Research Article Finite Element Analysis of Biot s

Research ArticleFinite Element Analysis of Biotrsquos Consolidation witha Coupled Nonlinear Flow Model

Yue-bao Deng1 Gan-bin Liu1 Rong-yue Zheng1 and Kang-he Xie2

1 Institute of Geotechnical Engineering Ningbo University Ningbo 315211 China2Research Center of Coastal and Urban Geotechnical Engineering Zhejiang University Hangzhou 310058 China

Correspondence should be addressed to Yue-bao Deng dengyuebaoaliyuncom

Received 24 September 2015 Revised 20 December 2015 Accepted 22 December 2015

Academic Editor Thibault Lemaire

Copyright copy 2016 Yue-bao Deng et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A nonlinear flow relationship which assumes that the fluid flow in the soil skeleton obeys the Hansbo non-Darcian flow and thatthe coefficient of permeability changes with void ratio was incorporated into Biotrsquos general consolidation theory for a consolidationsimulation of normally consolidated soft ground with or without vertical drains The governing equations with the couplednonlinear flow model were presented first for the force equilibrium condition and then for the continuity condition Based onthe weighted residual method the finite element (FE) formulations were then derived and an existing FE program was modifiedaccordingly to take the nonlinear flow model into consideration Comparative analyses using established theoretical solutions andnumerical solutions were completed and the results were satisfactory On this basis we investigated the effect of the couplednonlinear flow on consolidation development

1 Introduction

The theory of consolidation which forecasts the time-dependent deformation of a soil foundation is one of basicissues in the field of soil mechanics [1] Terzaghi [2] madeuse of the effective stress principle and first put forward theone-dimensional consolidation theory in the early 1920s Biot[3] further studied the interaction between the dissipation ofexcess pore pressures and the deformation of the soil skeletonand introduced a coupled consolidation theory Since thenBiotrsquos consolidation theory and its modifications have beenused extensively which hasmademore precise analysesmuchmore possible [4ndash7] It can be seen from these studies thatDarcyrsquos flow law which assumes a linear correlation betweenthe flow and hydraulic gradient has been commonly used forits simplicity

The validity of Darcyrsquos law has been widely recognizedhowever it has also been noted that the flow deviates fromDarcyrsquos law in the case of large plastic sticky soil with a smallhydraulic gradient Hansbo [8] reported that there are severalexperimental cases indicating that the flow may correlate toan exponential trend with the gradient at low gradient values

In addition Qi et al [9] also found the fit results with non-Darcy flow models to be relatively good for Xiaoshan clayThus the influence of non-Darcian flow which was utilizedto depict the deviation of water flow from Darcyrsquos law hasbeen assessed by a series of studies in the simulation of softground consolidation behavior Hansbo [10ndash13] conductedthe pioneering and systematic work in the application ofnon-Darcian flow to the uncoupled consolidation problemHis studies have aroused the attention of several researchersin regard to this issue Pascal et al [14] first consideredthreshold hydraulic gradient 119894

0in a 1D consolidation problem

and obtained the solution using the finite different method(FDM) Dubin and Moulin [15] extended Terzaghirsquos one-dimensional consolidation to consider the Hansbo non-Darcian flow model In recent years Xie et al [16] andLi et al [17 18] published a series of papers regardingthe analyses of one-dimensional consolidation with non-Darcian flow In general studies that consider non-Darcianflow for multidimensional consolidation analyses have beenrelatively rare Teh and Nie [19] derived the finite element(FE) equations for Biotrsquos axisymmetric consolidation problembased on the principle of virtual work and investigated the

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 3047213 13 pageshttpdxdoiorg10115520163047213

2 Mathematical Problems in Engineering

v

io

i0 il

= 120581im

= k(i minus i0)

Figure 1 The Hansbo non-Darcy flow model

effect of non-Darcian flow on soft ground consolidationbehavior However their work was not suitable for theconsolidation analysis of multidrain systems and had nottaken the variability of soil permeability into consideration

Based on these previous studies this paper aims toextend Biotrsquos general three-dimensional consolidation theoryto incorporate non-Darcian flow together with the consider-ation of the variation of the soil permeability during the con-solidation process Governing equations for the force equi-librium condition and the continuity condition will be pre-sented first and the finite element method (FEM) will be uti-lized here to solve these equations Some specific techniquesin the programming process for the newly derived formulaswill be stated in detail The validation of the program will beperformed for two cases

2 A Coupled Nonlinear Flow Model

Thefluid flow in the poremedia obeys theHansbo non-Darcyflow law (see Figure 1) which can be described as follows [8]

V = 120581119894119898 when 119894 lt 119894119897

V = 119896 (119894 minus 1198940) when 119894 ge 119894

119897

(1)

where V is flow velocity 119896 is permeability coefficient of thelinear flow at high gradient 120581 is a permeability parameterthat is applied in the exponential part 119898 is the exponent ofexponential flow at low gradient 119894

0is a threshold gradient

and 119894119897represents the gradient required to overcome the

maximum binding energy of mobile pore water [11] Severalstudies reported that the value of 119898 was within 10sim20 andthe typical value was 15 whereas the value of 119894

119897ranged in 0sim

40 [8] Note here that when119898 = 1 and 1198940= 0 (1) degenerates

to Darcyrsquos linear flow law that is V = 119896119894Combined with the conditions of the continuity and

differentiability at 119894 = 119894119897 the following expressions can be

derived from (1) [19]

V =119896119894119898

(119898119894119898minus1

119897)

when 119894 lt 119894119897

V = 119896 (119894 minus119898 minus 1

119898119894119897) when 119894 ge 119894

119897

(2)

In addition a nonconstant permeability is assumed forsoft soil The coefficient of permeability 119896 is one of the mostimportant parameters for studying nonlinear properties ofsoil and a linear correlation between the void ratio 119890 andlg 119896 is valid for the most common clays [20] which can beexpressed as follows

119890 = 1198900+ 119862119896lg( 119896

1198960

) (3)

where 1198900and 119896

0are the initial void ratio and the initial

permeability coefficient respectively and119862119896is seepage index

There is a linear relationship in 119890-lg1199011015840 for normallyconsolidated soft soils according to the oedometer test Thatis

119890 = 1198900minus 119862119888lg(

1199011015840

1199011015840

0

) (4)

where119862119888is compressive index 1199011015840 is the mean effective stress

1199011015840

0is initialmean effective stress themean effective stress1199011015840 =

(1205901015840

119909+ 1205901015840

119910+ 1205901015840

119911)3 1205901015840119909 1205901015840119910 and 1205901015840

119911are the effective stress in the

119909- 119910- and 119911-directions respectivelyCombining (3) and (4) it can be obtained that

119896 = 1198960(1199011015840

1199011015840

0

)

minus120572

(5)

where 120572 = 119862119888119862119896 120572 is in 05sim2 and the common value is in

05sim10 [20]A coupled nonlinear flow model which takes the non-

constancy of soil permeability and non-Darcian flow law intoaccount is then formed by a combination of (2) and (5)Here it should be pointed out that the settlement of soft soilfoundations will be affected by a series of other factors suchas the creep (viscous) behavior of clayey soil (eg [21 22]) theoverconsolidated condition and loading pattern This papermainly focuses on the study of the influence of varied soilpermeability and non-Darcian flow on consolidation process

3 Governing Equations

The basic assumptions on the coupled consolidation problemare the same as Biotrsquos general consolidation theory except forthe above newly presented flow model Equations for forceequilibrium and flow continuity relations are presented herefor comprehensive consideration

31 Force Equilibrium Equation The compact form of theforce equilibrium equation which is the same as it is for alinear flow model can be expressed as follows [23ndash25]

[120597]119879[D] [120597] f minus [120597]119879 M 119901 = 0 (6)

where the displacement vector f = [119906 V 119908]119879 119906 Vand 119908 are displacements in the 119909- 119910- and 119911-directionsrespectively119901 is the excess pore pressure (EPP) vector M =[1 1 1 0 0 0]

119879 the elastic matrix [D] about the isotropiclinear elastic material and the differential operator [120597] are thesame as those for a homogeneous element

Mathematical Problems in Engineering 3

y

z

o

xdy

dx

dzy

x

z

Figure 2 Schematic diagram of the soil unit

32 Flow Continuity Equation Figure 2 shows a typicalelement with widths d119909 and d119910 and thickness d119911 To satisfythe continuity condition in saturated soil the net flow rateshould be equal to the rate of volume change [3] Consider

120597V119909

120597119909+

120597V119910

120597119910+120597V119911

120597119911= minus

120597120576V

120597119905= minus

120597

120597119905(120597119906

120597119909+120597V120597119910+120597119908

120597119911) (7)

where V119909 V119910 and V

119911are flow velocities in the 119909- 119910- and 119911-

directions respectively 120576V is volumetric strain The matrixform of (7) can be rewritten as follows

M119879 [120597] k = minus 120597120597119905(M119879 120576) (8)

where the flow velocity vector k = [V119909V119910V119911]119879 the strain

vector 120576 = [120576119909120576119910120576119911120574119909119910120574119910119911120574119911119909]119879

Substituting the flow velocity of three directions into(8) the continuity equation can be obtained Note here thatit is necessary to consider the flow velocity componentsindividually because the correlation between the flow velocityV and hydraulic gradient 119894 in each direction may follow eithera linear or an exponential relationshipThere are nine possibleconditions when comparing the magnitudes of hydraulicgradients in the 119909- 119910- and 119911-directions with 119894

119897 To simplify

the following derivation process symbol Ψ is introduced todenote the 119909- 119910- or 119911-direction

For any direction Ψ it can be deduced from (2) that

VΨ=119878Ψ119896Ψ

1003816100381610038161003816119894Ψ1003816100381610038161003816

119898

(119898119894119898minus1

119897)

when 1003816100381610038161003816119894Ψ1003816100381610038161003816 lt 119894119897

VΨ= 119878Ψ119896Ψ(1003816100381610038161003816119894Ψ1003816100381610038161003816 minus119898 minus 1

119898119894119897) when 1003816100381610038161003816119894Ψ

1003816100381610038161003816 ge 119894119897

(9)

in which 119878Ψ(the subscript Ψ represents 119909 119910 or 119911) indicates

the direction of the hydraulic gradient When the pore waterflows to the positive direction of axis (see Figure 2) then thedirection of the hydraulic gradient is positive and 119878

Ψis +1

otherwise it is minus1 Additionally 119896Ψindicates the permeability

coefficient in the 119909- 119910- or 119911-direction According to (5)

119896Ψ= 119896Ψ0(1199011015840

1199011015840

0

)

minus120572

(10)

The derivation of (9) can be written as follows

120597VΨ

120597Ψ=

119896Ψ

120574119898119908119894119898minus1

119897

10038161003816100381610038161003816100381610038161003816

120597119901

120597Ψ

10038161003816100381610038161003816100381610038161003816

119898minus11205972119901

120597Ψ2when 1003816100381610038161003816119894Ψ

1003816100381610038161003816 lt 119894119897

120597VΨ

120597Ψ=119896Ψ

120574119908

1205972119901

120597Ψ2when 1003816100381610038161003816119894Ψ

1003816100381610038161003816 ge 119894119897

(11)

where 120574119908is the unit weight of water

According to the study of Teh and Nie [19] item |120597119901120597Ψ|

can be computed approximately by using the value of 119901 at theprevious time step Combined with (10) (11) can be rewrittenin a uniform expression as follows

120597VΨ

120597Ψ= 119867Ψ

119896Ψ0

120574119908

(1199011015840

1199011015840

0

)

minus120572

1205972119901

120597Ψ2(12)

in which

119867Ψ=

1

120574119898minus1119908119894119898minus1

119897

10038161003816100381610038161003816100381610038161003816

120597119901

120597Ψ

10038161003816100381610038161003816100381610038161003816

119898minus1

when 1003816100381610038161003816119894Ψ1003816100381610038161003816 lt 119894119897

1 when 1003816100381610038161003816119894Ψ1003816100381610038161003816 ge 119894119897

(13)

Substituting (12) into (8) the matrix form of flow continuityequation can be obtained

M119879 [120597] [H] [k] [120597]119879 M 119901 minus 120597

120597119905M119879 [120597] f = 0 (14)

where [H] is flow control matrix [k] is permeability matrixand

[H] = (1198671199090 0

0 1198671199100

0 0 119867119911

) (15)

[k] = 1

120574119908

(1199011015840

1199011015840

0

)

minus120572

(

1198961199090

0 0

0 1198961199100

0

0 0 1198961199110

) (16)

in which 119867119909 119867119910 and 119867

119911can be obtained from (13)

respectively 1198961199090 1198961199100 and 119896

1199110are the initial permeability

coefficients in the 119909- 119910- and 119911-directions respectivelyNote here that if parameter 119898 is set as 0 then the matrix

[H] degenerates to a unit matrix if the value of 120572 is set as 0the matrix [k] for nonconstant permeability degenerates tothe matrix for the constant case Further the flow continuityequation that is (14) will be reduced to Biotrsquos generalconsolidation equation

33 Boundary Conditions As for the coupled consolidationproblem of flow in saturated two-phase porous media thereare four categories of boundary conditions the displacementboundary the force boundary the EPP boundary and theflow velocity boundary The first two boundary conditionsare the same as general mechanical problems while the EPPboundary usually refers to the zero value of EPP on thedrainage surface In addition for a typical soil unit (see


y

z

o

x

2b

2a

2c

85

41

2 3

6 7

p

o998400120585

120577120578

u w

Figure 3 Spatial 8-node element

Figure 2) the flow velocity V119899 in the normal direction of a

random boundary surface 119878 can be expressed as follows

119897V119909+ 119898V119910+ 119899V119911= V119899 (17)

where 119897 119898 and 119899 denote the direction cosine of anglesbetween surface 119878 and the positive direction of 119909- 119910- and 119911-axis respectively A flow velocity of zero (V

119899= 0) is assumed

for completely impervious surfaces

4 Finite Element Approximations

The Galerkin weighted residual method (eg [23 25]) isadopted to transform the above governing equations into aform suitable for obtaining a solution by FEM In additionthe spatial 8-node element is utilized for spatial discretization(see Figure 3) Each node in this element has four degrees offreedom that is 119906 V 119908 and 119901

The FE approximations of displacements and EPP aregiven by

f asymp f = V 119879 = [N] 120575119890

119901 asymp = [N] p119890 (18)

where V and are approximate solutions of displace-ment in the 119909- 119910- and 119911-directions respectively is theapproximate solution of EPP the nodal displacement vectoris defined by 120575119890 = [120575119879

1120575119879

2sdot sdot sdot 120575

119879

8]119879 120575119894= [119906119894V119894119908119894]119879

119894 = 1 2 8 119906119894 V119894 and 119908

119894are nodal displacements in

the three directions respectively nodal EPP vector is definedby p119890 = [119901


