Excess pore pressure variation

Replacing the initial excess pore water pressure ${\overline{u}}_0$, where $\mathrm{\Delta }{\overline{h}}_0$ with ${\gamma }_w\mathrm{\Delta }{\overline{h}}_0$, where $\mathrm{\Delta }{\overline{h}}_0$ is the initial average hydraulic head increase, the variation of the hydraulic head increase h outside the zone of smear $\left(D/2\ge \right.\rho \ge$d_s/2), based on validity of Darcy’s law, becomes:

$\mathrm{\Delta }h=\frac{\mathrm{\Delta }{\overline{h}}_0}{\mu D^2}\left(1–{\overline{U}}_h\right)\left[\begin{array}{l}{D}^{2}ln\frac{2\rho }{d_s}–\frac{4{\rho }^2–d_s^2}{2}+\frac{k_h}{k_s}\left(D^{2}ln\frac{d_s}{d_w}–\frac{d_s^2–d_w^2}{2}\right)\\ +\frac{k_h}{q_w}\pi z\left(2l–z\right)\left(D^2–d_w^2\right)\end{array}\right]$

Inside the zone of smear for ρ = d_w/2 the correlation becomes:

$\mathrm{\Delta }h=\frac{\mathrm{\Delta }{\overline{h}}_0}{\mu D^2}\left(1–{\overline{U}}_h\right)\left[\begin{array}{l}{D}^{2}ln\frac{d_w^2}{d_s}–\frac{d_w^2–d_s^2}{2}+\frac{k_h}{k_s}\left(D^{2}ln\frac{d_s}{d_w}–\frac{d_s^2–d_w^2}{2}\right)\\ +\frac{k_h}{q_w}\pi z\left(2l–z\right)\left(D^2–d_w^2\right)\end{array}\right]$

The variation of hydraulic head increase outside the zone of smear in exponential flow ($D/2\ge \rho \ge d_s/2$), assuming that $i\le i_l$, becomes:

$\mathrm{\Delta }h=\frac{\mathrm{\Delta }{\overline{h}}_0}{2^n\beta n^2}\left(1–{\overline{U}}_h\right)\left\{\begin{array}{l}F\left(\frac{2\rho }{D}\right)–F\left(\frac{d_s}{D}\right)+\frac{{\kappa }_h}{{\kappa }_s}\left[F\left(\frac{d_s}{D}\right)–F\left(\frac{d_w}{D}\right)\right]\\ +\frac{{\kappa }_h}{q_w}\pi z\left(2l–z\right)\left(1–\frac{1}{n}\right){\left(1–\frac{d_w^2}{D^2}\right)}^{1/n}{\left(\frac{d_w}{D}\right)}^{\left(1–1/n\right)}\end{array}\right\}$

where

$F\left(x\right)=x^{1–1/n}\left[1–\frac{1–1/n}{3n–1}{x}^2–\frac{{\left(1–1/n\right)}^2}{2\left(5n–1\right)}{x}^4–\frac{{\left(1–1/n\right)}^2\left(2n–1\right)}{6n\left(7n–1\right)}{x}^6–\cdot \cdot \cdot \right]$

in which the variable x represents 2ρ/D, d_s/D, and d_w/D.

Inside the zone of smear for ρ = d_w/2 the correlation becomes

$\mathrm{\Delta }h=\frac{\mathrm{\Delta }{\overline{h}}_0}{2^n\beta n^2}\left(1–{\overline{U}}_h\right)\left\{\begin{array}{l}F\left(\frac{d_w}{D}\right)–F\left(\frac{d_s}{D}\right)+\frac{{\kappa }_h}{{\kappa }_s}\left[F\left(\frac{d_s}{D}\right)–F\left(\frac{d_w}{D}\right)\right]\\ +\frac{{\kappa }_s}{q_w}\pi z\left(2l–z\right)\left(1–\frac{1}{n}\right){\left(1–\frac{d_w^2}{D^2}\right)}^{1/n}{\left(\frac{d_w}{D}\right)}^{\left(1–1/n\right)}\end{array}\right\}$

Hydraulic gradient

The hydraulic gradient in Darcian flow outside the zone of smear ($D/2\ge \rho \ge d_s/2$) becomes:

$i=\frac{\mathrm{\Delta }{h}_0}{D}\left(1-{\overline{U}}_h\right)\frac{k_h/k_s}{\mu }\left(\frac{D}{\rho }-\frac{4\rho }{D}\right)$

The hydraulic gradient outside the zone of smear ($D/2\ge \rho \ge d_s/2$) in exponential flow, assuming that $i\le i_l$, becomes:

$i=\frac{\mathrm{\Delta }{\overline{h}}_0}{D}\left(1–{\overline{U}}_h\right){\left[\frac{{\kappa }_h/{\kappa }_s}{4\alpha \left(n–1\right)}\left(\frac{D}{2\rho }–\frac{2\rho }{D}\right)\right]}^{1/n}$

Influence of one-dimensional consolidation

Vertical drains are generally installed in places where the thickness of the fine-grained soil layer is so large that the influence on the consolidation process of one-dimensional consolidation by pore water escape in the vertical direction between the drains can be ignored. Because the difference in result in the beginning of the consolidation process between the analytical results based on Darcian and non-Darcian flow can be ignored (cf. Fig. 2.10), the contribution to the average consolidation by vertical pore water escape between the drains can be based on the classical Terzaghi solution with Darcian flow ${\overline{U}}_v=\left(2/l\right)\sqrt{c_{v}t/\pi }$, valid when ${\overline{U}}_v\le$ 50% (l according to Fig. 2.1).

According to Carillo’s relation ${\overline{U}}_{\mathrm{tot}}={\overline{U}}_v+{\overline{U}}_h–{\overline{U}}_v{\overline{U}}_h$, the total average consolidation, including the effect of vertical drainage (horizontal pore water flow) and one-dimensional consolidation (vertical pore water flow), can be expressed by the relation:

${\overline{U}}_{\mathrm{tot}}=1–\left(1–\frac{2}{l}\sqrt{\frac{c_{v}t}{\pi }}\right)exp\left(–\frac{8c_{h}t}{{\mu }_{\mathrm{a}\mathrm{v}}{D}^2}\right)$

  (2.16)

in Darcian flow, and

${\overline{U}}_{\mathrm{tot}}=1–\left(1–\frac{2}{l}\sqrt{\frac{c_{v}t}{\pi }}\right){\left[1+\frac{\lambda t}{{\alpha }_{\mathrm{a}\mathrm{v}}{D}^2}{\left(\frac{{\overline{u}}_0}{D{\gamma }_w}\right)}^{n–1}\right]}^{1/\left(1–n\right)}$

  (2.17)

in exponential flow.

A large number of case histories have been publicized (see, for example, the special issue of Géotechnique on vertical drains from March 1981 and the publications by Lo [1991] and Hansbo [1994, 1997a and 1997b]). In the following, results from well-instrumented test areas in different parts of the world will be analyzed on the basis of both Darcian and non-Darcian flow.

Equivalent diameter of band drains

The first type of band drains, the so-called cardboard wick, 100 mm in width and 3 mm in thickness, invented and introduced on the market by the Swedish Geotechnical Institute, was assumed by Kjellman (1947) to have an equivalent diameter of 50 mm. The author (Hansbo, 1979) showed that the process of consolidation for a circular drain and a band drain is very nearly the same if the circular drain is assumed to have a circumference equal to that of the band drain, that is,

$d_w=2\left(a+b\right)/\pi$

  (2.18)

where a = the thickness of the drain and b = the width of the drain.

