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Equations of anomalous absorption onto swelling porous media
Materials Letters 63 (2009) 2483–2485
Contents lists available at ScienceDirect
Materials Letters j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / m a t l e t
Equations of anomalous absorption onto swelling porous media Ninghu Su ⁎ School of Mathematics & Computational Science, Xiangtan University, Hunan 411105, China School of Earth & Environmental Sciences, James Cook University, Cairns, Qld. 4870, Australia Department of Environment and Resource Management, Mareeba, Qld. 4880, Australia
a r t i c l e
i n f o
Article history: Received 3 June 2009 Accepted 23 August 2009 Available online 1 September 2009 Keywords: Anomalous absorption Swelling porous media Fractional diffusion-wave equation Cumulative absorption Absorption rate
a b s t r a c t Absorption is a very common process which takes place on various types of materials ranging from porous media to new nano-materials and biological tissues. The majority of studies reported on absorption to date are concentrated on “rigid” porous media, which contradict the properties of real porous media which undergo swelling and shrinking changes. Here we present new absorption equations derived from a fractional diffusion-wave equation (fDWE) for absorption onto swelling porous media in a material coordinate. We show that the cumulative anomalous absorption is I(t) = St β/2 and the absorption rate iðtÞ = 12 βSt β = 2−1 , where S is the anomalous sorptivity and β the order of fractional derivative in fDWE. Using published data on cumulative absorption against time, the two adsorption parameters are determined: β = 1.2448 and S = 2.7775 cm2/h. The value of β = 1.2448 implies that absorption onto this swelling porous media belong to the category of super-diffusion, which is a phenomenon unknown to us before. In comparison, the traditional absorption equations do not have such features. When S is determined, the anomalous diffusivity, Dm, is calculated using its relation with S. We expect that the proposed new absorption equations will be valuable for explaining new phenomena and processes encountered in broader disciplines of science and engineering applications. © 2009 Elsevier B.V. All rights reserved.
1. Introduction In this paper, we outline the major findings of our research on anomalous absorption onto swelling media in a material coordinate, and the detailed derivations are provided in the Appendix. Absorption is the process in which one material enters the bulk space of another by either physico-chemical forces and/or gradients. The importance of absorption is evidenced by its diversity in crucial issues among many branches of sciences and engineering [1–7], which includes the creation of new products [3], nano-material [4], and renewal of global water resources through infiltration [8]. The majority of reported studies for analysing absorption to date are based on the classic theories developed for absorption onto rigid media [2,3,5,6]. These classic methods at best ignore the realistic swelling and shrinking properties of the natural media, and at worst miss crucial information about both sub-diffusion and super-diffusion processes [9] in porous media. Four decades ago, the equations governing the movement of liquids in swelling media were formulated in a material coordinate [1,10–14], however, attempts to modify the classic theories of flow in rigid media and apply them to swelling media are continually reported, including attempts to use the
