The drain that fails a season at a time

The building held down by its pump followed a basement whose water table had risen above its design, given back its design head by drains under its slab and a sump pump. While the pump runs, the slab sees what it was designed for; when the pump stops, the head comes back on the ground’s clock, and under a light building the first thing to go is not the slab’s cracks but the building’s weight against the water’s push. Under five storeys the building floats a few hours after the pump stops. The essay treated the failure as sudden: the pump runs, or it does not.

Drains fail slowly as well. The water they collect carries fines out of the ground, and the fines settle where the flow slows — in the drainage blanket under the slab, in the joints of the pipes, in the sump’s own inlet. The drains’ capacity falls year by year. The pump keeps running; its level switch keeps cycling; the sump stays at its set level. And the head under the slab creeps up, because the water reaches the slab faster than the silted drains can carry it away. Nothing alarms. This essay asks what the slow failure does to the margins the fast one was measured against, and whether the cheapest instrument in the building — a standpipe through the slab, read once a year — can see it coming.

The head the pump does not control #

The basement is the earlier essays’: 20 by 30 m in plan, dug 9 m, its 0.9 m slab held down at six columns and by five storeys above, designed for a water table 3 m below the ground. That table has risen, here by 2 m, and it swings by ±0.5 m over the year, highest in early spring. A sump pump holds the outlet of the drains at the design head, 6 m above the slab’s underside.

The pump does not hold the slab at that head. It holds the sump. Between the ground and the sump the water passes through two resistances in series: the ground itself, which it has to cross to reach the drainage blanket under the slab, and the drains, which carry it from the blanket to the sump. The head in the blanket — the head the slab feels — is wherever the two flows balance. Flow through each is in proportion to the head it loses across it, so with $C$ the drains’ conductance and $C_g$ the ground’s,

$$C_g\,(H - h) = C\,(h - h_0) \quad\Rightarrow\quad \Delta h = h - h_0 = \frac{H - h_0}{1 + \rho},\qquad \rho = \frac{C}{C_g}.$$

The head under the slab is the rise outside divided by one plus the ratio of the two conductances. New drains are far more conductive than the ground: here thirty times, which leaves the slab $2/31 = 0.06$ m above its design head. The pump looks as though it is holding the design head because, while the drains are new, it nearly is.

Silt lowers $\rho$, and the head climbs towards the full rise as $\rho$ falls. How fast drains silt depends on the soil, the filter, the flow and the maintenance, and it is not something a designer can know: the law here is an exponential loss of conductance with a time constant of eight years, the simplest that never quite reaches zero, and it is an assumption. What follows from the balance does not depend on it much, because the answers can be stated in the silt’s own time.

The curve has two clocks in it. The slow one is the silt’s: the ratio $\rho$ falls by a factor $e$ every eight years, and the head follows $1/(1 + \rho)$, which hardly moves while $\rho$ is large and then rises steeply once $\rho$ falls through a few. The fast one is the season’s: the water table swings ±0.5 m every year, and the head under the slab swings with it, divided by the same $1 + \rho$. While the drains are new the swing is a few centimetres. By year twenty it is the better part of a metre.

The first limit is reached at a seasonal peak. On the eight-year drains the head first touches flotation’s 0.97 m in the spring of year 24, when the table is at its highest, and falls back below it every autumn for another two or three years before it stays above. A building whose basement floats for a few weeks every spring is not a building that has failed gradually; it is one that fails suddenly, once a year, in a season that grows longer as the silt goes on.

One reading a year #

A standpipe through the slab is the obvious instrument: a tube open to the drainage blanket, capped at the floor, into which a dip meter is lowered. It reads the head the slab feels directly, and it costs almost nothing to read. Suppose it is read once a year, at the same date, and an alarm is raised at the first reading more than 0.25 m above the design head. The warning is the time between that reading and the moment the head first reaches flotation.

The warning is long in every case drawn — the silt gives its notice over years — but the month decides how long. Read in April, at the top of the season, the reading sees the head the slab will next feel at its worst, and the alarm at 0.25 m sounds in year 10.3, 13.9 years before flotation. Read in October, at the bottom, the reading sees the head a metre of table lower, divided by the same factor, and the alarm sounds four and a half years later. The staircase in the curves is the reading’s own resolution: an alarm can only sound on a reading day, so the warning drops by a year whenever the alarm date slips past one.

The month matters more as the alarm level rises. At an alarm of 0.5 m the April reading gives about eight years and the October reading two and a half, and the next figure follows the alarm level to where the October reading stops warning at all.

