The exponential relation you refer to is called Athy's law and is a well-known empirical law. There are many ways to express it in terms of burial depth, strain, or stress. This law is also the cornerstone of backstripping, which often expresses it as a function of burial depth. A google scholar search will give many useful references.

By the way, unconsolidated sediments under near surface conditions usually have porosities in the order of 50% (0.5)

The Petroleum Industry might be helpful here in that the porosity of rock

is very important in how oil flows, or, can not flow in various types of rock.

You might also investigate the volume and mass of the world's oceans,

and compare it to the entire surface area of the Earth to get an average

vertical flow rate of water per unit of time, in cubic meters of water per

year, or per million years.

The average depth is supposedly 3682.2 meters deep covering about 71%

of the planets surface area of 5.14 E 14 km^2 , and an elapsed time of

4544.40 million years.

Uniformly spread over the 100% surface the water would only be

about 0.7097 ( 3682.2 ) = 2613.3 meters deep,

so 2613.3 / 4544.40 = 0.575 meters of depth per million years.

You could do the same thing with the atmosphere, and calculate the

mass of the atmosphere per unit square meter of surface area of the

Earth's entire surface, then calculate the rate of increase in the mass

of the atmosphere per square meter per million years.

Of course this is a net number because materials like CO2 get

incorporated back into rock as CaCO3 ( Limestone ), and the

Hydrogen Sulfide combines with the CaCO3 to make Calcium

Sulfate ( Gypsum ), so that actual amount released is vastly

greater than the current remaining amount of atmosphere.

Also the released H2 and O2 ends up as water ( H2O ).

You can Google Search all the quantities, Volume of the

Earths Atmosphere ( wet ), and (dry ).

The current approach is focused on the ingection of nanoparticles composed of TiO2 to increase the porosity of the lithosphere ,and sooner or later I try to send you the article which is published by AAPG .