tl;dr: Oil wells produce some natural gas, and natural gas wells produce some oil, but natural gas wells tend to be more "specialized," meaning that as their quantity and productivity increases, the linkage between these two fuels diminishes.
This answer looks mainly at oil and gas production in the US (since that's what I'm more familiar with, and the U.S. leads global petroleum and natural gas production as of 2018).
Productivity by well type
The EIA provides data on production rate of oil and natural gas wells. The report explains how wells are classified as either a "gas well" or an "oil well":
EIA designates wells as either oil or natural gas wells based on a gas-oil ratio (GOR) of 6,000 cubic feet (cf) of natural gas to 1 barrel (b) of oil (cf/b) for each year’s production. If the GOR is equal to or less than 6,000 cf/b, then we classify the well as an oil well. If the GOR is greater than 6,000 cf/b, we classify the well as a natural gas well.
6000 cf of natural gas and 1 b of oil have roughly the same energy content -- about 1800 kWh. The FAQ in the report explains a bit more about how these wells are classified:
Does a natural gas well remain a natural gas well during its entire production history?
In this report, we sometimes classify a well as a natural gas well in one year and as an oil well in another year, and vice versa, depending on a well’s gas-oil ratio. We used this approach because the respective volumes of liquid and natural gas produced by a well can change significantly during its production history.
I downloaded the data and converted the numbers to terawatt-hours (TWh) to make it easier to compare:
Three important things stand out from this chart:
- There are more gas wells
- More fossil fuels are produced by gas wells
- The ratio of gas:oil in oil wells is greater than oil:gas in gas wells
This last point essentially means that gas wells are "better" at doing their main job, than oil wells are.
Most "new" natural gas is coming from shale gas wells
Data on production of natural gas in the US from various sources is available here from the Energy Information Agency.
I downloaded the data and put together this chart showing total historical withdrawals of natural gas, along with what is vented and flared*:
Modern shale gas wells (using hydraulic fracturing, or "fracking" methods) produce a higher proportion of gas compared to traditional methods. Growth in this area is massive over the last two decades, meaning the ratio of oil:gas production is falling.
However, this growth in natural gas tends to replace coal production, while total oil production continues to increase
From EIA's International Energy Statistics:
What is venting and flaring, and why is it done?
From the EIA:
In areas where natural gas is produced at oil wells but is not economical to transport for sale or contains high concentrations of hydrogen sulfide (a toxic gas), it is burned (flared) at well sites. Natural gas flaring produces CO2, carbon monoxide, sulfur dioxide, nitrogen oxides, and many other compounds, depending on the chemical composition of the natural gas and on how well the natural gas burns in the flare. However, flaring is safer than releasing natural gas into the air and results in lower overall greenhouse gas emissions because CO2 is not as strong a greenhouse gas as methane.
Over time, as natural gas has become a more important fuel, the quantities which are vented and flared have diminished, even as total production increases (source):