The water that comes back while the floors go up

A basement below the water table is a boat. The water pushes up on its base slab with a pressure set by the depth of water above the slab and nothing else, and what holds the basement down is weight: its own concrete, the building above it, and whatever ties anchor it to the ground. The check contains no strength. For a 20 × 30 m basement dug 6 m below a water table at the surface, the uplift is 35.3 MN and the slab and walls weigh 18.7 MN, a factor of 0.53, and four or five storeys of building are needed before the weight catches up.

That calculation is made on a finished state, and a basement is never built in one. It is built dry, inside an excavation whose water is drawn down by pumps or wells; the pumps keep the water under the slab while it is cast; and the building that will one day hold the basement down goes up afterwards, a floor at a time. At some point the pumps are switched off. From then on the water comes back, and the building goes on rising, and the basement is held down only if the weight is ahead of the water at every instant between the pumps stopping and the building being complete.

The basement’s first essay found that permanent drainage is usually refused, because a basement’s design life is a century and a pump’s is fifteen years. Temporary drainage is different: it is how every such basement is built, and the decision it forces is not whether to rely on a pump but for how long. The same shape of question — a weight arriving against something that changes with time — decides a tower built slowly on clay that gains strength as it drains, with the race run the other way round.

The race, drawn #

Two curves have to be compared, and they have different shapes.

The weight grows in steps. The basement box is complete at day zero; the frame above adds a floor of 8 kN/m² — 4.8 MN over the 600 m² plan — every seven days, and each floor’s weight arrives when it is cast, all at once. The weight that counts in the check is 0.9 of it, the factor EN 1997-1 applies to a permanent action that is holding something down, because a weight must be known before it can be relied on and might be a little less than drawn.

The water comes back smoothly. When the pumps stop, the drawn-down zone under and around the excavation refills by seepage through the ground, and the rate of refilling falls as the head still to be made up falls. That is a first-order recovery: the head under the slab rises as $h(\tau) = h_\infty(1 - e^{-\tau/T})$, where $\tau$ is the time since the pumps stopped and $T$, the recovery time, is set by the ground’s permeability, the storage of the drawn-down zone and how well any cut-off wall seals it. The uplift follows the head and counts at its full value.

The two curves cross twice. For six days after the pumps stop the water is still below the weight; then the uplift, rising fastest at the start, overtakes it and stays ahead for three weeks, until the frame has added enough floors to pull back in front. The finished building holds the basement down with 16.1 MN to spare. The problem is entirely in the middle — an instant in a construction programme that no check of any finished state can see, the kind of state in which a structure that was never complete is usually found to be at its weakest, at which the structure has neither the protection of the pumps nor the weight of the building.

The worst moment is just before a floor is cast, since the weight steps up and the water does not. Here it is the moment before the third floor, at day 21, and the shortfall is 5.05 MN.

The pumps left on for longer #

The same basement with the pumps run until the third floor is cast is a different picture.

Two more weeks of pumping and the race is not a race. When the pumps stop there are three floors on the slab, 0.9 of 33.1 MN against an uplift that begins at nothing and approaches 35.3 MN; the water rises fastest when it has the most to make up and the building has the most in hand, and by the time it is close to its full height two more floors have arrived. The closest approach is 3.6 MN, a fortnight later.

Neither figure needs the building to be heavy enough on its own when the pumps stop. Three floors on the box are 33.1 MN, which at 0.9 is 29.8 MN — less than the full uplift. The basement would float on the day the pumps stopped if the water came back that day. It is held down by the fact that the water takes time to come back, and the building does not stop growing while it does.

How long the pumps must run #

The question that decides a programme is the smallest number of floors at which the pumps may stop.

On the left of the figure the water returns almost at once and the answer is the one the finished-state check would give: five floors, the first number at which 0.9 of the weight exceeds the uplift. That rule — leave the pumps on until the building can hold itself down — is safe for any ground, and it is the rule that is right only when the ground refills in hours.

Moving right, the requirement falls in steps, and how fast it falls depends on how quickly the frame goes up. A frame that casts a floor every five days needs two floors at a 7-day recovery and none at 14; one that casts a floor every ten needs three at 7 and two at 14. The pumps’ running time is set by the ratio of two rates — how fast the ground refills against how fast the frame adds weight — and not by the weight of the building at all, except at the left-hand end where the ground wins every race.

The dashed curve is the same answer with the weight added continuously, a floor’s worth spread over its week, and it runs about a floor below the steps. The difference is real: a floor’s weight arrives when its concrete does, at the end of its cycle, and for the whole of the cycle before that the water is rising against the weight of the floor below.

