1 Introduction

The theory of linear consolidation of soils has been broadly studied in recent decades and is well established in many books [1, 2]. All of them present a complete study of 1-D rectangular and radial consolidation with universal curves of the average degree of consolidation, U(t), from which the settlements due to surface loads are derived. The solution of more complex scenarios has been carried out in recent years by analytical methods. These scenarios include multilayered soils, assuming drain resistance and smearing (Battaglio et al. [3]), multi-stage loading (Lu el at. [4]), vacuum assisted preloading (Indraratna el at. [5]) and other non-linear problems [6, 7, 8]. Numerical solutions have also been provided by other authors for scenarios with drains [9, 10].

As for the consolidation scenarios in 2-D domains, both in rectangular (use of infinite strip drains, of theoretical application) and cylindrical coordinates (use of vertical drains with radial geometry), the complexity of the problem, even in linear scenarios where the soil properties are assumed constant, has led to the emergence of a variety of solutions. On the one hand, the separate analysis of the vertical and horizontal (or radial) components of the process, with solutions in the form of series, and their subsequent superposition (Carrillo [11]) is the method generally taught in civil engineering schools [12] and that is also frequently used in the professional field. On the other hand, it is also possible to obtain closed-form solutions for the two-dimensional problem [13, 14], or even to reach analytical or semi-analytical expressions from spectral or approximate methods [15, 16]. All these formulations are presented as a very good approach for the solution of consolidation problems, but their handling is quite complicated since they require the use of numerous parameters, which usually leads to cumbersome calculations.

This paper focuses on the search for the average degree of consolidation in its most universal form, as a function of the lowest number of parameters of the problem, in 2-D anisotropic rectangular and cylindrical domains. For this purpose we i) derive the dimensionless groups that rule this problem by the discriminated nondimensionalization of the governing equations, using the most recent concepts involved in this technique, and ii) numerically verify the dependence of the average degree of consolidation on the above groups.

The concepts involved in the application of discriminated nondimensionalization will be explained in detail in the next section. Before this, let us make a historical revision of the time factor concept (by means of which most authors define the dimensionless time), its introduction in the context of consolidation and the connection between the reference used to make the time dimensionless in the time factor and the characteristic time of the consolidation process, both in 1-D and 2-D scenarios.

Regardless of the geometric domain, soil consolidation is an asymptotic process and, as such, there is no characteristic or particular value of time that can be assigned to its duration. However, for each percentage of fall in the excess pore pressure at a given point of the domain (or its average value in the whole domain, which defines the average degree of consolidation) there is only one time value. Most authors make no reference to a characteristic time to be used in the definition of the dimensionless time of the process, but solve the governing equation in finite 1-D domains analytically and give the name of time factor to the term c_vt/H² that emerges as argument in the solution. In this way, Terzaghi el at. [17] say “c_vt/H²is a pure number called the time factor”, while Berry and Reid [18] write “t′ is the dimensionless vertical time factor defined by t′ = C_vt/H²” and Atkinson [1] makes no reference to the subject. Later, Azizi [19] says “… c_vt/H²representing a time factor. … Because the (dimensionless) time factor is time dependent, the solution yields the precise dissipation with time of excess water pressure u at any depth z.” Finally, Juárez and Rico [2] write that the factors z/H and c_vt/H² that appear in the solution of soil consolidation are dimensionless quantities, calling the second as the time factor.

Somewhat more explicit as regards the subject are Sagaseta el at. [20] and Muir Wood [21]. After deriving the consolidation equation, the first authors introduce the dimensionless variables z′ = z/H, t′ = c_vt/H² and the consolidation degree (as they call) u′ = (u_o – u)/u_o. With these new variables they finally obtain a universal (normalized) equation. The second authors, in relation to the consolidation equation, affirm: “We can make this equation more universal by replacing our variables u, z and t by non-dimensional variables u′ = u/u_o (u_o being a reference pore pressure, typically the initial uniform excess pore pressure), z′ = z/H and t′ = c_vt/H², where t′ emerges as a dimensionless time factor.” In other words, the objective of these authors is to obtain a dimensionless governing equation following a correct procedure of nondimensionalization.

