# Derivation of Universal Curves for Nonlinear Soil Consolidation with Potential Constitutive Dependences

Authors: Gonzalo García-Ros; Iván Alhama; Manuel Cánovas; Francisco Alhama
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

Nonlinear consolidation scenarios, based on potential type constitutive dependences—like those proposed by Juárez-Badillo—and eliminating the more restrictive hypothesis of 1+e anddz constant, were characterized by the nondimensionalization process of the governing equations, providing the independent dimensionless groups that rule the main unknowns of interest. From these, universal curves have been depicted for both the characteristic time and the average degree of consolidation. The solutions were verified by numerical simulations and successfully compared in a case study, showing the simplicity of use of the curves and the high reliability of the solutions they provide.


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## Body

## 1. Introduction

Soil consolidation has been studied by a large number of authors for almost a hundred years. Since Terzaghi announced his linear consolidation theory [1], more complex models have been proposed. These include 2D and 3D geometries [2], as well as nonlinear problems (with a nonconstant coefficient of consolidation, due to the changes in the hydraulic conductivity and void ratio during the process [3]), for whose solutions numerical techniques are required. However, most nonlinear models focus on solving a particular scenario, i.e., with a very restrictive hypothesis, with no intention of reaching solutions of a universal character which depend of the lowest number of dimensionless parameters. Geng et al. [4] and Lu et al. [5] studied nonlinear soil consolidation problems, the first under cycling loadings, and the second under time-variable loadings and vertical drains, using logarithmic type constitutive relations for the main variables of the problem (void ratio, effective stress and hydraulic conductivity), obtaining that both the initial stress state and the ratio of the coefficients of the constitutive relations are the most influential parameters that govern the consolidation process. Other references to nonlinear consolidation are the papers of Wu et al. [6] and Brandenberg [7], which also handle logarithmic dependences with the void ratio. The first author presents an analytical solution for the electroosmotic consolidation with no allusion to the dimensionless groups that rule the problem, while the second approximates the secondary consolidation.Special attention deserve the works of Butterfield [8], Lancellotta and Preziosi [9], Zhuang et al. [10], and Conte and Troncone [11]. The first one describes the advantages of the potential dependences versus the logarithmic ones, which become evident in highly compressible soils for which, on the one hand, the slope of the e-log(σ′) dependence is far from being a constant value and, on the other, the assumption of the logarithmic relation can lead to negative values ​​for the void ratio when σ′f/σ′o takes a high value. Lancellotta and Preziosi, and Conte and Troncone make references to the potential type relations, although they choose to present solutions only for logarithmic dependences. The first ones present, among others, consolidation models with impermeable or draining boundaries, as well as solid particles sedimentation models in quiescent fluids, while the latter solve a problem of step loads under the hypothesis of small strains. Finally, Zhuang et al. obtain semianalytical solutions under the small strains hypothesis and using logarithmic constitutive dependences, characterizing the problem by means of a dimensionless parameter that is the quotient between the slopes of the e-log(σ′) and e-log(k) relations.For optimal dimensionless characterization of the solution patterns (essentially the characteristic time and the average degree of consolidation, the latter in terms of settlement and pressure dissipation) of the consolidation processes based on a generalized Juárez-Badillo (J-B hereinafter) model [12], the dimensionless groups are derived by the nondimensionalization of the governing equations, a process that requires bothersome mathematical steps due to the inherent nonlinearity of the model. For the original J-B problem, which assumes the strict hypothesis of an initial void ratio that is negligible in the term of soil contraction, the number of the dimensionless groups reported by J-B is larger than the number derived in this paper, so providing a less precise solution; in addition, the author does not present universal solutions since no characteristic time is proposed. For other extended J-B models, by deleting one or more of his restrictive hypotheses (for example, 1+eo=constant with eo≠0, 1+e≠constant anddz≠constant), new dimensionless groups are derived, verifying the solutions by numerical simulations for the most general and complex model. The general concept of characteristic time, which may be easily defined in lineal and isotropic soils, is extended herein to nonlinear consolidation and, after its introduction into the governing equation as an unknown reference to define the dimensionless time, a new group containing the characteristic time emerges from the nondimensionalization process, allowing the universal curves to be constructed.By nondimensionalization, the large number of isolated dimensionless parameters contained in the statement of the problem and in the constitutive relations, together with the dimensionless groups that can be formed from the relevant list of parameters and variables by applying simple rules of dimensional analysis [13], is reduced to the smallest number that best help researchers to manage the solution. As is known, the application of pi theorem—derived from the theory of homogeneous functions (Buckingham [14])—allows the unknowns of interest, expressed in their dimensionless form, to be set as a function of the mentioned dimensionless groups.There are two techniques whose purpose is the derivation of the dimensionless groups that rule the solution of a given problem: the dimensional analysis and the method of group transformations. In the first [13], the groups are derived by simple mathematical manipulations from a list of relevant variables expressed in terms of primary quantities (length, mass and time), while in the second [15], they are obtained from the mathematical model in its dimensionless form after some mathematical steps. The technique applied in this paper, which we call nondimensionalization of governing equations [16, 17], starts from the governing equation in order to deduce the dimensionless groups. In this, after normalizing the variables, these and their changes are averaged – in fact, the equation itself is averaged – and assumed to be of the order of unity, a valid hypothesis in problems with relatively smooth nonlinearities, thus allowing the coefficients of the equations to be of the same order of magnitude and unequivocally providing the most precise solution as demonstrated in many studies [18]. Based on this methodology, Manteca et al. [19] study the nonlinear consolidation problem with constitutive dependencies of logarithmic type, providing as a solution the universal curves and the dimensionless groups that govern the process.Classical nonlinear consolidation models (such as Davis and Raymond [20], Juárez-Badillo [12], and Cornetti and Battaglio [21]) differ from one another in the nature of the constitutive relations between the parameters, void ratio and permeability, and the effective soil pressure, when trying to reflect the behavior of real soils. In general, these dependences converge to provide the same results in problems with small changes in these parameters (quasilinear problems) but different results when the working range is wide. In addition, most nonlinear models assume restrictive hypotheses, such as a constant soil thickness in the contraction term of the governing equation, which distance the solutions from those obtained without these restrictions; these solutions are by the side of safety in consolidation and unsafety in swelling. In addition to the original J-B problem, an extended model with both void ratio and thickness of the volume element continuously changing during the consolidation process is analyzed and solved.To verify the obtained results, a comparison is made with numerical solutions based on the network simulation method [22]. This tool, widely used in other fields, is an efficient and computationally fast numerical method that has demonstrated its reliability in many linear and nonlinear engineering problems [23]. The dependences of characteristic time and average degree of consolidation on the rest of the dimensionless groups are checked; i.e., whenever the dimensionless parameters retain the same values, against changes in the individual parameters contained within the groups, neither the characteristic time nor average degree of consolidation change. After the verification and presentation of universal curves for a wide range of values of the parameters that sufficiently cover all real cases, contributions and conclusions are summarized.


## 2. The Original Juárez-Badillo Model

This author [12] presented his model twenty years after the nonlinear model of Davis and Raymond [20] and applied it to the odometer test to improve understanding of the primary consolidation phenomenon. Juárez-Badillo assumes incompressibility for water and soil particles and compressibility for the soil structure for which he sets the constitutive relation dV/V=-γv(dσ′/σ′). Doing so, it is immediate to write mv = γv/σ′. As regards permeability, following his deductions (Juárez-Badillo [24]), J-B assumes a constitutive k-V dependence in the form k/k1=V/V1κ, which is equivalent to a proportionality dependence between the relative deviations of these parameters dk/k=κ(dV/V). As a result, the dependence k-σ′ takes the form k/k1=σ′/σ′1-γvκ, or, in terms of unitary deviations, dk/k=-γvκ(dσ′/σ′).The original J-B model, under odometer conditions (small volume change and negligible specific weight of water), is defined by(1)dσ′dt=-1γwmv∂∂zk∂u∂zSubstituting the above dependences and using the parameters (2)λ=1-γvκ,k1σ1′γwγv=k1γwmv,1=cv,1,cv=kγwmv=k1σ1′γwγvσ′σ1′1-γvκ=cv,1σ′σ1′λ(1) reduces to (3)∂σ′∂t=-σ′cv,11σ1′∂∂zσ′σ1′-γvκ∂u∂zThe simplifications assumed by the odometer test, ∂u/∂z=-∂σ′/∂z and ∂u/∂t=-∂σ′/∂t allow us to write the last equation as (4)∂σ′∂t=σ′cv,1∂∂zσ′σ1′-γvκ1σ1′∂σ′∂zor, in terms of the excess pore pressure variable, as(5)∂u∂t=σ′cv,1∂∂zσ′σ1′-γvκ1σ1′∂u∂zIn this equation σ′1 can be substituted by any other pressure, for example σ′2, providing that cv is also substituted by cv,2. The case λ=0 (or γvκ=1), i.e., constant cv, is the nonlinear model of Davis and Raymond, which always provides an effective stress that is lower (or higher excess pore pressure) than that obtained with the Terzaghi model. However, the different dependences V-σ′ (or e-σ′) for J-B and Davis and Raymond result in a different average degree of consolidation (Ūs) for both models. In fact, for J-B [12], this unknown obeys the following expression: (6)Ūs=1-1/H1∫0H1σ′/σ′1-γvdz1-σ′2/σ′1-γvor, in terms of the new normalized variable v=σ′/σ′2λ(7)Ūs=1-H2/H1∫01v-γv/λdz′1-H2/H1Finally, J-B does not report analytical or numerical solutions for the excess pore pressure or for the effective pressure, except for the case λ=0, probably due to the large number of variables involved. He only reports [25] a consolidation abacus Ūs~T for the case λ=0 (γvκ=1) and a constant cv, using H2/H1 as parameter, and a set of abacuses for the case λ/γv=(1/γv)-κ ≠ 0, cv ≠ constant. In these abacuses the time factor is defined as T=tcv,2/H2. Despite these results, no physical meaning is attributed to λ, T, or v. In short, in view of (7), J-B concludes that his model depends on three groups, H2/H1, (1/γv)-κ, and the time factor T (since v is a function of T).