A comparison of the degree of consolidation obtained in the two cases, band drains and circular drains, is shown in Fig. 2.3. This comparison indicates that the equivalent diameter ought to be put somewhat smaller than according to Eq. (2.18). Therefore, Rixner et al. (1986) propose that the equivalent diameter should be put equal to $d_w=\left(a+b\right)/2$. However, the difference in result decreases with increasing time of consolidation and is insignificant in comparison with the influence on the result exerted by the choice of other consolidation parameters to be applied in the design.

Effect of well resistance

Because drains nowadays are frequently installed to great depths, well resistance has become a matter of increasing interest. This is understandable as well resistance in such cases can cause a serious delay in the consolidation process.

Before a certain drain type is accepted for a job, evidence ought to be presented that the drain fulfills the requirements on discharge capacity assumed in the design. Generally, appropriate laboratory testing can give sufficient evidence of the discharge capacity to be expected under field conditions, but in the case of a drain type never before used in practice, full-scale field tests are recommended. Full-scale field tests also serve the purpose of showing that the drains are strong enough to resist the strains subjected to them during installation. Moreover, compression of the soil in the course of consolidation settlement entails folding of the drain, which may bring about clogging of the channel system. This effect is difficult to discern in a laboratory test (cf. Lawrence and Koerner, 1988).

Discharge capacity tests on band drains installed in soil on a laboratory scale and subjected to increasing lateral pressure have resulted in the values presented in Fig. 2.4. In the National Authority for Electrical Energy (ENEL) tests (Jamiolkowski et al., 1983) the drains were tested in full scale. In the Chalmers Tekniska Högskola (CTH) tests (Hansbo, 1983) and in the Kyoto University (KU) tests (Kamon, 1984) the drains were tested with reduced width (40 and 30 mm, respectively). The results obtained reveal a great influence on the discharge capacity of the lateral consolidation pressure.

There are several reasons why the discharge capacity of a drain may become low: siltation of the channels in the core of band drains, unsatisfactory drain types with too low a discharge capacity, necking of drains, and so on. Back-calculated values of discharge capacity of drains under field conditions have been reported to be quite low for certain types of band drains without filters (Hansbo, 1986; Chai et al., 1996). If the drain installation is taking place in clay deposits, most of the band drains marketed today have a high enough discharge capacity (q_w > 150 m³/year) to become negligible in the design (cf. Hansbo, 1986; 1994). However, drain installation in more high-permeable soils than clay may require a much higher discharge capacity (cf. Fig. 2.5). The influence of well resistance decreases with increasing time of consolidation.

The filter sleeve surrounding the core must be strong enough to resist the tension and the wear and tear that takes place during drain installation. Moreover, the filter pore size should be small enough to prevent intrusion of fine material into the channels of the core, which leads to clogging effects. Filter deterioration in organic soils will cause a time-bound reduction of the discharge capacity (Koda et al., 1986; Hansbo, 1987).

According to Eq. (2.5), an acceptable delay t (%) in time of average consolidation requires a discharge capacity of minimum¹

$q_w=200\pi l^2{k}_h/\left\{3\mathrm{\Delta }t\left[ln\left(D/d_w\right)–0.75\right]\right\}$

  (2.19)

For drains with a discharge capacity of 500 m³/year, the delay in the time of consolidation at depth z = l (cf. Fig. 2.2) in various kinds of soil according to Eq. (2.5) is exemplified in Fig. 2.5 (Cortlever and Hansbo, 2004).

For partially penetrating drains, consolidation of the lower part of the layer that is penetrated by the drains will be somewhat delayed by inflow of pore water from the underlying part where the excess pore water pressure is higher. The delay will depend on the discharge capacity, the drain length, and the drain spacing (Runesson et al., 1985). It can be assumed to affect the degree of consolidation to a height above the drain tips approximately equal to the drain spacing.

As a rule, investigations on a laboratory scale of the discharge capacity of a drain ought to be carried out in a way that simulates field conditions as closely as possible, preferably embedded in the soil and at the soil temperature in question. Laboratory testing devices, which are presented in the Eurocode on vertical drainage under preparation (probably accepted and published in 2005/2006 as EN), will most probably be accepted as international standard. One of these devices (Apparatus No. 2, EN/ISO 12958) and also a device for testing the discharge capacity of buckled drains are illustrated by Cortlever and Hansbo (2004).

The test areas constructed at Skå-Edeby in 1957 and the depth of the soft clay deposits are shown in Fig. 2.6. Among the different test areas, area IV has no vertical drains while the others are provided with fully penetrating vertical sand drains, 0.18 m in diameter.

The soil consists of postglacial clay to a depth of about 6 m underlain by glacial, varved clay to a depth of about 12.5 m. The groundwater table is situated at a depth of approximately 1 m but varies with the time of year. The dry crust has a thickness of about 1.5 m. The geotechnical characteristics of the clay are shown in Fig. 2.7. According to the results from oedometer tests, the clay below the dry crust is normally consolidated. The compression ratio² $CR={\varepsilon }_2/log2=C_c/\left(1+e_0\right)$ according to the oedometer tests (Hansbo, 1960) was found equal to about 0.24 at 2 m depth, 0.35 (0.48) at 5 m depth, and 0.32 (0.63) at 8 m depth (the values in brackets are modified with regard to sample disturbance).

The corresponding oedometer modulus M for a normally consolidated clay can be found by the relation $CRlog\left[\left({\sigma }_c^{\prime }+\mathrm{\Delta }\sigma \right)/{\sigma }_c^{\prime }\right]=\mathrm{\Delta }\sigma /M$, where Δσ is the stress increase due to loading and σ_c′ is the preconsolidation pressure.

Excess pore pressure dissipation

The excess pore pressure distribution below the test area, observed at a depth of 5 m after termination of loading, is shown in Fig. 2.8. This can be used in determining the parameters A and B in Skempton’s pore pressure equation $u=B\left[{\sigma }_3+A\left({\sigma }_1–{\sigma }_3\right)\right]$ where σ₁ and σ₃ are the major and minor principle stresses induced by loading (Skempton, 1954). As we are dealing with water-saturated clay, B = 1. If the soil immediately after termination of loading behaves as an elastic medium with Poisson’s ratio ν = 0.5, we have at 5 m depth, according to the solution presented by Love (1929), σ₁ = 0.98q and σ₃ = 0.60q below the center, σ₁ = 0.95q and σ₃ = 0.49q 10 m from the center, and σ₁ = 0.50q and σ₃ = 0.06q 20 m from the center. Inserting q = 27 kN/m² and considering the observations made, this yields A ≈ 0.85.

The analysis of the excess pore pressure dissipation has been based on the consolidation characteristics shown in Table 2.1. In the case of non-Darcian flow, the limiting value of the hydraulic gradient i_l (where the flow turns from exponential to linear relation of the hydraulic gradient) is chosen equal to 5 (Fig. 2.9).

