Dimensionless characterization of the non-linear soil consolidation problem of Davis and Raymond. Extended models and universal curves

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Abstract

The dimensionless groups that govern the Davis and Raymond non-linear consolidation model, and its extended versions resulting from eliminating several restrictive hypotheses, were deduced. By means of the governing equations nondimensionalization technique and introducing the characteristic time concept, both in terms of settlement and pressures, was obtained (for the most general model) that the average degree of settlement only depends on the dimensionless time while the average degree of pressure dissipation does it, additionally, on the loading ratio. These results allowed the construction of universal curves expressing the solutions of the unknowns of interest in a direct and simple way.

Abstract

The dimensionless groups that govern the Davis and Raymond non-linear consolidation model, and its extended versions resulting from eliminating several restrictive hypotheses, were deduced. By means of the governing equations nondimensionalization technique and introducing the characteristic time concept, both in terms of settlement and pressures, was obtained (for the most general model) that the average degree of settlement only depends on the dimensionless time while the average degree of pressure dissipation does it, additionally, on the loading ratio. These results allowed the construction of universal curves expressing the solutions of the unknowns of interest in a direct and simple way.

1 Introduction

The soil consolidation problem under linear behaviour is governed by the diffusion equation of excess pore pressure, with well-known analytical solutions for the usual boundary conditions. However, under the hypotheses of non-linear dependencies of hydraulic conductivity and void ratio with the effective stress, as well as under the consideration of variable volume elements in thickness, numerical solutions are used. Among the most common non-linear models used in literature are those of Davis and Raymond [1, 2], Juárez-Badillo [3, 4, 5] and Cornetti and Battaglio [6, 7]. The interest of the non-linear models against the linear ones is consequence of the important deviations that emerge from their solutions, above 100% [2] both in relation to the time characteristic of the process and the evolution of the average degree of consolidation.

In this paper, we address, on the one hand, the deduction and verification of the dimensionless groups that govern the solutions of the Davis and Raymond model and the extensions derived from eliminating one or more of its restrictive hypotheses (constant 1+e, cv and dz) and, on the other hand, based on these groups, construction of universal curves that allow the engineer to obtain the solutions of the most important unknowns of the problem in a direct way. In order to verify the results obtained, a set of significant cases has been simulated, widely covering the range of values of the physical and geometric parameters and the loading ratio (final and initial effective stress) that can take place in real problems. In each case, the value of one or more of the parameters involved has been significantly altered, but the numerical value of the dimensionless groups has been retained so that it is verified that the same pattern of solutions is obtained for the whole set of cases; or the value of the groups has been modified appropriately to confirm that the form of the solutions changes.

As a deduction technique for these groups, the nondimensionalization of governing equations [8, 9, 10] has been used, a form of application of the pi theorem [11], thus allowing the dimensionless parameters involved in the problem (void ratio and compression index), and that they would form independent dimensionless groups, to be included in the inferred groups. In the nondimensionalization process, which has been carried out in terms of both excess pore pressure dissipation and settlement, the characteristic time of consolidation is introduced as a reference, an unknown that is incorporated into one of the resulting monomials and whose dependency with the rest of the groups is established by the pi theorem. Despite suppressing the restrictive original hypotheses of the authors [1], the resulting number of dimensionless groups is small enough to be able to represent their dependencies by means of universal curves obtained through numerical simulations using the network simulation method [12, 13].

For the most general and precise case (non-constant 1+e, cv and dz), the characteristic time in terms of settlement is defined by a single group, so it has a direct relationship with the parameters of the problem and a single test is sufficient to obtain the proportionality factor. However, in terms of pressure, the characteristic time is a function of the final and initial effective stress ratio. On the other hand, the average degree of settlement is a direct function of the dimensionless characteristic time, while the average degree of pressure dissipation depends, additionally, on the loading ratio, so its universal representation needs to be done by means of an abacus.

2 Davis and Raymond consolidation model

2.1 Davis and Raymond proposed model

The general consolidation equation is obtained by matching, in a soil element, the temporal change in water volume with the temporal change in void volume [14]. This balance, together with the constitutive equation that associates the variables flow and pressure gradient (Darcy’s law) and the empirical expressions that relate the parameters of the problem with the dependent variable (excess pore pressure, u, or effective pressure, σ′), allows to deduce the consolidation equation for the general case (linear or non-linear processes) expressed in terms of the independent variables position and time. In linear cases, this equation has a semi-analytical solution for all geometries, while for non-linear cases, in general, the use of numerical calculation is required. In short, equalling the expressions q˙w=dVwdt=Avzzdzandq˙v=dVvdt=Adzte1+e, we have

vzz=te1+e

Using Darcy’s Law, vz=kγwuz,

expression (1) is finally written in the form

zkγwuz=11+e2et

where k (hydraulic conductivity) and e (void ratio) are, in general, parameters that depend on the effective stress (σ′) through the so-called constitutive equations of the ground. In turn, the excess pore pressure (u) and the effective stress are related by the Terzaghi’s hypothesis under oedometric conditions, σ = σ′ + u, from which it follows that σt=utandσz=uz.

