An Equal-Strain Analytical Solution for the Radial Consolidation of Unsaturated Soils by Vertical Drains considering Drain Resistance

Developing an analytical solution for the consolidation of unsaturated soils remains a challenging task due to the complexity of coupled governing equations for air and water phases. *is paper presents an equal-strain model for the radial consolidation of unsaturated soils by vertical drains, and the effect of drain resistance is also considered. Simplified governing equations are established, and an analytical solution to calculate the excess pore-air and pore-water pressures is derived by using the methods of matrix analysis and eigenfunction expansion. *e average degrees of consolidation for air and water phases and the ground surface settlement are also given. *e solutions of the equal-strain model are verified by comparing the proposed free-strain model with the equal-strain model, and reasonably good agreement is obtained. Moreover, parametric studies regarding the drain resistance effect are graphically presented.


Introduction
As a common phenomenon of civil engineering, consolidation is a process of decreasing soil volume when soil is subjected to an increased stress.Understanding of this phenomenon is vital to designs of soft soil foundations, pavements, and other engineering structures.erefore, Terzaghi [1] established a classical theory for the analysis of consolidation in saturated soft soil foundation, which is well known in geotechnical engineering.Based on Terzaghi's theory, solutions of one-dimensional consolidation have been derived for multilayered soils [2,3], non-Darcy flow [4,5], time-dependent loading [6,7], arbitrary boundary [8,9], and nonlinear deformation [10,11].In practice, soft soils related to engineering are usually in a state of unsaturation.With the great progress made, some well-developed consolidation theories for unsaturated soft soils have been available [12][13][14][15].Among these theories, Fredlund and Rahardjo's [15] consolidation theory is well accepted for the consolidation of unsaturated soft soils.
Since the inception of the one-dimensional (1D) consolidation theory of unsaturated soft soils proposed by Fredlund and Hasan [16], the investigations for 1D consolidation have been significantly progressed.By assuming constant soil parameters during consolidation, Qin et al. [17,18] adopted the Laplace transform and Cayley-Hamilton methods to obtain an analytical solution.Shan et al. [19] and Zhou et al. [20] converted the nonlinear governing flow equations into traditional diffusion equations and proposed alternative solutions satisfying those converted equations.Ho et al. [21], on the other hand, introduced exact solutions using the eigenfunction expansion and Laplace transform techniques.In addition, some general solutions that consider various boundary conditions [9,19,20,22], initial pore-water and -air pressure distributions [9,20], and complex timedependent loadings [9,19,20,[22][23][24] were also obtained in a single soil layer.However, the soft soil foundation in practice is not always singly layered.For this reason, Shan [25] presented an analytical solution of unsaturated multilayered soil with typical boundary conditions.
In subgrade and pavement engineering, vertical drains are one of the most commonly used techniques to accelerate the consolidation process and increase the bearing capacity of the subgrade.It is essential to expand the consolidation theory of unsaturated soft soils from one-dimensional vertical consolidation to radial consolidation.For this purpose, Conte [26] developed a general formulation that can be advantageously used to analyze consolidation under plane-strain conditions.Ho et al. [27] discussed the excess pore-air and pore-water pressure dissipations in the plane-strain consolidation of unsaturated soft soils using eigenfunction expansion and Laplace transform techniques.Shortly afterwards, the solutions of unsaturated soft soils subjected to different time-dependent loadings were also obtained by Ho [28].
However, a lot of attempts have been made to solve the axisymmetric consolidation model of unsaturated soft soils particularly with analytical approaches.Among the pioneered studies, Conte [26] introduced the finite element technique to obtain a solution for the coupled consolidation under the axisymmetric conditions.Qin et al. [29] dealt with the vertical drain consolidation problem in unsaturated soft soils using the modified Bessel functions and the Laplace transformation.On the other hand, Zhou and Tu [30] and Zhou [31] presented the differential quadrature method (DQM) to estimate the axisymmetric consolidation behavior in the unsaturated soft soils.Ho et al. [32] gave an analytical solution for the axisymmetric consolidation of unsaturated soft soils subjected to constant external loading.e analytical procedure employs the variable separation and Laplace transformation techniques while capturing the uniform and linear initial excess pore pressure distributions with depth.
Most of the models mentioned above were obtained based on the free-strain assumption.However, the solutions of the free-strain model are complex and inconvenient to use in practice.In the research of saturated soft soils, Barron [33] firstly proposed the consolidation model for vertical drains under free-strain and equal-strain models.e mathematical expressions are as follows: where u and u are the excess pore-water pressure and average excess pore-water pressure, respectively; c r and c v are the coefficients of consolidation in the radial and vertical directions, respectively; r is the distance from the origin of the vertical drain in the radial direction; z is the distance from the surface of the foundation in the vertical direction; and t is the time.Barron [33] pointed out that there is negligible difference between the free-strain and equal-strain models.After this, equal-strain assumption is frequently used in most consolidation theories for vertical drains [34][35][36].
At present, there is no reference about the equal-strain model in the research of unsaturated soft soils.e objective of this paper is to propose an equal-strain consolidation model to investigate the consolidation behavior of unsaturated soft soils.In addition, drain resistance will also be considered in this study, and the accuracy of the equal-strain model will be verified by comparing with the free-strain model proposed by Qin et al. [29].Moreover, parametric studies regarding the drain resistance effect are also presented in this paper.