fractional diffusion equation formulated for rigid media for analysing anomalous absorption [2–6,15,16]. Despite extensive reports in the literature on fractional diffusionwave equations (fDWE) and their applications [17–23] in various fields to account for anomalous diffusion, no study has investigated their applications in anomalous swelling porous media. Here we show that very general equations of absorption onto swelling porous surfaces is derived from the general fDWE. These general equations of cumulative absorption, absorption rate and the anomalous sorptivity are presented in material coordinates. The key anomalous parameters are evaluated using published data [7]. When the order of derivative is unity, i.e., β = 1, these new anomalous equations degenerate to their classic counterparts [24]. As a result we reconcile the inconsistency of the methods developed for flow in both rigid and swelling media with both anomalous and classic flow mechanisms. In this letter we report these findings. 2. The fractional diffusion-wave equation (fDWE) for absorption on swelling porous media First, the material coordinate, m, is defined as follows [1,11–14] z
−1
m = ∫0 ð1 + eÞ ⁎ School of Earth & Environmental Sciences, James Cook University, Cairns, Qld. 4870, Australia. Tel.: +61 07 4042 1551; fax: +61 07 4042 1284. E-mail address: [email protected]. 0167-577X/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.matlet.2009.08.039
dz1
ð1Þ
where z is the usual spatial coordinate in a vertical direction, e is the void ratio given as e = ϑ/(1 − ϑ) with ϑ being the moisture ratio, i.e., ϑ = θl/θs, where θl and θs are the volume fractions of liquid and solid,
2484
N. Su / Materials Letters 63 (2009) 2483–2485
respectively. In the two-component system, the void ratio, e, is equal to the moisture ratio. Furthermore, researchers [18] prove that when the integer derivative with time in the classic diffusion equation is replaced by a fractional order, the anomalous diffusion can be explained. Based on Eq. (1) and the mapping from ∂ϑ/∂t to ∂ βϑ/∂t β, we arrive at a very generic fDWE for anomalous diffusion in swelling media in a material coordinate, ∂β ϑ ∂ ∂ϑ = D m ∂m ∂m ∂t β
0<β<2
ð2Þ
where Dm(ϑ) is the material diffusivity in the swelling–shrinking media and similar in its mathematical structure to that for nonswelling media [11–14], and related to the liquid-content-dependent hydraulic conductivity, K(ϑl). Dm is defined as [1] Dm =
Kðϑl Þ dψ ð1 + ϑÞ dϑ
ð3Þ
where ψ is the unloaded matrix potential. In this paper, we show (see Appendix A for more details) that the cumulative absorption onto swelling media in a material coordinate is given as IðtÞ = St
β=2
ð4Þ
where the anomalous sorptivity is 1 1= 2 S = Γ½1−β = 2ðϑ0 −ϑi ÞDm : 2
Fig. 1. The parameters derived from fitting the absorption equation to the data: S = 2.7775 cm/h1/2 and β = 1.2448.
ð5Þ
The absorption rate, i(t), is the derivation of Eq. (4) with respect to time, iðtÞ =
1 β = 2−1 βSt : 2
ð6Þ
In addition, the anomalous diffusivity, Dm, is derived by rearranging Eq. (5) once S is determined, Dm =
2 2S : Γ½1−β=2ðϑ0 −ϑi Þ
ð7Þ
For β = 1, Eq. (4) degenerates to the normal adsorption, IðtÞ = St
1=2
:
ð8Þ
Eq. (6) becomes the normal adsorption rate, iðtÞ =
1 −1 = 2 St : 2
ð9Þ
Eq. (5) becomes the normal sorptivity pffiffiffiffiffiffiffiffiffiffi 1 S = ðϑ0 −ϑi Þ πDm 2
2 1 2S : π ðϑ0 −ϑi Þ
4. Conclusion This paper presents new findings on absorption into swelling porous media using the time-fractional diffusion-wave equation formulated in a material coordinate. It is showed that with the published laboratory data [7], the key parameters such as the sorptivity, diffusivity and the order of fractional derivative can be derived easily. It is also shown that the value of the order of fractional derivative for the reported porous media is β = 1.2448, which means that absorption onto this swelling porous media belongs to the class of super-diffusion [9]. Acknowledgements
ð10Þ
where π = 3.1416, and Eq. (7) becomes the normal diffusivity Dm =
With S = 2.7775 cm/h1/2, β = 1.2448, ϑi = 0.107/2.65 = 0.04 [7, p. 124], and ϑ0 = 0.84/2.65 = 0.317 [7, p. 126], Eq. (7) yields Dm = 72.6 cm2/h. The value of β = 1.2448 means that absorption onto swelling porous media belongs to the class of super-diffusion [9], a phenomenon in swelling porous media unknown to us before. In comparison, the traditional absorption models do not have such features. The new absorption equations with the new parameter, β, provide new insights into absorption processes encountered in various fields of science and engineering applications.
