Secondary consolidation simulation of peat with vertical drains under time-dependent loading

C_c

compression index

C_h

coefficient of consolidation in the radial direction

C_S

recompression index

C_α

coefficient of secondary compression

D_e

diameter of the influence zone

e₀

initial void ratio

e_p

void ratio at the end of the primary consolidation

F(n)

function of n

H

thickness of the peat layer

k_h

coefficient of permeability in the horizontal direction

k_s

coefficient of permeability of the smear zone

m

function of n, S and k_h/k_s

N

number of time steps required for time t_c

N_I

number of incremental step loadings

P_m,t

applied loading

p_I

applied load of at the beginning of each incremental step loading

q₀

ultimate load

q_I

total loading of stage I

r

distance from the centre line of the vertical drain to the point of consideration of the pore water pressure variation

r_e

radius of the influence zone

r_s

radius of the smear zone

r_w

radius of the vertical drain

S

drain spacing

S_p,t

primary consolidation at a given time

S_T,t

total settlement at a given time

S*_s,t

secondary consolidation settlement at a given time

T_r

time factor for radial consolidation

t

time

t″

time at which the secondary consolidation is calculated

t_c

monitoring period

t_I

time at the end of stage 1

U_r

average degree of consolidation

U_r,t

average degree of consolidation at a given time for an incremental step loading

$Ut′$

average degree of consolidation due to the combined loading

u

excess pore water pressure at a point at a given time

u_av

average pore water pressure

u_I

pore pressure increment due to each incremental step loading

u_i

pore water pressure in the ith step

u_r,t

remaining pore pressure value at a given time for each incremental step loading

Δt

time difference in each step loading

Δσ_t

stress difference

μ_soil

fraction of the applied stress taken by the soil

σ₀

initial overburden pressure at the middle of the soft soil layer

σ_p

preconsolidation pressure

The blooming infrastructure development activities all over the world are often challenged by waterlogged areas and low-lying marshy lands with soft deposits such as peat and other organic soils. Weak compressible soft deposits of varying thickness often cause primary consolidation and long-term secondary compression settlements and failure due to insufficient shear strength under loading. In infrastructure development activities such as highway embankment construction, preloading with vertical drains is a common soft-soil-treatment method used to improve highly compressible, weak and thick organic soil deposits. As shown in Figure 1(a), a grid of vertical drains accelerates the rate of consolidation through radial and vertical drainage. The most common type of vertical drains used in construction is prefabricated vertical drains (PVDs), which create shorter drainage paths that will accelerate the consolidation process. At the same time, stiffer drains such as sand drains (sand compaction piles (SCPs)) and stone columns (gravel compaction piles (GCPs)) enhance the strength of the soft soil due to reinforcement action as well. The drain influence zone is a function of the drain spacing (S). The diameter of the influence zone (D_e) of the vertical drains installed in triangular and square patterns can be found as shown in Figures 1(b) and 1(c), respectively (Hansbo, 1980).

The smear effect and well resistance are also phenomena related to vertical drains. Once vertical drains are installed, the surrounding soil is disturbed. Hence, the horizonal permeability in that region is reduced; this condition is known as the smear effect. Well resistance mainly occurs due to the reduction of the drain cross-section (deterioration of the drain cross-section) and reduction of pore spaces by the intrusion of soil particles into the filter. This also delays the pore pressure generation and hence retards the consolidation process. In this study, the smear effect is taken into consideration, while well resistance is disregarded, as its effect is insignificant due to the relatively shorter drain length (Hansbo, 1997; Indraratna et al., 1994; Ratnayake, 1991).