One curve #

The continuous version has a closed form, and the closed form says more than any one basement can.

With weight added at a rate $\omega$ per day, the shortfall after the pumps stop is $U_\infty(1 - e^{-\tau/T}) - 0.9(W_s + \omega\tau)$, where $W_s$ is the weight on the slab when the pumps stop. The water’s rate of rise is $U_\infty e^{-\tau/T}/T$, largest at the start and falling; the weight’s is a constant $0.9\omega$. The shortfall is largest where the two rates are equal, at $e^{-\tau_1/T} = 0.9\omega T/U_\infty$, and making it zero there gives

$$W_s = \frac{U_\infty}{0.9} - \omega T\left[1 + \ln\frac{U_\infty}{0.9\,\omega T}\right].$$

The first term is the finished-state rule. The second is what the race is worth: the weight the frame adds in one recovery time, multiplied by a logarithm that grows slowly as that weight becomes a smaller share of the total. Divided through by $U_\infty/0.9$, everything collapses onto one variable.

The dimensionless number $x$ is the whole of the problem: how much weight the frame puts on in one recovery time, as a fraction of the weight it would take to resist the full uplift. It contains the plan area, the depth, the floor load, the cycle and the ground, and every basement with the same $x$ needs the same fraction of its full weight in place when the pumps stop. At $x = 1$ the frame outruns the water from the first day and the pumps may stop with nothing on the slab at all; a basement whose ground refills in three months under a frame that rises a floor a week is in that condition, which is why some basements are built with their pumps stopped almost as soon as the slab is cast and never come near floating.

The curve is steep near the left. At $x$ of a few hundredths — a fast recovery, a slow frame — almost the whole weight is needed, and the race is worth little; a basement in open gravel is in that corner. Between 0.1 and 1, which is where a recovery of days to weeks under a weekly cycle puts most basements, it falls fast, and each doubling of the recovery time or of the frame’s speed saves a large share of the floors.

Floors of pumping against piles in the ground #

The race does not have to be won outright. Where the pumps cannot run long enough — because the frame is slow, or the abstraction licence ends — ties can carry the shortfall, and the question becomes how many.

Each curve falls by a roughly constant amount per floor until it reaches zero, and the amount is nearly a floor’s factored weight: 4.3 MN, about ten piles, per week of pumping. So the trade between the two remedies is a rate, and a simple one. A week of pumping costs a week of pumps, power and discharge; ten tension piles cost ten tension piles, and remain in the ground for the life of the building doing nothing once the building is finished. Where the recovery is slow, the race makes the piles unnecessary after a floor or two; where it is fast, it saves a floor’s worth of piles per week and no more, and the piles that the basement’s tension-pile essay counted for the empty case — about forty — are the number the race starts from rather than the number it ends at.

The piles drawn here are the same 600 mm piles 15 m long, each holding 636 kN by shaft friction with a resistance factor of 1.4. For the numbers involved — a few dozen at most, at centres of four metres or more under a 20 × 30 m raft — the group is nowhere near close enough to lift its block of ground, which is the failure that limits tension piles packed tightly.

A recovery guessed too slow #

Every number so far needs the recovery time, and the recovery time is a property of the ground that is estimated before the excavation is dug, from permeability tests that scatter by an order of magnitude. A programme built on a guess has to survive the guess being wrong.

The curve for one floor is the one to read. A programme that stopped the pumps after the first floor on the strength of a 14-day recovery is safe if the ground behaves as assumed, and has a ratio of 1.09 at the worst instant. If the ground in fact refills in three days, the ratio at the worst instant is 0.66 — below 0.9, which means the weight is less than the uplift, unfactored, and the basement floats, held only by whatever friction its walls have on the soil around them. The partial factors give no protection against this, because they were applied to the weight and the water, and the error is in the time.

The three-floor curve shows what a margin in floors buys. It stays above the floating line for any recovery longer than about two and a half days and meets the check from four; a ground that refilled faster than that is a ground in which the dewatering would have needed so many wells that its speed of recovery would have been obvious from the pumping itself. That is the honest form of the advice: a programme should be judged by how fast a recovery it survives, not by whether it passes at the recovery assumed.

Measuring the recovery instead of guessing it #

The exponential has a property that makes the guess unnecessary. Its initial slope is $h_\infty/T$: the tangent to the recovery curve at the moment the pumps stop reaches the full head at exactly $\tau = T$. So a trial switch-off — the pumps turned off for a few hours, with piezometers under the slab read every half hour, and turned back on — measures $T$ directly, from the rate at which the head starts to come back, before any of the race has been run.

