Research ArticleDerivation of Universal Curves for Nonlinear Soil Consolidationwith Potential Constitutive Dependences

Gonzalo Garc-a-Ros 1 Ivaacuten Alhama1 Manuel Caacutenovas 2 and Francisco Alhama 3

1Technical University of Cartagena Civil Engineering Department Spain2Universidad Catolica del Norte Metallurgical and Mining Engineering Department Chile3Technical University of Cartagena Applied Physics Department Spain

Correspondence should be addressed to Gonzalo Garcıa-Ros gonzalogarciaupctes

Received 8 July 2018 Accepted 15 November 2018 Published 23 December 2018

Academic Editor Enrico Conte

Copyright copy 2018 Gonzalo Garcıa-Ros et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Nonlinear consolidation scenarios based on potential type constitutive dependencesmdashlike those proposed by Juarez-Badillomdashandeliminating the more restrictive hypothesis of 1+e and dz constant were characterized by the nondimensionalization process ofthe governing equations providing the independent dimensionless groups that rule the main unknowns of interest From theseuniversal curves have been depicted for both the characteristic time and the average degree of consolidation The solutions wereverified by numerical simulations and successfully compared in a case study showing the simplicity of use of the curves and thehigh reliability of the solutions they provide

1 Introduction

Soil consolidation has been studied by a large numberof authors for almost a hundred years Since Terzaghiannounced his linear consolidation theory [1] more com-plex models have been proposed These include 2D and3D geometries [2] as well as nonlinear problems (with anonconstant coefficient of consolidation due to the changesin the hydraulic conductivity and void ratio during theprocess [3]) for whose solutions numerical techniques arerequired However most nonlinear models focus on solvinga particular scenario ie with a very restrictive hypothesiswith no intention of reaching solutions of a universal char-acter which depend of the lowest number of dimensionlessparameters Geng et al [4] and Lu et al [5] studied nonlinearsoil consolidation problems the first under cycling loadingsand the second under time-variable loadings and verticaldrains using logarithmic type constitutive relations for themain variables of the problem (void ratio effective stressand hydraulic conductivity) obtaining that both the initialstress state and the ratio of the coefficients of the constitutiverelations are the most influential parameters that govern

the consolidation process Other references to nonlinearconsolidation are the papers ofWu et al [6] and Brandenberg[7] which also handle logarithmic dependences with thevoid ratio The first author presents an analytical solutionfor the electroosmotic consolidation with no allusion to thedimensionless groups that rule the problem while the secondapproximates the secondary consolidation

Special attention deserve the works of Butterfield [8]Lancellotta and Preziosi [9] Zhuang et al [10] and Conteand Troncone [11] The first one describes the advantages ofthe potential dependences versus the logarithmic ones whichbecome evident in highly compressible soils for which onthe one hand the slope of the e-log(1205901015840) dependence is farfrom being a constant value and on the other the assumptionof the logarithmic relation can lead to negative values forthe void ratio when 1205901015840f 1205901015840o takes a high value Lancellottaand Preziosi and Conte and Troncone make referencesto the potential type relations although they choose topresent solutions only for logarithmic dependences Thefirst ones present among others consolidation models withimpermeable or draining boundaries as well as solid particlessedimentation models in quiescent fluids while the latter

HindawiMathematical Problems in EngineeringVolume 2018 Article ID 5837592 15 pageshttpsdoiorg10115520185837592

2 Mathematical Problems in Engineering

solve a problem of step loads under the hypothesis of smallstrains Finally Zhuang et al obtain semianalytical solutionsunder the small strains hypothesis and using logarithmicconstitutive dependences characterizing the problem bymeans of a dimensionless parameter that is the quotientbetween the slopes of the e-log(1205901015840) and e-log(k) relations

For optimal dimensionless characterization of the solu-tion patterns (essentially the characteristic time and theaverage degree of consolidation the latter in terms of settle-ment and pressure dissipation) of the consolidation processesbased on a generalized Juarez-Badillo (J-B hereinafter)model[12] the dimensionless groups are derived by the nondi-mensionalization of the governing equations a process thatrequires bothersome mathematical steps due to the inherentnonlinearity of the model For the original J-B problemwhich assumes the strict hypothesis of an initial void ratiothat is negligible in the term of soil contraction the numberof the dimensionless groups reported by J-B is larger thanthe number derived in this paper so providing a less precisesolution in addition the author does not present universalsolutions since no characteristic time is proposed For otherextended J-Bmodels by deleting one ormore of his restrictivehypotheses (for example 1+eo=constant with eo = 0 1+e =constant and dz =constant) new dimensionless groups arederived verifying the solutions by numerical simulations forthe most general and complex model The general concept ofcharacteristic time which may be easily defined in lineal andisotropic soils is extended herein to nonlinear consolidationand after its introduction into the governing equation asan unknown reference to define the dimensionless time anew group containing the characteristic time emerges fromthe nondimensionalization process allowing the universalcurves to be constructed

By nondimensionalization the large number of isolateddimensionless parameters contained in the statement of theproblem and in the constitutive relations together with thedimensionless groups that can be formed from the relevantlist of parameters and variables by applying simple rules ofdimensional analysis [13] is reduced to the smallest numberthat best help researchers to manage the solution As isknown the application of pi theoremmdashderived from the the-ory of homogeneous functions (Buckingham [14])mdashallowsthe unknowns of interest expressed in their dimensionlessform to be set as a function of the mentioned dimensionlessgroups

There are two techniques whose purpose is the derivationof the dimensionless groups that rule the solution of a givenproblem the dimensional analysis and the method of grouptransformations In the first [13] the groups are derived bysimple mathematical manipulations from a list of relevantvariables expressed in terms of primary quantities (lengthmass and time) while in the second [15] they are obtainedfrom the mathematical model in its dimensionless form aftersomemathematical stepsThe technique applied in this paperwhich we call nondimensionalization of governing equations[16 17] starts from the governing equation in order to deducethe dimensionless groups In this after normalizing thevariables these and their changes are averaged ndash in fact theequation itself is averaged ndash and assumed to be of the order of

unity a valid hypothesis in problems with relatively smoothnonlinearities thus allowing the coefficients of the equationsto be of the same order of magnitude and unequivocallyproviding the most precise solution as demonstrated inmanystudies [18] Based on this methodology Manteca et al [19]study the nonlinear consolidation problem with constitutivedependencies of logarithmic type providing as a solution theuniversal curves and the dimensionless groups that governthe process

Classical nonlinear consolidation models (such as Davisand Raymond [20] Juarez-Badillo [12] and Cornetti andBattaglio [21]) differ from one another in the nature of theconstitutive relations between the parameters void ratio andpermeability and the effective soil pressure when trying toreflect the behavior of real soils In general these depen-dences converge to provide the same results in problems withsmall changes in these parameters (quasilinear problems) butdifferent results when the working range is wide In additionmost nonlinear models assume restrictive hypotheses suchas a constant soil thickness in the contraction term of thegoverning equation which distance the solutions from thoseobtained without these restrictions these solutions are bythe side of safety in consolidation and unsafety in swellingIn addition to the original J-B problem an extended modelwith both void ratio and thickness of the volume elementcontinuously changing during the consolidation process isanalyzed and solved

To verify the obtained results a comparison is madewith numerical solutions based on the network simulationmethod [22] This tool widely used in other fields is anefficient and computationally fast numerical method that hasdemonstrated its reliability in many linear and nonlinearengineering problems [23]Thedependences of characteristictime and average degree of consolidation on the rest ofthe dimensionless groups are checked ie whenever thedimensionless parameters retain the same values againstchanges in the individual parameters contained within thegroups neither the characteristic time nor average degree ofconsolidation change After the verification and presentationof universal curves for a wide range of values of the param-eters that sufficiently cover all real cases contributions andconclusions are summarized

2 The Original Juaacuterez-Badillo Model

This author [12] presented his model twenty years after thenonlinear model of Davis and Raymond [20] and appliedit to the odometer test to improve understanding of theprimary consolidation phenomenon Juarez-Badillo assumesincompressibility for water and soil particles and compress-ibility for the soil structure for which he sets the constitutiverelation dVV = minus120574v(d12059010158401205901015840) Doing so it is immediateto write mv = 120574v1205901015840 As regards permeability following hisdeductions (Juarez-Badillo [24]) J-B assumes a constitutivek-V dependence in the form kk1 = (VV1)120581 which is equiv-alent to a proportionality dependence between the relativedeviations of these parameters dkk = 120581(dVV) As a resultthe dependence k-1205901015840 takes the form kk1 = (120590101584012059010158401)minus120574v120581 orin terms of unitary deviations dkk = minus120574v120581(d12059010158401205901015840)

Mathematical Problems in Engineering 3

The original J-B model under odometer conditions(small volume change andnegligible specificweight ofwater)is defined by

d1205901015840dt

= minus 1120574wmv

120597120597z (k120597u120597z) (1)

Substituting the above dependences and using the parameters

120582 = 1-120574v120581k112059010158401120574w120574v = k1120574wmv1

= cv1cv = k120574wmv

= k112059010158401120574w120574v ( 120590101584012059010158401)1minus120574v120581 = cv1 ( 120590101584012059010158401)

120582

(2)

(1) reduces to

1205971205901015840120597t = minus1205901015840cv1 112059010158401 120597120597z [( 120590101584012059010158401)minus120574v120581 120597u120597z ] (3)

The simplifications assumed by the odometer test 120597u120597z =minus1205971205901015840120597z and 120597u120597t = minus1205971205901015840120597t allow us to write the lastequation as

1205971205901015840120597t = 1205901015840cv1 120597120597z [( 120590101584012059010158401)minus120574v120581 112059010158401 1205971205901015840120597z ] (4)

or in terms of the excess pore pressure variable as

120597u120597t = 1205901015840cv1 120597120597z [( 120590101584012059010158401)minus120574v120581 112059010158401 120597u120597z ] (5)

In this equation 12059010158401 can be substituted by any other pressurefor example 12059010158402 providing that cv is also substituted by cv2The case 120582=0 (or 120574v120581=1) ie constant cv is the nonlinearmodel of Davis and Raymond which always provides aneffective stress that is lower (or higher excess pore pressure)than that obtained with the Terzaghi model However thedifferent dependences V-1205901015840 (or e-1205901015840) for J-B and Davis andRaymond result in a different average degree of consolidation(Us) for bothmodels In fact for J-B [12] this unknown obeysthe following expression

Us = 1 minus (1H1) intH10

(120590101584012059010158401)minus120574v dz1 minus (1205901015840212059010158401)minus120574v (6)

or in terms of the new normalized variable V = (120590101584012059010158402)120582Us = 1 minus (H2H1) int1

0Vminus120574v120582dz10158401 minus H2H1 (7)

Finally J-B does not report analytical or numerical solutionsfor the excess pore pressure or for the effective pressureexcept for the case 120582=0 probably due to the large numberof variables involved He only reports [25] a consolidationabacus Us sim T for the case 120582=0 (120574v120581=1) and a constant cv

using H2H1 as parameter and a set of abacuses for the case120582120574v=(1120574v)-120581 = 0 cv = constant In these abacuses the timefactor is defined as T = tcv2H2 Despite these results nophysical meaning is attributed to 120582 T or V In short in viewof (7) J-B concludes that his model depends on three groupsH2H1 (1120574v)-120581 and the time factor T (since V is a function ofT)

3 Nondimensionalization of Originaland Extended Juaacuterez-Badillo ModelsDimensionless Groups

To improve identification of the parameters involved in theconstitutive dependences we redefine these as follows

dVV

= minus120574v1205901015840 d12059010158401205901015840 dkk

= minus120574k1205901015840 d12059010158401205901015840dkk

= 120574kv dVV(8)

or in their integral form

VVo

= ( 12059010158401205901015840o )minus120574v1205901015840kko

= ( 12059010158401205901015840o )minus120574k1205901015840kko

= ( VVo

)120574kv(9)

with

120574kv = 120574k1205901015840120574v1205901015840 (10)

31 Nondimensionalization of the J-B Model in Terms ofPressure Four cases are presented (1) original J-Bmodel (2)J-Bmodel assuming 1+e is not constant and dz is constant (3)J-B model assuming 1+e is constant (a simplification of theformer) and (4) the less restrictive model with both 1+e anddz not constant

311 Original J-BModel Theoriginal J-Bmodel [12] assumessmall deformations compared with the soil thickness and anegligible initial void ratio (eo asymp 0) Developing the termbetween brackets of (3) we can write

11205901015840 1205971205901015840120597t = minuscv1120574k1205901015840(12059010158401)2 ( 120590101584012059010158401)minus120574k1205901015840minus1 ( 1205971205901015840120597z )2

+ cv112059010158401 ( 120590101584012059010158401)minus120574k1205901015840 12059721205901015840120597z2

(11)

4 Mathematical Problems in Engineering

or by mathematical manipulation

1205971205901015840120597t= 1205901015840oko120574w120574v1205901015840 ( 12059010158401205901015840o )1minus120574k1205901015840 [minus 120574k12059010158401205901015840 ( 1205971205901015840120597z )2 + 12059721205901015840120597z2 ] (12)

To make this equation dimensionless the following variablesare used

(1205901015840)1015840 = 1205901015840 minus 1205901015840112059010158402 minus 12059010158401 t1015840 = t120591o1205901015840 z1015840 = z

H1

(13)

where 120591o1205901015840 is a characteristic time or unknown referencechosen as the time required for the effective pressure (onaverage along the whole domain) to reach a high percentageof its change for example 90 With these variables andprovided that (on average) d1205901015840 = d((1205901015840)1015840(12059010158402 minus 12059010158401) + 12059010158401) asymp12059010158402 minus 12059010158401 (12) can be written as

(12059010158402 minus 12059010158401)1205901015840m120591o1205901015840120597 (1205901015840)1015840120597t1015840

= minus cv1120574k1205901015840(12059010158401)2H21 ( 1205901015840m12059010158401 )minus120574k1205901015840minus1 (12059010158402 minus 12059010158401)2( 120597 (1205901015840)1015840120597z1015840 )2

+ cv112059010158401H21 ( 1205901015840m12059010158401 )minus120574k1205901015840 (12059010158402 minus 12059010158401) 1205972 (1205901015840)1015840120597z10158402

(14)

where 1205901015840m is an average value of the effective pressure whichwe will talk about later The dimensional coefficients of thisequation in terms of the parameters of the problem

1120591o1205901015840 ( k112059010158401120574w120574v1205901015840 )

120574k1205901015840H21

( 1205901015840212059010158401 minus 1) ( 1205901015840m12059010158401 )minus120574k1205901015840 ( k112059010158401120574w120574v1205901015840 ) 1

H21( 1205901015840m12059010158401 )1minus120574k1205901015840

(15)

give rise to two dimensionless independent groups

120587I = 120591o1205901015840k112059010158401120574w120574v1205901015840H21 ( 1205901015840m12059010158401 )1minus120574k1205901015840 120587II = 120574k1205901015840 ( 12059010158402 minus 120590101584011205901015840m )

(16)

Based on the pi theorem the solution of the order ofmagnitude of the characteristic time derived from 120587I =120595(120587II) is given by

120591o1205901015840sim ( 120574w120574v1205901015840H21

k112059010158401 ) ( 1205901015840m12059010158401 )120574k1205901015840minus1 Ψ120591 (120574k1205901015840 ( 12059010158402 minus 120590101584011205901015840m )) (17)

As regards U1205901015840 since it also depends on time the solution(dependent on two groups) is given by

U1205901015840 = ΨU [ t120591o1205901015840 120574k1205901015840 ( 12059010158402 minus 120590101584011205901015840m )] (18)

where Ψ120591 and ΨU are unknown functions of their arguments

312 J-B Model Assuming 1+e Is Not Constant and dz IsConstant Deleting the restrictive hypothesis eo asymp 0 usedby J-B for the solutions of 1205901015840 and U1205901015840 the latter through thedependence V-1205901015840 for which such a hypothesis is not satisfiedand assuming that 1+e =constant the governing equation is

120597120597z ( k120574w dudz

) = 120597120597t ( e1 + e) = 1(1 + e)2 ( 120597e120597t) (19)

Using the dependence VV1 = (120590101584012059010158401)minus120574v1205901015840 = (1 +e)(1 +eo)or 1 + e = (1 + eo)(12059010158401205901015840o)minus120574v1205901015840 and its derivative form 119889e =minus((1 + eo)120574v12059010158401205901015840o)(12059010158401205901015840o)minus120574v1205901015840minus11198891205901015840 the right and left termsof (19) can be written as

120597120597t ( e1 + e) = 1(1 + e)2 ( 120597e120597t)

= minus 120574v1205901015840(1 + eo) 1205901015840o ( 12059010158401205901015840o )120574v1205901015840minus1 1205971205901015840120597t(20)

On the other hand with kko = (12059010158401205901015840o)minus120574k1205901015840 the left term of(19) writes as

120597120597z ( k120574w 120597u120597z)= ko120574w ( 12059010158401205901015840o )minus120574k1205901015840 [ 120574k12059010158401205901015840 ( 120597u120597z)2 + 1205972u120597z2 ] (21)

Equating both terms (19) in terms of the effective pressuretakes the form

1205971205901015840120597t= (1 + eo) 1205901015840oko120574w120574v1205901015840 ( 12059010158401205901015840o )1minus120574v1205901015840minus120574k1205901015840 [minus 120574k12059010158401205901015840 ( 1205971205901015840120597z )2

+ 12059721205901015840120597z2 ](22)

Mathematical Problems in Engineering 5

Proceeding as in the previous case the coefficients of thedimensionless form of the former equation

1120591o1205901015840 ( (1 + eo) k112059010158401120574w120574v1205901015840 ) 120574k1205901015840

H21( 12059010158402 minus 120590101584011205901015840m ) ( 1205901015840m12059010158401 )1minus120574v1205901015840minus120574k1205901015840

( (1 + eo) k112059010158401120574w120574v1205901015840 ) 1H21

( 1205901015840m12059010158401 )1minus120574v1205901015840minus120574k1205901015840(23)

give rise to the dimensionless groups

120587I = 120591o1205901015840k112059010158401 (1 + eo)120574w120574v1205901015840H21 ( 1205901015840m12059010158401 )1minus120574k1205901015840minus120574v1205901015840 120587II = 120574k1205901015840 ( 12059010158402 minus 120590101584011205901015840m )

(24)

Thus the solutions for 120591o1205901015840 and U1205901015840 are given by

120591o1205901015840 = ( 120574v1205901015840120574wH21(1 + eo) k112059010158401) ( 1205901015840m12059010158401 )120574v1205901015840+120574k1205901015840minus1

sdot Ψ120591 (120574k1205901015840 ( 12059010158402 minus 120590101584011205901015840m ))(25)

U1205901015840 = ΨU [ t120591o1205901015840 120574k1205901015840 ( 12059010158402 minus 120590101584011205901015840m )] (26)

313 J-B Model Assuming 1+e Is Constant (A Simplificationof the Former) Based on the above it is easy to simplify thehypothesis to 1+e being constant With (120590101584012059010158401)minus120574v1205901015840 = 1 theconsolidation equation reduces to

1205971205901015840120597t = (1 + eo) 1205901015840oko120574w120574v1205901015840 ( 12059010158401205901015840o )1minus120574k1205901015840 [minus 120574k12059010158401205901015840 ( 1205971205901015840120597z )2

+ 12059721205901015840120597z2 ](27)

an equation dependent on the initial void ratio Using (13)and averaging this equation provides three-dimensionalcoefficients

1120591o1205901015840 ( (1 + eo) k112059010158401120574w120574v1205901015840 ) 120574k1205901015840

H21( 12059010158402 minus 120590101584011205901015840m ) ( 1205901015840m12059010158401 )1minus120574k1205901015840

( (1 + eo) k112059010158401120574w120574v1205901015840 ) 1H21

( 1205901015840m12059010158401 )1minus120574v1205901015840minus120574k1205901015840(28)

and two dimensionless independent groups

120587I = 120591o1205901015840k112059010158401 (1 + eo)120574w120574v1205901015840H21 ( 1205901015840m12059010158401 )1minus120574k1205901015840 120587II = 120574k1205901015840 ( 12059010158402 minus 120590101584011205901015840m )

(29)

Thus the solutions for 120591o1205901015840 and U1205901015840 are given by

120591o1205901015840 = ( 120574v1205901015840120574wH21(1 + eo) k112059010158401) ( 1205901015840m12059010158401 )120574k1205901015840minus1

sdot Ψ120591 (120574k1205901015840 ( 12059010158402 minus 120590101584011205901015840m ))(30)

U1205901015840 = ΨU [ t120591o1205901015840 120574k1205901015840 ( 12059010158402 minus 120590101584011205901015840m )] (31)

314 The Less Restrictive Model with Both 1+e and dz NotConstant For this model 1+e and dz not constants it isenough to consider (22) plus the condition dz not constantor dz = dzo(HH1) So after mathematical manipulation thegoverning equation writes as

1205971205901015840120597t= (1 + eo) 1205901015840oko120574w120574v1205901015840 ( 12059010158401205901015840o )1+120574v1205901015840minus120574k1205901015840 [minus 120574k12059010158401205901015840 ( 1205971205901015840120597z )2

+ 12059721205901015840120597z2 ](32)

The new coefficients

1120591o1205901015840 ( (1 + eo) k112059010158401120574w120574v1205901015840 ) 120574k1205901015840

H21( 12059010158402 minus 120590101584011205901015840m ) ( 1205901015840m12059010158401 )1+120574v1205901015840minus120574k1205901015840

( (1 + eo) k112059010158401120574w120574v1205901015840 ) 1H21

( 1205901015840m12059010158401 )1+120574v1205901015840minus120574k1205901015840(33)

provide the groups

120587I = 120591o1205901015840k112059010158401 (1 + eo)120574w120574v1205901015840H21 ( 1205901015840m12059010158401 )1minus120574k1205901015840+120574v1205901015840 120587II = 120574k1205901015840 ( 12059010158402 minus 120590101584011205901015840m )

(34)

6 Mathematical Problems in Engineering

and the solutions

120591o1205901015840 = ( 120574v1205901015840120574wH21(1 + eo) k112059010158401) ( 1205901015840m12059010158401 )120574k1205901015840minus120574v1205901015840minus1

sdot Ψ120591 (120574k1205901015840 ( 12059010158402 minus 120590101584011205901015840m ))(35)

U1205901015840 = ΨU [ t120591o1205901015840 120574k1205901015840 ( 12059010158402 minus 120590101584011205901015840m )] (36)

32 Nondimensionalization of the J-B Model in Terms ofSettlements Three cases defined by the hypotheses (1) 1+e =constant and dz constant (2) 1+e and dz constants and (3) 1+eand dz not being constant are considered For this study let usintroduce a new variable directly related with the settlementwith a clear physical meaning ldquo120577 = e - eordquo a kind of localdegree of settlement or differential void index

321 1+e =Constant and dz Constant For 1+e =constantusing the variable 120577 and the dependence V-1205901015840 it is straight-forward to write

120577 = e minus eo = (1 + eo) [( 120590101584012059010158401)minus120574v1205901015840 minus 1] (37)

Making use of the derivatives of (37)

( 1205971205901015840120597z )2 = 120590101584012(1 + eo)2 120574v12059010158402 ( 120590101584012059010158401)2120574v1205901015840+2 ( 120597120577120597z)2

12059721205901015840120597z2 = minus 12059010158401(1 + eo) 120574v1205901015840 ( 120590101584012059010158401)120574v1205901015840+1 1205972120577120597z2

+ 12059010158401 (120574v1205901015840 + 1)(1 + eo)2 120574v12059010158402 ( 120590101584012059010158401)2120574v1205901015840+1 ( 120597120577120597z)2

(38)

and the assumptions of the odometer test after cumbersomemathematical steps the nonlinear dimensional consolidation(19) is written in the form

120597120577120597t = ko1205901015840o (120574k1205901015840 minus 120574v1205901015840 minus 1)120574w120574v12059010158402 ( 12059010158401205901015840o )1minus120574k1205901015840 ( 120597120577120597z)2

+ ko1205901015840o (1 + eo)120574w120574v1205901015840 ( 12059010158401205901015840o )1minus120574v1205901015840minus120574k1205901015840 1205972120577120597z2(39)

Introducing the variables

(120577)1015840 = 120577 minus 12057711205772 minus 1205771 = 1205771 minus 1205771205771 z1015840 = z

H1

t1015840 = t120591os(40)

into the former equation with 120591os a reference chosen as thetime required for the settlement to reach a high percentageof its change along the whole domain its dimensionless formprovides three coefficients

1120591os ko1205901015840o (120574k1205901015840 minus 120574v1205901015840 minus 1)120574w120574v12059010158402 ( 1205901015840m12059010158401 )1minus120574k1205901015840 (ef minus eo)

H21

ko1205901015840o (1 + eo)120574w120574v1205901015840 ( 1205901015840m12059010158401 )1minus120574v1205901015840minus120574k1205901015840 1H21

(41)

Dividing by the last the resulting dimensionless groups are

120587I = 120591osko1205901015840o (1 + eo)120574w120574v1205901015840H21 ( 1205901015840m12059010158401 )1minus120574v1205901015840minus120574k1205901015840

120587II = (120574k1205901015840 minus 120574v1205901015840 minus 1)120574v1205901015840 (ef minus eo)(1 + eo) ( 1205901015840m12059010158401 )120574v1205901015840(42)

Making use of the constitutive dependence dVV =minus120574v1205901015840(d12059010158401205901015840) is possible to write ΔHH = minus120574v1205901015840(Δ12059010158401205901015840)and since (ef minus eo)(1 + eo) = ΔHHo 120587II finally takes theform

120587II = (1 minus 120574k1205901015840 + 120574v1205901015840) 12059010158402 minus 120590101584011205901015840m (43)

The solution for 120591os is120591os asymp 120574w120574v1205901015840H21

ko1205901015840o (1 + eo) ( 1205901015840m12059010158401 )120574v1205901015840+120574k1205901015840minus1Ψ120591 [(1 minus 120574k1205901015840+ 120574v1205901015840) 12059010158402 minus 120590101584011205901015840m ]

(44)

As for the average degree of consolidation the followingdependence arises

Us = ΨU [ t120591os (1 minus 120574k1205901015840 + 120574v1205901015840) 12059010158402 minus 120590101584011205901015840m ] (45)

322 1+e and dz Constants The hypothesis 1+e=constant isa simplification of the former case Proceeding in the sameway the resulting coefficients

1120591os ko1205901015840o (1 + 120574v1205901015840 minus 120574k1205901015840)120574w120574v12059010158402 ( 1205901015840m12059010158401 )1minus120574k1205901015840+120574v1205901015840 (ef minus eo)

H21

ko1205901015840o (1 + eo)120574w120574v1205901015840 ( 1205901015840m12059010158401 )1minus120574k1205901015840 1H21

(46)

Mathematical Problems in Engineering 7

provide the dimensionless groups

120587I = 120591osko1205901015840o (1 + eo)120574w120574v1205901015840H21 ( 1205901015840m12059010158401 )1minus120574k1205901015840

120587II = (1 minus 120574k1205901015840 + 120574v1205901015840) 12059010158402 minus 120590101584011205901015840m(47)

The solutions for 120591os and Us are

120591os asymp 120574w120574v1205901015840H21ko1205901015840o (1 + eo) ( 1205901015840m12059010158401 )120574k1205901015840minus1Ψ120591 [(1 minus 120574k1205901015840

+ 120574v1205901015840) 12059010158402 minus 120590101584011205901015840m ](48)

Us = ΨU [ t120591os (1 minus 120574k1205901015840 + 120574v1205901015840) 12059010158402 minus 120590101584011205901015840m ] (49)

The solutions for the original J-Bmodel are also (48) and (49)with the simplification of eo=0 The author however doesnot talk about the characteristic time and obtains the averagedegree of consolidation given by (7) in which this variabledepends on three groups (1minus120574k1205901015840)120574v1205901015840 H2H1 and tcv2H21undoubtedly a less precise solution

323 1+e and dz Not Being Constant When 1+e and dzare not constants substituting dz = dzo(HH1) in (39)after cumbersome mathematical manipulation the resultingcoefficients are 1120591os

ko1205901015840o (120574k1205901015840 minus 120574v1205901015840 minus 1)120574w120574v12059010158402 ( 1205901015840m12059010158401 )1+2120574v1205901015840minus120574k1205901015840 (ef minus eo)H21

ko1205901015840o (1 + eo)120574w120574v1205901015840 ( 1205901015840m12059010158401 )1+120574v1205901015840minus120574k1205901015840 1

H21

(50)

Dividing by the last the resulting dimensionless groups are

120587I = 120591osko1205901015840o (1 + eo)120574w120574v1205901015840H21 ( 1205901015840m12059010158401 )1+120574v1205901015840minus120574k1205901015840

120587II = (1 minus 120574k1205901015840 + 120574v1205901015840) 12059010158402 minus 120590101584011205901015840m(51)

The solutions for 120591os and Us are

120591os asymp 120574w120574v1205901015840H21ko1205901015840o (1 + eo) ( 1205901015840m12059010158401 )+120574k1205901015840minus120574v1205901015840minus1Ψ120591 [(1 minus 120574k1205901015840

+ 120574v1205901015840) 12059010158402 minus 120590101584011205901015840m ](52)

Us = ΨU [ t120591os (1 minus 120574k1205901015840 + 120574v1205901015840) 12059010158402 minus 120590101584011205901015840m ] (53)

Table 1 summarizes the results derived in this section

4 Verification of the Results byNumerical Simulations

This section is devoted to checking the solutions given inTable 1 To shorten the exposure we restrict the tests forthe more general model (1+e and dz not constants) andfor the unknowns 120591os and Us since they are the ones ofgreater interest in civil engineering Eight sets of simulationsarranged in three blocks have been run Table 2 In each onesome of the soil parameters or initial values of the problemhave been changed to give the same or different values asrequired to the dimensionless group 120587II in the search for thesame or different solutions of 120587I Changes in the values of theindividual parameters are sufficient to cover all real scenarios

Firstly a reference set (set 1) is established to which all theother sets can be referred and which permits them to be com-pared with each other The physical and geometrical charac-teristics that change are 120574v1205901015840 120574k1205901015840 eo H1 (m) 1205901015840o (Nm2)1205901015840f (Nm2) and ko (myear) The values of =(1205901015840f1205901015840o) H2(m) kf (myear) and 120587II are derived from them while 1205871is obtained once the characteristic time 120591os is read from thesimulation The criterion for the choice of 120591os is the timerequired by the soil to reach 90 of the total settlement

It is worth mentioning here the emergence of 1205901015840m anaveraged value of the effective pressure in the two dimen-sionless groups Undoubtedly the existence of 1205901015840m in themonomials can be explained not only by the nonlinearityof the problem but also by the kind of dependences in theconstitutive relations (there are other nonlinear problems inwhich averaged variables do not emerge [18]) How do wechoose 1205901015840m We could take the mean value 1205901015840m=(12059010158401+12059010158402)2or even 1205901015840m=12059010158401 or 1205901015840m=12059010158402 but expecting that the expres-sion 120587I = 120595(120587II) is slightly dependent on that choice (we willreturn to this question later) In this paper we have taken theapproximation 1205901015840m=(12059010158402 + 12059010158401)2

In block I the parameters that change in each set areeo H1 ko and 1205901015840o while maintaining the ratio =(1205901015840f1205901015840o) thus both 120587II and factor (1205901015840m1205901015840o)1minus120574k1205901015840+120574v1205901015840 of 120587I takethe same value in the four sets of the block As a resultwe expect a characteristic time proportional to the ratio120574w120574v1205901015840H21ko1205901015840o(1 + eo) and a same value of 120587I in all casesaccording to the solution 120587I = 120595(120587II) Indeed the resultsare consistent with expectations Block II contains an onlycase for which 120587II varies in comparison to block I as a result120587I also changes as expected Finally block III contains threecases with the same value of 120587II of block I but with differentvalues of the parameters 120574v1205901015840 120574k1205901015840 and the ratio =(1205901015840f 1205901015840o)The parameters of set 6 have been chosen in such a waythat the values 120574k1205901015840 and 120574v1205901015840 are compensated to maintainconstant (1 minus 120574k1205901015840 + 120574v1205901015840) we expect 120587I to be also unchangedIn effect simulation provides 120587I = 07139 a value quite closeto that of set 1 (07131) The negligible differences are due nodoubt to the different evolution of the effective pressure alongthe process caused by the different values of the coefficients120574k1205901015840 and 120574v1205901015840 Note that as 120574v1205901015840 duplicates the denominator ofthe factor 120591osko1205901015840o(1 + eo)120574w120574v1205901015840H21 the characteristic timehas to double too as indeed happens The last two sets studythe influence of a parameter whose change forces the value

8 Mathematical Problems in Engineering

Table1Dim

ensio

nlessg

roup

sthatcharacterizethe

originalandextend

edJ-Bmod

els

Juarez-Badillo

original

Juarez-Badillo

Juarez-Badillo

Juarez-Badillo

1+e=

constant

1+e

=consta

nt1+e

=consta

ntdz

constant

dzconstant

dzvaria

ble

120587 I120591 ok o

1205901015840 o120574 w120574 v1205901015840H2 1

(1205901015840 m 1205901015840 o)1minus120574k1205901015840

120591 ok o1205901015840 o(

1+eo)

120574 w120574 v1205901015840H2 1

(1205901015840 m 1205901015840 o)1minus120574k1205901015840

120591 ok o1205901015840 o(

1+eo)

120574 w120574 v1205901015840H2 1

(1205901015840 m 1205901015840 o)1minus120574k1205901015840minus120574 v1205901015840

120591 ok o1205901015840 o(

1+eo)

120574 w120574 v1205901015840H2 1

(1205901015840 m 1205901015840 o)1minus120574k1205901015840+120574 v1205901015840

120591 o1205901015840or

120591 os120587 II

120574 k1205901015840(1205901015840 2minus

1205901015840 1 1205901015840 m)for

pressure

(1minus120574 k1205901015840

+120574 v1205901015840)(1205901015840 2minus

1205901015840 1 1205901015840 m)for

settlem

ent

Mathematical Problems in Engineering 9

0010203040506070809

1

0 05 1 15 2 25 3time

Set 1Set 2Set 3Set 4

Set 5Set 6Set 7Set 8

Aver

age d

egre

e of c

onso

lidat

ion

Figure 1 Average degree of consolidation versus time (eight sets)

0010203040506070809

1

0 05 1 15 2 25 3

Set 1Set 2Set 3Set 4

Set 5Set 6Set 7Set 8

Aver

age d

egre

e of c

onso

lidat

ion

tos

Figure 2 Average degree of consolidation versus dimensionlesstime (eight sets)

(1 minus 120574k1205901015840 + 120574v1205901015840) to be also changed in order to maintain 120587IIconstant We expect that 120587I does not change as nearly occursThe small differences are again due to the existence of 1205901015840mwithin the monomials Note however that as diminishesthe influence of 1205901015840m in the results also diminishes since thevalues of 12059010158401 and 12059010158402 are closer

For a better understanding of the results Figures 1 and2 are presented In the first Us as a function of time isshown for the eight sets in Table 2 It is clear that 120591oscovers a wide range of values from 02283 (set 2) to 27968(set 3) the last more than ten times the first The secondfigure graphs Us in front of the dimensionless time t120591os Asshown the eight previous curves nearly converge to a single(or universal) graph having a common point at (Us=09t120591os=1) This means that monomial 120587II for the range ofvalues assigned to the parameters in contrast to 120587I scarcelyinfluences the value of the average degree of consolidationThis is not in contradiction with the results derived bynondimensionalization

0010203040506070809

1

0 05 1 15 2 25 3 35 4time

U (J-B)U (1+e cst)U (1+e not cst)U (1+e amp dz not cst)

Us (J-B)Us (1+e cst)Us (1+e not cst)Us (1+e amp dz not cst)

Aver

age d

egre

e of p

ress

ure d

issip

atio

nAv

erag

e deg

ree o

f con

solid

atio

n

Figure 3 Average degree of consolidation and pressure dissipation(set 6)

Although in this sectionwe have only checked the expres-sions of Us and 120591os for the most general model we considerit interesting to present both Us and U1205901015840 for the rest of themodels of Table 1 and for set 6 as a typical case Figure 3Theseresults emphasize the importance of using the less restrictive(more general) model in relation to the others in particularwith the original model of J-B in which the influence ofeo is despised Note that the smaller of the characteristictimes (related to settlement) corresponds to the more generalmodel (black and bold line) so the use of the others althoughon the safety side provides characteristic times oversizedfar from an optimal solution for the engineer For examplefor the case of Figure 3 in which eo=1 the error of J-Boriginal model (purple and bold line) in comparison with theextended model (black and bold line) for Us=90 is around150 This error increases appreciably as eo gets larger

5 Universal Curves

These easy to use and universal abacuses obtained by numeri-cal simulations allow engineers to read the characteristic time(120591os) and the average degree of consolidation (Us) in termsof the dimensionless groups given in Table 1 Only the mostgeneral (less restrictive) J-B model 1+e and dz not constantis presented

To obtain the characteristic time curves a large numberof simulations have been carried out studying separately theinfluence of the parameters 120574v1205901015840 and 120574k1205901015840 whose valuescover most real soils ranges from 101 to 8 120574v1205901015840 from 005to 03 and 120574k1205901015840 from 01 to 17 From these tests it has beendeduced as expected from the results of the previous sectionthat the relatively more influential parameter is while 120574v1205901015840and 120574k1205901015840 do not produce significant changes in the values ofthe characteristic timeThus the first universal group of linesshows the dependence of 120587I versus 120587II for different valuesof (with 120574v1205901015840=01) Figure 4 As the range of values of 120587IIdepends on according to (43) the range of values of eachline is different As shown in the figure there exists a universal

10 Mathematical Problems in Engineering

Table 2 Values of the parameters and dimensionless groups of simulations

Block Set 120574v1205901015840 120574k1205901015840 eoH1 ko 1205901015840o 1205901015840f H2 kf 120587II

120591os 120587I(m) (myr) (Nm2) (Nm2) 1205901015840f 1205901015840o (m) (myr) (yr)

I

1 01 0500 10 100 002 30000 60000 2 0933 00141 040 04566 071312 01 0500 10 100 002 60000 120000 2 0933 00141 040 02283 071313 01 0500 10 175 001 30000 60000 2 1633 00071 040 27968 071314 01 0500 13 142 004 22000 44000 2 1325 00283 040 05459 07131

II 5 01 1500 10 100 002 30000 90000 3 0896 00038 -040 10597 09834

III6 02 0600 10 100 002 30000 60000 2 0871 00132 040 09142 071397 01 0767 10 100 002 30000 120000 4 0871 00069 040 04360 072468 02 0920 10 100 002 30000 180000 6 0699 00038 040 08409 07312

line for each value of all converging in the central zone forvalues of 120587II around 0 This set of straight lines can be fittedby 120587I=m120587II+n being n=0846 and m given in Figure 5 as afunction of The curve that best fits the points in Figure 5 isgiven by

m = minus0235e(minus0868) minus 0306e(minus0012) (54)

Choosing the longest line corresponding to =8 as the onlyrepresentative line of the problem themaximumerrors in thereading of 120587I (and 120591os) are 38

In short the use of these curves is as follows from the dataof the problem Figure 4 allows us to read 120587I from 120587II and and from (51) the characteristic time 120591os is derived

As for the average degree of consolidation the universalcurves for 96 different scenarios of 120587II are shown in Figure 6It is appreciated that these curves are very close to each othershowing again that the Us only depends on the dimensionlesstime In the worst case the errors produced by changes invalue of the 120587II group are less than 28when using the curveof Figure 7

6 Case Study

Below is a practical application of the universal curvespresented in this paper For this purpose we take as a basisfor our study the data obtained experimentally byAbbasi et al[26] for different samples of clay whosemoisture in the liquidlimit is equal to 42 As can be seen (Table 3) it is a series ofdiscrete (tabulated) values that relate the void ratio with theeffective pressure and the hydraulic conductivity

We will address a real case of consolidation in which wehave a layer of soil 2meters thick (Ho) with an initial effectivepressure (1205901015840o) of 28 kNm2 on which a load of 31 kNm2 isapplied In this way the effective pressure at the end of theconsolidation process (1205901015840f ) will have the value of 59 kNm2From the tabulated data (e-1205901015840) it follows that eo = 105 andef = 095 and since the factor 1+e is proportional to the soilvolume from (9) we have 120574v1205901015840 = 0067

To obtain 120574k1205901015840 we first calculate the value of 120574kv from thetabulated data (e-k) to then obtain 120574k1205901015840 from (10) With thiswe have for the load step considered ko = 0035 myr 120574kv =8108 and 120574k1205901015840 = 0543

To solve the problem we will use the universal curvespresented in the previous section corresponding to the most

Table 3 Experimental values of e 1205901015840 and k obtained by Abbasi et al[26]

e 1205901015840o (Nm2)110 14000105 28000095 59000085 120000075 220000065 450000060 925000e k (myr)175 0683160 0315120 0084085 0021065 0011

general J-B model (analyzed in terms of settlement) Thusfrom the values of 1205901015840o 1205901015840f 120574v1205901015840 and 120574k1205901015840 by means of (51) itis obtained that 120587II = 0373 (with 1205901015840m = 435 kNm2) Then120587I is read from Figure 4 where for = 21 we have 120587I =072 An identical result is obtained if we use the proposedfitted equation 120587I=m120587II+n where n is a constant of value0846 andm is obtained from (54) or Figure 5 Finally knownthe values of Ho ko and eo from (51) we get 120591os = 0748years Likewise once the characteristic time of the problem(in terms of settlement) is known from the universal curve ofFigure 7 we easily obtain the evolution of the average degreeof consolidation along the whole process Figure 8

These solutions have been compared with those thatresult from the numerical simulation of the most generalJ-B model for which a value of 120591os = 075 years has beenobtained showing clearly that the relative error committedwhen applying the universal solutions is negligible (Table 4)The simulation of the problem has also been carried out fora model where 1+e and dz are also not constant but in whichthe constitutive relations of potential type have been replacedby pairs of data in tabulated form obtaining in this case 120591os= 0817 years The relative error which this time increasesto 82 finds its explanation in the strong nonlinearity ofthe problem which will always bring differences between the

Mathematical Problems in Engineering 11

040045050055060065070075080085090095100105110

minus07 minus06 minus05 minus04 minus03 minus02 minus01 0 01 02 03 04 05 06 07 08 09 1 11 12 13 14 15

1

2

ᴪ=8ᴪ=6ᴪ=4

ᴪ=2ᴪ=15ᴪ=101

Figure 4 120587I as a function of 120587II with as parameter

1 2 3 4 5 6 7 8minus042

minus04

minus038

minus036

minus034

minus032

minus03

minus028

minus026

m

datafitted curve

Figure 5 m as a function of

solutions provided by numerical models based on differentconstitutive relations (potential and tabulated in this case)Even so this value is totally acceptable within this field ofengineering demonstrating the strength and versatility of theuniversal solutions provided

7 Discussion

This work derives the dimensionless groups that rule thenonlinear soil consolidation process based on the original J-Bmodel following a formal nondimensionalization procedurethat starts from the governing equation and does not needany other assumption or mathematical manipulation Inaddition more real and precise extended J-B models whichassume both the void ratio and the thickness of the finitevolume element of the soil to be not constant have been

0010203040506070809

1

0 05 1 15 2 25 3

96 scenariosᴪ = [101 15 2 4 6 8]

Aver

age d

egre

e of c

onso

lidat

ion

tos

= [005 01 015 02]v= [02 05 1 15]k

Figure 6 Average degree of consolidation versus dimensionlesstime for different scenarios

0010203040506070809

1

0 05 1 15 2 25 3

Aver

age d

egre

e of c

onso

lidat

ion

tos

Figure 7 Proposed solution for the average degree of consolidation

nondimensionalized and their dimensionless groups derivedThe assumption of constitutive dependences (e-1205901015840 e-k and k-1205901015840) of potential type in these models is interesting since they

12 Mathematical Problems in Engineering

Table 4 Case study solution for the characteristic time 120591os Comparison with numerical models

Universal curves J-B most general model Tabulated model120591os (yr) 120591os (yr) Rel error 120591os (yr) Rel error 0748 0750 026 0817 820

0010203040506070809

1

0 025 05 075 1 125 15 175 2 225t ( years)

Aver

age d

egre

e of c

onso

lidat

ion

Figure 8 Case study solution for the average degree of consolida-tion Us from the universal curves

can be applied to highly compressible soils for which the e-log(1205901015840) relation moves away from being a straight line andin which negative values of the void ratio could be obtainedunder the application of high load ratios (1205901015840f 1205901015840o) [8]

The study has been developed both in terms of pres-sure dissipation and in terms of settlement resulting thatthe dimensionless groups derived from each model are ingeneral different The extension to more complex modelswhich implies the emergence of new parameters or equationswithin themathematical model does not necessarily give riseto additional dimensionless groups as might be expectedbut to new expressions of the groups or new regroupingof the parameters involved Thus the elimination of thehypothesis ldquodz constantrdquo in the most general consolidationmodel has not given rise to any new group both in pressureand in settlement (Table 1) but it has only modified oneof the groups obtained in the original model All thisrepresents an important advance in relation to the modelsand studies carried out by other authors and the attemptsof characterization found in the scientific literature [9ndash11]which try to lead to easy-to-use solutions for engineers Inno case had dimensionless groups been obtained as preciseas those presented in this work nor presented universalcurves of the characteristic time and the average degree ofconsolidation Authors such asConte andTroncone [11] applymore general load conditions (time-dependent loading) forwhich the dimensionless groups deduced in this work couldbe applicable for each load step

As regards time to make it dimensionless an unknownreference (120591o) is introduced in the form mentioned by Scott[27] and used in many other problems with asymptoticsolutions [28]Theprecise definition of 120591o necessarily involvesa criterion in relation to the percentage of average pressuredecrease (or settlement) associated with the end of theconsolidation processTheorder ofmagnitude of 120591o is derived

after nondimensionalization making 120591o only appear in oneof the independent dimensionless groups In this way theaverage degree of consolidation (in terms of pressure orsettlement) depends on the ldquotimecharacteristic timerdquo ratioin addition to the dimensionless groups (without unknowns)that arise

In short the procedure has led to a new very precisesolution that involves two monomials one (120587I) that containsthe characteristic time (120591o) and another (120587II) that is amathematical function of the soil parameters In this way120591o only depends on 120587II through the function 120587I = 120595(120587II)while U depends on 120587I and 120587II through the expression U= 120595(120587I 120587II) The results obtained have been verified forthe most general and complex consolidation model throughthe numerical simulation of a large number of scenariosin which the values of the dimensionless groups and theindividual parameters of the problem have been changing toconvenience The high errors produced by the original J-Bmodel with respect to the most general justify the use of thelatter in the search for more precise solutions

In addition the emergence of mean values of the effectivepressure when averaging the governing equation in orderto assign an order of magnitude unit to the dimensionlessvariables and their derivatives is an added difficulty to obtainthe dimensionless groups This is undoubtedly due to thenonlinearities (constitutive relations) involved in the consol-idation process This average effective pressure of referencehas been assumed as the semisum of its initial and finalvalues although other criteria could also be chosen Sincetwo scenarios can have equal120587II with the same effectivemeanpressure but different values of the parameters 120574v1205901015840 and 120574k1205901015840 we expect some minor deviations between the results of bothscenarios as indeed it has been checked Thus for the wholerange of parameters values in typical soils the errors due tothe above cause are negligible in the field of soil mechanicsOn the other hand when the value of 120587II does not changebut the mean effective pressure value does the errors in thesolutions of 120591os and Us are slightly larger although withinranges below 4

The brooch for all this work is the presentation ofuniversal curves for the most general model which allowfor the usual range of variation of the soil properties toobtain in a simple way the value of the characteristic timeof consolidation and the evolution of the average degreeof settlement throughout the entire process Thus startingfrom the nonlinear parameters of the soil (120574v1205901015840 and 120574k1205901015840)and the load ratio (1205901015840f 1205901015840o) the value of the monomial 120587IIis obtained from which the value of 120587I is directly deductedthanks to the universal curve 120587I = 120595(120587II) provided Finally120591os is easily obtained from the expression of 120587I On the otherhand the curve that represents the Us evolution along theprocess has turned out to be (practically) unique when it is