ð11Þ
The research reported in this paper is sponsored in part by the National Natural Science Foundation of China; Fok Ying Tung Education Foundation; the Youth Foundation of the Department of Education, Hunan Province, China, and a JCU Internal Research Grant. Appendix A
3. Results and discussion
We investigate absorption using Eq. (2) subject to the following initial and boundary conditions,
When the published laboratory data of Bridge and Collis-George [7] is fitted to Eq. (4), the two parameters in Eq. (4) can be derived, and the fitting is shown in Fig. 1.
ϑ = ϑi ; ϑ = ϑ0 f ðtÞ; ϑ→0;
t = 0; t > 0; m→∞
m > 0 m=0
g
:
ðS1Þ
N. Su / Materials Letters 63 (2009) 2483–2485
We use the following reduced variable to transform the initial and boundary conditions in Eq. (S1): θ=
ϑ−ϑi ϑ0 −ϑi
ðS2Þ
2485
i.e., k = 0,1, and use the second order approximation in Eq. (S7) to yield IðtÞ = St
β=2
ðS9Þ
which is Eq. (4) in the main text, and S is the anomalous sorptivity for absorption given as
where θ ϑi ϑ0 f(t)
the reduced moisture ratio; the initial value of the moisture ratio, ϑ; the value of ϑ at the surface, and a time-dependent function.
S=
Using the reduced variable, and application of Laplace transform to Eq. (2) and the conditions in (S1) yields 2˜
d θ s ˜ − θ=0 dm2 Dm
ðS3Þ
θ˜ = f˜ ðsÞ; m = 0 g: ˜ θ→0; m→∞
ðS4Þ
ðS10Þ
which is Eq. (5) in the main text. The absorption rate is the derivation of Eq. (S9) with respect to time, i.e., iðtÞ =
β
1 ðβ = 2Þ−1 βSt 2
ðS11Þ
which is Eq. (6) in the main text.
Eq. (S3) subject to Eq. (S4) has a solution of the form [21, p. 145], " # m β=2 : θ˜ = f ðsÞ exp − 1 = 2 s Dm
ðS5Þ
Here we consider the case for f(t) = 1, which corresponds to absorption into the surface with a constant surface value of ϑ0. In this case, Eq. (S5) yields 1 m β=2 θ˜ = exp − 1 = 2 s s Dm
! ðS6Þ
which has the following inverse Laplace transform [17, p. 394], 1 mt −β = 2 − 1=2 θ = ϕðρ; α; zÞ = ∑ k!Γ½1−βk = 2 k=0 Dm ∞
!k ðS7Þ
where ϕ(ρ,α;z) is the Wright function, and Γ[1 − βk/2] is the gamma function. The cumulative absorption, I(t), is defined as [12,13,24,25] ϑ0
IðtÞ = ∫ mdϑ:
pffiffiffiffiffiffiffi 1 Γ½1−β = 2ðϑ0 −ϑi Þ Dm 2
ðS8Þ
ϑi
Eq. (S8) requires Eq. (S7) to be expressed in m as a function of ϑ. Here we consider an approximation of Eq. (S7) up to the second order,
References [1] Philip JR. In: Karalis TK, editor. Mechanics of Swelling. Berlin: Springer-Verlag; 1992. [2] Tae G, Kornfield JA, Hubbell JA, Johannsmann D. Langmuir 2002;18:8241–5. [3] Chen CM, MacKintosh FC, Wiliams DRM. Langmuir 1995;7:2471–5. [4] Phang SW, Tadokoro M, Watanabe J, Kuramoto N. Current Appl Phys 2008;8:391–4. [5] Küntz M, Lavallee P. J Phys D Appl Phys 2001;34:2547–54. [6] Lockington DA, Parlange JY. J Phys D Appl Phys 2003;36:760–7. [7] Bridge BJ, Collis-George N. Aust J Soil Res 1973;11:121–32. [8] Betts RA, Boucher O, Collins M, Cox PM, Falloon PD, Gedney N, et al. Nature 2007;448: 1037–41. [9] Tsallis C, Bukman DJ. Phys Rev E 1996;54:R2197–200. [10] Raats PAC, Klute A. Soil Sci Soc Am Proc 1968;32:161–6. [11] Smiles DE, Rosenthal MJ. Aust J Soil Res 1968;6:237–48. [12] Philip JR. Water Resour Res 1969;5:1070–7. [13] Smiles DE. Soil Sci 1974;117:140–7. [14] Smiles