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                                                                 EXHIBIT 10.20




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                            SMITH INTERNATIONAL, INC.



                                SECOND AMENDMENT


                            Dated as of July 31, 1998



                                       TO


                                 NOTE AGREEMENT


                            Dated as of May 21, 1996



                        RE: 7.24% SENIOR GUARANTEED NOTES
                                DUE APRIL 2, 2001
                                       AND
                          7.63% SENIOR GUARANTEED NOTES
                                DUE APRIL 2, 2006



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         THIS SECOND AMENDMENT dated as of July 31, 1998 (the or this "Second
Amendment") to the several Note Agreements, each dated as of May 21, 1996
between SMITH INTERNATIONAL, INC., a Delaware corporation (the "Company"), and
each of the institutions identified in Schedule I thereto (collectively, the
"Purchasers"), as amended by the First Amendment and Waiver dated May 5, 1997,
is between the Company and each of the institutions which is a signatory to this
Second Amendment (the "Required Noteholders").

                                    RECITALS:

         A. The Company and each of the Purchasers have heretofore entered into
separate and several Note Agreements, each dated as of May 21, 1996 (the "Note
Agreements") as amended by the First Amendment and Waiver dated May 5, 1997 (the
"First Amendment"). The Required Noteholders are the holders of at least 66-2/3%
of the Company's 7.24% Senior Guaranteed Notes due April 2, 2001 in the
aggregate principal amount of $30,000,000 and the Company's 7.63% Senior
Guaranteed Notes due April 2, 2006 in the aggregate principal amount of
$20,000,000 (collectively, the "Notes") issued and sold under and pursuant to
the Note Agreements.

         B. The Company and the Required Noteholders now desire to amend the
Note Agreements effective on July 31, 1998 (the "Second Amendment Effective
Date") in the respects, but only in the respects, hereinafter set forth.
Pursuant to Section 7.1 of the Note Agreements, the consent of the Required
Noteholders is required to amend the Note Agreements as contemplated herein.

         C. Capitalized terms used herein but not otherwise defined shall have
the meanings assigned thereto in the Note Agreements.

         NOW, THEREFORE, the acceptance hereof by the Required Noteholders and
the full and complete satisfaction of the conditions precedent to the
effectiveness of this Second Amendment set forth in Section 3.1 hereof, will
confirm the understanding of the Company and the Required Noteholders that the
Note Agreements shall be and are hereby amended in the following respects:

SECTION 1. AMENDMENTS.

         SECTION 1.1. AMENDMENT OF SECTION 5.7(a). Section 5.7(a) of the
Existing Note Agreements shall be and is hereby amended in its entirety so that
the same shall henceforth read as follows:

                  "Section 5.7 Certain Ratios. (a) The Company will at all times
         maintain the ratio of Consolidated Indebtedness to Consolidated Total
         Capitalization at not more than 0.40 to 1.00.

         For purposes of determining compliance with this Section 5.7(a), the
         term "Consolidated Indebtedness" shall not include Indebtedness of M-I
         Drilling under the M-I Drilling Facility or other Indebtedness of M-I
         Drilling incurred within the limitations of Section 5.9(a)(5) 



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         if and to the extent the repayment of all of such Indebtedness is
         absolutely, irrevocably and unconditionally guaranteed by Halliburton
         and not by the Company."

         SECTION 1.2. AMENDMENT OF DEFINITIONS. The following definitions shall
be and are hereby added to Section 8.1 of the Existing Note Agreements in their
alphabetical order:

         ""Second Amendment" shall mean the Second Amendment dated as of July
         31, 1998 to the Note Agreement entered into by the Company and the
         Required Noteholders."

         ""Second Amendment Effective Date" shall mean July 31, 1998."

SECTION 2. REPRESENTATIONS AND WARRANTIES OF THE COMPANY.

         The Company hereby represents and warrants to each of you that all of
the representations and warranties set forth in Exhibit A attached hereto are
true and correct as of the date of acceptance set forth on the signature pages
to this Second Amendment and are incorporated herein by reference with the same
force and effect as though herein set forth in full.

