pH evolution depends on the chemical composition of the water but will decrease as CO2 increases. You can see my attached paper in "Discussion" section. One thing which is not in this paper is the temperature effect.

Article Major Ions in Marine Rainwater with Attention to Sources of ...

Remi is right. The pH depends on the CO2 partial pressure, the composition of the water, and the temperature. The following formula gives an estimate of the pH:

pH = log[Alk] - logKa1 - logKH - logPCO2

[Alk] is alkalinity, Ka1 is the first dissociation constant of carbonic acid, KH is the Henry's law constant for CO2, and PCO2 is the partial pressure in atmospheres. The estimate is better if Ka1 and KH are adjusted for temperature and ionic strength. This estimate is for a beaker of water. A real water hole probably has some calcium carbonate in its sediment and the alkalinity will change as some of the carbonate dissolves in response to the higher PCO2. See chapters 4 and 5 of "Aquatic Chemistry" by W. Stumm and J. J. Morgan.

In addition to Stumm and Morgan, a very thorough coverage of the topic is presented in "Carbon Dioxide Equilibria and their Applications" by J.N. Butler. 

Yes of course, pH depends on the chemical composition of the water and it will decrease as CO2 increases already Mr. Remi told you. see attached file may be it would be helpful for you.

Regards

The relationship between pH and dissolved CO2 concentration in aqueous natural environment, does not depends only on the dissolved salts or organic compounds content, as It tends to be also influenced by the dissolved O2 concentration and metabolic activity. I have briefly addressed such effect focusing the case of aerobic fermenters, while answering to a somewhat related question elsewhere at this forum: https://www.researchgate.net/post/What_are_pH_changes_as_air_is_dissolved_in_fermentation_medium

To take into account effects of solution composition, temperature, mineral equilibria, etc., I would put this script into PHREEQC:

SOLUTION # initial solution composition:

Temperature 10 

units mmol/L 

# Full chemistry can be stated. E.g., for a simple Ca-HCO3 water type:

Ca 1

Alkalinity 2

EQUILIBRIUM_PHASES

CO2(g) -3.40 # log(P_CO2 (atm)), e.g. log(0.000400 atm) = -3.40

                         # for a P_CO2 of 400 ppm = 0.000400 atm.

# Calcite 0      # Optional, to simulate a system in equilibrium with calcite

END

a publication about that:

Article Freshwater acidification: an example of an endangered crayfi...