Pore pressure and unsaturated strength

Because the methods of slices work in effective stresses, the factor of safety depends directly on the pore pressure at the base of each slice. OGR Slip2D can take that pore pressure from water surfaces, the ratio $r_u$, a constant value, a pressure grid or a finite-element seepage field. Three other effects of water also enter the analysis: water ponded on the slope, excess pore pressure from undrained loading and the strength that matric suction adds to an unsaturated soil.

3.6.1Pore pressure in the analysis#

The shear strength available on the base of a slice is written in effective stresses (Eq. (3.1.1)). For a Mohr–Coulomb material it is

where

$\sigma_n$

total normal stress on the base (kPa)

$u$

pore pressure at the base (kPa)

$c',\ \phi'$

effective cohesion (kPa) and effective friction angle (°)

The methods of slices carry it as forces: the pore-water force on a base of length $l$ is $U = u\,l$, subtracted from the total normal force $N$ (Eq. (3.1.2)). OGR Slip2D evaluates one pore pressure per slice, at the midpoint of its base, and every method reads that value in its own equilibrium equations (Section 3.2). The Ordinary method is the only one that does not subtract the whole of $u\,l$; it is treated in Section 3.6.10.

3.6.2Groundwater methods and pore-pressure types#

Pore pressure is defined at two levels. The project's groundwater method can install a field that covers the whole model (a pressure grid or a finite-element solution), and each material has a pore-pressure type, which decides where its pore pressure comes from whenever no such field governs it. Table 3.6.1 lists the project methods and Table 3.6.2 the material types.

3.6.3Water tables and piezometric lines#

A water table or a piezometric line is a polyline. At the base midpoint of a slice the program takes the vertical distance $h$ up to the surface assigned to the material and writes

where

$\gamma_w$

unit weight of the pore fluid (kN/m³)

$y_w(x_c)$

elevation of the water surface above the base midpoint, interpolated linearly along the polyline (m)

$y_b$

elevation of the base midpoint (m)

$H_u$

pressure-head factor, between 0 and 1 (—)

Where the surface lies below the point ($h \le 0$) the pore pressure is zero: a water surface never produces suction. With $H_u = 1$ the expression is the hydrostatic pressure below a horizontal free surface.

The Hu factor#

Below an inclined phreatic surface the water is not static, and the vertical depth overstates the pressure head. When the flow lines are straight and parallel to a phreatic line inclined at $\alpha$, as in an infinite slope with seepage parallel to its surface (Skempton & DeLory, 1957), the equipotentials are straight lines perpendicular to it. The equipotential through the base midpoint $P$ meets the phreatic line at $Q$ (Figure 3.6.1), where $u = 0$ and the total head equals the elevation $y_Q$. $P$ and $Q$ share the same total head, so the pressure head at $P$ is their difference of elevation. $P$ lies at a distance $h\cos\alpha$ from the line, measured along the equipotential, and $Q$ is higher than $P$ by that distance times $\cos\alpha$:

The automatic factor takes $\alpha$ as the inclination of the segment of the water surface above the base midpoint. It is exact only under the assumption that produced it (straight, parallel flow) and an approximation under a curved phreatic surface, where the equipotentials curve as well.

Water table or piezometric line#

Both give the pore pressure of Eq. (3.6.2). They differ in three respects, all of which follow from a water table being a free water surface and a piezometric line a measurement of pressure head:

• ponded water (Section 3.6.7) is created only by a water table drawn above the ground surface;
• only a water table separates the saturated unit weight from the unit weight: a material that uses a saturated unit weight takes it in the parts of each slice column that lie below the highest water table;
• a pressure grid is limited only by a water table: with a grid method, $u = 0$ above the highest water table (Section 3.6.5).

Several piezometric lines can coexist (two aquifers separated by a clay layer, for instance), and each material reads the one assigned to it.

3.6.4The pore-pressure ratio r_u#

Bishop & Morgenstern (1960) expressed the pore pressure in a slope as a fraction of the vertical stress, the pore-pressure ratio $r_u$. OGR Slip2D applies it at the base midpoint as

where

$r_u$

pore-pressure ratio of the material (—)

$\gamma$

unit weight of the material at the base, its saturated unit weight if the material uses one (kN/m³)

$z$

depth of the base midpoint below the midpoint of the slice top (m)

$d_w$

depth of water ponded on the ground above the slice, zero if there is none (m)

The vertical stress is that of a single unit weight over the whole depth (the one of the material at the base) plus the ponded water; external loads are not included. In a layered slope this differs from the slice weight, which is integrated band by band.

