Infinite Slope Factor of Safety with Slope-Parallel Seepage

Take the dry slope from the previous page and let the water table rise to the ground surface, with water seeping steadily down-slope parallel to the ground. Nothing about the soil grains changes. The factor of safety roughly halves.

The reason is Terzaghi's, and it is entirely about which stress friction responds to. Below the water table the total normal stress on the failure plane is set by the saturated unit weight, $\gamma_{sat} z\cos^{2}\beta$. But friction is mobilised by the effective stress, total less pore pressure. With flow parallel to the slope, the pore pressure on the plane is $u = \gamma_{w} z\cos^{2}\beta$, so the effective normal stress is $(\gamma_{sat} - \gamma_{w})z\cos^{2}\beta = \gamma' z\cos^{2}\beta$. Meanwhile the driving shear stress is still generated by the full saturated weight of the soil, because buoyancy does not lighten the down-slope pull. Resistance uses $\gamma'$; demand uses $\gamma_{sat}$. Divide, and the depth cancels once again, leaving:

$FS = \dfrac{\gamma'}{\gamma_{sat}}\cdot\dfrac{\tan\phi'}{\tan\beta}$

The ratio out front is close to one half for almost every real soil, and that single number explains a great deal of the world's landslide record. A saturated soil at 19.6 kN/m³ has a buoyant unit weight of 19.6 − 9.81 = 9.8, so $\gamma'/\gamma_{sat} = 0.50$ exactly. Saturated soils do not vary much: the ratio runs from about 0.45 for a heavy dense gravel to about 0.55 for a light organic silt. So a slope standing at $FS = 1.8$ in a dry August is at about 0.9 with the water table at the surface, and a hillside that has held for forty summers can go in one afternoon. This is why slopes fail during rain and not during the dry spell before it. Nothing about the soil changed; the pore pressure did.

It also explains the shape of the landslide record in mountain regions: the events cluster in the wettest weeks of the year, and within those weeks they cluster in the hours after the most intense rainfall, because pore pressure responds to infiltration with a lag set by the permeability of the mantle. The strength of the soil is not the variable. The water is.

What this form assumes, and where it is optimistic. It takes the water table exactly at the ground surface, seepage exactly parallel to the slope, and steady state. Real storms do none of those things. A wetting front descending into an unsaturated mantle produces transient pore pressures that can locally exceed the steady-state value; a permeable mantle over a tight substrate perches water and can generate pressures above hydrostatic; and artesian pressure from a deeper aquifer can beat anything a surface water table produces. All three are worse than this equation, not better.

It also has no cohesion in it at all. That is honest for a clean sand and wrong for most real colluvium, where root reinforcement and a little cementation supply a small $c'$. The general form on the neighbouring page carries both, and is the one to use once you have numbers for them.

FS = 1.0 is not safety; it is the definition of impending failure. A real slope is designed to the governing code, on a site investigation with real piezometer records, by a qualified practitioner. This page is a teaching aid.