When building bridges, it is always necessary to take into account the hydraulic effects of the structure. Piers in a river, for instance, always cause interesting water-related phenomena. Not only is a pier vulnerable to being undermined by the action of water, a pier can obstruct trees and other floating objects in the river until an effective dam is created, unless precautions are taken.
In extreme cases, like the old London Bridge, too many too-wide piers can pose a major obstruction of the waterway. In the case of London Bridge, the water at times was a full 5 feet higher on the upstream side of the bridge than the downstream side. (This and many other little factors is why London Bridge has gone down in history as an example of how not to build a bridge.)
Planning for Water
In stone arch bridges, the rounded shape of the arch tends to allow for ample waterway at low levels and potentially constricted waterway at high levels, as the arch presents much less of an opening at the top than the bottom. Dunkard Mill Bridge, Kansas’s largest stone arch bridge, had that problem, which led to the addition of a fourth arch (originally the bridge had three).
At the time these bridges were built, it was considered ideal practice to build the arches high enough above the waterway that only in the most extreme flooding the water would flow through the arch proper. Normally, even at high water, the stream being spanned would flow between vertical-sided piers and abutments on which the arch(es) would rest.
In the United States, this seemed to be a less popular method than, perhaps, one might expect. In Kansas, most of the arches of the stone bridges are at or a short distance above the river bed level. This can pose some problems, especially in multiple arch spans. The pier(s) in multiple-span bridges frequently present a rather large blockage to the river. Since the two arches meeting at the pier slope away from it, towards the top of the bridge there is a rather large solid wall presented to the stream.
Using more rounded arches greatly helps this, and building the bridge such that the top of the arches are at a level higher than the river banks significantly helps flow in times of flooding.

Cutwaters
To prevent debris from piling up on piers, cutwaters are frequently added. Curiously, when it comes to bridges in Kansas this seems to have been a frequently neglected point, many of the multi-span bridges have but blunt piers.
Another advantage of cutwaters is that they “cut the water” — allowing it to flow more smoothly under the arches rather than slamming into the pier. Because of the smoother flow of cutwaters, they help prevent scour.
Putting a cutwater on the downstream side of a bridge is an excellent means of preventing scour, and has been used even since medieval times. Oddly enough, in Kansas, even in the stone bridges with upstream cutwaters, cutwaters on the downstream side seem to have been rarely, if ever, used.
Avoiding Submersion
Finally, it is generally considered ideal for the bridge to always be kept above the water level, even in the highest floods. Obviously, being submerged does neither the bridge nor the road any good.
In Europe, this ideal of always keeping the bridges above the floodwaters seems to have been a more easily obtained ideal than in Kansas, at least. In Kansas, as the saying goes, “when it rains, it pours!” so that the stone arch bridges periodically spend time completely submerged.

Of course, if the truth be told, in the periodic heavy floods that come to Kansas, the streams are hardly large enough to handle the flow and the large floodplains become filled.
Reportedly, in a devastating flood in the 1920s, Cowley’s Grouse Creek became about a mile wide in spots. For perspective, when building the stone arch bridges over the Grouse Creek, a clear waterway of less than 200 feet was used — often the bridge’s waterway being closer to 100 feet. This flood width is comparable to the Mississippi in places!

That the Cowley stone arch bridges can repeatedly handle the abuse of this waterway, and have been for about a century now, is a testimony to the durability of the stone arch bridge.