early and final
stages of infiltration processes at the soil surface.
5. Water exchange between mobile and immobile zones
We continue to use the fractional model proposed for the water
exchange between the mobile and immobile zones (Su, 2012, p. 4),
for fractional mass derivatives enables Eq. (16) to be written
b1
@bh
D0 @ k hbþ1
@
¼
ðcn a 1Þ
ðK 0 hk Þ
b
@m
b þ 1 @mk
@t
ð18Þ
ð19Þ
Without considering immobile zones in soils, i.e., b1 = 0 as
b1 + b2 = 1, and b1 = 0, and by writing b2 = b, Eq. (18) simplifies to
b1
@ b3 him
¼ -ðhm him Þ
@t b3
ð28Þ
where b3 is the order of the fractional derivative for water mass
transfer between the mobile and immobile zones; - is the mass
transfer rate between the mobile and immobile zones, and b1 is
defined earlier. The solutions of Eq. (28) subject to two types of
initial conditions have been presented earlier (Su, 2012, p. 7), which
are directly applicable here.
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N. Su / Journal of Hydrology 519 (2014) 1792–1803
6. The transport exponent as the criterion for the types of water
movement in soils
Eqs. (18) and (22)are reminiscent of space-time fPDEs for solute
transport in porous media as reviewed by Zhang et al. (2009, Eqs.
(8) and (A5)). As with fractional models of solute transport in porous media, the time- and space-fractional derivative models for
water movement account for non-Fickian flow processes. The
time-fractional derivatives in the model account for partitioning
of water parcels on sticky porous surfaces resulting in slowing
processes (sub-diffusive) while the space-fractional derivatives
describe the flow processes in the media with higher velocity flow
paths of long spatial correlation leading to super-diffusion. The
resulting pattern of these two competing processes is measured
by the transport exponent, which is given by Zaslavsky (2002, p.
493) in the fDWE as
l ¼ ð2b=kÞ
ð29Þ
Analogous to Eq. (29) for fractional diffusion, the transport
exponent for the distributed-order mass-time or space-time fPDE
should be
l ¼ 2ðb2 þ b1 Þ=k
ð30Þ
The transport exponent applies to both swelling and non-swelling soils, which may have or without immobile zones. It can be
used to identify the pattern of diffusion: l < 1 is for sub-diffusion,
l = 1 for classic diffusion, and l > 1 for super-diffusion (Zaslavsky,
2002, p. 504–505). The classification using the transport exponent
is very different from that for time-fractional PDEs which is
measured by the temporal fractional order of derivatives only.
7. The scaling in the mass-time and space-time fPDEs for water
flow in soils
With the concept of the renormalisation group of kinetics
(RGK), and the space and time scaling parameters (Zaslasvky,
2002, p. 507–509) the space-time fPDE in Eq. (5) can be written
@ b hðx; tÞ
¼
@tb
c
Ll
!n
Lbt
@k
ðDhðx; tÞÞ
@xk
ð31Þ
where Ls and Lt are the space and time scaling parameters, respectively; n is the number of renormalisation transformation, and D is
the diffusivity.
When the space-time fPDE in Eq. (5) ‘‘survives’’ the RGK
transformation and if the condition limn!1 ðLs =Lt Þ ¼ 1 is met, the
following relationship results
b ln Ls l
¼
¼
k ln Lt 2
ð32Þ
which connects the orders of fractional derivatives and the scaling
parameters, Ls and Lt,
l¼
2 ln Ls
ln Lt
ð33Þ
The moment equation for Eq. (5) with the diffusivity included is
given by (Zaslavsky, 2002, p. 508)
hxk i ¼
Cð1 þ kÞ b
Dt
Cð1 þ bÞ
ð34Þ
Following the above relationships for non-swelling soils, we
map the RGK results for flow in swelling soils as
@ b hðm; tÞ
¼
@t b
Lkm
Lbt
!n
@k
ðDhðm; tÞÞ
@mk
ð35Þ
and
b ln Lm l
¼
¼
k ln Lt
2
ð36Þ
where Lm is the mass scaling parameter. Similar to Eq. (32), Eq. (36)
yields the following entities
l¼
2 ln Lm
ln Lt
ð37Þ
and
b ln Lm
¼
k ln Lt
ð38Þ
which establishes the connection between b and k and the scaling
parameters for swelling soils. Similarly the moment for flow in
swelling media is
hmk i ¼
Cð1 þ kÞD b
t
Cð1 þ bÞ
ð39Þ
8. Mechanisms represented by the fractional models explained
In Sections 2, 6 and 7, we have briefly discussed what the timeand space-fractional PDEs imply in terms of the CTRW theory in
the context of stochastic particle movement, and anomalous diffusion as well as the mass-time (or space-time) scaling, respectively,
which are unified in the form of an fPDE for the same mechanics of
flow in porous media.
