If you have topological maps, you can delineate the watershed by manually connecting the peaks of the hills, so to speak. This is relatively easy. It just takes some patience and care. And then you can use the scale of the map and clear plastic overlay with a grid to estimate the area.
There is software that can automate the process. If you have the GIS layer for the digital elevation model (DEM), you can use something like QGIS and GRASS. I've done this, and it's not simple or fool-proof.
For U.S. readers, the U.S.G.S. StreamStats website makes quick work of this. https://streamstats.usgs.gov/ss/ . Though I've had it fail in weird circumstances.
Dr. Mangiafico as the basics. Before GIS, approach 1 is the standard. I have done hundreds or maybe thousands of delineations this way. But it can be especially difficult in the coastal plain or other relatively flat terrain. Using a topographic contour map, you will note the contour interval scale, and the distance scale. You may note that the contour crenulations bend one way to signal streams, and the other way to signal ridges. If one starts at the Watershed mouth or location of interest, one follows perpendicular to the topographic contours to the nearest high point or ridge line, then follow the ridge line contours up to the highest point of the Watershed, then if you prefer, start from the outlet again in the other direction, follow up to that ridge line perpendicular to the contours, and then the ridgeline up to the highest point to meet the other high point. As you get used to this, you can follow the ridgeline from the peak elevation back to the outlet or point of interest. This becomes more complex when more than one topographic contour map is needed to delineate the entire area. For larger river systems, you might find that some of these may have been already done from their mouth at the ocean or at a confluence with another river.
Then once the boundaries are delineated, one must measure the area, such as in square inches, square centimeters or other appropriate. The contour map scale will be in distance, so develop the area factor to apply. In English units for example, one inch may equal one mile, so 1 square inch equals one square mile, which is 640 acres. Each acre is 43,560 square feet. For very small hydrologic units or catchments, dot grids may help if you have nothing else. Planimeters are another reasonably priced instrument that can be used to measure complex shaped areas. As suggested, GIS is another tool, but it often takes careful attention and meticulous editing. Several of my or coauthored papers in Researchgate suggest using LiDAR, when properly calibrated and georeferenced. If nothing else, find a watershed that has been delineated and determined for area, apply boundaries to map, use dot grid or planimeter to determine area, and then develop the adjustment factor for that scale map.
you might find this link interesting. https://www.ozarkundergroundlab.com/assets/oul_groundwater_tracing_handbook-2019-revised.pdf
I met Tom Aley in Missouri, and his work with dyes in karst terrain is exceptional.
Streams and rivers often move slowly, especially during baseflow periods. Tom Aley used the charcoal packets through time to site visits to absorb fluorescein dye, and it was released with alcohol I think for detection in fluorimeter. Karst terrain in Missouri had underground stream and spring connections that sometimes varied from the surface hydrologic boundaries.
I used Rhodamine dye in some studies to trace pesticides to help estimate issues with overspray of streams and samples to send to lab for analysis.
I am familiar with a few examples where mistakes of too much dye heavily tinted waters for miles, which can be very embarrassing.
I have worked on this topic, to determine the drainage area of a river, you will need to identify the boundaries of the area that drains into the river. This area is known as the river's watershed or catchment area.
There are several methods that you can use to determine the drainage area of a river, including:
Map-based methods: You can use topographic maps or satellite imagery to identify the boundaries of the watershed and draw a line around the area that drains into the river. You can then measure the area using a map scale or a geographic information system (GIS).
Field-based methods: You can also determine the drainage area by walking or driving around the perimeter of the watershed and measuring the area using a GPS device or a rangefinder.
Hydrologic models: You can also use computer models that simulate the flow of water in the watershed to estimate the drainage area. These models may use data on topography, soil type, vegetation, and other factors to predict the drainage area of the river.
It's worth noting that the drainage area of a river can vary over time due to changes in land use, vegetation, and other factors. As a result, it is important to periodically update the drainage area estimates to ensure they are accurate.
Here are a few references that provide more information about determining the drainage area of a river:
Shrestha, A., & Neupane, R. (2013). A review of methods for determining the drainage area of a river. Water and Environment Journal, 27(4), 466-475.
Smith, M. L., & Yevjevich, V. (1972). Methods for determination of drainage-basin characteristics. US Geological Survey Water-Supply Paper, 1849, 1-70.
Singh, V. P. (1995). Techniques of water-resources investigations of the United States Geological Survey. Chapter A2: Collection and analysis of data for headwater areas. US Geological Survey Book 5, Section A, 1-52.