SECTION 3. MISCELLANEOUS.

         SECTION 3.1. This Second Amendment shall become effective and binding
upon the Company and the holders of the Notes on the Second Amendment Effective
Date upon the acceptance hereof by the Required Noteholders in the space below
and upon the satisfaction in full of the following conditions:

         (a) each of their holders of the Notes shall have received a
certificate dated the date of acceptance set forth on the signature pages to
this Second Amendment, signed by the President or a Vice-President of the
Company, the truth and accuracy of which shall be a condition to their
obligations under this Second Amendment to the effect that: (1) the
representations and warranties of the Company set forth in Exhibit A attached
hereto are true and correct on and with respect to the Second Amendment
Effective Date and (2) no Default or Event of Default has occurred and is then
continuing, after giving effect to the amendments set forth herein;

         (b) the holders of the Notes shall have received, in form and substance
reasonably satisfactory to such holders and their special counsel, such
documents and evidence with respect to the Company as such holders may
reasonably request in order to establish the existence and good standing of the
Company and the authorization, execution and delivery of the Second Amendment;

         (c) the holders of the Notes shall have received an opinion of counsel
from counsel to the Company covering the matters set forth in clause (b) of this
Section 3.1 and such other matters relating to the Second Amendment as the
holders of the Notes may reasonably request;


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         (d) ACQCO shall have consented to the execution and delivery of this
Second Amendment and the holders of the Notes shall have received a true,
correct and complete originally executed copy thereof; and

         (e) any consents or approvals from any holder or holders of any
outstanding Security of the Company or ACQCO and any amendments of agreements
pursuant to which any Securities may have been issued which shall be necessary
to permit the consummation of the transactions contemplated hereby shall have
been obtained and all such consents or amendments shall be reasonably
satisfactory in form and substance to the holders of the Notes and their special
counsel and such holders and their special counsel shall have received true,
correct and complete copies of such executed consents and amendments.

         SECTION 3.2. The Company agrees to pay all reasonable fees and expenses
of the holders of the Notes and their special counsel in connection with the
preparation, execution and delivery of this Second Amendment.

         SECTION 3.3. Whenever in any certificate, letter, notice or other
instrument reference is made to the Note Agreements, such reference without more
shall include reference to this Second Amendment.

         SECTION 3.4. Except as modified and expressly amended by the Second
Amendment, the Note Agreements and the Notes are in all respects ratified,
confirmed and approved on all of the terms, provisions and conditions thereof
shall be and remain in full force and effect, including (without limitation),
Section 9.10, 9.11 and 9.13 of the Note Agreements. The descriptive headings of
the various Sections or parts of this Second Amendment are for convenience only
and shall not affect the meaning or construction of any of the provisions
hereof.

         SECTION 3.5. This Second Amendment may be executed and delivered in any
number of counterparts, each of such counterparts constituting an original, but
all together only one Second Amendment.


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                                          SMITH INTERNATIONAL, INC.


                                          By: /s/ JOHN J. KENNEDY
                                             ---------------------------------
                                             Its Senior Vice President, Chief
                                                 Financial Officer and Treasurer


                                          By: /s/ MARGARET K. DORMAN
                                             ---------------------------------
                                             Its Vice President, Controller
                                                 and Assistant Treasurer






Smith International, Inc. - Second Amendment to Note Agreement



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                                                     PRINCIPAL LIFE INSURANCE 
                                                          COMPANY, F/K/A

                                                     PRINCIPAL MUTUAL LIFE
                                                     INSURANCE COMPANY


                                                     By: /s/ THOMAS R. POSPISIL
                                                        ------------------------
                                                        Its Counsel



                                                     By: /s/ JAMES C. FIFIELD
                                                        ------------------------
                                                        Its Counsel


AGREED TO AND ACCEPTED
THIS 31st DAY OF JULY, 1998:




