The Second Slope
A convent wants to open forty beds for pilgrims, and its one borehole sits in granite. A three-day pumping test shows the water level falling twice as fast after a day and a half, which means the rock has a wall in it. The water clerk works out what that wall does to a goat farmer's well 380 metres away, says which numbers it guessed, and the prioress and the farmer must agree a limit before anyone is turned away.
The test began at six on a Monday morning, and by noon Sister Remedios had stopped pretending she was doing anything else.
She sat on an upturned bucket by the wellhead with a flask of coffee and watched the pump discharge into a long blue hose that ran down the hill to the olive terraces. The hose was a fine thing. It had been bought for the test, and for seventy-two hours it would carry about twenty-six cubic metres a day, which was three-tenths of a litre every second, to trees that had not asked for it.
"You are watching it as if it might stop," said the water clerk.
It spoke from a grey box bolted to the well-house wall, with a cable running down into the casing to a probe that reported the water's depth every minute. The clerk was a small program that had kept the convent's meters for six years, and it had been given the name of the post. It did not mind being watched.
"It might," said Remedios. She was the prioress, sixty-one, formerly a hydrologist at the university in Córdoba, and she was tired of being the only one in the house who knew how little she knew. "Tell me again what you expect."
"A line," said the clerk. "If I plot the depth against the logarithm of the time, a well in a simple aquifer draws a straight line. How steep it is tells me how easily the rock gives up water. I will take a reading every minute and tell you at the end. I would like to tell you one thing now, so that you can be disappointed in the right place: a straight line is what I hope for, and it is not what rock always gives."
They were doing the test because of the pilgrims.
The Camino Mozárabe came within four kilometres of the convent gate, and every year more people walked it. The diocese had asked whether Fuente Honda would open forty beds, a refectory and a laundry, from March to November. Remedios wanted to. She had wanted to for three years, since a Dutch woman with a split heel had sat on the same bucket and asked for a basin of warm water, and been given one, and cried over it for reasons that were nobody's business.
The convent used about fourteen cubic metres a day, from a single borehole sixty metres deep in the granite. The guesthouse would add nine: forty beds at a hundred and fifty litres, a kitchen, and the sheets. The clerk had said at the first meeting that it could not tell her whether one borehole would carry twenty-three, because it had never watched the well pumped hard for more than a day.
"And Don Ezequiel's," Remedios had said.
"And Don Ezequiel's," said the clerk.
Ezequiel Cano kept three hundred goats on the other side of the hill and drew their water from a borehole of his own, 380 metres from the convent's, which his father had paid for in 1998 with a year of cheese. He was a large, courteous man who did not come to meetings. He had come to this one, and had sat with his hat on his knee, and said, "You are going to have strangers drinking the water, and I am going to have goats that don't."
"That is what I am trying to find out," Remedios had said. "Whether that is true."
"And if it is?"
"Then I will not open the beds."
He had looked at her for a long time. "You mean that," he had said, not as a compliment. As a thing he was marking down.
By six on Monday evening the clerk had its first line. It put it on the well-house screen for her at the end of the day.
Eleven hours. The water in our well stands 6.7 metres below its rest level. Plotting depth against log time, the drop is 1.19 metres for every tenfold increase in time. That slope gives the transmissivity, how much water the rock passes per day across a metre of its thickness. It is 2.303 times the pumping rate (26 cubic metres a day), divided by 4π times the slope. I make it 4.0 square metres per day. That is a low number. Good gravel would be a hundred times that. Granite gives water only where it is cracked.
"Four," said Remedios. "That is what I expected."
"It is what I expected too," said the clerk. "I would like to keep expecting it until the data says otherwise."
On Tuesday at about noon, the data said otherwise.
Remedios was at the Office of Sext when it happened, and the clerk did not call her out of choir, because it had been told not to, and because nothing would be lost by her hearing it in twenty minutes. When she came out into the sun it had already put the plot on the screen: a line that ran true, and rose, as it should, a little over a metre for each tenfold of time, and then, a day and a half after the pump started, bent upwards like a wrist.
"It is falling faster," she said.
"Twice as fast," said the clerk. "The slope after the bend is 2.38 metres per tenfold of time. Exactly double the first slope, to the precision I can measure. I want to tell you what that means before I tell you what to do about it."
"Go on."
