It is the small hours of the morning in the Grand Canyon, pitch-black and with rain cascading down. Sixteen of us are huddled together in sodden sleeping bags, desperately trying to keep warm, on a cumbersome vessel that we have created by lashing together six inflatable rubber rafts. It is so dark that we have no idea whether the howling wind is blowing us up or down the swirling, turbulent Colorado River, though we do know that at one point we were stuck in a giant eddy for at least an hour. The storm is so loud that we can scarcely hear ourselves speak. To make matters worse, we must somehow keep well clear of the towering riverbanks because they are collapsing in the deluge. We are hours from civilisation, or any prospect of rescue.

As I pray for dawn, and vainly attempt to shelter my wife and myself with a flimsy tarpaulin, my mind is full of fearful thoughts – that some of us will soon be struck by hypothermia; that we will provide fodder for those lurid newspaper headlines about British tourists who come to grief in foreign parts; that we will, at the very least, miss our flights home from Las Vegas later in the day. I reflect ruefully that the south-western United States are meant to be suffering from their worst drought in modern times. I think of our son, Barney, who was unable to join his two sisters on the trip and is back in London, completely oblivious to his family’s plight. Above all I keep thinking that this nightmare cannot really be happening, for this was supposed to be the last and easiest leg of a rafting trip that has carried us more than 270 miles through the canyon during the previous 20 days, and down some of North America’s most formidable rapids.

I had rafted the Grand Canyon before, back in 2005. It was such a momentous experience, such a unique way of exploring one of the great natural wonders of the world, that I vowed one day to do it again – this time with my family. That is why I had repeatedly entered the annual lottery for the one private trip that the US National Park Service allows to set off down the canyon each day, and I had finally won a cherished permit. These private trips should not be confused with the commercial trips that whisk bevies of tourists down the canyon in motorised mega-rafts. The private trips are for the purists, the adventurers. You organise them yourselves. You have to row rather than use an outboard. You can take a maximum of 16 people, and you have up to 25 days to complete the journey, which gives you time to savour the majestic scenery But you must expect to be taken far outside your comfort zone.

After months of preparation my group duly assembled on our allotted day in April at Lee’s Ferry, the last crossing point above the canyon. I was accompanied by my apprehensive wife, Katy, and my two excited daughters, Hannah and Imogen, both in their 20s so far too old to be tempted on more conventional family holidays. Imogen’s boyfriend, Matt, had also flown out from London. We were joined by seven friends from Britain, the US and Canada and four experienced ‘river runners’ from the north-western US whom we had recruited through the grapevine, so we had at least one person in each raft who had done the trip before. It was an eclectic group – male and female – and included a former US ambassador, a former oil company executive, a London Blue Badge tour guide, a young technology whizz from Silicon Valley, a PhD student, a school librarian, a cyber warrior, a forester from Montana, a potato merchant from Washington State and a house painter from Wyoming. I had no idea whether it would gel.

A local outfitter provided the six rafts, and a small mountain of food and beer, which we packed into coolers containing great blocks of ice – three days before the end of the trip we were still eating fresh steak. We had a satellite phone for emergencies. We packed our clothes, sleeping bags, books and cameras into dry bags. We received an orientation talk from a park ranger named Peggy who reminded us that everything we took into the canyon had to come out again – even our excrement and the ashes from campfires. She told us how to prepare a landing site for a helicopter as that would be our only means of escape should we require emergency evacuation. ‘Just deal with it!’ she replied when asked what to do if stung by a scorpion or bitten by a rattlesnake. We laughed a little too loudly at her jokes.

Then, beneath a hot sun and perfect blue sky, our little armada set off down the broad green Colorado and entered the self-contained world of rock and water that would be our home for the next three weeks. We would have no communication with the outside world. Apart from a few fellow rafters we would see no other humans except at Phantom Ranch, 88 miles downriver, where there is a ranger station and a bridge for hikers. The canyon may be one of the world’s biggest attractions, but it is also one of the most remote and inaccessible. The vast majority of its five million or so annual visitors merely peek over the rim at one of two viewpoints.

