FIRST SUPPLEMENTAL INDENTURE

Exhibit 4.10

FIRST SUPPLEMENTAL INDENTURE

FIRST SUPPLEMENTAL INDENTURE (this “Supplemental Indenture”), dated as of April 1, 2011, is among Goodrich Petroleum Corporation, a Delaware corporation (herein called the “Company”), and Goodrich Petroleum Company, L.L.C., a Texas limited liability company (the “Subsidiary Guarantor”), and Wells Fargo Bank, National Association, as Trustee (herein called the “Trustee”) under the indenture, dated as of December 6, 2006, among the Company and the Trustee (the “Original Indenture,” and as amended and supplemented, including by this Supplemental Indenture, the “Indenture”) relating to the Company’s 3.25% Convertible Senior Notes due 2026 (the “Notes”).

RECITALS OF THE COMPANY AND THE SUBSIDIARY GUARANTOR

The Company, the Subsidiary Guarantor and the Trustee entered into an indenture, dated as of March 2, 2011 (the “Senior Notes Indenture”), relating to the Company’s 8.875% Senior Notes due 2019 (the “2019 Notes”), and the 2019 Notes.

Section 3.09 of the Original Indenture obligates the Company to cause certain subsidiaries that provide a guarantee with respect to any Public Debt Securities issued by the Company to guarantee the Notes by executing a supplemental indenture as provided in such Section.

Section 10.01(c) of the Original Indenture provides that the Company and the Trustee are authorized to execute and deliver this Supplemental Indenture to amend or supplement the Original Indenture without the consent of any Holder to add the Guarantees contemplated hereby.

This Supplemental Indenture is subject to the provisions of the Trust Indenture Act of 1939, as amended, that are required to be a part of this Supplemental Indenture and shall, to the extent applicable, be governed by such provisions.

All things necessary to make this Supplemental Indenture a valid agreement of each of the Company and the Subsidiary Guarantor, in accordance with its terms, have been done.

NOW, THEREFORE, THIS SUPPLEMENTAL INDENTURE WITNESSETH:

To comply with the provisions of the Original Indenture and in consideration of the foregoing and for other good and valuable consideration, the receipt of which is hereby acknowledged, the Subsidiary Guarantor, the Company and the Trustee mutually covenant and agree for the equal and ratable benefit of the Holders of the Notes as follows:

ARTICLE ONE

APPLICATION OF SUPPLEMENTAL INDENTURE

Section 101.     Interpretation.

Unless otherwise expressly specified, references in this Supplemental Indenture to specific Article numbers or Section numbers refer to Articles and Sections contained in this Supplemental Indenture, and not the Original Indenture or any other document. Capitalized terms used, but not defined, herein shall have the meanings assigned to them in the Original Indenture.

Section 102.     Effect of Supplemental Indenture.

The Original Indenture shall be supplemented and amended by replacing Article 13 of the Original Indenture in its entirety with the following:

ARTICLE 13

SUBSIDIARY GUARANTEES

Section 1301.     Subsidiary Guarantee.

(a) Subject to this Article 13, the Subsidiary Guarantor fully and unconditionally guarantees, on a senior unsecured basis, to each Holder of a Note authenticated and delivered by the Trustee and to the Trustee and its successors and assigns, irrespective of the validity and enforceability of this Indenture, the Notes or the obligations of the Company hereunder or thereunder, that: (i) the principal of, premium, if any, and interest on the Notes will be promptly paid in full when due, whether at Stated Maturity, by acceleration, redemption or otherwise, and interest on the overdue principal of, premium, if any, and interest on the Notes, if any, if lawful (subject in all cases to any applicable grace period provided herein), and all other monetary obligations of the Company to the Holders or the Trustee hereunder or thereunder will be promptly paid in full or performed, all in accordance with the terms hereof and thereof; and (ii) in case of any extension of time of payment or renewal of any Notes or any of such other obligations, that same will be promptly paid in full when due or performed in accordance with the terms of the extension or renewal, whether at stated maturity, by acceleration or otherwise. Failing payment when due of any amount so guaranteed or any performance so guaranteed for whatever reason, the Subsidiary Guarantor shall be obligated to pay the same immediately. The Subsidiary Guarantor agrees that this is a guarantee of payment and not a guarantee of collection.

