J. Cent. South Univ. (2016) 23: 934-943

DOI: 10.1007/s11771-016-3141-x

A general solution for vertical-drain consolidation with impeded drainage boundaries

FU Cui-wei(付崔伟), LEI Guo-hui(雷国辉)

Key Laboratory of Geomechanics and Embankment Engineering of Ministry of Education,

Geotechnical Research Institute, Hohai University, Nanjing 210098, China

 Central South University Press and Springer-Verlag Berlin Heidelberg 2016

Abstract: An analytical solution is derived from the generalized governing equations of equal-strain consolidation with vertical drains under multi-ramp surcharge preloading. The hydraulic boundary conditions at both top and bottom of the consolidating soil are modelled as impeded drainage. The impeded drainage is described by using the third type boundary condition with a characteristic factor of drainage efficiency. Fully drained and undrained boundary conditions can also be modelled by applying an infinite and a zero characteristic factor, respectively. Simultaneous radial and vertical flow conditions are considered, together with the effects of drain resistance and smear. An increase in total stress due to multi-ramp loading is reasonably modelled as a function of both time and depth. A solution to calculate excess pore-water pressure at any arbitrary point in soil is derived, and the overall average degree of consolidation is obtained. It shows that the proposed solution can be used to analyze not only vertical-drain consolidation but also one-dimensional consolidation under either one-way or two-way vertical drainage conditions. The characteristic factors of drainage efficiency of top and bottom boundaries have a potentially important influence on consolidation. The boundary may be considered fully drained when the characteristic factor is greater than 100 and fully undrained when the characteristic factor is less than 0.1. The stress distribution along depth induced by the surcharge loading has a limited effect on the overall average degree of consolidation. However, it has a significant effect on the dissipation of excess pore-water pressure.

Key words: consolidation; vertical drain; surcharge preloading; drainage boundary condition; multi-ramp loading

1 Introduction

Various analytical solutions have been derived for solving consolidation of soft soils under surcharge preloading. Nevertheless, most of the existing solutions were derived for consolidation of soils with fully drained or undrained top and bottom boundaries. Such hydraulic boundaries can be simply modelled using the first and second type boundary conditions, i.e., u_b=0 for drained boundary and u_b/z=0 for undrained boundary, where u_b is the excess pore-water pressure at the boundary and z is the vertical coordinate. In practice, however, impeded drainage is a matter of great concern to practicing engineers [1-2]. This may occur when a consolidating soil is overlain or underlain by a layer of relatively incompressible and insufficiently permeable soil. To allow for impeded drainage, some analytical solutions have been proposed for solving one-dimensional consolidation problems [3-13]. For consolidation with vertical drains, a limited number of the existing analytical solutions were developed only under instantaneous loading conditions [14-17]. However, in practical situations, surcharge loading is normally applied gradually and incrementally. Such loading conditions would be more appropriately modelled as multi-ramp loads.

In this work, a rigorous analytical solution is derived for consolidation with vertical drains under multi-ramp surcharge loading. Both the top and bottom hydraulic boundary conditions are modelled as impeded drainage, using the third type boundary condition developed by GRAY [13]. The validity and accuracy of the proposed solution are verified by comparing the special cases of the solution with two available analytical solutions. The effect of impeded drainage boundary on consolidation is investigated, together with the effect of the stress distribution.

2 Problem description

Figure 1 shows a cylindrical unit-cell model for consolidation with a vertical drain. The soil is subjected to a depth-varying and time-dependent increase in total stress under multi-ramp loading. The governing equations of equal-strain consolidation assuming constant material properties are given in full by [18]

                   (1)

                     (2)

where r and z are the radial and vertical coordinates, respectively; t is time; r_d, r_s and r_e are the radii of the vertical drain, the smear zone and the effective influence zone of the vertical drain, respectively; u and u_s are the excess pore-water pressure of undisturbed soil and smeared soil, respectively; s is the increase in total stress in soil due to surcharge loading;  and  are the average excess pore-water pressure at a given depth in the radial direction between r_s and r_e and between r_d and r_s, respectively; k_h, k_v and m_v are the horizontal and vertical hydraulic conductivity and volume compressibility of the undisturbed soil, respectively; k_sh, k_sv and m_sv are the horizontal and vertical hydraulic conductivity and volume compressibility of the smeared soil, respectively; and γ_w is the unit weight of water.

Fig. 1 A cylindrical unit-cell model of vertical-drain consolidation with impeded drainage boundaries

According to the continuity of the excess pore-water pressure and the flow rate at the interface between the vertical drain and the smeared soil, the drain resistance can be expressed as [19-20]

              (3)

where k_d is the hydraulic conductivity of the vertical drain.

