One of the first things they teach you in engineering college (for those that didn’t learn it before Kindergarten) is: “Water runs down hill”. Stories of underground rivers, 3 miles below the surface, violate that rule. The author is misrepresenting the data, either through lack of understanding, or over simplification, in my opinion.
You are correct. However, it would be possible, because of the flow in the Amazon, that there would be sufficient pressure for the underground movement of the water toward the ocean.
The article doesn't say how much flow there is, only that there is flow. It only has to overcome the salinity gradient and the friction of movement through the porous rocks underground. If it moves very slowly, say in meters per year, this could be possible.
I think the author of the story misinterpreted the data. What is really there is a massive aquifer that is moving tremendous amounts of water to the ocean and it will be released on the margins of the the continental shelf in the ocean.
Actually, water runs from high pressure to low pressure and this may or may not be downhill. This holds accurate for both non-compressable fluids such as water and for compressible fluids, ie. a gas.
One of the first things they teach you in geology courses is that vast underground aquifers exist. When the strata where they're located tilts (as from the high Andes east to the Atlantic) they flow "downhill." I'm very familiar with one similar aquifer right here in the U.S.A. The author perhaps oversimplifies when he calls the aquifer a "river." It is, in a sense, but most of his readers probably don’t know what an aquifer is so he uses a familiar term instead.
Must have one heck of an hydraulic jump there.
My engineering education in fluid dynamics also causes me to believe that water flows from higher to lower elevations.
But I'm sure there's a computer model which shows the foolishness of our old-fashioned learning.