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Not sure what you'd consider "close by" to Cheyenne, but ... 20 miles West,
there are wetlands areas in the Veedauwoo county park area, with several draws that have live streams there. Head West on Happy Jack road to any of the entrances on the South side of the road, almost to or at the county line. The park is free, you can take some of the "roads" in a ways and just park out of the roadway, or at some of the established clearings that have obviously been used to that purpose in the past. Contiune to walk in, heading mostly South. You'll drop down into a creek bottom pretty quickly.
You can also access this portion of the Veedauwoo Park from the Nat'l Park side from I-80. This campground is much more easily accessed from the pavement, and is substantially more "improved" with official campsites than the county park side. Being a Nat'l facility, there is a daily use fee or you'd need your annual pass.
At a longer distance away, the Snowy Range area West of Laramie has even wider open expanses of wetlands, with multiple streams, ponds, and lakes. Stop at the sporting goods stores in town and get a map of the area. As long as you don't use the official campground facilities, you don't need a day permit or pass ... but don't sit down at a campground table as you hike through the area, the rangers are very aggressive about enforcing the day use fee collection if you so much as stop for a minute at the facilities. A favorite area would be to head up to the meadows in the Brooklyn Lake area ... which is still snowbound in the higher reaches, now melting off at a pretty good rate. If you go during the week, it's most likely you'll have much of the acreage to yourself.
Of course, these areas see a substantially higher use on major holiday weekends ... Memorial Day, 4th of July, etc, and may be "crowded" compared to the normal daily use.
In other directions ... you could head out Hwy 85 to Hawk Springs res ....
Within the city limits of Cheyenne, I don't know of a place you can take your dog off leash and go through water features.
Be aware that in all the places I've mentioned, your dog must be under your physical control, which is generally be interpreted to be on a leash if there's a less than friendly confrontation with another party ....
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