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                                                                   EXHIBIT 10.21

                            Non-Dilution Agreement
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     This NON-DILUTION AGREEMENT dated December 13, 1999 (the "Agreement"), is
made and entered into by and between Titan Resources Holdings, Inc., a Delaware
corporation (the "Company"), and Union Oil Company of California, a California
corporation ("Union Oil").

                             W I T N E S S E T H:

     WHEREAS, Union Oil is a holder of shares of Common Stock (as defined) of
the Company;

     WHEREAS, Union Oil, the Company, TRH, Inc., a Delaware corporation and
wholly-owned subsidiary of the Company ("Sub"), and Titan Exploration Inc., a
Delaware corporation ("Titan"), are parties to that certain Agreement and Plan
of Merger dated as of the date hereof (the "Merger Agreement"), pursuant to
which Sub shall be merged with and into Titan, which shall become a wholly-owned
subsidiary of the Company; and

     WHEREAS, it is a condition to the consummation of the transactions
contemplated by the Merger Agreement that the Company and Union Oil enter into
this Agreement.

     NOW, THEREFORE, in consideration of the foregoing and the mutual agreements
set forth herein the parties to this Agreement hereby agree as follows:

     1.   Definitions of Certain Agreement Terms.  For purposes of this
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Agreement, the terms hereinafter set forth shall have the following definitions
unless otherwise specifically stated:

     "Board" means the Board of Directors of the Company.

     "Capital Stock" means the Common Stock or any other capital stock of the
Company which is convertible or exchangeable into Common Stock.

     "Common Stock" means the common stock, par value $.01 per share, of the
Company.

     "Exchange Act" means the Securities Exchange Act of 1934, as amended.

     "Excluded Securities" means Common Stock issued to officers, employees or
directors of, or consultants to, the Company or its subsidiaries pursuant to the
terms of any stock option or similar stock incentive plan adopted by the Board.

     "Maximum Ownership Percentage" means 65.4%.

     "Registration Rights Agreement" means the Registration Rights Agreement
dated of even date herewith between the Company and Union Oil.

     "SEC" means the United States Securities and Exchange Commission.
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     "Securities Act" means the Securities Act of 1933, as amended.

     "Union Oil Ownership Percentage" means the percentage, not to exceed the
Maximum Ownership Percentage, obtained by multiplying 100 by an amount equal to
(i) the number of shares of Common Stock and other Voting Stock beneficially
owned by Union Oil divided by (ii) the number of shares of Common Stock and
other Voting Stock issued and outstanding.

     "Voting Stock" means any shares of Capital Stock having the right to vote
generally with respect to the election of directors or convertible into Capital
Stock having the right to vote generally with respect to the election of
directors. For purposes of calculating Union Oil Ownership Percentage, the
voting power of Capital Stock shall be determined on a fully diluted basis, such
that (a) the total number of shares of Voting Stock outstanding shall be deemed
to be the number of votes entitled to be cast by the holders of outstanding
shares with respect to the election of directors or, if greater, the total
number of votes that would be entitled to be cast by the holders of all
outstanding shares upon conversion of all convertible Capital Stock that is
convertible into Capital Stock with greater voting power than the Capital Stock
so converted, and (b) the number of votes attributable to each share of Capital
Stock owned by Union Oil shall be the number of votes attributable to such share
or, if greater, attributable to the shares, if any, into which such share is
convertible.

     The terms "beneficial ownership" and "group" shall be used as defined in
Regulation 13D-G under the Exchange Act. The terms "affiliate" and "associate"
shall be used as defined in Rule 12b-2 under the Exchange Act.

     2.   Term; Termination. This Agreement shall become effective upon
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consummation of the transactions contemplated by the Merger Agreement and shall
terminate if the Merger Agreement is terminated in accordance with its terms.
Thereafter, this Agreement shall terminate upon the sale or transfer by Union
Oil or its affiliates of such number of shares of Capital Stock that causes the
Union Oil Ownership Percentage to decline to less than 10%. This Agreement may
also be terminated by the written agreement of all parties hereto. In the event
of the termination of this Agreement pursuant to this Section 2, this Agreement
shall become void and have no effect. Nothing contained in this Section 2 shall
relieve any party from liability for damages actually incurred as a result of
any breach of this Agreement. In addition to the foregoing, Union Oil shall have
the right at any time and from time to time to suspend its rights under this
Agreement for any such period of time as it shall request.