The two curves separate as the alarm rises, and the dry-season curve crosses zero at an alarm of 0.6 m. That is the alarm a reader would set by subtracting a comfortable margin from the 0.97 m limit, and read in October it never sounds in time: the October head at the year it would reach 0.6 m is already below a spring head that has floated the building. A dry-season reading spends the alarm level’s margin before the alarm does. The rule that follows is plain and is the one an annual reading has to obey: read at the season of the highest water table, or set the alarm at what the dry-season head will be when the wet-season head reaches the limit, which is the limit divided by the ratio of the table’s high to its low — here $0.97 \times 1.5/2.5 = 0.58$ m, the value at which the curve crosses.

The warning is the silt’s own time #

The silting time constant was assumed, and a reader who distrusts eight years is right to. The useful fact is that the warning does not depend on it in any way that matters, because every date in the problem is a time for $\rho$ to fall to some value, and every such time is a multiple of $\tau$.

The two lines are straight apart from the reading’s yearly steps. Without the season the arithmetic is exact: the head reaches a level $\ell$ when $1 + \rho = r/\ell$, at

$$t_\ell = \tau \ln\frac{\rho_0}{r/\ell - 1}.$$

For flotation at $\ell = 0.97$ m with the spring table $r = 2.5$ m, $t = \tau \ln(30/1.57) = 2.95\,\tau$; for the alarm at 0.25 m against the same spring table, $t = \tau \ln(30/9) = 1.20\,\tau$. The warning, read in spring, is about $1.75\,\tau$ — 14 years at $\tau = 8$, 7 at $\tau = 4$, 28 at $\tau = 16$. A drain that silts twice as fast gives half the warning, and the same alarm gives it in the same share of the silt’s life. The standpipe does not need to know how fast the drains silt; it measures the drains’ own clock.

That holds only while the silting is smooth. Drains also fail in steps: a collapsed pipe, a sump inlet blinded by one flood’s worth of fines, a filter fabric that clogs suddenly once a crust forms. A step loss of conductance moves the head by a large fraction of the remaining margin overnight, and an annual reading can miss a whole year of it. The smooth law is the case the standpipe is good at.

What the silt does to the pump #

The earlier essay measured the basement’s exposure to the pump by its deadline: the hours an outage may last before the head, returning towards the full table, reaches the first limit. A building whose pump fails twice a year for six hours on average, with a deadline of a few hours, floats some fraction of those times. The silt changes the deadline, because the head no longer starts its return from the design head.

The deadline falls the whole way. When the pump stops, the head in the blanket returns from wherever the drains were holding it towards the full table, $\Delta h(t) = r - (r - \Delta h_s)e^{-t/T}$, and reaches the limit $\ell$ after

$$t_\ell = T \ln\frac{r - \Delta h_s}{r - \ell},$$

where $\Delta h_s$ is the head the silted drains hold. New drains leave $\Delta h_s = 0.08$ m in spring, so with $T = 12$ hours and $r = 2.5$, $t_\ell = 12\ln(2.42/1.53) = 5.5$ hours. At year 14.8, when the October reading first alarms, the spring head is 0.44 m and the deadline is $12\ln(2.06/1.53) = 3.6$ hours. Every year of silt takes a fraction of the start’s distance from the limit away, and the deadline goes to zero when the silted head reaches the limit on its own.

The frequency is two stops a year times the chance a stop outlasts the deadline, $e^{-t_\ell/\bar t}$ with $\bar t = 6$ hours, averaged over the seasons because the deadline is longer in autumn. New, 0.55 a year: a building that, on these numbers, floats about once every two years already — which is the earlier essay’s finding, that a pump under a light building is a flotation risk, and not a new one. By the year the standpipe alarms it is 0.75 a year, up by a third. By the time the silt would float the building on its own in spring, nearly 1.5.

That is the answer to the question the earlier essay left. The basement loses its margins to the pump first and to the silt second, and the silt’s first effect is on the pump. Long before the head the drains hold reaches any limit by itself, it has shortened the outage the building can survive, and the slow failure arrives as a faster rate of the sudden one. A standpipe reading of 0.25 m does not mean the drains are nearly finished. It means the next stop of the pump is a third more likely to float the building than the first one was.