8]119879 119901119894is the nodal EPP [N] =

[1198731I 1198732I sdot sdot sdot 119873

8I] I denotes the cell matrix [N] =

[11987311198732sdot sdot sdot 119873

8]119873119894is the shape function and

119873119894=1

8(1 + 120585120585

119894) (1 + 120578120578

119894) (1 + 120577120577

119894) 119894 = 1 2 8 (19)

where 120585119894 120578119894 and 120577

119894are local coordinates of each node

41 Equilibrium Equation in FE Formulation According tothe weighted residual method equations eliminating theresiduals of the equilibrium equation and the force boundaryequation can be obtained when selecting 119873

119894as the weight

function After reorganization the following equilibriumequation can be obtained which is the same as it is for thelinear flow condition

[K119890] Δ120575

119890+ [K119888] Δp119890 = ΔR

119891119890 (20)

in which

[K119890] =∭

119881119890

[B]119879 [D] [B] d119881

[K119888] =∭

119881119890

[B]119879 M [N] d119881

ΔR119891119890

= ∬119878119890

[N]119879 ΔF119890 d119878

Δ120575119890= [Δ120575

119879

1Δ120575119879

2sdot sdot sdot Δ120575

119879

8]119879

Δp119890 = [Δ1199011 Δ1199012 sdot sdot sdot Δ1199018]119879

ΔF119890 = Δ119865119909 Δ119865119910 Δ119865119911119879

(21)

where [K119890] [K119888] and ΔR

119891 are the stiffness matrix the cou-

pling matrix and the equivalent incremental nodal load vec-tor respectively Δ120575 Δp and ΔF are incremental vectorsof the displacement EPP and force respectively the strain-displacement matrix [B] = minus[120597][N] = [Β

1B2sdot sdot sdot B

8] [B119894]

is a matrix with 6 times 3 order and [B119894] = minus[120597]119873

119894 119894 = 1 2 8

Δ119865119909 Δ119865119910 and Δ119865

119911are surface incremental forces in the 119909-

119910- and 119911-directions respectively In addition 119881119890 and 119878119890 arethe domain and boundary surface of the operating elementrespectively

42 Continuity Equation in FE Formulation Residuals of thecontinuity equation and flow velocity boundary condition are

119877119890

119868119902=120597

120597119905M119879 [120597] f minus M119879 [120597] [H] [k] [120597]119879 M

119877119890

119861119902= 119897V119909+ 119898V119910+ 119899V119911minus V119899

(22)

Equations eliminating the residuals are defined by the follow-ing expressions respectively

∭119881119890

119873119894(120597

120597119905M119879 [120597] f

minus M119879 [120597] [H] [k] [120597]119879 M ) d119881 = 0

119894 = 1 2 8

(23)

∬119878119890

119873119894(119897V119909+ 119898V119910+ 119899V119911minus V119899) d119878 = 0

119894 = 1 2 8

(24)


First substituting (18) into (23) we have

∭119881119890

119873119894(120597

120597119905M119879 [120597] f) d119881

=∭119881119890

119873119894M119879 [120597] [N] d119881

(25)

where 119890 = 120597120575119890120597119905 Combining the integration method

and the residual of the flow velocity boundary it can beobtained that

minus∭119881119890

119873119894(M119879 [120597] [H] [k] [120597]119879 M ) d119881

= ∬119878119890

119873119894V119899d119878

+∭119881119890

nabla119879119873119894 [H] [k] nabla [N] p

119890 d119881

(26)

where nabla = [120597120597119909 120597120597119910 120597120597119911]119879 Combining (23) (25) and

(26) we have

∭119881119890

119873119894 M119879[120597] [N] d119881

+∭119881119890

nabla119879119873119894 [H] [k] nabla [N] p

119890 d119881

= minus∬119878119890

119873119894V119899d119878 119894 = 1 2 8

(27)

Equation (27) is established fromnode 1 to node 8Thus thereare eight equationswhen (27) is expanded to eachnodeTheseequations can be expressed in a compact form as follows

[K119888]119879119890

minus [K119904] p119890 = R

119902119890

(28)

in which

[K119904] =∭

119881119890

[B119904]119879

[H] [k] [B119904] d119881

[B119904] =

120597

120597119909

120597

120597119910

120597

120597119911

119879

[1198731 1198732 sdot sdot sdot 1198738]

R119902119890

= ∬119878119890

[N]119879

V119899d119878

(29)

where [K119904] is the flow matrix R

119902119890 is the nodal equivalent

flow vector and specifically for the impermeable boundaryR119902119890= 0

Integrating both sides of (28) with respect to time from 119905119899

to 119905119899+1

we have

int

119905119895+1

119905119895

[K119888]119879119890

d119905 minus int119905119895+1

119905119895

[K119904] p119890 d119905

= int

119905119895+1

119905119895

R119902119890

d119905(30)

with the following approximation expression

int

119905119895+1

119905119895

p119890 d119905 = Δ119905 (p119890119899+ 120579 Δp119890)

int

119905119895+1

119905119895

R119902119890

d119905 = Δ119905 R119902119890

119899

(31)

The incremental form of continuity equation can be obtainedas follows

[K119888]119879

Δ120575119890minus 120579Δ119905 [K

119904] Δp119890 = ΔR

119901119890 (32)

in which

ΔR119901119890

= Δ119905 [R119902119890

119895+ [K119904] p119890119895] (33)

where ΔR119901119890 is the incremental nodal equivalent flow vector

andΔ119905 is the time increment Additionally 120579 is an integrationconstant A value of 120579 ge 05 is required to maintain thestability of the integration schemes [26]

Equations (20) and (32) are the FE formulations of Biotrsquosconsolidation theory considering the coupled nonlinear flowmodel in the current study Because themathematical deriva-tion of each item for general consolidation theory has beendescribed elsewhere (eg [23 25]) just the evaluation ofmatrices for nonlinear flow conditions will be discussed inthe following section

43 Matrix Evaluation and Programming Compared withthe element consolidation matrix for the linear flow modelmatrices [K

119890] and [K

119888] are the same while the flow matrix

[K119904] is different leading to a different value for ΔR

119901119890Matrix

evaluations of [K119904] and ΔR

119901119890 are the same as it is for general

consolidation theory but with an extra process for calculatingthe permeability matrix [k] and the flow control matrix [H]

According to (10) and (16) the permeability matrix atvarious time steps [k]

119894can be evaluated by iterative compu-

tation First the initial permeability matrix [k]0is formed

with the initial condition of soil permeability Simultaneouslythe initial stress field can be derived with the geostatic stressfield and the lateral pressure coefficient and then 1199011015840

0can be

calculated Substituting matrix [k]0into the FE iteration the

stress field of this step is obtained followed by ascertaining11990110158401

using a similar approach Substituting 11990110158401into (16) the flow

matrix [k]1is obtained The rest can be deduced by analogy

The evaluation of matrix [H] which can be consideredan adjusting matrix for determining the flow obeying eitherDarcyrsquos law or non-Darcian law depends on the magnitudecomparison between 119894

Ψand 119894119897 It will take on different forms

under different gradient conditions As referenced in theprevious section matrix [H] can be evaluated approximatelyby using the 119901 value of the previous time step Stated indetail under the local coordinate (see Figure 3) the hydraulicgradient of the next time step 119894119895+1 (the superscript 119895 denotes


the time step) can be calculated by the nodal EPP at currenttime step 119901119895 That is

119894119895+1

119909=1

120574119908

120597119901119895

120597119909

=

(119901119895

1+ 119901119895

4+ 119901119895

5+ 119901119895

8) minus (119901

119895

2+ 119901119895

3+ 119901119895

6+ 119901119895

7)

8120574119908119886

119894119895+1

119910=1

120574119908

120597119901119895

120597119910

=

(119901119895

1+ 119901119895

2+ 119901119895

5+ 119901119895

6) minus (119901

119895

3+ 119901119895

4+ 119901119895

7+ 119901119895

8)

8120574119908119887

119894119895+1

119911=1

120574119908

120597119901119895

120597119911

=

(119901119895

1+ 119901119895

2+ 119901119895

3+ 119901119895

4) minus (119901

119895

5+ 119901119895

6+ 119901119895

7+ 119901119895

8)

8120574119908119888

(34)

where 119886 119887 and 119888 are half sizes of the element in 119909- 119910- and 119911-directions respectively see Figure 3 The ascertained valuesof 119894Ψ119867Ψcan be calculated by (13) after a comparison between

119894Ψand 119894119897is completedThen the flow controlmatrix [H] of the

next time step is obtainedNote here that for Darcyrsquos flow which is a special

case of non-Darcian flow matrix [H] degenerates into aunit matrix Additionally two approaches can be used toascertain the initial flow control matrix [H]

0 Considering

that the hydraulic gradients decrease gradually from themaximum values at the initiation of consolidation to zerowhen consolidation has fully completed the first approach issetting matrix [H]

0as a unit matrix directly This approach

denotes that the flow at the initial condition follows the trendfor Darcy flow [19] Another approach is setting [H]

0as a

unit matrix at first and then evaluating it using an iterativeoperation The iterative process is the same as the processdescribed above which uses the EPP of the previous stepto calculate the matrix [H] of the current step The flowcontrol matrix [H]

119894for various time steps is also computed

by iteration in this approachBased on an existing numerical code [25 27 28] which

had been verified and applied in a series of studies a newprogram NDFEA (non-Darcy finite element analysis) wasdeveloped to analyze the consolidation process for both nat-ural and vertical drained foundations (either single-drain ormultidrain conditions) To check for the correct implemen-tation of the algorithms and formulations two comparativeanalyses for the analytical and numerical solutions will beperformed

5 Verification

51 Case 1 The first case is a fully penetrating single-drainfoundation with one clay layer which has also been used byHird et al [29] andTeh andNie [19]The cylindrical influencezone diameter is denoted by 119863

119890= 3m the smear zone

Z

Drain

Impervious

Impe

rvio

us

Impe

rvio

us

Impervious

Pervious

Impervious

Nature soil

Smear zonekw ks

kh k

bw = 886 cm

H = 5m

bs = 1772 cm

B e=2658

m

Figure 4 Calculation example for the first verification case analyses

diameter is 119889119904= 02m and the drain diameter is 119889

119908= 01m

According to the principle of equation area the cylindricalzone is translated into a block zone [25] see Figure 4 Thecorresponding sizes 119861

119890 119887119904 and 119887

119908for the equivalent system

are calculated by a simple function for example 119861119890=

0886119863119890 Foundation depth or drain length 119867 = 5m The

diagram in Figure 4 illustrates the FE discretization andboundary conditions the total number of mesh elements is3179 displacements at the outer boundaries are fixed in thehorizontal direction with only vertical displacement alloweddisplacement at the top surface is free whereas the bottomsurface is totally fixed regarding the drainage boundary theperiphery of the unit cell is set as impermeable except thedrain unit on the top surface which is assumed to havebeen drained Additionally the soil-drain system is assumedto be elastic with regard to 119864 and 120583 and 120583 = 00 is setfor 1D deformation analysis A surcharge loading of 119902

0is

also simulated by instantaneous application to the upperboundary and the permeability parameters 119896

119908 119896119904 119896ℎ and

119896V are used to capture the permeability differences of thedifferent parts respectively The values of these parametersare listed in Table 1 in which four calculation conditionswere set to examine the newly developed programThe abovesettings are aiming to be accordant with the assumptions ofthe available analytical solution

The program will be verified against the theoreticalsolutions based on both Darcy flow and non-Darcy flowAccording to the Hansbo consolidation theory including theeffects of both smear and well resistance [8] the averagedegree of consolidation at a given elevation can be written asfollows

For Darcy flow

119880ℎ= 1 minus exp(minus

8119888ℎ119905

1205831198632119890

) (35)


Table 1 Calculation conditions and results for the first verification case

Cases 119896ℎ(ms) 119896

ℎ119896V 119896

ℎ119896119904

119896119908119896ℎ

119864 (MPa) 120583 119898 119894119897

120572 ΔMax 1 ()Condition 1 1119864 minus 8 40 40 11198644 100 00 15 20 00 956Condition 2 1119864 minus 8 40 40 11198644 100 00 15 10 00 808Condition 3 1119864 minus 8 40 40 11198644 100 00 15 5 00 413Condition 4 1119864 minus 8 40 40 11198644 100 00 12 5 00 262ΔMax 1 the largest difference between the FE results and the Hansbo solution [11] for non-Darcy flow under different calculation conditions

For non-Darcian flow

119880ℎ= 1 minus [1 +

120582119905

1205721198632119890

(Δ1199060

119863119890120574119908

)

119898minus1

]

1(1minus119898)

(36)

in which 119905 is consolidation time for instantaneous appliedload the initial EPP is Δ119906

0= 1199020 Additionally

120583 = 1205831+ 1205832+ 1205833

1205831=

1198632

119890

1198632119890minus 1198892119908

[ln(119863119890

119889119904

) +119896ℎ

119896119904

ln(119889119904

119889119908

) minus3

4]

1205832=120587119896ℎ

119902119908

119911 (2119867 minus 119911)(1 minus1198892

119908

1198632119890

)

1205833=

1198892

119904

1198632119890minus 1198892119908

(1 minus1198892

119904

41198632119890

) +119896ℎ

119896119904

1198892

119908

1198632119890minus 1198892119908

(1198894

119904minus 1198894

119908

411986321198901198892119908

minus1198892

119904

1198892119908

+ 1)

119888ℎ=

119896ℎ

119898V120574119908

119898V =(1 + 120583) (1 minus 2120583)

119864 (1 minus 120583)

119902119908= 119896119908

1205871198892

119908

4

120581ℎ

120581119904

=119896ℎ

119896119904

120582 =120581ℎ

120574119908119898V

120572 =1198982119898120573119898

4 (119898 minus 1)119898+1

120573 = 1205731+ 1205732minus 1205733+ 1205734

1205731=

1

3119898 minus 1minus

119898 minus 1

119898 (3119898 minus 1) (5119898 minus 1)

minus(119898 minus 1)

2

21198982 (5119898 minus 1) (7119898 minus 1)

1205732=1

2119898[(120581ℎ

120581119904

minus 1)(119863119890

119889119904

)

(1119898minus1)

minus120581ℎ

120581119904

(119863119890

119889119908

)

(1119898minus1)

]

1205733= (

1

2119898minus

1

3119898 minus 1)[(

120581ℎ

120581119904

minus 1)(119863119890

119889119904

)

(1119898minus3)

minus120581ℎ

120581119904

(119863119890

119889119908

)

(1119898minus3)

]

1205734=120581ℎ

2119902119908

120587119911 (2119897 minus 119911) (1 minus1

119898)(

119863119890

119889119908

)

