What OAuth Scope for storing files

I have a question about the OAuth scope when apps that requests for access permissions on the fly as described in http://msdn.microsoft.com/en-us/library/jj687470.aspx

We would like to store large files using the REST API as described in http://msdn.microsoft.com/en-us/library/dn292553.aspx#LargeFiles

For writing files, I think we need at least the "Web.Write" scope for OAuth. However, when performing the OAuth flow, only Site Administrators (and above) could grant access successfully. Ordinary users fails with access denied errors from SharePointafter step #2 OAuthAuthorize.aspx.

Is there another or a more restricted OAuth scope that we should use?