## 3. Nondimensionalization of Original and Extended Juárez-Badillo Models. Dimensionless Groups

To improve identification of the parameters involved in the constitutive dependences we redefine these as follows:(8)dVV=-γv,σ′dσ′σ′,dkk=-γk,σ′dσ′σ′dkk=γk,vdVVor, in their integral form(9)VVo=σ′σ′o-γv,σ′kko=σ′σ′o-γk,σ′kko=VVoγk,vwith(10)γk,v=γk,σ′γv,σ′


### 3.1. Nondimensionalization of the J-B Model in Terms of Pressure

Four cases are presented:(1) original J-B model, (2) J-B model assuming 1+e is not constant anddz is constant, (3) J-B model assuming 1+e is constant (a simplification of the former), and (4) the less restrictive model with both 1+e anddz not constant.


#### 3.1.1. Original J-B Model

The original J-B model [12] assumes small deformations compared with the soil thickness and a negligible initial void ratio (eo≈0). Developing the term between brackets of (3), we can write(11)1σ′∂σ′∂t=-cv,1γk,σ′σ1′2σ′σ1′-γk,σ′-1∂σ′∂z2+cv,1σ1′σ′σ1′-γk,σ′∂2σ′∂z2or, by mathematical manipulation,(12)∂σ′∂t=σ′okoγwγv,σ′σ′σ′o1-γk,σ′-γk,σ′σ′∂σ′∂z2+∂2σ′∂z2To make this equation dimensionless, the following variables are used:(13)σ′′=σ′-σ1′σ2′-σ1′,t′=tτo,σ′,z′=zH1where τo,σ′ is a characteristic time or unknown reference, chosen as the time required for the effective pressure (on average along the whole domain) to reach a high percentage of its change, for example, 90%. With these variables and provided that (on average) dσ′=dσ′′σ2′-σ1′+σ1′≈σ2′-σ1′, (12) can be written as (14)σ2′-σ1′σm′τo,σ′∂σ′′∂t′=-cv,1γk,σ′σ1′2H12σm′σ1′-γk,σ′-1σ2′-σ1′2∂σ′′∂z′2+cv,1σ1′H12σm′σ1′-γk,σ′σ2′-σ1′∂2σ′′∂z′2where σm′ is an average value of the effective pressure which we will talk about later. The dimensional coefficients of this equation, in terms of the parameters of the problem(15)1τo,σ′,k1σ1′γwγv,σ′γk,σ′H12σ2′σ1′-1σm′σ1′-γk,σ′,k1σ1′γwγv,σ′1H12σm′σ1′1-γk,σ′give rise to two dimensionless independent groups: (16)πI=τo,σ′k1σ′1γwγv,σ′H12σm′σ1′1-γk,σ′,πII=γk,σ′σ2′-σ1′σm′Based on the pi theorem, the solution of the order of magnitude of the characteristic time, derived from πI=ψπII, is given by(17)τo,σ′~γwγv,σ′H12k1σ1′σm′σ1′γk,σ′-1Ψτγk,σ′σ2′-σ1′σm′As regards Ūσ′, since it also depends on time, the solution (dependent on two groups) is given by (18)Ūσ′=ΨUtτo,σ′,γk,σ′σ2′-σ1′σm′where Ψτ and ΨU are unknown functions of their arguments.


#### 3.1.2. J-B Model Assuming 1+e Is Not Constant anddz Is Constant

Deleting the restrictive hypothesiseo≈0 used by J-B for the solutions of σ′ and Ūσ′, the latter through the dependence V-σ′ for which such a hypothesis is not satisfied, and assuming that 1+e≠constant, the governing equation is(19)∂∂zkγwdudz=∂∂te1+e=11+e2∂e∂tUsing the dependence V/V1=σ′/σ′1-γv,σ′=(1+e)/(1+eo) or 1+e=1+eoσ′/σ′o-γv,σ′ and its derivative form de=-(1+eoγv,σ′/σ′o)σ′/σ′o-γv,σ′-1dσ′, the right and left terms of (19) can be written as(20)∂∂te1+e=11+e2∂e∂t=-γv,σ′1+eoσ′oσ′σ′oγv,σ′-1∂σ′∂tOn the other hand, with k/ko=σ′/σ′o-γk,σ′, the left term of (19) writes as (21)∂∂zkγw∂u∂z=koγwσ′σ′o-γk,σ′γk,σ′σ′∂u∂z2+∂2u∂z2Equating both terms, (19) in terms of the effective pressure takes the form (22)∂σ′∂t=1+eoσ′okoγwγv,σ′σ′σ′o1-γv,σ′-γk,σ′-γk,σ′σ′∂σ′∂z2+∂2σ′∂z2Proceeding as in the previous case, the coefficients of the dimensionless form of the former equation(23)1τo,σ′,1+eok1σ1′γwγv,σ′γk,σ′H12σ2′-σ1′σm′σm′σ1′1-γv,σ′-γk,σ′,1+eok1σ1′γwγv,σ′1H12σm′σ1′1-γv,σ′-γk,σ′give rise to the dimensionless groups (24)πI=τo,σ′k1σ′11+eoγwγv,σ′H12σm′σ1′1-γk,σ′-γv,σ′,πII=γk,σ′σ2′-σ1′σm′Thus, the solutions for τo,σ′ and Ūσ′ are given by(25)τo,σ′=γv,σ′γwH121+eok1σ1′σm′σ1′γv,σ′+γk,σ′-1Ψτγk,σ′σ2′-σ1′σm′(26)Ūσ′=ΨUtτo,σ′,γk,σ′σ2′-σ1′σm′


#### 3.1.3. J-B Model Assuming 1+e Is Constant (A Simplification of the Former)

Based on the above, it is easy to simplify the hypothesis to 1+e being constant. Withσ′/σ′1-γv,σ′=1, the consolidation equation reduces to(27)∂σ′∂t=1+eoσ′okoγwγv,σ′σ′σ′o1-γk,σ′-γk,σ′σ′∂σ′∂z2+∂2σ′∂z2an equation dependent on the initial void ratio. Using (13) and averaging, this equation provides three-dimensional coefficients (28)1τo,σ′,1+eok1σ1′γwγv,σ′γk,σ′H12σ2′-σ1′σm′σm′σ1′1-γk,σ′,1+eok1σ1′γwγv,σ′1H12σm′σ1′1-γv,σ′-γk,σ′and two dimensionless independent groups (29)πI=τo,σ′k1σ′11+eoγwγv,σ′H12σm′σ1′1-γk,σ′,πII=γk,σ′σ2′-σ1′σm′Thus, the solutions for τo,σ′ and Ūσ′ are given by(30)τo,σ′=γv,σ′γwH121+eok1σ1′σm′σ1′γk,σ′-1Ψτγk,σ′σ2′-σ1′σm′(31)Ūσ′=ΨUtτo,σ′,γk,σ′σ2′-σ1′σm′


#### 3.1.4. The Less Restrictive Model with Both 1+e anddz Not Constant

For this model, 1+e anddz not constants, it is enough to consider (22) plus the conditiondz not constant, or dz=dzo(H/H1). So, after mathematical manipulation, the governing equation writes as(32)∂σ′∂t=1+eoσ′okoγwγv,σ′σ′σ′o1+γv,σ′-γk,σ′-γk,σ′σ′∂σ′∂z2+∂2σ′∂z2The new coefficients(33)1τo,σ′,1+eok1σ1′γwγv,σ′γk,σ′H12σ2′-σ1′σm′σm′σ1′1+γv,σ′-γk,σ′,1+eok1σ1′γwγv,σ′1H12σm′σ1′1+γv,σ′-γk,σ′provide the groups(34)πI=τo,σ′k1σ′11+eoγwγv,σ′H12σm′σ1′1-γk,σ′+γv,σ′,πII=γk,σ′σ2′-σ1′σm′and the solutions(35)τo,σ′=γv,σ′γwH121+eok1σ1′σm′σ1′γk,σ′-γv,σ′-1Ψτγk,σ′σ2′-σ1′σm′(36)Ūσ′=ΨUtτo,σ′,γk,σ′σ2′-σ1′σm′


### 3.2. Nondimensionalization of the J-B Model in Terms of Settlements

Three cases, defined by the hypotheses(1) 1+e≠constant anddz constant, (2) 1+e anddz constants, and (3) 1+e anddz not being constant, are considered. For this study let us introduce a new variable directly related with the settlement, with a clear physical meaning, “ζ = e - eo”, a kind of local degree of settlement or differential void index.