In contrast with the above references, Scott [22] is the only author to follow a formal nondimensionalization procedure for 1-D rectangular scenarios, similar to that followed in this paper. After comparing heat conduction and soil consolidation equations, establishing the similarity between the parameters that rule these processes (consolidation coefficient and thermal diffusivity), he says: “Because of the linearity of the equation, it is appropriate to normalize the various parameters to make the equation nondimensional. In this way, the solution obtained in terms of dimensionless parameters is in a more suitable form for general applications. The normalization in one-dimensional terms, for example, is accomplished by relating the variables to the characteristic constants of the system as follows: An arbitrary constant value of pore pressure u_ois chosen and a dimensionless pressure variable (u′) is defined such that u′ = u/u_o. Next, a characteristic length H in the system is selected to give a dimensionless length variable, z′ = z/H. Finally, by choosing an arbitrary time constant t_o, we are able to obtain a dimensionless time variable, t′ = t/t_o. Substituting these dimensionless variables in the equation of consolidation gives1to∂u′∂t′=CvH2∂2u′∂z′2.It is apparent that one of the characteristic constants can be selected for convenience to make t_o = H²/c_vand therefore writes the former equation as∂u′∂t′=∂2u′∂z′2.t′,usually called time factor is redefined to be t′ = c_vt/H².” In this way, Scott first looks for three references (u_o, H and t_o, the last an unknown) to define the dimensionless variables u′, z′ and t′. Then, he substitutes these variables in the governing equation to derive its dimensionless form. Finally, ‘for convenience′ (what is to make unity the only dimensionless parameter that emerges in the equation), the unknown reference t_o is deduced.

The procedure applied in this paper to determine the consolidation time (t_o) is apparently the same as the one followed by Scott. However, the assignment of an order of magnitude unity to the dimensionless normalized variables (and their changes) provides a clear physical meaning for the reference time. Moreover, making use of the properties of homogeneous functions, the proposed procedure applied to 2-D scenarios provides more precise solutions for the unknown to than those that are possible to find with classical nondimensionalization.

This work takes the following form: After explaining the discriminated nondimensionalization procedure to determine the discriminated groups that derive from it, the technique is applied to the soil consolidation process in 2-D rectangular and cylindrical scenarios, looking for the dependence of the average degree of consolidation on these groups. Numerical simulations show this dependence by means of suitable abacuses. Then, following the steps involved in discriminated nondimensionalization, the precise concept of time of consolidation is introduced, obtaining its dependence on the parameters of the problem, based on the mathematical theory of homogeneous functions. Therefore, using this new concept and the dependence of the average degree of consolidation on the time of consolidation and other parameters, numerical simulations will allow the representation of this unknown through universal curves. Finally, a case study is addressed in order to compare the curves given in the manuscript to classical methods. In the last section, the main contributions of this work are summarized.

2 Discriminated nondimensionalization procedure

The discriminated nondimensionalization technique combines the scale analysis of Bejan [23] with the most general concepts of spatial discrimination introduced by Alhama and Madrid [24]. The first turns the dimensional governing equations into their dimensionless form from which the dimensionless groups are deduced, while the second, which applies both to governing equations and boundary conditions (turning these also into their dimensionless form), provides the smallest number of independent dimensionless groups that rule the problem. The steps involved in the application of this technique are:

1. Choose the references for both dependent and independent variables in order to define the dimensionless forms of them in such a way that they extend to the interval [0,1]. Discrimination forces the variables of vector character (coordinates, velocities...) to have, in general, a different reference for each spatial component. Furthermore, references are generally explicit in the statement of the problem but they may not be. In the last case, they are introduced as unknowns whose order of magnitude is found once the nondimensionalization procedure has finished.

2. Derive the dimensionless governing equations by replacing the old variables with the new dimensionless ones. Each term of these new equations is separated into two factors. The first, dimensionless, is formed by the normalized variables and their changes (derivatives), while the second, non-dimensionless, is a grouping of geometric and physical parameters.

3. Assuming that the first factor of ii) is of order of magnitude unity – a reasonable hypothesis in linear problems –, the other factors (also called coefficients) must be of the same order of magnitude (not necessarily unity) so providing a correct balance in the equation.

4. The independent ratios formed with these coefficients are the discriminated dimensionless groups that are sought. There are as many groups as coefficients minus one; they must be dimensionless and of order of magnitude unity.