Mathematical Problems in Engineering 13

represented as a function of the dimensionless time in theform t120591os regardless of the soil parameters and the loadsapplied All this supposes a clear advantage when solvingthe consolidation problem also showing the strength of thenondimensionalization method presented

Finally the case study addressed has demonstrated thesimplicity and reliability of the universal curves proposedFrom a series of tabulated data (e-1205901015840 and e-k) the parametersthat govern the nonlinear consolidation problem for themostextended J-B model have been adjusted with which thevalue of the monomial 120587II has been derived With this andusing the universal curves the solutions for the characteristictime 120591os and the average degree of consolidation Us havebeen obtained These results have been satisfactorily verifiedwith the solutions provided both by the numerical extendedmodel of J-B presented here and by another that substitutesthe constitutive dependences of potential type by data intabulated form

8 Conclusions

The application of the nondimensionalization technique hasallowed us to deduce the dimensionless groups that rule thenonlinear consolidation problem in which the constitutivedependencies e-1205901015840 and e-k have the form of potential func-tions This type of relations approximates the soil behaviorbetter than the logarithmic functions e-log(1205901015840) and e-log(k)particularly in very compressible soils and under high loadratios (1205901015840f 1205901015840o) in which inconsistent negative values of thevoid ratio may appear

Several scenarios have been addressed from the originalmodel of Juarez-Badillo to extended models based on thesame constitutive dependences but eliminating one or moreclassic (and restrictive) hypotheses in this process In allcases only two dimensionless groups emerge one of themconsisting of a grouping of the soil parameters and the othercontaining the unknown characteristic time Based on theseresults universal curves have been depicted the first to deter-mine the characteristic time of the process and the secondthat represents the evolution of average degree of settlement

The solutions have been verified by numerical simula-tions changing the values of the soil parameters and thedimensionless groups conveniently Finally the reliability andeffectiveness of the proposed solutions have been shownthrough a case study

Abbreviations

cv Consolidation coefficient (m2s or m2yr)cv1 Initial consolidation coefficient (m2s or

m2yr)cv2 Final consolidation coefficient (m2s or

m2yr)119889z Element of differential length in thedirection of the spatial coordinate z (m)119889zo Element of differential length in thedirection of the spatial coordinate z at theinitial time (m)

e Void ratio (dimensionless)

eo Initial void ratio (dimensionless)ef Final void ratio (dimensionless)H Soil thickness up to the impervious

boundary or drainage length (m)H1Ho Initial soil thickness (m)H2 Final soil thickness (m)k Hydraulic conductivity (or permeability)

(ms or myr)k1 ko Initial hydraulic conductivity (ms or

myr)kf Final hydraulic conductivity (ms or myr)mv Coefficient of volumetric compressibility

(m2N)mv1 Initial coefficient of volumetric

compressibility (m2N)t Time Independent variable (s or yr)T Juarez-Badillorsquos time factor

(dimensionless)u Excess pore pressure (Nm2)U Average degree of consolidation or

pressure dissipation (dimensionless)Us Average degree of consolidation

(dimensionless)U1205901015840 Average degree of pressure dissipation

(dimensionless)V Soil volume (m3)V1Vo Initial soil volume (m3)z Vertical spatial coordinate (m)120574v Nonlinear coefficient of compressibility of

Juarez-Badillo (dimensionless)120574kv Nonlinear coefficient of change ofpermeability with volume (dimensionless)120574k1205901015840 Nonlinear coefficient of change ofpermeability with effective pressure(dimensionless)120574v1205901015840 Nonlinear coefficient of compressibility orcoefficient of volumetric contraction(dimensionless)120574w Specific weight of water (Nm3)

Loading factor Ratio between the finaland initial effective pressure(dimensionless)120581 Nonlinear coefficient of change ofpermeability with volume ofJuarez-Badillo (dimensionless)120582 Parameter of Juarez-Badillo 120582 = 1 - 120574v120581(dimensionless)

V Auxiliary variable of Juarez-Badillo(dimensionless)120587i Dimensionless group or number(dimensionless)1205901015840 Effective pressure of the soil (Nm2)12059010158401 1205901015840o Initial effective pressure (Nm2)12059010158402 1205901015840f Final effective pressure (Nm2)1205901015840m Mean value of the effective pressure(Nm2)120591o Characteristic time (s or yr)

14 Mathematical Problems in Engineering

120591os Characteristic time which takes thesettlement to reach approximately its finalvalue (s or yr)120591o1205901015840 Characteristic time which takes the excesspore pressure to dissipate untilapproximately the value of zero (s or yr)Ψ Arbitrary mathematical function120577 Differential void ratio (dimensionless)1205771 1205772 Initial and final differential void ratio(dimensionless)

Subscripts

I II Denote different numbers

Superscripts

1015840 Denote dimensionless magnitude

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] K Terzaghi ldquoDie Berechnung der Durchlassigkeitszifferdestones aus dem verlauf der hydrodynamischen spanunngser-scheinnungenrdquo Technical report II a 132 N 34 125-138Akademie derWissenschaften inWien SitzungsberichteMath-naturwiss Klasse Abt 1923

[2] R T Walker ldquoVertical drain consolidation analysis in one twoand three dimensionsrdquo Computers amp Geosciences vol 38 no 8pp 1069ndash1077 2011

[3] G Y Zheng P Li and C Y Zhao ldquoAnalysis of non-linearconsolidation of soft clay by differential quadrature methodrdquoApplied Clay Science vol 79 pp 2ndash7 2013

[4] X Y Geng C J Xu and Y Q Cai ldquoNon-linear consolidationanalysis of soil with variable compressibility and permeabilityunder cyclic loadingsrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 30 no 8 pp 803ndash8212006

[5] M Lu S Wang S W Sloan B Indraratna and K Xie ldquoNon-linear radial consolidation of vertical drains under a generaltime-variable loadingrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 39 no 1 pp 51ndash622015

[6] HWu L HuW Qi and QWen ldquoAnalytical Solution for Elec-troosmotic Consolidation Considering Nonlinear Variation ofSoil Parametersrdquo International Journal of Geomechanics vol 17no 5 Article ID 06016032 2017

[7] S J Brandenberg ldquoiConsoljs JavaScript Implicit Finite-Differ-ence Code for Nonlinear Consolidation and Secondary Com-pressionrdquo International Journal of Geomechanics vol 17 no 6Article ID 04016149 2017

[8] R Butterfield ldquoA natural compression law for soils (an advanceon e-log prsquo)rdquo Geotechnique vol 29 no 4 pp 469ndash480 1979

[9] R Lancellotta and L Preziosi ldquoA general nonlinear mathemati-calmodel for soil consolidation problemsrdquo International Journalof Engineering Science vol 35 no 10-11 pp 1045ndash1063 1997

[10] Y-C Zhuang K-H Xie and X-B Li ldquoNonlinear analysis ofconsolidation with variable compressibility and permeabilityrdquoJournal of Zhejiang University (Engineering Science Edition) vol6 no 3 pp 181ndash187 2005

[11] E Conte andA Troncone ldquoNonlinear consolidation of thin lay-ers subjected to time-dependent loadingrdquo Canadian Geotechni-cal Journal vol 44 no 6 pp 717ndash725 2007

[12] R Yong and F Townsend ldquoGeneral Consolidation Theory forClaysrdquo Soil Mechanics Series Report No 8 Graduate School ofEngineering National University of Mexico 1983

[13] J C Gibbings Dimensional Analysis Springer-Verlag LondonUK 2011

[14] E Buckingham ldquoOn physically similar systems Illustrations ofthe use of dimensional equationsrdquo Physical Review A AtomicMolecular and Optical Physics vol 4 no 4 pp 345ndash376 1914

[15] R Seshadri and T Y Na ldquoA Survey ofMethods for DeterminingSimilarity Transformationsrdquo inGroup Invariance in EngineeringBoundary Value Problems pp 35ndash61 Springer New York NYUSA 1985

[16] V D Zimparov and V M Petkov ldquoApplication of discriminatedanalysis to low Reynolds number swirl flows in circular tubeswith twisted-tape inserts Pressure drop correlationsrdquo Interna-tional Review of Chemical Engineering vol 1 no 4 pp 346ndash3562009

[17] M Capobianchi and A Aziz ldquoA scale analysis for natural con-vective flows over vertical surfacesrdquo International Journal ofThermal Sciences vol 54 pp 82ndash88 2012

[18] M Conesa J F Sanchez Perez I Alhama and F AlhamaldquoOn the nondimensionalization of coupled nonlinear ordinarydifferential equationsrdquo Nonlinear Dynamics vol 84 no 1 pp91ndash105 2016

[19] I A Manteca G Garcıa-Ros and F Alhama ldquoUniversal solu-tion for the characteristic time and the degree of settlement innonlinear soil consolidation scenarios A deduction based onnondimensionalizationrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 57 pp 186ndash201 2018

[20] E H Davis and G P Raymond ldquoA non-linear theory ofconsolidationrdquo Geotechnique vol 15 no 2 pp 161ndash173 1965

[21] P Cornetti and M Battaglio ldquoNonlinear consolidation of soilmodeling and solution techniquesrdquo Mathematical and Com-puter Modelling vol 20 no 7 pp 1ndash12 1994

[22] C F Gonzalez-Fernandez ldquoApplications of the network simu-lation method to transport processesrdquo in Network SimulationMethod J Horno Ed Research Signpost Trivandrum India2002

[23] J Serna F J S Velasco and A Soto Meca ldquoApplication of net-work simulationmethod to viscous flowsThe nanofluid heatedlid cavity under pulsating flowrdquo Computers amp Fluids vol 91 pp10ndash20 2014

[24] E Juarez-Badillo ldquoGeneral Permeability Change Equationsfor Soilsrdquo in Proceedings of the International Conference onConstitutive Laws pp 205ndash209 University of Arizona TuesonAriz USA 1983

[25] E Juarez-Badillo and B Chen ldquoConsolidation curves for claysrdquoJournal of Geotechnical Engineering vol 109 no 10 pp 1303ndash1312 1983

Mathematical Problems in Engineering 15

[26] N Abbasi H Rahimi A A Javadi andA Fakher ldquoFinite differ-ence approach for consolidation with variable compressibilityand permeabilityrdquo Computers amp Geosciences vol 34 no 1 pp41ndash52 2007

[27] R F Scott Principles of Soils Mechanics Addison-Wesley Pub-lishing Company 1963

[28] F Alhama and C N Madrid Analisis Dimensional Discrimi-nado En Mecanica De Fluidos Y Transferencia De Calor 2012

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Mathematical Problems in Engineering 3

The original J-B model under odometer conditions(small volume change andnegligible specificweight ofwater)is defined by

d1205901015840dt

= minus 1120574wmv

120597120597z (k120597u120597z) (1)

Substituting the above dependences and using the parameters

120582 = 1-120574v120581k112059010158401120574w120574v = k1120574wmv1

= cv1cv = k120574wmv

= k112059010158401120574w120574v ( 120590101584012059010158401)1minus120574v120581 = cv1 ( 120590101584012059010158401)

120582

(2)

(1) reduces to

1205971205901015840120597t = minus1205901015840cv1 112059010158401 120597120597z [( 120590101584012059010158401)minus120574v120581 120597u120597z ] (3)

The simplifications assumed by the odometer test 120597u120597z =minus1205971205901015840120597z and 120597u120597t = minus1205971205901015840120597t allow us to write the lastequation as

1205971205901015840120597t = 1205901015840cv1 120597120597z [( 120590101584012059010158401)minus120574v120581 112059010158401 1205971205901015840120597z ] (4)

or in terms of the excess pore pressure variable as

120597u120597t = 1205901015840cv1 120597120597z [( 120590101584012059010158401)minus120574v120581 112059010158401 120597u120597z ] (5)

In this equation 12059010158401 can be substituted by any other pressurefor example 12059010158402 providing that cv is also substituted by cv2The case 120582=0 (or 120574v120581=1) ie constant cv is the nonlinearmodel of Davis and Raymond which always provides aneffective stress that is lower (or higher excess pore pressure)than that obtained with the Terzaghi model However thedifferent dependences V-1205901015840 (or e-1205901015840) for J-B and Davis andRaymond result in a different average degree of consolidation(Us) for bothmodels In fact for J-B [12] this unknown obeysthe following expression

Us = 1 minus (1H1) intH10

(120590101584012059010158401)minus120574v dz1 minus (1205901015840212059010158401)minus120574v (6)

or in terms of the new normalized variable V = (120590101584012059010158402)120582Us = 1 minus (H2H1) int1

0Vminus120574v120582dz10158401 minus H2H1 (7)

Finally J-B does not report analytical or numerical solutionsfor the excess pore pressure or for the effective pressureexcept for the case 120582=0 probably due to the large numberof variables involved He only reports [25] a consolidationabacus Us sim T for the case 120582=0 (120574v120581=1) and a constant cv

using H2H1 as parameter and a set of abacuses for the case120582120574v=(1120574v)-120581 = 0 cv = constant In these abacuses the timefactor is defined as T = tcv2H2 Despite these results nophysical meaning is attributed to 120582 T or V In short in viewof (7) J-B concludes that his model depends on three groupsH2H1 (1120574v)-120581 and the time factor T (since V is a function ofT)

3 Nondimensionalization of Originaland Extended Juaacuterez-Badillo ModelsDimensionless Groups

To improve identification of the parameters involved in theconstitutive dependences we redefine these as follows

dVV

= minus120574v1205901015840 d12059010158401205901015840 dkk

= minus120574k1205901015840 d12059010158401205901015840dkk

= 120574kv dVV(8)

or in their integral form

VVo

= ( 12059010158401205901015840o )minus120574v1205901015840kko

= ( 12059010158401205901015840o )minus120574k1205901015840kko

= ( VVo

)120574kv(9)

with

120574kv = 120574k1205901015840120574v1205901015840 (10)

31 Nondimensionalization of the J-B Model in Terms ofPressure Four cases are presented (1) original J-Bmodel (2)J-Bmodel assuming 1+e is not constant and dz is constant (3)J-B model assuming 1+e is constant (a simplification of theformer) and (4) the less restrictive model with both 1+e anddz not constant

311 Original J-BModel Theoriginal J-Bmodel [12] assumessmall deformations compared with the soil thickness and anegligible initial void ratio (eo asymp 0) Developing the termbetween brackets of (3) we can write

11205901015840 1205971205901015840120597t = minuscv1120574k1205901015840(12059010158401)2 ( 120590101584012059010158401)minus120574k1205901015840minus1 ( 1205971205901015840120597z )2

+ cv112059010158401 ( 120590101584012059010158401)minus120574k1205901015840 12059721205901015840120597z2

(11)

4 Mathematical Problems in Engineering

or by mathematical manipulation

1205971205901015840120597t= 1205901015840oko120574w120574v1205901015840 ( 12059010158401205901015840o )1minus120574k1205901015840 [minus 120574k12059010158401205901015840 ( 1205971205901015840120597z )2 + 12059721205901015840120597z2 ] (12)

To make this equation dimensionless the following variablesare used

(1205901015840)1015840 = 1205901015840 minus 1205901015840112059010158402 minus 12059010158401 t1015840 = t120591o1205901015840 z1015840 = z

H1

(13)

where 120591o1205901015840 is a characteristic time or unknown referencechosen as the time required for the effective pressure (onaverage along the whole domain) to reach a high percentageof its change for example 90 With these variables andprovided that (on average) d1205901015840 = d((1205901015840)1015840(12059010158402 minus 12059010158401) + 12059010158401) asymp12059010158402 minus 12059010158401 (12) can be written as

(12059010158402 minus 12059010158401)1205901015840m120591o1205901015840120597 (1205901015840)1015840120597t1015840

= minus cv1120574k1205901015840(12059010158401)2H21 ( 1205901015840m12059010158401 )minus120574k1205901015840minus1 (12059010158402 minus 12059010158401)2( 120597 (1205901015840)1015840120597z1015840 )2

+ cv112059010158401H21 ( 1205901015840m12059010158401 )minus120574k1205901015840 (12059010158402 minus 12059010158401) 1205972 (1205901015840)1015840120597z10158402

(14)

where 1205901015840m is an average value of the effective pressure whichwe will talk about later The dimensional coefficients of thisequation in terms of the parameters of the problem

1120591o1205901015840 ( k112059010158401120574w120574v1205901015840 )

120574k1205901015840H21

( 1205901015840212059010158401 minus 1) ( 1205901015840m12059010158401 )minus120574k1205901015840 ( k112059010158401120574w120574v1205901015840 ) 1

H21( 1205901015840m12059010158401 )1minus120574k1205901015840

(15)

give rise to two dimensionless independent groups

120587I = 120591o1205901015840k112059010158401120574w120574v1205901015840H21 ( 1205901015840m12059010158401 )1minus120574k1205901015840 120587II = 120574k1205901015840 ( 12059010158402 minus 120590101584011205901015840m )

(16)

Based on the pi theorem the solution of the order ofmagnitude of the characteristic time derived from 120587I =120595(120587II) is given by

120591o1205901015840sim ( 120574w120574v1205901015840H21

k112059010158401 ) ( 1205901015840m12059010158401 )120574k1205901015840minus1 Ψ120591 (120574k1205901015840 ( 12059010158402 minus 120590101584011205901015840m )) (17)

As regards U1205901015840 since it also depends on time the solution(dependent on two groups) is given by

U1205901015840 = ΨU [ t120591o1205901015840 120574k1205901015840 ( 12059010158402 minus 120590101584011205901015840m )] (18)

where Ψ120591 and ΨU are unknown functions of their arguments

312 J-B Model Assuming 1+e Is Not Constant and dz IsConstant Deleting the restrictive hypothesis eo asymp 0 usedby J-B for the solutions of 1205901015840 and U1205901015840 the latter through thedependence V-1205901015840 for which such a hypothesis is not satisfiedand assuming that 1+e =constant the governing equation is

120597120597z ( k120574w dudz

) = 120597120597t ( e1 + e) = 1(1 + e)2 ( 120597e120597t) (19)

Using the dependence VV1 = (120590101584012059010158401)minus120574v1205901015840 = (1 +e)(1 +eo)or 1 + e = (1 + eo)(12059010158401205901015840o)minus120574v1205901015840 and its derivative form 119889e =minus((1 + eo)120574v12059010158401205901015840o)(12059010158401205901015840o)minus120574v1205901015840minus11198891205901015840 the right and left termsof (19) can be written as

120597120597t ( e1 + e) = 1(1 + e)2 ( 120597e120597t)

= minus 120574v1205901015840(1 + eo) 1205901015840o ( 12059010158401205901015840o )120574v1205901015840minus1 1205971205901015840120597t(20)

On the other hand with kko = (12059010158401205901015840o)minus120574k1205901015840 the left term of(19) writes as

120597120597z ( k120574w 120597u120597z)= ko120574w ( 12059010158401205901015840o )minus120574k1205901015840 [ 120574k12059010158401205901015840 ( 120597u120597z)2 + 1205972u120597z2 ] (21)

Equating both terms (19) in terms of the effective pressuretakes the form

1205971205901015840120597t= (1 + eo) 1205901015840oko120574w120574v1205901015840 ( 12059010158401205901015840o )1minus120574v1205901015840minus120574k1205901015840 [minus 120574k12059010158401205901015840 ( 1205971205901015840120597z )2

+ 12059721205901015840120597z2 ](22)

Mathematical Problems in Engineering 5

Proceeding as in the previous case the coefficients of thedimensionless form of the former equation

1120591o1205901015840 ( (1 + eo) k112059010158401120574w120574v1205901015840 ) 120574k1205901015840

H21( 12059010158402 minus 120590101584011205901015840m ) ( 1205901015840m12059010158401 )1minus120574v1205901015840minus120574k1205901015840

( (1 + eo) k112059010158401120574w120574v1205901015840 ) 1H21

( 1205901015840m12059010158401 )1minus120574v1205901015840minus120574k1205901015840(23)

give rise to the dimensionless groups

120587I = 120591o1205901015840k112059010158401 (1 + eo)120574w120574v1205901015840H21 ( 1205901015840m12059010158401 )1minus120574k1205901015840minus120574v1205901015840 120587II = 120574k1205901015840 ( 12059010158402 minus 120590101584011205901015840m )

(24)

Thus the solutions for 120591o1205901015840 and U1205901015840 are given by

120591o1205901015840 = ( 120574v1205901015840120574wH21(1 + eo) k112059010158401) ( 1205901015840m12059010158401 )120574v1205901015840+120574k1205901015840minus1

sdot Ψ120591 (120574k1205901015840 ( 12059010158402 minus 120590101584011205901015840m ))(25)

U1205901015840 = ΨU [ t120591o1205901015840 120574k1205901015840 ( 12059010158402 minus 120590101584011205901015840m )] (26)

313 J-B Model Assuming 1+e Is Constant (A Simplificationof the Former) Based on the above it is easy to simplify thehypothesis to 1+e being constant With (120590101584012059010158401)minus120574v1205901015840 = 1 theconsolidation equation reduces to

1205971205901015840120597t = (1 + eo) 1205901015840oko120574w120574v1205901015840 ( 12059010158401205901015840o )1minus120574k1205901015840 [minus 120574k12059010158401205901015840 ( 1205971205901015840120597z )2

+ 12059721205901015840120597z2 ](27)

an equation dependent on the initial void ratio Using (13)and averaging this equation provides three-dimensionalcoefficients

1120591o1205901015840 ( (1 + eo) k112059010158401120574w120574v1205901015840 ) 120574k1205901015840

H21( 12059010158402 minus 120590101584011205901015840m ) ( 1205901015840m12059010158401 )1minus120574k1205901015840

( (1 + eo) k112059010158401120574w120574v1205901015840 ) 1H21

( 1205901015840m12059010158401 )1minus120574v1205901015840minus120574k1205901015840(28)

and two dimensionless independent groups

120587I = 120591o1205901015840k112059010158401 (1 + eo)120574w120574v1205901015840H21 ( 1205901015840m12059010158401 )1minus120574k1205901015840 120587II = 120574k1205901015840 ( 12059010158402 minus 120590101584011205901015840m )

(29)

Thus the solutions for 120591o1205901015840 and U1205901015840 are given by

120591o1205901015840 = ( 120574v1205901015840120574wH21(1 + eo) k112059010158401) ( 1205901015840m12059010158401 )120574k1205901015840minus1

sdot Ψ120591 (120574k1205901015840 ( 12059010158402 minus 120590101584011205901015840m ))(30)

U1205901015840 = ΨU [ t120591o1205901015840 120574k1205901015840 ( 12059010158402 minus 120590101584011205901015840m )] (31)

314 The Less Restrictive Model with Both 1+e and dz NotConstant For this model 1+e and dz not constants it isenough to consider (22) plus the condition dz not constantor dz = dzo(HH1) So after mathematical manipulation thegoverning equation writes as

1205971205901015840120597t= (1 + eo) 1205901015840oko120574w120574v1205901015840 ( 12059010158401205901015840o )1+120574v1205901015840minus120574k1205901015840 [minus 120574k12059010158401205901015840 ( 1205971205901015840120597z )2

+ 12059721205901015840120597z2 ](32)

The new coefficients

1120591o1205901015840 ( (1 + eo) k112059010158401120574w120574v1205901015840 ) 120574k1205901015840

H21( 12059010158402 minus 120590101584011205901015840m ) ( 1205901015840m12059010158401 )1+120574v1205901015840minus120574k1205901015840

( (1 + eo) k112059010158401120574w120574v1205901015840 ) 1H21

( 1205901015840m12059010158401 )1+120574v1205901015840minus120574k1205901015840(33)

provide the groups

120587I = 120591o1205901015840k112059010158401 (1 + eo)120574w120574v1205901015840H21 ( 1205901015840m12059010158401 )1minus120574k1205901015840+120574v1205901015840 120587II = 120574k1205901015840 ( 12059010158402 minus 120590101584011205901015840m )

(34)

6 Mathematical Problems in Engineering

and the solutions

120591o1205901015840 = ( 120574v1205901015840120574wH21(1 + eo) k112059010158401) ( 1205901015840m12059010158401 )120574k1205901015840minus120574v1205901015840minus1

sdot Ψ120591 (120574k1205901015840 ( 12059010158402 minus 120590101584011205901015840m ))(35)

U1205901015840 = ΨU [ t120591o1205901015840 120574k1205901015840 ( 12059010158402 minus 120590101584011205901015840m )] (36)

32 Nondimensionalization of the J-B Model in Terms ofSettlements Three cases defined by the hypotheses (1) 1+e =constant and dz constant (2) 1+e and dz constants and (3) 1+eand dz not being constant are considered For this study let usintroduce a new variable directly related with the settlementwith a clear physical meaning ldquo120577 = e - eordquo a kind of localdegree of settlement or differential void index

321 1+e =Constant and dz Constant For 1+e =constantusing the variable 120577 and the dependence V-1205901015840 it is straight-forward to write

120577 = e minus eo = (1 + eo) [( 120590101584012059010158401)minus120574v1205901015840 minus 1] (37)

Making use of the derivatives of (37)

( 1205971205901015840120597z )2 = 120590101584012(1 + eo)2 120574v12059010158402 ( 120590101584012059010158401)2120574v1205901015840+2 ( 120597120577120597z)2

12059721205901015840120597z2 = minus 12059010158401(1 + eo) 120574v1205901015840 ( 120590101584012059010158401)120574v1205901015840+1 1205972120577120597z2

+ 12059010158401 (120574v1205901015840 + 1)(1 + eo)2 120574v12059010158402 ( 120590101584012059010158401)2120574v1205901015840+1 ( 120597120577120597z)2

(38)

and the assumptions of the odometer test after cumbersomemathematical steps the nonlinear dimensional consolidation(19) is written in the form

120597120577120597t = ko1205901015840o (120574k1205901015840 minus 120574v1205901015840 minus 1)120574w120574v12059010158402 ( 12059010158401205901015840o )1minus120574k1205901015840 ( 120597120577120597z)2

+ ko1205901015840o (1 + eo)120574w120574v1205901015840 ( 12059010158401205901015840o )1minus120574v1205901015840minus120574k1205901015840 1205972120577120597z2(39)

Introducing the variables

(120577)1015840 = 120577 minus 12057711205772 minus 1205771 = 1205771 minus 1205771205771 z1015840 = z

H1

t1015840 = t120591os(40)

into the former equation with 120591os a reference chosen as thetime required for the settlement to reach a high percentageof its change along the whole domain its dimensionless formprovides three coefficients

1120591os ko1205901015840o (120574k1205901015840 minus 120574v1205901015840 minus 1)120574w120574v12059010158402 ( 1205901015840m12059010158401 )1minus120574k1205901015840 (ef minus eo)

H21

ko1205901015840o (1 + eo)120574w120574v1205901015840 ( 1205901015840m12059010158401 )1minus120574v1205901015840minus120574k1205901015840 1H21

(41)

Dividing by the last the resulting dimensionless groups are

120587I = 120591osko1205901015840o (1 + eo)120574w120574v1205901015840H21 ( 1205901015840m12059010158401 )1minus120574v1205901015840minus120574k1205901015840

120587II = (120574k1205901015840 minus 120574v1205901015840 minus 1)120574v1205901015840 (ef minus eo)(1 + eo) ( 1205901015840m12059010158401 )120574v1205901015840(42)

Making use of the constitutive dependence dVV =minus120574v1205901015840(d12059010158401205901015840) is possible to write ΔHH = minus120574v1205901015840(Δ12059010158401205901015840)and since (ef minus eo)(1 + eo) = ΔHHo 120587II finally takes theform

120587II = (1 minus 120574k1205901015840 + 120574v1205901015840) 12059010158402 minus 120590101584011205901015840m (43)

The solution for 120591os is120591os asymp 120574w120574v1205901015840H21

ko1205901015840o (1 + eo) ( 1205901015840m12059010158401 )120574v1205901015840+120574k1205901015840minus1Ψ120591 [(1 minus 120574k1205901015840+ 120574v1205901015840) 12059010158402 minus 120590101584011205901015840m ]

(44)

As for the average degree of consolidation the followingdependence arises

Us = ΨU [ t120591os (1 minus 120574k1205901015840 + 120574v1205901015840) 12059010158402 minus 120590101584011205901015840m ] (45)

322 1+e and dz Constants The hypothesis 1+e=constant isa simplification of the former case Proceeding in the sameway the resulting coefficients

1120591os ko1205901015840o (1 + 120574v1205901015840 minus 120574k1205901015840)120574w120574v12059010158402 ( 1205901015840m12059010158401 )1minus120574k1205901015840+120574v1205901015840 (ef minus eo)

H21

ko1205901015840o (1 + eo)120574w120574v1205901015840 ( 1205901015840m12059010158401 )1minus120574k1205901015840 1H21

(46)

Mathematical Problems in Engineering 7

provide the dimensionless groups

120587I = 120591osko1205901015840o (1 + eo)120574w120574v1205901015840H21 ( 1205901015840m12059010158401 )1minus120574k1205901015840

120587II = (1 minus 120574k1205901015840 + 120574v1205901015840) 12059010158402 minus 120590101584011205901015840m(47)

The solutions for 120591os and Us are

120591os asymp 120574w120574v1205901015840H21ko1205901015840o (1 + eo) ( 1205901015840m12059010158401 )120574k1205901015840minus1Ψ120591 [(1 minus 120574k1205901015840

+ 120574v1205901015840) 12059010158402 minus 120590101584011205901015840m ](48)

Us = ΨU [ t120591os (1 minus 120574k1205901015840 + 120574v1205901015840) 12059010158402 minus 120590101584011205901015840m ] (49)

The solutions for the original J-Bmodel are also (48) and (49)with the simplification of eo=0 The author however doesnot talk about the characteristic time and obtains the averagedegree of consolidation given by (7) in which this variabledepends on three groups (1minus120574k1205901015840)120574v1205901015840 H2H1 and tcv2H21undoubtedly a less precise solution

323 1+e and dz Not Being Constant When 1+e and dzare not constants substituting dz = dzo(HH1) in (39)after cumbersome mathematical manipulation the resultingcoefficients are 1120591os

ko1205901015840o (120574k1205901015840 minus 120574v1205901015840 minus 1)120574w120574v12059010158402 ( 1205901015840m12059010158401 )1+2120574v1205901015840minus120574k1205901015840 (ef minus eo)H21

ko1205901015840o (1 + eo)120574w120574v1205901015840 ( 1205901015840m12059010158401 )1+120574v1205901015840minus120574k1205901015840 1

H21

(50)

Dividing by the last the resulting dimensionless groups are

120587I = 120591osko1205901015840o (1 + eo)120574w120574v1205901015840H21 ( 1205901015840m12059010158401 )1+120574v1205901015840minus120574k1205901015840

120587II = (1 minus 120574k1205901015840 + 120574v1205901015840) 12059010158402 minus 120590101584011205901015840m(51)

The solutions for 120591os and Us are

120591os asymp 120574w120574v1205901015840H21ko1205901015840o (1 + eo) ( 1205901015840m12059010158401 )+120574k1205901015840minus120574v1205901015840minus1Ψ120591 [(1 minus 120574k1205901015840

+ 120574v1205901015840) 12059010158402 minus 120590101584011205901015840m ](52)

Us = ΨU [ t120591os (1 minus 120574k1205901015840 + 120574v1205901015840) 12059010158402 minus 120590101584011205901015840m ] (53)

Table 1 summarizes the results derived in this section

4 Verification of the Results byNumerical Simulations

This section is devoted to checking the solutions given inTable 1 To shorten the exposure we restrict the tests forthe more general model (1+e and dz not constants) andfor the unknowns 120591os and Us since they are the ones ofgreater interest in civil engineering Eight sets of simulationsarranged in three blocks have been run Table 2 In each onesome of the soil parameters or initial values of the problemhave been changed to give the same or different values asrequired to the dimensionless group 120587II in the search for thesame or different solutions of 120587I Changes in the values of theindividual parameters are sufficient to cover all real scenarios

Firstly a reference set (set 1) is established to which all theother sets can be referred and which permits them to be com-pared with each other The physical and geometrical charac-teristics that change are 120574v1205901015840 120574k1205901015840 eo H1 (m) 1205901015840o (Nm2)1205901015840f (Nm2) and ko (myear) The values of =(1205901015840f1205901015840o) H2(m) kf (myear) and 120587II are derived from them while 1205871is obtained once the characteristic time 120591os is read from thesimulation The criterion for the choice of 120591os is the timerequired by the soil to reach 90 of the total settlement

It is worth mentioning here the emergence of 1205901015840m anaveraged value of the effective pressure in the two dimen-sionless groups Undoubtedly the existence of 1205901015840m in themonomials can be explained not only by the nonlinearityof the problem but also by the kind of dependences in theconstitutive relations (there are other nonlinear problems inwhich averaged variables do not emerge [18]) How do wechoose 1205901015840m We could take the mean value 1205901015840m=(12059010158401+12059010158402)2or even 1205901015840m=12059010158401 or 1205901015840m=12059010158402 but expecting that the expres-sion 120587I = 120595(120587II) is slightly dependent on that choice (we willreturn to this question later) In this paper we have taken theapproximation 1205901015840m=(12059010158402 + 12059010158401)2

In block I the parameters that change in each set areeo H1 ko and 1205901015840o while maintaining the ratio =(1205901015840f1205901015840o) thus both 120587II and factor (1205901015840m1205901015840o)1minus120574k1205901015840+120574v1205901015840 of 120587I takethe same value in the four sets of the block As a resultwe expect a characteristic time proportional to the ratio120574w120574v1205901015840H21ko1205901015840o(1 + eo) and a same value of 120587I in all casesaccording to the solution 120587I = 120595(120587II) Indeed the resultsare consistent with expectations Block II contains an onlycase for which 120587II varies in comparison to block I as a result120587I also changes as expected Finally block III contains threecases with the same value of 120587II of block I but with differentvalues of the parameters 120574v1205901015840 120574k1205901015840 and the ratio =(1205901015840f 1205901015840o)The parameters of set 6 have been chosen in such a waythat the values 120574k1205901015840 and 120574v1205901015840 are compensated to maintainconstant (1 minus 120574k1205901015840 + 120574v1205901015840) we expect 120587I to be also unchangedIn effect simulation provides 120587I = 07139 a value quite closeto that of set 1 (07131) The negligible differences are due nodoubt to the different evolution of the effective pressure alongthe process caused by the different values of the coefficients120574k1205901015840 and 120574v1205901015840 Note that as 120574v1205901015840 duplicates the denominator ofthe factor 120591osko1205901015840o(1 + eo)120574w120574v1205901015840H21 the characteristic timehas to double too as indeed happens The last two sets studythe influence of a parameter whose change forces the value

8 Mathematical Problems in Engineering

Table1Dim

ensio

nlessg

roup

sthatcharacterizethe

originalandextend

edJ-Bmod

els

Juarez-Badillo

original

Juarez-Badillo

Juarez-Badillo

Juarez-Badillo

1+e=

constant

1+e

=consta

nt1+e

=consta

ntdz

constant

dzconstant

dzvaria

ble

120587 I120591 ok o

1205901015840 o120574 w120574 v1205901015840H2 1

(1205901015840 m 1205901015840 o)1minus120574k1205901015840

120591 ok o1205901015840 o(

1+eo)

120574 w120574 v1205901015840H2 1

(1205901015840 m 1205901015840 o)1minus120574k1205901015840

120591 ok o1205901015840 o(

1+eo)

120574 w120574 v1205901015840H2 1

(1205901015840 m 1205901015840 o)1minus120574k1205901015840minus120574 v1205901015840

120591 ok o1205901015840 o(

1+eo)

120574 w120574 v1205901015840H2 1

(1205901015840 m 1205901015840 o)1minus120574k1205901015840+120574 v1205901015840

120591 o1205901015840or

120591 os120587 II

120574 k1205901015840(1205901015840 2minus

1205901015840 1 1205901015840 m)for

pressure

(1minus120574 k1205901015840

+120574 v1205901015840)(1205901015840 2minus

1205901015840 1 1205901015840 m)for

settlem

ent

Mathematical Problems in Engineering 9

0010203040506070809

1

0 05 1 15 2 25 3time

Set 1Set 2Set 3Set 4

Set 5Set 6Set 7Set 8

Aver

age d

egre

e of c

onso

lidat

ion

Figure 1 Average degree of consolidation versus time (eight sets)

0010203040506070809

1

0 05 1 15 2 25 3

Set 1Set 2Set 3Set 4

Set 5Set 6Set 7Set 8

Aver

age d

egre

e of c

onso

lidat

ion

tos

Figure 2 Average degree of consolidation versus dimensionlesstime (eight sets)

(1 minus 120574k1205901015840 + 120574v1205901015840) to be also changed in order to maintain 120587IIconstant We expect that 120587I does not change as nearly occursThe small differences are again due to the existence of 1205901015840mwithin the monomials Note however that as diminishesthe influence of 1205901015840m in the results also diminishes since thevalues of 12059010158401 and 12059010158402 are closer

For a better understanding of the results Figures 1 and2 are presented In the first Us as a function of time isshown for the eight sets in Table 2 It is clear that 120591oscovers a wide range of values from 02283 (set 2) to 27968(set 3) the last more than ten times the first The secondfigure graphs Us in front of the dimensionless time t120591os Asshown the eight previous curves nearly converge to a single(or universal) graph having a common point at (Us=09t120591os=1) This means that monomial 120587II for the range ofvalues assigned to the parameters in contrast to 120587I scarcelyinfluences the value of the average degree of consolidationThis is not in contradiction with the results derived bynondimensionalization

0010203040506070809

1

0 05 1 15 2 25 3 35 4time

U (J-B)U (1+e cst)U (1+e not cst)U (1+e amp dz not cst)

Us (J-B)Us (1+e cst)Us (1+e not cst)Us (1+e amp dz not cst)

Aver

age d

egre

e of p

ress

ure d

issip

atio

nAv

erag

e deg

ree o

f con

solid

atio

n

Figure 3 Average degree of consolidation and pressure dissipation(set 6)

Although in this sectionwe have only checked the expres-sions of Us and 120591os for the most general model we considerit interesting to present both Us and U1205901015840 for the rest of themodels of Table 1 and for set 6 as a typical case Figure 3Theseresults emphasize the importance of using the less restrictive(more general) model in relation to the others in particularwith the original model of J-B in which the influence ofeo is despised Note that the smaller of the characteristictimes (related to settlement) corresponds to the more generalmodel (black and bold line) so the use of the others althoughon the safety side provides characteristic times oversizedfar from an optimal solution for the engineer For examplefor the case of Figure 3 in which eo=1 the error of J-Boriginal model (purple and bold line) in comparison with theextended model (black and bold line) for Us=90 is around150 This error increases appreciably as eo gets larger

5 Universal Curves

These easy to use and universal abacuses obtained by numeri-cal simulations allow engineers to read the characteristic time(120591os) and the average degree of consolidation (Us) in termsof the dimensionless groups given in Table 1 Only the mostgeneral (less restrictive) J-B model 1+e and dz not constantis presented

To obtain the characteristic time curves a large numberof simulations have been carried out studying separately theinfluence of the parameters 120574v1205901015840 and 120574k1205901015840 whose valuescover most real soils ranges from 101 to 8 120574v1205901015840 from 005to 03 and 120574k1205901015840 from 01 to 17 From these tests it has beendeduced as expected from the results of the previous sectionthat the relatively more influential parameter is while 120574v1205901015840and 120574k1205901015840 do not produce significant changes in the values ofthe characteristic timeThus the first universal group of linesshows the dependence of 120587I versus 120587II for different valuesof (with 120574v1205901015840=01) Figure 4 As the range of values of 120587IIdepends on according to (43) the range of values of eachline is different As shown in the figure there exists a universal

10 Mathematical Problems in Engineering

Table 2 Values of the parameters and dimensionless groups of simulations

Block Set 120574v1205901015840 120574k1205901015840 eoH1 ko 1205901015840o 1205901015840f H2 kf 120587II

120591os 120587I(m) (myr) (Nm2) (Nm2) 1205901015840f 1205901015840o (m) (myr) (yr)

I

1 01 0500 10 100 002 30000 60000 2 0933 00141 040 04566 071312 01 0500 10 100 002 60000 120000 2 0933 00141 040 02283 071313 01 0500 10 175 001 30000 60000 2 1633 00071 040 27968 071314 01 0500 13 142 004 22000 44000 2 1325 00283 040 05459 07131

II 5 01 1500 10 100 002 30000 90000 3 0896 00038 -040 10597 09834

III6 02 0600 10 100 002 30000 60000 2 0871 00132 040 09142 071397 01 0767 10 100 002 30000 120000 4 0871 00069 040 04360 072468 02 0920 10 100 002 30000 180000 6 0699 00038 040 08409 07312

line for each value of all converging in the central zone forvalues of 120587II around 0 This set of straight lines can be fittedby 120587I=m120587II+n being n=0846 and m given in Figure 5 as afunction of The curve that best fits the points in Figure 5 isgiven by

m = minus0235e(minus0868) minus 0306e(minus0012) (54)

Choosing the longest line corresponding to =8 as the onlyrepresentative line of the problem themaximumerrors in thereading of 120587I (and 120591os) are 38

In short the use of these curves is as follows from the dataof the problem Figure 4 allows us to read 120587I from 120587II and and from (51) the characteristic time 120591os is derived

As for the average degree of consolidation the universalcurves for 96 different scenarios of 120587II are shown in Figure 6It is appreciated that these curves are very close to each othershowing again that the Us only depends on the dimensionlesstime In the worst case the errors produced by changes invalue of the 120587II group are less than 28when using the curveof Figure 7

6 Case Study

Below is a practical application of the universal curvespresented in this paper For this purpose we take as a basisfor our study the data obtained experimentally byAbbasi et al[26] for different samples of clay whosemoisture in the liquidlimit is equal to 42 As can be seen (Table 3) it is a series ofdiscrete (tabulated) values that relate the void ratio with theeffective pressure and the hydraulic conductivity

We will address a real case of consolidation in which wehave a layer of soil 2meters thick (Ho) with an initial effectivepressure (1205901015840o) of 28 kNm2 on which a load of 31 kNm2 isapplied In this way the effective pressure at the end of theconsolidation process (1205901015840f ) will have the value of 59 kNm2From the tabulated data (e-1205901015840) it follows that eo = 105 andef = 095 and since the factor 1+e is proportional to the soilvolume from (9) we have 120574v1205901015840 = 0067

To obtain 120574k1205901015840 we first calculate the value of 120574kv from thetabulated data (e-k) to then obtain 120574k1205901015840 from (10) With thiswe have for the load step considered ko = 0035 myr 120574kv =8108 and 120574k1205901015840 = 0543

To solve the problem we will use the universal curvespresented in the previous section corresponding to the most

Table 3 Experimental values of e 1205901015840 and k obtained by Abbasi et al[26]

e 1205901015840o (Nm2)110 14000105 28000095 59000085 120000075 220000065 450000060 925000e k (myr)175 0683160 0315120 0084085 0021065 0011

general J-B model (analyzed in terms of settlement) Thusfrom the values of 1205901015840o 1205901015840f 120574v1205901015840 and 120574k1205901015840 by means of (51) itis obtained that 120587II = 0373 (with 1205901015840m = 435 kNm2) Then120587I is read from Figure 4 where for = 21 we have 120587I =072 An identical result is obtained if we use the proposedfitted equation 120587I=m120587II+n where n is a constant of value0846 andm is obtained from (54) or Figure 5 Finally knownthe values of Ho ko and eo from (51) we get 120591os = 0748years Likewise once the characteristic time of the problem(in terms of settlement) is known from the universal curve ofFigure 7 we easily obtain the evolution of the average degreeof consolidation along the whole process Figure 8

These solutions have been compared with those thatresult from the numerical simulation of the most generalJ-B model for which a value of 120591os = 075 years has beenobtained showing clearly that the relative error committedwhen applying the universal solutions is negligible (Table 4)The simulation of the problem has also been carried out fora model where 1+e and dz are also not constant but in whichthe constitutive relations of potential type have been replacedby pairs of data in tabulated form obtaining in this case 120591os= 0817 years The relative error which this time increasesto 82 finds its explanation in the strong nonlinearity ofthe problem which will always bring differences between the

Mathematical Problems in Engineering 11

040045050055060065070075080085090095100105110

minus07 minus06 minus05 minus04 minus03 minus02 minus01 0 01 02 03 04 05 06 07 08 09 1 11 12 13 14 15

1

2

ᴪ=8ᴪ=6ᴪ=4

ᴪ=2ᴪ=15ᴪ=101

Figure 4 120587I as a function of 120587II with as parameter

1 2 3 4 5 6 7 8minus042

minus04

minus038

minus036

minus034

minus032

minus03

minus028

minus026

m

datafitted curve

Figure 5 m as a function of

solutions provided by numerical models based on differentconstitutive relations (potential and tabulated in this case)Even so this value is totally acceptable within this field ofengineering demonstrating the strength and versatility of theuniversal solutions provided

7 Discussion

This work derives the dimensionless groups that rule thenonlinear soil consolidation process based on the original J-Bmodel following a formal nondimensionalization procedurethat starts from the governing equation and does not needany other assumption or mathematical manipulation Inaddition more real and precise extended J-B models whichassume both the void ratio and the thickness of the finitevolume element of the soil to be not constant have been

0010203040506070809

1

0 05 1 15 2 25 3

96 scenariosᴪ = [101 15 2 4 6 8]

Aver

age d

egre

e of c

onso

lidat

ion

tos

= [005 01 015 02]v= [02 05 1 15]k

Figure 6 Average degree of consolidation versus dimensionlesstime for different scenarios

0010203040506070809

1

0 05 1 15 2 25 3

Aver

age d

egre

e of c

onso

lidat

ion

tos

Figure 7 Proposed solution for the average degree of consolidation

nondimensionalized and their dimensionless groups derivedThe assumption of constitutive dependences (e-1205901015840 e-k and k-1205901015840) of potential type in these models is interesting since they

12 Mathematical Problems in Engineering

Table 4 Case study solution for the characteristic time 120591os Comparison with numerical models

Universal curves J-B most general model Tabulated model120591os (yr) 120591os (yr) Rel error 120591os (yr) Rel error 0748 0750 026 0817 820

0010203040506070809

1

0 025 05 075 1 125 15 175 2 225t ( years)

Aver

age d

egre

e of c

onso

lidat

ion

Figure 8 Case study solution for the average degree of consolida-tion Us from the universal curves

can be applied to highly compressible soils for which the e-log(1205901015840) relation moves away from being a straight line andin which negative values of the void ratio could be obtainedunder the application of high load ratios (1205901015840f 1205901015840o) [8]

The study has been developed both in terms of pres-sure dissipation and in terms of settlement resulting thatthe dimensionless groups derived from each model are ingeneral different The extension to more complex modelswhich implies the emergence of new parameters or equationswithin themathematical model does not necessarily give riseto additional dimensionless groups as might be expectedbut to new expressions of the groups or new regroupingof the parameters involved Thus the elimination of thehypothesis ldquodz constantrdquo in the most general consolidationmodel has not given rise to any new group both in pressureand in settlement (Table 1) but it has only modified oneof the groups obtained in the original model All thisrepresents an important advance in relation to the modelsand studies carried out by other authors and the attemptsof characterization found in the scientific literature [9ndash11]which try to lead to easy-to-use solutions for engineers Inno case had dimensionless groups been obtained as preciseas those presented in this work nor presented universalcurves of the characteristic time and the average degree ofconsolidation Authors such asConte andTroncone [11] applymore general load conditions (time-dependent loading) forwhich the dimensionless groups deduced in this work couldbe applicable for each load step

As regards time to make it dimensionless an unknownreference (120591o) is introduced in the form mentioned by Scott[27] and used in many other problems with asymptoticsolutions [28]Theprecise definition of 120591o necessarily involvesa criterion in relation to the percentage of average pressuredecrease (or settlement) associated with the end of theconsolidation processTheorder ofmagnitude of 120591o is derived

after nondimensionalization making 120591o only appear in oneof the independent dimensionless groups In this way theaverage degree of consolidation (in terms of pressure orsettlement) depends on the ldquotimecharacteristic timerdquo ratioin addition to the dimensionless groups (without unknowns)that arise