Problem Statements
2.1.Model Description.Figure 1 describes the details of the representative vertical drain within an unsaturated soft soil foundation.Dimensions of the soil system include the depth H and the radius of the influence zone r e .k w r and k a r are the coefficients of water and air permeability of the foundation in the radial direction, respectively.e circular vertical drain with the radius r w is installed at the center of the influence zone.k w and k a are the coefficients of water and air permeability of the vertical drain in the vertical direction, respectively.A constant surcharge q is instantaneously applied on the top surface of soft soil foundation.

Basic Assumptions.
e basic assumptions are the same as those of Fredlund's consolidation theory for unsaturated soft soils and Barron's [33] consolidation theory for vertical drains.e other three assumptions are listed as follows: (1) All the soil parameters are constants during consolidation.(2) e strains at the same depth of the foundation are equal (equal-strain assumption).(3) e distribution of excess pore-air pressure in unsaturated soft soils is uniform. 2 Advances in Civil Engineering

Derivation of Consolidation Governing Equations.
e governing equations for the consolidation of unsaturated soft soils with vertical drains consist of water phase and air phase equations under equal-strain conditions.In the polar coordinate system, considering a representative element of unsaturated soft soil foundation, with water and air flow in and out in the radial direction during consolidation, the derivation process is given as follows.

Water Phase.
e net flux of water through the element is computed from the volume of water entering and leaving the element within a period of time.By utilizing Darcy's law, the net flux of water per unit volume of the soil can be expressed in the polar coordinate system as where V 0 is the initial total volume of the soil element, V w is the volume of water in the soil element, c w is the unit weight of water, and u w is the excess pore-water pressure.e net flux of water per unit volume of the soil can be obtained by differentiating the water phase constitutive relation with respect to time: where m w 1k and m w 2 describe the coefficients of water volume change with respect to a change in the net normal stress d(q − u a ) and matric suction d(u a − u w ), respectively; and u w and u a are the average excess pore-water and pore-air pressures throughout the entire soil mass, respectively, which can be expressed as where u a is the excess pore-air pressure.Substituting (2) into (3), the governing equation for the water phase can be expressed as where C w r is the coefficient of consolidation with respect to the water phase in the radial direction (i.e., k w r /(c w m w 2 )) and C w a is the interactive constant associated with the water phase (i.e., (m w 1k − m w 2 )/m w 2 ).

Air Phase.
e net flux of air through the element is computed from the volume of air entering and leaving the element within a period of time.e flux of air per unit volume of the soil can be obtained by Fick's law as where ρ a is the density of air, g is the gravitational acceleration, and V a is the volume of air phase.
Based on assumption 3, the distribution of excess poreair pressure in unsaturated soft soils is uniform.e density of air is a function of air pressure in accordance with the ideal gas law.It can be expressed as where ω a is the molecular mass of air, R is the molar gas constant, T at is the absolute temperature,  u a is the absolute excess pore-air pressure (i.e., u a + u atm ), and u atm is the atmospheric pressure.
Replacing the air density, ρ a , in ( 6) with (7) gives where S s is the degree of saturation and n s is the porosity.e flux of air per unit volume of the soil due to changes in the net normal stress, d(q − u a ), and the matric suction, d(u a − u w ), can be expressed by the following equation: where m a 1k and m a 2 describe the coefficients of air volume change with respect to a change in the net normal stress d(q − u a ) and matric suction d(u a − u w ), respectively.
By substituting (8) into (9), the governing equation for the air phase can be given as where C a r is the coefficient of consolidation with respect to the air phase in the radial direction (i.e., and C a w is the interactive constant associated with the air phase (i.e., m a 2 . By rearranging the governing equations of water and air phases, the consolidation equations for water and air phases can be written in matrix form as and ( ) ,r and ( ) ,t denote the derivatives with respect to r and t, respectively.

Initial and Boundary
Conditions. e external radius, r e , is impervious or, because of symmetry, no water and air flow passes this boundary; that is,

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Excess pore pressures at the interface of soil stratum and vertical drain satisfy flow continuity condition.
e following boundary condition is considered: where e vertical boundary conditions are which indicate that the top surface of the soft soil foundation is permeable to water and air, while the bottom surface is impermeable to water and air.e initial excess pore pressures are uniform throughout the soil mass when t � 0. ey can be calculated according to the method given by Fredlund and Rahardjo [15].e initial condition is given as follows: where u 0 � u w 0 u a 0  .

