Exhibit 10.5

AMENDMENT

DUNE ENERGY, INC.

2012 STOCK INCENTIVE PLAN

NON-QUALIFIED STOCK OPTION AWARD AGREEMENT

(Non-Employee Directors)

WHEREAS, Dune Energy, Inc. (the “Company”) previously granted, pursuant to the “Dune Energy, Inc. 2012 Stock Incentive Plan Non-Qualified Stock Option Award Agreement” (the “Agreement”) a non-qualified stock option to             (“Participant”); and

WHEREAS, the Company desires to amend the Agreement to clarify (i) that in the event of a conflict between the terms of the Agreement and the Dune Energy, Inc. 2012 Stock Incentive Plan (“Plan”), the terms of the Agreement shall control, and (ii) that the Participant will have one year (not three months) to exercise the vested portion of such option following certain terminations of the Participant’s status as a Director; and

WHEREAS, pursuant to Section 2.5 of the Plan, the Committee (as defined in the Plan) has determined that such amendment may be made without the consent of the Participant.

NOW, THEREFORE, effective April 20, 2012, the Agreement is amended as follows:

1. The second to the last sentence of the first paragraph of Section 1 [previously—”In the event of any conflict between the terms of this Agreement and the terms of the Plan, the terms of the Plan shall control.”] shall be deleted and replaced in its entirety as follows:

“Solely for purposes of this Agreement, in the event of any conflict between the terms of this Agreement and the terms of the Plan, the terms of this Agreement shall control.”

2. Section 5(b) [previously—”(b) Termination for any other reason. In the event of the Participant’s Termination, resignation or removal for any reason (except for death, Disability or Cause), the vested portion of this Option shall remain exercisable until the earlier of (i) [three months from the date of such Termination], and (ii) the expiration of the stated term of the Option pursuant to Section 4 hereof.”] shall be deleted and replaced in its entirety as follows:

“(b) Termination for any other reason. In the event of the Participant’s Termination, resignation or removal for any reason (except for death, Disability or Cause), the vested portion of this Option shall remain exercisable until the earlier of (i) one year from the date of such Termination, and (ii) the expiration of the stated term of the Option pursuant to Section 4 hereof.”

IN WITNESS WHEREOF, the Company has executed this Amendment as of the date first written above.