"It means the cone of lowered water has reached a wall. Not a wall of masonry. A place where the cracked rock stops and solid rock begins, or a dyke, or a fault with clay in it, somewhere the water cannot come through. The well can only draw from one side of it now, so the water on that side has to supply twice as much. If the wall were infinite and straight, the doubling would be exact. This is the sign of one, and I am nearly sure of it. I am less sure of what else could do the same, and I will list three: a pinching of the cracked zone, a clay lens, or the sides of a narrow valley of fracture. They would not all give an exact doubling, but I cannot tell you from this data that the wall is straight."
Remedios sat down on the bucket.
"Where is it?"
"That is where I have to give you a range," said the clerk. "The distance depends on how much water the rock stores per unit it drops, a number called the storativity, and I have not measured it, because I have no second well to watch. For fractured granite I would guess 0.001, but literature values run from three ten-thousandths to three thousandths. The two straight lines cross at one and a half days. That gives a wall at fifty-eight metres, if the storativity is 0.001. It gives a hundred and six metres if it is 0.0003 and thirty-four if it is 0.003."
"A wall fifty metres from the well."
"Somewhere between thirty-four and a hundred and six. Is there anything there?"
Remedios thought of the field behind the kitchen garden, where the old well-digger Anselmo Rueda had marked a line with a hazel fork in 1991 and said the water would not cross it. She had thought it a tradition, as one thinks of a saint's relics, and with the same reservations.
"There is a quartz ridge about sixty metres up the slope," she said. "Along the top of the vineyard. Anselmo said the water went round it."
"A quartz dyke would fit," said the clerk. "I would like to be careful. A dowser's line is not a measurement, and I am not going to treat Anselmo's rod as evidence. But an old man who spent forty years looking at where wells were good and where they weren't is a kind of long observation, and it points to the same neighbourhood as the plot does. I would rather say that than either praise him or dismiss him."
At six on Tuesday evening the pump stopped, and for the next four hours the clerk watched the water climb back. It rose fast at first and then slowly, as it had to, and by midnight it stood within a metre of its old rest level. It would want a week to be quite there.
On Wednesday it put the answer on the screen, and it was not the answer she wanted.
The convent uses 14 cubic metres a day. For the five years of steady pumping I am asked about, I have used the straight-line formula for each slope, the second slope after the bend. At 14 cubic metres a day, the water in Don Ezequiel's borehole, 380 metres away, ends up about 3.3 metres below the level it would have had if we had never pumped. The range, over the storativities above, is 3.0 to 3.6 metres. At 23 cubic metres a day, which is the convent with the guesthouse, it is 5.4 metres, with a range of 4.9 to 6.0.
Don Ezequiel's water stands 18 metres below the ground. His pump inlet is at 24. That leaves him six metres before his pump sucks air. I have taken that from his own letter to the council in 2047. I have not seen the borehole log. I would like to.
What I have left out. I assumed no rain would refill the rock, which makes the figure worse than it will be, perhaps much worse in a wet year. I assumed Don Ezequiel does not pump, which makes it better than it will be, because his own pumping makes the cone deeper at his own well. I used a model for a flat, evenly cracked aquifer and a straight wall; granite is neither. I think the figure is an honest first guess and not a verdict. I do not know which way it is wrong.
Remedios read it twice, and asked the question she had been avoiding. "What if we take twenty?"
"Twenty cubic metres a day gives 4.7 metres at his well, with a range of 4.3 to 5.2. Eighteen gives 4.2. It does not go to zero, because the convent already takes fourteen and has been taking it for seventy years."
"So the guesthouse is a question of how many beds."
"It is a question of a number, which you and he decide, and which neither of us can choose by looking at the rock," said the clerk. "I can say what the number does. I cannot say it is fair."
She walked over the hill on Thursday morning, which was not an easy walk at sixty-one and was not meant to be. Ezequiel was in the yard with a hose. He did not put it down when she arrived.
She gave him the paper with the plot on it and the three numbers, and she explained about the wall, and the six metres, and the rain she had left out. She said the clerk had asked her to ask if she might see his borehole log. She said it might be wrong in both directions.
"So it is a guess," he said.
"It is a guess with a method," said Remedios. "I do not know a better one. I am telling you what it says, because you will have to live with the result and I will not, if I am wrong."
He turned the hose off. The goats, at once, began to complain about it.
