Exhibit 10.2

SPECIAL STANDBY RATE AND POTENTIAL SUSPENSION AGREEMENT

This Special Standby Rate and Potential Suspension Agreement (the “Agreement”) is made and entered into as of November 9, 2010 (the “Effective Date”), by and between Cobalt International Energy, L.P. (“Operator”), a Delaware limited partnership and ENSCO Offshore Company (“Contractor”), a Delaware corporation. Operator and Contractor may hereinafter be referred to individually as a “Party” or collectively as the “Parties”.

WHEREAS, Operator and Contractor entered into an Offshore Daywork Drilling Contract on the 5th day of May 2008 addressing utilization of the dynamically-positioned semi-submersible drilling rig ENSCO 8503 (the “Rig”) to conduct drilling and associated operations for Operator in federal waters on the Outer Continental Shelf in the U.S. Gulf of Mexico and otherwise as specified therein, which contract as heretofore amended is referred to herein as the “Drilling Contract”; and

WHEREAS, construction of the Rig has been completed and the Rig is being mobilized to the Area of Operations; and

WHEREAS, Operator has requested Contractor to consider various scenarios in the event Operator is unable to commence drilling operations upon the Term Commencement Date as contemplated in the Drilling Contract, including a potential extended standby period and/or alternative utilization of the Rig to engage in operations for another oil and gas company; and

WHEREAS, Operator may desire to defer commencement of drilling and associated operations under the Drilling Contract and to potentially allow for the deployment of the Rig to work for another oil and gas company under a separate drilling contract (as an alternative to assignment of the Drilling Contract) addressing utilization of the Rig by such other oil and gas company; and

WHEREAS, Contractor and Operator have entered into this Agreement for purposes of (i) addressing the possibility that Operator may not be in a position to commence operations under the Drilling Contract on the Term Commencement Date, (ii) specifying the manner in which the Parties will proceed to perform their respective obligations to effect the Term Commencement Date, (iii) agreeing on a Special Standby Rate to be applicable for certain periods as specified herein, and (iv) specifying the bases upon which the Drilling Contract may be suspended for potential Third Party Work as provided herein.

NOW, THEREFORE, the Operator and Contractor, for and in consideration of the mutual obligations, undertakings, premises and covenants herein contained, do hereby agree as follows:

Section 1.              Definitions.

A.           The defined terms as contained in Article I and elsewhere of the Drilling Contract as used herein (including the recitals hereinabove) shall be applicable to this Agreement with the same meaning.

1

B.             “Daily Upgrade Fee” has the meaning ascribed to it in the Drilling Contract as further defined in that certain letter agreement dated October 29, 2008 and executed by the Parties.

C.             “Drilling Mobilization for Operator” shall mean the date and time that the Rig is underway under tight tow or Rig propulsion to the first drilling location designated by Operator or its assignee.

D.            “Effective Date Regulations” shall mean any U.S. federal or state laws, rules or regulations applicable to operations under or contemplated by the Drilling Contract that are in effect as of the Effective Date.

E.              “Modification Deadline” shall mean November 30, 2010, unless Contractor has failed to provide Operator with applicable modification schedules and cost information sufficiently in advance of November 30, 2010 to enable Operator to make an informed request for any Rig Modifications or upgrades to the Rig, in which case “Modification Deadline” shall mean the date that is fourteen (14) days following receipt of such applicable modification schedules and cost information.

F.              “Rig Modifications” shall mean any modification or upgrade to the Rig’s blowout preventer and associated equipment, that are (i) requested by Operator prior to or on the Modification Deadline and/or (ii) required to comply with the Effective Date Regulations.

G.             “Daily Rig Modification Fee” shall have the meaning ascribed to it in Section 2.C. hereof.

H.            “Special Standby Period” shall mean the period of time from the Term Commencement Date until the earliest to occur of the following: (i) the date and time that is three hundred sixty five (365) days from the Term Commencement Date, or (ii) Drilling Mobilization for Operator. The aforesaid three hundred sixty-five (365) days shall continue to run during any Third Party Suspension Period.