DE, Raats PAC. In: Anderson MG, editor. Encyclopedia of Hydrological Sciences. Chichester: Wiley; 2005. [15] Camporese M, Ferraris S, Putti M, Salandin P, Teatini P. Water Resour Res 2006;42: W06420. [16] Gerolymatou E, Vardoulakis I, Hilfer R. J Phys D Appl Phys 2006;39:4104–10. [17] Gorenflo R, Luchko Y, Mainardi F. Fract Calc Appl Anal 1999;2:383–414. [18] Zhang Y, Benson DA, Reeves DM. Adv Water Resour 2009;32:561–81. [19] Kilbas AA, Srivastava HM, Trujillo JJ. Theory and Applications of Fractional Differential Equations. Amsterdam: Elsevier; 2006. [20] Podlubry I. Fractional Differential Equations. San Diego: Academic Press; 1999. [21] Debnath L. Frac Calc Appl Anal 2003;6:119–55. [22] Debnath L. Int J Math Math Sci 2003;54:3413–42. [23] Debnath L, Bhatta D. Integral Transforms and Their Applications. 2nd ed. Boca Raton: Chapman & Hall/CRC; 2007. [24] Philip JR. Adv Hydrosci 1969;5:215–96. [25] Smith RE. Infiltration Theory for Hydrologic Applications. Washington, D.C: American Geophysical Union; 2002.
Contents lists available at ScienceDirect
Materials Letters j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / m a t l e t
Equations of anomalous absorption onto swelling porous media Ninghu Su ⁎ School of Mathematics & Computational Science, Xiangtan University, Hunan 411105, China School of Earth & Environmental Sciences, James Cook University, Cairns, Qld. 4870, Australia Department of Environment and Resource Management, Mareeba, Qld. 4880, Australia
a r t i c l e
i n f o
Article history: Received 3 June 2009 Accepted 23 August 2009 Available online 1 September 2009 Keywords: Anomalous absorption Swelling porous media Fractional diffusion-wave equation Cumulative absorption Absorption rate
a b s t r a c t Absorption is a very common process which takes place on various types of materials ranging from porous media to new nano-materials and biological tissues. The majority of studies reported on absorption to date are concentrated on “rigid” porous media, which contradict the properties of real porous media which undergo swelling and shrinking changes. Here we present new absorption equations derived from a fractional diffusion-wave equation (fDWE) for absorption onto swelling porous media in a material coordinate. We show that the cumulative anomalous absorption is I(t) = St β/2 and the absorption rate iðtÞ = 12 βSt β = 2−1 , where S is the anomalous sorptivity and β the order of fractional derivative in fDWE. Using published data on cumulative absorption against time, the two adsorption parameters are determined: β = 1.2448 and S = 2.7775 cm2/h. The value of β = 1.2448 implies that absorption onto this swelling porous media belong to the category of super-diffusion, which is a phenomenon unknown to us before. In comparison, the traditional absorption equations do not have such features. When S is determined, the anomalous diffusivity, Dm, is calculated using its relation with S. We expect that the proposed new absorption equations will be valuable for explaining new phenomena and processes encountered in broader disciplines of science and engineering applications. © 2009 Elsevier B.V. All rights reserved.