What the alarm should set off #

An alarm that sounds ten or fourteen years before a limit is not an instruction to evacuate the basement. It is information about which failure the building is now closer to, and the last figure says what that is. At the alarm the drains still hold the head well below every limit; what has changed is the pump’s deadline, from 5.5 hours to 3.6 in spring. The response the alarm should set off is therefore the response to a shorter deadline: clean the drains, which moves the curve back towards its start, or shorten the outages — a second pump on a separate supply, a telemetered alarm on the sump level, a contract to restore the pump inside the new deadline rather than the old one. The earlier essay found that a standby pump bought most of the margin a light building had lost; the silt is the reason that margin has to be bought again every few years.

The reading also says something about the original design decision. A basement is a boat recorded why permanent drainage is usually refused at design: a pump’s life is a fraction of a building’s. The silt adds that the drains’ life is a fraction of the pump’s, measured in a time constant nobody can know in advance, and that the cheapest way to learn it is the one instrument the repair did not include. A basement relieved by drains without a standpipe has a failure whose warning exists and is not being read.

A heavier building, and a different first limit #

Under eight storeys the building cannot float on this rise: its weight exceeds the water’s push even with the head at the ground. The first limit is then the slab’s own cracks at 1.65 m of rise, and the same silting drains take the spring head there in year 33. The October reading still alarms at 0.25 m in year 14.8 — the alarm does not know how heavy the building is — and the warning becomes 18.4 years. The standpipe’s alarm date is a property of the drains, and the warning is a property of the building: a heavier building is given more warning by the same reading because its first limit is further up the same curve. A designer choosing an alarm level for a basement with several limits sets it against the lowest of them, which here moves from flotation to the cracks as the storeys are added — the order the leaking basement found, with the silt now supplying the rise.

An alternative to the drains altogether is to hold the building down rather than the water: tension piles or ground anchors under the slab, which the piles that would lift the ground found have their own way of failing, by lifting the block of soil they sit in. They do not silt and they do not stop. The comparison between the two is a comparison between a maintenance regime with a measurable rate and a capacity that is fixed on the day the piles are tested.

The free body the slab is #

Cut the basement free at its walls’ outer faces and its slab’s underside. Downward act the building’s weight, the basement’s own weight and whatever skin friction the walls carry, which the earlier essays took as nothing. Upward acts the water pressure on the underside, $\gamma_w (h_0 + \Delta h)$ over the whole plan. Flotation is the balance between the two, and $\Delta h$ is the only term that changes; the five storeys above set how much of it there is room for.

The standpipe reads $\Delta h$ directly, at one point. That point is enough only because the drainage blanket is a single connected layer whose head is nearly uniform; a blanket that silts unevenly — as it will, worst near the sump and the low points — has a head that varies across the plan, and the free body then wants the integral of a head the standpipe samples once. One standpipe at the far corner from the sump reads the worst of it; one beside the sump reads nearly the design head however silted the far side has become.

What a smooth silting law cannot promise #

The silting law is an assumption. An exponential loss of conductance is the simplest smooth law and makes every date a multiple of $\tau$, which is why the warning can be stated in the silt’s own time. A drain that silts linearly, or in steps, gives a different curve to the same limit, and the step case is the one an annual reading handles worst.

The two flows balance instantly. The ground answers in hours and the silt in years, so at every instant the head is the steady value for that day’s conductance. Through a single wet week the head under the slab lags the table outside by the ground’s time constant, which slightly delays the spring peaks; it does not change which year they first reach a limit.

The season is a sine. Real water tables rise steeply in a wet winter and fall slowly; a wet year can put the table half a metre above any average season and reach the limit years early, which is a statistical question about rainfall rather than about the drains.

The slab’s checks are the earlier essays’. The flotation limit of 0.97 m and the cracks’ 1.65 m are the basement that leaks before it floats’s numbers for this slab and this building, and a heavier building moves flotation past the cracks, as the essay on the pump found at six storeys and more.

Still open: whether the drains can be cleaned in time #

Every figure here lets the silt run. Drains can be jetted, sumps desilted and filters replaced, and a basement that depends on its drains will have a maintenance regime that does exactly that — on a schedule fixed by a contract rather than by the head. Cleaning restores some of the conductance, never all of it, because the fines that have moved into the blanket and the ground around the pipes stay there. Whether a regime of cleaning every few years holds the head under the slab below the alarm indefinitely, or only resets the clock to a point that is a little further along each time — so that the warning the standpipe gives shrinks with every cleaning, and a basement with a perfect maintenance record reaches the water that comes back while the floors go up on a schedule nobody wrote down — is the question that turns the silt from a failure into a rate, and the standpipe from an alarm into the measurement that sets the cleaning interval.