I.                 “Special Standby Rate” shall mean a day rate of US$210,000 per 24-hour day which shall be subject to the rate adjustment provisions in Section 3.2(b), (c), (e) and (f) of the Drilling Contract as respects variations in cost based on prevailing costs in the Area of Operations following the Effective Date.

J.                “Third Party Suspension Period” shall mean that period of time from when the Rig commences mobilization to a third party’s designated location to perform Third Party Work under tight tow or on a heavy-lift carrier until the time the applicable contract term for such Third Party Work has been completed and the Rig has demobilized from such third party’s designated location to the designated anchorage area in Grand Isle Block 105 in the U.S. Gulf of Mexico or such other location as may be mutually agreed by Operator and Contractor; provided, however, that the Third Party Suspension Period shall not conclude unless and until any modifications that were made to the Rig to enable the Third Party Work have been, if requested by Operator prior to the commencement of Rig demobilization from the last location of such Third Party Work, satisfactorily removed and the Rig is otherwise returned to Operator substantially in the same condition (exclusive of normal wear and tear) as originally delivered to Operator.

2

K.            “Third Party Work” shall mean the utilization of the Rig commencing during the Special Standby Period by an oil and gas company other than Operator pursuant to a separate drilling contract entered into between Contractor or Contractor’s affiliate and such other oil and gas company.

Section 2.              Compensation.

A.           The Special Standby Rate shall be payable by Operator to Contractor during the Special Standby Period, except during any Third Party Suspension Period for which a dayrate of zero ($0) shall be payable by Operator to Contractor. The Special Standby Rate shall be invoiced according to Article IV of the Drilling Contract.

B.             Notwithstanding Section 2.A, the Special Standby Rate shall also be payable by Operator to Contractor during any period in which Rig Acceptance Criteria, outstanding sea trials or operations cannot be conducted because the Rig is waiting to undergo or undergoing Rig Modifications or modifications required to comply with new laws, rules or regulations applicable to operations under or contemplated by the Drilling Contract that become effective after the Effective Date and before the Term Commencement Date (but not new interpretations of Effective Date Regulations which arise after the Effective Date). However, any incremental time solely required to perform any other remedial work necessary to meet the Rig Acceptance Criteria shall be at Zero Rate. All such time at the Special Standby Rate or Zero Rate that occurs prior to the Term Commencement Date shall extend the Secondary Delivery Date under the Drilling Contract.

C.             Upon completion of the Rig Modifications or modifications required to comply with new laws, rules or regulations applicable to operations under or contemplated by the Drilling Contract that become effective after the Effective Date and before the Term Commencement Date and any required related third party certifications, all documented costs incurred by Contractor to complete all such modifications shall be fully amortized and paid by Operator, without interest or premium, over a period of time equal to (i) the remaining Special Standby Period, if any, plus (ii) the two (2) year Primary Term (the “Daily Rig Modification Fee”). Upon commencement of the Primary Term, the unrecovered daily amortized amount shall be divided by seven hundred thirty-one (731) and that adjusted daily amount shall be paid by Operator thereafter for a maximum of seven hundred thirty-one (731) days. Contractor shall invoice Operator on a monthly basis for such amortized costs. In the event the Drilling Contract is terminated at any time prior to completion of the Primary Term for any reason, the unrecovered portion of any and all such amortized costs shall be due and payable to Contractor by Operator in a lump sum.

D.            Operator shall have the right, during normal business hours, to audit Contractor’s records with respect to the documented costs incurred by Contractor to complete the Rig Modifications or modifications required to comply with new laws, rules or regulations applicable to operations under or contemplated by the Drilling Contract that become effective after the Effective Date.