In their model [1], Davis and Raymond assumed the following classical hypotheses: i) secondary consolidation is ignored, ii) soil particles and pore water are incompressible, iii) soil is saturated, iv) soil weight is negligible and v) the thickness change is considered negligible compared to the initial thickness, this is, 1+e constant. In addition, they assumed a constant consolidation coefficient based on the fact that, in a real mass of soil, compressibility and hydraulic conductivity vary during the consolidation process, both decreasing while increasing the effective pressure but in such a way that the changes in both are compensated so that the consolidation coefficient cv = kmvγw remains more or less constant. On the other hand, they adopted an e∼ σ′ dependency governed by the following empirical law [15]:

e=eoIclog10σσo

where Ic is the compression index, a constant parameter.

From the definition of the volumetric compressibility coefficient, mv=eσ11+e, and the expression eσ=Icln10σ obtained from (3), we can write

mv=Icln101+eσ

which gives us the volumetric compressibility coefficient as a function of the effective stress.

Finally, the assumption of a constant value for both the consolidation coefficient and the factor 1+e allows writing

cv=kσ1+eln10Icγwkσ

an equation equivalent to assuming that k and σ′ change inversely during the consolidation process.

The Davis and Raymond consolidation equation [1] is derived from expression (1) and considering constant the factor 1+e, resulting in

zkγwuz=11+eet

an expression very similar to that of equation (2), with the exception that the factor 1+e does not appear squared (note that, in this case, e = eo). Since k and σ′ vary inversely (kσ = koσo) and σz=uz, equation (6) can be written as

koσoγwz1σσz=11+eoeσσt

which, after mathematical manipulation, results in

koσo1+eoln10Icγw2σz21σσz2=σt

and, from expression (5), since cv is constant we have cv=koσo1+eoln10Icγw, finally obtaining:

cv2σz21σσz2=σt

Expression (9a), a clearly non-linear equation, is the Davis and Raymond consolidation equation [1] in terms of the effective stress (σ′). Its form in terms of the excess pore pressure (u) is

cv2uz2+1σuuz2=ut

The authors, for the usual boundary conditions [14], obtained the analytical solution for the variable (u), which besides depending on position (z) and time (t) also depends on the ratio σf/σo [1]. In this way, the average degree of pressure dissipation throughout the domain (Ūσ) will depend only on time and the ratio between the final and initial effective pressures.

On the other hand, looking for a simplification (by means of a change in variable) that reduces the consolidation equation (9) to a linear one, the authors proposed the introduction of a new variable (w), to which they did not assign any explicit physical meaning, whose relation with the effective stress is in the form

w=log10σσf=log10σfuσf

The introduction of this new variable, through its spatial and temporal derivatives, into equation (9) leads to a pure diffusion equation

cv2wz2=wt

What is diffused in this linear equation, then, is a new local magnitude (w) that, although the authors did not mention, is directly related to the settlements (changes in the void ratio) of the problem. Thus, from equation (3), it is easy to obtain that

efeIc=log10σσf

It follows from equation (11) that the solution for the variable w depends on position and time, and therefore, as the authors conclude [1], the average degree of settlement in the domain (Ūs) is only function of time.

2.2 Extended models

Within this section, we will consider two models that eliminate some restrictive hypotheses considered by Davis and Raymond, giving rise to more general and, therefore, more precise models: i) the first one in which the value of 1+e is assumed to be non-constant but keeping the consolidation coefficient cv constant and ii) a more general second model in which both 1+e and cv are considered non-constant.

2.2.1 Model with non-constant 1+e and constant cv

In this case, we start from expression (2), which represents the general consolidation equation under the hypothesis of 1+e non-constant. For the changes in the void ratio with the effective stress, we maintain the constitutive relation (3). On the other hand, if we consider that cv is constant throughout the process, on this criterion it is true that

cv=koσo1+eoln10Icγw=kσ1+eln10Icγw

so that the relation between the hydraulic conductivity and the effective pressure, necessarily, becomes now

kσ=koσo1+eo1+e

that is, the changes in k are inversely proportional to those in σ′, but, in addition, they also depend on the changes in the factor 1+e. With all this, equation (2) can be written as

koσo1+eoγwz1σ1+eσz=11+e2eσσt

After mathematical operation, it is deduced that

koσo1+eo1+eln10Icγw2σz21σσz2+Icσ1+eln10σz2=σt

which, from the deduction of expression (13), finally leads to:

cv1+e2σz2+Icσln101+eσσz2=σt

2.2.2 Model with both non-constant 1+e and cv

For this extended model, we again begin from expressions (2) and (3). Maintaining the Davis and Raymond philosophy, we return to the relation kσ′ = koσo to represent the variation in the hydraulic conductivity versus the effective stress. In this way, now we have

cv=cvo1+e1+eo

With all this, equation (2) can be written as

koσo1+e2ln10Icγw2σz21σσz2=σt

which expressed in terms of the consolidation coefficient takes the form

cv1+e2σz21σσz2=σt

or, in terms of the initial consolidation coefficient (a constant parameter), results in

cvo1+e21+eo2σz21σσz2=σt

3 Dimensionless characterization of the Davis and Raymond model and its extended variants

In this section, we proceed to obtain the dimensionless groups that govern the solution of the non-linear consolidation scenarios presented in the previous section to subsequently address a universal representation of the main variables of interest: average degree of consolidation and characteristic time of the duration of the process. By defining the variables of the problem (effective stress, medium depth and characteristic time) in a dimensionless form, the non-linear equation is nondimensionalized. Given the different hypotheses assumed in the different models, the nondimensionalization process deduces different dependencies in each case and, at the same time, different solutions.