Table 2.1

Parameters used in the consolidation analysis

Depth (m)	0–1.0	1.0–1.5	1.5–3	3–5	5–7	7–9	9–11	11–12.5
σϖ (kPa)	27	27	27	26.5	26.5	26	26	25
σc′ (kPa)	OC	25	23	28	39	51	64	75
M (kPa)	10,000	400	250	240	250	300	400	500
s (m)	0.01	0.03	0.16	0.24	0.24	0.23	0.22	0.15
k (m/year)	0.031	0.031	0.025	0.022	0.018	0.018	0.017	0.016
κ (m/year)	0.020	0.020	0.016	0.014	0.0115	0.0115	0.011	0.0095

The k and M values given in Table 2.1 are chosen according to Larsson (1986). Based on trial and error, the κ value is chosen equal to about 0.65 times the k value.

The remaining excess pore water pressure distribution throughout the clay layer below the center of the circular embankment after 1.5, 14, 25, and 45 years of loading is compared with the analytical results obtained according to the two methods of analysis of one-dimensional consolidation (Figs. 2.10. and 2.11).

Initial excess pore water pressure is assumed equal to 25 kPa. Because of the width of the embankment, 35 m, the influence of two-dimensional consolidation at the center of the embankment has been ignored.

As can be seen, the result based on non-Darcian flow is in much better agreement with the excess pore pressure observations than the result based on Darcian flow. There is a possibility that the excess pore water pressure below the center of the load has been affected by pore pressure dissipation due to horizontal pore water flow, but this possibility only strengthens the assumption of non-Darcian flow.

The total primary consolidation settlement, based on the oedometer moduli M given in Table 2.1 and the stress increase σ_v due to loading, can be estimated at 1.28 m, while the total primary compression of the clay layer between 2.5 and 7.5 m can be estimated at 0.59 m. In the analysis, the effect of load reduction due to hydraulic uplift when part of the load sinks below the groundwater table has been ignored. Thus, owing to submergence of the fill during the course of settlement the load, in kN/m², will be reduced successively by about γ′s, where γ′ equals the effective unit weight, in kN/m³, of the soil above the groundwater table and of the part of the fill that sinks below the groundwater table and s is the settlement, in meters, of the soil surface. However, one must keep in mind that the groundwater level may vary considerably with time of the year and also that the groundwater level after completed consolidation may differ from its original position. The settlement reduction caused by submergence of the fill may also be fully compensated by secondary consolidation.

The settlement observed in the test area is shown in Fig. 2.12. The settlement of the ground surface, observed after 45 years of loading, equals 1.10 m, while the compression of the clay layer between 2.5 and 7.5 m of depth equals 0.50 m. According to Asaoka (1978), the total primary consolidation settlement s_p can be estimated from the relation $s_i=0.300+0761s_{i–1}$, which yields s_p = 1.26 m ($s_{i–1}\to s_i$), and the compression s_p of the clay layer between 2.5 and 7.5 m of depth from the relation $\mathrm{\Delta }{s}_i=0.119+0.815\mathrm{\Delta }{s}_{i–1}$, which yields s_p = 0.64 m.

Looking at the remaining excess pore water pressure after 25 and 45 years of consolidation, we find the average degree of consolidation equal to, respectively, 69% and 79%. This leads in both cases to a total consolidation settlement of 1.39 m, about 10% larger than the estimated primary consolidation settlement. For the layer between the depths 2.5 m and 7.5 m, the average degree of consolidation after 25 and 45 years of consolidation becomes, respectively, 62% and 73%, which in both cases would lead to a primary compression of 0.69 m, about 8% larger than the estimated compression according to Asaoka. Considering the excess pore pressure distribution after 100 years of loading, the average degree of consolidation would equal 99% in Darcian flow and 91% in non-Darcian flow. Based on excess pore pressure dissipation in non-Darcian flow, the total settlement achieved after 100 years of loading would become 1.26 m, while the compression of the layer between the depths 2.5 and 7.5 m would become 0.63 m. Ignoring the parameters of the dry crust, the input parameters given in Table 2.1 yield average values of the coefficient of consolidation kM/γ_w = c_v ≈ 0.61 m²/year and κM/γ_w = λ ≈ 0.39 m²/year.

The back-calculated value of c_v is almost 4 times the value determined by oedometer tests. In a case like this with a test area of diameter 35 m placed on a clay layer of 12.5 m thickness, one has to consider the contribution of pore water escape in outward radial direction. This can explain the difference between the oedometer value and the back-calculated value of c_v. Moreover, the prolonged consolidation process increases considerably the settlement contribution by secondary consolidation during the primary consolidation period.

Lilla Mellösa test area

The test area at Lilla Mellösa that has no vertical drains was installed in 1947 and the results obtained have been followed up ever since. Together with the test area at Skå-Edeby without vertical drains, presented in the preceding subsection, it belongs to the world’s oldest test areas.

The soil conditions at the Lilla Mellösa test field have been described in detail by Chang (1981). The soil description differs somewhat between Chang (1981) and Larsson (1986). Briefly, underneath a 0.8 m thick dry crust the soil consists of organic clay to a depth of approximately 6 m, underlain first by homogeneous gray clay to a depth of 10 m and then by gray, varved clay to a depth of 14 m, resting on a thin layer of sand on rock (Fig. 2.13).

The natural water content and the liquid limit decreases from about 100% to 130% at the top to about 70% to 90% at the bottom. The unit weight at 3 m depth is lowest, about 14 kN/m³, and increases almost linearly with depth to about 20 kN/m³ at 14 m depth. The undrained shear strength is lowest at 3 m depth, about 8 kPa, and increases almost linearly with depth to about 18 kPa at 14 m depth. The sensitivity increases with depth from about 10 to 20.

The ground surface level is situated at about + 7.0 m. According to the observations presented by Chang, the groundwater level has varied from about + 6.0 m to about + 6.55 m. The clay can be considered as normally consolidated.

The test area without vertical drains is square with a base width of 30 m. It was filled up with 2.5 m of gravel. The top width of the fill is 22.5 m. It has been provided with settlement meters and pore pressure meters. The unit weight of the gravel fill during the filling operation was determined and found equal to 17 kN/m³. This yields an initial load on the test area of 42.5 kN/m².

Excess pore pressure dissipation

The results obtained in the two test areas included in this paper, Lilla Mellösa and Skå-Edeby, have been analyzed by Larsson (1986) and the results obtained at Lilla Mellösa by Claesson (2003). In Larsson’s analysis, based on the effect of creep on excess pore water pressure dissipation and settlement in the area at Lilla Mellösa that has no drains, the results show good agreement with the observations after 21 and 32 years of loading.

For the analysis, it is important to apply the correct input parameters M, k, κ, n, and i_l, as well as the correct values of the initial excess pore pressure distribution. Since no observations of deviation from Darcy’s law at small hydraulic gradients has been made on Lilla Mellösa clay, the assumption used in this analysis will be based on the findings related to Skå-Edeby clay, that is, n = 1.5 and i_l = 5. The values of M are selected from the results of oedometer tests corrected with regard to ring friction, published by Chang (1981), taking into account the load increment obtained at various depths underneath the fill. The k values are in agreement with the values published by Larsson (1986). The ratio κ/k is chosen equal to 2/3, based on the experience from the undrained test area at Skå-Edeby. The values chosen and the initial vertical stress increase σ_v below the center of the test area are presented in Table 2.2.