E.              Upon completion of any and all Third Party Work, any time spent performing recertification of the Rig’s blowout preventer and associated equipment or consumed because the Rig is waiting to undergo or undergoing uncompleted Rig Modifications,

3

modifications required to comply with new laws, rules or regulations applicable to operations under or contemplated by the Drilling Contract that became effective after the Effective Date and/or normal seal and consumable replacement that occurs upon completion of Third Party Work shall be performed at the Special Standby Rate even if the Special Standby Period has elapsed.

F.              The Daily Upgrade Fee and Daily Rig Modification Fee shall become payable by Operator to Contractor beginning on the Term Commencement Date or, in the case of any Rig Modifications that are in progress or occur thereafter, on the later date of completion and certification, as the case may be, and continuing until the expiration of the Primary Term, including during the Special Standby Period or any Third Party Suspension Period.  If the Special Standby Period is less than three hundred sixty five (365) days from the Term Commencement Date, the Daily Upgrade Fee shall be adjusted accordingly with effect upon the beginning of the Primary Term as specified in Annex A hereof.  Notwithstanding anything in this Agreement or the Drilling Contract to the contrary, in the event of a termination of the Drilling Contract at any time prior to completion of the Primary Term for any reason, Operator shall pay Contractor a lump sum fee to reflect the unrecovered portion of the Daily Upgrade Fee calculated in the manner specified in Annex A hereto.

G.             During the Special Standby Period and the Third Party Suspension Period, if any, the rate adjustment provisions in Section 3.2(b), (c), (e) and (f) of the Drilling Contract shall continue to be applicable based upon prevailing costs in the Area of Operations.

Section 3.              Rig Acceptance.

Following arrival of the Rig in the U.S. Gulf of Mexico the Parties shall diligently proceed with their respective obligations under the Drilling Contract related to acceptance of the Rig including performance of the Rig Acceptance Criteria, and shall provide all personnel, services, and facilities associated therewith as contemplated in the Drilling Contract on a timely basis.  Upon completion of the Rig Acceptance Criteria and any outstanding deep sea trials, the Term Commencement Date shall commence notwithstanding anything to the contrary contained in the Drilling Contract.

Section 4.              Potential Suspension of the Drilling Contract

A.           Contractor shall inform Operator as soon as reasonably practicable of any proposed Third Party Work and the terms thereof, including, without limitation, the duration and location of such proposed Third Party Work and whether any permanent modifications to the Rig are required by such proposed Third Party Work.  Any proposed Third Party Work shall be subject to the prior written consent of Operator, which shall promptly be granted or denied.  Any material amendments or changes to the aforementioned terms of any Third Party Work previously consented to by Operator shall also be subject to the prior written consent of Operator, which shall promptly be granted or denied.

B.             Subject to the applicable third party’s willingness to grant access to the Rig, Operator personnel shall have the right to inspect the Rig during the Third Party Suspension Period(s), if any, in which case the provisions of Articles 8.9 and 8.10 of the Drilling Contract shall be applicable between Operator and Contractor notwithstanding Section 4.C. hereof.

4

C.             Except as specifically stated otherwise herein, the Drilling Contract shall be suspended with effect during any Third Party Suspension Period.

D.            It is expressly agreed that the provisions of this Agreement addressing the potential Third Party Work and the other provisions of this Agreement are mutually exclusive and not dependent upon one another.

Section 5.              Preservation of the Primary Term of the Drilling Contract.

A.           Operator and Contractor expressly agree that neither the Special Standby Period, any rig time paid at the Special Standby Rate, or the Third Party Suspension Period, if any, shall reduce the Primary Term of the Drilling Contract, which shall be preserved in its entirety and shall commence upon the conclusion of the Special Standby Period, any rig time paid at the Special Standby Rate or the Third Party Suspension Period(s), if any, whichever occurs last.