(1119898minus1)

(1

minus1198892

119908

1198632119890

)

1119898

(37)

FE analyses with different iteration times (eg iterationtime = 1 2 3 6) for the time step cycling were conducted tocheck the influence of the hydraulic gradient approximationon numerical results As Figure 5(a) shows the numericalresult of the second approach with six iteration times getsclose to the theoretical solution (non-Darcy condition) whencompared with the result with the first approach (iterationtime = 1) but this improvement is very limitedThe followingFE results for different conditions are based on the firstapproach in which [H]

0= I and [H]

119894are directly ascertained

using previous time step values without iterationFrom the plots in Figure 5 it can be found that the

results calculated by FEM are roughly accordant with theclosed-form solutions The last column in Table 1 shows thelargest differences for different calculation conditions Com-bining Figure 5 and Table 1 the following conclusions can beobtained (1) the largest differences for the three calculationconditions are between 2 and 10 (2) FE solutions aremostlybetween the analytical results based on Darcy flow and thosebased on non-Darcy flow (3) these differences increase withthe increase of the nonlinear control parameter 119894

119897 The cause

of the difference is stated as follows with the increase of 119894119897

the nonlinearity of water flow becomes more obvious (seeFigure 1) then there is the approximation that computing thehydraulic gradient of the next stepwith porewater pressure ofthe present step leads to a more apparent difference betweenthe actual value and calculating result by (34) However theerrors are not significant and the comparison results herematched perfectly with the previous comparative analyses byTeh and Nie [19]


001 01 1 10 100 1000100

80

60

40

20

0

Iteration times 6

t (day)

By (35) Darcy flowBy (36) non-Darcy flow

FE results non-Darcy flow (1)FE results non-Darcy flow (2)

The Hansbo non-Darcy flow model

Iteration times 1

Uh

()

m = 15 il = 200

(a) Condition 1

t (day)001 01 1 10 100 1000

100

80

60

40

20

0

By (35) Darcy flowBy (36) non-Darcy flowFE result non-Darcy flow


Uh

()

m = 15 il = 100

(b) Condition 2

t (day)001 01 1 10 100 1000

100

80

60

40

20

0



Uh

()

m = 15 il = 50

(c) Condition 3

t (day)

100

80

60

40

20

0


001 01 1 10 100 1000


Uh

()

m = 12 il = 50

(d) Condition 4

Figure 5 Comparison of FE results and analytical solutions under different conditions

52 Case 2 Another validation case will be conducted forthe analysis of the 1D consolidation problem which is asaturated homogeneous soft clay layer with a thickness of 119867(119867 = 5m) The top boundary is free and drained while thebottom boundary is fixed and impervious A surcharge 119902

0(1199020

= 100 kPa) is instantaneously applied at the ground surfaceThe permeability coefficient and mechanical parameters areassumed to be constant In addition the equation governingthe 1D consolidation of soil with non-Darcy flow is given asfollows [8 17]

119888V

119894119898minus1

119897

(1

120574119908

120597119901

120597119911)

119898minus11205972119901

1205971199112=120597119901

120597119905when 119894 lt 119894

119897

119888V1205972119901

1205971199112=120597119901

120597119905when 119894 ge 119894

119897

(38)

Table 2 Calculation conditions and parameter values and calcula-tion results for Case 2

Conditions 119898 119894119897

119896V (ms) 119864 (MPa) 120583 ΔMax 2 ()Condition 1 15 20 1119864 minus 8 30 035 296Condition 2 12 50 1119864 minus 8 30 035 247ΔMax 2 the largest difference between the results by FEM and FDM underdifferent conditions

Equation (38) is solved numerically using both FEM andFDM and the Crank-Nicolson difference scheme is adoptedhere for its relative stability The detailed scheme with theFDM has been presented by Li et al [17] and it has beenvalidated with good reliability for calculation accuracy


Table 3 Parameters and their values

119861 (m) 119867

(m)119889119904

(cm)119889119908

(cm)119864

(MPa)120583

()1205741015840

(kNm3)1206011015840

(∘)119896ℎ0

(ms)119896ℎ0119896V0()

119896ℎ01198961199040

()1198961199080119896ℎ0

()115 100 150 70 30 035 10 30 10minus9 10 40 104

001 01 1 10 100100

80

60

40

20

0

Aver

age d

egre

e of c

onso

lidat

ion

()

Time (days)

Condition 1 with FDMCondition 1 with FEM


1E minus 31E minus 41E minus 5

m = 15 il = 20

m = 12 il = 5

Figure 6 Comparative analysis of FEM and FDM for 1D consolida-tion with non-Darcy flow model

Two calculation conditions have been conducted forthis case The parameters and their values are listed inTable 2The comparison of results between the twonumericalmethods is shown in Figure 6 Additionally Table 2 presentsthe largest differences under different calculation conditionsIt can be noted that the largest difference between the twomethods is less than 3These contrasting analysis results aresatisfactory

6 Effect of Nonlinear Flow on Consolidation

61 Description of the Problem An equivalent block unit cellwill be studied to investigate the effect of nonlinear flow onconsolidation Information about the geometries boundaryconditions and FE discretization for the case can also be seenin Figure 4 The sizes of 119861

119890 119887119904 and 119887

119908are listed in Table 3

respectively Boundary conditions are the same as for thefirst case for validation except for the drainage conditionfor the top surface which is assumed to be drained Thetotal number of 1690 spatial eight-node block elements isemployed in the case after an initialmesh sensitivity studywasperformed In addition the soil and drain are still modeledwith a linear elastic model to highlight the influence of thenonlinear flowHere note that (4) which depicted a nonlinearcompression property of normally consolidation soil was justused for the derivation of a nonlinear flow expression thatis (5) The permeability property of the soil is assumed togenerally change with effective stress and the initial values119896ℎ0 119896V0 1198961199040 and 1198961199080 are listed in Table 3 in which parameters

Table 4 Calculation conditions and input parameters

Conditions 119898 119894119897

120572 = 119862119888119862119896

1199020(kPa)

Condition 1 10 0 00 100Condition 2 15 5 00 100Condition 3 15 5 05 100Condition 4 15 5 10 100Condition 5 15 5 05 50Condition 6 15 5 05 10Condition 7 15 10 05 100Condition 8 15 20 05 100Condition 9 15 20 10 100

corresponding to the initial stress fields 1205741015840 and 1206011015840 are alsopresented

As shown in Table 4 nine calculation conditions will beconducted in which 119898 is set as 15 because this value isvalidated by several studies whereas 119894

119897and 119886 are assumed

in a range of 0sim20 and 0ndash10 respectively Note here thatthe above nonlinear flow parameters are just applicable forthe soil while the permeability property of the drain isassumed constant In addition loading size 119902

0is related to

the hydraulic gradient The higher the value of 1199020 the more

the significant change of hydraulic gradient in the dissipationof EPP and thus the influence of non-Darcy flowwill bemoreobvious In this consideration a value range of 10 to 100 kPais adopted Among the nine calculation conditions the firstcase is the linear flow condition which is used to make acomparison with the other conditions

62 Parametric Studies The consolidation data are plottedin terms of average degree of consolidation 119880 (by (38)) withtime 119905 in Figure 7 The time taken to achieve a fixed degreeof consolidation and the largest differences for differentcalculation results are listed in Tables 5 and 6 respectively

The influence of the nonlinear flow parameter (120572 =

119862119888119862119896) which reflects the reduction rate of permeability

has been estimated by result comparison between conditions2 3 and 4 see Figure 7(a) As expected the consolidationrate slows down when the variability of 119896 is considered andthe retardation on consolidation becomes significant with anincreasing 120572 As shown in Table 5 under the non-Darcy flowconditions with 120572 = 00 05 and 10 consolidation times119880

50

(11988090) are approximately 23 days (102) 29 days (131) and 39

days (190) respectivelyThe time for the linear flow conditionis 22 days (92) in addition the largest differences for the threeconditions are in the range of 2 to 15 see Table 6 Amongthem the biggest value is 1458 under a condition where120572 = 10


01 1 10 100 1000100

80

60

40

20

0

Aver

age d

egre

e of c

onso

lidat

ion

()

Time (days)

Condition 1Condition 2


Non-Darcy flow

120572 = 10

120572 = 05

120572 = 00

Darcy flow 120572 = 00

m = 15 il = 50

(a) Assess the influence of nonlinear flow parameter 120572

01 1 10 100 1000100

80

60

40

20

0

Non-Darcy flow

Aver

age d

egre

e of c

onso

lidat

ion

()

Time (days)



120572 = 05


m = 15 il = 50

q0 = 10kPa

q0 = 50kPa

q0 = 100 kPa

(b) Assess the influence of surcharge intensity 1199020

01 1 10 100 1000100

80

60

40

20

0

Non-Darcy flow

Aver

age d

egre

e of c

onso

lidat

ion

()

Time (days)

Condition 1Condition 3Condition 7



m = 15 il = 200 120572 = 10

m = 15 il = 200 120572 = 05

m = 15 il = 100 120572 = 05

m = 15 il = 50 120572 = 05

(c) Assess the influence of non-Darcy flow parameter 119894119897

Figure 7 Result comparisons under different calculation conditions

Table 5 Time (days) taken to achieve specific degrees of consolidation under different calculation conditions

Conditions C1 C2 C3 C4 C5 C6 C7 C8 C911988050

22 23 29 39 30 35 31 33 4311988090

92 102 131 190 133 167 148 176 25911988050 and11988090 the time taken to achieve119880 = 50 and119880 = 90 respectively

Table 6 The largest difference of consolidation results for different calculation conditions

Conditions C1 C2 C3 C4 C5 C6 C7 C8 C9ΔMax 3 () 00 205 695 1458 708 1350 964 1379 2216ΔMax 3 the largest difference between the FE results for nonlinear flow under different conditions with linear flow model


Conditions 3 5 and 6 are used to analyze the effect ofloading size by fixing 120572 and varying 119902

0 The comparisons

are shown in Figure 7(b) and Table 5 Conclusions can beobtained that (1) with 119902

0reducing the consolidation rate is

retarded and the consolidation time is prolonged (2) theinfluence of 119902

0on consolidation is insignificant when 119902

0

changes from 100 kPa to 50 kPa whereas sharp retardationhappenswhen 119902

0further reduces to 10 kPa It is to be expected

that the fluid flow follows non-Darcy flowwhen 1199020is reduced

which results in a low flow velocity Thus it can be foreseenthat as the load further reduces the consolidation rate willslow further and the EPP gradients will be less than theinitial hydraulic gradient It can be observed fromTable 6 thatthe largest difference between condition 6 (119902

0= 10 kPa) and

condition 1 is 135Figure 7(c) shows the comparison of consolidation results

with different non-Darcy flow parameters For conditions 37 and 8 119894

119897varying from 5 to 20 consolidation times119880

50(11988090)

are approximately 29 days (131) 31 days (148) and 33 days(176) respectivelyThese data lend support to the observationthat as the values of 119898 andor 119894

119897increase consolidation

development slows down [19] However this effect is notso obvious as to be attributed simply to the influence ofthe nonlinear flow parameter The consolidation curve forcondition 9 (119898 = 15 119894

119897= 20 120572 = 10) is also plotted

in Figure 7(c) It can be obtained from the comparison ofconditions 8 and 9 that as the nonlinear flow degree increasesthe consolidation rate is further slowed down and the effectof 120572 is more obvious when compared with the influence of119894119897 11988050 for example changes from 29 days to 33 days for the

previous three conditions 3 7 and 8 whereas it is prolongedto 43 days which is almost twice as long as for the resultsbased on linear flow formulation The largest difference inthe results of the degree of consolidation under differentconditions also validates the above conclusion Table 6 showsthat the largest difference between conditions 1 and 9 is 22

7 Conclusions

Biotrsquos general consolidation theory was extended to incorpo-rate a coupled nonlinear flow model for the time-dependentsettlement analysis of normally consolidated soft soil foun-dations under static loading The presented model assumesthat fluid flow follows the Hansbo non-Darcian flow andthat the permeability coefficient of soil decreases with anexponential change of effective stress The governing differ-ential equations and the FE formulations were derived anda programming code was developed to solve the obtainedequations We implemented two calculation cases to checkthe rationality of the method and we found both casesto demonstrate that the consolidation results for the twocomparative methods were in good agreement Finally weinvestigated the effect of nonlinear flow on consolidation andwe drew the following conclusions

(1) The consolidation rate slows down when coefficientor permeability variability is considered With anincrease in the value of 120572 (= 119862

119888119862119896 value range is

05ndash20) the deduction of soil permeability in the

EPP dissipation process becomes obvious and theconsolidation rate slows down The effect of 120572 on theconsolidation development is somewhat significant

(2) The hydraulic gradient of EPP in the consolida-tion process becomes uniform due to the load sizedecrease which results as an apparent effect ofnon-Darcian flow on the degree of consolidationThus with the loading size decrease consolidationrate slows down and the time taken to achieve afixed degree of consolidation increases This trend ofdecreasing consolidation rate with decreasing loadingsize has a tendency to accelerate

(3) Increasing the values of non-Darcian flow parameters(119894119897and 119898) can cause the consolidation rate to slow

down In relative terms this effect is not as obviousas the influence of parameter 120572 on consolidationbehavior

Nomenclature

119886 119887 119888 Half sizes of the element in 119909- 119910- and119911-directions

119861119890 119887119904 119887119908 Widths of the equivalent unit cell the

smear zone and the drain119863119890 119889119904 119889119908 Diameters of the equivalent unit cell thesmear zone and the drain

119862119888 Compressive index

119862119896 Seepage index

1198900 Initial void ratio

119864 Youngrsquos modulus119867 Depth of the soft soil foundation119894 Hydraulic gradient1198940 119894119897 Threshold hydraulic gradient in the

Hansbo non-Darcian flow model119896 Permeability coefficient1198960 Initial permeability coefficient

119896ℎ 119896V Permeability coefficients in the horizontal

and in the vertical directions119896ℎ0 119896V0 Initial permeability coefficients in the

horizontal and in the vertical directions119896119904 119896119908 Permeability coefficients of the smear zone

and the drain1198961199040 1198961199080 Initial permeability coefficients of the

smear zone and the drain119897 Length of drain119898 The exponent of exponential flow at low

gradient119901 Excess pore water pressure1199010 Initial excess pore water pressure

1199011015840 Mean effective stress 1199011015840 = (1205901015840

119909+ 1205901015840

119910+ 1205901015840

119911)3

1199011015840

0 Initial mean effective stress

1199020 The instantaneous load applied at the top

surface of the ground119906 V 119908 Displacements in the three directions 119909 119910

and 119911120572 Ratio 119862

119888119862119896

120576V Volumetric strain120574119908 Unit weight of water


1205741015840 1206011015840 The effective weight and friction angle of

soil120581 Permeability parameter applied in the

exponential part120579 Integration constant120583 Poissonrsquos ratio1205901015840