Smith International, Inc. - Second Amendment to Note Agreement


<PAGE>   7



                                              JOHN HANCOCK MUTUAL LIFE
                                              INSURANCE COMPANY


                                              By: /s/ EUGENE R. HODGE, JR.
                                                 -------------------------------
                                                 Its Senior Investment Officer




AGREED TO AND ACCEPTED
THIS 31st DAY OF JULY, 1998:




Smith International, Inc. - Second Amendment to Note Agreement



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                                              JOHN HANCOCK VARIABLE LIFE
                                              INSURANCE COMPANY


                                              By: /s/ EUGENE R. HODGE, JR.
                                                 -------------------------------
                                                 Its Authorized Officer


AGREED TO AND ACCEPTED
THIS 31st DAY OF JULY, 1998:




Smith International, Inc. - Second Amendment to Note Agreement


<PAGE>   9



                                                      IDS CERTIFICATE COMPANY


                                                      By:
                                                         ----------------------
                                                         Its


AGREED TO AND ACCEPTED
THIS 31st DAY OF JULY, 1998:




Smith International, Inc. - Second Amendment to Note Agreement


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                                                     MELLON BANK, N.A., SOLELY
                                                     IN ITS CAPACITY AS TRUSTEE
                                                     FOR THE AT&T MASTER
                                                     PENSION TRUST (AS DIRECTED
                                                     BY JOHN HANCOCK MUTUAL
                                                     LIFE INSURANCE COMPANY),
                                                     AND NOT IN ITS INDIVIDUAL
                                                     CAPACITY


                                                    By: /s/ BERNADETTE RIST
                                                       -------------------------
                                                       Its Authorized Signatory


AGREED TO AND ACCEPTED
THIS 31st DAY OF JULY, 1998:




Smith International, Inc. - Second Amendment to Note Agreement



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                                               THE MARITIME LIFE ASSURANCE 
                                               COMPANY


                                               By: /s/ GARY MARTIN
                                                  ------------------------------
                                                  Its Vice President, Bonds and
                                                      Corporate Finance


                                               By: /s/ PETER A. STUART
                                                  ------------------------------
                                                  Its Senior Vice President
                                                      Chief Investment Officer


AGREED TO AND ACCEPTED
THIS 31st DAY OF JULY, 1998:




Smith International, Inc. - Second Amendment to Note Agreement


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                                                                     EXHIBIT A


                         REPRESENTATIONS AND WARRANTIES


The Company represents and warrants to you as follows:

         1.  Corporate Organization and Authority. The Company and each
Subsidiary

             (a) is a corporation duly organized, validly existing and in good
         standing under the laws of its jurisdiction of incorporation and has
         not commenced any proceedings to dissolve;

             (b) has all requisite power and authority and all necessary
         licenses and permits to own and operate its properties and to carry on
         its business as now conducted and as presently proposed to be
         conducted; and

             (c) is duly licensed or qualified and is in good standing as a
         foreign corporation in each jurisdiction wherein the nature of this
         business transacted by it or the nature of the property owned or leased
         by it makes such licensing or qualification necessary, except for such
         jurisdictions wherein the failure to be so licensed or qualified would
         not have a Material Adverse Effect.

         2.  Full Disclosure. Neither the Second Amendment nor any other written
statement furnished by the Company to you in connection with the negotiation of
the Second Amendment, contain any untrue statement of a material fact or omit a
material fact necessary to make the statement contained therein or herein not
misleading. There is no fact peculiar to the Company which the Company has not
disclosed to you in writing which materially affects adversely nor, so far as
the Company can now foresee, will materially affect adversely the properties,
business, prospects, profits or condition (financial or otherwise) of the
Company or the ability of the Company to enter into and perform the Second
Amendment.

