B.4.1.2 Rate of consolidation settlement p.II

Method of analysis

Theoretically, whatever pattern is used, each drain is considered to dewater a hypothetical soil cylinder whose cross-sectional area equals the cross-sectional area enclosed by four neighbouring drains, Figure B.2. The most efficient way of utilising the capacity of vertical drains for the purpose of speeding up the consolidation process is to install the drains in an equilateral triangular pattern. The consolidation process is mainly governed by pore water flow in the radial direction towards the drain and to a lesser extent by pore water flow in the vertical direction between the drains. Two methods of analysis exist, the so-called "free strain analysis" and the "equal strain analysis". As shown by Barron [3] the difference in results regarding average consolidation process obtained between the two methods of analysis is negligible. Therefore, because of its simplicity the equal strain analysis, Equation (1), has become routine [18], [28], [32], [35], [52].

In the methods of analysis used for determination of the influence of well resistance (limited discharge capacity), the consolidation characteristics of the soil are generally assumed to be constant throughout the soil layer. The influence of layers with different consolidation characteristics has been analysed by Onoue [41].

Another conventional assumption in analysis is the validity of Darcy’s law. Experience from a number of field tests [16], [21] and [46] and from laboratory tests on permeability [16] and [13] has shown that there is a deviation from Darcy’s law at small hydraulic gradients. Consolidation equations valid for both Darcian and non-Darcian flow have been developed [22].

The basic theory of vertical drainage used in routine analysis of most of vertical drainage projects was published by Hansbo [18] as an extension of Barron’s theory [3] for the case of drains with limited discharge capacity. Accordingly, the rate of consolidation follows the relation:

where, omitting terms of minor significance,

An important parameter in vertical drain analysis is the discharge capacity of the drains q_w, i.e. the amount of water flow per time unit that can take place in the vertical direction through the drain at a hydraulic gradient equal to one. (In EN 10318, the discharge capacity is equal to the transmissivity times the width of the drain.) Drains have appeared on the market with insufficient discharge capacity when installed to great depth. If the drains have insufficient discharge capacity, the degree of consolidation obtained by drain installation in homogeneous soil decreases with depth of installation.

The ratio of the time of consolidation t, considering the effect of well resistance (limited discharge capacity), and the time of consolidation t₁, neglecting the effect of well resistance, can be expressed by the relation t = t₁ (1 + ∆t), where the delay ∆t in time of consolidation follows the relation:

The most unfavourable case with regard to discharge capacity requirements is obtained when k_s = k_h which yields:

The average ∆t value becomes equal to two thirds of the value obtained at depth z = l.

The effect of well resistance (discharge capacity) depends on the depth of drain installation, the drain spacing and whether the drains are penetrating or not, Figure B.3. In the case shown in Figure B.3 the delay ∆t at 30 m depth becomes 1,46 (146 %) and the average ∆t value 0,97.

Provided that an increase of 10 % in the time of consolidation due to well resistance at the tip of partially penetrating drains (z = l, Figures B.2 and B.3) can be permitted relative to that obtained by using fully efficient drains, a conservative estimate of the required discharge capacity with regard to soil permeability and depth of installation is exemplified in Figure B.4. For penetrating drains (efficient drainage at top and bottom) the delay in consolidation takes place at mid-depth, see Figure B.3, and hence the depth values in Figure B.4 are doubled. The drain spacing in Figure B.4 is assumed equal to 0,9 m (drains placed in equilateral triangular pattern, i.e. D = 0,945 m, see Figure B.3) and the equivalent drain diameter d_w = 0,065 m.

The required discharge capacity according to Figure B.4 for partially penetrating drains, installed to a depth of 15 m in silty clay with a permeability of 0,25 m/year (0,8 × 10^-8 m/s) becomes 1000 m³/year, while the required discharge capacity for partially penetrating drains, installed to a depth of 15 m in clay with a permeability of 0,03 m/year (0,95 × 10^-9 m/s) becomes 110 m³/year.

The discharge capacity requirements decrease with increasing drain spacing. For a band drain spacing of, for example, 1,5 m (D = 1,575 m) and 2 m (D = 2,1 m), respectively, the required discharge capacities are 80 % and 70 %, respectively, of those presented in Figure B.4.

If the admissible delay in the time of consolidation is reduced to 5 %, the required discharge capacity given in Figure B.4 is doubled.

For penetrating drains, the depth values l given in Figure B.4 refer to half the depth of drain installation, see Figures B.2 and B.3.

Key

• 1 Partially penetrating drain (l = 30 m)
• 2 Penetrating drain (2l = 60 m)
• 3 Degree of consolidation %
• 4 Depth of drain installation, m
• 5 , average

Consolidation parameters: q_w = 100 m³/year (≈ 3,2 cm³/s), c_h = 1,0 m²/year (≈ 3,2 × 10^-8 m²/s), K_s = k_h = 0,1 m/year (≈ 3,2 × 10^-9 m/s), time of consolidation t = 0,5 year. Drain spacing 0,9 m (equilateral triangular pattern; D = 0,945 m), drain diameter d_w = 0,065 m.

With time a certain deterioration of the filter can be expected due to bacteriological activity or fungi attacks (see Annex A, Figure A.10). Deterioration generally reduces the discharge capacity towards the end of the consolidation process. Therefore, it has a relatively small influence on the rate of consolidation.

In highly compressible soils, the relative compression taking place during the consolidation process can lead to buckling or kinking of the drains (see Annex A, Figure A.5), which may seriously reduce the discharge capacity of certain types of drains [31].

Key

• 1 permeability (k_s = k_h), m/year (1 m/year = 3,17 x 10^-8 m/s)
• 2 q_w, m³/year
• 3 depth of installation, m

Drain spacing 0,9 m (equilateral triangular pattern; D = 0,945 m), drain diameter d_w = 0,065 m.

Key

Drain spacing 0,9 m (equilateral triangular pattern; D = 0,945 m), drain diameter d_w = 0,065 m, k_s = k_h.