You should be able to accomplish what you're looking to do with these two commands:
chown oracle:backups /app/dumps
chmod 2370 /app/dumps
chown sets the owner of the directory to the oracle user and the group to the backup user.
chmod does the following:
- Sets the setgid bit
- Gives the owner (oracle) write/execute access
- Gives the group (backups) read/write/execute access
Setting the setgid on a directory means that a file created in the directory will inherit the group of the directory and not of the creator. In your instance, this means that any files written there by oracle will have backups as a group.
Oracle can write to the dir but not read from it. If it needs to do so, change the permissions to 2770.
This also assumes that your script is creating the files with read/write access in the /app/dumps directory. If this isn't the case, you can run
setfacl -d -m g::rw /app/dumps to set the default group permissions of files created in /app/dumps to be read/write.