5. Based on the theory of homogeneous functions (Buckingham pi theorem, [25]), the solution of any unknown of the problem – correctly expressed in its dimensionless form – can be written in the form ψ(π₁, π₂ …, π_n), being π_i(1≤i≤n) the dimensionless groups derived from discriminated nondimensionalization and ψ an arbitrary function. This allows to determine the orders of magnitude of both the average degree of consolidation and the characteristic time.

This proposed method has been applied successfully in different engineering problems, such as fluid mechanics (Madrid and Alhama [26]), fluid flow and solute transport (Manteca el at. [27]), geothermics (Cánovas el at. [28, 29]) and mechanical engineering (Pérez el at. [30]), among others.

3 The governing equations of soil consolidation

Two physical models are considered: i) a 2-D rectangular anisotropic domain that drains towards the soil surface and towards the left boundary, due to the existence of a drain, Figure 1, and ii) a 2-D cylindrical anisotropic domain which drains towards the upper and inner boundary, also due to a drainage condition, Figure 1. The assumed hypotheses are: i) the soil is laterally confined and the drainage occurs in vertical and horizontal directions, ii) the excess pore pressure is generated by an external load constant in time, iii) Darcy’s law is applicable to the movement of the water through the soil, iv) the soil is fully saturated and the pore water and the soil particles behave as an incompressible medium with respect to the soil skeleton, v) the weight of the soil, grains and water can be ignored, vi) the deformation of the soil is free, without redistribution of stresses vii) the soil skeleton does not creep under a constant effective stress, and viii) the applied load increment produces only small strains, so that the total volume of the soil can be assumed to be constant. Under these hypothesis, the soil consolidation processes is ruled by the equations and boundary conditions (1-4) for rectangular coordinates and equations and boundary conditions (5-8) for cylindrical coordinates:

Figure 1

2-D physical models for rectangular (top) and radial (bottom) soil consolidation scenarios.

Rectangular coordinates:

Radial coordinates:

These equations, and their simplifications for 1-D cases, can be solved analytically. However, their solutions [2, 17], obtained in the form of series of functions, seem to be quite complex to manage.

4 The search for dimensionless parameters

4.1 Rectangular domains

The references for the variables x, z and u are explicit in the statement of the problem; these are L, H and u_o, respectively. As regards the time, the common references are H²/c_v,z or L²/c_v,x (in rectangular coordinates) and H²/c_v,z or rout2/c_v,r (in cylindrical). So, substituting

x′=xL,Z′=zH,u′=uuo,t′=tcv,zH2,(9)

in equation (1) and simplifying, yields

cv,zH2∂u′∂t′=cv,zH2∂2u′∂z′2+cv,xL2∂2u′∂x′2(10)

This dimensionless governing equation shows that its solution does not depend on the initial excess pore pressure u_o. Assuming that the changes in u′ at the whole domain are of order of magnitude unity, the coefficients of the terms of the equation – whose unit of measure is the inverse of time – are of the same order of magnitude. The only independent ratio formed from these coefficients is a dimensionless group that rules the solution of the problem:

π1=cv,zL2cv,xH2(11)

The dimensionless form of the unknowns of the problem will always depend on the group π₁, as well as on the group π2=tcv,zH2 if the unknowns are time dependent, and on x′ and z′ if they are a function of a given location. For example, the average degree of settlement (U) will depend on π₁ and π₂.

U¯(t)=ψ{tcv,zH2,cv,zL2cv,xH2}(12)

while the local degree of settlement (U) will depend on π_l, π₂, x′ and z′

U(x,z,t)=ψ{tcv,zH2,xL,zH,cv,zL2cv,xH2}(13)

By means of expression (12), it is possible to represent the average degree of settlement as an abacus in the form U (t) = ψ(c_v,_zt/H²), using π_l as parameter. This dependence has been solved numerically and the results are presented in Figure 2a. Figure 2b is a detail of this dependence for a better reading. The use of this abacus is immediate: starting from the parameters of the problem (L, H, c_v,x, c_v,z) and the time t for which U (t) is sought, dimensionless groups π_l and π₂ are determined; entering with these groups in the abacus, U (t) can be read.

Figure 2a

Average degree of settlement U (t) as a function of π₁ and π₂. π₁ = 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.8, 1, 1.25, 1.5, 2, 3, 5, 8 and 10.

