ANALYTICAL SOLUTIONS FOR ONE-DIMENSIONAL CONSOLIDATION IN UNSATURATED SOILS CONSIDERING THE NON-DARCY LAW OF WATER FLOW
J.W. LI and H.B. WANG
about the authors
Jiwei Li
Huazhong University of Science & Technology, School of Civil Engineering and Mechanics, Wuhan 430074, P. R. China E-mail: iamliji_007@126.com
Huabin Wang
Huazhong University of Science & Technology, School of Civil Engineering and Mechanics, Wuhan 430074, P. R. China E-mail: huabin@mail.hust.edu.cn
Abstract
Analytical solutions were derived for the non-linear, one-dimensional consolidation equations for unsaturated soils. The governing equations with a non-homogeneous mixed-boundary condition were presented, in which the water flow was assumed to be governed by a non-Darcy law, whereas the air flow followed the Darcy law. The non-Darcy law was actually the non-linear, flux-gradients relationship. The consolidation equations were thus present in a strong, non-linear way. In order to analytically solve the equation, a homotopy analysis method (HAM) was introduced in the study, which is an analytical technique for nonlinear problems. Firstly, a governing equation in a dimensionless form was derived for a one-dimensional consolidation under unsaturated soils. The method was then used for a mapping technique to transfer the original nonlinear differential equations to a number of linear differential equations. These differential equations were independent with respect to any small parameters, and were convenient for controlling the convergence region. After this transferring, a series solution to the equations was then obtained using the HAM by selecting the linear operator and the auxiliary parameters. Meanwhile, comparisons between the analytical solutions and the results of the finite-difference method indicate that the analytical solution is more efficient. Furthermore, our solutions indicate that the dissipation of air pressure is much faster than that of water pressure, and the values for the threshold gradient I have obvious effects on the dissipation values of the excess pore-water pressure, but no significant effect on that of the excess pore-air pressure.
Keywords
unsaturated soil, homotopy analysis method, analytical solutions, non-Darcy law, initial and boundary conditions
INTRODUCTION
The consolidation of unsaturated soils is a subject of great interest in geotechnical engineering practice [1-3]. In fact, the excess pore pressures dissipate with time and eventually return to their initial values in unsaturated soils, generated by external loading. The dissipation processes of excess pore pressures are called consolidation and result in a volume decrease [1]. Indeed, it is important to describe the dissipation of the excess pore pressures in understanding the consolidation of unsaturated soils, and the identification of the influencing internal mechanisms also plays an important role.
Several consolidation theories in unsaturated soils have been proposed over the past few years. The notable contributions have included the work of Blight [4], Scott [5], Barden [6] and Fredlund [7]. Fredlund and Hasan [7] proposed a one-dimensional consolidation theory, the most popular in the geotechnical engineering community, in which two partial differential equations were employed to describe the dissipation processes of excess pore pressures in unsaturated soils. Meanwhile, Qin et al. [2] gave an analytical solution for Fredlund's one-dimensional consolidation equation by applying the Laplace transform and Cayley-Hamilton mathematical methods in unsaturated soil with a finite thickness. Their boundary conditions were the top surface being permeable and the bottom surface being impermeable to air and water. Subjecting to the load exponentially varying with time, and using the same method and employing the same boundary conditions, Qin et al. [8] presented an analytical solution to the one-dimensional consolidation in unsaturated soils. Subjected to an arbitrary load, Shan et al. [9] employed a segregation variable method to obtain some exact solutions with three basic boundary conditions for unsaturated single-layer soils.
ACTA GeOTeCHNICA SLOVENICA, 2014/1 41.