119909 1205901015840

119910 1205901015840

119911 The effective stress in the 119909- 119910- and119911-directions

120585 120578 120577 The three directions of the localcoordinate system

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was sponsored by K C Wong Magna Fundin Ningbo University and by the National Natural ScienceFoundation of China (nos 51308309 and 51278256) whichare gratefully acknowledged

References

[1] L Ho B Fatahi and H Khabbaz ldquoAnalytical solution for one-dimensional consolidation of unsaturated soils using eigen-function expansion methodrdquo International Journal for Numeri-cal and Analytical Methods in Geomechanics vol 38 no 10 pp1058ndash1077 2014

[2] K TerzaghiTheoretical SoilMechanics JohnWileyamp SonsNewYork NY USA 1943

[3] M A Biot ldquoGeneral theory of three-dimensional consolida-tionrdquo Journal of Applied Physics vol 12 no 2 pp 155ndash164 1941

[4] RW Lewis andB A SchreflerTheFinite ElementMethod in theStatic and Dynamic Deformation and Consolidation of PorousMedia John Wiley amp Sons New York NY USA 2nd edition1998

[5] J Huang D V Griffiths and G A Fenton ldquoProbabilisticanalysis of coupled soil consolidationrdquo Journal of Geotechnicaland Geoenvironmental Engineering vol 136 no 3 pp 417ndash4302010

[6] C Menendez P J Nieto F A Ortega and A Bello ldquoNon-linear analysis of the consolidation of an elastic saturated soilwith incompressible fluid and variable permeability by FEMrdquoApplied Mathematics and Computation vol 216 no 2 pp 458ndash476 2010

[7] L Ho and B Fatahi ldquoAnalytical solution for the two-dimensional plane strain consolidation of an unsaturated soilstratum subjected to time-dependent loadingrdquo Computers andGeotechnics vol 67 pp 1ndash16 2015

[8] S Hansbo ldquoExperience of consolidation process from test areaswith and without vertical drainsrdquo in Ground ImprovementmdashCase Histories B Indraratna and J Chu Eds vol 3 pp 3ndash49Elsevier London UK 2005

[9] T Qi K-H Xie A-F Hu and Z-Q Zhang ldquoLaboratorial studyon non-Darcy seepage in Xiaoshan clayrdquo Journal of ZhejiangUniversity Engineering Science Edition vol 41 no 6 pp 1023ndash1028 2007 (Chinese)

[10] S Hansbo Consolidation of Clay with Special Reference toInfluence of Vertical Drains vol 18 of Swedish Geotechnical Insti-tute Proceeding Swedish Geotechnical Institute StockholmSweden 1960

[11] S Hansbo ldquoAspects of vertical drain design Darcian or non-Darcian flowrdquo Geotechnique vol 47 no 5 pp 983ndash992 1997

[12] S Hansbo ldquoConsolidation equation valid for both Darcian andnon-Darcian flowrdquo Geotechnique vol 51 no 1 pp 51ndash54 2001

[13] S Hansbo ldquoDeviation from Darcyrsquos law observed in one-dimensional consolidationrdquo Geotechnique vol 53 no 6 pp601ndash605 2003

[14] F Pascal H Pascal and D W Murray ldquoConsolidation withthreshold gradientsrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 5 no 3 pp 247ndash2611981

[15] B Dubin and G Moulin ldquoInfluence of a critical gradient onthe consolidation of claysrdquo in Consolidation of Soils Testing andEvaluation ASTM STP 892 pp 354ndash377 American Society forTesting and Materials West Conshohocken Pa USA 1986

[16] K-H Xie K Wang Y-L Wang and C-X Li ldquoAnalyticalsolution for one-dimensional consolidation of clayey soils witha threshold gradientrdquo Computers and Geotechnics vol 37 no 4pp 487ndash493 2010

[17] C-X Li K-H Xie and K Wang ldquoAnalysis of 1D consolidationwith non-Darcian flow described by exponent and thresholdgradientrdquo Journal of Zhejiang UniversitymdashScience A vol 11 no9 pp 656ndash667 2010

[18] C-X Li K-H Xie A-F Hu and B-X Hu ldquoOne-dimensionalconsolidation of double-layered soil with non-Darcian flowdescribed by exponent and threshold gradientrdquo Journal ofCentral South University of Technology vol 19 no 2 pp 562ndash571 2012

[19] C I Teh and X Y Nie ldquoCoupled consolidation theory withnon-Darcian flowrdquo Computers and Geotechnics vol 29 no 3pp 169ndash209 2002

[20] F Tavenas M Tremblay G Larouche and S Leroueil ldquoIn-situmeasurement of permeability in soft claysrdquo in ASCE SpecialConference on Use of In-situ Tests in Geotechnical Engineeringpp 1034ndash1048 ASCE 1986

[21] B Fatahi TM LeM Q Le andH Khabbaz ldquoSoil creep effectson ground lateral deformation and pore water pressure underembankmentsrdquoGeomechanics and Geoengineering vol 8 no 2pp 107ndash124 2013

[22] T M Le B Fatahi and H Khabbaz ldquoNumerical optimisationto obtain elastic viscoplastic model parameters for soft clayrdquoInternational Journal of Plasticity vol 65 pp 1ndash21 2015

[23] O C Zienkiewicz and R L Taylor The Finite Element Methodfor Solid and Structure Mechanics Elsevier Singapore 6thedition 2009

[24] R S Sandhu and E L Wilson ldquoFinite element analysis ofseepage in elastic mediardquo Proceedings of the American Societyof Civil Engineers vol 95 no 3 pp 641ndash652 1996

[25] K H Xie and J ZhouTheory and Application of Finite ElementMethod in Geotechnical Engineering Science Press BeijingChina 2002

[26] J R Booker and J C Small ldquoAn investigation of the stabilityof numerical solutions of Biotrsquos equations of consolidationrdquoInternational Journal of Solids and Structures vol 11 no 7-8 pp907ndash917 1975

[27] G X Zeng K H Xie and Z Y Shi ldquoConsolidation analysisof sand-drained ground by FEMrdquo in Proceedings of the 8th


Asian Regional Conference of Soil Mechanic and FoundationEngineering pp 139ndash142 Tokyo Japan 1987

[28] K-H Xie Y-B Deng K Wang and D-Z Huang ldquoA 3Dcomposite finite element used to model a vertical drain and itssurrounding remoulded soilrdquo Computers and Geotechnics vol38 no 8 pp 1078ndash1088 2011

[29] C C Hird I C Pyrah and D Russell ldquoFinite elementmodelling of vertical drains beneath embankments on softgroundrdquo Geotechnique vol 42 no 3 pp 499ndash511 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


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Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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OptimizationJournal of


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International Journal of


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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


v

io

i0 il

= 120581im

= k(i minus i0)

Figure 1 The Hansbo non-Darcy flow model

effect of non-Darcian flow on soft ground consolidationbehavior However their work was not suitable for theconsolidation analysis of multidrain systems and had nottaken the variability of soil permeability into consideration

Based on these previous studies this paper aims toextend Biotrsquos general three-dimensional consolidation theoryto incorporate non-Darcian flow together with the consider-ation of the variation of the soil permeability during the con-solidation process Governing equations for the force equi-librium condition and the continuity condition will be pre-sented first and the finite element method (FEM) will be uti-lized here to solve these equations Some specific techniquesin the programming process for the newly derived formulaswill be stated in detail The validation of the program will beperformed for two cases

2 A Coupled Nonlinear Flow Model

Thefluid flow in the poremedia obeys theHansbo non-Darcyflow law (see Figure 1) which can be described as follows [8]

V = 120581119894119898 when 119894 lt 119894119897

V = 119896 (119894 minus 1198940) when 119894 ge 119894

119897

(1)

where V is flow velocity 119896 is permeability coefficient of thelinear flow at high gradient 120581 is a permeability parameterthat is applied in the exponential part 119898 is the exponent ofexponential flow at low gradient 119894

0is a threshold gradient

and 119894119897represents the gradient required to overcome the

maximum binding energy of mobile pore water [11] Severalstudies reported that the value of 119898 was within 10sim20 andthe typical value was 15 whereas the value of 119894

119897ranged in 0sim

40 [8] Note here that when119898 = 1 and 1198940= 0 (1) degenerates

to Darcyrsquos linear flow law that is V = 119896119894Combined with the conditions of the continuity and

differentiability at 119894 = 119894119897 the following expressions can be

derived from (1) [19]

V =119896119894119898

(119898119894119898minus1

119897)

when 119894 lt 119894119897

V = 119896 (119894 minus119898 minus 1

119898119894119897) when 119894 ge 119894

119897

(2)

In addition a nonconstant permeability is assumed forsoft soil The coefficient of permeability 119896 is one of the mostimportant parameters for studying nonlinear properties ofsoil and a linear correlation between the void ratio 119890 andlg 119896 is valid for the most common clays [20] which can beexpressed as follows

119890 = 1198900+ 119862119896lg( 119896

1198960

) (3)

where 1198900and 119896

0are the initial void ratio and the initial

permeability coefficient respectively and119862119896is seepage index

There is a linear relationship in 119890-lg1199011015840 for normallyconsolidated soft soils according to the oedometer test Thatis

119890 = 1198900minus 119862119888lg(

1199011015840

1199011015840

0

) (4)

where119862119888is compressive index 1199011015840 is the mean effective stress

1199011015840

0is initialmean effective stress themean effective stress1199011015840 =

(1205901015840

119909+ 1205901015840

119910+ 1205901015840

119911)3 1205901015840119909 1205901015840119910 and 1205901015840

119911are the effective stress in the

119909- 119910- and 119911-directions respectivelyCombining (3) and (4) it can be obtained that

119896 = 1198960(1199011015840

1199011015840

0

)

minus120572

(5)

where 120572 = 119862119888119862119896 120572 is in 05sim2 and the common value is in

05sim10 [20]A coupled nonlinear flow model which takes the non-

constancy of soil permeability and non-Darcian flow law intoaccount is then formed by a combination of (2) and (5)Here it should be pointed out that the settlement of soft soilfoundations will be affected by a series of other factors suchas the creep (viscous) behavior of clayey soil (eg [21 22]) theoverconsolidated condition and loading pattern This papermainly focuses on the study of the influence of varied soilpermeability and non-Darcian flow on consolidation process

3 Governing Equations

The basic assumptions on the coupled consolidation problemare the same as Biotrsquos general consolidation theory except forthe above newly presented flow model Equations for forceequilibrium and flow continuity relations are presented herefor comprehensive consideration

31 Force Equilibrium Equation The compact form of theforce equilibrium equation which is the same as it is for alinear flow model can be expressed as follows [23ndash25]

[120597]119879[D] [120597] f minus [120597]119879 M 119901 = 0 (6)

where the displacement vector f = [119906 V 119908]119879 119906 Vand 119908 are displacements in the 119909- 119910- and 119911-directionsrespectively119901 is the excess pore pressure (EPP) vector M =[1 1 1 0 0 0]

119879 the elastic matrix [D] about the isotropiclinear elastic material and the differential operator [120597] are thesame as those for a homogeneous element


y

z

o

xdy

dx

dzy

x

z



120597V119909

120597119909+

120597V119910

120597119910+120597V119911

120597119911= minus

120597120576V

120597119905= minus

120597

120597119905(120597119906

120597119909+120597V120597119910+120597119908

120597119911) (7)




M119879 [120597] k = minus 120597120597119905(M119879 120576) (8)


vector 120576 = [120576119909120576119910120576119911120574119909119910120574119910119911120574119911119909]119879


119897 To simplify



VΨ=119878Ψ119896Ψ

1003816100381610038161003816119894Ψ1003816100381610038161003816

119898

(119898119894119898minus1

119897)

when 1003816100381610038161003816119894Ψ1003816100381610038161003816 lt 119894119897


119898119894119897) when 1003816100381610038161003816119894Ψ

1003816100381610038161003816 ge 119894119897

(9)



Ψis +1



119896Ψ= 119896Ψ0(1199011015840

1199011015840

0

)

minus120572

(10)


120597VΨ

120597Ψ=

119896Ψ

120574119898119908119894119898minus1

119897

10038161003816100381610038161003816100381610038161003816

120597119901

120597Ψ

10038161003816100381610038161003816100381610038161003816

119898minus11205972119901

120597Ψ2when 1003816100381610038161003816119894Ψ

1003816100381610038161003816 lt 119894119897

120597VΨ

120597Ψ=119896Ψ

120574119908

1205972119901

120597Ψ2when 1003816100381610038161003816119894Ψ

1003816100381610038161003816 ge 119894119897

(11)




120597VΨ

120597Ψ= 119867Ψ

119896Ψ0

120574119908

(1199011015840

1199011015840

0

)

minus120572

1205972119901

120597Ψ2(12)

in which

119867Ψ=

1

120574119898minus1119908119894119898minus1

119897

10038161003816100381610038161003816100381610038161003816

120597119901

120597Ψ

10038161003816100381610038161003816100381610038161003816

119898minus1

when 1003816100381610038161003816119894Ψ1003816100381610038161003816 lt 119894119897

1 when 1003816100381610038161003816119894Ψ1003816100381610038161003816 ge 119894119897

(13)


M119879 [120597] [H] [k] [120597]119879 M 119901 minus 120597

120597119905M119879 [120597] f = 0 (14)


[H] = (1198671199090 0

0 1198671199100

0 0 119867119911

) (15)

[k] = 1

120574119908

(1199011015840

1199011015840

0

)

minus120572

(

1198961199090

0 0

0 1198961199100

0

0 0 1198961199110

) (16)

in which 119867119909 119867119910 and 119867








y

z

o

x

2b

2a

2c

85

41

2 3

6 7

p

o998400120585

120577120578

u w




119897V119909+ 119898V119910+ 119899V119911= V119899 (17)







f asymp f = V 119879 = [N] 120575119890

119901 asymp = [N] p119890 (18)


1120575119879


119879

8]119879 120575119894= [119906119894V119894119908119894]119879

119894 = 1 2 8 119906119894 V119894 and 119908




8]119879 119901119894is the nodal EPP [N] =

[1198731I 1198732I sdot sdot sdot 119873


[11987311198732sdot sdot sdot 119873


119873119894=1

8(1 + 120585120585

119894) (1 + 120578120578

119894) (1 + 120577120577

119894) 119894 = 1 2 8 (19)

where 120585119894 120578119894 and 120577



119894as the weight


[K119890] Δ120575

119890+ [K119888] Δp119890 = ΔR

119891119890 (20)

in which

[K119890] =∭

119881119890

[B]119879 [D] [B] d119881

[K119888] =∭

119881119890

[B]119879 M [N] d119881

ΔR119891119890

= ∬119878119890

[N]119879 ΔF119890 d119878

Δ120575119890= [Δ120575

119879

1Δ120575119879


119879

8]119879


ΔF119890 = Δ119865119909 Δ119865119910 Δ119865119911119879

(21)

where [K119890] [K119888] and ΔR



1B2sdot sdot sdot B

8] [B119894]