In most research studies, embankment construction is considered to be an instantaneous process or a uniform ramp loading, as shown in Figure 2(a). However, in general, embankments are constructed over a period, gradually increasing the fill height similar to a loading condition, as shown in Figure 2(b). The loading is increased from 0 to q₀ over a time period of t_c, where q₀ and t_c represent the ultimate load and the monitoring period, respectively. However, the random load application shown in Figure 2(b) can be simulated approximately with linear loading in different stages, as shown in Figure 2(c). In this research, a practically applicable numerical method was successfully developed to handle the time-dependent loading and secondary consolidation using simple commonly carried-out geotechnical tests.

Barron (1948) introduced a mathematical model for radial consolidation with vertical drains for instantaneous loading. In his derivation, Barron extended Terzaghi’s one-dimensional (1D) consolidation theory to account for radial consolidation assuming equal-strain solution for the degree of consolidation and introduced complete formulae for radial consolidation, with or without peripheral smear and well resistance. Yoshikuni and Nakanado (1974) presented a numerical solution for the problem, and they stated that well resistance cannot be ignored in places where very soft, deep soil deposits are improved by vertical drains of a very small cross-sectional area. Hansbo (1980) presented a simple solution to the problem of smear and well resistance that gave results that were identical to those suggested by Barron (1948) and Yoshikuni and Nakanado (1974).

Hansbo et al. (1981) presented some case records related to different drain types and found that there was no specific significant difference concerning consolidation due to different installation techniques. Han and Ye (2002) developed a closed-form solution for computing the consolidation rates of stone-column-reinforced soft soil deposit with smear and well resistance. In their solution, they considered the stiffness difference between the stone columns and the surrounding soil, which was ignored in Barron’s solution.

Most researchers have analysed radial consolidation settlement by applying the theory of instantaneous loading. The general solution for 1D consolidation with time-dependent loading was first introduced by Schiffman (1958), and Olson (1977) extended Terzaghi’s conventional theory to predict the behaviour of soil under time-dependent loading for radial consolidation. Tang and Onitsuka (2000) also introduced a solution by the virtue of the impulse function method for consolidation by vertical drains under time-dependent loading considering both well resistance and smear action. Although Tang and Onitsuka (2000) developed a solution for single and multiple ramp loading cases, they were limited to single-layered ground. Hence, Tang and Onitsuka (2001) proposed a solution for double-layered ground with vertical drains under quasi-equal-strain conditions and also suggested that the method could further be extended to accommodate even multilayered ground conditions. Furthermore, studies by Zhu and Yin (2000) and Fox et al. (2003) are based on finite-element methods that incorporate time-dependent loading, large strains and complexity of soil layers. Indraratna et al. (2005a) introduced an analytical formulation of a modified consolidation theory that incorporates vacuum pressure and numerical modelling of soft clay stabilised with PVDs.

Recent research studies have been done on peat stabilisation using vertical drains and secondary consolidation in peaty soils. Yamazoe et al. (2020) did a study on the significant stress dependency of the coefficient of consolidation in peat, stabilised with vertical drains using Barron’s solution. It was concluded that for peaty soils with a high natural water content under significant consolidation pressure changes, Barron’s solution was less applicable, as it assumed a constant coefficient of consolidation. However, in the study done by Yamazoe et al. (2020), the simulation was also done by changing the coefficient of consolidation at different step loadings, and the results did not exhibit a greater deviation from field measurements, although a different coefficient of consolidation was used. Hence, Barron’s solution with a constant coefficient of consolidation was used in the suggested study and obtained reasonable results for the field measurements.

In this study, peat soils improved with sand columns, stone columns and PVDs were selected for a simulation. However, in a study done by Umaiyan and Muthukkumaran (2024), the effectiveness of a pervious concrete pile (PCP) in accelerating the rate of consolidation in unconsolidated soft clay deposits was explored. PCPs, sand drains and stone columns installed in two different soft soils (clay with high compressibility (CH) and clay with intermediate compressibility (CI)) were subjected to a set of laboratory consolidation tests, and the consolidation characteristics of the two soils were compared. It was found that, compared with the sand drain and stone column, the PCP provided a significant improvement in consolidation characteristics (coefficient of volume compressibility and coefficient of consolidation).