It will also be analyzed, for the models in which 1+e is non-constant, the influence of the elimination of another restrictive hypothesis, consequence of this, which is the consideration of the variation in the volume element size dz. This last question, which implies addressing a moving boundary problem (and even closer to the real consolidation problem), involves changes in the governing equations, which will result in different solutions for the variables of interest, as we will see in the following.

The consolidation problem describes an interesting phenomenon whereby as the excess pore pressure is relaxed (allowing water to escape from the system), the ground settlement takes place. Two processes closely linked but, due to the non-linearity of the problem, develop in a different way, with dependencies that are not always equal. Therefore, the characterization of the problem will be carried out, for all scenarios, both in terms of pressure and settlement.

3.1 Characterization of the model proposed by Davis and Raymond

3.1.1 Nondimensionalization of the model proposed by Davis and Raymond in terms of pressure

For the nondimensionalization of this model (9a), we will adopt the following dimensionless variables:

σ=σσoσfσo

normalized to the interval [0, 1]; τo,σ is the characteristic time that takes the excess pore pressure to relax to approximately the value of zero (throughout the domain). Substituting (22) in equation (9a), through mathematical manipulations, we get

cvσfσoHo22σz2σfσo2Ho21σσz2=σfσoτo,σσt

As they are normalized variables, all derivative terms of (σ′)′, z′ and t′ are averaged to the unit, while the value of σ′ is averaged to an arbitrary (characteristic) value σm. In this way, the coefficients of this equation (once simplified), of the same order of magnitude, are three:

cvHo2cvHo2σfσoσm1τo,σ

from which, dividing by the first, monomials result:

πI=Ho2cvτo,σor,alternatively,πI=τo,σcvHo2

πII=σfσoσm

that, by means of the pi theorem[11], provide the solution for the characteristic time (πI = Ψ[πII])

τo,σ=Ho2cvΨσfσoσm

where Ψ is an arbitrary and unknown function of its argument. Adopting for σm the value, for example, σo (it can also be the value σf or the average between them), the above equation is simplified to

τo,σ=Ho2cvΨσfσo

which means that the characteristic time taken by the excess pore pressure to relax to approximately the value of zero is a function of the ratio between the final and initial effective stresses, in addition to being directly proportional to Ho2cv.. In this way, and based on the result obtained, the evolution of the average degree of pressure dissipation, Ūσ, can be expressed as

U¯σ=Ψtτo,σ,σfσo

In short, the monomials that govern the solution of the characteristic time associated with the dissipation of interstitial pressure throughout the soil are two:

π1=τo,σcvHo2π2=σf/σo

and, therefore, the average degree of pressure dissipation (Ūσ) will also be a function dependent on π1 and π2.

At the local level, the characteristic time will depend, additionally, on the depth z, which when expressed in dimensionless form is z=zHo. The solution for (σ′)′ or σ′ is governed by the expression

σ=σσoσfσo=Ψtτo,σ,σfσo,zHo

where Ψ, an arbitrary function of its arguments, can also be written in the form

σ=σσoσfσo=ΨtcvHo2,σfσo,zHo

a result that coincides with the analytical solution (Davis and Raymond [1]) in terms of argument dependencies.

3.1.2 Nondimensionalization of the model proposed by Davis and Raymond in terms of settlement

Davis and Raymond did not seem to notice that the new local magnitude (w), equation (10), is proportional to the differential void ratio ‘ef – e’, equation (12) and, under this dependency, also the local settlement. It would have seemed more illustrative and didactic, although in essence it is the same approach, starting from the empirical relation e = e(σ′), equation (3), written in the form

e=efIclog10σσf

where ef denotes the final void ratio (corresponding to the final effective stress σf), and define the new variable ζ = e – ef, with a clear physical meaning (differential local void ratio, a kind of local degree of settlement, difference between the current void ratio and the final void ratio, in each position; a positive number). It is evident that the variable ζ thus defined must lead to a pure linear diffusion equation, with universal solutions. Indeed, we can rewrite (32) as

ζ=Iclog10σσf=Icln10lnσσf

while its partial derivatives with respect to position and time are

ζz=Icln101σσz

ζt=Icln101σσt

By calculating the partial derivatives of σ′ with respect to position and time, we obtain

σz=ln10Icσζz

σt=ln10Icσζt

expressions from which it is deduced that

σz2=ln10Ic2σ2ζz2

2σz2=ln10Ic2σζz2ln10Icσ2ζz2

Now, starting from (9a), it is immediate to reach

cv2ζz2=ζt

A pure diffusion equation of the local variable differential void ratio directly related to the local degree of settlement. For its nondimensionalization, the dimensionless variables are defined as

ζ=ζζoζfζo=ζoζζo=eeoefeoz=zHot=tτo,s

normalized to the interval [0, 1], with τo,s as the characteristic time taken by the variable ζ to cover approximately its entire range of values. Substituting in equation (40), through mathematical manipulations, we reach

cvζfζoHo22ζz2=ζfζoτo,sζt

The coefficients of this equation, once simplified, result in

cvHo21τo,s

that, by division, provides a single monomial solution

π1=τo,scvHo2

and a value for the characteristic time of settlement (τo,s) of the same order of magnitude as that obtained by Davis and Raymond [1], independent of the loading ratio σfσo.

