On the Border (Fence)

Consider this...
The Rio Grand river in South Texas, the border with Mexico, winds along changing positions like Hillary Clinton answering a question about drivers licenses for illegal aliens. Between Falcon Dam (the last dam on the river before the Gulf) and the mouth of the Rio Grande, the river travels about 275 miles. This distance by road is about 146 miles and "as the crow flies" (which, down there, are actually boat-tailed grackles) it is about 130 miles. There are several places where the river curves so wickedly that two bends are less than 50 feet apart for a 1 mile stretch of river.
Question...
Do you build a fence right along the river ... all 275 miles of it? Do you cut corners and spend half the money and build a straighter fence? If you build a straighter fence, what happens to the US land between the fence and the river? To whom does it belong? And what do you do when the river decides to change course around the fence (which it is wont to do)? To whom does the fence belong then? (hint: the land that shifts to the south side of the river then belongs to Mexico by treaty.)
Don't get me wrong, I am in favor of border security. I am just pointing out the obvious here. (Obvious to me, anyway.)