In short the procedure has led to a new very precisesolution that involves two monomials one (120587I) that containsthe characteristic time (120591o) and another (120587II) that is amathematical function of the soil parameters In this way120591o only depends on 120587II through the function 120587I = 120595(120587II)while U depends on 120587I and 120587II through the expression U= 120595(120587I 120587II) The results obtained have been verified forthe most general and complex consolidation model throughthe numerical simulation of a large number of scenariosin which the values of the dimensionless groups and theindividual parameters of the problem have been changing toconvenience The high errors produced by the original J-Bmodel with respect to the most general justify the use of thelatter in the search for more precise solutions

In addition the emergence of mean values of the effectivepressure when averaging the governing equation in orderto assign an order of magnitude unit to the dimensionlessvariables and their derivatives is an added difficulty to obtainthe dimensionless groups This is undoubtedly due to thenonlinearities (constitutive relations) involved in the consol-idation process This average effective pressure of referencehas been assumed as the semisum of its initial and finalvalues although other criteria could also be chosen Sincetwo scenarios can have equal120587II with the same effectivemeanpressure but different values of the parameters 120574v1205901015840 and 120574k1205901015840 we expect some minor deviations between the results of bothscenarios as indeed it has been checked Thus for the wholerange of parameters values in typical soils the errors due tothe above cause are negligible in the field of soil mechanicsOn the other hand when the value of 120587II does not changebut the mean effective pressure value does the errors in thesolutions of 120591os and Us are slightly larger although withinranges below 4

The brooch for all this work is the presentation ofuniversal curves for the most general model which allowfor the usual range of variation of the soil properties toobtain in a simple way the value of the characteristic timeof consolidation and the evolution of the average degreeof settlement throughout the entire process Thus startingfrom the nonlinear parameters of the soil (120574v1205901015840 and 120574k1205901015840)and the load ratio (1205901015840f 1205901015840o) the value of the monomial 120587IIis obtained from which the value of 120587I is directly deductedthanks to the universal curve 120587I = 120595(120587II) provided Finally120591os is easily obtained from the expression of 120587I On the otherhand the curve that represents the Us evolution along theprocess has turned out to be (practically) unique when it is

Mathematical Problems in Engineering 13

represented as a function of the dimensionless time in theform t120591os regardless of the soil parameters and the loadsapplied All this supposes a clear advantage when solvingthe consolidation problem also showing the strength of thenondimensionalization method presented

Finally the case study addressed has demonstrated thesimplicity and reliability of the universal curves proposedFrom a series of tabulated data (e-1205901015840 and e-k) the parametersthat govern the nonlinear consolidation problem for themostextended J-B model have been adjusted with which thevalue of the monomial 120587II has been derived With this andusing the universal curves the solutions for the characteristictime 120591os and the average degree of consolidation Us havebeen obtained These results have been satisfactorily verifiedwith the solutions provided both by the numerical extendedmodel of J-B presented here and by another that substitutesthe constitutive dependences of potential type by data intabulated form

8 Conclusions

The application of the nondimensionalization technique hasallowed us to deduce the dimensionless groups that rule thenonlinear consolidation problem in which the constitutivedependencies e-1205901015840 and e-k have the form of potential func-tions This type of relations approximates the soil behaviorbetter than the logarithmic functions e-log(1205901015840) and e-log(k)particularly in very compressible soils and under high loadratios (1205901015840f 1205901015840o) in which inconsistent negative values of thevoid ratio may appear

Several scenarios have been addressed from the originalmodel of Juarez-Badillo to extended models based on thesame constitutive dependences but eliminating one or moreclassic (and restrictive) hypotheses in this process In allcases only two dimensionless groups emerge one of themconsisting of a grouping of the soil parameters and the othercontaining the unknown characteristic time Based on theseresults universal curves have been depicted the first to deter-mine the characteristic time of the process and the secondthat represents the evolution of average degree of settlement

The solutions have been verified by numerical simula-tions changing the values of the soil parameters and thedimensionless groups conveniently Finally the reliability andeffectiveness of the proposed solutions have been shownthrough a case study

Abbreviations

cv Consolidation coefficient (m2s or m2yr)cv1 Initial consolidation coefficient (m2s or

m2yr)cv2 Final consolidation coefficient (m2s or

m2yr)119889z Element of differential length in thedirection of the spatial coordinate z (m)119889zo Element of differential length in thedirection of the spatial coordinate z at theinitial time (m)

e Void ratio (dimensionless)

eo Initial void ratio (dimensionless)ef Final void ratio (dimensionless)H Soil thickness up to the impervious

boundary or drainage length (m)H1Ho Initial soil thickness (m)H2 Final soil thickness (m)k Hydraulic conductivity (or permeability)

(ms or myr)k1 ko Initial hydraulic conductivity (ms or

myr)kf Final hydraulic conductivity (ms or myr)mv Coefficient of volumetric compressibility

(m2N)mv1 Initial coefficient of volumetric

compressibility (m2N)t Time Independent variable (s or yr)T Juarez-Badillorsquos time factor

(dimensionless)u Excess pore pressure (Nm2)U Average degree of consolidation or

pressure dissipation (dimensionless)Us Average degree of consolidation

(dimensionless)U1205901015840 Average degree of pressure dissipation

(dimensionless)V Soil volume (m3)V1Vo Initial soil volume (m3)z Vertical spatial coordinate (m)120574v Nonlinear coefficient of compressibility of

Juarez-Badillo (dimensionless)120574kv Nonlinear coefficient of change ofpermeability with volume (dimensionless)120574k1205901015840 Nonlinear coefficient of change ofpermeability with effective pressure(dimensionless)120574v1205901015840 Nonlinear coefficient of compressibility orcoefficient of volumetric contraction(dimensionless)120574w Specific weight of water (Nm3)

Loading factor Ratio between the finaland initial effective pressure(dimensionless)120581 Nonlinear coefficient of change ofpermeability with volume ofJuarez-Badillo (dimensionless)120582 Parameter of Juarez-Badillo 120582 = 1 - 120574v120581(dimensionless)

V Auxiliary variable of Juarez-Badillo(dimensionless)120587i Dimensionless group or number(dimensionless)1205901015840 Effective pressure of the soil (Nm2)12059010158401 1205901015840o Initial effective pressure (Nm2)12059010158402 1205901015840f Final effective pressure (Nm2)1205901015840m Mean value of the effective pressure(Nm2)120591o Characteristic time (s or yr)

14 Mathematical Problems in Engineering

120591os Characteristic time which takes thesettlement to reach approximately its finalvalue (s or yr)120591o1205901015840 Characteristic time which takes the excesspore pressure to dissipate untilapproximately the value of zero (s or yr)Ψ Arbitrary mathematical function120577 Differential void ratio (dimensionless)1205771 1205772 Initial and final differential void ratio(dimensionless)

Subscripts

I II Denote different numbers

Superscripts

1015840 Denote dimensionless magnitude

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] K Terzaghi ldquoDie Berechnung der Durchlassigkeitszifferdestones aus dem verlauf der hydrodynamischen spanunngser-scheinnungenrdquo Technical report II a 132 N 34 125-138Akademie derWissenschaften inWien SitzungsberichteMath-naturwiss Klasse Abt 1923

[2] R T Walker ldquoVertical drain consolidation analysis in one twoand three dimensionsrdquo Computers amp Geosciences vol 38 no 8pp 1069ndash1077 2011

[3] G Y Zheng P Li and C Y Zhao ldquoAnalysis of non-linearconsolidation of soft clay by differential quadrature methodrdquoApplied Clay Science vol 79 pp 2ndash7 2013

[4] X Y Geng C J Xu and Y Q Cai ldquoNon-linear consolidationanalysis of soil with variable compressibility and permeabilityunder cyclic loadingsrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 30 no 8 pp 803ndash8212006

[5] M Lu S Wang S W Sloan B Indraratna and K Xie ldquoNon-linear radial consolidation of vertical drains under a generaltime-variable loadingrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 39 no 1 pp 51ndash622015

[6] HWu L HuW Qi and QWen ldquoAnalytical Solution for Elec-troosmotic Consolidation Considering Nonlinear Variation ofSoil Parametersrdquo International Journal of Geomechanics vol 17no 5 Article ID 06016032 2017

[7] S J Brandenberg ldquoiConsoljs JavaScript Implicit Finite-Differ-ence Code for Nonlinear Consolidation and Secondary Com-pressionrdquo International Journal of Geomechanics vol 17 no 6Article ID 04016149 2017

[8] R Butterfield ldquoA natural compression law for soils (an advanceon e-log prsquo)rdquo Geotechnique vol 29 no 4 pp 469ndash480 1979

[9] R Lancellotta and L Preziosi ldquoA general nonlinear mathemati-calmodel for soil consolidation problemsrdquo International Journalof Engineering Science vol 35 no 10-11 pp 1045ndash1063 1997

[10] Y-C Zhuang K-H Xie and X-B Li ldquoNonlinear analysis ofconsolidation with variable compressibility and permeabilityrdquoJournal of Zhejiang University (Engineering Science Edition) vol6 no 3 pp 181ndash187 2005

[11] E Conte andA Troncone ldquoNonlinear consolidation of thin lay-ers subjected to time-dependent loadingrdquo Canadian Geotechni-cal Journal vol 44 no 6 pp 717ndash725 2007

[12] R Yong and F Townsend ldquoGeneral Consolidation Theory forClaysrdquo Soil Mechanics Series Report No 8 Graduate School ofEngineering National University of Mexico 1983

[13] J C Gibbings Dimensional Analysis Springer-Verlag LondonUK 2011

[14] E Buckingham ldquoOn physically similar systems Illustrations ofthe use of dimensional equationsrdquo Physical Review A AtomicMolecular and Optical Physics vol 4 no 4 pp 345ndash376 1914

[15] R Seshadri and T Y Na ldquoA Survey ofMethods for DeterminingSimilarity Transformationsrdquo inGroup Invariance in EngineeringBoundary Value Problems pp 35ndash61 Springer New York NYUSA 1985

[16] V D Zimparov and V M Petkov ldquoApplication of discriminatedanalysis to low Reynolds number swirl flows in circular tubeswith twisted-tape inserts Pressure drop correlationsrdquo Interna-tional Review of Chemical Engineering vol 1 no 4 pp 346ndash3562009

[17] M Capobianchi and A Aziz ldquoA scale analysis for natural con-vective flows over vertical surfacesrdquo International Journal ofThermal Sciences vol 54 pp 82ndash88 2012

[18] M Conesa J F Sanchez Perez I Alhama and F AlhamaldquoOn the nondimensionalization of coupled nonlinear ordinarydifferential equationsrdquo Nonlinear Dynamics vol 84 no 1 pp91ndash105 2016

[19] I A Manteca G Garcıa-Ros and F Alhama ldquoUniversal solu-tion for the characteristic time and the degree of settlement innonlinear soil consolidation scenarios A deduction based onnondimensionalizationrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 57 pp 186ndash201 2018

[20] E H Davis and G P Raymond ldquoA non-linear theory ofconsolidationrdquo Geotechnique vol 15 no 2 pp 161ndash173 1965

[21] P Cornetti and M Battaglio ldquoNonlinear consolidation of soilmodeling and solution techniquesrdquo Mathematical and Com-puter Modelling vol 20 no 7 pp 1ndash12 1994

[22] C F Gonzalez-Fernandez ldquoApplications of the network simu-lation method to transport processesrdquo in Network SimulationMethod J Horno Ed Research Signpost Trivandrum India2002

[23] J Serna F J S Velasco and A Soto Meca ldquoApplication of net-work simulationmethod to viscous flowsThe nanofluid heatedlid cavity under pulsating flowrdquo Computers amp Fluids vol 91 pp10ndash20 2014

[24] E Juarez-Badillo ldquoGeneral Permeability Change Equationsfor Soilsrdquo in Proceedings of the International Conference onConstitutive Laws pp 205ndash209 University of Arizona TuesonAriz USA 1983

[25] E Juarez-Badillo and B Chen ldquoConsolidation curves for claysrdquoJournal of Geotechnical Engineering vol 109 no 10 pp 1303ndash1312 1983

Mathematical Problems in Engineering 15

[26] N Abbasi H Rahimi A A Javadi andA Fakher ldquoFinite differ-ence approach for consolidation with variable compressibilityand permeabilityrdquo Computers amp Geosciences vol 34 no 1 pp41ndash52 2007

[27] R F Scott Principles of Soils Mechanics Addison-Wesley Pub-lishing Company 1963

[28] F Alhama and C N Madrid Analisis Dimensional Discrimi-nado En Mecanica De Fluidos Y Transferencia De Calor 2012

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4 Mathematical Problems in Engineering

or by mathematical manipulation

1205971205901015840120597t= 1205901015840oko120574w120574v1205901015840 ( 12059010158401205901015840o )1minus120574k1205901015840 [minus 120574k12059010158401205901015840 ( 1205971205901015840120597z )2 + 12059721205901015840120597z2 ] (12)

To make this equation dimensionless the following variablesare used

(1205901015840)1015840 = 1205901015840 minus 1205901015840112059010158402 minus 12059010158401 t1015840 = t120591o1205901015840 z1015840 = z

H1

(13)

where 120591o1205901015840 is a characteristic time or unknown referencechosen as the time required for the effective pressure (onaverage along the whole domain) to reach a high percentageof its change for example 90 With these variables andprovided that (on average) d1205901015840 = d((1205901015840)1015840(12059010158402 minus 12059010158401) + 12059010158401) asymp12059010158402 minus 12059010158401 (12) can be written as

(12059010158402 minus 12059010158401)1205901015840m120591o1205901015840120597 (1205901015840)1015840120597t1015840

= minus cv1120574k1205901015840(12059010158401)2H21 ( 1205901015840m12059010158401 )minus120574k1205901015840minus1 (12059010158402 minus 12059010158401)2( 120597 (1205901015840)1015840120597z1015840 )2

+ cv112059010158401H21 ( 1205901015840m12059010158401 )minus120574k1205901015840 (12059010158402 minus 12059010158401) 1205972 (1205901015840)1015840120597z10158402

(14)

where 1205901015840m is an average value of the effective pressure whichwe will talk about later The dimensional coefficients of thisequation in terms of the parameters of the problem

1120591o1205901015840 ( k112059010158401120574w120574v1205901015840 )

120574k1205901015840H21

( 1205901015840212059010158401 minus 1) ( 1205901015840m12059010158401 )minus120574k1205901015840 ( k112059010158401120574w120574v1205901015840 ) 1

H21( 1205901015840m12059010158401 )1minus120574k1205901015840

(15)

give rise to two dimensionless independent groups

120587I = 120591o1205901015840k112059010158401120574w120574v1205901015840H21 ( 1205901015840m12059010158401 )1minus120574k1205901015840 120587II = 120574k1205901015840 ( 12059010158402 minus 120590101584011205901015840m )

(16)

Based on the pi theorem the solution of the order ofmagnitude of the characteristic time derived from 120587I =120595(120587II) is given by

120591o1205901015840sim ( 120574w120574v1205901015840H21

k112059010158401 ) ( 1205901015840m12059010158401 )120574k1205901015840minus1 Ψ120591 (120574k1205901015840 ( 12059010158402 minus 120590101584011205901015840m )) (17)

As regards U1205901015840 since it also depends on time the solution(dependent on two groups) is given by

U1205901015840 = ΨU [ t120591o1205901015840 120574k1205901015840 ( 12059010158402 minus 120590101584011205901015840m )] (18)

where Ψ120591 and ΨU are unknown functions of their arguments

312 J-B Model Assuming 1+e Is Not Constant and dz IsConstant Deleting the restrictive hypothesis eo asymp 0 usedby J-B for the solutions of 1205901015840 and U1205901015840 the latter through thedependence V-1205901015840 for which such a hypothesis is not satisfiedand assuming that 1+e =constant the governing equation is

120597120597z ( k120574w dudz

) = 120597120597t ( e1 + e) = 1(1 + e)2 ( 120597e120597t) (19)

Using the dependence VV1 = (120590101584012059010158401)minus120574v1205901015840 = (1 +e)(1 +eo)or 1 + e = (1 + eo)(12059010158401205901015840o)minus120574v1205901015840 and its derivative form 119889e =minus((1 + eo)120574v12059010158401205901015840o)(12059010158401205901015840o)minus120574v1205901015840minus11198891205901015840 the right and left termsof (19) can be written as

120597120597t ( e1 + e) = 1(1 + e)2 ( 120597e120597t)

= minus 120574v1205901015840(1 + eo) 1205901015840o ( 12059010158401205901015840o )120574v1205901015840minus1 1205971205901015840120597t(20)

On the other hand with kko = (12059010158401205901015840o)minus120574k1205901015840 the left term of(19) writes as

120597120597z ( k120574w 120597u120597z)= ko120574w ( 12059010158401205901015840o )minus120574k1205901015840 [ 120574k12059010158401205901015840 ( 120597u120597z)2 + 1205972u120597z2 ] (21)

Equating both terms (19) in terms of the effective pressuretakes the form

1205971205901015840120597t= (1 + eo) 1205901015840oko120574w120574v1205901015840 ( 12059010158401205901015840o )1minus120574v1205901015840minus120574k1205901015840 [minus 120574k12059010158401205901015840 ( 1205971205901015840120597z )2

+ 12059721205901015840120597z2 ](22)

Mathematical Problems in Engineering 5

Proceeding as in the previous case the coefficients of thedimensionless form of the former equation

1120591o1205901015840 ( (1 + eo) k112059010158401120574w120574v1205901015840 ) 120574k1205901015840

H21( 12059010158402 minus 120590101584011205901015840m ) ( 1205901015840m12059010158401 )1minus120574v1205901015840minus120574k1205901015840

( (1 + eo) k112059010158401120574w120574v1205901015840 ) 1H21

( 1205901015840m12059010158401 )1minus120574v1205901015840minus120574k1205901015840(23)

give rise to the dimensionless groups

120587I = 120591o1205901015840k112059010158401 (1 + eo)120574w120574v1205901015840H21 ( 1205901015840m12059010158401 )1minus120574k1205901015840minus120574v1205901015840 120587II = 120574k1205901015840 ( 12059010158402 minus 120590101584011205901015840m )

(24)

Thus the solutions for 120591o1205901015840 and U1205901015840 are given by

120591o1205901015840 = ( 120574v1205901015840120574wH21(1 + eo) k112059010158401) ( 1205901015840m12059010158401 )120574v1205901015840+120574k1205901015840minus1

sdot Ψ120591 (120574k1205901015840 ( 12059010158402 minus 120590101584011205901015840m ))(25)

U1205901015840 = ΨU [ t120591o1205901015840 120574k1205901015840 ( 12059010158402 minus 120590101584011205901015840m )] (26)

313 J-B Model Assuming 1+e Is Constant (A Simplificationof the Former) Based on the above it is easy to simplify thehypothesis to 1+e being constant With (120590101584012059010158401)minus120574v1205901015840 = 1 theconsolidation equation reduces to

1205971205901015840120597t = (1 + eo) 1205901015840oko120574w120574v1205901015840 ( 12059010158401205901015840o )1minus120574k1205901015840 [minus 120574k12059010158401205901015840 ( 1205971205901015840120597z )2

+ 12059721205901015840120597z2 ](27)

an equation dependent on the initial void ratio Using (13)and averaging this equation provides three-dimensionalcoefficients

1120591o1205901015840 ( (1 + eo) k112059010158401120574w120574v1205901015840 ) 120574k1205901015840

H21( 12059010158402 minus 120590101584011205901015840m ) ( 1205901015840m12059010158401 )1minus120574k1205901015840

( (1 + eo) k112059010158401120574w120574v1205901015840 ) 1H21

( 1205901015840m12059010158401 )1minus120574v1205901015840minus120574k1205901015840(28)

and two dimensionless independent groups

120587I = 120591o1205901015840k112059010158401 (1 + eo)120574w120574v1205901015840H21 ( 1205901015840m12059010158401 )1minus120574k1205901015840 120587II = 120574k1205901015840 ( 12059010158402 minus 120590101584011205901015840m )

(29)

Thus the solutions for 120591o1205901015840 and U1205901015840 are given by

120591o1205901015840 = ( 120574v1205901015840120574wH21(1 + eo) k112059010158401) ( 1205901015840m12059010158401 )120574k1205901015840minus1

sdot Ψ120591 (120574k1205901015840 ( 12059010158402 minus 120590101584011205901015840m ))(30)

U1205901015840 = ΨU [ t120591o1205901015840 120574k1205901015840 ( 12059010158402 minus 120590101584011205901015840m )] (31)

314 The Less Restrictive Model with Both 1+e and dz NotConstant For this model 1+e and dz not constants it isenough to consider (22) plus the condition dz not constantor dz = dzo(HH1) So after mathematical manipulation thegoverning equation writes as

1205971205901015840120597t= (1 + eo) 1205901015840oko120574w120574v1205901015840 ( 12059010158401205901015840o )1+120574v1205901015840minus120574k1205901015840 [minus 120574k12059010158401205901015840 ( 1205971205901015840120597z )2

+ 12059721205901015840120597z2 ](32)

The new coefficients

1120591o1205901015840 ( (1 + eo) k112059010158401120574w120574v1205901015840 ) 120574k1205901015840

H21( 12059010158402 minus 120590101584011205901015840m ) ( 1205901015840m12059010158401 )1+120574v1205901015840minus120574k1205901015840

( (1 + eo) k112059010158401120574w120574v1205901015840 ) 1H21

( 1205901015840m12059010158401 )1+120574v1205901015840minus120574k1205901015840(33)

provide the groups

120587I = 120591o1205901015840k112059010158401 (1 + eo)120574w120574v1205901015840H21 ( 1205901015840m12059010158401 )1minus120574k1205901015840+120574v1205901015840 120587II = 120574k1205901015840 ( 12059010158402 minus 120590101584011205901015840m )

(34)

6 Mathematical Problems in Engineering

and the solutions

120591o1205901015840 = ( 120574v1205901015840120574wH21(1 + eo) k112059010158401) ( 1205901015840m12059010158401 )120574k1205901015840minus120574v1205901015840minus1

sdot Ψ120591 (120574k1205901015840 ( 12059010158402 minus 120590101584011205901015840m ))(35)

U1205901015840 = ΨU [ t120591o1205901015840 120574k1205901015840 ( 12059010158402 minus 120590101584011205901015840m )] (36)

32 Nondimensionalization of the J-B Model in Terms ofSettlements Three cases defined by the hypotheses (1) 1+e =constant and dz constant (2) 1+e and dz constants and (3) 1+eand dz not being constant are considered For this study let usintroduce a new variable directly related with the settlementwith a clear physical meaning ldquo120577 = e - eordquo a kind of localdegree of settlement or differential void index

321 1+e =Constant and dz Constant For 1+e =constantusing the variable 120577 and the dependence V-1205901015840 it is straight-forward to write

120577 = e minus eo = (1 + eo) [( 120590101584012059010158401)minus120574v1205901015840 minus 1] (37)

Making use of the derivatives of (37)

( 1205971205901015840120597z )2 = 120590101584012(1 + eo)2 120574v12059010158402 ( 120590101584012059010158401)2120574v1205901015840+2 ( 120597120577120597z)2

12059721205901015840120597z2 = minus 12059010158401(1 + eo) 120574v1205901015840 ( 120590101584012059010158401)120574v1205901015840+1 1205972120577120597z2

+ 12059010158401 (120574v1205901015840 + 1)(1 + eo)2 120574v12059010158402 ( 120590101584012059010158401)2120574v1205901015840+1 ( 120597120577120597z)2

(38)

and the assumptions of the odometer test after cumbersomemathematical steps the nonlinear dimensional consolidation(19) is written in the form

120597120577120597t = ko1205901015840o (120574k1205901015840 minus 120574v1205901015840 minus 1)120574w120574v12059010158402 ( 12059010158401205901015840o )1minus120574k1205901015840 ( 120597120577120597z)2

+ ko1205901015840o (1 + eo)120574w120574v1205901015840 ( 12059010158401205901015840o )1minus120574v1205901015840minus120574k1205901015840 1205972120577120597z2(39)

Introducing the variables

(120577)1015840 = 120577 minus 12057711205772 minus 1205771 = 1205771 minus 1205771205771 z1015840 = z

H1

t1015840 = t120591os(40)

into the former equation with 120591os a reference chosen as thetime required for the settlement to reach a high percentageof its change along the whole domain its dimensionless formprovides three coefficients

1120591os ko1205901015840o (120574k1205901015840 minus 120574v1205901015840 minus 1)120574w120574v12059010158402 ( 1205901015840m12059010158401 )1minus120574k1205901015840 (ef minus eo)

H21

ko1205901015840o (1 + eo)120574w120574v1205901015840 ( 1205901015840m12059010158401 )1minus120574v1205901015840minus120574k1205901015840 1H21

(41)

Dividing by the last the resulting dimensionless groups are

120587I = 120591osko1205901015840o (1 + eo)120574w120574v1205901015840H21 ( 1205901015840m12059010158401 )1minus120574v1205901015840minus120574k1205901015840

120587II = (120574k1205901015840 minus 120574v1205901015840 minus 1)120574v1205901015840 (ef minus eo)(1 + eo) ( 1205901015840m12059010158401 )120574v1205901015840(42)

Making use of the constitutive dependence dVV =minus120574v1205901015840(d12059010158401205901015840) is possible to write ΔHH = minus120574v1205901015840(Δ12059010158401205901015840)and since (ef minus eo)(1 + eo) = ΔHHo 120587II finally takes theform

120587II = (1 minus 120574k1205901015840 + 120574v1205901015840) 12059010158402 minus 120590101584011205901015840m (43)

The solution for 120591os is120591os asymp 120574w120574v1205901015840H21

ko1205901015840o (1 + eo) ( 1205901015840m12059010158401 )120574v1205901015840+120574k1205901015840minus1Ψ120591 [(1 minus 120574k1205901015840+ 120574v1205901015840) 12059010158402 minus 120590101584011205901015840m ]

(44)

As for the average degree of consolidation the followingdependence arises

Us = ΨU [ t120591os (1 minus 120574k1205901015840 + 120574v1205901015840) 12059010158402 minus 120590101584011205901015840m ] (45)

322 1+e and dz Constants The hypothesis 1+e=constant isa simplification of the former case Proceeding in the sameway the resulting coefficients

1120591os ko1205901015840o (1 + 120574v1205901015840 minus 120574k1205901015840)120574w120574v12059010158402 ( 1205901015840m12059010158401 )1minus120574k1205901015840+120574v1205901015840 (ef minus eo)

H21

ko1205901015840o (1 + eo)120574w120574v1205901015840 ( 1205901015840m12059010158401 )1minus120574k1205901015840 1H21

(46)

Mathematical Problems in Engineering 7

provide the dimensionless groups

120587I = 120591osko1205901015840o (1 + eo)120574w120574v1205901015840H21 ( 1205901015840m12059010158401 )1minus120574k1205901015840

120587II = (1 minus 120574k1205901015840 + 120574v1205901015840) 12059010158402 minus 120590101584011205901015840m(47)

The solutions for 120591os and Us are

120591os asymp 120574w120574v1205901015840H21ko1205901015840o (1 + eo) ( 1205901015840m12059010158401 )120574k1205901015840minus1Ψ120591 [(1 minus 120574k1205901015840

+ 120574v1205901015840) 12059010158402 minus 120590101584011205901015840m ](48)

Us = ΨU [ t120591os (1 minus 120574k1205901015840 + 120574v1205901015840) 12059010158402 minus 120590101584011205901015840m ] (49)

The solutions for the original J-Bmodel are also (48) and (49)with the simplification of eo=0 The author however doesnot talk about the characteristic time and obtains the averagedegree of consolidation given by (7) in which this variabledepends on three groups (1minus120574k1205901015840)120574v1205901015840 H2H1 and tcv2H21undoubtedly a less precise solution

323 1+e and dz Not Being Constant When 1+e and dzare not constants substituting dz = dzo(HH1) in (39)after cumbersome mathematical manipulation the resultingcoefficients are 1120591os

ko1205901015840o (120574k1205901015840 minus 120574v1205901015840 minus 1)120574w120574v12059010158402 ( 1205901015840m12059010158401 )1+2120574v1205901015840minus120574k1205901015840 (ef minus eo)H21

ko1205901015840o (1 + eo)120574w120574v1205901015840 ( 1205901015840m12059010158401 )1+120574v1205901015840minus120574k1205901015840 1

H21

(50)

Dividing by the last the resulting dimensionless groups are

120587I = 120591osko1205901015840o (1 + eo)120574w120574v1205901015840H21 ( 1205901015840m12059010158401 )1+120574v1205901015840minus120574k1205901015840

120587II = (1 minus 120574k1205901015840 + 120574v1205901015840) 12059010158402 minus 120590101584011205901015840m(51)

The solutions for 120591os and Us are

120591os asymp 120574w120574v1205901015840H21ko1205901015840o (1 + eo) ( 1205901015840m12059010158401 )+120574k1205901015840minus120574v1205901015840minus1Ψ120591 [(1 minus 120574k1205901015840

+ 120574v1205901015840) 12059010158402 minus 120590101584011205901015840m ](52)

Us = ΨU [ t120591os (1 minus 120574k1205901015840 + 120574v1205901015840) 12059010158402 minus 120590101584011205901015840m ] (53)

Table 1 summarizes the results derived in this section

4 Verification of the Results byNumerical Simulations

This section is devoted to checking the solutions given inTable 1 To shorten the exposure we restrict the tests forthe more general model (1+e and dz not constants) andfor the unknowns 120591os and Us since they are the ones ofgreater interest in civil engineering Eight sets of simulationsarranged in three blocks have been run Table 2 In each onesome of the soil parameters or initial values of the problemhave been changed to give the same or different values asrequired to the dimensionless group 120587II in the search for thesame or different solutions of 120587I Changes in the values of theindividual parameters are sufficient to cover all real scenarios

Firstly a reference set (set 1) is established to which all theother sets can be referred and which permits them to be com-pared with each other The physical and geometrical charac-teristics that change are 120574v1205901015840 120574k1205901015840 eo H1 (m) 1205901015840o (Nm2)1205901015840f (Nm2) and ko (myear) The values of =(1205901015840f1205901015840o) H2(m) kf (myear) and 120587II are derived from them while 1205871is obtained once the characteristic time 120591os is read from thesimulation The criterion for the choice of 120591os is the timerequired by the soil to reach 90 of the total settlement

It is worth mentioning here the emergence of 1205901015840m anaveraged value of the effective pressure in the two dimen-sionless groups Undoubtedly the existence of 1205901015840m in themonomials can be explained not only by the nonlinearityof the problem but also by the kind of dependences in theconstitutive relations (there are other nonlinear problems inwhich averaged variables do not emerge [18]) How do wechoose 1205901015840m We could take the mean value 1205901015840m=(12059010158401+12059010158402)2or even 1205901015840m=12059010158401 or 1205901015840m=12059010158402 but expecting that the expres-sion 120587I = 120595(120587II) is slightly dependent on that choice (we willreturn to this question later) In this paper we have taken theapproximation 1205901015840m=(12059010158402 + 12059010158401)2

In block I the parameters that change in each set areeo H1 ko and 1205901015840o while maintaining the ratio =(1205901015840f1205901015840o) thus both 120587II and factor (1205901015840m1205901015840o)1minus120574k1205901015840+120574v1205901015840 of 120587I takethe same value in the four sets of the block As a resultwe expect a characteristic time proportional to the ratio120574w120574v1205901015840H21ko1205901015840o(1 + eo) and a same value of 120587I in all casesaccording to the solution 120587I = 120595(120587II) Indeed the resultsare consistent with expectations Block II contains an onlycase for which 120587II varies in comparison to block I as a result120587I also changes as expected Finally block III contains threecases with the same value of 120587II of block I but with differentvalues of the parameters 120574v1205901015840 120574k1205901015840 and the ratio =(1205901015840f 1205901015840o)The parameters of set 6 have been chosen in such a waythat the values 120574k1205901015840 and 120574v1205901015840 are compensated to maintainconstant (1 minus 120574k1205901015840 + 120574v1205901015840) we expect 120587I to be also unchangedIn effect simulation provides 120587I = 07139 a value quite closeto that of set 1 (07131) The negligible differences are due nodoubt to the different evolution of the effective pressure alongthe process caused by the different values of the coefficients120574k1205901015840 and 120574v1205901015840 Note that as 120574v1205901015840 duplicates the denominator ofthe factor 120591osko1205901015840o(1 + eo)120574w120574v1205901015840H21 the characteristic timehas to double too as indeed happens The last two sets studythe influence of a parameter whose change forces the value

8 Mathematical Problems in Engineering

Table1Dim

ensio

nlessg

roup

sthatcharacterizethe

originalandextend

edJ-Bmod

els

Juarez-Badillo

original

Juarez-Badillo

Juarez-Badillo

Juarez-Badillo

1+e=

constant

1+e

=consta

nt1+e

=consta

ntdz

constant

dzconstant

dzvaria

ble

120587 I120591 ok o

1205901015840 o120574 w120574 v1205901015840H2 1

(1205901015840 m 1205901015840 o)1minus120574k1205901015840

120591 ok o1205901015840 o(

1+eo)

120574 w120574 v1205901015840H2 1

(1205901015840 m 1205901015840 o)1minus120574k1205901015840

120591 ok o1205901015840 o(

1+eo)

120574 w120574 v1205901015840H2 1

(1205901015840 m 1205901015840 o)1minus120574k1205901015840minus120574 v1205901015840

120591 ok o1205901015840 o(

1+eo)

120574 w120574 v1205901015840H2 1

(1205901015840 m 1205901015840 o)1minus120574k1205901015840+120574 v1205901015840

120591 o1205901015840or

120591 os120587 II

120574 k1205901015840(1205901015840 2minus

1205901015840 1 1205901015840 m)for

pressure

(1minus120574 k1205901015840

+120574 v1205901015840)(1205901015840 2minus

1205901015840 1 1205901015840 m)for

settlem

ent

Mathematical Problems in Engineering 9

0010203040506070809

1

0 05 1 15 2 25 3time

Set 1Set 2Set 3Set 4

Set 5Set 6Set 7Set 8

Aver

age d

egre

e of c

onso

lidat

ion

Figure 1 Average degree of consolidation versus time (eight sets)

0010203040506070809

1

0 05 1 15 2 25 3

Set 1Set 2Set 3Set 4

Set 5Set 6Set 7Set 8

Aver

age d

egre

e of c

onso

lidat

ion

tos

Figure 2 Average degree of consolidation versus dimensionlesstime (eight sets)

(1 minus 120574k1205901015840 + 120574v1205901015840) to be also changed in order to maintain 120587IIconstant We expect that 120587I does not change as nearly occursThe small differences are again due to the existence of 1205901015840mwithin the monomials Note however that as diminishesthe influence of 1205901015840m in the results also diminishes since thevalues of 12059010158401 and 12059010158402 are closer

For a better understanding of the results Figures 1 and2 are presented In the first Us as a function of time isshown for the eight sets in Table 2 It is clear that 120591oscovers a wide range of values from 02283 (set 2) to 27968(set 3) the last more than ten times the first The secondfigure graphs Us in front of the dimensionless time t120591os Asshown the eight previous curves nearly converge to a single(or universal) graph having a common point at (Us=09t120591os=1) This means that monomial 120587II for the range ofvalues assigned to the parameters in contrast to 120587I scarcelyinfluences the value of the average degree of consolidationThis is not in contradiction with the results derived bynondimensionalization

0010203040506070809

1

0 05 1 15 2 25 3 35 4time

U (J-B)U (1+e cst)U (1+e not cst)U (1+e amp dz not cst)

Us (J-B)Us (1+e cst)Us (1+e not cst)Us (1+e amp dz not cst)

Aver

age d

egre

e of p

ress

ure d

issip

atio

nAv

erag

e deg

ree o

f con

solid

atio

n

Figure 3 Average degree of consolidation and pressure dissipation(set 6)

Although in this sectionwe have only checked the expres-sions of Us and 120591os for the most general model we considerit interesting to present both Us and U1205901015840 for the rest of themodels of Table 1 and for set 6 as a typical case Figure 3Theseresults emphasize the importance of using the less restrictive(more general) model in relation to the others in particularwith the original model of J-B in which the influence ofeo is despised Note that the smaller of the characteristictimes (related to settlement) corresponds to the more generalmodel (black and bold line) so the use of the others althoughon the safety side provides characteristic times oversizedfar from an optimal solution for the engineer For examplefor the case of Figure 3 in which eo=1 the error of J-Boriginal model (purple and bold line) in comparison with theextended model (black and bold line) for Us=90 is around150 This error increases appreciably as eo gets larger

5 Universal Curves

These easy to use and universal abacuses obtained by numeri-cal simulations allow engineers to read the characteristic time(120591os) and the average degree of consolidation (Us) in termsof the dimensionless groups given in Table 1 Only the mostgeneral (less restrictive) J-B model 1+e and dz not constantis presented

To obtain the characteristic time curves a large numberof simulations have been carried out studying separately theinfluence of the parameters 120574v1205901015840 and 120574k1205901015840 whose valuescover most real soils ranges from 101 to 8 120574v1205901015840 from 005to 03 and 120574k1205901015840 from 01 to 17 From these tests it has beendeduced as expected from the results of the previous sectionthat the relatively more influential parameter is while 120574v1205901015840and 120574k1205901015840 do not produce significant changes in the values ofthe characteristic timeThus the first universal group of linesshows the dependence of 120587I versus 120587II for different valuesof (with 120574v1205901015840=01) Figure 4 As the range of values of 120587IIdepends on according to (43) the range of values of eachline is different As shown in the figure there exists a universal

10 Mathematical Problems in Engineering

Table 2 Values of the parameters and dimensionless groups of simulations

Block Set 120574v1205901015840 120574k1205901015840 eoH1 ko 1205901015840o 1205901015840f H2 kf 120587II

120591os 120587I(m) (myr) (Nm2) (Nm2) 1205901015840f 1205901015840o (m) (myr) (yr)

I

1 01 0500 10 100 002 30000 60000 2 0933 00141 040 04566 071312 01 0500 10 100 002 60000 120000 2 0933 00141 040 02283 071313 01 0500 10 175 001 30000 60000 2 1633 00071 040 27968 071314 01 0500 13 142 004 22000 44000 2 1325 00283 040 05459 07131

II 5 01 1500 10 100 002 30000 90000 3 0896 00038 -040 10597 09834

III6 02 0600 10 100 002 30000 60000 2 0871 00132 040 09142 071397 01 0767 10 100 002 30000 120000 4 0871 00069 040 04360 072468 02 0920 10 100 002 30000 180000 6 0699 00038 040 08409 07312

line for each value of all converging in the central zone forvalues of 120587II around 0 This set of straight lines can be fittedby 120587I=m120587II+n being n=0846 and m given in Figure 5 as afunction of The curve that best fits the points in Figure 5 isgiven by

m = minus0235e(minus0868) minus 0306e(minus0012) (54)

Choosing the longest line corresponding to =8 as the onlyrepresentative line of the problem themaximumerrors in thereading of 120587I (and 120591os) are 38

In short the use of these curves is as follows from the dataof the problem Figure 4 allows us to read 120587I from 120587II and and from (51) the characteristic time 120591os is derived

As for the average degree of consolidation the universalcurves for 96 different scenarios of 120587II are shown in Figure 6It is appreciated that these curves are very close to each othershowing again that the Us only depends on the dimensionlesstime In the worst case the errors produced by changes invalue of the 120587II group are less than 28when using the curveof Figure 7

6 Case Study

Below is a practical application of the universal curvespresented in this paper For this purpose we take as a basisfor our study the data obtained experimentally byAbbasi et al[26] for different samples of clay whosemoisture in the liquidlimit is equal to 42 As can be seen (Table 3) it is a series ofdiscrete (tabulated) values that relate the void ratio with theeffective pressure and the hydraulic conductivity

We will address a real case of consolidation in which wehave a layer of soil 2meters thick (Ho) with an initial effectivepressure (1205901015840o) of 28 kNm2 on which a load of 31 kNm2 isapplied In this way the effective pressure at the end of theconsolidation process (1205901015840f ) will have the value of 59 kNm2From the tabulated data (e-1205901015840) it follows that eo = 105 andef = 095 and since the factor 1+e is proportional to the soilvolume from (9) we have 120574v1205901015840 = 0067

To obtain 120574k1205901015840 we first calculate the value of 120574kv from thetabulated data (e-k) to then obtain 120574k1205901015840 from (10) With thiswe have for the load step considered ko = 0035 myr 120574kv =8108 and 120574k1205901015840 = 0543

To solve the problem we will use the universal curvespresented in the previous section corresponding to the most

Table 3 Experimental values of e 1205901015840 and k obtained by Abbasi et al[26]

e 1205901015840o (Nm2)110 14000105 28000095 59000085 120000075 220000065 450000060 925000e k (myr)175 0683160 0315120 0084085 0021065 0011

general J-B model (analyzed in terms of settlement) Thusfrom the values of 1205901015840o 1205901015840f 120574v1205901015840 and 120574k1205901015840 by means of (51) itis obtained that 120587II = 0373 (with 1205901015840m = 435 kNm2) Then120587I is read from Figure 4 where for = 21 we have 120587I =072 An identical result is obtained if we use the proposedfitted equation 120587I=m120587II+n where n is a constant of value0846 andm is obtained from (54) or Figure 5 Finally knownthe values of Ho ko and eo from (51) we get 120591os = 0748years Likewise once the characteristic time of the problem(in terms of settlement) is known from the universal curve ofFigure 7 we easily obtain the evolution of the average degreeof consolidation along the whole process Figure 8

These solutions have been compared with those thatresult from the numerical simulation of the most generalJ-B model for which a value of 120591os = 075 years has beenobtained showing clearly that the relative error committedwhen applying the universal solutions is negligible (Table 4)The simulation of the problem has also been carried out fora model where 1+e and dz are also not constant but in whichthe constitutive relations of potential type have been replacedby pairs of data in tabulated form obtaining in this case 120591os= 0817 years The relative error which this time increasesto 82 finds its explanation in the strong nonlinearity ofthe problem which will always bring differences between the

Mathematical Problems in Engineering 11

040045050055060065070075080085090095100105110

minus07 minus06 minus05 minus04 minus03 minus02 minus01 0 01 02 03 04 05 06 07 08 09 1 11 12 13 14 15

1

2

ᴪ=8ᴪ=6ᴪ=4

ᴪ=2ᴪ=15ᴪ=101

Figure 4 120587I as a function of 120587II with as parameter

1 2 3 4 5 6 7 8minus042

minus04

minus038

minus036

minus034

minus032

minus03

minus028

minus026

m

datafitted curve

Figure 5 m as a function of

solutions provided by numerical models based on differentconstitutive relations (potential and tabulated in this case)Even so this value is totally acceptable within this field ofengineering demonstrating the strength and versatility of theuniversal solutions provided

7 Discussion

This work derives the dimensionless groups that rule thenonlinear soil consolidation process based on the original J-Bmodel following a formal nondimensionalization procedurethat starts from the governing equation and does not needany other assumption or mathematical manipulation Inaddition more real and precise extended J-B models whichassume both the void ratio and the thickness of the finitevolume element of the soil to be not constant have been

0010203040506070809

1

0 05 1 15 2 25 3

96 scenariosᴪ = [101 15 2 4 6 8]

Aver

age d

egre

e of c

onso

lidat

ion

tos

= [005 01 015 02]v= [02 05 1 15]k

Figure 6 Average degree of consolidation versus dimensionlesstime for different scenarios

0010203040506070809

1

0 05 1 15 2 25 3

Aver

age d

egre

e of c

onso

lidat

ion

tos

Figure 7 Proposed solution for the average degree of consolidation

nondimensionalized and their dimensionless groups derivedThe assumption of constitutive dependences (e-1205901015840 e-k and k-1205901015840) of potential type in these models is interesting since they

12 Mathematical Problems in Engineering

Table 4 Case study solution for the characteristic time 120591os Comparison with numerical models

Universal curves J-B most general model Tabulated model120591os (yr) 120591os (yr) Rel error 120591os (yr) Rel error 0748 0750 026 0817 820

0010203040506070809

1

0 025 05 075 1 125 15 175 2 225t ( years)

Aver

age d

egre

e of c

onso

lidat

ion

Figure 8 Case study solution for the average degree of consolida-tion Us from the universal curves

can be applied to highly compressible soils for which the e-log(1205901015840) relation moves away from being a straight line andin which negative values of the void ratio could be obtainedunder the application of high load ratios (1205901015840f 1205901015840o) [8]

The study has been developed both in terms of pres-sure dissipation and in terms of settlement resulting thatthe dimensionless groups derived from each model are ingeneral different The extension to more complex modelswhich implies the emergence of new parameters or equationswithin themathematical model does not necessarily give riseto additional dimensionless groups as might be expectedbut to new expressions of the groups or new regroupingof the parameters involved Thus the elimination of thehypothesis ldquodz constantrdquo in the most general consolidationmodel has not given rise to any new group both in pressureand in settlement (Table 1) but it has only modified oneof the groups obtained in the original model All thisrepresents an important advance in relation to the modelsand studies carried out by other authors and the attemptsof characterization found in the scientific literature [9ndash11]which try to lead to easy-to-use solutions for engineers Inno case had dimensionless groups been obtained as preciseas those presented in this work nor presented universalcurves of the characteristic time and the average degree ofconsolidation Authors such asConte andTroncone [11] applymore general load conditions (time-dependent loading) forwhich the dimensionless groups deduced in this work couldbe applicable for each load step

As regards time to make it dimensionless an unknownreference (120591o) is introduced in the form mentioned by Scott[27] and used in many other problems with asymptoticsolutions [28]Theprecise definition of 120591o necessarily involvesa criterion in relation to the percentage of average pressuredecrease (or settlement) associated with the end of theconsolidation processTheorder ofmagnitude of 120591o is derived

after nondimensionalization making 120591o only appear in oneof the independent dimensionless groups In this way theaverage degree of consolidation (in terms of pressure orsettlement) depends on the ldquotimecharacteristic timerdquo ratioin addition to the dimensionless groups (without unknowns)that arise

In short the procedure has led to a new very precisesolution that involves two monomials one (120587I) that containsthe characteristic time (120591o) and another (120587II) that is amathematical function of the soil parameters In this way120591o only depends on 120587II through the function 120587I = 120595(120587II)while U depends on 120587I and 120587II through the expression U= 120595(120587I 120587II) The results obtained have been verified forthe most general and complex consolidation model throughthe numerical simulation of a large number of scenariosin which the values of the dimensionless groups and theindividual parameters of the problem have been changing toconvenience The high errors produced by the original J-Bmodel with respect to the most general justify the use of thelatter in the search for more precise solutions

In addition the emergence of mean values of the effectivepressure when averaging the governing equation in orderto assign an order of magnitude unit to the dimensionlessvariables and their derivatives is an added difficulty to obtainthe dimensionless groups This is undoubtedly due to thenonlinearities (constitutive relations) involved in the consol-idation process This average effective pressure of referencehas been assumed as the semisum of its initial and finalvalues although other criteria could also be chosen Sincetwo scenarios can have equal120587II with the same effectivemeanpressure but different values of the parameters 120574v1205901015840 and 120574k1205901015840 we expect some minor deviations between the results of bothscenarios as indeed it has been checked Thus for the wholerange of parameters values in typical soils the errors due tothe above cause are negligible in the field of soil mechanicsOn the other hand when the value of 120587II does not changebut the mean effective pressure value does the errors in thesolutions of 120591os and Us are slightly larger although withinranges below 4

The brooch for all this work is the presentation ofuniversal curves for the most general model which allowfor the usual range of variation of the soil properties toobtain in a simple way the value of the characteristic timeof consolidation and the evolution of the average degreeof settlement throughout the entire process Thus startingfrom the nonlinear parameters of the soil (120574v1205901015840 and 120574k1205901015840)and the load ratio (1205901015840f 1205901015840o) the value of the monomial 120587IIis obtained from which the value of 120587I is directly deductedthanks to the universal curve 120587I = 120595(120587II) provided Finally120591os is easily obtained from the expression of 120587I On the otherhand the curve that represents the Us evolution along theprocess has turned out to be (practically) unique when it is