J.W. LI & H.B. WANG: ANALYTICAL SOLUTIONS FOR ONE-DIMENSIONAL CONSOLIDATION IN UNSATURATED SOILS CONSIDERING THE NON-DARCY LAW OF WATER FLOW
However, the assumption focuses in the above contributions were based on Darcy's law, which is valid regardless of the magnitude of the hydraulic gradients. Indeed, some evidence shows that the flow of pore water in unsaturated soil may not obey Darcy's law. There are currently very few models of unsaturated soils that take into account the non-linear flux-gradient relationship. The non-Darcy law is actually a non-liner flux-gradients relationship. In particular, there are no analytical solutions of the consolidation in unsaturated soils that take into account the non-Darcy law. Cui et al. [10] reported non-Darcy behavior for a range of observed hydraulic gradients under unsaturated conditions. Considering water as a non-Newtonian fliud, Liu [11] deduced a constitutive model for unsaturated soils. Liu and Birkholzer [12] proposed a relationship by generalizing the existing non-Darcy law and Darcy law.
This paper aims to present a mathematical model of one-dimensional consolidation in unsaturated soils by considering the non-linear flux-gradient relationship, and derive its exact solution by a homotopy analysis method.
A homotopy analysis method (HAM) [13] was adapted to solve the non-linear model of one-dimensional consolidation under unsaturated conditions. The method was independent of any small or large parameters and was valid for most non-linear problems in science and engineering. The homotopy analysis method has been successfully applied to many nonlinear problems [14-16]. Finally, in order to verify the analytical solution, a comparison was carried out between the analytical solutions and the results of the finite-difference method in some cases. The results indicated that the analytical solution in the present study was reasonable. Moreover, the analytical solutions given here were valuable for understanding consolidation in unsaturated soil.
2 GOVERNING EQUATIONS 2.1 ASSUMPTIONS
The main assumptions for the one-dimensional consolidation governing equations are listed as follows:
(1)	the solid particles and water phase are incompressible.
(2)	the water flow is governed by a non-Darcy-type law, but air flow is governed by a Darcy-type law.
(3)	the effects of temperature change, air dissolved in water, air diffusion, and the generation and diffusion of vapor are ignored.
2.2 NON-DARCY LAW
A general relationship between water flux and hydraulic gradients under unsaturated conditions was proposed by Liu [12] based on Swartzendruber's work [13]. The relationship can be written as
v = -k(i - J' exp{-(X)-}dx) Jo	I
(1)
where v is the water velocity, k is the water permeability coefficient, i is the hydraulic gradient, I is the threshold gradient, and a is a constant parameter.
The one-dimensional differential form of Eq. (1) can be written as
du	o——+1	x
v = -k(—^ + 1-J 7wdz exp{-(X)a}dx) (2)
gw dz
I
where uw is the pore-water pressure, yw = pg , p is the water density, and g is the acceleration due to gravity.
The integrand of the third item in equation (2) can be expanded as a Taylor series
v-	.____r
exp{-( r )-} = E (-1)"-V I n=0 n!In-
(3)
Taking the first two of the Taylor series, integral terms
du
with the condition of v = 0 at-— +1 = 0 are simplified as gwdz
» duw +1 r	du
J gwdz +(1 - (X)- )dx = + 1 Jo	T	g dz
1 ( duw + 1)-+1
gwdz — + 1)I- gwdz
(4)
Substituting Eq. (4) into Eq. (2), the following equation can be obtained as
v = -k—^~ (-W + 1)a+1	(5)
(a + 1)Ia gwdz
The equation reduces to Darcy's law when a becomes zero (I # 0), i.e.,
v = -k(dUW + 1)	(6)
dz
2.3 CONSOLIDATION OF UNSATURATED SOILS
Following Fredlund and Morgenstern [18], the constitutive relation for the water and air phases are
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J.W. LI & H.B. WANG: ANALYTICAL SOLUTIONS FOR ONE-DIMENSIONAL CONSOLIDATION IN UNSATURATED SOILS CONSIDERING THE NON-DARCY LAW OF WATER FLOW
d(Vw / V) dt
d(Va / V) dt
d(s - Ua K w d(ua - Uw )

dt
dt
— mtk 9{a - Ua ) + ma d(Ua - )
dt
dt
(7)
(8)
d(V / V)
where,-- is the volume change of water in the
dt 8 soil, m— is the coefficient of water volume change with
respect to the change in the net normal stress ff-ua , mis the coefficient of water volume change with respect to
d(V / V)
the change in the matrix suction ua-uw ,—d— is the volume change of the air in the soil, m^ is the coefficient of air volume change with respect to the change in the net normal stress o-ua , and ma is the coefficient of air volume change with respect to the change in the matrix suction ua-uw . The subscript k stands for the K0-loading condition without lateral deformation.
