<PAGE>   1
                                                                    EXHIBIT 4.1
                                                                               
                                                                [EXECUTION COPY]


                                                                               
================================================================================

                       BAYARD DRILLING TECHNOLOGIES, INC.
                                        
                                      AND
                                        
                                   GUARANTORS
                                        
                                        
                                  $100,000,000
                           11% Senior Notes due 2005
                                        
                         ------------------------------
                                        
                          FIRST SUPPLEMENTAL INDENTURE
                                        
                           Dated as of April 7, 1999
                                        
                         ------------------------------
                                        
                       U.S. TRUST COMPANY OF TEXAS, N.A.
                                        
                                    Trustee


================================================================================


<PAGE>   2


  FIRST SUPPLEMENT INDENTURE, dated as of April 7, 1999, among Bayard Drilling
Technologies, Inc., a Delaware corporation (the "Company"), certain of the
Company's subsidiaries signatory hereto (each, a "Guarantor", collectively, the
"Guarantors") and U.S. Trust Company of Texas, N.A., as trustee (the
"Trustee").

                              W I T N E S S E T H:

   WHEREAS, the Company, the Guarantors and the Trustee are parties to that
certain Indenture, dated as of June 26, 1998 (the "Original Indenture"),
pursuant to which the Company issued $100,000,000 in aggregate principle amount
of its 11% Senior Subordinated Notes due 2005, Series B (the "Securities");

   WHEREAS, the Company intends to consummate a transaction involving the
merger (the "Merger") of Nabors Acquisition Corp. VII, a Delaware corporation
("Merger Sub"), with and into the Company pursuant to that certain Agreement
and Plan of Merger, dated as of October 19, 1998, as amended on January 15,
1999 and February 12, 1999 (as so amended, the "Merger Agreement"), by and
among Nabors Industries, Inc., a Delaware corporation, Merger Sub and the
Company;

   WHEREAS, in accordance with the provisions of the Original Indenture,
holders of more than a majority in aggregate principal amount of the Notes
outstanding have provided their written consents (the "Consents") to certain
amendments to the provisions of the Original Indenture (the "Amendments"),
which are described in the Consent and Modification Letter first distributed to
certain beneficial owners of the Notes on March 30, 1999; and

   WHEREAS, the Company and the Trustee, in accordance with Article 10 of the
Original Indenture and pursuant to appropriate resolutions of the Board of
Directors of the Company, have duly determined to make and execute this First
Supplemental Indenture in order to give effect to the Amendments;

   NOW THEREFORE, in consideration of the premises and other good and valuable
consideration, the receipt and adequacy of which are hereby acknowledged, the
parties hereto agree as follow:

                                  ARTICLE ONE

                      AMENDMENTS TO THE ORIGINAL INDENTURE

     Section 1.1          Effective as of the date hereof (the "Effective
Date"), the Original Indenture is amended as follows:





<PAGE>   3
                 (a)      Section 5.01 is amended to add a new Section 5.10(c)
to read in its entirety as follows:

                          "(c)    Sections 5.01 and 5.02 hereof shall not apply
                 to the merger of Nabors Acquisition Corp. VII, a Delaware
                 corporation ("Merger Sub"), with and into the Company pursuant
                 to that certain Agreement and Plan of Merger, dated as of
                 October 19, 1998, as amended on January 15, 1999 and February
                 12, 1999, by and among Nabors Industries, Inc., a Delaware
                 corporation, Merger Sub and the Company."

                                  ARTICLE TWO

                      AMENDMENTS TO THE ORIGINAL INDENTURE

                 Securities authenticated and delivered for transfer or
exchange of outstanding Securities after the Effective Date, and all Securities
presented or delivered to the Trustee on or after such date for the purpose of
being stamped, shall be stamped (unless textually revised as hereinafter
provided) by the Trustee with a notation substantially in the form as follows:

                          "The Indenture dated as of August 26, 1998 referred
                 to in this Security has been amended by a First Supplemental
                 Indenture dated as of April 7, 1999, pursuant to which certain
                 provisions of the Indenture have been amended as set forth in
                 such First Supplemental Indenture.  Copies of the First
                 Supplemental Indenture are on file with, and available on
                 request from, the Trustee and the Company."

                 Any securities hereafter authenticated or delivered may be 
textually revised, making changes in phraseology and form (but not in substance)
as may be appropriate so as to conform, in the opinion of the Trustee and the
Company, to the modifications made by this First Supplemental Indenture.

                                 ARTICLE THREE

                                 MISCELLANEOUS

     SECTION 3.1          DEFINITIONS.  Capitalized terms used but not defined
in this First Supplemental Indenture shall have the respective meanings
assigned to them in the Original Indenture.

     SECTION 3.2          CONFLICT WITH THE TRUST INDENTURE ACT.  If any
provision of this First Supplemental Indenture modifies or excludes any
provision of the Trust Indenture Act that is required under such Act to be part
of and govern this First Supplemental Indenture, the latter provision of the
Trust Indenture Act shall control.  If any provision hereof modifies or
excludes any provision of the Trust Indenture Act that may be so modified or
excluded, the latter




                                     -2-
<PAGE>   4

provision of the Trust Indenture Act shall be deemed to apply to this First
Supplemental Indenture, as so modified or excluded, as the case may be.