Table 2.2

Parameters used in the consolidation analysis

Depth (m)	0–1.0	1.0–1.5	1.5–3	3–5	5–7	7–9	9–11	11–14
M (kPa)	9000	200	200	190	160	240	280	200
Δσv (kPa)	42	42	41	41	40	39	38	33
s (m)	0.01	0.10	0.31	0.43	0.50	0.32	0.27	0.50
k (m/year)	0.020	0.020	0.025	0.018	0.015	0.022	0.03	0.03
κ (m/year)	0.013	0.013	0.017	0.012	0.010	0.015	0.02	0.02

In the analysis, the initial excess pore pressure distribution with depth has been based on the vertical stress distribution with depth below the center of the test area, published by Chang (1981). Of course, this assumption can be questioned. We do not know the magnitude of the pore pressure coefficient A in Skempton’s pore pressure equation. As in the case of the Skå-Edeby test area, we are dealing with water-saturated clay, and thus B = 1. In our case, A is also assumed equal to 1. In consequence, the initial excess pore water pressure has been considered equal to the vertical stress increase caused by the load, decreasing almost linearly with depth from 42 kPa at the ground surface to 32 kPa at the bottom of the clay layer. In reality, the value of A could be higher or lower depending on the loading conditions. As previously shown, the value of A was found equal to 0.85 in test area IV at Skå-Edeby with 1.5 m of fill, corresponding to a load of 27 kN/m². In our case the fill height is 2.5 m and, from experience of a number of observations, the value of A increases with increasing load. In, for example, the Gothenburg region, A values have been noticed that increase with increasing load from 0.7 to 1.5 (Hansbo, 1994).

The result of the analysis, assuming A = 1, is compared with pore pressure observations carried out 21, 32, and 56 years of loading (Fig. 2.14). The pore pressure observations after 21 years of loading indicate that in reality the pore pressure coefficient A > 1. It could also be that the groundwater level at the time of observation was higher than usual and that the oedometer modulus (representing the coefficient of consolidation) of the top layer, 1 m in thickness, is much lower than assumed. From Fig. 2.14 it can be concluded that the agreement between observations and the results obtained by assuming non-Darcian flow is far better than by assuming Darcian flow.

In the analysis of the pore pressure distribution, no consideration has been paid to the fact that part of the initial load will be subjected to hydraulic uplift due to settlement, causing load reduction. As previously pointed out, the groundwater level may vary considerably with time of the year and the groundwater level after completed consolidation may differ from its original position. In addition, the density of the fill may change with time. However, as previously mentioned, the excess pore water pressure is influenced by the magnitude of the pore pressure coefficient. It is therefore important to study the excess pore water pressure distribution during the loading operation to make an evaluation of the pore pressure coefficients possible. Otherwise, a direct comparison between settlement and pore pressure dissipation becomes rather intricate.

The Skå-Edeby test field was arranged for the main purpose of studying vertical sand drain and band drain behavior. As shown in Fig. 2.6, it originally contained four circular test areas, three of which were provided with sand drains, 0.18 m in diameter, with drain spacing varying from 0.9 to 2.2 m, and one without drains. Later on, a test area was arranged with band drains of type Geodrain, with drain spacing equal to 0.9 m. The test areas were loaded with 1.5 m of gravel (q = 27 kN/m²) except for test area III that was loaded with 2.2 m of gravel (q = 39 kN/m²). The geology at the site and the geotechnical properties of the different test areas were described in detail by Hansbo (1960). The results obtained have been presented in several publications (Holtz and Broms, 1972; Larsson, 1986).

The compression and consolidation characteristics of the clay in the test areas provided with vertical drains were determined on samples taken at depths 2, 5, and 8 m. The results obtained are shown in Table 2.3 (Hansbo, 1960).

The average value of the coefficient of consolidation c_h becomes 0.17 m²/year. The coefficient of consolidation in horizontal pore water flow c_h, which was determined in an oedometer test where drainage was allowed only through a central drain, was 0.7 m²/year. The corresponding permeability value k_h was estimated at 0.03 m/year.

The settlement versus time relationship for the test areas with sand drains and no drains is presented in Fig. 2.15. To facilitate the interpretation of the test results, the settlement curves have been revised to represent a common depth of 12.2 m.

The degree of consolidation in test area III, which was unloaded, is not as well defined as for test areas I and II where we have entered into a stage of secondary consolidation. All the same, a good estimate can be made, based on the shape of the settlement curve. Now the discharge capacity q_w of sand drains, 0.18 m in diameter, can be estimated at about 100 m³/year based on sand permeability. According to the results of the oedometer tests $c_h/c_{v,\mathrm{a}\mathrm{v}}=$ 4. Assuming a zone of smear d_s = 2d_w = 0.36 m and a ratio ${\kappa }_h/{\kappa }_s=k_h/k_s=$ 4, an acceptable agreement is obtained if a successively decreasing value of c_h is applied, as shown in Table 2.4.

The Ū_h values given in Table 2.4 are determined from the measured values of Ū_tot and Ū_v according to Carillo’s equation. The Ū_v values are estimated from the test area at Skå-Edeby without vertical drains, previously described, assuming a final primary settlement of 1.26 m. However, the assumption that Ū_v is unaffected by the disturbance due to drain installation, which undoubtedly causes a reduction of the coefficient of consolidation c_v, is most certainly overoptimistic.

The λ values are more consistent than the c_h values throughout the consolidation process. The c_h values are higher at the beginning of the consolidation process than at the end and the overall value somewhat lower than the coefficient of consolidation c_h determined in the laboratory (0.7 m²/year).

According to Eq. (2.19), the consolidation process in exponential flow depends on the initial excess pore water pressure ${\overline{u}}_0$. This is not the case in Darcian flow, Eq. (2.18). In test area III, the initial excess pore water pressure is higher than in test area II. From the test results in areas II and III, we find the best agreement by choosing the values given in Table 2.5. The values of c_h are higher in test area III than in test area II.

The efficiency of using occasional overload to eliminate secondary long-term settlement is demonstrated by the results obtained in test area III. With the load used in test area III, 39 kN/m², the primary settlement can be estimated at 1.5 m. As can be seen in Fig. 2.15, secondary consolidation did not start until several years after the removal of the occasional overload.

The excess pore pressure dissipation in test areas II and III with different loading condition is shown in Fig. 2.16. In the analysis based on Darcian flow, the consolidation process becomes independent of the magnitude of the load applied, which is not the case when the analysis is based on exponential flow. As can be seen, the best agreement between theory and observation is obtained by assuming exponential flow with the exponent n = 1.5.³

In test area V at Skå-Edeby (Fig. 2.17), band drains of type Geodrain with paper filters were installed with 0.9 m spacing. The initial load, 27 kN/m², was doubled after 3.5 years of consolidation. In a case like this, the following procedure of settlement analysis is followed.

The degree of consolidation Ū₁ and the settlement s₁ achieved under load q₁ at time t₁ when the additional load q₂ is being placed is calculated first. The part of the load q₁ that is still producing primary consolidation settlement (i.e., $\mathrm{\Delta }q=q_1\left(1–{\overline{U}}_1\right)$) is added to q₂ and the primary settlement process to be added after time t₁ under load q + q₂ is added to the settlement s₁. According to Eq. (2.20), the equivalent diameter of Geodrain (100 mm in width and 4 mm in thickness) is equal to d_w = 0.066 m. Assuming a zone of smear d_s = 0.19 m and $k_h/k_s={\kappa }_h/{\kappa }_s=4$, the best fit between observations and theory is obtained by applying the values c_h = 0.45 m²/year and λ = 0.23 m²/year with the exponent n = 1.5 (Fig. 2.18).