The continuity requirement leads to the following relations [3,18]:
mlk = m— + m"ik	(9)
mS, = m- + mi, (10)
where m^ and m2 are the coefficients of volume change of the soil with respect to a change in the net normal stress — — ua, and the matrix suction ua-uw, respectively. az is the total normal stress in the z direction.
According to the law of mass conservation for water, the change in water volume can be written as follows:
d(Vw / V«) dvn
dt
dz
(11)
Substituting Eq.(5) and Eq.(7) into Eq.(11), the following equation can be obtained as
duw dt
-C,
1-'
dua ' dt
_ _SIM
(-%- + 1)" (12)
where, Cw — -
n1k fW _
,CV =
d2uM
dz2 dz
k„,
gwmWl "
The air is considered to behave as ideal air, and based on Boyle's law and Darcy's law. The governing equation for the air phase yields
Ca — k _
CV = ka _0
RT
gu0M(m1k - ma2 - (1 - S)nuatm / U )2 )
u0 = ua + uatm > ka is the air conductivity, R is the universal air constant, T is the absolute temperature, M is the average molecular mass of the air phase, u°a is the initial excess air pressure, and uatm is the atmospheric pressure.
2.4 BOUNDARY AND INITIAL CONDITIONS
In the present study, an unsaturated soil layer was considered as an infinite horizontal extent and thickness H (as shown in Fig.1).The top surface is permeable to water and air, whereas the bottom is impermeable to water and air.
	
/	0
Top surface	
	Unsaturated soil
	layer
Bottom surface	
\	
//////////y	V/////////
Figure 1. One-dimensional consolidation in unsaturated soils with a permeable top surface and an impermeable bottom base.
Hence, the initial conditions and boundary conditions are respectively expressed as
ua(z, 0) = u1,uw(z,0) = uW
ua (0, t ) — u0, uw (0,t ) — uW dua (H, t) — 0 duw (H, t) — 0
(14)
(15)
dz ' dz
where u°a and u0 are the initial excess air and water pressures at t = 0, respectively. (0 < z < H , t > 0)
3. ANALYTICAL SOLUTION
dua dt
where Ca —
C
d^w dt
m
a
dz2
2
mak - ma - (1 - S)nuatm /(u°)2
(13) 3.1 THE ANALYTICAL SOLUTION OF THE EQUATIONS
The Eqs.(12)- (15) can now be rewritten in the dimen-sionless form as
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J.W. LI & H.B. WANG: ANALYTICAL SOLUTIONS FOR ONE-DIMENSIONAL CONSOLIDATION IN UNSATURATED SOILS CONSIDERING THE NON-DARCY LAW OF WATER FLOW
C d20w/ 96ws 96w	96a
(1 +	= + CwC2	(16)
Ia Qr]	9t	9T
C
d]2
_ C
96n
C
96„
9t 9t subject to the initial and boundary conditions
6a (r,0) = 1,6w (r,0) = 1 , in 0 < r < i
6a (0, T ) = 1,6? (0, T ) = 1
dea (1, T ) = 0, 96? (1, T ) = 0
, in t > 0
(17)
(18) (19)
9r
9r
where, the dimensionless parameters are defined by
k t
ua	uw	z
q __a 6 _ w r __ t = -
ua~ 0	uw~ 0	' u st t2
U0	U0	H lwm^kh
a	w
0
C0 _
gwH
, C1 _
m
1k
u0
, C2 _ — ,
mW? 2 uW
2w
ka u°
C _ a a
RT lwmlk
3 kw uW gü°a M(malk - ma2 - (1 - S)nuatm / (u°a )2 )
3.2 SGRieS SOLUTIONS GIVEN BY THE HAM
As a nonlinear analytical technique, the homotopy analysis method (HAM) is efficient in the selection of a series of basis functions and auxiliary linear operators, and easily makes the solution convergence. The technique is based on homotopy, which is an important part of topology. Using one interesting property of homotopy, we can transform any nonlinear problem into an infinite number of linear problems, no matter whether or not there exists a small or large parameter. These linear problems are not dependent on any small parameters, which is convenient for controlling the convergence region. After this transferring, a series solution to the nonlinear problem is then obtained by the HAM after the selection of auxiliary linear operator parameters. [13]
We chose the auxiliary linear operator,
L _
9r2
(20)
From Eq.(16), it is straightforward to define the nonlinear operator
C1 <9 6? 96?
