Does grass show in urine

The urine will most likely kill the grass because urine is an acid and consists of toxins your body has removed from your system.

I think it leaves brown patches

The true reason is because of the high amount of nitrogen found in dog urine it can kill the grass. In a lawn were no fertilizer has been applied, dog urine will increase plant growth and green-up the grass. The problem sometimes comes when your lawn has high nitrogen levels from fertilizer and the dog urinates. This would be an "overdose" of nitrogen and cause the turf to burn. It's a common misconception that "acid" in dog urine kills grass. The truth is that dog urine can have a pH of 6.0-8.0(slightly acidic to slightly alkaline) which is no where near acidic enough to kill grass. True dog urine spots will many times be brown in the middle and growing vigorously around the brown spot.

Dog Rocks are great, they reduce the amount of nitrate produced in the urine, which stops it killing off the grass. They are really easy to use, they just sit in your dogs water bowl.