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Short answer: A cooler containing more ice and less frozen food will last longer than a cooler containing all frozen food.

With frozen contents of a given mass, the length of time a cooler will stay cold is proportional to the latent heat of fusion of the contents. The latent heat of fusion of water ice is 334 joules per gram. ASHRAE (the American Society of Heating, Refrigerating and Air-Conditioning Engineers) gives the latent heat of fusion of various foods as:

Lean sirloin 239 J/g

Bacon 105 J/g

Italian sausage 171 J/g

Chicken 220 J/g

Additional foods can be found in the ASHRAE Handbook, Chapter 9, Table 3. The latent heat of fusion of water at 334 J/g exceeds that of any food, so the more ice and the less frozen food in the cooler, the longer it will last.

If you look for the manufacturer's promises for these "passive" boxes (there is absolutely no cooling, just the attempt to keep the cool in - called isolation), they usually state something around 24h to keep things cool.

You don't even have to start thinking how the manufacturer got to this number. There is no way to use it in the way you proposed because your plans are on another level.

I'm assuming the amount of food you want to take and your coolbox size is fixed, so the question you're asking is really whether you should fill every remaining gap with ice, or whether you should leave it as air. The answer to this is that air inside your coolbox is a big problem, and will majorly contribute to thawing.

The specific heat capacity of air is around 1kJ/kgK, compared to around 2kJ/kgK for ice. This tells us how much energy is needed to heat a kg of ice/water. We can change this to volumes by multiplying by density to get a volumetric heat capacity, giving us about 1kJ/m³K for air and 2000kJ/m³K for ice. This shows that ice requires 2000 times as much heat as air for the same temperature change. The benefit of ice is actually more than this, because it also takes in a lot of heat without changing temperature as it melts. Your frozen food, and basically any solid or liquid, is going to be way more similar to the ice than the air, so the big factor determining melting rate is going to be how much empty space there is. Basically, every space in coolbox should be filled with ice so there's as little air as possible, and when you remove food you should replace it with something solid.

The Yeti website backs this up here.

AIR IS THE ENEMY
Large areas of air inside your cooler will accelerate
ice-melt as the ice is consumed cooling the air. These spaces are best
filled with extra ice, towels, or crumpled newspaper if weight is a
concern.

18 days frozen is a very long time - if you know others who have done this please talk to them to find out exactly how they did it, and definitely test out your box before you go. You're going to need a really large coolbox for that to be feasible. This review of the Yeti Tundra 45 has it lasting 10-11 days but always in the shade and at temperatures of only 10-20 Celsius. Summer in the grand canyon you're looking at more like 20-40 Celsius, and you not going to be able to guarantee shade all the time.

The short answer is that you need to get rid of all the air from inside your ice box. So no, frozen food by themselves won't work. The easiest way to deal with this is to let an outfitter prepare the food. Even for a private unguided trip, you can get outfitters to do all the logistics work from put-in to take-out (that's what we did when I rafted the Grand).

The method all outfitters in the Grand Canyon use is with cavernous ice boxes like this Yeti model. The raft rigs are often made so the ice chest is used as the rower's chair in the middle of the boat.

Of course, in order to stay frozen for the entire trip, the ice boxes have to be filled with water and deep frozen:

By choosing [this outfitter], you’ve signed up for the most effective food pack on the river. That’s because of our unique cooler freezing system that makes the best use of your space. We don’t mess around with blocks of ice that don’t last very long on a hot trip. Instead, we freeze ice into a solid block in your cooler that allows you to keep your food fresh to the takeout. It takes a tremendous effort on our end, but this is the extra mile on which [the outfitter] has built our reputation.

I have experienced it through the outfitter we contracted, and it works. It needs decent preparation as you have to lay the food inside in the order you expect to use it and dig it out of the ice as you go. Every day we'd pick at the solid block and get only what we needed. Our meat boxes stayed frozen for 21 days (although it was in November, I know it works just as well in the summer) and could have lasted longer for sure. There was another even larger group that we caught up with as our take-out day was the same. They stayed on the river 28 days with the exact same setup we had (same outfitter).

The produce boxes, even without ice, stayed cool as the bottom of the rafts always were cold, the air inside being cooled by the 45°F or so water.

The ice boxes were way too heavy to move and they stayed rigged in the boats the entire trip. That ensures there's a constant cooling effect from the water as the boats stay at least partly in the water the entire trip.

Proper handling needs to be observed. Don't open the box needlessly and keep it open as briefly as possible.