􏽰
. General solution for u| r�r w can be written as products of functions with respect to dimensions z and time t.In addition, based on the homogeneous boundary conditions for the depth given in (14), the eigenfunction of u| r�r w is sin(Mz/H).Hence, where I is the identity matrix and M � [(2m − 1)/2]π.e average excess pore pressure u can be written as follows by substituting ( 21) into (20): By substituting ( 21) and ( 22) into ( 18), the characteristic equation can be obtained using the orthogonality of sine function, as shown below: Based on the orthogonality of the sine function, the initial condition of T can be obtained by the initial condition of u as According to the initial condition ( 24), ( 23) can be solved as where 21), (22), and (25) and substituting ( 21) and ( 22) into (17) give the solutions of u and u as shown below: After solving u, the average degree of consolidation for water and air phases (i.e., U e � U we U ae  ) can be obtained as follows: where Based on Fredlund's theory, the volume change is given by the following constitutive equation for unsaturated soft soils [15]: 4

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where ε v is the volumetric strain, m s 1k � m w 1k + m a 1k is the coefficient of volume change with respect to a change in net normal stress, and m s 2 � m w 2 + m a 2 is the coefficient of volume change with respect to a change in matric suction.
e ground surface settlement S e of the unsaturated soft soil can be computed as where . e normalized settlement, denoted as S ne , can be determined based on the volumetric strain ε v provided in (30): where S ∞ e is the ultimate ground surface settlement when the pore pressures dissipated entirely (i.e., S ∞ e � H[m s 1k u a 0 + m s 2 (u w 0 − u a 0 )]).As can be seen from ( 31), S ne is also the total degree of consolidation which is dominated by both air and water phases.

A Special Case.
When the effect drain resistance is neglected, the permeability of drain well is endless, videlicet k w and k a tend towards infinity, which yields [(κM/H) 2 B + I] −1 → 0. Parameter D can be rewritten as Based on the theory of series, the following can be obtained: By substituting (32) and ( 33) into (26-28) and ( 30) and ( 31), the solutions can be simplified to ) e equations from (34) to (38) are the simplified solutions in which the effect of drain resistance is neglected.

Analysis of Consolidation Behavior
In this study, the accuracy of the proposed equal-strain consolidation model in the unsaturated soft soils is investigated by comparing with the free-strain model.e consolidation behavior and influence factors are also discussed in this section.Following Fredlund and Rahardjo [15] and Conte [26], the consolidation parameters adopted in this study are listed as follows: (iv) Other parameters q � 100 kPa, u w 0 � 40 kPa, and u a 0 � 20 kPa.e above parameters are assumed to be constant during the consolidation process.An instantaneous compression induced by the external applied load q generates initial excess pore-water and pore-air pressures (i.e., u w 0 and u a 0 , resp.).Considering that the soil is loaded threedimensionally under isotropic conditions, changes in excess pore pressures can be determined using a method given by Fredlund and Hasan [16].

Verification.
e validity of the equal-strain model had been verified in the saturated soft soils.In this section, the validity of the proposed equal-strain model in the unsaturated soft soils will be verified as well.For this purpose, the special consolidation case without considering drain resistance is analyzed by using the proposed equal-strain solution and one available free-strain solution [29].
Figures 2(a) and 2(b), respectively, demonstrate changes in the average degrees of consolidation of air and water phases with different ratios of k a r /k w r under the equal-strain and free-strain conditions.It is worth noting that the air permeability k a r is varying while the water permeability k w r is kept constant at 10 −10 m/s.In addition, the time factor is defined as T � C w r t/H 2 .It can be observed from the figures that the results obtained from these two different models are in good agreement with each other.e slightly large difference between the two models is mainly found at the earlier stage of the consolidation process.Moreover, it can be seen that with an increase of k a r /k w r , the average degrees of consolidation increase gradually for both air phase and water phase.With an increase of consolidation time T, the curve of average degree of consolidation for water phase gradually tends to be consistent.is is because the pore-air pressure almost completely dissipates at the early stage of consolidation.
Figures 3(a) and 3(b) represent the difference of average consolidation degree between equal-strain model and freestrain model varying with T in different k a r /k w r values.As observed from Figure 3(a), the maximum difference (slightly more than 3%) between two models for air phase is mainly at Advances in Civil Engineering the earlier stage of consolidation before the excess porewater pressure begins to dissipate.Inversely, the maximum difference in the water phase is mainly near the end of the air dissipation, as shown in Figure 3(b).In brief, there is negligible influence on the difference between equal-strain condition and free-strain condition with different ratios of k a r /k w r .It shows that the equal-strain model is efficient in solving the unsaturated soil consolidation problem.Also, the solution is convenient for engineering application.