The continuity of the excess pore-water pressure and the flow rate at the interface between the smeared soil and the undisturbed soil can be described by

                                     (4)

                    (5)

According to the vertical coordinate system shown in Fig. 1, the impeded drainage at the top and bottom boundaries can be expressed by adopting the third type boundary condition developed by GRAY [13] as

                   (6)

             (7)

where h is the depth of the vertical drain; and R_t and R_b are the characteristic factors of drainage efficiency of the top and bottom boundaries, respectively. Fully drained and undrained boundary conditions can be modelled by applying an infinite and a zero characteristic factor, respectively.

The hydraulic boundary condition for the vertical boundary can be expressed as

                                   (8)

The initial condition is given by

                        (9)

Figure 2 schematically shows the depth-varying increase in total stress in soil due to multi-ramp surcharge loading. To facilitate the derivation of the analytical solution, a new single equation is constructed to accurately describe the increase in total stress as

                  (10)

where

     (11)

                        (12)

         (13)

where M is the total number of loading ramps; t_i,0 and t_i,1 are the start time and end time of the i-th ramp, respectively, as shown in Fig. 2(a); s_i is the increase in total stress in soil at the end time of the i-th ramp, and s₀=0; and s_a,i, s_b,i and s_c,i are coefficients describing the distribution of the increase in total stress as a function of depth. For rectangular, triangular and trapezoidal distributions, the values of s_a,i, s_b,i and s_c,i can be readily derived from Eq. (12) according to the values of stress increase at z=0 and z=h, as presented in Figs. 2(b) to (d). For a parabolic distribution shown in Fig. 2(e), s_a,i≠0 and s_c,i=s_i at z=0, and s_b,i can be specified.

Fig. 2 Time-dependent and depth-varying increase in total stress in soil under multi-ramp loading:

The equations above describe the unit-cell consolidation problem to be solved.

3 analytical solution

3.1 Consolidation of undisturbed soil

The excess pore-water pressure of undisturbed soil can be expressed by introducing the Fourier sine and cosine series as follows:

         (14)

where u_n is the corresponding Fourier coefficient; b_n and ω_n are coefficients to be determined by the hydraulic boundary conditions. Substituting Eq. (14) into Eqs. (6) and (7) yields

        (15)

                  (16)

where ω_n is the successive positive root of the transcendental Eq. (16).

The average excess pore-water pressure at a given depth and the increase in total stress (i.e. Eqs. (10) and (12)) in undisturbed soil can be expressed, similar to  Eq. (15), as follows:

          (17)

             (18)

               (19)

where  and s_n,i are their corresponding Fourier coefficients; and the latter can be derived as

(20)

where

   (21)

     (22)

                    (23)

                  (24)

                              (25)

       (26)

Substituting Eqs. (15), (17) and (19) along with Eq. (11) into the governing Eq. (1) yields

                         (27)

where

     (28)

Using the method of separation of variables, the following equation can be written:

                               (29)

where A and B are functions of radial coordinate and time, respectively. Substituting Eq. (29) into Eq. (27) gives

                 (30)

where λ_n is a constant. A solution [21] to Eq. (30) is

             (31)

where I₀ and K₀ are the modified Bessel functions of the first and second kind of zero order, respectively; c_1n and c_2n are the constants of integration to be determined; and

                                      (32)

                                     (33)

The average excess pore-water pressure at a given depth is given by

                    (34)

From Eqs. (17) and (34), the following equation can be derived:

                  (35)

Substituting Eq. (31) into Eq. (35) yields

                                   (36)

where

                 (37)

where I₁ and K₁ are the modified Bessel functions of the first and second kind of order one, respectively.

Substituting Eq. (36) into Eq. (30) yields

             (38)

A solution of Eq. (38) is

                  (39)

where a_n is the constant of integration to be determined; T_h is the time factor; e is the base of the natural logarithm; and

                               (40)

                                  (41)

Based on Eqs. (29), (31) and (39), Eq. (15) can be rewritten as

    (42)

3.2 Consolidation of smeared soil

Again, by introducing the Fourier sine and cosine series, the excess pore-water pressure at any arbitrary point and the average excess pore-water pressure at a given depth of smeared soil can be expressed in accordance with Eqs. (6) and (7) of the top and bottom hydraulic boundary conditions as follows:

      (43)

        (44)

where u_sn and  are their corresponding Fourier coefficients.

Using the method of separation of variables, the following equation can be written:

                           (45)

Following the same derivation procedures as above for the consolidation of undisturbed soil, the following solution to Eq. (45) for the consolidation of smeared soil can be obtained:

       (46)

                (47)

where λ_sn is the separation constant; c_3n, c_4n and a_sn are the constants of integration to be determined; and

                                   (48)

                                   (49)

                             (50)

                                (51)

          (52)

Thus, Eq. (43) can be rewritten as

     (53)

In the following sections, the constants of integration in Eqs. (42) and (53) are determined according to the initial conditions and the vertical hydraulic boundary conditions, together with the equations of drain resistance and interface flow continuity.