Mathematical Problems in Engineering 13

represented as a function of the dimensionless time in theform t120591os regardless of the soil parameters and the loadsapplied All this supposes a clear advantage when solvingthe consolidation problem also showing the strength of thenondimensionalization method presented

Finally the case study addressed has demonstrated thesimplicity and reliability of the universal curves proposedFrom a series of tabulated data (e-1205901015840 and e-k) the parametersthat govern the nonlinear consolidation problem for themostextended J-B model have been adjusted with which thevalue of the monomial 120587II has been derived With this andusing the universal curves the solutions for the characteristictime 120591os and the average degree of consolidation Us havebeen obtained These results have been satisfactorily verifiedwith the solutions provided both by the numerical extendedmodel of J-B presented here and by another that substitutesthe constitutive dependences of potential type by data intabulated form

8 Conclusions

The application of the nondimensionalization technique hasallowed us to deduce the dimensionless groups that rule thenonlinear consolidation problem in which the constitutivedependencies e-1205901015840 and e-k have the form of potential func-tions This type of relations approximates the soil behaviorbetter than the logarithmic functions e-log(1205901015840) and e-log(k)particularly in very compressible soils and under high loadratios (1205901015840f 1205901015840o) in which inconsistent negative values of thevoid ratio may appear

Several scenarios have been addressed from the originalmodel of Juarez-Badillo to extended models based on thesame constitutive dependences but eliminating one or moreclassic (and restrictive) hypotheses in this process In allcases only two dimensionless groups emerge one of themconsisting of a grouping of the soil parameters and the othercontaining the unknown characteristic time Based on theseresults universal curves have been depicted the first to deter-mine the characteristic time of the process and the secondthat represents the evolution of average degree of settlement

The solutions have been verified by numerical simula-tions changing the values of the soil parameters and thedimensionless groups conveniently Finally the reliability andeffectiveness of the proposed solutions have been shownthrough a case study

Abbreviations

cv Consolidation coefficient (m2s or m2yr)cv1 Initial consolidation coefficient (m2s or

m2yr)cv2 Final consolidation coefficient (m2s or

m2yr)119889z Element of differential length in thedirection of the spatial coordinate z (m)119889zo Element of differential length in thedirection of the spatial coordinate z at theinitial time (m)

e Void ratio (dimensionless)

eo Initial void ratio (dimensionless)ef Final void ratio (dimensionless)H Soil thickness up to the impervious

boundary or drainage length (m)H1Ho Initial soil thickness (m)H2 Final soil thickness (m)k Hydraulic conductivity (or permeability)

(ms or myr)k1 ko Initial hydraulic conductivity (ms or

myr)kf Final hydraulic conductivity (ms or myr)mv Coefficient of volumetric compressibility

(m2N)mv1 Initial coefficient of volumetric

compressibility (m2N)t Time Independent variable (s or yr)T Juarez-Badillorsquos time factor

(dimensionless)u Excess pore pressure (Nm2)U Average degree of consolidation or

pressure dissipation (dimensionless)Us Average degree of consolidation

(dimensionless)U1205901015840 Average degree of pressure dissipation

(dimensionless)V Soil volume (m3)V1Vo Initial soil volume (m3)z Vertical spatial coordinate (m)120574v Nonlinear coefficient of compressibility of

Juarez-Badillo (dimensionless)120574kv Nonlinear coefficient of change ofpermeability with volume (dimensionless)120574k1205901015840 Nonlinear coefficient of change ofpermeability with effective pressure(dimensionless)120574v1205901015840 Nonlinear coefficient of compressibility orcoefficient of volumetric contraction(dimensionless)120574w Specific weight of water (Nm3)

Loading factor Ratio between the finaland initial effective pressure(dimensionless)120581 Nonlinear coefficient of change ofpermeability with volume ofJuarez-Badillo (dimensionless)120582 Parameter of Juarez-Badillo 120582 = 1 - 120574v120581(dimensionless)

V Auxiliary variable of Juarez-Badillo(dimensionless)120587i Dimensionless group or number(dimensionless)1205901015840 Effective pressure of the soil (Nm2)12059010158401 1205901015840o Initial effective pressure (Nm2)12059010158402 1205901015840f Final effective pressure (Nm2)1205901015840m Mean value of the effective pressure(Nm2)120591o Characteristic time (s or yr)

14 Mathematical Problems in Engineering

120591os Characteristic time which takes thesettlement to reach approximately its finalvalue (s or yr)120591o1205901015840 Characteristic time which takes the excesspore pressure to dissipate untilapproximately the value of zero (s or yr)Ψ Arbitrary mathematical function120577 Differential void ratio (dimensionless)1205771 1205772 Initial and final differential void ratio(dimensionless)

Subscripts

I II Denote different numbers

Superscripts

1015840 Denote dimensionless magnitude

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] K Terzaghi ldquoDie Berechnung der Durchlassigkeitszifferdestones aus dem verlauf der hydrodynamischen spanunngser-scheinnungenrdquo Technical report II a 132 N 34 125-138Akademie derWissenschaften inWien SitzungsberichteMath-naturwiss Klasse Abt 1923

[2] R T Walker ldquoVertical drain consolidation analysis in one twoand three dimensionsrdquo Computers amp Geosciences vol 38 no 8pp 1069ndash1077 2011

[3] G Y Zheng P Li and C Y Zhao ldquoAnalysis of non-linearconsolidation of soft clay by differential quadrature methodrdquoApplied Clay Science vol 79 pp 2ndash7 2013

[4] X Y Geng C J Xu and Y Q Cai ldquoNon-linear consolidationanalysis of soil with variable compressibility and permeabilityunder cyclic loadingsrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 30 no 8 pp 803ndash8212006

[5] M Lu S Wang S W Sloan B Indraratna and K Xie ldquoNon-linear radial consolidation of vertical drains under a generaltime-variable loadingrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 39 no 1 pp 51ndash622015

[6] HWu L HuW Qi and QWen ldquoAnalytical Solution for Elec-troosmotic Consolidation Considering Nonlinear Variation ofSoil Parametersrdquo International Journal of Geomechanics vol 17no 5 Article ID 06016032 2017

[7] S J Brandenberg ldquoiConsoljs JavaScript Implicit Finite-Differ-ence Code for Nonlinear Consolidation and Secondary Com-pressionrdquo International Journal of Geomechanics vol 17 no 6Article ID 04016149 2017

[8] R Butterfield ldquoA natural compression law for soils (an advanceon e-log prsquo)rdquo Geotechnique vol 29 no 4 pp 469ndash480 1979

[9] R Lancellotta and L Preziosi ldquoA general nonlinear mathemati-calmodel for soil consolidation problemsrdquo International Journalof Engineering Science vol 35 no 10-11 pp 1045ndash1063 1997

[10] Y-C Zhuang K-H Xie and X-B Li ldquoNonlinear analysis ofconsolidation with variable compressibility and permeabilityrdquoJournal of Zhejiang University (Engineering Science Edition) vol6 no 3 pp 181ndash187 2005

[11] E Conte andA Troncone ldquoNonlinear consolidation of thin lay-ers subjected to time-dependent loadingrdquo Canadian Geotechni-cal Journal vol 44 no 6 pp 717ndash725 2007

[12] R Yong and F Townsend ldquoGeneral Consolidation Theory forClaysrdquo Soil Mechanics Series Report No 8 Graduate School ofEngineering National University of Mexico 1983

[13] J C Gibbings Dimensional Analysis Springer-Verlag LondonUK 2011

[14] E Buckingham ldquoOn physically similar systems Illustrations ofthe use of dimensional equationsrdquo Physical Review A AtomicMolecular and Optical Physics vol 4 no 4 pp 345ndash376 1914

[15] R Seshadri and T Y Na ldquoA Survey ofMethods for DeterminingSimilarity Transformationsrdquo inGroup Invariance in EngineeringBoundary Value Problems pp 35ndash61 Springer New York NYUSA 1985

[16] V D Zimparov and V M Petkov ldquoApplication of discriminatedanalysis to low Reynolds number swirl flows in circular tubeswith twisted-tape inserts Pressure drop correlationsrdquo Interna-tional Review of Chemical Engineering vol 1 no 4 pp 346ndash3562009

[17] M Capobianchi and A Aziz ldquoA scale analysis for natural con-vective flows over vertical surfacesrdquo International Journal ofThermal Sciences vol 54 pp 82ndash88 2012

[18] M Conesa J F Sanchez Perez I Alhama and F AlhamaldquoOn the nondimensionalization of coupled nonlinear ordinarydifferential equationsrdquo Nonlinear Dynamics vol 84 no 1 pp91ndash105 2016

[19] I A Manteca G Garcıa-Ros and F Alhama ldquoUniversal solu-tion for the characteristic time and the degree of settlement innonlinear soil consolidation scenarios A deduction based onnondimensionalizationrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 57 pp 186ndash201 2018

[20] E H Davis and G P Raymond ldquoA non-linear theory ofconsolidationrdquo Geotechnique vol 15 no 2 pp 161ndash173 1965

[21] P Cornetti and M Battaglio ldquoNonlinear consolidation of soilmodeling and solution techniquesrdquo Mathematical and Com-puter Modelling vol 20 no 7 pp 1ndash12 1994

[22] C F Gonzalez-Fernandez ldquoApplications of the network simu-lation method to transport processesrdquo in Network SimulationMethod J Horno Ed Research Signpost Trivandrum India2002

[23] J Serna F J S Velasco and A Soto Meca ldquoApplication of net-work simulationmethod to viscous flowsThe nanofluid heatedlid cavity under pulsating flowrdquo Computers amp Fluids vol 91 pp10ndash20 2014

[24] E Juarez-Badillo ldquoGeneral Permeability Change Equationsfor Soilsrdquo in Proceedings of the International Conference onConstitutive Laws pp 205ndash209 University of Arizona TuesonAriz USA 1983

[25] E Juarez-Badillo and B Chen ldquoConsolidation curves for claysrdquoJournal of Geotechnical Engineering vol 109 no 10 pp 1303ndash1312 1983

Mathematical Problems in Engineering 15

[26] N Abbasi H Rahimi A A Javadi andA Fakher ldquoFinite differ-ence approach for consolidation with variable compressibilityand permeabilityrdquo Computers amp Geosciences vol 34 no 1 pp41ndash52 2007

[27] R F Scott Principles of Soils Mechanics Addison-Wesley Pub-lishing Company 1963

[28] F Alhama and C N Madrid Analisis Dimensional Discrimi-nado En Mecanica De Fluidos Y Transferencia De Calor 2012

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Mathematical Problems in Engineering 5

Proceeding as in the previous case the coefficients of thedimensionless form of the former equation

1120591o1205901015840 ( (1 + eo) k112059010158401120574w120574v1205901015840 ) 120574k1205901015840

H21( 12059010158402 minus 120590101584011205901015840m ) ( 1205901015840m12059010158401 )1minus120574v1205901015840minus120574k1205901015840

( (1 + eo) k112059010158401120574w120574v1205901015840 ) 1H21

( 1205901015840m12059010158401 )1minus120574v1205901015840minus120574k1205901015840(23)

give rise to the dimensionless groups

120587I = 120591o1205901015840k112059010158401 (1 + eo)120574w120574v1205901015840H21 ( 1205901015840m12059010158401 )1minus120574k1205901015840minus120574v1205901015840 120587II = 120574k1205901015840 ( 12059010158402 minus 120590101584011205901015840m )

(24)

Thus the solutions for 120591o1205901015840 and U1205901015840 are given by

120591o1205901015840 = ( 120574v1205901015840120574wH21(1 + eo) k112059010158401) ( 1205901015840m12059010158401 )120574v1205901015840+120574k1205901015840minus1

sdot Ψ120591 (120574k1205901015840 ( 12059010158402 minus 120590101584011205901015840m ))(25)

U1205901015840 = ΨU [ t120591o1205901015840 120574k1205901015840 ( 12059010158402 minus 120590101584011205901015840m )] (26)

313 J-B Model Assuming 1+e Is Constant (A Simplificationof the Former) Based on the above it is easy to simplify thehypothesis to 1+e being constant With (120590101584012059010158401)minus120574v1205901015840 = 1 theconsolidation equation reduces to

1205971205901015840120597t = (1 + eo) 1205901015840oko120574w120574v1205901015840 ( 12059010158401205901015840o )1minus120574k1205901015840 [minus 120574k12059010158401205901015840 ( 1205971205901015840120597z )2

+ 12059721205901015840120597z2 ](27)

an equation dependent on the initial void ratio Using (13)and averaging this equation provides three-dimensionalcoefficients

1120591o1205901015840 ( (1 + eo) k112059010158401120574w120574v1205901015840 ) 120574k1205901015840

H21( 12059010158402 minus 120590101584011205901015840m ) ( 1205901015840m12059010158401 )1minus120574k1205901015840

( (1 + eo) k112059010158401120574w120574v1205901015840 ) 1H21

( 1205901015840m12059010158401 )1minus120574v1205901015840minus120574k1205901015840(28)

and two dimensionless independent groups

120587I = 120591o1205901015840k112059010158401 (1 + eo)120574w120574v1205901015840H21 ( 1205901015840m12059010158401 )1minus120574k1205901015840 120587II = 120574k1205901015840 ( 12059010158402 minus 120590101584011205901015840m )

(29)

Thus the solutions for 120591o1205901015840 and U1205901015840 are given by

120591o1205901015840 = ( 120574v1205901015840120574wH21(1 + eo) k112059010158401) ( 1205901015840m12059010158401 )120574k1205901015840minus1

sdot Ψ120591 (120574k1205901015840 ( 12059010158402 minus 120590101584011205901015840m ))(30)

U1205901015840 = ΨU [ t120591o1205901015840 120574k1205901015840 ( 12059010158402 minus 120590101584011205901015840m )] (31)

314 The Less Restrictive Model with Both 1+e and dz NotConstant For this model 1+e and dz not constants it isenough to consider (22) plus the condition dz not constantor dz = dzo(HH1) So after mathematical manipulation thegoverning equation writes as

1205971205901015840120597t= (1 + eo) 1205901015840oko120574w120574v1205901015840 ( 12059010158401205901015840o )1+120574v1205901015840minus120574k1205901015840 [minus 120574k12059010158401205901015840 ( 1205971205901015840120597z )2

+ 12059721205901015840120597z2 ](32)

The new coefficients

1120591o1205901015840 ( (1 + eo) k112059010158401120574w120574v1205901015840 ) 120574k1205901015840

H21( 12059010158402 minus 120590101584011205901015840m ) ( 1205901015840m12059010158401 )1+120574v1205901015840minus120574k1205901015840

( (1 + eo) k112059010158401120574w120574v1205901015840 ) 1H21

( 1205901015840m12059010158401 )1+120574v1205901015840minus120574k1205901015840(33)

provide the groups

120587I = 120591o1205901015840k112059010158401 (1 + eo)120574w120574v1205901015840H21 ( 1205901015840m12059010158401 )1minus120574k1205901015840+120574v1205901015840 120587II = 120574k1205901015840 ( 12059010158402 minus 120590101584011205901015840m )

(34)

6 Mathematical Problems in Engineering

and the solutions

120591o1205901015840 = ( 120574v1205901015840120574wH21(1 + eo) k112059010158401) ( 1205901015840m12059010158401 )120574k1205901015840minus120574v1205901015840minus1

sdot Ψ120591 (120574k1205901015840 ( 12059010158402 minus 120590101584011205901015840m ))(35)

U1205901015840 = ΨU [ t120591o1205901015840 120574k1205901015840 ( 12059010158402 minus 120590101584011205901015840m )] (36)

32 Nondimensionalization of the J-B Model in Terms ofSettlements Three cases defined by the hypotheses (1) 1+e =constant and dz constant (2) 1+e and dz constants and (3) 1+eand dz not being constant are considered For this study let usintroduce a new variable directly related with the settlementwith a clear physical meaning ldquo120577 = e - eordquo a kind of localdegree of settlement or differential void index

321 1+e =Constant and dz Constant For 1+e =constantusing the variable 120577 and the dependence V-1205901015840 it is straight-forward to write

120577 = e minus eo = (1 + eo) [( 120590101584012059010158401)minus120574v1205901015840 minus 1] (37)

Making use of the derivatives of (37)

( 1205971205901015840120597z )2 = 120590101584012(1 + eo)2 120574v12059010158402 ( 120590101584012059010158401)2120574v1205901015840+2 ( 120597120577120597z)2

12059721205901015840120597z2 = minus 12059010158401(1 + eo) 120574v1205901015840 ( 120590101584012059010158401)120574v1205901015840+1 1205972120577120597z2

+ 12059010158401 (120574v1205901015840 + 1)(1 + eo)2 120574v12059010158402 ( 120590101584012059010158401)2120574v1205901015840+1 ( 120597120577120597z)2

(38)

and the assumptions of the odometer test after cumbersomemathematical steps the nonlinear dimensional consolidation(19) is written in the form

120597120577120597t = ko1205901015840o (120574k1205901015840 minus 120574v1205901015840 minus 1)120574w120574v12059010158402 ( 12059010158401205901015840o )1minus120574k1205901015840 ( 120597120577120597z)2

+ ko1205901015840o (1 + eo)120574w120574v1205901015840 ( 12059010158401205901015840o )1minus120574v1205901015840minus120574k1205901015840 1205972120577120597z2(39)

Introducing the variables

(120577)1015840 = 120577 minus 12057711205772 minus 1205771 = 1205771 minus 1205771205771 z1015840 = z

H1

t1015840 = t120591os(40)

into the former equation with 120591os a reference chosen as thetime required for the settlement to reach a high percentageof its change along the whole domain its dimensionless formprovides three coefficients

1120591os ko1205901015840o (120574k1205901015840 minus 120574v1205901015840 minus 1)120574w120574v12059010158402 ( 1205901015840m12059010158401 )1minus120574k1205901015840 (ef minus eo)

H21

ko1205901015840o (1 + eo)120574w120574v1205901015840 ( 1205901015840m12059010158401 )1minus120574v1205901015840minus120574k1205901015840 1H21

(41)

Dividing by the last the resulting dimensionless groups are

120587I = 120591osko1205901015840o (1 + eo)120574w120574v1205901015840H21 ( 1205901015840m12059010158401 )1minus120574v1205901015840minus120574k1205901015840

120587II = (120574k1205901015840 minus 120574v1205901015840 minus 1)120574v1205901015840 (ef minus eo)(1 + eo) ( 1205901015840m12059010158401 )120574v1205901015840(42)

Making use of the constitutive dependence dVV =minus120574v1205901015840(d12059010158401205901015840) is possible to write ΔHH = minus120574v1205901015840(Δ12059010158401205901015840)and since (ef minus eo)(1 + eo) = ΔHHo 120587II finally takes theform

120587II = (1 minus 120574k1205901015840 + 120574v1205901015840) 12059010158402 minus 120590101584011205901015840m (43)

The solution for 120591os is120591os asymp 120574w120574v1205901015840H21

ko1205901015840o (1 + eo) ( 1205901015840m12059010158401 )120574v1205901015840+120574k1205901015840minus1Ψ120591 [(1 minus 120574k1205901015840+ 120574v1205901015840) 12059010158402 minus 120590101584011205901015840m ]

(44)

As for the average degree of consolidation the followingdependence arises

Us = ΨU [ t120591os (1 minus 120574k1205901015840 + 120574v1205901015840) 12059010158402 minus 120590101584011205901015840m ] (45)

322 1+e and dz Constants The hypothesis 1+e=constant isa simplification of the former case Proceeding in the sameway the resulting coefficients

1120591os ko1205901015840o (1 + 120574v1205901015840 minus 120574k1205901015840)120574w120574v12059010158402 ( 1205901015840m12059010158401 )1minus120574k1205901015840+120574v1205901015840 (ef minus eo)

H21

ko1205901015840o (1 + eo)120574w120574v1205901015840 ( 1205901015840m12059010158401 )1minus120574k1205901015840 1H21

(46)

Mathematical Problems in Engineering 7

provide the dimensionless groups

120587I = 120591osko1205901015840o (1 + eo)120574w120574v1205901015840H21 ( 1205901015840m12059010158401 )1minus120574k1205901015840

120587II = (1 minus 120574k1205901015840 + 120574v1205901015840) 12059010158402 minus 120590101584011205901015840m(47)

The solutions for 120591os and Us are

120591os asymp 120574w120574v1205901015840H21ko1205901015840o (1 + eo) ( 1205901015840m12059010158401 )120574k1205901015840minus1Ψ120591 [(1 minus 120574k1205901015840

+ 120574v1205901015840) 12059010158402 minus 120590101584011205901015840m ](48)

Us = ΨU [ t120591os (1 minus 120574k1205901015840 + 120574v1205901015840) 12059010158402 minus 120590101584011205901015840m ] (49)

The solutions for the original J-Bmodel are also (48) and (49)with the simplification of eo=0 The author however doesnot talk about the characteristic time and obtains the averagedegree of consolidation given by (7) in which this variabledepends on three groups (1minus120574k1205901015840)120574v1205901015840 H2H1 and tcv2H21undoubtedly a less precise solution

323 1+e and dz Not Being Constant When 1+e and dzare not constants substituting dz = dzo(HH1) in (39)after cumbersome mathematical manipulation the resultingcoefficients are 1120591os

ko1205901015840o (120574k1205901015840 minus 120574v1205901015840 minus 1)120574w120574v12059010158402 ( 1205901015840m12059010158401 )1+2120574v1205901015840minus120574k1205901015840 (ef minus eo)H21

ko1205901015840o (1 + eo)120574w120574v1205901015840 ( 1205901015840m12059010158401 )1+120574v1205901015840minus120574k1205901015840 1

H21

(50)

Dividing by the last the resulting dimensionless groups are

120587I = 120591osko1205901015840o (1 + eo)120574w120574v1205901015840H21 ( 1205901015840m12059010158401 )1+120574v1205901015840minus120574k1205901015840

120587II = (1 minus 120574k1205901015840 + 120574v1205901015840) 12059010158402 minus 120590101584011205901015840m(51)

The solutions for 120591os and Us are

120591os asymp 120574w120574v1205901015840H21ko1205901015840o (1 + eo) ( 1205901015840m12059010158401 )+120574k1205901015840minus120574v1205901015840minus1Ψ120591 [(1 minus 120574k1205901015840

+ 120574v1205901015840) 12059010158402 minus 120590101584011205901015840m ](52)

Us = ΨU [ t120591os (1 minus 120574k1205901015840 + 120574v1205901015840) 12059010158402 minus 120590101584011205901015840m ] (53)

Table 1 summarizes the results derived in this section

4 Verification of the Results byNumerical Simulations

This section is devoted to checking the solutions given inTable 1 To shorten the exposure we restrict the tests forthe more general model (1+e and dz not constants) andfor the unknowns 120591os and Us since they are the ones ofgreater interest in civil engineering Eight sets of simulationsarranged in three blocks have been run Table 2 In each onesome of the soil parameters or initial values of the problemhave been changed to give the same or different values asrequired to the dimensionless group 120587II in the search for thesame or different solutions of 120587I Changes in the values of theindividual parameters are sufficient to cover all real scenarios

Firstly a reference set (set 1) is established to which all theother sets can be referred and which permits them to be com-pared with each other The physical and geometrical charac-teristics that change are 120574v1205901015840 120574k1205901015840 eo H1 (m) 1205901015840o (Nm2)1205901015840f (Nm2) and ko (myear) The values of =(1205901015840f1205901015840o) H2(m) kf (myear) and 120587II are derived from them while 1205871is obtained once the characteristic time 120591os is read from thesimulation The criterion for the choice of 120591os is the timerequired by the soil to reach 90 of the total settlement

It is worth mentioning here the emergence of 1205901015840m anaveraged value of the effective pressure in the two dimen-sionless groups Undoubtedly the existence of 1205901015840m in themonomials can be explained not only by the nonlinearityof the problem but also by the kind of dependences in theconstitutive relations (there are other nonlinear problems inwhich averaged variables do not emerge [18]) How do wechoose 1205901015840m We could take the mean value 1205901015840m=(12059010158401+12059010158402)2or even 1205901015840m=12059010158401 or 1205901015840m=12059010158402 but expecting that the expres-sion 120587I = 120595(120587II) is slightly dependent on that choice (we willreturn to this question later) In this paper we have taken theapproximation 1205901015840m=(12059010158402 + 12059010158401)2

In block I the parameters that change in each set areeo H1 ko and 1205901015840o while maintaining the ratio =(1205901015840f1205901015840o) thus both 120587II and factor (1205901015840m1205901015840o)1minus120574k1205901015840+120574v1205901015840 of 120587I takethe same value in the four sets of the block As a resultwe expect a characteristic time proportional to the ratio120574w120574v1205901015840H21ko1205901015840o(1 + eo) and a same value of 120587I in all casesaccording to the solution 120587I = 120595(120587II) Indeed the resultsare consistent with expectations Block II contains an onlycase for which 120587II varies in comparison to block I as a result120587I also changes as expected Finally block III contains threecases with the same value of 120587II of block I but with differentvalues of the parameters 120574v1205901015840 120574k1205901015840 and the ratio =(1205901015840f 1205901015840o)The parameters of set 6 have been chosen in such a waythat the values 120574k1205901015840 and 120574v1205901015840 are compensated to maintainconstant (1 minus 120574k1205901015840 + 120574v1205901015840) we expect 120587I to be also unchangedIn effect simulation provides 120587I = 07139 a value quite closeto that of set 1 (07131) The negligible differences are due nodoubt to the different evolution of the effective pressure alongthe process caused by the different values of the coefficients120574k1205901015840 and 120574v1205901015840 Note that as 120574v1205901015840 duplicates the denominator ofthe factor 120591osko1205901015840o(1 + eo)120574w120574v1205901015840H21 the characteristic timehas to double too as indeed happens The last two sets studythe influence of a parameter whose change forces the value

8 Mathematical Problems in Engineering

Table1Dim

ensio

nlessg

roup

sthatcharacterizethe

originalandextend

edJ-Bmod

els

Juarez-Badillo

original

Juarez-Badillo

Juarez-Badillo

Juarez-Badillo

1+e=

constant

1+e

=consta

nt1+e

=consta

ntdz

constant

dzconstant

dzvaria

ble

120587 I120591 ok o

1205901015840 o120574 w120574 v1205901015840H2 1

(1205901015840 m 1205901015840 o)1minus120574k1205901015840

120591 ok o1205901015840 o(

1+eo)

120574 w120574 v1205901015840H2 1

(1205901015840 m 1205901015840 o)1minus120574k1205901015840

120591 ok o1205901015840 o(

1+eo)

120574 w120574 v1205901015840H2 1

(1205901015840 m 1205901015840 o)1minus120574k1205901015840minus120574 v1205901015840

120591 ok o1205901015840 o(

1+eo)

120574 w120574 v1205901015840H2 1

(1205901015840 m 1205901015840 o)1minus120574k1205901015840+120574 v1205901015840

120591 o1205901015840or

120591 os120587 II

120574 k1205901015840(1205901015840 2minus

1205901015840 1 1205901015840 m)for

pressure

(1minus120574 k1205901015840

+120574 v1205901015840)(1205901015840 2minus

1205901015840 1 1205901015840 m)for

settlem

ent

Mathematical Problems in Engineering 9

0010203040506070809

1

0 05 1 15 2 25 3time

Set 1Set 2Set 3Set 4

Set 5Set 6Set 7Set 8

Aver

age d

egre

e of c

onso

lidat

ion

Figure 1 Average degree of consolidation versus time (eight sets)

0010203040506070809

1

0 05 1 15 2 25 3

Set 1Set 2Set 3Set 4

Set 5Set 6Set 7Set 8

Aver

age d

egre

e of c

onso

lidat

ion

tos

Figure 2 Average degree of consolidation versus dimensionlesstime (eight sets)

(1 minus 120574k1205901015840 + 120574v1205901015840) to be also changed in order to maintain 120587IIconstant We expect that 120587I does not change as nearly occursThe small differences are again due to the existence of 1205901015840mwithin the monomials Note however that as diminishesthe influence of 1205901015840m in the results also diminishes since thevalues of 12059010158401 and 12059010158402 are closer

For a better understanding of the results Figures 1 and2 are presented In the first Us as a function of time isshown for the eight sets in Table 2 It is clear that 120591oscovers a wide range of values from 02283 (set 2) to 27968(set 3) the last more than ten times the first The secondfigure graphs Us in front of the dimensionless time t120591os Asshown the eight previous curves nearly converge to a single(or universal) graph having a common point at (Us=09t120591os=1) This means that monomial 120587II for the range ofvalues assigned to the parameters in contrast to 120587I scarcelyinfluences the value of the average degree of consolidationThis is not in contradiction with the results derived bynondimensionalization

0010203040506070809

1

0 05 1 15 2 25 3 35 4time

U (J-B)U (1+e cst)U (1+e not cst)U (1+e amp dz not cst)

Us (J-B)Us (1+e cst)Us (1+e not cst)Us (1+e amp dz not cst)

Aver

age d

egre

e of p

ress

ure d

issip

atio

nAv

erag

e deg

ree o

f con

solid

atio

n

Figure 3 Average degree of consolidation and pressure dissipation(set 6)

Although in this sectionwe have only checked the expres-sions of Us and 120591os for the most general model we considerit interesting to present both Us and U1205901015840 for the rest of themodels of Table 1 and for set 6 as a typical case Figure 3Theseresults emphasize the importance of using the less restrictive(more general) model in relation to the others in particularwith the original model of J-B in which the influence ofeo is despised Note that the smaller of the characteristictimes (related to settlement) corresponds to the more generalmodel (black and bold line) so the use of the others althoughon the safety side provides characteristic times oversizedfar from an optimal solution for the engineer For examplefor the case of Figure 3 in which eo=1 the error of J-Boriginal model (purple and bold line) in comparison with theextended model (black and bold line) for Us=90 is around150 This error increases appreciably as eo gets larger

5 Universal Curves

These easy to use and universal abacuses obtained by numeri-cal simulations allow engineers to read the characteristic time(120591os) and the average degree of consolidation (Us) in termsof the dimensionless groups given in Table 1 Only the mostgeneral (less restrictive) J-B model 1+e and dz not constantis presented

To obtain the characteristic time curves a large numberof simulations have been carried out studying separately theinfluence of the parameters 120574v1205901015840 and 120574k1205901015840 whose valuescover most real soils ranges from 101 to 8 120574v1205901015840 from 005to 03 and 120574k1205901015840 from 01 to 17 From these tests it has beendeduced as expected from the results of the previous sectionthat the relatively more influential parameter is while 120574v1205901015840and 120574k1205901015840 do not produce significant changes in the values ofthe characteristic timeThus the first universal group of linesshows the dependence of 120587I versus 120587II for different valuesof (with 120574v1205901015840=01) Figure 4 As the range of values of 120587IIdepends on according to (43) the range of values of eachline is different As shown in the figure there exists a universal

10 Mathematical Problems in Engineering

Table 2 Values of the parameters and dimensionless groups of simulations

Block Set 120574v1205901015840 120574k1205901015840 eoH1 ko 1205901015840o 1205901015840f H2 kf 120587II

120591os 120587I(m) (myr) (Nm2) (Nm2) 1205901015840f 1205901015840o (m) (myr) (yr)

I

1 01 0500 10 100 002 30000 60000 2 0933 00141 040 04566 071312 01 0500 10 100 002 60000 120000 2 0933 00141 040 02283 071313 01 0500 10 175 001 30000 60000 2 1633 00071 040 27968 071314 01 0500 13 142 004 22000 44000 2 1325 00283 040 05459 07131

II 5 01 1500 10 100 002 30000 90000 3 0896 00038 -040 10597 09834

III6 02 0600 10 100 002 30000 60000 2 0871 00132 040 09142 071397 01 0767 10 100 002 30000 120000 4 0871 00069 040 04360 072468 02 0920 10 100 002 30000 180000 6 0699 00038 040 08409 07312

line for each value of all converging in the central zone forvalues of 120587II around 0 This set of straight lines can be fittedby 120587I=m120587II+n being n=0846 and m given in Figure 5 as afunction of The curve that best fits the points in Figure 5 isgiven by

m = minus0235e(minus0868) minus 0306e(minus0012) (54)

Choosing the longest line corresponding to =8 as the onlyrepresentative line of the problem themaximumerrors in thereading of 120587I (and 120591os) are 38

In short the use of these curves is as follows from the dataof the problem Figure 4 allows us to read 120587I from 120587II and and from (51) the characteristic time 120591os is derived

As for the average degree of consolidation the universalcurves for 96 different scenarios of 120587II are shown in Figure 6It is appreciated that these curves are very close to each othershowing again that the Us only depends on the dimensionlesstime In the worst case the errors produced by changes invalue of the 120587II group are less than 28when using the curveof Figure 7

6 Case Study

Below is a practical application of the universal curvespresented in this paper For this purpose we take as a basisfor our study the data obtained experimentally byAbbasi et al[26] for different samples of clay whosemoisture in the liquidlimit is equal to 42 As can be seen (Table 3) it is a series ofdiscrete (tabulated) values that relate the void ratio with theeffective pressure and the hydraulic conductivity

We will address a real case of consolidation in which wehave a layer of soil 2meters thick (Ho) with an initial effectivepressure (1205901015840o) of 28 kNm2 on which a load of 31 kNm2 isapplied In this way the effective pressure at the end of theconsolidation process (1205901015840f ) will have the value of 59 kNm2From the tabulated data (e-1205901015840) it follows that eo = 105 andef = 095 and since the factor 1+e is proportional to the soilvolume from (9) we have 120574v1205901015840 = 0067

To obtain 120574k1205901015840 we first calculate the value of 120574kv from thetabulated data (e-k) to then obtain 120574k1205901015840 from (10) With thiswe have for the load step considered ko = 0035 myr 120574kv =8108 and 120574k1205901015840 = 0543

To solve the problem we will use the universal curvespresented in the previous section corresponding to the most

Table 3 Experimental values of e 1205901015840 and k obtained by Abbasi et al[26]

e 1205901015840o (Nm2)110 14000105 28000095 59000085 120000075 220000065 450000060 925000e k (myr)175 0683160 0315120 0084085 0021065 0011

general J-B model (analyzed in terms of settlement) Thusfrom the values of 1205901015840o 1205901015840f 120574v1205901015840 and 120574k1205901015840 by means of (51) itis obtained that 120587II = 0373 (with 1205901015840m = 435 kNm2) Then120587I is read from Figure 4 where for = 21 we have 120587I =072 An identical result is obtained if we use the proposedfitted equation 120587I=m120587II+n where n is a constant of value0846 andm is obtained from (54) or Figure 5 Finally knownthe values of Ho ko and eo from (51) we get 120591os = 0748years Likewise once the characteristic time of the problem(in terms of settlement) is known from the universal curve ofFigure 7 we easily obtain the evolution of the average degreeof consolidation along the whole process Figure 8

These solutions have been compared with those thatresult from the numerical simulation of the most generalJ-B model for which a value of 120591os = 075 years has beenobtained showing clearly that the relative error committedwhen applying the universal solutions is negligible (Table 4)The simulation of the problem has also been carried out fora model where 1+e and dz are also not constant but in whichthe constitutive relations of potential type have been replacedby pairs of data in tabulated form obtaining in this case 120591os= 0817 years The relative error which this time increasesto 82 finds its explanation in the strong nonlinearity ofthe problem which will always bring differences between the

Mathematical Problems in Engineering 11

040045050055060065070075080085090095100105110

minus07 minus06 minus05 minus04 minus03 minus02 minus01 0 01 02 03 04 05 06 07 08 09 1 11 12 13 14 15

1

2

ᴪ=8ᴪ=6ᴪ=4

ᴪ=2ᴪ=15ᴪ=101

Figure 4 120587I as a function of 120587II with as parameter

1 2 3 4 5 6 7 8minus042

minus04

minus038

minus036

minus034

minus032

minus03

minus028

minus026

m

datafitted curve

Figure 5 m as a function of

solutions provided by numerical models based on differentconstitutive relations (potential and tabulated in this case)Even so this value is totally acceptable within this field ofengineering demonstrating the strength and versatility of theuniversal solutions provided

7 Discussion

This work derives the dimensionless groups that rule thenonlinear soil consolidation process based on the original J-Bmodel following a formal nondimensionalization procedurethat starts from the governing equation and does not needany other assumption or mathematical manipulation Inaddition more real and precise extended J-B models whichassume both the void ratio and the thickness of the finitevolume element of the soil to be not constant have been

0010203040506070809

1

0 05 1 15 2 25 3

96 scenariosᴪ = [101 15 2 4 6 8]

Aver

age d

egre

e of c

onso

lidat

ion

tos

= [005 01 015 02]v= [02 05 1 15]k

Figure 6 Average degree of consolidation versus dimensionlesstime for different scenarios

0010203040506070809

1

0 05 1 15 2 25 3

Aver

age d

egre

e of c

onso

lidat

ion

tos

Figure 7 Proposed solution for the average degree of consolidation

nondimensionalized and their dimensionless groups derivedThe assumption of constitutive dependences (e-1205901015840 e-k and k-1205901015840) of potential type in these models is interesting since they

12 Mathematical Problems in Engineering

Table 4 Case study solution for the characteristic time 120591os Comparison with numerical models

Universal curves J-B most general model Tabulated model120591os (yr) 120591os (yr) Rel error 120591os (yr) Rel error 0748 0750 026 0817 820

0010203040506070809

1

0 025 05 075 1 125 15 175 2 225t ( years)

Aver

age d

egre

e of c

onso

lidat

ion

Figure 8 Case study solution for the average degree of consolida-tion Us from the universal curves

can be applied to highly compressible soils for which the e-log(1205901015840) relation moves away from being a straight line andin which negative values of the void ratio could be obtainedunder the application of high load ratios (1205901015840f 1205901015840o) [8]

The study has been developed both in terms of pres-sure dissipation and in terms of settlement resulting thatthe dimensionless groups derived from each model are ingeneral different The extension to more complex modelswhich implies the emergence of new parameters or equationswithin themathematical model does not necessarily give riseto additional dimensionless groups as might be expectedbut to new expressions of the groups or new regroupingof the parameters involved Thus the elimination of thehypothesis ldquodz constantrdquo in the most general consolidationmodel has not given rise to any new group both in pressureand in settlement (Table 1) but it has only modified oneof the groups obtained in the original model All thisrepresents an important advance in relation to the modelsand studies carried out by other authors and the attemptsof characterization found in the scientific literature [9ndash11]which try to lead to easy-to-use solutions for engineers Inno case had dimensionless groups been obtained as preciseas those presented in this work nor presented universalcurves of the characteristic time and the average degree ofconsolidation Authors such asConte andTroncone [11] applymore general load conditions (time-dependent loading) forwhich the dimensionless groups deduced in this work couldbe applicable for each load step

As regards time to make it dimensionless an unknownreference (120591o) is introduced in the form mentioned by Scott[27] and used in many other problems with asymptoticsolutions [28]Theprecise definition of 120591o necessarily involvesa criterion in relation to the percentage of average pressuredecrease (or settlement) associated with the end of theconsolidation processTheorder ofmagnitude of 120591o is derived

after nondimensionalization making 120591o only appear in oneof the independent dimensionless groups In this way theaverage degree of consolidation (in terms of pressure orsettlement) depends on the ldquotimecharacteristic timerdquo ratioin addition to the dimensionless groups (without unknowns)that arise

In short the procedure has led to a new very precisesolution that involves two monomials one (120587I) that containsthe characteristic time (120591o) and another (120587II) that is amathematical function of the soil parameters In this way120591o only depends on 120587II through the function 120587I = 120595(120587II)while U depends on 120587I and 120587II through the expression U= 120595(120587I 120587II) The results obtained have been verified forthe most general and complex consolidation model throughthe numerical simulation of a large number of scenariosin which the values of the dimensionless groups and theindividual parameters of the problem have been changing toconvenience The high errors produced by the original J-Bmodel with respect to the most general justify the use of thelatter in the search for more precise solutions

In addition the emergence of mean values of the effectivepressure when averaging the governing equation in orderto assign an order of magnitude unit to the dimensionlessvariables and their derivatives is an added difficulty to obtainthe dimensionless groups This is undoubtedly due to thenonlinearities (constitutive relations) involved in the consol-idation process This average effective pressure of referencehas been assumed as the semisum of its initial and finalvalues although other criteria could also be chosen Sincetwo scenarios can have equal120587II with the same effectivemeanpressure but different values of the parameters 120574v1205901015840 and 120574k1205901015840 we expect some minor deviations between the results of bothscenarios as indeed it has been checked Thus for the wholerange of parameters values in typical soils the errors due tothe above cause are negligible in the field of soil mechanicsOn the other hand when the value of 120587II does not changebut the mean effective pressure value does the errors in thesolutions of 120591os and Us are slightly larger although withinranges below 4

The brooch for all this work is the presentation ofuniversal curves for the most general model which allowfor the usual range of variation of the soil properties toobtain in a simple way the value of the characteristic timeof consolidation and the evolution of the average degreeof settlement throughout the entire process Thus startingfrom the nonlinear parameters of the soil (120574v1205901015840 and 120574k1205901015840)and the load ratio (1205901015840f 1205901015840o) the value of the monomial 120587IIis obtained from which the value of 120587I is directly deductedthanks to the universal curve 120587I = 120595(120587II) provided Finally120591os is easily obtained from the expression of 120587I On the otherhand the curve that represents the Us evolution along theprocess has turned out to be (practically) unique when it is

Mathematical Problems in Engineering 13

represented as a function of the dimensionless time in theform t120591os regardless of the soil parameters and the loadsapplied All this supposes a clear advantage when solvingthe consolidation problem also showing the strength of thenondimensionalization method presented

Finally the case study addressed has demonstrated thesimplicity and reliability of the universal curves proposedFrom a series of tabulated data (e-1205901015840 and e-k) the parametersthat govern the nonlinear consolidation problem for themostextended J-B model have been adjusted with which thevalue of the monomial 120587II has been derived With this andusing the universal curves the solutions for the characteristictime 120591os and the average degree of consolidation Us havebeen obtained These results have been satisfactorily verifiedwith the solutions provided both by the numerical extendedmodel of J-B presented here and by another that substitutesthe constitutive dependences of potential type by data intabulated form

8 Conclusions

The application of the nondimensionalization technique hasallowed us to deduce the dimensionless groups that rule thenonlinear consolidation problem in which the constitutivedependencies e-1205901015840 and e-k have the form of potential func-tions This type of relations approximates the soil behaviorbetter than the logarithmic functions e-log(1205901015840) and e-log(k)particularly in very compressible soils and under high loadratios (1205901015840f 1205901015840o) in which inconsistent negative values of thevoid ratio may appear

Several scenarios have been addressed from the originalmodel of Juarez-Badillo to extended models based on thesame constitutive dependences but eliminating one or moreclassic (and restrictive) hypotheses in this process In allcases only two dimensionless groups emerge one of themconsisting of a grouping of the soil parameters and the othercontaining the unknown characteristic time Based on theseresults universal curves have been depicted the first to deter-mine the characteristic time of the process and the secondthat represents the evolution of average degree of settlement

The solutions have been verified by numerical simula-tions changing the values of the soil parameters and thedimensionless groups conveniently Finally the reliability andeffectiveness of the proposed solutions have been shownthrough a case study

Abbreviations

cv Consolidation coefficient (m2s or m2yr)cv1 Initial consolidation coefficient (m2s or

m2yr)cv2 Final consolidation coefficient (m2s or

m2yr)119889z Element of differential length in thedirection of the spatial coordinate z (m)119889zo Element of differential length in thedirection of the spatial coordinate z at theinitial time (m)

e Void ratio (dimensionless)

eo Initial void ratio (dimensionless)ef Final void ratio (dimensionless)H Soil thickness up to the impervious

boundary or drainage length (m)H1Ho Initial soil thickness (m)H2 Final soil thickness (m)k Hydraulic conductivity (or permeability)

(ms or myr)k1 ko Initial hydraulic conductivity (ms or

myr)kf Final hydraulic conductivity (ms or myr)mv Coefficient of volumetric compressibility

(m2N)mv1 Initial coefficient of volumetric

compressibility (m2N)t Time Independent variable (s or yr)T Juarez-Badillorsquos time factor

(dimensionless)u Excess pore pressure (Nm2)U Average degree of consolidation or

pressure dissipation (dimensionless)Us Average degree of consolidation

(dimensionless)U1205901015840 Average degree of pressure dissipation

(dimensionless)V Soil volume (m3)V1Vo Initial soil volume (m3)z Vertical spatial coordinate (m)120574v Nonlinear coefficient of compressibility of

Juarez-Badillo (dimensionless)120574kv Nonlinear coefficient of change ofpermeability with volume (dimensionless)120574k1205901015840 Nonlinear coefficient of change ofpermeability with effective pressure(dimensionless)120574v1205901015840 Nonlinear coefficient of compressibility orcoefficient of volumetric contraction(dimensionless)120574w Specific weight of water (Nm3)

Loading factor Ratio between the finaland initial effective pressure(dimensionless)120581 Nonlinear coefficient of change ofpermeability with volume ofJuarez-Badillo (dimensionless)120582 Parameter of Juarez-Badillo 120582 = 1 - 120574v120581(dimensionless)

V Auxiliary variable of Juarez-Badillo(dimensionless)120587i Dimensionless group or number(dimensionless)1205901015840 Effective pressure of the soil (Nm2)12059010158401 1205901015840o Initial effective pressure (Nm2)12059010158402 1205901015840f Final effective pressure (Nm2)1205901015840m Mean value of the effective pressure(Nm2)120591o Characteristic time (s or yr)

14 Mathematical Problems in Engineering

120591os Characteristic time which takes thesettlement to reach approximately its finalvalue (s or yr)120591o1205901015840 Characteristic time which takes the excesspore pressure to dissipate untilapproximately the value of zero (s or yr)Ψ Arbitrary mathematical function120577 Differential void ratio (dimensionless)1205771 1205772 Initial and final differential void ratio(dimensionless)

Subscripts

I II Denote different numbers

Superscripts

1015840 Denote dimensionless magnitude

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] K Terzaghi ldquoDie Berechnung der Durchlassigkeitszifferdestones aus dem verlauf der hydrodynamischen spanunngser-scheinnungenrdquo Technical report II a 132 N 34 125-138Akademie derWissenschaften inWien SitzungsberichteMath-naturwiss Klasse Abt 1923

[2] R T Walker ldquoVertical drain consolidation analysis in one twoand three dimensionsrdquo Computers amp Geosciences vol 38 no 8pp 1069ndash1077 2011

[3] G Y Zheng P Li and C Y Zhao ldquoAnalysis of non-linearconsolidation of soft clay by differential quadrature methodrdquoApplied Clay Science vol 79 pp 2ndash7 2013

[4] X Y Geng C J Xu and Y Q Cai ldquoNon-linear consolidationanalysis of soil with variable compressibility and permeabilityunder cyclic loadingsrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 30 no 8 pp 803ndash8212006

[5] M Lu S Wang S W Sloan B Indraratna and K Xie ldquoNon-linear radial consolidation of vertical drains under a generaltime-variable loadingrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 39 no 1 pp 51ndash622015

[6] HWu L HuW Qi and QWen ldquoAnalytical Solution for Elec-troosmotic Consolidation Considering Nonlinear Variation ofSoil Parametersrdquo International Journal of Geomechanics vol 17no 5 Article ID 06016032 2017

[7] S J Brandenberg ldquoiConsoljs JavaScript Implicit Finite-Differ-ence Code for Nonlinear Consolidation and Secondary Com-pressionrdquo International Journal of Geomechanics vol 17 no 6Article ID 04016149 2017

[8] R Butterfield ldquoA natural compression law for soils (an advanceon e-log prsquo)rdquo Geotechnique vol 29 no 4 pp 469ndash480 1979

[9] R Lancellotta and L Preziosi ldquoA general nonlinear mathemati-calmodel for soil consolidation problemsrdquo International Journalof Engineering Science vol 35 no 10-11 pp 1045ndash1063 1997

