SJ,
I've appreciated your comments. Again looking at Geneis 2:14 I see some translations say the Tigris "flows east of Assyria", some say the Tigris "runs eastward to Assyria" and some e.g. the Revised Standard Version say the Tigris "runs along the east side of Asshur". Of course I'd prefer to use "runs along the east side of Asshur".
But rivers do change their courses over time. I've found a map in the book 'Babylon' by Joan Oates which shows the courses of the Tigris and Euphrates from the 3rd millenium BC compared to modern times, and they are quite different. I'll just quote a little from the book:
"Archaeological surveys and historical sources both confirm this pattern of shifting water-courses and associated concentrations of population... It is clear too that at no time did the Euphrates occupy one single channel. In the earliest period for which we have evidence there seem to have been three major branches running through Kish, Cutha and Jamdat Nasr, settlement having been heaviest along the easternmost branch. By the 3rd millenium the Kish channel was undoubtedly the most important, just as Kish was the major city. The earliest documentation for the then apparently minor Babylon branch comes later in the 3rd millenium; it was not until the end of the 2nd millenium that this channel became the most important of the Euphrates courses. It remains today an impressive river, although the main Euphrates channel now flows even further to the west."