τo,sHo2cv

In this way, and based on the result obtained, the average degree of settlement (Ūs) is expressed as

U¯s=Ψtτo,s

a result also coinciding with that obtained by Davis and Raymond in terms of argument dependencies.

3.2 Characterization of the model with non-constant 1+e and constant cv

3.2.1 Nondimensionalization of the model with non-constant 1+e and constant cv in terms of pressure

In this case, starting from equation (17), which we repeat here for convenience

cv1+e2σz2+Icσln101+eσσz2=σt

and with the dimensionless variables of the expression (22), we reach

cv1+eσfσoHo22σz2+Icσln101+eσσfσoHo2σz2=σfσoτo,σσt

from which the following coefficients result in:

cvIcσfσoln10Ho2σmcv(1+em)(σfσo)Ho2σmcv(1+em)Ho21τo,σ

where the values of σ′ and e have been averaged to arbitrary characteristic values (σm and em). Dividing the coefficients by the third coefficient, we obtain the monomials

πI=τo,σcv1+emHo2πII=σfσoσmπIII=Ic(σfσo)ln(10)(1+em)σm

that provide the solution for the characteristic time (πI = Ψ[πII, πIII]).

Adopting for σm and em, for instance, the values σo and eo, the previous monomials can be simplified to

π1=τo,σcv1+eoHo2π2=σfσoπ3=Ic1+eo

allowing to write now more comfortably (π1 = Ψ[π2, π3])

τo,σ=Ho2cv1+eoΨσfσo,Ic1+eo

On this occasion, the evolution of the average degree of pressure dissipation (Ūσ) can be expressed as

U¯σ=tτo,σ,σfσo,Ic1+eo

If, in addition, we take into account the variation of the volume element size dz throughout the consolidation process, considering the relation HHo=ΔzΔzo=1+e1+eo, we just have to add to equation (47) the dependency

1Δz2=1Δzo21+eo21+e2

resulting in the coefficients

cvIcσfσoln10Ho2σm1+eo21+em2cv(σfσo)Ho2σm(1+eo)2(1+em)cvHo2(1+eo)2(1+em)1τo,σ

and, dividing by the third coefficient, the monomials

πI=τo,σcv1+eo2Ho21+emπII=σfσoσmπIII=Ic(σfσo)ln(10)(1+em)σm

Once we adopt the values σo and eo for σm and em, the resulting monomials are on this occasion the same (50) as for the case of constant dz. Thus, the fact of adopting the more general (and more precise) hypothesis, dz variable, does not increase the number of argument dependencies of the problem, which is governed by three monomials. Therefore, the expressions for the characteristic time (τo,σ) and the average degree of pressure dissipation (Ūσ′) are those already expressed in equations (51) and (52).

3.2.2 Nondimensionalization of the model with non-constant 1+e and constant cv in terms of settlement

In this case, starting from (17) and through the expressions (3739), it is immediate to reach

cv1+e2ζz2cvζz2=ζt

and with the dimensionless variables of the expression (41), we obtain

cv1+eζfζoHo22ζz2cvζfζoHo2ζz2=ζfζoτo,sζt

The coefficients of this equation, once simplified, result in

cv1+emHo2cv(ζfζo)Ho21τo,s

that, dividing by the second, provide the monomials

πI=τo,scvζoHo2πIIζo1+em

of which it is easy to reach

πI=τo,scv1+eoHo2Iclog10σfσo1+eo=τo,scv1+eoHo2HoHfHoπII=Iclog10σfσo1+em=HoHfHm

what allows, finally, to write

π1=τo,scv1+eoHo2π2=HfHo

For the case of variable dz, proceeding as in the previous section, the coefficients obtained are

cvHo21+eo21+emcv(ζfζo)Ho2(1+eo)2(1+em)1τo,s

providing the monomials

πI=τo,scvζoHo21+eo21+em2πII=ζo1+em

from which it is easy to reach the same monomials (61) as for the case of constant dz, allowing to write now for the characteristic time of settlement (τo,s), from π1 = Ψ[π2]

τo,s=Ho2cv1+eoΨHfHo

whereas the evolution of the average degree of settlement (Ūs) can be expressed a

U¯s=Ψtτo,s,HfHo

3.3 Characterization of the model with both non-constant 1+e and cv

3.3.1 Nondimensionalization of the model with both non-constant 1+e and cv in terms of pressure

In this case, starting from equation (21), which we repeat here for convenience

cvo1+e21+eo2σz21σσz2=σt

and with the dimensionless variables of expression (22), we have

cvo1+e21+eoσfσoHo22σz21σσfσoHo2σz2=σfσoτo,σσt

from which the following coefficients result:

cvo1+em2Ho21+eocvo(1+em)2(σfσo)Ho2(1+eo)σm1τo,σ

Dividing these coefficients by the first, we get the groups

πI=τo,σcvo1+em2Ho21+eoπII=σfσoσm

Adopting for σm and em, again, the values σo and eo, the previous monomials can be simplified to (after the necessary partition of the group πI)

π1=τo,σcvo1+eoHo2π2=σfσoπ3=HfHo

or since HfHo=1Iclog10σfσo1+eo

π1=τo,σcvo1+eoHo2π2=σfσoπ3=Ic1+eo

Now we can write for the characteristic time in terms of pressure (τo,σ), from π1 = Ψ [π2, π3]

