NPS-CW-S&SS shows:

11th Regiment, Virginia Infantry
11th Infantry Regiment was organized at Lynchburg, Virginia, in May, 1861, and accepted into Confederate service in July. Its members were raised in the counties of Campbell, Botetourt, Montgomery, Fauquier, Culpeper, and Rockbridge. The unit fought at First Manassas under General Longstreet and at Dranesville under J.E.B. Stuart. Later it was assigned to General A.P. Hill's, Kemper's, and W.R. Terry's Brigade, Army of Northern Virginia. It served with the army from Williamsburg to Gettysburg except when it was at Suffolk with Longstreet. The 11th was engaged at Plymouth in North Carolina and after returning to Virginia saw action at Drewry's Bluff and Cold Harbor. It went on to fight in the Petersburg trenches south and north of the James River and ended the war at Appomattox. This regiment reported 6 killed and 15 wounded at Dranesville, totalled 750 men in April, 1862, and lost 134 at Williamsburg and 100 at Frayser's Farm. It sustained 63 casualties at Second Manassas, had about forty percent disabled of the 359 engaged at Gettysburg, and lost 15 killed and 94 wounded at Drewry's Bluff. Many were captured at Sayler's Creek, and only 1 officer and 28 men surrendered. The field officers were Colonels David Funsten, Samuel Garland, Jr., Maurice S. Langhorne, and Kirkwood Otey; and Majors Adam Clement, Carter H. Harrison, and J.R. Hutter.

Co I = Rough and Ready Rifles, Fauquier and Culpeper Counties; riflemen; enlisted May 25, 1861, for 1 year; mustered June 4, 1861; reorganized April 26, 1862.
Captains: James Henry Jameson [resigned, April 26, 1862], Andrew J. Jones

Records show two (2) Blankinship's, but not John Archer Blankinship:

Blankinship, C.T.
Co. C Private Private

Blankinship, L.C.
Co. C Private Private

The records do show:

John Blankinship, 1st Lt, CO C, 16th Va Cav

John Blankinship, Pvt, Co B, 45th Batt'n Va Inf

John Blankinship, Pvt, Capt. Cayce's Co., Virginia Light Artillery (Purcell Artillery)

John Blankinship, Pvt, Co K, 45th Virginia Infantry

but I don't see a John Archer Blankinship.