#### 3.2.1. 1+e≠Constant anddz Constant

For 1+e≠constant, using the variableζ and the dependence V-σ′, it is straightforward to write (37)ζ=e-eo=1+eoσ′σ′1-γv,σ′-1Making use of the derivatives of (37)(38)∂σ′∂z2=σ′121+eo2γv,σ′2σ′σ′12γv,σ′+2∂ζ∂z2∂2σ′∂z2=-σ′11+eoγv,σ′σ′σ′1γv,σ′+1∂2ζ∂z2+σ′1γv,σ′+11+eo2γv,σ′2σ′σ′12γv,σ′+1∂ζ∂z2and the assumptions of the odometer test, after cumbersome mathematical steps, the nonlinear dimensional consolidation (19) is written in the form (39)∂ζ∂t=koσ′oγk,σ′-γv,σ′-1γwγv,σ′2σ′σ′o1-γk,σ′∂ζ∂z2+koσ′o1+eoγwγv,σ′σ′σ′o1-γv,σ′-γk,σ′∂2ζ∂z2Introducing the variables (40)ζ′=ζ-ζ1ζ2-ζ1=ζ1-ζζ1,z′=zH1,t′=tτo,sinto the former equation, with τo,s, a reference chosen as the time required for the settlement to reach a high percentage of its change along the whole domain, its dimensionless form provides three coefficients(41)1τo,s,koσ′oγk,σ′-γv,σ′-1γwγv,σ′2σm′σ1′1-γk,σ′ef-eoH12,koσ′o1+eoγwγv,σ′σm′σ1′1-γv,σ′-γk,σ′1H12Dividing by the last, the resulting dimensionless groups are(42)πI=τo,skoσ′o1+eoγwγv,σ′H12σm′σ1′1-γv,σ′-γk,σ′πII=γk,σ′-γv,σ′-1γv,σ′ef-eo1+eoσm′σ1′γv,σ′Making use of the constitutive dependence dV/V=-γv,σ′(dσ′/σ′) is possible to write ΔH/H=-γv,σ′(Δσ′/σ′), and since ef-eo/1+eo=ΔH/Ho, πII finally takes the form(43)πII=1-γk,σ′+γv,σ′σ2′-σ1′σm′The solution for τo,s is(44)τo,s≈γwγv,σ′H12koσ′o1+eoσm′σ1′γv,σ′+γk,σ′-1Ψτ1-γk,σ′+γv,σ′σ2′-σ1′σm′As for the average degree of consolidation, the following dependence arises:(45)Ūs=ΨUtτo,s,1-γk,σ′+γv,σ′σ2′-σ1′σm′


#### 3.2.2. 1+e anddz Constants

The hypothesis 1+e=constant is a simplification of the former case. Proceeding in the same way, the resulting coefficients(46)1τo,s,koσ′o1+γv,σ′-γk,σ′γwγv,σ′2σm′σ1′1-γk,σ′+γv,σ′ef-eoH12,koσ′o1+eoγwγv,σ′σm′σ1′1-γk,σ′1H12,provide the dimensionless groups(47)πI=τo,skoσ′o1+eoγwγv,σ′H12σm′σ1′1-γk,σ′πII=1-γk,σ′+γv,σ′σ2′-σ1′σm′The solutions for τo,s and Ūs are(48)τo,s≈γwγv,σ′H12koσ′o1+eoσm′σ1′γk,σ′-1Ψτ1-γk,σ′+γv,σ′σ2′-σ1′σm′(49)Ūs=ΨUtτo,s,1-γk,σ′+γv,σ′σ2′-σ1′σm′The solutions for the original J-B model are also (48) and (49) with the simplification of eo=0. The author, however, does not talk about the characteristic time and obtains the average degree of consolidation given by (7), in which this variable depends on three groups: (1-γk,σ′)/γv,σ′, H2/H1, and tcv,2/H12, undoubtedly a less precise solution.


#### 3.2.3. 1+e anddz Not Being Constant

When 1+e anddz are not constants, substituting dz=dzo(H/H1) in (39), after cumbersome mathematical manipulation, the resulting coefficients are(50)1τo,s,koσ′oγk,σ′-γv,σ′-1γwγv,σ′2σm′σ1′1+2γv,σ′-γk,σ′ef-eoH12,koσ′o1+eoγwγv,σ′σm′σ1′1+γv,σ′-γk,σ′1H12Dividing by the last, the resulting dimensionless groups are(51)πI=τo,skoσ′o1+eoγwγv,σ′H12σm′σ1′1+γv,σ′-γk,σ′πII=1-γk,σ′+γv,σ′σ2′-σ1′σm′The solutions for τo,s and Ūs are(52)τo,s≈γwγv,σ′H12koσ′o1+eoσm′σ1′+γk,σ′-γv,σ′-1Ψτ1-γk,σ′+γv,σ′σ2′-σ1′σm′(53)Ūs=ΨUtτo,s,1-γk,σ′+γv,σ′σ2′-σ1′σm′Table 1 summarizes the results derived in this section.Table 1
Dimensionless groups that characterize the original and extended J-B models.
Juárez-Badillooriginal Juárez-Badillo Juárez-Badillo Juárez-Badillo 1+e=constant 1+e≠constant 1+e≠constant dz constant dz constant dz variable π I τ o k o σ ′ o γ w γ v , σ ′ H 1 2 σ ′ m σ ′ o 1 - γ k , σ ′ τ o k o σ ′ o 1 + e o γ w γ v , σ ′ H 1 2 σ ′ m σ ′ o 1 - γ k , σ ′ τ o k o σ ′ o 1 + e o γ w γ v , σ ′ H 1 2 σ ′ m σ ′ o 1 - γ k , σ ′ - γ v , σ ′ τ o k o σ ′ o 1 + e o γ w γ v , σ ′ H 1 2 σ ′ m σ ′ o 1 - γ k , σ ′ + γ v , σ ′ τ o , σ ′ or τo,s   ⁢πII⁢ ⁢γk,σ′σ2′-σ1′σm′ for pressure ⁢1-γk,σ′+γv,σ′σ2′-σ1′σm′ for settlement


## 3.1. Nondimensionalization of the J-B Model in Terms of Pressure

Four cases are presented:(1) original J-B model, (2) J-B model assuming 1+e is not constant anddz is constant, (3) J-B model assuming 1+e is constant (a simplification of the former), and (4) the less restrictive model with both 1+e anddz not constant.