τo,σ=Ho2cvo1+eoΨσfσo,Ic1+eo

and for the evolution of the average degree of pressure dissipation, Ūσ

U¯σ=Ψtτo,σ,σfσo,Ic1+eo

For the variant dz, proceeding as in the previous sections, we have the coefficients

cvo1+eoHo2cvo(1+eo)(σfσo)Ho2σm1τo,σ

which, dividing by the first, provide the monomials

πI=τo,σcvo1+eoHo2πII=σfσoσm

that can be simplified to

π1=τo,σcvo1+eoHo2π2=σfσo

being able to write for the characteristic time in terms of pressure (τo,σ), from π1 = Ψ [π2] more comfortably

τo,σ=Ho2cvo1+eoσfσo

and for the evolution of the average degree of pressure dissipation, Ūσ

U¯σ=Ψtτo,σ,σfσo

Thus, in view of the expressions (6971) and (7476), the fact of adopting the more general (and more precise) hypothesis, dz variable, decreases in this case the number of argument dependencies of the problem, which is reduced to only two monomials.

3.3.2 Nondimensionalization of the model with both non-constant 1+e and cv in terms of settlement

In this case, starting from (21) and through expressions (3739), it is immediate to reach

cvo1+e21+eo2ζz2=ζt

and with the dimensionless variables of expression (41), we have

cvo1+e21+eoζfζoHo22ζz2=ζfζoτo,sζt

The coefficients of this equation, once simplified, result in

cvoHo21+em21+eo1τo,s

that, by dividing, provide the monomial

πI=τo,scvo1+em2Ho21+eo

and, proceeding as in the previous sections, we reach

π1=τo,scvo1+eoHo2π2=HfHo

So the expressions for the characteristic time (τo,s) and the average degree of settlement (Ūs) remain

τo,s=Ho2cvo1+eoΨHfHo

U¯s=Ψtτo,s,HfHo

Finally, adding the condition of variable dz in expression (78), the following coefficients are obtained:

cvo1+eoHo21τo,s

These coefficients provide the monomial

π1=τo,scvo1+eoHo2

and the expressions for the characteristic time (τo,s) and the average degree of settlement (Ūs) are

τo,sHo2cvo1+eo

U¯s=Ψtτo,s

Thus, in view of the expressions (8183) and (8587), the fact of adopting the more general (and more precise) hypothesis, dz variable, decreases in this case the number of argument dependencies of the problem, which is reduced to a single monomial.

Table 1 summarizes the expressions that govern each of the models addressed in this section, whereas Table 2 shows the monomials that rule their solution patterns.

Table 1

Governing equations for the different variants of the Davis and Raymond model

PressureSettlement
Davis and Raymondcv2σz21σσz2=σtcv2ζz2=ζt
Davis and Raymond 1+e ≠ constant cv constantcv1+e2σz2+Icσln101+eσσz2=σtcv1+e2ζz2cvζz2=ζt
Davis and Raymond 1+e ≠ constant cv constantcvo1+e21+eo2σz21σσz2=σtcvo1+e21+eo2ζz2=ζt

Table 2

Dimensionless groups that characterize the solutions for the different variants of the Davis and Raymond model

PressureSettlement
Davis and Raymondπ1=τo,σcvHo2π2=σfσoπ1=τo,scvHo2
Davis and Raymond 1+e ≠ constant cv constant dz constant and dz ≠ constantπ1=τo,σcv1+eoHo2π2=σfσoπ3=Ic1+eoπ1=τo,scv1+eoHo2π2=HfHo
Davis and Raymond 1+e ≠ constant cv ≠ constant dz constantπ1=τo,σcvo1+eoHo2π2=σfσoπ3=Ic1+eoπ1=τo,scvo1+eoHo2π2=HfHo
Davis and Raymond 1+e ≠ constant cv ≠ constant dz ≠ constantπ1=τo,σcvo1+eoHo2π2=σfσoπ1=τo,scvo1+eoHo2

4 Verification of results and universal curves

The objective of this section is the verification of the deduced dimensionless groups that govern the consolidation problem of Davis and Raymond and, once tested, the obtention of universal curves based on these groups for the variables of greatest interest to geotechnical engineers: characteristic time and average degree of consolidation.

In order not to make this section too extensive, we exclusively stick to the most general and precise Davis and Raymond model, that is, the one that considers both 1+e and cv non-constant, with the added hypothesis of variable dz. For the rest of the models, of less interest, universal solutions can be found (based on the monomials obtained ) in the works of Davis and Raymond [1] (for the original model of the authors with constant cv and 1+e) and García-Ros [16] (for the extended models with constant cv and non-constant 1+e).

4.1 Verification of results and universal curves for the model with both non-constant cv and 1+e and variable dz

Table 3 shows a series of nine simulations for the most general Davis and Raymond model (non-constant 1+e and cv, with variable dz) in which several parameters or initial values are modified in order to check and verify the dependencies of the different solutions of the models with respect to the monomials of Table 2: π1 and π2 in terms of pressure (hereinafter π1σ and π2σ) and π1 in terms of settlement (henceforth π1s). The objective is to show (and verify) that, independently of the values that the particular parameters of the problem take, the monomials π1σ, π2σ and π1s govern the solution pattern of the problem.

Table 3

Verification of the dimensionless groups for the extended Davis and Raymond model with both non-constant cv and 1+e and variable dz.