119894 119894 = 1 2 8

Δ119865119909 Δ119865119910 and Δ119865




119877119890

119868119902=120597

120597119905M119879 [120597] f minus M119879 [120597] [H] [k] [120597]119879 M

119877119890

119861119902= 119897V119909+ 119898V119910+ 119899V119911minus V119899

(22)


∭119881119890

119873119894(120597

120597119905M119879 [120597] f

minus M119879 [120597] [H] [k] [120597]119879 M ) d119881 = 0

119894 = 1 2 8

(23)

∬119878119890

119873119894(119897V119909+ 119898V119910+ 119899V119911minus V119899) d119878 = 0

119894 = 1 2 8

(24)



∭119881119890

119873119894(120597

120597119905M119879 [120597] f) d119881

=∭119881119890

119873119894M119879 [120597] [N] d119881

(25)



minus∭119881119890

119873119894(M119879 [120597] [H] [k] [120597]119879 M ) d119881

= ∬119878119890

119873119894V119899d119878

+∭119881119890

nabla119879119873119894 [H] [k] nabla [N] p

119890 d119881

(26)


(26) we have

∭119881119890

119873119894 M119879[120597] [N] d119881

+∭119881119890

nabla119879119873119894 [H] [k] nabla [N] p

119890 d119881

= minus∬119878119890

119873119894V119899d119878 119894 = 1 2 8

(27)


[K119888]119879119890

minus [K119904] p119890 = R

119902119890

(28)

in which

[K119904] =∭

119881119890

[B119904]119879

[H] [k] [B119904] d119881

[B119904] =

120597

120597119909

120597

120597119910

120597

120597119911

119879

[1198731 1198732 sdot sdot sdot 1198738]

R119902119890

= ∬119878119890

[N]119879

V119899d119878

(29)





to 119905119899+1

we have

int

119905119895+1

119905119895

[K119888]119879119890

d119905 minus int119905119895+1

119905119895

[K119904] p119890 d119905

= int

119905119895+1

119905119895

R119902119890

d119905(30)


int

119905119895+1

119905119895

p119890 d119905 = Δ119905 (p119890119899+ 120579 Δp119890)

int

119905119895+1

119905119895

R119902119890

d119905 = Δ119905 R119902119890

119899

(31)


[K119888]119879

Δ120575119890minus 120579Δ119905 [K

119904] Δp119890 = ΔR

119901119890 (32)

in which

ΔR119901119890

= Δ119905 [R119902119890

119895+ [K119904] p119890119895] (33)





119890] and [K



119901119890Matrix








0can be










119894119895+1

119909=1

120574119908

120597119901119895

120597119909

=

(119901119895

1+ 119901119895

4+ 119901119895

5+ 119901119895

8) minus (119901

119895

2+ 119901119895

3+ 119901119895

6+ 119901119895

7)

8120574119908119886

119894119895+1

119910=1

120574119908

120597119901119895

120597119910

=

(119901119895

1+ 119901119895

2+ 119901119895

5+ 119901119895

6) minus (119901

119895

3+ 119901119895

4+ 119901119895

7+ 119901119895

8)

8120574119908119887

119894119895+1

119911=1

120574119908

120597119901119895

120597119911

=

(119901119895

1+ 119901119895

2+ 119901119895

3+ 119901119895

4) minus (119901

119895

5+ 119901119895

6+ 119901119895

7+ 119901119895

8)

8120574119908119888

(34)





0 Considering




0as a





5 Verification



Z

Drain

Impervious

Impe

rvio

us

Impe

rvio

us

Impervious

Pervious

Impervious

Nature soil

Smear zonekw ks

kh k

bw = 886 cm

H = 5m

bs = 1772 cm

B e=2658

m



119908= 01m


119890 119887119904 and 119887





0is


119908 119896119904 119896ℎ and



For Darcy flow


8119888ℎ119905

1205831198632119890

) (35)



Cases 119896ℎ(ms) 119896

ℎ119896V 119896

ℎ119896119904

119896119908119896ℎ

119864 (MPa) 120583 119898 119894119897



119880ℎ= 1 minus [1 +

120582119905

1205721198632119890

(Δ1199060

119863119890120574119908

)

119898minus1

]

1(1minus119898)

(36)



120583 = 1205831+ 1205832+ 1205833

1205831=

1198632

119890

1198632119890minus 1198892119908

[ln(119863119890

119889119904

) +119896ℎ

119896119904

ln(119889119904

119889119908

) minus3

4]

1205832=120587119896ℎ

119902119908

119911 (2119867 minus 119911)(1 minus1198892

119908

1198632119890

)

1205833=

1198892

119904

1198632119890minus 1198892119908

(1 minus1198892

119904

41198632119890

) +119896ℎ

119896119904

1198892

119908

1198632119890minus 1198892119908

(1198894

119904minus 1198894

119908

411986321198901198892119908

minus1198892

119904

1198892119908

+ 1)

119888ℎ=

119896ℎ

119898V120574119908

119898V =(1 + 120583) (1 minus 2120583)

119864 (1 minus 120583)

119902119908= 119896119908

1205871198892

119908

4

120581ℎ

120581119904

=119896ℎ

119896119904

120582 =120581ℎ

120574119908119898V

120572 =1198982119898120573119898

4 (119898 minus 1)119898+1

120573 = 1205731+ 1205732minus 1205733+ 1205734

1205731=

1


119898 minus 1

119898 (3119898 minus 1) (5119898 minus 1)


2

21198982 (5119898 minus 1) (7119898 minus 1)

1205732=1

2119898[(120581ℎ

120581119904

minus 1)(119863119890

119889119904

)

(1119898minus1)

minus120581ℎ

120581119904

(119863119890

119889119908

)

(1119898minus1)

]

1205733= (

1

2119898minus

1

3119898 minus 1)[(

120581ℎ

120581119904

minus 1)(119863119890

119889119904

)

(1119898minus3)

minus120581ℎ

120581119904

(119863119890

119889119908

)

(1119898minus3)

]

1205734=120581ℎ

2119902119908

120587119911 (2119897 minus 119911) (1 minus1

119898)(

119863119890

119889119908

)

(1119898minus1)

(1

minus1198892

119908

1198632119890

)

1119898

(37)


0= I and [H]




119897 The cause




001 01 1 10 100 1000100

80

60

40

20

0

Iteration times 6

t (day)




Iteration times 1

Uh

()

m = 15 il = 200

(a) Condition 1

t (day)001 01 1 10 100 1000

100

80

60

40

20

0



Uh

()

m = 15 il = 100

(b) Condition 2

t (day)001 01 1 10 100 1000

100

80

60

40

20

0



Uh

()

m = 15 il = 50

(c) Condition 3

t (day)

100

80

60

40

20

0


001 01 1 10 100 1000


Uh

()

m = 12 il = 50

(d) Condition 4



0(1199020


119888V

119894119898minus1

119897

(1

120574119908

120597119901

120597119911)

119898minus11205972119901

1205971199112=120597119901

120597119905when 119894 lt 119894

119897

119888V1205972119901

1205971199112=120597119901

120597119905when 119894 ge 119894

119897

(38)


Conditions 119898 119894119897





119861 (m) 119867

(m)119889119904

(cm)119889119908

(cm)119864

(MPa)120583

()1205741015840

(kNm3)1206011015840

(∘)119896ℎ0

(ms)119896ℎ0119896V0()

119896ℎ01198961199040

()1198961199080119896ℎ0

()115 100 150 70 30 035 10 30 10minus9 10 40 104

001 01 1 10 100100

80

60

40

20

0

Aver

age d

egre

e of c

onso

lidat

ion

()

Time (days)




m = 15 il = 20

m = 12 il = 5





119890 119887119904 and 119887




Conditions 119898 119894119897

120572 = 119862119888119862119896

1199020(kPa)






0is related to







50




01 1 10 100 1000100

80

60

40

20

0

Aver

age d

egre

e of c

onso

lidat

ion

()

Time (days)



Non-Darcy flow

120572 = 10

120572 = 05

120572 = 00


m = 15 il = 50


01 1 10 100 1000100

80

60

40

20

0

Non-Darcy flow

Aver

age d

egre

e of c

onso

lidat

ion

()

Time (days)



120572 = 05


m = 15 il = 50

q0 = 10kPa

q0 = 50kPa

q0 = 100 kPa


01 1 10 100 1000100

80

60

40

20

0

Non-Darcy flow

Aver

age d

egre

e of c

onso

lidat

ion

()

Time (days)




m = 15 il = 200 120572 = 10

m = 15 il = 200 120572 = 05

m = 15 il = 100 120572 = 05

m = 15 il = 50 120572 = 05





22 23 29 39 30 35 31 33 4311988090






0 The comparisons





0





0= 10 kPa) and




50(11988090)




119897= 20 120572 = 10) is also plotted



7 Conclusions



119888119862119896 value range is






Nomenclature
















119909+ 1205901015840

119910+ 1205901015840

119911)3

1199011015840




and 119911120572 Ratio 119862

119888119862119896






119909 1205901015840

119910 1205901015840





Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










y

z

o

xdy

dx

dzy

x

z



120597V119909

120597119909+

120597V119910

120597119910+120597V119911

120597119911= minus

120597120576V

120597119905= minus

120597

120597119905(120597119906

120597119909+120597V120597119910+120597119908

120597119911) (7)




M119879 [120597] k = minus 120597120597119905(M119879 120576) (8)


vector 120576 = [120576119909120576119910120576119911120574119909119910120574119910119911120574119911119909]119879


119897 To simplify



VΨ=119878Ψ119896Ψ

1003816100381610038161003816119894Ψ1003816100381610038161003816

119898

(119898119894119898minus1

119897)

when 1003816100381610038161003816119894Ψ1003816100381610038161003816 lt 119894119897


119898119894119897) when 1003816100381610038161003816119894Ψ

1003816100381610038161003816 ge 119894119897

(9)



Ψis +1



119896Ψ= 119896Ψ0(1199011015840

1199011015840

0

)

minus120572

(10)


120597VΨ

120597Ψ=

119896Ψ

120574119898119908119894119898minus1

119897

10038161003816100381610038161003816100381610038161003816

120597119901

120597Ψ

10038161003816100381610038161003816100381610038161003816

119898minus11205972119901

120597Ψ2when 1003816100381610038161003816119894Ψ

1003816100381610038161003816 lt 119894119897

120597VΨ

120597Ψ=119896Ψ

120574119908

1205972119901

120597Ψ2when 1003816100381610038161003816119894Ψ

1003816100381610038161003816 ge 119894119897

(11)




120597VΨ

120597Ψ= 119867Ψ

119896Ψ0

120574119908

(1199011015840

1199011015840

0

)

minus120572

1205972119901

120597Ψ2(12)

in which

119867Ψ=

1

120574119898minus1119908119894119898minus1

119897

10038161003816100381610038161003816100381610038161003816

120597119901

120597Ψ

10038161003816100381610038161003816100381610038161003816

119898minus1

when 1003816100381610038161003816119894Ψ1003816100381610038161003816 lt 119894119897

1 when 1003816100381610038161003816119894Ψ1003816100381610038161003816 ge 119894119897

(13)


M119879 [120597] [H] [k] [120597]119879 M 119901 minus 120597

120597119905M119879 [120597] f = 0 (14)


[H] = (1198671199090 0

0 1198671199100

0 0 119867119911

) (15)

[k] = 1

120574119908

(1199011015840

1199011015840

0

)

minus120572

(

1198961199090

0 0

0 1198961199100

0

0 0 1198961199110

) (16)

in which 119867119909 119867119910 and 119867








y

z

o

x

2b

2a

2c

85

41

2 3

6 7

p

o998400120585

120577120578

u w




119897V119909+ 119898V119910+ 119899V119911= V119899 (17)







f asymp f = V 119879 = [N] 120575119890

119901 asymp = [N] p119890 (18)


1120575119879


119879

8]119879 120575119894= [119906119894V119894119908119894]119879

119894 = 1 2 8 119906119894 V119894 and 119908




8]119879 119901119894is the nodal EPP [N] =

[1198731I 1198732I sdot sdot sdot 119873


[11987311198732sdot sdot sdot 119873


119873119894=1

8(1 + 120585120585

119894) (1 + 120578120578

119894) (1 + 120577120577

119894) 119894 = 1 2 8 (19)

where 120585119894 120578119894 and 120577



119894as the weight


[K119890] Δ120575

119890+ [K119888] Δp119890 = ΔR

119891119890 (20)

in which

[K119890] =∭

119881119890

[B]119879 [D] [B] d119881

[K119888] =∭

119881119890

[B]119879 M [N] d119881

ΔR119891119890

= ∬119878119890

[N]119879 ΔF119890 d119878

Δ120575119890= [Δ120575

119879

1Δ120575119879


119879

8]119879


ΔF119890 = Δ119865119909 Δ119865119910 Δ119865119911119879

(21)

where [K119890] [K119888] and ΔR



1B2sdot sdot sdot B

8] [B119894]


119894 119894 = 1 2 8

Δ119865119909 Δ119865119910 and Δ119865




119877119890

119868119902=120597

120597119905M119879 [120597] f minus M119879 [120597] [H] [k] [120597]119879 M

119877119890

119861119902= 119897V119909+ 119898V119910+ 119899V119911minus V119899

(22)


∭119881119890

119873119894(120597

120597119905M119879 [120597] f

minus M119879 [120597] [H] [k] [120597]119879 M ) d119881 = 0

119894 = 1 2 8

(23)

∬119878119890

119873119894(119897V119909+ 119898V119910+ 119899V119911minus V119899) d119878 = 0

119894 = 1 2 8

(24)



∭119881119890

119873119894(120597

120597119905M119879 [120597] f) d119881

=∭119881119890

119873119894M119879 [120597] [N] d119881

(25)



minus∭119881119890

119873119894(M119879 [120597] [H] [k] [120597]119879 M ) d119881

= ∬119878119890

119873119894V119899d119878

+∭119881119890

nabla119879119873119894 [H] [k] nabla [N] p

119890 d119881

(26)


(26) we have

∭119881119890

119873119894 M119879[120597] [N] d119881

+∭119881119890

nabla119879119873119894 [H] [k] nabla [N] p

119890 d119881

= minus∬119878119890

119873119894V119899d119878 119894 = 1 2 8

(27)