### 3.1.1. Original J-B Model

The original J-B model [12] assumes small deformations compared with the soil thickness and a negligible initial void ratio (eo≈0). Developing the term between brackets of (3), we can write(11)1σ′∂σ′∂t=-cv,1γk,σ′σ1′2σ′σ1′-γk,σ′-1∂σ′∂z2+cv,1σ1′σ′σ1′-γk,σ′∂2σ′∂z2or, by mathematical manipulation,(12)∂σ′∂t=σ′okoγwγv,σ′σ′σ′o1-γk,σ′-γk,σ′σ′∂σ′∂z2+∂2σ′∂z2To make this equation dimensionless, the following variables are used:(13)σ′′=σ′-σ1′σ2′-σ1′,t′=tτo,σ′,z′=zH1where τo,σ′ is a characteristic time or unknown reference, chosen as the time required for the effective pressure (on average along the whole domain) to reach a high percentage of its change, for example, 90%. With these variables and provided that (on average) dσ′=dσ′′σ2′-σ1′+σ1′≈σ2′-σ1′, (12) can be written as (14)σ2′-σ1′σm′τo,σ′∂σ′′∂t′=-cv,1γk,σ′σ1′2H12σm′σ1′-γk,σ′-1σ2′-σ1′2∂σ′′∂z′2+cv,1σ1′H12σm′σ1′-γk,σ′σ2′-σ1′∂2σ′′∂z′2where σm′ is an average value of the effective pressure which we will talk about later. The dimensional coefficients of this equation, in terms of the parameters of the problem(15)1τo,σ′,k1σ1′γwγv,σ′γk,σ′H12σ2′σ1′-1σm′σ1′-γk,σ′,k1σ1′γwγv,σ′1H12σm′σ1′1-γk,σ′give rise to two dimensionless independent groups: (16)πI=τo,σ′k1σ′1γwγv,σ′H12σm′σ1′1-γk,σ′,πII=γk,σ′σ2′-σ1′σm′Based on the pi theorem, the solution of the order of magnitude of the characteristic time, derived from πI=ψπII, is given by(17)τo,σ′~γwγv,σ′H12k1σ1′σm′σ1′γk,σ′-1Ψτγk,σ′σ2′-σ1′σm′As regards Ūσ′, since it also depends on time, the solution (dependent on two groups) is given by (18)Ūσ′=ΨUtτo,σ′,γk,σ′σ2′-σ1′σm′where Ψτ and ΨU are unknown functions of their arguments.


### 3.1.2. J-B Model Assuming 1+e Is Not Constant anddz Is Constant

Deleting the restrictive hypothesiseo≈0 used by J-B for the solutions of σ′ and Ūσ′, the latter through the dependence V-σ′ for which such a hypothesis is not satisfied, and assuming that 1+e≠constant, the governing equation is(19)∂∂zkγwdudz=∂∂te1+e=11+e2∂e∂tUsing the dependence V/V1=σ′/σ′1-γv,σ′=(1+e)/(1+eo) or 1+e=1+eoσ′/σ′o-γv,σ′ and its derivative form de=-(1+eoγv,σ′/σ′o)σ′/σ′o-γv,σ′-1dσ′, the right and left terms of (19) can be written as(20)∂∂te1+e=11+e2∂e∂t=-γv,σ′1+eoσ′oσ′σ′oγv,σ′-1∂σ′∂tOn the other hand, with k/ko=σ′/σ′o-γk,σ′, the left term of (19) writes as (21)∂∂zkγw∂u∂z=koγwσ′σ′o-γk,σ′γk,σ′σ′∂u∂z2+∂2u∂z2Equating both terms, (19) in terms of the effective pressure takes the form (22)∂σ′∂t=1+eoσ′okoγwγv,σ′σ′σ′o1-γv,σ′-γk,σ′-γk,σ′σ′∂σ′∂z2+∂2σ′∂z2Proceeding as in the previous case, the coefficients of the dimensionless form of the former equation(23)1τo,σ′,1+eok1σ1′γwγv,σ′γk,σ′H12σ2′-σ1′σm′σm′σ1′1-γv,σ′-γk,σ′,1+eok1σ1′γwγv,σ′1H12σm′σ1′1-γv,σ′-γk,σ′give rise to the dimensionless groups (24)πI=τo,σ′k1σ′11+eoγwγv,σ′H12σm′σ1′1-γk,σ′-γv,σ′,πII=γk,σ′σ2′-σ1′σm′Thus, the solutions for τo,σ′ and Ūσ′ are given by(25)τo,σ′=γv,σ′γwH121+eok1σ1′σm′σ1′γv,σ′+γk,σ′-1Ψτγk,σ′σ2′-σ1′σm′(26)Ūσ′=ΨUtτo,σ′,γk,σ′σ2′-σ1′σm′


### 3.1.3. J-B Model Assuming 1+e Is Constant (A Simplification of the Former)

Based on the above, it is easy to simplify the hypothesis to 1+e being constant. Withσ′/σ′1-γv,σ′=1, the consolidation equation reduces to(27)∂σ′∂t=1+eoσ′okoγwγv,σ′σ′σ′o1-γk,σ′-γk,σ′σ′∂σ′∂z2+∂2σ′∂z2an equation dependent on the initial void ratio. Using (13) and averaging, this equation provides three-dimensional coefficients (28)1τo,σ′,1+eok1σ1′γwγv,σ′γk,σ′H12σ2′-σ1′σm′σm′σ1′1-γk,σ′,1+eok1σ1′γwγv,σ′1H12σm′σ1′1-γv,σ′-γk,σ′and two dimensionless independent groups (29)πI=τo,σ′k1σ′11+eoγwγv,σ′H12σm′σ1′1-γk,σ′,πII=γk,σ′σ2′-σ1′σm′Thus, the solutions for τo,σ′ and Ūσ′ are given by(30)τo,σ′=γv,σ′γwH121+eok1σ1′σm′σ1′γk,σ′-1Ψτγk,σ′σ2′-σ1′σm′(31)Ūσ′=ΨUtτo,σ′,γk,σ′σ2′-σ1′σm′


### 3.1.4. The Less Restrictive Model with Both 1+e anddz Not Constant

For this model, 1+e anddz not constants, it is enough to consider (22) plus the conditiondz not constant, or dz=dzo(H/H1). So, after mathematical manipulation, the governing equation writes as(32)∂σ′∂t=1+eoσ′okoγwγv,σ′σ′σ′o1+γv,σ′-γk,σ′-γk,σ′σ′∂σ′∂z2+∂2σ′∂z2The new coefficients(33)1τo,σ′,1+eok1σ1′γwγv,σ′γk,σ′H12σ2′-σ1′σm′σm′σ1′1+γv,σ′-γk,σ′,1+eok1σ1′γwγv,σ′1H12σm′σ1′1+γv,σ′-γk,σ′provide the groups(34)πI=τo,σ′k1σ′11+eoγwγv,σ′H12σm′σ1′1-γk,σ′+γv,σ′,πII=γk,σ′σ2′-σ1′σm′and the solutions(35)τo,σ′=γv,σ′γwH121+eok1σ1′σm′σ1′γk,σ′-γv,σ′-1Ψτγk,σ′σ2′-σ1′σm′(36)Ūσ′=ΨUtτo,σ′,γk,σ′σ2′-σ1′σm′


## 3.1.1. Original J-B Model

The original J-B model [12] assumes small deformations compared with the soil thickness and a negligible initial void ratio (eo≈0). Developing the term between brackets of (3), we can write(11)1σ′∂σ′∂t=-cv,1γk,σ′σ1′2σ′σ1′-γk,σ′-1∂σ′∂z2+cv,1σ1′σ′σ1′-γk,σ′∂2σ′∂z2or, by mathematical manipulation,(12)∂σ′∂t=σ′okoγwγv,σ′σ′σ′o1-γk,σ′-γk,σ′σ′∂σ′∂z2+∂2σ′∂z2To make this equation dimensionless, the following variables are used:(13)σ′′=σ′-σ1′σ2′-σ1′,t′=tτo,σ′,z′=zH1where τo,σ′ is a characteristic time or unknown reference, chosen as the time required for the effective pressure (on average along the whole domain) to reach a high percentage of its change, for example, 90%. With these variables and provided that (on average) dσ′=dσ′′σ2′-σ1′+σ1′≈σ2′-σ1′, (12) can be written as (14)σ2′-σ1′σm′τo,σ′∂σ′′∂t′=-cv,1γk,σ′σ1′2H12σm′σ1′-γk,σ′-1σ2′-σ1′2∂σ′′∂z′2+cv,1σ1′H12σm′σ1′-γk,σ′σ2′-σ1′∂2σ′′∂z′2where σm′ is an average value of the effective pressure which we will talk about later. The dimensional coefficients of this equation, in terms of the parameters of the problem(15)1τo,σ′,k1σ1′γwγv,σ′γk,σ′H12σ2′σ1′-1σm′σ1′-γk,σ′,k1σ1′γwγv,σ′1H12σm′σ1′1-γk,σ′give rise to two dimensionless independent groups: (16)πI=τo,σ′k1σ′1γwγv,σ′H12σm′σ1′1-γk,σ′,πII=γk,σ′σ2′-σ1′σm′Based on the pi theorem, the solution of the order of magnitude of the characteristic time, derived from πI=ψπII, is given by(17)τo,σ′~γwγv,σ′H12k1σ1′σm′σ1′γk,σ′-1Ψτγk,σ′σ2′-σ1′σm′As regards Ūσ′, since it also depends on time, the solution (dependent on two groups) is given by (18)Ūσ′=ΨUtτo,σ′,γk,σ′σ2′-σ1′σm′where Ψτ and ΨU are unknown functions of their arguments.


