Rapid drawdown

When the water against an embankment or a natural slope is lowered faster than the soil can drain, a zone of low permeability keeps much of its pore pressure while losing the support of the water outside. Rapid drawdown is therefore one of the governing cases for the upstream slopes of dams and for the banks of reservoirs. OGR Slip2D offers four procedures for this case (an effective-stress analysis with Skempton's $\bar B$ and three multi-stage procedures with undrained strengths), a sweep of the drawdown level and, as an alternative to the assumption of no drainage, a transient seepage analysis.

3.8.1The problem#

Before a drawdown the slope is in equilibrium under the water outside it, which loads its face, and under the pore pressures inside it. A rapid drawdown removes the external water, but the pore pressures in a soil of low permeability cannot follow at once: the soil is unloaded in undrained conditions, and its strength after the drawdown depends on the effective stresses it had been consolidated to before it (Lowe & Karafiath, 1960). A free-draining soil, in contrast, follows the new water level. The procedures in use differ in how they estimate the strength of the undrained zones (Wright & Duncan, 1987; Duncan, Wright & Wong, 1990; USACE, 2003).

OGR Slip2D represents the two reservoir levels with two water surfaces: the water table is the initial level and the drawdown line the final, lower one. Without a drawdown line the drawdown is complete. Every procedure works on copies of the model at the two levels, so the model itself is left unchanged. At the final level the drawdown line takes the place of the water table, so that the pore pressures, the ponded water, the saturated unit weights and the slice boundaries all follow it; piezometric lines are not moved.

3.8.2The four procedures#

The procedures differ only in the strength they give the materials marked undrained after the drawdown (Table 3.8.1). Freely draining materials keep their effective strength and take the pore pressures of the final level in all four.

Since the drained cap can only lower a strength, on any slip surface the procedure of Duncan, Wright and Wong never gives a higher factor of safety than that of Lowe and Karafiath.

3.8.3Effective stress with B-bar#

The $\bar B$ procedure is a single effective-stress analysis at the final level. The materials marked undrained keep the pore pressure they had at the initial level, changed by the reduction of vertical stress that the removal of the water causes, through Skempton's coefficient:

where

$u_\text{ini}$

pore pressure at the base midpoint with the initial level (kPa)

$d_\text{ini},\ d_\text{fin}$

depth of water ponded on the ground above the slice before and after the drawdown (m)

$\bar B$

coefficient of the material (—)

$\Delta\sigma_v$ is the weight of the column of water removed above the slice, limited by the ground surface. Under the face of the slope, where the ground lies between the two levels, only the water that stood on the ground is removed (Figure 3.8.1). Because $\Delta\sigma_v$ is negative, a larger $\bar B$ gives a lower pore pressure after the drawdown: $\bar B = 1$ follows the unloading completely and $\bar B = 0$ keeps the initial pore pressure unchanged. The result is not allowed below zero, since a negative value would be suction created by the drawdown.

With $\bar B = 1$, a complete drawdown and an initial level above the whole slope, the pore pressure after the drawdown is $\gamma_w$ times the depth below the ground surface, that is $r_u = \gamma_w/\gamma$. Morgenstern (1963) prepared his stability charts for rapid drawdown on the same assumption, with $\bar B = 1$, and the $\bar B$ procedure of OGR Slip2D is checked against two of his cases and against that identity.

3.8.4Undrained strength envelopes#

The multi-stage procedures need the undrained strength of the soil as a function of its consolidation stress, which is obtained from isotropically consolidated undrained (IC-U) triaxial tests. Those tests can be plotted in two ways (Duncan, Wright & Wong, 1990). The R envelope is the line tangent to the total-stress Mohr circles at failure, each drawn from its consolidation pressure $\sigma'_{3c}$:

The $K_c = 1$ envelope plots the shear stress on the failure plane at failure against the effective consolidation pressure $\sigma'_{fc} = \sigma'_{3c}$:

Both describe the same tests, so either can be computed from the other. A Mohr circle that starts at $\sigma'_{3c}$ and has radius $r = \tfrac{1}{2}(\sigma_1 - \sigma_3)_f$ is tangent to the R envelope when the distance from its centre to the line equals its radius:

and with $\tau_{ff} = r\cos\phi'$ the two parameters of the $K_c = 1$ envelope follow:

The inverse, used when the $K_c = 1$ envelope is given and the R envelope is needed, follows with $T = \tan\psi/\cos\phi'$:

In all of these $\phi'$ is the effective friction angle of the soil, from its drained envelope, not the angle of the R envelope.

3.8.5Multi-stage procedures#

Stage 1: before the drawdown#

The slope is analysed in effective stresses at the initial level with the chosen method of slices. Its solution gives, for each slice, the effective normal stress and the shear stress on the base, which are taken as the consolidation stresses of the soil before the drawdown:

where

$N'$

effective normal force on the base in the stage-1 solution (kN/m)

$l$

length of the base (m)

$F_1$

factor of safety of stage 1 (—)

$\tau_{fc}$ is the mobilised shear stress, not the strength. The procedures assume that the slope is stable before the drawdown: a slip surface with $F_1 < 1$, or whose stage-1 calculation did not converge, is not analysed further.