The λ value 0.23 m²/year agrees fairly well with the value obtained in test area I with 0.9 m drain spacing (λ = 0.26 m²/year) and is an indication of increasing average disturbance of the clay with decreasing drain spacing.

The results obtained at Skå-Edeby show that the values of the coefficients of consolidation are smaller than can be expected from the values arrived at in the undrained area. This is the case in spite of the fact that the coefficient of consolidation is normally higher in horizontal pore water flow than in vertical pore water flow due to the structural features of the soil (e.g., due to the existence of sand and silt layers). This obviously depends on the fact that the drain installation causes disturbance effects leading to a reduction of the coefficient of consolidation. The disturbance is evidenced by the fact that the sand drain installation at Skå-Edeby in itself induced excess pore water pressure of maximum 40 kPa at 0.9 m drain spacing, 35 kPa at 1.5 m drain spacing, and 20 kPa at 2.2 m drain spacing (Hansbo, 1960).

Örebro test field

Another example of the calculated versus observed consolidation process based on settlement is given by the results obtained at the Örebro test field, Sweden. This test field, 125 m by 45 m, was arranged just outside Örebro for comparing the efficiency of prefabricated band-shaped drains (of types Geodrain and Alidrain) with that of displacement type sand drains. The test field was divided into three sections of equal size with drain spacing 1.1, 1.4, and 1.6 m in an equilateral triangular pattern. The test area was loaded with 2.2 m of sand and gravel (load q = 40 kN/m²).

The soil consists of gray, lightly overconsolidated clay, about 8 m in thickness. The clay layer is slightly organic to a depth of 1–2 m and is varved, with silt seams, in its lower part below a depth of about 5 m. The average virgin compression ratio increases from about CR = 0.4 just below the dry crust to about CR = 0.65 in the lower part of the clay layer. The primary settlement, calculated based on the compression ratios and an overconsolidation of about 5 kPa (below the dry crust), was estimated at 1.1 m. The coefficient of consolidation according to oedometer tests varies from a minimum of c_v = 0.06 m²/year to a maximum of 1 m²/year. An estimated average is c_v = 0.2 m²/year.

The coefficient of consolidation in the case of horizontal pore water flow was not determined.

The settlement process observed in the test sections provided with Geodrain and Alidrain was very nearly the same. The results obtained in the test sections provided with Geodrains and sand drains are shown in Fig. 2.19.

Using Asaoka’s method for determination of the primary consolidation settlement, we find in the test area provided with Geodrains $s_i=$ 0.1809 + 0.8343s_i–1, which yields ($s_{i–1}\to s_i$)s_p = 1.09 m, and for the test area provided with sand drains $s_i=$ 0.4312 + 0.6119s_i–1, which yields $s_p=$ 1.11 m. These values are in excellent agreement with the settlement estimated on the basis of the oedometer tests.

The theoretical analysis of the consolidation process in the case of sand drains is based on the following assumptions: d_w = 0.18 m, d_s = 0.36 m. The corresponding values for Geodrain are: d_w = 0.066 m, d_s = 0.19 m. Inserting the parameters $k_h/k_s={\kappa }_h/{\kappa }_s=$ 4, l = 4 m, and c_v = 0.2 m²/year, the best fit of c_h and λ with observations is given in Table 2.6.

The agreement between field observations is better according to Darcian flow at 1.1 m drain spacing, while the agreement according to exponential flow with n = 1.5 is better at 1.6 m drain spacing.

The result of the back-analysis shown in Table 2.6 indicates that the flow condition prevailing in the soil of the Örebro test area follows Darcy’s law better than the exponential correlation. However, as can be seen in Fig. 2.19, the theoretical correlations based on the average of the consolidation parameters shown in Table 2.6, that is, c_h = 1.15 m²/year and λ = 0.58 m²/year, show better general agreement between observations and the exponential flow theory.

Bangkok test field

In connection with the planning of a new international airfield in Bangkok, Thailand, three test areas were arranged to form a basis for the design of soil improvement by preloading in combination with vertical drains. The results of the settlement observations in two of these test areas, TS 1 and TS 3, (Hansbo, 1997b) showed a better agreement with Eq. (2.6) than with Eq. (2.5). In this subsection, the results obtained in test area TS 3 will be examined in detail.

The crest width of TS 3 is 14.8 m (square) and the bottom width is 40 m. It is provided with an approximately 10 m wide loading berm of 1.5 m thick. The fill placed on the area amounts to a maximum of about 4.2 m, corresponding to a load of about 80 kN/m².

The drains of type Mebradrain were installed to a depth of 12 m in a square pattern with a spacing of 1.0 m, which yields D = 1.13 m. The equivalent drain diameter determined according to Eq. (2.20) becomes d_w = 0.066 m. The equivalent diameter of the mandrel d_m = 0.10 m. The smear zone is estimated at d_s = 0.20 m. The permeability ratios k_h/k_s and κ_h/κ_s are assumed equal to the ratio c_h/c_v.

The consolidation characteristics of the clay deposit, determined by oedometer tests, can be summarized as follows (DMJM International, 1996): average coefficient above the preconsolidation pressure c_v = 1.06 m²/year (standard deviation = 0.061 m²/year), c_h = 1.37 m²/year (standard deviation = 0.050 m²/year). This yields $k_h/k_s=$ ${\kappa }_h/{\kappa }_s=1.3$ (in a paper previously published by the author [Hansbo, 1997b] this ratio was assumed equal to 2). The virgin compression ratio CR varies from 0.3 to 0.55 (average 0.43; standard deviation 0.1) and the average recompression ratio RR = 0.03 (standard deviation = 0.007). The clay penetrated by the vertical drains is slightly overconsolidated with a preconsolidation pressure about 15–50 kPa higher than the effective overburden pressure. The clay below the tip of the drains is heavily overconsolidated. In this case, the influence on the consolidation process of one-dimensional vertical consolidation will be ignored owing to difficulties in assessing the drainage conditions.

The monitoring system consisted of vertical settlement meters placed on the soil surface and at different depths, and of inclinometers to study the horizontal displacements. Unfortunately, the results of the settlement observations at various depths are contradictory and, therefore, only the surface settlement observations can be trusted. The contribution to the vertical settlement of horizontal deformations is analyzed based on the inclinometers placed 7.8 m from the center of the test area. The total settlement observed, including the settlement caused by horizontal deformations, and the corrected settlement curve with regard to the influence on vertical settlement by horizontal deformations (representing the mere effect of consolidation) are shown in Fig. 2.20.

A follow-up of the course of consolidation settlement according to Asaoka’s method, based on settlement observations at equal time intervals, results in the following relation $s_i=0.2625+0.8195s_{i–1}$, which yields the primary settlement s_p = 1.45 m.