     3.   Issuances for other than Cash.
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     (a)  If the Company issues any Capital Stock in exchange for property other
than cash or credit, it shall give Union Oil written notice of such issuance not
later than 20 days prior to such issuance, describing the number of shares to be
issued and the terms and conditions upon which the Company proposes to issue the
same. Following any such issuance, Union Oil shall have 30 days from the date of
such issuance to elect to exercise its rights under this Section 3 by giving
written notice to the Company.

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     (b)  If Union Oil provides the notice specified in Section 3(a), Union Oil
shall have the right to purchase from the Company, and the Company shall be
obligated to issue and sell to Union Oil, up to such additional number of shares
of such Capital Stock as is necessary to cause the Union Oil Ownership
Percentage immediately following the applicable issuance to equal the Union Oil
Ownership Percentage immediately preceding such issuance, upon the terms and
conditions set forth herein. The closing of such purchase and sale shall take
place at the offices of the Company on such business day and time as Union Oil
shall specify in its notice to the Company; provided, that in no event, unless
otherwise agreed by the parties, shall such closing be earlier than five (5)
business days after such notice or later than thirty (30) days after such
notice.

     (c)  The cash price per share to be paid by Union Oil pursuant to this
Section 3 shall be the Market Value Per Share determined as of the close of
business on the date of the issuance giving rise to Union Oil's purchase right
hereunder. "Market Value Per Share" means, with respect to a share of Common
Stock, the average of the closing price of the Common Stock on all domestic
securities exchanges on which such security may at the time be listed, or, if
there have been no sales on any such exchange on any day, the average of the
highest bid and lowest asked prices on all such exchanges at the end of such
day, or, if on any day such security is not so listed, the average of the
representative bid and asked prices quoted in the NASDAQ System as of 4:00 P.M.,
New York time, on such day, or, if on any day such security is not quoted in the
NASDAQ System, the average of the highest bid and lowest asked prices on such
day in the domestic over-the-counter market as reported by the National
Quotation Bureau, Incorporated, or any similar successor organization. With
respect to Capital Stock other than Common Stock, the Market Value Per Share
shall be as determined in good faith by the Board of Directors.

     4.   Issuances for Cash.  If the Company issues Capital Stock (other than
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Excluded Securities) for cash or credit (the "Offered Securities"), Union Oil
shall have a preemptive right to purchase or subscribe for the number or amount
of such Offered Securities up to the Union Oil Ownership Percentage (determined
as of the time of the public announcement of such issuance) of the total number
or amount of Offered Securities proposed to be issued. The Company shall provide
Union Oil with notice of an issuance subject to this preemptive right at least
10 days prior to such issuance specifying the number of shares and terms of such
issuance and, if Union Oil elects to exercise the right, it shall do so in such
a way as not to delay pricing and closing of the issuance. If Union Oil
exercises its right, it shall only be required to pay the same cash
consideration per share as being paid by other purchasers for the Offered
Securities. The preemptive right given by the Company shall terminate if
unexercised within 10 days after receipt of the notice of the issuance of such
Capital Stock.

     5.   Certain Limitations.  Notwithstanding anything to the contrary in this
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Agreement, the Company shall not be required to offer or sell any Capital Stock
to Union Oil, and Union Oil shall have no preemptive rights, pursuant to this
Agreement (i) to the extent that after a purchase by Union Oil pursuant to this
Agreement the Union Oil Ownership Percentage would be greater than the Maximum
Ownership Percentage or (ii) if such action would cause the Company to be in
violation of applicable federal securities laws by virtue of such offer or sale.
All shares of Capital Stock issued hereunder shall be entitled to the benefits
of the Registration Rights Agreement. The Company will cause any shares of
Capital Stock issued pursuant to this Agreement to be listed for

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trading or quotation on any securities exchanges or quotation systems on which
the securities of that class are then listed for trading or quotation.