The rafting was thrilling. We would float gently down the placid river, admiring our spectacular surroundings and spotting bighorn sheep on the shore, until we heard the roar of a rapid and saw a ribbon of white foam dancing across the water ahead of us. We would then tighten our life jackets, make sure everything was strapped down and plunge into the furious maelstrom – whooping, hollering and rowing frantically to avoid the giant ‘holes’, the vertical walls of icy water, the vicious lateral waves and hidden rocks that could flip our half-ton rafts in an instant as a river flowing at 9,000 cubic feet a second waged its eternal battle with the huge boulders in its bed. Invariably we emerged drenched but elated.

In all we negotiated nearly 90 major rapids without flipping, but we broke two oars and punctured a raft in the process. Three of us were hurled into the raging torrents. The river also swallowed a dozen plastic bowls, several hats, a pair of purple rubber gloves, a wine glass and a copy of Jonathan Franzen’s novel The Corrections. Given the sheer volume of water, it was extraordinary to think that the Colorado peters out before it reaches Mexico’s Gulf of California because of excessive irrigation.

But the rafting was also a means to an end. It was the only way other than on foot of exploring a canyon that is 18 miles across at its widest point and would stretch from London to Newcastle if overlaid on Britain.

We marvelled as the great ramparts of rock on each side of us grew steadily higher, reducing our rafts to mere specks of red and yellow at the base of a giant gorge. Then new ramparts would appear behind the first ones, and behind them an astonishing array of buttes and buttresses, caverns and arches, spires and chimneys. The canyon grew in size so gradually that we only grasped its true magnitude when a helicopter flew into it near the end of our trip. The machine looked no bigger than a wasp.

The canyon played havoc with our sense of time as well as space. Occasionally, carved on rock faces, we saw the names of 19th-century pioneers who drowned while trying to raft down it – John Wesley Powell, a one-armed Civil War veteran, was the first to successfully navigate what was then called the ‘Great Unknown’, in 1869. We thought those inscriptions were old until we found 1,000-year-old Native American granaries built into the cliffs, and petroglyphs of primitive humans painted on rock faces. But even they were young compared with the rocks themselves.

As we progressed, and as the river cut steadily deeper, ever more strata were exposed – layer upon layer of red, yellow, ochre, green and black, more than 20 in all, most the petrified sediments of ancient seas bearing names such as Kaibab, Toroweap, Coconino and Supai. In one side canyon we were actually able to touch the two-inch-thick Great Unconformity, the last remnants of nearly a billion years of soft rock crushed between the Tapeats sandstone above and the granite and schist below. At the canyon’s deepest point, almost a mile down, we encountered ebony-black Vishnu schist, which, at nearly two billion years old, is some of the oldest exposed rock on the planet.

The canyon is a geologist’s dream, a history of the Earth in petrified form. Powell called it a ‘library of the gods’ and ‘the most sublime spectacle on Earth’, though he and his nine colleagues narrowly escaped death by drowning and starvation. Three died when they gave up and tried to escape on foot.

We would spend a few hours on the water each day, then hike. We climbed up to distant viewpoints along seldom-used trails that were, in places, so narrow, crumbly and vertiginous that I could scarcely bear to watch my daughters negotiating them. At Whitmore Wash we climbed right to the rim, using an old stockmen’s trail that passed displays of hexagonal basalt columns every bit as dramatic as those of the Giant’s Causeway in Northern Ireland. We scattered lizards as we went. We admired the flowering cacti, and were particularly taken by the occasional lone century plant standing sentinel in the arid landscape like an outsize hollyhock. It flowers only once in 20 or 30 years then dies.

We clambered up side canyons that had been sculpted into fantastical forms by millions of years of rushing water. Some led to huge bowls and vast natural amphitheatres of smooth, swirling rock – one named Womb of Woman. Some tapered into narrow, labyrinthine passages with vertical walls that were hundreds of feet high but so close that you could touch both at once. Others had their own gentle microclimates, with green ferns, lush hanging gardens and clear streams, and provided a lovely contrast to the harsh desert landscape all around.

