The average length of time which water spends in a lake is quite simply the volume of the lake divided by the inflow of water from all sources. The flushing rate is the inverse of detention time, or the amount of time it takes to replace the volume of a lake. This is best understood as an example.
A 100 acre lake with an average depth of 10 feet holds 1000 acre-feet of water, or 43.56 million cubic feet. If surface water inflows average 10 cubic feet per second, that adds 864,000 cubic feet per day. If direct precipitation is 46 inches per year, that adds another 383 acre-feet of water per year, or about 46,000 cubic feet per day. If groundwater in-seepage contributes another 20,000 cubic feet per day, the grand total of water inputs would be 930,000 cubic feet per day, with the surface inflow being dominant in this case (which is not always the case, so adding in direct precipitation and groundwater is important). The volume of 43,560,000 cubic feet divided by an average inflow of 930,000 cubic feet per day yields a detention time of just under 47 days. The flushing rate would be just under 8 times per year.
In the example above, if the inflow was less, the detention time would increase and the flushing rate would decrease. If more water was somehow put in the lake, the detention time would decrease and the flushing rate would increase. One could use sources of outflow to get detention time and flushing rate, with surface outflow, evaporation, and out-seepage of groundwater as the primary outflows (unless there is a major withdrawal of some kind), but the inflow and outflow have to match over the long run for the lake to remain stable, and it is easier to get values for inflow sources than outflows.
A key point is that it is only the volume of the lake and the total inflow or outflow that matter to detention time and flushing rate. One cannot change these values by altering an outlet structure, dredging a channel, or other physical manipulations within the lake. If no more water is added or subtracted from a lake with the same volume, the detention time and flushing rate do not change. Dredging and other lake manipulations may alter the circulation pattern, and if the lake is subdivided into parts, this may affect detention and flushing of one part, but the overall detention time and flushing rate cannot change without a change in lake volume or inflow quantity.
Detention and flushing rate are important to lake function and many management options. A lake with a detention time of less than about two weeks is unlikely to develop algae blooms, as the water does not stay around long enough to let blooms form. Lakes with very long detention times, more than a year, are less subject to watershed influences on a day to day or even season to season basis; there is simply not enough inflow to alter water quality over a short space of time. And most water quality models that allow us to predict changes with specified management actions depend on detention time or flushing rate as an important term in the mathematical model; being off by even a small amount affects calculations and reliability.
Detention time and flushing rate are not constants, however, and vary over time with changing inflows. For lake with long average detention times, this is not a major influence, but for lakes with average detention times of days to a few months, the variation within a year can be meaningful. A lake with an average detention time of a month could experience much lower summer inflows and have the same water present for 3 months, while spring thaw and related snowmelt and rain reduce the detention time to a matter of days in April.
Detention time and flushing rate can vary over space as well. A “dead end” part of a lake may have a much longer detention time and be flushed much less than some area in the main path of inflows, leading to stagnation and possible water quality problems in that dead end cove or segment of the lake. Rerouting water through the dead end may improve circulation and reduced detention time for that area but unless this is new inflow to the system, it will not change the average detention time for the whole lake.
One other related concept is important to understand when managing lakes. If a lake is long and linear, with most inputs at one end and the outlet at the other end, as with certain human-made reservoirs, the movement of water is called “plug-flow” in engineering terms, and is similar to water moving through a garden hose. Each increment of inflow pushes the next increment forward, and while there will be some mixing, water entering the lake at any time will have more or less similar detention time to water entering at any other time, with variation due mainly to changes in inflow rate. An algae bloom that forms in such a lake will be removed after about one flushing, or replacement of the equivalent of about one lake bottom.
But for the more typical lake configuration of a bowl with various input sources and an outlet somewhere along shore, the engineering model is called a “continuously stirred tank reactor”. In this model, each increment of water added is completely mixed with all other water in the lake, and while average detention time is calculable, the actual detention time for any increment of water can be highly variable. Because of the mixing, it takes much longer to get any undesirable water mass flushed out of the system. Just replacing the water volume of the lake one time (one flushing) will not get rid of all of an algae bloom or a spike of phosphorus or an oil spill; it takes 3 to 5 flushings to clear the lake.