## 3.1.2. J-B Model Assuming 1+e Is Not Constant anddz Is Constant

Deleting the restrictive hypothesiseo≈0 used by J-B for the solutions of σ′ and Ūσ′, the latter through the dependence V-σ′ for which such a hypothesis is not satisfied, and assuming that 1+e≠constant, the governing equation is(19)∂∂zkγwdudz=∂∂te1+e=11+e2∂e∂tUsing the dependence V/V1=σ′/σ′1-γv,σ′=(1+e)/(1+eo) or 1+e=1+eoσ′/σ′o-γv,σ′ and its derivative form de=-(1+eoγv,σ′/σ′o)σ′/σ′o-γv,σ′-1dσ′, the right and left terms of (19) can be written as(20)∂∂te1+e=11+e2∂e∂t=-γv,σ′1+eoσ′oσ′σ′oγv,σ′-1∂σ′∂tOn the other hand, with k/ko=σ′/σ′o-γk,σ′, the left term of (19) writes as (21)∂∂zkγw∂u∂z=koγwσ′σ′o-γk,σ′γk,σ′σ′∂u∂z2+∂2u∂z2Equating both terms, (19) in terms of the effective pressure takes the form (22)∂σ′∂t=1+eoσ′okoγwγv,σ′σ′σ′o1-γv,σ′-γk,σ′-γk,σ′σ′∂σ′∂z2+∂2σ′∂z2Proceeding as in the previous case, the coefficients of the dimensionless form of the former equation(23)1τo,σ′,1+eok1σ1′γwγv,σ′γk,σ′H12σ2′-σ1′σm′σm′σ1′1-γv,σ′-γk,σ′,1+eok1σ1′γwγv,σ′1H12σm′σ1′1-γv,σ′-γk,σ′give rise to the dimensionless groups (24)πI=τo,σ′k1σ′11+eoγwγv,σ′H12σm′σ1′1-γk,σ′-γv,σ′,πII=γk,σ′σ2′-σ1′σm′Thus, the solutions for τo,σ′ and Ūσ′ are given by(25)τo,σ′=γv,σ′γwH121+eok1σ1′σm′σ1′γv,σ′+γk,σ′-1Ψτγk,σ′σ2′-σ1′σm′(26)Ūσ′=ΨUtτo,σ′,γk,σ′σ2′-σ1′σm′


## 3.1.3. J-B Model Assuming 1+e Is Constant (A Simplification of the Former)

Based on the above, it is easy to simplify the hypothesis to 1+e being constant. Withσ′/σ′1-γv,σ′=1, the consolidation equation reduces to(27)∂σ′∂t=1+eoσ′okoγwγv,σ′σ′σ′o1-γk,σ′-γk,σ′σ′∂σ′∂z2+∂2σ′∂z2an equation dependent on the initial void ratio. Using (13) and averaging, this equation provides three-dimensional coefficients (28)1τo,σ′,1+eok1σ1′γwγv,σ′γk,σ′H12σ2′-σ1′σm′σm′σ1′1-γk,σ′,1+eok1σ1′γwγv,σ′1H12σm′σ1′1-γv,σ′-γk,σ′and two dimensionless independent groups (29)πI=τo,σ′k1σ′11+eoγwγv,σ′H12σm′σ1′1-γk,σ′,πII=γk,σ′σ2′-σ1′σm′Thus, the solutions for τo,σ′ and Ūσ′ are given by(30)τo,σ′=γv,σ′γwH121+eok1σ1′σm′σ1′γk,σ′-1Ψτγk,σ′σ2′-σ1′σm′(31)Ūσ′=ΨUtτo,σ′,γk,σ′σ2′-σ1′σm′


## 3.1.4. The Less Restrictive Model with Both 1+e anddz Not Constant

For this model, 1+e anddz not constants, it is enough to consider (22) plus the conditiondz not constant, or dz=dzo(H/H1). So, after mathematical manipulation, the governing equation writes as(32)∂σ′∂t=1+eoσ′okoγwγv,σ′σ′σ′o1+γv,σ′-γk,σ′-γk,σ′σ′∂σ′∂z2+∂2σ′∂z2The new coefficients(33)1τo,σ′,1+eok1σ1′γwγv,σ′γk,σ′H12σ2′-σ1′σm′σm′σ1′1+γv,σ′-γk,σ′,1+eok1σ1′γwγv,σ′1H12σm′σ1′1+γv,σ′-γk,σ′provide the groups(34)πI=τo,σ′k1σ′11+eoγwγv,σ′H12σm′σ1′1-γk,σ′+γv,σ′,πII=γk,σ′σ2′-σ1′σm′and the solutions(35)τo,σ′=γv,σ′γwH121+eok1σ1′σm′σ1′γk,σ′-γv,σ′-1Ψτγk,σ′σ2′-σ1′σm′(36)Ūσ′=ΨUtτo,σ′,γk,σ′σ2′-σ1′σm′


## 3.2. Nondimensionalization of the J-B Model in Terms of Settlements

Three cases, defined by the hypotheses(1) 1+e≠constant anddz constant, (2) 1+e anddz constants, and (3) 1+e anddz not being constant, are considered. For this study let us introduce a new variable directly related with the settlement, with a clear physical meaning, “ζ = e - eo”, a kind of local degree of settlement or differential void index.


### 3.2.1. 1+e≠Constant anddz Constant

For 1+e≠constant, using the variableζ and the dependence V-σ′, it is straightforward to write (37)ζ=e-eo=1+eoσ′σ′1-γv,σ′-1Making use of the derivatives of (37)(38)∂σ′∂z2=σ′121+eo2γv,σ′2σ′σ′12γv,σ′+2∂ζ∂z2∂2σ′∂z2=-σ′11+eoγv,σ′σ′σ′1γv,σ′+1∂2ζ∂z2+σ′1γv,σ′+11+eo2γv,σ′2σ′σ′12γv,σ′+1∂ζ∂z2and the assumptions of the odometer test, after cumbersome mathematical steps, the nonlinear dimensional consolidation (19) is written in the form (39)∂ζ∂t=koσ′oγk,σ′-γv,σ′-1γwγv,σ′2σ′σ′o1-γk,σ′∂ζ∂z2+koσ′o1+eoγwγv,σ′σ′σ′o1-γv,σ′-γk,σ′∂2ζ∂z2Introducing the variables (40)ζ′=ζ-ζ1ζ2-ζ1=ζ1-ζζ1,z′=zH1,t′=tτo,sinto the former equation, with τo,s, a reference chosen as the time required for the settlement to reach a high percentage of its change along the whole domain, its dimensionless form provides three coefficients(41)1τo,s,koσ′oγk,σ′-γv,σ′-1γwγv,σ′2σm′σ1′1-γk,σ′ef-eoH12,koσ′o1+eoγwγv,σ′σm′σ1′1-γv,σ′-γk,σ′1H12Dividing by the last, the resulting dimensionless groups are(42)πI=τo,skoσ′o1+eoγwγv,σ′H12σm′σ1′1-γv,σ′-γk,σ′πII=γk,σ′-γv,σ′-1γv,σ′ef-eo1+eoσm′σ1′γv,σ′Making use of the constitutive dependence dV/V=-γv,σ′(dσ′/σ′) is possible to write ΔH/H=-γv,σ′(Δσ′/σ′), and since ef-eo/1+eo=ΔH/Ho, πII finally takes the form(43)πII=1-γk,σ′+γv,σ′σ2′-σ1′σm′The solution for τo,s is(44)τo,s≈γwγv,σ′H12koσ′o1+eoσm′σ1′γv,σ′+γk,σ′-1Ψτ1-γk,σ′+γv,σ′σ2′-σ1′σm′As for the average degree of consolidation, the following dependence arises:(45)Ūs=ΨUtτo,s,1-γk,σ′+γv,σ′σ2′-σ1′σm′


### 3.2.2. 1+e anddz Constants

The hypothesis 1+e=constant is a simplification of the former case. Proceeding in the same way, the resulting coefficients(46)1τo,s,koσ′o1+γv,σ′-γk,σ′γwγv,σ′2σm′σ1′1-γk,σ′+γv,σ′ef-eoH12,koσ′o1+eoγwγv,σ′σm′σ1′1-γk,σ′1H12,provide the dimensionless groups(47)πI=τo,skoσ′o1+eoγwγv,σ′H12σm′σ1′1-γk,σ′πII=1-γk,σ′+γv,σ′σ2′-σ1′σm′The solutions for τo,s and Ūs are(48)τo,s≈γwγv,σ′H12koσ′o1+eoσm′σ1′γk,σ′-1Ψτ1-γk,σ′+γv,σ′σ2′-σ1′σm′(49)Ūs=ΨUtτo,s,1-γk,σ′+γv,σ′σ2′-σ1′σm′The solutions for the original J-B model are also (48) and (49) with the simplification of eo=0. The author, however, does not talk about the characteristic time and obtains the average degree of consolidation given by (7), in which this variable depends on three groups: (1-γk,σ′)/γv,σ′, H2/H1, and tcv,2/H12, undoubtedly a less precise solution.