[10] Y-C Zhuang K-H Xie and X-B Li ldquoNonlinear analysis ofconsolidation with variable compressibility and permeabilityrdquoJournal of Zhejiang University (Engineering Science Edition) vol6 no 3 pp 181ndash187 2005

[11] E Conte andA Troncone ldquoNonlinear consolidation of thin lay-ers subjected to time-dependent loadingrdquo Canadian Geotechni-cal Journal vol 44 no 6 pp 717ndash725 2007

[12] R Yong and F Townsend ldquoGeneral Consolidation Theory forClaysrdquo Soil Mechanics Series Report No 8 Graduate School ofEngineering National University of Mexico 1983

[13] J C Gibbings Dimensional Analysis Springer-Verlag LondonUK 2011

[14] E Buckingham ldquoOn physically similar systems Illustrations ofthe use of dimensional equationsrdquo Physical Review A AtomicMolecular and Optical Physics vol 4 no 4 pp 345ndash376 1914

[15] R Seshadri and T Y Na ldquoA Survey ofMethods for DeterminingSimilarity Transformationsrdquo inGroup Invariance in EngineeringBoundary Value Problems pp 35ndash61 Springer New York NYUSA 1985

[16] V D Zimparov and V M Petkov ldquoApplication of discriminatedanalysis to low Reynolds number swirl flows in circular tubeswith twisted-tape inserts Pressure drop correlationsrdquo Interna-tional Review of Chemical Engineering vol 1 no 4 pp 346ndash3562009

[17] M Capobianchi and A Aziz ldquoA scale analysis for natural con-vective flows over vertical surfacesrdquo International Journal ofThermal Sciences vol 54 pp 82ndash88 2012

[18] M Conesa J F Sanchez Perez I Alhama and F AlhamaldquoOn the nondimensionalization of coupled nonlinear ordinarydifferential equationsrdquo Nonlinear Dynamics vol 84 no 1 pp91ndash105 2016

[19] I A Manteca G Garcıa-Ros and F Alhama ldquoUniversal solu-tion for the characteristic time and the degree of settlement innonlinear soil consolidation scenarios A deduction based onnondimensionalizationrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 57 pp 186ndash201 2018

[20] E H Davis and G P Raymond ldquoA non-linear theory ofconsolidationrdquo Geotechnique vol 15 no 2 pp 161ndash173 1965

[21] P Cornetti and M Battaglio ldquoNonlinear consolidation of soilmodeling and solution techniquesrdquo Mathematical and Com-puter Modelling vol 20 no 7 pp 1ndash12 1994

[22] C F Gonzalez-Fernandez ldquoApplications of the network simu-lation method to transport processesrdquo in Network SimulationMethod J Horno Ed Research Signpost Trivandrum India2002

[23] J Serna F J S Velasco and A Soto Meca ldquoApplication of net-work simulationmethod to viscous flowsThe nanofluid heatedlid cavity under pulsating flowrdquo Computers amp Fluids vol 91 pp10ndash20 2014

[24] E Juarez-Badillo ldquoGeneral Permeability Change Equationsfor Soilsrdquo in Proceedings of the International Conference onConstitutive Laws pp 205ndash209 University of Arizona TuesonAriz USA 1983

[25] E Juarez-Badillo and B Chen ldquoConsolidation curves for claysrdquoJournal of Geotechnical Engineering vol 109 no 10 pp 1303ndash1312 1983

Mathematical Problems in Engineering 15

[26] N Abbasi H Rahimi A A Javadi andA Fakher ldquoFinite differ-ence approach for consolidation with variable compressibilityand permeabilityrdquo Computers amp Geosciences vol 34 no 1 pp41ndash52 2007

[27] R F Scott Principles of Soils Mechanics Addison-Wesley Pub-lishing Company 1963

[28] F Alhama and C N Madrid Analisis Dimensional Discrimi-nado En Mecanica De Fluidos Y Transferencia De Calor 2012

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6 Mathematical Problems in Engineering

and the solutions

120591o1205901015840 = ( 120574v1205901015840120574wH21(1 + eo) k112059010158401) ( 1205901015840m12059010158401 )120574k1205901015840minus120574v1205901015840minus1

sdot Ψ120591 (120574k1205901015840 ( 12059010158402 minus 120590101584011205901015840m ))(35)

U1205901015840 = ΨU [ t120591o1205901015840 120574k1205901015840 ( 12059010158402 minus 120590101584011205901015840m )] (36)

32 Nondimensionalization of the J-B Model in Terms ofSettlements Three cases defined by the hypotheses (1) 1+e =constant and dz constant (2) 1+e and dz constants and (3) 1+eand dz not being constant are considered For this study let usintroduce a new variable directly related with the settlementwith a clear physical meaning ldquo120577 = e - eordquo a kind of localdegree of settlement or differential void index

321 1+e =Constant and dz Constant For 1+e =constantusing the variable 120577 and the dependence V-1205901015840 it is straight-forward to write

120577 = e minus eo = (1 + eo) [( 120590101584012059010158401)minus120574v1205901015840 minus 1] (37)

Making use of the derivatives of (37)

( 1205971205901015840120597z )2 = 120590101584012(1 + eo)2 120574v12059010158402 ( 120590101584012059010158401)2120574v1205901015840+2 ( 120597120577120597z)2

12059721205901015840120597z2 = minus 12059010158401(1 + eo) 120574v1205901015840 ( 120590101584012059010158401)120574v1205901015840+1 1205972120577120597z2

+ 12059010158401 (120574v1205901015840 + 1)(1 + eo)2 120574v12059010158402 ( 120590101584012059010158401)2120574v1205901015840+1 ( 120597120577120597z)2

(38)

and the assumptions of the odometer test after cumbersomemathematical steps the nonlinear dimensional consolidation(19) is written in the form

120597120577120597t = ko1205901015840o (120574k1205901015840 minus 120574v1205901015840 minus 1)120574w120574v12059010158402 ( 12059010158401205901015840o )1minus120574k1205901015840 ( 120597120577120597z)2

+ ko1205901015840o (1 + eo)120574w120574v1205901015840 ( 12059010158401205901015840o )1minus120574v1205901015840minus120574k1205901015840 1205972120577120597z2(39)

Introducing the variables

(120577)1015840 = 120577 minus 12057711205772 minus 1205771 = 1205771 minus 1205771205771 z1015840 = z

H1

t1015840 = t120591os(40)

into the former equation with 120591os a reference chosen as thetime required for the settlement to reach a high percentageof its change along the whole domain its dimensionless formprovides three coefficients

1120591os ko1205901015840o (120574k1205901015840 minus 120574v1205901015840 minus 1)120574w120574v12059010158402 ( 1205901015840m12059010158401 )1minus120574k1205901015840 (ef minus eo)

H21

ko1205901015840o (1 + eo)120574w120574v1205901015840 ( 1205901015840m12059010158401 )1minus120574v1205901015840minus120574k1205901015840 1H21

(41)

Dividing by the last the resulting dimensionless groups are

120587I = 120591osko1205901015840o (1 + eo)120574w120574v1205901015840H21 ( 1205901015840m12059010158401 )1minus120574v1205901015840minus120574k1205901015840

120587II = (120574k1205901015840 minus 120574v1205901015840 minus 1)120574v1205901015840 (ef minus eo)(1 + eo) ( 1205901015840m12059010158401 )120574v1205901015840(42)

Making use of the constitutive dependence dVV =minus120574v1205901015840(d12059010158401205901015840) is possible to write ΔHH = minus120574v1205901015840(Δ12059010158401205901015840)and since (ef minus eo)(1 + eo) = ΔHHo 120587II finally takes theform

120587II = (1 minus 120574k1205901015840 + 120574v1205901015840) 12059010158402 minus 120590101584011205901015840m (43)

The solution for 120591os is120591os asymp 120574w120574v1205901015840H21

ko1205901015840o (1 + eo) ( 1205901015840m12059010158401 )120574v1205901015840+120574k1205901015840minus1Ψ120591 [(1 minus 120574k1205901015840+ 120574v1205901015840) 12059010158402 minus 120590101584011205901015840m ]

(44)

As for the average degree of consolidation the followingdependence arises

Us = ΨU [ t120591os (1 minus 120574k1205901015840 + 120574v1205901015840) 12059010158402 minus 120590101584011205901015840m ] (45)

322 1+e and dz Constants The hypothesis 1+e=constant isa simplification of the former case Proceeding in the sameway the resulting coefficients

1120591os ko1205901015840o (1 + 120574v1205901015840 minus 120574k1205901015840)120574w120574v12059010158402 ( 1205901015840m12059010158401 )1minus120574k1205901015840+120574v1205901015840 (ef minus eo)

H21

ko1205901015840o (1 + eo)120574w120574v1205901015840 ( 1205901015840m12059010158401 )1minus120574k1205901015840 1H21

(46)

Mathematical Problems in Engineering 7

provide the dimensionless groups

120587I = 120591osko1205901015840o (1 + eo)120574w120574v1205901015840H21 ( 1205901015840m12059010158401 )1minus120574k1205901015840

120587II = (1 minus 120574k1205901015840 + 120574v1205901015840) 12059010158402 minus 120590101584011205901015840m(47)

The solutions for 120591os and Us are

120591os asymp 120574w120574v1205901015840H21ko1205901015840o (1 + eo) ( 1205901015840m12059010158401 )120574k1205901015840minus1Ψ120591 [(1 minus 120574k1205901015840

+ 120574v1205901015840) 12059010158402 minus 120590101584011205901015840m ](48)

Us = ΨU [ t120591os (1 minus 120574k1205901015840 + 120574v1205901015840) 12059010158402 minus 120590101584011205901015840m ] (49)

The solutions for the original J-Bmodel are also (48) and (49)with the simplification of eo=0 The author however doesnot talk about the characteristic time and obtains the averagedegree of consolidation given by (7) in which this variabledepends on three groups (1minus120574k1205901015840)120574v1205901015840 H2H1 and tcv2H21undoubtedly a less precise solution

323 1+e and dz Not Being Constant When 1+e and dzare not constants substituting dz = dzo(HH1) in (39)after cumbersome mathematical manipulation the resultingcoefficients are 1120591os

ko1205901015840o (120574k1205901015840 minus 120574v1205901015840 minus 1)120574w120574v12059010158402 ( 1205901015840m12059010158401 )1+2120574v1205901015840minus120574k1205901015840 (ef minus eo)H21

ko1205901015840o (1 + eo)120574w120574v1205901015840 ( 1205901015840m12059010158401 )1+120574v1205901015840minus120574k1205901015840 1

H21

(50)

Dividing by the last the resulting dimensionless groups are

120587I = 120591osko1205901015840o (1 + eo)120574w120574v1205901015840H21 ( 1205901015840m12059010158401 )1+120574v1205901015840minus120574k1205901015840

120587II = (1 minus 120574k1205901015840 + 120574v1205901015840) 12059010158402 minus 120590101584011205901015840m(51)

The solutions for 120591os and Us are

120591os asymp 120574w120574v1205901015840H21ko1205901015840o (1 + eo) ( 1205901015840m12059010158401 )+120574k1205901015840minus120574v1205901015840minus1Ψ120591 [(1 minus 120574k1205901015840

+ 120574v1205901015840) 12059010158402 minus 120590101584011205901015840m ](52)

Us = ΨU [ t120591os (1 minus 120574k1205901015840 + 120574v1205901015840) 12059010158402 minus 120590101584011205901015840m ] (53)

Table 1 summarizes the results derived in this section

4 Verification of the Results byNumerical Simulations

This section is devoted to checking the solutions given inTable 1 To shorten the exposure we restrict the tests forthe more general model (1+e and dz not constants) andfor the unknowns 120591os and Us since they are the ones ofgreater interest in civil engineering Eight sets of simulationsarranged in three blocks have been run Table 2 In each onesome of the soil parameters or initial values of the problemhave been changed to give the same or different values asrequired to the dimensionless group 120587II in the search for thesame or different solutions of 120587I Changes in the values of theindividual parameters are sufficient to cover all real scenarios

Firstly a reference set (set 1) is established to which all theother sets can be referred and which permits them to be com-pared with each other The physical and geometrical charac-teristics that change are 120574v1205901015840 120574k1205901015840 eo H1 (m) 1205901015840o (Nm2)1205901015840f (Nm2) and ko (myear) The values of =(1205901015840f1205901015840o) H2(m) kf (myear) and 120587II are derived from them while 1205871is obtained once the characteristic time 120591os is read from thesimulation The criterion for the choice of 120591os is the timerequired by the soil to reach 90 of the total settlement

It is worth mentioning here the emergence of 1205901015840m anaveraged value of the effective pressure in the two dimen-sionless groups Undoubtedly the existence of 1205901015840m in themonomials can be explained not only by the nonlinearityof the problem but also by the kind of dependences in theconstitutive relations (there are other nonlinear problems inwhich averaged variables do not emerge [18]) How do wechoose 1205901015840m We could take the mean value 1205901015840m=(12059010158401+12059010158402)2or even 1205901015840m=12059010158401 or 1205901015840m=12059010158402 but expecting that the expres-sion 120587I = 120595(120587II) is slightly dependent on that choice (we willreturn to this question later) In this paper we have taken theapproximation 1205901015840m=(12059010158402 + 12059010158401)2

In block I the parameters that change in each set areeo H1 ko and 1205901015840o while maintaining the ratio =(1205901015840f1205901015840o) thus both 120587II and factor (1205901015840m1205901015840o)1minus120574k1205901015840+120574v1205901015840 of 120587I takethe same value in the four sets of the block As a resultwe expect a characteristic time proportional to the ratio120574w120574v1205901015840H21ko1205901015840o(1 + eo) and a same value of 120587I in all casesaccording to the solution 120587I = 120595(120587II) Indeed the resultsare consistent with expectations Block II contains an onlycase for which 120587II varies in comparison to block I as a result120587I also changes as expected Finally block III contains threecases with the same value of 120587II of block I but with differentvalues of the parameters 120574v1205901015840 120574k1205901015840 and the ratio =(1205901015840f 1205901015840o)The parameters of set 6 have been chosen in such a waythat the values 120574k1205901015840 and 120574v1205901015840 are compensated to maintainconstant (1 minus 120574k1205901015840 + 120574v1205901015840) we expect 120587I to be also unchangedIn effect simulation provides 120587I = 07139 a value quite closeto that of set 1 (07131) The negligible differences are due nodoubt to the different evolution of the effective pressure alongthe process caused by the different values of the coefficients120574k1205901015840 and 120574v1205901015840 Note that as 120574v1205901015840 duplicates the denominator ofthe factor 120591osko1205901015840o(1 + eo)120574w120574v1205901015840H21 the characteristic timehas to double too as indeed happens The last two sets studythe influence of a parameter whose change forces the value

8 Mathematical Problems in Engineering

Table1Dim

ensio

nlessg

roup

sthatcharacterizethe

originalandextend

edJ-Bmod

els

Juarez-Badillo

original

Juarez-Badillo

Juarez-Badillo

Juarez-Badillo

1+e=

constant

1+e

=consta

nt1+e

=consta

ntdz

constant

dzconstant

dzvaria

ble

120587 I120591 ok o

1205901015840 o120574 w120574 v1205901015840H2 1

(1205901015840 m 1205901015840 o)1minus120574k1205901015840

120591 ok o1205901015840 o(

1+eo)

120574 w120574 v1205901015840H2 1

(1205901015840 m 1205901015840 o)1minus120574k1205901015840

120591 ok o1205901015840 o(

1+eo)

120574 w120574 v1205901015840H2 1

(1205901015840 m 1205901015840 o)1minus120574k1205901015840minus120574 v1205901015840

120591 ok o1205901015840 o(

1+eo)

120574 w120574 v1205901015840H2 1

(1205901015840 m 1205901015840 o)1minus120574k1205901015840+120574 v1205901015840

120591 o1205901015840or

120591 os120587 II

120574 k1205901015840(1205901015840 2minus

1205901015840 1 1205901015840 m)for

pressure

(1minus120574 k1205901015840

+120574 v1205901015840)(1205901015840 2minus

1205901015840 1 1205901015840 m)for

settlem

ent

Mathematical Problems in Engineering 9

0010203040506070809

1

0 05 1 15 2 25 3time

Set 1Set 2Set 3Set 4

Set 5Set 6Set 7Set 8

Aver

age d

egre

e of c

onso

lidat

ion

Figure 1 Average degree of consolidation versus time (eight sets)

0010203040506070809

1

0 05 1 15 2 25 3

Set 1Set 2Set 3Set 4

Set 5Set 6Set 7Set 8

Aver

age d

egre

e of c

onso

lidat

ion

tos

Figure 2 Average degree of consolidation versus dimensionlesstime (eight sets)

(1 minus 120574k1205901015840 + 120574v1205901015840) to be also changed in order to maintain 120587IIconstant We expect that 120587I does not change as nearly occursThe small differences are again due to the existence of 1205901015840mwithin the monomials Note however that as diminishesthe influence of 1205901015840m in the results also diminishes since thevalues of 12059010158401 and 12059010158402 are closer

For a better understanding of the results Figures 1 and2 are presented In the first Us as a function of time isshown for the eight sets in Table 2 It is clear that 120591oscovers a wide range of values from 02283 (set 2) to 27968(set 3) the last more than ten times the first The secondfigure graphs Us in front of the dimensionless time t120591os Asshown the eight previous curves nearly converge to a single(or universal) graph having a common point at (Us=09t120591os=1) This means that monomial 120587II for the range ofvalues assigned to the parameters in contrast to 120587I scarcelyinfluences the value of the average degree of consolidationThis is not in contradiction with the results derived bynondimensionalization

0010203040506070809

1

0 05 1 15 2 25 3 35 4time

U (J-B)U (1+e cst)U (1+e not cst)U (1+e amp dz not cst)

Us (J-B)Us (1+e cst)Us (1+e not cst)Us (1+e amp dz not cst)

Aver

age d

egre

e of p

ress

ure d

issip

atio

nAv

erag

e deg

ree o

f con

solid

atio

n

Figure 3 Average degree of consolidation and pressure dissipation(set 6)

Although in this sectionwe have only checked the expres-sions of Us and 120591os for the most general model we considerit interesting to present both Us and U1205901015840 for the rest of themodels of Table 1 and for set 6 as a typical case Figure 3Theseresults emphasize the importance of using the less restrictive(more general) model in relation to the others in particularwith the original model of J-B in which the influence ofeo is despised Note that the smaller of the characteristictimes (related to settlement) corresponds to the more generalmodel (black and bold line) so the use of the others althoughon the safety side provides characteristic times oversizedfar from an optimal solution for the engineer For examplefor the case of Figure 3 in which eo=1 the error of J-Boriginal model (purple and bold line) in comparison with theextended model (black and bold line) for Us=90 is around150 This error increases appreciably as eo gets larger

5 Universal Curves

These easy to use and universal abacuses obtained by numeri-cal simulations allow engineers to read the characteristic time(120591os) and the average degree of consolidation (Us) in termsof the dimensionless groups given in Table 1 Only the mostgeneral (less restrictive) J-B model 1+e and dz not constantis presented

To obtain the characteristic time curves a large numberof simulations have been carried out studying separately theinfluence of the parameters 120574v1205901015840 and 120574k1205901015840 whose valuescover most real soils ranges from 101 to 8 120574v1205901015840 from 005to 03 and 120574k1205901015840 from 01 to 17 From these tests it has beendeduced as expected from the results of the previous sectionthat the relatively more influential parameter is while 120574v1205901015840and 120574k1205901015840 do not produce significant changes in the values ofthe characteristic timeThus the first universal group of linesshows the dependence of 120587I versus 120587II for different valuesof (with 120574v1205901015840=01) Figure 4 As the range of values of 120587IIdepends on according to (43) the range of values of eachline is different As shown in the figure there exists a universal

10 Mathematical Problems in Engineering

Table 2 Values of the parameters and dimensionless groups of simulations

Block Set 120574v1205901015840 120574k1205901015840 eoH1 ko 1205901015840o 1205901015840f H2 kf 120587II

120591os 120587I(m) (myr) (Nm2) (Nm2) 1205901015840f 1205901015840o (m) (myr) (yr)

I

1 01 0500 10 100 002 30000 60000 2 0933 00141 040 04566 071312 01 0500 10 100 002 60000 120000 2 0933 00141 040 02283 071313 01 0500 10 175 001 30000 60000 2 1633 00071 040 27968 071314 01 0500 13 142 004 22000 44000 2 1325 00283 040 05459 07131

II 5 01 1500 10 100 002 30000 90000 3 0896 00038 -040 10597 09834

III6 02 0600 10 100 002 30000 60000 2 0871 00132 040 09142 071397 01 0767 10 100 002 30000 120000 4 0871 00069 040 04360 072468 02 0920 10 100 002 30000 180000 6 0699 00038 040 08409 07312

line for each value of all converging in the central zone forvalues of 120587II around 0 This set of straight lines can be fittedby 120587I=m120587II+n being n=0846 and m given in Figure 5 as afunction of The curve that best fits the points in Figure 5 isgiven by

m = minus0235e(minus0868) minus 0306e(minus0012) (54)

Choosing the longest line corresponding to =8 as the onlyrepresentative line of the problem themaximumerrors in thereading of 120587I (and 120591os) are 38

In short the use of these curves is as follows from the dataof the problem Figure 4 allows us to read 120587I from 120587II and and from (51) the characteristic time 120591os is derived

As for the average degree of consolidation the universalcurves for 96 different scenarios of 120587II are shown in Figure 6It is appreciated that these curves are very close to each othershowing again that the Us only depends on the dimensionlesstime In the worst case the errors produced by changes invalue of the 120587II group are less than 28when using the curveof Figure 7

6 Case Study

Below is a practical application of the universal curvespresented in this paper For this purpose we take as a basisfor our study the data obtained experimentally byAbbasi et al[26] for different samples of clay whosemoisture in the liquidlimit is equal to 42 As can be seen (Table 3) it is a series ofdiscrete (tabulated) values that relate the void ratio with theeffective pressure and the hydraulic conductivity

We will address a real case of consolidation in which wehave a layer of soil 2meters thick (Ho) with an initial effectivepressure (1205901015840o) of 28 kNm2 on which a load of 31 kNm2 isapplied In this way the effective pressure at the end of theconsolidation process (1205901015840f ) will have the value of 59 kNm2From the tabulated data (e-1205901015840) it follows that eo = 105 andef = 095 and since the factor 1+e is proportional to the soilvolume from (9) we have 120574v1205901015840 = 0067

To obtain 120574k1205901015840 we first calculate the value of 120574kv from thetabulated data (e-k) to then obtain 120574k1205901015840 from (10) With thiswe have for the load step considered ko = 0035 myr 120574kv =8108 and 120574k1205901015840 = 0543

To solve the problem we will use the universal curvespresented in the previous section corresponding to the most

Table 3 Experimental values of e 1205901015840 and k obtained by Abbasi et al[26]

e 1205901015840o (Nm2)110 14000105 28000095 59000085 120000075 220000065 450000060 925000e k (myr)175 0683160 0315120 0084085 0021065 0011

general J-B model (analyzed in terms of settlement) Thusfrom the values of 1205901015840o 1205901015840f 120574v1205901015840 and 120574k1205901015840 by means of (51) itis obtained that 120587II = 0373 (with 1205901015840m = 435 kNm2) Then120587I is read from Figure 4 where for = 21 we have 120587I =072 An identical result is obtained if we use the proposedfitted equation 120587I=m120587II+n where n is a constant of value0846 andm is obtained from (54) or Figure 5 Finally knownthe values of Ho ko and eo from (51) we get 120591os = 0748years Likewise once the characteristic time of the problem(in terms of settlement) is known from the universal curve ofFigure 7 we easily obtain the evolution of the average degreeof consolidation along the whole process Figure 8

These solutions have been compared with those thatresult from the numerical simulation of the most generalJ-B model for which a value of 120591os = 075 years has beenobtained showing clearly that the relative error committedwhen applying the universal solutions is negligible (Table 4)The simulation of the problem has also been carried out fora model where 1+e and dz are also not constant but in whichthe constitutive relations of potential type have been replacedby pairs of data in tabulated form obtaining in this case 120591os= 0817 years The relative error which this time increasesto 82 finds its explanation in the strong nonlinearity ofthe problem which will always bring differences between the

Mathematical Problems in Engineering 11

040045050055060065070075080085090095100105110

minus07 minus06 minus05 minus04 minus03 minus02 minus01 0 01 02 03 04 05 06 07 08 09 1 11 12 13 14 15

1

2

ᴪ=8ᴪ=6ᴪ=4

ᴪ=2ᴪ=15ᴪ=101

Figure 4 120587I as a function of 120587II with as parameter

1 2 3 4 5 6 7 8minus042

minus04

minus038

minus036

minus034

minus032

minus03

minus028

minus026

m

datafitted curve

Figure 5 m as a function of

solutions provided by numerical models based on differentconstitutive relations (potential and tabulated in this case)Even so this value is totally acceptable within this field ofengineering demonstrating the strength and versatility of theuniversal solutions provided

7 Discussion

This work derives the dimensionless groups that rule thenonlinear soil consolidation process based on the original J-Bmodel following a formal nondimensionalization procedurethat starts from the governing equation and does not needany other assumption or mathematical manipulation Inaddition more real and precise extended J-B models whichassume both the void ratio and the thickness of the finitevolume element of the soil to be not constant have been

0010203040506070809

1

0 05 1 15 2 25 3

96 scenariosᴪ = [101 15 2 4 6 8]

Aver

age d

egre

e of c

onso

lidat

ion

tos

= [005 01 015 02]v= [02 05 1 15]k

Figure 6 Average degree of consolidation versus dimensionlesstime for different scenarios

0010203040506070809

1

0 05 1 15 2 25 3

Aver

age d

egre

e of c

onso

lidat

ion

tos

Figure 7 Proposed solution for the average degree of consolidation

nondimensionalized and their dimensionless groups derivedThe assumption of constitutive dependences (e-1205901015840 e-k and k-1205901015840) of potential type in these models is interesting since they

12 Mathematical Problems in Engineering

Table 4 Case study solution for the characteristic time 120591os Comparison with numerical models

Universal curves J-B most general model Tabulated model120591os (yr) 120591os (yr) Rel error 120591os (yr) Rel error 0748 0750 026 0817 820

0010203040506070809

1

0 025 05 075 1 125 15 175 2 225t ( years)

Aver

age d

egre

e of c

onso

lidat

ion

Figure 8 Case study solution for the average degree of consolida-tion Us from the universal curves

can be applied to highly compressible soils for which the e-log(1205901015840) relation moves away from being a straight line andin which negative values of the void ratio could be obtainedunder the application of high load ratios (1205901015840f 1205901015840o) [8]

The study has been developed both in terms of pres-sure dissipation and in terms of settlement resulting thatthe dimensionless groups derived from each model are ingeneral different The extension to more complex modelswhich implies the emergence of new parameters or equationswithin themathematical model does not necessarily give riseto additional dimensionless groups as might be expectedbut to new expressions of the groups or new regroupingof the parameters involved Thus the elimination of thehypothesis ldquodz constantrdquo in the most general consolidationmodel has not given rise to any new group both in pressureand in settlement (Table 1) but it has only modified oneof the groups obtained in the original model All thisrepresents an important advance in relation to the modelsand studies carried out by other authors and the attemptsof characterization found in the scientific literature [9ndash11]which try to lead to easy-to-use solutions for engineers Inno case had dimensionless groups been obtained as preciseas those presented in this work nor presented universalcurves of the characteristic time and the average degree ofconsolidation Authors such asConte andTroncone [11] applymore general load conditions (time-dependent loading) forwhich the dimensionless groups deduced in this work couldbe applicable for each load step

As regards time to make it dimensionless an unknownreference (120591o) is introduced in the form mentioned by Scott[27] and used in many other problems with asymptoticsolutions [28]Theprecise definition of 120591o necessarily involvesa criterion in relation to the percentage of average pressuredecrease (or settlement) associated with the end of theconsolidation processTheorder ofmagnitude of 120591o is derived

after nondimensionalization making 120591o only appear in oneof the independent dimensionless groups In this way theaverage degree of consolidation (in terms of pressure orsettlement) depends on the ldquotimecharacteristic timerdquo ratioin addition to the dimensionless groups (without unknowns)that arise

In short the procedure has led to a new very precisesolution that involves two monomials one (120587I) that containsthe characteristic time (120591o) and another (120587II) that is amathematical function of the soil parameters In this way120591o only depends on 120587II through the function 120587I = 120595(120587II)while U depends on 120587I and 120587II through the expression U= 120595(120587I 120587II) The results obtained have been verified forthe most general and complex consolidation model throughthe numerical simulation of a large number of scenariosin which the values of the dimensionless groups and theindividual parameters of the problem have been changing toconvenience The high errors produced by the original J-Bmodel with respect to the most general justify the use of thelatter in the search for more precise solutions

In addition the emergence of mean values of the effectivepressure when averaging the governing equation in orderto assign an order of magnitude unit to the dimensionlessvariables and their derivatives is an added difficulty to obtainthe dimensionless groups This is undoubtedly due to thenonlinearities (constitutive relations) involved in the consol-idation process This average effective pressure of referencehas been assumed as the semisum of its initial and finalvalues although other criteria could also be chosen Sincetwo scenarios can have equal120587II with the same effectivemeanpressure but different values of the parameters 120574v1205901015840 and 120574k1205901015840 we expect some minor deviations between the results of bothscenarios as indeed it has been checked Thus for the wholerange of parameters values in typical soils the errors due tothe above cause are negligible in the field of soil mechanicsOn the other hand when the value of 120587II does not changebut the mean effective pressure value does the errors in thesolutions of 120591os and Us are slightly larger although withinranges below 4

The brooch for all this work is the presentation ofuniversal curves for the most general model which allowfor the usual range of variation of the soil properties toobtain in a simple way the value of the characteristic timeof consolidation and the evolution of the average degreeof settlement throughout the entire process Thus startingfrom the nonlinear parameters of the soil (120574v1205901015840 and 120574k1205901015840)and the load ratio (1205901015840f 1205901015840o) the value of the monomial 120587IIis obtained from which the value of 120587I is directly deductedthanks to the universal curve 120587I = 120595(120587II) provided Finally120591os is easily obtained from the expression of 120587I On the otherhand the curve that represents the Us evolution along theprocess has turned out to be (practically) unique when it is

Mathematical Problems in Engineering 13

represented as a function of the dimensionless time in theform t120591os regardless of the soil parameters and the loadsapplied All this supposes a clear advantage when solvingthe consolidation problem also showing the strength of thenondimensionalization method presented

Finally the case study addressed has demonstrated thesimplicity and reliability of the universal curves proposedFrom a series of tabulated data (e-1205901015840 and e-k) the parametersthat govern the nonlinear consolidation problem for themostextended J-B model have been adjusted with which thevalue of the monomial 120587II has been derived With this andusing the universal curves the solutions for the characteristictime 120591os and the average degree of consolidation Us havebeen obtained These results have been satisfactorily verifiedwith the solutions provided both by the numerical extendedmodel of J-B presented here and by another that substitutesthe constitutive dependences of potential type by data intabulated form

8 Conclusions

The application of the nondimensionalization technique hasallowed us to deduce the dimensionless groups that rule thenonlinear consolidation problem in which the constitutivedependencies e-1205901015840 and e-k have the form of potential func-tions This type of relations approximates the soil behaviorbetter than the logarithmic functions e-log(1205901015840) and e-log(k)particularly in very compressible soils and under high loadratios (1205901015840f 1205901015840o) in which inconsistent negative values of thevoid ratio may appear

Several scenarios have been addressed from the originalmodel of Juarez-Badillo to extended models based on thesame constitutive dependences but eliminating one or moreclassic (and restrictive) hypotheses in this process In allcases only two dimensionless groups emerge one of themconsisting of a grouping of the soil parameters and the othercontaining the unknown characteristic time Based on theseresults universal curves have been depicted the first to deter-mine the characteristic time of the process and the secondthat represents the evolution of average degree of settlement

The solutions have been verified by numerical simula-tions changing the values of the soil parameters and thedimensionless groups conveniently Finally the reliability andeffectiveness of the proposed solutions have been shownthrough a case study

Abbreviations

cv Consolidation coefficient (m2s or m2yr)cv1 Initial consolidation coefficient (m2s or

m2yr)cv2 Final consolidation coefficient (m2s or

m2yr)119889z Element of differential length in thedirection of the spatial coordinate z (m)119889zo Element of differential length in thedirection of the spatial coordinate z at theinitial time (m)

e Void ratio (dimensionless)

eo Initial void ratio (dimensionless)ef Final void ratio (dimensionless)H Soil thickness up to the impervious

boundary or drainage length (m)H1Ho Initial soil thickness (m)H2 Final soil thickness (m)k Hydraulic conductivity (or permeability)

(ms or myr)k1 ko Initial hydraulic conductivity (ms or

myr)kf Final hydraulic conductivity (ms or myr)mv Coefficient of volumetric compressibility

(m2N)mv1 Initial coefficient of volumetric

compressibility (m2N)t Time Independent variable (s or yr)T Juarez-Badillorsquos time factor

(dimensionless)u Excess pore pressure (Nm2)U Average degree of consolidation or

pressure dissipation (dimensionless)Us Average degree of consolidation

(dimensionless)U1205901015840 Average degree of pressure dissipation

(dimensionless)V Soil volume (m3)V1Vo Initial soil volume (m3)z Vertical spatial coordinate (m)120574v Nonlinear coefficient of compressibility of

Juarez-Badillo (dimensionless)120574kv Nonlinear coefficient of change ofpermeability with volume (dimensionless)120574k1205901015840 Nonlinear coefficient of change ofpermeability with effective pressure(dimensionless)120574v1205901015840 Nonlinear coefficient of compressibility orcoefficient of volumetric contraction(dimensionless)120574w Specific weight of water (Nm3)

Loading factor Ratio between the finaland initial effective pressure(dimensionless)120581 Nonlinear coefficient of change ofpermeability with volume ofJuarez-Badillo (dimensionless)120582 Parameter of Juarez-Badillo 120582 = 1 - 120574v120581(dimensionless)

V Auxiliary variable of Juarez-Badillo(dimensionless)120587i Dimensionless group or number(dimensionless)1205901015840 Effective pressure of the soil (Nm2)12059010158401 1205901015840o Initial effective pressure (Nm2)12059010158402 1205901015840f Final effective pressure (Nm2)1205901015840m Mean value of the effective pressure(Nm2)120591o Characteristic time (s or yr)

14 Mathematical Problems in Engineering

120591os Characteristic time which takes thesettlement to reach approximately its finalvalue (s or yr)120591o1205901015840 Characteristic time which takes the excesspore pressure to dissipate untilapproximately the value of zero (s or yr)Ψ Arbitrary mathematical function120577 Differential void ratio (dimensionless)1205771 1205772 Initial and final differential void ratio(dimensionless)

Subscripts

I II Denote different numbers

Superscripts

1015840 Denote dimensionless magnitude

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] K Terzaghi ldquoDie Berechnung der Durchlassigkeitszifferdestones aus dem verlauf der hydrodynamischen spanunngser-scheinnungenrdquo Technical report II a 132 N 34 125-138Akademie derWissenschaften inWien SitzungsberichteMath-naturwiss Klasse Abt 1923

[2] R T Walker ldquoVertical drain consolidation analysis in one twoand three dimensionsrdquo Computers amp Geosciences vol 38 no 8pp 1069ndash1077 2011

[3] G Y Zheng P Li and C Y Zhao ldquoAnalysis of non-linearconsolidation of soft clay by differential quadrature methodrdquoApplied Clay Science vol 79 pp 2ndash7 2013

[4] X Y Geng C J Xu and Y Q Cai ldquoNon-linear consolidationanalysis of soil with variable compressibility and permeabilityunder cyclic loadingsrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 30 no 8 pp 803ndash8212006

[5] M Lu S Wang S W Sloan B Indraratna and K Xie ldquoNon-linear radial consolidation of vertical drains under a generaltime-variable loadingrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 39 no 1 pp 51ndash622015

[6] HWu L HuW Qi and QWen ldquoAnalytical Solution for Elec-troosmotic Consolidation Considering Nonlinear Variation ofSoil Parametersrdquo International Journal of Geomechanics vol 17no 5 Article ID 06016032 2017

[7] S J Brandenberg ldquoiConsoljs JavaScript Implicit Finite-Differ-ence Code for Nonlinear Consolidation and Secondary Com-pressionrdquo International Journal of Geomechanics vol 17 no 6Article ID 04016149 2017

[8] R Butterfield ldquoA natural compression law for soils (an advanceon e-log prsquo)rdquo Geotechnique vol 29 no 4 pp 469ndash480 1979

[9] R Lancellotta and L Preziosi ldquoA general nonlinear mathemati-calmodel for soil consolidation problemsrdquo International Journalof Engineering Science vol 35 no 10-11 pp 1045ndash1063 1997

[10] Y-C Zhuang K-H Xie and X-B Li ldquoNonlinear analysis ofconsolidation with variable compressibility and permeabilityrdquoJournal of Zhejiang University (Engineering Science Edition) vol6 no 3 pp 181ndash187 2005

[11] E Conte andA Troncone ldquoNonlinear consolidation of thin lay-ers subjected to time-dependent loadingrdquo Canadian Geotechni-cal Journal vol 44 no 6 pp 717ndash725 2007

[12] R Yong and F Townsend ldquoGeneral Consolidation Theory forClaysrdquo Soil Mechanics Series Report No 8 Graduate School ofEngineering National University of Mexico 1983

[13] J C Gibbings Dimensional Analysis Springer-Verlag LondonUK 2011

[14] E Buckingham ldquoOn physically similar systems Illustrations ofthe use of dimensional equationsrdquo Physical Review A AtomicMolecular and Optical Physics vol 4 no 4 pp 345ndash376 1914

[15] R Seshadri and T Y Na ldquoA Survey ofMethods for DeterminingSimilarity Transformationsrdquo inGroup Invariance in EngineeringBoundary Value Problems pp 35ndash61 Springer New York NYUSA 1985

[16] V D Zimparov and V M Petkov ldquoApplication of discriminatedanalysis to low Reynolds number swirl flows in circular tubeswith twisted-tape inserts Pressure drop correlationsrdquo Interna-tional Review of Chemical Engineering vol 1 no 4 pp 346ndash3562009

[17] M Capobianchi and A Aziz ldquoA scale analysis for natural con-vective flows over vertical surfacesrdquo International Journal ofThermal Sciences vol 54 pp 82ndash88 2012

[18] M Conesa J F Sanchez Perez I Alhama and F AlhamaldquoOn the nondimensionalization of coupled nonlinear ordinarydifferential equationsrdquo Nonlinear Dynamics vol 84 no 1 pp91ndash105 2016

[19] I A Manteca G Garcıa-Ros and F Alhama ldquoUniversal solu-tion for the characteristic time and the degree of settlement innonlinear soil consolidation scenarios A deduction based onnondimensionalizationrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 57 pp 186ndash201 2018

[20] E H Davis and G P Raymond ldquoA non-linear theory ofconsolidationrdquo Geotechnique vol 15 no 2 pp 161ndash173 1965

[21] P Cornetti and M Battaglio ldquoNonlinear consolidation of soilmodeling and solution techniquesrdquo Mathematical and Com-puter Modelling vol 20 no 7 pp 1ndash12 1994

[22] C F Gonzalez-Fernandez ldquoApplications of the network simu-lation method to transport processesrdquo in Network SimulationMethod J Horno Ed Research Signpost Trivandrum India2002

[23] J Serna F J S Velasco and A Soto Meca ldquoApplication of net-work simulationmethod to viscous flowsThe nanofluid heatedlid cavity under pulsating flowrdquo Computers amp Fluids vol 91 pp10ndash20 2014

[24] E Juarez-Badillo ldquoGeneral Permeability Change Equationsfor Soilsrdquo in Proceedings of the International Conference onConstitutive Laws pp 205ndash209 University of Arizona TuesonAriz USA 1983

[25] E Juarez-Badillo and B Chen ldquoConsolidation curves for claysrdquoJournal of Geotechnical Engineering vol 109 no 10 pp 1303ndash1312 1983

Mathematical Problems in Engineering 15

[26] N Abbasi H Rahimi A A Javadi andA Fakher ldquoFinite differ-ence approach for consolidation with variable compressibilityand permeabilityrdquo Computers amp Geosciences vol 34 no 1 pp41ndash52 2007

[27] R F Scott Principles of Soils Mechanics Addison-Wesley Pub-lishing Company 1963

[28] F Alhama and C N Madrid Analisis Dimensional Discrimi-nado En Mecanica De Fluidos Y Transferencia De Calor 2012

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Submit your manuscripts atwwwhindawicom

Mathematical Problems in Engineering 7

provide the dimensionless groups

120587I = 120591osko1205901015840o (1 + eo)120574w120574v1205901015840H21 ( 1205901015840m12059010158401 )1minus120574k1205901015840

120587II = (1 minus 120574k1205901015840 + 120574v1205901015840) 12059010158402 minus 120590101584011205901015840m(47)

The solutions for 120591os and Us are

120591os asymp 120574w120574v1205901015840H21ko1205901015840o (1 + eo) ( 1205901015840m12059010158401 )120574k1205901015840minus1Ψ120591 [(1 minus 120574k1205901015840

+ 120574v1205901015840) 12059010158402 minus 120590101584011205901015840m ](48)

Us = ΨU [ t120591os (1 minus 120574k1205901015840 + 120574v1205901015840) 12059010158402 minus 120590101584011205901015840m ] (49)

The solutions for the original J-Bmodel are also (48) and (49)with the simplification of eo=0 The author however doesnot talk about the characteristic time and obtains the averagedegree of consolidation given by (7) in which this variabledepends on three groups (1minus120574k1205901015840)120574v1205901015840 H2H1 and tcv2H21undoubtedly a less precise solution

323 1+e and dz Not Being Constant When 1+e and dzare not constants substituting dz = dzo(HH1) in (39)after cumbersome mathematical manipulation the resultingcoefficients are 1120591os

ko1205901015840o (120574k1205901015840 minus 120574v1205901015840 minus 1)120574w120574v12059010158402 ( 1205901015840m12059010158401 )1+2120574v1205901015840minus120574k1205901015840 (ef minus eo)H21

ko1205901015840o (1 + eo)120574w120574v1205901015840 ( 1205901015840m12059010158401 )1+120574v1205901015840minus120574k1205901015840 1

H21

(50)

Dividing by the last the resulting dimensionless groups are

120587I = 120591osko1205901015840o (1 + eo)120574w120574v1205901015840H21 ( 1205901015840m12059010158401 )1+120574v1205901015840minus120574k1205901015840

120587II = (1 minus 120574k1205901015840 + 120574v1205901015840) 12059010158402 minus 120590101584011205901015840m(51)

The solutions for 120591os and Us are

120591os asymp 120574w120574v1205901015840H21ko1205901015840o (1 + eo) ( 1205901015840m12059010158401 )+120574k1205901015840minus120574v1205901015840minus1Ψ120591 [(1 minus 120574k1205901015840

+ 120574v1205901015840) 12059010158402 minus 120590101584011205901015840m ](52)

Us = ΨU [ t120591os (1 minus 120574k1205901015840 + 120574v1205901015840) 12059010158402 minus 120590101584011205901015840m ] (53)

Table 1 summarizes the results derived in this section

4 Verification of the Results byNumerical Simulations

This section is devoted to checking the solutions given inTable 1 To shorten the exposure we restrict the tests forthe more general model (1+e and dz not constants) andfor the unknowns 120591os and Us since they are the ones ofgreater interest in civil engineering Eight sets of simulationsarranged in three blocks have been run Table 2 In each onesome of the soil parameters or initial values of the problemhave been changed to give the same or different values asrequired to the dimensionless group 120587II in the search for thesame or different solutions of 120587I Changes in the values of theindividual parameters are sufficient to cover all real scenarios

Firstly a reference set (set 1) is established to which all theother sets can be referred and which permits them to be com-pared with each other The physical and geometrical charac-teristics that change are 120574v1205901015840 120574k1205901015840 eo H1 (m) 1205901015840o (Nm2)1205901015840f (Nm2) and ko (myear) The values of =(1205901015840f1205901015840o) H2(m) kf (myear) and 120587II are derived from them while 1205871is obtained once the characteristic time 120591os is read from thesimulation The criterion for the choice of 120591os is the timerequired by the soil to reach 90 of the total settlement

It is worth mentioning here the emergence of 1205901015840m anaveraged value of the effective pressure in the two dimen-sionless groups Undoubtedly the existence of 1205901015840m in themonomials can be explained not only by the nonlinearityof the problem but also by the kind of dependences in theconstitutive relations (there are other nonlinear problems inwhich averaged variables do not emerge [18]) How do wechoose 1205901015840m We could take the mean value 1205901015840m=(12059010158401+12059010158402)2or even 1205901015840m=12059010158401 or 1205901015840m=12059010158402 but expecting that the expres-sion 120587I = 120595(120587II) is slightly dependent on that choice (we willreturn to this question later) In this paper we have taken theapproximation 1205901015840m=(12059010158402 + 12059010158401)2

In block I the parameters that change in each set areeo H1 ko and 1205901015840o while maintaining the ratio =(1205901015840f1205901015840o) thus both 120587II and factor (1205901015840m1205901015840o)1minus120574k1205901015840+120574v1205901015840 of 120587I takethe same value in the four sets of the block As a resultwe expect a characteristic time proportional to the ratio120574w120574v1205901015840H21ko1205901015840o(1 + eo) and a same value of 120587I in all casesaccording to the solution 120587I = 120595(120587II) Indeed the resultsare consistent with expectations Block II contains an onlycase for which 120587II varies in comparison to block I as a result120587I also changes as expected Finally block III contains threecases with the same value of 120587II of block I but with differentvalues of the parameters 120574v1205901015840 120574k1205901015840 and the ratio =(1205901015840f 1205901015840o)The parameters of set 6 have been chosen in such a waythat the values 120574k1205901015840 and 120574v1205901015840 are compensated to maintainconstant (1 minus 120574k1205901015840 + 120574v1205901015840) we expect 120587I to be also unchangedIn effect simulation provides 120587I = 07139 a value quite closeto that of set 1 (07131) The negligible differences are due nodoubt to the different evolution of the effective pressure alongthe process caused by the different values of the coefficients120574k1205901015840 and 120574v1205901015840 Note that as 120574v1205901015840 duplicates the denominator ofthe factor 120591osko1205901015840o(1 + eo)120574w120574v1205901015840H21 the characteristic timehas to double too as indeed happens The last two sets studythe influence of a parameter whose change forces the value

8 Mathematical Problems in Engineering

Table1Dim

ensio

nlessg

roup

sthatcharacterizethe

originalandextend

edJ-Bmod

els

Juarez-Badillo

original

Juarez-Badillo

Juarez-Badillo

Juarez-Badillo

1+e=

constant

1+e

=consta

nt1+e

=consta

ntdz

constant

dzconstant

dzvaria

ble

120587 I120591 ok o

1205901015840 o120574 w120574 v1205901015840H2 1

(1205901015840 m 1205901015840 o)1minus120574k1205901015840

120591 ok o1205901015840 o(

1+eo)

120574 w120574 v1205901015840H2 1

(1205901015840 m 1205901015840 o)1minus120574k1205901015840

120591 ok o1205901015840 o(

1+eo)

120574 w120574 v1205901015840H2 1

(1205901015840 m 1205901015840 o)1minus120574k1205901015840minus120574 v1205901015840

120591 ok o1205901015840 o(

1+eo)

120574 w120574 v1205901015840H2 1

(1205901015840 m 1205901015840 o)1minus120574k1205901015840+120574 v1205901015840

120591 o1205901015840or

120591 os120587 II

120574 k1205901015840(1205901015840 2minus

1205901015840 1 1205901015840 m)for

pressure

(1minus120574 k1205901015840

+120574 v1205901015840)(1205901015840 2minus

1205901015840 1 1205901015840 m)for

settlem

ent

Mathematical Problems in Engineering 9

0010203040506070809

1

0 05 1 15 2 25 3time

Set 1Set 2Set 3Set 4

Set 5Set 6Set 7Set 8

Aver

age d

egre

e of c

onso

lidat

ion

Figure 1 Average degree of consolidation versus time (eight sets)