     SECTION 3.3          SECURITIES DEEMED CONFORMED.  As of the Effective
Date, the provisions of each Security then outstanding shall be deemed to be
conformed, without the necessity for any reissuance or exchange of such
Security or any other action on the part of the Holders, the Company, the
Guarantors or the Trustee, so as to reflect this First Supplemental Indenture.

     SECTION 3.4          SUCCESSORS.  All agreements of the Company, the
Guarantors and the Trustee in this First Supplemental Indenture and in the
Original Indenture shall bind their respective successors.

     SECTION 3.5          BENEFITS OF SUPPLEMENTAL INDENTURE  Nothing in this
First Supplemental Indenture, express or implied, shall give to any person,
other than the parties hereto and their successors hereunder and the Holders,
any benefit or any legal or equitable right, remedy or claim under this First
Supplemental Indenture or the Original Indenture.

     SECTION 3.6          SEPARABILITY.  In case any provision in this First
Supplemental Indenture, or in the Original Indenture, shall be invalid, illegal
or unenforceable, the validity, legality and enforceability of the remaining
provisions shall not in any way be affected or impaired thereby, it being
intended that all of the provisions hereof shall be enforceable to the full
extent permitted by law.

     SECTION 3.7          TRUSTEE RESPONSIBILITY.  The Trustee assumes no
duties, responsibilities or liabilities by reason of this First Supplemental
Indenture other than as set forth in the Original Indenture.  The Trustee
assumes no responsibility for the correctness of the statements herein
contained, which shall be taken as statements of the Company and the
Guarantors.  The First Supplemental Indenture is executed and accepted by the
Trustee subject to all of the terms and conditions of its acceptance of the
trust under the Original Indenture, as fully as if said terms and conditions
were herein set forth in full.

     SECTION 3.8          HEADINGS.  The Section headings of this First
Supplemental Indenture have been inserted for convenience of reference only,
are not to be considered a part of this First Supplemental Indenture and shall
not modify or restrict any of the terms or provisions hereof.

     SECTION 3.9          MULTIPLE ORIGINALS.  The parties may sign any number
of copies of this First Supplemental Indenture.  Each signed copy shall be an
original, but all of them together represent the same agreement.  One signed
copy is enough to prove this First Supplemental Indenture.  This First
Supplemental Indenture may be executed in any number of counterparts, each of
which shall be deemed an original but all such counterparts shall together
constitute but one and the same instrument.

     SECTION 3.10         GOVERNING LAW.  THIS FIRST SUPPLEMENTAL INDENTURE
SHALL BE GOVERNED BY, AND CONSTRUED IN ACCORDANCE WITH THE LAWS OF THE STATE OF
NEW YORK BUT WITHOUT GIVING EFFECT TO APPLICABLE





                                    - 3 -
<PAGE>   5


PRINCIPLES OF CONFLICTS OF LAW TO THE EXTENT THAT THE APPLICATION OF THE LAWS
OF ANOTHER JURISDICTION WOULD BE REQUIRED THEREBY.





                                    - 4 -
<PAGE>   6

                 IN WITNESS HEREOF, the parties hereto have caused this First
Supplemental Indenture to be duly executed as of the date first above written.


                                  COMPANY:
                                  BAYARD DRILLING TECHNOLOGIES, INC.
                                  
                                  
                                  By: /s/ David E. Grose                        
                                     -------------------------------------------
                                      David E. Grose
                                      Vice President and Chief Financial Officer
                                  
                                  
                                  GUARANTORS:
                                  BAYARD DRILLING, L.L.C.
                                  
                                  By: /s/ David E. Grose                       
                                     -------------------------------------------
                                      David E. Grose
                                      Vice President and Chief Financial Officer
                                  
                                  
                                  BAYARD DRILLING, L.P.
                                  
                                  By:   BAYARD DRILLING, L.L.C.,
                                        its general partner


                                  By: /s/ David E. Grose                       
                                     -------------------------------------------
                                      David E. Grose
                                      Vice President and Chief Financial Officer
                                  
                                  
                                  BONRAY DRILLING CORPORATION


                                  By: /s/ David E. Grose                       
                                     -------------------------------------------
                                      David E. Grose
                                      Vice President and Chief Financial Officer
                                  
                                  
                                  TREND DRILLING CO.


                                  By: /s/ David E. Grose                       
                                     -------------------------------------------
                                      David E. Grose
                                      Vice President and Chief Financial Officer





                                    - 5 -
<PAGE>   7

                                  TRUSTEE:
                                  U.S. TRUST COMPANY OF TEXAS, N.A.
                                  