CaseKo (m/yr)eoIcσo (N/m2)Ho (m)σf (N/m2)cvo (m2/yr)τo,σ (yr)τo,s (yr)π1σπ2σπ1s
010.021.50.45300001600000.7830.49410.43280.9672.00.847
020.041.50.45150001300000.7830.49410.43280.9672.00.847
030.020.250.1125300001600001.5660.49410.43280.9672.00.847
040.041.50.456000021200003.1330.49410.43280.9672.00.847
050.0310.3250001.5500001.1750.9260.8110.9672.00.847
060.021.50.453000011200000.7830.55010.43281.0774.00.847
070.021.50.453000021200000.7832.20041.73121.0774.00.847
080.021.50.453000012400000.7830.60010.43281.1758.00.847
090.021.50.453000014800000.7830.64440.432826216.00.847

For this purpose, a first reference case is established, on the basis of which we will modify the different parameters or initial values to define the other cases. For all models, the potentially variable physical and geometric characteristics are ko (m/yr), eo, Ic, σo (N/m2), Ho (m) and σf (N/m2). The values of cvo (m2/yr) and π2σ are deduced from them, whereas π1σ and π1s are obtained once we know the values of τo,σ and τo,s (yr), characteristic times corresponding to pressure or settlement, respectively; these times are recorded once the simulation has been performed. As a criterion for the choice of the value of the characteristic time, it has been considered to take this time value for which 90% of the definitive settlement (τo,s), or 90% of an average pressure dissipation (τo,σ), has been reached.

In Table 3, cases 01–05 represent different consolidation scenarios, but the monomial remains π2σ = σfσo constant. In view of the results, it is observed how each scenario can present different values for both the initial consolidation coefficient (cvo) and the characteristic time (τo,σ); however while the value of the monomial π2σ (σfσo equals to 2) remains unchanged, the dimensionless expression of the characteristic time π1σ will also not vary. Scenarios 06 and 07 show how by varying the loading ratio (σfσo now equal to 4), the value of the monomial π1σ changes with respect to the first 5 cases, having the same value for cases 06 and 07 despite having very different characteristic times (τo,σ). Cases 08 and 09 complete the verification, showing how the value of π1σ changes (growing) as σfσo increases.

Regarding the dimensionless form π1s of the characteristic time of settlement (τo,s), it is verified, as it has been deduced from the nondimensionalization process, that its value does not depend on any group of parameters of the problem, remaining constant at all times (π1s equals to 0.847). In this way, from expression (85), where π1 is of the order of magnitude of the unit, once the problem has been numerically solved and the veracity of the proposed dimensionless groups verified, the universal solution for the characteristic time in terms of settlement is reached:

τo,s=0.847Ho2cvo1+eo.

Once the expression for the characteristic time in terms of settlement (τo,s) has been obtained, we can represent the universal curve for the average degree of settlement (Ūs) as a function of the dimensionless time t/τo,s (Figure 1). Considering the reader that, to obtain this curve, only a single simulation (of any consolidation scenario) has been necessary.

Fig. 1
Fig. 1

Average degree of settlement evolution for the extended Davis and Raymond model with both non-constant cv and 1+e and variable dz.

Citation: Applied Mathematics and Nonlinear Sciences 4, 1; 10.2478/AMNS.2019.1.00008

Regarding the problem in terms of pressure, as can be deduced from the expressions (7476), the dimensionless form of the characteristic time is a function of σfσo. On the other hand, in view of expressions (75) and (86), the characteristic time in terms of both pressure and settlement is proportional to the factor Ho2cvo1+eo. In addition, in practice, although what interests the geotechnical engineer about the consolidation process is the evolution of settlements, in many cases, it is easier to track in situ the evolution of interstitial pressure. For this reason, it seems interesting to know the relation between the two characteristic times for the different loading ratios σfσo (Figure 2).

Fig. 2
Fig. 2

Ratio τo,s/τo,σ as a function of σf/σo for the extended model of Davis and Raymond with both non-constant cv and 1+e and variable dz.

Citation: Applied Mathematics and Nonlinear Sciences 4, 1; 10.2478/AMNS.2019.1.00008

Finally, once the value for the characteristic time in terms of pressure (τo,σ′) is known, we can represent the universal curve for the average degree of pressure dissipation (Ūσ) as a function of the dimensionless time t/τo,σ and the monomial σfσo (Figure 3).

Fig. 3
Fig. 3

Evolution of average degree of pressure dissipation for the extended Davis and Raymond model with both non-constant cv and 1+e and variable dz.

Citation: Applied Mathematics and Nonlinear Sciences 4, 1; 10.2478/AMNS.2019.1.00008

5 Final comments and conclusions

The search for the dimensionless groups that govern the non-linear consolidation problem based on the Davis and Raymond original and extended models, by means of the nondimensionalization technique for governing equations, has led to simple solutions despite the enormous set of physical and geometrical parameters, in addition to those referred to the boundary conditions, involved in the problem.

By introducing as reference different characteristic times of consolidation (parameters of great interest in ground engineering) in order to nondimensionalize the real time, the groups have been deduced by means of the dimensional coefficients derived from the mathematical treatment of the governing equations. In this way, these same characteristic times can be expressed as a function of the emerging dimensionless groups.

The comparison between the most complex model (non-constant 1+e and cv and variable dz) and the original (whose groups can be deduced from the analytical expressions reported by Davis and Raymond) has given rise, curiously, to the same number of monomials, one for settlement and two for pressure (despite having introduced two new parameters in the extended model: initial void ratio and initial consolidation coefficient), which can be considered a contribution of great interest given the higher precision of the extended model.