3.3 Initial conditions

Without loss of generality, the initial average excess pore-water pressures for undisturbed soil and smeared soil are assumed to be

               (54)

       (55)

Substituting Eq. (42) into Eq. (54) and substituting Eq. (53) into Eq. (55) yield

                           (56)

                        (57)

where s_n,0 is the Fourier coefficient of Fourier series expansions of the initial increase in total vertical stress s₀ as shown in Fig. 2(a).

In order to ensure continuity of pore-water pressure and flow rate at all times, the time functions for the consolidation of undisturbed soil and smeared soil must be the same, i.e.,

                                (58)

Substituting Eqs. (39) and (47) into Eq. (58) yields

          (59)

Equation (59) requires that

                                  (60)

                                      (61)

                                  (62)

It can be readily proved that by Eqs. (56), (57), (60) and (61), Eq. (62) is satisfied.

For the initial conditions specified in Eq. (9) and Fig. 2, i.e., s₀=0 and s_n,0=0, Eq. (56) becomes

                               (63)

By substituting Eqs. (28) and (63) into Eq. (39), the following generalized time function can be derived:

(64)

where

                    (65)

3.4 Drain resistance

Substituting Eq. (53) into Eq. (3) yields

                  (66)

                                   (67)

                  (68)

For an ideal drain without drain resistance, k_d=∞, and hence Δ₁=0.

3.5 Interface flow continuity

Substituting Eqs. (42) and (53) into Eqs. (4) and (5) and considering Eq. (59) yield

          (69)

             (70)

Substituting Eq. (66) into Eqs. (69) and (70) gives

                           (71)

where

                (72)

                  (73)

                    (74)

          (75)

3.6 Vertical hydraulic boundary conditions

Substituting Eq. (42) into Eq. (8) yields

                    (76)

The following equation can be derived from    Eqs. (71) and (76):

                               (77)

                                (78)

                     (79)

Substituting Eqs. (66), (77) and (78) into Eq. (69) leads to

        (80)

3.7 Solutions for excess pore-water pressure

Based on Eqs. (15), (29), (31) and (64), a solution is obtained for calculating the excess pore-water pressure at any arbitrary point in the undisturbed soil:

          (81)

Similarly, based on Eqs. (43), (45), (46), (58), (61) and (64), a solution is obtained for calculating the excess pore-water pressure at any arbitrary point in the smeared soil:

      (82)

where

                      (83)

3.8 Degree of consolidation

As usual, the overall average degree of consolidation is defined in terms of settlement as follows:

   (84)

where s_M is the maximum increase in total stress in soil at the end time t_M,1 of the M-th ramp of surcharge loading, as shown in Fig. 2(a); and  is the overall average excess pore-water pressure.

Based on Eqs. (10) to (13), the following expressions can be derived:

     (85)

      (86)

Based on Eqs. (81) and (82),  can be derived as

    (87)

Thus, by substituting Eqs. (85) to (87) into Eq. (84), the overall average degree of consolidation can be obtained. For ease of application of the proposed solution, a simple Fortran program that solves the modified Bessel functions with freeware subroutines has been developed. The results are obtained through double-precision arithmetic calculation.

The degree of consolidation under instantaneous loading can be obtained by simply inputting M=1 and t_1,1=t_1,0=0 into Eqs. (84)-(87). Similarly, the degree of consolidation under single-ramp loading can be obtained by simply inputting M=1 and t_1,1>t_1,0=0 into Eqs. (84)- (87).

4 Verification

It is worth noting that according to Eqs. (6) and (7), consolidation with vertical drains under one-way fully drained conditions at the top or bottom boundary can be analyzed using the proposed solution by simply letting R_t or R_b=∞. A fully undrained boundary can be modelled by applying R_t or R_b=0. Consolidation under two-way fully drained boundary conditions can be analyzed by letting R_t=R_b=∞. Apart from this, the proposed solution can also be used to analyze one-dimensional consolidation without vertical drains, by applying an extremely low r_d value (e.g., 0.001 m), letting k_d, k_h, k_sh and k_sv be equal to k_v, and letting m_sv be equal to m_v.