The CTRW approach explains the partitioning of water parcels
on sticky porous surfaces which results in the time-fractional property in the PDE while the speeding flow on its paths leads to the
space-fractional property in the PDE. These competing processes
parameterised by the fractional time and space derivatives, b and
k, in the PDE determines whether the transport process is subdiffusion, super-diffusion or classic diffusion, which can be measured by the transport exponent, l ¼ ð2b=kÞ. The two parameters
b and k are also ‘‘the critical exponents’’ that characterise the fractal structures of the mass-time (or space-time) process (Zaslavsky,
2002, p. 507). Furthermore it is seen that the transport exponent is
also connected to the scaling parameters by l ¼ 2lnlnLLtm as in Eq. (33)
for the space-time and Eq. (37) for the mass-time fPDE.
The two fractional derivatives in the fPDEs are a consequence of
the non-locality property of the transport processes in heterogeneous media (Morales-Casique et al., 2006; Benson et al., 2013, p.
479; Zhang et al., 2009, p. 561–562). Physically, the term nonlocality is used to explain that the concentrations of a tracer (or
moisture ratio or any other quantities) at previous times and/or
larger upstream locations contribute to the variation of the concentration at the point of observation due to the uncertain velocity
field, and/or physicochemically due to reactions such as absorption/desorption. The spatial non-locality means that the concentration change at the point of observation depends on upstream
concentrations while the temporal non-locality implies that the
concentration change at the point of observation depends on the
prior concentration loading (Benson et al., 2013).
Mathematically, the time- and space-fractional PDEs such as
Eqs. (20) and (26), when the drift term is included, describe the
non-local, non-classic advection–dispersion in soils, which account
for the history- and scale-dependent process. Numerically, the
non-classic diffusion can be measured by the moment i.e.,
ð1þkÞ
ð1þkÞ
hmk i ¼ CCð1þbÞ
Dt b in Eq. (39) for swelling soils and hxk i ¼ CCð1þbÞ
Dt b
in Eq. (34) for non-swelling soils. For classic diffusion, k ¼ 2 and
b = 1, the classic moment is recovered from Eqs. (34) and (39).
Examples of the effects on the flow in soils by the orders of
fractional derivatives can be seen in Pachepsky et al. (2003) who
demonstrate the variability of b – 1 for k ¼ 2 in a time-fractional
1797
N. Su / Journal of Hydrology 519 (2014) 1792–1803
diffusion equation for moisture movement, and Voller (2011) who
shows the effect of spatial fractional derivatives.
While the fractional calculus-based approach is explained here,
non-fractional models with functional parameters can also account
for heterogeneity such as the scale-dependent process. Voller (2011,
@
p. 260–261) shows that the non-fractional PDE @z
KðzÞ @h
¼ 0 with
@z
KðzÞ ¼ 1k Cð1 þ kÞz1k yields an identical result for the wetting front
derived using a fractional model under the same boundary conditions. This fact indirectly justifies the approach to modelling solute
transport processes in the subsurface using the classic PDE with a
space- and time-dependent dispersion coefficient (Su et al., 2005),
and also means that the connection between the fPDEs and classic
PDEs for water movement in soils deserves further investigation.
In this paper we present the fractional calculus-based models to
investigate the stochastic moisture movement in soils because the
parameters k and b in the fPDEs have multiple implications which
can provide alternative methods for their determination and
verification.
9. Solutions of the distributed-order mass-time and space-time
fPDEs
As examples of applications of the mass-time and space-time
fPDEs, we present solutions of Eqs. (18) and (20) for swelling soils,
and Eqs. (22) and (26) in one dimension for non-swelling soils,
which will be used to derive equations for infiltration.
The following initial and boundary conditions are defined in our
example,
9
m>0 =
>
h ¼ hs ; t > 0; m ¼ 0
>
;
h ! hi ; t > 0; m ! 1
h ¼ hi ;
t ¼ 0;
ð40Þ
where hi is the initial moisture ratio, and hs is the saturated
moisture ratio at the surface. For non-swelling soils, one only needs
to change m to z in Eq. (40) to derive solutions of the fPDEs.