Li et al. (2021) conducted a series of conventional consolidation tests on undisturbed peat soil samples taken from Kunming for consolidation and secondary consolidation characteristics. They found out that with an increase in consolidation pressure, the coefficient of secondary compression (C_α) values of peat soil and general soft clay first rapidly reached the peak value and then decreased gradually. However, different from that of general soft clay, the peak C_α value of peat soil was higher due to the high organic content and the abundance of micropores in the organic matter. In this study, a constant C_α value was used for simulation, and it was recommended that variation in C_α be incorporated to obtain more accurate results.

Meng et al. (2021) and Qin et al. (2022) presented analytical solutions for the radial consolidation of a PVD foundation under unsaturated conditions by considering the smear effect, drain resistance and time-dependent loading. However, the method proposed in this paper was conducted on radial consolidation of saturated soils stabilised with vertical drains under time-dependent loading, on which fewer research studies had been done during the past few years.

Although the methods proposed for predicting predict settlement variation in soils under time-dependent loading are reasonably accurate, owing to the difficulties in practical implementations, designers prefer to use relatively simple solutions with normally conducted field test results. In current practice, the design of stone columns reinforced embankment is commonly based on the theoretical solution from Barron (1948), assuming equal-strain conditions and instantaneous loading. However, in the actual loading scenario, the surcharge loading is an arbitrary time-dependent loading. Therefore, a simple solution procedure for the radial consolidation theory of soils with multiple ramp loadings is essential for design engineers for practical applications. In most of past research studies, the original Barron equation is extended to accommodate various time-dependent loading conditions, resulting in a complicated solution process requiring special field and laboratory testing. However, this study uses the original Barron equation and introduces a simple methodology to be used in time-dependent loading.

The proposed methodology uses the original consolidation theory by Barron (1948) to calculate the variation of settlement, pore water pressure and degree of consolidation due to incremental step loading of embankments, constructed on compressible soft organic soil, stabilised with vertical drains. The following assumptions are made in the proposed methodology.

• The effect of vertical consolidation is not considered, as vertical drains accelerate the consolidation process through rapid radial drainage (Jamiolkowski et al., 1983).

• Well resistance is neglected, as its effect is insignificant due to the relatively shorter drain length (Hansbo, 1997; Indraratna et al., 1994; Ratnayake, 1991).

In the proposed method, Equations 1 and 2 by Barron (1948) for equal-strain conditions are used to find the average degree of consolidation (U_r) due to radial drainage without and with the smear effect, respectively. Equation 3, which is the extended equation by Barron (1948) for equal strain, is used to find the excess pore water pressure (u) at a point at a given time, with or without the smear effect.

where

C_h is the coefficient of consolidation in the radial direction; r_e is the radius of the influence zone; r_w is the radius of the vertical drain; and t is time.

r is the distance from the centre line of the vertical drain to the point of consideration of the pore water pressure variation; r_s is the radius of the smear zone; k_h is the coefficient of permeability in the horizontal direction; and k_s is the coefficient of permeability of the smear zone.

Theoretically, Equations 1–3 are valid only if the loading is instantaneous. However, in most practical applications, the applied load varies with time. Hence, as the initial step, the embankment load variation with time that was obtained from the actual field data was divided into stages, as shown in Figure 3. Each loading stage was then considered separately and discretised into several small instantaneous loadings applied at equal time intervals for all the stages, similar to the dynamic loading described by Thilakasiri et al. (1996) and Gunaratne et al. (1996). The procedure followed to analyse a single stage (stage I) is shown in Figure 4.