(b) The Subsidiary Guarantor agrees that, to the maximum extent permitted under applicable law, their obligations hereunder shall be unconditional, irrespective of the validity, regularity or enforceability of the Notes or this Indenture, the absence of any action to enforce the same, any waiver or consent by any Holder of the Notes with respect to any provisions hereof or thereof, the recovery of any judgment against the Company, any action to enforce the same or any other circumstance which might otherwise constitute a legal or equitable discharge or defense of a Subsidiary Guarantor. The Subsidiary Guarantor waives diligence, presentment, demand of payment, filing of claims with a court in the event of insolvency or bankruptcy of the Company, any right to require a proceeding first against the Company, protest, notice and all demands whatsoever and covenant that this Guarantee shall not be discharged except by complete performance of the obligations contained in the Notes and this Indenture.

(c) If any Holder or the Trustee is required by any court or otherwise to return to the Company, the Subsidiary Guarantor or any custodian, trustee, liquidator or other similar official acting in relation to either of the Company or the Subsidiary Guarantor, any amount paid by either to the Trustee or such Holder, this Guarantee, to the extent theretofore discharged, shall be reinstated in full force and effect.

(d) The Subsidiary Guarantor agrees that it shall not be entitled to any right of subrogation in relation to the Holders in respect of any obligations guaranteed hereby until payment in full of all obligations guaranteed hereby. The Subsidiary Guarantor further agrees that, as between the Subsidiary Guarantor, on the one hand, and the Holders and the Trustee, on the other hand, (x) the maturity of the obligations guaranteed hereby may be accelerated as provided in Article 7 of the Indenture for the purposes of this Guarantee, notwithstanding any stay, injunction or other prohibition preventing such acceleration in respect of the obligations

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guaranteed hereby, and (y) in the event of any declaration of acceleration of such obligations as provided in Article 7 of the Indenture, such obligations (whether or not due and payable) shall forthwith become due and payable by the Subsidiary Guarantor for the purpose of this Guarantee.

Section 1302.     Limitation on Subsidiary Guarantor Liability.

Each Subsidiary Guarantor hereby confirms that it is the intention of all such parties that the Guarantee of such Subsidiary Guarantor not constitute a fraudulent transfer or conveyance for purposes of Bankruptcy Law, the Uniform Fraudulent Conveyance Act, the Uniform Fraudulent Transfer Act or any similar Federal or state law to the extent applicable to any Guarantee. To effectuate the foregoing intention, the Trustee and the Subsidiary Guarantor hereby irrevocably agree that the obligations of such Subsidiary Guarantor will be limited to the maximum amount which, after giving effect to all other contingent and fixed liabilities of such Subsidiary Guarantor, and after giving effect to any collections from or payments made by or on behalf of any other Subsidiary Guarantor in respect of the obligations of such other Subsidiary Guarantor under its Guarantee or pursuant to its contribution obligations under this Article 13, will result in the obligations of such Subsidiary Guarantor under its Guarantee not constituting a fraudulent conveyance or fraudulent transfer under Federal or state law. Until such time as the Notes are paid in full, the Subsidiary Guarantor waives all rights of subrogation or contribution, whether arising by contract or operation of law (including, without limitation, any such right arising under Federal Bankruptcy Law) or otherwise by reason of any payment by it pursuant to the provisions of this Article 13.

Section 1303.     Release of the Subsidiary Guarantor.

The Subsidiary Guarantor will be deemed automatically and unconditionally released and discharged from all of its obligations under its Guarantee without any further action on the part of the Trustee or any Holder of the Notes:

ARTICLE TWO

MISCELLANEOUS

Section 201.     Capitalized Terms.

Capitalized terms used herein without definition shall have the meanings assigned to them in the Indenture.

Section 202.     NEW YORK LAW TO GOVERN.

THE LAWS OF THE STATE OF NEW YORK SHALL GOVERN AND BE USED TO CONSTRUE AND ENFORCE THIS SUPPLEMENTAL INDENTURE.

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Section 203.     Counterparts.

The parties may sign any number of copies of this Supplemental Indenture. Each signed copy shall be an original, but all of them together represent the same agreement. This Supplemental Indenture may be executed in multiple counterparts which, when taken together, shall constitute one instrument.

Section 204.     Effect of Headings.

The Section headings herein are for convenience only and shall not affect the construction hereof.

Section 205.     The Trustee.

Except as otherwise expressly provided herein, no duties, responsibilities or liabilities are assumed, or shall be construed to be assumed, by the Trustee by reason of this Supplemental Indenture. This Supplemental Indenture is executed and accepted by the Trustee subject to all the terms and conditions set forth in the Indenture with the same force and effect as if those terms and conditions were repeated at length herein and made applicable to the Trustee with respect hereto.

[Remainder of Page Intentionally Left Blank. Signature Page Follows.]

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IN WITNESS WHEREOF, the parties hereto have caused this Supplemental Indenture to be duly executed and attested, all as of this 1^st day of April 2011.

[Signature Page to Supplemental Indenture]