9.1. Solutions for water movement into swelling soils
9.1.1. Swelling soils with mobile and immobile zones
With Eq. (40), the derivation of solutions of Eq. (18) is detailed
in Appendix A with the following two-term approximation given in
Eq. (A23), i.e.,
ðhs hi Þðcn a 1ÞK 0 m2k1
b1
b2
h ¼ hi þ
þ
ð41Þ
C½1 b1 tb1 C½1 b2 tb2
C½2kD20
which is a solution for swelling soils with mobile–immobile zones,
and is rearranged to give
"
m¼
#1=ð2k1Þ
D20 C½1 b1 C½1 b2 C½2ktb2 þb1
ðh hi Þ
ðcn a 1ÞK 0 ðb1 C½1 b2 tb2 þ b2 C½1 b1 tb1 Þ ðhs hi Þ
ð42Þ
9.1.2. Swelling soils without immobile zones
For the uniform porosity without distinguishing between
mobile and immobile zones in a swelling soil, b1 = 0, b2 = 1 and
b1 = 0, and by writing b = b2, Eq. (41) becomes
h ¼ hi þ
ðhs hi Þðcn a 1ÞK 0 m2k1
ð43Þ
C½2kD20 C½1 btb
D20 C½1 bC½2kt b ðh hi Þ
m¼
ðcn a 1ÞK 0 ðC½1 bÞ ðhs hi Þ
9.2.1. Non-swelling soils with mobile and immobile zones
Procedures similar to those for deriving solutions of Eq. (18) are
used to derive solutions of Eqs. (22) and (26) for flow into nonswelling soils in one dimension. The two-term approximation for
moisture ratio in a non-swelling soil is given by Eq. (B4),
h ¼ hi þ
ðhs hi ÞK 0 z2k1
C½2kD20
b1
b2
þ
C½1 b1 t b1 C½1 b2 t b2
ð45Þ
which is rearranged to give the following result as in Eq. (B5)
"
z¼
#1=ð2k1Þ
D20 C½1 b1 C½1 b2 C½2kt b2 þb1 ðh hi Þ
K 0 ðb1 C½1 b2 tb2 þ b2 C½1 b1 tb1 Þ ðhs hi Þ
ð46Þ
9.2.2. Solutions for non-swelling soils without immobile zones
For the uniform porosity without distinguishing between
mobile and immobile zones in a non-swelling soil, i.e., b1 = 0 as
b1 + b2 = 1, and b1 = 0, and by writing b = b2, Eq. (45) becomes
h ¼ hi þ
ðhs hi ÞK 0 z2k1
C½2kD20 C½1 btb
ð47Þ
and Eq. (46), which is from Eq. (B7), becomes
z¼
"
#1=ð2k1Þ
D20 C½1 bC½2ktb ðh hi Þ
ðhs hi Þ
K 0 ðC½1 bÞ
ð48Þ
It is seen from the above that for swelling soils the term (cna 1)
is present in the formulations with the material coordinate m used.
For non-swelling soils, the solutions do not contain the term
(cna 1) and the conventional physical coordinate z is used.
10. New equations of infiltration and their parameters
In this section we use Eqs. (42) and (44) to derive equations of
infiltration into swelling soils with and without immobile zones,
and Eqs. (46) and (48) to derive equations of infiltration into
non-swelling soils with and without immobile zones, respectively.
The cumulative infiltration, I(t), is given as (Philip, 1969b;
Smiles, 1974)
IðtÞ ¼ At þ
Z
hs
mdh
ð49Þ
hi
where A represents the final infiltration rate for large time.
10.1. Equations for cumulative infiltration into swelling soils
Here we present different forms for equations of infiltration
with different approximate solutions of Eq. (18). The procedures
for integrating Eq. (49) are detailed in Appendix C, which has
two forms of equations of cumulative infiltration.
10.1.1. Equations for infiltration into swelling soils with mobile and
immobile zones
In this case, we have from Eq. (C1) the equation of cumulative
infiltration into swelling soils,
IðtÞ ¼ At þ S
t b2 þb1
b2
S1 t þ S2 t b 1
1=ð2k1Þ
ð50Þ
where
and Eq. (42), which is from Eq. (A25), becomes
"
9.2. Solutions for water movement into non-swelling soils
#1=ð2k1Þ
ð44Þ
S¼
"
#1=ð2k1Þ
ð2k 1Þðhs hi Þ D20 C½1 b1 C½1 b2 C½2k
2k
ðcn a 1ÞK 0
ð51Þ
1798
N. Su / Journal of Hydrology 519 (2014) 1792–1803
is the sorptivity, and
S1 ¼ b1 C½1 b2 ð52Þ
and
S2 ¼ b2 C½1 b1 ð53Þ
Eq. (50) can also be rewritten as
IðtÞ ¼ At þ S
S1 S2
þ
tb1 t b2
1=ð12kÞ
ð54Þ
which means that the dimensions (or units) of S are
1=ð12kÞ
L=½T b1 þ T b2 in order to ensure that the dimension of
cumulative infiltration has the unit of length [L]. In Eqs. (50)–(54),
the values of the parameters are defined earlier, namely, 0 < b1
6 2, 0 < b2 6 2, b1 + b2 = 1, and 0 < k 6 2.