Each evening, as the sinking sun set the canyon walls ablaze, we pitched camp on pristine sandy beaches – most rafters scrupulously observe the park service’s edict about leaving nothing behind. We ate hugely, and royally, from our well-stocked coolers to sate our prodigious appetites, and drank plenty of wine and beer.

Every day seemed to produce a fresh surprise – another mini-wonder within the giant wonder that is the canyon – that had us gasping in amazement. We frolicked beneath waterfalls so magical that no set designer could ever have created them. We wallowed in the brilliant, turquoise waters of the Little Colorado where it joined the emerald waters of the main river in a stunning juxtaposition of colours. With our hands on one wall and our bottoms on the other, we inched up the deep sinuous channel carved through the soft rock by the milky, aquamarine waters of Havasu Creek. We soaked in Pumpkin Spring, a spring-fed pool overlooking the river whose arsenic-laced waters contain algae that have turned its walls bright yellow and orange

Where Kanab Creek joined the Colorado we smeared ourselves head-to-toe in rich mud washed down from distant Utah, let it dry then peeled it off – a treatment that might have cost hundreds of dollars in a metropolitan spa. We played an impromptu game of football on the sandy floor of Redwall Cavern – a vast cave that would, Powell claimed somewhat hyperbolically, seat 50,000 people.

My fears about the group not gelling proved unfounded. The joshing, joking and inevitable water fights had begun on day one, and only intensified. The Brits were soon conspiring to steal the Stars and Stripes from one of the American rafts. People sparked off one another intellectually. The excitement of running rapids, our constant discovery of new delights and our shared awe at our surroundings cemented our sense of unity. The imported boatmen were full of fun and enthusiasm, and swiftly became fast friends. The sound of laughter dominated our long, happy and largely carefree days.

Not even the twentysomethings missed their smartphones. On the contrary, we revelled in our digital detox to such an extent that when we stopped after a week to collect fresh water at Phantom Ranch nobody wanted to use the payphone for fear of ‘breaking the spell’. So absolute was our isolation that we would not have known had World War Three erupted.

Deprived of Facebook and Twitter, we created our own entertainment. We read books, played games, swam, fished for trout and – late one afternoon – watched a large beaver swim nonchalantly around our moored rafts. We staged a quiz and a debate (‘This house believes China will be the world’s leading superpower by 2030’) around the campfire, and had different members of the group give mini-lectures on their particular areas of expertise. Using tin plates and cooking pots, we created a percussion band whose dire, cacophonous output will assuredly not go viral.

We dispensed with our watches because they had no use. When we grew tired we laid our mats on the sand and fell asleep beneath a spectacular firmament of stars and a halogen moon. One of our group erected a ‘moonbrella’ to block its glare – except on the night when we lay on our backs and watched, awestruck, a total lunar eclipse. We woke at dawn each day to blue skies, the chirrup of canyon wrens and the exquisite pleasure of hot, strong coffee.

There were hardships of course. One day we endured a vicious sandstorm that drove grit into eyes, nostrils, books, clothes, cameras – everything. On another we faced headwinds so strong that our rafts were blown backwards no matter how hard we rowed. Our hands and feet were soon cracked and blistered by constant exposure to water and the dry desert air. Our lips were chapped, our nails broken, our toes stubbed, and I had to drop my trousers rather abruptly one afternoon to pick scores of tiny cactus thorns out of my left buttock. Inserting contact lenses amid so much fine sand was agony.

There were real dangers too, and not just from the rafting. One day a sizeable boulder came tumbling down a mountainside as we hiked up. It bounced right through the middle of our group, narrowly missing Katy’s head. Had it hit anyone it would undoubtedly have killed them. We encountered several rattlesnakes, including one that had somehow found its way into a raft. James, the Silicon Valley whizz, prudently shook out his clothes before dressing one morning and a scorpion scuttled away. And then, of course, there was that final night.

