Exhibit 10.2

SECOND AMENDMENT TO

ATWOOD OCEANICS, INC.

2013 LONG-TERM INCENTIVE PLAN

WHEREAS, Atwood Oceanics, Inc., a Texas corporation (the “Company”), has established and maintains the Atwood Oceanics, Inc. 2013 Long-Term Incentive Plan (the “Plan”); and

WHEREAS, pursuant to Section 13 of the Plan, the Company has the right to amend the Plan at any time by action of the Board of Directors of the Company (the “Board”), subject to prior approval by the Company’s stockholders to the extent such approval is determined to be required by applicable legal and/or stock exchange requirements; and

WHEREAS, the Company desires to amend the Plan to remove the minimum vesting requirement for time-vested stock awards and add a minimum vesting requirement to stock options and stock appreciation rights; and

WHEREAS, this amendment is not a material modification that is subject to prior stockholder approval;

NOW, THEREFORE, the Plan is amended as follows:

1. Section 6(c) of the Plan is hereby replaced in its entirety by the following:

“(c) Stock Award (including Restricted Stock and Restricted Stock Units). An Award may consist of Common Stock or may be denominated in units of Common Stock. With respect to a Stock Award that is a Performance Award, the minimum Restriction Period shall be one year from the Grant Date. A Stock Award may provide for accelerated vesting and lapse of restrictions in the event of a Change of Control or a Participant’s termination of service due to death, disability or retirement. All other terms, conditions and limitations applicable to any Stock Award pursuant to this Plan shall be determined by the Committee.”

2. Section 6 of the Plan is hereby amended by adding subsection (g) to the end thereof:

“(g) Minimum Vesting Requirements. All Employee Awards in the form of Options or SARs shall have a minimum vesting period of one year from the Grant Date; provided, however, that Employee Awards in the form of Options and SARs with respect to 5% of the total shares of Common Stock authorized to be issued under the Plan pursuant to Section 5 hereof may have a vesting period of less than one year.”

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This Second Amendment to the Plan shall be effective as of the date it is originally approved by the Board.