What you should do:

In your particular case, if you wanted to carry extras for yourself, you could get a smaller ice box and use the same technique. All you need is a chest freezer large enough to accomodate your box while it freezes. Then, to make sure it stays frozen as long as possible, keep the box in the water at all times.

You'll already be using drag bags for your beer, and you could do the same for your personal ice box. Just remember to haul it back inside the boat for the bigger rapids and it should be fine. Make sure your icebox is watertight too: getting water in and out of it as it's dragged in the river behind the boat is a sureshot way to melt it all in record time.

I don't expect a smaller ice box will stay frozen as long as the larger ones, but you'll be good for at least a week if you're taking care of it properly.

The main issue is carrying your frozen box to the put-in. That's the only leg where it will be entirely in hot air.

I will not comment on the general feasibility of your plan. However, I did a little research about the specific heat of various materials, and the results are actually pretty interesting.

We are looking at how long a given item will stay cold. Said another way, we are interested in how much heat energy the food or ice can absorb while remaining cold. The relevant property is called specific heat, or how much energy must be added to a given quantity of material to raise it by a given temperature. This value changes based on a number of different factors. One factor is the phase the material is in, such as solid or liquid.

As a reference, the specific heat of liquid water is 4.187kJ/kgK, while the specific heat of solid water (ice) is 2.108 kJ/kgK.

I found a fascinating study and a pdf excerpt (warning: does download a file) from a refrigeration handbook.

I will fully admit that the some of the math in the pdf goes over my head, but I understand enough to feel comfortable writing this answer. The authors note that one of the most important factors in determining the behavior of foodstuffs is the fraction of ice present.

One of the more interesting things that I noticed when looking at the tables of specific heat in both sources was that while water has a higher specific heat capacity than essentially any food item, ice does not show the same trend, especially in meats. Based on that, I would say that frozen meat should take more energy per unit mass to heat up than ice, and meat is also more dense than ice. Based on that, a solid cube of meat should stay cool longer.

However, there are a couple wrinkles in this. You are not dealing with a solid chunk of meat, but a bunch of smaller chunks which while "packed full", will probably not be truly packed as a solid piece of ice. Also, while various meat products are better at staying cold, that's with the caveat that we are talking about the temperature regime in which they are frozen. Once thawed, they have appreciably lower specific heat capacity than liquid water. This is relevant, since meat needn't stay 100% frozen to be safe (as long as it is eaten within a day or two), and so liquid water will need more heat than the food will to warm up.

This answer brings up a good point in discussing the heat of fusion, which takes significant amounts of energy. Looking at chicken, for example, 1 kg will take in 220 kJ to that, while it takes 3.32 kJ to raise the temperature of that same chicken by 1 Kelvin. As the other user stated, the heat of fusion of water is 334 kJ/kg, a difference of 114 kJ. This is equivalent to the energy required to raise the temperature of the chicken by 34.3 K.

In Summary

I've done a little revising of my ideas while writing this answer, after reading one of the other answers which was posted while I was in the middle of mine.

1. While fully frozen, meats tend to have a higher specific heat
   capacity than water




2. Water still has a higher latent heat of fusion than meat




3. While it is harder to heat frozen meat than ice, it is far harder to
   melt ice than thaw meat

Therefore:

• A cooler full of meat will stay cold longer (will take longer to get to the point where it will thaw, but will melt/thaw faster)

But

• A cooler full of ice will stay frozen longer (will heat up to its melting point, but will take longer to fully melt)

Also:

When you take out your food to use/eat it, you remove some of the cooling ability of the contents of your cooler. Therefore, the more food you eat, the faster the contents will heat up. If you add a significant quantity of ice, you mitigate that by having a baseline quantity of cold material that will not be removed.

TL;DR

Use ice. It stays as a heat sink even when the cooler is almost empty of food. Also, while it heats faster from -40 °C to 0 °C, it makes up for it in the energy required to actually melt the ice, which will keep your food colder in the long run.

I'm doubtful of things lasting 18 days, but definitely pre-chill the frozen stuff as cold as you can beforehand. Frozen is frozen, but there's a significant difference between -5 °C and -50 °C in terms of how long stuff will stay frozen.

To give two more numbers that can serve as rough guesstimates which make clear that cooling for 18 days on a tour is likely unfeasible. I'm assuming hiking tour as it isn't specified.

• One rule of thumb for shipping biological (research) material on dry ice in the type of styrofoam containers used for this is to pack 1 kg of dry ice per day.
  
  Which would be 18 kg in your case. In that order of magnitude you probably need a bit less because the surface area : volume ratio gets better (and these shipping rules are meant to ensure the material will stay deep frozen). OTOH, those packages are not opened during shipping.
  
  In any case, say, 15 kg of coolant is not feasible for hiking.




• I've had occasion to do some experiments for someone who had to take medication with them on a 2 week hiking tour. I took such a shipping box of styrofoam and added further styrofoam to improve insulation (to about 7 cm wall thickness - the medication didn't take much space, would be different with food). 250 g of (water) ice took ≈ 1 day to melt and heat up to 7 °C in that box with 19 °C outside temp, that's a bit more than 1 W thermal. Again, no opening of the lid in between - my guess is that opening can easily double those thermal losses.
  
  So depending on how often you open the box and what your outside temperature is, you may do OK with, say, 10 kg of ice for 18 days (more mass => more surface => more losses, even if the loss doesn't increase proportionally). It wouldn't matter whether those are inside your meat or extra as long as you always only eat meat that thawed already in the box.
  
  (For the medication the solution was that it turned out to be OK with up to 3 weeks at up to room temperature, so keeping it insulated and/or slightly cooled during the day was sufficient. A moist towel to use evaporation cooling around the insulation box was sufficient.)




• OTOH, if you can have a cache in the river you have a refrigerator at hand according to the temperature @ivanivan cites.




• For practical purposes, we can assume that the latent heat in frozen water inside the food is the same as that of the same amount of ice outside the food.
  
  So from that perspective, it doesn't matter whether you pack your latent heat sink inside or outside the meat.
  
  But of course, if you want refrigerate a given amount of food but not more, for a specified duration, you may need additional ice.

• (In any case, hamburger or other ground meat is the wrong idea as this will spoil easily: go for meat that isn't that sensitive to cooling and storage time, e.g. beef or venison, no ground meat, no pork, no chicken, no fish. Note that spoiled meat is not only spoiled but under outdoor hygiene conditions you should also avoid contaminating your fingers with spoiled meat)




• Unless the conditions of your tour are so that you anyways have to carry all your water for those 18 days (and even then: would you be willing to drink that melt water?), is carrying around 10ish kg of water or other coolant plus the bulk of the insulation for having best case ca. 2 kg of protein really practicable?




• If you move to dried meat (jerky/biltong/...) or summer sausage, you neither need cooling nor do you need to carry around that much water.
  
  You coudl also think about other sources of macronutrients that don't need cooling and don't have too much water. There's a wide variety ranging from hard cheese over milk/egg/whey powder to vegetable or whey protein powders for protein. Fat/lipids and carbohydrates are also easily available without much water and without the need for cooling.

The water temperature, which used to get as warm as 80 degrees F, is now icy-cold all year and averages around 42 degrees F

And hey, what do you know, refrigerators are often set at 42F...

So... multiple smaller high quality coolers, with dry ice and your frozen meals. Do not open them until needed, and try not to need them until the previous cooler is empty or at least mostly empty. You'll keep stuff frozen for a long time in a sealed cooler with dry ice, and then use a drag bag or two as a refrigerator.

Note that for longest storage, you'll want to vacuum seal and make sure the food is truly completly frozen at well below 32F/0c

** EDIT as response to comments **

Temperature quote is from http://www.bobspixels.com/kaibab.org/misc/gc_coriv.htm

Similar temps or ranges of temps that indicate "refrigerator temp for many folks" are indicated on multiple other sources.

By "smaller cooler" I mean break things into 3 or 4 "3 day" size coolers instead of 1 "10 day" size cooler. With one larger cooler all the foods are being exposed to the changes in temp caused by opening and closing the cooler to access the food within. With smaller coolers, only the food you are planning on consuming in the next couple of days is exposed.

Another factor to keep in mind: Heat rises--the warmest part of the ice chest is on top. Thus you have ice there and food underneath it so it's ice that melts rather than food.

That being said, you're asking more of an ice chest than it's capable of. Even stuff like the Yeti chests can't do this.

As others have pointed out, you cannot do so with a portable cooler.

It is, however, possible to keep ice for months, passively. You will need to dig an ice house, which should be the size of a typical basement/cellar, and fill it with several tons of ice during the winter. The Grand Canyon North Rim should have plenty, you can either dig your pit on the rim or dig it further down, and carry your ice. Pack animals may come in handy here. Store your food with this ice, and it should remain quite good.

You're unlikely to get a permit to dig such a construction in the National Park or the neighbouring National Forest. Perhaps you'll have some luck on neighbouring BLM land (maybe just outside the Kanab Creek Wilderness Area). Team up with some history enthusiasts for this exercise in enacting pre-modern cooling methods. Maybe you can contact National Geographic or so for a grant and they can make a documentary of your efforts, such that the process is funded. Should be fun!