3.6.5Water pressure grids#

A water pressure grid is a set of points $(x_i,\ y_i,\ v_i)$, either piezometer readings or the output of another seepage analysis, from which a value $v$ is interpolated at each base midpoint. The project's groundwater method states what the values are, and the pore pressure follows from it:

The values are interpolated with a thin-plate spline (Harder & Desmarais, 1972; Duchon, 1977), the radial-basis surface

whose $n + 3$ coefficients solve the symmetric system

The spline passes through every data point and reproduces a planar field exactly, so a hydrostatic field given as total heads (constant) or as pore pressures (linear in $y$) is recovered without error. Outside the cloud of points it extrapolates, and a slip surface near the crest of a slope often has bases there; the points should cover the whole region the slip surfaces cross.

When the spline cannot be used the program falls back to inverse-distance weighting (Shepard, 1968) over the nearest points:

where

$d_i$

distance from the point to the $i$-th nearest data point (m)

$k$

number of neighbours, 8 by default (—)

3.6.6Pore pressure from a seepage analysis#

When the groundwater method is a finite-element analysis (Section 3.7), the solution is a total head $H_i$ at each node of the mesh, and the nodal pore pressures are $u_i = \gamma_w (H_i - y_i)$. At a base midpoint the program finds the triangle that contains it and interpolates with the linear shape functions of that triangle:

where

$N_i$

linear shape functions of the triangle, equal to the barycentric coordinates of the point (—)

$u_i$

pore pressures at its three nodes (kPa)

The interpolation is exact for the piecewise-linear field the solver computes. Above the free surface the field is negative (the pore water is in suction), and those values are passed to the treatment of unsaturated strength of Section 3.6.9.

3.6.7Ponded water#

Where a water table lies above the ground surface, the water standing on the slope is a load, not a pore pressure. Its hydrostatic pressure $p = \gamma_w d$ acts normal to the top of each slice it covers (Figure 3.6.2). On a slice of width $b$ whose top has slope $m = \mathrm{d}y/\mathrm{d}x$, the top is $b\sqrt{1 + m^2}$ long and the resultant of that pressure has the components

where

$d$

depth of water above the midpoint of the slice top (m)

$W_w$

vertical component, downwards: the weight of the water column standing on the slice (kN/m)

$H_w$

horizontal component, positive towards $+x$; it pushes the slope towards the side where the ground rises (kN/m)

The two components belong to a single force. Summed over a submerged face, the horizontal components give the classical thrust $\tfrac{1}{2}\gamma_w h_w^2$ on its vertical projection. Both act at the midpoint of the slice top.

Only a water table produces ponded water. A piezometric line drawn above the ground records a pressure head somewhere in the ground and implies no standing water; a drawdown line ponds only at the final level of a rapid-drawdown analysis, when it takes the place of the water table (Section 3.8). With a finite-element groundwater method the water is also read from the boundary conditions: it stands above every boundary node whose prescribed total head or pressure head exceeds the elevation of the node, in runs that are contiguous along the boundary and never extrapolated beyond their last submerged node. Where drawn and prescribed water overlap, the higher level governs.

Ponded water carries no pseudo-static seismic force. The seismic coefficients multiply the weight of the soil only (Section 3.9): water has no shear strength, and its inertia is not carried by the sliding mass.

3.6.8Excess pore pressure from undrained loading#

A soil of low permeability loaded faster than it can drain responds with a rise in pore pressure. Skempton (1954) related that rise to the change of stress through the pore-pressure coefficients; OGR Slip2D uses the one-dimensional form with the coefficient $\bar B$:

where

$u_0$

pore pressure given by the groundwater method and the material's type (kPa)

$\bar B$

Skempton's coefficient of the material (—)

$\Delta\sigma_v$

increase of vertical stress at the base midpoint (kPa)

The increase of vertical stress is the sum of three contributions, each of which has to be switched on; all are transmitted straight down, with no spreading with depth:

where

$\gamma_j,\ t_j$

unit weight and thickness of each band above the point whose material is set to load the materials below it; the column is cut at material boundaries and water tables as for the slice weight (kN/m³, m)

$q_v$

vertical component of the distributed loads set to create excess pore pressure (kPa)

$P_v$

vertical component of a line load set to create excess pore pressure and applied within the slice (kN/m)

$b$

slice width (m)

$k_v$

vertical seismic coefficient, when the seismic load is on and set to create excess pore pressure (—)

Excess develops only in materials with $\bar B > 0$; $\bar B = 0$ is a free-draining material. Whether a material's weight loads the materials below it is a separate setting: an embankment over a clay foundation would typically load the clay while developing no excess itself. A line load is a concentrated force, so the stress it produces depends on the width it is spread over (here, the slice), and $\Delta u$ beneath it grows as the slices are made narrower; a distributed load is the better model for a surcharge whose excess pore pressure matters.