[K119888]119879119890

minus [K119904] p119890 = R

119902119890

(28)

in which

[K119904] =∭

119881119890

[B119904]119879

[H] [k] [B119904] d119881

[B119904] =

120597

120597119909

120597

120597119910

120597

120597119911

119879

[1198731 1198732 sdot sdot sdot 1198738]

R119902119890

= ∬119878119890

[N]119879

V119899d119878

(29)





to 119905119899+1

we have

int

119905119895+1

119905119895

[K119888]119879119890

d119905 minus int119905119895+1

119905119895

[K119904] p119890 d119905

= int

119905119895+1

119905119895

R119902119890

d119905(30)


int

119905119895+1

119905119895

p119890 d119905 = Δ119905 (p119890119899+ 120579 Δp119890)

int

119905119895+1

119905119895

R119902119890

d119905 = Δ119905 R119902119890

119899

(31)


[K119888]119879

Δ120575119890minus 120579Δ119905 [K

119904] Δp119890 = ΔR

119901119890 (32)

in which

ΔR119901119890

= Δ119905 [R119902119890

119895+ [K119904] p119890119895] (33)





119890] and [K



119901119890Matrix








0can be










119894119895+1

119909=1

120574119908

120597119901119895

120597119909

=

(119901119895

1+ 119901119895

4+ 119901119895

5+ 119901119895

8) minus (119901

119895

2+ 119901119895

3+ 119901119895

6+ 119901119895

7)

8120574119908119886

119894119895+1

119910=1

120574119908

120597119901119895

120597119910

=

(119901119895

1+ 119901119895

2+ 119901119895

5+ 119901119895

6) minus (119901

119895

3+ 119901119895

4+ 119901119895

7+ 119901119895

8)

8120574119908119887

119894119895+1

119911=1

120574119908

120597119901119895

120597119911

=

(119901119895

1+ 119901119895

2+ 119901119895

3+ 119901119895

4) minus (119901

119895

5+ 119901119895

6+ 119901119895

7+ 119901119895

8)

8120574119908119888

(34)





0 Considering




0as a





5 Verification



Z

Drain

Impervious

Impe

rvio

us

Impe

rvio

us

Impervious

Pervious

Impervious

Nature soil

Smear zonekw ks

kh k

bw = 886 cm

H = 5m

bs = 1772 cm

B e=2658

m



119908= 01m


119890 119887119904 and 119887





0is


119908 119896119904 119896ℎ and



For Darcy flow


8119888ℎ119905

1205831198632119890

) (35)



Cases 119896ℎ(ms) 119896

ℎ119896V 119896

ℎ119896119904

119896119908119896ℎ

119864 (MPa) 120583 119898 119894119897



119880ℎ= 1 minus [1 +

120582119905

1205721198632119890

(Δ1199060

119863119890120574119908

)

119898minus1

]

1(1minus119898)

(36)



120583 = 1205831+ 1205832+ 1205833

1205831=

1198632

119890

1198632119890minus 1198892119908

[ln(119863119890

119889119904

) +119896ℎ

119896119904

ln(119889119904

119889119908

) minus3

4]

1205832=120587119896ℎ

119902119908

119911 (2119867 minus 119911)(1 minus1198892

119908

1198632119890

)

1205833=

1198892

119904

1198632119890minus 1198892119908

(1 minus1198892

119904

41198632119890

) +119896ℎ

119896119904

1198892

119908

1198632119890minus 1198892119908

(1198894

119904minus 1198894

119908

411986321198901198892119908

minus1198892

119904

1198892119908

+ 1)

119888ℎ=

119896ℎ

119898V120574119908

119898V =(1 + 120583) (1 minus 2120583)

119864 (1 minus 120583)

119902119908= 119896119908

1205871198892

119908

4

120581ℎ

120581119904

=119896ℎ

119896119904

120582 =120581ℎ

120574119908119898V

120572 =1198982119898120573119898

4 (119898 minus 1)119898+1

120573 = 1205731+ 1205732minus 1205733+ 1205734

1205731=

1


119898 minus 1

119898 (3119898 minus 1) (5119898 minus 1)


2

21198982 (5119898 minus 1) (7119898 minus 1)

1205732=1

2119898[(120581ℎ

120581119904

minus 1)(119863119890

119889119904

)

(1119898minus1)

minus120581ℎ

120581119904

(119863119890

119889119908

)

(1119898minus1)

]

1205733= (

1

2119898minus

1

3119898 minus 1)[(

120581ℎ

120581119904

minus 1)(119863119890

119889119904

)

(1119898minus3)

minus120581ℎ

120581119904

(119863119890

119889119908

)

(1119898minus3)

]

1205734=120581ℎ

2119902119908

120587119911 (2119897 minus 119911) (1 minus1

119898)(

119863119890

119889119908

)

(1119898minus1)

(1

minus1198892

119908

1198632119890

)

1119898

(37)


0= I and [H]




119897 The cause




001 01 1 10 100 1000100

80

60

40

20

0

Iteration times 6

t (day)




Iteration times 1

Uh

()

m = 15 il = 200

(a) Condition 1

t (day)001 01 1 10 100 1000

100

80

60

40

20

0



Uh

()

m = 15 il = 100

(b) Condition 2

t (day)001 01 1 10 100 1000

100

80

60

40

20

0



Uh

()

m = 15 il = 50

(c) Condition 3

t (day)

100

80

60

40

20

0


001 01 1 10 100 1000


Uh

()

m = 12 il = 50

(d) Condition 4



0(1199020


119888V

119894119898minus1

119897

(1

120574119908

120597119901

120597119911)

119898minus11205972119901

1205971199112=120597119901

120597119905when 119894 lt 119894

119897

119888V1205972119901

1205971199112=120597119901

120597119905when 119894 ge 119894

119897

(38)


Conditions 119898 119894119897





119861 (m) 119867

(m)119889119904

(cm)119889119908

(cm)119864

(MPa)120583

()1205741015840

(kNm3)1206011015840

(∘)119896ℎ0

(ms)119896ℎ0119896V0()

119896ℎ01198961199040

()1198961199080119896ℎ0

()115 100 150 70 30 035 10 30 10minus9 10 40 104

001 01 1 10 100100

80

60

40

20

0

Aver

age d

egre

e of c

onso

lidat

ion

()

Time (days)




m = 15 il = 20

m = 12 il = 5





119890 119887119904 and 119887




Conditions 119898 119894119897

120572 = 119862119888119862119896

1199020(kPa)






0is related to







50




01 1 10 100 1000100

80

60

40

20

0

Aver

age d

egre

e of c

onso

lidat

ion

()

Time (days)



Non-Darcy flow

120572 = 10

120572 = 05

120572 = 00


m = 15 il = 50


01 1 10 100 1000100

80

60

40

20

0

Non-Darcy flow

Aver

age d

egre

e of c

onso

lidat

ion

()

Time (days)



120572 = 05


m = 15 il = 50

q0 = 10kPa

q0 = 50kPa

q0 = 100 kPa


01 1 10 100 1000100

80

60

40

20

0

Non-Darcy flow

Aver

age d

egre

e of c

onso

lidat

ion

()

Time (days)




m = 15 il = 200 120572 = 10

m = 15 il = 200 120572 = 05

m = 15 il = 100 120572 = 05

m = 15 il = 50 120572 = 05





22 23 29 39 30 35 31 33 4311988090






0 The comparisons





0





0= 10 kPa) and




50(11988090)




119897= 20 120572 = 10) is also plotted



7 Conclusions



119888119862119896 value range is






Nomenclature
















119909+ 1205901015840

119910+ 1205901015840

119911)3

1199011015840




and 119911120572 Ratio 119862

119888119862119896






119909 1205901015840

119910 1205901015840





Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










y

z

o

x

2b

2a

2c

85

41

2 3

6 7

p

o998400120585

120577120578

u w




119897V119909+ 119898V119910+ 119899V119911= V119899 (17)







f asymp f = V 119879 = [N] 120575119890

119901 asymp = [N] p119890 (18)


1120575119879


119879

8]119879 120575119894= [119906119894V119894119908119894]119879

119894 = 1 2 8 119906119894 V119894 and 119908




8]119879 119901119894is the nodal EPP [N] =

[1198731I 1198732I sdot sdot sdot 119873


[11987311198732sdot sdot sdot 119873


119873119894=1

8(1 + 120585120585

119894) (1 + 120578120578

119894) (1 + 120577120577

119894) 119894 = 1 2 8 (19)

where 120585119894 120578119894 and 120577



119894as the weight


[K119890] Δ120575

119890+ [K119888] Δp119890 = ΔR

119891119890 (20)

in which

[K119890] =∭

119881119890

[B]119879 [D] [B] d119881

[K119888] =∭

119881119890

[B]119879 M [N] d119881

ΔR119891119890

= ∬119878119890

[N]119879 ΔF119890 d119878

Δ120575119890= [Δ120575

119879

1Δ120575119879


119879

8]119879


ΔF119890 = Δ119865119909 Δ119865119910 Δ119865119911119879

(21)

where [K119890] [K119888] and ΔR



1B2sdot sdot sdot B

8] [B119894]


119894 119894 = 1 2 8

Δ119865119909 Δ119865119910 and Δ119865




119877119890

119868119902=120597

120597119905M119879 [120597] f minus M119879 [120597] [H] [k] [120597]119879 M

119877119890

119861119902= 119897V119909+ 119898V119910+ 119899V119911minus V119899

(22)


∭119881119890

119873119894(120597

120597119905M119879 [120597] f

minus M119879 [120597] [H] [k] [120597]119879 M ) d119881 = 0

119894 = 1 2 8

(23)

∬119878119890

119873119894(119897V119909+ 119898V119910+ 119899V119911minus V119899) d119878 = 0

119894 = 1 2 8

(24)



∭119881119890

119873119894(120597

120597119905M119879 [120597] f) d119881

=∭119881119890

119873119894M119879 [120597] [N] d119881

(25)



minus∭119881119890

119873119894(M119879 [120597] [H] [k] [120597]119879 M ) d119881

= ∬119878119890

119873119894V119899d119878

+∭119881119890

nabla119879119873119894 [H] [k] nabla [N] p

119890 d119881

(26)


(26) we have

∭119881119890

119873119894 M119879[120597] [N] d119881

+∭119881119890

nabla119879119873119894 [H] [k] nabla [N] p

119890 d119881

= minus∬119878119890

119873119894V119899d119878 119894 = 1 2 8

(27)


[K119888]119879119890

minus [K119904] p119890 = R

119902119890

(28)

in which

[K119904] =∭

119881119890

[B119904]119879

[H] [k] [B119904] d119881

[B119904] =

120597

120597119909

120597

120597119910

120597

120597119911

119879

[1198731 1198732 sdot sdot sdot 1198738]

R119902119890

= ∬119878119890

[N]119879

V119899d119878

(29)





to 119905119899+1

we have

int

119905119895+1

119905119895

[K119888]119879119890

d119905 minus int119905119895+1

119905119895

[K119904] p119890 d119905

= int

119905119895+1

119905119895

R119902119890

d119905(30)


int

119905119895+1

119905119895

p119890 d119905 = Δ119905 (p119890119899+ 120579 Δp119890)

int

119905119895+1

119905119895

R119902119890

d119905 = Δ119905 R119902119890

119899

(31)


[K119888]119879

Δ120575119890minus 120579Δ119905 [K

119904] Δp119890 = ΔR

119901119890 (32)

in which

ΔR119901119890

= Δ119905 [R119902119890

119895+ [K119904] p119890119895] (33)





119890] and [K



119901119890Matrix








0can be










119894119895+1

119909=1

120574119908

120597119901119895

120597119909

=

(119901119895

1+ 119901119895

4+ 119901119895

5+ 119901119895

8) minus (119901

119895

2+ 119901119895

3+ 119901119895

6+ 119901119895

7)

8120574119908119886

119894119895+1

119910=1

120574119908

120597119901119895

120597119910

=

(119901119895

1+ 119901119895

2+ 119901119895

5+ 119901119895

6) minus (119901

119895

3+ 119901119895

4+ 119901119895

7+ 119901119895

8)

8120574119908119887

119894119895+1

119911=1

120574119908

120597119901119895

120597119911

=

(119901119895

1+ 119901119895

2+ 119901119895

3+ 119901119895

4) minus (119901

119895

5+ 119901119895

6+ 119901119895

7+ 119901119895

8)

8120574119908119888

(34)





0 Considering




0as a





5 Verification



Z

Drain

Impervious

Impe

rvio

us

Impe

rvio

us

Impervious

Pervious

Impervious

Nature soil

Smear zonekw ks

kh k

bw = 886 cm

H = 5m

bs = 1772 cm

B e=2658

m



119908= 01m


119890 119887119904 and 119887





0is


119908 119896119904 119896ℎ and



For Darcy flow


8119888ℎ119905

1205831198632119890

) (35)



Cases 119896ℎ(ms) 119896

ℎ119896V 119896

ℎ119896119904

119896119908119896ℎ

119864 (MPa) 120583 119898 119894119897



119880ℎ= 1 minus [1 +

120582119905

1205721198632119890

(Δ1199060

119863119890120574119908

)

119898minus1

]

1(1minus119898)

(36)



120583 = 1205831+ 1205832+ 1205833

1205831=

1198632

119890

1198632119890minus 1198892119908

[ln(119863119890

119889119904

) +119896ℎ

119896119904

ln(119889119904

119889119908

) minus3

4]

1205832=120587119896ℎ

119902119908

119911 (2119867 minus 119911)(1 minus1198892

119908

1198632119890

)

1205833=

1198892

119904

1198632119890minus 1198892119908

(1 minus1198892

119904

41198632119890

) +119896ℎ

119896119904

1198892

119908

1198632119890minus 1198892119908

(1198894

119904minus 1198894

119908

411986321198901198892119908

minus1198892

119904

1198892119908

+ 1)

119888ℎ=

119896ℎ

119898V120574119908

119898V =(1 + 120583) (1 minus 2120583)

119864 (1 minus 120583)

119902119908= 119896119908

1205871198892

119908

4

120581ℎ

120581119904

=119896ℎ

119896119904

120582 =120581ℎ

120574119908119898V

120572 =1198982119898120573119898

4 (119898 minus 1)119898+1

120573 = 1205731+ 1205732minus 1205733+ 1205734

1205731=

1


119898 minus 1

119898 (3119898 minus 1) (5119898 minus 1)


2

21198982 (5119898 minus 1) (7119898 minus 1)

1205732=1

2119898[(120581ℎ

120581119904

minus 1)(119863119890

119889119904

)

(1119898minus1)

minus120581ℎ

120581119904

(119863119890

119889119908

)

(1119898minus1)

]

1205733= (

1

2119898minus

1

3119898 minus 1)[(

120581ℎ

120581119904

minus 1)(119863119890

119889119904

)

(1119898minus3)

minus120581ℎ

120581119904

(119863119890

119889119908

)