1. Introduction In this paper, we outline the major findings of our research on anomalous absorption onto swelling media in a material coordinate, and the detailed derivations are provided in the Appendix. Absorption is the process in which one material enters the bulk space of another by either physico-chemical forces and/or gradients. The importance of absorption is evidenced by its diversity in crucial issues among many branches of sciences and engineering [1–7], which includes the creation of new products [3], nano-material [4], and renewal of global water resources through infiltration [8]. The majority of reported studies for analysing absorption to date are based on the classic theories developed for absorption onto rigid media [2,3,5,6]. These classic methods at best ignore the realistic swelling and shrinking properties of the natural media, and at worst miss crucial information about both sub-diffusion and super-diffusion processes [9] in porous media. Four decades ago, the equations governing the movement of liquids in swelling media were formulated in a material coordinate [1,10–14], however, attempts to modify the classic theories of flow in rigid media and apply them to swelling media are continually reported, including attempts to use the
fractional diffusion equation formulated for rigid media for analysing anomalous absorption [2–6,15,16]. Despite extensive reports in the literature on fractional diffusionwave equations (fDWE) and their applications [17–23] in various fields to account for anomalous diffusion, no study has investigated their applications in anomalous swelling porous media. Here we show that very general equations of absorption onto swelling porous surfaces is derived from the general fDWE. These general equations of cumulative absorption, absorption rate and the anomalous sorptivity are presented in material coordinates. The key anomalous parameters are evaluated using published data [7]. When the order of derivative is unity, i.e., β = 1, these new anomalous equations degenerate to their classic counterparts [24]. As a result we reconcile the inconsistency of the methods developed for flow in both rigid and swelling media with both anomalous and classic flow mechanisms. In this letter we report these findings. 2. The fractional diffusion-wave equation (fDWE) for absorption on swelling porous media First, the material coordinate, m, is defined as follows [1,11–14] z
−1
m = ∫0 ð1 + eÞ ⁎ School of Earth & Environmental Sciences, James Cook University, Cairns, Qld. 4870, Australia. Tel.: +61 07 4042 1551; fax: +61 07 4042 1284. E-mail address: [email protected]. 0167-577X/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.matlet.2009.08.039
dz1
ð1Þ
where z is the usual spatial coordinate in a vertical direction, e is the void ratio given as e = ϑ/(1 − ϑ) with ϑ being the moisture ratio, i.e., ϑ = θl/θs, where θl and θs are the volume fractions of liquid and solid,
2484
N. Su / Materials Letters 63 (2009) 2483–2485
respectively. In the two-component system, the void ratio, e, is equal to the moisture ratio. Furthermore, researchers [18] prove that when the integer derivative with time in the classic diffusion equation is replaced by a fractional order, the anomalous diffusion can be explained. Based on Eq. (1) and the mapping from ∂ϑ/∂t to ∂ βϑ/∂t β, we arrive at a very generic fDWE for anomalous diffusion in swelling media in a material coordinate, ∂β ϑ ∂ ∂ϑ = D m ∂m ∂m ∂t β
0<β<2
ð2Þ
where Dm(ϑ) is the material diffusivity in the swelling–shrinking media and similar in its mathematical structure to that for nonswelling media [11–14], and related to the liquid-content-dependent hydraulic conductivity, K(ϑl). Dm is defined as [1] Dm =
Kðϑl Þ dψ ð1 + ϑÞ dϑ
ð3Þ
where ψ is the unloaded matrix potential. In this paper, we show (see Appendix A for more details) that the cumulative absorption onto swelling media in a material coordinate is given as IðtÞ = St
β=2
ð4Þ
where the anomalous sorptivity is 1 1= 2 S = Γ½1−β = 2ðϑ0 −ϑi ÞDm : 2
Fig. 1. The parameters derived from fitting the absorption equation to the data: S = 2.7775 cm/h1/2 and β = 1.2448.