96n
N(qw)_r-¡rr? -C?C2-Qt (21)
From Eq.(17), we define the operator
92q _ rq ~dt
9r2
and the initial approximation
) _ Cs^rf - C2^-Ca-^ (22)
96?
9t
C = =1 + h(1 - h)2 exp(-0.4t) + V(1 - h)2 exp(-0.2t) (23)
The zero-order deformation equation is constructed as (Liao 2004)
(1- p)L(f(t, h ; P) - O0W ) = phHfN (fw (t, h; p)) (24)
(1 - p)L(f (t,h;p) - 0°) = phHfLa (f (t,h;p)) (25) subject to the conditions
f (h,0) = 0 , in 0 < h < 1 , (0, t ; p) = 0
^ (1, t ; p)
, in t > 0
(26) (27)
9r
_0
where, i = m, a , p G (0,1) is the embedding parameter, h is a non-zero auxiliary parameter, Hf is a auxiliary function, and $ is an unknown function of t, q, p respectively. It is obviously that when p = 0 and p = 1, it respectively holds that fi (t, r,0) = 0'0 ,f(t, r,1) = $ (t, r). Then when p increases from 0 to 1, q, p) varies from $0 to $ (t, r). With respect to p due to the Taylors' series, q, p) can be expanded, i.e.

■■ (t, r, p) _E 6k (t , r)pk
k_0
where
6k(T, r) _
1 9kf (t, r, p).
k !
9pk
p_0
(28)
(29)
If h is chosen in such a way that this series is convergent at p = 1, so we have
6 (t , r) _E 6'm (t , r)
m_0
(30)
Differentiating the zero-order deformation equations m times with respect to p, then dividing by m!, and finally setting p = 0, we have the mh-order deformation equations, i.e.,
L($w (t, r) - XmC1 (t, r)) = hHfRm (t, r) (31) L(C (t, r) - Xm$m-1(t, r)) = hHfRm (t, r) (32)
subject to the conditions
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J.W. LI & H.B. WANG: ANALYTICAL SOLUTIONS FOR ONE-DIMENSIONAL CONSOLIDATION IN UNSATURATED SOILS CONSIDERING THE NON-DARCY LAW OF WATER FLOW
0'm(h,0) = 0 in 0 < h < 1
OL (0, t ) = 0
dem (1, t )
, in t > 0
dh
=0
where
and
cm
0,	m < 1,
1,	m > 1
(33)
(34)
(35)
q d2em-1 + a^ m-1 d2qT dem-1
Rm (t, h) =
10
dh2
j m—1—j
j=0
de
m—1
d2q a
Ram (t, h)=C3 —m-1—C dh
dt
de
—CC
de
m—1
m—1
dt
— Ca
dt de
m—1
dt
(36)
(37)
To satisfy the initial condition Eq.(33), we chose the auxiliary function as: Hf = t.