The settlement analysis based on Eq. (2.19) is carried out in 4 successive steps: load step 1 with load q₁ = 20 kN/m²; load step 2 with load q₂ = 30 kN/m²; load step 3 with q₃ = 10 kN/m², and load step 4 with q₄ = 20 kN/m². The primary consolidation settlement caused by a load intensity of 80 kN/m², determined by the compression characteristics, becomes equal to 1.4 m. By slightly modifying the compression characteristics to yield a final primary consolidation settlement of 1.45 m, we find s₁ = 0.15 m; s₂ = 0.6 m; s₃ = 0.2 m, and s₄ = 0.5 m (in total, 1.45 m).

The analysis of the consolidation process according to Eq. (2.19) has to be carried out in the following way. The degree of consolidation Ū₁, inserting $\mathrm{\Delta }{\overline{h}}_1=\mathrm{\Delta }{q}_1/{\gamma }_w$, determines the course of settlement in the first load step. When calculating the course of settlement in the second load step we have to apply the value $\mathrm{\Delta }{\overline{h}}_2=\left(1–{\overline{U}}_1\right)\mathrm{\Delta }{q}_1/{\gamma }_w+\mathrm{\Delta }{q}_2/{\gamma }_w$. The settlement at the end of the load step is obtained from $\mathrm{\Delta }s=\mathrm{\Delta }{s}_1{\overline{U}}_1+\left[\mathrm{\Delta }{s}_1\left(1–{\overline{U}}_1\right)+\mathrm{\Delta }{s}_2\right]\overline{U}$, and so on.

Now, 50 days after the start of loading (consolidation time t = 50 − 15 = 35 days; $\mathrm{\Delta }\overline{h}$ = 2 m; λ = 0.37 m²/year, n = 1.5) we find Ū = 0.21, which yields s = 0.03 m. In load step 2, the load q₂ = 30 kN/m² has to be increased by 0.79 × 20 = 16 kN/m² corresponding to $\mathrm{\Delta }{\overline{h}}_{2,\mathrm{corr}}$ = 4.6 m and s_2,corr = 0.6 + 0.12 = 0.72 m. After 75 days when load step 2 is completed, we find [t = (75 − 50)/2]Ū = 0.12, which yields s = 0.09 m and s = 0.12 m. After 140 days when load step 3 is being applied, we have (t = 12.5 + 65 = 77.5 days)Ū = 0.50, from which s = 0.36 m and s = 0.39 m. This yields h_3,corr = 3.3 m and s_3,corr = 0.2 + 0.36 = 0.56 m. After 220 days when load step 4 is being applied, we find (t = 80 days)Ū = 0.46, from which s = 0.26 m and s = 0.26 + 0.39 = 0.65 m. This yields h_4,corr = 1.8 + 2.0 = 3.8 m and s_4,corr = 0.5 + 0.3 = 0.8 m. After 250 days when load step 4 is completed, we have (t = 15 days)Ū₂ = 0.13, which yields s = 0.10 m and s = 0.75 m. Then 100, 200, and 400 days later we have Ū = 0.59, Ū = 0.74, and Ū = 0.89, from which s = 0.47 (s = 0.47 + 0.65 = 1.12 m), 0.67 (s = 1.24 m), and 0.71 m (s = 1.36 m), respectively.

The theoretical course of settlement determined in the conventional way, Eq. (2.18), is less complicated in that the total consolidation curve can be determined for each load step separately and added to each other. Assuming c_h = 0.93 m²/year we find, to give an example, 400 days after the start of the loading process (t₁ = 385 days, Ū = 0.92; t₂ = 340 days, Ū = 0.89; t₃ = 260 days, Ū = 0.82; t₄ = 170 days, Ū = 0.67) the settlement s = 0.92 · 0.15 + 0.89 · 0.6 + 0.82 · 0.2 + 0.67 · 0.5 = 1.17 m.

The results obtained by the two methods of analysis are shown in Fig. 2.20.

The values of $\mathrm{\Delta }\overline{h}$ applied in the different load steps agree quite well with observations of the excess pore water pressure (cf. Hansbo, 1997a and 1997b).

Inserting the maximum value ${{\overline{h}}_0}_v$ = 4.6 m into Eq. (2.14) we find outside the zone of smear i_max = 17.7, and inserting this into Eq. (2.16), the correlation between c_h = 0.93 m²/year and λ = 0.37 m²/year correspond to i_l = 3.7.

Vagnhärad vacuum test

Torstensson (1984) reported an interesting full-scale test in which consolidation of the clay was achieved by the vacuum method. The subsoil at the test site consists of postglacial clay to a depth of 3 m and below this of varved glacial clay to a depth of 9 m underlain by silt. The clay is slightly overconsolidated with a preconsolidation pressure about 5–20 kPa higher than the effective overburden pressure. The coefficient of consolidation c_h was found equal to 0.95 m²/year and the average virgin compression ratio CR equal to 0.7 (max. 1.0).

The vacuum area, 12 m square, was first covered by a sand/gravel layer 0.2 m in thickness, and then by a Baracuda membrane, which was buried to 1.5 m depth along the border of the test area and sealed by means of a mixture of bentonite and silt.

Mebradrains (d_w = 0.066 m) were installed in a square pattern with 1.0 m spacing to a depth of 10 m. The equivalent diameter of the mandrel d_m = 0.096 m. The average under-pressure achieved by the vacuum pump was 85 kPa. After 67 days, the vacuum process was stopped and then resumed after 6 months of rest. From the shape of the settlement curve (Fig. 2.21), Asaoka’s method yields the correlation s_i = 0.0756 + 0.9075s_i–1, from which s_p = 0.82 m. This value is low with regard to the loading conditions and the compression characteristics. This is mainly because the applied vacuum effect is not fully achieved in the drains. Thus, the primary settlement of 0.82 m corresponds to a vacuum effect of about 35 kPa (h = 3.5 m). Another reason may be that the test area is too small as compared to the thickness of the clay layer.

The theoretical settlement curve in this case has to be determined in two steps, the first one up to a loading time of 67 days leading to a then settlement $s_1={\overline{U}}_{h1}{s}_p$. In the next load step, starting again from the time of resumption of the application of vacuum, the remaining primary settlement is obtained from the relation $\mathrm{\Delta }s={\overline{U}}_h\left(s_p–s_1\right)$, such that the remaining settlement equals $s_t=s_1+{\overline{U}}_{ht}\left(s_p–s_1\right)$ where t starts from the time of resumption of the application of vacuum. In this case, where vacuum is applied to create underpressure in the drains, the effect of vertical one-dimensional consolidation is eliminated.

Inserting the values D = 1.13 m, d_w = 0.066 m, d_s = 0.19 m, $k_h/k_s={\kappa }_h/{\kappa }_s=4$, and $\mathrm{\Delta }\overline{h}$ = 3.5 m into Eqs. (2.18) and (2.19), the best agreement between theory and observations is found for c_h = 2.4 m²/year and λ = 0.95 m²/year (Fig. 2.21). Even in this case, the λ theory agrees better with observations than the classical theory.

Inserting the maximum value $\mathrm{\Delta }\overline{h}$ = 3.5 m into Eq. (2.13) yields i_max = 7.3, and inserting the values c_h = 2.4 m²/year and i_max = 7.3 into Eq. (2.15) yields λ = 1.1 m²/year. The value λ = 0.95 m²/year corresponds according to Eq. (2.15) to i_max = 9.9. Assuming i_l = 5, the value of i_max, derived from Eq. (2.16), becomes respectively 7.7 instead of 7.3 and 11.8 instead of 9.9.