ð5Þ
The absorption rate, i(t), is the derivation of Eq. (4) with respect to time, iðtÞ =
1 β = 2−1 βSt : 2
ð6Þ
In addition, the anomalous diffusivity, Dm, is derived by rearranging Eq. (5) once S is determined, Dm =
2 2S : Γ½1−β=2ðϑ0 −ϑi Þ
ð7Þ
For β = 1, Eq. (4) degenerates to the normal adsorption, IðtÞ = St
1=2
:
ð8Þ
Eq. (6) becomes the normal adsorption rate, iðtÞ =
1 −1 = 2 St : 2
ð9Þ
Eq. (5) becomes the normal sorptivity pffiffiffiffiffiffiffiffiffiffi 1 S = ðϑ0 −ϑi Þ πDm 2
2 1 2S : π ðϑ0 −ϑi Þ
4. Conclusion This paper presents new findings on absorption into swelling porous media using the time-fractional diffusion-wave equation formulated in a material coordinate. It is showed that with the published laboratory data [7], the key parameters such as the sorptivity, diffusivity and the order of fractional derivative can be derived easily. It is also shown that the value of the order of fractional derivative for the reported porous media is β = 1.2448, which means that absorption onto this swelling porous media belongs to the class of super-diffusion [9]. Acknowledgements
ð10Þ
where π = 3.1416, and Eq. (7) becomes the normal diffusivity Dm =
With S = 2.7775 cm/h1/2, β = 1.2448, ϑi = 0.107/2.65 = 0.04 [7, p. 124], and ϑ0 = 0.84/2.65 = 0.317 [7, p. 126], Eq. (7) yields Dm = 72.6 cm2/h. The value of β = 1.2448 means that absorption onto swelling porous media belongs to the class of super-diffusion [9], a phenomenon in swelling porous media unknown to us before. In comparison, the traditional absorption models do not have such features. The new absorption equations with the new parameter, β, provide new insights into absorption processes encountered in various fields of science and engineering applications.
ð11Þ
The research reported in this paper is sponsored in part by the National Natural Science Foundation of China; Fok Ying Tung Education Foundation; the Youth Foundation of the Department of Education, Hunan Province, China, and a JCU Internal Research Grant. Appendix A
3. Results and discussion
We investigate absorption using Eq. (2) subject to the following initial and boundary conditions,
When the published laboratory data of Bridge and Collis-George [7] is fitted to Eq. (4), the two parameters in Eq. (4) can be derived, and the fitting is shown in Fig. 1.
ϑ = ϑi ; ϑ = ϑ0 f ðtÞ; ϑ→0;
t = 0; t > 0; m→∞
m > 0 m=0
g
:
ðS1Þ
N. Su / Materials Letters 63 (2009) 2483–2485
We use the following reduced variable to transform the initial and boundary conditions in Eq. (S1): θ=
ϑ−ϑi ϑ0 −ϑi
ðS2Þ
2485
i.e., k = 0,1, and use the second order approximation in Eq. (S7) to yield IðtÞ = St
β=2
ðS9Þ
which is Eq. (4) in the main text, and S is the anomalous sorptivity for absorption given as
where θ ϑi ϑ0 f(t)
the reduced moisture ratio; the initial value of the moisture ratio, ϑ; the value of ϑ at the surface, and a time-dependent function.
S=
Using the reduced variable, and application of Laplace transform to Eq. (2) and the conditions in (S1) yields 2˜
d θ s ˜ − θ=0 dm2 Dm
ðS3Þ
θ˜ = f˜ ðsÞ; m = 0 g: ˜ θ→0; m→∞
ðS4Þ
ðS10Þ
which is Eq. (5) in the main text. The absorption rate is the derivation of Eq. (S9) with respect to time, i.e., iðtÞ =
β
1 ðβ = 2Þ−1 βSt 2
ðS11Þ
which is Eq. (6) in the main text.