Figure 2b

Average degree of settlement U (t) as a function of π_l and π₂. π_l = 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.8, and 1.

As shown, there are significant deviations in the average degree of settlement for a given time and different scenarios (or values of π_l). Note that when π_l > 1, which means that (H²/c_v,z) < (L²/c_v,x), the reference for dimensionless time is smaller than the characteristic time for excess pore pressure to be released by draining the water horizontally. This means that small values of π₂ do not allow pressure to be released rapidly by vertical or horizontal drainage, which results in small values of the average degree of consolidation. The opposite occurs when π_l < 1 or (H²/c_v,z) > (L²/c_v,x); in this case, the small values of π₂ do not allow pressure to be released by vertical drainage but do so by horizontal drainage, giving higher values of the average degree of consolidation.

The alternative choice of the reference L²/c_v,x to make time dimensionless (making π₂ = c_v,xt/L²) provides an abacus quite similar to that of Figures 2a and 2b, but substituting the parameter of the curves π_l = 0.1 by π_l = 10, 0.2 by 5, 0.5 by 2 and so on.

The value π_l = 10 may be considered as a limit curve that provides a good approximation of the 1-D vertical consolidation, for which c_v,x = 0. Therefore, π_l (whose theoretical value tends to infinity in this case) does not emerge as a dimensionless group in the nondimensionalization process of the governing equation.

4.2 Cylindrical domains

In the general anisotropic case, the problem is defined by the geometrical parameters r_in, r_outand H, and the consolidation coefficients c_v,r and c_v,z. From this scenario, a direct dimensionless group based on the boundary conditions or the domain geometry can be written: π₃ = r_out/r_in.

As in the rectangular case, references for r, z and u are also explicit: r_out, H and u_o, respectively, while for time the reference H²/c_v,_z is assumed. Therefore, the dimensionless variables are

r′=rroutZ′=zHu′=uuot′=tcv,zH2(14)

These, by introducing them into equation (5), give rise to the dimensionless governing equation whose solution, as in the rectangular case, does not depend on the initial excess pore pressure

cv,zH2∂u′∂t′=cv,zH2∂2u′∂z′2+cv,rrout2∂2u′∂r′2+1r′∂2u′∂r′(15)

The coefficients of this equation, cv,zH2andcv,rrout2, provide one dimensionless group

π1=cv,zrout2cv,rH2(16)

The solution for the average degree of consolidation for a given time will depend on the dimensionless form of t, π2=tcv,zH2, and the groups π₁ and π₃. This is,

U¯(t)=ψ{tcv,zH2,cv,zrout2cv,rH2,routrin}(17)

The local degree of consolidation depends, in addition, on the dimensionless coordinates

U(r,z,t)=(18)

ψ{rrout,zH,tcv,zH2cv,zrout2cv,rH2,routrin}

The dependence of the average degree of consolidation may now be represented by a set of abacuses, each of them for a given value of the group π₃. As in the rectangular case, within each abacus U (t) = ψ(c_v,zt/H²) = ψ(π₂) and each curve is related to a value of π₁. Figures 3a to 3c show four abacuses, each one for the values π₃ = 5, 10 and 20, respectively, covering a wide range of real cases.

Figure 3a

Average degree of settlement U (t) for π₃ = 5. π₁ = 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.8, 1, 1.25, 1.5, 2, 3, 5, 8 and 10.

Figure 3b

Average degree of settlement U (t) for π₃ = 10. π₁ = 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.8, 1, 1.25, 1.5, 2, 3, 5, 8 and 10.

Figure 3c

Average degree of settlement U (t) for π₃ = 20. π₁ = 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.8, 1, 1.25, 1.5, 2, 3, 5, 8 and 10.

Starting from the parameters of the problem (r_in, r_out, H, c_v,r, c_v,z) and the time for which U (t) is sought, the values of the groups π₁, π₂ and π₃ are determined. With π₁ and π₃, the corresponding abacus and curve – interpolating if required – are chosen; finally, introducing π₂ in the curve, U (t) can be obtained. Similar comments to those made for rectangular domains may apply to these abacuses, regarding the aspect of the curves and the evolution of the average degree of consolidation. However, the slopes of the curves are reduced as π₃ increases, providing a lower degree of consolidation for the same time and the same value of π₁. Note that an increase in π₃, while the rest of the parameters of the problem remain constant, leads to a smaller drainage surface inside the cylinder and a lower degree of consolidation for the same time.

