Combine the carrots, celery, onion and rice in a slow cooker. Add the broth and water and stir to combine.
Cook on low for 6-8 hours, or until the rice is tender.
In a saucepan, melt the butter. In a small bowl, combine the flour, salt and pepper. Whisk the flour into the butter. It will look clumpy, but continue to cook and stir for a minute or two. Slowly whisk in the milk, constantly whisking to work out any clumps.
Stir the milk mixture into the slow cooker. Add the chicken, then continue to cook on low for an additional 15 minutes to heat through.
Serve warm.
Notes
Nutrition information provided as an estimate only. Various brands and products can change the counts. Any nutritional information should be used as a general guide.