In order to verify the validity and accuracy of the proposed analytical solution, the results calculated from the proposed solution for one-dimensional consolidation with impeded drainage boundaries are compared with those given by the analytical solution of MESRI [11]. He developed an analytical solution to one-dimensional consolidation with impeded drainage boundaries under single-ramp loading conditions. Figure 3 compares the degrees of consolidation calculated from the special case of the proposed solution and the solution of MESRI [11]. The duration of a single-ramp load of s_M=100 kPa is assumed to be 0.5 year. The following calculation parameters are assumed: h=10 m, k_v=1×10^{-8 m/s, and m}_v=1×10^-3 kPa^-1. It can be seen from Fig. 3 that there is no discernible difference between the results calculated by this study and the solution of MESRI [11]. For one-dimensional consolidation without vertical drains, the boundary may be considered fully drained when R_tor R_b is greater than 100 and fully undrained when R_t or R_b is less than 0.1.

Fig. 3 A comparison between solution proposed in this study and solution developed by MESRI [11]:

Under multi-ramp loading conditions, the validity and accuracy of the proposed analytical solution is verified against the solution of TANG and ONITSUKA [22]. They developed a solution to the equal-strain consolidation with vertical drains under one-way fully drained conditions. For comparison purposes, the following calculation parameters presented by TANG and ONITSUKA [22] are adopted: h=11.2 m, r_d=0.035 m, r_s=0.07 m, r_e=0.7 m, k_d=10^{-5 m/s, k}_v=k_sv=k_h=2×10^{-9 m/s, k}_sh=2×10^{-10 m/s, m}_v=m_sv=10^-3 kPa^-1. Figure 4 compares the degrees of consolidation calculated from the special case (R_t=∞ and R_b=0) of the proposed solution and the solution of TANG and ONITSUKA [22]. Excellent agreement is obtained, as shown by the solid line and the open circles. To investigate the effect of impeded drainage boundary on consolidation, R_t=1, 10 and 100 is also employed to obtain the results. It can be seen that for consolidation with vertical drains, the boundary may be considered fully drained when R_t is greater than 100, similar to the case for one-dimensional consolidation without vertical drains. The higher the value of the characteristic factor of drainage efficiency R_t, the higher the consolidation rate, as expected. It is evident that the impeded drainage boundary has a potentially important influence on consolidation.

Fig. 4 A comparison between solution proposed in this study and solution developed by TANG and ONITSUKA [22]:

5 Effect of depth-dependent stress distribution on consolidation

To investigate the effect of stress distribution along depth on consolidation, five different distributions of the increase in total stress in soil due to single-ramp loading are considered, as shown in Fig. 5(a). The following calculation parameters are adopted: h=15 m, r_d=0.035 m, r_s=0.14 m, r_e=0.5 m, k_d=10^{-5 m/s, k}_h=2×10^{-9 m/s, k}_sh=2×10^{-10 m/s, k}_v=k_sv=1×10^{-9 m/s, m}_v=m_sv=10^-3 kPa^-1, s_M(0)=100 kPa.

Figures 5(b) and 5(c) show the calculated degrees of consolidation under two-way and one-way fully drained conditions, respectively. It can be seen that under two-way fully drained conditions, the stress distribution has a negligible effect on the degree of consolidation. Under one-way fully drained conditions (herein R_t=∞ and R_b=0), the consolidation rate for rectangular stress distribution is lower than that for trapezoidal and triangular stress distributions but higher than that for inverse trapezoidal and triangular stress distributions. This can be explained by the fact that the higher the stress at the drainage boundary, the faster the dissipation of excess pore-water pressure, as expected. This can be visualized by the isochrones of excess pore-water pressure normalized by the maximum total stress applied, as shown in Fig. 6. It can also be observed from Fig. 6 that the stress distribution has a significant effect on the dissipation of excess pore-water pressure.

Fig. 5 Effect of stress distribution on degree of consolidation:

Fig. 6 Effect of stress distribution on isochrones of excess pore-water pressure for consolidation under one-way fully drained conditions:

6 Conclusions

1) An analytical solution is proposed for consolidation with vertical drains under generalized drainage boundary conditions including impeded drainage, fully drained and undrained conditions. Multi-ramp surcharge preloading, drain resistance and smear effect are considered. The proposed solution can also be used to analyze one-dimensional consolidation.

2) The characteristic factor of drainage efficiency of the boundary has a potentially important influence on consolidation. The boundary may be considered a fully drained when the characteristic factor is greater than 100 and fully undrained when the characteristic factor is less than 0.1.

3) The stress distribution along depth induced by the surcharge preloading has a limited effect on the overall average degree of consolidation. However, it has a significant effect on the dissipation of excess pore- water pressure.

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(Edited by YANG Hua)

Foundation item: Project(51278171) supported by the National Natural Science Foundation of China; Project(B13024) supported by Program of Introducing Talents of Discipline to Universities (“111” Project), China; Project(2014B04914) supported by the Fundamental Research Funds for the Central Universities of China

Received date: 2015-01-21; Accepted date: 2015-04-10

Corresponding author: LEI Guo-hui, Professor, PhD; Tel: +86-13851922201; E-mail: leiguohui@hhu.edu.cn