The solution to the above equations was obtained using Maple software. It was seen that 6^ , 6^ can be respectively expressed by
2m+2 2m+3 2m+1
= EE E am (h)
i=0 j=0 k=0
1
—'T ■ 1 5 I k ; 5 ht
2m 2m+3 2m
=EEE j (h)
i=0 j=0 k=0
1
—it . , 5 I k
e 5 ht
(38)
(39)
where aj (h),bjk (h) are dependent upon h. And j (h),bjjk (h) can be easily obtained, substituting Eqs. (38)-(39) into the m-order deformation Eqs. (31)-(35) and all coefficients ajk , bjjk (see appendix A) of the solution can be obtained one by one from the first coefficients. The first coefficients were given by the initial approximation Eq.(23), i.e.,
0 (h) = 1, a0n(h) = a?3!(h) = 1, a°n(h) = a2Vh) = 1,
0 n.\ „0
000
a°21(h) = a^(h) = —2 So the solution can be given by
2m+22m+32m+1
(40)
1
--It
Ow (t, h) =EE E E j (h)e5 hj tk (41)
m=0 i=0 j=0 k=0
2m 2m+3 2m
.(t, h) =EEEEbmk (h)e
m=0 i=0 j=0 k=0
—ii	. j
5 I k 5 ht
(42)
4 EXAMPLE AND VeRIFICflTION
It is important to ensure that the solution series can be converging. Fortunately, the convergence and the rate of approximation for the HAM solutions strongly depend on the values of the auxiliary parameter h. It is found that h must be negative (h £ (—1,0)) to ensure that the solution series converges.
In order to validate the analytical solution of one-dimensional consolidation for unsaturated soil, a typical example is computed by using the analytical solution and a comparison is made between the analytical solution of the mathematic model and the numerical results obtained by the finite-difference method using the typical example. The material parameters for the example are adopted as the same parameters in Table 1 and Table 2 [3].
Table 1. The material parameters for the solution of the examples.
Layer thickness (m)	H	10
Initial pore-air pressure (kPa)	U0	20
Initial pore-water pressure (kPa)	uW	40
Porosity	n	0.5
Saturation	S	0.8
Coefficient of volume change (kPa-1)	m1k	-2.5x10-4
Ratio of two coefficients of volume change	m2/mii	0.4
Ratio of two coefficients of volume change	mWk/m1k	0.2
Ratio of two coefficients of volume change	mWlmxk	4
Table 2. The values of the parameters Cj (i = 0,1,2,3,w,a).		
Parameter of equation	C0	0.4
Parameter of equation	C1	1.25
Parameter of equation	C2	0.5
Parameter of equation	C3	80.6
Interactive constant for water phase	Cw	-0.75
Interactive constant for air phase	Ca	-0.00775
Fig.2 and Fig.3 show that the change in the values of Qw, 9a at a = 2, I = 10, h = -0.001 with t under different n From the dimensionless definitions of the parameters, i.e., the change in the values of pore water pressure and pore air pressure at a = 2, I = 10, h = -0.001 with time under different depth z, we can find that the excess pore-water
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J.W. LI & H.B. WANG: ANALYTICAL SOLUTIONS FOR ONE-DIMENSIONAL CONSOLIDATION IN UNSATURATED SOILS CONSIDERING THE NON-DARCY LAW OF WATER FLOW
pressure is gradually increasing at beginning. It reaches the highest value at about t = 5 (i.e., t = 1250/kw) and then begins to dissipate. But the excess pore-air pressure reaches the highest value at about t = 1 (i.e., t = 250/kw) and then begins to dissipate.
Fig.4 and Fig.5 show that the change in the values of Qw Qa at a = 2, I = 10, h = -0.001 with n under different t. That is the excess pore-water pressure and the excess pore-air pressure increase initially, and then decrease with the depth.
Figure 2. The 8th-order HAM approximations of dw at n = 0.1, 0.3, 0.5, a = 2, I = 10, h = -0.001 open circle: the numerical results.
0	2	4	6	8	10
T
Figure 3. The 8th-order HAM approximations of da at n = 0.1, 0.3, 0.5, a = 2, I = 10, h = -0.001 open circle: the numerical results.
52. ACTA GeûTeCHNICA SLOVENICA, 2014/1
J.W. LI & H.B. WANG: ANALYTICAL SOLUTIONS FOR ONE-DIMENSIONAL CONSOLIDATION IN UNSATURATED SOILS CONSIDERING THE NON-DARCY LAW OF WATER FLOW
Figs. 2-5 show the comparison of the 8th-order HAM approximations and numerical solutions. It is clear that the computed results using the two methods are almost
the same, which proves that the analytical solution proposed in this paper is credible.