The rate of infiltration is given by differentiating Eq. (50) with
respect to time, which is Eq. (C5),
iðtÞ ¼ A
1=ð2k1Þ
þ
þ St a1 a S1 tb2 þ S2 tb1
1
S1 b2 t b2 þ S2 b1 t b1
ð2k 1Þ
ð55Þ
Fig. 1. The new equation of distributed mass-time fractional infiltration into
swelling soils with mobile and immobile zones.
where
a¼
b2 þ b1
< 1:0
2k 1
ð56Þ
A ¼ ðcn a 1ÞK 0
ð57Þ
with a = 1 for the saturated soil (Smiles and Raats, 2005), and the
particle specific gravity of minerals being cn 2.65 g/cm3. We have
previously discussed how A and K0 relate (Su, 2010).
The field data on infiltration (Talsma and van der Lelij, 1976,
converted from Fig. 1) is fitted to Eq. (50) to derive the parameters
as shown in Fig. 1 here.
The commercial code TableCurve2D™ was used to perform the
data fitting shown in Fig. 1. In the fitting, it was assumed that
the total porosity for this soil was 0.4 with 0.05 being the immobile
porosity yielding b1 = 0.125, and 0.35 being the mobile porosity
giving b2 = 0.875. The derived values of parameters are shown to
b=ð2k1Þ
be A = 1.30 mm/day, S = 54.02 mm=day
, b2 = 0.3495,
b1 = 0.3352, and k ¼ 1:9658. The transport exponent determined
by Eq. (30) is l = 0.697, which implies that the mechanism in the
infiltration process belongs to the category of sub-diffusion or slow
diffusion.
In the sorptivity equation Eq. (51), there are measurable parameters and also unknown parameters: hs, hi and cn can be measured,
and b, k, A and S can be derived by fitting Eqs. (50) or (55) to the
data. Rearranging Eq. (57) yields K0 = A/(cna 1), note that for a
saturated surface, a = 1.0. Then the only unknown parameter in
the sorptivity in Eq. (51) is the diffusion coefficient, D0, which
can be derived by rearranging Eq. (51),
D0 ¼
2kS
ð2k 1Þðhs hi Þ
ð59Þ
where the sorptivity S is given as
As 0 < b2 6 2, 0 < b1 6 2, and 0 < k 6 2, Eq. (55) means that
i(t) = A as t ? 1. For swelling soils, we then have
"
IðtÞ ¼ At þ St b=ð2k1Þ
2k1
ðcn a 1ÞK 0
C½1 b1 C½1 b2 C½2k
#1=2
ð58Þ
which offers an alternative method for determining the diffusion
parameter.
10.1.2. Equations of infiltration into swelling soils without immobile
zones
In this case, we have b1 = 0, b2 = 1 as b1 + b2 = 1, b1 = 0, and by
writing b = b2, the equation of cumulative infiltration is given by
Eq. (C7) as
"
#1=ð2k1Þ
ð2k 1Þðhs hi Þ D20 C½1 bC½2k
S¼
2k
ðcn a 1ÞK 0
ð60Þ
In Eqs. (59) and (60), the ranges of parameters are given earlier,
i.e., 0 < b1 6 2, 0 < b2 6 2, b1 + b2 = 1, and 0 < k 6 2. The dimensions
(or units) of S are bL=T b=ð2k1Þc .
The rate of infiltration into the soil is given by differentiating Eq.
(59) with respect to time, which is Eq. (C9),
iðtÞ ¼ A þ
Sb
t½b=ð2k1Þ1
ð2k 1Þ
ð61Þ
The same data in Fig. 1 from field experiments published by
Talsma and van der Lelij (1976) is fitted to Eq. (59) and demonstrated in Fig. 2. The fitting process results in A = 1.30 mm/day,
b=ð2k1Þ
S = 48.64 mm=day
, b = 0.3445, and k ¼ 1:9523. The transport
exponent determined by Eq. (29) is l = 0.353.
Similar to procedures leading to Eq. (58), the constant in the diffusivity in Eq. (14) for soils without immobile zones can be derived
by rearranging Eq. (60),
D0 ¼
"
#1=2
2k1
2kS
ðcn a 1ÞK 0
ð2k 1Þðhs hi Þ
C½1 bC½2k
ð62Þ
10.2. Equations of infiltration into non-swelling soils
10.2.1. Equations of infiltration into non-swelling soils with mobile
and immobile zones
The procedures used above for deriving equations of infiltration
into swelling soils can also be used to derive equations of infiltration into non-swelling soils using Eqs. (46) and (48). The cumulative infiltration equation so derived is identical to Eqs. (50) and
(59) in the structure for swelling soils except for a different sorptivity without the term (cna 1), i.e.,
IðtÞ ¼ At þ S
where
t b2 þb1
b2
S1 t þ S2 t b 1
1=ð2k1Þ
ð63Þ
N. Su / Journal of Hydrology 519 (2014) 1792–1803
1799
11. Conclusion and discussions
Fig. 2. The new equation of mass-time fractional infiltration into swelling soils
without an immobile zone.