B.             Notwithstanding anything to the contrary in this Agreement, Operator may elect by submission of written notice to Contractor at any time and for any reason to commence the Primary Term of the Drilling Contract and to thereafter pay the applicable rates under the Drilling Contract, at which time the Special Standby Rate shall cease to be applicable.

Section 6.              Miscellaneous.

A.           From and after the Term Commencement Date, all Rig service obligations shall be for Operator’s account as specified in Appendix D of the Drilling Contract, except during any Third Party Suspension Period(s).

B.             Within thirty (30) days from the Effective Date, Contractor, Operator and JP Morgan Chase Bank, National Association shall enter into an amendment of the Escrow Agreement entered into as of 9 December 2009 as may be agreed to and necessary to reflect the terms, conditions and impact of this Agreement which shall incorporate the principles listed in Annex B hereto.

C.             The Drilling Contract, as amended, shall be and is hereby deemed to be further amended to comport with the provisions of this Agreement and shall otherwise remain in full force and effect. Except as expressly amended hereby, the terms and conditions of the Drilling Contract shall otherwise be applicable to this Agreement.  In the event the terms and conditions of this Agreement conflict with the terms and conditions of the Drilling Contract, as amended, the terms of this Agreement shall prevail.

D.            This Agreement may be executed in counterparts via facsimile or electronically transmitted signature, each of which will be considered an original and all of which together will constitute one and the same agreement.  At the request of a Party, the other Party will confirm its facsimile or electronically transmitted signature page by delivering an original signature page to the requesting Party.

5

IN WITNESS WHEREOF, the Parties hereto have executed this Special Standby Rate and Potential Suspension Agreement by their duly authorized officers or representatives as of the Effective Date.

6

ANNEX A

The following principles shall be applied in administering the Daily Upgrade Fee:

DAILY UPGRADE FEE

The Daily Upgrade Fee shall reflect the intention of Article 3 of the Drilling Contract as amended by Amendment #1, dated October 29, 2008.  The fee is based on Contractor receiving a 12% return on payments amortized over the term of the Drilling Contract.   The Daily Upgrade Fee is a factor of the Rig Upgrade Cost divided by the number of days of anticipated term then multiplied by an Upgrade Premium factor.  The Upgrade Premium factor is calculated by determining the multiplier required to produce a 12% Internal Rate of Return (IRR) over the anticipated payment period.

The following examples show how the Daily Upgrade Fee is calculated for two term periods.

Term — 1096 days (12 months Special Standby Period plus two year Primary Term)

Term — 913 days (6 months Special Standby Period plus two year Primary Term)

For the purpose of this Agreement, the Daily Upgrade Fee shall initially be set assuming the term of payment is 1096 days (12 months Special Standby Period plus two year Primary Term) and the current estimate of the Total Upgrade Cost, $10,867,652, resulting in a Daily Upgrade Fee of $11,761.  If, at a later date, the payment term is altered, as would be the case if the Special Standby Period is less than or greater than 365 days, the Daily Upgrade Fee shall be recalculated to maintain a 12% IRR and full repayment of the Rig Upgrade Cost over the new anticipated term.

The provisions of Article 3.3 of the Drilling Contract shall apply in the event of termination prior to completion of term.

7

ANNEX B

With respect to Section 6.C. of the Contract Amendment and Potential Suspension Agreement, the principal aspects of the contemplated amendment to the Escrow Agreement entered into on 9 December 2009 are as follows:

·                  Invoices for Special Standby Rate shall be paid from the escrow account following submission to Operator, subject to a maximum draw of $62MM. Operator to pay Company directly for all other invoices during the Special Standby Period, including but not limited to accruals for Rig Upgrades, Rig Modifications, reimbursables, etc.

·                  The remaining escrow balance continues until the first anniversary of the Primary Term, at which time 50% of the remaining escrow fund shall be released to Operator minus any prior escrow draw-downs due to Operator’s default in payments.

·                  The remaining balance of escrow account shall be released to Operator upon termination of the Drilling Contract.

8