(1119898minus3)

]

1205734=120581ℎ

2119902119908

120587119911 (2119897 minus 119911) (1 minus1

119898)(

119863119890

119889119908

)

(1119898minus1)

(1

minus1198892

119908

1198632119890

)

1119898

(37)


0= I and [H]




119897 The cause




001 01 1 10 100 1000100

80

60

40

20

0

Iteration times 6

t (day)




Iteration times 1

Uh

()

m = 15 il = 200

(a) Condition 1

t (day)001 01 1 10 100 1000

100

80

60

40

20

0



Uh

()

m = 15 il = 100

(b) Condition 2

t (day)001 01 1 10 100 1000

100

80

60

40

20

0



Uh

()

m = 15 il = 50

(c) Condition 3

t (day)

100

80

60

40

20

0


001 01 1 10 100 1000


Uh

()

m = 12 il = 50

(d) Condition 4



0(1199020


119888V

119894119898minus1

119897

(1

120574119908

120597119901

120597119911)

119898minus11205972119901

1205971199112=120597119901

120597119905when 119894 lt 119894

119897

119888V1205972119901

1205971199112=120597119901

120597119905when 119894 ge 119894

119897

(38)


Conditions 119898 119894119897





119861 (m) 119867

(m)119889119904

(cm)119889119908

(cm)119864

(MPa)120583

()1205741015840

(kNm3)1206011015840

(∘)119896ℎ0

(ms)119896ℎ0119896V0()

119896ℎ01198961199040

()1198961199080119896ℎ0

()115 100 150 70 30 035 10 30 10minus9 10 40 104

001 01 1 10 100100

80

60

40

20

0

Aver

age d

egre

e of c

onso

lidat

ion

()

Time (days)




m = 15 il = 20

m = 12 il = 5





119890 119887119904 and 119887




Conditions 119898 119894119897

120572 = 119862119888119862119896

1199020(kPa)






0is related to







50




01 1 10 100 1000100

80

60

40

20

0

Aver

age d

egre

e of c

onso

lidat

ion

()

Time (days)



Non-Darcy flow

120572 = 10

120572 = 05

120572 = 00


m = 15 il = 50


01 1 10 100 1000100

80

60

40

20

0

Non-Darcy flow

Aver

age d

egre

e of c

onso

lidat

ion

()

Time (days)



120572 = 05


m = 15 il = 50

q0 = 10kPa

q0 = 50kPa

q0 = 100 kPa


01 1 10 100 1000100

80

60

40

20

0

Non-Darcy flow

Aver

age d

egre

e of c

onso

lidat

ion

()

Time (days)




m = 15 il = 200 120572 = 10

m = 15 il = 200 120572 = 05

m = 15 il = 100 120572 = 05

m = 15 il = 50 120572 = 05





22 23 29 39 30 35 31 33 4311988090






0 The comparisons





0





0= 10 kPa) and




50(11988090)




119897= 20 120572 = 10) is also plotted



7 Conclusions



119888119862119896 value range is






Nomenclature
















119909+ 1205901015840

119910+ 1205901015840

119911)3

1199011015840




and 119911120572 Ratio 119862

119888119862119896






119909 1205901015840

119910 1205901015840





Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











∭119881119890

119873119894(120597

120597119905M119879 [120597] f) d119881

=∭119881119890

119873119894M119879 [120597] [N] d119881

(25)



minus∭119881119890

119873119894(M119879 [120597] [H] [k] [120597]119879 M ) d119881

= ∬119878119890

119873119894V119899d119878

+∭119881119890

nabla119879119873119894 [H] [k] nabla [N] p

119890 d119881

(26)


(26) we have

∭119881119890

119873119894 M119879[120597] [N] d119881

+∭119881119890

nabla119879119873119894 [H] [k] nabla [N] p

119890 d119881

= minus∬119878119890

119873119894V119899d119878 119894 = 1 2 8

(27)


[K119888]119879119890

minus [K119904] p119890 = R

119902119890

(28)

in which

[K119904] =∭

119881119890

[B119904]119879

[H] [k] [B119904] d119881

[B119904] =

120597

120597119909

120597

120597119910

120597

120597119911

119879

[1198731 1198732 sdot sdot sdot 1198738]

R119902119890

= ∬119878119890

[N]119879

V119899d119878

(29)





to 119905119899+1

we have

int

119905119895+1

119905119895

[K119888]119879119890

d119905 minus int119905119895+1

119905119895

[K119904] p119890 d119905

= int

119905119895+1

119905119895

R119902119890

d119905(30)


int

119905119895+1

119905119895

p119890 d119905 = Δ119905 (p119890119899+ 120579 Δp119890)

int

119905119895+1

119905119895

R119902119890

d119905 = Δ119905 R119902119890

119899

(31)


[K119888]119879

Δ120575119890minus 120579Δ119905 [K

119904] Δp119890 = ΔR

119901119890 (32)

in which

ΔR119901119890

= Δ119905 [R119902119890

119895+ [K119904] p119890119895] (33)





119890] and [K



119901119890Matrix








0can be










119894119895+1

119909=1

120574119908

120597119901119895

120597119909

=

(119901119895

1+ 119901119895

4+ 119901119895

5+ 119901119895

8) minus (119901

119895

2+ 119901119895

3+ 119901119895

6+ 119901119895

7)

8120574119908119886

119894119895+1

119910=1

120574119908

120597119901119895

120597119910

=

(119901119895

1+ 119901119895

2+ 119901119895

5+ 119901119895

6) minus (119901

119895

3+ 119901119895

4+ 119901119895

7+ 119901119895

8)

8120574119908119887

119894119895+1

119911=1

120574119908

120597119901119895

120597119911

=

(119901119895

1+ 119901119895

2+ 119901119895

3+ 119901119895

4) minus (119901

119895

5+ 119901119895

6+ 119901119895

7+ 119901119895

8)

8120574119908119888

(34)





0 Considering




0as a





5 Verification



Z

Drain

Impervious

Impe

rvio

us

Impe

rvio

us

Impervious

Pervious

Impervious

Nature soil

Smear zonekw ks

kh k

bw = 886 cm

H = 5m

bs = 1772 cm

B e=2658

m



119908= 01m


119890 119887119904 and 119887





0is


119908 119896119904 119896ℎ and



For Darcy flow


8119888ℎ119905

1205831198632119890

) (35)



Cases 119896ℎ(ms) 119896

ℎ119896V 119896

ℎ119896119904

119896119908119896ℎ

119864 (MPa) 120583 119898 119894119897



119880ℎ= 1 minus [1 +

120582119905

1205721198632119890

(Δ1199060

119863119890120574119908

)

119898minus1

]

1(1minus119898)

(36)



120583 = 1205831+ 1205832+ 1205833

1205831=

1198632

119890

1198632119890minus 1198892119908

[ln(119863119890

119889119904

) +119896ℎ

119896119904

ln(119889119904

119889119908

) minus3

4]

1205832=120587119896ℎ

119902119908

119911 (2119867 minus 119911)(1 minus1198892

119908

1198632119890

)

1205833=

1198892

119904

1198632119890minus 1198892119908

(1 minus1198892

119904

41198632119890

) +119896ℎ

119896119904

1198892

119908

1198632119890minus 1198892119908

(1198894

119904minus 1198894

119908

411986321198901198892119908

minus1198892

119904

1198892119908

+ 1)

119888ℎ=

119896ℎ

119898V120574119908

119898V =(1 + 120583) (1 minus 2120583)

119864 (1 minus 120583)

119902119908= 119896119908

1205871198892

119908

4

120581ℎ

120581119904

=119896ℎ

119896119904

120582 =120581ℎ

120574119908119898V

120572 =1198982119898120573119898

4 (119898 minus 1)119898+1

120573 = 1205731+ 1205732minus 1205733+ 1205734

1205731=

1


119898 minus 1

119898 (3119898 minus 1) (5119898 minus 1)


2

21198982 (5119898 minus 1) (7119898 minus 1)

1205732=1

2119898[(120581ℎ

120581119904

minus 1)(119863119890

119889119904

)

(1119898minus1)

minus120581ℎ

120581119904

(119863119890

119889119908

)

(1119898minus1)

]

1205733= (

1

2119898minus

1

3119898 minus 1)[(

120581ℎ

120581119904

minus 1)(119863119890

119889119904

)

(1119898minus3)

minus120581ℎ

120581119904

(119863119890

119889119908

)

(1119898minus3)

]

1205734=120581ℎ

2119902119908

120587119911 (2119897 minus 119911) (1 minus1

119898)(

119863119890

119889119908

)

(1119898minus1)

(1

minus1198892

119908

1198632119890

)

1119898

(37)


0= I and [H]




119897 The cause




001 01 1 10 100 1000100

80

60

40

20

0

Iteration times 6

t (day)




Iteration times 1

Uh

()

m = 15 il = 200

(a) Condition 1

t (day)001 01 1 10 100 1000

100

80

60

40

20

0



Uh

()

m = 15 il = 100

(b) Condition 2

t (day)001 01 1 10 100 1000

100

80

60

40

20

0



Uh

()

m = 15 il = 50

(c) Condition 3

t (day)

100

80

60

40

20

0


001 01 1 10 100 1000


Uh

()

m = 12 il = 50

(d) Condition 4



0(1199020


119888V

119894119898minus1

119897

(1

120574119908

120597119901

120597119911)

119898minus11205972119901

1205971199112=120597119901

120597119905when 119894 lt 119894

119897

119888V1205972119901

1205971199112=120597119901

120597119905when 119894 ge 119894

119897

(38)


Conditions 119898 119894119897





119861 (m) 119867

(m)119889119904

(cm)119889119908

(cm)119864

(MPa)120583

()1205741015840

(kNm3)1206011015840

(∘)119896ℎ0

(ms)119896ℎ0119896V0()

119896ℎ01198961199040

()1198961199080119896ℎ0

()115 100 150 70 30 035 10 30 10minus9 10 40 104

001 01 1 10 100100

80

60

40

20

0

Aver

age d

egre

e of c

onso

lidat

ion

()

Time (days)




m = 15 il = 20

m = 12 il = 5





119890 119887119904 and 119887




Conditions 119898 119894119897

120572 = 119862119888119862119896

1199020(kPa)






0is related to







50




01 1 10 100 1000100

80

60

40

20

0

Aver

age d

egre

e of c

onso

lidat

ion

()

Time (days)



Non-Darcy flow

120572 = 10

120572 = 05

120572 = 00


m = 15 il = 50


01 1 10 100 1000100

80

60

40

20

0

Non-Darcy flow

Aver

age d

egre

e of c

onso

lidat

ion

()

Time (days)



120572 = 05


m = 15 il = 50

q0 = 10kPa

q0 = 50kPa

q0 = 100 kPa


01 1 10 100 1000100

80

60

40

20

0

Non-Darcy flow

Aver

age d

egre

e of c

onso

lidat

ion

()

Time (days)




m = 15 il = 200 120572 = 10

m = 15 il = 200 120572 = 05

m = 15 il = 100 120572 = 05

m = 15 il = 50 120572 = 05





22 23 29 39 30 35 31 33 4311988090






0 The comparisons





0





0= 10 kPa) and




50(11988090)




119897= 20 120572 = 10) is also plotted



7 Conclusions



119888119862119896 value range is






Nomenclature
















119909+ 1205901015840

119910+ 1205901015840

119911)3

1199011015840




and 119911120572 Ratio 119862

119888119862119896






119909 1205901015840

119910 1205901015840





Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119894119895+1

119909=1

120574119908

120597119901119895

120597119909

=

(119901119895

1+ 119901119895

4+ 119901119895

5+ 119901119895

8) minus (119901

119895

2+ 119901119895

3+ 119901119895

6+ 119901119895

7)

8120574119908119886

119894119895+1

119910=1

120574119908

120597119901119895

120597119910

=

(119901119895

1+ 119901119895

2+ 119901119895

5+ 119901119895

6) minus (119901

119895

3+ 119901119895

4+ 119901119895

7+ 119901119895

8)

8120574119908119887

119894119895+1

119911=1

120574119908

120597119901119895

120597119911

=

(119901119895

1+ 119901119895

2+ 119901119895

3+ 119901119895

4) minus (119901

119895

5+ 119901119895

6+ 119901119895

7+ 119901119895

8)

8120574119908119888

(34)





0 Considering




0as a





5 Verification



Z

Drain

Impervious

Impe

rvio

us

Impe

rvio

us

Impervious

Pervious

Impervious

Nature soil

Smear zonekw ks

kh k

bw = 886 cm

H = 5m

bs = 1772 cm

B e=2658

m



119908= 01m


119890 119887119904 and 119887





0is


119908 119896119904 119896ℎ and



For Darcy flow


8119888ℎ119905

1205831198632119890

) (35)



Cases 119896ℎ(ms) 119896

ℎ119896V 119896

ℎ119896119904

119896119908119896ℎ

119864 (MPa) 120583 119898 119894119897



119880ℎ= 1 minus [1 +

120582119905

1205721198632119890

(Δ1199060

119863119890120574119908

)

119898minus1

]

1(1minus119898)

(36)



120583 = 1205831+ 1205832+ 1205833

1205831=

1198632

119890

1198632119890minus 1198892119908

[ln(119863119890

119889119904

) +119896ℎ

119896119904

ln(119889119904

119889119908

) minus3

4]

1205832=120587119896ℎ

119902119908

119911 (2119867 minus 119911)(1 minus1198892

119908

1198632119890

)

1205833=

1198892

119904

1198632119890minus 1198892119908

(1 minus1198892

119904

41198632119890

) +119896ℎ

119896119904

1198892

119908

1198632119890minus 1198892119908

(1198894

119904minus 1198894

119908

411986321198901198892119908

minus1198892

119904

1198892119908

+ 1)

119888ℎ=

119896ℎ

119898V120574119908

119898V =(1 + 120583) (1 minus 2120583)

119864 (1 minus 120583)

119902119908= 119896119908

1205871198892

119908

4

120581ℎ

120581119904

=119896ℎ

119896119904

120582 =120581ℎ

120574119908119898V

120572 =1198982119898120573119898

4 (119898 minus 1)119898+1

120573 = 1205731+ 1205732minus 1205733+ 1205734

1205731=

1


119898 minus 1

119898 (3119898 minus 1) (5119898 minus 1)


2

21198982 (5119898 minus 1) (7119898 minus 1)

1205732=1

2119898[(120581ℎ

120581119904

minus 1)(119863119890

119889119904

)

(1119898minus1)

minus120581ℎ

120581119904

(119863119890

119889119908

)

(1119898minus1)

]

1205733= (

1

2119898minus

1

3119898 minus 1)[(

120581ℎ

120581119904

minus 1)(119863119890

119889119904

)