"
#1=ð2k1Þ
ð2k 1Þðhs hi Þ D20 C½1 b1 C½1 b2 C½2k
S¼
2k
K0
ð64Þ
is the sorptivity, and
S1 ¼ b1 C½1 b2 ð65Þ
and
S2 ¼ b2 C½1 b1 ð66Þ
The rate of infiltration into soils is derived by differentiating
Eq. (63) with respect to time,
1=ð2k1Þ
iðtÞ ¼ A þ St a1 a S1 tb2 þ S2 tb1
þ
1
S1 b2 t b2 þ S2 b1 tb1
ð2k 1Þ
ð67Þ
where
a¼
b2 þ b1
< 1:0
2k 1
ð68Þ
which is identical to Eq. (55) for swelling soils except for a different
sorptivity. Eq. (67) means that at large time, i.e., t ? 1, the final
infiltration rate is A.
10.2.2. Equations of infiltration into non-swelling soils without
immobile zones
For the uniform porosity model, b1 = 0, b2 = 1, b1 = 0, and by
writing b = b2, Eq. (63) reduces to
IðtÞ ¼ At þ St b=ð2k1Þ
ð69Þ
where
"
#1=ð2k1Þ
ð2k 1Þðhs hi Þ D20 C½1 bC½2k
S¼
2k
K0
ð70Þ
and the equation for the infiltration rate becomes
iðtÞ ¼ A þ
Sb
t ½b=ð2k1Þ1
ð2k 1Þ
ð71Þ
Rearranging the sorptivity equations Eqs. (64) and (70) also
yields the constant in the diffusivity function given the measured
and fitted values of the other parameters.
In this paper we present mass-time and space-time fractional
partial differential equations (fPDEs) of water movement in soils.
The mass-time fPDE is for water movement in swelling soils while
the space-time fPDE is for water movement in non-swelling soils,
and both formulations have the option for soils with or without
immobile zones. This set of models is an extension to the previous
models (Su, 2012) by introducing fractional mass-time and spacetime derivatives, and also power functions for the diffusivity and
hydraulic conductivity.
The fDWE results from the asymptotic or long-time approximation of the CTRW model with power laws as the two transitional
probability distribution functions for the length of jumps and waiting time intervals, respectively. The connection between the CTRW
concept and the fPDE establishes a conceptual link for the same
flow processes without resorting to the traditional mass balance
method for the derivation of the fPDEs. The two parameters, b
and k, have three meanings: the orders of temporal fractional
and spatial fractional derivatives, respectively, the exponents for
the two probability density functions in the CTRW model, and
the critical exponents characterising the fractal structures of
space-time fractals. Thus these approaches offer alternative methods for determining the orders of fractional derivatives.
While we have justified the fPDEs with a sufficient background
of the fractional models, and demonstrated their applications, the
challenge now is to establish the relationships between the orders
of mass-time and space-time fractional derivatives and other
parameters of flow and soils, including parameters for infiltration
and flow under other conditions.
It should be noted that the ranges of the orders of time-fractional derivatives, b1 and b2, are defined differently. For example,
Meerschaert et al. (2010) citing published reports indicate that
for b1 = 1 and 0 < b1 6 2 the fractional diffusion equation governs
the scaling limit of a decoupled (or independent) CTRW with
power-law waiting times while Meerschaert (2011) gives
0 < b1 6 1. The fractional derivative in time explains the sticky or
trapping mechanism (Meerschaert, 2011) by a media which is soil
particles here. The nature of fractal porous media means that the
porous media is self-similar at all levels. The trapping processes
operate at all levels on the irregular surfaces of pores in both
mobile and immobile zones. In these formulations, the only
difference between mobile and immobile zones is the lack of
convection (or advection) in immobile zones. The consequence of
the sticky or trapping processes results in the reduction in the
values of b1 and b2, which have been shown to be less than 1.0
for the reported soil.
An important feature of the new fPDEs and the equations of
infiltration is the transport exponent, which is the term
l ¼ 2ðb2 þ b1 Þ=k for soils with mobile and immobile zones, and
l ¼ 2b=k for soils without immobile zones. These transport exponents are applicable to both swelling and non-swelling soils and
can be used to classify flow patterns.
Based on the solutions of the fPDEs, we present equations for
cumulative infiltration and infiltration rates. We demonstrate the
use of these infiltration equations by applying them to the published data, and derive the values of the parameters in the two
forms of the infiltration equations. These two forms of the equations of infiltration fitted to the published data show that the final
infiltration rates, A, sorptivity, S, and the orders of b1 and b2 for
both immobile and mobile zones are close and the same types of
diffusion mechanisms (i.e., sub-diffusion) operate in both mobile
and immobile zones, which is measured by the transport exponent.