The above results are simplified for 1-D radial coordinates, which supposes that drainage only occurs at the boundary r = r_in. Group π₁ disappears and the average degree of consolidation only depends on π₂ and π₃.

U¯(t)=ψ{tcv,zH2,routrin}(19)

5 Characteristic time of consolidation, t_o

5.1 Rectangular domains

Following Alhama and Madrid [24], it is possible to assume an unknown reference – not explicit in the statement of the problem -, instead of the ratios H²/c_v,z or L²/c_v,x formed by parameters of the problem, to make time dimensionless in the governing equations. The order of magnitude of this reference, once it has been well defined and introduced in the process of nondimensionalization, can be obtained as a function of the rest of the dimensionless groups.

The reference time, called characteristic time (t_o), is of the order of the time required to reduce the initial value of the excess pore pressure (u_o) to a certain (small) percentage, at a given point of the domain (far from the drainage boundaries). In this way, t_o is associated to the time for which the average degree of consolidation is sufficiently high. As a consequence, the range of values of t′ = t/t_o can be approached to [0,1], as occurs with the other dimensionless variables.

By substituting x′, z′, u′and t′ in (1), the resulting dimensionless governing equation is

1to∂u′∂t′=cv,zH2∂2u′∂z′2+cv,xL2∂2u′∂x′2(20)

The two independent ratios formed from the coefficients of this equation are the dimensionless groups sought:

π1=cv,zL2cv,xH2π2,to=tocv,zH2(21)

These groups have a clear physical meaning, as occurs with all those derived by discrimination. The first group, written in the form π1=L2cv,x/H2cv,z, is the ratio between two characteristic times: i) the time that takes the process to reach a significant reduction in the excess pore pressure along the horizontal axis, and ii) the time needed to reach a significant reduction in the excess pore pressure along the vertical axis. The second group, π2=to/H2cv,z, compares the characteristic time needed to reach a significant reduction in the excess pore pressure in the whole domain with the reference ii). Note that this last group could have been presented in the form to/L2cv,x, comparing the characteristic time of the whole domain with the reference i). As we will see later, both ways reach the same universal curves for the average degree of consolidation.

From the resulting groups (21), the value of t_o is deduced from the solution π₂,t_o= F(π₁),

to=H2cv,zψcv,zL2cv,xH2(22)

where F is an arbitrary unknown function of the argument π₁. This solution depends on the location where the decrease in excess pore pressure is measured, as well as on the percentage of falling selected. So, we have chosen the coordinates of the domain x′ = 0.9 and z′ = 0.1 and fixed the percentage of falling at different values: 50, 80, 90 and 95%. Numerical simulations were carried out for values of π₁ within the interval [0.01,15], in order to cover the totality of practical cases, changing c_v,x or L since these parameters only appear in π₁. For each simulation, to is read and π₂ deduced from π₂,t_otocv,zH2, providing in this way one point (π₁, π₂,t_o) of the dependence π₂,t_o = F (π₁), shown in Figure 4. The curves show a notable change in to for π₁ values below 4, but a slow increase for higher values. This behaviour is coherent, since if we maintain (H²/t_oC_v,z) constant, being (L²/t_oC_v,x) >> (H²/t_oC_v,z) – which means that water drainage essentially takes place vertically –, an increase in (L²/t_oC_v,x) hardly influences the consolidation time to; however, if (H²/t_oC_v,z) is maintained constant and (L²/t_oC_v,x) >> (H²/t_oC_v,z) – which means that water drainage essentially occurs in a horizontal direction –, an increase in (L²/t_oc_v,x) produces a significant rise in the consolidation time t_o.

Figure 4

Universal curves π₂,t_o = ψ(π₁) for 2-D rectangular anisotropic domains. Reference location: x′ = 0.9, z′ = 0.1. Percentage of fall in the excess pore pressure: 50, 80, 90 and 95%.