Stage 2: undrained strength after the drawdown#

At the final level, every slice whose base lies in a material marked undrained receives an undrained strength $\tau_{ff}$ from its consolidation state and is analysed in total stresses, with $c = \tau_{ff}$, $\phi = 0$ and $u = 0$. Freely draining slices keep $c'$, $\phi'$ and the pore pressures of the final level. The consolidation state of each slice is taken from the stage-1 slice at the same abscissa, because the slices of the two stages need not coincide.

In the procedures of Lowe & Karafiath (1960) and Duncan, Wright & Wong (1990) the undrained strength is interpolated between two bounds, both evaluated at the same $\sigma'_{fc}$, according to the anisotropy of the consolidation stresses. The consolidation stress ratio of the slice, and the one at which the soil would fail while consolidating, are

and the strength is interpolated linearly in the ratio:

Here $\tau_1$ and $\tau_f$ are the strengths at $K_c = 1$ and at $K_c = K_f$. At $K_c = 1$ the strength is that of the $K_c = 1$ envelope; at $K_c = K_f$ the soil would fail during consolidation, and the strength is the drained one (Figure 3.8.3).

In the two-stage procedure of the Corps of Engineers (USACE, 1970; USACE, 2003) the undrained strength is read directly from the R envelope at the consolidation stress:

Stage 3: the drained cap#

An undrained strength above the drained one can only be sustained by negative pore pressures, which cavitation or partial drainage may prevent. Duncan, Wright and Wong therefore add a third stage, and the Corps of Engineers use a composite envelope, the lower of the R envelope and the drained one. In both procedures OGR Slip2D limits the strength of each undrained slice by the drained strength at the normal stress of the solution after the drawdown, and repeats the analysis:

where

$\tau^{(k)}$

strength of the slice in pass $k$ (kPa)

$N^{(k)}$

normal force on the base in the solution of pass $k$, in which the undrained slices carry no pore pressure (kN/m)

$\omega$

relaxation factor (—)

Capping a slice lowers the factor of safety, which changes the normal forces at which the cap is read; the cap is therefore iterated to a fixed point. The factor of safety of the procedure is the lower of those of stages 2 and 3. Reading the drained strength at the stress after the drawdown, rather than at the consolidation stress $\sigma'_{fc}$, is the choice with which the program reproduces both procedures in the worked example of USACE (2003, Appendix G).

3.8.6Sweep of drawdown levels#

The complete drawdown is not always the critical one. On a homogeneous slope the factor of safety falls steadily as the reservoir is lowered, and the complete drawdown governs (Morgenstern, 1963). In a zoned dam whose upstream shell drains freely, a complete drawdown leaves the shell with no pore pressure, while a partial one leaves water in it near the toe; the minimum can then fall at an intermediate level.

The drawdown level sweep repeats the whole analysis, search included, at a series of final levels. The search is repeated because the critical surface moves with the level: a surface held fixed can miss the minimum the sweep looks for. By default the levels are eleven, equally spaced from the initial level at the middle of the model down to the lowest point of its external boundary, plus the complete drawdown. At each level the drawdown line is moved rigidly so that its mean elevation is that level, or a horizontal line is created if the model has none.

3.8.7Transient seepage as an alternative#

The procedures above assume that the undrained materials do not drain at all during the drawdown, and that the free-draining ones drain completely. When the rate of drawdown and the permeabilities are known, the drawdown can instead be modelled as a transient seepage problem (Section 3.7.8). The reservoir level of each stage is lowered through its boundary conditions, and the stability is computed in effective stresses at every stage flagged Calculate SF, with the pore pressures of that stage (suction included, where the material parameters allow it) and with its own ponded water. The result is the factor of safety as a function of time, whose minimum need not fall at the end of the drawdown.

This route needs no undrained envelope, but its answer depends on the permeabilities and storage parameters of the materials, which are often less certain than their strengths.

3.8.8References#

• Duncan, J.M., Wright, S.G. & Wong, K.S. (1990). Slope stability during rapid drawdown. Proceedings of the H. Bolton Seed Memorial Symposium, vol. 2, 253–272. BiTech Publishers, Vancouver.
• Lowe, J. & Karafiath, L. (1960). Stability of earth dams upon drawdown. Proceedings of the 1st Pan-American Conference on Soil Mechanics and Foundation Engineering, Mexico City, vol. 2, 537–552.
• Morgenstern, N. (1963). Stability charts for earth slopes during rapid drawdown. Géotechnique, 13(2), 121–131.
• Skempton, A.W. (1954). The pore-pressure coefficients A and B. Géotechnique, 4(4), 143–147.
• U.S. Army Corps of Engineers (1970). Engineering and Design: Stability of Earth and Rock-Fill Dams. Engineer Manual EM 1110-2-1902. Washington, DC.
• U.S. Army Corps of Engineers (2003). Engineering and Design: Slope Stability. Engineer Manual EM 1110-2-1902. Washington, DC.
• Wright, S.G. & Duncan, J.M. (1987). An Examination of Slope Stability Computation Procedures for Sudden Drawdown. Miscellaneous Paper GL-87-25, U.S. Army Engineer Waterways Experiment Station, Vicksburg.