0010203040506070809

1

0 05 1 15 2 25 3

Set 1Set 2Set 3Set 4

Set 5Set 6Set 7Set 8

Aver

age d

egre

e of c

onso

lidat

ion

tos

Figure 2 Average degree of consolidation versus dimensionlesstime (eight sets)

(1 minus 120574k1205901015840 + 120574v1205901015840) to be also changed in order to maintain 120587IIconstant We expect that 120587I does not change as nearly occursThe small differences are again due to the existence of 1205901015840mwithin the monomials Note however that as diminishesthe influence of 1205901015840m in the results also diminishes since thevalues of 12059010158401 and 12059010158402 are closer

For a better understanding of the results Figures 1 and2 are presented In the first Us as a function of time isshown for the eight sets in Table 2 It is clear that 120591oscovers a wide range of values from 02283 (set 2) to 27968(set 3) the last more than ten times the first The secondfigure graphs Us in front of the dimensionless time t120591os Asshown the eight previous curves nearly converge to a single(or universal) graph having a common point at (Us=09t120591os=1) This means that monomial 120587II for the range ofvalues assigned to the parameters in contrast to 120587I scarcelyinfluences the value of the average degree of consolidationThis is not in contradiction with the results derived bynondimensionalization

0010203040506070809

1

0 05 1 15 2 25 3 35 4time

U (J-B)U (1+e cst)U (1+e not cst)U (1+e amp dz not cst)

Us (J-B)Us (1+e cst)Us (1+e not cst)Us (1+e amp dz not cst)

Aver

age d

egre

e of p

ress

ure d

issip

atio

nAv

erag

e deg

ree o

f con

solid

atio

n

Figure 3 Average degree of consolidation and pressure dissipation(set 6)

Although in this sectionwe have only checked the expres-sions of Us and 120591os for the most general model we considerit interesting to present both Us and U1205901015840 for the rest of themodels of Table 1 and for set 6 as a typical case Figure 3Theseresults emphasize the importance of using the less restrictive(more general) model in relation to the others in particularwith the original model of J-B in which the influence ofeo is despised Note that the smaller of the characteristictimes (related to settlement) corresponds to the more generalmodel (black and bold line) so the use of the others althoughon the safety side provides characteristic times oversizedfar from an optimal solution for the engineer For examplefor the case of Figure 3 in which eo=1 the error of J-Boriginal model (purple and bold line) in comparison with theextended model (black and bold line) for Us=90 is around150 This error increases appreciably as eo gets larger

5 Universal Curves

These easy to use and universal abacuses obtained by numeri-cal simulations allow engineers to read the characteristic time(120591os) and the average degree of consolidation (Us) in termsof the dimensionless groups given in Table 1 Only the mostgeneral (less restrictive) J-B model 1+e and dz not constantis presented

To obtain the characteristic time curves a large numberof simulations have been carried out studying separately theinfluence of the parameters 120574v1205901015840 and 120574k1205901015840 whose valuescover most real soils ranges from 101 to 8 120574v1205901015840 from 005to 03 and 120574k1205901015840 from 01 to 17 From these tests it has beendeduced as expected from the results of the previous sectionthat the relatively more influential parameter is while 120574v1205901015840and 120574k1205901015840 do not produce significant changes in the values ofthe characteristic timeThus the first universal group of linesshows the dependence of 120587I versus 120587II for different valuesof (with 120574v1205901015840=01) Figure 4 As the range of values of 120587IIdepends on according to (43) the range of values of eachline is different As shown in the figure there exists a universal

10 Mathematical Problems in Engineering

Table 2 Values of the parameters and dimensionless groups of simulations

Block Set 120574v1205901015840 120574k1205901015840 eoH1 ko 1205901015840o 1205901015840f H2 kf 120587II

120591os 120587I(m) (myr) (Nm2) (Nm2) 1205901015840f 1205901015840o (m) (myr) (yr)

I

1 01 0500 10 100 002 30000 60000 2 0933 00141 040 04566 071312 01 0500 10 100 002 60000 120000 2 0933 00141 040 02283 071313 01 0500 10 175 001 30000 60000 2 1633 00071 040 27968 071314 01 0500 13 142 004 22000 44000 2 1325 00283 040 05459 07131

II 5 01 1500 10 100 002 30000 90000 3 0896 00038 -040 10597 09834

III6 02 0600 10 100 002 30000 60000 2 0871 00132 040 09142 071397 01 0767 10 100 002 30000 120000 4 0871 00069 040 04360 072468 02 0920 10 100 002 30000 180000 6 0699 00038 040 08409 07312

line for each value of all converging in the central zone forvalues of 120587II around 0 This set of straight lines can be fittedby 120587I=m120587II+n being n=0846 and m given in Figure 5 as afunction of The curve that best fits the points in Figure 5 isgiven by

m = minus0235e(minus0868) minus 0306e(minus0012) (54)

Choosing the longest line corresponding to =8 as the onlyrepresentative line of the problem themaximumerrors in thereading of 120587I (and 120591os) are 38

In short the use of these curves is as follows from the dataof the problem Figure 4 allows us to read 120587I from 120587II and and from (51) the characteristic time 120591os is derived

As for the average degree of consolidation the universalcurves for 96 different scenarios of 120587II are shown in Figure 6It is appreciated that these curves are very close to each othershowing again that the Us only depends on the dimensionlesstime In the worst case the errors produced by changes invalue of the 120587II group are less than 28when using the curveof Figure 7

6 Case Study

Below is a practical application of the universal curvespresented in this paper For this purpose we take as a basisfor our study the data obtained experimentally byAbbasi et al[26] for different samples of clay whosemoisture in the liquidlimit is equal to 42 As can be seen (Table 3) it is a series ofdiscrete (tabulated) values that relate the void ratio with theeffective pressure and the hydraulic conductivity

We will address a real case of consolidation in which wehave a layer of soil 2meters thick (Ho) with an initial effectivepressure (1205901015840o) of 28 kNm2 on which a load of 31 kNm2 isapplied In this way the effective pressure at the end of theconsolidation process (1205901015840f ) will have the value of 59 kNm2From the tabulated data (e-1205901015840) it follows that eo = 105 andef = 095 and since the factor 1+e is proportional to the soilvolume from (9) we have 120574v1205901015840 = 0067

To obtain 120574k1205901015840 we first calculate the value of 120574kv from thetabulated data (e-k) to then obtain 120574k1205901015840 from (10) With thiswe have for the load step considered ko = 0035 myr 120574kv =8108 and 120574k1205901015840 = 0543

To solve the problem we will use the universal curvespresented in the previous section corresponding to the most

Table 3 Experimental values of e 1205901015840 and k obtained by Abbasi et al[26]

e 1205901015840o (Nm2)110 14000105 28000095 59000085 120000075 220000065 450000060 925000e k (myr)175 0683160 0315120 0084085 0021065 0011

general J-B model (analyzed in terms of settlement) Thusfrom the values of 1205901015840o 1205901015840f 120574v1205901015840 and 120574k1205901015840 by means of (51) itis obtained that 120587II = 0373 (with 1205901015840m = 435 kNm2) Then120587I is read from Figure 4 where for = 21 we have 120587I =072 An identical result is obtained if we use the proposedfitted equation 120587I=m120587II+n where n is a constant of value0846 andm is obtained from (54) or Figure 5 Finally knownthe values of Ho ko and eo from (51) we get 120591os = 0748years Likewise once the characteristic time of the problem(in terms of settlement) is known from the universal curve ofFigure 7 we easily obtain the evolution of the average degreeof consolidation along the whole process Figure 8

These solutions have been compared with those thatresult from the numerical simulation of the most generalJ-B model for which a value of 120591os = 075 years has beenobtained showing clearly that the relative error committedwhen applying the universal solutions is negligible (Table 4)The simulation of the problem has also been carried out fora model where 1+e and dz are also not constant but in whichthe constitutive relations of potential type have been replacedby pairs of data in tabulated form obtaining in this case 120591os= 0817 years The relative error which this time increasesto 82 finds its explanation in the strong nonlinearity ofthe problem which will always bring differences between the

Mathematical Problems in Engineering 11

040045050055060065070075080085090095100105110

minus07 minus06 minus05 minus04 minus03 minus02 minus01 0 01 02 03 04 05 06 07 08 09 1 11 12 13 14 15

1

2

ᴪ=8ᴪ=6ᴪ=4

ᴪ=2ᴪ=15ᴪ=101

Figure 4 120587I as a function of 120587II with as parameter

1 2 3 4 5 6 7 8minus042

minus04

minus038

minus036

minus034

minus032

minus03

minus028

minus026

m

datafitted curve

Figure 5 m as a function of

solutions provided by numerical models based on differentconstitutive relations (potential and tabulated in this case)Even so this value is totally acceptable within this field ofengineering demonstrating the strength and versatility of theuniversal solutions provided

7 Discussion

This work derives the dimensionless groups that rule thenonlinear soil consolidation process based on the original J-Bmodel following a formal nondimensionalization procedurethat starts from the governing equation and does not needany other assumption or mathematical manipulation Inaddition more real and precise extended J-B models whichassume both the void ratio and the thickness of the finitevolume element of the soil to be not constant have been

0010203040506070809

1

0 05 1 15 2 25 3

96 scenariosᴪ = [101 15 2 4 6 8]

Aver

age d

egre

e of c

onso

lidat

ion

tos

= [005 01 015 02]v= [02 05 1 15]k

Figure 6 Average degree of consolidation versus dimensionlesstime for different scenarios

0010203040506070809

1

0 05 1 15 2 25 3

Aver

age d

egre

e of c

onso

lidat

ion

tos

Figure 7 Proposed solution for the average degree of consolidation

nondimensionalized and their dimensionless groups derivedThe assumption of constitutive dependences (e-1205901015840 e-k and k-1205901015840) of potential type in these models is interesting since they

12 Mathematical Problems in Engineering

Table 4 Case study solution for the characteristic time 120591os Comparison with numerical models

Universal curves J-B most general model Tabulated model120591os (yr) 120591os (yr) Rel error 120591os (yr) Rel error 0748 0750 026 0817 820

0010203040506070809

1

0 025 05 075 1 125 15 175 2 225t ( years)

Aver

age d

egre

e of c

onso

lidat

ion

Figure 8 Case study solution for the average degree of consolida-tion Us from the universal curves

can be applied to highly compressible soils for which the e-log(1205901015840) relation moves away from being a straight line andin which negative values of the void ratio could be obtainedunder the application of high load ratios (1205901015840f 1205901015840o) [8]

The study has been developed both in terms of pres-sure dissipation and in terms of settlement resulting thatthe dimensionless groups derived from each model are ingeneral different The extension to more complex modelswhich implies the emergence of new parameters or equationswithin themathematical model does not necessarily give riseto additional dimensionless groups as might be expectedbut to new expressions of the groups or new regroupingof the parameters involved Thus the elimination of thehypothesis ldquodz constantrdquo in the most general consolidationmodel has not given rise to any new group both in pressureand in settlement (Table 1) but it has only modified oneof the groups obtained in the original model All thisrepresents an important advance in relation to the modelsand studies carried out by other authors and the attemptsof characterization found in the scientific literature [9ndash11]which try to lead to easy-to-use solutions for engineers Inno case had dimensionless groups been obtained as preciseas those presented in this work nor presented universalcurves of the characteristic time and the average degree ofconsolidation Authors such asConte andTroncone [11] applymore general load conditions (time-dependent loading) forwhich the dimensionless groups deduced in this work couldbe applicable for each load step

As regards time to make it dimensionless an unknownreference (120591o) is introduced in the form mentioned by Scott[27] and used in many other problems with asymptoticsolutions [28]Theprecise definition of 120591o necessarily involvesa criterion in relation to the percentage of average pressuredecrease (or settlement) associated with the end of theconsolidation processTheorder ofmagnitude of 120591o is derived

after nondimensionalization making 120591o only appear in oneof the independent dimensionless groups In this way theaverage degree of consolidation (in terms of pressure orsettlement) depends on the ldquotimecharacteristic timerdquo ratioin addition to the dimensionless groups (without unknowns)that arise

In short the procedure has led to a new very precisesolution that involves two monomials one (120587I) that containsthe characteristic time (120591o) and another (120587II) that is amathematical function of the soil parameters In this way120591o only depends on 120587II through the function 120587I = 120595(120587II)while U depends on 120587I and 120587II through the expression U= 120595(120587I 120587II) The results obtained have been verified forthe most general and complex consolidation model throughthe numerical simulation of a large number of scenariosin which the values of the dimensionless groups and theindividual parameters of the problem have been changing toconvenience The high errors produced by the original J-Bmodel with respect to the most general justify the use of thelatter in the search for more precise solutions

In addition the emergence of mean values of the effectivepressure when averaging the governing equation in orderto assign an order of magnitude unit to the dimensionlessvariables and their derivatives is an added difficulty to obtainthe dimensionless groups This is undoubtedly due to thenonlinearities (constitutive relations) involved in the consol-idation process This average effective pressure of referencehas been assumed as the semisum of its initial and finalvalues although other criteria could also be chosen Sincetwo scenarios can have equal120587II with the same effectivemeanpressure but different values of the parameters 120574v1205901015840 and 120574k1205901015840 we expect some minor deviations between the results of bothscenarios as indeed it has been checked Thus for the wholerange of parameters values in typical soils the errors due tothe above cause are negligible in the field of soil mechanicsOn the other hand when the value of 120587II does not changebut the mean effective pressure value does the errors in thesolutions of 120591os and Us are slightly larger although withinranges below 4

The brooch for all this work is the presentation ofuniversal curves for the most general model which allowfor the usual range of variation of the soil properties toobtain in a simple way the value of the characteristic timeof consolidation and the evolution of the average degreeof settlement throughout the entire process Thus startingfrom the nonlinear parameters of the soil (120574v1205901015840 and 120574k1205901015840)and the load ratio (1205901015840f 1205901015840o) the value of the monomial 120587IIis obtained from which the value of 120587I is directly deductedthanks to the universal curve 120587I = 120595(120587II) provided Finally120591os is easily obtained from the expression of 120587I On the otherhand the curve that represents the Us evolution along theprocess has turned out to be (practically) unique when it is

Mathematical Problems in Engineering 13

represented as a function of the dimensionless time in theform t120591os regardless of the soil parameters and the loadsapplied All this supposes a clear advantage when solvingthe consolidation problem also showing the strength of thenondimensionalization method presented

Finally the case study addressed has demonstrated thesimplicity and reliability of the universal curves proposedFrom a series of tabulated data (e-1205901015840 and e-k) the parametersthat govern the nonlinear consolidation problem for themostextended J-B model have been adjusted with which thevalue of the monomial 120587II has been derived With this andusing the universal curves the solutions for the characteristictime 120591os and the average degree of consolidation Us havebeen obtained These results have been satisfactorily verifiedwith the solutions provided both by the numerical extendedmodel of J-B presented here and by another that substitutesthe constitutive dependences of potential type by data intabulated form

8 Conclusions

The application of the nondimensionalization technique hasallowed us to deduce the dimensionless groups that rule thenonlinear consolidation problem in which the constitutivedependencies e-1205901015840 and e-k have the form of potential func-tions This type of relations approximates the soil behaviorbetter than the logarithmic functions e-log(1205901015840) and e-log(k)particularly in very compressible soils and under high loadratios (1205901015840f 1205901015840o) in which inconsistent negative values of thevoid ratio may appear

Several scenarios have been addressed from the originalmodel of Juarez-Badillo to extended models based on thesame constitutive dependences but eliminating one or moreclassic (and restrictive) hypotheses in this process In allcases only two dimensionless groups emerge one of themconsisting of a grouping of the soil parameters and the othercontaining the unknown characteristic time Based on theseresults universal curves have been depicted the first to deter-mine the characteristic time of the process and the secondthat represents the evolution of average degree of settlement

The solutions have been verified by numerical simula-tions changing the values of the soil parameters and thedimensionless groups conveniently Finally the reliability andeffectiveness of the proposed solutions have been shownthrough a case study

Abbreviations

cv Consolidation coefficient (m2s or m2yr)cv1 Initial consolidation coefficient (m2s or

m2yr)cv2 Final consolidation coefficient (m2s or

m2yr)119889z Element of differential length in thedirection of the spatial coordinate z (m)119889zo Element of differential length in thedirection of the spatial coordinate z at theinitial time (m)

e Void ratio (dimensionless)

eo Initial void ratio (dimensionless)ef Final void ratio (dimensionless)H Soil thickness up to the impervious

boundary or drainage length (m)H1Ho Initial soil thickness (m)H2 Final soil thickness (m)k Hydraulic conductivity (or permeability)

(ms or myr)k1 ko Initial hydraulic conductivity (ms or

myr)kf Final hydraulic conductivity (ms or myr)mv Coefficient of volumetric compressibility

(m2N)mv1 Initial coefficient of volumetric

compressibility (m2N)t Time Independent variable (s or yr)T Juarez-Badillorsquos time factor

(dimensionless)u Excess pore pressure (Nm2)U Average degree of consolidation or

pressure dissipation (dimensionless)Us Average degree of consolidation

(dimensionless)U1205901015840 Average degree of pressure dissipation

(dimensionless)V Soil volume (m3)V1Vo Initial soil volume (m3)z Vertical spatial coordinate (m)120574v Nonlinear coefficient of compressibility of

Juarez-Badillo (dimensionless)120574kv Nonlinear coefficient of change ofpermeability with volume (dimensionless)120574k1205901015840 Nonlinear coefficient of change ofpermeability with effective pressure(dimensionless)120574v1205901015840 Nonlinear coefficient of compressibility orcoefficient of volumetric contraction(dimensionless)120574w Specific weight of water (Nm3)

Loading factor Ratio between the finaland initial effective pressure(dimensionless)120581 Nonlinear coefficient of change ofpermeability with volume ofJuarez-Badillo (dimensionless)120582 Parameter of Juarez-Badillo 120582 = 1 - 120574v120581(dimensionless)

V Auxiliary variable of Juarez-Badillo(dimensionless)120587i Dimensionless group or number(dimensionless)1205901015840 Effective pressure of the soil (Nm2)12059010158401 1205901015840o Initial effective pressure (Nm2)12059010158402 1205901015840f Final effective pressure (Nm2)1205901015840m Mean value of the effective pressure(Nm2)120591o Characteristic time (s or yr)

14 Mathematical Problems in Engineering

120591os Characteristic time which takes thesettlement to reach approximately its finalvalue (s or yr)120591o1205901015840 Characteristic time which takes the excesspore pressure to dissipate untilapproximately the value of zero (s or yr)Ψ Arbitrary mathematical function120577 Differential void ratio (dimensionless)1205771 1205772 Initial and final differential void ratio(dimensionless)

Subscripts

I II Denote different numbers

Superscripts

1015840 Denote dimensionless magnitude

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] K Terzaghi ldquoDie Berechnung der Durchlassigkeitszifferdestones aus dem verlauf der hydrodynamischen spanunngser-scheinnungenrdquo Technical report II a 132 N 34 125-138Akademie derWissenschaften inWien SitzungsberichteMath-naturwiss Klasse Abt 1923

[2] R T Walker ldquoVertical drain consolidation analysis in one twoand three dimensionsrdquo Computers amp Geosciences vol 38 no 8pp 1069ndash1077 2011

[3] G Y Zheng P Li and C Y Zhao ldquoAnalysis of non-linearconsolidation of soft clay by differential quadrature methodrdquoApplied Clay Science vol 79 pp 2ndash7 2013

[4] X Y Geng C J Xu and Y Q Cai ldquoNon-linear consolidationanalysis of soil with variable compressibility and permeabilityunder cyclic loadingsrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 30 no 8 pp 803ndash8212006

[5] M Lu S Wang S W Sloan B Indraratna and K Xie ldquoNon-linear radial consolidation of vertical drains under a generaltime-variable loadingrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 39 no 1 pp 51ndash622015

[6] HWu L HuW Qi and QWen ldquoAnalytical Solution for Elec-troosmotic Consolidation Considering Nonlinear Variation ofSoil Parametersrdquo International Journal of Geomechanics vol 17no 5 Article ID 06016032 2017

[7] S J Brandenberg ldquoiConsoljs JavaScript Implicit Finite-Differ-ence Code for Nonlinear Consolidation and Secondary Com-pressionrdquo International Journal of Geomechanics vol 17 no 6Article ID 04016149 2017

[8] R Butterfield ldquoA natural compression law for soils (an advanceon e-log prsquo)rdquo Geotechnique vol 29 no 4 pp 469ndash480 1979

[9] R Lancellotta and L Preziosi ldquoA general nonlinear mathemati-calmodel for soil consolidation problemsrdquo International Journalof Engineering Science vol 35 no 10-11 pp 1045ndash1063 1997

[10] Y-C Zhuang K-H Xie and X-B Li ldquoNonlinear analysis ofconsolidation with variable compressibility and permeabilityrdquoJournal of Zhejiang University (Engineering Science Edition) vol6 no 3 pp 181ndash187 2005

[11] E Conte andA Troncone ldquoNonlinear consolidation of thin lay-ers subjected to time-dependent loadingrdquo Canadian Geotechni-cal Journal vol 44 no 6 pp 717ndash725 2007

[12] R Yong and F Townsend ldquoGeneral Consolidation Theory forClaysrdquo Soil Mechanics Series Report No 8 Graduate School ofEngineering National University of Mexico 1983

[13] J C Gibbings Dimensional Analysis Springer-Verlag LondonUK 2011

[14] E Buckingham ldquoOn physically similar systems Illustrations ofthe use of dimensional equationsrdquo Physical Review A AtomicMolecular and Optical Physics vol 4 no 4 pp 345ndash376 1914

[15] R Seshadri and T Y Na ldquoA Survey ofMethods for DeterminingSimilarity Transformationsrdquo inGroup Invariance in EngineeringBoundary Value Problems pp 35ndash61 Springer New York NYUSA 1985

[16] V D Zimparov and V M Petkov ldquoApplication of discriminatedanalysis to low Reynolds number swirl flows in circular tubeswith twisted-tape inserts Pressure drop correlationsrdquo Interna-tional Review of Chemical Engineering vol 1 no 4 pp 346ndash3562009

[17] M Capobianchi and A Aziz ldquoA scale analysis for natural con-vective flows over vertical surfacesrdquo International Journal ofThermal Sciences vol 54 pp 82ndash88 2012

[18] M Conesa J F Sanchez Perez I Alhama and F AlhamaldquoOn the nondimensionalization of coupled nonlinear ordinarydifferential equationsrdquo Nonlinear Dynamics vol 84 no 1 pp91ndash105 2016

[19] I A Manteca G Garcıa-Ros and F Alhama ldquoUniversal solu-tion for the characteristic time and the degree of settlement innonlinear soil consolidation scenarios A deduction based onnondimensionalizationrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 57 pp 186ndash201 2018

[20] E H Davis and G P Raymond ldquoA non-linear theory ofconsolidationrdquo Geotechnique vol 15 no 2 pp 161ndash173 1965

[21] P Cornetti and M Battaglio ldquoNonlinear consolidation of soilmodeling and solution techniquesrdquo Mathematical and Com-puter Modelling vol 20 no 7 pp 1ndash12 1994

[22] C F Gonzalez-Fernandez ldquoApplications of the network simu-lation method to transport processesrdquo in Network SimulationMethod J Horno Ed Research Signpost Trivandrum India2002

[23] J Serna F J S Velasco and A Soto Meca ldquoApplication of net-work simulationmethod to viscous flowsThe nanofluid heatedlid cavity under pulsating flowrdquo Computers amp Fluids vol 91 pp10ndash20 2014

[24] E Juarez-Badillo ldquoGeneral Permeability Change Equationsfor Soilsrdquo in Proceedings of the International Conference onConstitutive Laws pp 205ndash209 University of Arizona TuesonAriz USA 1983

[25] E Juarez-Badillo and B Chen ldquoConsolidation curves for claysrdquoJournal of Geotechnical Engineering vol 109 no 10 pp 1303ndash1312 1983

Mathematical Problems in Engineering 15

[26] N Abbasi H Rahimi A A Javadi andA Fakher ldquoFinite differ-ence approach for consolidation with variable compressibilityand permeabilityrdquo Computers amp Geosciences vol 34 no 1 pp41ndash52 2007

[27] R F Scott Principles of Soils Mechanics Addison-Wesley Pub-lishing Company 1963

[28] F Alhama and C N Madrid Analisis Dimensional Discrimi-nado En Mecanica De Fluidos Y Transferencia De Calor 2012

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8 Mathematical Problems in Engineering

Table1Dim

ensio

nlessg

roup

sthatcharacterizethe

originalandextend

edJ-Bmod

els

Juarez-Badillo

original

Juarez-Badillo

Juarez-Badillo

Juarez-Badillo

1+e=

constant

1+e

=consta

nt1+e

=consta

ntdz

constant

dzconstant

dzvaria

ble

120587 I120591 ok o

1205901015840 o120574 w120574 v1205901015840H2 1

(1205901015840 m 1205901015840 o)1minus120574k1205901015840

120591 ok o1205901015840 o(

1+eo)

120574 w120574 v1205901015840H2 1

(1205901015840 m 1205901015840 o)1minus120574k1205901015840

120591 ok o1205901015840 o(

1+eo)

120574 w120574 v1205901015840H2 1

(1205901015840 m 1205901015840 o)1minus120574k1205901015840minus120574 v1205901015840

120591 ok o1205901015840 o(

1+eo)

120574 w120574 v1205901015840H2 1

(1205901015840 m 1205901015840 o)1minus120574k1205901015840+120574 v1205901015840

120591 o1205901015840or

120591 os120587 II

120574 k1205901015840(1205901015840 2minus

1205901015840 1 1205901015840 m)for

pressure

(1minus120574 k1205901015840

+120574 v1205901015840)(1205901015840 2minus

1205901015840 1 1205901015840 m)for

settlem

ent

Mathematical Problems in Engineering 9

0010203040506070809

1

0 05 1 15 2 25 3time

Set 1Set 2Set 3Set 4

Set 5Set 6Set 7Set 8

Aver

age d

egre

e of c

onso

lidat

ion

Figure 1 Average degree of consolidation versus time (eight sets)

0010203040506070809

1

0 05 1 15 2 25 3

Set 1Set 2Set 3Set 4

Set 5Set 6Set 7Set 8

Aver

age d

egre

e of c

onso

lidat

ion

tos

Figure 2 Average degree of consolidation versus dimensionlesstime (eight sets)

(1 minus 120574k1205901015840 + 120574v1205901015840) to be also changed in order to maintain 120587IIconstant We expect that 120587I does not change as nearly occursThe small differences are again due to the existence of 1205901015840mwithin the monomials Note however that as diminishesthe influence of 1205901015840m in the results also diminishes since thevalues of 12059010158401 and 12059010158402 are closer

For a better understanding of the results Figures 1 and2 are presented In the first Us as a function of time isshown for the eight sets in Table 2 It is clear that 120591oscovers a wide range of values from 02283 (set 2) to 27968(set 3) the last more than ten times the first The secondfigure graphs Us in front of the dimensionless time t120591os Asshown the eight previous curves nearly converge to a single(or universal) graph having a common point at (Us=09t120591os=1) This means that monomial 120587II for the range ofvalues assigned to the parameters in contrast to 120587I scarcelyinfluences the value of the average degree of consolidationThis is not in contradiction with the results derived bynondimensionalization

0010203040506070809

1

0 05 1 15 2 25 3 35 4time

U (J-B)U (1+e cst)U (1+e not cst)U (1+e amp dz not cst)

Us (J-B)Us (1+e cst)Us (1+e not cst)Us (1+e amp dz not cst)

Aver

age d

egre

e of p

ress

ure d

issip

atio

nAv

erag

e deg

ree o

f con

solid

atio

n

Figure 3 Average degree of consolidation and pressure dissipation(set 6)

Although in this sectionwe have only checked the expres-sions of Us and 120591os for the most general model we considerit interesting to present both Us and U1205901015840 for the rest of themodels of Table 1 and for set 6 as a typical case Figure 3Theseresults emphasize the importance of using the less restrictive(more general) model in relation to the others in particularwith the original model of J-B in which the influence ofeo is despised Note that the smaller of the characteristictimes (related to settlement) corresponds to the more generalmodel (black and bold line) so the use of the others althoughon the safety side provides characteristic times oversizedfar from an optimal solution for the engineer For examplefor the case of Figure 3 in which eo=1 the error of J-Boriginal model (purple and bold line) in comparison with theextended model (black and bold line) for Us=90 is around150 This error increases appreciably as eo gets larger

5 Universal Curves

These easy to use and universal abacuses obtained by numeri-cal simulations allow engineers to read the characteristic time(120591os) and the average degree of consolidation (Us) in termsof the dimensionless groups given in Table 1 Only the mostgeneral (less restrictive) J-B model 1+e and dz not constantis presented

To obtain the characteristic time curves a large numberof simulations have been carried out studying separately theinfluence of the parameters 120574v1205901015840 and 120574k1205901015840 whose valuescover most real soils ranges from 101 to 8 120574v1205901015840 from 005to 03 and 120574k1205901015840 from 01 to 17 From these tests it has beendeduced as expected from the results of the previous sectionthat the relatively more influential parameter is while 120574v1205901015840and 120574k1205901015840 do not produce significant changes in the values ofthe characteristic timeThus the first universal group of linesshows the dependence of 120587I versus 120587II for different valuesof (with 120574v1205901015840=01) Figure 4 As the range of values of 120587IIdepends on according to (43) the range of values of eachline is different As shown in the figure there exists a universal

10 Mathematical Problems in Engineering

Table 2 Values of the parameters and dimensionless groups of simulations

Block Set 120574v1205901015840 120574k1205901015840 eoH1 ko 1205901015840o 1205901015840f H2 kf 120587II

120591os 120587I(m) (myr) (Nm2) (Nm2) 1205901015840f 1205901015840o (m) (myr) (yr)

I

1 01 0500 10 100 002 30000 60000 2 0933 00141 040 04566 071312 01 0500 10 100 002 60000 120000 2 0933 00141 040 02283 071313 01 0500 10 175 001 30000 60000 2 1633 00071 040 27968 071314 01 0500 13 142 004 22000 44000 2 1325 00283 040 05459 07131

II 5 01 1500 10 100 002 30000 90000 3 0896 00038 -040 10597 09834

III6 02 0600 10 100 002 30000 60000 2 0871 00132 040 09142 071397 01 0767 10 100 002 30000 120000 4 0871 00069 040 04360 072468 02 0920 10 100 002 30000 180000 6 0699 00038 040 08409 07312

line for each value of all converging in the central zone forvalues of 120587II around 0 This set of straight lines can be fittedby 120587I=m120587II+n being n=0846 and m given in Figure 5 as afunction of The curve that best fits the points in Figure 5 isgiven by

m = minus0235e(minus0868) minus 0306e(minus0012) (54)

Choosing the longest line corresponding to =8 as the onlyrepresentative line of the problem themaximumerrors in thereading of 120587I (and 120591os) are 38

In short the use of these curves is as follows from the dataof the problem Figure 4 allows us to read 120587I from 120587II and and from (51) the characteristic time 120591os is derived

As for the average degree of consolidation the universalcurves for 96 different scenarios of 120587II are shown in Figure 6It is appreciated that these curves are very close to each othershowing again that the Us only depends on the dimensionlesstime In the worst case the errors produced by changes invalue of the 120587II group are less than 28when using the curveof Figure 7

6 Case Study

Below is a practical application of the universal curvespresented in this paper For this purpose we take as a basisfor our study the data obtained experimentally byAbbasi et al[26] for different samples of clay whosemoisture in the liquidlimit is equal to 42 As can be seen (Table 3) it is a series ofdiscrete (tabulated) values that relate the void ratio with theeffective pressure and the hydraulic conductivity

We will address a real case of consolidation in which wehave a layer of soil 2meters thick (Ho) with an initial effectivepressure (1205901015840o) of 28 kNm2 on which a load of 31 kNm2 isapplied In this way the effective pressure at the end of theconsolidation process (1205901015840f ) will have the value of 59 kNm2From the tabulated data (e-1205901015840) it follows that eo = 105 andef = 095 and since the factor 1+e is proportional to the soilvolume from (9) we have 120574v1205901015840 = 0067

To obtain 120574k1205901015840 we first calculate the value of 120574kv from thetabulated data (e-k) to then obtain 120574k1205901015840 from (10) With thiswe have for the load step considered ko = 0035 myr 120574kv =8108 and 120574k1205901015840 = 0543

To solve the problem we will use the universal curvespresented in the previous section corresponding to the most

Table 3 Experimental values of e 1205901015840 and k obtained by Abbasi et al[26]

e 1205901015840o (Nm2)110 14000105 28000095 59000085 120000075 220000065 450000060 925000e k (myr)175 0683160 0315120 0084085 0021065 0011

general J-B model (analyzed in terms of settlement) Thusfrom the values of 1205901015840o 1205901015840f 120574v1205901015840 and 120574k1205901015840 by means of (51) itis obtained that 120587II = 0373 (with 1205901015840m = 435 kNm2) Then120587I is read from Figure 4 where for = 21 we have 120587I =072 An identical result is obtained if we use the proposedfitted equation 120587I=m120587II+n where n is a constant of value0846 andm is obtained from (54) or Figure 5 Finally knownthe values of Ho ko and eo from (51) we get 120591os = 0748years Likewise once the characteristic time of the problem(in terms of settlement) is known from the universal curve ofFigure 7 we easily obtain the evolution of the average degreeof consolidation along the whole process Figure 8

These solutions have been compared with those thatresult from the numerical simulation of the most generalJ-B model for which a value of 120591os = 075 years has beenobtained showing clearly that the relative error committedwhen applying the universal solutions is negligible (Table 4)The simulation of the problem has also been carried out fora model where 1+e and dz are also not constant but in whichthe constitutive relations of potential type have been replacedby pairs of data in tabulated form obtaining in this case 120591os= 0817 years The relative error which this time increasesto 82 finds its explanation in the strong nonlinearity ofthe problem which will always bring differences between the

Mathematical Problems in Engineering 11

040045050055060065070075080085090095100105110

minus07 minus06 minus05 minus04 minus03 minus02 minus01 0 01 02 03 04 05 06 07 08 09 1 11 12 13 14 15

1

2

ᴪ=8ᴪ=6ᴪ=4

ᴪ=2ᴪ=15ᴪ=101

Figure 4 120587I as a function of 120587II with as parameter

1 2 3 4 5 6 7 8minus042

minus04

minus038

minus036

minus034

minus032

minus03

minus028

minus026

m

datafitted curve

Figure 5 m as a function of

solutions provided by numerical models based on differentconstitutive relations (potential and tabulated in this case)Even so this value is totally acceptable within this field ofengineering demonstrating the strength and versatility of theuniversal solutions provided

7 Discussion

This work derives the dimensionless groups that rule thenonlinear soil consolidation process based on the original J-Bmodel following a formal nondimensionalization procedurethat starts from the governing equation and does not needany other assumption or mathematical manipulation Inaddition more real and precise extended J-B models whichassume both the void ratio and the thickness of the finitevolume element of the soil to be not constant have been

0010203040506070809

1

0 05 1 15 2 25 3

96 scenariosᴪ = [101 15 2 4 6 8]

Aver

age d

egre

e of c

onso

lidat

ion

tos

= [005 01 015 02]v= [02 05 1 15]k

Figure 6 Average degree of consolidation versus dimensionlesstime for different scenarios

0010203040506070809

1

0 05 1 15 2 25 3

Aver

age d

egre

e of c

onso

lidat

ion

tos

Figure 7 Proposed solution for the average degree of consolidation

nondimensionalized and their dimensionless groups derivedThe assumption of constitutive dependences (e-1205901015840 e-k and k-1205901015840) of potential type in these models is interesting since they

12 Mathematical Problems in Engineering

Table 4 Case study solution for the characteristic time 120591os Comparison with numerical models

Universal curves J-B most general model Tabulated model120591os (yr) 120591os (yr) Rel error 120591os (yr) Rel error 0748 0750 026 0817 820

0010203040506070809

1

0 025 05 075 1 125 15 175 2 225t ( years)

Aver

age d

egre

e of c

onso

lidat

ion

Figure 8 Case study solution for the average degree of consolida-tion Us from the universal curves

can be applied to highly compressible soils for which the e-log(1205901015840) relation moves away from being a straight line andin which negative values of the void ratio could be obtainedunder the application of high load ratios (1205901015840f 1205901015840o) [8]

The study has been developed both in terms of pres-sure dissipation and in terms of settlement resulting thatthe dimensionless groups derived from each model are ingeneral different The extension to more complex modelswhich implies the emergence of new parameters or equationswithin themathematical model does not necessarily give riseto additional dimensionless groups as might be expectedbut to new expressions of the groups or new regroupingof the parameters involved Thus the elimination of thehypothesis ldquodz constantrdquo in the most general consolidationmodel has not given rise to any new group both in pressureand in settlement (Table 1) but it has only modified oneof the groups obtained in the original model All thisrepresents an important advance in relation to the modelsand studies carried out by other authors and the attemptsof characterization found in the scientific literature [9ndash11]which try to lead to easy-to-use solutions for engineers Inno case had dimensionless groups been obtained as preciseas those presented in this work nor presented universalcurves of the characteristic time and the average degree ofconsolidation Authors such asConte andTroncone [11] applymore general load conditions (time-dependent loading) forwhich the dimensionless groups deduced in this work couldbe applicable for each load step

As regards time to make it dimensionless an unknownreference (120591o) is introduced in the form mentioned by Scott[27] and used in many other problems with asymptoticsolutions [28]Theprecise definition of 120591o necessarily involvesa criterion in relation to the percentage of average pressuredecrease (or settlement) associated with the end of theconsolidation processTheorder ofmagnitude of 120591o is derived

after nondimensionalization making 120591o only appear in oneof the independent dimensionless groups In this way theaverage degree of consolidation (in terms of pressure orsettlement) depends on the ldquotimecharacteristic timerdquo ratioin addition to the dimensionless groups (without unknowns)that arise

In short the procedure has led to a new very precisesolution that involves two monomials one (120587I) that containsthe characteristic time (120591o) and another (120587II) that is amathematical function of the soil parameters In this way120591o only depends on 120587II through the function 120587I = 120595(120587II)while U depends on 120587I and 120587II through the expression U= 120595(120587I 120587II) The results obtained have been verified forthe most general and complex consolidation model throughthe numerical simulation of a large number of scenariosin which the values of the dimensionless groups and theindividual parameters of the problem have been changing toconvenience The high errors produced by the original J-Bmodel with respect to the most general justify the use of thelatter in the search for more precise solutions

In addition the emergence of mean values of the effectivepressure when averaging the governing equation in orderto assign an order of magnitude unit to the dimensionlessvariables and their derivatives is an added difficulty to obtainthe dimensionless groups This is undoubtedly due to thenonlinearities (constitutive relations) involved in the consol-idation process This average effective pressure of referencehas been assumed as the semisum of its initial and finalvalues although other criteria could also be chosen Sincetwo scenarios can have equal120587II with the same effectivemeanpressure but different values of the parameters 120574v1205901015840 and 120574k1205901015840 we expect some minor deviations between the results of bothscenarios as indeed it has been checked Thus for the wholerange of parameters values in typical soils the errors due tothe above cause are negligible in the field of soil mechanicsOn the other hand when the value of 120587II does not changebut the mean effective pressure value does the errors in thesolutions of 120591os and Us are slightly larger although withinranges below 4

The brooch for all this work is the presentation ofuniversal curves for the most general model which allowfor the usual range of variation of the soil properties toobtain in a simple way the value of the characteristic timeof consolidation and the evolution of the average degreeof settlement throughout the entire process Thus startingfrom the nonlinear parameters of the soil (120574v1205901015840 and 120574k1205901015840)and the load ratio (1205901015840f 1205901015840o) the value of the monomial 120587IIis obtained from which the value of 120587I is directly deductedthanks to the universal curve 120587I = 120595(120587II) provided Finally120591os is easily obtained from the expression of 120587I On the otherhand the curve that represents the Us evolution along theprocess has turned out to be (practically) unique when it is

Mathematical Problems in Engineering 13

represented as a function of the dimensionless time in theform t120591os regardless of the soil parameters and the loadsapplied All this supposes a clear advantage when solvingthe consolidation problem also showing the strength of thenondimensionalization method presented

Finally the case study addressed has demonstrated thesimplicity and reliability of the universal curves proposedFrom a series of tabulated data (e-1205901015840 and e-k) the parametersthat govern the nonlinear consolidation problem for themostextended J-B model have been adjusted with which thevalue of the monomial 120587II has been derived With this andusing the universal curves the solutions for the characteristictime 120591os and the average degree of consolidation Us havebeen obtained These results have been satisfactorily verifiedwith the solutions provided both by the numerical extendedmodel of J-B presented here and by another that substitutesthe constitutive dependences of potential type by data intabulated form

8 Conclusions

The application of the nondimensionalization technique hasallowed us to deduce the dimensionless groups that rule thenonlinear consolidation problem in which the constitutivedependencies e-1205901015840 and e-k have the form of potential func-tions This type of relations approximates the soil behaviorbetter than the logarithmic functions e-log(1205901015840) and e-log(k)particularly in very compressible soils and under high loadratios (1205901015840f 1205901015840o) in which inconsistent negative values of thevoid ratio may appear

Several scenarios have been addressed from the originalmodel of Juarez-Badillo to extended models based on thesame constitutive dependences but eliminating one or moreclassic (and restrictive) hypotheses in this process In allcases only two dimensionless groups emerge one of themconsisting of a grouping of the soil parameters and the othercontaining the unknown characteristic time Based on theseresults universal curves have been depicted the first to deter-mine the characteristic time of the process and the secondthat represents the evolution of average degree of settlement

The solutions have been verified by numerical simula-tions changing the values of the soil parameters and thedimensionless groups conveniently Finally the reliability andeffectiveness of the proposed solutions have been shownthrough a case study

Abbreviations

cv Consolidation coefficient (m2s or m2yr)cv1 Initial consolidation coefficient (m2s or

m2yr)cv2 Final consolidation coefficient (m2s or

m2yr)119889z Element of differential length in thedirection of the spatial coordinate z (m)119889zo Element of differential length in thedirection of the spatial coordinate z at theinitial time (m)

e Void ratio (dimensionless)

eo Initial void ratio (dimensionless)ef Final void ratio (dimensionless)H Soil thickness up to the impervious

boundary or drainage length (m)H1Ho Initial soil thickness (m)H2 Final soil thickness (m)k Hydraulic conductivity (or permeability)

(ms or myr)k1 ko Initial hydraulic conductivity (ms or

myr)kf Final hydraulic conductivity (ms or myr)mv Coefficient of volumetric compressibility

(m2N)mv1 Initial coefficient of volumetric

compressibility (m2N)t Time Independent variable (s or yr)T Juarez-Badillorsquos time factor

(dimensionless)u Excess pore pressure (Nm2)U Average degree of consolidation or

pressure dissipation (dimensionless)Us Average degree of consolidation

(dimensionless)U1205901015840 Average degree of pressure dissipation

(dimensionless)V Soil volume (m3)V1Vo Initial soil volume (m3)z Vertical spatial coordinate (m)120574v Nonlinear coefficient of compressibility of

Juarez-Badillo (dimensionless)120574kv Nonlinear coefficient of change ofpermeability with volume (dimensionless)120574k1205901015840 Nonlinear coefficient of change ofpermeability with effective pressure(dimensionless)120574v1205901015840 Nonlinear coefficient of compressibility orcoefficient of volumetric contraction(dimensionless)120574w Specific weight of water (Nm3)

Loading factor Ratio between the finaland initial effective pressure(dimensionless)120581 Nonlinear coefficient of change ofpermeability with volume ofJuarez-Badillo (dimensionless)120582 Parameter of Juarez-Badillo 120582 = 1 - 120574v120581(dimensionless)

V Auxiliary variable of Juarez-Badillo(dimensionless)120587i Dimensionless group or number(dimensionless)1205901015840 Effective pressure of the soil (Nm2)12059010158401 1205901015840o Initial effective pressure (Nm2)12059010158402 1205901015840f Final effective pressure (Nm2)1205901015840m Mean value of the effective pressure(Nm2)120591o Characteristic time (s or yr)

14 Mathematical Problems in Engineering

120591os Characteristic time which takes thesettlement to reach approximately its finalvalue (s or yr)120591o1205901015840 Characteristic time which takes the excesspore pressure to dissipate untilapproximately the value of zero (s or yr)Ψ Arbitrary mathematical function120577 Differential void ratio (dimensionless)1205771 1205772 Initial and final differential void ratio(dimensionless)

Subscripts

I II Denote different numbers

Superscripts

1015840 Denote dimensionless magnitude

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] K Terzaghi ldquoDie Berechnung der Durchlassigkeitszifferdestones aus dem verlauf der hydrodynamischen spanunngser-scheinnungenrdquo Technical report II a 132 N 34 125-138Akademie derWissenschaften inWien SitzungsberichteMath-naturwiss Klasse Abt 1923

[2] R T Walker ldquoVertical drain consolidation analysis in one twoand three dimensionsrdquo Computers amp Geosciences vol 38 no 8pp 1069ndash1077 2011

[3] G Y Zheng P Li and C Y Zhao ldquoAnalysis of non-linearconsolidation of soft clay by differential quadrature methodrdquoApplied Clay Science vol 79 pp 2ndash7 2013

[4] X Y Geng C J Xu and Y Q Cai ldquoNon-linear consolidationanalysis of soil with variable compressibility and permeabilityunder cyclic loadingsrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 30 no 8 pp 803ndash8212006

[5] M Lu S Wang S W Sloan B Indraratna and K Xie ldquoNon-linear radial consolidation of vertical drains under a generaltime-variable loadingrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 39 no 1 pp 51ndash622015

[6] HWu L HuW Qi and QWen ldquoAnalytical Solution for Elec-troosmotic Consolidation Considering Nonlinear Variation ofSoil Parametersrdquo International Journal of Geomechanics vol 17no 5 Article ID 06016032 2017

[7] S J Brandenberg ldquoiConsoljs JavaScript Implicit Finite-Differ-ence Code for Nonlinear Consolidation and Secondary Com-pressionrdquo International Journal of Geomechanics vol 17 no 6Article ID 04016149 2017

[8] R Butterfield ldquoA natural compression law for soils (an advanceon e-log prsquo)rdquo Geotechnique vol 29 no 4 pp 469ndash480 1979

[9] R Lancellotta and L Preziosi ldquoA general nonlinear mathemati-calmodel for soil consolidation problemsrdquo International Journalof Engineering Science vol 35 no 10-11 pp 1045ndash1063 1997

[10] Y-C Zhuang K-H Xie and X-B Li ldquoNonlinear analysis ofconsolidation with variable compressibility and permeabilityrdquoJournal of Zhejiang University (Engineering Science Edition) vol6 no 3 pp 181ndash187 2005

[11] E Conte andA Troncone ldquoNonlinear consolidation of thin lay-ers subjected to time-dependent loadingrdquo Canadian Geotechni-cal Journal vol 44 no 6 pp 717ndash725 2007

[12] R Yong and F Townsend ldquoGeneral Consolidation Theory forClaysrdquo Soil Mechanics Series Report No 8 Graduate School ofEngineering National University of Mexico 1983

[13] J C Gibbings Dimensional Analysis Springer-Verlag LondonUK 2011

[14] E Buckingham ldquoOn physically similar systems Illustrations ofthe use of dimensional equationsrdquo Physical Review A AtomicMolecular and Optical Physics vol 4 no 4 pp 345ndash376 1914

[15] R Seshadri and T Y Na ldquoA Survey ofMethods for DeterminingSimilarity Transformationsrdquo inGroup Invariance in EngineeringBoundary Value Problems pp 35ndash61 Springer New York NYUSA 1985

[16] V D Zimparov and V M Petkov ldquoApplication of discriminatedanalysis to low Reynolds number swirl flows in circular tubeswith twisted-tape inserts Pressure drop correlationsrdquo Interna-tional Review of Chemical Engineering vol 1 no 4 pp 346ndash3562009