### 3.2.3. 1+e anddz Not Being Constant

When 1+e anddz are not constants, substituting dz=dzo(H/H1) in (39), after cumbersome mathematical manipulation, the resulting coefficients are(50)1τo,s,koσ′oγk,σ′-γv,σ′-1γwγv,σ′2σm′σ1′1+2γv,σ′-γk,σ′ef-eoH12,koσ′o1+eoγwγv,σ′σm′σ1′1+γv,σ′-γk,σ′1H12Dividing by the last, the resulting dimensionless groups are(51)πI=τo,skoσ′o1+eoγwγv,σ′H12σm′σ1′1+γv,σ′-γk,σ′πII=1-γk,σ′+γv,σ′σ2′-σ1′σm′The solutions for τo,s and Ūs are(52)τo,s≈γwγv,σ′H12koσ′o1+eoσm′σ1′+γk,σ′-γv,σ′-1Ψτ1-γk,σ′+γv,σ′σ2′-σ1′σm′(53)Ūs=ΨUtτo,s,1-γk,σ′+γv,σ′σ2′-σ1′σm′Table 1 summarizes the results derived in this section.Table 1
Dimensionless groups that characterize the original and extended J-B models.
Juárez-Badillooriginal Juárez-Badillo Juárez-Badillo Juárez-Badillo 1+e=constant 1+e≠constant 1+e≠constant dz constant dz constant dz variable π I τ o k o σ ′ o γ w γ v , σ ′ H 1 2 σ ′ m σ ′ o 1 - γ k , σ ′ τ o k o σ ′ o 1 + e o γ w γ v , σ ′ H 1 2 σ ′ m σ ′ o 1 - γ k , σ ′ τ o k o σ ′ o 1 + e o γ w γ v , σ ′ H 1 2 σ ′ m σ ′ o 1 - γ k , σ ′ - γ v , σ ′ τ o k o σ ′ o 1 + e o γ w γ v , σ ′ H 1 2 σ ′ m σ ′ o 1 - γ k , σ ′ + γ v , σ ′ τ o , σ ′ or τo,s   ⁢πII⁢ ⁢γk,σ′σ2′-σ1′σm′ for pressure ⁢1-γk,σ′+γv,σ′σ2′-σ1′σm′ for settlement


## 3.2.1. 1+e≠Constant anddz Constant

For 1+e≠constant, using the variableζ and the dependence V-σ′, it is straightforward to write (37)ζ=e-eo=1+eoσ′σ′1-γv,σ′-1Making use of the derivatives of (37)(38)∂σ′∂z2=σ′121+eo2γv,σ′2σ′σ′12γv,σ′+2∂ζ∂z2∂2σ′∂z2=-σ′11+eoγv,σ′σ′σ′1γv,σ′+1∂2ζ∂z2+σ′1γv,σ′+11+eo2γv,σ′2σ′σ′12γv,σ′+1∂ζ∂z2and the assumptions of the odometer test, after cumbersome mathematical steps, the nonlinear dimensional consolidation (19) is written in the form (39)∂ζ∂t=koσ′oγk,σ′-γv,σ′-1γwγv,σ′2σ′σ′o1-γk,σ′∂ζ∂z2+koσ′o1+eoγwγv,σ′σ′σ′o1-γv,σ′-γk,σ′∂2ζ∂z2Introducing the variables (40)ζ′=ζ-ζ1ζ2-ζ1=ζ1-ζζ1,z′=zH1,t′=tτo,sinto the former equation, with τo,s, a reference chosen as the time required for the settlement to reach a high percentage of its change along the whole domain, its dimensionless form provides three coefficients(41)1τo,s,koσ′oγk,σ′-γv,σ′-1γwγv,σ′2σm′σ1′1-γk,σ′ef-eoH12,koσ′o1+eoγwγv,σ′σm′σ1′1-γv,σ′-γk,σ′1H12Dividing by the last, the resulting dimensionless groups are(42)πI=τo,skoσ′o1+eoγwγv,σ′H12σm′σ1′1-γv,σ′-γk,σ′πII=γk,σ′-γv,σ′-1γv,σ′ef-eo1+eoσm′σ1′γv,σ′Making use of the constitutive dependence dV/V=-γv,σ′(dσ′/σ′) is possible to write ΔH/H=-γv,σ′(Δσ′/σ′), and since ef-eo/1+eo=ΔH/Ho, πII finally takes the form(43)πII=1-γk,σ′+γv,σ′σ2′-σ1′σm′The solution for τo,s is(44)τo,s≈γwγv,σ′H12koσ′o1+eoσm′σ1′γv,σ′+γk,σ′-1Ψτ1-γk,σ′+γv,σ′σ2′-σ1′σm′As for the average degree of consolidation, the following dependence arises:(45)Ūs=ΨUtτo,s,1-γk,σ′+γv,σ′σ2′-σ1′σm′


## 3.2.2. 1+e anddz Constants

The hypothesis 1+e=constant is a simplification of the former case. Proceeding in the same way, the resulting coefficients(46)1τo,s,koσ′o1+γv,σ′-γk,σ′γwγv,σ′2σm′σ1′1-γk,σ′+γv,σ′ef-eoH12,koσ′o1+eoγwγv,σ′σm′σ1′1-γk,σ′1H12,provide the dimensionless groups(47)πI=τo,skoσ′o1+eoγwγv,σ′H12σm′σ1′1-γk,σ′πII=1-γk,σ′+γv,σ′σ2′-σ1′σm′The solutions for τo,s and Ūs are(48)τo,s≈γwγv,σ′H12koσ′o1+eoσm′σ1′γk,σ′-1Ψτ1-γk,σ′+γv,σ′σ2′-σ1′σm′(49)Ūs=ΨUtτo,s,1-γk,σ′+γv,σ′σ2′-σ1′σm′The solutions for the original J-B model are also (48) and (49) with the simplification of eo=0. The author, however, does not talk about the characteristic time and obtains the average degree of consolidation given by (7), in which this variable depends on three groups: (1-γk,σ′)/γv,σ′, H2/H1, and tcv,2/H12, undoubtedly a less precise solution.


## 3.2.3. 1+e anddz Not Being Constant

When 1+e anddz are not constants, substituting dz=dzo(H/H1) in (39), after cumbersome mathematical manipulation, the resulting coefficients are(50)1τo,s,koσ′oγk,σ′-γv,σ′-1γwγv,σ′2σm′σ1′1+2γv,σ′-γk,σ′ef-eoH12,koσ′o1+eoγwγv,σ′σm′σ1′1+γv,σ′-γk,σ′1H12Dividing by the last, the resulting dimensionless groups are(51)πI=τo,skoσ′o1+eoγwγv,σ′H12σm′σ1′1+γv,σ′-γk,σ′πII=1-γk,σ′+γv,σ′σ2′-σ1′σm′The solutions for τo,s and Ūs are(52)τo,s≈γwγv,σ′H12koσ′o1+eoσm′σ1′+γk,σ′-γv,σ′-1Ψτ1-γk,σ′+γv,σ′σ2′-σ1′σm′(53)Ūs=ΨUtτo,s,1-γk,σ′+γv,σ′σ2′-σ1′σm′Table 1 summarizes the results derived in this section.Table 1
Dimensionless groups that characterize the original and extended J-B models.
Juárez-Badillooriginal Juárez-Badillo Juárez-Badillo Juárez-Badillo 1+e=constant 1+e≠constant 1+e≠constant dz constant dz constant dz variable π I τ o k o σ ′ o γ w γ v , σ ′ H 1 2 σ ′ m σ ′ o 1 - γ k , σ ′ τ o k o σ ′ o 1 + e o γ w γ v , σ ′ H 1 2 σ ′ m σ ′ o 1 - γ k , σ ′ τ o k o σ ′ o 1 + e o γ w γ v , σ ′ H 1 2 σ ′ m σ ′ o 1 - γ k , σ ′ - γ v , σ ′ τ o k o σ ′ o 1 + e o γ w γ v , σ ′ H 1 2 σ ′ m σ ′ o 1 - γ k , σ ′ + γ v , σ ′ τ o , σ ′ or τo,s   ⁢πII⁢ ⁢γk,σ′σ2′-σ1′σm′ for pressure ⁢1-γk,σ′+γv,σ′σ2′-σ1′σm′ for settlement