(1119898minus3)

minus120581ℎ

120581119904

(119863119890

119889119908

)

(1119898minus3)

]

1205734=120581ℎ

2119902119908

120587119911 (2119897 minus 119911) (1 minus1

119898)(

119863119890

119889119908

)

(1119898minus1)

(1

minus1198892

119908

1198632119890

)

1119898

(37)


0= I and [H]




119897 The cause




001 01 1 10 100 1000100

80

60

40

20

0

Iteration times 6

t (day)




Iteration times 1

Uh

()

m = 15 il = 200

(a) Condition 1

t (day)001 01 1 10 100 1000

100

80

60

40

20

0



Uh

()

m = 15 il = 100

(b) Condition 2

t (day)001 01 1 10 100 1000

100

80

60

40

20

0



Uh

()

m = 15 il = 50

(c) Condition 3

t (day)

100

80

60

40

20

0


001 01 1 10 100 1000


Uh

()

m = 12 il = 50

(d) Condition 4



0(1199020


119888V

119894119898minus1

119897

(1

120574119908

120597119901

120597119911)

119898minus11205972119901

1205971199112=120597119901

120597119905when 119894 lt 119894

119897

119888V1205972119901

1205971199112=120597119901

120597119905when 119894 ge 119894

119897

(38)


Conditions 119898 119894119897





119861 (m) 119867

(m)119889119904

(cm)119889119908

(cm)119864

(MPa)120583

()1205741015840

(kNm3)1206011015840

(∘)119896ℎ0

(ms)119896ℎ0119896V0()

119896ℎ01198961199040

()1198961199080119896ℎ0

()115 100 150 70 30 035 10 30 10minus9 10 40 104

001 01 1 10 100100

80

60

40

20

0

Aver

age d

egre

e of c

onso

lidat

ion

()

Time (days)




m = 15 il = 20

m = 12 il = 5





119890 119887119904 and 119887




Conditions 119898 119894119897

120572 = 119862119888119862119896

1199020(kPa)






0is related to







50




01 1 10 100 1000100

80

60

40

20

0

Aver

age d

egre

e of c

onso

lidat

ion

()

Time (days)



Non-Darcy flow

120572 = 10

120572 = 05

120572 = 00


m = 15 il = 50


01 1 10 100 1000100

80

60

40

20

0

Non-Darcy flow

Aver

age d

egre

e of c

onso

lidat

ion

()

Time (days)



120572 = 05


m = 15 il = 50

q0 = 10kPa

q0 = 50kPa

q0 = 100 kPa


01 1 10 100 1000100

80

60

40

20

0

Non-Darcy flow

Aver

age d

egre

e of c

onso

lidat

ion

()

Time (days)




m = 15 il = 200 120572 = 10

m = 15 il = 200 120572 = 05

m = 15 il = 100 120572 = 05

m = 15 il = 50 120572 = 05





22 23 29 39 30 35 31 33 4311988090






0 The comparisons





0





0= 10 kPa) and




50(11988090)




119897= 20 120572 = 10) is also plotted



7 Conclusions



119888119862119896 value range is






Nomenclature
















119909+ 1205901015840

119910+ 1205901015840

119911)3

1199011015840




and 119911120572 Ratio 119862

119888119862119896






119909 1205901015840

119910 1205901015840





Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Cases 119896ℎ(ms) 119896

ℎ119896V 119896

ℎ119896119904

119896119908119896ℎ

119864 (MPa) 120583 119898 119894119897



119880ℎ= 1 minus [1 +

120582119905

1205721198632119890

(Δ1199060

119863119890120574119908

)

119898minus1

]

1(1minus119898)

(36)



120583 = 1205831+ 1205832+ 1205833

1205831=

1198632

119890

1198632119890minus 1198892119908

[ln(119863119890

119889119904

) +119896ℎ

119896119904

ln(119889119904

119889119908

) minus3

4]

1205832=120587119896ℎ

119902119908

119911 (2119867 minus 119911)(1 minus1198892

119908

1198632119890

)

1205833=

1198892

119904

1198632119890minus 1198892119908

(1 minus1198892

119904

41198632119890

) +119896ℎ

119896119904

1198892

119908

1198632119890minus 1198892119908

(1198894

119904minus 1198894

119908

411986321198901198892119908

minus1198892

119904

1198892119908

+ 1)

119888ℎ=

119896ℎ

119898V120574119908

119898V =(1 + 120583) (1 minus 2120583)

119864 (1 minus 120583)

119902119908= 119896119908

1205871198892

119908

4

120581ℎ

120581119904

=119896ℎ

119896119904

120582 =120581ℎ

120574119908119898V

120572 =1198982119898120573119898

4 (119898 minus 1)119898+1

120573 = 1205731+ 1205732minus 1205733+ 1205734

1205731=

1


119898 minus 1

119898 (3119898 minus 1) (5119898 minus 1)


2

21198982 (5119898 minus 1) (7119898 minus 1)

1205732=1

2119898[(120581ℎ

120581119904

minus 1)(119863119890

119889119904

)

(1119898minus1)

minus120581ℎ

120581119904

(119863119890

119889119908

)

(1119898minus1)

]

1205733= (

1

2119898minus

1

3119898 minus 1)[(

120581ℎ

120581119904

minus 1)(119863119890

119889119904

)

(1119898minus3)

minus120581ℎ

120581119904

(119863119890

119889119908

)

(1119898minus3)

]

1205734=120581ℎ

2119902119908

120587119911 (2119897 minus 119911) (1 minus1

119898)(

119863119890

119889119908

)

(1119898minus1)

(1

minus1198892

119908

1198632119890

)

1119898

(37)


0= I and [H]




119897 The cause




001 01 1 10 100 1000100

80

60

40

20

0

Iteration times 6

t (day)




Iteration times 1

Uh

()

m = 15 il = 200

(a) Condition 1

t (day)001 01 1 10 100 1000

100

80

60

40

20

0



Uh

()

m = 15 il = 100

(b) Condition 2

t (day)001 01 1 10 100 1000

100

80

60

40

20

0



Uh

()

m = 15 il = 50

(c) Condition 3

t (day)

100

80

60

40

20

0


001 01 1 10 100 1000


Uh

()

m = 12 il = 50

(d) Condition 4



0(1199020


119888V

119894119898minus1

119897

(1

120574119908

120597119901

120597119911)

119898minus11205972119901

1205971199112=120597119901

120597119905when 119894 lt 119894

119897

119888V1205972119901

1205971199112=120597119901

120597119905when 119894 ge 119894

119897

(38)


Conditions 119898 119894119897





119861 (m) 119867

(m)119889119904

(cm)119889119908

(cm)119864

(MPa)120583

()1205741015840

(kNm3)1206011015840

(∘)119896ℎ0

(ms)119896ℎ0119896V0()

119896ℎ01198961199040

()1198961199080119896ℎ0

()115 100 150 70 30 035 10 30 10minus9 10 40 104

001 01 1 10 100100

80

60

40

20

0

Aver

age d

egre

e of c

onso

lidat

ion

()

Time (days)




m = 15 il = 20

m = 12 il = 5





119890 119887119904 and 119887




Conditions 119898 119894119897

120572 = 119862119888119862119896

1199020(kPa)






0is related to







50




01 1 10 100 1000100

80

60

40

20

0

Aver

age d

egre

e of c

onso

lidat

ion

()

Time (days)



Non-Darcy flow

120572 = 10

120572 = 05

120572 = 00


m = 15 il = 50


01 1 10 100 1000100

80

60

40

20

0

Non-Darcy flow

Aver

age d

egre

e of c

onso

lidat

ion

()

Time (days)



120572 = 05


m = 15 il = 50

q0 = 10kPa

q0 = 50kPa

q0 = 100 kPa


01 1 10 100 1000100

80

60

40

20

0

Non-Darcy flow

Aver

age d

egre

e of c

onso

lidat

ion

()

Time (days)




m = 15 il = 200 120572 = 10

m = 15 il = 200 120572 = 05

m = 15 il = 100 120572 = 05

m = 15 il = 50 120572 = 05





22 23 29 39 30 35 31 33 4311988090






0 The comparisons





0





0= 10 kPa) and




50(11988090)




119897= 20 120572 = 10) is also plotted



7 Conclusions



119888119862119896 value range is






Nomenclature
















119909+ 1205901015840

119910+ 1205901015840

119911)3

1199011015840




and 119911120572 Ratio 119862

119888119862119896






119909 1205901015840

119910 1205901015840





Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










001 01 1 10 100 1000100

80

60

40

20

0

Iteration times 6

t (day)




Iteration times 1

Uh

()

m = 15 il = 200

(a) Condition 1

t (day)001 01 1 10 100 1000

100

80

60

40

20

0



Uh

()

m = 15 il = 100

(b) Condition 2

t (day)001 01 1 10 100 1000

100

80

60

40

20

0



Uh

()

m = 15 il = 50

(c) Condition 3

t (day)

100

80

60

40

20

0


001 01 1 10 100 1000


Uh

()

m = 12 il = 50

(d) Condition 4



0(1199020


119888V

119894119898minus1

119897

(1

120574119908

120597119901

120597119911)

119898minus11205972119901

1205971199112=120597119901

120597119905when 119894 lt 119894

119897

119888V1205972119901

1205971199112=120597119901

120597119905when 119894 ge 119894

119897

(38)


Conditions 119898 119894119897





119861 (m) 119867

(m)119889119904

(cm)119889119908

(cm)119864

(MPa)120583

()1205741015840

(kNm3)1206011015840

(∘)119896ℎ0

(ms)119896ℎ0119896V0()

119896ℎ01198961199040

()1198961199080119896ℎ0

()115 100 150 70 30 035 10 30 10minus9 10 40 104

001 01 1 10 100100

80

60

40

20

0

Aver

age d

egre

e of c

onso

lidat

ion

()

Time (days)




m = 15 il = 20

m = 12 il = 5





119890 119887119904 and 119887




Conditions 119898 119894119897

120572 = 119862119888119862119896

1199020(kPa)






0is related to







50




01 1 10 100 1000100

80

60

40

20

0

Aver

age d

egre

e of c

onso

lidat

ion

()

Time (days)



Non-Darcy flow

120572 = 10

120572 = 05

120572 = 00


m = 15 il = 50


01 1 10 100 1000100

80

60

40

20

0

Non-Darcy flow

Aver

age d

egre

e of c

onso

lidat

ion

()

Time (days)



120572 = 05


m = 15 il = 50

q0 = 10kPa

q0 = 50kPa

q0 = 100 kPa


01 1 10 100 1000100

80

60

40

20

0

Non-Darcy flow

Aver

age d

egre

e of c

onso

lidat

ion

()

Time (days)




m = 15 il = 200 120572 = 10

m = 15 il = 200 120572 = 05

m = 15 il = 100 120572 = 05

m = 15 il = 50 120572 = 05





22 23 29 39 30 35 31 33 4311988090






0 The comparisons





0





0= 10 kPa) and




50(11988090)




119897= 20 120572 = 10) is also plotted



7 Conclusions



119888119862119896 value range is






Nomenclature
















119909+ 1205901015840

119910+ 1205901015840

119911)3

1199011015840




and 119911120572 Ratio 119862

119888119862119896






119909 1205901015840

119910 1205901015840





Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119861 (m) 119867

(m)119889119904

(cm)119889119908

(cm)119864

(MPa)120583

()1205741015840

(kNm3)1206011015840

(∘)119896ℎ0

(ms)119896ℎ0119896V0()

119896ℎ01198961199040

()1198961199080119896ℎ0

()115 100 150 70 30 035 10 30 10minus9 10 40 104

001 01 1 10 100100

80

60

40

20

0

Aver

age d

egre

e of c

onso

lidat

ion

()

Time (days)




m = 15 il = 20

m = 12 il = 5





119890 119887119904 and 119887




Conditions 119898 119894119897

120572 = 119862119888119862119896

1199020(kPa)






0is related to







50




01 1 10 100 1000100

80

60

40

20

0

Aver

age d

egre

e of c

onso

lidat

ion

()

Time (days)



Non-Darcy flow

120572 = 10

120572 = 05

120572 = 00


m = 15 il = 50


01 1 10 100 1000100

80

60

40

20

0

Non-Darcy flow

Aver

age d

egre

e of c

onso

lidat

ion

()

Time (days)



120572 = 05


m = 15 il = 50

q0 = 10kPa

q0 = 50kPa

q0 = 100 kPa


01 1 10 100 1000100

80

60

40

20

0

Non-Darcy flow

Aver

age d

egre

e of c

onso

lidat

ion

()

Time (days)




m = 15 il = 200 120572 = 10

m = 15 il = 200 120572 = 05

m = 15 il = 100 120572 = 05

m = 15 il = 50 120572 = 05





22 23 29 39 30 35 31 33 4311988090






0 The comparisons





0





0= 10 kPa) and




50(11988090)




119897= 20 120572 = 10) is also plotted



7 Conclusions



119888119862119896 value range is






Nomenclature
















119909+ 1205901015840

119910+ 1205901015840

119911)3

1199011015840




and 119911120572 Ratio 119862

119888119862119896






119909 1205901015840

119910 1205901015840





Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










01 1 10 100 1000100

80

60

40

20

0

Aver

age d

egre

e of c

onso

lidat

ion

()

Time (days)



Non-Darcy flow

120572 = 10

120572 = 05

120572 = 00


m = 15 il = 50


01 1 10 100 1000100

80

60

40

20

0

Non-Darcy flow

Aver

age d

egre

e of c

onso

lidat

ion

()

Time (days)



120572 = 05


m = 15 il = 50

q0 = 10kPa

q0 = 50kPa

q0 = 100 kPa


01 1 10 100 1000100

80

60

40

20

0

Non-Darcy flow

Aver

age d

egre

e of c

onso

lidat

ion

()

Time (days)




m = 15 il = 200 120572 = 10

m = 15 il = 200 120572 = 05

m = 15 il = 100 120572 = 05

m = 15 il = 50 120572 = 05





22 23 29 39 30 35 31 33 4311988090






0 The comparisons





0





0= 10 kPa) and




50(11988090)




119897= 20 120572 = 10) is also plotted



7 Conclusions



119888119862119896 value range is






Nomenclature
















119909+ 1205901015840

119910+ 1205901015840

119911)3

1199011015840




and 119911120572 Ratio 119862

119888119862119896






119909 1205901015840

119910 1205901015840





Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











0 The comparisons





0





0= 10 kPa) and




50(11988090)




119897= 20 120572 = 10) is also plotted



7 Conclusions



119888119862119896 value range is






Nomenclature
















119909+ 1205901015840

119910+ 1205901015840

119911)3

1199011015840




and 119911120572 Ratio 119862

119888119862119896






119909 1205901015840

119910 1205901015840





Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













119909 1205901015840

119910 1205901015840





Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

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