The total time duration of stage I is divided into N_I number of incremental step loadings, as shown in Figure 4. The total loading q_I of stage I is considered to be applied in equal incremental step loadings, which will result in an applied load of p_I at the beginning of each incremental step loading. p_I is estimated using Equation 11, and it is assumed that the applied loading increment is equal to the pore pressure increment (u_I) due to each incremental step loading. However, with increasing time, the pore pressure generated at the beginning (u_I) of each incremental step loading will start to dissipate due to the consolidation process. Hence, the remaining pore pressure value at a given time for each incremental step loading can be calculated as u_r,t using Equation 12. The corresponding average degree of consolidation (U_r,t) at a given time for an incremental step loading can be calculated using Equation 1 or 2. After each Δt time step, the next incremental loading step is applied within a loading stage (in this case, stage I). It is assumed that every incremental step loading is an instantaneous loading and the dissipation of the pore water pressure with time is independent of the next incremental step loading. Once the u_r,t value for each incremental step loading at a given time is obtained, the remaining excess pore water pressure from the previous loading increment is added after time Δt using the method of superposition, to obtain ∑u_r,t values (as shown in the flow chart in Figure 5) at a given time, where ∑u_r,t is the total remaining excess pore pressure at a given time of the entire layer under the effect of all the incremental step loadings of a given stage (in this case, stage I). The following equations were used in step loading calculations:

0 < t < t_c, where t_c is the monitoring period, which is any time after the start of the loading. However, Figure 4 shows that a t_c after the entire loading is applied. Once the total remaining excess pore water pressure at a given time under the effect of all the incremental step loadings up to that time (t_c) is obtained, the applied loading (P_m,t) at the same time is calculated using Equation 13. P_m,t will remain constant after reaching the maximum loading corresponding to that specific stage (for stage I, it would be q_I).

The analysis procedure adopted for loading stage I that is described above is shown in Figure 5 as a flow chart. The same method is incorporated into other stages (even stages such as stage II where the loading is kept constant) to obtain the total remaining excess pore water pressure (total u_r,t) and the total applied load (total P_t) of the entire layer at a given time due to the effect of all the stages (from stage I up to stage i, based on monitoring time, t_c),which can be calculated using Equations 15 and 16, respectively.

• j is any positive integer.

• m is the number of step loadings.

• The number of time steps required for time t_c is N, N = t_c/Δt.

• If N > N_I, same calculation procedure should be repeated for other stages.

Once the total remaining excess pore water pressure (total u_r,t) and the total applied load (total P_t) at a given time under the effect of all the stages up to the monitoring period (t_c) are calculated, the primary consolidation at that time is calculated from Terzaghi’s 1D consolidation theory. Equation 17 is used if the soil is in the over-consolidated state, and Equation 18 is used if the soil is in the normally consolidated state.

C_s is the recompression index; C_c is the compression index; e₀ is the initial void ratio; H is the thickness of the peat layer; σ₀ is the initial overburden pressure at the middle of the soft soil layer; and σ_p is the preconsolidation pressure.

As the next step, the average degree of consolidation (⁠$Ut′$⁠) due to the combined loading at a given time is calculated using the following equation:

Once the average degree of consolidation $Ut′$ of the layer exceeds a certain threshold value at time t′, the secondary consolidation is assumed to take place simultaneously with the primary consolidation. In the suggested method, the threshold value could be varied according to the soil type and the available data. For mineral soils, this value is considered to be in the range of 90%, and in organic soils, the secondary consolidation initiates much earlier than in inorganic soils (Mitchell et al., 2006). The secondary consolidation settlement at a given time is estimated using the log-time method (Equation 21) and then added to the corresponding primary consolidation settlement value at the same time. However, if the degree of consolidation value goes below the specified threshold value, the secondary consolidation ceases.

C_α is the coefficient of secondary compression; H is the thickness of the peat layer; e_p is the void ratio at the end of the primary consolidation; and t″ is the time at which the secondary consolidation is calculated.

The total settlement (S_T,t) at a given time is calculated using the following equation:

S*_s,t is applicable only if the average degree of consolidation (⁠$Ut′$⁠) is above the specified threshold value.