## 4. Verification of the Results by Numerical Simulations

This section is devoted to checking the solutions given in Table1. To shorten the exposure, we restrict the tests for the more general model (1+e anddz not constants) and for the unknowns τo,s and Ūs since they are the ones of greater interest in civil engineering. Eight sets of simulations, arranged in three blocks, have been run, Table 2. In each one, some of the soil parameters or initial values of the problem have been changed to give the same or different values, as required, to the dimensionless group πII, in the search for the same or different solutions of πI. Changes in the values of the individual parameters are sufficient to cover all real scenarios.Table 2
Values of the parameters and dimensionless groups of simulations.
Block Set γ v , σ ′ γ k , σ ′ e o H 1 k o σ ′ o σ ′ f ᴪ H 2 k f π I I τ o , s π I (m) (m/yr) (N/m2) (N/m2) σ ′ f/σ′o (m) (m/yr) (yr) I 1 0.1 0.500 1.0 1.00 0.02 30000 60000 2 0.933 0.0141 0.40 0.4566 0.7131 2 0.1 0.500 1.0 1.00 0.02 60000 120000 2 0.933 0.0141 0.40 0.2283 0.7131 3 0.1 0.500 1.0 1.75 0.01 30000 60000 2 1.633 0.0071 0.40 2.7968 0.7131 4 0.1 0.500 1.3 1.42 0.04 22000 44000 2 1.325 0.0283 0.40 0.5459 0.7131  II 5 0.1 1.500 1.0 1.00 0.02 30000 90000 3 0.896 0.0038 -0.40 1.0597 0.9834  III 6 0.2 0.600 1.0 1.00 0.02 30000 60000 2 0.871 0.0132 0.40 0.9142 0.7139 7 0.1 0.767 1.0 1.00 0.02 30000 120000 4 0.871 0.0069 0.40 0.4360 0.7246 8 0.2 0.920 1.0 1.00 0.02 30000 180000 6 0.699 0.0038 0.40 0.8409 0.7312Firstly, a reference set (set 1) is established to which all the other sets can be referred and which permits them to be compared with each other. The physical and geometrical characteristics that change are:γv,σ′, γk,σ′, eo, H1 (m), σ′o (N/m2), σ′f (N/m2), and ko (m/year). The values ofᴪ =(σ′f/σ′o), H2 (m), kf (m/year), and πII are derived from them, while π1 is obtained once the characteristic time τo,s is read from the simulation. The criterion for the choice of τo,s is the time required by the soil to reach 90 % of the total settlement.It is worth mentioning here the emergence ofσ′m, an averaged value of the effective pressure, in the two dimensionless groups. Undoubtedly, the existence of σ′m in the monomials can be explained not only by the nonlinearity of the problem but also by the kind of dependences in the constitutive relations (there are other nonlinear problems in which averaged variables do not emerge [18]). How do we choose σ′m? We could take the mean value σ′m=(σ′1+σ′2)/2, or even σ′m=σ′1 or σ′m=σ′2, but expecting that the expression πI=ψπII is slightly dependent on that choice (we will return to this question later). In this paper, we have taken the approximation σ′m=(σ′2+σ′1)/2.In block I, the parameters that change in each set are:eo, H1, ko and σ′o, while maintaining the ratio ᴪ =(σ′f /σ′o); thus, both πII and factor σ′m/σ′o1-γk,σ′+γv,σ′ of πI take the same value in the four sets of the block. As a result, we expect a characteristic time proportional to the ratio γwγv,σ′H12/koσ′o1+eo and a same value of πI in all cases, according to the solution πI=ψπII. Indeed, the results are consistent with expectations. Block II contains an only case for which πII varies in comparison to block I; as a result, πI also changes, as expected. Finally, block III contains three cases with the same value of πII of block I, but with different values of the parameters γv,σ′, γk,σ′, and the ratio ᴪ =(σ′f/σ′o). The parameters of set 6 have been chosen in such a way that the values γk,σ′ and γv,σ′ are compensated to maintain constant 1-γk,σ′+γv,σ′; we expect πI to be also unchanged. In effect, simulation provides πI = 0.7139, a value quite close to that of set 1 (0.7131). The negligible differences are due, no doubt, to the different evolution of the effective pressure along the process caused by the different values of the coefficients γk,σ′ and γv,σ′. Note that as γv,σ′ duplicates the denominator of the factor τo,skoσ′o1+eo/γwγv,σ′H12, the characteristic time has to double too, as indeed happens. The last two sets study the influence of ᴪ, a parameter whose change forces the value 1-γk,σ′+γv,σ′ to be also changed in order to maintain πII constant. We expect that πI does not change as nearly occurs. The small differences are again due to the existence of σ′m within the monomials. Note, however, that as ᴪ diminishes the influence of σ′m in the results also diminishes, since the values of σ′1 and σ′2 are closer.For a better understanding of the results, Figures1 and 2 are presented. In the first, Ūs as a function of time is shown for the eight sets in Table 2. It is clear that τo,s covers a wide range of values, from 0.2283 (set 2) to 2.7968 (set 3), the last more than ten times the first. The second figure graphs Ūs in front of the dimensionless time t/τo,s. As shown, the eight previous curves nearly converge to a single (or universal) graph, having a common point at (Ūs=0.9, t/τo,s=1). This means that monomial πII, for the range of values assigned to the parameters, in contrast to πI, scarcely influences the value of the average degree of consolidation. This is not in contradiction with the results derived by nondimensionalization.Figure 1
Average degree of consolidation versus time (eight sets).Figure 2
Average degree of consolidation versus dimensionless time (eight sets).Although in this section we have only checked the expressions ofŪs and τo,s for the most general model, we consider it interesting to present both Ūs and Ūσ′ for the rest of the models of Table 1 and for set 6 as a typical case, Figure 3. These results emphasize the importance of using the less restrictive (more general) model in relation to the others, in particular with the original model of J-B, in which the influence of eo is despised. Note that the smaller of the characteristic times (related to settlement) corresponds to the more general model (black and bold line), so the use of the others, although on the safety side, provides characteristic times oversized, far from an optimal solution for the engineer. For example, for the case of Figure 3 in which eo=1, the error of J-B original model (purple and bold line) in comparison with the extended model (black and bold line) for Ūs=90% is around 150%. This error increases appreciably as eo gets larger.Figure 3
Average degree of consolidation and pressure dissipation (set 6).


## 5. Universal Curves

These easy to use and universal abacuses obtained by numerical simulations allow engineers to read the characteristic time (τo,s) and the average degree of consolidation (Ūs) in terms of the dimensionless groups given in Table 1. Only the most general (less restrictive) J-B model, 1+e anddz not constant, is presented.To obtain the characteristic time curves a large number of simulations have been carried out studying, separately, the influence of the parametersᴪ, γv,σ′, and γk,σ′, whose values ​​cover most real soils: ᴪ ranges from 1.01 to 8, γv,σ′ from 0.05 to 0.3, and γk,σ′ from 0.1 to 1.7. From these tests, it has been deduced, as expected from the results of the previous section, that the relatively more influential parameter is ᴪ, while γv,σ′ and γk,σ′ do not produce significant changes in the values of the characteristic time. Thus, the first universal group of lines shows the dependence of πI versus πII for different values of ᴪ (with γv,σ′=0.1), Figure 4. As the range of values of πII depends on ᴪ, according to (43), the range of values of each line is different. As shown in the figure, there exists a universal line for each value of ᴪ, all converging in the central zone for values of πII around 0. This set of straight lines can be fitted by πI=mπII+n, being n=0.846 and m given in Figure 5, as a function of ᴪ. The curve that best fits the points in Figure 5 is given by(54)m=-0.235e-0.868ᴪ-0.306e-0.012ᴪChoosing the longest line corresponding toᴪ=8 as the only representative line of the problem, the maximum errors in the reading of πI (and τo,s) are 3.8%.Figure 4
π I as a function of πII with ᴪ as parameter.Figure 5
m as a function ofᴪ.In short, the use of these curves is as follows: from the data of the problem, Figure4 allows us to read πI from πII andᴪ, and from (51) the characteristic time τo,s is derived.As for the average degree of consolidation, the universal curves for 96 different scenarios ofπII are shown in Figure 6. It is appreciated that these curves are very close to each other, showing again that the Ūs only depends on the dimensionless time. In the worst case, the errors produced by changes in value of the πII group are less than 2.8% when using the curve of Figure 7.Figure 6
Average degree of consolidation versus dimensionless time for different scenarios.Figure 7
Proposed solution for the average degree of consolidation.