It is worth mentioning that in the less general extended models, case i) non-constant 1+e and constant cv and dz, and case ii) non-constant 1+e and cv and constant dz, one more group has emerged in both settlement and pressure, which is undoubtedly due to the inconsistencies resulting in, the first case, assuming 1+e non-constant but keeping constant cv and, the second case, assuming both 1+e and cv non-constant but dz constant.

The application of the pi theorem has allowed to represent the results as a function of the dimensionless characteristic time, for both the average degree of settlement and the average degree of pressure dissipation, in the second case by means of an abacus that used the loading ratio as a parameter. It has been observed that the characteristic time of settlement is always lower than the characteristic time in terms of pressure and that this decreases depending on the loading ratio. This is, undoubtedly, due to the non-linear nature of the e∼σ′ constitutive relation, according to which as the effective soil stress is more, an increase of this leads to a diminution of the void ratio every time smaller.

Communicated by Juan Luis García Guirao
Nomenclature(m2)

A cross section area

cv

consolidation coefficient (m2/s or m2/yr)

cvo

initial consolidation coefficient (m2/s or m2/yr)

dz

volume element size (m)

dzo

initial volume element size (m)

e

void ratio (dimensionless)

ef

final void ratio (dimensionless)

em

mean void ratio (dimensionless)

eo

initial void ratio (dimensionless)

H

soil thickness (m)

Hf

final soil thickness (m)

Hm

mean soil thickness (m)

Ho

initial soil thickness (m)

Ic

compression index (dimensionless)

k

hydraulic conductivity (m/s)

ko

initial hydraulic conductivity (m/s)

mv

volumetric compressibility coefficient (m2/N)

v

void volume temporal change (m3/s)

w

water volume temporal change (m3/s)

t

time (s or yr)

t′

dimensionless time (dimensionless)

u

excess pore or interstitial pressure (N/m2)

average degree of consolidation (dimensionless)

s

average degree of settlement (dimensionless)

σ

average degree of pressure dissipation (dimensionless)

Vv

void volume (m3)

Vw

water volume (m3)

vz

water velocity in the vertical spatial direction z (m/s)

w

auxiliary variable of Davis and Raymond (dimensionless)

z

spatial direction z (m)

z′

dimensionless spatial direction z (dimensionless)

Ψ

unknown arbitrary function

Δz

volume element thickness (m)

Δzo

initial volume element thickness (m)

γw

water-specific weight (N/m3)

σ

total pressure or stress (N/m2)

σ

effective pressure or stress (N/m2)

(σ′)′

dimensionless effective pressure or stress (dimensionless)

σf

final effective pressure or stress (N/m2)

σm

mean effective pressure or stress (N/m2)

σo

initial effective pressure or stress (N/m2)

τo

characteristic time (s or yr)

τo,s

characteristic time of settlement (s or yr)

τo,σ

characteristic time of pressure dissipation (s or yr)

ζ

differential local void ratio (dimensionless)

ζf

final differential local void ratio (dimensionless)

ζo

initial differential local void ratio (dimensionless)

References

  • [1]

    Davis E. H. & Raymond G. P. (1965). A non-linear theory of consolidation. Geotechnique 15(2) 161-173.

    • Crossref
    • Export Citation
  • [2]

    García-Ros G. Alhama I. & Morales J. L. (2019). Numerical simulation of nonlinear consolidation problems by models based on the network method. Applied Mathematical Modelling 69 604-620. https://doi.org/10.1016/j.apm.2019.01.003

    • Crossref
    • Export Citation
  • [3]

    Juárez-Badillo E. (1983). General Consolidation Theory for Clays. Soil Mechanics Series (No. 8). Report.

  • [4]

    Juárez-Badillo E. & Chen B. (1983). Consolidation curves for clays. Journal of Geotechnical Engineering 109(10) 1303-1312.

    • Crossref
    • Export Citation
  • [5]

    García-Ros G. Alhama I. Cánovas M. & Alhama F. (2018). Derivation of Universal Curves for Nonlinear Soil Consolidation with Potential Constitutive Dependences. Mathematical Problems in Engineering 2018. Article ID 5837592 15 pages. https://doi.org/10.1155/2018/5837592

  • [6]

    Cornetti P. & Battaglio M. (1994). Nonlinear consolidation of soil modeling and solution techniques. Mathematical and computer modelling 20(7) 1-12.

    • Crossref
    • Export Citation
  • [7]

    Manteca I. A. García-Ros G. & López F. A. (2018). Universal solution for the characteristic time and the degree of settlement in nonlinear soil consolidation scenarios. A deduction based on nondimensionalization. Communications in Nonlinear Science and Numerical Simulation 57 186-201. https://doi.org/10.1016/j.cnsns.2017.09.007

    • Crossref
    • Export Citation
  • [8]

    Seco-Nicolás M. Alarcón M. & Alhama F. (2018). Thermal behavior of fluid within pipes based on discriminated dimensional analysis. An improved approach to universal curves. Applied Thermal Engineering 131 54-69. https://doi.org/10.1016/j.applthermaleng.2017.11.091

  • [9]

    Conesa M. Pérez J. S. Alhama I. & Alhama F. (2016). On the nondimensionalization of coupled nonlinear ordinary differential equations. Nonlinear Dynamics 84(1) 91-105. https://doi.org/10.1007/s11071-015-2233-8

    • Crossref
    • Export Citation
  • [10]

    Gibbings J. C. (1980). On dimensional analysis. Journal of Physics A: Mathematical and General 13(1) 75.