Once the total settlement (S_T,t) at a given time is calculated, graphs can be plotted to obtain the settlement variation with time.

Equation 3 is used to calculate the pore water pressure at a point r away from the centre line of the vertical drain, considering the presence or the absence of the smear effect. The pore water pressure value at a given time is calculated as described above in the section headed ‘Settlement calculation procedure: stage I’, and then the method of superposition is used to obtain the total excess pore water pressure at that point due to the combined loading of all the stages. A separate calculation procedure could be developed for this analysis similar to what is shown in Figure 5.

The study was performed to calculate the settlement variation of three embankment sections at the Colombo–Katunayaka Expressway (CKE) project and the Outer Circular Highway (OCH) project in Sri Lanka. The embankments were constructed on organic clayey deposits, and preloading with vertical drains (PVDs in section K5+600, SCPs in section K2+600 and stone columns (GCPs) in section FK0+560) was used to improve the soil properties of the soft soil layers. The field data monitored at the site using settlement plates were compared with the settlement variation obtained from the proposed method considering the presence and the absence of smear effects. The reduction in the stresses applied on the soft soil due to GCPs and SCPs was also taken into consideration by incorporating the stress concentration factor (SCF).

The embankment section FK0+560 at the OCH project was analysed. The subsoil of the selected embankment was obtained from borehole investigation reports and was composed of 12.5 m of organic clay layer overlying a medium sand layer. The above cohesive layers were normally consolidated, and the consolidation parameters C_s = 0.1063, C_c = 0.79,e₀ = 2.13 and C_h = 1.8 m²/year were obtained by performing conventional oedometer tests on undisturbed samples.

An embankment fill with a density of 20 kN/m³ and a 6.0 m height was constructed over 127 days with a settlement monitoring period of 400 days, as shown in Figure 6(a). Ground improvement was carried out with 0.7 m dia. GCPs installed in a square grid pattern with 1.5 m spacing. The general SCF used in practice depends on area replacement ratio; hence, an SCF of 5 was used for this section (Han and Ye, 2001; Mitchell and Huber, 1985). Depending on the SCF value, the fraction of the applied stress taken by the soil, μ_soil, was calculated as 0.5938. Furthermore, the used S value, which is the ratio between the radius of the smear zone and the radius of the vertical drain, was 2.5 based on the findings of Hansbo (1986) and Chai et al. (2001). A k_h/k_s value (k_h is the coefficient of horizontal permeability, and k_s is the coefficient of horizontal permeability of the smear zone) of 3 was used in the analysis based on the observations of Saye (2003).

The duration (Δt) of each incremental step loading was considered as 2 days, and the analysis was conducted by dividing the ramp loading into ten stages. Figure 6(a) compares the field-measured settlement and model-predicted settlement. Figures 6(b) and 6(c) show the predicted average pore water pressure variation and the degree of consolidation variation of the cohesive soil layer, respectively. However, the simulated pore water pressure could not be compared with field measurements, as the pore pressure measurements were not available for this embankment section. It is seen that the actual settlement variation agrees well with the predicted settlement obtained from the analysis. As shown in Figures 6(a)–6(c), the settlement, average excess pore pressure variation and the degree of consolidation with smear are less than the settlement without the smear effect as expected, due to the lower horizontal coefficient of consolidation in the smear zone.

Cases 2 and 3 were also analysed as described for case 1, and the basic input parameters used in the above two cases are listed in Table 1.

Figures 7 and 8 compare the field-measured and predicted results of the settlement variation of case 2 (K2+600 at CKE with SCP) and case 3 (K5+600 at CKE with PVD), respectively. In these two cases, it is also seen that the actual settlement variation agrees well with the predicted settlement obtained from the analysis for practical purposes. As shown in Figures 7 and 8, the settlement with smear is less than the settlement without the smear effect as expected, due to the lower horizontal coefficient of consolidation in the smear zone.