[17] M Capobianchi and A Aziz ldquoA scale analysis for natural con-vective flows over vertical surfacesrdquo International Journal ofThermal Sciences vol 54 pp 82ndash88 2012

[18] M Conesa J F Sanchez Perez I Alhama and F AlhamaldquoOn the nondimensionalization of coupled nonlinear ordinarydifferential equationsrdquo Nonlinear Dynamics vol 84 no 1 pp91ndash105 2016

[19] I A Manteca G Garcıa-Ros and F Alhama ldquoUniversal solu-tion for the characteristic time and the degree of settlement innonlinear soil consolidation scenarios A deduction based onnondimensionalizationrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 57 pp 186ndash201 2018

[20] E H Davis and G P Raymond ldquoA non-linear theory ofconsolidationrdquo Geotechnique vol 15 no 2 pp 161ndash173 1965

[21] P Cornetti and M Battaglio ldquoNonlinear consolidation of soilmodeling and solution techniquesrdquo Mathematical and Com-puter Modelling vol 20 no 7 pp 1ndash12 1994

[22] C F Gonzalez-Fernandez ldquoApplications of the network simu-lation method to transport processesrdquo in Network SimulationMethod J Horno Ed Research Signpost Trivandrum India2002

[23] J Serna F J S Velasco and A Soto Meca ldquoApplication of net-work simulationmethod to viscous flowsThe nanofluid heatedlid cavity under pulsating flowrdquo Computers amp Fluids vol 91 pp10ndash20 2014

[24] E Juarez-Badillo ldquoGeneral Permeability Change Equationsfor Soilsrdquo in Proceedings of the International Conference onConstitutive Laws pp 205ndash209 University of Arizona TuesonAriz USA 1983

[25] E Juarez-Badillo and B Chen ldquoConsolidation curves for claysrdquoJournal of Geotechnical Engineering vol 109 no 10 pp 1303ndash1312 1983

Mathematical Problems in Engineering 15

[26] N Abbasi H Rahimi A A Javadi andA Fakher ldquoFinite differ-ence approach for consolidation with variable compressibilityand permeabilityrdquo Computers amp Geosciences vol 34 no 1 pp41ndash52 2007

[27] R F Scott Principles of Soils Mechanics Addison-Wesley Pub-lishing Company 1963

[28] F Alhama and C N Madrid Analisis Dimensional Discrimi-nado En Mecanica De Fluidos Y Transferencia De Calor 2012

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Mathematical Problems in Engineering 9

0010203040506070809

1

0 05 1 15 2 25 3time

Set 1Set 2Set 3Set 4

Set 5Set 6Set 7Set 8

Aver

age d

egre

e of c

onso

lidat

ion

Figure 1 Average degree of consolidation versus time (eight sets)

0010203040506070809

1

0 05 1 15 2 25 3

Set 1Set 2Set 3Set 4

Set 5Set 6Set 7Set 8

Aver

age d

egre

e of c

onso

lidat

ion

tos

Figure 2 Average degree of consolidation versus dimensionlesstime (eight sets)

(1 minus 120574k1205901015840 + 120574v1205901015840) to be also changed in order to maintain 120587IIconstant We expect that 120587I does not change as nearly occursThe small differences are again due to the existence of 1205901015840mwithin the monomials Note however that as diminishesthe influence of 1205901015840m in the results also diminishes since thevalues of 12059010158401 and 12059010158402 are closer

For a better understanding of the results Figures 1 and2 are presented In the first Us as a function of time isshown for the eight sets in Table 2 It is clear that 120591oscovers a wide range of values from 02283 (set 2) to 27968(set 3) the last more than ten times the first The secondfigure graphs Us in front of the dimensionless time t120591os Asshown the eight previous curves nearly converge to a single(or universal) graph having a common point at (Us=09t120591os=1) This means that monomial 120587II for the range ofvalues assigned to the parameters in contrast to 120587I scarcelyinfluences the value of the average degree of consolidationThis is not in contradiction with the results derived bynondimensionalization

0010203040506070809

1

0 05 1 15 2 25 3 35 4time

U (J-B)U (1+e cst)U (1+e not cst)U (1+e amp dz not cst)

Us (J-B)Us (1+e cst)Us (1+e not cst)Us (1+e amp dz not cst)

Aver

age d

egre

e of p

ress

ure d

issip

atio

nAv

erag

e deg

ree o

f con

solid

atio

n

Figure 3 Average degree of consolidation and pressure dissipation(set 6)

Although in this sectionwe have only checked the expres-sions of Us and 120591os for the most general model we considerit interesting to present both Us and U1205901015840 for the rest of themodels of Table 1 and for set 6 as a typical case Figure 3Theseresults emphasize the importance of using the less restrictive(more general) model in relation to the others in particularwith the original model of J-B in which the influence ofeo is despised Note that the smaller of the characteristictimes (related to settlement) corresponds to the more generalmodel (black and bold line) so the use of the others althoughon the safety side provides characteristic times oversizedfar from an optimal solution for the engineer For examplefor the case of Figure 3 in which eo=1 the error of J-Boriginal model (purple and bold line) in comparison with theextended model (black and bold line) for Us=90 is around150 This error increases appreciably as eo gets larger

5 Universal Curves

These easy to use and universal abacuses obtained by numeri-cal simulations allow engineers to read the characteristic time(120591os) and the average degree of consolidation (Us) in termsof the dimensionless groups given in Table 1 Only the mostgeneral (less restrictive) J-B model 1+e and dz not constantis presented

To obtain the characteristic time curves a large numberof simulations have been carried out studying separately theinfluence of the parameters 120574v1205901015840 and 120574k1205901015840 whose valuescover most real soils ranges from 101 to 8 120574v1205901015840 from 005to 03 and 120574k1205901015840 from 01 to 17 From these tests it has beendeduced as expected from the results of the previous sectionthat the relatively more influential parameter is while 120574v1205901015840and 120574k1205901015840 do not produce significant changes in the values ofthe characteristic timeThus the first universal group of linesshows the dependence of 120587I versus 120587II for different valuesof (with 120574v1205901015840=01) Figure 4 As the range of values of 120587IIdepends on according to (43) the range of values of eachline is different As shown in the figure there exists a universal

10 Mathematical Problems in Engineering

Table 2 Values of the parameters and dimensionless groups of simulations

Block Set 120574v1205901015840 120574k1205901015840 eoH1 ko 1205901015840o 1205901015840f H2 kf 120587II

120591os 120587I(m) (myr) (Nm2) (Nm2) 1205901015840f 1205901015840o (m) (myr) (yr)

I

1 01 0500 10 100 002 30000 60000 2 0933 00141 040 04566 071312 01 0500 10 100 002 60000 120000 2 0933 00141 040 02283 071313 01 0500 10 175 001 30000 60000 2 1633 00071 040 27968 071314 01 0500 13 142 004 22000 44000 2 1325 00283 040 05459 07131

II 5 01 1500 10 100 002 30000 90000 3 0896 00038 -040 10597 09834

III6 02 0600 10 100 002 30000 60000 2 0871 00132 040 09142 071397 01 0767 10 100 002 30000 120000 4 0871 00069 040 04360 072468 02 0920 10 100 002 30000 180000 6 0699 00038 040 08409 07312

line for each value of all converging in the central zone forvalues of 120587II around 0 This set of straight lines can be fittedby 120587I=m120587II+n being n=0846 and m given in Figure 5 as afunction of The curve that best fits the points in Figure 5 isgiven by

m = minus0235e(minus0868) minus 0306e(minus0012) (54)

Choosing the longest line corresponding to =8 as the onlyrepresentative line of the problem themaximumerrors in thereading of 120587I (and 120591os) are 38

In short the use of these curves is as follows from the dataof the problem Figure 4 allows us to read 120587I from 120587II and and from (51) the characteristic time 120591os is derived

As for the average degree of consolidation the universalcurves for 96 different scenarios of 120587II are shown in Figure 6It is appreciated that these curves are very close to each othershowing again that the Us only depends on the dimensionlesstime In the worst case the errors produced by changes invalue of the 120587II group are less than 28when using the curveof Figure 7

6 Case Study

Below is a practical application of the universal curvespresented in this paper For this purpose we take as a basisfor our study the data obtained experimentally byAbbasi et al[26] for different samples of clay whosemoisture in the liquidlimit is equal to 42 As can be seen (Table 3) it is a series ofdiscrete (tabulated) values that relate the void ratio with theeffective pressure and the hydraulic conductivity

We will address a real case of consolidation in which wehave a layer of soil 2meters thick (Ho) with an initial effectivepressure (1205901015840o) of 28 kNm2 on which a load of 31 kNm2 isapplied In this way the effective pressure at the end of theconsolidation process (1205901015840f ) will have the value of 59 kNm2From the tabulated data (e-1205901015840) it follows that eo = 105 andef = 095 and since the factor 1+e is proportional to the soilvolume from (9) we have 120574v1205901015840 = 0067

To obtain 120574k1205901015840 we first calculate the value of 120574kv from thetabulated data (e-k) to then obtain 120574k1205901015840 from (10) With thiswe have for the load step considered ko = 0035 myr 120574kv =8108 and 120574k1205901015840 = 0543

To solve the problem we will use the universal curvespresented in the previous section corresponding to the most

Table 3 Experimental values of e 1205901015840 and k obtained by Abbasi et al[26]

e 1205901015840o (Nm2)110 14000105 28000095 59000085 120000075 220000065 450000060 925000e k (myr)175 0683160 0315120 0084085 0021065 0011

general J-B model (analyzed in terms of settlement) Thusfrom the values of 1205901015840o 1205901015840f 120574v1205901015840 and 120574k1205901015840 by means of (51) itis obtained that 120587II = 0373 (with 1205901015840m = 435 kNm2) Then120587I is read from Figure 4 where for = 21 we have 120587I =072 An identical result is obtained if we use the proposedfitted equation 120587I=m120587II+n where n is a constant of value0846 andm is obtained from (54) or Figure 5 Finally knownthe values of Ho ko and eo from (51) we get 120591os = 0748years Likewise once the characteristic time of the problem(in terms of settlement) is known from the universal curve ofFigure 7 we easily obtain the evolution of the average degreeof consolidation along the whole process Figure 8

These solutions have been compared with those thatresult from the numerical simulation of the most generalJ-B model for which a value of 120591os = 075 years has beenobtained showing clearly that the relative error committedwhen applying the universal solutions is negligible (Table 4)The simulation of the problem has also been carried out fora model where 1+e and dz are also not constant but in whichthe constitutive relations of potential type have been replacedby pairs of data in tabulated form obtaining in this case 120591os= 0817 years The relative error which this time increasesto 82 finds its explanation in the strong nonlinearity ofthe problem which will always bring differences between the

Mathematical Problems in Engineering 11

040045050055060065070075080085090095100105110

minus07 minus06 minus05 minus04 minus03 minus02 minus01 0 01 02 03 04 05 06 07 08 09 1 11 12 13 14 15

1

2

ᴪ=8ᴪ=6ᴪ=4

ᴪ=2ᴪ=15ᴪ=101

Figure 4 120587I as a function of 120587II with as parameter

1 2 3 4 5 6 7 8minus042

minus04

minus038

minus036

minus034

minus032

minus03

minus028

minus026

m

datafitted curve

Figure 5 m as a function of

solutions provided by numerical models based on differentconstitutive relations (potential and tabulated in this case)Even so this value is totally acceptable within this field ofengineering demonstrating the strength and versatility of theuniversal solutions provided

7 Discussion

This work derives the dimensionless groups that rule thenonlinear soil consolidation process based on the original J-Bmodel following a formal nondimensionalization procedurethat starts from the governing equation and does not needany other assumption or mathematical manipulation Inaddition more real and precise extended J-B models whichassume both the void ratio and the thickness of the finitevolume element of the soil to be not constant have been

0010203040506070809

1

0 05 1 15 2 25 3

96 scenariosᴪ = [101 15 2 4 6 8]

Aver

age d

egre

e of c

onso

lidat

ion

tos

= [005 01 015 02]v= [02 05 1 15]k

Figure 6 Average degree of consolidation versus dimensionlesstime for different scenarios

0010203040506070809

1

0 05 1 15 2 25 3

Aver

age d

egre

e of c

onso

lidat

ion

tos

Figure 7 Proposed solution for the average degree of consolidation

nondimensionalized and their dimensionless groups derivedThe assumption of constitutive dependences (e-1205901015840 e-k and k-1205901015840) of potential type in these models is interesting since they

12 Mathematical Problems in Engineering

Table 4 Case study solution for the characteristic time 120591os Comparison with numerical models

Universal curves J-B most general model Tabulated model120591os (yr) 120591os (yr) Rel error 120591os (yr) Rel error 0748 0750 026 0817 820

0010203040506070809

1

0 025 05 075 1 125 15 175 2 225t ( years)

Aver

age d

egre

e of c

onso

lidat

ion

Figure 8 Case study solution for the average degree of consolida-tion Us from the universal curves

can be applied to highly compressible soils for which the e-log(1205901015840) relation moves away from being a straight line andin which negative values of the void ratio could be obtainedunder the application of high load ratios (1205901015840f 1205901015840o) [8]

The study has been developed both in terms of pres-sure dissipation and in terms of settlement resulting thatthe dimensionless groups derived from each model are ingeneral different The extension to more complex modelswhich implies the emergence of new parameters or equationswithin themathematical model does not necessarily give riseto additional dimensionless groups as might be expectedbut to new expressions of the groups or new regroupingof the parameters involved Thus the elimination of thehypothesis ldquodz constantrdquo in the most general consolidationmodel has not given rise to any new group both in pressureand in settlement (Table 1) but it has only modified oneof the groups obtained in the original model All thisrepresents an important advance in relation to the modelsand studies carried out by other authors and the attemptsof characterization found in the scientific literature [9ndash11]which try to lead to easy-to-use solutions for engineers Inno case had dimensionless groups been obtained as preciseas those presented in this work nor presented universalcurves of the characteristic time and the average degree ofconsolidation Authors such asConte andTroncone [11] applymore general load conditions (time-dependent loading) forwhich the dimensionless groups deduced in this work couldbe applicable for each load step

As regards time to make it dimensionless an unknownreference (120591o) is introduced in the form mentioned by Scott[27] and used in many other problems with asymptoticsolutions [28]Theprecise definition of 120591o necessarily involvesa criterion in relation to the percentage of average pressuredecrease (or settlement) associated with the end of theconsolidation processTheorder ofmagnitude of 120591o is derived

after nondimensionalization making 120591o only appear in oneof the independent dimensionless groups In this way theaverage degree of consolidation (in terms of pressure orsettlement) depends on the ldquotimecharacteristic timerdquo ratioin addition to the dimensionless groups (without unknowns)that arise

In short the procedure has led to a new very precisesolution that involves two monomials one (120587I) that containsthe characteristic time (120591o) and another (120587II) that is amathematical function of the soil parameters In this way120591o only depends on 120587II through the function 120587I = 120595(120587II)while U depends on 120587I and 120587II through the expression U= 120595(120587I 120587II) The results obtained have been verified forthe most general and complex consolidation model throughthe numerical simulation of a large number of scenariosin which the values of the dimensionless groups and theindividual parameters of the problem have been changing toconvenience The high errors produced by the original J-Bmodel with respect to the most general justify the use of thelatter in the search for more precise solutions

In addition the emergence of mean values of the effectivepressure when averaging the governing equation in orderto assign an order of magnitude unit to the dimensionlessvariables and their derivatives is an added difficulty to obtainthe dimensionless groups This is undoubtedly due to thenonlinearities (constitutive relations) involved in the consol-idation process This average effective pressure of referencehas been assumed as the semisum of its initial and finalvalues although other criteria could also be chosen Sincetwo scenarios can have equal120587II with the same effectivemeanpressure but different values of the parameters 120574v1205901015840 and 120574k1205901015840 we expect some minor deviations between the results of bothscenarios as indeed it has been checked Thus for the wholerange of parameters values in typical soils the errors due tothe above cause are negligible in the field of soil mechanicsOn the other hand when the value of 120587II does not changebut the mean effective pressure value does the errors in thesolutions of 120591os and Us are slightly larger although withinranges below 4

The brooch for all this work is the presentation ofuniversal curves for the most general model which allowfor the usual range of variation of the soil properties toobtain in a simple way the value of the characteristic timeof consolidation and the evolution of the average degreeof settlement throughout the entire process Thus startingfrom the nonlinear parameters of the soil (120574v1205901015840 and 120574k1205901015840)and the load ratio (1205901015840f 1205901015840o) the value of the monomial 120587IIis obtained from which the value of 120587I is directly deductedthanks to the universal curve 120587I = 120595(120587II) provided Finally120591os is easily obtained from the expression of 120587I On the otherhand the curve that represents the Us evolution along theprocess has turned out to be (practically) unique when it is

Mathematical Problems in Engineering 13

represented as a function of the dimensionless time in theform t120591os regardless of the soil parameters and the loadsapplied All this supposes a clear advantage when solvingthe consolidation problem also showing the strength of thenondimensionalization method presented

Finally the case study addressed has demonstrated thesimplicity and reliability of the universal curves proposedFrom a series of tabulated data (e-1205901015840 and e-k) the parametersthat govern the nonlinear consolidation problem for themostextended J-B model have been adjusted with which thevalue of the monomial 120587II has been derived With this andusing the universal curves the solutions for the characteristictime 120591os and the average degree of consolidation Us havebeen obtained These results have been satisfactorily verifiedwith the solutions provided both by the numerical extendedmodel of J-B presented here and by another that substitutesthe constitutive dependences of potential type by data intabulated form

8 Conclusions

The application of the nondimensionalization technique hasallowed us to deduce the dimensionless groups that rule thenonlinear consolidation problem in which the constitutivedependencies e-1205901015840 and e-k have the form of potential func-tions This type of relations approximates the soil behaviorbetter than the logarithmic functions e-log(1205901015840) and e-log(k)particularly in very compressible soils and under high loadratios (1205901015840f 1205901015840o) in which inconsistent negative values of thevoid ratio may appear

Several scenarios have been addressed from the originalmodel of Juarez-Badillo to extended models based on thesame constitutive dependences but eliminating one or moreclassic (and restrictive) hypotheses in this process In allcases only two dimensionless groups emerge one of themconsisting of a grouping of the soil parameters and the othercontaining the unknown characteristic time Based on theseresults universal curves have been depicted the first to deter-mine the characteristic time of the process and the secondthat represents the evolution of average degree of settlement

The solutions have been verified by numerical simula-tions changing the values of the soil parameters and thedimensionless groups conveniently Finally the reliability andeffectiveness of the proposed solutions have been shownthrough a case study

Abbreviations

cv Consolidation coefficient (m2s or m2yr)cv1 Initial consolidation coefficient (m2s or

m2yr)cv2 Final consolidation coefficient (m2s or

m2yr)119889z Element of differential length in thedirection of the spatial coordinate z (m)119889zo Element of differential length in thedirection of the spatial coordinate z at theinitial time (m)

e Void ratio (dimensionless)

eo Initial void ratio (dimensionless)ef Final void ratio (dimensionless)H Soil thickness up to the impervious

boundary or drainage length (m)H1Ho Initial soil thickness (m)H2 Final soil thickness (m)k Hydraulic conductivity (or permeability)

(ms or myr)k1 ko Initial hydraulic conductivity (ms or

myr)kf Final hydraulic conductivity (ms or myr)mv Coefficient of volumetric compressibility

(m2N)mv1 Initial coefficient of volumetric

compressibility (m2N)t Time Independent variable (s or yr)T Juarez-Badillorsquos time factor

(dimensionless)u Excess pore pressure (Nm2)U Average degree of consolidation or

pressure dissipation (dimensionless)Us Average degree of consolidation

(dimensionless)U1205901015840 Average degree of pressure dissipation

(dimensionless)V Soil volume (m3)V1Vo Initial soil volume (m3)z Vertical spatial coordinate (m)120574v Nonlinear coefficient of compressibility of

Juarez-Badillo (dimensionless)120574kv Nonlinear coefficient of change ofpermeability with volume (dimensionless)120574k1205901015840 Nonlinear coefficient of change ofpermeability with effective pressure(dimensionless)120574v1205901015840 Nonlinear coefficient of compressibility orcoefficient of volumetric contraction(dimensionless)120574w Specific weight of water (Nm3)

Loading factor Ratio between the finaland initial effective pressure(dimensionless)120581 Nonlinear coefficient of change ofpermeability with volume ofJuarez-Badillo (dimensionless)120582 Parameter of Juarez-Badillo 120582 = 1 - 120574v120581(dimensionless)

V Auxiliary variable of Juarez-Badillo(dimensionless)120587i Dimensionless group or number(dimensionless)1205901015840 Effective pressure of the soil (Nm2)12059010158401 1205901015840o Initial effective pressure (Nm2)12059010158402 1205901015840f Final effective pressure (Nm2)1205901015840m Mean value of the effective pressure(Nm2)120591o Characteristic time (s or yr)

14 Mathematical Problems in Engineering

120591os Characteristic time which takes thesettlement to reach approximately its finalvalue (s or yr)120591o1205901015840 Characteristic time which takes the excesspore pressure to dissipate untilapproximately the value of zero (s or yr)Ψ Arbitrary mathematical function120577 Differential void ratio (dimensionless)1205771 1205772 Initial and final differential void ratio(dimensionless)

Subscripts

I II Denote different numbers

Superscripts

1015840 Denote dimensionless magnitude

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] K Terzaghi ldquoDie Berechnung der Durchlassigkeitszifferdestones aus dem verlauf der hydrodynamischen spanunngser-scheinnungenrdquo Technical report II a 132 N 34 125-138Akademie derWissenschaften inWien SitzungsberichteMath-naturwiss Klasse Abt 1923

[2] R T Walker ldquoVertical drain consolidation analysis in one twoand three dimensionsrdquo Computers amp Geosciences vol 38 no 8pp 1069ndash1077 2011

[3] G Y Zheng P Li and C Y Zhao ldquoAnalysis of non-linearconsolidation of soft clay by differential quadrature methodrdquoApplied Clay Science vol 79 pp 2ndash7 2013

[4] X Y Geng C J Xu and Y Q Cai ldquoNon-linear consolidationanalysis of soil with variable compressibility and permeabilityunder cyclic loadingsrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 30 no 8 pp 803ndash8212006

[5] M Lu S Wang S W Sloan B Indraratna and K Xie ldquoNon-linear radial consolidation of vertical drains under a generaltime-variable loadingrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 39 no 1 pp 51ndash622015

[6] HWu L HuW Qi and QWen ldquoAnalytical Solution for Elec-troosmotic Consolidation Considering Nonlinear Variation ofSoil Parametersrdquo International Journal of Geomechanics vol 17no 5 Article ID 06016032 2017

[7] S J Brandenberg ldquoiConsoljs JavaScript Implicit Finite-Differ-ence Code for Nonlinear Consolidation and Secondary Com-pressionrdquo International Journal of Geomechanics vol 17 no 6Article ID 04016149 2017

[8] R Butterfield ldquoA natural compression law for soils (an advanceon e-log prsquo)rdquo Geotechnique vol 29 no 4 pp 469ndash480 1979

[9] R Lancellotta and L Preziosi ldquoA general nonlinear mathemati-calmodel for soil consolidation problemsrdquo International Journalof Engineering Science vol 35 no 10-11 pp 1045ndash1063 1997

[10] Y-C Zhuang K-H Xie and X-B Li ldquoNonlinear analysis ofconsolidation with variable compressibility and permeabilityrdquoJournal of Zhejiang University (Engineering Science Edition) vol6 no 3 pp 181ndash187 2005

[11] E Conte andA Troncone ldquoNonlinear consolidation of thin lay-ers subjected to time-dependent loadingrdquo Canadian Geotechni-cal Journal vol 44 no 6 pp 717ndash725 2007

[12] R Yong and F Townsend ldquoGeneral Consolidation Theory forClaysrdquo Soil Mechanics Series Report No 8 Graduate School ofEngineering National University of Mexico 1983

[13] J C Gibbings Dimensional Analysis Springer-Verlag LondonUK 2011

[14] E Buckingham ldquoOn physically similar systems Illustrations ofthe use of dimensional equationsrdquo Physical Review A AtomicMolecular and Optical Physics vol 4 no 4 pp 345ndash376 1914

[15] R Seshadri and T Y Na ldquoA Survey ofMethods for DeterminingSimilarity Transformationsrdquo inGroup Invariance in EngineeringBoundary Value Problems pp 35ndash61 Springer New York NYUSA 1985

[16] V D Zimparov and V M Petkov ldquoApplication of discriminatedanalysis to low Reynolds number swirl flows in circular tubeswith twisted-tape inserts Pressure drop correlationsrdquo Interna-tional Review of Chemical Engineering vol 1 no 4 pp 346ndash3562009

[17] M Capobianchi and A Aziz ldquoA scale analysis for natural con-vective flows over vertical surfacesrdquo International Journal ofThermal Sciences vol 54 pp 82ndash88 2012

[18] M Conesa J F Sanchez Perez I Alhama and F AlhamaldquoOn the nondimensionalization of coupled nonlinear ordinarydifferential equationsrdquo Nonlinear Dynamics vol 84 no 1 pp91ndash105 2016

[19] I A Manteca G Garcıa-Ros and F Alhama ldquoUniversal solu-tion for the characteristic time and the degree of settlement innonlinear soil consolidation scenarios A deduction based onnondimensionalizationrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 57 pp 186ndash201 2018

[20] E H Davis and G P Raymond ldquoA non-linear theory ofconsolidationrdquo Geotechnique vol 15 no 2 pp 161ndash173 1965

[21] P Cornetti and M Battaglio ldquoNonlinear consolidation of soilmodeling and solution techniquesrdquo Mathematical and Com-puter Modelling vol 20 no 7 pp 1ndash12 1994

[22] C F Gonzalez-Fernandez ldquoApplications of the network simu-lation method to transport processesrdquo in Network SimulationMethod J Horno Ed Research Signpost Trivandrum India2002

[23] J Serna F J S Velasco and A Soto Meca ldquoApplication of net-work simulationmethod to viscous flowsThe nanofluid heatedlid cavity under pulsating flowrdquo Computers amp Fluids vol 91 pp10ndash20 2014

[24] E Juarez-Badillo ldquoGeneral Permeability Change Equationsfor Soilsrdquo in Proceedings of the International Conference onConstitutive Laws pp 205ndash209 University of Arizona TuesonAriz USA 1983

[25] E Juarez-Badillo and B Chen ldquoConsolidation curves for claysrdquoJournal of Geotechnical Engineering vol 109 no 10 pp 1303ndash1312 1983

Mathematical Problems in Engineering 15

[26] N Abbasi H Rahimi A A Javadi andA Fakher ldquoFinite differ-ence approach for consolidation with variable compressibilityand permeabilityrdquo Computers amp Geosciences vol 34 no 1 pp41ndash52 2007

[27] R F Scott Principles of Soils Mechanics Addison-Wesley Pub-lishing Company 1963

[28] F Alhama and C N Madrid Analisis Dimensional Discrimi-nado En Mecanica De Fluidos Y Transferencia De Calor 2012

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10 Mathematical Problems in Engineering

Table 2 Values of the parameters and dimensionless groups of simulations

Block Set 120574v1205901015840 120574k1205901015840 eoH1 ko 1205901015840o 1205901015840f H2 kf 120587II

120591os 120587I(m) (myr) (Nm2) (Nm2) 1205901015840f 1205901015840o (m) (myr) (yr)

I

1 01 0500 10 100 002 30000 60000 2 0933 00141 040 04566 071312 01 0500 10 100 002 60000 120000 2 0933 00141 040 02283 071313 01 0500 10 175 001 30000 60000 2 1633 00071 040 27968 071314 01 0500 13 142 004 22000 44000 2 1325 00283 040 05459 07131

II 5 01 1500 10 100 002 30000 90000 3 0896 00038 -040 10597 09834

III6 02 0600 10 100 002 30000 60000 2 0871 00132 040 09142 071397 01 0767 10 100 002 30000 120000 4 0871 00069 040 04360 072468 02 0920 10 100 002 30000 180000 6 0699 00038 040 08409 07312

line for each value of all converging in the central zone forvalues of 120587II around 0 This set of straight lines can be fittedby 120587I=m120587II+n being n=0846 and m given in Figure 5 as afunction of The curve that best fits the points in Figure 5 isgiven by

m = minus0235e(minus0868) minus 0306e(minus0012) (54)

Choosing the longest line corresponding to =8 as the onlyrepresentative line of the problem themaximumerrors in thereading of 120587I (and 120591os) are 38

In short the use of these curves is as follows from the dataof the problem Figure 4 allows us to read 120587I from 120587II and and from (51) the characteristic time 120591os is derived

As for the average degree of consolidation the universalcurves for 96 different scenarios of 120587II are shown in Figure 6It is appreciated that these curves are very close to each othershowing again that the Us only depends on the dimensionlesstime In the worst case the errors produced by changes invalue of the 120587II group are less than 28when using the curveof Figure 7

6 Case Study

Below is a practical application of the universal curvespresented in this paper For this purpose we take as a basisfor our study the data obtained experimentally byAbbasi et al[26] for different samples of clay whosemoisture in the liquidlimit is equal to 42 As can be seen (Table 3) it is a series ofdiscrete (tabulated) values that relate the void ratio with theeffective pressure and the hydraulic conductivity

We will address a real case of consolidation in which wehave a layer of soil 2meters thick (Ho) with an initial effectivepressure (1205901015840o) of 28 kNm2 on which a load of 31 kNm2 isapplied In this way the effective pressure at the end of theconsolidation process (1205901015840f ) will have the value of 59 kNm2From the tabulated data (e-1205901015840) it follows that eo = 105 andef = 095 and since the factor 1+e is proportional to the soilvolume from (9) we have 120574v1205901015840 = 0067

To obtain 120574k1205901015840 we first calculate the value of 120574kv from thetabulated data (e-k) to then obtain 120574k1205901015840 from (10) With thiswe have for the load step considered ko = 0035 myr 120574kv =8108 and 120574k1205901015840 = 0543

To solve the problem we will use the universal curvespresented in the previous section corresponding to the most

Table 3 Experimental values of e 1205901015840 and k obtained by Abbasi et al[26]

e 1205901015840o (Nm2)110 14000105 28000095 59000085 120000075 220000065 450000060 925000e k (myr)175 0683160 0315120 0084085 0021065 0011

general J-B model (analyzed in terms of settlement) Thusfrom the values of 1205901015840o 1205901015840f 120574v1205901015840 and 120574k1205901015840 by means of (51) itis obtained that 120587II = 0373 (with 1205901015840m = 435 kNm2) Then120587I is read from Figure 4 where for = 21 we have 120587I =072 An identical result is obtained if we use the proposedfitted equation 120587I=m120587II+n where n is a constant of value0846 andm is obtained from (54) or Figure 5 Finally knownthe values of Ho ko and eo from (51) we get 120591os = 0748years Likewise once the characteristic time of the problem(in terms of settlement) is known from the universal curve ofFigure 7 we easily obtain the evolution of the average degreeof consolidation along the whole process Figure 8

These solutions have been compared with those thatresult from the numerical simulation of the most generalJ-B model for which a value of 120591os = 075 years has beenobtained showing clearly that the relative error committedwhen applying the universal solutions is negligible (Table 4)The simulation of the problem has also been carried out fora model where 1+e and dz are also not constant but in whichthe constitutive relations of potential type have been replacedby pairs of data in tabulated form obtaining in this case 120591os= 0817 years The relative error which this time increasesto 82 finds its explanation in the strong nonlinearity ofthe problem which will always bring differences between the

Mathematical Problems in Engineering 11

040045050055060065070075080085090095100105110

minus07 minus06 minus05 minus04 minus03 minus02 minus01 0 01 02 03 04 05 06 07 08 09 1 11 12 13 14 15

1

2

ᴪ=8ᴪ=6ᴪ=4

ᴪ=2ᴪ=15ᴪ=101

Figure 4 120587I as a function of 120587II with as parameter

1 2 3 4 5 6 7 8minus042

minus04

minus038

minus036

minus034

minus032

minus03

minus028

minus026

m

datafitted curve

Figure 5 m as a function of

solutions provided by numerical models based on differentconstitutive relations (potential and tabulated in this case)Even so this value is totally acceptable within this field ofengineering demonstrating the strength and versatility of theuniversal solutions provided

7 Discussion

This work derives the dimensionless groups that rule thenonlinear soil consolidation process based on the original J-Bmodel following a formal nondimensionalization procedurethat starts from the governing equation and does not needany other assumption or mathematical manipulation Inaddition more real and precise extended J-B models whichassume both the void ratio and the thickness of the finitevolume element of the soil to be not constant have been

0010203040506070809

1

0 05 1 15 2 25 3

96 scenariosᴪ = [101 15 2 4 6 8]

Aver

age d

egre

e of c

onso

lidat

ion

tos

= [005 01 015 02]v= [02 05 1 15]k

Figure 6 Average degree of consolidation versus dimensionlesstime for different scenarios

0010203040506070809

1

0 05 1 15 2 25 3

Aver

age d

egre

e of c

onso

lidat

ion

tos

Figure 7 Proposed solution for the average degree of consolidation

nondimensionalized and their dimensionless groups derivedThe assumption of constitutive dependences (e-1205901015840 e-k and k-1205901015840) of potential type in these models is interesting since they

12 Mathematical Problems in Engineering

Table 4 Case study solution for the characteristic time 120591os Comparison with numerical models

Universal curves J-B most general model Tabulated model120591os (yr) 120591os (yr) Rel error 120591os (yr) Rel error 0748 0750 026 0817 820

0010203040506070809

1

0 025 05 075 1 125 15 175 2 225t ( years)

Aver

age d

egre

e of c

onso

lidat

ion

Figure 8 Case study solution for the average degree of consolida-tion Us from the universal curves

can be applied to highly compressible soils for which the e-log(1205901015840) relation moves away from being a straight line andin which negative values of the void ratio could be obtainedunder the application of high load ratios (1205901015840f 1205901015840o) [8]

The study has been developed both in terms of pres-sure dissipation and in terms of settlement resulting thatthe dimensionless groups derived from each model are ingeneral different The extension to more complex modelswhich implies the emergence of new parameters or equationswithin themathematical model does not necessarily give riseto additional dimensionless groups as might be expectedbut to new expressions of the groups or new regroupingof the parameters involved Thus the elimination of thehypothesis ldquodz constantrdquo in the most general consolidationmodel has not given rise to any new group both in pressureand in settlement (Table 1) but it has only modified oneof the groups obtained in the original model All thisrepresents an important advance in relation to the modelsand studies carried out by other authors and the attemptsof characterization found in the scientific literature [9ndash11]which try to lead to easy-to-use solutions for engineers Inno case had dimensionless groups been obtained as preciseas those presented in this work nor presented universalcurves of the characteristic time and the average degree ofconsolidation Authors such asConte andTroncone [11] applymore general load conditions (time-dependent loading) forwhich the dimensionless groups deduced in this work couldbe applicable for each load step

As regards time to make it dimensionless an unknownreference (120591o) is introduced in the form mentioned by Scott[27] and used in many other problems with asymptoticsolutions [28]Theprecise definition of 120591o necessarily involvesa criterion in relation to the percentage of average pressuredecrease (or settlement) associated with the end of theconsolidation processTheorder ofmagnitude of 120591o is derived

after nondimensionalization making 120591o only appear in oneof the independent dimensionless groups In this way theaverage degree of consolidation (in terms of pressure orsettlement) depends on the ldquotimecharacteristic timerdquo ratioin addition to the dimensionless groups (without unknowns)that arise

In short the procedure has led to a new very precisesolution that involves two monomials one (120587I) that containsthe characteristic time (120591o) and another (120587II) that is amathematical function of the soil parameters In this way120591o only depends on 120587II through the function 120587I = 120595(120587II)while U depends on 120587I and 120587II through the expression U= 120595(120587I 120587II) The results obtained have been verified forthe most general and complex consolidation model throughthe numerical simulation of a large number of scenariosin which the values of the dimensionless groups and theindividual parameters of the problem have been changing toconvenience The high errors produced by the original J-Bmodel with respect to the most general justify the use of thelatter in the search for more precise solutions

In addition the emergence of mean values of the effectivepressure when averaging the governing equation in orderto assign an order of magnitude unit to the dimensionlessvariables and their derivatives is an added difficulty to obtainthe dimensionless groups This is undoubtedly due to thenonlinearities (constitutive relations) involved in the consol-idation process This average effective pressure of referencehas been assumed as the semisum of its initial and finalvalues although other criteria could also be chosen Sincetwo scenarios can have equal120587II with the same effectivemeanpressure but different values of the parameters 120574v1205901015840 and 120574k1205901015840 we expect some minor deviations between the results of bothscenarios as indeed it has been checked Thus for the wholerange of parameters values in typical soils the errors due tothe above cause are negligible in the field of soil mechanicsOn the other hand when the value of 120587II does not changebut the mean effective pressure value does the errors in thesolutions of 120591os and Us are slightly larger although withinranges below 4

The brooch for all this work is the presentation ofuniversal curves for the most general model which allowfor the usual range of variation of the soil properties toobtain in a simple way the value of the characteristic timeof consolidation and the evolution of the average degreeof settlement throughout the entire process Thus startingfrom the nonlinear parameters of the soil (120574v1205901015840 and 120574k1205901015840)and the load ratio (1205901015840f 1205901015840o) the value of the monomial 120587IIis obtained from which the value of 120587I is directly deductedthanks to the universal curve 120587I = 120595(120587II) provided Finally120591os is easily obtained from the expression of 120587I On the otherhand the curve that represents the Us evolution along theprocess has turned out to be (practically) unique when it is

Mathematical Problems in Engineering 13

represented as a function of the dimensionless time in theform t120591os regardless of the soil parameters and the loadsapplied All this supposes a clear advantage when solvingthe consolidation problem also showing the strength of thenondimensionalization method presented

Finally the case study addressed has demonstrated thesimplicity and reliability of the universal curves proposedFrom a series of tabulated data (e-1205901015840 and e-k) the parametersthat govern the nonlinear consolidation problem for themostextended J-B model have been adjusted with which thevalue of the monomial 120587II has been derived With this andusing the universal curves the solutions for the characteristictime 120591os and the average degree of consolidation Us havebeen obtained These results have been satisfactorily verifiedwith the solutions provided both by the numerical extendedmodel of J-B presented here and by another that substitutesthe constitutive dependences of potential type by data intabulated form

8 Conclusions

The application of the nondimensionalization technique hasallowed us to deduce the dimensionless groups that rule thenonlinear consolidation problem in which the constitutivedependencies e-1205901015840 and e-k have the form of potential func-tions This type of relations approximates the soil behaviorbetter than the logarithmic functions e-log(1205901015840) and e-log(k)particularly in very compressible soils and under high loadratios (1205901015840f 1205901015840o) in which inconsistent negative values of thevoid ratio may appear

Several scenarios have been addressed from the originalmodel of Juarez-Badillo to extended models based on thesame constitutive dependences but eliminating one or moreclassic (and restrictive) hypotheses in this process In allcases only two dimensionless groups emerge one of themconsisting of a grouping of the soil parameters and the othercontaining the unknown characteristic time Based on theseresults universal curves have been depicted the first to deter-mine the characteristic time of the process and the secondthat represents the evolution of average degree of settlement

The solutions have been verified by numerical simula-tions changing the values of the soil parameters and thedimensionless groups conveniently Finally the reliability andeffectiveness of the proposed solutions have been shownthrough a case study

Abbreviations

cv Consolidation coefficient (m2s or m2yr)cv1 Initial consolidation coefficient (m2s or

m2yr)cv2 Final consolidation coefficient (m2s or

m2yr)119889z Element of differential length in thedirection of the spatial coordinate z (m)119889zo Element of differential length in thedirection of the spatial coordinate z at theinitial time (m)

e Void ratio (dimensionless)

eo Initial void ratio (dimensionless)ef Final void ratio (dimensionless)H Soil thickness up to the impervious

boundary or drainage length (m)H1Ho Initial soil thickness (m)H2 Final soil thickness (m)k Hydraulic conductivity (or permeability)

(ms or myr)k1 ko Initial hydraulic conductivity (ms or

myr)kf Final hydraulic conductivity (ms or myr)mv Coefficient of volumetric compressibility

(m2N)mv1 Initial coefficient of volumetric

compressibility (m2N)t Time Independent variable (s or yr)T Juarez-Badillorsquos time factor

(dimensionless)u Excess pore pressure (Nm2)U Average degree of consolidation or

pressure dissipation (dimensionless)Us Average degree of consolidation

(dimensionless)U1205901015840 Average degree of pressure dissipation

(dimensionless)V Soil volume (m3)V1Vo Initial soil volume (m3)z Vertical spatial coordinate (m)120574v Nonlinear coefficient of compressibility of

Juarez-Badillo (dimensionless)120574kv Nonlinear coefficient of change ofpermeability with volume (dimensionless)120574k1205901015840 Nonlinear coefficient of change ofpermeability with effective pressure(dimensionless)120574v1205901015840 Nonlinear coefficient of compressibility orcoefficient of volumetric contraction(dimensionless)120574w Specific weight of water (Nm3)

Loading factor Ratio between the finaland initial effective pressure(dimensionless)120581 Nonlinear coefficient of change ofpermeability with volume ofJuarez-Badillo (dimensionless)120582 Parameter of Juarez-Badillo 120582 = 1 - 120574v120581(dimensionless)

V Auxiliary variable of Juarez-Badillo(dimensionless)120587i Dimensionless group or number(dimensionless)1205901015840 Effective pressure of the soil (Nm2)12059010158401 1205901015840o Initial effective pressure (Nm2)12059010158402 1205901015840f Final effective pressure (Nm2)1205901015840m Mean value of the effective pressure(Nm2)120591o Characteristic time (s or yr)

14 Mathematical Problems in Engineering

120591os Characteristic time which takes thesettlement to reach approximately its finalvalue (s or yr)120591o1205901015840 Characteristic time which takes the excesspore pressure to dissipate untilapproximately the value of zero (s or yr)Ψ Arbitrary mathematical function120577 Differential void ratio (dimensionless)1205771 1205772 Initial and final differential void ratio(dimensionless)

Subscripts

I II Denote different numbers

Superscripts

1015840 Denote dimensionless magnitude

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] K Terzaghi ldquoDie Berechnung der Durchlassigkeitszifferdestones aus dem verlauf der hydrodynamischen spanunngser-scheinnungenrdquo Technical report II a 132 N 34 125-138Akademie derWissenschaften inWien SitzungsberichteMath-naturwiss Klasse Abt 1923

[2] R T Walker ldquoVertical drain consolidation analysis in one twoand three dimensionsrdquo Computers amp Geosciences vol 38 no 8pp 1069ndash1077 2011

[3] G Y Zheng P Li and C Y Zhao ldquoAnalysis of non-linearconsolidation of soft clay by differential quadrature methodrdquoApplied Clay Science vol 79 pp 2ndash7 2013

[4] X Y Geng C J Xu and Y Q Cai ldquoNon-linear consolidationanalysis of soil with variable compressibility and permeabilityunder cyclic loadingsrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 30 no 8 pp 803ndash8212006

[5] M Lu S Wang S W Sloan B Indraratna and K Xie ldquoNon-linear radial consolidation of vertical drains under a generaltime-variable loadingrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 39 no 1 pp 51ndash622015

[6] HWu L HuW Qi and QWen ldquoAnalytical Solution for Elec-troosmotic Consolidation Considering Nonlinear Variation ofSoil Parametersrdquo International Journal of Geomechanics vol 17no 5 Article ID 06016032 2017

[7] S J Brandenberg ldquoiConsoljs JavaScript Implicit Finite-Differ-ence Code for Nonlinear Consolidation and Secondary Com-pressionrdquo International Journal of Geomechanics vol 17 no 6Article ID 04016149 2017

[8] R Butterfield ldquoA natural compression law for soils (an advanceon e-log prsquo)rdquo Geotechnique vol 29 no 4 pp 469ndash480 1979

[9] R Lancellotta and L Preziosi ldquoA general nonlinear mathemati-calmodel for soil consolidation problemsrdquo International Journalof Engineering Science vol 35 no 10-11 pp 1045ndash1063 1997

[10] Y-C Zhuang K-H Xie and X-B Li ldquoNonlinear analysis ofconsolidation with variable compressibility and permeabilityrdquoJournal of Zhejiang University (Engineering Science Edition) vol6 no 3 pp 181ndash187 2005

[11] E Conte andA Troncone ldquoNonlinear consolidation of thin lay-ers subjected to time-dependent loadingrdquo Canadian Geotechni-cal Journal vol 44 no 6 pp 717ndash725 2007

[12] R Yong and F Townsend ldquoGeneral Consolidation Theory forClaysrdquo Soil Mechanics Series Report No 8 Graduate School ofEngineering National University of Mexico 1983

[13] J C Gibbings Dimensional Analysis Springer-Verlag LondonUK 2011

[14] E Buckingham ldquoOn physically similar systems Illustrations ofthe use of dimensional equationsrdquo Physical Review A AtomicMolecular and Optical Physics vol 4 no 4 pp 345ndash376 1914

[15] R Seshadri and T Y Na ldquoA Survey ofMethods for DeterminingSimilarity Transformationsrdquo inGroup Invariance in EngineeringBoundary Value Problems pp 35ndash61 Springer New York NYUSA 1985

[16] V D Zimparov and V M Petkov ldquoApplication of discriminatedanalysis to low Reynolds number swirl flows in circular tubeswith twisted-tape inserts Pressure drop correlationsrdquo Interna-tional Review of Chemical Engineering vol 1 no 4 pp 346ndash3562009

[17] M Capobianchi and A Aziz ldquoA scale analysis for natural con-vective flows over vertical surfacesrdquo International Journal ofThermal Sciences vol 54 pp 82ndash88 2012

[18] M Conesa J F Sanchez Perez I Alhama and F AlhamaldquoOn the nondimensionalization of coupled nonlinear ordinarydifferential equationsrdquo Nonlinear Dynamics vol 84 no 1 pp91ndash105 2016

[19] I A Manteca G Garcıa-Ros and F Alhama ldquoUniversal solu-tion for the characteristic time and the degree of settlement innonlinear soil consolidation scenarios A deduction based onnondimensionalizationrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 57 pp 186ndash201 2018

[20] E H Davis and G P Raymond ldquoA non-linear theory ofconsolidationrdquo Geotechnique vol 15 no 2 pp 161ndash173 1965

[21] P Cornetti and M Battaglio ldquoNonlinear consolidation of soilmodeling and solution techniquesrdquo Mathematical and Com-puter Modelling vol 20 no 7 pp 1ndash12 1994

[22] C F Gonzalez-Fernandez ldquoApplications of the network simu-lation method to transport processesrdquo in Network SimulationMethod J Horno Ed Research Signpost Trivandrum India2002

[23] J Serna F J S Velasco and A Soto Meca ldquoApplication of net-work simulationmethod to viscous flowsThe nanofluid heatedlid cavity under pulsating flowrdquo Computers amp Fluids vol 91 pp10ndash20 2014

[24] E Juarez-Badillo ldquoGeneral Permeability Change Equationsfor Soilsrdquo in Proceedings of the International Conference onConstitutive Laws pp 205ndash209 University of Arizona TuesonAriz USA 1983

[25] E Juarez-Badillo and B Chen ldquoConsolidation curves for claysrdquoJournal of Geotechnical Engineering vol 109 no 10 pp 1303ndash1312 1983

Mathematical Problems in Engineering 15

[26] N Abbasi H Rahimi A A Javadi andA Fakher ldquoFinite differ-ence approach for consolidation with variable compressibilityand permeabilityrdquo Computers amp Geosciences vol 34 no 1 pp41ndash52 2007

[27] R F Scott Principles of Soils Mechanics Addison-Wesley Pub-lishing Company 1963

[28] F Alhama and C N Madrid Analisis Dimensional Discrimi-nado En Mecanica De Fluidos Y Transferencia De Calor 2012

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Mathematical Problems in Engineering 11