         3. Transactions Legal and Authorized. The execution and delivery of the
Second Amendment

             (a) are within the corporate powers of the Company;

             (b) will not violate any material provisions of any law or any
         order of any court or governmental authority or agency and will not
         conflict with or result in any breach of any of the terms, conditions
         of provisions of, or constitute a default under, the Certificate of
         Incorporation of By-laws of the Company or any loan agreement,
         indenture or other agreement or instrument to which the Company will be
         a party or by which it may be bound on the Second Amendment Effective
         Date or result in the imposition of any Lien on any property of the
         Company; and


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                  (c) have been duly authorized by proper corporate action on
         the part of the Company (no action by the stockholders of the Company
         being required by law, by the Certificate of Incorporation or the
         By-laws of the Company or otherwise), executed and delivered by the
         Company and the Second Amendment constitutes the legal, valid and
         binding obligation, contract and agreement of the Company enforceable
         in accordance with its terms.

         4.  Pending Litigation. There are no proceedings pending or, to the
knowledge of the Company, threatened against or affecting the Company in any
court or before any governmental authority or arbitration board or tribunal
which, if adversely determined, could in the aggregate have a material adverse
effect on the properties, business, prospects, profits or condition (financial
or otherwise) of the Company and its Subsidiaries. The Company is not in default
with respect to any order of any court or governmental authority or arbitration
board or tribunal.

         5.  No Defaults. Neither the Company nor any Subsidiary is in default
(a) in the payment of principal or interest on any Indebtedness for borrowed
money (b) under any instrument or instruments or agreements under and subject to
which any Indebtedness for borrowed money has been issued, and no event has
occurred and is continuing under the provisions of any such instrument or
agreement which with the lapse of time or the giving of notice, or both, would
constitute an event of default thereunder. Neither the Company nor any
Subsidiary is in violation of any term of any agreement, charter instrument,
regulation or other instrument to which it is a party or by which it may be
bound which violation would in the aggregate have a material adverse effect on
the business or the financial condition of the Company.

         6.  Consents. No approval, consent or withholding of objection on the
part of any regulatory body, state, Federal or local, is necessary in connection
with the execution and delivery by the Company of the Second Amendment or
compliance by the Company with any of the provisions thereof. All approval sand
consents required from any other Person for the due execution and delivery by
the Company of the Second Amendment and compliance by the Company with any of
the provisions thereof have been obtained and are in full force and effect.

         7.  Compliance with Law. Neither the Company nor any Subsidiary (a) is
in violation of any law, ordinance, franchise, governmental rule or regulation
to which it is subject; or (b) has failed to obtain any license, permit,
franchise or other governmental authorization necessary to the ownership of its
property or to the conduct of its business, which violation or failure to obtain
would, individually or in the aggregate, materially adversely affect the
business, prospects, profits, properties or condition (financial or otherwise)
of the Company and its Subsidiaries, taken as a whole, or impair the ability of
the Company to perform its obligations contained in the Second Amendment.
Neither the Company nor any Subsidiary is in default with respect to any order
of any court or governmental authority or arbitration board or tribunal.



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                       CONFIRMATION OF GUARANTY AGREEMENT


         Smith International Acquisition Corp. (the "GUARANTOR") pursuant to
that certain Guaranty Agreement dated as of March 1, 1994 (the "GUARANTY"),
which has been executed and delivered by the Guarantor to the Noteholders (as
defined in the Guaranty), hereby agrees that its liabilities thereunder and the
Security Documents to which such Guarantor is a party are and shall remain
enforceable against such Guarantor in accordance with their terms, and are, and
shall continue to be, in full force and effect and shall not be reduced,
altered, limited, lessened or in any way affected by the execution and delivery
of the Second Amendment to Note Agreement dated as of even date herewith and to
which this Confirmation of Guaranty Agreement is attached, executed by the
Company (as defined in the Second Amendment) and the Required Noteholders (as
defined in the Second Amendment), and are hereby confirmed and ratified in all
respects.

         WITNESS THE EXECUTION HEREOF, effective as of the 31st day of July,
1998.


                              SMITH INTERNATIONAL ACQUISITION CORP.



                              By:  /s/ NEAL S. SUTTON
                                  --------------------------------------------
                                  Name:   Neal S. Sutton
                                  Title:  Sr. VP-Admin., General Counsel & Sec.