That is the observational method in its plainest form: design for a range of the uncertain quantity, measure the quantity at the start of the work, and decide on the measurement. It works here because the quantity can be measured at no risk — a few hours of rising head is a few per cent of the full head, far below the 53 per cent at which the box alone would float — and because the decision it informs, how many more weeks to pump, can be changed every week. The same trial repeated after each floor tests whether the recovery has changed, which it does if a cut-off wall is leaking more than it did, or a neighbouring excavation has started drawing on the same ground.

The whole of it, by hand #

For the basement drawn, with a floor every seven days and a recovery time of seven days, the numbers fit on a line each.

The full uplift is $6 \times 9.81 \times 600 = 35{,}300$ kN; divided by 0.9 it is 39,200 kN, the weight that would hold the basement down by itself. The frame adds $8 \times 600 = 4{,}800$ kN a week, $\omega = 686$ kN a day. The dimensionless number is

$$x = \frac{0.9 \times 686 \times 7}{35{,}300} = 0.122,$$

and $1 - x(1 + \ln(1/x)) = 1 - 0.122 \times 3.10 = 0.62$. So the pumps must leave $0.62 \times 39{,}200 = 24{,}400$ kN on the slab. The box weighs 18,700, so the frame must add 5,700 kN — 1.2 floors, two when rounded up, and three once the floors are counted as arriving at the end of their weeks rather than through them. The worst instant comes $T\ln(1/x) = 7 \times 2.10 = 14.7$ days after the pumps stop, which is why stopping after the first floor at day 7 put the worst moment at the third floor’s casting, day 21.

Uniform ground, a fixed water table and a frame that keeps to its programme #

The calculation rests on several simplifications, and each can be wrong in a direction worth knowing.

The recovery is first-order. Water returning to a drawn-down zone through uniform ground does approximately follow an exponential, but the radial flow towards a single well or a long trench has a recovery that is faster at first and slower later — closer to a logarithm in time than an exponential. An exponential fitted to the first hours’ steep rise then overstates how fast the head approaches its final value, which errs on the safe side; one fitted to a long, slow tail of readings understates the first rise, which is where the race is lost.

The water returns to the level it started at. A wet season during the work, or a river in flood, can put the water table above the level the basement was designed for — a load whose size depends on something outside the structure — and the head under the slab is the only quantity in the check that the structure cannot do anything about.

The frame keeps to its cycle. A strike, a late steel delivery or a crane breakdown stops the weight and not the water. A programme that stops the pumps on the strength of the next three floors arriving on time has made the building’s construction schedule a part of the basement’s stability, and a week’s delay at the wrong moment is a week in which the water catches up a floor. That is a reason to stop the pumps with the margin the last figure shows, not exactly at the number the race allows.

The walls carry no friction. Shear on the outside of the retaining walls adds to the weight, and it is left out here as it is in most designs, because it depends on how the backfill was placed and how much the walls have moved. It is a reserve that the calculation does not count, and in the floating case above it is the only thing between the basement and the water.

The slab between the walls, and the sand beneath it #

The figures cannot show the slab itself. The check here is on the whole basement as a rigid body, and the uplift is applied over its whole plan; the slab between the walls and any piles is a plate spanning under that pressure, and its bending is a separate check whose worst moment arrives with the same water. A slab sized for the finished building, with columns bearing on it at every bay, can be carrying the full uplift with only three floors of those columns’ load above it.

They also cannot show the ground under the slab during the race. While the head is rising, the effective stress in the soil beneath falls, and a fine sand or silt with an upward gradient through it can lose its strength before the slab has moved at all. That is a different failure, at the formation rather than at the structure, and it is the reason dewatering specifications control the rate at which the pumps are turned down as well as the moment they are turned off.

Still open: the basement that leaks back #

Everything here has the water returning to its original level and staying there. A basement in a city is often built beside another basement already pumping, or under a site where the long-term water table has been rising for decades as industrial abstraction has stopped. In both, the head under the slab is not a fixed number but a slow trend under the fast recovery — a few centimetres a year, measured in boreholes, and not part of any design load until somebody adds it. Whether a basement that won its race during construction is still holding itself down fifty years later, when the water table it was designed for has risen a metre, is a question about a load that changes on the timescale of the building’s life rather than its construction, and the margin the finished building was given is the only thing that answers it.