## 6. Case Study

Below is a practical application of the universal curves presented in this paper. For this purpose, we take as a basis for our study the data obtained experimentally by Abbasi et al. [26] for different samples of clay whose moisture in the liquid limit is equal to 42. As can be seen (Table 3), it is a series of discrete (tabulated) values ​​that relate the void ratio with the effective pressure and the hydraulic conductivity.Table 3
Experimental values of e,σ′ and k obtained by Abbasi et al. [26].
e σ ′ o (N/m2) 1.10 14000 1.05 28000 0.95 59000 0.85 120000 0.75 220000 0.65 450000 0.60 925000  e k (m/yr)  1.75 0.683 1.60 0.315 1.20 0.084 0.85 0.021 0.65 0.011We will address a real case of consolidation, in which we have a layer of soil 2 meters thick (Ho), with an initial effective pressure (σ′o) of 28 kN/m2, on which a load of 31 kN/m2 is applied. In this way, the effective pressure at the end of the consolidation process (σ′f) will have the value of 59 kN/m2. From the tabulated data (e-σ′) it follows that eo = 1.05 and ef = 0.95 and, since the factor 1+e is proportional to the soil volume, from (9) we have γv,σ′ = 0.067.To obtainγk,σ′ we first calculate the value of γk,v from the tabulated data (e-k), to then obtain γk,σ′ from (10). With this, we have, for the load step considered, ko = 0.035 m/yr, γk,v = 8.108, and γk,σ′ = 0.543.To solve the problem we will use the universal curves presented in the previous section, corresponding to the most general J-B model (analyzed in terms of settlement). Thus, from the values ​​ofσ′o, σ′f, γv,σ′, and γk,σ′, by means of (51) it is obtained that πII = 0.373 (with σ′m = 43.5 kN/m2). Then, πI is read from Figure 4, where for ᴪ = 2.1 we have πI = 0.72. An identical result is obtained if we use the proposed fitted equation πI=mπII+n, where n is a constant of value 0.846 and m is obtained from (54) or Figure 5. Finally, known the values of Ho, ko and eo, from (51) we get τo,s = 0.748 years. Likewise, once the characteristic time of the problem (in terms of settlement) is known, from the universal curve of Figure 7 we easily obtain the evolution of the average degree of consolidation along the whole process, Figure 8.Figure 8
Case study solution for the average degree of consolidationŪs from the universal curves.These solutions have been compared with those that result from the numerical simulation of the most general J-B model, for which a value ofτo,s = 0.75 years has been obtained, showing clearly that the relative error committed when applying the universal solutions is negligible (Table 4). The simulation of the problem has also been carried out for a model where 1+e anddz are also not constant but in which the constitutive relations of potential type have been replaced by pairs of data in tabulated form, obtaining in this case τo,s = 0.817 years. The relative error, which this time increases to 8.2%, finds its explanation in the strong nonlinearity of the problem, which will always bring differences between the solutions provided by numerical models based on different constitutive relations (potential and tabulated, in this case). Even so, this value is totally acceptable within this field of engineering, demonstrating the strength and versatility of the universal solutions provided.Table 4
Case study solution for the characteristic timeτo,s. Comparison with numerical models.
Universal curves J-B most general model Tabulated model τ o , s (yr) τ o , s (yr) Rel. error % τ o , s (yr) Rel. error % 0.748 0.750 0.26 0.817 8.20


## 7. Discussion

This work derives the dimensionless groups that rule the nonlinear soil consolidation process based on the original J-B model, following a formal nondimensionalization procedure that starts from the governing equation and does not need any other assumption or mathematical manipulation. In addition, more real and precise extended J-B models, which assume both the void ratio and the thickness of the finite volume element of the soil to be not constant, have been nondimensionalized and their dimensionless groups derived. The assumption of constitutive dependences (e-σ′, e-k and k-σ′) of potential type in these models is interesting since they can be applied to highly compressible soils, for which the e-log(σ′) relation moves away from being a straight line and in which negative values ​​of the void ratio could be obtained under the application of high load ratios (σ′f/σ′o) [8].The study has been developed both in terms of pressure dissipation and in terms of settlement, resulting that the dimensionless groups derived from each model are, in general, different. The extension to more complex models, which implies the emergence of new parameters or equations within the mathematical model, does not necessarily give rise to additional dimensionless groups, as might be expected, but to new expressions of the groups or new regrouping of the parameters involved. Thus, the elimination of the hypothesis “dz constant” in the most general consolidation model has not given rise to any new group, both in pressure and in settlement (Table 1), but it has only modified one of the groups obtained in the original model. All this represents an important advance in relation to the models and studies carried out by other authors and the attempts of characterization found in the scientific literature [9–11], which try to lead to easy-to-use solutions for engineers. In no case had dimensionless groups been obtained as precise as those presented in this work nor presented universal curves of the characteristic time and the average degree of consolidation. Authors such as Conte and Troncone [11] apply more general load conditions (time-dependent loading) for which the dimensionless groups deduced in this work could be applicable for each load step.As regards time, to make it dimensionless, an unknown reference (τo) is introduced in the form mentioned by Scott [27] and used in many other problems with asymptotic solutions [28]. The precise definition of τo necessarily involves a criterion in relation to the percentage of average pressure decrease (or settlement) associated with the end of the consolidation process. The order of magnitude of τo is derived after nondimensionalization, making τo only appear in one of the independent dimensionless groups. In this way, the average degree of consolidation (in terms of pressure or settlement) depends on the “time/characteristic time” ratio, in addition to the dimensionless groups (without unknowns) that arise.In short, the procedure has led to a new, very precise, solution that involves two monomials: one (πI) that contains the characteristic time (τo) and another (πII) that is a mathematical function of the soil parameters. In this way, τo only depends on πII through the function πI=ψπII, while Ū depends on πI and πII through the expression Ū = ψ(πI, πII). The results obtained have been verified for the most general and complex consolidation model through the numerical simulation of a large number of scenarios in which the values of the dimensionless groups and the individual parameters of the problem have been changing to convenience. The high errors produced by the original J-B model with respect to the most general justify the use of the latter in the search for more precise solutions.In addition, the emergence of mean values of the effective pressure when averaging the governing equation, in order to assign an order of magnitude unit to the dimensionless variables and their derivatives, is an added difficulty to obtain the dimensionless groups. This is, undoubtedly, due to the nonlinearities (constitutive relations) involved in the consolidation process. This average effective pressure of reference has been assumed as the semisum of its initial and final values, although other criteria could also be chosen. Since two scenarios can have equalπII with the same effective mean pressure but different values of the parameters γv,σ′ and γk,σ′, we expect some minor deviations between the results of both scenarios, as indeed it has been checked. Thus, for the whole range of parameters values in typical soils, the errors due to the above cause are negligible in the field of soil mechanics. On the other hand, when the value of πII does not change but the mean effective pressure value does, the errors in the solutions of τo,s and Ūs are slightly larger, although within ranges below 4%.The brooch for all this work is the presentation of universal curves for the most general model, which allow, for the usual range of variation of the soil properties, to obtain in a simple way the value of the characteristic time of consolidation and the evolution of the average degree of settlement throughout the entire process. Thus, starting from the nonlinear parameters of the soil (γv,σ′ and γk,σ′) and the load ratio (σ′f/σ′o), the value of the monomial πII is obtained, from which the value of πI is directly deducted thanks to the universal curve πI=ψπII provided. Finally, τo,s is easily obtained from the expression of πI. On the other hand, the curve that represents the Ūs evolution along the process has turned out to be (practically) unique when it is represented as a function of the dimensionless time in the form t/τo,s, regardless of the soil parameters and the loads applied. All this supposes a clear advantage when solving the consolidation problem, also showing the strength of the nondimensionalization method presented.Finally, the case study addressed has demonstrated the simplicity and reliability of the universal curves proposed. From a series of tabulated data (e-σ′ and e-k) the parameters that govern the nonlinear consolidation problem for the most extended J-B model have been adjusted, with which the value of the monomial πII has been derived. With this, and using the universal curves, the solutions for the characteristic time τo,s and the average degree of consolidation Ūs have been obtained. These results have been satisfactorily verified with the solutions provided both by the numerical extended model of J-B presented here and by another that substitutes the constitutive dependences of potential type by data in tabulated form.


## 8. Conclusions

The application of the nondimensionalization technique has allowed us to deduce the dimensionless groups that rule the nonlinear consolidation problem in which the constitutive dependencies e-σ′ and e-k have the form of potential functions. This type of relations approximates the soil behavior better than the logarithmic functions e-log(σ′) and e-log(k), particularly in very compressible soils and under high load ratios (σ′f/σ′o) in which inconsistent negative values of the void ratio may appear.Several scenarios have been addressed, from the original model of Juárez-Badillo to extended models based on the same constitutive dependences but eliminating one or more classic (and restrictive) hypotheses in this process. In all cases, only two dimensionless groups emerge, one of them consisting of a grouping of the soil parameters and the other containing the unknown characteristic time. Based on these results, universal curves have been depicted: the first to determine the characteristic time of the process and the second that represents the evolution of average degree of settlement.The solutions have been verified by numerical simulations, changing the values of the soil parameters and the dimensionless groups conveniently. Finally, the reliability and effectiveness of the proposed solutions have been shown through a case study.



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