                                  
                                  By:                                          
                                     -------------------------------------------
                                  Name:                                        
                                       -----------------------------------------
                                  Title:                                       
                                        ----------------------------------------
                                  




                                     - 6 -
<PAGE>   8


                                  TREND DRILLING CO.
                                  
                                  
                                  By:                                          
                                     -------------------------------------------
                                      David E. Grose
                                      Vice President and Chief Financial Officer
                                  

                                  TRUSTEE:
                                  U.S. TRUST COMPANY OF TEXAS, N.A.
                                  
                                  
                                  By: /s/ Melissa Scott                        
                                     -------------------------------------------
                                      Melissa Scott
                                      Vice President





                                     - 7 -
<PAGE>   9



STATE OF OKLAHOMA   )
                    )    ss.:
COUNTY OF OKLAHOMA  )


          On the 6th day of April, 1999, before me personally came David E.
Grose, to me known, who, being by me duly sworn, did depose and say that he is
Vice President and Chief Financial Officer of Bayard Drilling Technologies,
Inc., one of the corporations described in and which executed the foregoing
instrument, and that he signed his name thereto by authority of the Board of
Directors of said corporation.


                                        /s/ Signature Illegible
                                        ----------------------------------------
My Commission Expires:  6-21-99         Notary Public



STATE OF OKLAHOMA   )
                    )    ss.:
COUNTY OF OKLAHOMA  )

          On the 6th day of April, 1999, before me personally came David E.
Grose, to me known, who, being by me duly sworn, did depose and say that he is
Vice President and Chief Financial Officer of Bayard Drilling, L.L.C., one of
the entities described in and which executed the foregoing instrument, and that
he signed his name thereto by authority of the ________________ of such entity.


                                        /s/ Signature Illegible
                                        ----------------------------------------
My Commission Expires:  6-21-99         Notary Public





                                     - 8 -
<PAGE>   10



STATE OF OKLAHOMA   )
                    )  ss.:
COUNTY OF OKLAHOMA  )


          On the 6th day of April, 1999, before me personally came David E.
Grose, to me known, who, being by me duly sworn, did depose and say that he is
Vice President and Chief Financial Officer of Bayard Drilling, L.L.C., the
general partner of Bayard Drilling, L.P., which is one of the entities
described in and which executed the foregoing instrument, and that he signed
his name thereto by authority of the ________________ of Bayard Drilling, L.L.C.


                                        /s/ Signature Illegible
                                        ----------------------------------------
My Commission Expires:  6-21-99         Notary Public

STATE OF OKLAHOMA   )
                    )  ss.:
COUNTY OF OKLAHOMA  )


          On the 6th day of April, 1999, before me personally came David E.
Grose, to me known, who, being by me duly sworn, did depose and say that he is
Vice President and Chief Financial Officer of Bonray Drilling Corporation, one
of the corporations described in and which executed the foregoing instrument,
and that he signed his name thereto by authority of the Board of Directors of
said corporation.


                                        /s/ Signature Illegible
                                        ----------------------------------------
My Commission Expires:  6-21-99         Notary Public





                                     - 9 -
<PAGE>   11



STATE OF OKLAHOMA   )
                    )  ss.:
COUNTY OF OKLAHOMA  )


          On the 6th day of April, 1999, before me personally came David E.
Grose, to me known, who, being by me duly sworn, did depose and say that he is
Vice President and Chief Financial Officer of Trend Drilling Co., one of the
corporations described in and which executed the foregoing instrument, and that
he signed his name there by authority of the Board of Directors of said
corporation.

                                        /s/ Signature Illegible
                                        ----------------------------------------
My Commission Expires:  6-21-99         Notary Public

STATE OF OKLAHOMA   )
                    )  ss.:
COUNTY OF OKLAHOMA  )


          On the ____ day of April, 1999, before me personally came
_________________, to me known, who, being by me duly sworn, did depose and say
that he is __________________ of U.S. Trust Company of Texas, N.A., one of the
entities described in and which executed the foregoing instrument, and that he
signed his name there by authority of the ______________ of such entity.




                                        ----------------------------------------
                                        Notary Public





                                     - 10 -
<PAGE>   12



STATE OF _______________  )
                          )  ss.:
COUNTY OF ______________  )


          On the ____ day of April, 1999, before me personally came David E.
Grose, to me known, who, being by me duly sworn, did depose and say that he is
Vice President and Chief Financial Officer of Trend Drilling Co., one of the
corporations described in and which executed the foregoing instrument, and that
he signed his name thereto by authority of the Board of Directors of said
corporation.

                                        
                                        ----------------------------------------
                                        Notary Public


STATE OF TEXAS    )
                  )  ss.:
COUNTY OF DALLAS  )


          On the ____ day of April, 1999, before me personally came Melissa
Scott, to me known, who, being by me duly sworn, did depose and say that he is
Vice President of U.S. Trust Company of Texas, N.A., one of the entities
described in and which executed the foregoing instrument, and that she signed
her name thereto on behalf of such entity.


                                        /s/ Carrie D. Oates
                                        ----------------------------------------
                                        Notary Public





                                    - 11 -