Influence of Drain
Resistance. Figure 4 represents the degrees of consolidation of air and water phases and normalized settlement varying with T under different values of r w .e influence radius r e � 1.8 m is adopted in this investigation.Figure 4(a) shows the degrees of consolidation of air phase varying with different values of r w as time elapses.It is obvious that the consolidation proceeds more quickly with the increase of r w .As the radius of vertical drain increases, the radial drainage area will be indirectly increased, which accelerates the process of consolidation. Figure 4(b) shows the degrees of consolidation of the water phase with different values of r w .As observed, rate of consolidation of water phase at the initial stage proceeds more quickly because of increasing r w .After the excess pore-air pressure dissipated almost completely, a plateau period may occur in the excess pore-water pressure patterns.At the later stage, the rate of water phase consolidation is similar to the initial stage.Besides, it can also be found that the excess pore-air pressure dissipated faster than that of the excess pore-water pressure because the permeability of air phase is greater than water phase both in the vertical drain and unsaturated soft soil foundation.Figure 4(c) presents the normalized settlement against time factor T with different values of r w .e settlement process is similar to the consolidation process of water phase.
Figure 5 demonstrates changes in excess pore pressures and normalized settlement with different ratios of k a /k w .Figure 5(a) illustrates the dissipation curves of excess poreair pressure varying with T in different ratios of k a /k w .As observed, when k a /k w increases, the excess pore-air pressure tends to dissipate faster.When k a /k w is very high, for instance, k a /k w � 10 3 , the excess pore-air pressure may dissipate instantaneously.Figure 5(b) shows the dissipation curves of excess pore-water pressure varying with k a /k w when time elapses.It is evident that the dissipation at the initial stage of consolidation proceeds more quickly as 6 Advances in Civil Engineering a result of increasing k a /k w .After the excess pore-air pressure diminished almost fully, a plateau period may occur in the excess pore-water pressure patterns when k a /k w > 1.It should be noted that the greater the ratio of k a /k w is, the longer the plateau gets.Lastly, the curves of excess pore-water pressure dissipation converge to a single curve, and complete dissipation occurs at approximately the same time.Figure 5(c) shows the variations of the normalized settlement against time factor T with different ratios of k a /k w .It is worth noting that S ne patterns consist of double inverse S curves when k a /k w > 0.1, similar to the excess porewater pressure dissipation process.e initial stage of consolidation is governed by the simultaneous dissipation of both excess pore-air and pore-water pressures, and once the excess pore-air pressure is fully dissipated in the soft soil system, S ne patterns converge to only one curve and gradually approach 1 at the later stages.More remarkable, after the plateau period, the consolidation behavior is dominated by single phase (water phase) and the consolidation resembles to the classical Terzaghi's consolidation [1].
By comparing Figure 4 with Figure 5, the variation of vertical drain radius makes the normalized settlement S ne a great change in the whole period of consolidation.In contrast, the difference of normalized settlement S ne between different air permeabilities k a is mainly observed before the plateau period.is is because the change of the radius of vertical drain influenced all period of consolidation process.However, the change of the air permeability of vertical drain  Advances in Civil Engineering only had significant influence before the plateau period.After the plateau period, the consolidation process is dominated by water phase.

Summary and Conclusions
e equal-strain model established in this paper has a negligible difference for average degree of consolidation compared with the free-strain model.Parametric studies regarding wellresistance effect are graphically presented and discussed, and the key findings are summarized as follows: (1) e solutions calculated from the equal-strain model show good agreement with that obtained from the free-strain model.e main difference between the two models is mainly found at the earlier stage of the consolidation.(2) e higher the ratio of k a r /k w r is, the faster the consolidation is.With an increase of consolidation time T, the curve of average degree of consolidation for water phase gradually tends to be consistent.
(3) With the increase of k a /k w , the excess pore-air and -water pressures tend to dissipate faster.After the excess pore-air pressure dissipated almost completely, a plateau period occurs in the excess porewater pressure patterns when k a /k w > 1.It should be noted that the greater the ratio of k a /k w is, the longer the plateau gets.(4) Both the water and air phases of consolidation proceed more quickly with the increase of r w , and the change of the radius of vertical drain influenced all period of consolidation process.

Figure 1 :
Figure 1: Calculation diagram of vertical drain in an unsaturated soil stratum.

Figure 2 :Figure 3 :
Figure 2: Average degrees of consolidation of (a) air phase and (b) water phase varying with different ratios of k a r /k w r .

Figure 4 :
Figure 4: Degrees of consolidation of (a) air phase, (b) water phase, and (c) normalized settlement varying with different values of r w .

Figure 5 :
Figure 5: Degrees of consolidation of (a) air phase, (b) water phase, and (c) normalized settlement varying with different ratios of k a /k w .