Just pulled military records from Fold3.com regarding your 2xG-Grandfather, William Welcome Wilhelm, and his brother, Archey. Arch's birth name was very likely John Archibald Wilhelm. One doc pictured below shows "Archible," for goodness' sake. Both the 1850 and 1860 censuses indicate a birth year of 1845--which made him 17 when he enlisted at Horse Head, Arkansas. Google the locations mentioned to learn the many places their unit walked and fought. Where Welcome fought, that is. I'm not sure Arch made it out of state. I've found no record of when Archey died or where he is buried. He had allegedly become seriously ill and was released from service. Sent home, he never recovered. As you know, more Civil War soldiers died from sickness than from injury in battle. Below are Archey's records. Click on the four images within each gallery:
Here are 18 images from Welcome Wilhelm's military records. Sadly, I am inept at getting these pictures to appear consecutively--starting with the first page. My apologies. Please click and scroll over to the right and begin reading the LAST pic. As a reminder, Welk Wilhelm was born September 14, 1843 in Hamilton County, Tennessee. Shortly before 1850, his parents migrated to northwest Arkansas. He was 18 years old when he enlisted September 26, 1861 at Camp Jackson in Arkansas. Welk survived the War and married Mary Cowan in Florence, Texas in 1866. Together they had eight children before Welcome left his family in 1882 for parts unknown. (UPDATE: As of August 2017, I've changed the year 1883 to 1882 as to when Welk fled for parts unknown. Why? Because a newspaper article said so.) Some say he left after killing another man in self defense. I call it abandoning your family.
It was in the late 1990s that I first found info on our Wilhelm ancestors' service with Company C, 15th Northwest Arkansas Infantry. I am grateful to the late Edward G. Gerdes for sharing his research: http://www.couchgenweb.com/civilwar/15infcoc.html
The National Park Service has a brief summary of this unit's service:
https://www.nps.gov/civilwar/search-battle-units-detail.htm?battleUnitCode=CAR0015RI04
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Researchers Beatrice Willhelm Steeves (1907-1995) and Ola Irene Johnson (1914-1988) also have 1845 as Archey Wilhelm's birth year, along with 1843 for Welcome.