Disk Utility shows me three hard drives with the same name. How do I find the physical hard drives they refer to?

You can do this from Terminal with the system_profiler command:

$ system_profiler SPSerialATADataType

It will generate a list of all SATA drives attached to your Mac. You will get an output similar to the following (yours will be longer as you have more drives):

Intel 7 Series Chipset:

  Vendor: Intel
  Product: 7 Series Chipset
  Link Speed: 6 Gigabit
  Negotiated Link Speed: 6 Gigabit
  Physical Interconnect: SATA
  Description: AHCI Version 1.30 Supported

    APPLE SSD SM128E:

      Capacity: 121.33 GB (121,332,826,112 bytes)
      Model: APPLE SSD SM128E                        
      Revision: CXM09A1Q
      Serial Number: xxxxxxxxxxxxxx      
      Native Command Queuing: Yes
      Queue Depth: 32
      Removable Media: No
      Detachable Drive: No
      BSD Name: disk0
      Medium Type: Solid State
      TRIM Support: Yes
      Partition Map Type: GPT (GUID Partition Table)
      S.M.A.R.T. status: Verified
      Volumes:
        EFI:
          Capacity: 209.7 MB (209,715,200 bytes)
          BSD Name: disk0s1
          Content: EFI
          Volume UUID: 0E239BC6-F960-3107-89CF-1C97F78BB46B
        disk0s2:
          Capacity: 120.99 GB (120,988,852,224 bytes)
          BSD Name: disk0s2
          Content: Apple_CoreStorage
        Boot OS X:
          Capacity: 134.2 MB (134,217,728 bytes)
          BSD Name: disk0s3
          Content: Apple_Boot
          Volume UUID: BFD92A4F-346D-3CCD-911F-48D03D8FD8AE

What you’re going to look for is the disk identifier and the serial number of any drive where the SMART status doesn’t say “Verified”

Once you have the serial number, match it with the label on the drive itself

The method I use is to grab the serial number from the drive. This information is usually printed by the factory on a label on the physical disk so that it is easy to identify.

Start by ensuring you have HomeBrew installed. Then run the following command to install smartmontools:

brew install smartmontools

Then grab the serial number from the failing drive:

smartctl -a /dev/disk2

This assumes that the failing disk is /dev/disk2 - obviously change that to reflect the disk you have ben reported as failing.

In the output you'll find something like:

Serial Number: C8AF902EAB39

(format varies depending on manufacturer)

You should be able to match that up with the label on the drive.