The way in which these curves are used is as follows: Starting from the parameters c_v,x, c_v,z, L and H, and a given percentage of excess pore pressure fall, π₁ is evaluated from π1=cv,zL2cv,xH2 and π_2,to from the dependence π_2,to = ψ(π₁), Figure 4. Finally t_o is determined using equation (21).

The reduction of these results to the 1-D case is immediate. Then, equation (10) only contains two addends, whose coefficients give rise to one dimensionless group, π2,to=tocv,zH2 – of order of magnitude unity–, which pro vides the solution for t_o.

to∼H2cv,z(23)

Again, by giving the same physical meaning for t_o, one numerical simulation for each percentage of fall in excess pore pressure provides the constant that changes the above expression to an equality, to=CoH2cv,z.Table 1 shows the values of this constant for different fall percentages, at the measurement point z′ = 0.1.

Table 1

Vertical 1-D consolidation. Values of the constant C_o that relate t_o and H²/_cv,z for different falls in excess pore pressure. Measurement point: z′ = 0.1.

	u = 0.1 uo (90% consolidation)	u = 0.2 uo (80% consolidation)	u = 0.3 uo (70% consolidation)	u = 0.5 uo (50% consolidation)
Co	1.026	0.7452	0.5809	0.3737

5.2 Cylindrical domains

Again, assuming t_o as the time required to dissipate the initial pore pressure to a negligible value in a location far from the drainage boundaries, the dimensionless governing equation is written in the form

1to∂u′∂t′=cv,zH2∂2u′∂z′2+cv,rrout2∂2u′∂r′2+1r′∂u′∂r′(24)

where t′ = t/t_o. In addition to the aforementioned π₃, the dimensionless groups that can be formed from this equation are

π1=cv,zrout2cv,rH2π2,to=tocv,zH2(25)

and the solution of t_o, obtained from π₂,t_o – ψ (π₁, π₃), results

to=H2cv,zψcv,zrout2cv,rH2,routrin(26)

To represent this dependence it is necessary to fix a measurement point and a percentage of fall for the excess pore pressure. Numerical simulations were carried out t_o read to for a set of values of π₁ within the interval [0.01,15], π₃ = 5, 10, 20 and 50, and a 90% excess pore pressure fall at z′=0.1, r′=0.9. Figure 5 shows the abacus of this dependence using the group π₃ as parameter. The curves reflect the notable influence of π₁ and π₃ on t_o, but for π₁ >> 1, which means that vertical consolidation prevails against horizontal, t_o only depends on H and c_v,z, doing constant (≈ 1) the group π₂,t_o whatever the value of π₃. As in rectangular domains, the use of these curves starts from the geometrical and physical parameters of the problem: c_v,r, c_v,z, H, r_in and r_out. From these, π₁ and π₃ are determined and, by using Figure 5, the corresponding value of π₂,t_ocan be read or interpolated. Finally, t_o is determined from equation (25).

Figure 5

Universal curves π₂,t_o = ψ(π₁, π₃) for 2-D cylindrical anisotropic domains. Reference location: z′ = 0.1, r′ = 0.9. π₃= 5, 10, 20 and 50. Percentage of fall in the excess pore pressure: 90%.

To obtain the 1-D radial coordinates dependences, which means deleting the vertical consolidation, the characteristic time simplifies to the expression π₂,t_o = ψ(π₃), where π2,to=tocv,rrout2, or

to=rout2cv,rψroutrin(27)

Figure 6 shows this dependence for a 90% fall measured at r′ = 0.9.

Figure 6

Radial 1-D domains. Universal curves π₂,t_o = ψ(π₃). Reference location: r′ =0.9, percentage of fall in the excess pore pressure: 90%.

6 Universal curves based on the characteristic consolidation time t_o

The introduction of the quantity t_o provides the opportunity to investigate more universal curves of U (t) based on this time. Hereinafter, we assume that to defines the time required to reduce the initial excess pore pressure to the value 0.1uo, at the point z′ = 0.1 and x′ = 0.9 (z′ = 0.1 and r′ = 0.9 in cylindrical coordinates), so covering a significant release of the excess pore pressure in the whole domain. The first step to obtain these new curves is to find t_o using the curves presented in the above section. With the reference t′ = t/t_o (instead of t′ = c_v,zt/H²), the resulting expressions of the average degree of consolidation are:

Rectangular 2-D coordinates:

Cylindrical 2-D coordinates:

Radial 1-D coordinates:

A significant number of simulations were carried out to represent the dependences of the above expressions. For the 2-D rectangular case,Figure 7a shows the results for a range of values of π₁ = cv,zL2cv,xH2 that sufficiently covers real scenarios, π₁ = 0.1-10 (an order of magnitude above and below unity). Note that the curves π₁ = 0.1 and π₁ = 10 are the same, and also π₁ = 0.2 and π₁ =5 (in general the curves of parameters π₁ = C_o and π₁ = Co−1). As shown, the curves are so close that they allow any of them to read U (t) with an error below 10%. Taking the curve π₁ =1 (or π₁ = 0.1) to read U (t) for all π₁ values, the reading is in the side of unsafety (safety), since real values of U (t) are below (above) the readings.

Figure 7a

2-D rectangular domains. U (t) as a function of t′ tto and π1=cv,zL2cv,xH2.

For 2-D cylindrical domains, equation (29), the existence of three dimensionless groups requires a group of abacuses in order to obtain U (t), one for each value of the group π₃=r_0Ut /r_in. Figures 7b to 7d show the abacus for π₃ = 5, 10 and 20, respectively. Note that the curves π₁ ≥ 0.8 are so close that they can be considered an only one.

Figure 7b

2-D radial domains. U (t) as a function of t′ tto and π1=cv,zrout2cv,rH2.π₃ = 5. π₁ = 0.1, 0.2, 0.3, 0.5, 0.8, 1, 1.5, 2, 3, 5, 8 and 10.

Figure 7c

2-D radial domains. U (t) as a function of t′ tto and π1=cv,zrout2cv,rH2.π₃ = 10. π₁ = 0.1, 0.2, 0.3, 0.5, 0.8, 1, 1.5, 2, 3, 5, 8 and 10.

Figure 7d

2-D radial domains. U (t) as a function of t′ tto and π1=cv,zrout2cv,rH2.π₃ = 20. π₁ = 0.1, 0.2, 0.3, 0.5, 0.8, 1, 1.5, 2, 3, 5, and 10.

For radial 1-D scenarios, π₁ does not influence the solution and, from numerical simulations, the dependence (26) is shown in Figure 7e.

Figure 7e

1-D radial domain. U(t) as a function of t′ tto and π3=routrin.

7 Discriminated nondimensionalization dimensionless groups versus classical groups

It should be mentioned that, compared with discriminated nondimensionalization, the classical procedure leads to poorer results. If the same reference, as in the classical method, is used to make horizontal and vertical coordinates dimensionless, the values of their dimensionless forms and their derivatives are not constrained in the interval [0,1] but only one of them. This means that the coefficients of the terms of the dimensionless governing equations cannot be of the same order of magnitude (even if measured in the same unity) and, consequently, the final dimensionless groups are not of the order of magnitude unity. In addition, classical nondimensionalization generally leads to a large number of dimensionless groups, particularly in anisotropic scenarios, which leads to a less precise result. For example, in 2-D radial domains, the solution for t_o provided by classical nondimensionalization (which is the same as that obtained by dimensional analysis) is

while for U (t)

Note that while the ratio r_out/r_in emerges in both classical and discriminated nondimensionalization, the ratios c_v,z/c_v,r and H/r_out are not assumed as dimensionless when discrimination is applied but join in a new group to provide a kind of corrected conductivity ratio. For radial 1- D domains, the above solutions simplify to others less precise than those obtained by discrimination.

Finally, in rectangular coordinates, the classical method leads to solutions

whereby all functions depend on one more argument than discriminated solutions.

8 Case study

In this section a case study is addressed to compare the solutions given in the manuscript with those provided by the classical methods. In the problem to be analyzed, a fully saturated clay soil is subjected to a wide extension load of constant value and permanent over time, giving rise to a consolidation phenomenon. In order to accelerate this process, vertical drains of circular section are installed in the soil before applying the load, following a scheme similar to that of Figure 1 (bottom). The geotechnical properties of the soil to be studied are summarized in Table 2.