A separate spreadsheet was developed with macro coding to predict the pore pressure variation at a point at a given time, using the extended procedure by Barron (1948) described in the section headed ‘Proposed methodology’. Since excess pore water measurements were not available in the OCH and CKE projects in Sri Lanka, the results were compared with the field data measurements in the case study presented by Indraratna et al. (2005b). Figure 9 shows a cross-section of the embankment considered in the case study.

From the data obtained from the case study, it was found that the PVDs were installed in a triangular pattern with a spacing of 1.3 m and the embankment construction was done in two stages, where in stage I, the embankment was raised to a height of 2.57 m in 14 days. Then, following a rest period of 105 days, stage II was placed until the total embankment height was raised to 4.74 m in 24 days with a monitoring period of 400 days. The compacted unit weight of the fill was 20.5 kN/m³. The depth of the clay layer (very soft silty clay and soft silty clay) was 14 m. Piezometers were installed at a depth of 11.2 m below the ground surface, at a location 0.65 m away from the centre line. The C_c, e₀ and K_h values at a depth of 11.2 m were 0.83, 1.86 and 0.6 × 10⁻⁹ m/s, respectively. The average C_c, e₀ and γ values of the clay layer were 0.95, 2.455 and 15.5 kN/m³, respectively. The smear zone data S value and the k_h/k_s value were assumed as 4 (Chai et al., 2001; Hansbo, 1986) and 5 (Saye, 2003), respectively. The duration (Δt) of each incremental step loading was considered as 2 days, and the analysis was done by dividing the ramp loading into three stages. Figure 10 compares the pore water pressure results obtained from the proposed method, from the field study and the simulation process proposed by Indraratna et al. (2005b). Figure 11 compares the settlement variation obtained from the proposed method with the field data and the settlement variation proposed by Indraratna et al. (2005b). The basic data presented in the case study were used to predict the settlement along with few assumptions such as secondary compression (C_α) to determine the coefficient of secondary consolidation settlement. It was seen that the simulated pore water pressure variation and the predicted settlement variation matched with an acceptable accuracy with the field measurements and the results of the method proposed by Indraratna et al. (2005b).

This paper explains the effect of radial consolidation under arbitrary time-dependent loading of a soft layer with or without the smear effect using a simple methodology based on the theory proposed by Barron (1948). The proposed method is applied in comparing the measured settlements of three embankment sections stabilised with GCPs, SCPs and PVDs. Further, pore pressure measurement at a point and the settlement variation from the proposed method are compared with a field measurement reported in a case study by Indraratna et al. (2005b). It is seen that the simulated pore water pressure variation at a point and the average settlement variation match with an acceptable accuracy with the field measurements using this simple proposed methodology.

One of the best advantages of using the proposed method is that various time-dependent loading conditions can be considered in the formulation even when the loading is kept constant in the loading process. This can be considered a relatively simple, methodical procedure using the superposition concept for consolidation with radial drainage that can be implemented in field applications involved in practical ground improvements at any time during the loading process or afterwards. The method can be applied for various embankment sections with different ground improvement techniques (SCPs, GCPs or PVDs), agreeing with the field measurements for computing pore water pressure and settlement.

The major importance of this method is the adequacy of the geotechnical parameters obtained from a simple ground investigation programme and the ability to incorporate easily the effect of secondary consolidation, which is significant in very soft organic soils. However, this simulation can also be applied to soils that do not undergo secondary consolidation settlement. It can be concluded that the smear effect is not very much significant for the variation of settlement, as shown in Figure 6(a), whereas the variation of pore water pressure with and without the smear effect is significant, as shown in Figure 6(b). However, the proposed methodology can be further modified in the future for application to more than one single cohesive layered ground. Moreover, if the variation of C_α can be modelled (Li et al., 2021), it could also be easily incorporated into the present methodology, giving more accurate results.

The authors would like to acknowledge Metallurgical Group Corporation, China, for funding this project.