In different types of soils, l for infiltration and water movement in
soils could be very different.
1800
N. Su / Journal of Hydrology 519 (2014) 1792–1803
It should be noted that the initial and boundary conditions used
to derive the equations of infiltration as in Eq. (40) incorporate the
initial moisture content, hi, and surface moisture, hs. When the
equations of infiltration are used in the continuous numerical simulation of complex rainfall events, the values of hi and hs need to be
updated for successive rainfall events – this is one of conventional
steps in numerical simulations.
The author wishes to acknowledge that the research presented
here was partly supported by the Australian Department of Industry (ACSRF01065), the Australian Government’s ‘‘Reef Rescue
Research and Development program’’ (RRRD004), and the ‘‘Distinguished Expert of Ningxia’’ programme. The author is very grateful
to the anonymous reviewers for their comments.
Appendix A. Solutions of distributed-order mass-time fPDEs for
water movement into swelling soils
To aid in the analysis, we introduce the reduced moisture ratio, #
h hi
hs hi
ðA1Þ
then Eq. (18) is written
b1
k bþ1
b2
@ #
@ #
D0 @ #
@#
þ b 2 b2 ¼
ðcn a 1ÞK 0 k#k1
@m
b þ 1 @mk
@t b1
@t
ðA2Þ
Following Debnath (2003, p. 144–145) for solving the fPDEs,
taking the Laplace transform of Eq. (A2) and using the conditions
in Eq. (40) yield
k
D0 d #~bþ1
d#~
ðcn a 1ÞK 0 k#~k1
k
dm
b þ 1 dm
)
ðb1 sb1 þ b2 sb2 Þ#~ ¼
ðA3Þ
#~ ¼ 1s ; m ¼ 0
~h ! 0; m ! 1
ðA4Þ
where s is the Laplace transform variable. Eq. (A3) is similar to the
fractional Basset equation in fluid mechanics, which has been analysed by Debnath (2003) for 0 < k < 1, and closed-form analytical
solutions are given for k ¼ 12.
We look for solutions for b = 0, and k = 1 as discussed in Su
(2010, 2012), which allow the linearisation of the right-hand side
of Eq. (A3) as
a0
k
d#~
d #~
þ b0
þ c0 #~ ¼ 0;
k
dm
dm
0<k62
ðA10Þ
~ The
which can be inverted to yield solutions of Eq. (A5) in #.
inversion can be completed by following procedures in Haubold
et al. (2011, Eq. (17.6)), which is
1 k1 X
k
a0
c0 k
mðk1Þk Ekþ1
m
k;kþðk1Þk
b0
b0
k¼0
k
1
X
1
a0
c0 k
mðk1Þk Ekþ1
m
þ
k;ðk1Þkþ1
s k¼0
b0
b0
ðA11Þ
where Ecg;n ðzÞ is the Generalised Mittag–Leffler (GML) type function
(Kilbas et al., 2006, p. 45)
Eqg;n ðzÞ ¼
1
X
ðqÞk zk
n!C½gk þ n
k¼0
ðA12Þ
with the Pochhammer symbol being ðqÞk ¼ qðq þ 1Þ ðq þ k 1Þ
, and in particular, (q)0 = 1 and q – 0. For q = 1, the GML
¼ CC½q½qþk
function becomes the Mittag–Leffler function (MLF),
A.1. Solutions for swelling soils with mobile and immobile zones
b1
a0 s1
s1 pk1
þ
b0 pk þ ba00 p þ bc00
pk þ ba00 p þ bc00
a0 m
#~ ¼
b0 s
Acknowledgements
#¼
~~ ¼
#
ðA5Þ
where
a0 ¼ ðcn a 1ÞK 0
ðA6Þ
b0 ¼ D0
ðA7Þ
c0 ¼ ðb1 sb1 þ b2 sb2 Þ
ðA8Þ
E1g;n ðzÞ ¼ Eg;n ðzÞ ¼
1
X
zk
C½gk þ n
k¼0
ðA13Þ
Eq. (A11), after restoring the original variables using Eqs. A6, A7,
A8, gives
k
1 ð1 cn aÞK 0 mk1 X
ðcn a 1ÞK 0
b1 sb1 þ b2 sb2 k
kþ1
mðk1Þk Ek;kþðk1Þk
m
#~ ¼
D0 s
D0
D0
k¼0
k
1
b
b
1 X ðcn a 1ÞK 0
b1 s 1 þ b2 s 2 k
þ
mðk1Þk Ekþ1
m
k;ðk1Þkþ1 s k¼0
D0
D0
ðA14Þ
The inverse Laplace transform of Eq. (A14) yields a solution of
Eq. (18) subject to the conditions in Eq. (40). Here, we are only
interested in the leading terms in Eq. (A14) which contain all the
key parameters in the model. For k = 0, Eq. (A14) through Eq.