3.6.9Negative pore pressures and unsaturated strength#

Above the free surface a seepage analysis returns negative pore pressures: the matric suction $s = u_a - u_w$, with the pore-air pressure $u_a$ atmospheric and taken as zero. Entering them unchanged in Eq. (3.6.1) would credit the whole suction with the saturated friction angle. Fredlund, Morgenstern & Widger (1978) wrote the strength of an unsaturated soil with two independent stress variables:

where

$u_a,\ u_w$

pore-air and pore-water pressures (kPa)

$\phi^b$

angle of the increase of shear strength with matric suction (°)

OGR Slip2D applies a bilinear form of it, controlled by two parameters of each material: $\phi^b$ and the air-entry value $s_e$. Below $s_e$ the soil is still effectively saturated and the suction acts through $\phi'$; beyond it only the excess over $s_e$ acts, through $\phi^b$. The program expresses this as an effective pore pressure $u_\text{eff}$, which the methods subtract as usual, and a suction cohesion $c_s$, which they add to the cohesion of the base:

so that the strength of the base becomes

Before this, the suction can be limited by the project's negative pore pressure cutoff $s_\text{max}$, which replaces $s$ by $\min(s,\ s_\text{max})$. Figure 3.6.3 shows the three cases. With the default parameters, $\phi^b = 0$ and $s_e = 0$, any negative pore pressure is set to zero and suction adds no strength, which is the conservative choice.

3.6.10Pore pressure in the Ordinary method#

The Ordinary method ignores the interslice forces and resolves the forces on each slice normal to its base. Subtracting the whole pore-water force $u\,l$ from $W\cos\alpha$ in that resolution gives low, often negative, effective normal forces on steep wet bases. OGR Slip2D uses the correction of Turnbull & Hvorslev (1967), which takes the pore-water force over the vertical projection of the base:

where

$W$

weight of the slice (kN/m)

$\alpha,\ l$

inclination (°) and length (m) of the base

$N'$

effective normal force on the base, to which the program adds the components of the other external forces (kN/m)

The term $u\,l\cos^2\alpha = u\,b\cos\alpha$ is the component normal to the base of the vertical force $u\,b$. In the same spirit, the normal force at which the Ordinary method reads the strength leaves out the horizontal thrust of ponded water, so that water enters it through its vertical components only and a submerged slope gives the factor of safety of the same slope with buoyant unit weights. The other eight methods take the whole force $U = u\,l$ in their own equilibrium (Section 3.2).

3.6.11References#

• Bishop, A.W. & Morgenstern, N. (1960). Stability coefficients for earth slopes. Géotechnique, 10(4), 129–153.
• Duchon, J. (1977). Splines minimizing rotation-invariant semi-norms in Sobolev spaces. In W. Schempp & K. Zeller (eds), Constructive Theory of Functions of Several Variables, Lecture Notes in Mathematics 571, 85–100. Springer, Berlin.
• Fredlund, D.G., Morgenstern, N.R. & Widger, R.A. (1978). The shear strength of unsaturated soils. Canadian Geotechnical Journal, 15(3), 313–321.
• Harder, R.L. & Desmarais, R.N. (1972). Interpolation using surface splines. Journal of Aircraft, 9(2), 189–191.
• Shepard, D. (1968). A two-dimensional interpolation function for irregularly-spaced data. Proceedings of the 1968 23rd ACM National Conference, 517–524.
• Skempton, A.W. (1954). The pore-pressure coefficients A and B. Géotechnique, 4(4), 143–147.
• Skempton, A.W. & DeLory, F.A. (1957). Stability of natural slopes in London Clay. Proceedings of the 4th International Conference on Soil Mechanics and Foundation Engineering, London, 2, 378–381.
• Turnbull, W.J. & Hvorslev, M.J. (1967). Special problems in slope stability. Journal of the Soil Mechanics and Foundations Division, ASCE, 93(SM4), 499–528.