     6.   Successors, Assigns and Transferees.  The terms and provisions of this
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Agreement shall not bind, inure to the benefit of or be enforceable by or
against the successors, assigns or transferees of each of the parties hereto.
No party hereto may assign its rights under this Agreement; provided that Union
Oil may at any time and from time to time assign any or all of its rights under
this Agreement to an affiliate of Union Oil by giving written notice thereof to
the Company.

     7.   Entire Agreement; Amendments.  This Agreement, and such additional
          ----------------------------
instruments as may be concurrently executed and delivered pursuant to this
Agreement, constitutes the entire understanding of the parties with respect to
its subject matter. There are no restrictions, agreements, promises,
representations, warranties, covenants or undertakings other than those
expressly set forth herein or in the documents delivered concurrently herewith.
This Agreement may be amended only by a written instrument duly executed by all
the parties hereto.

     8.   Headings.  The section headings contained in this Agreement are for
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reference purposes only and shall not effect in any way the meaning or
interpretation of this Agreement.

     9.   Notices.  All notices, requests, claims, demands and other
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communications hereunder shall be in writing and shall be given (and shall be
deemed to have been duly given if so given) by hand delivery, facsimile or by
mail (registered or certified, postage prepaid, return receipt requested) to the
respective parties as follows:

          If to Union Oil:

               Union Oil Company of California
               c/o Mr. Phil Ballard
               Manager, Corporate Development
               Bengal Company of California
               One Sugar Creek Place
               14141 Southwest Freeway
               Sugar Land, TX 77478
               Fax: (281) 287-5170

          And another copy to:

               Union Oil Company of California
               2141 Rosecrans Avenue, Suite 4000
               El Segundo, CA 90245
               Attn: (1) General Counsel, and
                     (2) Vice President, Corporate Development
               Fax: (310) 726-7819

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          If to the Company:

               Titan Resources Holdings, Inc.
               500 West Texas
               Suite 200
               Midland, Texas 79701
               Attention: Mr. Jack D. Hightower
               Fax: (915) 687-3863

          with a copy to:

               Thompson & Knight L.L.P.
               1700 Pacific Avenue
               Suite 3300
               Dallas, Texas 75201
               Attention: Mr. Joe Dannenmaier
               Fax: (214) 969-1751

or to such other address as the person to whom notice is given may have
previously furnished to the others in writing in the manner set forth above.

     10.  Governing Law.  This Agreement shall be governed by and construed and
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enforced in accordance with the laws of the State of Delaware, without reference
to the conflict of laws principles thereof.

     11.  Waiver.  Any waiver by any party of a breach of any provision of this
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Agreement shall not operate as or be construed to be a waiver of any other
breach of such provision or of any breach of any other provision of this
Agreement. The failure of a party to insist upon strict adherence to any term of
this Agreement on one or more occasions shall not be considered a waiver or
deprive that party of the right thereafter to insist upon strict adherence to
that term or any other term of this Agreement.

     12.  Specific Performance.  The parties hereto acknowledge and agree that
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Union Oil would not have an adequate remedy at law for money damages in the
event that this Agreement were not performed in accordance with its specific
terms. It is accordingly agreed that Union Oil shall be entitled to enforce
specifically the provisions of this Agreement, in any court of the United States
or any state thereof having jurisdiction, in addition to any other remedy to
which Union Oil may be entitled under this Agreement or at law or in equity.

     13.  Counterparts.  This Agreement may be executed in counterparts, each of
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which shall be an original, but each of which together shall constitute one and
the same Agreement.

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     IN WITNESS WHEREOF, and intending to be legally bound hereby, each of the
undersigned parties has executed or caused this Agreement to be executed on the
date first above written.

                                        TITAN RESOURCES HOLDINGS, INC.



                                        By: /s/ Phillip Ballard
                                            ------------------------------------
                                            Phillip Ballard
                                            Vice President


                                        UNION OIL COMPANY OF CALIFORNIA



                                        By: /s/ Timothy H. Ling
                                            ------------------------------------
                                            Timothy H. Ling
                                            Executive Vice President, North
                                            American Energy Operations and Chief
                                            Financial Officer

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