The study focuses on the analysis of the evolution of the average degree of settlement, (U), in order to determine the duration of the consolidation process. To get the value of U, the solutions in the form of series provided by the classical methods can be used, which treat the problem by superposing the vertical (U_v) and radial (U_r) solutions[11], equations (37) and (38), respectively.

where (T_v) and (T_r) are, respectively, the vertical and radial time factors, whose expressions are:

The above solutions lead to the time-dependent average degree of consolidation given by:

This means that, in order to determine the time for which the consolidation process can be considered finished, it will be necessary to test different values of t until the characteristic time is found, which means a laborious task.

A value of t = 2.7 years can be initially taken. By means of equations (37) and (38), U_v = 0.52 and U_r = 0.37 are obtained, finally resulting U = 0.70. At this point, it is worth mentioning that there are graphic solutions in the form of abacuses for expressions (37) and (38) [18], which saves the cumbersome task of calculating the series of addends. However, this method requires: i) obtaining two time factors, (T_v = 0.20, T_r = 0.51) and the parameter n (of value 20), ii) the use of two abacuses to determine U_v and U_r, and iii) use the expression (40) to get U.

On the other hand, the use of the solutions proposed in this article is, by comparison, much simpler and requires fewer steps. First, we must obtain the values of π₁, π₂ and π₃, the dimensionless groups that govern the solution pattern of the proposed problem. These are:

From these values, the average degree of consolidation is read directly using Figure 3c, obtaining the same result as with the previous method (U = 0.70).

To determine the characteristic time of the problem, it would be necessary to repeat the previous procedure (testing with different times) as many times as necessary until reaching, for example, a high average degree of settlement. However, with the characteristic time curves provided in this paper, the process is greatly simplified. As can be deduced from expression (26)

the characteristic time, expressed in a dimensionless form π₂,t_o, is a function of the groups π₁ and π₃. In this way, based on the values π₁ = 0.4 and π₃ = 20, the value of π₂,t_ois obtained directly from Figure 5, resulting π₂,t_o = 0.6. Finally, from the expression π₂,t_o = π2,to=tocv,zH2 the value of to = 8 years is obtained, a time for which a fall in the excess pore pressure of 90% is reached at a point in the domain that is far away from both the surface drainage boundary and the vertical drain (z′ = 0.1, r′ = 0.9). This criterion of defining the characteristic time ensures high values for the average degree of consolidation; indeed, when we calculate this for t_o =8 years – applying again equations (37-40) or Figure 3c – we obtain a value of U = 0.95. A result that, on the other hand, can also be obtained from Figure 7d for a value of t′ = 1.

9 Conclusions

The nondimensionalization of governing equations, in conjunction with the concept of spatial discrimination in anisotropic media, provides expressions for the average degree of consolidation as a function of the smallest number of dimensionless groups that enable the standard representation of this quantity by an abacus, for rectangular coordinates, and by a group of abacuses for cylindrical coordinates. This technique is applied in the paper to the consolidation process in 2-D, rectangular and cylindrical anisotropic scenarios.

Compared to the classical nondimensionalization, the use of discrimination, in its wider meaning, has two advantages. Firstly, as regards spatial directions, discrimination forces us to separate the references of the coordinate variables, deleting the form factor – or geometrical aspect ratio – of the domain as independent group and joining it with the ratio of the vertical and horizontal consolidation coefficients. The reduction from three classical independent groups to two discriminated groups makes it easier to represent the dependence of the average degree of consolidation on these groups by abacuses. Secondly, the introduction of the quantity ‘characteristic time’, defined as the time at which the excess pore pressure is reduced to a small fraction of its initial value – at a given point of the domain –, allows the expression of this quantity to be determined as a function of the other dimensionless groups of the problem.

When both the different references for spatial coordinates and characteristic time are used to define the dimensionless variables of the problem, the resulting dimensionless discriminated governing equations provide groups on the basis of which the new universal curves of the average degree of consolidation can be represented. These curves differ very little from each other, to such an extent that one of them can be used to read the average degree of consolidation with an acceptable percentage of error for engineering purposes.

The dependences of the characteristic time and the average degree of consolidation are solved numerically by the network method and shown finally in the form of universal abacuses.

A case study has been addressed to show that the solutions provided by the proposed curves are as precise as those obtained by classical methods, but they present two important advantages: i) the procedure to obtain the solutions is much quicker and easier, with no mathematical manipulation, and ii) the determination of the characteristic consolidation time is direct, and not through successive trials.