We naively believed that the climax to our trip had come two days before when we reached Lava – the last really big rapid on the Colorado, whose fearsome reputation had haunted us from the outset. On the scale of one to 10 used to grade the rapids it warrants a nine.

‘It sounds like a 747 taking off,’ Craig, one of the boatmen, exclaimed as we rounded a corner of the river and heard Lava’s roar. We stopped to scout it from the riverbank, attempting to plot the safest course through its fearsome maze of ‘holes’ and ‘hydraulic haystacks’ and ‘laterals’ and rocks as big as garages. ‘It’s what we call a crapshoot,’ John, our leading boatman, declared.

There was a nervous silence as we returned to our boats, donned helmets if we had them, and secured our gear. ‘Good luck’ and ‘Have a good run’, we shouted as the rafts pushed off, one by one. Katy went first, in Craig’s boat, and made it safely down. I went last, with Hannah and Imogen as my passengers, and our hearts were pounding.

For a moment the river remained calm as we glided towards the boiling foam. Then suddenly we were past the point of no return. We were being swept down a tongue of smooth water into the very heart of that foaming white turbulence. What followed was 20 seconds of sheer exhilaration. The raft reared and plunged like a bucking bronco. Water crashed over us from all sides. My rowlocked oars were ripped from my hands by the strength of the waves, or scythed through air where there should have been water, as I desperately tried to keep us on course. The noise was deafening – but not loud enough to drown out my daughters’ cries of delight.

And then it was over. We emerged to the cheers of our fellow rafters, and were soon celebrating our deliverance on what is informally known as ‘Tequila Beach’ just below Lava. ‘That was amazing!’ Imogen exclaimed. ‘Can we do it again?’ Hannah laughed. ‘I was so scared for you,’ Katy said.

Thereafter the rapids gradually diminished until there were no more left and the river’s rage was spent. Our plan was to lash the rafts together late on our final afternoon, then float 40 miles out of the canyon overnight – most of us asleep – towards the distant lights of Las Vegas. We had arranged to be collected at the top of Lake Mead early the next morning. It was a plan with pleasing echoes of Huckleberry Finn. It was a plan that would, I reckoned, provide the perfect end to this article, allowing me to compare the gaudy, man-made wonders of Vegas with the superior natural ones of the canyon.

The night started well enough as we relived the highlights of our trip, toasted its success with the last of our alcohol and dined on barbecued chicken, chocolate cake and cheese. We held a mock awards ceremony, presenting John with the Order of the Broken Oar for guiding us safely down the river. We then drifted happily off to sleep – only to be woken sometime after midnight by gusting winds and the first rain to fall in the canyon in four months.

It was not just rain. It was a deluge. For hours, armed with head torches whose feeble beams were no match for such a pitch-black night, we struggled to keep our vessel on course, periodically crashing into rocks and trees that had toppled into the river. For hours we fought an increasingly futile battle against the bitter cold and wet. The night seemed interminable. We thought that dawn would never come. But it did, eventually, shortly before 5am.

As weary heads emerged from those sodden heaps of sleeping bags I expected a collective cry of misery, a litany of complaints, but I underestimated the extraordinary team spirit our group had developed. Someone remembered that Imogen turned 25 that day and started singing Happy Birthday. Everyone laughed and joined in. The hilarity grew as we sang rounds of Row, Row, Row Your Boat and London’s Burning. Hannah rummaged in a cooler and produced cold bacon sandwiches and energy bars. The sky lightened. The rain eased. We started rowing hard. We were hailed by another group of rafters camped beside the river, and shortly before 8am the last cliffs fell away. We rounded a bend in the river and cheered in triumph as we saw the trucks of our outfitting company waiting for us at Pearce Ferry.

Hours later we were catching our flights home – filthy, bedraggled and exhausted, but utterly euphoric after the ultimate family adventure.