Figure 4. The 8 -order HAM approximations of 8w at t = 0.2, 0.5, 0.6, a = 2, I = 10, h = -0.001 open circle: the numerical results.
0	0.2	0.4	0.6	0.8	I
il
-The 8th-order HAM approximations
° The numerical results_
Figure 5. The 8th-order HAM approximations of da at t = 0.2, 0.5, 0.6, a = 2, I = 10, h = -0.001 open circle: the numerical results.
52. ACTA GeûTeCHNICA SLOVENICA, 2014/1
J.W. LI & H.B. WANG: ANALYTICAL SOLUTIONS FOR ONE-DIMENSIONAL CONSOLIDATION IN UNSATURATED SOILS CONSIDERING THE NON-DARCY LAW OF WATER FLOW
Figs. 6-7 show that the values of I have obvious effects on pore-water pressure. However, Fig. 6 shows that the the dissipation values of the excess pore-water pressure.	values of I have no significant effect on the dissipation
It indicates that for smaller I, there is a smaller excess	values of the excess pore-air pressure.
Figure 6. Change in dw with t under different I at q = 0.1, 0.3, 0.6, respectively.
0	2	4	6	8	10
T
Figure 7. Change in da with t under different I at q = 0.1, 0.3, 0.6, respectively.
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J.W. LI & H.B. WANG: ANALYTICAL SOLUTIONS FOR ONE-DIMENSIONAL CONSOLIDATION IN UNSATURATED SOILS CONSIDERING THE NON-DARCY LAW OF WATER FLOW
5 CONCLUSIONS
In this paper the water phase was assumed to obey the non-Darcy law. The simplified form of the governing equations for the one-dimensional consolidation of unsaturated soil was adopted. The series solutions based on the HAM for the excess pore-air pressure and the excess pore-water pressure were first obtained for the dimensionless consolidation in unsaturated soils. The solutions were applicable to the unsaturated soil layer with pore water and pore air pressure equivalent for the initial value at the top surface and the bottom impermeable to water and air. In addition, the validity of the present solutions was confirmed through a comparison with the numerical results.
Based on the solutions, the changes in the excess poreair pressure and excess pore-water pressure with time were analyzed at different I and for different depths. In addition, it was found that the dissipation of air pressure was much faster than the dissipation of water pressure. Furthermore, the values of I had obvious effects on the dissipation values of the excess pore-water pressure but no significant effect on that of the excess pore-air pres-
ACKNOML£DG£M£NTS
This research was supported by funding from the Natural National Science Foundation of China (41372296). In addition, partial support from the Doctoral Fund from the Ministry of Education of China (20100142110059) is acknowledged. A special note of appreciation is extended to the Ministry of National Science and Technology for their funding support (2012BAK10B00).
RGFGRGNCGS
[1]	Fredlund, D.G., Rahardjo, H. (1993). Soil Mechanics for Unsaturated Soil. John Wiley and Sons, New York.
[2]	Qin, A.F., Chen, G.J., Tan Y.W., Sun, D.A. (2008). Analytical solution to one-dimensional consolidation in unsaturated soils. Appl. Math. Mech., 29:1329-1340.
[3]	Conte, E. (2004). Consolidation analysis for unsaturated soils. Canadian Geotechnical Journal, 41, 599-612.
[4]	Blight G.E. (1961). Strength and consolidation characteristics of compacted soils. Dissertation. England: University of London.
[5]	Scott, R.F. (1963). Principles of soil mechanics. Addison Wesley Publishing Company.
[6]	Barden L. (1965). Consolidation of compacted and unsaturated clays. Geotechnique,15, 267-86.
[7]	Fredlund D.G., Hasan J.U. (1979). One-dimensional consolidation theory unsaturated soils. Canadian Geotechnical Journal,17, 521-31.
[8]	Qin, A.F., Sun, D.A., Tan, Y.W. (2010). Analytical solution to one-dimensional consolidation in unsaturated soils under load varying exponentially with time. Computers and Geotechnics, 37, 233-238.
[9]	Shan, Z.D., Ling, D.S., Ding, H.J. (2012). Exact solutions for one dimensional consolidation of single layer unsaturated soil. International Journal for Numerical and Analytical Methods in Geome-chanics, 36 (6), 708-722.
