From Canadian Geotechnical Journal, ©2000 National Research Council Canada. Used with permission. This page was created with Microsoft Word and converted to HTML with MathML using MathType's MathPage technology with the "HTML+MathML" option.

# Equivalent plane strain solution

The theory of consolidation of vertical drains under axisymmetric conditions has gained wide acceptance in geotechnical engineering because of its simplicity. According to Hansbo (1981), for axisymmetric flow the average degree of consolidation ${\overline{U}}_{\text{h}}$ on a horizontal plane at a depth z and at time t may be predicted from

 ${\overline{U}}_{\text{h}}=1-\mathrm{exp}\left(-\frac{8{T}_{\text{h}}}{\mu }\right)$ [1]

where ${T}_{\text{h}}$ is the time factor, and the effect of smear and well resistance (μ) is given by

 $\mu =\mathrm{ln}\left(\frac{n}{s}\right)+\left(\frac{{k}_{\text{h}}}{{{k}^{\prime }}_{\text{h}}}\right)\mathrm{ln}\left(s\right)-0.75+\pi \left(2lz-{z}^{2}\right)\frac{{k}_{\text{h}}}{{q}_{\text{w}}}$ [2]

In the above, $n=R/{r}_{\text{w}},\text{​}\text{​}$ where R is the radius of the influence zone of the drain and ${r}_{\text{w}}$ is the radius of the drain; $s={r}_{\text{s}}/{r}_{\text{w}},$ where ${r}_{\text{s}}$ is the radius of the smear; l is the length of the drain having one-way drainage or half this value for two-way drainage; z is the depth of the drain under consideration; ${q}_{\text{w}}$ is the discharge capacity of the drain; and  are the coefficients of horizontal permeability outside and inside the smeared zone, respectively.

In equation [2], the final term represents the well resistance, which is a function of the drain length l and depth z for $l>z$ and is inversely proportional to the drain discharge capacity $\left({q}_{\text{w}}\right)$.

Indraratna and Redana (1997) showed that the degree of consolidation at a depth z in plane strain condition can be represented by

 ${\overline{U}}_{\text{hp}}=1-\frac{\overline{u}}{{\overline{u}}_{0}}=1-\mathrm{exp}\left(\frac{{}^{-}8{T}_{\text{hp}}}{{\mu }_{\text{p}}}\right)$ [3]

where ${\overline{u}}_{0}$ is the initial pore pressure, $\overline{u}$ is the pore pressure at time t (average values), ${T}_{\text{hp}}$ is time factor in plane strain, and

 ${\mu }_{\text{p}}=\left[\alpha +\beta \frac{{k}_{\text{hp}}}{{{k}^{\prime }}_{\text{hp}}}+\theta \left(2lz-{z}^{2}\right)\right]$ [4]

where ${k}_{\text{hp}}$ and ${{k}^{\prime }}_{\text{hp}}$ are the undisturbed horizontal and corresponding smear zone permeabilities, respectively. Ignoring higher order terms, the geometric parameters α and β and the flow term θ are given by

 $\alpha =\frac{2}{3}-\frac{2{b}_{\text{s}}}{B}\left(1-\frac{{b}_{\text{s}}}{B}+\frac{{b}_{\text{s}}^{\text{2}}}{3{B}^{2}}\right)$ [5]
 $\beta =\frac{1}{{B}^{2}}{\left({b}_{\text{s}}-{b}_{\text{w}}\right)}^{2}+\frac{{b}_{\text{s}}}{3{B}^{3}}\left(3{b}_{\text{w}}^{\text{2}}-{b}_{\text{s}}^{\text{2}}\right)$ [6]

and

 $\theta =\frac{2{k}_{\text{hp}}^{\text{2}}}{{{k}^{\prime }}_{\text{hp}}B{q}_{z}}\left(1-\frac{{b}_{\text{w}}}{B}\right)$ [7]

where ${q}_{\text{z}}$ is the equivalent plane strain discharge capacity of the drain, and dimensions  are defined in Figure 1.

At a given stress level, to maintain the same degree of consolidation at each time step, the average degree of consolidation for both axisymmetric $\left({\overline{U}}_{\text{h}}\right)$ and equivalent plain strain $\left({\overline{U}}_{\text{hp}}\right)$ conditions are made equal, hence

 ${\overline{U}}_{\text{h}}={\overline{U}}_{\text{hp}}$ [8]

Combining equations [3] and [8] with the original Hansbo (1981) theory (equation [1]) defines the time factor ratio by the following equation:

 $\frac{{T}_{\text{hp}}}{{T}_{\text{h}}}=\frac{{k}_{\text{hp}}}{{k}_{\text{h}}}\frac{{R}^{2}}{{B}^{2}}=\frac{{\mu }_{\text{p}}}{\mu }$ [9]

Indraratna and Redana (1997) showed that if the radius of the axisymmetric influence zone of a single drain (R) were taken to be the same as the half-width (B) in plain strain, then the relationship between ${k}_{\text{hp}}$ and ${{k}^{\prime }}_{\text{hp}}$ is given by

 ${k}_{\text{hp}}=\frac{{k}_{\text{h}}\left[\alpha +\beta \frac{{k}_{\text{hp}}}{{{k}^{\prime }}_{\text{hp}}}+\theta \left(2lz-{z}^{2}\right)\right]}{\left[\mathrm{ln}\left(\frac{n}{s}\right)+\left(\frac{{k}_{\text{h}}}{{{k}^{\prime }}_{\text{h}}}\right)\mathrm{ln}\left(s\right)-0.75+\pi \left(2lz-{z}^{2}\right)\frac{{k}_{\text{h}}}{{q}_{\text{w}}}\right]}$

If both the smear and well resistance are ignored, then the simplified ratio of plane strain to axisymmetric horizontal permeability ${k}_{\text{h}}$ is represented by

 $\frac{{k}_{\text{hp}}}{{k}_{\text{h}}}=\frac{0.67}{\left[\mathrm{ln}\left(n\right)-0.75\right]}$ [11]

If the effect of well resistance is ignored, the permeability in the smear zone can be isolated by neglecting the final terms of the denominator and numerator in equation [10] to give

 $\frac{{{k}^{\prime }}_{\text{hp}}}{{k}_{\text{hp}}}=\frac{\beta }{\frac{{k}_{\text{hp}}}{{k}_{\text{h}}}\left[\mathrm{ln}\left(\frac{n}{s}\right)+\left(\frac{{k}_{\text{h}}}{{{k}^{\prime }}_{\text{h}}}\right)\mathrm{ln}\left(s\right)-0.75\right]-\alpha }$ [12]