(A13) yields
ð1 cn aÞK 0 mk1
ðb1 sb1 þ b2 sb2 Þ k
m
#~ ¼
Ek;k D0
D0 s
1
ðb1 sb1 þ b2 sb2 Þ k
þ Ek;1 m
s
D0
ðA15Þ
where the two-parameter MLFs are respectively given as
X
n
1
ðb1 sb1 þ b2 sb2 Þ k
1
ðb1 sb1 þ b2 sb2 Þ k
Ek;k m ¼
m
D0
D0
C½kðn þ 1Þ
n¼0
ðA16Þ
and
X
n
1
ðb1 sb1 þ b2 sb2 Þ k
1
ðb1 sb1 þ b2 sb2 Þ k
Ek;1 m ¼
m
D0
D0
C½ðkn þ 1Þ
n¼0
ðA17Þ
We retain only two terms in the first MLF in Eq. (A15) to yield
Eq. (A5) can be solved by using a further Laplace transform with
respect to m (Debnath, 2003, p. 131) with p as the second Laplace
transform variable, and the conditions in Eq. (A4),
s1 ða0 þ b0 pk1 Þ
~~
#
¼
ða0 p þ b0 pk þ c0 Þ
Eq. (A9) can be written in the following form
ðA9Þ
ðb1 sb1 þ b2 sb2 Þ k
1
Ek;k m ¼
D0
C½k
1
ðb1 sb1 þ b2 sb2 Þ k
m
D0
C½2k
ðA18Þ
which is used in Eq. (A15) to give
ðc a 1ÞK 0 m
#~ ¼ n
D0 s
k1
1
ðb1 sb1 þ b2 sb2 Þ k
1
m D0
C½2k
C½k
ðA19Þ
1801
N. Su / Journal of Hydrology 519 (2014) 1792–1803
The inverse Laplace transform of Eq. (A19) yields
k ðc a 1ÞK 0 mk1
1
b1
b2
m
1
#¼ n
þ
b1
b2
D0
C½2k C½1 b1 t
D0
C½k
C½1 b2 t
1 k1 X
k
K0
b1 sb1 þ b2 sb2 k
kþ1
zðk1Þk Ek;kþðk1Þk
z
D0
D0
k¼0
k
1
b
1 X K0
b1 s 1 þ b2 sb2 k
zðk1Þk Ekþ1
z
þ
k;ðk1Þkþ1 s k¼0 D0
D0
K0z
#~ ¼
D0 s
ðA20Þ
For large m for swelling soils, the first term in Eq. (A20) is much
larger than the second term so that C1½k can be ignored, then
Eq. (A20) becomes
#¼
ðcn a 1ÞK 0 m2k1
C
½2kD20
b1
b2
þ
C½1 b1 tb1 C½1 b2 tb2
ðA21Þ
D20
h ¼ hi þ
ðhs hi ÞK 0 z2k1
C½2kD20
b1
b2
þ
C½1 b1 t b1 C½1 b2 t b2
ðB4Þ
"
#1=ð2k1Þ
b2 þb1
C½1 b1 C½1 b2 C½2k#t
ðcn a 1ÞK 0 ðb1 C½1 b2 tb2 þ b2 C½1 b1 tb1 Þ
m¼
The two-term approximation to Eq. (B3) for non-swelling soils,
after restoring the original variables using Eq. (A1), is
Rearranging Eq. (B4) yields
which can be rearranged to yield
"
ðB3Þ
ðA22Þ
z¼
#1=ð2k1Þ
D20 C½1 b1 C½1 b2 C½2kt b2 þb1 ðh hi Þ
K 0 ðb1 C½1 b2 tb2 þ b2 C½1 b1 tb1 Þ ðhs hi Þ
ðB5Þ
When the original variable is restored using Eq. (A1), Eq. (A21)
becomes
B.2. Solutions for non-swelling soils without immobile zones
h ¼ hi
For the uniform porosity model, b1 = 0 and b2 = 1 as b1 + b2 = 1,
then b1 = 0. By writing b = b2, Eq. (B4) reduces to
þ
ðhs hi Þðcn a 1ÞK 0 m2k1
C½2kD20
b1
C½1 b1 tb1
þ
b2
C½1 b2 tb2
h ¼ hi þ
ðA23Þ
For the uniform porosity model without distinguishing the
mobile and immobile zones, b1 = 0 and b2 = 1 as b1 + b2 = 1, then
b1 = 0. By writing b = b2, Eq. (A23) becomes
ðhs hi Þðcn a 1ÞK 0 m2k1
ðA24Þ
C½2kD20 C½1 btb
and Eq. (A22), after restoring the original variables using Eq. (A1),
becomes
"
D20 C½1 bC½2kt b ðh hi Þ
m¼
ðcn a 1ÞK 0 ðC½1 bÞ ðhs hi Þ
C½2kD20 C½1 btb
ðB6Þ
and Eq. (B5), after restoring the original variables using Eq. (A1),
becomes
A.2. Solutions for swelling soils without immobile zones
h ¼ hi þ
ðhs hi ÞK 0 z2k1
#1=ð2k1Þ
"
#1=ð2k1Þ
D20 C½1 bC½2ktb ðh hi Þ
z¼
ðhs hi Þ
K 0 ðC½1 bÞ
ðB7Þ
Appendix C. Equations of infiltration into swelling soils
The solutions presented in Appendix A are used in Eq. (49) to
derive equations of infiltration into swelling soils. Here we consider two cases: a dual porosity soil with mobile and immobile
zones, and a uniform porosity soil without immobile zones.