Porto Tolle test site, Italy

The Porto Tolle experimental embankment had a crest width and length of 30 and 300 m, respectively, and a bottom width of 65 m. The load placed on the embankment was about 100 kN/m². It was divided into four sections with different types of vertical drains. Here, two types of drains will be analyzed, namely one with band drains of type Geodrain and another with jetted sand drains, 0.3 m in diameter. The drains were placed in an equilateral triangular pattern, Geodrains with 3.8 m spacing and sand drains with 5.0 m spacing.

The soil consists of 7 m sand and silty sand underlain by about 20 m of soft, normally consolidated light-brown silty clay resting on dense, silty sand. The consolidation characteristics of the clay were determined by means of oedometer tests and field investigations. According to the oedometer tests, the coefficient of consolidation c_v was found equal to 3–9 m²/year and the coefficient of consolidation c_h equal to 6–16 m²/year. From the results of a trial embankment without vertical drains, c_v was found equal to 9 m²/year. The virgin compression ratio CR was not given. The primary consolidation settlement of the section with Geodrains, shown in Fig. 2.22, estimated according to Asaoka’s method, yields $s_i=0.249+0.756s_{i–1}$, corresponding to a total primary consolidation settlement of 1.0 m, while the settlement of the section with sand drains yields $s_i=0.259+0.703s_{i–1}$, corresponding to a total primary consolidation settlement of 0.9 m. The vertical stress increase in the middle of the clay layer due to the load of the embankment can be estimated as σ_v = 100[(30 + 65)/2] /[17 + (30 + 65)/2] ≈ 74 kPa.

The observations are compared with Eqs. (2.18) and (2.19) assuming that the primary consolidation settlement is in accordance with the preceding values, that is, equal to 1.0 m in the section provided with Geodrain and 0.9 m in the section provided with sand drains. Inserting the values D = 3.99 m, d_w = 0.066 m, and d_s = 0.19 m, $k_h/k_s={\kappa }_h/{\kappa }_s=2$ for Geodrains, and D = 5.25 m, d_w = 0.3 m, and d_s = 0.4 m, $k_h/k_s={\kappa }_h/{\kappa }_s=2$ for sand drains, and an initial hydraulic head $\mathrm{\Delta }{\overline{h}}_0$ = 7 m (assuming $\mathrm{\Delta }{\overline{u}}_0={\gamma }_w\mathrm{\Delta }{\overline{h}}_0=\mathrm{\Delta }\sigma$) into Eqs. (2.18) and (2.19), the best agreement between theory and observations is found for c_h = 24 m²/year and λ = 14 m²/year with the exponent n = 1.5 (Fig. 2.22). In both cases the coefficient of consolidation c_v is assumed equal to 9 m²/year.

According to Eq. (2.18), the average degree of consolidation when the load step leading to full load is completed (consolidation time t = 3.4 months; full load during 1.7 months) $\overline{U}=$41% for Geodrains and 42% for sand drains. At t = 6, 9, and 12 months we find $\overline{U}=$ 68%, 85%, and 93%, respectively, for Geodrains, and $\overline{U}=$72%, 87%, and 94%, respectively, for sand drains. According to Eq. (2.19), the average degree of consolidation when the load step leading to full load is completed (consolidation time t = 3.4 months; full load during 1.7 months) $\overline{U}=$ 51% for Geodrains and 47% for sand drains. At t = 6, 9, and 12 months we find $\overline{U}=$ 76%, 87%, and 92%, respectively, for Geodrains, and $\overline{U}=$ 73%, 85%, and 91%, respectively, for sand drains. Even in this case the λ theory agrees better with observations than the classical theory.

Stockholm Arlanda project

Various types of soil improvement techniques have been used at Stockholm Arlanda Airport, among these, vertical drainage in combination with preloading along the runway/taxiway in an area of 250,000 m² containing clay and organic deposits.

The requirements placed on the completed project allowed a maximum settlement of 30 mm. The design of the vertical drain project aimed at creating an effective preconsolidation pressure of at least 1.25 times the effective vertical pressure reached in the soil after completion of the project. This was done to eliminate the risk of too large creep settlement after the primary consolidation period (cf. test area III at Skå-Edeby, Fig. 2.15). The drain spacing and the preloading conditions were selected to achieve 95% primary consolidation within 12 months of loading. The Arlanda project is interesting from the point of view of the extreme loading conditions. Thus, to achieve the goal setup, preloading has been carried out by means of 14–20 m of fill. The load of the fill has caused a relative compression of 22–29%, corresponding to total settlements of 1.6–2.6 m.

According to the design requirements, the drains had to be installed in an equilateral triangular pattern with a drain spacing of 0.9 m. Furthermore, depending on the large relative compression to be expected, the drains had to be capable of undergoing a vertical compression in the order of 35% without losing their function. Prefabricated drains of type Mebradrain 88 were selected for the project. The installation of the drains was made by means of steel mandrel with the dimensions 60 × 120 mm.

The clay at one of the sites of observation presented here, site K, was medium to highly sensitive and normally consolidated with water content of mainly 60–90% and a liquid limit of 40–80% (the results at other points of observation and consequential analysis are presented by Eriksson et al., 2000). The undrained shear strength was fairly constant, mainly about 7–9 kPa, irrespective of depth. The groundwater level was situated just below the ground level. The results of the oedometer tests, interpreted in the way shown in (see Fig. 2.23), and the settlements derived accordingly are given in Table 2.7.

Table 2.7

Parameters used in the settlement analysis (see Fig. 2.23) at Stockholm Arlanda Airport

Depth (m)	Mij (kPa)	Mj′	${\mathit{\sigma }}_{\mathit{i}\mathit{j}}^{\prime }-{\mathit{\sigma }}_{\mathit{c}\mathit{j}}^{\prime }$ (kPa)	Δhj (m)	Load (kN/m2)	Settlement (m)
2	80	12.9	9	2.5	0–80	1.8
4	165	19.3	12	2.0	80–215	0.7
6	188	15.3	21	2.0	215–390	0.4
8	279	17.5	20	2.5	0–390	2.9

Note: h_j = hydraulic head of excess pore water pressure.

The settlement values calculated from the oedometer test results were compared with those determined from the course of settlement shown in Fig. 2.24. Thus, according to Asaoka’s method, the primary consolidation settlement is obtained from the correlation $s_i=0.590+0.775s_{i–1}$, which yields s_p = 2.63 m. The discrepancy can be explained by the fact that the oedometer results given in Table 2.7 represent mean values of oedometer tests carried out on samples taken as far as 70–165 m away from the point of settlement observation. The settlements obtained in the various load steps were adjusted by multiplication by a factor obtained by dividing the final settlement values obtained by Asaoka’s method with the final settlement values evaluated based on the oedometer tests. The adjusted values are given in Table 2.8.

According to the oedometer tests, the coefficient of consolidation c_v varies from about 0.2–0.3 m²/year at preconsolidation pressure to about 0.5–1.0 m²/year (maximum 2.5 m²/year) at the end of primary compression under the maximum load applied (390 kN/m²). The following assumptions are applied in the analysis: D = 1.05 × 0.9 = 0.945 m; d_w = 0.066 m; d_s = 0.19 m; $k_h/k_s={\kappa }_h/{\kappa }_s=c_h/c_v=3$; l = 4.5 m; c_h = 2.6 m²/year; and λ = 0.7 m²/year with the exponent n = 1.5.