    • Crossref
    • Export Citation
  • [11]

    Buckingham E. (1914). On physically similar systems; illustrations of the use of dimensional equations. Physical review 4(4) 345.

    • Crossref
    • Export Citation
  • [12]

    Jordán J. Z. (2007). Network simulation method applied to radiation and viscous dissipation effects on MHD unsteady free convection over vertical porous plate. Applied Mathematical Modelling 31(9) 2019-2033. https://doi.org/10.1016/j.apm.2006.08.004

    • Crossref
    • Export Citation
  • [13]

    González-Fernández C. F. (2002). Applications of the network simulation method to transport processes.

  • [14]

    Wood D. M. (2009). Soil mechanics: a one-dimensional introduction. Cambridge University Press.

  • [15]

    Taylor D. W. (1942). Research on consolidation of clays (Vol. 82). Massachusetts Institute of Technology.

  • [16]

    García-Ros G. (2016). Caracterización adimensional y simulación numérica de procesos lineales y no lineales de consolidación de suelos. Doctoral thesis.

Footnotes

the assumption of variable dz adds dz=dzo1+e1+eo
the assumption of variable dz adds dz=dzo1+e1+eo

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [1]

    Davis E. H. & Raymond G. P. (1965). A non-linear theory of consolidation. Geotechnique 15(2) 161-173.

    • Crossref
    • Export Citation
  • [2]

    García-Ros G. Alhama I. & Morales J. L. (2019). Numerical simulation of nonlinear consolidation problems by models based on the network method. Applied Mathematical Modelling 69 604-620. https://doi.org/10.1016/j.apm.2019.01.003

    • Crossref
    • Export Citation
  • [3]

    Juárez-Badillo E. (1983). General Consolidation Theory for Clays. Soil Mechanics Series (No. 8). Report.

  • [4]

    Juárez-Badillo E. & Chen B. (1983). Consolidation curves for clays. Journal of Geotechnical Engineering 109(10) 1303-1312.

    • Crossref
    • Export Citation
  • [5]

    García-Ros G. Alhama I. Cánovas M. & Alhama F. (2018). Derivation of Universal Curves for Nonlinear Soil Consolidation with Potential Constitutive Dependences. Mathematical Problems in Engineering 2018. Article ID 5837592 15 pages. https://doi.org/10.1155/2018/5837592

  • [6]

    Cornetti P. & Battaglio M. (1994). Nonlinear consolidation of soil modeling and solution techniques. Mathematical and computer modelling 20(7) 1-12.

    • Crossref
    • Export Citation
  • [7]

    Manteca I. A. García-Ros G. & López F. A. (2018). Universal solution for the characteristic time and the degree of settlement in nonlinear soil consolidation scenarios. A deduction based on nondimensionalization. Communications in Nonlinear Science and Numerical Simulation 57 186-201. https://doi.org/10.1016/j.cnsns.2017.09.007

    • Crossref
    • Export Citation
  • [8]

    Seco-Nicolás M. Alarcón M. & Alhama F. (2018). Thermal behavior of fluid within pipes based on discriminated dimensional analysis. An improved approach to universal curves. Applied Thermal Engineering 131 54-69. https://doi.org/10.1016/j.applthermaleng.2017.11.091

  • [9]

    Conesa M. Pérez J. S. Alhama I. & Alhama F. (2016). On the nondimensionalization of coupled nonlinear ordinary differential equations. Nonlinear Dynamics 84(1) 91-105. https://doi.org/10.1007/s11071-015-2233-8

    • Crossref
    • Export Citation
  • [10]

    Gibbings J. C. (1980). On dimensional analysis. Journal of Physics A: Mathematical and General 13(1) 75.

    • Crossref
    • Export Citation
  • [11]

    Buckingham E. (1914). On physically similar systems; illustrations of the use of dimensional equations. Physical review 4(4) 345.

    • Crossref
    • Export Citation
  • [12]

    Jordán J. Z. (2007). Network simulation method applied to radiation and viscous dissipation effects on MHD unsteady free convection over vertical porous plate. Applied Mathematical Modelling 31(9) 2019-2033. https://doi.org/10.1016/j.apm.2006.08.004

    • Crossref
    • Export Citation
  • [13]

    González-Fernández C. F. (2002). Applications of the network simulation method to transport processes.

  • [14]

    Wood D. M. (2009). Soil mechanics: a one-dimensional introduction. Cambridge University Press.

  • [15]

    Taylor D. W. (1942). Research on consolidation of clays (Vol. 82). Massachusetts Institute of Technology.

  • [16]

    García-Ros G. (2016). Caracterización adimensional y simulación numérica de procesos lineales y no lineales de consolidación de suelos. Doctoral thesis.

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    Average degree of settlement evolution for the extended Davis and Raymond model with both non-constant cv and 1+e and variable dz.

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    Ratio τo,s/τo,σ as a function of σf/σo for the extended model of Davis and Raymond with both non-constant cv and 1+e and variable dz.

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    Evolution of average degree of pressure dissipation for the extended Davis and Raymond model with both non-constant cv and 1+e and variable dz.

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