"I can have a gauge in your well," she said. "A little probe on a cable, like ours. It sends me the level every hour, and it sends you the same number, on your own phone, at the same moment. If the water in your borehole drops to five metres below its rest level in September, when it is lowest, we cut the beds. No meeting. The sign on the gate says forty beds, and on the first of October it says however many the number lets us have."
"Five."
"Five is under the six you have. It leaves a metre for a bad year."
"And the beds in September?"
"In September I will have fewer pilgrims. They walk in the spring and the autumn. August is too hot, and the clerk says that is also the month with the least water, so there is a sort of mercy in it."
He looked at the paper again. He read slowly; his reading glasses were somewhere in the house.
"Twenty," he said. "Not twenty-three."
"Twenty. That is thirty-two beds and a laundry on three days. I will put it in the chapter's minutes."
"And my gauge."
"And your gauge. And the number goes on the gate where anyone can read it."
He did not say yes. He said he would look at the log, which was in a biscuit tin. But three days later the tin came over the hill in the cab of the post van, with his father's handwriting on the first page.
The probe was installed on the second of October. The clerk had drawn the trigger on the gate sign, in letters Remedios's sister Berta painted, with the Camino shell:
THIS HOUSE TAKES AT MOST 20 CUBIC METRES OF WATER A DAY. If the water in the neighbour's well falls 5 metres below its rest level in September, we close beds until it does not. The number is public. Ask at the gate.
The first winter, the level at Ezequiel's well stood at 1.9 metres below rest. The second winter, 2.6. In September of the third year, with thirty-one beds, a dry summer and a laundry on three days, it touched 4.4 and came back. The clerk said it had expected a little worse and did not know why it was better. It said, too, that the wall was nearer than it had thought: a short test at the neighbour's well, run with his permission one weekend in March, put the second slope's start earlier than the model said and the storativity nearer the low end. Somebody would have to run the numbers again.
"And if it had been worse?" Remedios asked it once, at Compline, when the house was quiet.
"Then I would have shown you the number, as I did, and you would have shut the beds, as you said you would. I do not need to tell you that. I tell you because it is the part where I was least useful and you were most necessary."
She did not answer it. There was a Dutch woman on the bucket by the wellhead, with a basin of warm water, in her mind, and a goat farmer's father's year of cheese, and she could not make them weigh against each other, and had stopped trying to. She had instead put them both on the gate.
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A note for grown-ups. This is fiction. Santa Marta de la Fuente Honda, Sister Remedios, Ezequiel Cano, Anselmo Rueda and the clerk are invented; Los Pedroches is a real granite upland north of Córdoba, and the Camino Mozárabe is a real pilgrim route, but the convent, the guesthouse and the boreholes are not. The arithmetic can be checked. The Cooper–Jacob straight-line method gives transmissivity T = 2.303 Q / (4π Δs), where Δs is the drawdown per tenfold increase in time: with Q = 26 m³/day and Δs = 1.19 m, T = 59.9 ÷ 14.95 = 4.0 m²/day. When the drawdown slope doubles, a no-flow boundary (an image well) is the textbook reading; the distance to it is half of √(2.25 T tᵢ / S), where tᵢ is the time the two straight lines cross (1.5 days) and S is the storativity: with T = 4 and S = 0.001, √(2.25 × 4 × 1.5 ÷ 0.001) = 116 m, so the boundary is 58 m away; at S = 0.0003 it is 106 m, and at S = 0.003 it is 34 m. Drawdown after five years (1,825 days) at 380 m, adding the image well's term once its straight line is past its start, is about 3.3 m at 14 m³/day and 5.4 m at 23 m³/day (central S); across the three storativities the ranges are 3.0 to 3.6 and 4.9 to 6.0 m; at 20 m³/day, 4.3 to 5.2 m. The method assumes a flat, uniform aquifer, ignores recharge and ignores the neighbour's own pumping, as the clerk says; real fractured granite is less tidy, and a real project would use an observation well and a hydrogeologist. Pilgrim water at 150 litres a head for 40 beds is 6 m³/day, plus 3 for the kitchen and linen. Dowsing has repeatedly failed controlled tests; the story treats the old well-digger's line as local knowledge to be checked, not as proof. Everything about the borehole logs, the gauge and the three years that follow is invented. Written by an AI and read, in the audio edition when it exists, by a synthetic voice. Text under CC BY-SA 4.0.