Inserting the consolidation time t₁ = 1 month (the length of time that corresponds to full loading condition in load step 1), we find ${\overline{U}}_1=$ 0.42 at the completion of load step 1, and, consequently, s = 0.42 × 1.63 = 0.69 m. At the completion of load step 2 (t₁ = 2 months and t₂ = 0 · 5 months) we find ${\overline{U}}_1=$ 0.65 and ${\overline{U}}_2=$ 0.25, where s = 0.65 × 1.63 + 0.25 × 0.63 = 1.22 m. Finally, at the completion of load step 3 (t₁ = 4.5 months, t₂ = 3 months, and t₃ = 1.25 month) we find ${\overline{U}}_1=$ 0.90, ${\overline{U}}_2=$ 0.79, and ${\overline{U}}_3=$ 0.49, that is, s = 0.90 × 1.63 + 0.79 × 0.63 + 0.49 × 0.36 = 2.15 m. Three months later we find ${\overline{U}}_1=$ 0.98, ${\overline{U}}_2=$ 0.95, and ${\overline{U}}_3=$ 0.89, that is, s = 0.98 × 1.63 + 0.95 × 0.63 + 0.89 × 0.36 = 2.52 m. Another three months later we find ${\overline{U}}_1=$ 0.995, ${\overline{U}}_2=$ 0.99, and ${\overline{U}}_3=$ 0.98, that is, s = 2.60 m.

The course of settlement in non-Darcian flow:

In this case, according to Eq. (2.19), the average piezometric head in each load step has a great influence on the rate of consolidation.

Assuming that the average excess pore water pressure is equal to the load placed on the ground, we have in load step 1 (where the time of full loading is 1 month) the hydraulic head $\mathrm{\Delta }{\overline{h}}_1$ = 8 m, which yields ${\overline{U}}_1=$0.46. This yields s = Δs₁Ū₁ = 0.46 × 1.63 = 0.76 m and a hydraulic head $\mathrm{\Delta }{\overline{h}}_2=\left(1–{\overline{U}}_1\right)$ × Δq₁/γ_w + Δq₂/γ_w = 0.54 × 8 + 13.5 = 17.8 m. At the completion of load step 2, inserting a time of consolidation of half a month (the time of full loading), we have ${\overline{U}}_2=$ 0.38, which yields the settlement $s=\mathrm{\Delta }{s}_1{\overline{U}}_1+{\overline{U}}_2\left[\mathrm{\Delta }{s}_1+\mathrm{\Delta }{s}_2–\mathrm{\Delta }{s}_1{\overline{U}}_1\right]$ = 0.76 + 0.38(2.27 − 0.76) = 1.33 m. The hydraulic head now becomes $\mathrm{\Delta }{\overline{h}}_3$ = 0.62 × 17.8 + 17.5 = 28.5 m. Finally, at the completion of load step 3, when the loading operation has been completed, we have (where the time of consolidation under full load is 1.25 month) ${\overline{U}}_3=$ 0.70, which yields s = 1.33 + 0.70(2.63 − 1.33) = 2.24 m. Three months later we have ${\overline{U}}_3=$ 0.93, from which s = 2.45 m.

A comparison between analytical results and observations are made in Fig. 2.24.

The course of excess pore pressure dissipation:

In this case, the piezometers were placed inside concrete tubes with an outer diameter of 1 m before the installation of the drains. The existence of these tubes created an obstacle to keeping up the intended drain pattern. Thus, the distance from a piezometer to the nearest drain was estimated at 0.9 m. The average D value in the vicinity of a piezometer was estimated at 1.335 m instead of the normal value of 0.945 m.

The excess pore water pressure u caused by loading is analyzed by Skempton’s pore pressure equation $u=B\left[{\sigma }_3+A\left({\sigma }_1–{\sigma }_3\right)\right]$. In our case the pore pressure coefficients can be assumed equal to B = 1 (water-saturated clay) and A = 1.2 (load increase very high in comparison with the preconsolidation pressure). According to the theory of elasticity, the principal stress increase becomes ${\sigma }_1=\mathrm{\Delta }q$ and ${\sigma }_3=0.9\mathrm{\Delta }q$. This yields u₀ = 1.02q where the hydraulic head increase $\mathrm{\Delta }{h}_0=u_0/{\gamma }_w=0.104\mathrm{\Delta }q$.

Now, assuming that the piezometers are placed in the center between the drains we have ρ/D = 0.5. For Ū_h = 0 and D = 1.335 m, we then have, according to Eq. (2.7), h = 1.06 × 0.104q = 0.11q. When dealing with the excess pore pressure dissipation, the excess pore pressure developed during the placement of each load step is assumed to correspond to the total load increase (no reduction of the pore pressure caused during the placement of the load step). This can be justified by the dynamic disturbance effects caused by the filling operation. Therefore, the time of consolidation is chosen with reference to the time when the filling operation in a load step is completed. The effect of the contribution by one-dimensional vertical consolidation on the excess pore pressure values is not taken into account.

Assuming Darcian flow, the value of h during the course of consolidation can be found by Eqs. (2.7) and (2.5). Proceeding along the mentioned principle, taking into account only the effect of the drains (c_v = 0), we find, at the completion of load step 1, h = 0.11 × 80 = 8.8 m. At the completion of load step 2 we have t₁ = 1 month which yields Ū_h1 = 0.19. Hence h = 0.81 × 8.8 + 0.11 × 135 = 22.0 m. Finally, at the completion of load step 3 we have t₁ = 3.5 months and t₂ = 2.5 months, from which Ū_h1 = 0.53 and Ū_h2 = 0.42; from this we obtain h = 0.47 × 8.8 + 0.58 × 0.11 × 135 + 0.11 × 175 = 32.0 m. An easier way of finding the value of Δh, applied in the following, is to start the analysis only from the time when load step 2 is completed, which yields the same result: h = 0.58 × 22.0 + 0.11 × 175 = 32.0 m. Three months later we have Ū_h3 = 0.48, where h = 0.52 × 32.0 = 16.7 m, and another 3 months later, Ū_h3 = 0.73, that is, h = 8.7 m.

Assuming non-Darcian flow, we find, according to Eq. (2.9), for Ū_h = 0 and ρ/D = 0.5 the value of h = 1.11 × 0.104q = 0.115h₀. At the completion of load step 1 we now have h = 0.115 × 80 = 9.2 m. At the completion of load step 2 we have, according to Eq. (2.6), Ū_h1 = 0.24, where h = 0.76 × 9.2 + 0.115 × 135 = 22.5 m. At the completion of load step 3 we have Ū_h2 = 0.59, from which h = 0.41 × 22.5 + 0.115 × 175 = 29.4 m. Three months later we have Ū_h3 = 0.68, where h = 0.32 × 29.4 = 9.4 m, and another 3 months later, Ū_h3 = 0.84, that is, h = 4.6 m.

The results of the analysis are compared with the results of pore pressure observations in Fig. 2.25. In reality, the observational results were obtained through measurements where the horizontal distance between the piezometers and the nearest drain was 0.8 m (ρ = D/2 = 0.8). The values of ρ and D, however, chosen in the analysis, are based on the average drain spacing nearest to the piezometers.