ðA25Þ
C.1. Model 1: Infiltration into swelling soils with mobile and immobile
zones
Appendix B. Solutions of space-time fPDEs for water movement
into non-swelling soils
In this case, Eq. (A22) is used in Eq. (49) to derive an equation of
infiltration. As the reduced moisture ratio, #, is used, the integration limits need to be changed in Eq. (49) to yield an equation of
cumulative infiltration
B.1. Solutions for non-swelling soils with mobile and immobile zones
With the reduced moisture ratio in Eq. (A1), Eq. (22) for nonswelling soils is written
b1
@ b1 #
@ b2 #
D0 @ k #bþ1
@#
þ b 2 b2 ¼
K 0 k#k1
b1
@z
b þ 1 @zk
@t
@t
9
z>0 =
>
# ¼ 1; t > 0; z ¼ 0
>
;
# ! 0; t > 0; z ! 1
t b2 þb1
S1 t b 2 þ S2 t b 1
1=ð2k1Þ
ðC1Þ
where
ðB1Þ
which is to be solved with the following initial and boundary conditions (these reduced initial and boundary conditions are identical
to those used for swelling soils when the reduced variable is used in
Eq. (40)),
# ¼ 0;
IðtÞ ¼ At þ S
"
#1=ð2k1Þ
ð2k 1Þðhs hi Þ D20 C½1 b1 C½1 b2 C½2k
S¼
2k
ðcn a 1ÞK 0
ðC2Þ
is the sorptivity,
S1 ¼ b1 C½1 b2 ðC3Þ
and
t ¼ 0;
ðB2Þ
Following the procedures leading to the solutions of Eq. (A14),
the solution of Eq. (B1) is given as follows (by replacing m by z,
and removing the term (cna - 1) with b = 0 and k = 1),
S2 ¼ b2 C½1 b1 ðC4Þ
To ensure that the infiltration equations have non-negative values, it is essential that S P 0, which means that in Eq. (C2) the relationship ð2k1Þ
P 0 holds, which implies k P 12. The rate of
2k
infiltration into soils is derived by differentiating Eq. (C1) with
respect to time,
1802
N. Su / Journal of Hydrology 519 (2014) 1792–1803
1=ð2k1Þ
iðtÞ ¼ A þ Sta1 aðS1 t b2 þ S2 tb1 Þ
þ
1
S1 b2 t b2 þ S2 b1 t b1
ð2k 1Þ
ðC5Þ
where the combined mass-time orders of fractional derivatives is
integrated in one parameter
a¼
b2 þ b1
< 1:0
2k 1
ðC6Þ
with 0 < b2 6 2, 0 < b1 6 2, and 0 < k 6 2.
C.2. Model 2: Infiltration into swelling soils without immobile zones
In this case, we have b1 = 0 and b2 = 1 as b1 + b2 = 1, b1 = 0, and
by writing b = b2, Eq. (A25) is used in Eq. (49) to yield an equation
of cumulative infiltration,
IðtÞ ¼ At þ St b=ð2k1Þ
ðC7Þ
where S is the sorptivity defined as
S¼
"
#1=ð2k1Þ
ð2k 1Þðhs hi Þ D20 C½1 bC½2k
2k
ðcn a 1ÞK 0
ðC8Þ
Eq. (C7) can also be derived from Eq. (C1) by setting b1 = 0,
b2 = 1, b1 = 0, and b = b2.
The rate of infiltration into swelling soils is given by differentiating Eq. (C7) with respect to time,
iðtÞ ¼ A þ
Sb
t ½b=ð2k1Þ1
ð2k 1Þ
ðC9Þ
As for soils with immobile zones, the relationship k P 12 also
holds for soils without immobile zones.
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