[10]	Cui, Y.J., Tang, A.M., Loiseau, C., Delage, P. (2008). Determining the unsaturated hydraulic conductivity of a compacted sand-bentonite mixture under constant-volume and free-swell conditions. Phys. Chem. Earth, 33, S462-S471.
[11]	Liu, H.H. (2011). A conductivity relationship for steady-state unsaturated flow processes under optimal flow conditions. Vadose Zone J., 10, 736-740.
[12]	Liu, H.H., Jens, Birkholzer. (2012). On the relationship between water flux and hydraulic gradient for unsaturated and saturated clay. Journal of Hydrology, 475, 242-247.
[13]	Liao, S.J. (2003). Beyond Perturbation: Introduction to Homotopy Analysis Method. Chapman & Hall/CRC Press, Boca Raton.
[14]	Liao, S.J., Pop, I. (2004). Explicit analytic solution for similarity boundary layer equations, Int. J. Heat Mass Transfer, 47, 75-85.
[15]	Liao, S.J. (2005). A new branch of solutions of boundary-layer flows over an impermeable stretched plate. Int. J. Heat Mass Transfer, 48, 2529-2539.
[16]	Liao, S.J. (2006). An analytic solution of unsteady boundary-layer flows caused by an impulsively stretching plate. Commun. Nonlinear Sci. Numer. Simul.,11, 326-339.
[17]	Swartzendruber, D. (1961). Modification of Darcy's law for the flow of water in soils. Soil Sci.,93, 22-29 (Soil Science Society of America Journal, 69, 328-342.)
[18]	Fredlund, D.G., Morgenstern, N.R. (1976). Constitutive relation for volume change in unsaturated soils. Canadian Geotechnical Journal, 13, 386-396.
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J.W. LI & H.B. WANG: ANALYTICAL SOLUTIONS FOR ONE-DIMENSIONAL CONSOLIDATION IN UNSATURATED SOILS CONSIDERING THE NON-DARCY LAW OF WATER FLOW
AP6NDIX A
The values for amk (h),
bijk(h) satisfy the recursive formula
2m+22m+32m+1	—1it	2m 2m+12m-1	—1it
L( EE E aj (h)e 5 hj tk - Xm EE E aj (h)e 5 hj tk) = hHfRj (t, h)
i=0 j=0 k=0 i=0 j=0 k=0
2m 2m+3 2m	——it	2m-22m+12m-2	——^
L(EEEbi^ (h)e 5 hjtk — Cm E E E j (h)e 5 hjtk) = hHfRm(t,h)
i=0 j=0 k=0 i=0 j=0 k=0
where
|0, m < 1, ¡1, m > 1
and
c —2 2m 2m+12m—1	—1it
Rm (t,h) = £—VEE E am, (h)e 5 hjtk)
1 —h i=0 j=0 k=0
aC C m—1 d2ew 2j+22j+32j+1	—lit —0 , ■ 2m—2j2m—2j+12m—2j—1	—
+^ £-2-(E EEj (h)e 5 h tk)-O^E EE) EE) a^ (h)e5hltk)
1 j=0 —h i=0 l=0 k=0 — i=0 l=0 k=0
— 2m 2m+12m—1	—1it	— 2m—2 2m+12m—2	—1it
—--(EE E a,^ (h)e 5 hj tk) — CwC2 — ( E E E (h)e 5 hj tk)
81 i=0 j=0 k=0 —t i=0 j=0 k=0
—2 2m—22m+12m—2	—1it	— 2m—22m+12m—2	—1it
Rm(t,h) = C3fy( E E E j(h)e 5 hjtk) — C2 — ( E E E bi^(h)e 5 hjtk)
8 h i=0 j=0 k=0 81 i=0 j=0 k=0
— 2m 2m+12m—1	—1it
— Ca f- (EE E ^ (h)e 5 hj tk)
—t i=0 j=0 k=0
cm
60. ACTA GeOTeCHNICA SLOVeNICA, 2014/1