On my morning walks by the river I take note of how much water appears to be flowing in the river. Lately the flows have not been that high because most of the snow has already melted and there has not been that much rain or snow for the last few weeks. Previously, when the snow melted the riverbanks overflowed onto roads and it was a very different river.

How do you explain and/or predict the flow of a river?

To predict flow you need to start by having a good measurement of existing flow. A standard technique is to measure the width of the stream and the depths of the stream at various regular intervals then sum over these trapezoidal area estimates (A = Σ a_i). You would then have to measure the velocity v of the water perhaps by floating a cork in water between two markers and timing how long it takes. Once you had area and velocity measurements you could compute a flow volume (Q = A x v). This flow volume would vary from day to day.

What factors might you use to explain and predict a river flow?

Explaining a river flow is different than predicting a river flow. A big factor in explaining a river flow volume is the number and size of tributaries leading into it. This factor stays fairly constant from day to day so is not very useful in predicting the daily variation in river flow. Other factors like precipitation, ground saturation, ground permeability, evaporation, etc. might be more useful in predicting the day to day expected flows.

Most cities are built along a river. Around half of those cities withdraw a major part of their water supply from upriver. Depending on the size of the city and its seasonal demand for water, this extracted volume could be a significant factor influencing flow rate.

A home property can also be the focus of an investigation into daily flow volumes. What factors explain and predict the amount of water you use on your property on a daily basis? Those on metered water have an advantage over non-metered users in that they can figure out those factors better because they have accurate flow measurements to go by (depending on how that usage is reported).

Cashflow is another type of flow that concerns entrepreneurs and investors. What are the factors that explain and predict the cashflow of a company? What is the time frame of concern in our cashflow projections - a day, a week, a month, quarterly, etc... The time frame determines how frequently we would have to measure cashflow to determine if the cashflow model is correct. Comparing cashflow models to riverflow models offers potential insights.

Stock and flow diagrams are commonly used in systems theory to model systems dynamics. The simplist stock and flow diagram looks like this bathtub model used to explain and predict the level of water in a bathtub:

Donnella Meadows in her book Thinking In Systems: A Primer (3rd Edition, 2008) uses a slightly more complex stock and flow diagram to explain and predict the volume of living wood in a forest and also the lumber inventory associated with that forest:

There are many mathematical and graphical techniques you can use to explain and predict flows. The study of river flows offers a useful foundational metaphor for thinking about other types of flows (e.g., the flow of electricity is often understood in terms of water flows). The techniques needed to explain and predict stream flows might also be used to explain and predict these other types of flows as well. Something to think about the next time you are walking beside a river and looking for something to occupy your mind.