Eq. (S3) subject to Eq. (S4) has a solution of the form [21, p. 145], " # m β=2 : θ˜ = f ðsÞ exp − 1 = 2 s Dm
ðS5Þ
Here we consider the case for f(t) = 1, which corresponds to absorption into the surface with a constant surface value of ϑ0. In this case, Eq. (S5) yields 1 m β=2 θ˜ = exp − 1 = 2 s s Dm
! ðS6Þ
which has the following inverse Laplace transform [17, p. 394], 1 mt −β = 2 − 1=2 θ = ϕðρ; α; zÞ = ∑ k!Γ½1−βk = 2 k=0 Dm ∞
!k ðS7Þ
where ϕ(ρ,α;z) is the Wright function, and Γ[1 − βk/2] is the gamma function. The cumulative absorption, I(t), is defined as [12,13,24,25] ϑ0
IðtÞ = ∫ mdϑ:
pffiffiffiffiffiffiffi 1 Γ½1−β = 2ðϑ0 −ϑi Þ Dm 2
ðS8Þ
ϑi
Eq. (S8) requires Eq. (S7) to be expressed in m as a function of ϑ. Here we consider an approximation of Eq. (S7) up to the second order,
References [1] Philip JR. In: Karalis TK, editor. Mechanics of Swelling. Berlin: Springer-Verlag; 1992. [2] Tae G, Kornfield JA, Hubbell JA, Johannsmann D. Langmuir 2002;18:8241–5. [3] Chen CM, MacKintosh FC, Wiliams DRM. Langmuir 1995;7:2471–5. [4] Phang SW, Tadokoro M, Watanabe J, Kuramoto N. Current Appl Phys 2008;8:391–4. [5] Küntz M, Lavallee P. J Phys D Appl Phys 2001;34:2547–54. [6] Lockington DA, Parlange JY. J Phys D Appl Phys 2003;36:760–7. [7] Bridge BJ, Collis-George N. Aust J Soil Res 1973;11:121–32. [8] Betts RA, Boucher O, Collins M, Cox PM, Falloon PD, Gedney N, et al. Nature 2007;448: 1037–41. [9] Tsallis C, Bukman DJ. Phys Rev E 1996;54:R2197–200. [10] Raats PAC, Klute A. Soil Sci Soc Am Proc 1968;32:161–6. [11] Smiles DE, Rosenthal MJ. Aust J Soil Res 1968;6:237–48. [12] Philip JR. Water Resour Res 1969;5:1070–7. [13] Smiles DE. Soil Sci 1974;117:140–7. [14] Smiles DE, Raats PAC. In: Anderson MG, editor. Encyclopedia of Hydrological Sciences. Chichester: Wiley; 2005. [15] Camporese M, Ferraris S, Putti M, Salandin P, Teatini P. Water Resour Res 2006;42: W06420. [16] Gerolymatou E, Vardoulakis I, Hilfer R. J Phys D Appl Phys 2006;39:4104–10. [17] Gorenflo R, Luchko Y, Mainardi F. Fract Calc Appl Anal 1999;2:383–414. [18] Zhang Y, Benson DA, Reeves DM. Adv Water Resour 2009;32:561–81. [19] Kilbas AA, Srivastava HM, Trujillo JJ. Theory and Applications of Fractional Differential Equations. Amsterdam: Elsevier; 2006. [20] Podlubry I. Fractional Differential Equations. San Diego: Academic Press; 1999. [21] Debnath L. Frac Calc Appl Anal 2003;6:119–55. [22] Debnath L. Int J Math Math Sci 2003;54:3413–42. [23] Debnath L, Bhatta D. Integral Transforms and Their Applications. 2nd ed. Boca Raton: Chapman & Hall/CRC; 2007. [24] Philip JR. Adv Hydrosci 1969;5:215–96. [25] Smith RE. Infiltration Theory for Hydrologic Applications. Washington, D.C: American Geophysical Union; 2002.