040045050055060065070075080085090095100105110

minus07 minus06 minus05 minus04 minus03 minus02 minus01 0 01 02 03 04 05 06 07 08 09 1 11 12 13 14 15

1

2

ᴪ=8ᴪ=6ᴪ=4

ᴪ=2ᴪ=15ᴪ=101

Figure 4 120587I as a function of 120587II with as parameter

1 2 3 4 5 6 7 8minus042

minus04

minus038

minus036

minus034

minus032

minus03

minus028

minus026

m

datafitted curve

Figure 5 m as a function of

solutions provided by numerical models based on differentconstitutive relations (potential and tabulated in this case)Even so this value is totally acceptable within this field ofengineering demonstrating the strength and versatility of theuniversal solutions provided

7 Discussion

This work derives the dimensionless groups that rule thenonlinear soil consolidation process based on the original J-Bmodel following a formal nondimensionalization procedurethat starts from the governing equation and does not needany other assumption or mathematical manipulation Inaddition more real and precise extended J-B models whichassume both the void ratio and the thickness of the finitevolume element of the soil to be not constant have been

0010203040506070809

1

0 05 1 15 2 25 3

96 scenariosᴪ = [101 15 2 4 6 8]

Aver

age d

egre

e of c

onso

lidat

ion

tos

= [005 01 015 02]v= [02 05 1 15]k

Figure 6 Average degree of consolidation versus dimensionlesstime for different scenarios

0010203040506070809

1

0 05 1 15 2 25 3

Aver

age d

egre

e of c

onso

lidat

ion

tos

Figure 7 Proposed solution for the average degree of consolidation

nondimensionalized and their dimensionless groups derivedThe assumption of constitutive dependences (e-1205901015840 e-k and k-1205901015840) of potential type in these models is interesting since they

12 Mathematical Problems in Engineering

Table 4 Case study solution for the characteristic time 120591os Comparison with numerical models

Universal curves J-B most general model Tabulated model120591os (yr) 120591os (yr) Rel error 120591os (yr) Rel error 0748 0750 026 0817 820

0010203040506070809

1

0 025 05 075 1 125 15 175 2 225t ( years)

Aver

age d

egre

e of c

onso

lidat

ion

Figure 8 Case study solution for the average degree of consolida-tion Us from the universal curves

can be applied to highly compressible soils for which the e-log(1205901015840) relation moves away from being a straight line andin which negative values of the void ratio could be obtainedunder the application of high load ratios (1205901015840f 1205901015840o) [8]

The study has been developed both in terms of pres-sure dissipation and in terms of settlement resulting thatthe dimensionless groups derived from each model are ingeneral different The extension to more complex modelswhich implies the emergence of new parameters or equationswithin themathematical model does not necessarily give riseto additional dimensionless groups as might be expectedbut to new expressions of the groups or new regroupingof the parameters involved Thus the elimination of thehypothesis ldquodz constantrdquo in the most general consolidationmodel has not given rise to any new group both in pressureand in settlement (Table 1) but it has only modified oneof the groups obtained in the original model All thisrepresents an important advance in relation to the modelsand studies carried out by other authors and the attemptsof characterization found in the scientific literature [9ndash11]which try to lead to easy-to-use solutions for engineers Inno case had dimensionless groups been obtained as preciseas those presented in this work nor presented universalcurves of the characteristic time and the average degree ofconsolidation Authors such asConte andTroncone [11] applymore general load conditions (time-dependent loading) forwhich the dimensionless groups deduced in this work couldbe applicable for each load step

As regards time to make it dimensionless an unknownreference (120591o) is introduced in the form mentioned by Scott[27] and used in many other problems with asymptoticsolutions [28]Theprecise definition of 120591o necessarily involvesa criterion in relation to the percentage of average pressuredecrease (or settlement) associated with the end of theconsolidation processTheorder ofmagnitude of 120591o is derived

after nondimensionalization making 120591o only appear in oneof the independent dimensionless groups In this way theaverage degree of consolidation (in terms of pressure orsettlement) depends on the ldquotimecharacteristic timerdquo ratioin addition to the dimensionless groups (without unknowns)that arise

In short the procedure has led to a new very precisesolution that involves two monomials one (120587I) that containsthe characteristic time (120591o) and another (120587II) that is amathematical function of the soil parameters In this way120591o only depends on 120587II through the function 120587I = 120595(120587II)while U depends on 120587I and 120587II through the expression U= 120595(120587I 120587II) The results obtained have been verified forthe most general and complex consolidation model throughthe numerical simulation of a large number of scenariosin which the values of the dimensionless groups and theindividual parameters of the problem have been changing toconvenience The high errors produced by the original J-Bmodel with respect to the most general justify the use of thelatter in the search for more precise solutions

In addition the emergence of mean values of the effectivepressure when averaging the governing equation in orderto assign an order of magnitude unit to the dimensionlessvariables and their derivatives is an added difficulty to obtainthe dimensionless groups This is undoubtedly due to thenonlinearities (constitutive relations) involved in the consol-idation process This average effective pressure of referencehas been assumed as the semisum of its initial and finalvalues although other criteria could also be chosen Sincetwo scenarios can have equal120587II with the same effectivemeanpressure but different values of the parameters 120574v1205901015840 and 120574k1205901015840 we expect some minor deviations between the results of bothscenarios as indeed it has been checked Thus for the wholerange of parameters values in typical soils the errors due tothe above cause are negligible in the field of soil mechanicsOn the other hand when the value of 120587II does not changebut the mean effective pressure value does the errors in thesolutions of 120591os and Us are slightly larger although withinranges below 4

The brooch for all this work is the presentation ofuniversal curves for the most general model which allowfor the usual range of variation of the soil properties toobtain in a simple way the value of the characteristic timeof consolidation and the evolution of the average degreeof settlement throughout the entire process Thus startingfrom the nonlinear parameters of the soil (120574v1205901015840 and 120574k1205901015840)and the load ratio (1205901015840f 1205901015840o) the value of the monomial 120587IIis obtained from which the value of 120587I is directly deductedthanks to the universal curve 120587I = 120595(120587II) provided Finally120591os is easily obtained from the expression of 120587I On the otherhand the curve that represents the Us evolution along theprocess has turned out to be (practically) unique when it is

Mathematical Problems in Engineering 13

represented as a function of the dimensionless time in theform t120591os regardless of the soil parameters and the loadsapplied All this supposes a clear advantage when solvingthe consolidation problem also showing the strength of thenondimensionalization method presented

Finally the case study addressed has demonstrated thesimplicity and reliability of the universal curves proposedFrom a series of tabulated data (e-1205901015840 and e-k) the parametersthat govern the nonlinear consolidation problem for themostextended J-B model have been adjusted with which thevalue of the monomial 120587II has been derived With this andusing the universal curves the solutions for the characteristictime 120591os and the average degree of consolidation Us havebeen obtained These results have been satisfactorily verifiedwith the solutions provided both by the numerical extendedmodel of J-B presented here and by another that substitutesthe constitutive dependences of potential type by data intabulated form

8 Conclusions

The application of the nondimensionalization technique hasallowed us to deduce the dimensionless groups that rule thenonlinear consolidation problem in which the constitutivedependencies e-1205901015840 and e-k have the form of potential func-tions This type of relations approximates the soil behaviorbetter than the logarithmic functions e-log(1205901015840) and e-log(k)particularly in very compressible soils and under high loadratios (1205901015840f 1205901015840o) in which inconsistent negative values of thevoid ratio may appear

Several scenarios have been addressed from the originalmodel of Juarez-Badillo to extended models based on thesame constitutive dependences but eliminating one or moreclassic (and restrictive) hypotheses in this process In allcases only two dimensionless groups emerge one of themconsisting of a grouping of the soil parameters and the othercontaining the unknown characteristic time Based on theseresults universal curves have been depicted the first to deter-mine the characteristic time of the process and the secondthat represents the evolution of average degree of settlement

The solutions have been verified by numerical simula-tions changing the values of the soil parameters and thedimensionless groups conveniently Finally the reliability andeffectiveness of the proposed solutions have been shownthrough a case study

Abbreviations

cv Consolidation coefficient (m2s or m2yr)cv1 Initial consolidation coefficient (m2s or

m2yr)cv2 Final consolidation coefficient (m2s or

m2yr)119889z Element of differential length in thedirection of the spatial coordinate z (m)119889zo Element of differential length in thedirection of the spatial coordinate z at theinitial time (m)

e Void ratio (dimensionless)

eo Initial void ratio (dimensionless)ef Final void ratio (dimensionless)H Soil thickness up to the impervious

boundary or drainage length (m)H1Ho Initial soil thickness (m)H2 Final soil thickness (m)k Hydraulic conductivity (or permeability)

(ms or myr)k1 ko Initial hydraulic conductivity (ms or

myr)kf Final hydraulic conductivity (ms or myr)mv Coefficient of volumetric compressibility

(m2N)mv1 Initial coefficient of volumetric

compressibility (m2N)t Time Independent variable (s or yr)T Juarez-Badillorsquos time factor

(dimensionless)u Excess pore pressure (Nm2)U Average degree of consolidation or

pressure dissipation (dimensionless)Us Average degree of consolidation

(dimensionless)U1205901015840 Average degree of pressure dissipation

(dimensionless)V Soil volume (m3)V1Vo Initial soil volume (m3)z Vertical spatial coordinate (m)120574v Nonlinear coefficient of compressibility of

Juarez-Badillo (dimensionless)120574kv Nonlinear coefficient of change ofpermeability with volume (dimensionless)120574k1205901015840 Nonlinear coefficient of change ofpermeability with effective pressure(dimensionless)120574v1205901015840 Nonlinear coefficient of compressibility orcoefficient of volumetric contraction(dimensionless)120574w Specific weight of water (Nm3)

Loading factor Ratio between the finaland initial effective pressure(dimensionless)120581 Nonlinear coefficient of change ofpermeability with volume ofJuarez-Badillo (dimensionless)120582 Parameter of Juarez-Badillo 120582 = 1 - 120574v120581(dimensionless)

V Auxiliary variable of Juarez-Badillo(dimensionless)120587i Dimensionless group or number(dimensionless)1205901015840 Effective pressure of the soil (Nm2)12059010158401 1205901015840o Initial effective pressure (Nm2)12059010158402 1205901015840f Final effective pressure (Nm2)1205901015840m Mean value of the effective pressure(Nm2)120591o Characteristic time (s or yr)

14 Mathematical Problems in Engineering

120591os Characteristic time which takes thesettlement to reach approximately its finalvalue (s or yr)120591o1205901015840 Characteristic time which takes the excesspore pressure to dissipate untilapproximately the value of zero (s or yr)Ψ Arbitrary mathematical function120577 Differential void ratio (dimensionless)1205771 1205772 Initial and final differential void ratio(dimensionless)

Subscripts

I II Denote different numbers

Superscripts

1015840 Denote dimensionless magnitude

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] K Terzaghi ldquoDie Berechnung der Durchlassigkeitszifferdestones aus dem verlauf der hydrodynamischen spanunngser-scheinnungenrdquo Technical report II a 132 N 34 125-138Akademie derWissenschaften inWien SitzungsberichteMath-naturwiss Klasse Abt 1923

[2] R T Walker ldquoVertical drain consolidation analysis in one twoand three dimensionsrdquo Computers amp Geosciences vol 38 no 8pp 1069ndash1077 2011

[3] G Y Zheng P Li and C Y Zhao ldquoAnalysis of non-linearconsolidation of soft clay by differential quadrature methodrdquoApplied Clay Science vol 79 pp 2ndash7 2013

[4] X Y Geng C J Xu and Y Q Cai ldquoNon-linear consolidationanalysis of soil with variable compressibility and permeabilityunder cyclic loadingsrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 30 no 8 pp 803ndash8212006

[5] M Lu S Wang S W Sloan B Indraratna and K Xie ldquoNon-linear radial consolidation of vertical drains under a generaltime-variable loadingrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 39 no 1 pp 51ndash622015

[6] HWu L HuW Qi and QWen ldquoAnalytical Solution for Elec-troosmotic Consolidation Considering Nonlinear Variation ofSoil Parametersrdquo International Journal of Geomechanics vol 17no 5 Article ID 06016032 2017

[7] S J Brandenberg ldquoiConsoljs JavaScript Implicit Finite-Differ-ence Code for Nonlinear Consolidation and Secondary Com-pressionrdquo International Journal of Geomechanics vol 17 no 6Article ID 04016149 2017

[8] R Butterfield ldquoA natural compression law for soils (an advanceon e-log prsquo)rdquo Geotechnique vol 29 no 4 pp 469ndash480 1979

[9] R Lancellotta and L Preziosi ldquoA general nonlinear mathemati-calmodel for soil consolidation problemsrdquo International Journalof Engineering Science vol 35 no 10-11 pp 1045ndash1063 1997

[10] Y-C Zhuang K-H Xie and X-B Li ldquoNonlinear analysis ofconsolidation with variable compressibility and permeabilityrdquoJournal of Zhejiang University (Engineering Science Edition) vol6 no 3 pp 181ndash187 2005

[11] E Conte andA Troncone ldquoNonlinear consolidation of thin lay-ers subjected to time-dependent loadingrdquo Canadian Geotechni-cal Journal vol 44 no 6 pp 717ndash725 2007

[12] R Yong and F Townsend ldquoGeneral Consolidation Theory forClaysrdquo Soil Mechanics Series Report No 8 Graduate School ofEngineering National University of Mexico 1983

[13] J C Gibbings Dimensional Analysis Springer-Verlag LondonUK 2011

[14] E Buckingham ldquoOn physically similar systems Illustrations ofthe use of dimensional equationsrdquo Physical Review A AtomicMolecular and Optical Physics vol 4 no 4 pp 345ndash376 1914

[15] R Seshadri and T Y Na ldquoA Survey ofMethods for DeterminingSimilarity Transformationsrdquo inGroup Invariance in EngineeringBoundary Value Problems pp 35ndash61 Springer New York NYUSA 1985

[16] V D Zimparov and V M Petkov ldquoApplication of discriminatedanalysis to low Reynolds number swirl flows in circular tubeswith twisted-tape inserts Pressure drop correlationsrdquo Interna-tional Review of Chemical Engineering vol 1 no 4 pp 346ndash3562009

[17] M Capobianchi and A Aziz ldquoA scale analysis for natural con-vective flows over vertical surfacesrdquo International Journal ofThermal Sciences vol 54 pp 82ndash88 2012

[18] M Conesa J F Sanchez Perez I Alhama and F AlhamaldquoOn the nondimensionalization of coupled nonlinear ordinarydifferential equationsrdquo Nonlinear Dynamics vol 84 no 1 pp91ndash105 2016

[19] I A Manteca G Garcıa-Ros and F Alhama ldquoUniversal solu-tion for the characteristic time and the degree of settlement innonlinear soil consolidation scenarios A deduction based onnondimensionalizationrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 57 pp 186ndash201 2018

[20] E H Davis and G P Raymond ldquoA non-linear theory ofconsolidationrdquo Geotechnique vol 15 no 2 pp 161ndash173 1965

[21] P Cornetti and M Battaglio ldquoNonlinear consolidation of soilmodeling and solution techniquesrdquo Mathematical and Com-puter Modelling vol 20 no 7 pp 1ndash12 1994

[22] C F Gonzalez-Fernandez ldquoApplications of the network simu-lation method to transport processesrdquo in Network SimulationMethod J Horno Ed Research Signpost Trivandrum India2002

[23] J Serna F J S Velasco and A Soto Meca ldquoApplication of net-work simulationmethod to viscous flowsThe nanofluid heatedlid cavity under pulsating flowrdquo Computers amp Fluids vol 91 pp10ndash20 2014

[24] E Juarez-Badillo ldquoGeneral Permeability Change Equationsfor Soilsrdquo in Proceedings of the International Conference onConstitutive Laws pp 205ndash209 University of Arizona TuesonAriz USA 1983

[25] E Juarez-Badillo and B Chen ldquoConsolidation curves for claysrdquoJournal of Geotechnical Engineering vol 109 no 10 pp 1303ndash1312 1983

Mathematical Problems in Engineering 15

[26] N Abbasi H Rahimi A A Javadi andA Fakher ldquoFinite differ-ence approach for consolidation with variable compressibilityand permeabilityrdquo Computers amp Geosciences vol 34 no 1 pp41ndash52 2007

[27] R F Scott Principles of Soils Mechanics Addison-Wesley Pub-lishing Company 1963

[28] F Alhama and C N Madrid Analisis Dimensional Discrimi-nado En Mecanica De Fluidos Y Transferencia De Calor 2012

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

12 Mathematical Problems in Engineering

Table 4 Case study solution for the characteristic time 120591os Comparison with numerical models

Universal curves J-B most general model Tabulated model120591os (yr) 120591os (yr) Rel error 120591os (yr) Rel error 0748 0750 026 0817 820

0010203040506070809

1

0 025 05 075 1 125 15 175 2 225t ( years)

Aver

age d

egre

e of c

onso

lidat

ion

Figure 8 Case study solution for the average degree of consolida-tion Us from the universal curves

can be applied to highly compressible soils for which the e-log(1205901015840) relation moves away from being a straight line andin which negative values of the void ratio could be obtainedunder the application of high load ratios (1205901015840f 1205901015840o) [8]

The study has been developed both in terms of pres-sure dissipation and in terms of settlement resulting thatthe dimensionless groups derived from each model are ingeneral different The extension to more complex modelswhich implies the emergence of new parameters or equationswithin themathematical model does not necessarily give riseto additional dimensionless groups as might be expectedbut to new expressions of the groups or new regroupingof the parameters involved Thus the elimination of thehypothesis ldquodz constantrdquo in the most general consolidationmodel has not given rise to any new group both in pressureand in settlement (Table 1) but it has only modified oneof the groups obtained in the original model All thisrepresents an important advance in relation to the modelsand studies carried out by other authors and the attemptsof characterization found in the scientific literature [9ndash11]which try to lead to easy-to-use solutions for engineers Inno case had dimensionless groups been obtained as preciseas those presented in this work nor presented universalcurves of the characteristic time and the average degree ofconsolidation Authors such asConte andTroncone [11] applymore general load conditions (time-dependent loading) forwhich the dimensionless groups deduced in this work couldbe applicable for each load step

As regards time to make it dimensionless an unknownreference (120591o) is introduced in the form mentioned by Scott[27] and used in many other problems with asymptoticsolutions [28]Theprecise definition of 120591o necessarily involvesa criterion in relation to the percentage of average pressuredecrease (or settlement) associated with the end of theconsolidation processTheorder ofmagnitude of 120591o is derived

after nondimensionalization making 120591o only appear in oneof the independent dimensionless groups In this way theaverage degree of consolidation (in terms of pressure orsettlement) depends on the ldquotimecharacteristic timerdquo ratioin addition to the dimensionless groups (without unknowns)that arise

In short the procedure has led to a new very precisesolution that involves two monomials one (120587I) that containsthe characteristic time (120591o) and another (120587II) that is amathematical function of the soil parameters In this way120591o only depends on 120587II through the function 120587I = 120595(120587II)while U depends on 120587I and 120587II through the expression U= 120595(120587I 120587II) The results obtained have been verified forthe most general and complex consolidation model throughthe numerical simulation of a large number of scenariosin which the values of the dimensionless groups and theindividual parameters of the problem have been changing toconvenience The high errors produced by the original J-Bmodel with respect to the most general justify the use of thelatter in the search for more precise solutions

In addition the emergence of mean values of the effectivepressure when averaging the governing equation in orderto assign an order of magnitude unit to the dimensionlessvariables and their derivatives is an added difficulty to obtainthe dimensionless groups This is undoubtedly due to thenonlinearities (constitutive relations) involved in the consol-idation process This average effective pressure of referencehas been assumed as the semisum of its initial and finalvalues although other criteria could also be chosen Sincetwo scenarios can have equal120587II with the same effectivemeanpressure but different values of the parameters 120574v1205901015840 and 120574k1205901015840 we expect some minor deviations between the results of bothscenarios as indeed it has been checked Thus for the wholerange of parameters values in typical soils the errors due tothe above cause are negligible in the field of soil mechanicsOn the other hand when the value of 120587II does not changebut the mean effective pressure value does the errors in thesolutions of 120591os and Us are slightly larger although withinranges below 4

The brooch for all this work is the presentation ofuniversal curves for the most general model which allowfor the usual range of variation of the soil properties toobtain in a simple way the value of the characteristic timeof consolidation and the evolution of the average degreeof settlement throughout the entire process Thus startingfrom the nonlinear parameters of the soil (120574v1205901015840 and 120574k1205901015840)and the load ratio (1205901015840f 1205901015840o) the value of the monomial 120587IIis obtained from which the value of 120587I is directly deductedthanks to the universal curve 120587I = 120595(120587II) provided Finally120591os is easily obtained from the expression of 120587I On the otherhand the curve that represents the Us evolution along theprocess has turned out to be (practically) unique when it is

Mathematical Problems in Engineering 13

represented as a function of the dimensionless time in theform t120591os regardless of the soil parameters and the loadsapplied All this supposes a clear advantage when solvingthe consolidation problem also showing the strength of thenondimensionalization method presented

Finally the case study addressed has demonstrated thesimplicity and reliability of the universal curves proposedFrom a series of tabulated data (e-1205901015840 and e-k) the parametersthat govern the nonlinear consolidation problem for themostextended J-B model have been adjusted with which thevalue of the monomial 120587II has been derived With this andusing the universal curves the solutions for the characteristictime 120591os and the average degree of consolidation Us havebeen obtained These results have been satisfactorily verifiedwith the solutions provided both by the numerical extendedmodel of J-B presented here and by another that substitutesthe constitutive dependences of potential type by data intabulated form

8 Conclusions

The application of the nondimensionalization technique hasallowed us to deduce the dimensionless groups that rule thenonlinear consolidation problem in which the constitutivedependencies e-1205901015840 and e-k have the form of potential func-tions This type of relations approximates the soil behaviorbetter than the logarithmic functions e-log(1205901015840) and e-log(k)particularly in very compressible soils and under high loadratios (1205901015840f 1205901015840o) in which inconsistent negative values of thevoid ratio may appear

Several scenarios have been addressed from the originalmodel of Juarez-Badillo to extended models based on thesame constitutive dependences but eliminating one or moreclassic (and restrictive) hypotheses in this process In allcases only two dimensionless groups emerge one of themconsisting of a grouping of the soil parameters and the othercontaining the unknown characteristic time Based on theseresults universal curves have been depicted the first to deter-mine the characteristic time of the process and the secondthat represents the evolution of average degree of settlement

The solutions have been verified by numerical simula-tions changing the values of the soil parameters and thedimensionless groups conveniently Finally the reliability andeffectiveness of the proposed solutions have been shownthrough a case study

Abbreviations

cv Consolidation coefficient (m2s or m2yr)cv1 Initial consolidation coefficient (m2s or

m2yr)cv2 Final consolidation coefficient (m2s or

m2yr)119889z Element of differential length in thedirection of the spatial coordinate z (m)119889zo Element of differential length in thedirection of the spatial coordinate z at theinitial time (m)

e Void ratio (dimensionless)

eo Initial void ratio (dimensionless)ef Final void ratio (dimensionless)H Soil thickness up to the impervious

boundary or drainage length (m)H1Ho Initial soil thickness (m)H2 Final soil thickness (m)k Hydraulic conductivity (or permeability)

(ms or myr)k1 ko Initial hydraulic conductivity (ms or

myr)kf Final hydraulic conductivity (ms or myr)mv Coefficient of volumetric compressibility

(m2N)mv1 Initial coefficient of volumetric

compressibility (m2N)t Time Independent variable (s or yr)T Juarez-Badillorsquos time factor

(dimensionless)u Excess pore pressure (Nm2)U Average degree of consolidation or

pressure dissipation (dimensionless)Us Average degree of consolidation

(dimensionless)U1205901015840 Average degree of pressure dissipation

(dimensionless)V Soil volume (m3)V1Vo Initial soil volume (m3)z Vertical spatial coordinate (m)120574v Nonlinear coefficient of compressibility of

Juarez-Badillo (dimensionless)120574kv Nonlinear coefficient of change ofpermeability with volume (dimensionless)120574k1205901015840 Nonlinear coefficient of change ofpermeability with effective pressure(dimensionless)120574v1205901015840 Nonlinear coefficient of compressibility orcoefficient of volumetric contraction(dimensionless)120574w Specific weight of water (Nm3)

Loading factor Ratio between the finaland initial effective pressure(dimensionless)120581 Nonlinear coefficient of change ofpermeability with volume ofJuarez-Badillo (dimensionless)120582 Parameter of Juarez-Badillo 120582 = 1 - 120574v120581(dimensionless)

V Auxiliary variable of Juarez-Badillo(dimensionless)120587i Dimensionless group or number(dimensionless)1205901015840 Effective pressure of the soil (Nm2)12059010158401 1205901015840o Initial effective pressure (Nm2)12059010158402 1205901015840f Final effective pressure (Nm2)1205901015840m Mean value of the effective pressure(Nm2)120591o Characteristic time (s or yr)

14 Mathematical Problems in Engineering

120591os Characteristic time which takes thesettlement to reach approximately its finalvalue (s or yr)120591o1205901015840 Characteristic time which takes the excesspore pressure to dissipate untilapproximately the value of zero (s or yr)Ψ Arbitrary mathematical function120577 Differential void ratio (dimensionless)1205771 1205772 Initial and final differential void ratio(dimensionless)

Subscripts

I II Denote different numbers

Superscripts

1015840 Denote dimensionless magnitude

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] K Terzaghi ldquoDie Berechnung der Durchlassigkeitszifferdestones aus dem verlauf der hydrodynamischen spanunngser-scheinnungenrdquo Technical report II a 132 N 34 125-138Akademie derWissenschaften inWien SitzungsberichteMath-naturwiss Klasse Abt 1923

[2] R T Walker ldquoVertical drain consolidation analysis in one twoand three dimensionsrdquo Computers amp Geosciences vol 38 no 8pp 1069ndash1077 2011

[3] G Y Zheng P Li and C Y Zhao ldquoAnalysis of non-linearconsolidation of soft clay by differential quadrature methodrdquoApplied Clay Science vol 79 pp 2ndash7 2013

[4] X Y Geng C J Xu and Y Q Cai ldquoNon-linear consolidationanalysis of soil with variable compressibility and permeabilityunder cyclic loadingsrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 30 no 8 pp 803ndash8212006

[5] M Lu S Wang S W Sloan B Indraratna and K Xie ldquoNon-linear radial consolidation of vertical drains under a generaltime-variable loadingrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 39 no 1 pp 51ndash622015

[6] HWu L HuW Qi and QWen ldquoAnalytical Solution for Elec-troosmotic Consolidation Considering Nonlinear Variation ofSoil Parametersrdquo International Journal of Geomechanics vol 17no 5 Article ID 06016032 2017

[7] S J Brandenberg ldquoiConsoljs JavaScript Implicit Finite-Differ-ence Code for Nonlinear Consolidation and Secondary Com-pressionrdquo International Journal of Geomechanics vol 17 no 6Article ID 04016149 2017

[8] R Butterfield ldquoA natural compression law for soils (an advanceon e-log prsquo)rdquo Geotechnique vol 29 no 4 pp 469ndash480 1979

[9] R Lancellotta and L Preziosi ldquoA general nonlinear mathemati-calmodel for soil consolidation problemsrdquo International Journalof Engineering Science vol 35 no 10-11 pp 1045ndash1063 1997

[10] Y-C Zhuang K-H Xie and X-B Li ldquoNonlinear analysis ofconsolidation with variable compressibility and permeabilityrdquoJournal of Zhejiang University (Engineering Science Edition) vol6 no 3 pp 181ndash187 2005

[11] E Conte andA Troncone ldquoNonlinear consolidation of thin lay-ers subjected to time-dependent loadingrdquo Canadian Geotechni-cal Journal vol 44 no 6 pp 717ndash725 2007

[12] R Yong and F Townsend ldquoGeneral Consolidation Theory forClaysrdquo Soil Mechanics Series Report No 8 Graduate School ofEngineering National University of Mexico 1983

[13] J C Gibbings Dimensional Analysis Springer-Verlag LondonUK 2011

[14] E Buckingham ldquoOn physically similar systems Illustrations ofthe use of dimensional equationsrdquo Physical Review A AtomicMolecular and Optical Physics vol 4 no 4 pp 345ndash376 1914

[15] R Seshadri and T Y Na ldquoA Survey ofMethods for DeterminingSimilarity Transformationsrdquo inGroup Invariance in EngineeringBoundary Value Problems pp 35ndash61 Springer New York NYUSA 1985

[16] V D Zimparov and V M Petkov ldquoApplication of discriminatedanalysis to low Reynolds number swirl flows in circular tubeswith twisted-tape inserts Pressure drop correlationsrdquo Interna-tional Review of Chemical Engineering vol 1 no 4 pp 346ndash3562009

[17] M Capobianchi and A Aziz ldquoA scale analysis for natural con-vective flows over vertical surfacesrdquo International Journal ofThermal Sciences vol 54 pp 82ndash88 2012

[18] M Conesa J F Sanchez Perez I Alhama and F AlhamaldquoOn the nondimensionalization of coupled nonlinear ordinarydifferential equationsrdquo Nonlinear Dynamics vol 84 no 1 pp91ndash105 2016

[19] I A Manteca G Garcıa-Ros and F Alhama ldquoUniversal solu-tion for the characteristic time and the degree of settlement innonlinear soil consolidation scenarios A deduction based onnondimensionalizationrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 57 pp 186ndash201 2018

[20] E H Davis and G P Raymond ldquoA non-linear theory ofconsolidationrdquo Geotechnique vol 15 no 2 pp 161ndash173 1965

[21] P Cornetti and M Battaglio ldquoNonlinear consolidation of soilmodeling and solution techniquesrdquo Mathematical and Com-puter Modelling vol 20 no 7 pp 1ndash12 1994

[22] C F Gonzalez-Fernandez ldquoApplications of the network simu-lation method to transport processesrdquo in Network SimulationMethod J Horno Ed Research Signpost Trivandrum India2002

[23] J Serna F J S Velasco and A Soto Meca ldquoApplication of net-work simulationmethod to viscous flowsThe nanofluid heatedlid cavity under pulsating flowrdquo Computers amp Fluids vol 91 pp10ndash20 2014

[24] E Juarez-Badillo ldquoGeneral Permeability Change Equationsfor Soilsrdquo in Proceedings of the International Conference onConstitutive Laws pp 205ndash209 University of Arizona TuesonAriz USA 1983

[25] E Juarez-Badillo and B Chen ldquoConsolidation curves for claysrdquoJournal of Geotechnical Engineering vol 109 no 10 pp 1303ndash1312 1983

Mathematical Problems in Engineering 15

[26] N Abbasi H Rahimi A A Javadi andA Fakher ldquoFinite differ-ence approach for consolidation with variable compressibilityand permeabilityrdquo Computers amp Geosciences vol 34 no 1 pp41ndash52 2007

[27] R F Scott Principles of Soils Mechanics Addison-Wesley Pub-lishing Company 1963

[28] F Alhama and C N Madrid Analisis Dimensional Discrimi-nado En Mecanica De Fluidos Y Transferencia De Calor 2012

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Mathematical Problems in Engineering 13

represented as a function of the dimensionless time in theform t120591os regardless of the soil parameters and the loadsapplied All this supposes a clear advantage when solvingthe consolidation problem also showing the strength of thenondimensionalization method presented

Finally the case study addressed has demonstrated thesimplicity and reliability of the universal curves proposedFrom a series of tabulated data (e-1205901015840 and e-k) the parametersthat govern the nonlinear consolidation problem for themostextended J-B model have been adjusted with which thevalue of the monomial 120587II has been derived With this andusing the universal curves the solutions for the characteristictime 120591os and the average degree of consolidation Us havebeen obtained These results have been satisfactorily verifiedwith the solutions provided both by the numerical extendedmodel of J-B presented here and by another that substitutesthe constitutive dependences of potential type by data intabulated form

8 Conclusions

The application of the nondimensionalization technique hasallowed us to deduce the dimensionless groups that rule thenonlinear consolidation problem in which the constitutivedependencies e-1205901015840 and e-k have the form of potential func-tions This type of relations approximates the soil behaviorbetter than the logarithmic functions e-log(1205901015840) and e-log(k)particularly in very compressible soils and under high loadratios (1205901015840f 1205901015840o) in which inconsistent negative values of thevoid ratio may appear

Several scenarios have been addressed from the originalmodel of Juarez-Badillo to extended models based on thesame constitutive dependences but eliminating one or moreclassic (and restrictive) hypotheses in this process In allcases only two dimensionless groups emerge one of themconsisting of a grouping of the soil parameters and the othercontaining the unknown characteristic time Based on theseresults universal curves have been depicted the first to deter-mine the characteristic time of the process and the secondthat represents the evolution of average degree of settlement

The solutions have been verified by numerical simula-tions changing the values of the soil parameters and thedimensionless groups conveniently Finally the reliability andeffectiveness of the proposed solutions have been shownthrough a case study

Abbreviations

cv Consolidation coefficient (m2s or m2yr)cv1 Initial consolidation coefficient (m2s or

m2yr)cv2 Final consolidation coefficient (m2s or

m2yr)119889z Element of differential length in thedirection of the spatial coordinate z (m)119889zo Element of differential length in thedirection of the spatial coordinate z at theinitial time (m)

e Void ratio (dimensionless)

eo Initial void ratio (dimensionless)ef Final void ratio (dimensionless)H Soil thickness up to the impervious

boundary or drainage length (m)H1Ho Initial soil thickness (m)H2 Final soil thickness (m)k Hydraulic conductivity (or permeability)

(ms or myr)k1 ko Initial hydraulic conductivity (ms or

myr)kf Final hydraulic conductivity (ms or myr)mv Coefficient of volumetric compressibility

(m2N)mv1 Initial coefficient of volumetric

compressibility (m2N)t Time Independent variable (s or yr)T Juarez-Badillorsquos time factor

(dimensionless)u Excess pore pressure (Nm2)U Average degree of consolidation or

pressure dissipation (dimensionless)Us Average degree of consolidation

(dimensionless)U1205901015840 Average degree of pressure dissipation

(dimensionless)V Soil volume (m3)V1Vo Initial soil volume (m3)z Vertical spatial coordinate (m)120574v Nonlinear coefficient of compressibility of

Juarez-Badillo (dimensionless)120574kv Nonlinear coefficient of change ofpermeability with volume (dimensionless)120574k1205901015840 Nonlinear coefficient of change ofpermeability with effective pressure(dimensionless)120574v1205901015840 Nonlinear coefficient of compressibility orcoefficient of volumetric contraction(dimensionless)120574w Specific weight of water (Nm3)

Loading factor Ratio between the finaland initial effective pressure(dimensionless)120581 Nonlinear coefficient of change ofpermeability with volume ofJuarez-Badillo (dimensionless)120582 Parameter of Juarez-Badillo 120582 = 1 - 120574v120581(dimensionless)

V Auxiliary variable of Juarez-Badillo(dimensionless)120587i Dimensionless group or number(dimensionless)1205901015840 Effective pressure of the soil (Nm2)12059010158401 1205901015840o Initial effective pressure (Nm2)12059010158402 1205901015840f Final effective pressure (Nm2)1205901015840m Mean value of the effective pressure(Nm2)120591o Characteristic time (s or yr)

14 Mathematical Problems in Engineering

120591os Characteristic time which takes thesettlement to reach approximately its finalvalue (s or yr)120591o1205901015840 Characteristic time which takes the excesspore pressure to dissipate untilapproximately the value of zero (s or yr)Ψ Arbitrary mathematical function120577 Differential void ratio (dimensionless)1205771 1205772 Initial and final differential void ratio(dimensionless)

Subscripts

I II Denote different numbers

Superscripts

1015840 Denote dimensionless magnitude

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] K Terzaghi ldquoDie Berechnung der Durchlassigkeitszifferdestones aus dem verlauf der hydrodynamischen spanunngser-scheinnungenrdquo Technical report II a 132 N 34 125-138Akademie derWissenschaften inWien SitzungsberichteMath-naturwiss Klasse Abt 1923

[2] R T Walker ldquoVertical drain consolidation analysis in one twoand three dimensionsrdquo Computers amp Geosciences vol 38 no 8pp 1069ndash1077 2011

[3] G Y Zheng P Li and C Y Zhao ldquoAnalysis of non-linearconsolidation of soft clay by differential quadrature methodrdquoApplied Clay Science vol 79 pp 2ndash7 2013

[4] X Y Geng C J Xu and Y Q Cai ldquoNon-linear consolidationanalysis of soil with variable compressibility and permeabilityunder cyclic loadingsrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 30 no 8 pp 803ndash8212006

[5] M Lu S Wang S W Sloan B Indraratna and K Xie ldquoNon-linear radial consolidation of vertical drains under a generaltime-variable loadingrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 39 no 1 pp 51ndash622015

[6] HWu L HuW Qi and QWen ldquoAnalytical Solution for Elec-troosmotic Consolidation Considering Nonlinear Variation ofSoil Parametersrdquo International Journal of Geomechanics vol 17no 5 Article ID 06016032 2017

[7] S J Brandenberg ldquoiConsoljs JavaScript Implicit Finite-Differ-ence Code for Nonlinear Consolidation and Secondary Com-pressionrdquo International Journal of Geomechanics vol 17 no 6Article ID 04016149 2017

[8] R Butterfield ldquoA natural compression law for soils (an advanceon e-log prsquo)rdquo Geotechnique vol 29 no 4 pp 469ndash480 1979

[9] R Lancellotta and L Preziosi ldquoA general nonlinear mathemati-calmodel for soil consolidation problemsrdquo International Journalof Engineering Science vol 35 no 10-11 pp 1045ndash1063 1997

[10] Y-C Zhuang K-H Xie and X-B Li ldquoNonlinear analysis ofconsolidation with variable compressibility and permeabilityrdquoJournal of Zhejiang University (Engineering Science Edition) vol6 no 3 pp 181ndash187 2005

[11] E Conte andA Troncone ldquoNonlinear consolidation of thin lay-ers subjected to time-dependent loadingrdquo Canadian Geotechni-cal Journal vol 44 no 6 pp 717ndash725 2007

[12] R Yong and F Townsend ldquoGeneral Consolidation Theory forClaysrdquo Soil Mechanics Series Report No 8 Graduate School ofEngineering National University of Mexico 1983

[13] J C Gibbings Dimensional Analysis Springer-Verlag LondonUK 2011

[14] E Buckingham ldquoOn physically similar systems Illustrations ofthe use of dimensional equationsrdquo Physical Review A AtomicMolecular and Optical Physics vol 4 no 4 pp 345ndash376 1914

[15] R Seshadri and T Y Na ldquoA Survey ofMethods for DeterminingSimilarity Transformationsrdquo inGroup Invariance in EngineeringBoundary Value Problems pp 35ndash61 Springer New York NYUSA 1985

[16] V D Zimparov and V M Petkov ldquoApplication of discriminatedanalysis to low Reynolds number swirl flows in circular tubeswith twisted-tape inserts Pressure drop correlationsrdquo Interna-tional Review of Chemical Engineering vol 1 no 4 pp 346ndash3562009

[17] M Capobianchi and A Aziz ldquoA scale analysis for natural con-vective flows over vertical surfacesrdquo International Journal ofThermal Sciences vol 54 pp 82ndash88 2012

[18] M Conesa J F Sanchez Perez I Alhama and F AlhamaldquoOn the nondimensionalization of coupled nonlinear ordinarydifferential equationsrdquo Nonlinear Dynamics vol 84 no 1 pp91ndash105 2016

[19] I A Manteca G Garcıa-Ros and F Alhama ldquoUniversal solu-tion for the characteristic time and the degree of settlement innonlinear soil consolidation scenarios A deduction based onnondimensionalizationrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 57 pp 186ndash201 2018

[20] E H Davis and G P Raymond ldquoA non-linear theory ofconsolidationrdquo Geotechnique vol 15 no 2 pp 161ndash173 1965

[21] P Cornetti and M Battaglio ldquoNonlinear consolidation of soilmodeling and solution techniquesrdquo Mathematical and Com-puter Modelling vol 20 no 7 pp 1ndash12 1994

[22] C F Gonzalez-Fernandez ldquoApplications of the network simu-lation method to transport processesrdquo in Network SimulationMethod J Horno Ed Research Signpost Trivandrum India2002

[23] J Serna F J S Velasco and A Soto Meca ldquoApplication of net-work simulationmethod to viscous flowsThe nanofluid heatedlid cavity under pulsating flowrdquo Computers amp Fluids vol 91 pp10ndash20 2014

[24] E Juarez-Badillo ldquoGeneral Permeability Change Equationsfor Soilsrdquo in Proceedings of the International Conference onConstitutive Laws pp 205ndash209 University of Arizona TuesonAriz USA 1983

[25] E Juarez-Badillo and B Chen ldquoConsolidation curves for claysrdquoJournal of Geotechnical Engineering vol 109 no 10 pp 1303ndash1312 1983

Mathematical Problems in Engineering 15

[26] N Abbasi H Rahimi A A Javadi andA Fakher ldquoFinite differ-ence approach for consolidation with variable compressibilityand permeabilityrdquo Computers amp Geosciences vol 34 no 1 pp41ndash52 2007

[27] R F Scott Principles of Soils Mechanics Addison-Wesley Pub-lishing Company 1963

[28] F Alhama and C N Madrid Analisis Dimensional Discrimi-nado En Mecanica De Fluidos Y Transferencia De Calor 2012

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

14 Mathematical Problems in Engineering

120591os Characteristic time which takes thesettlement to reach approximately its finalvalue (s or yr)120591o1205901015840 Characteristic time which takes the excesspore pressure to dissipate untilapproximately the value of zero (s or yr)Ψ Arbitrary mathematical function120577 Differential void ratio (dimensionless)1205771 1205772 Initial and final differential void ratio(dimensionless)

Subscripts

I II Denote different numbers

Superscripts

1015840 Denote dimensionless magnitude

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] K Terzaghi ldquoDie Berechnung der Durchlassigkeitszifferdestones aus dem verlauf der hydrodynamischen spanunngser-scheinnungenrdquo Technical report II a 132 N 34 125-138Akademie derWissenschaften inWien SitzungsberichteMath-naturwiss Klasse Abt 1923

[2] R T Walker ldquoVertical drain consolidation analysis in one twoand three dimensionsrdquo Computers amp Geosciences vol 38 no 8pp 1069ndash1077 2011

[3] G Y Zheng P Li and C Y Zhao ldquoAnalysis of non-linearconsolidation of soft clay by differential quadrature methodrdquoApplied Clay Science vol 79 pp 2ndash7 2013

[4] X Y Geng C J Xu and Y Q Cai ldquoNon-linear consolidationanalysis of soil with variable compressibility and permeabilityunder cyclic loadingsrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 30 no 8 pp 803ndash8212006

[5] M Lu S Wang S W Sloan B Indraratna and K Xie ldquoNon-linear radial consolidation of vertical drains under a generaltime-variable loadingrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 39 no 1 pp 51ndash622015

[6] HWu L HuW Qi and QWen ldquoAnalytical Solution for Elec-troosmotic Consolidation Considering Nonlinear Variation ofSoil Parametersrdquo International Journal of Geomechanics vol 17no 5 Article ID 06016032 2017

[7] S J Brandenberg ldquoiConsoljs JavaScript Implicit Finite-Differ-ence Code for Nonlinear Consolidation and Secondary Com-pressionrdquo International Journal of Geomechanics vol 17 no 6Article ID 04016149 2017

[8] R Butterfield ldquoA natural compression law for soils (an advanceon e-log prsquo)rdquo Geotechnique vol 29 no 4 pp 469ndash480 1979

[9] R Lancellotta and L Preziosi ldquoA general nonlinear mathemati-calmodel for soil consolidation problemsrdquo International Journalof Engineering Science vol 35 no 10-11 pp 1045ndash1063 1997

[10] Y-C Zhuang K-H Xie and X-B Li ldquoNonlinear analysis ofconsolidation with variable compressibility and permeabilityrdquoJournal of Zhejiang University (Engineering Science Edition) vol6 no 3 pp 181ndash187 2005

[11] E Conte andA Troncone ldquoNonlinear consolidation of thin lay-ers subjected to time-dependent loadingrdquo Canadian Geotechni-cal Journal vol 44 no 6 pp 717ndash725 2007

[12] R Yong and F Townsend ldquoGeneral Consolidation Theory forClaysrdquo Soil Mechanics Series Report No 8 Graduate School ofEngineering National University of Mexico 1983

[13] J C Gibbings Dimensional Analysis Springer-Verlag LondonUK 2011

[14] E Buckingham ldquoOn physically similar systems Illustrations ofthe use of dimensional equationsrdquo Physical Review A AtomicMolecular and Optical Physics vol 4 no 4 pp 345ndash376 1914

[15] R Seshadri and T Y Na ldquoA Survey ofMethods for DeterminingSimilarity Transformationsrdquo inGroup Invariance in EngineeringBoundary Value Problems pp 35ndash61 Springer New York NYUSA 1985

[16] V D Zimparov and V M Petkov ldquoApplication of discriminatedanalysis to low Reynolds number swirl flows in circular tubeswith twisted-tape inserts Pressure drop correlationsrdquo Interna-tional Review of Chemical Engineering vol 1 no 4 pp 346ndash3562009

[17] M Capobianchi and A Aziz ldquoA scale analysis for natural con-vective flows over vertical surfacesrdquo International Journal ofThermal Sciences vol 54 pp 82ndash88 2012

[18] M Conesa J F Sanchez Perez I Alhama and F AlhamaldquoOn the nondimensionalization of coupled nonlinear ordinarydifferential equationsrdquo Nonlinear Dynamics vol 84 no 1 pp91ndash105 2016

[19] I A Manteca G Garcıa-Ros and F Alhama ldquoUniversal solu-tion for the characteristic time and the degree of settlement innonlinear soil consolidation scenarios A deduction based onnondimensionalizationrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 57 pp 186ndash201 2018

[20] E H Davis and G P Raymond ldquoA non-linear theory ofconsolidationrdquo Geotechnique vol 15 no 2 pp 161ndash173 1965

[21] P Cornetti and M Battaglio ldquoNonlinear consolidation of soilmodeling and solution techniquesrdquo Mathematical and Com-puter Modelling vol 20 no 7 pp 1ndash12 1994

[22] C F Gonzalez-Fernandez ldquoApplications of the network simu-lation method to transport processesrdquo in Network SimulationMethod J Horno Ed Research Signpost Trivandrum India2002

[23] J Serna F J S Velasco and A Soto Meca ldquoApplication of net-work simulationmethod to viscous flowsThe nanofluid heatedlid cavity under pulsating flowrdquo Computers amp Fluids vol 91 pp10ndash20 2014

[24] E Juarez-Badillo ldquoGeneral Permeability Change Equationsfor Soilsrdquo in Proceedings of the International Conference onConstitutive Laws pp 205ndash209 University of Arizona TuesonAriz USA 1983

[25] E Juarez-Badillo and B Chen ldquoConsolidation curves for claysrdquoJournal of Geotechnical Engineering vol 109 no 10 pp 1303ndash1312 1983

Mathematical Problems in Engineering 15

[26] N Abbasi H Rahimi A A Javadi andA Fakher ldquoFinite differ-ence approach for consolidation with variable compressibilityand permeabilityrdquo Computers amp Geosciences vol 34 no 1 pp41ndash52 2007

[27] R F Scott Principles of Soils Mechanics Addison-Wesley Pub-lishing Company 1963

[28] F Alhama and C N Madrid Analisis Dimensional Discrimi-nado En Mecanica De Fluidos Y Transferencia De Calor 2012

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Mathematical Problems in Engineering 15

[26] N Abbasi H Rahimi A A Javadi andA Fakher ldquoFinite differ-ence approach for consolidation with variable compressibilityand permeabilityrdquo Computers amp Geosciences vol 34 no 1 pp41ndash52 2007

[27] R F Scott Principles of Soils Mechanics Addison-Wesley Pub-lishing Company 1963

[28] F Alhama and C N Madrid Analisis Dimensional Discrimi-nado En Mecanica De Fluidos Y Transferencia De Calor 2012

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MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

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Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom