You'd need some massive pits and you'd have to displace way more than 31.5 million cubic meters of soil as it isn't all going to evaporate in a day. Ideally you'd want really shallow pools so you'd eat up an insane surface area.
For some reference an Olympic sized swimming pool is about 2,500 cubic meters so you'd need 12,600 (pools) just to hold one day worth of the contaminated water.
For a fun little compassion (that amuses me and might someone else) it is estimated that all of the silver ever mined adds up to about 166,375 cubic meters Source: https://www.usgs.gov/faqs/how-much-silver-has-been-found-wor...
And only about 8000 cubic meters of gold has been mined. Source: https://www.jmbullion.com/investing-guide/james/gold-supply/
Edit: derpity-derp-derp fix
Averaged over a year, the energy available to evaporate water over a given area is roughly equal to insolation from the sun. Saudi Arabia has excellent solar resources averaging 2200 kWh/m^2/year [2], e.g. 6 kWh/m^2/day, e.g. 21.7 MJ/m^2/day. With perfect sunlight absorption you could evaporate 9 liters per m^2 of evaporation-pit per day.
They're currently discharging 31.5 billion liters of liquid per day, per the article. That translates to about 3,500,000,000 m^2 (3500 square kilometers) of pit area. So that's probably one reason why nobody deals with desalination brine this way. It would require very large evaporation pits even if you have excellent sun levels to evaporate the waste. It seems like it would be technically possible in SA, as they have a low population density and lots of unused land, but it sounds expensive. Some countries that rely on desalination, like Israel, don't have the spare land area even in theory.
[1] https://en.wikipedia.org/wiki/Enthalpy_of_vaporization#/medi...
[2] https://www.sciencedirect.com/science/article/pii/S101836391...
I don’t think you can just look at insolation over a given area. Wind and even the cooling itself will cause the air over the area to redistribute with warmer dryer air that can continue picking up water.
Besides that it’s more important to consider how saturated the air in the region is. If it can hold more water it will.
Other things to consider
Some of the water would seep into the ground instead of waiting around to evaporate. Maybe there are ways to speed that up too.
Finally Another option is the spay the water into the air to speed up evaporation a bit.
Seepage into the ground -- good point. If there is existing groundwater, seepage would taint it. If there is no existing groundwater, but ground is permeable, that helps.
Spray water into air? That costs more energy. Just waiting for natural evaporation is nice because it doesn't require any more human-added energy.
Of course you'd want to analyze more than the One Big Factor if you were pondering implementing this in practice. But I think that the one-factor approach illustrates why this hasn't been an "obvious" disposal solution for waste brine.
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